2 Chapter 1 THE MARKET (Ch. 1) NAME The Market Introduction. The problems in this chapter examine some variations on Price the apartment market described in the text. In most of the problems we work with the true demand curve constructed from the reservation prices of the consumers rather than the “smoothed” demand curve that we used in the text. Remember that the reservation price of a consumer is that price where he is just indiﬀerent between renting or not renting the apartment. At any price below the reservation price the consumer will demand one apartment, at any price above the reservation price the consumer will demand zero apartments, and exactly at the reservation price the consumer will be indiﬀerent between having zero or one apartment. You should also observe that when demand curves have the “staircase” shape used here, there will typically be a range of prices where supply equals demand. Thus we will ask for the the highest and lowest price in the range. 60 50 40 30 20 10 0 2 1 3 4 5 6 7 8 Apartments (b) Suppose the supply of apartments is ﬁxed at 5 units. In this case there is a whole range of prices that will be equilibrium prices. What is the highest price that would make the demand for apartments equal to 5 1.1 (3) Suppose that we have 8 people who want to rent an apartment. Their reservation prices are given below. (To keep the numbers small, think of these numbers as being daily rent payments.) units? $18. (c) What is the lowest price that would make the market demand equal Person Price = A = 40 B 25 C D 30 35 E 10 F 18 G 15 H 5 to 5 units? $15. (d) With a supply of 4 apartments, which of the people A–H end up (a) Plot the market demand curve in the following graph. (Hint: When the market price is equal to some consumer i’s reservation price, there will be two diﬀerent quantities of apartments demanded, since consumer i will be indiﬀerent between having or not having an apartment.) getting apartments? A, B, C, D. (e) What if the supply of apartments increases to 6 units. What is the range of equilibrium prices? $10 to $15. 1.2 (3) Suppose that there are originally 5 units in the market and that 1 of them is turned into a condominium. (a) Suppose that person A decides to buy the condominium. What will be the highest price at which the demand for apartments will equal the supply of apartments? What will be the lowest price? Enter your answers in column A, in the table. Then calculate the equilibrium prices of apartments if B, C, . . . , decide to buy the condominium. NAME 3 4 Person A B C D E F G H High price 18 18 18 18 25 25 25 25 15 Low price 15 15 15 18 15 18 18 18. (d) How does this compare to the market outcome? 1 2 3 4 5 6 7 40 35 30 25 18 15 10 5 Revenue 40 70 90 100 90 90 70 40 A, B, C, D. (c) If the monopolist were required by law to rent exactly 5 apartments, 8 It’s the same. 1.5 (2) In the text we argued that a tax on landlords would not get passed along to the renters. What would happen if instead the tax was imposed on renters? (a) To answer this question, consider the group of people in Problem 1.1. What is the maximum that they would be willing to pay to the landlord if they each had to pay a $5 tax on apartments to the city? Fill in the box below with these reservation prices. $18. A, B, C, D, F. (e) If this landlord could charge each individual a diﬀerent price, and he knew the reservation prices of all the individuals, what is the maximum $148. (f ) If 5 apartments were rented, which individuals would get the apartments? A, B, C, D, F. Price revenue he could make if he rented all 5 apartments? $18. the consumers described above will end up in the 5 apartments? (a) Fill in the box with the maximum price and revenue that the monopolist can make if he rents 1, 2, . . ., 8 apartments. (Assume that he must charge one price for all apartments.) (d) Who would get apartments? F, (c) If you have rent control with unlimited subletting allowed, which of 1.3 (2) Suppose now that a monopolist owns all the apartments and that he is trying to determine which price and quantity maximize his revenues. what price would he charge to maximize his revenue? E, who is willing to pay only $10 for an apartment would sublet to payment? Yes. (b) Which of the people A–F would get apartments? control restrictions.) (b) What will be the maximum amount that can be charged for the sublet Suppose that one of the apartments was turned into a condo- Number (Ch. 1) who is willing to pay $18. (b) Suppose that there were two people at each reservation price and 10 apartments. What is the highest price at which demand equals supply? minium. Is that price still an equilibrium price? THE MARKET (a) If subletting is legal—or, at least, practiced—who will sublet to whom in equilibrium? (Assume that people who sublet can evade the city rent- A, B, C, D, F. 1.4 (2) Suppose that there are 5 apartments to be rented and that the city rent-control board sets a maximum rent of $9. Further suppose that people A, B, C, D, and E manage to get an apartment, while F, G, and H are frozen out. Person A B C D E F G H Reservation Price 35 20 25 30 5 13 10 0 (b) Using this information determine the maximum equilibrium price if there are 5 apartments to be rented. $13. (c) Of course, the total price a renter pays consists of his or her rent plus the tax. This amount is $18. (d) How does this compare to what happens if the tax is levied on the landlords? It’s the same. 6 Chapter 2 NAME Budget Constraint Introduction. These workouts are designed to build your skills in describing economic situations with graphs and algebra. Budget sets are a good place to start, because both the algebra and the graphing are very easy. Where there are just two goods, a consumer who consumes x1 units of good 1 and x2 units of good 2 is said to consume the consumption bundle, (x1 , x2 ). Any consumption bundle can be represented by a point on a two-dimensional graph with quantities of good 1 on the horizontal axis and quantities of good 2 on the vertical axis. If the prices are p1 for good 1 and p2 for good 2, and if the consumer has income m, then she can aﬀord any consumption bundle, (x1 , x2 ), such that p1 x1 +p2 x2 ≤ m. On a graph, the budget line is just the line segment with equation p1 x1 + p2 x2 = m and with x1 and x2 both nonnegative. The budget line is the boundary of the budget set. All of the points that the consumer can aﬀord lie on one side of the line and all of the points that the consumer cannot aﬀord lie on the other. If you know prices and income, you can construct a consumer’s budget line by ﬁnding two commodity bundles that she can “just aﬀord” and drawing the straight line that runs through both points. BUDGET CONSTRAINT (Ch. 2) ale Laurel can aﬀord if she spends all of her money on ale. Since ale costs 1 dollar a bottle, her income in dollars is equal to the largest number of bottles she can aﬀord. Alternatively, you can reason as follows. Since the bundles (20, 5) and (10, 10) cost the same, it must be that giving up 10 bottles of ale makes her able to aﬀord an extra 5 loaves of bread. So bread costs twice as much as ale. The price of ale is 1 dollar, so the price of bread is 2 dollars. The bundle (20, 5) costs as much as her income. Therefore her income must be 20 × 1 + 5 × 2 = 30. When you have completed this workout, we hope that you will be able to do the following: • Write an equation for the budget line and draw the budget set on a graph when you are given prices and income or when you are given two points on the budget line. • Graph the eﬀects of changes in prices and income on budget sets. • Understand the concept of numeraire and know what happens to the budget set when income and all prices are multiplied by the same positive amount. • Know what the budget set looks like if one or more of the prices is negative. • See that the idea of a “budget set” can be applied to constrained choices where there are other constraints on what you can have, in addition to a constraint on money expenditure. Example: Myrtle has 50 dollars to spend. She consumes only apples and bananas. Apples cost 2 dollars each and bananas cost 1 dollar each. You are to graph her budget line, where apples are measured on the horizontal axis and bananas on the vertical axis. Notice that if she spends all of her income on apples, she can aﬀord 25 apples and no bananas. Therefore her budget line goes through the point (25, 0) on the horizontal axis. If she spends all of her income on bananas, she can aﬀord 50 bananas and no apples. Therfore her budget line also passes throught the point (0, 50) on the vertical axis. Mark these two points on your graph. Then draw a straight line between them. This is Myrtle’s budget line. What if you are not told prices or income, but you know two commodity bundles that the consumer can just aﬀord? Then, if there are just two commodities, you know that a unique line can be drawn through two points, so you have enough information to draw the budget line. Example: Laurel consumes only ale and bread. If she spends all of her income, she can just aﬀord 20 bottles of ale and 5 loaves of bread. Another commodity bundle that she can aﬀord if she spends her entire income is 10 bottles of ale and 10 loaves of bread. If the price of ale is 1 dollar per bottle, how much money does she have to spend? You could solve this problem graphically. Measure ale on the horizontal axis and bread on the vertical axis. Plot the two points, (20, 5) and (10, 10), that you know to be on the budget line. Draw the straight line between these points and extend the line to the horizontal axis. This point denotes the amount of NAME 7 2.1 (0) You have an income of $40 to spend on two commodities. Commodity 1 costs $10 per unit, and commodity 2 costs $5 per unit. 8 BUDGET CONSTRAINT (Ch. 2) (a) p1 = 1, p2 = 1, m = 15. (Use blue ink.) (b) p1 = 1, p2 = 2, m = 20. (Use red ink.) (a) Write down your budget equation. 10x1 + 5x2 = 40. (b) If you spent all your income on commodity 1, how much could you (c) p1 = 0, p2 = 1, m = 10. (Use black ink.) (d) p1 = p2 , m = 15p1 . (Use pencil or black ink. Hint: How much of good 1 could you aﬀord if you spend your entire budget on good 1?) 4. buy? x2 (c) If you spent all of your income on commodity 2, how much could you buy? below. 8. 20 Use blue ink to draw your budget line in the graph 15 Blue Line Black Line x2 8 6 4 2 0 ;;;;;; ;;;;;; Blue Line ;;;;;; ;;;;;; ;;;;;; Red Line ;;;;;; ;;;;;; ;;;;;; ;;;;;;Black Shading ;;;;;; ;;;;;; ;;;;;;;;;;;;; ;;;;;; ;;;;;;;;;;;;; Black Line ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; Blue ;;;;;;;;;;;;; ;;;;;;;;;;;;; Shading ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; 2 4 6 10 Red Line 5 0 8 x1 (d) Suppose that the price of commodity 1 falls to $5 while everything else stays the same. Write down your new budget equation. 40. 5x1 +5x2 = On the graph above, use red ink to draw your new budget line. (e) Suppose that the amount you are allowed to spend falls to $30, while the prices of both commodities remain at $5. Write down your budget equation. line. 5x1 + 5x2 = 30. 5 10 15 20 x1 2.3 (0) Your budget is such that if you spend your entire income, you can aﬀord either 4 units of good x and 6 units of good y or 12 units of x and 2 units of y. (a) Mark these two consumption bundles and draw the budget line in the graph below. y 16 Use black ink to draw this budget 12 (f ) On your diagram, use blue ink to shade in the area representing commodity bundles that you can aﬀord with the budget in Part (e) but could not aﬀord to buy with the budget in Part (a). Use black ink or pencil to shade in the area representing commodity bundles that you could aﬀord with the budget in Part (a) but cannot aﬀord with the budget in Part (e). 8 2.2 (0) On the graph below, draw a budget line for each case. 0 4 4 8 12 16 x NAME 9 (b) What is the ratio of the price of x to the price of y? 1/2. (c) If you spent all of your income on x, how much x could you buy? 10 BUDGET CONSTRAINT (Ch. 2) 2.6 (0) In a small country near the Baltic Sea, there are only three commodities: potatoes, meatballs, and jam. Prices have been remarkably stable for the last 50 years or so. Potatoes cost 2 crowns per sack, meatballs cost 4 crowns per crock, and jam costs 6 crowns per jar. (a) Write down a budget equation for a citizen named Gunnar who has an income of 360 crowns per year. Let P stand for the number of sacks of potatoes, M for the number of crocks of meatballs, and J for the number 16. (d) If you spent all of your income on y, how much y could you buy? 8. (e) Write a budget equation that gives you this budget line, where the x + 2y = 16. price of x is 1. (f ) Write another budget equation that gives you the same budget line, but where the price of x is 3. 3x + 6y = 48. 2.4 (1) Murphy was consuming 100 units of X and 50 units of Y . The price of X rose from 2 to 3. The price of Y remained at 4. 2P + 4M + 6J = of jars of jam consumed by Gunnar in a year. 360. (b) The citizens of this country are in general very clever people, but they are not good at multiplying by 2. This made shopping for potatoes excruciatingly diﬃcult for many citizens. Therefore it was decided to introduce a new unit of currency, such that potatoes would be the numeraire. A sack of potatoes costs one unit of the new currency while the same relative prices apply as in the past. In terms of the new currency, what is 2 crowns. the price of meatballs? (a) How much would Murphy’s income have to rise so that he can still exactly aﬀord 100 units of X and 50 units of Y ? (c) In terms of the new currency, what is the price of jam? $100. 2.5 (1) If Amy spent her entire allowance, she could aﬀord 8 candy bars and 8 comic books a week. She could also just aﬀord 10 candy bars and 4 comic books a week. The price of a candy bar is 50 cents. Draw her budget line in the box below. What is Amy’s weekly allowance? $6. (d) What would Gunnar’s income in the new currency have to be for him to be exactly able to aﬀord the same commodity bundles that he could aﬀord before the change? 180 crowns. Comic books (e) Write down Gunnar’s new budget equation. 32 3 crowns. 180. P + 2M + 3J = Is Gunnar’s budget set any diﬀerent than it was before the change? No. 24 2.7 (0) Edmund Stench consumes two commodities, namely garbage and punk rock video cassettes. He doesn’t actually eat the former but keeps it in his backyard where it is eaten by billy goats and assorted vermin. The reason that he accepts the garbage is that people pay him $2 per sack for taking it. Edmund can accept as much garbage as he wishes at that price. He has no other source of income. Video cassettes cost him $6 each. 16 8 (a) If Edmund accepts zero sacks of garbage, how many video cassettes 0 8 16 12 24 NAME 11 (b) If he accepts 15 sacks of garbage, how many video cassettes can he 5. buy? (c) Write down an equation for his budget line. 0. can he buy? 32 Candy bars 6C − 2G = 0. (d) Draw Edmund’s budget line and shade in his budget set. 12 BUDGET CONSTRAINT (Ch. 2) (a) Write down a budget equation stating those combinations of the three commodities, Good X, hours of speeches by politicians (P ), and hours of speeches by university administrators (A) that Emmett could aﬀord to consume per week. 15X − 1P − 2A = 50. (b) On the graph above, draw a two-dimensional diagram showing the locus of consumptions of the two kinds of speeches that would be possible for Emmett if he consumed 10 units of Good X per week. Garbage 20 15 10 5 0 ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;;Budget Line ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; Budget Set ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; 5 10 15 20 Video cassettes 2.8 (0) If you think Edmund is odd, consider his brother Emmett. Emmett consumes speeches by politicians and university administrators. He is paid $1 per hour for listening to politicians and $2 per hour for listening to university administrators. (Emmett is in great demand to help ﬁll empty chairs at public lectures because of his distinguished appearance and his ability to refrain from making rude noises.) Emmett consumes one good for which he must pay. We have agreed not to disclose what that good is, but we can tell you that it costs $15 per unit and we shall call it Good X. In addition to what he is paid for consuming speeches, Emmett receives a pension of $50 per week. 2.9 (0) Jonathan Livingstone Yuppie is a prosperous lawyer. He has, in his own words, “outgrown those conﬁning two-commodity limits.” Jonathan consumes three goods, unblended Scotch whiskey, designer tennis shoes, and meals in French gourmet restaurants. The price of Jonathan’s brand of whiskey is $20 per bottle, the price of designer tennis shoes is $80 per pair, and the price of gourmet restaurant meals is $50 per meal. After he has paid his taxes and alimony, Jonathan has $400 a week to spend. (a) Write down a budget equation for Jonathan, where W stands for the number of bottles of whiskey, T stands for the number of pairs of tennis shoes, and M for the number of gourmet restaurant meals that he consumes. 20W + 80T + 50M = 400. (b) Draw a three-dimensional diagram to show his budget set. Label the intersections of the budget set with each axis. M 8 5 Administrator speeches 20 T W 100 (c) Suppose that he determines that he will buy one pair of designer tennis shoes per week. What equation must be satisﬁed by the combinations of 75 restaurant meals and whiskey that he could aﬀord? 50 20W +50M = 320. 25 0 25 50 75 100 Politician speeches 2.10 (0) Martha is preparing for exams in economics and sociology. She has time to read 40 pages of economics and 30 pages of sociology. In the same amount of time she could also read 30 pages of economics and 60 pages of sociology. NAME 13 14 BUDGET CONSTRAINT (Ch. 2) (a) Assuming that the number of pages per hour that she can read of either subject does not depend on how she allocates her time, how many pages of sociology could she read if she decided to spend all of her time (e) Let M stand for the number of instances of an ad being read by an M.B.A. and L stand for the number of instances of an ad being read by a lawyer. This budget line is a line segment that lies on the line with on sociology and none on economics? 150 pages. (Hint: You have two points on her budget line, so you should be able to determine the entire line.) equation M + 2L = 16. With a ﬁxed advertising budget, how many readings by M.B.A.’s must he sacriﬁce to get an additional reading 2. by a lawyer with a hot tub? (b) How many pages of economics could she read if she decided to spend MBA's x 1000 50 pages. all of her time reading economics? 16 2.11 (1) Harry Hype has $5,000 to spend on advertising a new kind of dehydrated sushi. Market research shows that the people most likely to buy this new product are recent recipients of M.B.A. degrees and lawyers who own hot tubs. Harry is considering advertising in two publications, a boring business magazine and a trendy consumer publication for people who wish they lived in California. Fact 1: Ads in the boring business magazine cost $500 each and ads in the consumer magazine cost $250 each. Fact 2: Each ad in the business magazine will be read by 1,000 recent M.B.A.’s and 300 lawyers with hot tubs. Fact 3: Each ad in the consumer publication will be read by 300 recent M.B.A.’s and 250 lawyers who own hot tubs. Fact 4: Nobody reads more than one ad, and nobody who reads one magazine reads the other. (a) If Harry spends his entire advertising budget on the business publication, his ad will be read by 3,000 10,000 recent M.B.A.’s and by lawyers with hot tubs. (b) If he spends his entire advertising budget on the consumer publication, 6,000 his ad will be read by lawyers with hot tubs. recent M.B.A.’s and by 5,000 (c) Suppose he spent half of his advertising budget on each publication. His ad would be read by lawyers with hot tubs. 8,000 recent M.B.A.’s and by 12 10 8 6 4 0 NAME 15 Gum 20 15 12 16 Lawyers x 1000 (a) On the graph below, draw the red budget (with red ink) and the blue budget (with blue ink) for a Mungoan named Harold whose blue income is 10 and whose red income is 30. Shade in the “budget set” containing all of the commodity bundles that Harold can aﬀord, given 16 BUDGET CONSTRAINT (Ch. 2) Harold contemplates the set of commodity bundles that it could aﬀord by making one declaration or the other. Let us call a commodity bundle “attainable” if Harold can aﬀord it by declaring itself to be a “Blue” and buying the bundle with blue money or if Harold can aﬀord the bundle by declaring itself to be a “Red” and buying it with red money. On the diagram below, shade in all of the attainable bundles. Gum 10 20 Blue Lines 5 ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; 0 5 10 Red Line 15 15 20 Ambrosia 10 (b) Another Mungoan, Gladys, faces the same prices that Harold faces and has the same red income as Harold, but Gladys has a blue income of 20. Explain how it is that Gladys will not spend its entire blue income no matter what its tastes may be. (Hint: Draw Gladys’s budget lines.) 5 The blue budget line lies strictly outside the red budget line, so to satisfy both budgets, one must be strictly inside the red budget line. (c) A group of radical economic reformers on Mungo believe that the currency rules are unfair. “Why should everyone have to pay two prices for everything?” they ask. They propose the following scheme. Mungo will continue to have two currencies, every good will have a blue price and a red price, and every Mungoan will have a blue income and a red income. But nobody has to pay both prices. Instead, everyone on Mungo must declare itself to be either a Blue-Money Purchaser (a “Blue”) or a RedMoney Purchaser (a “Red”) before it buys anything at all. Blues must make all of their purchases in blue money at the blue prices, spending only their blue incomes. Reds must make all of their purchases in red money, spending only their red incomes. Suppose that Harold has the same income after this reform, and that prices do not change. Before declaring which kind of purchaser it will be, ∗ 8 • There are just two consumer goods on Mungo, ambrosia and bubble gum. All Mungoans prefer more of each good to less. • The blue prices are 1 bcu (bcu stands for blue currency unit) per unit of ambrosia and 1 bcu per unit of bubble gum. • The red prices are 2 rcus (red currency units) per unit of ambrosia and 6 rcus per unit of bubble gum. (d) Draw a “budget line” showing the combinations of number of readings by recent M.B.A.’s and by lawyers with hot tubs that he can obtain if he spends his entire advertising budget. Does this line extend all the way its∗ two budget constraints. Remember, Harold has to have enough blue money and enough red money to pay both the blue-money cost and the red-money cost of a bundle of goods. 4 2 2.12 (0) On the planet Mungo, they have two kinds of money, blue money and red money. Every commodity has two prices—a red-money price and a blue-money price. Every Mungoan has two incomes—a red income and a blue income. In order to buy an object, a Mungoan has to pay that object’s redmoney price in red money and its blue-money price in blue money. (The shops simply have two cash registers, and you have to pay at both registers to buy an object.) It is forbidden to trade one kind of money for the other, and this prohibition is strictly enforced by Mungo’s ruthless and eﬃcient monetary police. 4,000 to the axes? No. Sketch, shade in, and label the budget set, which includes all the combinations of MBA’s and lawyers he can reach if he spends no more than his budget. Budget line a ;;;;;;;; ;;;;;;;; ;;;;;;;; c ;;;;;;;; ;;;;;;;; ;;;;;;;; b ;;;;;;;; ;;;;;;;; Budget ;;;;;;;; ;;;;;;;; Set ;;;;;;;; ;;;;;;;; ;;;;;;;; ;;;;;;;; ;;;;;;;; ;;;;;;;; 3 5 We refer to all Mungoans by the gender-neutral pronoun, “it.” Although Mungo has two sexes, neither of them is remotely like either of ours. ;;;;;;;;;;;;; Blue Line ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;; Red Line ;;;;;;;;;;;;; ;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;; 0 5 10 15 20 Ambrosia 2.13 (0) Are Mungoan budgets really so fanciful? Can you think of situations on earth where people must simultaneously satisfy more than one budget constraint? Is money the only scarce resource that people use up when consuming? Consumption of many commodities takes time as well as money. People have to simultaneously satisfy a time budget and a money budget. Other examples--people may have a calorie budget or a cholesterol budget or an alcohol-intake budget. 18 Chapter 3 PREFERENCES (Ch. 3) paper. Draw an upward-sloping curve passing through the point (0, 4) and getting steeper as one moves to the right. NAME Preferences When you have completed this workout, we hope that you will be able to do the following: • Given the formula for an indiﬀerence curve, draw this curve, and ﬁnd its slope at any point on the curve. Introduction. In the previous section you learned how to use graphs to show the set of commodity bundles that a consumer can aﬀord. In this section, you learn to put information about the consumer’s preferences on the same kind of graph. Most of the problems ask you to draw indiﬀerence curves. Sometimes we give you a formula for the indiﬀerence curve. Then all you have to do is graph a known equation. But in some problems, we give you only “qualitative” information about the consumer’s preferences and ask you to sketch indiﬀerence curves that are consistent with this information. This requires a little more thought. Don’t be surprised or disappointed if you cannot immediately see the answer when you look at a problem, and don’t expect that you will ﬁnd the answers hiding somewhere in your textbook. The best way we know to ﬁnd answers is to “think and doodle.” Draw some axes on scratch paper and label them, then mark a point on your graph and ask yourself, “What other points on the graph would the consumer ﬁnd indiﬀerent to this point?” If possible, draw a curve connecting such points, making sure that the shape of the line you draw reﬂects the features required by the problem. This gives you one indiﬀerence curve. Now pick another point that is preferred to the ﬁrst one you drew and draw an indiﬀerence curve through it. • Determine whether a consumer prefers one bundle to another or is indiﬀerent between them, given speciﬁc indiﬀerence curves. • Draw indiﬀerence curves for the special cases of perfect substitutes and perfect complements. • Draw indiﬀerence curves for someone who dislikes one or both commodities. • Draw indiﬀerence curves for someone who likes goods up to a point but who can get “too much” of one or more goods. • Identify weakly preferred sets and determine whether these are convex sets and whether preferences are convex. • Know what the marginal rate of substitution is and be able to determine whether an indiﬀerence curve exhibits “diminishing marginal rate of substitution.” • Determine whether a preference relation or any other relation between pairs of things is transitive, whether it is reﬂexive, and whether it is complete. Example: Jocasta loves to dance and hates housecleaning. She has strictly convex preferences. She prefers dancing to any other activity and never gets tired of dancing, but the more time she spends cleaning house, the less happy she is. Let us try to draw an indiﬀerence curve that is consistent with her preferences. There is not enough information here to tell us exactly where her indiﬀerence curves go, but there is enough information to determine some things about their shape. Take a piece of scratch paper and draw a pair of axes. Label the horizontal axis “Hours per day of housecleaning.” Label the vertical axis “Hours per day of dancing.” Mark a point a little ways up the vertical axis and write a 4 next to it. At this point, she spends 4 hours a day dancing and no time housecleaning. Other points that would be indiﬀerent to this point would have to be points where she did more dancing and more housecleaning. The pain of the extra housekeeping should just compensate for the pleasure of the extra dancing. So an indiﬀerence curve for Jocasta must be upward sloping. Because she loves dancing and hates housecleaning, it must be that she prefers all the points above this indiﬀerence curve to all of the points on or below it. If Jocasta has strictly convex preferences, then it must be that if you draw a line between any two points on the same indiﬀerence curve, all the points on the line (except the endpoints) are preferred to the endpoints. For this to be the case, it must be that the indiﬀerence curve slopes upward ever more steeply as you move to the right along it. You should convince yourself of this by making some drawings on scratch NAME 19 Bananas 40 30 20 10 ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; Red Curve ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; Pencil Shading ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; Blue Curve ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; Blue Shading ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; 0 10 20 30 40 Apples For each of the following statements about Charlie’s preferences, write “true” or “false.” (c) (30, 5) ∼ (10, 15). True. (d) (10, 15) (20, 5). True. (e) (20, 5) (10, 10). True. (f ) (24, 4) (11, 9.1). False. (g) (11, 14) (2, 49). True. (a) On the graph below, plot several points that lie on the indiﬀerence curve that passes through the point (20, 5), and sketch this curve, using blue ink. Do the same, using red ink, for the indiﬀerence curve passing through the point (10, 15). (b) Use pencil to shade in the set of commodity bundles that Charlie weakly prefers to the bundle (10, 15). Use blue ink to shade in the set of commodity bundles such that Charlie weakly prefers (20, 5) to these bundles. 20 PREFERENCES (Ch. 3) (k) Remember that Charlie’s indiﬀerence curve through the point (10, 10) has the equation xB = 100/xA . Those of you who know calculus will remember that the slope of a curve is just its derivative, which in this case is −100/x2A . (If you don’t know calculus, you will have to take our word for this.) Find Charlie’s marginal rate of substitution at the point, (10, 10). −1. (l) What is his marginal rate of substitution at the point (5, 20)? −4. (m) What is his marginal rate of substitution at the point (20, 5)? (−.25). (n) Do the indiﬀerence curves you have drawn for Charlie exhibit diminishing marginal rate of substitution? Yes. 3.2 (0) Ambrose consumes only nuts and berries. Fortunately, he likes both goods. The consumption bundle where Ambrose consumes x1 units of nuts per week and x2 units of berries per week is written as (x1 , x2 ). The set of consumption bundles (x1 , x2 ) such that Ambrose is indiﬀerent between (x1 , x2 ) and (1, 16) is the set of bundles such that x1 ≥ 0, x2 ≥ 0, √ and x2 = 20 − 4 x1 . The set of bundles (x1 , x2 ) such that (x1 , x2 ) ∼ √ (36, 0) is the set of bundles such that x1 ≥ 0, x2 ≥ 0 and x2 = 24 − 4 x1 . (a) On the graph below, plot several points that lie on the indiﬀerence curve that passes through the point (1, 16), and sketch this curve, using blue ink. Do the same, using red ink, for the indiﬀerence curve passing through the point (36, 0). (h) A set is convex if for any two points in the set, the line segment between them is also in the set. Is the set of bundles that Charlie weakly prefers to (20, 5) a convex set? 3.1 (0) Charlie likes both apples and bananas. He consumes nothing else. The consumption bundle where Charlie consumes xA bushels of apples per year and xB bushels of bananas per year is written as (xA , xB ). Last year, Charlie consumed 20 bushels of apples and 5 bushels of bananas. It happens that the set of consumption bundles (xA , xB ) such that Charlie is indiﬀerent between (xA , xB ) and (20, 5) is the set of all bundles such that xB = 100/xA . The set of bundles (xA , xB ) such that Charlie is just indiﬀerent between (xA , xB ) and the bundle (10, 15) is the set of bundles such that xB = 150/xA . Yes. (b) Use pencil to shade in the set of commodity bundles that Ambrose weakly prefers to the bundle (1, 16). Use red ink to shade in the set of all commodity bundles (x1 , x2 ) such that Ambrose weakly prefers (36, 0) to these bundles. Is the set of bundles that Ambrose prefers to (1, 16) a convex set? Yes. (i) Is the set of bundles that Charlie considers inferior to (20, 5) a convex set? No. (j) The slope of Charlie’s indiﬀerence curve through a point, (xA , xB ), is known as his marginal rate of substitution at that point. (c) What is the slope of Ambrose’s indiﬀerence curve at the point (9, 8)? (Hint: Recall from calculus the way to calculate the slope of a curve. If you don’t know calculus, you will have to draw your diagram carefully and estimate the slope.) −2/3. NAME 21 22 PREFERENCES (Ch. 3) −1. (d) What is the slope of his indiﬀerence curve at the point (4, 12)? 8-ounce Berries 40 30 20 10 8 ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Pencil Shading ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Red Curve ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Red ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Blue Curve ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Shading ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; 0 10 20 30 6 Blue Lines ; 4 Red Lines 2 40 Nuts 0 (e) What is the slope of his indiﬀerence curve at the point (9, 12)? 2 4 6 −2/3 8 16-ounce −1. at the point (4, 16)? (f ) Do the indiﬀerence curves you have drawn for Ambrose exhibit dimin- Yes. ishing marginal rate of substitution? (g) Does Ambrose have convex preferences? Yes. 3.3 (0) Shirley Sixpack is in the habit of drinking beer each evening while watching “The Best of Bowlerama” on TV. She has a strong thumb and a big refrigerator, so she doesn’t care about the size of the cans that beer comes in, she only cares about how much beer she has. (a) On the graph below, draw some of Shirley’s indiﬀerence curves between 16-ounce cans and 8-ounce cans of beer. Use blue ink to draw these indiﬀerence curves. NAME 23 ;;;;;;;;;;;;;;;;;;; ;;;;;;; ;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; 8 0 24 PREFERENCES curves in words. ;; Blue ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; Red ;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ; shading ;;;;;;;;;;;;;; shading ; 4 2 3.4 (0) Elmo ﬁnds himself at a Coke machine on a hot and dusty Sunday. The Coke machine requires exact change—two quarters and a dime. No other combination of coins will make anything come out of the machine. No stores are open; no one is in sight. Elmo is so thirsty that the only thing he cares about is how many soft drinks he will be able to buy with the change in his pocket; the more he can buy, the better. While Elmo searches his pockets, your task is to draw some indiﬀerence curves that describe Elmo’s preferences about what he ﬁnds. (Ch. 3) as if Elmo’s fractional quarters and fractional dimes are accepted at the corresponding fraction of their value.) Describe these new indiﬀerence Dimes 6 (b) Lorraine Quiche likes to have a beer while she watches “Masterpiece Theatre.” She only allows herself an 8-ounce glass of beer at any one time. Since her cat doesn’t like beer and she hates stale beer, if there is more than 8 ounces in the can she pours the excess into the sink. (She has no moral scruples about wasting beer.) On the graph above, use red ink to draw some of Lorraine’s indiﬀerence curves. Black lines 3.5 (0) Randy Ratpack hates studying both economics and history. The more time he spends studying either subject, the less happy he is. But Randy has strictly convex preferences. ;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;; ;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; 2 4 6 Line segments with slope −2.5. (a) Sketch an indiﬀerence curve for Randy where the two commodities are hours per week spent studying economics and hours per week spent studying history. Will the slope of an indiﬀerence curve be positive or Negative. negative? (b) Do Randy’s indiﬀerence curves get steeper or ﬂatter as you move from 8 Quarters left to right along one of them? (a) If Elmo has 2 quarters and a dime in his pockets, he can buy 1 soft drink. How many soft drinks can he buy if he has 4 quarters and 2 dimes? 2. Steeper. Hours studying history 8 (b) Use red ink to shade in the area on the graph consisting of all combinations of quarters and dimes that Elmo thinks are just indiﬀerent to having 2 quarters and 1 dime. (Imagine that it is possible for Elmo to have fractions of quarters or of dimes, but, of course, they would be useless in the machine.) Now use blue ink to shade in the area consisting of all combinations that Elmo thinks are just indiﬀerent to having 4 quarters and 2 dimes. Notice that Elmo has indiﬀerence “bands,” not indiﬀerence curves. Preference direction 6 4 (c) Does Elmo have convex preferences between dimes and quarters? 2 Yes. (d) Does Elmo always prefer more of both kinds of money to less? (e) Does Elmo have a bliss point? No. No. (f ) If Elmo had arrived at the Coke machine on a Saturday, the drugstore across the street would have been open. This drugstore has a soda fountain that will sell you as much Coke as you want at a price of 4 cents an ounce. The salesperson will take any combination of dimes and quarters in payment. Suppose that Elmo plans to spend all of the money in his pocket on Coke at the drugstore on Saturday. On the graph above, use pencil or black ink to draw one or two of Elmo’s indiﬀerence curves between quarters and dimes in his pocket. (For simplicity, draw your graph 0 2 4 6 8 Hours studying economics 3.6 (0) Flossy Toothsome likes to spend some time studying and some time dating. In fact her indiﬀerence curves between hours per week spent studying and hours per week spent dating are concentric circles around her favorite combination, which is 20 hours of studying and 15 hours of dating per week. The closer she is to her favorite combination, the happier she is. NAME 25 (a) Suppose that Flossy is currently studying 25 hours a week and dating 3 hours a week. Would she prefer to be studying 30 hours a week and dating 8 hours a week? Yes. (Hint: Remember the formula for the distance between two points in the plane?) (b) On the axes below, draw a few of Flossy’s indiﬀerence curves and use your diagram to illustrate which of the two time allocations discussed above Flossy would prefer. Hours dating Preference direction 20 (Ch. 3) (a) Nancy Lerner wants to maximize her grade in this course. Let x1 be her score on the ﬁrst midterm and x2 be her score on the second midterm. Which combination of scores would Nancy prefer, x1 = 20 and x2 = 70 or x1 = 60 and x2 = 60? (20,70). ; (20,15) 10 PREFERENCES 3.8 (0) Professor Goodheart always gives two midterms in his communications class. He only uses the higher of the two scores that a student gets on the midterms when he calculates the course grade. (b) On the graph below, use red ink to draw an indiﬀerence curve showing all of the combinations of scores that Nancy likes exactly as much as x1 = 20 and x2 = 70. Also use red ink to draw an indiﬀerence curve showing the combinations that Nancy likes exactly as much as x1 = 60 and x2 = 60. 40 30 26 (30,8) (c) Does Nancy have convex preferences over these combinations? (25,3) 0 10 20 No. 30 40 Hours studying Grade on second midterm 80 3.7 (0) Joan likes chocolate cake and ice cream, but after 10 slices of cake, she gets tired of cake, and eating more cake makes her less happy. Joan always prefers more ice cream to less. Joan’s parents require her to eat everything put on her plate. In the axes below, use blue ink to draw a set of indiﬀerence curves that depict her preferences between plates with diﬀerent amounts of cake and ice cream. Be sure to label the axes. 60 40 Red curves Blue curves ; (a) Suppose that Joan’s preferences are as before, but that her parents allow her to leave anything on her plate that she doesn’t want. On the graph below, use red ink to draw some indiﬀerence curves depicting her preferences between plates with diﬀerent amounts of cake and ice cream. 20 Preference direction 0 20 40 60 80 Grade on first midterm Ice cream (d) Nancy is also taking a course in economics from Professor Stern. Professor Stern gives two midterms. Instead of discarding the lower grade, Professor Stern discards the higher one. Let x1 be her score on the ﬁrst midterm and x2 be her score on the second midterm. Which combination of scores would Nancy prefer, x1 = 20 and x2 = 70 or x1 = 60 and Blue curves x2 = 50? (60,50). (e) On the graph above, use blue ink to draw an indiﬀerence curve showing all of the combinations of scores on her econ exams that Nancy likes exactly as well as x1 = 20 and x2 = 70. Also use blue ink to draw an indiﬀerence curve showing the combinations that Nancy likes exactly as well as x1 = 60 and x2 = 50. Does Nancy have convex preferences over Red curves Preference direction 10 Chocolate cake these combinations? NAME 27 3.9 (0) Mary Granola loves to consume two goods, grapefruits and avocados. (a) On the graph below, the slope of an indiﬀerence curve through any point where she has more grapefruits than avocados is −2. This means that when she has more grapefruits than avocados, she is willing to give up 2 grapefruit(s) to get one avocado. (b) On the same graph, the slope of an indiﬀerence curve at points where she has fewer grapefruits than avocados is −1/2. This means that when she has fewer grapefruits than avocados, she is just willing to give up 1/2 grapefruit(s) to get one avocado. 28 PREFERENCES Yes. (Ch. 3) (b) How much money per day would he have left for other stuﬀ if he ate at 2 P.M.? $12. (c) On the graph below, use blue ink to draw the broken line that shows combinations of meal time and money for other stuﬀ that Ralph can just aﬀord. On this same graph, draw some indiﬀerence curves that would be consistent with Ralph choosing to eat his lunch at 11 A.M. Money 20 15 (c) On this graph, draw an indiﬀerence curve for Mary through bundle (10A, 10G). Draw another indiﬀerence curve through (20A, 20G). 10 Grapefruits 40 5 0 30 Slope -2 10 11 12 1 2 Time 20 Slope -1/2 10 45 0 Cherry Coke 10 20 (d) Does Mary have convex preferences? 30 40 Avocados Yes. 3.10 (2) Ralph Rigid likes to eat lunch at 12 noon. However, he also likes to save money so he can buy other consumption goods by attending the “early bird specials” and “late lunchers” promoted by his local diner. Ralph has 15 dollars a day to spend on lunch and other stuﬀ. Lunch at noon costs $5. If he delays his lunch until t hours after noon, he is able to buy his lunch for a price of $5 − t. Similarly if he eats his lunch t hours before noon, he can buy it for a price of $5 − t. (This is true for fractions of hours as well as integer numbers of hours.) (a) If Ralph eats lunch at noon, how much money does he have per day to spend on other stuﬀ? 3.11 (0) Henry Hanover is currently consuming 20 cheeseburgers and 20 Cherry Cokes a week. A typical indiﬀerence curve for Henry is depicted below. $10. 40 30 20 10 0 10 20 30 40 Cheeseburgers NAME 29 (a) If someone oﬀered to trade Henry one extra cheeseburger for every No. Coke he gave up, would Henry want to do this? (b) What if it were the other way around: for every cheeseburger Henry gave up, he would get an extra Coke. Would he accept this oﬀer? Yes. (c) At what rate of exchange would Henry be willing to stay put at his current consumption level? 2 cheeseburgers for 1 Coke. 3.12 (1) Tommy Twit is happiest when he has 8 cookies and 4 glasses of milk per day. Whenever he has more than his favorite amount of either food, giving him still more makes him worse oﬀ. Whenever he has less than his favorite amount of either food, giving him more makes him better oﬀ. His mother makes him drink 7 glasses of milk and only allows him 2 cookies per day. One day when his mother was gone, Tommy’s sadistic sister made him eat 13 cookies and only gave him 1 glass of milk, despite the fact that Tommy complained bitterly about the last 5 cookies that she made him eat and begged for more milk. Although Tommy complained later to his mother, he had to admit that he liked the diet that his sister forced on him better than what his mother demanded. (a) Use black ink to draw some indiﬀerence curves for Tommy that are consistent with this story. Milk 30 PREFERENCES (Ch. 3) (b) Tommy’s mother believes that the optimal amount for him to consume is 7 glasses of milk and 2 cookies. She measures deviations by absolute values. If Tommy consumes some other bundle, say, (c, m), she measures his departure from the optimal bundle by D = |7 − m| + |2 − c|. The larger D is, the worse oﬀ she thinks Tommy is. Use blue ink in the graph above to sketch a few of Mrs. Twit’s indiﬀerence curves for Tommy’s consumption. (Hint: Before you try to draw Mrs. Twit’s indiﬀerence curves, we suggest that you take a piece of scrap paper and draw a graph of the locus of points (x1 , x2 ) such that |x1 | + |x2 | = 1.) 3.13 (0) Coach Steroid likes his players to be big, fast, and obedient. If player A is better than player B in two of these three characteristics, then Coach Steroid prefers A to B, but if B is better than A in two of these three characteristics, then Steroid prefers B to A. Otherwise, Steroid is indiﬀerent between them. Wilbur Westinghouse weighs 340 pounds, runs very slowly, and is fairly obedient. Harold Hotpoint weighs 240 pounds, runs very fast, and is very disobedient. Jerry Jacuzzi weighs 150 pounds, runs at average speed, and is extremely obedient. (a) Does Steroid prefer Westinghouse to Hotpoint or vice versa? He prefers Westinghouse to Hotpoint. He (b) Does Steroid prefer Hotpoint to Jacuzzi or vice versa? prefers Hotpoint to Jacuzzi. He (c) Does Steroid prefer Westinghouse to Jacuzzi or vice versa? 12 prefers Jacuzzi to Westinghouse. 11 10 9 (d) Does Coach Steroid have transitive preferences? No. 8 (e) After several losing seasons, Coach Steroid decides to change his way of judging players. According to his new preferences, Steroid prefers player A to player B if player A is better in all three of the characteristics that Steroid values, and he prefers B to A if player B is better at all three things. He is indiﬀerent between A and B if they weigh the same, are equally fast, and are equally obedient. In all other cases, Coach Steroid simply says “A and B are not comparable.” 7 (2,7) 6 5 4 (8,4) 3 2 1 (13,1) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 NAME 31 Yes. 3.14 (0) The Bear family is trying to decide what to have for dinner. Baby Bear says that his ranking of the possibilities is (honey, grubs, Goldilocks). Mama Bear ranks the choices (grubs, Goldilocks, honey), while Papa Bear’s ranking is (Goldilocks, honey, grubs). They decide to take each pair of alternatives and let a majority vote determine the family rankings. (a) Papa suggests that they ﬁrst consider honey vs. grubs, and then the winner of that contest vs. Goldilocks. Which alternative will be chosen? Goldilocks. (b) Mama suggests instead that they consider honey vs. Goldilocks and then the winner vs. grubs. Which gets chosen? (g) Are Coach Steroid’s new preferences transitive? Yes. 32 PREFERENCES (Ch. 3) (c) B ∼ C. True. (d) A ∼ C. False. (e) C ∼ A. False. (f ) A B. True. (g) B A. True. (h) B C. True. (i) A C. False. (j) C A. True. (k) A B. False. (l) B A. False. Grubs. (c) What order should Baby Bear suggest if he wants to get his favorite food for dinner? No. 16 Cookies (h) Are Coach Steroid’s new preferences reﬂexive? (f ) Are Coach Steroid’s new preferences complete? Grubs versus Goldilocks, then Honey versus the winner. (d) Are the Bear family’s “collective preferences,” as determined by voting, transitive? No. 3.15 (0) Olson likes strong coﬀee, the stronger the better. But he can’t distinguish small diﬀerences. Over the years, Mrs. Olson has discovered that if she changes the amount of coﬀee by more than one teaspoon in her six-cup pot, Olson can tell that she did it. But he cannot distinguish diﬀerences smaller than one teaspoon per pot. Where A and B are two diﬀerent cups of coﬀee, let us write A B if Olson prefers cup A to cup B. Let us write A B if Olson either prefers A to B, or can’t tell the diﬀerence between them. Let us write A ∼ B if Olson can’t tell the diﬀerence between cups A and B. Suppose that Olson is oﬀered cups A, B, and C all brewed in the Olsons’ six-cup pot. Cup A was brewed using 14 teaspoons of coﬀee in the pot. Cup B was brewed using 14.75 teaspoons of coﬀee in the pot and cup C was brewed using 15.5 teaspoons of coﬀee in the pot. For each of the following expressions determine whether it is true of false. (a) A ∼ B. True. (b) B ∼ A. True. (m) B C. False. (n) A C. False. (o) C A. True. (p) Is Olson’s “at-least-as-good-as” relation, , transitive? No. (q) Is Olson’s “can’t-tell-the-diﬀerence” relation, ∼, transitive? (r) is Olson’s “better-than” relation, , transitive. Yes. No. 34 Chapter 4 NAME Utility Introduction. In the previous chapter, you learned about preferences and indiﬀerence curves. Here we study another way of describing preferences, the utility function. A utility function that represents a person’s preferences is a function that assigns a utility number to each commodity bundle. The numbers are assigned in such a way that commodity bundle (x, y) gets a higher utility number than bundle (x , y ) if and only if the consumer prefers (x, y) to (x , y ). If a consumer has the utility function U (x1 , x2 ), then she will be indiﬀerent between two bundles if they are assigned the same utility. If you know a consumer’s utility function, then you can ﬁnd the indiﬀerence curve passing through any commodity bundle. Recall from the previous chapter that when good 1 is graphed on the horizontal axis and good 2 on the vertical axis, the slope of the indiﬀerence curve passing through a point (x1 , x2 ) is known as the marginal rate of substitution. An important and convenient fact is that the slope of an indiﬀerence curve is minus the ratio of the marginal utility of good 1 to the marginal utility of good 2. For those of you who know even a tiny bit of calculus, calculating marginal utilities is easy. To ﬁnd the marginal utility of either good, you just take the derivative of utility with respect to the amount of that good, treating the amount of the other good as a constant. (If you don’t know any calculus at all, you can calculate an approximation to marginal utility by the method described in your textbook. Also, at the beginning of this section of the workbook, we list the marginal utility functions for commonly encountered utility functions. Even if you can’t compute these yourself, you can refer to this list when later problems require you to use marginal utilities.) Example: Arthur’s utility function is U (x1 , x2 ) = x1 x2 . Let us ﬁnd the indiﬀerence curve for Arthur that passes through the point (3, 4). First, calculate U (3, 4) = 3 × 4 = 12. The indiﬀerence curve through this point consists of all (x1 , x2 ) such that x1 x2 = 12. This last equation is equivalent to x2 = 12/x1 . Therefore to draw Arthur’s indiﬀerence curve through (3, 4), just draw the curve with equation x2 = 12/x1 . At the point (x1 , x2 ), the marginal utility of good 1 is x2 and the marginal utility of good 2 is x1 . Therefore Arthur’s marginal rate of substitution at the point (3, 4) is −x2 /x1 = −4/3. Example: Arthur’s uncle, Basil, has the utility function U ∗ (x1 , x2 ) = 3x1 x2 − 10. Notice that U ∗ (x1 , x2 ) = 3U (x1 , x2 ) − 10, where U (x1 , x2 ) is Arthur’s utility function. Since U ∗ is a positive multiple of U minus a constant, it must be that any change in consumption that increases U will also increase U ∗ (and vice versa). Therefore we say that Basil’s utility function is a monotonic increasing transformation of Arthur’s utility function. Let NAME u(x1 , x2 ) M U1 (x1 , x2 ) 35 M U2 (x1 , x2 ) M RS(x1 , x2 ) UTILITY (Ch. 4) us ﬁnd Basil’s indiﬀerence curve through the point (3, 4). First we ﬁnd that U ∗ (3, 4) = 3×3×4−10 = 26. The indiﬀerence curve passing through this point consists of all (x1 , x2 ) such that 3x1 x2 − 10 = 26. Simplify this last expression by adding 10 to both sides of the equation and dividing both sides by 3. You ﬁnd x1 x2 = 12, or equivalently, x2 = 12/x1 . This is exactly the same curve as Arthur’s indiﬀerence curve through (3, 4). We could have known in advance that this would happen, because if two consumers’ utility functions are monotonic increasing transformations of each other, then these consumers must have the same preference relation between any pair of commodity bundles. When you have ﬁnished this workout, we hope that you will be able to do the following: • Draw an indiﬀerence curve through a speciﬁed commodity bundle when you know the utility function. • Calculate marginal utilities and marginal rates of substitution when you know the utility function. • Determine whether one utility function is just a “monotonic transformation” of another and know what that implies about preferences. • Find utility functions that represent preferences when goods are perfect substitutes and when goods are perfect complements. • Recognize utility functions for commonly studied preferences such as perfect substitutes, perfect complements, and other kinked indiﬀerence curves, quasilinear utility, and Cobb-Douglas utility. 4.0 Warm Up Exercise. This is the ﬁrst of several “warm up exercises” that you will ﬁnd in Workouts. These are here to help you see how to do calculations that are needed in later problems. The answers to all warm up exercises are in your answer pages. If you ﬁnd the warm up exercises easy and boring, go ahead—skip them and get on to the main problems. You can come back and look at them if you get stuck later. This exercise asks you to calculate marginal utilities and marginal rates of substitution for some common utility functions. These utility functions will reappear in several chapters, so it is a good idea to get to know them now. If you know calculus, you will ﬁnd this to be a breeze. Even if your calculus is shaky or nonexistent, you can handle the ﬁrst three utility functions just by using the deﬁnitions in the textbook. These three are easy because the utility functions are linear. If you do not know any calculus, ﬁll in the rest of the answers from the back of the workbook and keep a copy of this exercise for reference when you encounter these utility functions in later problems. 36 UTILITY (Ch. 4) 4.1 (0) Remember Charlie from Chapter 3? Charlie consumes apples and bananas. We had a look at two of his indiﬀerence curves. In this problem we give you enough information so you can ﬁnd all of Charlie’s indiﬀerence curves. We do this by telling you that Charlie’s utility function happens to be U (xA , xB ) = xA xB . 2x1 + 3x2 2 3 −2/3 4x1 + 6x2 4 6 −2/3 ax1 + bx2 a b −a/b dle (40, 5) is U (40, 5) = √ 2 x1 + x2 √1 x1 1 − √1x1 includes all commodity bundles (xA , xB ) such that xA xB = ln x1 + x2 1/x1 1 −1/x1 v(x1 ) + x2 v (x1 ) 1 −v (x1 ) −x2 /x1 x1 x2 x2 x1 xa1 xb2 b axa−1 1 x2 bxa1 xb−1 2 (x1 + 2)(x2 + 1) x2 + 1 x1 + 2 − (x1 + a)(x2 + b) x2 + b x1 + a xa1 + xa2 axa−1 1 axa−1 2 +b − xx12+a a−1 − xx12 2 − ax bx1 (a) Charlie has 40 apples and 5 bananas. Charlie’s utility for the bun- 200. The indiﬀerence curve through (40, 5) 200. So 200 the indiﬀerence curve through (40, 5) has the equation xB = . On xA the graph below, draw the indiﬀerence curve showing all of the bundles that Charlie likes exactly as well as the bundle (40, 5). Bananas 40 x2 +1 x1 +2 30 20 10 0 10 20 30 40 Apples (b) Donna oﬀers to give Charlie 15 bananas if he will give her 25 apples. Would Charlie have a bundle that he likes better than (40, 5) if he makes this trade? Yes. What is the largest number of apples that Donna could demand from Charlie in return for 15 bananas if she expects him to be willing to trade or at least indiﬀerent about trading? 30. (Hint: If Donna gives Charlie 15 bananas, he will have a total of 20 bananas. If he has 20 bananas, how many apples does he need in order to be as well-oﬀ as he would be without trade?) 4.2 (0) Ambrose, whom you met in the last chapter, continues to thrive on nuts and berries. You saw two of his indiﬀerence curves. One indif√ ference curve had the equation x2 = 20 − 4 x1 , and another indiﬀerence √ curve had the equation x2 = 24 − 4 x1 , where x1 is his consumption of NAME 37 38 UTILITY (Ch. 4) nuts and x2 is his consumption of berries. Now it can be told that Ambrose has quasilinear utility. In fact, his preferences can be represented √ by the utility function U (x1 , x2 ) = 4 x1 + x2 . 4.3 (0) Burt’s utility function is U (x1 , x2 ) = (x1 + 2)(x2 + 6), where x1 is the number of cookies and x2 is the number of glasses of milk that he consumes. (a) Ambrose originally consumed 9 units of nuts and 10 units of berries. His consumption of nuts is reduced to 4 units, but he is given enough berries so that he is just as well-oﬀ as he was before. After the change, (a) What is the slope of Burt’s indiﬀerence curve at the point where he 14. how many units of berries does Ambrose consume? (b) On the graph below, indicate Ambrose’s original consumption and sketch an indiﬀerence curve passing through this point. As you can verify, Ambrose is indiﬀerent between the bundle (9,10) and the bundle (25,2). If you doubled the amount of each good in each bundle, you would have bundles (18,20) and (50,4). Are these two bundles on the same indiﬀerence curve? No. (Hint: How do you check whether two bundles are indiﬀerent when you know the utility function?) is consuming the bundle (4, 6)? −2. Use pencil or black ink to draw a line with this slope through the point (4, 6). (Try to make this graph fairly neat and precise, since details will matter.) The line you just drew is the tangent line to the consumer’s indiﬀerence curve at the point (4, 6). (b) The indiﬀerence curve through the point (4, 6) passes through the points ( 10 ,0), (7, 2 ), and (2, 12 ). Use blue ink to sketch in this indiﬀerence curve. Incidentally, the equation for Burt’s indiﬀerence curve through the point (4, 6) is x2 = 72/(x1 + 2)−6. Glasses of milk Berries 16 20 b 12 15 (9,10) 10 Red Line 8 a 5 4 Blue curve Black Line 0 5 10 15 20 Nuts 0 4 8 12 16 Cookies (c) What is Ambrose’s marginal rate of substitution, M RS(x1 , x2 ), when he is consuming the bundle (9, 10)? (Give a numerical answer.) −2/3. What is Ambrose’s marginal rate of substitution when he is consuming he would have the bundle −2/3. the bundle (9, 20)? (c) Burt currently has the bundle (4, 6). Ernie oﬀers to give Burt 9 glasses of milk if Burt will give Ernie 3 cookies. If Burt makes this trade, (d) We can write a general expression for Ambrose’s marginal rate of substitution when he is consuming commodity bundle (x1 , x2 ). This is √ M RS(x1 , x2 ) = −2/ x1 . Although we always write M RS(x1 , x2 ) as a function of the two variables, x1 and x2 , we see that Ambrose’s utility function has the special property that his marginal rate of substitution x2 does not change when the variable changes. NAME 39 (1, 15). Burt refuses to trade. Was this a wise decision? Yes, U (1, 15) = Mark the bundle (1, 15) on your graph. 63 < U (4, 6) = 72. (d) Ernie says to Burt, “Burt, your marginal rate of substitution is −2. That means that an extra cookie is worth only twice as much to you as an extra glass of milk. I oﬀered to give you 3 glasses of milk for every cookie you give me. If I oﬀer to give you more than your marginal rate of substitution, then you should want to trade with me.” Burt replies, 40 UTILITY (Ch. 4) “Ernie, you are right that my marginal rate of substitution is −2. That means that I am willing to make small trades where I get more than 2 glasses of milk for every cookie I give you, but 9 glasses of milk for 3 cookies is too big a trade. My indiﬀerence curves are not straight lines, 4.5 (0) As you may recall, Nancy Lerner is taking Professor Stern’s economics course. She will take two examinations in the course, and her score for the course is the minimum of the scores that she gets on the two exams. Nancy wants to get the highest possible score for the course. you see.” Would Burt be willing to give up 1 cookie for 3 glasses of milk? (a) Write a utility function that represents Nancy’s preferences over alternative combinations of test scores x1 and x2 on tests 1 and 2 re- Yes, U (3, 9) = 75 > U (4, 6) = 72. giving up 2 cookies for 6 glasses of milk? Would Burt object to No, U (2, 12) = 72 = U (4, 6). (e) On your graph, use red ink to draw a line with slope −3 through the point (4, 6). This line shows all of the bundles that Burt can achieve by trading cookies for milk (or milk for cookies) at the rate of 1 cookie for every 3 glasses of milk. Only a segment of this line represents trades that make Burt better oﬀ than he was without trade. Label this line segment on your graph AB. 4.4 (0) Phil Rupp’s utility function is U (x, y) = max{x, 2y}. (a) On the graph below, use blue ink to draw and label the line whose equation is x = 10. Also use blue ink to draw and label the line whose equation is 2y = 10. (b) If x = 10 and 2y < 10, then U (x, y) = then U (x, y) = 10. If x < 10 and 2y = 10, spectively. U (x1 , x2 ) = 4.6 (0) Remember Shirley Sixpack and Lorraine Quiche from the last chapter? Shirley thinks a 16-ounce can of beer is just as good as two 8-ounce cans. Lorraine only drinks 8 ounces at a time and hates stale beer, so she thinks a 16-ounce can is no better or worse than an 8-ounce can. (a) Write a utility function that represents Shirley’s preferences between commodity bundles comprised of 8-ounce cans and 16-ounce cans of beer. Let X stand for the number of 8-ounce cans and Y stand for the number of 16-ounce cans. (c) Now use red ink to sketch in the indiﬀerence curve along which U (x, y) = 10. Does Phil have convex preferences? No. u(X, Y ) = X + 2Y . (b) Now write a utility function that represents Lorraine’s preferences. u(X, Y ) = X + Y . (c) Would the function utility U (X, Y ) = 100X +200Y represent Shirley’s preferences? 10. min{x1 , x2 }, or any monotonic transformation. Yes. Would the utility function U (x, y) = (5X + 10Y )2 represent her preferences? Yes. X + 3Y represent her preferences? Would the utility function U (x, y) = No. (d) Give an example of two commodity bundles such that Shirley likes the ﬁrst bundle better than the second bundle, while Lorraine likes the y 20 second bundle better than the ﬁrst bundle. 15 (0,2) to (3,0). Blue lines 10 5 0 2y=10 Red indifference curve 5 4.7 (0) Harry Mazzola has the utility function u(x1 , x2 ) = min{x1 + 2x2 , 2x1 + x2 }, where x1 is his consumption of corn chips and x2 is his consumption of french fries. x=10 10 15 Shirley prefers Lorraine disagrees. 20 x (a) On the graph below, use a pencil to draw the locus of points along which x1 + 2x2 = 2x1 + x2 . Use blue ink to show the locus of points for which x1 + 2x2 = 12, and also use blue ink to draw the locus of points for which 2x1 + x2 = 12. NAME 41 (b) On the graph you have drawn, shade in the region where both of the following inequalities are satisﬁed: x1 + 2x2 ≥ 12 and 2x1 + x2 ≥ 12. At the bundle (x1 , x2 ) = (8, 2), one sees that 2x1 + x2 = x1 + 2x2 = 12. Therefore u(8, 2) = 18 and 12. (c) Use black ink to sketch in the indiﬀerence curve along which Harry’s utility is 12. Use red ink to sketch in the indiﬀerence curve along which Harry’s utility is 6. (Hint: Is there anything about Harry Mazzola that reminds you of Mary Granola?) French fries Blue lines 6 4 2 (Ch. 4) there are (more men than women, more women than men)? More women. Draw a squiggly red line over the part of the blue line for which U (x, y) = min{2x − y, 2y − x} = 2y − x. This shows all the combinations that Vanna thinks are just as good as (10, 10) but where there are (more men than women, more women than men)? More women. Now draw a blue line along which 2x − y = 10. Draw a squiggly red line over the part of this new blue line for which min{2x − y, 2y − x} = 2x − y. Use pencil to shade in the area on the graph that represents all combinations that Vanna likes at least as well as (10, 10). to her party? Worse. (c) If Vanna has 16 women at her party and more men than women, and if she thinks the party is exactly as good as having 10 men and 10 women, how many men does she have at the party? 22. If Vanna has 16 women at her party and more women than men, and if she thinks the party is exactly as good as having 10 men and 10 women, how many men does she have at her party? 13. (d) Vanna’s indiﬀerence curves are shaped like what letter of the alpha- Blue lines Pencil line UTILITY (b) Suppose that there are 9 men and 10 women at Vanna’s party. Would Vanna think it was a better party or a worse party if 5 more men came ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; Black line ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; Red ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; line ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; 8 42 V. bet? y 0 2 4 6 8 Corn chips 20 15 (d) At the point where Harry is consuming 5 units of corn chips and 2 units of french fries, how many units of corn chips would he be willing to 2. trade for one unit of french fries? 10 4.8 (1) Vanna Boogie likes to have large parties. She also has a strong preference for having exactly as many men as women at her parties. In fact, Vanna’s preferences among parties can be represented by the utility function U (x, y) = min{2x − y, 2y − x} where x is the number of women and y is the number of men at the party. On the graph below, let us try to draw the indiﬀerence curve along which Vanna’s utility is 10. 5 ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Pencil ;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; line ;;;;;;;;;;;;;;;;;;;;;;; Squiggly ;;;;;;;;;;;;;;;;;;;;;;; red ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; lines ;;;;;;;;;;;;;;;;;;;;;;; Blue lines 0 5 10 15 20 x (a) Use pencil to draw the locus of points at which x = y. What point on this gives Vanna a utility of 10? (10, 10). Use blue ink to draw the line along which 2y − x = 10. When min{2x − y, 2y − x} = 2y − x, NAME 43 calculus users: A diﬀerentiable function f (u) is an increasing function of u if its derivative is positive.) Yes. (a) f (u) = 3.141592u. (b) f (u) = 5, 000 − 23u. No. (c) f (u) = u − 100, 000. Yes. 4.9 (0) Suppose that the utility functions u(x, y) and v(x, y) are related by v(x, y) = f (u(x, y)). In each case below, write “Yes” if the function f is a positive monotonic transformation and “No” if it is not. (Hint for 44 UTILITY (Ch. 4) (b) Bertha Brassy has preferences represented by the utility function V (a, b) = 1, 000a2 b2 , where a is the number of ounces of animal crackers that she consumes and b is the number of ounces of beans that she consumes. On the graph below, sketch the locus of points that Bertha ﬁnds indiﬀerent to having 8 ounces of animal crackers and 2 ounces of beans. Also sketch the locus of points that she ﬁnds indiﬀerent to having 6 ounces of animal crackers and 4 ounces of beans. Beans 8 6 (d) f (u) = log10 u. Yes. (6,4) 4 (e) f (u) = −e−u . (f ) f (u) = 1/u. (g) f (u) = −1/u. Yes. 2 (8,2) No. 0 2 4 6 8 Animal crackers Yes. (c) Are Martha’s preferences convex? 4.10 (0) Martha Modest has preferences represented by the utility function U (a, b) = ab/100, where a is the number of ounces of animal crackers that she consumes and b is the number of ounces of beans that she consumes. Yes. Are Bertha’s? (d) What can you say about the diﬀerence between the indiﬀerence curves you drew for Bertha and those you drew for Martha? (a) On the graph below, sketch the locus of points that Martha ﬁnds indiﬀerent to having 8 ounces of animal crackers and 2 ounces of beans. Also sketch the locus of points that she ﬁnds indiﬀerent to having 6 ounces of animal crackers and 4 ounces of beans. Beans Yes. There is no difference. (e) How could you tell this was going to happen without having to draw the curves? Their utility functions only differ 8 by a monotonic transformation. 6 (6,4) 4 4.11 (0) Willy Wheeler’s preferences over bundles that contain nonnegative amounts of x1 and x2 are represented by the utility function U (x1 , x2 ) = x21 + x22 . 2 (8,2) (a) Draw a few of his indiﬀerence curves. What kind of geometric ﬁg0 2 4 6 8 Animal crackers ure are they? origin. Quarter circles centered at the Does Willy have convex preferences? No. NAME 45 UTILITY (Ch. 4) home runs, then A = S +D +T +H +W and B = S +W +2D +3T +4H.) Let N be the number of times the batter bats. Then his index of runs created in the season is deﬁned to be AB/N and will be called his RC. x2 8 (a) In 1987, George Bell batted 649 times. He had 39 walks, 105 singles, 32 doubles, 4 triples, and 47 home runs. In 1987, Wade Boggs batted 656 times. He had 105 walks, 130 singles, 40 doubles, 6 triples, and 24 home runs. In 1987, Alan Trammell batted 657 times. He had 60 walks, 140 singles, 34 doubles, 3 triples, and 28 home runs. In 1987, Tony Gwynn batted 671 times. He had 82 walks, 162 singles, 36 doubles, 13 triples, and 7 home runs. We can calculate A, the number of hits plus walks, B the number of total bases, and RC, the runs created index for each of these players. For Bell, A = 227, B = 408, RC = 143. For Boggs, A = 305, B = 429, RC = 199. For Trammell, A = 265, B = 389, RC = 157. For 6 4 2 0 Calculus 46 2 4 6 Gwynn, A = 8 x1 4.12 (0) Joe Bob has a utility function given by u(x1 , x2 ) = x21 +2x1 x2 + x22 . 300 , B = 383 , RC = 171. (b) If somebody has a preference ordering among these players, based only on the runs-created index, which player(s) would she prefer to Trammell? Boggs and Gwynn. (a) Compute Joe Bob’s marginal rate of substitution: M RS(x1 , x2 ) = −1. (b) Joe Bob’s straight cousin, Al, has a utility function v(x1 , x2 ) = x2 +x1 . Compute Al’s marginal rate of substitution. M RS(x1 , x2 ) = −1. (c) The diﬀerences in the number of times at bat for these players are small, and we will ignore them for simplicity of calculation. On the graph below, plot the combinations of A and B achieved by each of the players. Draw four “indiﬀerence curves,” one through each of the four points you have plotted. These indiﬀerence curves should represent combinations of A and B that lead to the same number of runs-created. Number of total bases (c) Do u(x1 , x2 ) and v(x1 , x2 ) represent the same preferences? Yes. Can you show that Joe Bob’s utility function is a monotonic transformation of Al’s? (Hint: Some have said that Joe Bob is square.) 480 Boggs Notice Bell 400 Gwynn Trammell that u(x1, x2 ) = [v(x1 , x2 )]2 . 320 4.13 (0) The idea of assigning numerical values to determine a preference ordering over a set of objects is not limited in application to commodity bundles. The Bill James Baseball Abstract argues that a baseball player’s batting average is not an adequate measure of his oﬀensive productivity. Batting averages treat singles just the same as extra base hits. Furthermore they do not give credit for “walks,” although a walk is almost as good as a single. James argues that a double in two at-bats is better than a single, but not as good as two singles. To reﬂect these considerations, James proposes the following index, which he calls “runs created.” Let A be the number of hits plus the number of walks that a batter gets in a season. Let B be the number of total bases that the batter gets in the season. (Thus, if a batter has S singles, W walks, D doubles, T triples, and H NAME 47 4.14 (0) This problem concerns the runs-created index discussed in the preceding problem. Consider a batter who bats 100 times and always either makes an out, hits for a single, or hits a home run. (a) Let x be the number of singles and y be the number of home runs in 100 at-bats. Suppose that the utility function U (x, y) by which we evaluate alternative combinations of singles and home runs is the runscreated index. Then the formula for the utility function is U (x, y) = (x + y)(x + 4y)/100. (b) Let’s try to ﬁnd out about the shape of an indiﬀerence curve between singles and home runs. Hitting 10 home runs and no singles would give him the same runs-created index as hitting 20 singles and no home runs. Mark the points (0, 10) and (x, 0), where U (x, 0) = U (0, 10). (c) Where x is the number of singles you solved for in the previous part, mark the point (x/2, 5) on your graph. Is U (x/2, 5) greater than or less than or equal to U (0, 10)? Greater than. Is this consistent with the batter having convex preferences between singles and home runs? Yes. Home runs 20 15 10 Preference direction (0,10) (10,5) 5 (20,0) 0 5 10 15 20 Singles 240 160 80 0 60 120 180 240 300 360 Number of hits plus walks 48 UTILITY (Ch. 4) 50 Chapter 5 CHOICE (Ch. 5) x2 = 20. Therefore we know that the consumer chooses the bundle (x1 , x2 ) = (120, 20). NAME Choice Introduction. You have studied budgets, and you have studied preferences. Now is the time to put these two ideas together and do something with them. In this chapter you study the commodity bundle chosen by a utility-maximizing consumer from a given budget. Given prices and income, you know how to graph a consumer’s budget. If you also know the consumer’s preferences, you can graph some of his indiﬀerence curves. The consumer will choose the “best” indiﬀerence curve that he can reach given his budget. But when you try to do this, you have to ask yourself, “How do I ﬁnd the most desirable indiﬀerence curve that the consumer can reach?” The answer to this question is “look in the likely places.” Where are the likely places? As your textbook tells you, there are three kinds of likely places. These are: (i) a tangency between an indiﬀerence curve and the budget line; (ii) a kink in an indiﬀerence curve; (iii) a “corner” where the consumer specializes in consuming just one good. Here is how you ﬁnd a point of tangency if we are told the consumer’s utility function, the prices of both goods, and the consumer’s income. The budget line and an indiﬀerence curve are tangent at a point (x1 , x2 ) if they have the same slope at that point. Now the slope of an indiﬀerence curve at (x1 , x2 ) is the ratio −M U1 (x1 , x2 )/M U2 (x1 , x2 ). (This slope is also known as the marginal rate of substitution.) The slope of the budget line is −p1 /p2 . Therefore an indiﬀerence curve is tangent to the budget line at the point (x1 , x2 ) when M U1 (x1 , x2 )/M U2 (x1 , x2 ) = p1 /p2 . This gives us one equation in the two unknowns, x1 and x2 . If we hope to solve for the x’s, we need another equation. That other equation is the budget equation p1 x1 + p2 x2 = m. With these two equations you can solve for (x1 , x2 ).∗ For equilibrium at kinks or at corners, we don’t need the slope of the indiﬀerence curves to equal the slope of the budget line. So we don’t have the tangency equation to work with. But we still have the budget equation. The second equation that you can use is an equation that tells you that you are at one of the kinky points or at a corner. You will see exactly how this works when you work a few exercises. Example: A consumer has the utility function U (x1 , x2 ) = min{x1 , 3x2 }. The price of x1 is 2, the price of x2 is 1, and her income is 140. Her indiﬀerence curves are L-shaped. The corners of the L’s all lie along the line x1 = 3x2 . She will choose a combination at one of the corners, so this gives us one of the two equations we need for ﬁnding the unknowns x1 and x2 . The second equation is her budget equation, which is 2x1 + x2 = 140. Solve these two equations to ﬁnd that x1 = 60 and x2 = 20. So we know that the consumer chooses the bundle (x1 , x2 ) = (60, 20). When you have ﬁnished these exercises, we hope that you will be able to do the following: • Calculate the best bundle a consumer can aﬀord at given prices and income in the case of simple utility functions where the best aﬀordable bundle happens at a point of tangency. • Find the best aﬀordable bundle, given prices and income for a consumer with kinked indiﬀerence curves. • Recognize standard examples where the best bundle a consumer can aﬀord happens at a corner of the budget set. • Draw a diagram illustrating each of the above types of equilibrium. Example: A consumer has the utility function U (x1 , x2 ) = x21 x2 . The price of good 1 is p1 = 1, the price of good 2 is p2 = 3, and his income is 180. Then, M U1 (x1 , x2 ) = 2x1 x2 and M U2 (x1 , x2 ) = x21 . Therefore his marginal rate of substitution is −M U1 (x1 , x2 )/M U2 (x1 , x2 ) = −2x1 x2 /x21 = −2x2 /x1 . This implies that his indiﬀerence curve will be tangent to his budget line when −2x2 /x1 = −p1 /p2 = −1/3. Simplifying this expression, we have 6x2 = x1 . This is one of the two equations we need to solve for the two unknowns, x1 and x2 . The other equation is the budget equation. In this case the budget equation is x1 + 3x2 = 180. Solving these two equations in two unknowns, we ﬁnd x1 = 120 and ∗ Some people have trouble remembering whether the marginal rate of substitution is −M U1 /M U2 or −M U2 /M U1 . It isn’t really crucial to remember which way this goes as long as you remember that a tangency happens when the marginal utilities of any two goods are in the same proportion as their prices. NAME 51 • Apply the methods you have learned to choices made with some kinds of nonlinear budgets that arise in real-world situations. 5.1 (0) We begin again with Charlie of the apples and bananas. Recall that Charlie’s utility function is U (xA , xB ) = xA xB . Suppose that the price of apples is 1, the price of bananas is 2, and Charlie’s income is 40. (a) On the graph below, use blue ink to draw Charlie’s budget line. (Use a ruler and try to make this line accurate.) Plot a few points on the indiﬀerence curve that gives Charlie a utility of 150 and sketch this curve with red ink. Now plot a few points on the indiﬀerence curve that gives Charlie a utility of 300 and sketch this curve with black ink or pencil. 52 CHOICE (Ch. 5) (g) The best bundle that Charlie can aﬀord must lie somewhere on the line you just penciled in. It must also lie on his budget line. If the point is outside of his budget line, he can’t aﬀord it. If the point lies inside of his budget line, he can aﬀord to do better by buying more of both goods. On your graph, label this best aﬀordable bundle with an E. This Bananas 40 Red curves happens where xA = 20 and xB = 10. Verify your answer by solving the two simultaneous equations given by his budget equation and the tangency condition. 30 (h) What is Charlie’s utility if he consumes the bundle (20, 10)? Black curve Pencil line 20 it, or never touch it? 10 200. (i) On the graph above, use red ink to draw his indiﬀerence curve through (20,10). Does this indiﬀerence curve cross Charlie’s budget line, just touch Just touch it. e Blue budget line 5.2 (0) Clara’s utility function is U (X, Y ) = (X + 2)(Y + 1), where X is her consumption of good X and Y is her consumption of good Y . a (a) Write an equation for Clara’s indiﬀerence curve that goes through the 0 10 20 30 36 point (X, Y ) = (2, 8). Y = X+2 − Clara’s indiﬀerence curve for U = 36. 40 Apples (b) Can Charlie aﬀord any bundles that give him a utility of 150? (c) Can Charlie aﬀord any bundles that give him a utility of 300? Y U=36 12 No. 11 8 (e) Neither of the indiﬀerence curves that you drew is tangent to Charlie’s budget line. Let’s try to ﬁnd one that is. At any point, (xA , xB ), Charlie’s marginal rate of substitution is a function of xA and xB . In fact, if you calculate the ratio of marginal utilities for Charlie’s utility function, you will ﬁnd that Charlie’s marginal rate of substitution is M RS(xA , xB ) = −xB /xA . This is the slope of his indiﬀerence curve at (xA , xB ). The −1/2 On the axes below, sketch 16 Yes. (d) On your graph, mark a point that Charlie can aﬀord and that gives him a higher utility than 150. Label that point A. slope of Charlie’s budget line is 1. 4 0 (give a numerical answer). 8 11 12 16 X (f ) Write an equation that implies that the budget line is tangent to an indiﬀerence curve at (xA , xB ). −xB /xA = −1/2. There are many solutions to this equation. Each of these solutions corresponds to a point on a diﬀerent indiﬀerence curve. Use pencil to draw a line that passes through all of these points. 4 (b) Suppose that the price of each good is 1 and that Clara has an income of 11. Draw in her budget line. Can Clara achieve a utility of 36 with this budget? Yes. NAME 53 54 CHOICE (Ch. 5) (c) At the commodity bundle, (X, Y ), Clara’s marginal rate of substitution is Y +1 − X+2 . Berries 20 (d) If we set the absolute value of the MRS equal to the price ratio, we have the equation Y +1 X+2 15 = 1. 10 X + Y = 11. (e) The budget equation is Red curve Blue line 5 Red curve (f ) Solving these two equations for the two unknowns, X and Y , we ﬁnd X= 5 Pencil line Blue line 6. and Y = 0 5.3 (0) Ambrose, the nut and berry consumer, has a utility function √ U (x1 , x2 ) = 4 x1 + x2 , where x1 is his consumption of nuts and x2 is his consumption of berries. (a) The commodity bundle (25, 0) gives Ambrose a utility of 20. Other points that give him the same utility are (16, 4), (9, 8 ), (4, ), (1, 16 ), and (0, 20 ). Plot these points on the axes below and draw a red indiﬀerence curve through them. 10 5 15 20 25 30 Nuts (f ) Now let us explore a case where there is a “boundary solution.” Suppose that the price of nuts is still 1 and the price of berries is 2, but Ambrose’s income is only 9. Draw his budget line (in blue). Sketch the indiﬀerence curve that passes through the point (9, 0). What is the slope of his indiﬀerence curve at the point (9, 0)? −2/3. 12 (b) Suppose that the price of a unit of nuts is 1, the price of a unit of berries is 2, and Ambrose’s income is 24. Draw Ambrose’s budget line with blue ink. How many units of nuts does he choose to buy? 16 units. (h) Which is steeper at this point, the budget line or the indiﬀerence curve? 4 units. (d) Find some points on the indiﬀerence curve that gives him a utility of 25 and sketch this indiﬀerence curve (in red). (e) Now suppose that the prices are as before, but Ambrose’s income is 34. Draw his new budget line (with pencil). How many units of nuts will 16 units. he choose? How many units of berries? Indifference curve. (i) Can Ambrose aﬀord any bundles that he likes better than the point (9, 0)? (c) How many units of berries? −1/2. (g) What is the slope of his budget line at this point? No. 5.4 (1) Nancy Lerner is trying to decide how to allocate her time in studying for her economics course. There are two examinations in this course. Her overall score for the course will be the minimum of her scores on the two examinations. She has decided to devote a total of 1,200 minutes to studying for these two exams, and she wants to get as high an overall score as possible. She knows that on the ﬁrst examination if she doesn’t study at all, she will get a score of zero on it. For every 10 minutes that she spends studying for the ﬁrst examination, she will increase her score by one point. If she doesn’t study at all for the second examination she will get a zero on it. For every 20 minutes she spends studying for the second examination, she will increase her score by one point. 9 units. NAME 55 (a) On the graph below, draw a “budget line” showing the various combinations of scores on the two exams that she can achieve with a total of 1,200 minutes of studying. On the same graph, draw two or three “indifference curves” for Nancy. On your graph, draw a straight line that goes through the kinks in Nancy’s indiﬀerence curves. Label the point where this line hits Nancy’s budget with the letter A. Draw Nancy’s indiﬀerence curve through this point. Score on test 2 56 CHOICE (Ch. 5) for the ﬁrst examination, her score on this exam will be x1 = m1 /5. If she spends m2 minutes studying for the second examination, her score on this exam will be x2 = m2 /10. (a) On the graph below, draw a “budget line” showing the various combinations of scores on the two exams that she can achieve with a total of 400 minutes of studying. On the same graph, draw two or three “indiﬀerence curves” for Nancy. On your graph, ﬁnd the point on Nancy’s budget line that gives her the best overall score in the course. (b) Given that she spends a total of 400 minutes studying, Nancy will 80 maximize her overall score by achieving a score of "L" shaped indifference curves 60 40 examination and a 0 80 on the ﬁrst on the second examination. (c) Her overall score for the course will then be 80. 20 Score on test 2 80 Budget line 0 20 40 60 80 100 120 60 Score on test 1 40 (b) Write an equation for the line passing through the kinks of Nancy’s indiﬀerence curves. x1 = x2 . Max. overall score 20 ; Preference direction (c) Write an equation for Nancy’s budget line. 10x1 + 20x2 = 1, 200. (d) Solve these two equations in two unknowns to determine the intersection of these lines. This happens at the point (x1 , x2 ) = (40, 40). 0 20 40 60 80 Score on test 1 5.6 (0) Elmer’s utility function is U (x, y) = min{x, y 2 }. (a) If Elmer consumes 4 units of x and 3 units of y, his utility is 4. (b) If Elmer consumes 4 units of x and 2 units of y, his utility is 4. (c) If Elmer consumes 5 units of x and 2 units of y, his utility is 4. (e) Given that she spends a total of 1,200 minutes studying, Nancy will maximize her overall score by spending ﬁrst examination and tion. 800 400 minutes studying for the minutes studying for the second examina- 5.5 (1) In her communications course, Nancy also takes two examinations. Her overall grade for the course will be the maximum of her scores on the two examinations. Nancy decides to spend a total of 400 minutes studying for these two examinations. If she spends m1 minutes studying (d) On the graph below, use blue ink to draw the indiﬀerence curve for Elmer that contains the bundles that he likes exactly as well as the bundle (4, 2). NAME 57 58 CHOICE (Ch. 5) (e) On the same graph, use blue ink to draw the indiﬀerence curve for Elmer that contains bundles that he likes exactly as well as the bundle (1, 1) and the indiﬀerence curve that passes through the point (16, 5). Y (f ) On your graph, use black ink to show the locus of points at which Elmer’s indiﬀerence curves have kinks. What is the equation for this curve? 16 x = y2. 12 (g) On the same graph, use black ink to draw Elmer’s budget line when the price of x is 1, the price of y is 2, and his income is 8. What bundle does Elmer choose in this situation? 8 (4,2). Red line Black (3,3) curve 4 y Blue curve 16 Blue curve 12 0 4 8 12 16 Blue curves X 8 Black line Chosen bundle 4 (16,5) (b) On the same graph, use red ink to draw Linus’s budget line if the price of x is 1 and the price of y is 2 and his income is 8. What bundle Black curve does Linus choose in this situation? 0 8 4 12 16 20 (0,4). 24 x (h) Suppose that the price of x is 10 and the price of y is 15 and Elmer buys 100 units of x. What is Elmer’s income? 1,150. (Hint: At ﬁrst you might think there is too little information to answer this question. But think about how much y he must be demanding if he chooses 100 units of x.) 5.7 (0) Linus has the utility function U (x, y) = x + 3y. (a) On the graph below, use blue ink to draw the indiﬀerence curve passing through the point (x, y) = (3, 3). Use black ink to sketch the indiﬀerence curve connecting bundles that give Linus a utility of 6. NAME 59 (c) What bundle would Linus choose if the price of x is 1, the price of y is 4, and his income is 8? (8,0). 5.8 (2) Remember our friend Ralph Rigid from Chapter 3? His favorite diner, Food for Thought, has adopted the following policy to reduce the crowds at lunch time: if you show up for lunch t hours before or after 12 noon, you get to deduct t dollars from your bill. (This holds for any fraction of an hour as well.) 60 CHOICE (Ch. 5) Food Money 120 20 100 December 15 Red curves 10 5 0 yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; Blue budget set yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;;;;; 10 11 12 September August 80 1 2 Time Blue budget constraint Black budget constraint Red budget constraint 60 40 20 0 10 20 30 40 50 60 70 80 90 100 Temperature (a) Use blue ink to show Ralph’s budget set. On this graph, the horizontal axis measures the time of day that he eats lunch, and the vertical axis measures the amount of money that he will have to spend on things other than lunch. Assume that he has $20 total to spend and that lunch at noon costs $10. (Hint: How much money would he have left if he ate at noon? at 1 P.M.? at 11 A.M.?) (b) Recall that Ralph’s preferred lunch time is 12 noon, but that he is willing to eat at another time if the food is suﬃciently cheap. Draw some red indiﬀerence curves for Ralph that would be consistent with his choosing to eat at 11 A.M. 5.9 (0) Joe Grad has just arrived at the big U. He has a fellowship that covers his tuition and the rent on an apartment. In order to get by, Joe has become a grader in intermediate price theory, earning $100 a month. Out of this $100 he must pay for his food and utilities in his apartment. His utilities expenses consist of heating costs when he heats his apartment and air-conditioning costs when he cools it. To raise the temperature of his apartment by one degree, it costs $2 per month (or $20 per month to raise it ten degrees). To use air-conditioning to cool his apartment by a degree, it costs $3 per month. Whatever is left over after paying the utilities, he uses to buy food at $1 per unit. (a) When Joe ﬁrst arrives in September, the temperature of his apartment is 60 degrees. If he spends nothing on heating or cooling, the temperature in his room will be 60 degrees and he will have $100 left to spend on food. If he heated the room to 70 degrees, he would have $80 left to spend on food. If he cooled the room to 50 degrees, he would have $70 left to spend on food. On the graph below, show Joe’s September budget constraint (with black ink). (Hint: You have just found three points that Joe can aﬀord. Apparently, his budget set is not bounded by a single straight line.) (b) In December, the outside temperature is 30 degrees and in August poor Joe is trying to understand macroeconomics while the temperature outside is 85 degrees. On the same graph you used above, draw Joe’s budget constraints for the months of December (in blue ink) and August (in red ink). (c) Draw a few smooth (unkinky) indiﬀerence curves for Joe in such a way that the following are true. (i) His favorite temperature for his apartment would be 65 degrees if it cost him nothing to heat it or cool it. (ii) Joe chooses to use the furnace in December, air-conditioning in August, and neither in September. (iii) Joe is better oﬀ in December than in August. (d) In what months is the slope of Joe’s budget constraint equal to the slope of his indiﬀerence curve? August and December. NAME 61 (e) In December Joe’s marginal rate of substitution between food and degrees Fahrenheit is -2. In August, his MRS is 3. (f ) Since Joe neither heats nor cools his apartment in September, we cannot determine his marginal rate of substitution exactly, but we do know that it must be no smaller than 3. -2 and no larger than (Hint: Look carefully at your graph.) 62 (Ch. 5) things. Therefore its budget line has the equation Use blue ink to draw this budget line. 20,000 and expenditure on other instructional materials is 50,000. −.5. (f ) The slope of the ﬂatter line segment is −1. steeper segment is Plan A: This plan would give a grant of $10,000 to each high school in the state that the school could spend as it wished. Plan B: This plan would give a $10,000 grant to any high school, so long as the school spent at least $10,000 more than it currently spends on computers. Any high school can choose not to participate, in which case it does not receive the grant, but it doesn’t have to increase its expenditure on computers. Plan C: Plan C is a “matching grant.” For every dollar’s worth of computers that a high school orders, the state will give the school 50 cents. Plan D: This plan is like plan C, except that the maximum amount of matching funds that any high school could get from the state would be limited to $10,000. The slope of the Use pencil to draw this budget line. Thousands of dollars worth of other things 60 Black budget line (plan A) Pencil budget line (plan D) 50 Blue budget line (plan C) 40 Red budget line (plan B) 30 (a) Write an equation for Central High School’s budget if plan A is 20 adopted. C + X = 70, 000. Use black ink to draw the budget line for Central High School if plan A is adopted. 10 (b) If plan B is adopted, the boundary of Central High School’s budget set has two separate downward-sloping line segments. One of these segments describes the cases where C.H.S. spends at least $30,000 on computers. This line segment runs from the point (C, X) = (70, 000, 0) to the point 10 0 20 (30,000, 40,000). (C, X) = .5C+X=60,000. (e) If plan D is adopted, the school district’s budget consists of two line segments that intersect at the point where expenditure on computers is 5.10 (0) Central High School has $60,000 to spend on computers and other stuﬀ, so its budget equation is C + X = 60, 000, where C is expenditure on computers and X is expenditures on other things. C.H.S. currently plans to spend $20,000 on computers. The State Education Commission wants to encourage “computer literacy” in the high schools under its jurisdiction. The following plans have been proposed. 30 40 50 60 Thousands of dollars worth of computers (c) Another line segment corresponds to the cases where C.H.S. spends less than $30,000 on computers. This line segment runs from (C, X) = (30,000,30,000) to the point (C, X) = (0, 60, 000). Use red ink to draw these two line segments. NAME 63 (a) If the state adopts none of the new plans, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. CHOICE (d) If plan C is adopted and Central High School spends C dollars on computers, then it will have X = 60, 000 − .5C dollars left to spend on other 5.11 (0) Suppose that Central High School has preferences that can be represented by the utility function U (C, X) = CX 2 . Let us try to determine how the various plans described in the last problem will aﬀect the amount that C.H.S. spends on computers. 64 CHOICE (Ch. 5) Other goods 16 20,000. Pencil curve 12 Blue curve (b) If plan A is adopted, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. 23,333. 8 Red line Black line 4 (c) On your graph, sketch the indiﬀerence curve that passes through the point (30,000, 40,000) if plan B is adopted. At this point, which is steeper, the indiﬀerence curve or the budget line? The budget line. 0 20 40 60 80 100 120 Local phone calls (d) If plan B is adopted, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. (Hint: Look at your graph.) 30,000. (e) If plan C is adopted, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. 40,000. (f ) If plan D is adopted, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. 23,333. 5.12 (0) The telephone company allows one to choose between two diﬀerent pricing plans. For a fee of $12 per month you can make as many local phone calls as you want, at no additional charge per call. Alternatively, you can pay $8 per month and be charged 5 cents for each local phone call that you make. Suppose that you have a total of $20 per month to spend. (a) On the graph below, use black ink to sketch a budget line for someone who chooses the ﬁrst plan. Use red ink to draw a budget line for someone who chooses the second plan. Where do the two budget lines cross? (80, 8). (b) On the graph above, use pencil to draw indiﬀerence curves for someone who prefers the second plan to the ﬁrst. Use blue ink to draw an indiﬀerence curve for someone who prefers the ﬁrst plan to the second. 5.13 (1) This is a puzzle—just for fun. Lewis Carroll (1832-1898), author of Alice in Wonderland and Through the Looking Glass, was a mathematician, logician, and political scientist. Carroll loved careful reasoning about puzzling things. Here Carroll’s Alice presents a nice bit of economic analysis. At ﬁrst glance, it may seem that Alice is talking nonsense, but, indeed, her reasoning is impeccable. “I should like to buy an egg, please.” she said timidly. “How do you sell them?” “Fivepence farthing for one—twopence for two,” the Sheep replied. “Then two are cheaper than one?” Alice said, taking out her purse. “Only you must eat them both if you buy two,” said the Sheep. “Then I’ll have one please,” said Alice, as she put the money down on the counter. For she thought to herself, “They mightn’t be at all nice, you know.” (a) Let us try to draw a budget set and indiﬀerence curves that are consistent with this story. Suppose that Alice has a total of 8 pence to spend and that she can buy either 0, 1, or 2 eggs from the Sheep, but no fractional eggs. Then her budget set consists of just three points. The point where she buys no eggs is (0, 8). Plot this point and label it A. On your graph, the point where she buys 1 egg is (1, 2 34 ). (A farthing is 1/4 of a penny.) Plot this point and label it B. (b) The point where she buys 2 eggs is (2, 6). Plot this point and label it C. If Alice chooses to buy 1 egg, she must like the bundle B better than either the bundle A or the bundle C. Draw indiﬀerence curves for Alice that are consistent with this behavior. NAME 65 66 CHOICE 68 DEMAND (Ch. 5) Other goods 8 a 6 c 4 2 b 0 1 Chapter 6 2 3 4 Eggs NAME Demand Introduction. In the previous chapter, you found the commodity bundle that a consumer with a given utility function would choose in a speciﬁc price-income situation. In this chapter, we take this idea a step further. We ﬁnd demand functions, which tell us for any prices and income you might want to name, how much of each good a consumer would want. In general, the amount of each good demanded may depend not only on its own price, but also on the price of other goods and on income. Where there are two goods, we write demand functions for Goods 1 and 2 as x1 (p1 , p2 , m) and x2 (p1 , p2 , m).∗ When the consumer is choosing positive amounts of all commodities and indiﬀerence curves have no kinks, the consumer chooses a point of tangency between her budget line and the highest indiﬀerence curve that it touches. Example: Consider a consumer with utility function U (x1 , x2 ) = (x1 + 2)(x2 + 10). To ﬁnd x1 (p1 , p2 , m) and x2 (p1 , p2 , m), we need to ﬁnd a commodity bundle (x1 , x2 ) on her budget line at which her indiﬀerence curve is tangent to her budget line. The budget line will be tangent to the indiﬀerence curve at (x1 , x2 ) if the price ratio equals the marginal rate of substitution. For this utility function, M U1 (x1 , x2 ) = x2 + 10 and M U2 (x1 , x2 ) = x1 + 2. Therefore the “tangency equation” is p1 /p2 = (x2 + 10)/(x1 + 2). Cross-multiplying the tangency equation, one ﬁnds p1 x1 + 2p1 = p2 x2 + 10p2 . The bundle chosen must also satisfy the budget equation, p1 x1 + p2 x2 = m. This gives us two linear equations in the two unknowns, x1 and x2 . You can solve these equations yourself, using high school algebra. You will ﬁnd that the solution for the two “demand functions” is m − 2p1 + 10p2 2p1 m + 2p1 − 10p2 x2 = . 2p2 x1 = There is one thing left to worry about with the “demand functions” we just found. Notice that these expressions will be positive only if m−2p1 + 10p2 > 0 and m+2p1 −10p2 > 0. If either of these expressions is negative, then it doesn’t make sense as a demand function. What happens in this ∗ For some utility functions, demand for a good may not be aﬀected by all of these variables. For example, with Cobb-Douglas utility, demand for a good depends on the good’s own price and on income but not on the other good’s price. Still, there is no harm in writing demand for Good 1 as a function of p1 , p2 , and m. It just happens that the derivative of x1 (p1 , p2 , m) with respect to p2 is zero. (Ch. 6) case is that the consumer will choose a “boundary solution” where she consumes only one good. At this point, her indiﬀerence curve will not be tangent to her budget line. When a consumer has kinks in her indiﬀerence curves, she may choose a bundle that is located at a kink. In the problems with kinks, you will be able to solve for the demand functions quite easily by looking at diagrams and doing a little algebra. Typically, instead of ﬁnding a tangency equation, you will ﬁnd an equation that tells you “where the kinks are.” With this equation and the budget equation, you can then solve for demand. You might wonder why we pay so much attention to kinky indiﬀerence curves, straight line indiﬀerence curves, and other “funny cases.” Our reason is this. In the funny cases, computations are usually pretty easy. But often you may have to draw a graph and think about what you are doing. That is what we want you to do. Think and ﬁddle with graphs. Don’t just memorize formulas. Formulas you will forget, but the habit of thinking will stick with you. When you have ﬁnished this workout, we hope that you will be able to do the following: • Find demand functions for consumers with Cobb-Douglas and other similar utility functions. • Find demand functions for consumers with quasilinear utility functions. • Find demand functions for consumers with kinked indiﬀerence curves and for consumers with straight-line indiﬀerence curves. • Recognize complements and substitutes from looking at a demand curve. • Recognize normal goods, inferior goods, luxuries, and necessities from looking at information about demand. • Calculate the equation of an inverse demand curve, given a simple demand equation. 6.1 (0) Charlie is back—still consuming apples and bananas. His utility function is U (xA , xB ) = xA xB . We want to ﬁnd his demand function for apples, xA (pA , pB , m), and his demand function for bananas, xB (pA , pB , m). (a) When the prices are pA and pB and Charlie’s income is m, the equation for Charlie’s budget line is pA xA +pB xB = m. The slope of Charlie’s indifference curve at the bundle (xA , xB ) is −M U1 (xA , xB )/M U2 (xA , xB ) = −xB /xA . The slope of Charlie’s budget line is −pA /pB . Charlie’s indiﬀerence curve will be tangent to his budget line at the point (xA , xB ) if the following equation is satisﬁed: pA /pB = xB /xA . NAME 69 (b) You now have two equations, the budget equation and the tangency equation, that must be satisﬁed by the bundle demanded. Solve these two equations for xA and xB . Charlie’s demand function for apm ples is xA (pA , pB , m) = 2p , and his demand function for bananas is A m xB (pA , pB , m) = 2p . B (c) In general, the demand for both commodities will depend on the price of both commodities and on income. But for Charlie’s utility function, the demand function for apples depends only on income and the price of apples. Similarly, the demand for bananas depends only on income and the price of bananas. Charlie always spends the same fraction of his income on bananas. What fraction is this? 1/2. 6.2 (0) Douglas Cornﬁeld’s preferences are represented by the utility function u(x1 , x2 ) = x21 x32 . The prices of x1 and x2 are p1 and p2 . 70 DEMAND (Ch. 6) (b) Let us ﬁnd his demand for berries. Now we need the budget equation. In Part (a), you solved for the amount of x1 that he will demand. The budget equation tells us that p1 x1 + p2 x2 = M . Plug the solution that you found for x1 into the budget equation and solve for x2 as a function p M of income and prices. The answer is x2 = p − 4 p2 . 2 1 (c) When we visited Ambrose in Chapter 5, we looked at a “boundary solution,” where Ambrose consumed only nuts and no berries. In that example, p1 = 1, p2 = 2, and M = 9. If you plug these numbers into the if and only if M > (b) If Cornﬁeld’s budget line is tangent to his indiﬀerence curve at (x1 , x2 ), then pp12 xx12 = 2/3. (Hint: Look at the equation that equates the slope of his indiﬀerence curve with the slope of his budget line.) When he is consuming the best bundle he can aﬀord, what fraction of his income does Douglas spend on x1 ? 16. 6.4 (0) Donald Fribble is a stamp collector. The only things other than stamps that Fribble consumes are Hostess Twinkies. It turns out that Fribble’s preferences are represented by the utility function u(s, t) = s + ln t where s is the number of stamps he collects and t is the number of Twinkies he consumes. The price of stamps is ps and the price of Twinkies is pt . Donald’s income is m. (a) Write an expression that says that the ratio of Fribble’s marginal utility for Twinkies to his marginal utility for stamps is equal to the ratio 2/5. (c) Other members of Doug’s family have similar utility functions, but the exponents may be diﬀerent, or their utilities may be multiplied by a positive constant. If a family member has a utility function U (x, y) = cxa1 xb2 where a, b, and c are positive numbers, what fraction of his or her income will that family member spend on x1 ? a/(a+b). 6.3 (0) Our thoughts return to Ambrose and his nuts and berries. Am√ brose’s utility function is U (x1 , x2 ) = 4 x1 + x2 , where x1 is his consumption of nuts and x2 is his consumption of berries. (a) Let us ﬁnd his demand function for nuts. The slope of Ambrose’s indiﬀerence curve at (x1 , x2 ) is , and x2 = −3.5 . Since we get a negative solution for x2 , it must be that the budget line x1 + 2x2 = 9 is not tangent to an indiﬀerence curve when x2 ≥ 0. The best that Ambrose can do with this budget is to spend all of his income on nuts. Looking at the formulas, we see that at the prices p1 = 1 and p2 = 2, Ambrose will demand a positive amount of both goods (a) The slope of Cornﬁeld’s indiﬀerence curve at the point (x1 , x2 ) is −2x2 /3x1 . 16 formulas we found in Parts (a) and (b), you ﬁnd x1 = − √2x1 . Setting this slope equal to the slope of the budget line, you can solve for x1 without even using the 2 2p2 budget equation. The solution is x1 = . p 1 NAME 71 of the price of Twinkies to the price of stamps. 1/t = pt /ps . (Hint: The derivative of ln t with respect to t is 1/t, and the derivative of s with respect to s is 1.) (b) You can use the equation you found in the last part to show that if he buys both goods, Donald’s demand function for Twinkies depends only on the price ratio and not on his income. Donald’s demand function for Twinkies is (c) Notice that for this special utility function, if Fribble buys both goods, then the total amount of money that he spends on Twinkies has the peculiar property that it depends on only one of the three variables m, pt , and ps , namely the variable ps . (Hint: The amount of money that he spends on Twinkies is pt t(ps , pt , m).) 72 DEMAND (d) Since there are only two goods, any money that is not spent on Twinkies must be spent on stamps. Use the budget equation and Donald’s demand function for Twinkies to ﬁnd an expression for the number of stamps he will buy if his income is m, the price of stamps is ps and the m price of Twinkies is pt . s = p − 1. s What would his demand for Twinkies be? (Hint: Recall the discussion of boundary optimum.) (Ch. 6) 8-ounce cans 40 Blue budget line 30 (e) The expression you just wrote down is negative if m < ps . Surely it makes no sense for him to be demanding negative amounts of postage stamps. If m < ps , what would Fribble’s demand for postage stamps be? s=0 t(ps , pt , m) = ps /pt . 20 t = m/pt . Red curves 10 (f ) Donald’s wife complains that whenever Donald gets an extra dollar, Red curve he always spends it all on stamps. Is she right? (Assume that m > ps .) 0 10 20 30 40 16-ounce cans Yes. (a) At these prices, which size can will she buy, or will she buy some of (g) Suppose that the price of Twinkies is $2 and the price of stamps is $1. On the graph below, draw Fribble’s Engel curve for Twinkies in red ink and his Engel curve for stamps in blue ink. (Hint: First draw the Engel curves for incomes greater than $1, then draw them for incomes less than $1.) Income each? 16-ounce cans. (b) Suppose that the price of 16-ounce beers remains $1 and the price of 8-ounce beers falls to $.55. Will she buy more 8-ounce beers? 8 beers will she buy then? Red line 75 cans. (d) If the price of 16-ounce beers is $1 each and if Shirley chooses some 8-ounce beers and some 16-ounce beers, what must be the price of 8-ounce 6 Blue line beers? 4 $.50. (e) Now let us try to describe Shirley’s demand function for 16-ounce beers as a function of general prices and income. Let the prices of 8-ounce and 16-ounce beers be p8 and p16 , and let her income be m. If p16 < 2p8 , then 2 the number of 16-ounce beers she will demand is 1 m/p16 . then the number of 16-ounce beers she will demand is 0.5 0 No. (c) What if the price of 8-ounce beers falls to $.40? How many 8-ounce 2 4 6 8 Quantities 6.5 (0) Shirley Sixpack, as you will recall, thinks that two 8-ounce cans of beer are exactly as good as one 16-ounce can of beer. Suppose that these are the only sizes of beer available to her and that she has $30 to spend on beer. Suppose that an 8-ounce beer costs $.75 and a 16-ounce beer costs $1. On the graph below, draw Shirley’s budget line in blue ink, and draw some of her indiﬀerence curves in red. 2 If p16 > 2p8 , 0. If p16 = p8 , she will be indiﬀerent between any aﬀordable combinations. 6.6 (0) Miss Muﬀet always likes to have things “just so.” In fact the only way she will consume her curds and whey is in the ratio of 2 units of whey per unit of curds. She has an income of $20. Whey costs $.75 per unit. Curds cost $1 per unit. On the graph below, draw Miss Muﬀet’s budget line, and plot some of her indiﬀerence curves. (Hint: Have you noticed something kinky about Miss Muﬀet?) NAME 73 (a) How many units of curds will Miss Muﬀet demand in this situation? 8 units. 16 units. How many units of whey? 74 DEMAND (Ch. 6) (c) If Mary had only 144 square feet in her garden, how many cockle 36. shells would she grow? Whey (d) If Mary grows both silver bells and cockle shells, then we know that 32 w = 2c the number of square feet in her garden must be greater than 24 16 Indifference curves 8 Budget line 0 8 16 24 192. 6.8 (0) Casper consumes cocoa and cheese. He has an income of $16. Cocoa is sold in an unusual way. There is only one supplier and the more cocoa one buys from him, the higher the price one has to pay per unit. In fact, x units of cocoa will cost Casper a total of x2 dollars. Cheese is sold in the usual way at a price of $2 per unit. Casper’s budget equation, therefore, is x2 + 2y = 16 where x is his consumption of cocoa and y is his consumption of cheese. Casper’s utility function is U (x, y) = 3x + y. (a) On the graph below, draw the boundary of Casper’s budget set in blue ink. Use red ink to sketch two or three of his indiﬀerence curves. 32 Curds Cheese (b) Write down Miss Muﬀet’s demand function for whey as a function of the prices of curds and whey and of her income, where pc is the price 16 of curds, pw is the price of whey, and m is her income. D(pc , pw , m) = m Red indifference curves 12 . (Hint: You can solve for her demands by solving two equa- pw +pc /2 tions in two unknowns. One equation tells you that she consumes twice as much whey as curds. The second equation is her budget equation.) 8 Blue budget line 4 6.7 (1) Mary’s utility function is U (b, c) = b + 100c − c , where b is the number of silver bells in her garden and c is the number of cockle shells. She has 500 square feet in her garden to allocate between silver bells and cockle shells. Silver bells each take up 1 square foot and cockle shells each take up 4 square feet. She gets both kinds of seeds for free. 2 0 4 8 12 16 Cocoa (a) To maximize her utility, given the size of her garden, Mary should plant 308 silver bells and 48 cockle shells. (Hint: Write down her “budget constraint” for space. Solve the problem as if it were an ordinary demand problem.) (b) Write an equation that says that at the point (x, y), the slope of Casper’s budget “line” equals the slope of his indiﬀerence “curve.” 2x/2 = 3/1. (b) If she suddenly acquires an extra 100 square feet for her garden, how much should she increase her planting of silver bells? silver bells. cockle shells? 100 extra How much should she increase her planting of Not at all. NAME 75 3 Casper demands units of cocoa and 3.5 units of cheese. 6.9 (0) Perhaps after all of the problems with imaginary people and places, you would like to try a problem based on actual fact. The U.S. government’s Bureau of Labor Statistics periodically makes studies of family budgets and uses the results to compile the consumer price index. These budget studies and a wealth of other interesting economic data can be found in the annually published Handbook of Labor Statistics. The 76 DEMAND (Ch. 6) tables below report total current consumption expenditures and expenditures on certain major categories of goods for 5 diﬀerent income groups in the United States in 1961. People within each of these groups all had similar incomes. Group A is the lowest income group and Group E is the highest. (d) Which of these goods satisfy your textbook’s deﬁnition of necessity Table 6.1 (e) On the graph below, use the information from Table 6.1 to draw “Engel curves.” (Use total expenditure on current consumption as income for purposes of drawing this curve.) Use red ink to draw the Engel curve for food prepared at home. Use blue ink to draw an Engel curve for food away from home. Use pencil to draw an Engel curve for clothing. How does the shape of an Engel curve for a luxury diﬀer from the shape of Expenditures by Category for Various Income Groups in 1961 Income Group Food Prepared at Home Food Away from Home Housing Clothing Transportation Other Total Expenditures A 465 68 626 119 139 364 1781 B 783 171 1090 328 519 745 3636 C 1078 213 1508 508 826 1039 5172 D 1382 384 2043 830 1222 1554 7415 E 1848 872 4205 1745 2048 3490 14208 goods at most income levels? Food prepared at home, housing. an Engel curve for a necessity? The curve for a luxury gets flatter as income rises, the curve for a necessity gets steeper. Total expenditures (thousands of dollars) 12 Table 6.2 Blue line Percentage Allocation of Family Budget Red line Pencil line 9 Income Group A B D E Food Prepared at Home 26 22 C 21 19 13 Food Away from Home 3.8 4.7 4.1 5.2 6.1 Housing 35 30 29 28 30 Clothing 6.7 9.0 9.8 11 12 Transportation 7.8 14 16 17 14 (a) Complete Table 6.2. (b) Which of these goods are normal goods? All of them. (c) Which of these goods satisfy your textbook’s deﬁnition of luxury goods at most income levels? Food away from home, clothing, transportation. 6 3 0 750 1500 2250 3000 Expenditure on specific goods 6.10 (0) Percy consumes cakes and ale. His demand function for cakes is qc = m − 30pc + 20pa , where m is his income, pa is the price of ale, pc is the price of cakes, and qc is his consumption of cakes. Percy’s income is $100, and the price of ale is $1 per unit. (a) Is ale a substitute for cakes or a complement? Explain. substitute. A An increase in the price of ale increases demand for cakes. NAME 77 78 (b) Write an equation for Percy’s demand function for cakes where income and the price of ale are held ﬁxed at $100 and $1. DEMAND (Ch. 6) (b) If the Stouts have homothetic preferences, how much more food will qc = 120−30pc . they buy once they enter the food stamp program? 5 units. (c) Write an equation for Percy’s inverse demand function for cakes where income is $100 and the price of ale remains at $1. At what price would Percy buy 30 cakes? Percy’s inverse demand curve for cakes. $3. Dollars worth of other things pc = 4 − qc /30. 120 Use blue ink to draw 100 (d) Suppose that the price of ale rises to $2.50 per unit and remains there. Write an equation for Percy’s inverse demand for cakes. 5 − qc /30. pc = 80 Use red ink to draw in Percy’s new inverse demand curve for cakes. 60 Price New consumption point a 40 4 Red budget line 20 Red Line 3 Black budget line 2 Blue Line 20 0 45 40 60 80 100 120 Food Calculus 1 0 30 60 90 120 Number of cakes 6.11 (0) Richard and Mary Stout have fallen on hard times, but remain rational consumers. They are making do on $80 a week, spending $40 on food and $40 on all other goods. Food costs $1 per unit. On the graph below, use black ink to draw a budget line. Label their consumption bundle with the letter A. (a) The Stouts suddenly become eligible for food stamps. This means that they can go to the agency and buy coupons that can be exchanged for $2 worth of food. Each coupon costs the Stouts $1. However, the maximum number of coupons they can buy per week is 10. On the graph, draw their new budget line with red ink. NAME Yes. 79 Write a general formula for this student’s overall score for the M course as a function of the three variables, P1 , P2 , and M : S = P +P . 1 2 If this student wants to get an overall score of S, with the smallest possible total amount of studying, this student must spend studying for the ﬁrst exam and P2 S P1 S minutes studying for the second exam. (b) Suppose that a student has the utility function U (S, M ) = S − A 2 M , 2 where S is the student’s overall score for the course, M is the number of minutes the student spends studying, and A is a variable that reﬂects how much the student dislikes studying. In Part (a) of this problem, you found that a student who studies for M minutes and allocates this time wisely between the two exams will get an overall score of S = P1M +P2 . Substitute P1M +P2 for S in the utility function and then diﬀerentiate with respect to M to ﬁnd the amount of study time, M , that maximizes the 1 student’s utility. M = A(P +P ) . Your answer will be a function of 1 2 the variables P1 , P2 , and A. If the student chooses the utility-maximizing amount of study time and allocates it wisely between the two exams, he 1 or she will have an overall score for the course of S = A(P +P )2 . 1 2 (c) Nancy Lerner has a utility function like the one presented above. She chose the utility-maximizing amount of study time for herself. For Nancy, P1 = 10 and P2 = 20. She spent a total of M = 1, 200 minutes studying for the two exams. This gives us enough information to solve for the 1 variable A in Nancy’s utility function. In fact, for Nancy, A = 36,000 . (d) Ed Fungus is a student in Nancy’s class. Ed’s utility function is just like Nancy’s, with the same value of A. But Ed learns more slowly than Nancy. In fact it takes Ed exactly twice as long to learn anything as it takes Nancy, so that for him, P1 = 20 and P2 = 40. Ed also chooses his amount of study time so as to maximize his utility. Find the ratio of the amount of time Ed spends studying to the amount of time Nancy spends studying. 1/2. Will his score for the course be greater than half, equal to half, or less than half of Nancy’s? Less than half. 6.13 (1) Here is a puzzle for you. At ﬁrst glance, it would appear that there is not nearly enough information to answer this question. But when you graph the indiﬀerence curve and think about it a little, you will see that there is a neat, easily calculated solution. 6.12 (2) As you may remember, Nancy Lerner is taking an economics course in which her overall score is the minimum of the number of correct answers she gets on two examinations. For the ﬁrst exam, each correct answer costs Nancy 10 minutes of study time. For the second exam, each correct answer costs her 20 minutes of study time. In the last chapter, you found the best way for her to allocate 1200 minutes between the two exams. Some people in Nancy’s class learn faster and some learn slower than Nancy. Some people will choose to study more than she does, and some will choose to study less than she does. In this section, we will ﬁnd a general solution for a person’s choice of study times and exam scores as a function of the time costs of improving one’s score. (a) Suppose that if a student does not study for an examination, he or she gets no correct answers. Every answer that the student gets right on the ﬁrst examination costs P1 minutes of studying for the ﬁrst exam. Every answer that he or she gets right on the second examination costs P2 minutes of studying for the second exam. Suppose that this student spends a total of M minutes studying for the two exams and allocates the time between the two exams in the most eﬃcient possible way. Will the student have the same number of correct answers on both exams? 80 DEMAND (Ch. 6) Kinko spends all his money on whips and leather jackets. Kinko’s utility function is U (x, y) = min{4x, 2x + y}, where x is his consumption of whips and y is his consumption of leather jackets. Kinko is consuming 15 whips and 10 leather jackets. The price of whips is $10. You are to ﬁnd Kinko’s income. (a) Graph the indiﬀerence curve for Kinko that passes through the point (15, 10). What is the slope of this indiﬀerence curve at (15, 10)? −2. What must be the price of leather jackets if Kinko chooses this point? $5. Now, what is Kinko’s income? 15 × 10 + 10 × 5 = 200. Leather jackets 40 Indifference curve 30 20 2x + y = 40 10 (15,10) 4x = 40 0 10 20 30 40 Whips 82 Chapter 7 NAME Revealed Preference Introduction. In the last section, you were given a consumer’s preferences and then you solved for his or her demand behavior. In this chapter we turn this process around: you are given information about a consumer’s demand behavior and you must deduce something about the consumer’s preferences. The main tool is the weak axiom of revealed preference. This axiom says the following. If a consumer chooses commodity bundle A when she can aﬀord bundle B, then she will never choose bundle B from any budget in which she can also aﬀord A. The idea behind this axiom is that if you choose A when you could have had B, you must like A better than B. But if you like A better than B, then you will never choose B when you can have A. If somebody chooses A when she can aﬀord B, we say that for her, A is directly revealed preferred to B. The weak axiom says that if A is directly revealed preferred to B, then B is not directly revealed preferred to A. Example: Let us look at an example of how you check whether one bundle is revealed preferred to another. Suppose that a consumer buys the bundle A A A A A (xA 1 , x2 ) = (2, 3) at prices (p1 , p2 ) = (1, 4). The cost of bundle (x1 , x2 ) at these prices is (2 × 1) + (3 × 4) = 14. Bundle (2, 3) is directly revealed preferred to all the other bundles that she can aﬀord at prices (1, 4), when she has an income of 14. For example, the bundle (5, 2) costs only 13 at prices (1, 4), so we can say that for this consumer (2, 3) is directly revealed preferred to (1, 4). You will also have some problems about price and quantity indexes. A price index is a comparison of average price levels between two diﬀerent times or two diﬀerent places. If there is more than one commodity, it is not necessarily the case that all prices changed in the same proportion. Let us suppose that we want to compare the price level in the “current year” with the price level in some “base year.” One way to make this comparison is to compare the costs in the two years of some “reference” commodity bundle. Two reasonable choices for the reference bundle come to mind. One possibility is to use the current year’s consumption bundle for the reference bundle. The other possibility is to use the bundle consumed in the base year. Typically these will be diﬀerent bundles. If the baseyear bundle is the reference bundle, the resulting price index is called the Laspeyres price index. If the current year’s consumption bundle is the reference bundle, then the index is called the Paasche price index. Example: Suppose that there are just two goods. In 1980, the prices were (1, 3) and a consumer consumed the bundle (4, 2). In 1990, the prices were (2, 4) and the consumer consumed the bundle (3, 3). The cost of the 1980 bundle at 1980 prices is (1 × 4) + (3 × 2) = 10. The cost of this same bundle at 1990 prices is (2 × 4) + (4 × 2) = 16. If 1980 is treated as the base year and 1990 as the current year, the Laspeyres price ratio NAME 83 REVEALED PREFERENCE (Ch. 7) is 16/10. To calculate the Paasche price ratio, you ﬁnd the ratio of the cost of the 1990 bundle at 1990 prices to the cost of the same bundle at 1980 prices. The 1990 bundle costs (2 × 3) + (4 × 3) = 18 at 1990 prices. The same bundle cost (1 × 3) + (3 × 3) = 12 at 1980 prices. Therefore the Paasche price index is 18/12. Notice that both price indexes indicate that prices rose, but because the price changes are weighted diﬀerently, the two approaches give diﬀerent price ratios. Making an index of the “quantity” of stuﬀ consumed in the two periods presents a similar problem. How do you weight changes in the amount of good 1 relative to changes in the amount of good 2? This time we could compare the cost of the two periods’ bundles evaluated at some reference prices. Again there are at least two reasonable possibilities, the Laspeyres quantity index and the Paasche quantity index. The Laspeyres quantity index uses the base-year prices as the reference prices, and the Paasche quantity index uses current prices as reference prices. Example: In the example above, the Laspeyres quantity index is the ratio of the cost of the 1990 bundle at 1980 prices to the cost of the 1980 bundle at 1980 prices. The cost of the 1990 bundle at 1980 prices is 12 and the cost of the 1980 bundle at 1980 prices is 10, so the Laspeyres quantity index is 12/10. The cost of the 1990 bundle at 1990 prices is 18 and the cost of the 1980 bundle at 1990 prices is 16. Therefore the Paasche quantity index is 18/16. When you have completed this section, we hope that you will be able to do the following: • Decide from given data about prices and consumption whether one commodity bundle is preferred to another. • Given price and consumption data, calculate Paasche and Laspeyres price and quantity indexes. • Use the weak axiom of revealed preferences to make logical deductions about behavior. • Use the idea of revealed preference to make comparisons of well-being across time and across countries. 7.1 (0) When prices are (4, 6), Goldie chooses the bundle (6, 6), and when prices are (6, 3), she chooses the bundle (10, 0). (a) On the graph below, show Goldie’s ﬁrst budget line in red ink and her second budget line in blue ink. Mark her choice from the ﬁrst budget with the label A, and her choice from the second budget with the label B. (b) Is Goldie’s behavior consistent with the weak axiom of revealed preference? 84 No. REVEALED PREFERENCE (Ch. 7) (e) Use pencil to shade the bundles of goods on Freddy’s new red budget line that he deﬁnitely will not purchase with this budget. Is it possible that he would increase his consumption of tomatoes when his budget Good 2 20 changes from the blue line to the red one? 15 No. Tomatoes Blue line 40 10 30 5 a 20 Red line b 0 5 10 10 15 20 Good 1 7.2 (0) Freddy Frolic consumes only asparagus and tomatoes, which are highly seasonal crops in Freddy’s part of the world. He sells umbrellas for a living, which provides a ﬂuctuating income depending on the weather. But Freddy doesn’t mind; he never thinks of tomorrow, so each week he spends as much as he earns. One week, when the prices of asparagus and tomatoes were each $1 a pound, Freddy consumed 15 pounds of each. Use blue ink to show the budget line in the diagram below. Label Freddy’s consumption bundle with the letter A. (a) What is Freddy’s income? $30. (b) The next week the price of tomatoes rose to $2 a pound, but the price of asparagus remained at $1 a pound. By chance, Freddy’s income had changed so that his old consumption bundle of (15,15) was just aﬀordable at the new prices. Use red ink to draw this new budget line on the graph below. Does your new budget line go through the point A? What is the slope of this line? Yes. −1/2. (c) How much asparagus can he aﬀord now if he spent all of his income on asparagus? 45 pounds. (d) What is Freddy’s income now? $45. Blue line Pencil ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; shading ;;;;;;;;;;;;;;;;;; a ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; Red line ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; 0 10 20 30 40 Asparagus 7.3 (0) Pierre consumes bread and wine. For Pierre, the price of bread is 4 francs per loaf, and the price of wine is 4 francs per glass. Pierre has an income of 40 francs per day. Pierre consumes 6 glasses of wine and 4 loaves of bread per day. Bob also consumes bread and wine. For Bob, the price of bread is 1/2 dollar per loaf and the price of wine is 2 dollars per glass. Bob has an income of $15 per day. (a) If Bob and Pierre have the same tastes, can you tell whether Bob is better oﬀ than Pierre or vice versa? Explain. off. Bob is better He can afford Pierre’s bundle and still have income left. (b) Suppose prices and incomes for Pierre and Bob are as above and that Pierre’s consumption is as before. Suppose that Bob spends all of his income. Give an example of a consumption bundle of wine and bread such that, if Bob bought this bundle, we would know that Bob’s tastes are not the same as Pierre’s tastes. 7.5 wine and 0 bread, for example. If they had the same preferences, NAME 85 this violates WARP, since each can afford but rejects the other’s bundle. 7.4 (0) Here is a table of prices and the demands of a consumer named Ronald whose behavior was observed in 5 diﬀerent price-income situations. Situation A B C D E p1 1 1 1 3 1 p2 1 2 1 1 2 x1 5 35 10 5 10 x2 35 10 15 15 10 REVEALED PREFERENCE Horst prefers Nigel’s budget to his own. With Nigel’s budget, he can afford his own (b) Is Ronald’s behavior consistent with the Weak Axiom of Revealed bundle with money left over. (b) Would Nigel prefer to have Horst’s income and prices or his own, or (c) Shade lightly in red ink all of the points that you are certain are worse for Ronald than the bundle C. (d) Suppose that you are told that Ronald has convex and monotonic preferences and that he obeys the strong axiom of revealed preference. Shade lightly in blue ink all of the points that you are certain are at least as good as the bundle C. information to tell. 7.6 (0) Here is a table that illustrates some observed prices and choices for three diﬀerent goods at three diﬀerent prices in three diﬀerent situations. Situation A B C x2 30 20 10 There is too little is there too little information to tell? Yes. 40 (Ch. 7) 7.5 (0) Horst and Nigel live in diﬀerent countries. Possibly they have diﬀerent preferences, and certainly they face diﬀerent prices. They each consume only two goods, x and y. Horst has to pay 14 marks per unit of x and 5 marks per unit of y. Horst spends his entire income of 167 marks on 8 units of x and 11 units of y. Good x costs Nigel 9 quid per unit and good y costs him 7 quid per unit. Nigel buys 10 units of x and 9 units of y. (a) Which prices and income would Horst prefer, Nigel’s income and prices or his own, or is there too little information to tell? Explain your answer. (a) Sketch each of his budget lines and label the point chosen in each case by the letters A, B, C, D, and E. Preference? 86 ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy a ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy Blue shading ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; c ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; d ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; b ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; e ;;;;;;;;;;;;;;;;;;;;; yyyyyyyyyyyyyyyyyyyyy ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; Red shading ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;; ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;; ;; ;;;;;;;;;;;;;;;;;;;;;;;; 0 10 20 30 Prices/Quantities 87 A B C A 28 27 30 B 33 32 30 C 13 17 16 (b) Fill in the entry in Row i and Column j of the table below with a D if the Situation-i bundle is directly revealed preferred to the Situation-j bundle. For example, in Situation A the consumer’s expenditure is $28. We see that at Situation-A prices, he could also aﬀord the Situation-B bundle, which cost 27. Therefore the Situation-A bundle is directly revealed preferred to the Situation-B bundle, so we put a D in Row A, Column B. Now let us consider Row B, Column A. The cost of the Situation-B bundle at Situation-B prices is 32. The cost of the Situation-A bundle at Situation-B prices is 33. So, in Situation B, the consumer could not aﬀord the Situation-A bundle. Therefore Situation B is not directly revealed preferred to Situation A. So we leave the entry in Row B, Column A blank. Generally, there is a D in Row i Column j if the number in the ij entry of the table in part (a) is less than or equal to the entry in Row i, Column i. There will be a violation of WARP if for some i and j, there is a D in Row i Column j and also a D in Row j, Column i. Do these observations violate WARP? p2 2 1 1 p3 8 8 2 x1 2 3 2 x2 1 4 6 x3 3 2 2 (a) We will ﬁll in the table below as follows. Where i and j stand for any of the letters A, B, and C in Row i and Column j of the matrix, write the value of the Situation-j bundle at the Situation-i prices. For example, in Row A and Column A, we put the value of the bundle purchased in Situation A at Situation A prices. From the table above, we see that in Situation A, the consumer bought bundle (2, 1, 3) at prices (1, 2, 8). The cost of this bundle A at prices A is therefore (1×2)+(2×1)+(8×3) = 28, so we put 28 in Row A, Column A. In Situation B the consumer bought bundle (3, 4, 2). The value of the Situation-B bundle, evaluated at the situation-A prices is (1 × 3) + (2 × 4) + (8 × 2) = 27, so put 27 in Row A, Column B. We have ﬁlled in some of the boxes, but we leave a few for you to do. 40 x1 NAME p1 1 4 3 88 REVEALED PREFERENCE (Ch. 7) thanks her and says he will let her know tomorrow. Joe puts his earmuﬀs back on and muses. If he accepts Mandy’s oﬀer, he will still have to pay rent on his current apartment but he won’t have to heat it. If he moved, heating would be cheaper, but he would have the $40 service charge. The outdoor temperature averages 20 degrees Fahrenheit in February, and it costs him $2 per month to raise his apartment temperature by 1 degree. Joe is still grading homework and has $100 a month left to spend on food and utilities after he has paid the rent on his apartment. The price of food is still $1 per unit. (a) Draw Joe’s budget line for February if he moves to Mandy’s apartment and on the same graph, draw his budget line if he doesn’t move. (b) After drawing these lines himself, Joe decides that he would be better oﬀ not moving. From this, we can tell, using the principle of revealed preference that Joe must plan to keep his apartment at a temperature of 60 degrees. less than (c) Joe calls Mandy and tells her his decision. Mandy oﬀers to pay half the service charge. Draw Joe’s budget line if he accepts Mandy’s new oﬀer. Joe now accepts Mandy’s oﬀer. From the fact that Joe accepted this oﬀer we can tell that he plans to keep the temperature in Mandy’s apartment above 40 degrees. No. Situation A B C A — D I B I — D C D I — Food 120 (c) Now ﬁll in Row i, Column j with an I if observation i is indirectly revealed preferred to j. Do these observations violate the Strong Axiom of Revealed Preference? Yes. 100 Don't move budget line 80 60 Move budget line 40 'New offer' budget line 20 7.7 (0) It is January, and Joe Grad, whom we met in Chapter 5, is shivering in his apartment when the phone rings. It is Mandy Manana, one of the students whose price theory problems he graded last term. Mandy asks if Joe would be interested in spending the month of February in her apartment. Mandy, who has switched majors from economics to political science, plans to go to Aspen for the month and so her apartment will be empty (alas). All Mandy asks is that Joe pay the monthly service charge of $40 charged by her landlord and the heating bill for the month of February. Since her apartment is much better insulated than Joe’s, it only costs $1 per month to raise the temperature by 1 degree. Joe 0 10 20 30 40 50 60 70 80 Temperature 7.8 (0) Lord Peter Pommy is a distinguished criminologist, schooled in the latest techniques of forensic revealed preference. Lord Peter is investigating the disappearance of Sir Cedric Pinchbottom who abandoned his aging mother on a street corner in Liverpool and has not been seen NAME 89 90 REVEALED PREFERENCE since. Lord Peter has learned that Sir Cedric left England and is living under an assumed name somewhere in the Empire. There are three suspects, R. Preston McAfee of Brass Monkey, Ontario, Canada, Richard Manning of North Shag, New Zealand, and Richard Stevenson of Gooey Shoes, Falkland Islands. Lord Peter has obtained Sir Cedric’s diary, which recorded his consumption habits in minute detail. By careful observation, he has also discovered the consumption behavior of McAfee, Manning, and Stevenson. All three of these gentlemen, like Sir Cedric, spend their entire incomes on beer and sausage. Their dossiers reveal the following: Sausage 40 30 McAfee 20 • Sir Cedric Pinchbottom — In the year before his departure, Sir Cedric consumed 10 kilograms of sausage and 20 liters of beer per week. At that time, beer cost 1 English pound per liter and sausage cost 1 English pound per kilogram. • R. Preston McAfee — McAfee is known to consume 5 liters of beer and 20 kilograms of sausage. In Brass Monkey, Ontario beer costs 1 Canadian dollar per liter and sausage costs 2 Canadian dollars per kilogram. • Richard Manning — Manning consumes 5 kilograms of sausage and 10 liters of beer per week. In North Shag, a liter of beer costs 2 New Zealand dollars and sausage costs 2 New Zealand dollars per kilogram. • Richard Stevenson — Stevenson consumes 5 kilograms of sausage and 30 liters of beer per week. In Gooey Shoes, a liter of beer costs 10 Falkland Island pounds and sausage costs 20 Falkland Island pounds per kilogram. (a) Draw the budget line for each of the three fugitives, using a diﬀerent color of ink for each one. Label the consumption bundle that each chooses. On this graph, superimpose Sir Cedric’s budget line and the bundle he chose. NAME 91 no food? $150. Use blue ink to draw their budget line if they choose the coupon option. (Ch. 7) 10 Pinchbottom Stevenson Manning 0 10 20 30 40 Beer (b) After pondering the dossiers for a few moments, Lord Peter announced. “Unless Sir Cedric has changed his tastes, I can eliminate one of the suspects. Revealed preference tells me that one of the suspects is McAfee. innocent.” Which one? (c) After thinking a bit longer, Lord Peter announced. “If Sir Cedric left voluntarily, then he would have to be better oﬀ than he was before. Therefore if Sir Cedric left voluntarily and if he has not changed his tastes, he must be living in Falklands. 7.9 (1) The McCawber family is having a tough time making ends meet. They spend $100 a week on food and $50 on other things. A new welfare program has been introduced that gives them a choice between receiving a grant of $50 per week that they can spend any way they want, and buying any number of $2 food coupons for $1 apiece. (They naturally are not allowed to resell these coupons.) Food is a normal good for the McCawbers. As a family friend, you have been asked to help them decide on which option to choose. Drawing on your growing fund of economic knowledge, you proceed as follows. (a) On the graph below, draw their old budget line in red ink and label their current choice C. Now use black ink to draw the budget line that they would have with the grant. If they chose the coupon option, how much food could they buy if they spent all their money on food coupons? $300. 92 How much could they spend on other things if they bought REVEALED PREFERENCE (Ch. 7) Other things 180 Other things 150 180 Black budget line 120 Black budget line 150 a 90 Blue budget line 120 Red budget line 60 a c 90 30 Blue budget line Red budget line 60 b 0 c 30 60 90 120 150 180 b 0 210 240 Food 30 30 60 90 120 150 180 210 240 Food (b) Using the fact that food is a normal good for the McCawbers, and knowing what they purchased before, darken the portion of the black budget line where their consumption bundle could possibly be if they chose the lump-sum grant option. Label the ends of this line segment A and B. (c) After studying the graph you have drawn, you report to the McCawbers. “I have enough information to be able to tell you which choice to make. You should choose the coupon because you can 7.10 (0) In 1933, the Swedish economist Gunnar Myrdal (who later won a Nobel prize in economics) and a group of his associates at Stockholm University collected a fantastically detailed historical series of prices and price indexes in Sweden from 1830 until 1930. This was published in a book called The Cost of Living in Sweden. In this book you can ﬁnd 100 years of prices for goods such as oat groats, hard rye bread, salted codﬁsh, beef, reindeer meat, birchwood, tallow candles, eggs, sugar, and coﬀee. There are also estimates of the quantities of each good consumed by an average working-class family in 1850 and again in 1890. The table below gives prices in 1830, 1850, 1890, and 1913, for ﬂour, meat, milk, and potatoes. In this time period, these four staple foods accounted for about 2/3 of the Swedish food budget. Prices of Staple Foods in Sweden Prices are in Swedish kronor per kilogram, except for milk, which is in Swedish kronor per liter. get more food even when other expenditure is constant. (d) Mr. McCawber thanks you for your help and then asks, “Would you have been able to tell me what to do if you hadn’t known whether food was a normal good for us?” On the axes below, draw the same budget lines you drew on the diagram above, but draw indiﬀerence curves for which food is not a normal good and for which the McCawbers would be better oﬀ with the program you advised them not to take. Grain Flour Meat Milk Potatoes 1830 .14 .28 .07 .032 1850 .14 .34 .08 .044 1890 .16 .66 .10 .051 1913 .19 .85 .13 .064 Based on the tables published in Myrdal’s book, typical consumption bundles for a working-class Swedish family in 1850 and 1890 are listed below. (The reader should be warned that we have made some NAME 93 approximations and simpliﬁcations to draw these simple tables from the much more detailed information in the original study.) Quantities Consumed by a Typical Swedish Family Quantities are measured in kilograms per year, except for milk, which is measured in liters per year. Grain Flour Meat Milk Potatoes 1850 165 22 120 200 1890 220 42 180 200 Cost of 1850 and 1890 Bundles at Various Years’ Prices 1850 bundle REVEALED PREFERENCE 1890 bundle Cost at 1830 Prices 44.1 61.6 Cost at 1850 Prices 49.0 68.3 Cost at 1890 Prices 63.1 91.1 foods in 1850, he would have to spend 1.39 times as much on these foods as does the typical Swedish worker of 1850. (g) If a Swede in 1890 decided to purchase the same bundle of food staples that was consumed by typical 1850 workers, he would spend the fraction Cost at 1913 Prices 78.5 113.7 7.11 (0) This question draws from the tables in the previous question. Let us try to get an idea of what it would cost an American family at today’s prices to purchase the bundle consumed by an average Swedish family in 1850. In the United States today, the price of ﬂour is about $.40 per kilogram, the price of meat is about $3.75 per kilogram, the price of milk is about $.50 per liter, and the price of potatoes is about $1 per kilogram. We can also compute a Laspeyres price index across time and across countries and use it to estimate the value of a current US dollar relative to the value of an 1850 Swedish kronor. (a) How much would it cost an American at today’s prices to buy the bundle of staple food commodities purchased by an average Swedish workingclass family in 1850? Yes. (b) Is the 1890 bundle revealed preferred to the 1850 bundle? (c) The Laspeyres quantity index for 1890 with base year 1850 is the ratio of the value of the 1890 bundle at 1850 prices to the value of the 1850 bundle at 1850 prices. Calculate the Laspeyres quantity index of staple (d) The Paasche quantity index for 1890 with base year 1850 is the ratio of the value of the 1890 bundle at 1890 prices to the value of the 1850 bundle at 1890 prices. Calculate the Paasche quantity index for 1890 with 1.44. (b) Myrdal estimates that in 1850, about 2/3 of the average family’s budget was spent on food. In turn, the four staples discussed in the last question constitute about 2/3 of the average family’s food budget. If the prices of other goods relative to the price of the food staples are similar in the United States today to what they were in Sweden in 1850, about how much would it cost an American at current prices to consume the same overall consumption bundle consumed by a Swedish working-class family in 1850? $919. (c) Using the Swedish consumption bundle of staple foods in 1850 as weights, calculate a Laspeyres price index to compare prices in current American dollars relative to prices in 1850 Swedish kronor. 8.35. If we use this to estimate the value of current dollars relative to 1850 Swedish kronor, we would say that a U.S. dollar today is worth about Swedish kronor. (e) The Laspeyres price index for 1890 with base year 1850 is calculated using 1850 quantities for weights. Calculate the Laspeyres price index for 1890 with base year 1850 for this group of four staple foods. 1.29. NAME 95 (a) Would the Laspeyres price index for 1985, with base year 1960 be less Exactly 2. than 2, greater than 2, or exactly equal to 2? about the Paasche price index? What Exactly 2. (b) If bananas are a normal good, will total banana consumption increase? Yes. If everybody has homothetic preferences, can you determine by what percentage total banana consumption must have increased? Explain. $408. 1.39. food consumption for 1890 with base year 1850. base year 1850. (Ch. 7) (f ) If a Swede were rich enough in 1850 to aﬀord the 1890 bundle of staple .69 of the amount that the typical Swedish worker of 1890 spends on these goods. (a) Complete the table below, which reports the annual cost of the 1850 and 1890 bundles of staple foods at various years’ prices. Cost 94 Yes, by 50%. Everybody’s budget 96 REVEALED PREFERENCE (Ch. 7) (a) They used to buy 30 cans of beer per week and spent the rest of their income on meat pies. How many meat pies did they buy? (c) The sales tax on beer induced Norm and Sheila to reduce their beer consumption to 20 cans per week. What happened to their consumption Stayed the same--10. preferences, the consumption of each good 7.13 (1) Norm and Sheila consume only meat pies and beer. Meat pies used to cost $2 each and beer was $1 per can. Their gross income used to be $60 per week, but they had to pay an income tax of $10. Use red ink to sketch their old budget line for meat pies and beer. 60 (d) This part of the problem will require some careful thinking. Suppose that instead of just taxing beer, the government decided to tax both beer and meat pies at the same percentage rate, and suppose that the price of beer and the price of meat pies each went up by the full amount of the tax. The new tax rate for both goods was set high enough to raise exactly the same amount of money from Norm and Sheila as the tax on beer used to raise. This new tax collects $ .50 for every bottle of beer on beer? $20. Now you can ﬁgure out how big a tax on each good is equivalent to an income tax of the amount you just found. 50 (e) Use black ink to draw the budget line for Norm and Sheila that corresponds to the tax in the last section. Are Norm and Sheila better oﬀ having just beer taxed or having both beer and meat pies taxed if both Red budget line 40 sets of taxes raise the same revenue? principle of revealed preference.) 30 20 0 How much revenue $20. sold and $ 1 for every meat pie sold. (Hint: If both goods are taxed at the same rate, the eﬀect is the same as an income tax.) How large an income tax would it take to raise the same revenue as the $1 tax Beer 10 10. (b) The government decided to eliminate the income tax and to put a sales tax of $1 per can on beer, raising its price to $2 per can. Assuming that Norm and Sheila’s pre-tax income and the price of meat pies did not change, draw their new budget line in blue ink. did this tax raise from Norm and Sheila? increases in the same proportion. 1850 7.12 (0) Suppose that between 1960 and 1985, the price of all goods exactly doubled while every consumer’s income tripled. of meat pies? line shifted out by 50%. With homothetic .12 Blue budget line Black budget line 10 20 30 40 50 60 Pies Both. (Hint: Try to use the 98 Chapter 8 SLUTSKY EQUATION NAME • Slutsky Equation • Introduction. It is useful to think of a price change as having two distinct eﬀects, a substitution eﬀect and an income eﬀect. The substitution eﬀect of a price change is the change that would have happened if income changed at the same time in such a way that the consumer could exactly aﬀord her old consumption bundle. The rest of the change in the consumer’s demand is called the income eﬀect. Why do we bother with breaking a real change into the sum of two hypothetical changes? Because we know things about the pieces that we wouldn’t know about the whole without taking it apart. In particular, we know that the substitution effect of increasing the price of a good must reduce the demand for it. We also know that the income eﬀect of an increase in the price of a good is equivalent to the eﬀect of a loss of income. Therefore if the good whose price has risen is a normal good, then both the income and substitution eﬀect operate to reduce demand. But if the good is an inferior good, income and substitution eﬀects act in opposite directions. Example: A consumer has the utility function U (x1 , x2 ) = x1 x2 and an income of $24. Initially the price of good 1 was $1 and the price of good 2 was $2. Then the price of good 2 rose to $3 and the price of good 1 stayed at $1. Using the methods you learned in Chapters 5 and 6, you will ﬁnd that this consumer’s demand function for good 1 is D1 (p1 , p2 , m) = m/2p1 and her demand function for good 2 is D2 (p1 , p2 , m) = m/2p2 . Therefore initially she will demand 12 units of good 1 and 6 units of good 2. If, when the price of good 2 rose to $3, her income had changed enough so that she could exactly aﬀord her old bundle, her new income would have to be (1 × 12) + (3 × 6) = $30. At an income of $30, at the new prices, she would demand D2 (1, 3, 30) = 5 units of good 2. Before the change she bought 6 units of 2, so the substitution eﬀect of the price change on her demand for good 2 is 5 − 6 = −1 units. Our consumer’s income didn’t really change. Her income stayed at $24. Her actual demand for good 2 after the price change was D2 (1, 3, 24) = 4. The diﬀerence between what she actually demanded after the price change and what she would have demanded if her income had changed to let her just aﬀord the old bundle is the income eﬀect. In this case the income eﬀect is 4 − 5 = −1 units of good 2. Notice that in this example, both the income eﬀect and the substitution eﬀect of the price increase worked to reduce the demand for good 2. When you have completed this workout, we hope that you will be able to do the following: • Find Slutsky income eﬀect and substitution eﬀect of a speciﬁc price change if you know the demand function for a good. • Show the Slutsky income and substitution eﬀects of a price change NAME 99 blue line parallel to the red line or the black line that you drew before? Red line. • • • (Ch. 8) on an indiﬀerence curve diagram. Show the Hicks income and substitution eﬀects of a price change on an indiﬀerence curve diagram. Find the Slutsky income and substitution eﬀects for special utility functions such as perfect substitutes, perfect complements, and Cobb-Douglas. Use an indiﬀerence-curve diagram to show how the case of a Giﬀen good might arise. Show that the substitution eﬀect of a price increase unambiguously decreases demand for the good whose price rose. Apply income and substitution eﬀects to draw some inferences about behavior. 8.1 (0) Gentle Charlie, vegetarian that he is, continues to consume apples and bananas. His utility function is U (xA , xB ) = xA xB . The price of apples is $1, the price of bananas is $2, and Charlie’s income is $40 a day. The price of bananas suddenly falls to $1. (a) Before the price change, Charlie consumed apples and bananas per day. On the graph below, use black ink to draw Charlie’s original budget line and put the label A on his chosen consumption bundle. (b) If, after the price change, Charlie’s income had changed so that he could exactly aﬀord his old consumption bundle, his new income would 30. have been With this income and the new prices, Charlie would consume 15 apples and 15 bananas. Use red ink to draw the budget line corresponding to this income and these prices. Label the bundle that Charlie would choose at this income and the new prices with the letter B. (c) Does the substitution eﬀect of the fall in the price of bananas make him buy more bananas or fewer bananas? More bananas. How 5 more. many more or fewer? (d) After the price change, Charlie actually buys 20 apples and 20 bananas. Use blue ink to draw Charlie’s actual budget line after the price change. Put the label C on the bundle that he actually chooses after the price change. Draw 3 horizontal lines on your graph, one from A to the vertical axis, one from B to the vertical axis, and one from C to the vertical axis. Along the vertical axis, label the income eﬀect, the substitution eﬀect, and the total eﬀect on the demand for bananas. Is the 100 SLUTSKY EQUATION (Ch. 8) (a) How many bottles of claret will Neville buy? 90. (b) If the price of claret rose to 40 pounds, how much income would Neville have to have in order to be exactly able to aﬀord the amount of claret and the amount of other goods that he bought before the price change? Bananas 40 8,400 pounds. At this income, and a price of 40 pounds, how Blue line 88 bottles. many bottles would Neville buy? Red line 30 20 10 (c) At his original income of 7,500 and a price of 40, how much claret c 20 Income would Neville demand? b Total Substitution 10 a (d) When the price of claret rose from 30 to 40, the number of bottles that Neville demanded decreased by Black line creased, reduced) 0 10 20 30 40 Apples in his income of $ 10 per day. Does the income eﬀect make him consume more bananas or fewer? more or how many fewer? More. reduced How many 5 more. 20. The substitution eﬀect (in- his demand by the income eﬀect (increased, reduced) (e) The income eﬀect of the fall in the price of bananas on Charlie’s demand for bananas is the same as the eﬀect of an (increase, decrease) increase 70 bottles. reduced 2 bottles and his demand by 8.3 (0) Note: Do this problem only if you have read the section entitled “Another Substitution Eﬀect” that describes the “Hicks substitution effect”. Consider the ﬁgure below, which shows the budget constraint and the indiﬀerence curves of good King Zog. Zog is in equilibrium with an income of $300, facing prices pX = $4 and pY = $10. Y (f ) Does the substitution eﬀect of the fall in the price of bananas make Charlie consume more apples or fewer? fewer? 5 fewer. Fewer. How many more or Does the income eﬀect of the fall in the price of bananas make Charlie consume more apples or fewer? More. What is the total eﬀect of the change in the price of bananas on the demand for apples? Zero. 8.2 (0) Neville’s passion is ﬁne wine. When the prices of all other goods are ﬁxed at current levels, Neville’s demand function for highquality claret is q = .02m − 2p, where m is his income, p is the price of claret (in British pounds), and q is the number of bottles of claret that he demands. Neville’s income is 7,500 pounds, and the price of a bottle of suitable claret is 30 pounds. 18 bottles. 30 22.5 E C F 30 35 43 75 90 120 X NAME 101 102 SLUTSKY EQUATION (Ch. 8) 30. (a) How much X does Zog consume? Price (b) If the price of X falls to $2.50, while income and the price of Y stay constant, how much X will Zog consume? 35. 4 (c) How much income must be taken away from Zog to isolate the Hicksian income and substitution eﬀects (i.e., to make him just able to aﬀord to 3 2.5 $75. reach his old indiﬀerence curve at the new prices)? 2 (d) The total eﬀect of the price change is to change consumption from E the point 1 C. to the point 30 (e) The income eﬀect corresponds to the movement from the point F C to the point 35 x while the substitution eﬀect corre- sponds to the movement from the point E to the point F. An inferior (f ) Is X a normal good or an inferior good? 8.4 (0) Maude spends all of her income on delphiniums and hollyhocks. She thinks that delphiniums and hollyhocks are perfect substitutes; one delphinium is just as good as one hollyhock. Delphiniums cost $4 a unit and hollyhocks cost $5 a unit. good. (g) On the axes below, sketch an Engel curve and a demand curve for Good X that would be reasonable given the information in the graph above. Be sure to label the axes on both your graphs. Income (a) If the price of delphiniums decreases to $3 a unit, will Maude buy 300 more of them? Yes. What part of the change in consumption is due to the income eﬀect and what part is due to the substitution eﬀect? 225 All due to income effect. 30 (b) If the prices of delphiniums and hollyhocks are respectively pd = $4 and ph = $5 and if Maude has $120 to spend, draw her budget line in blue ink. Draw the highest indiﬀerence curve that she can attain in red ink, and label the point that she chooses as A. 43 x NAME 103 SLUTSKY EQUATION (Ch. 8) (b) The substitution eﬀect is a change in demand from Delphiniums 40 30 104 92. 92 a 80 to The income eﬀect of the price change is a change in demand from to 100. Black line 20 (c) On the axes below, use blue ink to draw Douglas Cornﬁeld’s budget line before the price change. Locate the bundle he chooses at these prices on your graph and label this point A. Use black ink to draw Douglas Cornﬁeld’s budget line after the price change. Label his consumption bundle after the change by B. Red curves 10 Blue line b 0 10 20 30 40 Hollyhocks y (c) Now let the price of hollyhocks fall to $3 a unit, while the price of delphiniums does not change. Draw her new budget line in black ink. Draw the highest indiﬀerence curve that she can now reach with red ink. Label the point she chooses now as B. (d) How much would Maude’s income have to be after the price of hollyhocks fell, so that she could just exactly aﬀord her old commodity bundle A? 80 60 40 b Black line $120. a Blue line (e) When the price of hollyhocks fell to $3, what part of the change in Maude’s demand was due to the income eﬀect and what part was due to 0 the substitution eﬀect? All substitution effect. Black line c 20 40 80 120 160 200 240 280 320 x 8.5 (1) Suppose that two goods are perfect complements. If the price of one good changes, what part of the change in demand is due to the substitution eﬀect, and what part is due to the income eﬀect? All income effect. 8.6 (0) Douglas Cornﬁeld’s demand function for good x is x(px , py , m) = 2m/5px . His income is $1,000, the price of x is $5, and the price of y is $20. If the price of x falls to $4, then his demand for x will change from 80 to 100. (d) On the graph above, use black ink to draw a budget line with the new prices but with an income that just allows Douglas to buy his old bundle, A. Find the bundle that he would choose with this budget line and label this bundle C. 8.7 (1) Mr. Consumer allows himself to spend $100 per month on cigarettes and ice cream. Mr. C’s preferences for cigarettes and ice cream are unaﬀected by the season of the year. (a) If his income were to change at the same time so that he could exactly aﬀord his old commodity bundle at px = 4 and py = 20, what would his new income be? 920. What would be his demand for x at this new level of income, at prices px = 4 and py = 20? 92. (a) In January, the price of cigarettes was $1 per pack, while ice cream cost $2 per pint. Faced with these prices, Mr. C bought 30 pints of ice cream and 40 packs of cigarettes. Draw Mr. C’s January budget line with blue ink and label his January consumption bundle with the letter J. NAME 105 106 SLUTSKY EQUATION decrease Ice cream (Ch. 8) in his income. Therefore the information we have suggests normal that ice cream is a(n) (normal, inferior, neutral) good. 100 90 (c) In March, Mr. C again had $100 to spend. Ice cream was on sale for $1 per pint. Cigarette prices, meanwhile, increased to $1.50 per pack. Draw his March budget line with black ink. Is he better oﬀ than in January, 80 worse oﬀ, or can you not make such a comparison? Better off. How does your answer to the last question change if the price of cigarettes 60 Pencil budget line 8.8 (1) This problem continues with the adventures of Mr. Consumer from the previous problem. 50 (a) In April, cigarette prices rose to $2 per pack and ice cream was still on sale for $1 per pint. Mr. Consumer bought 34 packs of cigarettes and 32 pints of ice cream. Draw his April budget line with pencil and label his April bundle with the letter A. Was he better oﬀ or worse oﬀ than Black budget line 40 30 Red budget line J A in January? 10 20 30 40 50 60 70 80 90 100 Cigarettes (b) In February, Mr. C again had $100 to spend and ice cream still cost $2 per pint, but the price of cigarettes rose to $1.25 per pack. Mr. C consumed 30 pints of ice cream and 32 packs of cigarettes. Draw Mr. C’s February budget line with red ink and mark his February bundle with the letter F . The substitution eﬀect of this price change would make him buy (less, more, the same amount of) less cigarettes and (less, more, the same amount of) more ice cream. Since this is true and the total change in his ice cream consumption was zero, it must be that the income eﬀect of this price change on his consumption of ice cream makes him buy (more, less, the same amount of) less ice cream. The income eﬀect of this price change is like the eﬀect of an (increase, decrease) NAME 107 Was he better oﬀ or worse oﬀ than in Better off. February, or can’t one tell? Blue budget line 10 Worse off. F 20 0 Now you can’t tell. had increased to $2 per pack? 70 (b) In May, cigarettes stayed at $2 per pack and as the sale on ice cream ended, the price returned to $2 per pint. On the way to the store, however, Mr. C found $30 lying in the street. He then had $130 to spend on cigarettes and ice cream. Draw his May budget with a dashed line. Without knowing what he purchased, one can determine whether he is better oﬀ than he was in at least one previous month. Which month or months? He is better off in May than in February. (c) In fact, Mr. C buys 40 packs of cigarettes and 25 pints of ice cream in May. Does he satisfy WARP? No. 8.9 (2) In the last chapter, we studied a problem involving food prices and consumption in Sweden in 1850 and 1890. (a) Potato consumption was the same in both years. Real income must have gone up between 1850 and 1890, since the amount of food staples purchased, as measured by either the Laspeyres or the Paasche quantity index, rose. The price of potatoes rose less rapidly than the price of either meat or milk, and at about the same rate as the price of grain ﬂour. So real income went up and the price of potatoes went down relative to other goods. From this information, determine whether potatoes were 108 SLUTSKY EQUATION most likely a normal or an inferior good. Explain your answer. (Ch. 8) First-class miles If potatoes were a normal good, both the 1600 fall in potato price and the rise in income would increase the demand for potatoes. But potato consumption did not increase. Pencil line 1200 Blue line So potatoes must be an inferior good. 800 (b) Can one also tell from these data whether it is likely that pota- 400 Red line b a toes were a Giﬀen good? Black line c If potatoes were a Giffen good, then the fall in the price of potatoes would decrease demand and the 0 400 rise in income would also decrease demand 800 1200 1600 Second-class miles constant. So potatoes were probably not a (b) Let m1 be the number of miles she travels by ﬁrst-class coach and m2 be the number of miles she travels by second-class coach. Write down two equations that you can solve to ﬁnd the number of miles she chooses to travel by ﬁrst-class coach and the number of miles she chooses to travel Giffen good. by second-class coach. for potatoes. But potato demand stayed .2m1 + .1m2 = 200, m1 + m2 = 1, 500. 8.10 (1) Agatha must travel on the Orient Express from Istanbul to Paris. The distance is 1,500 miles. A traveler can choose to make any fraction of the journey in a ﬁrst-class carriage and travel the rest of the way in a second-class carriage. The price is 10 cents a mile for a secondclass carriage and 20 cents a mile for a ﬁrst-class carriage. Agatha much prefers ﬁrst-class to second-class travel, but because of a misadventure in an Istanbul bazaar, she has only $200 left with which to buy her tickets. Luckily, she still has her toothbrush and a suitcase full of cucumber sandwiches to eat on the way. Agatha plans to spend her entire $200 on her tickets for her trip. She will travel ﬁrst class as much as she can aﬀord to, but she must get all the way to Paris, and $200 is not enough money to get her all the way to Paris in ﬁrst class. (a) On the graph below, use red ink to show the locus of combinations of ﬁrst- and second-class tickets that Agatha can just aﬀord to purchase with her $200. Use blue ink to show the locus of combinations of ﬁrstand second-class tickets that are suﬃcient to carry her the entire distance from Istanbul to Paris. Locate the combination of ﬁrst- and second-class miles that Agatha will choose on your graph and label it A. (c) The number of miles that she travels by second-class coach is 1,000. (d) Just before she was ready to buy her tickets, the price of second-class tickets fell to $.05 while the price of ﬁrst-class tickets remained at $.20. On the graph that you drew above, use pencil to show the combinations of ﬁrst-class and second-class tickets that she can aﬀord with her $200 at these prices. On your graph, locate the combination of ﬁrst-class and second-class tickets that she would now choose. (Remember, she is going to travel as much ﬁrst-class as she can aﬀord to and still make the 1,500 mile trip on $200.) Label this point B. How many miles does she travel by second class now? 666.66. (Hint: For an exact solution you will have to solve two linear equations in two unknowns.) Is second-class travel a normal good for Agatha? Yes. No. Is it a Giﬀen good for her? NAME 109 110 SLUTSKY EQUATION (Ch. 8) 112 BUYING AND SELLING 8.11 (0) We continue with the adventures of Agatha, from the previous problem. Just after the price change from $.10 per mile to $.05 per mile for second-class travel, and just before she had bought any tickets, Agatha misplaced her handbag. Although she kept most of her money in her sock, the money she lost was just enough so that at the new prices, she could exactly aﬀord the combination of ﬁrst- and second-class tickets that she would have purchased at the old prices. How much money did she lose? $50. On the graph you started in the previous problem, use black ink to draw the locus of combinations of ﬁrst- and second-class tickets that she can just aﬀord after discovering her loss. Label the point that she chooses with a C. How many miles will she travel by second class now? 1,000. (a) Finally, poor Agatha ﬁnds her handbag again. How many miles will she travel by second class now (assuming she didn’t buy any tickets before she found her lost handbag)? 666.66. When the price of secondclass tickets fell from $.10 to $.05, how much of a change in Agatha’s demand for second-class tickets was due to a substitution eﬀect? How much of a change was due to an income eﬀect? Chapter 9 None. −333.33. NAME Buying and Selling Introduction. In previous chapters, we studied the behavior of consumers who start out without owning any goods, but who had some money with which to buy goods. In this chapter, the consumer has an initial endowment, which is the bundle of goods the consumer owns before any trades are made. A consumer can trade away from his initial endowment by selling one good and buying the other. The techniques that you have already learned will serve you well here. To ﬁnd out how much a consumer demands at given prices, you ﬁnd his budget line and then ﬁnd a point of tangency between his budget line and an indiﬀerence curve. To determine a budget line for a consumer who is trading from an initial endowment and who has no source of income other than his initial endowment, notice two things. First, the initial endowment must lie on the consumer’s budget line. This is true because, no matter what the prices are, the consumer can always aﬀord his initial endowment. Second, if the prices are p1 and p2 , the slope of the budget line must be −p1 /p2 . This is true, since for every unit of good 1 the consumer gives up, he can get exactly p1 /p2 units of good 2. Therefore if you know the prices and you know the consumer’s initial endowment, then you can always write an equation for the consumer’s budget line. After all, if you know one point on a line and you know its slope, you can either draw the line or write down its equation. Once you have the budget equation, you can ﬁnd the bundle the consumer chooses, using the same methods you learned in Chapter 5. (Ch. 9) Example: Sherwin has 18 hours a day which he divides between labor and leisure. He can work as many hours a day as he wishes for a wage of $5 per hour. He also receives a pension that gives him $10 a day whether he works or not. The price of other goods is $1 per unit. If Sherwin makes no trades at all, he will have 18 hours of leisure and 10 units of other goods. Therefore Sherwin’s initial endowment is 18 hours of leisure a day and $10 a day for other goods. Let R be the amount of leisure that he has per day, and let C be the number of dollars he has to spend per day on other goods. If his wage is $5 an hour, he can aﬀord to consume bundle (R, C) if it costs no more per day than the value of his initial endowment. The value of his initial endowment (his full income) is $10 + ($5 × 18) = $100 per day. Therefore Sherwin’s budget equation is 5R + C = 100. 9.1 (0) Abishag Appleby owns 20 quinces and 5 kumquats. She has no income from any other source, but she can buy or sell either quinces or kumquats at their market prices. The price of kumquats is four times the price of quinces. There are no other commodities of interest. (a) How many quinces could she have if she was willing to do without kumquats? 40. How many kumquats could she have if she was willing to do without quinces? 10. Kumquats 40 30 Example: A peasant consumes only rice and ﬁsh. He grows some rice and some ﬁsh, but not necessarily in the same proportion in which he wants to consume them. Suppose that if he makes no trades, he will have 20 units of rice and 5 units of ﬁsh. The price of rice is 1 yuan per unit, and the price of ﬁsh is 2 yuan per unit. The value of the peasant’s endowment is (1 × 20) + (2 × 5) = 30. Therefore the peasant can consume any bundle (R, F ) such that (1 × R) + (2 × F ) = 30. Perhaps the most interesting application of trading from an initial endowment is the theory of labor supply. To study labor supply, we consider the behavior of a consumer who is choosing between leisure and other goods. The only thing that is at all new or “tricky” is ﬁnding the appropriate budget constraint for the problem at hand. To study labor supply, we think of the consumer as having an initial endowment of leisure, some of which he may trade away for goods. In most applications we set the price of “other goods” at 1. The wage rate is the price of leisure. The role that is played by income in the ordinary consumer-good model is now played by “full income.” A worker’s full income is the income she would have if she chose to take no leisure. 20 Red line 10 c Squiggly line 0 10 Blue line e 20 30 40 Quinces (b) Draw Abishag’s budget set, using blue ink, and label the endowment bundle with the letter E. If the price of quinces is 1 and the price of kumquats is 4, write Abishag’s budget equation. Q + 4K = 40. If the price of quinces is 2 and the price of kumquats is 8, write Abishag’s budget equation. 2Q + 8K = 80. What eﬀect does doubling both NAME 113 114 BUYING AND SELLING (Ch. 9) prices have on the set of commodity bundles that Abishag can aﬀord? (e) Assuming that the price of tomatoes rose to $15 from $5 before Mario made any transactions, the change in the demand for tomatoes due to No effect. the substitution eﬀect was (c) Suppose that Abishag decides to sell 10 quinces. Label her ﬁnal consumption bundle in your graph with the letter C. tomatoes due to the ordinary income eﬀect was (d) Now, after she has sold 10 quinces and owns the bundle labeled C, suppose that the price of kumquats falls so that kumquats cost the same as quinces. On the diagram above, draw Abishag’s new budget line, using red ink. +5. 0. The change in the demand for The total change in the demand for tomatoes was Red line 30 20 a Blue line Black line 10 (a) What is the monetary value of Mario’s endowment of vegetables? 0 10 20 30 40 Tomatoes $200. (c) Suppose that before Mario makes any trades, the price of tomatoes rises to $15 a pound, while the price of eggplant stays at $5 a pound. What is the value of Mario’s endowment now? $300. Draw his new budget line, using red ink. He will now choose a consumption bundle 15 consisting of 15 tomatoes and 9.3 (0) Lucetta consumes only two goods, A and B. Her only source of income is gifts of these commodities from her many admirers. She doesn’t always get these goods in the proportions in which she wants to consume them, but she can always buy or sell A at the price pA = 1 and B at the price pB = 2. Lucetta’s utility function is U (a, b) = ab, where a is the amount of A she consumes and b is the amount of B she consumes. (a) Suppose that Lucetta’s admirers give her 100 units of A and 200 units of B. In the graph below, use red ink to draw her budget line. Label her initial endowment E. eggplants. (b) What are Lucetta’s gross demands for A? (d) Suppose that Mario had sold his entire crop at the market for a total of $200, intending to buy back some tomatoes and eggplant for his own consumption. Before he had a chance to buy anything back, the price of tomatoes rose to $15, while the price of eggplant stayed at $5. Draw his budget line, using pencil or black ink. Mario will now consume pounds of tomatoes and 10 10 NAME 115 (e) Does Lucetta’s consumption of good B rise or fall? of good A? 25 units. B? 250 units. And for 125 units. (c) What are Lucetta’s net demands? 150 of A and −75 of B. pounds of eggplant. (d) Suppose that before Lucetta has made any trades, the price of good B falls to 1, and the price of good A stays at 1. Draw Lucetta’s budget line at these prices on your graph, using blue ink. By how much? −5. Eggplant 40 9.2 (0) Mario has a small garden where he raises eggplant and tomatoes. He consumes some of these vegetables, and he sells some in the market. Eggplants and tomatoes are perfect complements for Mario, since the only recipes he knows use them together in a 1:1 ratio. One week his garden yielded 30 pounds of eggplant and 10 pounds of tomatoes. At that time the price of each vegetable was $5 per pound. 20 pounds of tomatoes and 20 pounds ends up consuming of eggplant. Draw the indiﬀerence curve through the consumption bundle that Mario chooses and label this bundle A. The change in the demand for tomatoes due to the endowment income eﬀect was (e) If Abishag obeys the weak axiom of revealed preference, then there are some points on her red budget line that we can be sure Abishag will not choose. On the graph, make a squiggly line over the portion of Abishag’s red budget line that we can be sure she will not choose. (b) On the graph below, use blue ink to draw Mario’s budget line. Mario −10. It rises. What happens to Lucetta’s consumption It decreases by 100 units. 116 BUYING AND SELLING (Ch. 9) change. 9.4 (0) Priscilla ﬁnds it optimal not to engage in trade at the going prices and just consumes her endowment. Priscilla has no kinks in her indiﬀerence curves, and she is endowed with positive amounts of both goods. Use pencil or black ink to draw a budget line and an indiﬀerence curve for Priscilla that would be consistent with these facts. Suppose that the price of good 2 stays the same, but the price of good 1 falls. Use blue ink to show her new budget line. Priscilla satisﬁes the weak axiom of revealed preference. Could it happen that Priscilla will consume Good B less of good 1 than before? Explain. 600 No. If p1 falls, then with the new budget, she can still 500 afford her old bundle. She could afford the bundles with less of good 1 than her 400 endowment at the old prices. By WARP, she 300 Blue budget line won’t choose them now. Red budget line 200 X2 e 100 0 75 150 225 e 300 Blue budget line Good A (f ) Suppose that before the price of good B fell, Lucetta had exchanged all of her gifts for money, planning to use the money to buy her consumption bundle later. How much of good B will she choose to consume? units. How much of good A? 250 X1 250 units. (g) Explain why her consumption is diﬀerent depending on whether she was holding goods or money at the time of the price change. Black budget line In the former case, the fall in pB makes her poorer because she is a net seller of good B. In the latter case, her income doesn’t 9.5 (0) Potatoes are a Giﬀen good for Paddy, who has a small potato farm. The price of potatoes fell, but Paddy increased his potato consumption. At ﬁrst this astonished the village economist, who thought that a decrease in the price of a Giﬀen good was supposed to reduce demand. But then he remembered that Paddy was a net supplier of potatoes. With the help of a graph, he was able to explain Paddy’s behavior. In the axes below, show how this could have happened. Put “potatoes” on the horizontal axis and “all other goods” on the vertical axis. Label the old equilibrium A and the new equilibrium B. Draw a point C so that the NAME 117 118 BUYING AND SELLING Slutsky substitution eﬀect is the movement from A to C and the Slutsky income eﬀect is the movement from C to B. On this same graph, you are also going to have to show that potatoes are a Giﬀen good. To do this, draw a budget line showing the eﬀect of a fall in the price of potatoes if Paddy didn’t own any potatoes, and only had money income. Label the new consumption point under these circumstances by D. (Warning: You probably will need to make a few dry runs on some scratch paper to get the whole story straight.) (Ch. 9) First-class miles 1600 1200 Pencil line Blue line 800 a 400 Red line All other goods d 0 400 a 800 1200 1600 Second-class miles c (b) Use red ink to draw a line showing all of the combinations of ﬁrst-class and second-class travel that she can aﬀord when she is on the train, by trading her endowment of tickets at the new prices that apply on board the train. b e (c) On your graph, show the point that she chooses after ﬁnding out about the price change. Does she choose more, less, or the same amount Potatoes of second-class tickets? 9.6 (0) Recall the travails of Agatha, from the previous chapter. She had to travel 1,500 miles from Istanbul to Paris. She had only $200 with which to buy ﬁrst-class and second-class tickets on the Orient Express when the price of ﬁrst-class tickets was $.20 a mile and the price of second-class tickets was $.10 a mile. She bought tickets that enabled her to travel all the way to Paris, with as many miles of ﬁrst class as she could aﬀord. After she boarded the train, she discovered to her amazement that the price of second-class tickets had fallen to $.05 a mile while the price of ﬁrst-class tickets remained at $.20 a mile. She also discovered that on the train it was possible to buy or sell ﬁrst-class tickets for $.20 a mile and to buy or sell second-class tickets for $.05 a mile. Agatha had no money left to buy either kind of ticket, but she did have the tickets that she had already bought. (a) On the graph below, use pencil to show the combinations of tickets that she could aﬀord at the old prices. Use blue ink to show the combinations of tickets that would take her exactly 1,500 miles. Mark the point that she chooses with the letter A. NAME 119 The same. 9.7 (0) Mr. Cog works in a machine factory. He can work as many hours per day as he wishes at a wage rate of w. Let C be the number of dollars he spends on consumer goods and let R be the number of hours of leisure that he chooses. (a) Mr. Cog earns $8 an hour and has 18 hours per day to devote to labor or leisure, and he has $16 of nonlabor income per day. Write an equation for his budget between consumption and leisure. C + 8R = 160. Use blue ink to draw his budget line in the graph below. His initial endowment is the point where he does no work and enjoys 18 hours of leisure per day. Mark this point on the graph below with the letter A. (Remember that although Cog can choose to work and thereby “sell” some of his endowment of leisure, he cannot “buy leisure” by paying somebody else to loaf for him.) If Mr. Cog has the utility function U (R, C) = CR, how many hours of leisure per day will he choose? hours per day will he work? 120 BUYING AND SELLING 10. How many 8. (Ch. 9) Mr. Cog’s supply function for labor is therefore 18 − R(w, m) = Consumption 9− m/2w. 240 9.8 (0) Fred has just arrived at college and is trying to ﬁgure out how to supplement the meager checks that he gets from home. “How can anyone live on $50 a week for spending money?” he asks. But he asks to no avail. “If you want more money, get a job,” say his parents. So Fred glumly investigates the possibilities. The amount of leisure time that he has left after allowing for necessary activities like sleeping, brushing teeth, and studying for economics classes is 50 hours a week. He can work as many hours per week at a nearby Taco Bell for $5 an hour. Fred’s utility function for leisure and money to spend on consumption is U (C, L) = CL. 200 Red budget line 160 120 Black budget line 80 (a) Fred has an endowment that consists of $50 of money to spend on Blue budget line consumption and 50 hours of leisure, some of which he might “sell” for money. The money value of Fred’s endowment bundle, including both 40 a 4 0 8 12 his money allowance and the market value of his leisure time is therefore 16 20 24 Leisure (b) Suppose that Mr. Cog’s wage rate rises to $12 an hour. Use red ink to draw his new budget line. (He still has $16 a day in nonlabor income.) If he continued to work exactly as many hours as he did before the wage increase, how much more money would he have each day to spend on consumption? 8 13 $32. But with his new budget line, he chooses to work hours, and so his consumption increases by $36. (c) Suppose that Mr. Cog still receives $8 an hour but that his nonlabor income rises to $48 per day. Use black ink to draw his budget line. How many hours does he choose to work? 6. (d) Suppose that Mr. Cog has a wage of $w per hour, a nonlabor income of $m, and that he has 18 hours a day to divide between labor and leisure. Cog’s budget line has the equation C + wR = m + 18w. Using the same methods you used in the chapter on demand functions, ﬁnd the amount of leisure that Cog will demand as a function of wages and of nonlabor income. (Hint: Notice that this is the same as ﬁnding the demand for R when the price of R is w, the price of C is 1, and income is m + 18w.) Mr. Cog’s demand function for leisure is R(w, m) = 9 + (m/2w). $300. Fred’s “budget line” for leisure and consumption is like a budget line for someone who can buy these two goods at a price of $1 per unit of consumption and a price of $5 per unit of leisure. The only diﬀerence is that this budget line doesn’t run all the way to the horizontal axis. (b) On the graph below, use black ink to show Fred’s budget line. (Hint: Find the combination of leisure and consumption expenditures that he could have if he didn’t work at all. Find the combination he would have if he chose to have no leisure at all. What other points are on your graph?) On the same graph, use blue ink to sketch the indiﬀerence curves that give Fred utility levels of 3,000, 4,500, and 7,500. (c) If you maximized Fred’s utility subject to the above budget, how much consumption would he choose? $150. (Hint: Remember how to solve for the demand function of someone with a Cobb-Douglas utility function?) (d) The amount of leisure that Fred will choose to consume is hours. This means that his optimal labor supply will be 20 30 hours. NAME 121 Blue indifference curve (7500) 250 Black budget line 200 (Ch. 9) (c) If an individual is not working, he receives a payment of $100. If he works he does not receive the $100, and all wages are subject to a 50% income tax. (Use blue ink on the graph below.) 150 100 BUYING AND SELLING (b) All individuals receive a lump sum payment of $100 per week from the government. There is no tax on the ﬁrst $100 per week of labor income. But all labor income above $100 per week is subject to a 50% income tax. (Use red ink on the graph above.) Consumption 300 122 Blue indifference curve (4500) Consumption 50 Blue indifference curve (3000) 10 0 30 20 40 400 50 60 300 Leisure Red budget line 9.9 (0) George Johnson earns $5 per hour in his job as a truﬄe sniffer. After allowing time for all of the activities necessary for bodily upkeep, George has 80 hours per week to allocate between leisure and labor. Sketch the budget constraints for George resulting from the following government programs. 200 100 (a) There is no government subsidy or taxation of labor income. (Use blue ink on the graph below.) Blue budget line Consumption 400 Blue budget line 0 20 40 60 Red budget line 300 80 Leisure 200 (d) The same conditions as in Part (c) apply, except that the ﬁrst 20 hours of labor are exempt from the tax. (Use red ink on the graph above.) 100 0 20 40 60 (e) All wages are taxed at 50%, but as an incentive to encourage work, the government gives a payment of $100 to anyone who works more than 20 hours a week. (Use blue ink on the graph below.) 80 Leisure NAME 123 124 BUYING AND SELLING (Ch. 9) (a) Use these data to plot a “labor supply curve” on the graph below. Consumption Hourly wage rate (in 1983 dollars) 400 12 10 300 8 200 6 4 100 2 Blue budget line 0 0 20 40 60 10 30 20 80 Leisure 40 50 60 Hours of work per week (b) At wage rates below $4 an hour, does the workweek get longer or 9.10 (0) In the United States, real wage rates in manufacturing have risen steadily from 1890 to the present. In the period from 1890 to 1930, the length of the workweek was reduced dramatically. But after 1930, despite the continuing growth of real wage rates, the length of the work week has stayed remarkably constant at about 40 hours per week. shorter as the wage rate rises? Hourly Wages and Length of Work Week in U.S. Manufacturing, 1890–1983 Sources: Handbook of Labor Statistics, 1983 and U.S. Economic History, by Albert Niemi (p. 274). Wages are in 1983 dollars. substitution eﬀect alone would make for a (longer, shorter) longer workweek. If leisure is a normal good, the endowment income eﬀect tends (c) The data in this table could be consistent with workers choosing various hours a week to work, given the wage rate. An increase in wages has both an endowment income eﬀect and a substitution eﬀect. The to make people choose (more, less) shorter Year 1890 1909 1920 1930 1940 1950 1960 1970 1983 Wage 1.89 2.63 3.11 3.69 5.27 6.86 8.56 9.66 10.74 Hours Worked 59.0 51.0 47.4 42.1 38.1 40.5 39.7 39.8 40.1 Shorter. more leisure and a (longer, shorter) workweek. At wage rates below $4 an hour, the (substi- tution, endowment income) endowment income eﬀect appears to dominate. How would you explain what happens at wages above $4 an hour? Substitution and endowment income effects cancel each other out, so the workweek stays roughly constant. (d) Between 1890 and 1909, wage rates rose by weekly earnings rose by only 20 39 percent, but percent. For this period, the NAME gain in earnings (overstates, understates) 125 understates in workers’ wealth, since they chose to take (more, less) in 1909 than they took in 1890. 126 the gain more BUYING AND SELLING (Ch. 9) Consumption leisure 400 Red part of line 300 9.11 (0) Professor Mohamed El Hodiri of the University of Kansas, in a classic tongue-in-cheek article “The Economics of Sleeping,” Manifold, 17 (1975), oﬀered the following analysis. “Assume there are 24 hours in a day. Daily consumption being x and hours of sleep s, the consumer maximizes a utility function of the form u = x2 s, where x = w(24 − s), with w being the wage rate.” 200 100 Blue part of line (a) In El Hodiri’s model, does the optimal amount of sleeping increase, Stays the same. decrease, or stay the same as wages increase? 0 20 40 60 80 Leisure (c) The blue line segment that you drew lies on a line with equation (b) How many hours of sleep per day is best in El Hodiri’s model? 8. c + 4r = 320. The red line that you drew lies on a line with equation c + 6r = 400. (Hint: For the red line, you know one point on the line and you know its slope.) 9.12 (0) Wendy and Mac work in fast food restaurants. Wendy gets $4 an hour for the ﬁrst 40 hours that she works and $6 an hour for every hour beyond 40 hours a week. Mac gets $5 an hour no matter how many hours he works. Each has 80 hours a week to allocate between work and leisure and neither has any income from sources other than labor. Each has a utility function U = cr, where c is consumption and r is leisure. Each can choose the number of hours to work. (d) If Wendy was paid $4 an hour no matter how many hours she worked, she would work 40 hours and earn a total of $160 a week. On your graph, use black ink to draw her indiﬀerence curve through this point. (e) Will Wendy choose to work overtime? Yes. What is the best (200, 33.3). choice for Wendy from the red budget line? (c, r) = (a) How many hours will Mac choose to work? 40. How many hours a week will she work? (b) Wendy’s budget “line” has a kink in it at the point where r = 40 and c = 160. Use blue ink for the part of her budget line where she would be if she does not work overtime. Use red ink for the part where she would be if she worked overtime. NAME 127 9.13 (1) Wally Piper is a plumber. He charges $10 per hour for his work and he can work as many hours as he likes. Wally has no source of income other than his labor. He has 168 hours per week to allocate between labor and leisure. On the graph below, draw Wally’s budget set, showing the various combinations of weekly leisure and income that Wally can aﬀord. Income (f ) Suppose that the jobs are equally agreeable in all other respects. Since Wendy and Mac have the same preferences, they will be able to agree about who has the better job. Who has the better job? Mac. (Hint: Calculate Wendy’s utility when she makes her best choice. Calculate what her utility would be if she had Mac’s job and chose the best amount of time to work.) 128 BUYING AND SELLING (Ch. 9) “overtime” that he worked beyond 40 hours per week. On the graph that you drew above, use red ink to sketch in Wally’s budget line with this pay schedule. Draw the indiﬀerence curve through the point that Wally chooses with this pay schedule. Will Wally work more than 40 hours or less than 40 hours per week with this pay schedule? More. 9.14 (1) Felicity loves her job. She is paid $10 an hour and can work as many hours a day as she wishes. She chooses to work only 5 hours a day. She says the job is so interesting that she is happier working at this job than she would be if she made the same income without working at all. A skeptic asks, “If you like the job better than not working at all, why don’t you work more hours and earn more money?” Felicity, who is entirely rational, patiently explains that work may be desirable on average but undesirable on the margin. The skeptic insists that she show him her indiﬀerence curves and her budget line. 2400 2000 46.6. Red budget line 1600 1200 (a) On the axes below, draw a budget line and indiﬀerence curves that are consistent with Felicity’s behavior and her remarks. Put leisure on the horizontal axis and income on the vertical axis. (Hint: Where does the indiﬀerence curve through her actual choice hit the vertical line l = 24?) 800 Income 400 0 240 40 80 128 120 160 200 240 Leisure (a) Write down Wally’s budget equation. I + 10R = 1, 680. (b) While self-employed, Wally chose to work 40 hours per week. The construction ﬁrm, Glitz and Drywall, had a rush job to complete. They oﬀered Wally $20 an hour and said that he could work as many hours as he liked. Wally still chose to work only 40 hours per week. On the graph you drew above, draw in Wally’s new budget line. (c) Wally has convex preferences and no kinks in his indiﬀerence curves. On the graph, draw indiﬀerence curves that are consistent with his choice of working hours when he was self-employed and when he worked for Glitz and Drywall. (d) Glitz and Drywall were in a great hurry to complete their project and wanted Wally to work more than 40 hours. They decided that instead of paying him $20 per hour, they would pay him only $10 an hour for the ﬁrst 40 hours that he worked per week and $20 an hour for every hour of 50 24 Leisure 9.15 (2) Dudley’s utility function is U (C, R) = C − (12 − R)2 , where R is the amount of leisure he has per day. He has 16 hours a day to divide between work and leisure. He has an income of $20 a day from nonlabor sources. The price of consumption goods is $1 per unit. (a) If Dudley can work as many hours a day as he likes but gets zero wages for his labor, how many hours of leisure will he choose? 12. NAME 129 130 BUYING AND SELLING (Ch. 9) 132 INTERTEMPORAL CHOICE (b) If Dudley can work as many hours a day as he wishes for a wage rate of $10 an hour, how many hours of leisure will he choose? 7. How many hours will he work? 9. (Hint: Write down Dudley’s budget constraint. Solve for the amount of leisure that maximizes his utility subject to this constraint. Remember that the amount of labor he wishes to supply is 16 minus his demand for leisure.) (c) If Dudley’s nonlabor income decreased to $5 a day, while his wage rate remained at $10, how many hours would he choose to work? 9. (d) Suppose that Dudley has to pay an income tax of 20 percent on all of his income, and suppose that his before-tax wage remained at $10 an hour and his before-tax nonlabor income was $20 per day. How many 8. hours would he choose to work? Chapter 10 NAME Intertemporal Choice Introduction. The theory of consumer saving uses techniques that you have already learned. In order to focus attention on consumption over time, we will usually consider examples where there is only one consumer good, but this good can be consumed in either of two time periods. We will be using two “tricks.” One trick is to treat consumption in period 1 and consumption in period 2 as two distinct commodities. If you make period-1 consumption the numeraire, then the “price” of period-2 consumption is the amount of period-1 consumption that you have to give up to get an extra unit of period-2 consumption. This price turns out to be 1/(1 + r), where r is the interest rate. The second trick is in the way you treat income in the two diﬀerent periods. Suppose that a consumer has an income of m1 in period 1 and m2 in period 2 and that there is no inﬂation. The total amount of period1 consumption that this consumer could buy, if he borrowed as much m2 money as he could possibly repay in period 2, is m1 + 1+r . As you work the exercises and study the text, it should become clear that the consumer’s budget equation for choosing consumption in the two periods is always c2 m2 c1 + = m1 + . 1+r 1+r This budget constraint looks just like the standard budget constraint that you studied in previous chapters, where the price of “good 1” is 1, the m2 price of “good 2” is 1/(1 + r), and “income” is m1 + (1+r) . Therefore if you are given a consumer’s utility function, the interest rate, and the consumer’s income in each period, you can ﬁnd his demand for consumption in periods 1 and 2 using the methods you already know. Having solved for consumption in each period, you can also ﬁnd saving, since the consumer’s saving is just the diﬀerence between his period-1 income and his period-1 consumption. Example: A consumer has the utility function U (c1 , c2 ) = c1 c2 . There is no inﬂation, the interest rate is 10%, and the consumer has income 100 in period 1 and 121 in period 2. Then the consumer’s budget constraint c1 + c2 /1.1 = 100 + 121/1.1 = 210. The ratio of the price of good 1 to the price of good 2 is 1 + r = 1.1. The consumer will choose a consumption bundle so that M U1 /M U2 = 1.1. But M U1 = c2 and M U2 = c1 , so the consumer must choose a bundle such that c2 /c1 = 1.1. Take this equation together with the budget equation to solve for c1 and c2 . The solution is c1 = 105 and c2 = 115.50. Since the consumer’s period-1 income is only 100, he must borrow 5 in order to consume 105 in period 1. To pay back principal and interest in period 2, he must pay 5.50 out of his period-2 income of 121. This leaves him with 115.50 to consume. You will also be asked to determine the eﬀects of inﬂation on con- (Ch. 10) sumer behavior. The key to understanding the eﬀects of inﬂation is to see what happens to the budget constraint. Example: Suppose that in the previous example, there happened to be an inﬂation rate of 6%, and suppose that the price of period-1 goods is 1. Then if you save $1 in period 1 and get it back with 10% interest, you will get back $1.10 in period 2. But because of the inﬂation, goods in period 2 cost 1.06 dollars per unit. Therefore the amount of period-1 consumption that you have to give up to get a unit of period-2 consumption is 1.06/1.10 = .964 units of period-2 consumption. If the consumer’s money income in each period is unchanged, then his budget equation is c1 + .964c2 = 210. This budget constraint is the same as the budget constraint would be if there were no inﬂation and the interest rate were r, where .964 = 1/(1 + r). The value of r that solves this equation is known as the real rate of interest. In this case the real rate of interest is about .038. When the interest rate and inﬂation rate are both small, the real rate of interest is closely approximated by the diﬀerence between the nominal interest rate, (10% in this case) and the inﬂation rate (6% in this case), that is, .038 ∼ .10 − .06. As you will see, this is not such a good approximation if inﬂation rates and interest rates are large. 10.1 (0) Peregrine Pickle consumes (c1 , c2 ) and earns (m1 , m2 ) in periods 1 and 2 respectively. Suppose the interest rate is r. (a) Write down Peregrine’s intertemporal budget constraint in present value terms. c1 + c2 m2 = m1 + (1+r) . (1+r) (b) If Peregrine does not consume anything in period 1, what is the most he can consume in period 2? m1 (1 + r) + m2 . value, present value) of his endowment. This is the (future Future value. (c) If Peregrine does not consume anything in period 2, what is the most m2 he can consume in period 1? m1 + (1+r) . This is the (future value, present value) of his endowment. Present value. slope of Peregrine’s budget line? −(1 + r). What is the 10.2 (0) Molly has a Cobb-Douglas utility function U (c1 , c2 ) = ca1 c21−a , where 0 < a < 1 and where c1 and c2 are her consumptions in periods 1 and 2 respectively. We saw earlier that if utility has the form u(x1 , x2 ) = xa1 x1−a and the budget constraint is of the “standard” form p1 x1 +p2 x2 = 2 m, then the demand functions for the goods are x1 = am/p1 and x2 = (1 − a)m/p2 . NAME 133 134 INTERTEMPORAL CHOICE (Ch. 10) (a) Suppose that Molly’s income is m1 in period 1 and m2 in period 2. Write down her budget constraint in terms of present values. Consumption next year in 1,000s c1 + 4 c2 /(1 + r) = m1 + m2 /(1 + r). 3 Red line 2 (b) We want to compare this budget constraint to one of the standard form. In terms of Molly’s budget constraint, what is p1 ? is p2 ? 1/(1 + r). 1. Blue line What m1 + m2 /(1 + r). What is m? a e 1 Squiggly line 0 1 2 3 4 Consumption this year in 1,000s (c) If a = .2, solve for Molly’s demand functions for consumption in each period as a function of m1 , m2 , and r. Her demand function for consumption in period 1 is c1 = .2m1 + .2m2 /(1 + r). demand function for consumption in period 2 is c2 = (a) What is the present value of Nickleby’s endowment? Her .8(1+ r)m1 + .8m2 . $3,000. What is the future value of his endowment? $3,300. With blue ink, show the combinations of consumption this year and consumption next year that he can aﬀord. Label Nickelby’s endowment with the letter E. (b) Suppose that Nickleby has the utility function U (C1 , C2 ) = C1 C2 . Write an expression for Nickleby’s marginal rate of substitution between consumption this year and consumption next year. (Your answer will be a function of the variables C1 , C2 .) (d) An increase in the interest rate will consumption. It will increase increase decrease her period-1 her period-2 consumption and MRS = −C2/C1 . (c) What is the slope of Nickleby’s budget line? −1.1. Write an equation that states that the slope of Nickleby’s indiﬀerence curve is equal to the slope of his budget line when the interest rate is 10%. her savings in period 1. C2 /C1 . Also write down Nickleby’s budget equation. 1.1 = C1 + C2 /1.1 = 3, 000. (d) Solve these two equations. Nickleby will consume 10.3 (0) Nickleby has an income of $2,000 this year, and he expects an income of $1,100 next year. He can borrow and lend money at an interest rate of 10%. Consumption goods cost $1 per unit this year and there is no inﬂation. NAME (e) Will he borrow or save in the ﬁrst period? 135 Save. How much? 500. 136 1,650 INTERTEMPORAL CHOICE 1,500 units units in period 2. Label this point A on your (Ch. 10) (a) Use blue ink to illustrate his budget set in the graph below. (Hint: The boundary of the budget set is not a single straight line.) C2 (f ) On your graph use red ink to show what Nickleby’s budget line would be if the interest rate rose to 20%. Knowing that Nickleby chose the point A at a 10% interest rate, even without knowing his utility function, you can determine that his new choice cannot be on certain parts of his new budget line. Draw a squiggly mark over the part of his new budget line where that choice can not be. (Hint: Close your eyes and think of WARP.) (g) Solve for Nickleby’s optimal choice when the interest rate is 20%. Nickleby will consume units in period 2. in period 1 and diagram. 1,458.3 units in period 1 and Save. 20 10 Blue line Black line 1,750 How much? 541.7. 10.4 (0) Decide whether each of the following statements is true or false. Then explain why your answer is correct, based on the Slutsky decomposition into income and substitution eﬀects. (a) “If both current and future consumption are normal goods, an increase in the interest rate will necessarily make a saver save more.” Red line 30 0 (h) Will he borrow or save in the ﬁrst period? False. Substitution effect makes him consume less in period 1 and save more. For a saver, income effect works in opposite direction. Either effect could dominate. (b) “If both current and future consumption are normal goods, an increase in the interest rate will necessarily make a saver choose more consumption in the second period.” 40 True. The income and substitution effects both lead to more consumption in the second period. 10.5 (1) Laertes has an endowment of $20 each period. He can borrow money at an interest rate of 200%, and he can lend money at a rate of 0%. (Note: If the interest rate is 0%, for every dollar that you save, you get back $1 in the next period. If the interest rate is 200%, then for every dollar you borrow, you have to pay back $3 in the next period.) 10 20 30 40 C1 (b) Laertes could invest in a project that would leave him with m1 = 30 and m2 = 15. Besides investing in the project, he can still borrow at 200% interest or lend at 0% interest. Use red ink to draw the new budget set in the graph above. Would Laertes be better oﬀ or worse oﬀ by investing in this project given his possibilities for borrowing or lending? Or can’t one tell without knowing something about his preferences? Explain. Better off. If he invests in the project, he can borrow or lend to get any bundle he could afford without investing. (c) Consider an alternative project that would leave Laertes with the endowment m1 = 15, m2 = 30. Again suppose he can borrow and lend as above. But if he chooses this project, he can’t do the ﬁrst project. Use pencil or black ink to draw the budget set available to Laertes if he chooses this project. Is Laertes better oﬀ or worse oﬀ by choosing this project than if he didn’t choose either project? Or can’t one tell without knowing more about his preferences? Explain. Can’t tell. He can afford some things he couldn’t afford originally. But some things he could afford before, he can’t afford if he invests in this project. 10.6 (0) The table below reports the inﬂation rate and the annual rate of return on treasury bills in several countries for the years 1984 and 1985. NAME 137 138 INTERTEMPORAL CHOICE (Ch. 10) Inﬂation Rate and Interest Rate for Selected Countries Real Rates of Interest in 1984 and 1985 % Inﬂation Rate, 1984 3.6 304.6 3.1 2.2 9.2 90.0 0.6 Country United States Israel Switzerland W. Germany Italy Argentina Japan % Inﬂation Rate, 1985 1.9 48.1 0.8 −0.2 5.8 672.2 2.0 % Interest Rate, 1984 9.6 217.3 3.6 5.3 15.3 NA NA % Interest Rate, 1985 7.5 210.1 4.1 4.2 13.9 NA NA (a) In the table below, use the formula that your textbook gives for the exact real rate of interest to compute the exact real rates of interest. (b) What would the nominal rate of return on a bond in Argentina have to be to give a real rate of return of 5% in 1985? 710.8%. What would the nominal rate of return on a bond in Japan have to be to give a real rate of return of 5% in 1985? 7.1%. (c) Subtracting the inﬂation rate from the nominal rate of return gives a good approximation to the real rate for countries with a low rate of inﬂation. For the United States in 1984, the approximation gives you 6% Country 1984 United States 5.7 1985 5.5 Israel −21.57 109.4 3.33 Switzerland 0.5 W. Germany 3.0 4.4 Italy 5.6 7.6 10.7 (0) We return to the planet Mungo. On Mungo, macroeconomists and bankers are jolly, clever creatures, and there are two kinds of money, red money and blue money. Recall that to buy something in Mungo you have to pay for it twice, once with blue money and once with red money. Everything has a blue-money price and a red-money price, and nobody is ever allowed to trade one kind of money for the other. There is a bluemoney bank where you can borrow and lend blue money at a 50% annual interest rate. There is a red-money bank where you can borrow and lend red money at a 25% annual interest rate. A Mungoan named Jane consumes only one commodity, ambrosia, but it must decide how to allocate its consumption between this year and next year. Jane’s income this year is 100 blue currency units and no red currency units. Next year, its income will be 100 red currency units and no blue currency units. The blue currency price of ambrosia is one b.c.u. per ﬂagon this year and will be two b.c.u.’s per ﬂagon next year. The red currency price of ambrosia is one r.c.u. per ﬂagon this year and will be the same next year. while the more exact method suggested by the text gives you 5.79%. (a) If Jane spent all of its blue income in the ﬁrst period, it would be But for countries with very high inﬂation this is a poor approximation. The approximation gives you −87.3% for Israel in 1984, while the more exact formula gives you −21.57%. For Argentina in 1985, the approximation would tell us that a bond yielding 677.7% a nominal rate of contrasts with the answer would yield a real interest rate of 5%. This 710.8% that you found above. NAME 139 enough to pay the blue price for 100 ﬂagons of ambrosia. If Jane saved all of this year’s blue income at the blue-money bank, it would have 150 b.c.u.’s next year. This would give it enough blue currency to pay the blue price for 75 ﬂagons of ambrosia. On the graph below, draw Jane’s blue budget line, depicting all of those combinations of current and next period’s consumption that it has enough blue income to buy. 140 INTERTEMPORAL CHOICE (Ch. 10) Ambrosia next period 100 Ambrosia next period 100 75 Red line 75 50 50 25 25 0 25 50 75 100 Ambrosia this period 0 ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; c ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; Blue ;;;;;;;;;;;;;;;;;;;; line ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; 25 50 75 100 Ambrosia this period (b) If Jane planned to spend no red income in the next period and to borrow as much red currency as it can pay back with interest with next period’s red income, how much red currency could it borrow? 80. (e) It turns out that Jane ﬁnds it optimal to operate on its blue budget line and beneath its red budget line. Find such a point on your graph and mark it with a C. (c) The (exact) real rate of interest on blue money is real rate of interest on red money is −25%. The 25%. (d) On the axes below, draw Jane’s blue budget line and its red budget line. Shade in all of those combinations of current and future ambrosia consumption that Jane can aﬀord given that it has to pay with both currencies. (f ) On the following graph, show what happens to Jane’s original budget set if the blue interest rate rises and the red interest rate does not change. On your graph, shade in the part of the new budget line where Jane’s new demand could possibly be. (Hint: Apply the principle of revealed preference. Think about what bundles were available but rejected when Jane chose to consume at C before the change in blue interest rates.) NAME 141 142 INTERTEMPORAL CHOICE (Ch. 10) (a) Write Harvey’s budget constraint in terms of future value, assuming Ambrosia next period 1.21c1 + c2 = 3, 520. no inﬂation. 100 (b) How much bread does Harvey consume in the ﬁrst period and how Red 75 50 25 Shaded region much money does he save? (The answer is not necessarily an integer.) line ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; c ;;;;;;;;;;;;;;;;;;;; New blue ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; line ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; Blue ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;line ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;; 0 25 He picks c1 = c2 . Substitute into the budget to find c1 = 3, 520/2.21 = 1, 592.8. He saves 2, 000 − 3, 520/2.21 = 407.2. (c) Suppose that Harvey’s money income in both periods is the same as before, the interest rate is still 21%, but there is a 10% inﬂation rate. Then in period 2, a loaf of bread will cost $ 1.10. Write down Harvey’s budget equation for period-1 and period-2 bread, given this new 50 75 100 Ambrosia this period 10.8 (0) Mr. O. B. Kandle will only live for two periods. In the ﬁrst period he will earn $50,000. In the second period he will retire and live on his savings. His utility function is U (c1 , c2 ) = c1 c2 , where c1 is consumption in period 1 and c2 is consumption in period 2. He can borrow and lend at the interest rate r = .10. (a) If the interest rate rises, will his period-1 consumption increase, decrease, or stay the same? 1.21c1 + 1.1c2 = 3, 520. information. 10.10 (2) In an isolated mountain village, the only crop is corn. Good harvests alternate with bad harvests. This year the harvest will be 1,000 bushels. Next year it will be 150 bushels. There is no trade with the outside world. Corn can be stored from one year to the next, but rats will eat 25% of what is stored in a year. The villagers have Cobb-Douglas utility functions, U (c1 , c2 ) = c1 c2 where c1 is consumption this year, and c2 is consumption next year. (a) Use red ink to draw a “budget line,” showing consumption possibilities for the village, with this year’s consumption on the horizontal axis and next year’s consumption on the vertical axis. Put numbers on your graph to show where the budget line hits the axes. Stay the same. His demand for c1 is .5(m1 + m2 /(1 + r)) and m2 = 0. (b) Would an increase in the interest rate make him consume more or less in the second period? Next year's consumption More. He saves the same 1250 amount, but with higher interest rates, he 1150 gets more back next period. 900 Blue line Black line (c) If Mr. Kandle’s income is zero in period 1, and $ 55,000 in period 2, would an increase in the interest rate make him consume more, less, or the same amount in period 1? Red line Less. 150 10.9 (1) Harvey Habit’s utility function is U (c1 , c2 ) = min{c1 , c2 }, where c1 is his consumption of bread in period 1 and c2 is his consumption of bread in period 2. The price of bread is $1 per loaf in period 1. The interest rate is 21%. Harvey earns $2,000 in period 1 and he will earn $1,100 in period 2. NAME How much will the rats eat? This year's consumption 143 (b) How much corn will the villagers consume this year? bushels. 1111 1000 1136 How (c) Suppose that a road is built to the village so that now the village is able to trade with the rest of the world. Now the villagers are able to buy and sell corn at the world price, which is $1 per bushel. They are also able to borrow and lend money at an interest rate of 10%. On your graph, use blue ink to draw the new budget line for the villagers. Solve for the and the second period (Ch. 10) saving to buy 1.05 units of consumption goods at the beginning of 1986. If you gave up a unit of consumption goods at the beginning of 1978 and saved your money at interest, you would be able to use the 450 bushels. amount they would now consume in the ﬁrst period INTERTEMPORAL CHOICE (b) If you gave up a unit of consumption goods at the beginning of 1985 and saved your money at interest, you could use the proceeds of your 600 100 bushels. much corn will the villagers consume next year? 144 proceeds of your saving to buy beginning of 1979. .96 units of consumption goods at the 10.12 (1) Marsha Mellow doesn’t care whether she consumes in period 1 or in period 2. Her utility function is simply U (c1 , c2 ) = c1 + c2 . Her initial endowment is $20 in period 1 and $40 in period 2. In an antique shop, she discovers a cookie jar that is for sale for $12 in period 1 and that she is certain she can sell for $20 in period 2. She derives no consumption beneﬁts from the cookie jar, and it costs her nothing to store it for one period. 568 bushels 624 bushels. (d) Suppose that all is as in the last part of the question except that there is a transportation cost of $.10 per bushel for every bushel of grain hauled into or out of the village. On your graph, use black ink or pencil to draw the budget line for the village under these circumstances. (a) On the graph below, label her initial endowment, E, and use blue ink to draw the budget line showing combinations of period-1 and period-2 consumption that she can aﬀord if she doesn’t buy the cookie jar. On the same graph, label the consumption bundle, A, that she would have if she did not borrow or lend any money but bought the cookie jar in period 1, sold it in period 2, and used the proceeds to buy period-2 consumption. 10.11 (0) The table below records percentage interest rates and inﬂation rates for the United States in some recent years. Complete this table. Inﬂation and Interest in the United States, 1965-1985 If she cannot borrow or lend, should Marsha invest in the cookie jar? Year 1965 1970 1975 1978 1980 1985 Yes. (b) Suppose that Marsha can borrow and lend at an interest rate of 50%. On the graph where you labelled her initial endowment, draw the budget line showing all of the bundles she can aﬀord if she invests in the cookie jar and borrows or lends at the interest rate of 50%. On the same graph use red ink to draw one or two of Marsha’s indiﬀerence curves. CPI, Start of Year 38.3 47.1 66.3 79.2 100.0 130.0 CPI, End of Year 39.4 49.2 69.1 88.1 110.4 133 2.3 % Inﬂation Rate 2.9 4.3 4.2 11.3 10.4 Nominal Int. Rate 4.0 6.4 5.8 7.2 11.6 7.5 Real Int. Rate 1.1 2.1 1.6 −3.7 1.09 5.07 (a) People complained a great deal about the high interest rates in the late 70s. In fact, interest rates had never reached such heights in modern times. Explain why such complaints are misleading. Nominal interest rates were high, but so was inflation. Period-2 consumption 80 60 e ; 20 Red curves Blue line Real interest rates were not high. (They were negative in 1978.) a 40 0 20 40 60 80 Period-1 consumption NAME 145 146 INTERTEMPORAL CHOICE 148 ASSET MARKETS (Ch. 10) (c) Suppose that instead of consumption in the two periods being perfect substitutes, they are perfect complements, so that Marsha’s utility function is min{c1 , c2 }. If she cannot borrow or lend, should she buy the cookie jar? No. If she can borrow and lend at an interest rate of 50%, should she invest in the cookie jar? Yes. If she can borrow or lend as much at an interest rate of 100%, should she invest in the cookie jar? No. Chapter 11 NAME Asset Markets Introduction. The fundamental equilibrium condition for asset markets is that in equilibrium the rate of return on all assets must be the same. Thus if you know the rate of interest and the cash ﬂow generated by an asset, you can predict what its market equilibrium price will be. This condition has many interesting implications for the pricing of durable assets. Here you will explore several of these implications. Example: A drug manufacturing ﬁrm owns the patent for a new medicine. The patent will expire on January 1, 1996, at which time anyone can produce the drug. Whoever owns the patent will make a proﬁt of $1,000,000 per year until the patent expires. For simplicity, let us suppose that profits for any year are all collected on December 31. The interest rate is 5%. Let us ﬁgure out what the selling price of the patent rights will be on January 1, 1993. On January 1, 1993, potential buyers realize that owning the patent will give them $1,000,000 every year starting 1 year from now and continuing for 3 years. The present value of this cash ﬂow is 1, 000, 000 1, 000, 000 1, 000, 000 $ + ∼ $2, 723, 248. + (1.05) (1.05)2 (1.05)3 Nobody would pay more than this amount for the patent since if you put $2,723,248 at 5% interest, you could collect $1,000,000 a year from the bank for 3 years, starting 1 year from now. The patent wouldn’t sell for less than $2,723,248, since if it sold for less, one would get a higher rate of return by investing in this patent than one could get from investing in anything else. What will the price of the patent be on January 1, 1994? At that time, the patent is equivalent to a cash ﬂow of $1,000,000 in 1 year and another $1,000,000 in 2 years. The present value of this ﬂow, viewed from the standpoint of January 1, 1994, will be $ 1, 000, 000 1, 000, 000 ∼ $1, 859, 310. + (1.05) (1.05)2 A slightly more diﬃcult problem is one where the cash ﬂow from an asset depends on how the asset is used. To ﬁnd the price of such an asset, one must ask what will be the present value of the cash ﬂow that the asset yields if it is managed in such a way as to maximize its present value. Example: People will be willing to pay $15 a bottle to drink a certain wine this year. Next year they would be willing to pay $25, and the year after that they would be willing to pay $26. After that, it starts to deteriorate and the amount people are willing to pay to drink it falls. The interest rate is 5%. We can determine not only what the wine will sell for but also when it will be drunk. If the wine is drunk in the ﬁrst year, it would have to sell for $15. But no rational investor is going to sell the wine for (Ch. 11) $15 in the ﬁrst year, because it will sell for $25 one year later. This is a 66.66% rate of return, which is better than the rate of interest. When the interest rate is 5%, investors are willing to pay at least $25/1.05 = $23.81 for the wine. So investors must outbid drinkers, and none will be drunk this year. Will investors want to hold onto the wine for 2 years? In 2 years, the wine will be worth $26, so the present value of buying the wine and storing it for 2 years is $26/(1.05)2 = $23.58. This is less than the present value of holding the wine for 1 year and selling it for $25. So, we conclude that the wine will be drunk after 1 year. Its current selling price will be $23.81, and 1 year from now, it will sell for $25. 11.0 Warm Up Exercise. Here are a few problems on present values. In all of the following examples, assume that you can both borrow and lend at an annual interest rate of r and that the interest rate will remain the same forever. (a) You would be indiﬀerent between getting $1 now and 1+r dollars, one year from now, because if you put the dollar in the bank, then one year from now you could get back 1+r dollars from the bank. (b) You would be indiﬀerent between getting 1 dollar(s) one year from now and getting $1/(1 + r) dollars now, because 1/(1 + r) deposited in the bank right now would enable you to withdraw principal and interest worth $1. (c) For any X > 0, you would be indiﬀerent between getting r) X/(1 + dollars right now and $X one year from now. The present value of $X received one year from now is X/(1 + r) dollars. (d) The present value of an obligation to pay $X one year from now is −X/(1 + r) dollars. (e) The present value of $X, to be received 2 years from now, is X/(1 + r)2 dollars. (f ) The present value of an asset that pays Xt dollars t years from now is Xt /(1 + r)t dollars. NAME 149 150 (g) The present value of an asset that pays $X1 one year from now, $X2 in two years, and $X10 ten years from now is r)2 + X10 /(1 + r)10 X1 /(1+r)+X2 /(1+ dollars. (h) The present value of an asset that pays a constant amount, $X per year forever can be computed in two diﬀerent ways. One way is to ﬁgure out the amount of money you need in the bank so that the bank would give you $X per year, forever, without ever exhausting your principal. The annual interest received on a bank account of X/r be $X. Therefore having getting $X a year forever. dollars will 11.2 (0) Publicity agents for the Detroit Felines announce the signing of a phenomenal new quarterback, Archie Parabola. They say that the contract is worth $1,000,000 and will be paid in 20 installments of $50,000 per year starting one year from now and with one new installment each year for next 20 years. The contract contains a clause that guarantees he will get all of the money even if he is injured and cannot play a single game. Sports writers declare that Archie has become an “instant millionaire.” dollars right now is just as good as (i) Another way to calculate the present value of $X a year forever is to ∞ i i=1 X/(1 + r) . This series is known as a geometric series. Whenever r > 0, this sum is well deﬁned (a) Archie’s brother, Fenwick, who majored in economics, explains to Archie that he is not a millionaire. In fact, his contract is worth less than half a million dollars. Explain in words why this is so. X/r. The present value of $50,000 a year for 20 (j) If the interest rate is 10%, the present value of receiving $1,000 one year from now will be, to the nearest dollar, $909. What will be the market price of an acre of Still $463. land? X/r (Ch. 11) After 10 years. evaluate the inﬁnite series and is equal to ASSET MARKETS (b) Suppose that the Christmas trees do not have to be sold after 10 years, but could be sold in any year. Their value if they are cut before they are 10 years old is zero. After the trees are 10 years old, an acre of trees is worth $1,000 and its value will increase by $100 per year for the next 20 years. After the trees are cut, the land on which the trees stood can always be sold for $200 an acre. When should the trees be cut to maximize the present value of the payments received for trees and land? The present years is less than $1,000,000, since the value of receiving $1,000 a year forever, will be, to the nearest dollar, present value of a dollar received in the $10,000. future is less than $1. (k) If the interest rate is 10%, what is the present value of an asset that requires you to pay out $550 one year from now and will pay you back Archie’s college course on “Sports Management” didn’t cover present values. So his brother tried to reason out the calculation for him. Here is how it goes: $1,210 two years from now? 500 −550/(1.1) + 1, 210/(1.1)2 = (b) Suppose that the interest rate is 10% and is expected to remain at 10% forever. How much would it cost the team to buy Archie a perpetuity that would pay him and his heirs $1 per year forever, starting in 1 year? dollars. $10. 11.1 (0) An area of land has been planted with Christmas trees. On December 1, ten years from now, the trees will be ready for harvest. At that time, the standing Christmas trees can be sold for $1,000 per acre. The land, after the trees have been removed, will be worth $200 per acre. There are no taxes or operating expenses, but also no revenue from this land until the trees are harvested. The interest rate is 10%. (c) How much would it cost to buy a perpetuity that paid $50,000 a year forever, starting in one year? $500,000. In the last part, you found the present value of Archie’s contract if he were going to get $50,000 a year forever. But Archie is not going to get $50, 000 a year forever. The payments stop after 20 years. The present value of Archie’s actual contract is the same as the present value of a contract that pays him $50,000 a year forever, but makes him pay back $50,000 each year, forever, starting 21 years from now. Therefore you can ﬁnd the present value of Archie’s contract by subtracting the (a) What can we expect the market price of the land to be? $1, 200/(1.1)10 ∼ $463 per acre. NAME 151 152 present value of $50,000 a year forever, starting 21 years from now from the present value of $50,000 a year forever. ASSET MARKETS (d) If the interest rate is and will remain at 10%, a stream of payments of $50,000 a year, starting 21 years from now has the same present value as a lump sum of $ years from now. 500,000 (Ch. 11) (b) The two professors have also forgotten the formula for the present value of a stream of $1 per year starting next year and continuing for K years. They agree to call this number S(K) and they see that S(K) = to be received all at once, exactly 20 1 1 1 + ... + . + (1 + r) (1 + r)2 (1 + r)K Professor Thesis notices that if each of the payments came 1 year earlier, the present value of the resulting stream of payments would be (e) If the interest rate is and will remain at 10%, what is the present value 1+ of $50,000 per year forever, starting 21 years from now? $75,000. (Hint: The present value of $1 to be paid in 20 years is 1/(1 + r)20 = .15.) (f ) Now calculate the present value of Archie’s contract. Professor Antithesis points out that speeding up any stream of payments by a year is also equivalent to multiplying its present value by (1 + r). Putting their two observations together, the two professors noticed an equation that could be solved for S(K). This equation is S(K) + 1 − 8.50 × 50, 000 = $425, 000. 11.3 (0) Professor Thesis is puzzling over the formula for the present value of a stream of payments of $1 a year, starting 1 year from now and continuing forever. He knows that the value of this stream is expressed by the inﬁnite series S= 1 1 1 + + ..., + 1 + r (1 + r)2 (1 + r)3 but he can’t remember the simpliﬁed formula for this sum. All he knows is that if the ﬁrst payment were to arrive today, rather than a year from now, the present value of the sum would be $1 higher. So he knows that S+1=1+ 1 1 1 + + .... + (1 + r) (1 + r)2 (1 + r)3 1 1 1 1 + ... + = S(K) + 1 − . + (1 + r) (1 + r)2 (1 + r)K−1 (1 + r)K 1 (1+r)K = S(K)(1 + r). that the formula for S(K) is Calculus Solving this equation for S(K), they ﬁnd S(K) = (1 − 1 (1+r)K )/r. 11.4 (0) You are the business manager of P. Bunyan Forests, Inc., and are trying to decide when you should cut your trees. The market value of the lumber that you will get if you let your trees reach the age of t years 2 is given by the function W (t) = e.20t−.001t . Mr. Bunyan can earn an interest rate of 5% per year on money in the bank. The rate of growth of the market value of the trees will be greater than 5% until the trees reach 75 years of age. (Hint: It follows from elementary calculus that if F (t) = eg(t) , then F (t)/F (t) = g (t).) (a) If he is only interested in the trees as an investment, how old should Professor Antithesis suﬀers from a similar memory lapse. He can’t remember the formula for S either. But, he knows that the present value of $1 a year forever, starting right now has to be 1 + r times as large as the present value of $1 a year, starting a year from now. (This is true because if you advance any income stream by a year, you multiply its present value by 1+r.) That is, 1+ 1 1 1 + + . . . = (1 + r)S. + (1 + r) (1 + r)2 (1 + r)3 (a) If Professor Thesis and Professor Antithesis put their knowledge together, they can express a simple equation involving only the variable S. This equation is S + 1 = that S = 1/r. (1 + r)S. Solving this equation, they ﬁnd Mr. Bunyan let the trees get? 75 years. (b) At what age do the trees have the greatest market value? 100 years. 11.5 (0) You expect the price of a certain painting to rise by 8% per year forever. The market interest rate for borrowing and lending is 10%. Assume there are no brokerage costs in purchasing or selling. (a) If you pay a price of $x for the painting now and sell it in a year, how much has it cost you to hold the painting rather than to have loaned the $x at the market interest rate? It has cost .02x. NAME 153 (b) You would be willing to pay $100 a year to have the painting on your walls. Write an equation that you can solve for the price x at which you would be just willing to buy the painting .02x = 100. 154 ASSET MARKETS (Ch. 11) (e) What is the most that Ashley would be willing to pay today for a case of Wine C? (Hint: What is the present value of his investment if he sells it to a drinker at the optimal time?) Express your answer in exponential notation without calculating it out. (c) How much should you be willing to pay to buy the painting? $5,000. 11.6 (2) Ashley is thinking of buying a truckload of wine for investment purposes. He can borrow and lend as much as he likes at an annual interest rate of 10%. He is looking at three kinds of wine. To keep our calculations simple, let us assume that handling and storage costs are negligible. • Wine drinkers would pay exactly $175 a case to drink Wine A today. But if Wine A is allowed to mature for one year, it will improve. In fact wine drinkers will be willing to pay $220 a case to drink this wine one year from today. After that, the wine gradually deteriorates and becomes less valuable every year. • From now until one year from now, Wine B is indistinguishable from Wine A. But instead of deteriorating after one year, Wine B will improve. In fact the amount that wine drinkers would be willing to pay to drink Wine B will be $220 a case in one year and will rise by $10 per case per year for the next 30 years. • Wine drinkers would be willing to pay $100 per case to drink Wine C right now. But one year from now, they will be willing to pay $250 per case to drink it and the amount they will be willing to pay to drink it will rise by $50 per case per year for the next 20 years. (a) What is the most Ashley would be willing to pay per case for Wine A? $200. (b) What is the most Ashley would be willing to pay per case for Wine B? $200. (Hint: When will Wine B be drunk?) $500/1.16 . 11.7 (0) Fisher Brown is taxed at 40% on his income from ordinary bonds. Ordinary bonds pay 10% interest. Interest on municipal bonds is not taxed at all. (a) If the interest rate on municipal bonds is 7%, should he buy municipal bonds or ordinary bonds? Brown should buy municipal bonds. (b) Hunter Black makes less money than Fisher Brown and is taxed at only 25% on his income from ordinary bonds. Which kind of bonds should he buy? Black should buy ordinary bonds. (c) If Fisher has $1,000,000 in bonds and Hunter has $10,000 in bonds, 0. how much tax does Fisher pay on his interest from bonds? How much tax does Hunter pay on his interest from bonds? $250. (d) The government is considering a new tax plan under which no interest income will be taxed. If the interest rates on the two types of bonds do not change, and Fisher and Hunter are allowed to adjust their portfolios, how much will Fisher’s after-tax income be increased? $30,000. How much will Hunter’s after-tax income be increased? $250. (e) What would the change in the tax law do to the demand for municipal (c) How old will Wine C be when it ﬁrst becomes worthwhile for investors to sell oﬀ their holdings and for drinkers to drink it? 6 years. (Hint: When does the rate of return on holding wine get to 10%?) bonds if the interest rates did not change? It would reduce it to zero. (f ) What interest rate will new issues of municipal bonds have to pay (d) What will the price of Wine C be at the time it is ﬁrst drunk? in order to attract purchasers? $500 per case. 10%. NAME 155 (g) What do you think will happen to the market price of the old municipal bonds, which had a 7% yield originally? The price of the old bonds will fall until their yield 156 ASSET MARKETS They will have to pay (Ch. 11) (a) How much is the James bond 007 worth to Dr. No at an interest rate of 10%? 200/1.1+200/1.12 +200/1.13 +2, 000/1.13 = 2, 000. equals 10%. (b) How valuable is James bond 007 at an interest rate of 5%? 11.8 (0) In the text we discussed the market for oil assuming zero production costs, but now suppose that it is costly to get the oil out of the ground. Suppose that it costs $5 dollars per barrel to extract oil from the ground. Let the price in period t be denoted by pt and let r be the interest rate. 200/1.05 + 200/1.052 + 200/1.053 + 2, 000/1.053 = 2, 272.32. (c) Ms. Yes oﬀers Dr. No $2,200 for the James bond 007. Should Dr. No (a) If a ﬁrm extracts a barrel of oil in period t, how much proﬁt does it make in period t? say yes or no to Ms. Yes if the interest rate is 10%? pt − 5. the interest rate is 5%? Yes. What if No. (b) If a ﬁrm extracts a barrel of oil in period t + 1, how much proﬁt does it make in period t + 1? pt+1 − 5. (c) What is the present value of the proﬁts from extracting a barrel of oil in period t+1? (pt+1 −5)/(1+r)t+1 . What is the present value of proﬁt from extracting a barrel of oil in period t? (pt −5)/(1+r)t. (d) If the ﬁrm is willing to supply oil in each of the two periods, what must be true about the relation between the present value of proﬁts from sale of a barrel of oil in the two periods? values must be equal. pt+1 −5 (1+r)t+1 = The present Express this relation as an equation. pt −5 (1+r)t . pt+1 = (1 + r)pt − 5r. (f ) Is the percentage rate of price increase between periods larger or smaller than the interest rate? need today to ﬁnance this dastardly program? $2,272.32, which is the present value calculated in the first part of the problem. If the interest rate were 10%, would the world be in more or less danger from Dr. No? More danger, since the dastardly plan is now cheaper. (e) Solve the equation in the above part for pt+1 as a function of pt and r. (d) In order to destroy the world, Dr. No hires Professor Know to develop a nasty zap beam. In order to lure Professor Know from his university position, Dr. No will have to pay the professor $200 a year. The nasty zap beam will take three years to develop, at the end of which it can be built for $2,000. If the interest rate is 5%, how much money will Dr. No The percent change in price is smaller. 11.9 (0) Dr. No owns a bond, serial number 007, issued by the James Company. The bond pays $200 for each of the next three years, at which time the bond is retired and pays its face value of $2,000. 11.10 (0) Chillingsworth owns a large, poorly insulated home. His annual fuel bill for home heating averages $300 per year. An insulation contractor suggests to him the following options. Plan A. Insulate just the attic. If he does this, he will permanently reduce his fuel consumption by 15%. Total cost of insulating the attic is $300. Plan B. Insulate the attic and the walls. If he does this, he will permanently reduce his fuel consumption by 20%. Total cost of insulating the attic and the walls is $500. Plan C. Insulate the attic and the walls, and install a solar heating unit. If he does this, he will permanently reduce his fuel costs to zero. Total cost of this option is $7,000 for the solar heater and $500 for the insulating. NAME 157 (a) Assume for simplicity of calculations that the house and the insulation will last forever. Calculate the present value of the dollars saved on fuel from each of the three options if the interest rate is 10%. The present values are: Plan A? $450. Plan B? $600. Plan C? $3,000. (b) Each plan requires an expenditure of money to undertake. The diﬀerence between the present value and the present cost of each plan is: Plan A? 450 − 300 = 150. C? 3, 000 − 7, 500 = −4, 500. Plan B? 600 − 500 = 100. Plan (c) If the price of fuel is expected to remain constant, which option should he choose if he can borrow and lend at an annual interest rate of 10%? A. (Ch. 11) (a) Suppose that you have just graduated from high school at age 18. You want to estimate the present value of your lifetime earnings if you do not go to college but take a job immediately. To do this, you have to make some assumptions. Assume that you would work for 47 years, until you are 65 and then retire. Assume also that you would make $13,000 a year for the rest of your life. (If you were going to do this more carefully, you would want to take into account that people’s wages vary with their age, but let’s keep things simple for this problem.) Assume that the interest rate is 5%. Find the present value of your lifetime earnings. (Hint: First ﬁnd out the present value of $13,000 a year forever. Subtract from this (e) Suppose that the government oﬀers to pay half of the cost of any insulation or solar heating device. Which option would he now choose at interest rates 10%? B. C. 5%? (f ) Suppose that there is no government subsidy but that fuel prices are expected to rise by 5% per year. What is the present value of fuel savings from each of the three proposals if interest rates are 10%? (Hint: If a stream of income is growing at x% and being discounted at y%, its present value should be the same as that of a constant stream of income discounted at (y − x)%.) Plan A? $6,000. interest rates are 10%? $233,753. B. rate of 5%? $900. Plan B? $1,200. Which proposal should Chillingsworth choose if B. C. 5%? 11.11 (1) Have you ever wondered if a college education is ﬁnancially worthwhile? The U.S. Census Bureau collects data on income and education that throws some light on this question. A recent census publication (Current Population Reports, Series P-70, No. 11) reports the average annual wage income in 1984 of persons aged 35–44 by the level of schooling achieved. The average wage income of high school graduates was $13,000 per year. The average wage income of persons with bachelor’s degrees was $24,000 per year. The average wage income of persons with master’s degrees was $28,000 per year. The average wage income of persons with Ph.D.’s was $40,000 per year. These income diﬀerences probably overstate the return to education itself, because it is likely that those people who get more education tend to be more able than those who get less. NAME 159 (f ) What would be the present value of her lifetime earnings if she goes to college for 4 years and then takes a job until she is 65? $213,990. (g) If college tuition is $5,000 per year, is it ﬁnancially worthwhile for her to go to college? Explain. Yes, the gain in the present value of her income exceeds the present value of tuition. 11.12 (0) As you may have noticed, economics is a diﬃcult major. Are their any rewards for all this eﬀort? The U.S. census publication discussed in the last problem suggests that there might be. There are tables reporting wage income by the ﬁeld in which one gets a degree. For bachelor’s degrees, the most lucrative majors are economics and engineering. The average wage incomes for economists are about $28,000 per year and for engineers are about $27,000. Psychology majors average about $15,000 a year and English majors about $14,000 per year. (a) Can you think of any explanation for these diﬀerences? Some might say that economics, like accounting or mortuary science, is so boring the pay has to be high to get you to do it. ASSET MARKETS Some of the income diﬀerence is, therefore, a return to ability rather than to education. But just to get a rough idea of returns to education, let us see what would be the return if the reported wage diﬀerences are all due to education. the present value of $13,000 a year forever, starting 47 years from now.) (d) Which option should he choose if he can borrow and lend at an annual Plan C? 158 Others would suggest that the ability to do well in economics is scarce and is valued by the marketplace. Perhaps the English majors and psychology majors include disproportionately many persons who are not full-time participants in the labor force. No doubt there are several other good partial explanations. (b) The same table shows that the average person with an advanced degree in business earns $38,000 per year and the average person with a degree in medicine earns $45,000 per year. Suppose that an advanced degree in business takes 2 years after one spends 4 years getting a bachelor’s degree and that a medical degree takes 4 years after getting a bachelor’s (b) Again, supposing you have just graduated from high school at age 18, and you want to estimate the present value of your life time earnings if you go to college for 4 years and do not earn any wages until you graduate from college. Assume that after graduating from college, you would work for 43 years at $24,000 per year. What would be the present value of your lifetime earnings? $346,441. (c) Now calculate the present value of your lifetime earnings if you get a master’s degree. Assume that if you get a master’s, you have no earnings for 6 years and then you work for 41 years at $28,000 per year. What would be the present value of your lifetime income? $361,349. (d) Finally calculate the present value of your lifetime earnings if you get a Ph.D. Assume that if you get a Ph.D., you will have no earnings for 8 years and then you work for 39 years at $40,000 per year. What would be the present value of your lifetime income? $460,712. (e) Consider the case of someone who married right after ﬁnishing high school and stopped her education at that point. Suppose that she is now 45 years old. Her children are nearly adults, and she is thinking about going back to work or going to college. Assuming she would earn the average wage for her educational level and would retire at age 65, what would be the present value of her lifetime earnings if she does not go to college? 160 $162,000. ASSET MARKETS (Ch. 11) degree. Suppose that you are 22 years old and have just ﬁnished college. If r = .05, ﬁnd the present value of lifetime earnings for a graduating senior who will get an advanced degree in business and earn the average wage rate for someone with this degree until retiring at 65. Make a similar calculation for medicine. $596,000. $630,000. 11.13 (0) On the planet Stinko, the principal industry is turnip growing. For centuries the turnip ﬁelds have been fertilized by guano which was deposited by the now-extinct giant scissor-billed kiki-bird. It costs $5 per ton to mine kiki-bird guano and deliver it to the ﬁelds. Unfortunately, the country’s stock of kiki-bird guano is about to be exhausted. Fortunately the scientists on Stinko have devised a way of synthesizing kiki-guano from political science textbooks and swamp water. This method of production makes it possible to produce a product indistinguishable from kiki-guano and to deliver it to the turnip ﬁelds at a cost of $30 per ton. The interest rate on Stinko is 10%. There are perfectly competitive markets for all commodities. (a) Given the current price and the demand function for kiki-guano, the last of the deposits on Stinko will be exhausted exactly one year from now. Next year, the price of kiki-guano delivered to the ﬁelds will have to be $30, so that the synthetic kiki-guano industry will just break even. The owners of the guano deposits know that next year, they would get a net return of $25 a ton for any guano they have left to sell. In equilibrium, what must be the current price of kiki-guano delivered to the turnip ﬁelds? The price of guano delivered to the field must be the $5 + the present value of $25. This is 5 + 25/1.1 = 27.73. (Hint: In equilibrium, sellers must be indiﬀerent between selling their kiki-guano right now or at any other time before the total supply is exhausted. But we know that they must be willing to sell it right up until the day, one year from now, when the supply will be exhausted and the price will be $30, the cost of synthetic guano.) (b) Suppose that everything is as we have said previously except that the deposits of kiki-guano will be exhausted 10 years from now. What must be the current price of kiki-guano? (Hint: 1.110 = 2.59.) 5 + 25/(1.1)10 = 14.65. 162 Chapter 12 NAME Uncertainty Introduction. In Chapter 11, you learned some tricks that allow you to use techniques you already know for studying intertemporal choice. Here you will learn some similar tricks, so that you can use the same methods to study risk taking, insurance, and gambling. One of these new tricks is similar to the trick of treating commodities at diﬀerent dates as diﬀerent commodities. This time, we invent new commodities, which we call contingent commodities. If either of two events A or B could happen, then we deﬁne one contingent commodity as consumption if A happens and another contingent commodity as consumption if B happens. The second trick is to ﬁnd a budget constraint that correctly speciﬁes the set of contingent commodity bundles that a consumer can aﬀord. This chapter presents one other new idea, and that is the notion of von Neumann-Morgenstern utility. A consumer’s willingness to take various gambles and his willingness to buy insurance will be determined by how he feels about various combinations of contingent commodities. Often it is reasonable to assume that these preferences can be expressed by a utility function that takes the special form known as von NeumannMorgenstern utility. The assumption that utility takes this form is called the expected utility hypothesis. If there are two events, 1 and 2 with probabilities π1 and π2 , and if the contingent consumptions are c1 and c2 , then the von Neumann-Morgenstern utility function has the special functional form, U (c1 , c2 ) = π1 u(c1 ) + π2 u(c2 ). The consumer’s behavior is determined by maximizing this utility function subject to his budget constraint. Example: You are thinking of betting on whether the Cincinnati Reds will make it to the World Series this year. A local gambler will bet with you at odds of 10 to 1 against the Reds. You think the probability that the Reds will make it to the World Series is π = .2. If you don’t bet, you are certain to have $1,000 to spend on consumption goods. Your behavior satisﬁes the expected utility hypothesis and your von Neumann√ √ Morgenstern utility function is π1 c1 + π2 c2 . The contingent commodities are dollars if the Reds make the World Series and dollars if the Reds don’t make the World Series. Let cW be your consumption contingent on the Reds making the World Series and cN W be your consumption contingent on their not making the Series. Betting on the Reds at odds of 10 to 1 means that if you bet $x on the Reds, then if the Reds make it to the Series, you make a net gain of $10x, but if they don’t, you have a net loss of $x. Since you had $1,000 before betting, if you bet $x on the Reds and they made it to the Series, you would have cW = 1, 000 + 10x to spend on consumption. If you bet $x on the Reds and they didn’t make it to the Series, you would lose $x, NAME 163 amount he can consume, contingent on the system not being approved, is cN A = 50, 000 − 5x. (b) You can solve for Condor’s budget constraint on contingent commodity bundles (cA , cN A ) by eliminating x from these two equations. His budget constraint can be written as .5 cA + .5 cN A = 50, 000. (c) Buzz Condor has no moral qualms about trading on inside information, nor does he have any concern that he will be caught and punished. To decide how much stock to buy, he simply maximizes his von NeumannMorgenstern utility function subject to his budget. If he sets his marginal rate of substitution between the two contingent commodities equal to their relative prices and simpliﬁes the equation, he ﬁnds that cA /cN A = 3. (Reminder: Where a is any constant, the derivative of a ln x with respect to x is a/x.) (d) Condor ﬁnds that his optimal contingent commodity bundle is (cA , cN A ) = UNCERTAINTY (Ch. 12) and you would have cN W = 1, 000 − x. By increasing the amount $x that you bet, you can make cW larger and cN W smaller. (You could also bet against the Reds at the same odds. If you bet $x against the Reds and they fail to make it to the Series, you make a net gain of .1x and if they make it to the Series, you lose $x. If you work through the rest of this discussion for the case where you bet against the Reds, you will see that the same equations apply, with x being a negative number.) We can use the above two equations to solve for a budget equation. From the second equation, we have x = 1, 000 − cN W . Substitute this expression for x into the ﬁrst equation and rearrange terms to ﬁnd cW + 10cN W = 11, 000, or equivalently, .1cW + cN W = 1, 100. (The same budget equation can be written in many equivalent ways by multiplying both sides by a positive constant.) Then you will choose your contingent consumption bundle (cW , cN W ) √ √ to maximize U (cW , cN W ) = .2 cW + .8 cN W subject to the budget constraint, .1cW + cN W = 1, 100. Using techniques that are now familiar, you can solve this consumer problem. From the budget constraint, you see that consumption contingent on the Reds making the World Series costs 1/10 as much as consumption contingent on their not making it. If you set the marginal rate of substitution between cW and cN W equal to the price ratio and simplify the resulting expression, you will ﬁnd that cN W = .16cW . This equation, together with the budget equation implies that cW = $4, 230.77 and cN W = $676.92. You achieve this bundle by betting $323.08 on the Reds. If the Reds make it to the Series, you will have $1, 000 + 10 × 323.08 = $4, 230.80. If not, you will have $676.92. (We rounded the solutions to the nearest penny.) 12.1 (0) In the next few weeks, Congress is going to decide whether or not to develop an expensive new weapons system. If the system is approved, it will be very proﬁtable for the defense contractor, General Statics. Indeed, if the new system is approved, the value of stock in General Statics will rise from $10 per share to $15 a share, and if the project is not approved, the value of the stock will fall to $5 a share. In his capacity as a messenger for Congressman Kickback, Buzz Condor has discovered that the weapons system is much more likely to be approved than is generally thought. On the basis of what he knows, Condor has decided that the probability that the system will be approved is 3/4 and the probability that it will not be approved is 1/4. Let cA be Condor’s consumption if the system is approved and cN A be his consumption if the system is not approved. Condor’s von Neumann-Morgenstern utility function is U (cA , cN A ) = .75 ln cA + .25 ln cN A . Condor’s total wealth is $50,000, all of which is invested in perfectly safe assets. Condor is about to buy stock in General Statics. (a) If Condor buys x shares of stock, and if the weapons system is approved, he will make a proﬁt of $5 per share. Thus the amount he can consume, contingent on the system being approved, is cA = $50, 000 + 5x. If Condor buys x shares of stock, and if the weapons system is not approved, then he will make a loss of $ 164 UNCERTAINTY 5 per share. Thus the (Ch. 12) (c) You can eliminate x from the two equations for cF and cN F that you found above. This gives you a budget equation for Willy. Of course there are many equivalent ways of writing the same budget equation, since multiplying both sides of a budget equation by a positive constant yields an equivalent budget equation. The form of the budget equation in which the “price” of cN F is 1 can be written as .9cN F + .1 cF = 455,000. (d) Willy’s marginal rate of substitution between the two contingent commodities, dollars if there is no ﬂood and dollars if there is a ﬂood, is √ .1 c M RS(cF , cN F ) = − .9√cNFF . To ﬁnd his optimal bundle of contingent commodities, you must set this marginal rate of substitution equal to the number = −9. Solving this equation, you ﬁnd that Willy will choose to consume the two contingent commodities in the ratio cN F /cF = 1. (75,000,25,000). To acquire this contingent com- modity bundle, he must buy 5,000 shares of stock in General Statics. 12.2 (0) Willy owns a small chocolate factory, located close to a river that occasionally ﬂoods in the spring, with disastrous consequences. Next summer, Willy plans to sell the factory and retire. The only income he will have is the proceeds of the sale of his factory. If there is no ﬂood, the factory will be worth $500,000. If there is a ﬂood, then what is left of the factory will be worth only $50,000. Willy can buy ﬂood insurance at a cost of $.10 for each $1 worth of coverage. Willy thinks that the probability that there will be a ﬂood this spring is 1/10. Let cF denote the contingent commodity dollars if there is a ﬂood and cN F denote dollars if there is no ﬂood. Willy’s von Neumann-Morgenstern utility function is √ √ U (cF , cN F ) = .1 cF + .9 cN F . (a) If he buys no insurance, then in each contingency, Willy’s consumption will equal the value of his factory, so Willy’s contingent commodity bundle will be (cF , cN F ) = (50, 000, 500, 000). (b) To buy insurance that pays him $x in case of a ﬂood, Willy must pay an insurance premium of .1x. (The insurance premium must be paid whether or not there is a ﬂood.) If Willy insures for $x, then if there is a ﬂood, he gets $x in insurance beneﬁts. Suppose that Willy has contracted for insurance that pays him $x in the event of a ﬂood. Then after paying his insurance premium, he will be able to consume cF = 50, 000 + .9x. If Willy has this amount of insurance and there is no ﬂood, then he will be able to consume cN F = 500, 000 − .1x. (e) Since you know the ratio in which he will consume cF and cN F , and you know his budget equation, you can solve for his optimal consumption bundle, which is (cF , cN F )= (455, 000, 455, 000). buy an insurance policy that will pay him $450,000 Willy will if there is a ﬂood. The amount of insurance premium that he will have to pay is $45,000. 12.3 (0) Clarence Bunsen is an expected utility maximizer. His preferences among contingent commodity bundles are represented by the expected utility function √ √ u(c1 , c2 , π1 , π2 ) = π1 c1 + π2 c2 . Clarence’s friend, Hjalmer Ingqvist, has oﬀered to bet him $1,000 on the outcome of the toss of a coin. That is, if the coin comes up heads, Clarence must pay Hjalmer $1,000 and if the coin comes up tails, Hjalmer must pay Clarence $1,000. The coin is a fair coin, so that the probability of heads and the probability of tails are both 1/2. If he doesn’t accept the bet, Clarence will have $10,000 with certainty. In the privacy of his car dealership oﬃce over at Bunsen Motors, Clarence is making his decision. (Clarence uses the pocket calculator that his son, Elmer, gave him last Christmas. You will ﬁnd that it will be helpful for you to use a calculator too.) Let Event 1 be “coin comes up heads” and let Event 2 be “coin comes up tails.” NAME 165 9,000 (a) If Clarence accepts the bet, then in Event 1, he will have dollars and in Event 2, he will have 11,000 dollars. (b) Since the probability of each event is 1/2, Clarence’s expected utility for a gamble in which he gets c1 in Event 1 and c2 in Event 2 can be 1√ 1√ described by the formula 2 c1 + 2 c2 . Therefore Clarence’s 99.8746. expected utility if he accepts the bet with Hjalmer will be (Use that calculator.) dollars and in Event 2, he will have 10,000 Therefore if he doesn’t bet, his expected utility will be dollars. 100. (d) Having calculated his expected utility if he bets and if he does not bet, Clarence determines which is higher and makes his decision accordingly. Does Clarence take the bet? UNCERTAINTY (Ch. 12) (c) Clarence later asks himself, “If I make a bet where I lose my $10,000 if the coin comes up heads, what is the smallest amount that I would have to win in the event of tails in order to make the bet a good one for me to take?” After some trial and error, Clarence found the answer. You, too, might want to ﬁnd the answer by trial and error, but it is easier to ﬁnd the answer by solving an equation. On the left side of your equation, you would write down Clarence’s utility if he doesn’t bet. On the right side of the equation, you write down an expression for Clarence’s utility if he makes a bet such that he is left with zero consumption in Event 1 and x in Event 2. Solve this equation for x. The answer to Clarence’s question is where x = 10, 000. The equation that you should write is (c) If Clarence decides not to bet, then in Event 1, he will have 10,000 166 No. 100 = 1√ 2 x. The solution is x = 40, 000. (d) Your answer to the last part gives you two points on Clarence’s indiﬀerence curve between the contingent commodities, money in Event 1 and money in Event 2. (Poor Clarence has never heard of indiﬀerence curves or contingent commodities, so you will have to work this part for him, while he heads over to the Chatterbox Cafe for morning coﬀee.) One of these points is where money in both events is $10,000. On the graph below, label this point A. The other is where money in Event 1 is zero 40,000. and money in Event 2 is point B. 12.4 (0) It is a slow day at Bunsen Motors, so since he has his calculator warmed up, Clarence Bunsen (whose preferences toward risk were described in the last problem) decides to study his expected utility function more closely. On the graph below, label this Money in Event 2 (x 1,000) 40 b 30 (a) Clarence ﬁrst thinks about really big gambles. What if he bet his entire $10,000 on the toss of a coin, where he loses if heads and wins if tails? Then if the coin came up heads, he would have 0 dollars and if it came up tails, he would have $20,000. His expected utility if he took the bet would be 70.71 , while his expected utility if he didn’t take the bet would be such a bet. 100. 20 d 10 a 0 10 Therefore he concludes that he would not take (b) Clarence then thinks, “Well, of course, I wouldn’t want to take a chance on losing all of my money on just an ordinary bet. But, what if somebody oﬀered me a really good deal. Suppose I had a chance to bet where if a fair coin came up heads, I lost my $10,000, but if it came up tails, I would win $50,000. Would I take the bet? If I took the bet, my expected utility would be expected utility would be 122.5. 100. If I didn’t take the bet, my Therefore I should take the bet.” NAME 167 c 20 30 40 Money in Event 1 (x 1,000) (e) You can quickly ﬁnd a third point on this indiﬀerence curve. The coin is a fair coin, and Clarence cares whether heads or tails turn up only because that determines his prize. Therefore Clarence will be indiﬀerent between two gambles that are the same except that the assignment of prizes to outcomes are reversed. In this example, Clarence will be indifferent between point B on the graph and a point in which he gets zero if Event 2 happens and 40,000 if Event 1 happens. Find this point on the Figure above and label it C. 168 UNCERTAINTY (Ch. 12) (f ) Another gamble that is on the same indiﬀerence curve for Clarence as not gambling at all is the gamble where he loses $5,000 if heads turn up and where he wins 6,715.73 dollars if tails turn up. (Hint: To solve this problem, put the utility of not betting on the left side of an equation and on the right side of the equation, put the utility of having $10, 000 − $5, 000 in Event 1 and $10, 000 + x in Event 2. Then solve the resulting equation for x.) On the axes above, plot this point and label it D. Now sketch in the entire indiﬀerence curve through the points that you have labeled. Money in Event 2 200 150 100 12.5 (0) Hjalmer Ingqvist’s son-in-law, Earl, has not worked out very well. It turns out that Earl likes to gamble. His preferences over contingent commodity bundles are represented by the expected utility function Blue curve 50 Red curves u(c1 , c2 , π1 , π2 ) = π1 c21 + π2 c22 . 0 (a) Just the other day, some of the boys were down at Skoog’s tavern when Earl stopped in. They got to talking about just how bad a bet they could get him to take. At the time, Earl had $100. Kenny Olson shuﬄed a deck of cards and oﬀered to bet Earl $20 that Earl would not cut a spade from the deck. Assuming that Earl believed that Kenny wouldn’t cheat, the probability that Earl would win the bet was 1/4 and the probability that Earl would lose the bet was 3/4. If he won the bet, Earl would have 120 dollars and if he lost the bet, he would have and his expected utility if he did not take the bet would be Therefore he refused the bet. 80 8,400 , dollars. Earl’s expected utility if he took the bet would be 10,000. (b) Just when they started to think Earl might have changed his ways, Kenny oﬀered to make the same bet with Earl except that they would bet $100 instead of $20. What is Earl’s expected utility if he takes that bet? 10,000. Would Earl be willing to take this bet? He is 50 100 150 200 Money in Event 1 (d) On the same graph, let us draw Hjalmer’s son-in-law Earl’s indifference curves between contingent commodities where the probabilities are diﬀerent. Suppose that a card is drawn from a fair deck of cards. Let Event 1 be the event that the card is black. Let event 2 be the event that the card drawn is red. Suppose each event has probability 1/2. Then Earl’s preferences between income contingent on Event 1 and income con1 2 1 2 tingent on Event 2 are represented by the formula u = 2 c1 + 2 c2 . On the graph, use red ink to show two of Earl’s indiﬀerence curves, including the one that passes through (100, 100). 12.6 (1) Sidewalk Sam makes his living selling sunglasses at the boardwalk in Atlantic City. If the sun shines Sam makes $30, and if it rains Sam only makes $10. For simplicity, we will suppose that there are only two kinds of days, sunny ones and rainy ones. just indifferent about taking it or not. (c) Let Event 1 be the event that a card drawn from a fair deck of cards is a spade. Let Event 2 be the event that the card is not a spade. Earl’s preferences between income contingent on Event 1, c1 , and income contingent 1 2 3 2 on Event 2, c2 , can be represented by the equation u = 4 c1 + 4 c2 . Use blue ink on the graph below to sketch Earl’s indiﬀerence curve passing through the point (100, 100). (a) One of the casinos in Atlantic City has a new gimmick. It is accepting bets on whether it will be sunny or rainy the next day. The casino sells dated “rain coupons” for $1 each. If it rains the next day, the casino will give you $2 for every rain coupon you bought on the previous day. If it doesn’t rain, your rain coupon is worthless. In the graph below, mark Sam’s “endowment” of contingent consumption if he makes no bets with the casino, and label it E. NAME 169 170 UNCERTAINTY (Ch. 12) (a) How many units of consumption is it optimal for Sam to consume Cr (b) How many rain coupons is it optimal for Sam to buy? 30 20 20 units. conditional on rain? 40 Blue line 12.8 (0) Sidewalk Sam’s brother Morgan von Neumanstern is an expected utility maximizer. His von Neumann-Morgenstern utility function for wealth is u(c) = ln c. Sam’s brother also sells sunglasses on another beach in Atlantic City and makes exactly the same income as Sam does. He can make exactly the same deal with the casino as Sam can. a e 10 (a) If Morgan believes that there is a 50% chance of rain and a 50% chance of sun every day, what would his expected utility of consuming (cs , cr ) Red line 0 10 20 10. 30 be? 40 Cs u = 12 ln cs + 12 ln cr . (b) How does Morgan’s utility function compare to Sam’s? Is one a (b) On the same graph, mark the combination of consumption contingent on rain and consumption contingent on sun that he could achieve by buying 10 rain coupons from the casino. Label it A. (c) On the same graph, use blue ink to draw the budget line representing all of the other patterns of consumption that Sam can achieve by buying rain coupons. (Assume that he can buy fractional coupons, but not negative amounts of them.) What is the slope of Sam’s budget line at points above and to the left of his initial endowment? The slope is −1. Morgan’s utility monotonic transformation of the other? function is just the natural log of Sam’s, so the answer is yes. (c) What will Morgan’s optimal pattern of consumption be? Answer: Morgan will consume 20 on the sunny days and 20 the rainy days. How does this compare to Sam’s consumption? (d) Suppose that the casino also sells sunshine coupons. These tickets also cost $1. With these tickets, the casino gives you $2 if it doesn’t rain and nothing if it does. On the graph above, use red ink to sketch in the budget line of contingent consumption bundles that Sam can achieve by buying sunshine tickets. (e) If the price of a dollar’s worth of consumption when it rains is set equal to 1, what is the price of a dollar’s worth of consumption if it shines? The price is 1. 12.7 (0) Sidewalk Sam, from the previous problem, has the utility function for consumption in the two states of nature u(cs , cr , π) = c1−π cπr , s NAME 171 (b) If Billy John pays $p for an insurance policy that would give him $1,000,000 if he suﬀered a career-ending injury while in college, then he would be sure to have an income of $1, 000, 000 − p no matter what happened to him. Write an equation that can be solved to ﬁnd the largest price that Billy John would be willing to pay for such an insurance policy. The equation is 910 = is the same as Sam’s consumption. 12.9 (0) Billy John Pigskin of Mule Shoe, Texas, √ has a von NeumannMorgenstern utility function of the form u(c) = c. Billy John also weighs about 300 pounds and can outrun jackrabbits and pizza delivery trucks. Billy John is beginning his senior year of college football. If he is not seriously injured, he will receive a $1,000,000 contract for playing professional football. If an injury ends his football career, he will receive a $10,000 contract as a refuse removal facilitator in his home town. There is a 10% chance that Billy John will be injured badly enough to end his career. (a) What is Billy John’s expected utility? where cs is the dollar value of his consumption if it shines, cr is the dollar value of his consumption if it rains, and π is the probability that it will rain. The probability that it will rain is π = .5. 1, 000, 000 − p. on This √ √ .1 10, 000 + .9 1, 000, 000 = 910. 172 UNCERTAINTY We calculate (Ch. 12) Money if the Boars lose 400 Red line 300 200 e Blue line d c 100 (c) Solve this equation for p. p = 171, 900. 0 100 200 300 400 Money if the Boars win (b) Label the point on this graph where you would be if you did not bet at all with an E. 12.10 (1) You have $200 and are thinking about betting on the Big Game next Saturday. Your team, the Golden Boars, are scheduled to play their traditional rivals the Robber Barons. It appears that the going odds are 2 to 1 against the Golden Boars. That is to say if you want to bet $10 on the Boars, you can ﬁnd someone who will agree to pay you $20 if the Boars win in return for your promise to pay him $10 if the Robber Barons win. Similarly if you want to bet $10 on the Robber Barons, you can ﬁnd someone who will pay you $10 if the Robber Barons win, in return for your promise to pay him $20 if the Robber Barons lose. Suppose that you are able to make as large a bet as you like, either on the Boars or on the Robber Barons so long as your gambling losses do not exceed $200. (To avoid tedium, let us ignore the possibility of ties.) (c) After careful thought you decide to bet $50 on the Boars. Label the point you have chosen on the graph with a C. Suppose that after you have made this bet, it is announced that the star Robber Baron quarterback suﬀered a sprained thumb during a tough economics midterm examination and will miss the game. The market odds shift from 2 to 1 against the Boars to “even money” or 1 to 1. That is, you can now bet on either team and the amount you would win if you bet on the winning team is the same as the amount that you would lose if you bet on the losing team. You cannot cancel your original bet, but you can make new bets at the new odds. Suppose that you keep your ﬁrst bet, but you now also bet $50 on the Robber Barons at the new odds. If the Boars win, then after you collect your winnings from one bet and your losses from the other, how much money will you have left? $250. If the Robber Barons win, how much money will you have left after collecting your winnings and paying oﬀ your losses? (a) If you do not bet at all, you will have $200 whether or not the Boars win. If you bet $50 on the Boars, then after all gambling obligations are settled, you will have a total of 300 dollars if the Boars win and $200. (d) Use red ink to draw a line on the diagram you made above, showing the combinations of “money if the Boars win” and “money if the Robber Barons win” that you could arrange for yourself by adding possible bets at the new odds to the bet you made before the news of the quarterback’s misfortune. On this graph, label the point D that you reached by making the two bets discussed above. 150 dollars if they lose. On the graph below, use blue ink to draw a line that represents all of the combinations of “money if the Boars win” and “money if the Robber Barons win” that you could have by betting from your initial $200 at these odds. 12.11 (2) The certainty equivalent of a lottery is the amount of money you would have to be given with certainty to be just as well-oﬀ with that lottery. Suppose that your von Neumann-Morgenstern utility function NAME 173 over lotteries that give you an amount x√ if Event 1 happens and y if √ Event 1 does not happen is U (x, y, π) = π x + (1 − π) y, where π is the probability that Event 1 happens and 1 − π is the probability that Event 1 does not happen. (a) If π = .5, calculate the utility of a lottery that gives you $10,000 if Event 1 happens and $100 if Event 1 does not happen. 55 = .5 × 100 + .5 × 10. 174 UNCERTAINTY (Ch. 12) (d) Therefore if he buys full insurance, the certainty equivalent of his wealth is $248,400 , and his expected utility is √ 248, 800. 12.13 (1) Portia has been waiting a long time for her ship to come in and has concluded that there is a 25% chance that it will arrive today. If it does come in today, she will receive $1,600. If it does not come in today, it will never come and her wealth will be zero. Portia has a von Neumann-Morgenstern utility such that she wants to maximize the √ expected value of c, where c is total wealth. What is the minimum price $100. at which she will sell the rights to her ship? (b) If you were sure to receive $4,900, what would your utility be? 70. (Hint: If you receive $4,900 with certainty, then you receive $4,900 in both events.) (c) Given this utility function and π = .5, write a general formula for the certainty equivalent of a lottery that gives you $x if Event 1 happens and $y if Event 1 does not happen. (.5x1/2 + .5y 1/2 )2. (d) Calculate the certainty equivalent of receiving $10,000 if Event 1 hap- $3,025. pens and $100 if Event 1 does not happen. 12.12 (0) Dan Partridge is a risk averter who tries to maximize the √ expected value of c, where c is his wealth. Dan has $50,000 in safe assets and he also owns a house that is located in an area where there are lots of forest ﬁres. If his house burns down, the remains of his house and the lot it is built on would be worth only $40,000, giving him a total wealth of $90,000. If his home doesn’t burn, it will be worth $200,000 and his total wealth will be $250,000. The probability that his home will burn down is .01. (a) Calculate his expected utility if he doesn’t buy ﬁre insurance. $498. (b) Calculate the certainty equivalent of the lottery he faces if he doesn’t buy ﬁre insurance. $248,004. (c) Suppose that he can buy insurance at a price of $1 per $100 of insurance. For example if he buys $100,000 worth of insurance, he will pay $1,000 to the company no matter what happens, but if his house burns, he will also receive $100,000 from the company. If Dan buys $160,000 worth of insurance, he will be fully insured in the sense that no matter what happens his after-tax wealth will be $248,400. 176 Chapter 13 NAME Risky Assets Introduction. Here you will solve the problems of consumers who wish to divide their wealth optimally between a risky asset and a safe asset. The expected rate of return on a portfolio is just a weighted average of the rate of return on the safe asset and the expected rate of return on the risky asset, where the weights are the fractions of the consumer’s wealth held in each. The standard deviation of the portfolio return is just the standard deviation of the return on the risky asset times the fraction of the consumer’s wealth held in the risky asset. Sometimes you will look at the problem of a consumer who has preferences over the expected return and the risk of her portfolio and who faces a budget constraint. Since a consumer can always put all of her wealth in the safe asset, one point on this budget constraint will be the combination of the safe rate of return and no risk (zero standard deviation). Now as the consumer puts x percent of her wealth into the risky asset, she gains on that amount the diﬀerence between the expected rate of return for the risky asset and the rate of return on the safe asset. But she also absorbs some risk. So the slope of the budget line will be the diﬀerence between the two returns divided by the standard deviation of the portfolio that has x percent of the consumer’s wealth invested in the risky asset. You can then apply the usual indiﬀerence curve–budget line analysis to ﬁnd the consumer’s optimal choice of risk and expected return given her preferences. (Remember that if the standard deviation is plotted on the horizontal axis and if less risk is preferred to more, the better bundles will lie to the northwest.) You will also be asked to apply the result from the Capital Asset Pricing Model that the expected rate of return on any asset is equal to the sum of the risk-free rate of return plus the risk adjustment. Remember too that the expected rate of return on an asset is its expected change in price divided by its current price. RISKY ASSETS (Ch. 13) (b) By solving the second equation above for x and substituting the result into the ﬁrst equation, derive an expression for the rate of return on the rm −rf portfolio in terms of the portfolio’s riskiness. rx = σ σx + rf . m (c) Suppose that Ms. Lynch can borrow money at the interest rate rf and invest it in the risky asset. If rm = 20, rf = 10, and σm = 10, what will be Ms. Lynch’s expected return if she borrows an amount equal to 100% of her initial wealth and invests it in the risky asset? (Hint: This is just like investing 200% of her wealth in the risky asset.) Apply the formula rx = xrm + (1 − x)rf with x = 2 to get rx = 2 × 20 − 1 × 10 = 30. (d) Suppose that Ms. Lynch can borrow or lend at the risk-free rate. If rf is 10%, rm is 20%, and σm is 10%, what is the formula for the “budget line” Ms. Lynch faces? graph below. rx = σx + 10. Plot this budget line in the Expected return 40 U=10 30 Budget line 20 U=5 10 U=0 13.1 (3) Ms. Lynch has a choice of two assets: The ﬁrst is a risk-free asset that oﬀers a rate of return of rf , and the second is a risky asset (a china shop that caters to large mammals) that has an expected rate of return of rm and a standard deviation of σm . (a) If x is the percent of wealth Ms. Lynch invests in the risky asset, what is the equation for the expected rate of return on the portfolio? rx = xrm + (1 − x)rf . deviation of the portfolio? What is the equation for the standard σx = xσm . 0 10 20 30 40 Standard deviation (e) Which of the following risky assets would Ms. Lynch prefer to her present risky asset, assuming she can only invest in one risky asset at a time and that she can invest a fraction of her wealth in whichever risky asset she chooses? Write the word “better,” “worse,” or “same” after each of the assets. Asset A with ra =17% and σa = 5%. Better. Asset B with rb =30% and σb = 25%. Worse. NAME Asset C with rc =11% and σc = 1%. 177 She will not invest anything in the risky asset. 20 , and his optimal value of σx is 5. (Hint: You will need to solve two equations in two unknowns. One of the equations is the budget constraint.) (f ) Plot Mr. Smith’s optimal choice and an indiﬀerence curve through it in the graph. (g) What fraction of his wealth should Mr. Smith invest in the risky asset? Using the answer to Part (a), we find an x 13.2 (3) Fenner Smith is contemplating dividing his portfolio between two assets, a risky asset that has an expected return of 30% and a standard deviation of 10%, and a safe asset that has an expected return of 10% and a standard deviation of 0%. (a) If Mr. Smith invests x percent of his wealth in the risky asset, what rx = 30x + 10(1 − x). will be his expected return? (Ch. 13) Mr. Smith’s optimal value of rx is Better. (f ) Suppose Ms. Lynch’s utility function has the form u(rx , σx ) = rx −2σx . How much of her portfolio will she invest in the original risky asset? (You might want to graph a few of Ms. Lynch’s indiﬀerence curves before answering; e.g., graph the combinations of rx and σx that imply u(rx , σx ) = 0, 1, . . .). RISKY ASSETS (e) If Mr. Smith’s utility function is u(rx , σx ) = min{rx , 30 − 2σx }, then Same. Asset D with rd =25% and σd = 14%. 178 that solves 20 = rx = 30x + 10(1 − x). 13.3 (2) Assuming that the Capital Asset Pricing Model is valid, complete the following table. In this table p0 is the current price of asset i and Ep1 is the expected price of asset i in the next period. (b) If Mr. Smith invests x percent of his wealth in the risky asset, what will be the standard deviation of his wealth? σx = 10x. (c) Solve the above two equations for the expected return on Mr. Smith’s wealth as a function of the standard deviation he accepts. The answer is x = .5. The budget line is rx = 2σx + 10. rf rm ri βi p0 Ep1 10 20 10 0 100 110 10 20 25 10 15 20 0 30 10 22 (d) Plot this “budget line” on the graph below. 100 125 2 200 240 20 2/3 40 48 10 0 80 88 1.5 Expected return 13.4 (2) Farmer Alf Alpha has a pasture located on a sandy hill. The return to him from this pasture is a random variable depending on how much rain there is. In rainy years the yield is good; in dry years the yield is poor. The market value of this pasture is $5,000. The expected return from this pasture is $500 with a standard deviation of $100. Every inch of rain above average means an extra $100 in proﬁt and every inch of rain below average means another $100 less proﬁt than average. Farmer Alf has another $5,000 that he wants to invest in a second pasture. There are two possible pastures that he can buy. 40 30 Budget line 20 Optimal choice 10 (a) One is located on low land that never ﬂoods. This pasture yields an expected return of $500 per year no matter what the weather is like. What is Alf Alpha’s expected rate of return on his total investment if he Indifference curves buys this pasture for his second pasture? 0 5 10 15 20 Standard deviation NAME deviation of his rate of return in this case? 179 (b) Another pasture that he could buy is located on the very edge of the river. This gives very good yields in dry years but in wet years it ﬂoods. This pasture also costs $5,000. The expected return from this pasture is $500 and the standard deviation is $100. Every inch of rain below average means an extra $100 in proﬁt and every inch of rain above average means another $100 less proﬁt than average. If Alf buys this pasture and keeps his original pasture on the sandy hill, what is his expected rate of return on his total investment? 10%. What is the standard deviation of the rate of return on his total investment in this case? 0%. (c) If Alf is a risk averter, which of these two pastures should he buy and why? He should choose the second pasture since it has the same expected return and lower risk. 180 RISKY ASSETS (Ch. 13) 10%. What is the standard 10%. 182 Chapter 14 CONSUMER’S SURPLUS (Ch. 14) the area of this trapezoid by applying the formula NAME Consumer’s Surplus Area of a trapezoid = base 1 × (height1 + height2 ). 2 In this case we have A = 5 × 12 (100 + 50) = $375. Introduction. In this chapter you will study ways to measure a consumer’s valuation of a good given the consumer’s demand curve for it. The basic logic is as follows: The height of the demand curve measures how much the consumer is willing to pay for the last unit of the good purchased—the willingness to pay for the marginal unit. Therefore the sum of the willingnesses-to-pay for each unit gives us the total willingness to pay for the consumption of the good. In geometric terms, the total willingness to pay to consume some amount of the good is just the area under the demand curve up to that amount. This area is called gross consumer’s surplus or total beneﬁt of the consumption of the good. If the consumer has to pay some amount in order to purchase the good, then we must subtract this expenditure in order to calculate the (net) consumer’s surplus. When the utility function takes the quasilinear form, u(x) + m, the area under the demand curve measures u(x), and the area under the demand curve minus the expenditure on the other good measures u(x) + m. Thus in this case, consumer’s surplus serves as an exact measure of utility, and the change in consumer’s surplus is a monetary measure of a change in utility. If the utility function has a diﬀerent form, consumer’s surplus will not be an exact measure of utility, but it will often be a good approximation. However, if we want more exact measures, we can use the ideas of the compensating variation and the equivalent variation. Recall that the compensating variation is the amount of extra income that the consumer would need at the new prices to be as well oﬀ as she was facing the old prices; the equivalent variation is the amount of money that it would be necessary to take away from the consumer at the old prices to make her as well oﬀ as she would be, facing the new prices. Although diﬀerent in general, the change in consumer’s surplus and the compensating and equivalent variations will be the same if preferences are quasilinear. In this chapter you will practice: • Calculating consumer’s surplus and the change in consumer’s surplus • Calculating compensating and equivalent variations Example: Suppose that the inverse demand curve is given by P (q) = 100 − 10q and that the consumer currently has 5 units of the good. How much money would you have to pay him to compensate him for reducing his consumption of the good to zero? Answer: The inverse demand curve has a height of 100 when q = 0 and a height of 50 when q = 5. The area under the demand curve is a trapezoid with a base of 5 and heights of 100 and 50. We can calculate NAME 183 (a) If the equilibrium rent for an apartment turns out to be $20, which consumers will get apartments? A,B,C,D. Example: Suppose now that the consumer is purchasing the 5 units at a price of $50 per unit. If you require him to reduce his purchases to zero, how much money would be necessary to compensate him? In this case, we saw above that his gross beneﬁts decline by $375. On the other hand, he has to spend 5 × 50 = $250 less. The decline in net surplus is therefore $125. Example: Suppose that a consumer has a utility function u(x1 , x2 ) = x1 + x2 . Initially the consumer faces prices (1, 2) and has income 10. If the prices change to (4, 2), calculate the compensating and equivalent variations. Answer: Since the two goods are perfect substitutes, the consumer will initially consume the bundle (10, 0) and get a utility of 10. After the prices change, she will consume the bundle (0, 5) and get a utility of 5. After the price change she would need $20 to get a utility of 10; therefore the compensating variation is 20 − 10 = 10. Before the price change, she would need an income of 5 to get a utility of 5. Therefore the equivalent variation is 10 − 5 = 5. 14.1 (0) Sir Plus consumes mead, and his demand function for tankards of mead is given by D(p) = 100 − p, where p is the price of mead in shillings. (a) If the price of mead is 50 shillings per tankard, how many tankards of (b) How much gross consumer’s surplus does he get from this consumption? 3,750. (c) How much money does he spend on mead? 2,500. (d) What is his net consumer’s surplus from mead consumption? 1,250. 14.2 (0) Here is the table of reservation prices for apartments taken from Chapter 1: Person Price 184 = A = 40 B 25 CONSUMER’S SURPLUS C D 30 35 E 10 F 18 G 15 H 5 (Ch. 14) (f ) What is his total utility for earplugs and other things if the price of earplugs is $80? (b) If the equilibrium rent for an apartment turns out to be $20, what 50. mead will he consume? $4,200. 1,050 is the consumer’s (net) surplus generated in this market for person A? (g) Utility decreases by $80. 20. (h) What is the change in (net) consumer’s surplus when the price changes 5. For person B? (c) If the equilibrium rent is $20, what is the total net consumers’ surplus generated in the market? 50. (d) If the equilibrium rent is $20, what is the total gross consumers’ 130. surplus in the market? from $50 to $80? when the price changes from $50 to 1,050. 14.4 (2) In the graph below, you see a representation of Sarah Gamp’s indiﬀerence curves between cucumbers and other goods. Suppose that the reference price of cucumbers and the reference price of “other goods” are both 1. Other goods (e) If the rent declines to $19, how much does the gross surplus increase? 40 A 0. (f ) If the rent declines to $19, how much does the net surplus increase? 30 4. Calculus 14.3 (0) Quasimodo consumes earplugs and other things. His utility function for earplugs x and money to spend on other goods y is given by u(x, y) = 100x − B 10 x2 + y. 2 (a) What kind of utility function does Quasimodo have? (b) What is his inverse demand curve for earplugs? 20 Quasilinear. p = 100 − x. (c) If the price of earplugs is $50, how many earplugs will he consume? 0 10 20 30 40 Cucumbers (a) What is the minimum amount of money that Sarah would need in order to purchase a bundle that is indiﬀerent to A? 50. order to purchase a bundle that is indiﬀerent to B? (d) If the price of earplugs is $80, how many earplugs will he consume? 20. $5,250. 30. (c) Suppose that the reference price for cucumbers is 2 and the reference price for other goods is 1. How much money does she need in order to purchase a bundle that is indiﬀerent to bundle A? (e) Suppose that Quasimodo has $4,000 in total to spend a month. What is his total utility for earplugs and money to spend on other things if the price of earplugs is $50? 20. (b) What is the minimum amount of money that Sarah would need in 30. (d) What is the minimum amount of money that Sarah would need to purchase a bundle that is indiﬀerent to B using these new prices? 40. NAME 185 186 (e) No matter what prices Sarah faces, the amount of money she needs to purchase a bundle indiﬀerent to A must be (higher, lower) than the amount she needs to purchase a bundle indiﬀerent to B. CONSUMER’S SURPLUS increase is lower. income. 14.5 (2) Bernice’s preferences can be represented by u(x, y) = min{x, y}, where x is pairs of earrings and y is dollars to spend on other things. She faces prices (px , py ) = (2, 1) and her income is 12. of earrings and 4 4 $3. This is the (compensating, equivalent) variation in Equivalent. (e) What bundle would Bernice choose if she faced the new prices and had just enough income to reach her original indiﬀerence curve? (4, 4). Draw with black ink the budget line that passes through this bundle at the new prices. How much income would Bernice have with this budget? (a) Draw in pencil on the graph below some of Bernice’s indiﬀerence curves and her budget constraint. Her optimal bundle is (Ch. 14) (d) The maximum amount that Bernice would pay to avoid the price pairs $16. dollars to spend on other things. (f ) In order to be as well-oﬀ as she was with her original bundle, Bernice’s Dollars for other things $4. original income would have to rise by 16 Black line Calculus 12 Pencil lines Blue lines 8 This is the (compensating, Compensating. equivalent) variation in income. 14.6 (0) Ulrich likes video games and sausages. In fact, his preferences can be represented by u(x, y) = ln(x + 1) + y where x is the number of video games he plays and y is the number of dollars that he spends on sausages. Let px be the price of a video game and m be his income. (a) Write an expression that says that Ulrich’s marginal rate of substitution equals the price ratio. ( Hint: Remember Donald Fribble from 4 1/(x + 1) = px . Chapter 6?) Red line (b) Since Ulrich has quasilinear preferences, you can solve this equation alone to get his demand function for video games, which is 0 4 8 x = 1/px − 1. 12 16 Pairs of earrings (b) The price of a pair of earrings rises to $3 and Bernice’s income stays the same. Using blue ink, draw her new budget constraint on the graph 3 above. Her new optimal bundle is 3 pairs of earrings and dollars to spend on other things. (c) What bundle would Bernice choose if she faced the original prices and had just enough income to reach the new indiﬀerence curve? (3, 3). Draw with red ink the budget line that passes through this bundle at the original prices. How much income would Bernice need at the original prices to have this (red) budget line? NAME 187 on in full to consumers. With the tax in place, Ulrich demands bundle is 9.5 10.19. 1 dollars’ worth of sausages. His utility from this (Round oﬀ to two decimal places.) (f ) Now if we took away all of Ulrich’s video games, how much money would he have to have to spend on sausages to be just as well-oﬀ as with the bundle he purchased after the tax was in place? $10.19. (g) What is the change in Ulrich’s consumer surplus due to the tax? −.45 How much money did the government collect from Ulrich by means of the tax? Calculus $.25. 14.7 (1) Lolita, an intelligent and charming Holstein cow, consumes only two goods, cow feed (made of ground corn and oats) and hay. Her preferences are represented by the utility function U (x, y) = x − x2 /2 + y, where x is her consumption of cow feed and y is her consumption of hay. Lolita has been instructed in the mysteries of budgets and optimization and always maximizes her utility subject to her budget constraint. Lolita has an income of $m that she is allowed to spend as she wishes on cow feed and hay. The price of hay is always $1, and the price of cow feed will be denoted by p, where 0 < p ≤ 1. (a) Write Lolita’s inverse demand function for cow feed. (Hint: Lolita’s utility function is quasilinear. When y is the numeraire and the price of x is p, the inverse demand function for someone with quasilinear utility f (x) + y is found by simply setting p = f (x).) p = 1 − x. (b) If the price of cow feed is p and her income is m, how much hay does Lolita choose? (Hint: The money that she doesn’t spend on feed is used to buy hay.) m − p(1 − p). (c) Plug these numbers into her utility function to ﬁnd out the utility level that she enjoys at this price and this income. u = m+(1−p)2 /2. (d) Suppose that Lolita’s daily income is $3 and that the price of feed is $.50. What bundle does she buy? (1/2, 11/4). she buy if the price of cow feed rose to $1? His demand function for the dollars to spend on y = m − 1 + p. (c) Video games cost $.25 and Ulrich’s income is $10. Then Ulrich demands 3 video games and 9.25 10.64. His utility from this bundle is places.) dollars’ worth of sausages. (Round oﬀ to two decimal (d) If we took away all of Ulrich’s video games, how much money would he need to have to spend on sausages to be just as well-oﬀ as before? $10.64. $9. (e) Now an amusement tax of $.25 is put on video games and is passed video game and sausages is What bundle would (0, 3). 188 CONSUMER’S SURPLUS (Ch. 14) (e) How much money would Lolita be willing to pay to avoid having the price of cow feed rise to $1? equivalent 1/8. This amount is known as the variation. (f ) Suppose that the price of cow feed rose to $1. How much extra money would you have to pay Lolita to make her as well-oﬀ as she was at the old prices? 1/8. This amount is known as the compensating variation. Which is bigger, the compensating or the equivalent variation, or are they the same? Same. (g) At the price $.50 and income $3, how much (net) consumer’s surplus is Lolita getting? 1/8. 14.8 (2) F. Flintstone has quasilinear preferences and his inverse demand function for Brontosaurus Burgers is P (b) = 30 − 2b. Mr. Flintstone is currently consuming 10 burgers at a price of 10 dollars. (a) How much money would he be willing to pay to have this amount $200. rather than no burgers at all? consumer’s surplus? What is his level of (net) $100. (b) The town of Bedrock, the only supplier of Brontosaurus Burgers, decides to raise the price from $10 a burger to $14 a burger. What is Mr. Flintstone’s change in consumer’s surplus? At price $10, consumer’s surplus is $100. At $14, he demands 8 burgers, for net consumer’s surplus of 1 2 (16 × 8) = 64. The change in consumer’s surplus is −$36. 14.9 (1) Karl Kapitalist is willing to produce p/2 − 20 chairs at every price, p > 40. At prices below 40, he will produce nothing. If the price of chairs is $100, Karl will produce much is his producer’s surplus? 30 1 2 (60 chairs. At this price, how × 30) = 900. 14.10 (2) Ms. Q. Moto loves to ring the church bells for up to 10 hours a day. Where m is expenditure on other goods, and x is hours of bell ringing, her utility is u(m, x) = m + 3x for x ≤ 10. If x > 10, she develops painful blisters and is worse oﬀ than if she didn’t ring the bells. NAME 189 190 CONSUMER’S SURPLUS 192 MARKET DEMAND (Ch. 14) Her income is equal to $100 and the sexton allows her to ring the bell for 10 hours. (a) Due to complaints from the villagers, the sexton has decided to restrict Ms. Moto to 5 hours of bell ringing per day. This is bad news for Ms. Moto. In fact she regards it as just as bad as losing income. $15 dollars of (b) The sexton relents and oﬀers to let her ring the bells as much as she likes so long as she pays $2 per hour for the privilege. How much ringing does she do now? 10 hours. as a loss of how much income? This tax on her activities is as bad $20. (c) The villagers continue to complain. The sexton raises the price of bell ringing to $4 an hour. How much ringing does she do now? hours. 0 This tax, as compared to the situation in which she could ring the bells for free, is as bad as a loss of how much income? Chapter 15 $30. NAME −b (f ) D(p) = (p + a) . (Ch. 15) −bp/(p + a). Market Demand Introduction. Some problems in this chapter will ask you to construct the market demand curve from individual demand curves. The market demand at any given price is simply the sum of the individual demands at that price. The key thing to remember in going from individual demands to the market demand is to add quantities. Graphically, you sum the individual demands horizontally to get the market demand. The market demand curve will have a kink in it whenever the market price is high enough that some individual demand becomes zero. Sometimes you will need to ﬁnd a consumer’s reservation price for a good. Recall that the reservation price is the price that makes the consumer indiﬀerent between having the good at that price and not having the good at all. Mathematically, the reservation price p∗ satisﬁes u(0, m) = u(1, m − p∗ ), where m is income and the quantity of the other good is measured in dollars. Finally, some of the problems ask you to calculate price and/or income elasticities of demand. These problems are especially easy if you know a little calculus. If the demand function is D(p), and you want to calculate the price elasticity of demand when the price is p, you only need to calculate dD(p)/dp and multiply it by p/q. 15.0 Warm Up Exercise. (Calculating elasticities.) Here are some drills on price elasticities. For each demand function, ﬁnd an expression for the price elasticity of demand. The answer will typically be a function of the price, p. As an example, consider the linear demand curve, D(p) = 30 − 6p. Then dD(p)/dp = −6 and p/q = p/(30 − 6p), so the price elasticity of demand is −6p/(30 − 6p). (a) D(p) = 60 − p. −p/(60 − p). (b) D(p) = a − bp. −bp/(a − bp). (c) D(p) = 40p−2 . −2. (d) D(p) = Ap−b . −b. 15.1 (0) In Gas Pump, South Dakota, there are two kinds of consumers, Buick owners and Dodge owners. Every Buick owner has a demand function for gasoline DB (p) = 20 − 5p for p ≤ 4 and DB (p) = 0 if p > 4. Every Dodge owner has a demand function DD (p) = 15 − 3p for p ≤ 5 and DD (p) = 0 for p > 5. (Quantities are measured in gallons per week and price is measured in dollars.) Suppose that Gas Pump has 150 consumers, 100 Buick owners, and 50 Dodge owners. (a) If the price is $3, what is the total amount demanded by each individual Buick Owner? 5. And by each individual Dodge owner? 6. (b) What is the total amount demanded by all Buick owners? What is the total amount demanded by all Dodge owners? 500. 300. (c) What is the total amount demanded by all consumers in Gas Pump at a price of 3? 800. (d) On the graph below, use blue ink to draw the demand curve representing the total demand by Buick owners. Use black ink to draw the demand curve representing total demand by Dodge owners. Use red ink to draw the market demand curve for the whole town. (e) At what prices does the market demand curve have kinks? At p = 4 and p = 5. (f ) When the price of gasoline is $1 per gallon, how much does weekly demand fall when price rises by 10 cents? 65 gallons. (g) When the price of gasoline is $4.50 per gallon, how much does weekly (e) D(p) = (p + 3)−2 . −2p/(p + 3). demand fall when price rises by 10 cents? 15 gallons. (h) When the price of gasoline is $10 per gallon, how much does weekly demand fall when price rises by 10 cents? Remains at zero. NAME 193 194 MARKET DEMAND (Ch. 15) (b) At what price is the dog breeders’ elasticity equal to −1? Dollars per gallon At what price is the pet owners’ elasticity equal to −1? 6 $100. $11.25. 5 4 3 Blue line (c) On the graph below, draw the dog breeders’ demand curve in blue ink, the pet owners’ demand curve in red ink, and the market demand curve in pencil. 2 Red line 1 Black line 500 0 1000 1500 2000 2500 3000 Gallons per week 15.2 (0) For each of the following demand curves, compute the inverse demand curve. (d) Find a nonzero price at which there is positive total demand for dog polishers and at which there is a kink in the demand curve. p(q) = 5 − q/2 if q < 10. (a) D(p) = max{10 − 2p, 0}. $22.50. 290− What is the market demand function for prices below the kink? √ (b) D(p) = 100/ p. p(q) = 10, 000/q 2 . 5p. What is the market demand function for prices above the kink? 200 − p. p(q) = (10 − ln q)/4. (d) ln D(p) = ln 20 − 2 ln p. p(q) = 20/q. (c) ln D(p) = 10 − 4p. 15.3 (0) The demand function of dog breeders for electric dog polishers is qb = max{200−p, 0}, and the demand function of pet owners for electric dog polishers is qo = max{90 − 4p, 0}. (e) Where on the market demand curve is the price elasticity equal to (a) At price p, what is the price elasticity of dog breeders’ demand for electric dog polishers? −p/(200 − p). of pet owners’ demand? −1? What is the price elasticity $100. At what price will the revenue from the sale of electric dog polishers be maximized? $100. If the goal of the sellers is to maximize revenue, will electric dog polishers be sold to breeders only, to −4p/(90 − 4p). pet owners only, or to both? NAME 195 196 MARKET DEMAND Breeders only. (Ch. 15) (c) What is the price elasticity of demand when price is p and income is m? Price 300 Calculus −p. The income elasticity of demand? 15.5 (0) The demand function for drangles is q(p) = (p + 1)−2 . (a) What is the price elasticity of demand at price p? 250 1. −2p/(p+1). (b) At what price is the price elasticity of demand for drangles equal to 200 −1? (c) Write an expression for total revenue from the sale of drangles as 150 100 When the price equals 1. a function of their price. R(p) = pq = p/(p + 1)2 . Use calculus to ﬁnd the revenue-maximizing price. Don’t forget to check the Blue line second-order condition. Differentiating and solving gives p = 1. 50 Pencil line 22.5 (d) Suppose that the demand function for drangles takes the more general form q(p) = (p + a)−b where a > 0 and b > 1. Calculate an expression for Red line 90 100 50 0 the price elasticity of demand at price p. 150 200 250 290 300 Quantity Calculus 15.4 (0) The demand for kitty litter, in pounds, is ln D(p) = 1, 000 − p + ln m, where p is the price of kitty litter and m is income. −bp/(p + a). price is the price elasticity of demand equal to −1? At what p = a/(b − 1). 15.6 (0) Ken’s utility function is uK (x1 , x2 ) = x1 + x2 and Barbie’s utility function is uB (x1 , x2 ) = (x1 + 1)(x2 + 1). A person can buy 1 unit of good 1 or 0 units of good 1. It is impossible for anybody to buy fractional units or to buy more than 1 unit. Either person can buy any quantity of good 2 that he or she can aﬀord at a price of $1 per unit. (a) What is the price elasticity of demand for kitty litter when p = 2 and m = 500? m = 1, 500? −2. When p = 3 and m = 500? −3. When p = 4 and −4. (b) What is the income elasticity of demand for kitty litter when p = 2 and m = 500? 1. When p = 3 and m = 1, 500? When p = 2 and m = 1, 000? 1. 1. (a) Where m is Barbie’s wealth and p1 is the price of good 1, write an equation that can be solved to ﬁnd Barbie’s reservation price for good 1. (m − p1 + 1)2 = m + 1. What is Barbie’s reservation price p = (m + 1)/2. What is Ken’s reservation price for for good 1? good 1? $1. (b) If Ken and Barbie each have a wealth of 3, plot the market demand curve for good 1. NAME 197 198 MARKET DEMAND (Ch. 15) R(q) = 20q−q 2 /10, 000 Price 4 (b) Write expressions for total revenue 3 (c) On the graph below, use blue ink to draw the inverse demand function and use red ink to draw the marginal revenue function. On your graph, also draw a vertical blue line representing the capacity of the stadium. MR = 20 − q/5, 000 and marginal revenue number of tickets sold. as a function of the 2 Price 30 1 Red line 25 0 1 2 3 4 Quantity Stadium capacity 20 1 15.7 (0) The demand function for yo-yos is D(p, M ) = 4 − 2p + 100 M, where p is the price of yo-yos and M is income. If M is 100 and p is 1, Black line 15 Blue line 10 1/3. (a) What is the income elasticity of demand for yo-yos? 5 (b) What is the price elasticity of demand for yo-yos? −2/3. Red line 15.8 (0) If the demand function for zarfs is P = 10 − Q, 0 20 40 60 80 100 120 imum? P = 5. (d) What price will generate the maximum revenue? quantity will be sold at this price? (b) How many zarfs will be sold at that price? p(q) = 20 − q/10, 000. NAME 199 (g) Write an expression for marginal revenue as a function of output. M R(q) = 30 − q/5, 000. Use red ink to draw the new demand function and use black ink to draw the new marginal revenue function. (h) Ignoring stadium capacity, what price would generate maximum What quantity would be sold at this price? 150,000. (i) As you noticed above, the quantity that would maximize total revenue given the new higher demand curve is greater than the capacity of the stadium. Clever though the athletic director is, he cannot sell seats he hasn’t got. He notices that his marginal revenue is positive for any number of seats that he sells up to the capacity of the stadium. Therefore, in order to maximize his revenue, he should sell of 100,000 tickets at a price $20. (j) When he does this, his marginal revenue from selling an extra seat is 10. The elasticity of demand for tickets at this price quantity combination is 6 = −2. 15.10 (0) The athletic director discussed in the last problem is considering the extra revenue he would gain from three proposals to expand the size of the football stadium. Recall that the demand function he is now facing is given by q(p) = 300, 000 − 10, 000p. (a) How much could the athletic director increase the total revenue per game from ticket sales if he added 1,000 new seats to the stadium’s capacity and adjusted the ticket price to maximize his revenue? 9,900. (b) How much could he increase the revenue per game by adding 50,000 new seats? what is the price elasticity of demand? $250,000. 60,000 new seats? (Hint: The athletic director still wants to maximize revenue.) $250,000. (c) A zealous alumnus oﬀers to build as large a stadium as the athletic director would like and donate it to the university. There is only one hitch. The athletic director must price his tickets so as to keep the stadium full. If the athletic director wants to maximize his revenue from ticket sales, how large a stadium should he choose? What 150,000 seats. −1. 0. At this quantity, Will the stadium be full? Yes. (f ) A series of winning seasons caused the demand curve for football tickets to shift upward. The new demand function is q(p) = 300, 000 − 10, 000p. What is the new inverse demand function? q/10, 000. $15. $10. 100,000. (e) At this quantity, what is marginal revenue? revenue? 160 Q = 5. 15.9 (0) The demand function for football tickets for a typical game at a large midwestern university is D(p) = 200, 000 − 10, 000p. The university has a clever and avaricious athletic director who sets his ticket prices so as to maximize revenue. The university’s football stadium holds 100,000 spectators. (a) Write down the inverse demand function. 140 Quantity x 1000 (a) At what price will total revenue realized from their sale be at a max- 200 MARKET DEMAND (Ch. 15) p(q) = 30 − 202 Chapter 16 EQUILIBRIUM (Ch. 16) NAME Price Equilibrium 80 60 Introduction. Supply and demand problems are bread and butter for economists. In the problems below, you will typically want to solve for equilibrium prices and quantities by writing an equation that sets supply equal to demand. Where the price received by suppliers is the same as the price paid by demanders, one writes supply and demand as functions of the same price variable, p, and solves for the price that equalizes supply and demand. But if, as happens with taxes and subsidies, suppliers face diﬀerent prices from demanders, it is a good idea to denote these two prices by separate variables, ps and pd . Then one can solve for equilibrium by solving a system of two equations in the two unknowns ps and pd . The two equations are the equation that sets supply equal to demand and the equation that relates the price paid by demanders to the net price received by suppliers. 40 Red line p2 p1 20 Blue line q2 q1 20 0 40 60 80 100 120 Yak butter (b) Write down the equation that you would solve to ﬁnd the equilibrium Solve 120 − 4p = 2p − 30. price. $25. Example: The demand function for commodity x is q = 1, 000 − 10pd , (c) What is the equilibrium price of yak butter? where pd is the price paid by consumers. The supply function for x is q = 100 + 20ps , where ps is the price received by suppliers. For each unit sold, the government collects a tax equal to half of the price paid by consumers. Let us ﬁnd the equilibrium prices and quantities. In equilibrium, supply must equal demand, so that 1, 000 − 10pd = 100 + 20ps . Since the government collects a tax equal to half of the price paid by consumers, it must be that the sellers only get half of the price paid by consumers, so it must be that ps = pd /2. Now we have two equations in the two unknowns, ps and pd . Substitute the expression pd /2 for ps in the ﬁrst equation, and you have 1, 000 − 10pd = 100 + 10pd . Solve this equation to ﬁnd pd = 45. Then ps = 22.5 and q = 550. equilibrium quantity? 20. Locate the equilibrium price and quantity on the graph, and label them p1 and q1 . What is the (d) A terrible drought strikes the central Ohio steppes, traditional homeland of the yaks. The supply schedule shifts to 2ps − 60. The demand schedule remains as before. Draw the new supply schedule. Write down the equation that you would solve to ﬁnd the new equilibrium price of 120 − 4p = 2p − 60. yak butter. (e) The new equilibrium price is 30 and the quantity is 0. Locate the new equilibrium price and quantity on the graph and label them p2 and q2 . 16.1 (0) The demand for yak butter is given by 120 − 4pd and the supply is 2ps − 30, where pd is the price paid by demanders and ps is the price received by suppliers, measured in dollars per hundred pounds. Quantities demanded and supplied are measured in hundred-pound units. (f ) The government decides to relieve stricken yak butter consumers and producers by paying a subsidy of $5 per hundred pounds of yak butter to producers. If pd is the price paid by demanders for yak butter, what is the total amount received by producers for each unit they produce? pd + 5. (a) On the axes below, draw the demand curve (with blue ink) and the supply curve (with red ink) for yak butter. NAME 203 (g) Write down an equation that can be solved for the equilibrium price 2pd − 50 = paid by consumers, given the subsidy program. 120 − 4pd . What are the equilibrium price paid by consumers pd = 170/6, and the equilibrium quantity of yak butter now? q = 170/3 − 50 = 20/3. 170/6. 204 2pd − 50. EQUILIBRIUM (Ch. 16) (c) The government wants to make sure that only 20 throstles are bought, but it doesn’t want the ﬁrms in the industry to receive more than the minimum price that it would take to have them supply 20 throstles. One way to do this is for the government to issue 20 ration coupons. Then in order to buy a throstle, a consumer would need to present a ration coupon along with the necessary amount of money to pay for the good. If the ration coupons were freely bought and sold on the open market, what would be the equilibrium price of these coupons? (h) Suppose the government had paid the subsidy to consumers rather than producers. What would be the equilibrium net price paid by consumers? When the price paid by consumers is pd , how much yak butter is produced? The equilibrium quantity would be 20/3. 16.2 (0) Here are the supply and demand equations for throstles, where p is the price in dollars: D(p) = 40 − p S(p) = 10 + p. On the axes below, draw the demand and supply curves for throstles, using blue ink. Price 40 $10. (d) On the graph above, shade in the area that represents the deadweight loss from restricting the supply of throstles to 20. How much is this expressed in dollars? (Hint: What is the formula for the area of a triangle?) $25. 16.3 (0) The demand curve for ski lessons is given by D(pD ) = 100−2pD and the supply curve is given by S(pS ) = 3pS . (a) What is the equilibrium price? quantity? Demand $20. What is the equilibrium 60. (b) A tax of $10 per ski lesson is imposed on consumers. Write an equation that relates the price paid by demanders to the price received by suppliers. 30 pD = pS + 10. 20 Deadweight loss demand. Write an equation that states that supply equals 100 − 2pD = 3pS . 10 (c) Solve these two equations for the two unknowns pS and pD . With Supply the $10 tax, the equilibrium price pD paid by consumers would be 0 10 20 (a) The equilibrium price of throstles is quantity is 30 40 Throstles 15 per lesson. The total number of lessons given would be and the equilibrium 25. (b) Suppose that the government decides to restrict the industry to selling only 20 throstles. At what price would 20 throstles be demanded? How many throstles would suppliers supply at that price? price would the suppliers supply only 20 units? $10. 30. 20. At what $26 48. (d) A senator from a mountainous state suggests that although ski lesson consumers are rich and deserve to be taxed, ski instructors are poor and deserve a subsidy. He proposes a $6 subsidy on production while maintaining the $10 tax on consumption of ski lessons. Would this policy have any diﬀerent eﬀects for suppliers or for demanders than a tax of $4 per lesson? No. 16.4 (0) The demand curve for salted codﬁsh is D(P ) = 200 − 5P and the supply curve S(P ) = 5P . NAME 205 (a) On the graph below, use blue ink to draw the demand curve and the supply curve. The equilibrium market price is $20 206 EQUILIBRIUM (Ch. 16) (b) What is the equilibrium quantity? 10. and the equilibrium 100. quantity sold is (c) An ad valorem tax of 300% is imposed on merino ewes so that the price paid by demanders is four times the price received by suppliers. What is the equilibrium price paid by the demanders for merino ewes Price 40 now? Demand $20. What is the equilibrium price received by the suppliers for merino ewes? $5. 5. What is the equilibrium quantity? 30 20 10 16.6 (0) Schrecklich and LaMerde are two justiﬁably obscure nineteenthcentury impressionist painters. The world’s total stock of paintings by Schrecklich is 100, and the world’s stock of paintings by LaMerde is 150. The two painters are regarded by connoisseurs as being very similar in style. Therefore the demand for either painter’s work depends both on its own price and the price of the other painter’s work. The demand function for Schrecklichs is DS (P ) = 200−4PS −2PL , and the demand function for LaMerdes is DL (P ) = 200 − 3PL − PS , where PS and PL are respectively the price in dollars of a Schrecklich painting and a LaMerde painting. Deadweight loss Red supply Blue Supply 0 50 (a) Write down two simultaneous equations that state the equilibrium condition that the demand for each painter’s work equals supply. 100 150 200 Quantity of codfish (b) A quantity tax of $2 per unit sold is placed on salted codﬁsh. Use red ink to draw the new supply curve, where the price on the vertical axis remains the price per unit paid by demanders. The new equilibrium price paid by the demanders will be $19. suppliers will be $21 and the new price received by the The equilibrium quantity sold will be 95. (c) The deadweight loss due to this tax will be 5 = 2 × 5/2. On your graph, shade in the area that represents the deadweight loss. 16.5 (0) The demand function for merino ewes is D(P ) = 100/P , and the supply function is S(P ) = P . (a) What is the equilibrium price? The equations are 200 − 4PS − 2PL = 100 and 200 − 3PL − PS = 150. $10. NAME 207 (b) Solving these two equations, one ﬁnds that the equilibrium price of 20 Schrecklichs is and the equilibrium price of LaMerdes is 10. (c) On the diagram below, draw a line that represents all combinations of prices for Schrecklichs and LaMerdes such that the supply of Schrecklichs equals the demand for Schrecklichs. Draw a second line that represents those price combinations at which the demand for LaMerdes equals the supply of LaMerdes. Label the unique price combination at which both markets clear with the letter E. 208 EQUILIBRIUM (Ch. 16) Price 20 Pl 40 Schrecklich 15 Red lines 30 10 Red line 20 e Blue lines 5 La Mendes 10 e e' 0 2 4 6 8 10 12 Quantity (thousands) (b) If the supply is perfectly inelastic at 5,000 units, what is the equilib0 10 20 30 40 Ps rium price? $12. Show the supply curve on your graph and label the equilibrium with an E. (c) Suppose that the demand curve shifts outward by 10%. Write down (d) A ﬁre in a bowling alley in Hamtramck, Michigan, destroyed one of the world’s largest collections of works by Schrecklich. The ﬁre destroyed a total of 10 Schrecklichs. After the ﬁre, the equilibrium price of Schrecklichs was 23 and the equilibrium price of LaMerdes was 9. the new equation for the demand function. q = 66, 000/p. Suppose that the supply curve remains vertical but shifts to the right by 5%. Solve for the new equilibrium price 12.51 and quantity 5, 250. (d) By what percentage approximately did the equilibrium price rise? (e) On the diagram you drew above, use red ink to draw a line that shows the locus of price combinations at which the demand for Schrecklichs equals the supply of Schrecklichs after the ﬁre. On your diagram, label the new equilibrium combination of prices E . It rose by about 5 percent. Use red ink to draw the new demand curve and the new supply curve on your graph. (e) Suppose that in the above problem the demand curve shifts outward by x% and the supply curve shifts right by y%. By approximately what percentage will the equilibrium price rise? 16.7 (0) The price elasticity of demand for oatmeal is constant and equal to −1. When the price of oatmeal is $10 per unit, the total amount demanded is 6,000 units. (a) Write an equation for the demand function. q = 60, 000/p. Graph this demand function below with blue ink. (Hint: If the demand curve has a constant price elasticity equal to 6, then D(p) = ap" for some constant a. You have to use the data of the problem to solve for the constants a and 6 that apply in this particular case.) By about (x − y) percent. 16.8 (0) An economic historian* reports that econometric studies indicate for the pre–Civil War period, 1820–1860, the price elasticity of demand for cotton from the American South was approximately −1. Due to the rapid expansion of the British textile industry, the demand curve for American cotton is estimated to have shifted outward by about 5% per year during this entire period. * Gavin Wright, The Political Economy of the Cotton South, W. W. Norton, 1978. NAME 209 (a) If during this period, cotton production in the United States grew by 3% per year, what (approximately) must be the rate of change of the price It would rise by about 2% of cotton during this period? a year. (b) Assuming a constant price elasticity of −1, and assuming that when the price is $20, the quantity is also 20, graph the demand curve for cotton. What is the total revenue when the price is $20? 400. What 400. is the total revenue when the price is $10? 210 EQUILIBRIUM (Ch. 16) (e) What would be the eﬀect on total revenue of cotton farmers in the South? Since the demand has elasticity of −1, the revenue would stay the same. (f ) The expansion of the British textile industry ended in the 1860s, and for the remainder of the nineteenth century, the demand curve for American cotton remained approximately unchanged. By about 1900, the South approximately regained its prewar output level. What do you think happened to cotton prices then? They would recover to their old levels. Price of cotton 40 16.9 (0) The number of bottles of chardonnay demanded per year is $1, 000, 000 − 60, 000P , where P is the price per bottle (in U.S. dollars). The number of bottles supplied is 40, 000P . 30 (a) What is the equilibrium price? $10. What is the equilibrium 400, 000. 20 quantity? 10 (b) Suppose that the government introduces a new tax such that the wine maker must pay a tax of $5 per bottle for every bottle that he produces. What is the new equilibrium price paid by consumers? 0 10 the new price received by suppliers? 20 30 40 Quantity of cotton $7. $12. What is What is the new equilibrium 280,000. quantity? (c) If the change in the quantity of cotton supplied by the United States is to be interpreted as a movement along an upward-sloping long-run supply curve, what would the elasticity of supply have to be? (Hint: From 1820 16.10 (0) The inverse demand function for bananas is Pd = 18 − 3Qd and the inverse supply function is Ps = 6 + Qs , where prices are measured in cents. to 1860 quantity rose by about 3% per year and price rose by 2 % per year. [See your earlier answer.] If the quantity change is a movement along the long-run supply curve, then the long-run price elasticity must (a) If there are no taxes or subsidies, what is the equilibrium quantity? be what?) 1.5 %. (d) The American Civil War, beginning in 1861, had a devastating eﬀect on cotton production in the South. Production fell by about 50% and remained at that level throughout the war. What would you predict would be the eﬀect on the price of cotton? It would double if demand didn’t change. NAME 211 -16.66%. If the cross-elasticity of demand between bananas and apples is +.5, what will happen to the quantity of apples demanded as a consequence of the banana subsidy, if the price of apples stays constant? (State your answer in terms of percentage change.) -8.33%. 16.11 (1) King Kanuta rules a small tropical island, Nutting Atoll, whose primary crop is coconuts. If the price of coconuts is P , then King Kanuta’s subjects will demand D(P ) = 1, 200 − 100P coconuts per week for their own use. The number of coconuts that will be supplied per week by the island’s coconut growers is S(p) = 100P . (a) The equilibrium price of coconuts will be 6 and the equilib- 600. (b) One day, King Kanuta decided to tax his subjects in order to collect coconuts for the Royal Larder. The king required that every subject who consumed a coconut would have to pay a coconut to the king as a tax. Thus, if a subject wanted 5 coconuts for himself, he would have to purchase 10 coconuts and give 5 to the king. When the price that is received by the sellers is pS , how much does it cost one of the king’s subjects to get an extra coconut for himself? 2pS . (c) When the price paid to suppliers is pS , how many coconuts will the king’s subjects demand for their own consumption? (Hint: Express pD in terms of pS and substitute into the demand function.) Since pD = 2pS , they consume 1, 200 − 200pS . (d) Since the king consumes a coconut for every coconut consumed by the subjects, the total amount demanded by the king and his subjects is twice the amount demanded by the subjects. Therefore, when the price received by suppliers is pS , the total number of coconuts demanded per week by Kanuta and his subjects is 2, 400 − 400pS . (e) Solve for the equilibrium value of pS number of coconuts produced What is the equilibrium market price? 9 cents. 3.5. What is the new equilibrium price received by suppliers? cents. What is the new equilibrium price paid by demanders? 9.5 7.5 cents. (c) Express the change in price as a percentage of the original price. rium quantity supplied will be 3. (b) If a subsidy of 2 cents per pound is paid to banana growers, then in equilibrium it still must be that the quantity demanded equals the quantity supplied, but now the price received by sellers is 2 cents higher than the price paid by consumers. What is the new equilibrium quantity? 24/5 , the equilibrium total 480 , and the equilibrium total number of coconuts consumed by Kanuta’s subjects. 240. 212 EQUILIBRIUM (Ch. 16) (f ) King Kanuta’s subjects resented paying the extra coconuts to the king, and whispers of revolution spread through the palace. Worried by the hostile atmosphere, the king changed the coconut tax. Now, the shopkeepers who sold the coconuts would be responsible for paying the tax. For every coconut sold to a consumer, the shopkeeper would have to pay one coconut to the king. This plan resulted in 480/2 = 240 coconuts being sold to the consumers. The shopkeepers got 24/5 per coconut after paying their tax to the king, and the consumers paid a price of 48/5 per coconut. 214 Chapter 17 NAME Auctions Introduction. An auction is described by a set of rules. The rules specify bidding procedures for participants and the way in which the array of bids made determines who gets the object being sold and how much each bidder pays. Those who are trying to sell an object by auction typically do not know the willingness to pay of potential buyers but have some probabilistic expectations. Sellers are interested in ﬁnding rules that maximize their expected revenue from selling the object. Social planners are often interested not only in the revenue generated from an auction method, but also in its eﬃciency. In the absence of externalities, an auction for a single object will be eﬃcient only if the object is sold to the buyer who values it most highly. 17.1 (1) At Toivo’s auction house in Ishpemming, Michigan, a beautiful stuﬀed moosehead is being sold by auction. There are 5 bidders in attendance: Aino, Erkki, Hannu, Juha, and Matti. The moosehead is worth $100 to Aino, $20 to Erkki, and $5 to each of the others. The bidders do not collude and they don’t know each others’ valuations. (a) If the auctioneer sells it in an English auction, who would get the moosehead and approximately how much would the buyer pay? Aino AUCTIONS (Ch. 17) more, while a weekend recreational user would be willing to pay $4, 500 but no more. Charlie puts a sign in his window. “Bulldozer Sale Today.” Charlie is disappointed to discover that only two potential buyers have come to his auction. These two buyers evidently don’t know each other. Charlie believes that the probability that either is a professional bulldozer operator is independent of the other’s type and he believes that each of them has a probability of 1/2 of being a professional bulldozer operator and a probability of 1/2 of being a recreational user. Charlie considers the following three ways of selling the bulldozer: • Method 1. Post a price of $6,000, and if nobody takes the bulldozer at that price, sell it to the wholesaler. • Method 2. Post a price equal to a recreational bulldozer user’s buyer value and sell it to anyone who oﬀers that price. • Method 3. Run a sealed-bid auction and sell the bulldozer to the high bidder at the second highest bid (if there is a tie, choose one of the high bidders at random and sell the bulldozer to this bidder at the price bid by both bidders.) (a) What is the probability that both potential buyers are professional bulldozer operators? 1/4. ational bulldozer users? is of each type? What is the probability that both are recre- 1/4. What is the probability that one of them 1/2. (b) If Charlie sells by method 1, what is the probability that he will be able to sell the bulldozer to one of the two buyers? would get it for $20. 3/4. What is the probability that he will have to sell the bulldozer to the wholesaler? (b) If the auctioneer sells it in a sealed-bid, second-price auction and if no bidder knows the others’ values for the moosehead, how much should $100. Aino bid in order to maximize his expected gain? should Erkki bid? $5. $20. How much How much would each of the others bid? 1/4. $4,500. (d) Suppose that Charlie sells by method 3 and that both potential buyers bid rationally. If both bidders are professional bulldozer oper- 17.2 (2) Charlie Plopp sells used construction equipment in a quiet Oklahoma town. He has run short of cash and needs to raise money quickly by selling an old bulldozer. If he doesn’t sell his bulldozer to a customer today, he will have to sell it to a wholesaler for $1,000. Two kinds of people are interested in buying bulldozers. These are professional bulldozer operators and people who use bulldozers only for recreational purposes on weekends. Charlie knows that a professional bulldozer operator would be willing to pay $6,000 for his bulldozer but no NAME the bulldozer? 215 The professional. Charlie get for his bulldozer? $4,500. How much money will If both bidders are recre- ational bulldozer users, how much will each bid? much will Charlie receive for his bulldozer? (3/4)×$6, 000+(1/4)× (c) If Charlie sells by method 2, how much will he receive for his bulldozer? Who would get the moosehead and how much would he pay? Aino would get it for $20. What is his expected revenue? $1, 000 = $4, 750. $4,500. $4,500. How What will be ators, how much will each bid? $6,000. How much will Char- lie receive for his bulldozer? $6,000. If one bidder is a professional bulldozer operator and one is a recreational user, what bids will Professional bids $6,000. Charlie receive? Recreational user bids $4,500. 216 AUCTIONS Who will get (Ch. 17) $1, 500+(3/4)C. With method 2 he gets $C. But $1, 500 + (3/4)C > C whenever C < $6, 000. (e) For what values of C is Charlie better oﬀ selling by method 2 than by method 1? C > $(3/4)6, 000 + (1/4)1, 000 = 4, 759. Charlie’s expected revenue from selling the bulldozer by method 3? (f ) For what values of C is Charlie better oﬀ selling by method 1 than (1/4) × $6, 000 + (3/4) × $4, 500 = $4, 875. by method 3? (e) Which of the three methods will give Charlie the highest expected revenue? Method 3. (a) What will Charlie’s expected revenue be if he posts a price equal to the reservation price of professional bulldozer operators? (3/4) × (b) If Charlie posts a price equal to the reservation price C of recreational bulldozer operators, what is his expected revenue? $C. (c) If Charlie sells his bulldozer by method 3, the second-price sealed-bid auction, what is his expected revenue? (The answer is a function of C.) (1/4) × $6, 000 + (3/4) × $C = $1, 500 + $(3/4)C. (d) Show that selling by method 3 will give Charlie a higher expected pay- With method 3, each bidder will bid his true valuation. (a) With three buyers, Charlie’s expected revenue from using method 1 5375 , his expected revenue from using method 2 is his expected revenue from using method 3 is C , and $3, 000 + (C/2). (b) At which values of C would method 1 give Charlie a higher expected revenue than either of the other two methods of selling proposed above? C < $4, 750. (c) At which values of C (if any) would method 2 give Charlie a higher expected revenue than either of the other two methods of selling proposed above? None. (d) At which values of C would method 3 give Charlie a higher expected revenue than either of the other two methods of selling proposed above? C > $4, 750. valuations of $6,000, he will get $6,000. will get 17.4 (3) Yet again we tread the dusty streets of Charlie Plopp’s home town. Everything is as in the previous problem. Professional bulldozer operators are willing to pay $6,000 for a bulldozer and recreational users are willing to pay $C. Charlie is just about to sell his bulldozer when a third potential buyer appears. Charlie believes that this buyer, like the other two, is equally likely to be a professional bulldozer operator as a recreational bulldozer operator and that this probability is independent of the types of the other two. is $6, 000 + (1/4) × $1, 000 = $4, 750. If both bidders have which is the case whenever C < 4, 333.33. 17.3 (2) We revisit our ﬁnancially aﬄicted friend, Charlie Plopp. This time we will look at a slightly generalized version of the same problem. All else is as before, but the willingness to pay of recreational bulldozers is an amount C < $6, 000 which is known to Charlie. In the previous problem we dealt with the special case where C = $4, 500. Now we want to explore the way in which the sales method that gives Charlie the highest expected revenue depends on the size of C. oﬀ than selling by method 2 if C < $6, 000. This happens when 4, 750 > 1, 500 + 3/4C, Otherwise, he $C. His expected payoff is then 17.5 (2) General Scooters has decided to replace its old assembly line with a new one that makes extensive use of robots. There are two contractors who would be able to build the new assembly line. General Scooters does not know exactly what it would cost either of the contractors to do NAME 217 this job. However its engineers and investigators have discovered that for either contractor, this cost will take one of three possible values: H, M , and L, where H > M > L. The best information that General Scooters’s investigators have been able to give it is that for each contractor the probability is 1/3 that the cost is H, 1/3 that the cost is M , and 1/3 that the cost is L and that the probability distribution of costs is independent between the two contractors. Each contractor knows its own costs but thinks that the other’s costs are equally likely to be H, M , or L. General Scooters is conﬁdent that the contractors will not collude. Accountants at General Scooters suggested that General Scoooters accept sealed bids from the two contractors for constructing the assembly line and that it announce that it will award the contract to the low bidder but will pay the low bidder the amount bid by the other contractor. (If there is a tie for low bidder, one of the bidders will be selected at random to get the contract.) In this case, as your textbook shows for the Vickrey auction, each contractor would ﬁnd it in his own interest to bid his true valuation. (a) Suppose that General Scooters uses the bidding mechanism suggested by the accountants. What is the probability that it will have to pay L? 1/9 (Hint: The only case where it pays L is when both contractors have a cost of L.) What is the probability that it will have to pay H to get the job done? 5/9 (Hint: Notice that it has to pay H if at least one of the two contractors has costs of H.) What is the probability that it will have to pay M ? 1/3. Write an expression in terms of the variables H, M , and L for the expected cost of the project to General Scooters. H 5 + M 3 + L1 9 9 depends on the rules of the auction. If you contract to the low bidder at the low bid, then all bidders will bid NAME 219 your costs are M , your expected proﬁts will be positive if you bid H and negative or zero if you bid L or M . What if your costs are L? For what values of H, M , and L will the best strategy available to you be to bid H? 5M − 4H > L. 17.6 (3) Late in the day at an antique rug auction there are only two bidders left, April and Bart. The last rug is brought out and each bidder takes a look at it. The seller says that she will accept sealed bids from each bidder and will sell the rug to the highest bidder at the highest bidder’s bid. Each bidder believes that the other is equally likely to value the rug at any amount between 0 and $1,000. Therefore for any number X between 0 and 1,000, each bidder believes that the probability that the other bidder values the rug at less than X is X/1, 000. The rug is actually worth $800 to April. If she gets the rug, her proﬁt will be the diﬀerence between $800 and what she pays for it, and if she doesn’t get the rug, her proﬁt will be zero. She wants to make her bid in such a way as to maximize her expected proﬁt. (a) Suppose that April thinks that Bart will bid exactly what the rug is worth to him. If she bids $700 for the rug, what is the probability that she will get the rug? proﬁt? $100. 7/10. If she gets the rug for $700, what is her What is her expected proﬁt if she bids $700? $70. (b) Suppose that Bart will pay exactly what the rug is worth to him. If April bids $600 for the rug, what is the probability that she will get the rug? $200. 6/10. (Ch. 17) a higher amount than they would if you contract at the second-lowest bid. (c) The chairman ignored the accountants and proposed the following plan. “Let us award the contract by means of sealed bids, but let us do it wisely. Since we know that the contractors’ costs are either H, M , or L, we will accept only bids of H, M , or L, and we will award the contract to the low bidder at the price he himself bids. (If there is a tie, we will randomly select one of the bidders and award it to him at his bid.)” If the chairman’s scheme is adopted, would it ever be worthwhile for a contractor with costs of L to bid L? (Hint: What are the contractor’s proﬁts if he bids L and costs are L. Does he have a chance of a positive No. If he bids L, he proﬁt if he bids M ?) is sure to make zero profits. If he bids higher than L there is a chance that he might get the contract and make a profit. (d) Suppose that the chairman’s bidding scheme is adopted and that both contractors use the strategy of padding their bids in the following way. A contractor will bid M if her costs are L, and she will bid H if her costs are H or M . If contractors use this strategy, what is the expected cost of the 2 1 project to General Scooters? H + M Which of the two schemes 3 3 will result in a lower expected cost for General Scooters, the accountants’ The accountants’ scheme. The amount that contractors will bid bidder’s AUCTIONS scheme or the chairman’s scheme?* 9 (b) When the distinguished-looking, silver-haired chairman of General Scooters was told of the accountants’ suggested bidding scheme, he was outraged. “What a stupid bidding system! Any fool can see that it is more proﬁtable for us to pay the lower of the two bids. Why on earth would you ever want to pay the higher bid rather than the lower one?” he roared. A timid-looking accountant summoned up his courage and answered the chairman’s question. What answer would you suggest that he make? 218 What is her proﬁt if she gets the rug for $600? What is her expected proﬁt if she bids $600? $120. (c) Again suppose that Bart will bid exactly what the rug is worth to him. If April bids $x for the rug (where x is a number between 0 and 1,000) what is the probability that she will get the rug? What is her proﬁt if she gets the rug? $800-x x/1, 000 Write a formula for her expected proﬁt if she bids $x. $(800 − x)(x/1, 000). Find the bid x that maximizes her expected proﬁt. (Hint: Take a derivative.) x = 400. (d) Now let us go a little further toward ﬁnding a general answer. Suppose that the value of the rug to April is $V and she believes that Bart will bid exactly what the rug is worth to him. Write a formula that expresses her expected proﬁt in terms of the variables V and x if she bids $x. (e) We have not yet demonstrated that the bid-padding strategies proposed above are equilibrium strategies for bidders. Here we will show that this is the case for some (but not all) values of H, M , and L. Suppose that you are one of the two contractors. You believe that the other contractor is equally likely to have costs of H, M , or L and that he will bid H when his costs are M or H and he will bid M when his costs are L. Obviously if your costs are H, you can do no better than to bid H. If * The chairman’s scheme might not have worked out so badly for General Scooters if he had not insisted that the only acceptable bids are H, M , and L. If bidders had been allowed to bid any number between L and H, then the only equilibrium in bidding strategies would involve the use of mixed strategies, and if the contractors used these strategies, the expected cost of the project to General Scooters would be the same as it is with the second-bidder auction proposed by the accountants. 220 AUCTIONS (Ch. 17) $(V − x)(x/1, 000) . Now calculate the bid $x that will maximize her expected proﬁt. (Same hint: Take a derivative.) x = V /2. 17.7 (3) If you did the previous problem correctly, you found that if April believes that Bart will bid exactly as much as the rug is worth to him, then she will bid only half as much as the rug is worth to her. If this is the case, it doesn’t seem reasonable for April to believe that Bart will bid his full value. Let’s see what would the best thing for April to do if she believed that Bart would bid only half as much as the rug is worth to him. (a) If Bart always bids half of what the rug is worth to him, what is the highest amount that Bart would ever bid? $500. never pay for April to bid more than $500.01? She can get it for sure Why would it by bidding just over $500, since Bart will never bid more than $500. (b) Suppose that the the rug is worth $800 to April and she bids $300 for it. April will only get the rug if the value of the rug to Bart is less than $600. $300 for it? rug? What is the probability that she will get the rug if she bids 6/10. $500. What is her proﬁt if she bids $300 and gets the What is her expected proﬁt if she bids $300? $300. (c) Suppose that the rug is worth $800 to April. What is the probability that she will get it if she bids $x where $x < $500? 2x/1, 000. Write a formula for her expected proﬁt as a function of her bid $x when the rug is worth $800 to her. $(800− x)2x/1, 000. her expected proﬁt in this case? What bid maximizes $400. (d) Suppose that April values the rug at $V and she believes that Bart will bid half of his true value. Show that the best thing for April is to bid half of her own true value. Maximize (V − x)2x/1, 000. The derivative with respect to x is when x = V /2. 0 NAME 221 (e) Suppose that April believes that Bart will bid half of his actual value and Bart believes that April will bid half of her actual value. Suppose also that they both act to maximize their expected proﬁt given these beliefs. Will these beliefs be self-conﬁrming in the sense that given these beliefs, each will take the action that the other expects? Yes. 222 AUCTIONS (Ch. 17) (e) Assuming that Al always bids his expected value for a car, given the result of his taste test, how much will Al bid for a car that tastes sweet? 3/5 × 1, 000 + 2/5 × 100 = $640. 17.8 (2) Rod’s Auction House in Bent Crankshaft, Oregon, holds sealedbid used-car auctions every Tuesday. Each used car is sold to the highest bidder at the second-highest bidder’s bid. On average, half of the cars sold at Rod’s Auction House are lemons and half are good used cars. A good used car is worth $1,000 to any buyer and a lemon is worth only $100. Buyers are allowed to look over the used cars for a few minutes before they are auctioned. Almost all of the buyers who attend the auctions can do no better than random choice at picking good cars from among the lemons. The only exception is Al Crankcase. Al can sometimes, but not always, detect a lemon by licking the oil oﬀ of the dipstick. To Al, the oil from a good car’s dipstick invariably has a sweet, lingering taste. On the other hand, the oil from the dipsticks of 1/3 of the lemons has a sour, acidic taste, while the oil from the dipsticks of the remaining 2/3 of the lemons has the same sweet taste as the oil from the good cars. Al attends every auction, licks every dipstick, and taking into account the results of his taste test, bids his expected value for every car. How much will he bid $100. for a car that tastes sour? (f ) Consider a naive bidder at Rod’s Auction House, who knows that half of the cars are good and half are lemons, but has no clue at all about which ones are good. If this individual bids his expected value for a randomly selected car, how much would he bid? $550. (g) Given that Al bids his expected value for every used car and the naive bidders bid the expected value of a randomly selected car, will a naive bidder ever get a car whose oil tasted sweet to Al? No. (h) What is the expected value of cars that naive bidders get if they always bid the expected value of a randomly selected car? $100. Will naive bidders make money, lose money, or break even if they follow this policy? Lose money. (a) This auction environment is an example of a (common, private) (i) Suppose that bidders other than Al realize that they will get only the cars that taste sour to Al. If they bid the expected value of such a car, common how much will they bid? value auction. (b) Suppose that Al licks the dipsticks of 900 cars, half of which are good cars and half of which are lemons. Suppose that the dipstick oil from all of the good cars and from 2/3 of the lemons tastes sweet and the oil from 1/3 of the lemons tastes sour to Al. How many good cars will there be whose oil tastes sweet to him? 450. How many lemons will there be second-highest bid.) whose oil tastes sweet to Al? 300. How many of the 900 cars will have oil that tastes good to Al? 750. What fraction of the cars whose oil tastes sweet to Al will be good cars? 450/750=3/5. 3/5. (d) If Al ﬁnds that the oil on a car’s dipstick tastes sour, what is the probability that it is a good used car? 0. NAME 223 inspect the paintings brieﬂy and then must place their bids. Because they are allowed only a brief inspection, Steve and Leroy each try to guess whether the paintings are fakes by smelling them. Steve ﬁnds that if a painting fails his sniﬀ test, then it is certainly a fake. However, he cannot detect all fakes. In fact the probability that a fake passes Steve’s sniﬀ test is 1/2. Leroy detects fakes in the same way as Steve. Half of the fakes fail his sniﬀ test and half of them pass his sniﬀ test. Genuine paintings are sure to pass Leroy’s sniﬀ test. For any fake, the probability that Steve recognizes it as a fake is independent of the probability that Leroy recognizes it as a fake. The auction house posts a price for each painting. Potential buyers can submit a written oﬀer to buy at the posted price on the day of the sale. If more than one person oﬀers to buy the painting, the auction house will select one of them at random and sell it to that person at the posted price. (a) One day, as the auction house is about to close, Steve arrives and discovers that neither Leroy nor any other bidders have appeared. He sniﬀs a painting, and it passes his test. Given that it has passed his test, what is the probability that it is a good painting? (Hint: Since fakes are much more common than good paintings, the number of fakes that pass Steve’s test will exceed the number of genuine antiques that pass his test.) 1/3. Steve realizes that he can buy the painting for the posted price if he wants it. What is the highest posted price at which he would be willing to buy the painting? $333.33. (b) On another day, Steve and Leroy see each other at the auction, sniﬃng all of the paintings. No other customers have appeared at the auction house. In deciding how much to bid for a painting that passes his sniﬀ test, Steve considers the following: If a painting is selected at random and sniﬀed by both Steve and Leroy, there are ﬁve possible outcomes. Fill in the blanks for the probability of each. • Genuine and passes both dealers’ tests. Probability: • Fake and passes both dealers’ tests. Probability: .2. the good cars and he will pay $100 for them. $540. 17.9 (3) Steve and Leroy buy antique paintings at an art gallery in Fresno, California. Eighty percent of the paintings that are sold at the gallery are fakes, and the rest are genuine. After a painting is purchased, it will be carefully analyzed, and then everybody will know for certain whether it is genuine or a fake. A genuine antique is worth $1,000. A fake is worthless. Before they place their bids, buyers are allowed to 224 AUCTIONS (Ch. 17) own sniﬀ test, what is the probability that a randomly selected painting is genuine and that Steve is able to buy it? .1. What is the probability that a randomly selected painting is a fake and that Steve will bid on it and get it? .3. If Steve oﬀers to pay $300 for every painting that passes his sniﬀ test, will his expected proﬁt be positive or negative? Negative. Suppose that Steve knows that Leroy is willing to pay the reserve price for any painting that passes Leroy’s sniﬀ test. What is the highest reserve price that Steve should be willing to pay for a painting that passes his own sniﬀ test? $250. 17.10 (2) Every day the Repo ﬁnance company holds a sealed-bid, second-price auction in which it sells a repossessed automobile. There are only three bidders who bid on these cars, Arnie, Barney, and Carny. Each of these bidders is a used-car dealer whose willingness to pay for another used car ﬂuctuates randomly from day to day in response to the variation in demand at his car lot. The value of one of these used cars to any dealer, on any given day is a random variable which takes a high value $H with probability 1/2 and a low value $L with probability 1/2. The value that each dealer places on a car on a given day is independent of the values placed by the other dealers. Each day the used-car dealers submit written bids for the used car being auctioned. The Repo ﬁnance company will sell the car to the dealer with the highest bid at the price bid by the second-highest bidder. If there is a tie for the highest bid, then the second-highest bid is equal to the highest bid and so that day’s car will be sold to a randomly selected top bidder at the price bid by all top bidders. (a) How much should a dealer bid for a used car on a day when he places a value of $H on a used car? $H. How much should a dealer bid for a used car on a day when he places a value of $L on a used car? $L. (b) If the dealers do not collude, how much will Repo get for a used car .2. • Fake and passes Steve’s test but fails Leroy’s. Probability: .2. • Fake and passes Leroy’s test but fails Steve’s. Probability: .2. • Fake and fails both dealers’ tests. Probability: Al will get all of (k) What will Al’s expected proﬁt be on a car that passes his test? (c) If Al ﬁnds that the oil on a car’s dipstick tastes sweet, what is the probability that it is a good used car? $100. (j) Suppose that bidders other than Al believe that they will only get cars whose oil tastes sour to Al and suppose that they bid their expected value of such cars. Suppose also that for every car, Al bids his expected value, given the results of his taste test. Who will get the good cars and at what price? (Recall that cars are sold to the highest bidder at the .2. on days when two or three dealers value the car at $H? $H. How much will Repo get for a used car on days when fewer than two dealers value the car at $H? that day’s used car? (c) On the day when Steve and Leroy are the only customers, the auction house sets a reserve price of $300. Suppose that Steve believes that Leroy will oﬀer to buy any painting that passes his sniﬀ test. Recall that if Steve and Leroy both bid on a painting, the probability that Steve gets it is only 1/2. If Steve decides to bid on every painting that passes his $L. (c) On any given day, what is the probability that Repo receives $H for 1/2. $L for that day’s used car? from the sale? What is the probability that Repo receives 1/2. $(H + L)/2. What is Repo’s expected revenue NAME 225 (d) If there is no collusion and every dealer bids his actual valuation for every used car, what is the probability on any given day that Arnie gets a car for a lower price than the value he places on it? (Hint: This will happen only if the car is worth $H to Arnie and $L to the other dealers.) 1/8. Suppose that we measure a car dealer’s proﬁt by the diﬀerence between what a car is worth to him and what he pays for it. On a randomly selected day, what is Arnie’s expected proﬁt? (H −L)/8. (e) The expected total proﬁt of all participants in the market is the sum of the expected proﬁts of the three car dealers and the expected revenue realized by Repo. Used cars are sold by a sealed-bid, second-price auction and the dealers do not collude. What is the sum of the expected proﬁts of all participants in the market? (3(H −L)/8)+((H +L)/2) = (7/8)H + (1/8)L. 17.11 (3) This problem (and the two that follow) concerns collusion among bidders in sealed-bid auctions. Many writers have found evidence that collusive bidding occurs. The common name for a group that practices collusive bidding is a “bidding ring.”* Arnie, Barney, and Carny of the previous problem happened to meet at a church social and got to talking about the high prices they were paying for used cars and the low proﬁts they were making. Carny complained, “About half the time the used cars go for $H, and when that happens, none of us makes any money.” Arnie got a thoughtful look and then whispered, “Why don’t we agree to always bid $L in Repo’s used-car auctions?” Barney said, “I’m not so sure that’s a good idea. If we all bid $L, then we will save some money, but the trouble is, when we all bid the same, we are just as likely to get the car if we have a low value as we are to get it if we have a high value. When we bid what we think its worth, then it always goes to one of the people who value it most.” (a) If Arnie, Barney, and Carny agree to always bid $L, then on any given day, what is the probability that Barney gets the car for $L when it is actually worth $H to him? day? 1/6. What is Barney’s expected proﬁt per $(H − L)/6. agreement than they would if they did not collude? Explain. Yes. (H − L)/6 > (H − L)/8. * Our discussion draws extensively on the paper, “Collusive Bidder Behavior at Single-Object, Second-Price, and English Auctions” by Daniel Graham and Robert Marshall in the Journal of Political Economy, 1987. NAME the knockout auction and the other two dealers will bid 227 0 In this case, in the knockout auction, the dealer pays 0 for the right to be the only high bidder in Repo’s auction. In this case, the day’s used car will go to the only dealer with value $H and he pays Repo $L for it. On this day, the dealer with the high buyer value makes a total proﬁt $H − L (c) We continue to assume that in the knockout auction, dealers bid their actual values of winning the knockout. On days when two or more buyers value the used car at $H, the winner of the knockout auction pays H −L for the right to be the only high bidder in Repo’s auction. (d) If Arnie’s scheme is adopted, what is the expected total proﬁt of each of the three car dealers? (Remember to include each dealer’s share of the revenue from the knockout auction.) 7( H-L)/8 17.13 (2) After the passage of several weeks during which Repo never got more than one high bid for a car, the Repo folks guessed that something was amiss. Some members of the board of directors proposed hiring a hit man to punish Arnie, Barney, and Carny, but cooler heads prevailed and they decided instead to hire an economist who had studied Intermediate Microeconomics. The economist suggested: “Why don’t you set a reserve price $R which is just a little bit lower than $H (but of course much larger than $L)? If you get at least one bid of $R, sell it for $R to one of these bidders, and if you don’t get a bid as large as $R, then just dump that day’s car into the river. (Sadly, the environmental protection authorities in Repo’s hometown are less than vigilant.) “But what a waste,” said a Repo oﬃcial. “Just do the math,” replied the economist. (a) The economist continued. “If Repo sticks to its guns and refuses to sell at any price below $R, then even if Arnie, Barney, and Carny collude, the best they can do is for each to bid $R when they value a car at $H and to bid nothing when they value it at $L.” If they follow this strategy, the probability that Repo can sell a given car for $R is expected proﬁt will be 7/8 , so Repo’s $(7/8)R. (b) Setting a reserve price that is just slightly below $H and destroying cars for which it gets no bid will be more proﬁtable for Repo than setting no reservation price if the ratio H/L is greater than proﬁtable if H/L is less than AUCTIONS (Ch. 17) (c) Calculate the expected total proﬁts of all participants in the market (including Repo as well as the three dealers) in the case where the dealers collude. 3(H − L)/6 + L = (H + L)/2. Are these expected total proﬁts larger or smaller than they are when the dealers do not collude? Smaller. (d) The cars are said to be allocated eﬃciently if a car never winds up in the hands of a dealer who values it less than some other dealer values it. With a sealed-bid, second-price auction, if there is no collusion, are the cars allocated eﬃciently? Yes. If the dealers collude as in this problem, are the cars allocated eﬃciently? No. 17.12 (2) Arnie, Barney, and Carny happily practiced the strategy of “always bid low” for several weeks, until one day Arnie had another idea. Arnie proposed to the others: “When we all bid $L, it sometimes happens that the one who gets the week’s car values it at only $L although it is worth $H to somebody else. I’ve thought of a scheme that will increase proﬁts for all of us.” Here is Arnie’s scheme. Every day, before Repo holds its auction, Arnie, Barney and Carny will hold a sealed-bid, second-price preauction auction among themselves in which they bid for the right to be the only high bidder in that day’s auction. The dealer who wins this preauction bidding can bid anything he likes, while the other two bidders must bid $L. A preauction auction like this is known is a “knockout.” The revenue that is collected from the “knockout” auction is divided equally among Arnie, Barney, and Carny. For this problem, assume that in the knockout auction, each bidder bids his actual value of winning the knockout auction.* (a) If the winner of the knockout auction values the day’s used car at $H, then he knows that he can bid $H for this car in Repo’s second-price sealed-bid auction and he will get it for a price of $L. Therefore the value of winning the knockout auction to someone who values a used car at $H must be $H − L. The value of winning the knockout auction to someone who values a used car at $L is (b) Do the three dealers make higher expected proﬁts with this collusive of 226 7/8. 7/8. and less 0 (b) On a day when one dealer values the used car at $H and the other two value it at $L, the dealer with value $H will bid $H − L in * It is not necessarily the case that this is the best strategy in the knockout auction, since one’s bids aﬀect the revenue redistributed from the auction as well as who gets the right to bid. Graham and Marshall present a variation on this mechanism that ensures “honest” bidding in the knockout auction. 228 AUCTIONS (Ch. 17) 230 Chapter 18 TECHNOLOGY (Ch. 18) Example: Let the production function be f (x1 , x2 ) = min{x1 , x2 }. Then NAME Technology f (tx1 , tx2 ) = min{tx1 , tx2 } = min t{x1 , x2 } = t min{x1 , x2 } = tf (x1 , x2 ). Therefore when all inputs are multiplied by t, output is also multiplied by t. It follows that this production function has constant returns to scale. Introduction. In this chapter you work with production functions, re- You will also be asked to determine whether the marginal product of each single factor of production increases or decreases as you increase the amount of that factor without changing the amount of other factors. Those of you who know calculus will recognize that the marginal product of a factor is the ﬁrst derivative of output with respect to the amount of that factor. Therefore the marginal product of a factor will decrease, increase, or stay constant as the amount of the factor increases depending on whether the second derivative of the production function with respect to the amount of that factor is negative, positive, or zero. lating output of a ﬁrm to the inputs it uses. This theory will look familiar to you, because it closely parallels the theory of utility functions. In utility theory, an indiﬀerence curve is a locus of commodity bundles, all of which give a consumer the same utility. In production theory, an isoquant is a locus of input combinations, all of which give the same output. In consumer theory, you found that the slope of an indiﬀerence curve at the bundle (x1 , x2 ) is the ratio of marginal utilities, M U1 (x1 , x2 )/M U2 (x1 , x2 ). In production theory, the slope of an isoquant at the input combination (x1 , x2 ) is the ratio of the marginal products, M P1 (x1 , x2 )/M P2 (x1 , x2 ). Most of the functions that we gave as examples of utility functions can also be used as examples of production functions. There is one important diﬀerence between production functions and utility functions. Remember that utility functions were only “unique up to monotonic transformations.” In contrast, two diﬀerent production functions that are monotonic transformations of each other describe diﬀerent technologies. 3/4 Example: Consider the production function f (x1 , x2 ) = x1/2 1 x2 . The −1/2 3/4 marginal product of factor 1 is 12 x1 x2 . This is a decreasing function of x1 , as you can verify by taking the derivative of the marginal product with respect to x1 . Similarly, you can show that the marginal product of x2 decreases as x2 increases. 18.0 Warm Up Exercise. The ﬁrst part of this exercise is to calculate marginal products and technical rates of substitution for several frequently encountered production functions. As an example, consider √ the production function f (x1 , x2 ) = 2x1 + x2 . The marginal product of x1 is the derivative of f (x1 , x2 ) with respect to x1 , holding x2 ﬁxed. This is just 2. The marginal product of x2 is the derivative of f (x1 , x2 ) with respect to x2 , holding x1 ﬁxed, which in this case is 2√1x2 . The T RS is √ −M P1 /M P2 = −4 x2 . Those of you who do not know calculus should ﬁll in this table from the answers in the back. The table will be a useful reference for later problems. Example: If the utility function U (x1 , x2 ) = x1 + x2 represents a person’s ∗ 2 preferences, then so would the utility function U (x1 , x2 ) = (x1 + x2 ) . A person who had the utility function U ∗ (x1 , x2 ) would have the same indiﬀerence curves as a person with the utility function U (x1 , x2 ) and would make the same choices from every budget. But suppose that one ﬁrm has the production function f (x1 , x2 ) = x1 + x2 , and another has the production function f ∗ (x1 , x2 ) = (x1 + x2 )2 . It is true that the two ﬁrms will have the same isoquants, but they certainly do not have the same technology. If both ﬁrms have the input combination (x1 , x2 ) = (1, 1), then the ﬁrst ﬁrm will have an output of 2 and the second ﬁrm will have an output of 4. Now we investigate “returns to scale.” Here we are concerned with the change in output if the amount of every input is multiplied by a number t > 1. If multiplying inputs by t multiplies output by more than t, then there are increasing returns to scale. If output is multiplied by exactly t, there are constant returns to scale. If output is multiplied by less than t, then there are decreasing returns to scale. 3/4 Example: Consider the production function f (x1 , x2 ) = x1/2 1 x2 . If we multiply the amount of each input by t, then output will be f (tx1 , tx2 ) = (tx1 )1/2 (tx2 )3/4 . To compare f (tx1 , tx2 ) to f (x1 , x2 ), factor out the expressions involving t from the last equation. You get f (tx1 , tx2 ) = 1/2 3/4 t5/4 x1 x2 = t5/4 f (x1 , x2 ). Therefore when you multiply the amounts of all inputs by t, you multiply the amount of output by t5/4 . This means there are increasing returns to scale. NAME 231 232 TECHNOLOGY (Ch. 18) Returns to Scale and Changes in Marginal Products Marginal Products and Technical Rates of Substitution f (x1 , x2 ) M P1 (x1 , x2 ) M P2 (x1 , x2 ) T RS(x1 , x2 ) 1 2 −1/2 ax1 + bx2 a b −a/b 50x1 x2 50x2 50x1 − xx21 1/4 3/4 x1 x2 1 −3/4 3/4 x2 4 x1 x1 + 2x2 Cxa1 xb2 Caxa−1 xb2 1 3 1/4 −1/4 4 x1 x2 Cbxa1 xb−1 2 (x1 + 2)(x2 + 1) x2 + 1 x1 + 2 − (x1 + a)(x2 + b) x2 + b x1 + a − √ ax1 + b x2 a √b 2 x2 xa1 + xa2 axa−1 1 axa−1 2 (xa1 + xa2 )b baxa−1 (xa1 + xa2 )b−1 1 baxa−1 (xa1 + xa2 )b−1 2 For each production function in the table below, put an I, C, or D in the ﬁrst column if the production function has increasing, constant, or decreasing returns to scale. Put an I, C, or D in the second (third) column, depending on whether the marginal product of factor 1 (factor 2) is increasing, constant, or decreasing, as the amount of that factor alone is varied. x2 − 3x 1 2 − ax bx1 x2 +1 x1 +2 x2 +b x1 +a √ 2a x2 − b a−1 − xx12 a−1 − xx12 f (x1 , x2 ) Scale M P1 M P2 x1 + 2x2 √ x1 + 2x2 C C C D D D .2x1 x22 I C I 1/4 3/4 C D D D C D D D D C D D x 1 x2 x1 + √ x2 (x1 + 1).5 (x2 ).5 3 1/3 1/3 x1 + x2 18.1 (0) Prunella raises peaches. Where L is the number of units of labor she uses and T is the number of units of land she uses, her output 1 1 is f (L, T ) = L 2 T 2 bushels of peaches. (a) On the graph below, plot some input combinations that give her an output of 4 bushels. Sketch a production isoquant that runs through these points. The points on the isoquant that gives her an output of 4 bushels all satisfy the equation T = 16/L. NAME 233 234 TECHNOLOGY (Ch. 18) (d) Assuming she has 1 unit of land, how much extra output does she get from adding an extra unit of labor when she previously used 1 unit of T √ √ labor? 2 − 1 ≈ .41. 4 units of labor? 5 − 2 ≈ .24. If you know calculus, compute the marginal product of labor at the input combination (1, 1) and compare it with the result from the unit increase 8 6 in labor output found above. the MP is .5 when L = 1 and .25 when L = 4. 4 (e) In the long run, Prunella can change her input of land as well as of labor. Suppose that she increases the size of her orchard to 4 units of land. Use red ink to draw a new curve on the graph above showing output as a function of labor input. Also use red ink to draw a curve showing marginal product of labor as a function of labor input when the amount of land is ﬁxed at 4. 2 0 4 8 12 16 L 18.2 (0) Suppose x1 and x2 are used in ﬁxed proportions and f (x1 , x2 ) = min{x1 , x2 }. (b) This production function exhibits (constant, increasing, decreasing) Constant returns to scale. returns to scale. (a) Suppose that x1 < x2 . The marginal product for x1 is (c) In the short run, Prunella cannot vary the amount of land she uses. On the graph below, use blue ink to draw a curve showing Prunella’s output as a function of labor input if she has 1 unit of land. Locate the points on your graph at which the amount of labor is 0, 1, 4, 9, and 16 and label them. The slope of this curve is known as the marginal product √ Derivative is 1/2 L, so labor . of as the amount of labor increase? and (increases, remains constant, decreases) 1 remains constant 0 for small increases in x1 . For x2 the marginal product is , Is this curve getting steeper or ﬂatter and (increases, remains constant, decreases) remains constant for small increases in x2 . The technical rate of substitution between x2 Flatter. and x1 is infinity. constant, decreasing) Output 8 This technology demonstrates (increasing, constant returns to scale. (b) Suppose that f (x1 , x2 ) = min{x1 , x2 } and x1 = x2 = 20. What is 0. the marginal product of a small increase in x1 ? Red MPL line What is the 6 marginal product of a small increase in x2 ? Red line 0. product of x1 will (increase, decrease, stay constant) the amount of x2 is increased by a little bit. 4 2 Calculus Blue line 0 4 8 12 The marginal increase if 18.3 (0) Suppose the production function is Cobb-Douglas and 1/2 3/2 f (x1 , x2 ) = x1 x2 . (a) Write an expression for the marginal product of x1 at the point 1 −1/2 3/2 (x1 , x2 ). 2 x1 x2 . 16 Labour NAME 235 236 TECHNOLOGY (Ch. 18) (b) The marginal product of x1 (increases, decreases, remains constant) decreases for small increases in x1 , holding x2 ﬁxed. (c) The marginal product of factor 2 is 1/2 1/2 3/2x1 x2 , and it (in- Fragles for small increases 8 (d) An increase in the amount of x2 (increases, leaves unchanged, de- 6 creases, remains constant, decreases) in x2 . creases) increases increases Blue line the marginal product of x1 . 4 (e) The technical rate of substitution between x2 and x1 is −x2 /3x1 . 2 Black line (f ) Does this technology have diminishing technical rate of substitution? Yes. Red line (g) This technology demonstrates (increasing, constant, decreasing) increasing 0 4 8 12 16 Labour returns to scale. √ 18.4 (0) The production function for fragles is f (K, L) = L/2 + K, where L is the amount of labor used and K the amount of capital used. (a) There are (constant, increasing, decreasing) decreasing turns to scale. The marginal product of labor is stant, increasing, decreasing). constant re- (con- (b) In the short run, capital is ﬁxed at 4 units. Labor is variable. On the graph below, use blue ink to draw output as a function of labor input in the short run. Use red ink to draw the marginal product of labor as a function of labor input in the short run. The average product of labor is deﬁned as total output divided by the amount of labor input. Use black ink to draw the average product of labor as a function of labor input in the short run. 18.5 (0) General Monsters Corporation has two plants for producing juggernauts, one in Flint and one in Inkster. The Flint plant produces according to fF (x1 , x2 ) = min{x1 , 2x2 } and the Inkster plant produces according to fI (x1 , x2 ) = min{2x1 , x2 }, where x1 and x2 are the inputs. (a) On the graph below, use blue ink to draw the isoquant for 40 juggernauts at the Flint plant. Use red ink to draw the isoquant for producing 40 juggernauts at the Inkster plant. NAME 237 238 TECHNOLOGY (Ch. 18) the combinations of A and B that could be produced with 160 workers. (Assume that it is also possible for some workers to do nothing at all.) X2 80 B ;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;; Red ;;;;;;;;;;; ;;;;;;;;;;; shading ;;;;;;;;;;; ;;;;;;;;;;; Black ;;;;;;;;;;; ;;;;;;;;;;; shading ;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;; ; Blue ;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;;;a shading ;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;; 80 60 Red isoquant 60 40 b 40 Blue isoquant a ; c 20 20 Black isoquant 0 0 20 40 60 (b) Suppose that the ﬁrm wishes to produce 20 juggernauts at each plant. How much of each input will the ﬁrm need to produce 20 juggernauts at the Flint plant? x1 = 20, x2 = 10. How much of each input will the ﬁrm need to produce 20 juggernauts at the Inkster plant? x1 = 10, x2 = 20. Label with an a on the graph, the point representing the total amount of each of the two inputs that the ﬁrm needs to produce a total of 40 juggernauts, 20 at the Flint plant and 20 at the Inkster plant. (c) Label with a b on your graph the point that shows how much of each of the two inputs is needed in toto if the ﬁrm is to produce 10 juggernauts in the Flint plant and 30 juggernauts in the Inkster plant. Label with a c the point that shows how much of each of the two inputs that the ﬁrm needs in toto if it is to produce 30 juggernauts in the Flint plant and 10 juggernauts in the Inkster plant. Use a black pen to draw the ﬁrm’s isoquant for producing 40 units of output if it can split production in any manner between the two plants. Is the technology available to this ﬁrm Yes. convex? 40 60 80 A (b) Suppose now that every unit of product A that is produced requires the use of 4 shovels as well as 2 workers and that every unit of product B produced requires 2 shovels and 4 workers. On the graph you have just drawn, use red ink to shade in the area depicting combinations of A and B that could be produced with 180 shovels if there were no worries about the labor supply. Write down an equation for the set of combinations of A and B that require exactly 180 shovels. 4A + 2B = 180. (c) On the same diagram, use black ink to shade the area that represents possible output combinations when one takes into account both the limited supply of labor and the limited supply of shovels. (d) On your diagram locate the feasible combination of inputs that use up all of the labor and all of the shovels. If you didn’t have the graph, what equations would you solve to determine this point? 160 2A + 4B = and 4A + 2B = 180. (e) If you have 160 workers and 180 shovels, what is the largest amount of 18.6 (0) You manage a crew of 160 workers who could be assigned to make either of two products. Product A requires 2 workers per unit of output. Product B requires 4 workers per unit of output. (a) Write an equation to express the combinations of products A and B that could be produced using exactly 160 workers. 160. 20 80 X1 ; 2A + 4B = On the diagram below, use blue ink to shade in the area depicting NAME 239 Both do. On the same graph, use black ink to draw a couple of isoquants for the second ﬁrm. product A that you could produce? 45 units. If you produce this amount, you will not use your entire supply of one of the inputs. Which one? Workers. How many will be left unused? 70. 18.7 (0) A ﬁrm has the production function f (x, y) = min{2x, x + y}. On the graph below, use red ink to sketch a couple of production isoquants for this ﬁrm. A second ﬁrm has the production function f (x, y) = x + min{x, y}. Do either or both of these ﬁrms have constant returns to scale? 240 TECHNOLOGY (Ch. 18) (c) For what positive values of a, b, and C is there diminishing technical rate of substitution? For all positive values. 18.10 (0) Suppose that the production function is f (x1 , x2 ) = b (xa1 + xa2 ) , where a and b are positive constants. y 40 (a) For what positive values of a and b are there decreasing returns to 30 Red isoquants scale? ab < 1. returns to scale? 20 10 20 30 40 x How can this be? where a + b + c > 1. Prove that there are increasing returns to scale. For any t > 1, f (tx1 , tx2 , tx3 ) = A(tx1 )a(tx2 )b (tx3 )c = f (x1 , x2 , x3 ) > tf (x1 , x2 , x3 ). Cxa1 xb2 , (a) For what positive values of a, b, and C are there decreasing returns All C > 0 and a + b < 1. scale? All C > 0 and a + b = 1. scale? All C > 0 and a + b > 1. constant returns to increasing returns to (b) For what positive values of a, b, and C is there decreasing marginal product for factor 1? Returns to scale are different depending on the ratio in which the factors are used. Find a combination of inputs such that doubling the amount of both inputs will more than double the amount of output. x1 = 1, x2 = 4, for example. Find a combination of inputs such that doubling the amount of both inputs will less than double 18.9 (0) Suppose that the production function is f (x1 , x2 ) = where a, b, and C are positive constants. to scale? increases (b) This production function does not satisfy the deﬁnition of increasing returns to scale, constant returns to scale, or decreasing returns to scale. f (x1 , x2 , x3 ) = Axa1 xb2 xc3 , t as the amount of factor 1 increases. The marginal product of factor 2 (increases, decreases, stays constant) as the amount of factor 2 increases. 18.8 (0) Suppose the production function has the form a+b+c Increasing (a) The marginal product of factor 1 (increases, decreases, stays constant) decreases 0 ab = 1. 18.11 (0) Suppose that a ﬁrm has the production function f (x1 , x2 ) = √ x1 + x22 . Black isoquants 10 Constant returns to scale? ab > 1. All C > 0 and b > 0 and a < 1. output. x1 = 4, x2 = 0, for example. 242 Chapter 19 Proﬁt Maximization Introduction. A ﬁrm in a competitive industry cannot charge more than the market price for its output. If it also must compete for its inputs, then it has to pay the market price for inputs as well. Suppose that a proﬁtmaximizing competitive ﬁrm can vary the amount of only one factor and that the marginal product of this factor decreases as its quantity increases. Then the ﬁrm will maximize its proﬁts by hiring enough of the variable factor so that the value of its marginal product is equal to the wage. Even if a ﬁrm uses several factors, only some of them may be variable in the short run. Example: A ﬁrm has the production function f (x1 , x2 ) = 1/2 1/2 x 1 x2 . Suppose that this ﬁrm is using 16 units of factor 2 and is unable to vary this quantity in the short run. In the short run, the only thing that is left for the ﬁrm to choose is the amount of factor 1. Let the price of the ﬁrm’s output be p, and let the price it pays per unit of factor 1 be w1 . We want to ﬁnd the amount of x1 that the ﬁrm will use and the amount of output it will produce. Since the amount of factor 2 used in the short run 1/2 must be 16, we have output equal to f (x1 , 16) = 4x1 . The marginal product of x1 is calculated by taking the derivative of output with respect −1/2 to x1 . This marginal product is equal to 2x1 . Setting the value of the −1/2 = w1 . marginal product of factor 1 equal to its wage, we have p2x1 Now we can solve this for x1 . We ﬁnd x1 = (2p/w1 )2 . Plugging this into the production function, we see that the ﬁrm will choose to produce 1/2 4x1 = 8p/w1 units of output. NAME 243 (c) Suppose that the wage of labor falls to 4, and the price of output remains at p. On the graph, use red ink to draw the new isoproﬁt line for the ﬁrm that passes through its old choice of input and output. Will referring to your diagram. Calculus (Ch. 19) 19.1 (0) The short-run production function of a competitive ﬁrm is given by f (L) = 6L2/3 , where L is the amount of labor it uses. (For those who do not know calculus—if total output is aLb , where a and b are constants, and where L is the amount of some factor of production, then the marginal product of L is given by the formula abLb−1 .) The cost per unit of labor is w = 6 and the price per unit of output is p = 3. (a) Plot a few points on the graph of this ﬁrm’s production function and sketch the graph of the production function, using blue ink. Use black ink to draw the isoproﬁt line that passes through the point (0, 12), the isoproﬁt line that passes through (0, 8), and the isoproﬁt line that passes through the point (0, 4). What is the slope of each of the isoproﬁt lines? They all have slope 2. How many points on the isoproﬁt line through (0, 12) consist of input-output points that are actually pos- sible? None. Make a squiggly line over the part of the isoproﬁt line through (0, 4) that consists of outputs that are actually possible. (b) How many units of labor will the ﬁrm hire? output will it produce? 24. will its total proﬁts be? 24. Yes. Explain why, As the diagram shows, the 8. How much If the ﬁrm has no other costs, how much Output 48 In the long run, a ﬁrm is able to vary all of its inputs. Consider the case of a competitive ﬁrm that uses two inputs. Then if the ﬁrm is maximizing its proﬁts, it must be that the value of the marginal product of each of the two factors is equal to its wage. This gives two equations in the two unknown factor quantities. If there are decreasing returns to scale, these two equations are enough to determine the two factor quantities. If there are constant returns to scale, it turns out that these two equations are only suﬃcient to determine the ratio in which the factors are used. In the problems on the weak axiom of proﬁt maximization, you are asked to determine whether the observed behavior of ﬁrms is consistent with proﬁt-maximizing behavior. To do this you will need to plot some of the ﬁrm’s isoproﬁt lines. An isoproﬁt line relates all of the input-output combinations that yield the same amount of proﬁt for some given input and output prices. To get the equation for an isoproﬁt line, just write down an equation for the ﬁrm’s proﬁts at the given input and output prices. Then solve it for the amount of output produced as a function of the amount of the input chosen. Graphically, you know that a ﬁrm’s behavior is consistent with proﬁt maximization if its input-output choice the ﬁrm increase its output at the new price? PROFIT MAXIMIZATION in each period lies below the isoproﬁt lines of the other periods. NAME Black lines 36 Red line 24 Squiggly line 13.3 12 8 4 0 Blue curve 8 4 12 16 20 24 Labour input 244 PROFIT MAXIMIZATION (Ch. 19) persons produced, h is the number of heathens who attend Jed’s sermons, and p is the number of hours of preaching. For every person converted, Jed receives a payment of s from the grateful convert. Sad to say, heathens do not ﬂock to Jed’s sermons of their own accord. Jed must oﬀer heathens a payment of w to attract them to his sermons. Suppose the amount of preaching is ﬁxed at p̄ and that Jed is a proﬁt-maximizing prophet. firm can reach a higher isoprofit line by (a) If h < p̄, what is the marginal product of heathens? increasing output. is the value of the marginal product of an additional heathen? 19.2 (0) A Los Angeles ﬁrm uses a single input to produce √a recreational commodity according to a production function f (x) = 4 x, where x is the number of units of input. The commodity sells for $100 per unit. The input costs $50 per unit. (b) If h > p̄, what is the marginal product of heathens? 0. What is the value of the marginal product of an additional heathen in this case? 1. What s. 0. (a) Write down a function that states the ﬁrm’s proﬁt as a function of the amount of input. √ π = 400 x − 50x. (c) Sketch the shape of this production function in the graph below. Label the axes, and indicate the amount of the input where h = p̄. rp (b) What is the proﬁt-maximizing amount of input? 16. 16. of output? How much proﬁts does it make when it maximizes proﬁts? $800. (c) Suppose that the ﬁrm is taxed $20 per unit of its output and the price of its input is subsidized by $10. What is its new input level? What is its new output level? 16. 16. How much proﬁt does it make now? $640. (Hint: A good way to solve this is to write an expression for the ﬁrm’s proﬁt as a function of its input and solve for the proﬁt-maximizing amount of input.) (d) Suppose that instead of these taxes and subsidies, the ﬁrm is taxed at 50% of its proﬁts. Write down its after-tax proﬁts as a function of the amount of input. √ π = .50 × (400 x − 50x). proﬁt-maximizing amount of output? make after taxes? 16. What is the How much proﬁt does it $400. 19.3 (0) Brother Jed takes heathens and reforms them into righteous individuals. There are two inputs needed in this process: heathens (who are widely available) and preaching. The production function has the following form: rp = min{h, p}, where rp is the number of righteous _ p h (d) If w < s, how many heathens will be converted? how many heathens will be converted? p̄. If w > s, 0. 19.4 (0) Allie’s Apples, Inc. purchases apples in bulk and sells two products, boxes of apples and jugs of cider. Allie’s has capacity limitations of three kinds: warehouse space, crating facilities, and pressing facilities. A box of apples requires 6 units of warehouse space, 2 units of crating facilities, and no pressing facilities. A jug of cider requires 3 units of warehouse space, 2 units of crating facilities, and 1 unit of pressing facilities. The total amounts available each day are: 1,200 units of warehouse space, 600 units of crating facilities, and 250 units of pressing facilities. (a) If the only capacity limitations were on warehouse facilities, and if all warehouse space were used for the production of apples, how many boxes of apples could be produced in one day? 200. How many jugs of cider could be produced each day if, instead, all warehouse space were used in NAME 245 the production of cider and there were no other capacity constraints? 400. Draw a blue line in the following graph to represent the warehouse space constraint on production combinations. 246 PROFIT MAXIMIZATION (b) Following the same reasoning, draw a red line to represent the constraints on output to limitations on crating capacity. How many boxes of apples could Allie produce if he only had to worry about crating capacity? 300. and no other constraints? An infinite number. How many 250. jugs of cider? Period 1 2 3 300. How many jugs of cider? (c) Finally draw a black line to represent constraints on output combinations due to limitations on pressing facilities. How many boxes of apples could Allie produce if he only had to worry about the pressing capacity (d) Now shade the area that represents feasible combinations of daily production of apples and cider for Allie’s Apples. Cider (Ch. 19) 19.5 (0) A proﬁt-maximizing ﬁrm produces one output, y, and uses one input, x, to produce it. The price per unit of the factor is denoted by w and the price of the output is denoted by p. You observe the ﬁrm’s behavior over three periods and ﬁnd the following: y 1 2.5 4 x 1 3 8 w 1 .5 .25 p 1 1 1 (a) Write an equation that gives the ﬁrm’s proﬁts, π, as a function of the amount of input x it uses, the amount of output y it produces, the per-unit cost of the input w, and the price of output p. π = py − wx. (b) In the diagram below, draw an isoproﬁt line for each of the three periods, showing combinations of input and output that would yield the same proﬁts that period as the combination actually chosen. What are the equations for these three lines? y = x, y = 1 + .5x, y = 2 + .25x. Using the theory of revealed proﬁtability, shade in the region on the graph that represents input-output combinations that could be feasible as far as one can tell from the evidence that is available. 600 500 How would you describe this region in words? Black revenue line The region that is below all 3 isoprofit lines. 400 Blue line Output 300 200 100 12 ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; 100 0 200 Black line 8 Red line Period 2 6 300 400 500 600 Apples 4 (e) Allie’s can sell apples for $5 per box of apples and cider for $2 per jug. Draw a black line to show the combinations of sales of apples and cider that would generate a revenue of $1,000 per day. At the proﬁt- 200 maximizing production plan, Allie’s is producing and 0 Period 1 10 jugs of cider. Total revenues are 2 boxes of apples 0 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; Period 3 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; 2 4 6 8 $1,000. NAME 247 (a) Using black ink, plot Shill Oil’s isoproﬁt lines and choices for the three diﬀerent periods. Label them 10, 20, and 40. Using red ink draw Golf Oil’s isoproﬁt line and production choice. Label it with a 40 in red ink. Million barrels of gasoline 10 12 Input 19.6 (0) T-bone Pickens is a corporate raider. This means that he looks for companies that are not maximizing proﬁts, buys them, and then tries to operate them at higher proﬁts. T-bone is examining the ﬁnancial records of two reﬁneries that he might buy, the Shill Oil Company and the Golf Oil Company. Each of these companies buys oil and produces gasoline. During the time period covered by these records, the price of gasoline ﬂuctuated signiﬁcantly, while the cost of oil remained constant at $10 a barrel. For simplicity, we assume that oil is the only input to gasoline production. Shill Oil produced 1 million barrels of gasoline using 1 million barrels of oil when the price of gasoline was $10 a barrel. When the price of gasoline was $20 a barrel, Shill produced 3 million barrels of gasoline using 4 million barrels of oil. Finally, when the price of gasoline was $40 a barrel, Shill used 10 million barrels of oil to produce 5 million barrels of gasoline. Golf Oil (which is managed by Martin E. Lunch III) did exactly the same when the price of gasoline was $10 and $20, but when the price of gasoline hit $40, Golf produced 3.5 million barrels of gasoline using 8 million barrels of oil. 12 10 248 PROFIT MAXIMIZATION (Ch. 19) (b) How much proﬁts could Golf Oil have made when the price of gasoline was $40 a barrel if it had chosen to produce the same amount that it did when the price was $20 a barrel? $80 million. What proﬁts did Golf actually make when the price of gasoline was $40? $60 million. (c) Is there any evidence that Shill Oil is not maximizing proﬁts? Explain. No. The data satisfy WAPM. (d) Is there any evidence that Golf Oil is not maximizing proﬁts? Explain. Yes. When price of gas was $40, Golf could have made more money by acting as it did when price of gas was $20. 19.7 (0) After carefully studying Shill Oil, T-bone Pickens decides that it has probably been maximizing its proﬁts. But he still is very interested in buying Shill Oil. He wants to use the gasoline they produce to fuel his delivery ﬂeet for his chicken farms, Capon Truckin’. In order to do this Shill Oil would have to be able to produce 5 million barrels of gasoline from 8 million barrels of oil. Mark this point on your graph. Assuming that Shill always maximizes proﬁts, would it be technologically feasible for it to produce this input-output combination? Why or why not? No. If it could, then it would have made 8 more profits by choosing this combination than what it chose when 40 6 price of oil was $40. 4 20 Red 40 2 10 0 2 4 6 8 10 12 Million barrels of oil 19.8 (0) Suppose that ﬁrms operate in a competitive market, attempt to maximize proﬁts, and only use one factor of production. Then we know that for any changes in the input and output price, the input choice and the output choice must obey the Weak Axiom of Proﬁt Maximization, ∆p∆y − ∆w∆x ≥ 0. Which of the following propositions can be proven by the Weak Axiom of Proﬁt Maximizing Behavior (WAPM)? Respond yes or no, and give a short argument. NAME 249 (a) If the price of the input does not change, then a decrease in the price of the output will imply that the ﬁrm will produce the same amount or less output. change, Yes. If price of input doesn’t 250 PROFIT MAXIMIZATION (Ch. 19) (d) When Hoglund and Skoglund are both maximizing proﬁts, will Skoglund’s output be more than twice as much, less than twice as much or exactly twice as much as Hoglund’s? Explain. ∆w = 0, so WAPM says ∆p∆y ≥ 0. More than twice as much. S. would produce twice as much as H. if they used equal amounts of (b) If the price of the output remains constant, then a decrease in the input price will imply that the ﬁrm will use the same amount or more of the input. change, Yes. If price of output doesn’t ∆p = 0, so WAPM says −∆w∆x ≥ 0. (c) If both the price of the output and the input increase and the ﬁrm produces less output, then the ﬁrm will use more of the input. No. Sign pattern is (+)(−) − (+)(+) ≥ 0, which cannot happen. 19.9 (1) Farmer Hoglund has discovered that on his farm, he can get 30 bushels of corn per acre if he applies no fertilizer. When he applies N pounds of fertilizer to an acre of land, the marginal product of fertilizer is 1 − N/200 bushels of corn per pound of fertilizer. (a) If the price of corn is $3 a bushel and the price of fertilizer is $p per pound (where p < 3), how many pounds of fertilizer should he use per acre in order to maximize proﬁts? 200 − 66.66p. (b) (Only for those who remember a bit of easy integral calculus.) Write down a function that states Farmer Hoglund’s yield per acre as a function of the amount of fertilizer he uses. 30 + N − N 2 /400. fertilizer, but S. uses more fertilizer than H. does. (e) Explain how someone who looked at Hoglund’s and Skoglund’s corn yields and their fertilizer inputs but couldn’t observe the quality of their land, would get a misleading idea of the productivity of fertil- Fertilizer did not cause the entire izer. difference in yield. The best land got the most fertilizer. 19.10 (0) A ﬁrm has two variable factors and a production function, 1/2 1/4 f (x1 , x2 ) = x1 x2 . The price of its output is 4. Factor 1 receives a wage of w1 and factor 2 receives a wage of w2 . (a) Write an equation that says that the value of the marginal product −1/2 1/4 of factor 1 is equal to the wage of factor 1 2x1 x2 = w1 and an equation that says that the value of the marginal product of factor 1/2 −3/4 2 is equal to the wage of factor 2. x1 x2 = w2 . Solve two equations in the two unknowns, x1 and x2 , to give the amounts of factors 1 and 2 that maximize the ﬁrm’s proﬁts as a function of w1 and w2 . This 3 2 2 gives x1 = 8/(w1 w2 ) and x2 = 4/(w1 w2 ). (Hint: You could use the ﬁrst equation to solve for x1 as a function of x2 and of the factor wages. Then substitute the answer into the second equation and solve for x2 as a function of the two wage rates. Finally use your solution for x2 to ﬁnd the solution for x1 .) (b) If the wage of factor 1 is 2, and the wage of factor 2 is 1, how many (c) Hoglund’s neighbor, Skoglund, has better land than Hoglund. In fact, for any amount of fertilizer that he applies, he gets exactly twice as much corn per acre as Hoglund would get with the same amount of fertilizer. How much fertilizer will Skoglund use per acre when the price of corn is $3 a bushel and the price of fertilizer is $p a pound? 200 − 33.33p. (Hint: Start by writing down Skoglund’s marginal product of fertilizer as a function of N .) NAME 251 (a) Write the two equations that say that the value of the marginal prod−1/2 1/2 uct of each factor is equal to its wage. 2x1 x2 = w1 and 1/2 −1/2 2x1 x2 = w2 . If w1 = 2w2 , these two equations imply that x1 /x2 = 1/2. (b) For this production function, is it possible to solve the two marginal No. productivity equations uniquely for x1 and x2 ? 19.12 (1) √A ﬁrm has two variable factors and a production function f (x1 , x2 ) = 2x1 + 4x2 . On the graph below, draw production isoquants corresponding to an ouput of 3 and to an output of 4. (a) If the price of the output good is 4, the price of factor 1 is 2, and the price of factor 2 is 3, ﬁnd the proﬁt-maximizing amount of factor 1 0 , the proﬁt-maximizing amount of factor 2 proﬁt-maximizing output 16/9 , and the 8/3. Factor 2 16 12 8 4 9 _ 4 0 4 8 12 16 Factor 1 units of factor 1 will the ﬁrm demand? factor 2 will it demand? 1. 1. 1. How many units of How much output will it produce? How much proﬁt will it make? 1. 19.11 (0) A ﬁrm has two variable factors and a production function 1/2 1/2 f (x1 , x2 ) = x1 x2 . The price of its output is 4, the price of factor 1 is w1 , and the price of factor 2 is w2 . 252 PROFIT MAXIMIZATION (Ch. 19) 254 Chapter 20 Cost Minimization Introduction. In the chapter on consumer choice, you studied a con- (a) In the graph below, draw a production isoquant representing input combinations that will produce 20 units of output. Draw another isoquant representing input combinations that will produce 40 units of output. x2 40 30 20 NAME 10 20 30 40 x1 (b) Does this production function exhibit increasing, decreasing, or constant returns to scale? Constant. (c) If Nadine uses only unskilled labor, how much unskilled labor would y. she need in order to produce y units of output? (d) If Nadine uses only skilled labor to produce output, how much skilled y labor would she need in order to produce y units of output? 2. (e) If Nadine faces factor prices (1, 1), what is the cheapest way for her to produce 20 units of output? x1 = 255 256 COST MINIMIZATION 0 , x2 = 10. (Ch. 20) (d) If the ﬁrm faces factor prices (1, 1), what is the cheapest way for it 0. to produce 10 eﬀronteries? How much will this cost? only (g) If Nadine faces factor prices (w1 , w2 ), what will be the minimal cost of producing 20 units of output? 20 units 0 (f ) If Nadine faces factor prices (1, 3), what is the cheapest way for her , x2 = 40 units 10 In consumer theory, you also dealt with cases where the consumer’s indiﬀerence “curves” were straight lines and with cases where there were 20 (Ch. 20) 20.1 (0) Nadine sells user-friendly software. Her ﬁrm’s production function is f (x1 , x2 ) = x1 + 2x2 , where x1 is the amount of unskilled labor and x2 is the amount of skilled labor that she employs. sumer who tries to maximize his utility subject to the constraint that he has a ﬁxed amount of money to spend. In this chapter you study the behavior of a ﬁrm that is trying to produce a ﬁxed amount of output in the cheapest possible way. In both theories, you look for a point of tangency between a curved line and a straight line. In consumer theory, there is an “indiﬀerence curve” and a “budget line.” In producer theory, there is a “production isoquant” and an “isocost line.” As you recall, in consumer theory, ﬁnding a tangency gives you only one of the two equations you need to locate the consumer’s chosen point. The second equation you used was the budget equation. In cost-minimization theory, again the tangency condition gives you one equation. This time you don’t know in advance how much the producer is spending; instead you are told how much output he wants to produce and must ﬁnd the cheapest way to produce it. So your second equation is the equation that tells you that the desired amount is being produced. √ Example. A ﬁrm has the production function f (x1 , x2 ) = ( x1 + √ 3 x2 )2 . The price of factor 1 is w1 = 1 and the price of factor 2 is w2 = 1. Let us ﬁnd the cheapest way to produce 16 units of output. We will be looking for a point where the technical rate of substitution equals −w1 /w2 . If you calculate the technical rate of substitution (or look it up from the warm up exercise in Chapter 18), you ﬁnd T RS(x1 , x2 ) = −(1/3)(x2 /x1 )1/2 . Therefore we must have −(1/3)(x2 /x1 )1/2 = −w1 /w2 = −1. This equation can be simpliﬁed to x2 = 9x1 . So we know that the combination of inputs chosen has to lie somewhere on the line x2 = 9x1 . We are looking for the cheapest way to produce 16 units of output. So the point we are looking for must sat√ √ √ √ isfy the equation ( x1 + 3 x2 )2 = 16, or equivalently x1 + 3 x2 = 4. Since x2√= 9x1 , we can substitute for x2 in the previous equation to get √ √ x1 + 3 9x1 = 4. This equation simpliﬁes further to 10 x1 = 4. Solving this for x1 , we have x1 = 16/100. Then x2 = 9x1 = 144/100. The amounts x1 and x2 that we solved for in the previous paragraph are known as the conditional factor demands for factors 1 and 2, conditional on the wages w1 = 1, w2 = 1, and output y = 16. We express this by saying x1 (1, 1, 16) = 16/100 and x2 (1, 1, 16) = 144/100. Since we know the amount of each factor that will be used to produce 16 units of output and since we know the price of each factor, we can now calculate the cost of producing 16 units. This cost is c(w1 , w2 , 16) = w1 x1 (w1 , w2 , 16)+w2 x2 (w1 , w2 , 16). In this instance since w1 = w2 = 1, we have c(1, 1, 16) = x1 (1, 1, 16) + x2 (1, 1, 16) = 160/100. to produce 20 units of output? x1 = COST MINIMIZATION kinks in the indiﬀerence curves. Then you found that the consumer’s choice might occur at a boundary or at a kink. Usually a careful look at the diagram would tell you what is going on. The story with kinks and boundary solutions is almost exactly the same in the case of costminimizing ﬁrms. You will ﬁnd some exercises that show how this works. NAME c = min{20w1 , 10w2 } = It can produce 10 units of output by using the bundle (10, 5), so this is the cheapest way. 10 min{2w1 , w2 }. It will cost $15. (h) If Nadine faces factor prices (w1 , w2 ), what will be the mini- (e) If the ﬁrm faces factor prices (w1 , w2 ), what is the cheapest cost to c(w1 , w2 , y) = mal cost of producing y units of output? produce 10 eﬀronteries? c(w1, w2 , 10) = 10w1 + 5w2 . min{w1 , w2 /2}y. (f ) If the ﬁrm faces factor prices (w1 , w2 ), what will be the minimal cost 20.2 (0) The Ontario Brassworks produces brazen eﬀronteries. As you know brass is an alloy of copper and zinc, used in ﬁxed proportions. The production function is given by: f (x1 , x2 ) = min{x1 , 2x2 }, where x1 is the amount of copper it uses and x2 is the amount of zinc that it uses in production. (a) Illustrate a typical isoquant for this production function in the graph below. of producing y eﬀronteries? Calculus (w1 + w2/2)y. 20.3 (0) A ﬁrm uses labor and machines to produce output according to the production function f (L, M ) = 4L1/2 M 1/2 , where L is the number of units of labor used and M is the number of machines. The cost of labor is $40 per unit and the cost of using a machine is $10. (a) On the graph below, draw an isocost line for this ﬁrm, showing combinations of machines and labor that cost $400 and another isocost line showing combinations that cost $200. What is the slope of these isocost x2 40 lines? -4. 30 (b) Suppose that the ﬁrm wants to produce its output in the cheapest possible way. Find the number of machines it would use per worker. (Hint: The ﬁrm will produce at a point where the slope of the production 20 isoquant equals the slope of the isocost line.) 4. 10 (c) On the graph, sketch the production isoquant corresponding to an x = 1_ x 2 2 1 0 10 20 output of 40. Calculate the amount of labor 30 40 x1 (b) Does this production function exhibit increasing, decreasing, or constant returns to scale? and the cost of producing 40 units at these factor prices: c(40, 10, 40) = 400. Constant. (c) If the ﬁrm wanted to produce 10 eﬀronteries, how much copper would it need? 5 units number of machines 20 that are used to produce 40 units of output in the cheapest possible way, given the above factor prices. Calculate the 10 units. How much zinc would it need? 5 units. (d) How many units of labor y/8 and how many machines y/2 would the ﬁrm use to produce y units in the cheapest possible way? How much would this cost? returns to scale.) 10y. (Hint: Notice that there are constant NAME 257 258 COST MINIMIZATION (Ch. 20) (a) If the production function is given by f (x1 , x2 ) = min{x1 , x2 }, what Machines 40 is the minimum cost of producing one unit of output? $5. (b) If the production function is given by f (x3 , x4 ) = x3 + x4 , what is the 30 minimum cost of producing one unit of output? 20 (c) If the production function is given by f (x1 , x2 , x3 , x4 ) = min{x1 + x2 , x3 + x4 }, what is the minimum cost of producing one unit of output? $200 isocost line 10 $2. $3. (d) If the production function is given by f (x1 , x2 ) = min{x1 , x2 } + min{x3 , x4 }, what is the minimum cost of producing one unit of output? $400 isocost line $5. 0 10 20 30 40 Labour 20.4 (0) Earl sells lemonade in a competitive market on a busy street 1/3 1/3 corner in Philadelphia. His production function is f (x1 , x2 ) = x1 x2 , where output is measured in gallons, x1 is the number of pounds of lemons he uses, and x2 is the number of labor-hours spent squeezing them. 20.6 (0) Joe Grow, an avid indoor gardener, has found that the number of happy plants, h, depends on the amount of light, l, and water, w. In fact, Joe noticed that plants require twice as much light as water, and any more or less is wasted. Thus, Joe’s production function is h = min{l, 2w}. (a) Suppose Joe is using 1 unit of light, what is the least amount of water he can use and still produce a happy plant? 1/2 unit of (a) Does Earl have constant returns to scale, decreasing returns to scale, or increasing returns to scale? water. Decreasing. (b) If Suppose Joe wants to produce 4 happy plants, what are the mini(b) Where w1 is the cost of a pound of lemons and w2 is the wage rate for lemon-squeezers, the cheapest way for Earl to produce lemonade is to mum amounts of light and water required? use w1 /w2 hours of labor per pound of lemons. (Hint: Set the slope of his isoquant equal to the slope of his isocost line.) (c) Joe’s conditional factor demand function for light is l(w1 , w2 , h) = h (4, 2). and his conditional factor demand function for water is (c) If he is going to produce y units in the cheapest way possible, w(w1 , w2 , h) = h/2. then the number of pounds of lemons he will use is x1 (w1 , w2 , y) = 1/2 1/2 w2 y 3/2 /w1 and the number of hours of labor that he will use 1/2 3/2 1/2 is x2 (w1 , w2 , y) = w1 y /w2 . (Hint: Use the production function and the equation you found in the last part of the answer to solve for the input quantities.) (d) The cost to Earl of producing y units at factor prices w1 and w2 is 1/2 1/2 3/2 c(w1 , w2 , y) = w1 x1 (w1 , w2 , y) + w2 x2 (w1 , w2 , y) = 2w1 w2 y . 20.5 (0) The prices of inputs (x1 , x2 , x3 , x4 ) are (4, 1, 3, 2). NAME 259 (a) If Flo uses no fertilizer, how many hours of talk must she devote if she wants one happy plant? 1 hour. If she doesn’t talk to her plants at all, how many bags of fertilizer will she need for one happy plant? (b) If wt < wf /2, would it be cheaper for Flo to use fertilizer or talk to It would be cheaper to talk. (c) Flo’s cost function is c(wf , wt , h) = 0 wt < wf /2 and h if if wt > wf /2. 20.8 (0) Remember T-bone Pickens, the corporate raider? Now he’s concerned about his chicken farms, Pickens’s Chickens. He feeds his chickens on a mixture of soybeans and corn, depending on the prices of each. According to the data submitted by his managers, when the price of soybeans was $10 a bushel and the price of corn was $10 a bushel, they used 50 bushels of corn and 150 bushels of soybeans for each coop of chickens. When the price of soybeans was $20 a bushel and the price of corn was $10 a bushel, they used 300 bushels of corn and no soybeans per coop of chickens. When the price of corn was $20 a bushel and the price of soybeans was $10 a bushel, they used 250 bushels of soybeans and no corn for each coop of chickens. (a) Graph these three input combinations and isocost lines in the following diagram. Corn 400 260 COST MINIMIZATION (Ch. 20) (b) How much money did Pickens’ managers spend per coop of chickens when the prices were (10, 10)? $2,500. $2,000. When the prices were When the prices were (20, 10)? $3,000. (c) Is there any evidence that Pickens’s managers were not minimizing costs? Why or why not? There is no such evidence, since the data satisfy WACM. w min{ 2f , wt }h. (d) Her conditional factor demand for talk is t(wf , wt , h) = 20.7 (1) Joe’s sister, Flo Grow, is a university administrator. She uses an alternative method of gardening. Flo has found that happy plants only need fertilizer and talk. (Warning: Frivolous observations about university administrators’ talk being a perfect substitute for fertilizer is in extremely poor taste.) Where f is the number of bags of fertilizer used and t is the number of hours she talks to her plants, the number of happy plants produced is exactly h = t + 2f . Suppose fertilizer costs wf per bag and talk costs wt per hour. (10, 20)? 1/2 bag. raise one happy plant? (d) If each unit of light costs w1 and each unit of water costs w2 , Joe’s w cost function is c(w1 , w2 , h) = w1 h + 22 h. (d) Pickens wonders whether there are any prices of corn and soybeans at which his managers will use 150 bushels of corn and 50 bushels of soybeans to produce a coop of chickens. How much would this production method cost per coop of chickens if the prices were ps = 10 and pc = 10? $2,000. if the prices were ps = 10, pc = 20? prices were ps = 20, pc = 10? $3,500. if the $2,500. (e) If Pickens’s managers were always minimizing costs, can it be possible to produce a coop of chickens using 150 bushels and 50 bushels of soybeans? No. At prices (20, 10), this bundle costs less than the bundle actually used at prices (20, 10). If it produced as much as that bundle, the chosen bundle wouldn’t have been chosen. 20.9 (0) A genealogical ﬁrm called Roots produces its output using only √ one input. Its production function is f (x) = x. 300 (a) Does the ﬁrm have increasing, constant, or decreasing returns to scale? 200 Decreasing. 125 100 (b) How many units of input does it take to produce 10 units of output? 100 units. If the input costs w per unit, what does it cost to produce 10 units of output? 0 100 200 300 400 Soybeans 100w. NAME 261 262 (c) How many units of input does it take to produce y units of output? y2. (Ch. 20) x2 40 If the input costs w per unit, what does it cost to produce y units of output? COST MINIMIZATION y 2 w. Output of 4 deer 30 Output of 5 deer (d) If the input costs w per unit, what is the average cost of producing y units? AC(w, y) = 20 yw. 20.10 (0) A university cafeteria produces square meals, using only one input and a rather remarkable production process. We are not allowed to say what that ingredient is, but an authoritative kitchen source says that “fungus is involved.” The cafeteria’s production function is f (x) = x2 , where x is the amount of input and f (x) is the number of square meals produced. 10 x = 1_ x 2 2 1 0 10 20 30 40 x1 (a) Does the cafeteria have increasing, constant, or decreasing returns to (b) Does this production function exhibit increasing, decreasing, or con- Increasing. scale? stant returns to scale? Decreasing returns to scale. (b) How many units of input does it take to produce 144 square meals? (c) If Irma faces factor prices (1, 1), what is the cheapest way for her to 12. If the input costs w per unit, what does it cost to produce 144 produce 4 deer? 12w. square meals? Use (16,8). How much does this cost? $24. (d) At the factor prices (1, 1), what is the cheapest way to produce 5 deer? (c) How many units of input does it take to produce y square meals? √ y. Use (25,12.5). How much does this cost? $37.50. If the input costs w per unit, what does it cost to produce y √ w y. square meals? (e) At the factor prices (1, 1), the cost of producing y deer with this technology is c(1, 1, y) = 3y 2 /2. (d) If the input costs w per unit, what is the average cost of producing y √ w/ y. square meals? AC(w, y) = (f ) At the factor prices (w1 , w2 ), the cost of producing y deer with this technology is c(w1 , w2 , y) = 20.11 (0) Irma’s Handicrafts produces plastic deer for lawn ornaments. “It’s hard work,” says Irma, “but anything to make a buck.” Her production function is given by f (x1 , x2 ) = (min{x1 , 2x2 })1/2 , where x1 is the amount of plastic used, x2 is the amount of labor used, and f (x1 , x2 ) is the number of deer produced. (a) In the graph below, draw a production isoquant representing input combinations that will produce 4 deer. Draw another production isoquant representing input combinations that will produce 5 deer. NAME 263 (a) In the graph below, draw a production isoquant representing input combinations that will produce 4 deer. Draw another production isoquant representing input combinations that will produce 6 deer. x2 40 Output of 6 deer 20 16 Output of 4 deer 10 8 0 10 18 20 30 40 x1 (b) Does this production function exhibit increasing, decreasing, or constant returns to scale? Decreasing returns to scale. (c) If Al faces factor prices (1, 1), what is the cheapest way for him to (8, 0). How much does this cost? $8. (d) At the factor prices (1, 1), what is the cheapest way to produce 6 (18, 0). How much does this cost? $18. (e) At the factor prices (1, 1), the cost of producing y deer with this technology is c(1, 1, y) = y 2/2. (f ) At the factor prices (3, 1), the cost of producing y deer with this technology is c(3, 1, y) = y 2. 20.13 (0) Suppose that Al Deardwarf from the last problem cannot vary the amount of wood that he uses in the short run and is stuck with using 20 units of wood. Suppose that he can change the amount of plastic that he uses, even in the short run. (a) How much plastic would Al need in order to make 100 deer? units. 264 COST MINIMIZATION (Ch. 20) (b) If the cost of plastic is $1 per unit and the cost of wood is $1 per unit, how much would it cost Al to make 100 deer? c(1, 1, y) = 20 + (y 2 − 20)/2. 30 deer? 20.12 (0) Al Deardwarf also makes plastic deer for lawn ornaments. Al has found a way to automate the production process completely. He doesn’t use any labor–only wood and plastic. Al says he likes the business “because I need the doe.” Al’s production function is given by f (x1 , x2 ) = (2x1 + x2 )1/2 , where x1 is the amount of plastic used, x2 is the amount of wood used, and f (x1 , x2 ) is the number of deer produced. $5,010. (c) Write down Al’s short-run cost function at these factor prices. 36 produce 4 deer? (w1 + w2 /2)y 2 . 4,990 266 Chapter 21 COST CURVES (Ch. 21) NAME AC, MC 40 Cost Curves Red line 30 Introduction. Here you continue to work on cost functions. Total cost can be divided into ﬁxed cost, the part that doesn’t change as output changes, and variable cost. To get the average (total) cost, average ﬁxed cost, and average variable cost, just divide the appropriate cost function by y, the level of output. The marginal cost function is the derivative of the total cost function with respect to output—or the rate of increase in cost as output increases, if you don’t know calculus. Remember that the marginal cost curve intersects both the average cost curve and the average variable cost curve at their minimum points. So to ﬁnd the minimum point on the average cost curve, you simply set marginal cost equal to average cost and similarly for the minimum of average variable cost. 20 AC=MC=20 10 0 10 20 30 Example: A ﬁrm has the total cost function C(y) = 100 + 10y. Let us ﬁnd the equations for its various cost curves. Total ﬁxed costs are 100, so the equation of the average ﬁxed cost curve is 100/y. Total variable costs are 10y, so average variable costs are 10y/y = 10 for all y. Marginal cost is 10 for all y. Average total costs are (100 + 10y)/y = 10 + 10/y. Notice that for this ﬁrm, average total cost decreases as y increases. Notice also that marginal cost is less than average total cost for all y. (d) Suppose Otto has to pay $b a year to produce obnoxious television commercials. Otto’s total cost curve is now T C(y) = (a) What is his total cost if he sells 10 cars? cars? 20c. 10c. What if he sells 20 Write down the equation for Otto’s total costs assuming he sells y cars: T C(y) = cy. c. c. 21.2 (0) Otto’s brother, Dent Carr, is in the auto repair business. Dent recently had little else to do and decided to calculate his cost conditions. He found that the total cost of repairing s cars is T C(s) = 2s2 + 10. But Dent’s attention was diverted to other things . . . and that’s where you come in. Please complete the following: c. (c) In the graph below draw Otto’s average and marginal cost curves if c = 20. NAME 267 21.3 (0) A third brother, Rex Carr, owns a junk yard. Rex can use one of two methods to destroy cars. The ﬁrst involves purchasing a hydraulic car smasher that costs $200 a year to own and then spending $1 for every car smashed into oblivion; the second method involves purchasing a shovel that will last one year and costs $10 and paying the last Carr brother, Scoop, to bury the cars at a cost of $5 each. Average Fixed Costs: 10/s. Average Total Costs: 2s + 10/s. Marginal Costs: 268 COST CURVES imized? 500. output per year: T C1 (y) = y + 200 , T C2 (y) = 5y + 10. 4s. (Ch. 21) At this level of output, how much is average cost? $2. (c) If she has 1,000 square feet of ﬂoor space, write down her marginal cost function: (a) Write down the total cost functions for the two methods, where y is M C = y/500 marginal cost function 5 + 10/y and 1. 1 + 200/y and a For the second method these costs are 5. (c) If Rex wrecks 40 cars per year, which method should he use? Method 2. he use? If Rex wrecks 50 cars per year, which method should Method 1. What is the smallest number of cars per year for which it would pay him to buy the hydraulic smasher? (a) If she has 200 square feet of ﬂoor space, write down her marginal cost y function: M C = and her average cost function: AC = 100 y 200 + 200 . At what amount of output is average cost minimized? y At this level of output, how much is average cost? $2. 1,000. At what amount of output is At this level of output, how much $2. (d) Use red ink to show Mary’s average cost curve and her marginal cost curves if she has 200 square feet. Use blue ink to show her average cost curve and her marginal cost curve if she has 500 square feet. Use black ink to show her average cost curve and her marginal cost curve if she has 1,000 square feet. Label the average cost curves AC and the marginal cost curves M C. Dollars 4 mc 3 21.4 (0) Mary Magnolia wants to open a ﬂower shop, the Petal Pusher, in a new mall. She has her choice of three diﬀerent ﬂoor sizes, 200 square feet, 500 square feet, or 1,000 square feet. The monthly rent will be $1 a square foot. Mary estimates that if she has F square feet of ﬂoor space and sells y bouquets a month, her variable costs will be cv (y) = y 2 /F per month. 200. is average cost? 48 cars per year. and her average cost function: AC = (1, 000/y) + y/1, 000. average cost minimized? (b) The ﬁrst method has an average cost function 2s. Average Variable Costs: c. 2s2 . 10. Total Fixed Costs: For every additional auto Otto sells, by how much do his costs increase? Write down Otto’s marginal cost function: M C(y) = his (e) If b = $100, use red ink to draw Otto’s average cost curve on the graph above. Dent’s Total Variable Costs: (b) What is Otto’s average cost function? AC(y) = cy + b , c + b/y , and his marginal cost average cost curve is now AC(y) = curve is M C(y) = 21.1 (0) Mr. Otto Carr, owner of Otto’s Autos, sells cars. Otto buys autos for $c each and has no other costs. 40 Output Red lines mc Blue lines ac ac Black lines mc ac 2 LRMC=LRAC (yellow line) 1 0 200 400 600 800 1000 1200 Bouquents (e) Use yellow marker to show Mary’s long-run average cost curve and her long-run marginal cost curve in your graph. Label them LRAC and LRMC. (b) If she has 500 square feet, write down her marginal cost function: M C = y/250 (500/y) + y/500. and her average cost function: AC = At what amount of output is average cost min- 21.5 (0) Touchie MacFeelie publishes comic books. The only inputs he needs are old jokes and cartoonists. His production function is 1 Q = .1J 2 L3/4 , NAME 269 270 where J is the number of old jokes used, L the number of hours of cartoonists’ labor used as inputs, and Q is the number of comic books produced. Explain your answer. t It exhibits dollars. comic books produced is multiplied by f (J, L) > tf (J, L). 25/4 . 21.8 (0) Consider the cost function c(y) = 4y 2 + 16. (b) If the number of old jokes used is 100, write an expression for the marginal product of cartoonists’ labor as a function of L. (b) The marginal cost function is Decreasing. M C = 8y. (c) The level of output that yields the minimum average cost of production is 21.6 (0) Touchie MacFeelie’s irascible business manager, Gander MacGrope, announces that old jokes can be purchased for $1 each and that the wage rate of cartoonists’ labor is $2. y = 2. (d) The average variable cost function is (a) Suppose that in the short run, Touchie is stuck with exactly 100 old jokes (for which he paid $1 each) but is able to hire as much labor as he wishes. How much labor would he have to hire in order produce Q comic books? AV C = 4y. (e) At what level of output does average variable cost equal marginal At y = 0. cost? Q4/3 . 21.9 (0) A competitive ﬁrm has a production function of the form Y = 2L + 5K. If w = $2 and r = $3, what will be the minimum cost of (b) Write down Touchie’s short-run total cost as a function of his output 2Q4/3 + 100. producing 10 units of output? $6. 8Q1/3 /3. (c) His short-run marginal cost function is 2Q1/3 + 100/Q. (d) His short-run average cost function is Calculus 16 y. AC = 4y + (a) The average cost function is MP = 3 Is the marginal product of labor decreasing or increasing as the 4L1/4 amount of labor increases? 18.7 (c) Given our production function, the cheapest proportions in which to use jokes and labor are the same no matter how many comic books we print. But when we double the amount of both inputs, the number of increasing returns to scale since f (tJ, tL) = 5/4 (Ch. 21) (b) This would cost (a) Does this production process exhibit increasing, decreasing, or constant returns to scale? COST CURVES 21.7 (1) Touchie asks his brother, Sir Francis MacFeelie, to study the long-run picture. Sir Francis, who has carefully studied the appendix to Chapter 19 in your text, prepared the following report. (a) If all inputs are variable, and if old jokes cost $1 each and cartoonist labor costs $2 per hour, the cheapest way to produce exactly 4/5 one comic book is to use 10 (4/3)3/5 ≈ 7.4 jokes and 104/5 (3/4)2/5 ≈ 5.6 hours of labor. (Fractional jokes are cer- tainly allowable.) 272 Chapter 22 FIRM SUPPLY (Ch. 22) NAME Dollars Firm Supply 80 Supply 60 ac 40 20 ;;;;;;;;;;;; ;;;;;;;;;;;; Profit ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; Costs ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; 0 5 Revenue Introduction. The short-run supply curve of a competitive ﬁrm is the portion of its short-run marginal cost curve that is upward sloping and lies above its average variable cost curve. The long-run supply curve of a competitive ﬁrm is the portion of its short-run marginal cost curve that is upward-sloping and lies above its long-run average cost curve. Example: A ﬁrm has the long-run cost function c(y) = 2y 2 + 200 for y > 0 and c(0) = 0. Let us ﬁnd its long-run supply curve. The ﬁrm’s marginal cost when its output is y is M C(y) = 4y. If we graph output on the horizontal axis and dollars on the vertical axis, then we ﬁnd that the long-run marginal cost curve is an upward-sloping straight line through the origin with slope 4. The long-run supply curve is the portion of this curve that lies above the long-run average cost curve. When output is y, long-run average costs of this ﬁrm are AC(y) = 2y + 200/y. This is a Ushaped curve. As y gets close to zero, AC(y) becomes very large because 200/y becomes very large. When y is very large, AC(y) becomes very large because 2y is very large. When is it true that AC(y) < M C(y)? This happens when 2y + 200/y < 4y. Simplify this inequality to ﬁnd that AC(y) < M C(y) when y > 10. Therefore the long-run supply curve is the piece of the long-run marginal cost curve for which y > 10. So the long-run supply curve has the equation p = 4y for y > 10. If we want to ﬁnd quantity supplied as a function of price, we just solve this expression for y as a function of p. Then we have y = p/4 whenever p > 40. Suppose that p < 40. For example, what if p = 20, how much will the ﬁrm supply? At a price of 20, if the ﬁrm produces where price equals long-run marginal cost, it will produce 5 = 20/4 units of output. When the ﬁrm produces only 5 units, its average costs are 2 × 5 + 200/5 = 50. Therefore when the price is 20, the best the ﬁrm can do if it produces a positive amount is to produce 5 units. But then it will have total costs of 5 × 50 = 250 and total revenue of 5 × 20 = 100. It will be losing money. It would be better oﬀ producing nothing at all. In fact, for any price p < 40, the ﬁrm will choose to produce zero output. 22.1 (0) Remember Otto’s brother Dent Carr, who is in the auto repair business? Dent found that the total cost of repairing s cars is c(s) = 2s2 + 100. 10 mc avc 15 20 Output (b) If the market price is $20, how many cars will Dent be willing to repair? repair? 5. If the market price is $40, how many cars will Dent 10. (c) Suppose the market price is $40 and Dent maximizes his proﬁts. On the above graph, shade in and label the following areas: total costs, total revenue, and total proﬁts. Calculus 22.2 (0) A competitive ﬁrm has the following short-run cost function: c(y) = y 3 − 8y 2 + 30y + 5. (a) The ﬁrm’s marginal cost function is M C(y) = 3y 2 − 16y + 30. (b) The ﬁrm’s average variable cost function is AV C(y) = 30. y 2 − 8y + (Hint: Notice that total variable costs equal c(y) − c(0).) (c) On the axes below, sketch and label a graph of the marginal cost function and of the average variable cost function. (d) Average variable cost is falling as output rises if output is less than (a) This implies that Dent’s average cost is equal to his average variable cost is equal to 2s 2s + 100/s , 4 and rising as output rises if output is greater than 4. (e) Marginal cost equals average variable cost when output is 4. , and his marginal cost is equal to 4s. On the graph below, plot the above curves, and also plot Dent’s supply curve. NAME 273 (f ) The ﬁrm will supply zero output if the price is less than 274 14. (g) The smallest positive amount that the ﬁrm will ever supply at any 4. price is of output? FIRM SUPPLY (Ch. 22) (d) The price of fertilizer and of cabbages remain as before, but Mr. McGregor learns that he could ﬁnd summer jobs for Flopsy and Peter in a local sweatshop. Flopsy and Peter would together earn $300 for the summer, which Mr. McGregor could pocket, but they would have no time to work in the cabbage patch. Without their labor, he would get no cabbages. Now what is Mr. McGregor’s total cost of producing y cabbages? At what price would the ﬁrm supply exactly 6 units 42. c(y) = 300 + (y/10)2 . Costs (e) Should he continue to grow cabbages or should he put Flopsy and 40 mc Peter to work in the sweatshop? 30 avc 20 Sweatshop. 22.4 (0) Severin, the herbalist, is famous for his hepatica. His total cost function is c(y) = y 2 + 10 for y > 0 and c(0) = 0. (That is, his cost of producing zero units of output is zero.) (a) What is his marginal cost function? 10 function? 2y. What is his average cost y + 10/y. (b) At what quantity is his marginal cost equal to his average cost? 0 Calculus 2 4 6 √ 10. 8 y supply a positive quantity in long-run equilibrium? y 2 /100. (b) If the only way that Mr. McGregor can vary his output is by varying the amount of fertilizer applied to his cabbage patch, write an expression y/50. (c) If the price of cabbages is $2 each, how many cabbages will Mr. Mc- 100. Gregor produce? 100. a mountain is $100. NAME 275 276 (b) On the graph below, show Stanley Ford’s marginal cost curve (in blue ink) and his average cost curve (in red ink). 4000 $2,000 FIRM SUPPLY if he produces more than 20 mountains. (Ch. 22) (a) Lady Wellesleigh can either make silk purses or she can earn $5 an hour as a seamstress in a sweatshop. If she worked in the sweat shop, how Pencil mc curve Red ac curve How sows’ ears as she likes for $1 each. Lady Wellesleigh has no other source of income than her labor. Her utility function is a Cobb-Douglas function U (c, r) = c1/3 r 2/3 , where c is the amount of money per day that she has to spend on consumption goods and r is the amount of leisure that she has. Lady Wellesleigh has 24 hours a day that she can devote either to leisure or to working. Dollars 3000 √ 2 10. (a) The marginal cost of producing a mountain for Stanley is $1,000 if he produces fewer than 20 mountains. The marginal cost of producing How many sacks of fertilizer will he buy? How much proﬁt will he make? 10. 22.5 (1) Stanley Ford makes mountains out of molehills. He can do this with almost no eﬀort, so for the purposes of this problem, let us assume that molehills are the only input used in the production of mountains. Suppose mountains are produced at constant returns to scale and that it takes 100 molehills to make 1 mountain. The current market price of molehills is $20 each. A few years ago, Stan bought an “option” that permits him to buy up to 2,000 molehills at $10 each. His option contract explicitly says that he can buy fewer than 2,000 molehills if he wishes, but he can not resell the molehills that he buys under this contract. In order to get governmental permission to produce mountains from molehills, Stanley would have to pay $10,000 for a molehill-masher’s license. What is the total cost of the amount of fertilizer needed to for his marginal cost, as a function of y. M C(y) = √ 10. much would he supply at that price? (a) What is the total cost of the fertilizer needed to produce 100 cabbages? $100. √ (c) In a competitive market, what is the lowest price at which he will 22.3 (0) Mr. McGregor owns a 5-acre cabbage patch. He forces his wife, Flopsy, and his son, Peter, to work in the cabbage patch without wages. Assume for the time being that the land can be used for nothing other than cabbages and that Flopsy and Peter can ﬁnd no alternative employment. The only input that Mr. McGregor pays for is fertilizer. √ If he uses x sacks of fertilizer, the amount of cabbages that he gets is 10 x. Fertilizer costs $1 per sack. produce y cabbages? At what quantity is his average cost minimized? many hours would she work? 8. (Hint: To solve for this amount, write down Lady Wellesleigh’s budget constraint and recall how to ﬁnd the demand function for someone with a Cobb-Douglas utility function.) 2000 Blue mc curve (b) If she could earn a wage of $w an hour as a seamstress, how much 1000 would she work? 8 hours. (c) If the price of silk purses is $p, how much money will Lady Wellesleigh 0 10 20 30 40 Output earn per purse after she pays for the sows’ ears that she uses? p − 1. (d) If she can earn $5 an hour as a seamstress, what is the lowest price (c) If the price of mountains is $1,600, how many mountains will Stanley produce? at which she will make any silk purses? 20 mountains. $6. (d) The government is considering raising the price of a molehill-masher’s license to $11,000. Stanley claims that if it does so he will have to go out (e) What is the supply function for silk purses? (Hint: The price of silk purses determines the “wage rate” that Lady W. can earn by making silk purses. This determines the number of hours she will choose to work and of business. Is Stanley telling the truth? No. What is the highest fee for a license that the government could charge without driving him hence the supply of silk purses.) out of business? The maximum they could charge is the amount of his profits excluding the license fee, $12,000. (e) Stanley’s lawyer, Eliot Sleaze, has discovered a clause in Stanley’s option contract that allows him to resell the molehills that he purchased under the option contract at the market price. On the graph above, use a pencil to draw Stanley’s new marginal cost curve. If the price of mountains remains $1,600, how many mountains will Stanley produce now? He will sell all of his molehills and produce zero mountains. 22.6 (1) Lady Wellesleigh makes silk purses out of sows’ ears. She is the only person in the world who knows how to do so. It takes one sow’s ear and 1 hour of her labor to make a silk purse. She can buy as many S(p) = 8 for p > 6, 0 otherwise. Calculus 22.7 (0) Remember Earl, who sells lemonade in Philadelphia? You met him in the chapter on cost functions. Earl’s production function is 1/3 1/3 f (x1 , x2 ) = x1 x2 , where x1 is the number of pounds of lemons he uses and x2 is the number of hours he spends squeezing them. As you 1/2 1/2 found out, his cost function is c(w1 , w2 , y) = 2w1 w2 y 3/2 , where y is the number of units of lemonade produced. (a) If lemons cost $1 per pound, the wage rate is $1 per hour, and the price of lemonade is p, Earl’s marginal cost function is M C(y) = and his supply function is S(p) = 2 p /9. 3y 1/2 If lemons cost $4 per pound and the wage rate is $9 per hour, his supply function will be S(p) = p2 /324. NAME 277 (b) In general, Earl’s marginal cost depends on the price of lemons and the wage rate. At prices w1 for lemons and w2 for labor, his marginal cost when he is producing y units of lemonade is M C(w1 , w2 , y) = 1/2 1/2 The amount that Earl will supply depends on the three variables, p, w1 , w2 . As a function of these three variables, Earl’s 3w1 w2 y 1/2 . supply is S(p, w1 , w2 ) = Calculus p2 /9w1 w2. (w1 + w2 /2)y 2 . (a) Irma’s cost function is c(w1 , w2 , y) = The number of units of output that she would supply at price p is S(p) = p/3. At these factor prices, her average cost per unit of output would be AC(y) = (Ch. 22) y = (c) What is Jack’s supply function when stones cost $8 each? (p/3)1/2 . When stones cost $w each? y = (8p/3w)1/2 . (d) If Jack has 19 relatives who can also get blood from a stone in the same way, what is the aggregate supply function for blood when stones Y = 20(8p/3w)1/2 . 22.10 (1) The Miss Manners Reﬁnery in Dry Rock, Oklahoma, converts crude oil into gasoline. It takes 1 barrel of crude oil to produce 1 barrel of gasoline. In addition to the cost of oil there are some other costs involved in reﬁning gasoline. Total costs of producing y barrels of gasoline are described by the cost function c(y) = y 2 /2 + po y, where po is the price of a barrel of crude oil. (a) Express the marginal cost of producing gasoline as a function of po (b) If w1 = w2 = 1, then Irma’s marginal cost of producing y units of 3y. FIRM SUPPLY cost $w each? 22.8 (0) As you may recall from the chapter on cost functions, Irma’s handicrafts has the production function f (x1 , x2 ) = (min{x1 , 2x2 })1/2 , where x1 is the amount of plastic used, x2 is the amount of labor used, and f (x1 , x2 ) is the number of lawn ornaments produced. Let w1 be the price per unit of plastic and w2 be the wage per unit of labor. output is M C(y) = 278 and y. y + po . (b) Suppose that the reﬁnery can buy 50 barrels of crude oil for $5 a barrel but must pay $15 a barrel for any more that it buys beyond 50 barrels. The marginal cost curve for gasoline will be 3y/2. barrels of gasoline and y + 15 y+5 up to 50 thereafter. (c) If the competitive price of the lawn ornaments she sells is p = 48, and w1 = w2 = 1, how many will she produce? 16. How much proﬁt will (c) Plot Miss Manners’ supply curve in the diagram below using blue ink. Price of gasoline 384. she make? 80 (d) More generally, at factor prices w1 and w2 , her marginal cost is a function M C(w1 , w2 , y) = (2w1 + w2 )y. At these factor prices and an output price of p, the number of units she will choose to supply is S(p, w1 , w2 ) = 60 Blue lines Black line p/(2w1 + w2 ). Red line 40 22.9 (0) Jack Benny can get blood from a stone. If he has x stones, the 1 number of pints of blood he can extract from them is f (x) = 2x 3 . Stones cost Jack $w each. Jack can sell each pint of blood for $p. 30 20 (a) How many stones does Jack need to extract y pints of blood? y 3 /8. (b) What is the cost of extracting y pints of blood? 0 wy 3/8. NAME 279 (d) Suppose that Miss Manners faces a horizontal demand curve for gasoline at a price of $30 per barrel. Plot this demand curve on the graph 25 above using red ink. How much gasoline will she supply? 280 FIRM SUPPLY (e) If Miss Manners could no longer get the ﬁrst 50 barrels of crude for $5, but had to pay $15 a barrel for all crude oil, how would her output 4 (f ) Now suppose that an entitlement program is introduced that permits reﬁneries to buy one barrel of oil at $5 for each barrel of oil that they (Ch. 22) Red line 3 2 S(p) = 1 Assume that it can buy fractions of a barrel in the same manner. Plot this supply curve on the graph above using black ink. If the demand curve is horizontal at $30 a barrel, how much gasoline will 0 buy for $15. What will Miss Manners’ supply curve be now? 75 100 Barrels of gasoline Price 5 It would decrease to 15 barrels. 50 6 barrels. change? 25 p − 10. Miss Manners supply now? 10 20 30 20 barrels. 40 50 60 Bushels of corn 22.11 (2) Suppose that a farmer’s cost of growing y bushels of corn is given by the cost function c(y) = (y 2 /20) + y. (e) If p = $2, how many bushels of corn will he produce? 0. many bushels will he get from the government stockpiles? 20. (a) If the price of corn is $5 a bushel, how much corn will this farmer (f ) If p = $5, how much corn will he supply? 15 bushels. How many bushels of corn will he get from the government stockpiles, assuming grow? 40 bushels. he chooses to be in the PIK program? (b) What is the farmer’s supply curve of corn as a function of the price of corn? S(p) = 10p − 10. into account the value of payments in kind received. py − c(y) + p(40 − y)/2 = py − y 2 /20 − y + p(40 − y)/2. (d) At the market price p, what will be the farmer’s proﬁt-maximizing output of corn? the graph below. S(p) = 5p − 10. Plot a supply curve for corn in $12.50. (g) At any price between p = $2 and p = $5, write a formula for the size of the PIK payment. (c) The government now introduces a Payment in Kind (PIK) program. If the farmer decides to grow y bushels of corn, he will get (40−y)/2 bushels from the government stockpiles. Write an expression for the farmer’s proﬁts as a function of his output and the market price of corn, taking How His supply curve is S(p) = 5p − 10, and his payment is (40 − y)/2. So he gets 25 − 2.5p. (h) How much corn will he supply to the market, counting both production and PIK payment, as a function of the market price p? Sum supply curve and PIK payment to get T S(p) = 2.5p + 15. (i) Use red ink to illustrate the total supply curve of corn (including the corn from the PIK payment) in your graph above. 282 Chapter 23 NAME INDUSTRY SUPPLY (Ch. 23) (d) s1 (p) = 3p − 12, s2 (p) = 2p − 8, s3 (p) = p − 4. Industry Supply S(p) = 6p − 24 for p > 4. Introduction. To ﬁnd the industry supply of output, just add up the supply of output coming from each individual ﬁrm. Remember to add quantities, not prices. The industry supply curve will have a kink in it where the market price becomes low enough that some ﬁrm reduces its quantity supplied to zero. The series of problems about the garden gnome industry are designed to help you to understand the distinction between the long run and the short run. To solve these problems, you need to pay careful attention to the timing of decisions. In particular, in this problem, units of capital (gnome molds) can be produced and delivered only one year after they are ordered. The last three questions of this chapter apply supply and demand analysis to some problems in the economics of illegal activities. In these examples, you will make use of your knowledge of where supply functions come from. 23.0 Warm Up Exercise. Here are some drills for you on ﬁnding market supply functions from linear ﬁrm supply functions. The trick here is to remember that the market supply function may have kinks in it. For example, if the ﬁrm supply functions are s1 (p) = p and s2 (p) = p − 2, then the market supply function is S(p) = p for p ≤ 2 and S(p) = 2p − 2 for p > 2; that is, only the ﬁrst ﬁrm supplies a positive output at prices below $2, and both ﬁrms supply output at prices above $2. Now try to construct the market supply function in each of the following cases. S(p) = 6p. (a) s1 (p) = p, s2 (p) = 2p, s3 (p) = 3p. 23.1 (1) Al Deardwarf’s cousin, Zwerg, makes plaster garden gnomes. The technology in the garden gnome business is as follows. You need a gnome mold, plaster, and labor. A gnome mold is a piece of equipment that costs $1,000 and will last exactly one year. After a year, a gnome mold is completely worn out and has no scrap value. With a gnome mold, you can make 500 gnomes per year. For every gnome that you make, you also have to use a total of $7 worth of plaster and labor. The total amounts of plaster and labor used are variable in the short run. If you want to produce only 100 gnomes a year with a gnome mold, you spend only $700 a year on plaster and labor, and so on. The number of gnome molds in the industry cannot be changed in the short run. To get a newly built one, you have to special-order it from the gnome-mold factory. The gnome-mold factory only takes orders on January 1 of any given year, and it takes one whole year from the time a gnome mold is ordered until it is delivered on the next January 1. When a gnome mold is installed in your plant, it is stuck there. To move it would destroy it. Gnome molds are useless for anything other than making garden gnomes. For many years, the demand function facing the garden-gnome industry has been D(p) = 60, 000 − 5, 000p, where D(p) is the total number of garden gnomes sold per year and p is the price. Prices of inputs have been constant for many years and the technology has not changed. Nobody expects any changes in the future, and the industry is in long-run equilibrium. The interest rate is 10%. When you buy a new gnome mold, you have to pay for it when it is delivered. For simplicity of calculations, we will assume that all of the gnomes that you build during the one-year life of the gnome mold are sold at Christmas and that the employees and plaster suppliers are paid only at Christmas for the work they have done during the past year. Also for simplicity of calculations, let us approximate the date of Christmas by December 31. (a) If you invested $1,000 in the bank on January 1, how much money S(p) = 2p for p ≤ 1, S(p) = (b) s1 (p) = 2p, s2 (p) = p − 1. 3p − 1 for p > 1. (c) 200 ﬁrms each have a supply function s1 (p) = 2p − 8 and 100 ﬁrms each have a supply function s2 (p) = p − 3. S(p) = 0 for p < 3, S(p) = 100p − 300 for 3 ≤ p ≤ 4, S(p) = 500p − 1, 900 for p > 4. could you expect to get out of the bank one year later? $1,100. If you received delivery of a gnome mold on January 1 and paid for it at that time, by how much would your revenue have to exceed the costs of plaster and labor if it is to be worthwhile to buy the machine? (Remember that the machine will be worn out and worthless at the end of the year.) $1,100. (b) Suppose that you have exactly one newly installed gnome mold in your plant; what is your short-run marginal cost of production if you produce up to 500 gnomes? NAME for producing up to 500 gnomes? 283 $7. If you have only one gnome mold, is it possible in the short run to produce more than 500 gnomes? 284 INDUSTRY SUPPLY $7. What is your average variable cost (Ch. 23) (a) On the graph below, draw the short run industry supply curve for garden gnomes that applies in the year 2001, after the discovery of the new plaster is announced. On the same graph, show the demand curve for garden gnomes. No. (c) If you have exactly one newly installed gnome mold, you would pro- 7 duce 500 gnomes if the price of gnomes is above dollars. You would produce no gnomes if the price of gnomes is below 7 dollars. You would be indiﬀerent between producing any number of gnomes 7 between 0 and 500 if the price of gnomes is dollars. (d) If you could sell as many gnomes as you liked for $10 each and none at a higher price, what rate of return would you make on your $1,000 by investing in a gnome mold? 50%. Is this higher than the return from putting your money in the bank? Yes. What is the lowest price for gnomes such that the rate of return you get from ivesting $1000 in a gnome mold is as at least 10%? $9.20. equilibrium price of gnomes be lower than this? Could the long-run No. (e) At the price you found in the last section, how many gnomes would be demanded each year? 14,000. How many molds would be (b) In 2001, what is the short run equilibrium total output of garden gnomes, 28. and what is the short run equilibrium price of the suppliers of plaster, he received a net cash ﬂow of $ 2,100. Did he make more than a 10% rate of return on his investment in the gnome mold? purchased each year? 14,000. garden gnomes? $9.20. (Hint: Look at the intersection of the supply and demand curves you just drew. Cousin Zwerg bought a gnome mold that was delivered on January 1, 2001, and, as had been agreed, he paid $1,000 for it on that day. On January 1, 2002, when he sold the gnomes he had made during the year and when he paid the workers and Yes. What rate of return did he make? 110%. Is this a long-run equilibrium price? Yes. 23.2 (1) We continue our study of the garden-gnome industry. Suppose that initially everything was as described in the previous problem. To the complete surprise of everyone in the industry, on January 1, 2001, the invention of a new kind of plaster was announced. This new plaster made it possible to produce garden gnomes using the same molds, but it reduced the cost of the plaster and labor needed to produce a gnome from $7 to $5 per gnome. Assume that consumers’ demand function for gnomes in 2001 was not changed by this news. The announcement came early enough in the day for everybody to change his order for gnome molds to be delivered on January 1, 2002, but of course, but the total number of molds available to be used in 2001 was just the 28 molds that had been ordered the previous year. The manufacturer of garden gnome molds contracted to sell them for $1,000 when they were ordered, so it can’t change the price it charges on delivery. (c) Zwerg’s neighbor, Munchkin, also makes garden gnomes, and he has a gnome mold that is to be delivered on January 1, 2001. On this day, Zwerg, who is looking for a way to invest some more money, is considering buying Munchkin’s new mold from Munchkin and installing it in his own plant. If Munchkin keeps his mold, he will get a net cash ﬂow of $ 2,100 in one year. If the interest rate that Munchkin faces, both for borrowing and lending is 10%, then should he be willing to sell his mold for $1,000? No. What is the lowest price that he would be willing to sell it for? $1,909. If the best rate of return that Zwerg can make on alternative investments of additional funds is 10%, what is the most that Zwerg would be willing to pay for Munchkin’s new mold? $1,909. NAME 285 (d) What do you think will happen to the number of garden gnomes ordered for delivery on January 1, 2002? Will it be larger, smaller, or the same as the number ordered the previous year? Larger. After the passage of suﬃcient time, the industry will reach a new long-run equilibrium. What will be the new equilibrium price of gnomes? $7.20. 23.3 (1) In the previous problem, we studied the eﬀects of a cost-saving invention. For this problem, we suppose that there was no such invention, but that a tax is introduced. Suppose that on January 1, 2001, the industry was as described in the previous problem (without the invention of the new kind of plaster). On this day, the government surprised the garden gnome industry by introducing a tax on the production of garden gnomes. For every garden gnome produced, the manufacturer must pay a $1 tax. The announcement came early enough in the day so that there was time for gnome producers to change their orders of gnome molds for 2002. But the gnome molds available to be used in 2001 are those that had been ordered a year previously. Gnome makers had signed contracts promising to pay $1,000 for each gnome mold that they ordered, and they couldn’t back out of these promises. Thus in the short run, during the year 2001, the number of gnome molds is stuck at 28. (a) On the graph below, draw the short run industry supply curve for garden gnomes that applies in the year 2001, after the new tax is introduced. On the same graph, show the demand curve for garden gnomes. 286 INDUSTRY SUPPLY (Ch. 23) (d) In 2001, what will be the total output of garden gnomes? 14,000. What will be the price of garden gnomes? $9.20. What rate of return will Deardwarf’s cousin Zwerg make on his investment in a garden gnome mold that he ordered a year ago and paid $1,000 for at that time? −40%. (e) Remember that Zwerg’s neighbor, Munchkin, also has a gnome mold that is to be delivered on January 1, 2001. Knowing about the tax makes Munchkin’s mold a less attractive investment than it was without the tax, but still Zwerg would buy it if he can get it cheap enough so that he makes a 10% rate of return on his investment. How much should he be $545.45. willing to pay for Munchkin’s new mold? (f ) What do you think will happen to the number of gnome molds ordered for delivery on January 1, 2002? Will it be larger, smaller, or the same Smaller. as the number ordered the previous year? (g) The tax on garden gnomes was left in place for many years, and nobody expected any further changes in the tax or in demand or supply conditions. After the passage of suﬃcient time, the industry reached a new long-run equilibrium. What was the new equilibrium price of gnomes? $10.20. (h) In the short run, who would end up paying the tax on garden gnomes, the producers or the consumers? Producers. In the long run, did the price of gnomes go up by more, less, or the same amount as the tax per gnome? (b) In 2001, after the tax is introduced, what is the short run equilibrium total output of garden gnomes, 14,000. and what is the short run equilibrium price of garden gnomes? $9.20. (Hint: Look at the intersection of the supply and demand curves you just drew.) (c) If you have a garden gnome mold, the marginal cost of producing a garden gnome, including the tax, is (will, will not) will $8. Therefore all gnome molds (i) Suppose that early in the morning of January 1, 2001, the government had announced that there would be a $1 tax on garden gnomes, but that the tax would not go into eﬀect until January 1, 2002. Would the producers of garden gnomes necessarily be worse oﬀ than if there were no tax? Why or why not? No. The producers would anticipate the tax increase and restrict supply, thereby raising prices. be used up to capacity in 2001. NAME 287 (j) Is it reasonable to suppose that the government could introduce “surprise” taxes without making ﬁrms suspicious that there would be similar “surprises” in the future? Suppose that the introduction of the tax in January 2001 makes gnome makers suspicious that there will be more taxes introduced in later years. Will this aﬀect equilibrium prices and supplies? How? Same amount. If a surprise tax makes gnome makers expect similar ‘‘surprises’’ in future, it will take a higher current price to get them to enter the industry. This will raise 288 INDUSTRY SUPPLY (Ch. 23) (f ) Now suppose that the demand curve shifts to D(p) = 52.5 − p. What will be the equilibrium number of ﬁrms? (Hint: Can a new ﬁrm enter the market and make nonnegative proﬁts?) If a new firm entered, there would be 51 firms. The supply-demand equation would be 52.5 − p = 51p/2. Solve for p∗ = 105/53 < 2. A new firm would lose money. Therefore in equilibrium there would be 50 firms. the price paid by consumers. 23.4 (0) Consider a competitive industry with a large number of ﬁrms, all of which have identical cost functions c(y) = y 2 + 1 for y > 0 and c(0) = 0. Suppose that initially the demand curve for this industry is given by D(p) = 52 − p. (The output of a ﬁrm does not have to be an integer number, but the number of ﬁrms does have to be an integer.) (a) What is the supply curve of an individual ﬁrm? S(p) = p/2. If there are n ﬁrms in the industry, what will be the industry supply curve? ∗ 50p/2 to get p = 2.02. What will be the equilibrium output of each ﬁrm? y ∗ = 1.01. What will be the equilibrium proﬁts of each ﬁrm? Around .02. (h) Now suppose that the demand curve shifts to D(p) = 53−p. What will be the equilibrium number of ﬁrms? 51. price? 2. Y = np/2. (b) What is the smallest price at which the product can be sold? p∗ = 2. What will be the equilibrium (i) What will be the equilibrium output of each ﬁrm? will be the equilibrium proﬁts of each ﬁrm? (c) What will be the equilibrium number of ﬁrms in the industry? (Hint: Take a guess at what the industry price will be and see if it works.) Guess at p∗ = 2. Solve 52.5 − p = (g) What will be the equilibrium price? This gives D(p) = 52 − 2 = n2/2, which says n∗ = 50. (d) What will be the equilibrium price? equilibrium output of each ﬁrm? p∗ = 2. What will be the y ∗ = 1. Zero. (a) What is the competitive equilibrium price per ride? (Hint: In competitive equilibrium, price must equal marginal cost.) Y ∗ = 50. What 23.5 (3) In 1990, the town of Ham Harbor had a more-or-less free market in taxi services. Any respectable ﬁrm could provide taxi service as long as the drivers and cabs satisﬁed certain safety standards. Let us suppose that the constant marginal cost per trip of a taxi ride is $5, and that the average taxi has a capacity of 20 trips per day. Let the demand function for taxi rides be given by D(p) = 1, 200 − 20p, where demand is measured in rides per day, and price is measured in dollars. Assume that the industry is perfectly competitive. is the equilibrium number of rides per day? (e) What will be the equilibrium output of the industry? y = 1. taxicabs will there be in equilibrium? 55. 5. 1,100. What How many NAME 289 (b) In 1990 the city council of Ham Harbor created a taxicab licensing board and issued a license to each of the existing cabs. The board stated that it would continue to adjust the taxicab fares so that the demand for rides equals the supply of rides, but no new licenses will be issued in the future. In 1995 costs had not changed, but the demand curve for taxicab rides had become D(p) = 1, 220 − 20p. What was the equilibrium price $6. of a ride in 1995? with acquiring a taxicab license? 20. INDUSTRY SUPPLY (Ch. 23) (a) If a farm is located t miles from the center of town, write down a formula for its proﬁt per bushel of wheat transported to market. Profit per bushel = 5 − .10t. (b) Suppose you can grow 1,000 bushels on an acre of land. How much will an acre of land located t miles from the market rent for? (c) What was the proﬁt per ride in 1995, neglecting any costs associated license per day? 290 $1. Rent = 5, 000 − 100t. What was the proﬁt per taxicab (c) How far away from the market do you have to be for land to be worth If the taxi operated every day, what was the 50 miles. zero? $7,300. proﬁt per taxicab license per year? (d) If the interest rate was 10% and costs, demand, and the number of licenses were expected to remain constant forever, what would be the $73,000. market price of a taxicab license? 23.7 (1) Consider an industry with three ﬁrms. Suppose the ﬁrms have the following supply functions: S1 (p) = p, S2 (p) = p − 5, and S3 (p) = 2p respectively. On the graph below plot each of the three supply curves and the resulting industry supply curve. Price (e) Suppose that the commission decided in 1995 to issue enough new licenses to reduce the taxicab price per ride to $5. How many more 20 S 2 S 3 S 1 1. licenses would this take? 15 (f ) Assuming that demand in Ham Harbor is not going to grow any more, how much would a taxicab license be worth at this new fare? Industry supply 10 Nothing. 5 (g) How much money would each current taxicab owner be willing to pay to prevent any new licenses from being issued? $73,000 each. 0 What is the total amount that all taxicab owners together would be willing to pay to prevent any new licences from ever being issued? $4,015,000. The total amount that consumers would be willing to pay to have another taxicab license issued would be (more than, less than, the same as) more than 10 5. 291 Output? 5. level for ﬁrm 1 at this price? NAME 30 40 Quantity (a) If the market demand curve has the form D(p) = 15, what is the resulting market price? this amount. 23.6 (2) In this problem, we will determine the equilibrium pattern of agricultural land use surrounding a city. Think of the city as being located in the middle of a large featureless plain. The price of wheat at the market at the center of town is $10 a bushel, and it only costs $5 a bushel to grow wheat. However, it costs 10 cents a mile to transport a bushel of wheat to the center of town. 20 15. What is the output 0. Firm 2? Firm 3? 10. 23.8 (0) Suppose all ﬁrms in a given industry have the same supply curve given by Si (p) = p/2. Plot and label the four industry supply curves generated by these ﬁrms if there are 1, 2, 3, or 4 ﬁrms operating in the industry. 292 INDUSTRY SUPPLY (Ch. 23) Price 20 Price 20 Red mc S 1 S 2 S 4 S 3 15 15 P+t Red ac 10 P+l 10 5 5 0 10 20 30 $3.50 , quantity = 3.5. exist in such a market? How many ﬁrms would 2. Blue ac Blue mc 0 40 Quantity (a) If all of the ﬁrms had a cost structure such that if the price was below $3, they would be losing money, what would be the equilibrium price and output in the industry if the market demand was equal to D(p) = 3.5? Answer: price = Black ac P 10 20 30 40 Quantity (b) Suppose the government imposes a tax, t, on every unit of output sold by the industry. Use red ink to draw the new conditions on the above graph. After the industry has adjusted to the imposition of the tax, the competitive model would predict the following: the market price would (increase, decrease) increase be (more, the same, fewer) fewer t by amount , there would ﬁrms operating in the industry, and the output level for each ﬁrm operating in the industry would Stay the same. (increase, stay the same, decrease). (b) What if the identical conditions as above held except that the market demand was equal to D(p) = 8 − p? Now, what would be the equilibrium price and output? $3.20 and 4.8. operate in such a market? How many ﬁrms would 3. (c) What if the government imposes a tax, l, on every ﬁrm in the industry. Draw the new cost conditions on the above graph using black ink. After the industry has adjusted to the imposition of the tax the competitive model would predict the following: the market price would (increase, decrease) fewer) fewer increase , there would be (more, the same, ﬁrms operating in the industry, and the output level for each ﬁrm operating in the industry would stay the same, decrease). 23.9 (0) There is free entry into the pollicle industry. Anybody can enter this industry and have the same U-shaped average cost curve as all of the other ﬁrms in the industry. (a) On the diagram below, draw a representative ﬁrm’s average and marginal cost curves using blue ink. Also, indicate the long-run equilibrium level of the market price. increase (increase, 23.10 (0) In many communities, a restaurant that sells alcoholic beverages is required to have a license. Suppose that the number of licenses is limited and that they may be easily transferred to other restaurant owners. Suppose that the conditions of this industry closely approximate perfect competition. If the average restaurant’s revenue is $100,000 a year, and if a liquor license can be leased for a year for $85,000 from an existing restaurant, what is the average variable cost in the industry? $15,000. NAME 293 23.11 (2) In order to protect the wild populations of cockatoos, the Australian authorities have outlawed the export of these large parrots. An illegal market in cockatoos has developed. The cost of capturing an Australian cockatoo and shipping him to the United States is about $40 per bird. Smuggled parrots are drugged and shipped in suitcases. This is extremely traumatic for the birds and about 50% of the cockatoos shipped die in transit. Each smuggled cockatoo has a 10% chance of being discovered, in which case the bird is conﬁscated and a ﬁne of $500 is charged. Conﬁscated cockatoos that are alive are returned to the wild. Conﬁscated cockatoos that are found dead are donated to university cafeterias.∗ 294 INDUSTRY SUPPLY (Ch. 23) (f ) Suppose that the trade in cockatoos is legalized. Suppose that it costs about $40 to capture and ship a cockatoo to the United States in a comfortable cage and that the number of deaths in transit by this method is negligible. What would be the equilibrium price of cockatoos $40. How many cockatoos would be sold in 6,400. How many cockatoos would have to be in the United States? the United States? caught in Australia for the U.S. market? 6,400. (a) The probability that a smuggled parrot will reach the buyer alive and unconﬁscated is .45. Therefore when the price of smuggled parrots is p, what is the expected gross revenue to a parrot-smuggler from shipping a parrot? .45p. (b) What is the expected cost, including expected ﬁnes and the cost of $.10 × 500 + 40 = $90. capturing and shipping, per parrot? (c) The supply schedule for smuggled parrots will be a horizontal line at the market price just break even?) $200. (Hint: At what price does a parrot-smuggler (d) The demand function for smuggled cockatoos in the United States is D(p) = 7, 200 − 20p per year. How many smuggled cockatoos will be sold in the United States per year at the equilibrium price? 3,200. How many cockatoos must be caught in Australia in order that this number of live birds reaches U.S. buyers? 3, 200/.45 = 7, 111. (e) Suppose that instead of returning live conﬁscated cockatoos to the wild, the customs authorities sold them in the American market. The proﬁts from smuggling a cockatoo do not change from this policy change. Since the supply curve is horizontal, it must be that the equilibrium price of smuggled cockatoos will have to be the same as the equilibrium price when the conﬁscated cockatoos were returned to nature. How many live 23.12 (0) The horn of the rhinoceros is prized in Japan and China for its alleged aphrodisiac properties. This has proved to be most unfortunate for the rhinoceroses of East Africa. Although it is illegal to kill rhinoceroses in the game parks of Kenya, the rhinoceros population of these parks has been almost totally depleted by poachers. The price of rhinoceros horns in recent years has risen so high that a poacher can earn half a year’s wages by simply killing one rhinoceros. Such high rewards for poaching have made laws against poaching almost impossible to enforce in East Africa. There are also large game parks with rhinoceros populations in South Africa. Game wardens there were able to prevent poaching almost completely and the rhinoceros population of South Africa has prospered. In a recent program from the television series Nova, a South African game warden explained that some rhinoceroses even have to be “harvested” in order to prevent overpopulation of rhinoceroses. “What then,” asked the interviewer, “do you do with the horns from the animals that are harvested or that die of natural causes?” The South African game warden proudly explained that since international trade in rhinoceros horns was illegal, South Africa did not contribute to international crime by selling these horns. Instead the horns were either destroyed or stored in a warehouse. cockatoos will be sold in the United States in equilibrium? 3,200. How many cockatoos will be permanently removed from the Australian wild? 6,400. ∗ The story behind this problem is based on actual fact, but the numbers we use are just made up for illustration. It would be very interesting to have some good estimates of the actual demand functions and cost functions. NAME 295 Price S (Blue) S (Red) (a) Suppose that all of the rhinoceros horns produced in South Africa are destroyed. Label the axes below and draw world supply and demand curves for rhinoceros horns with blue ink. Label the equilibrium price and quantity. 296 INDUSTRY SUPPLY (Ch. 23) that marijuana is seized is .3 and that the ﬁne if you are caught is $10 per ounce. (a) If the “street price” is $p per ounce, what is the expected revenue net of ﬁnes to a dealer from selling an ounce of marijuana? Pa What then would be the equilibrium price of marijuana? P b .7p − 3. $11.4. (b) Suppose that the demand function for marijuana has the equation Q = A − Bp. If all conﬁscated marijuana is destroyed, what will be the D (Blue) equilibrium consumption of marijuana? A − 11.4B. Suppose that conﬁscated marijuana is not destroyed but sold on the open market. What will be the equilibrium consumption of marijuana? Q a Q b Quantity (b) If South Africa were to sell its rhinoceros horns on the world market, which of the curves in your diagram would shift and in what di- A − 11.4B. (c) The price of marijuana will (increase, decrease, stay the same) Stay the same. rection? Supply curve to the right. Use red ink to illustrate the shifted curve or curves. If South Africa were to do this, (d) If there were increasing rather than constant marginal cost in marijuana production, do you think that consumption would be greater if conﬁscated marijuana were sold than if it were destroyed? Ex- would world consumption of rhinoceros horns be increased or decreased? plain. Increased. the supply curve will shift to the right, or decreased? Would the world price of rhinoceros horns be increased Decreased. ing be increased or decreased? Would the amount of rhinoceros poach- Decreased. 23.13 (1) The sale of rhinoceros horns is not prohibited because of concern about the wicked pleasures of aphrodisiac imbibers, but because the supply activity is bad for rhinoceroses. Similarly, the Australian reason for restricting the exportation of cockatoos to the United States is not because having a cockatoo is bad for you. Indeed it is legal for Australians to have cockatoos as pets. The motive for the restriction is simply to protect the wild populations from being overexploited. In the case of other commodities, it appears that society has no particular interest in restricting the supply activities but wishes to restrict consumption. A good example is illicit drugs. The growing of marijuana, for example, is a simple pastoral activity, which in itself is no more harmful than growing sweet corn or brussels sprouts. It is the consumption of marijuana to which society objects. Suppose that there is a constant marginal cost of $5 per ounce for growing marijuana and delivering it to buyers. But whenever the marijuana authorities ﬁnd marijuana growing or in the hands of dealers, they seize the marijuana and ﬁne the supplier. Suppose that the probability Consumption will increase because lowering the price. 298 Chapter 24 MONOPOLY (Ch. 24) NAME Cents Monopoly 100 75 Introduction. The proﬁt-maximizing output of a monopolist is found by 50 solving for the output at which marginal revenue is equal to marginal cost. Having solved for this output, you ﬁnd the monopolist’s price by plugging the proﬁt-maximizing output into the demand function. In general, the marginal revenue function can be found by taking the derivative of the total revenue function with respect to the quantity. But in the special case of linear demand, it is easy to ﬁnd the marginal revenue curve graphically. With a linear inverse demand curve, p(y) = a − by, the marginal revenue curve always takes the form M R(y) = a − 2by. 24.1 (0) Professor Bong has just written the ﬁrst textbook in Punk Economics. It is called Up Your Isoquant. Market research suggests that the demand curve for this book will be Q = 2, 000 − 100P , where P is its price. It will cost $1,000 to set the book in type. This setup cost is necessary before any copies can be printed. In addition to the setup cost, there is a marginal cost of $4 per book for every book printed. (a) The total revenue function for Professor Bong’s book is R(Q) = 25 0 Blue line Black lines Red line 25 50 75 100 125 150 Pigeon pies (b) What level of output will maximize Peter’s proﬁts? price will Peter charge per pie? What (c) After Peter had been in business for several months, he noticed that the demand curve had shifted to p(y) = 75 − y/2. Use blue ink to plot this curve in the graph above. Plot the new marginal revenue curve on the same graph with black ink. 20Q − Q2 /100. (d) What is his proﬁt-maximizing output at this new price? (b) The total cost function for producing Professor Bong’s book is C(Q) = is the new proﬁt-maximizing price? 75. What 37.5 cents per pie. 24.3 (0) Suppose that the demand function for Japanese cars in the United States is such that annual sales of cars (in thousands of cars) will be 250− 2P , where P is the price of Japanese cars in thousands of dollars. 1, 000 + 4Q. (c) The marginal revenue function is M R(Q) = 4. the marginal cost function is M C(Q) = 20 − Q/50 and The proﬁt-maximizing quantity of books for professor Bong to sell is Q∗ = (a) If the supply schedule is horizontal at a price of $5,000 what will be the equilibrium number of Japanese cars sold in the United States? 240 800. thousand. How much money will Americans spend in total on Japanese cars? 24.2 (0) Peter Morgan sells pigeon pies from a pushcart in Central Park. Morgan is the only supplier of this delicacy in Central Park. His costs are zero due to the abundant supplies of raw materials available in the park. (a) When he ﬁrst started his business, the inverse demand curve for pigeon pies was p(y) = 100 − y, where the price is measured in cents and y measures the number of pies sold. Use black ink to plot this curve in the graph below. On the same graph, use red ink to plot the marginal revenue curve. NAME 299 (c) How much revenue will the U.S. government collect with this tariﬀ? 472 50. 50 cents. 1.2 billion dollars. (b) Suppose that in response to pressure from American car manufacturers, the United States imposes an import duty on Japanese cars in such a way that for every car exported to the United States the Japanese manufacturers must pay a tax to the U.S. government of $2,000. How many 236 Japanese automobiles will now be sold in the United States? 7 thousand. At what price will they be sold? 300 MONOPOLY thousand dollars. (Ch. 24) (h) How much money will Americans spend on Japanese cars? billion dollars. 1.652 million dollars. (d) On the graph below, the price paid by American consumers is measured on the vertical axis. Use blue ink to show the demand and supply schedules before the import duty is imposed. After the import duty is imposed, the supply schedule shifts and the demand schedule stays as before. Use red ink to draw the new supply schedule. (i) Why might the Japanese “voluntarily” submit to export controls? Total revenue of Japanese companies and government is greater with export controls Price (thousands) than without them. 8 output, costs are lower. 7 Supply with duty 5 24.4 (0) A monopolist has an inverse demand curve given by p(y) = 12 − y and a cost curve given by c(y) = y 2 . Supply 4 3. (a) What will be its proﬁt-maximizing level of output? 2 Demand 0 Higher revenue, lower costs imply more profit. Blue lines Red line 6 Since there is less 50 100 150 200 250 300 Japanese autos (thousands) (e) Suppose that instead of imposing an import duty, the U.S. government persuades the Japanese government to impose “voluntary export restrictions” on their exports of cars to the United States. Suppose that the Japanese agree to restrain their exports by requiring that every car exported to the United States must have an export license. Suppose further that the Japanese government agrees to issue only 236,000 export licenses and sells these licenses to the Japanese ﬁrms. If the Japanese ﬁrms know the American demand curve and if they know that only 236,000 Japanese cars will be sold in America, what price will they be able to charge in America for their cars? 7 thousand dollars. (f ) How much will a Japanese ﬁrm be willing to pay the Japanese government for an export license? 2 thousand dollars. (Hint: Think about what it costs to produce a car and how much it can be sold for if you have an export license.) (g) How much will be the Japanese government’s total revenue from the sale of export licenses? 472 million dollars. (b) Suppose the government decides to put a tax on this monopolist so that for each unit it sells it has to pay the government $2. What will be its output under this form of taxation? 2.5. (c) Suppose now that the government puts a lump sum tax of $10 on the proﬁts of the monopolist. What will be its output? 3. 24.5 (1) In Gomorrah, New Jersey, there is only one newspaper, the Daily Calumny. The demand for the paper depends on the price and the amount of scandal reported. The demand function is Q = 15S 1/2 P −3 , where Q is the number of issues sold per day, S is the number of column inches of scandal reported in the paper, and P is the price. Scandals are not a scarce commodity in Gomorrah. However, it takes resources to write, edit, and print stories of scandal. The cost of reporting S units of scandal is $10S. These costs are independent of the number of papers sold. In addition it costs money to print and deliver the paper. These cost $.10 per copy and the cost per unit is independent of the amount of scandal reported in the paper. Therefore the total cost of printing Q copies of the paper with S column inches of scandal is $10S + .10Q. (a) Calculate the price elasticity of demand for the Daily Calumny. −3. ported? Does the price elasticity depend on the amount of scandal re- No. Is the price elasticity constant over all prices? Yes. NAME 301 302 (b) Remember that M R = P (1 + 1" ). To maximize proﬁts, the Daily Calumny will set marginal revenue equal to marginal cost. Solve for MONOPOLY (Ch. 24) (c) The monopolist would have (larger, smaller) proﬁts at the new demand curve than it had at the original demand curve. Larger. the proﬁt-maximizing price for the Calumny to charge per newspaper. Price $.15. When the newspaper charges this price, the diﬀerence between the price and the marginal cost of printing and delivering each newspaper is 200 $.05. Black Line 150 (c) If the Daily Calumny charges the proﬁt-maximizing price and prints 100 column inches of scandal, how many copies would it sell? (Round 44,444. to the nearest integer.) for the number of copies sold as a function of S: Q(S) = Red Line 100 Write a general expression Q = 50 15S 1/2 (.15)−3 = 4, 444.44S 1/2 . (d) Assuming that the paper charges the proﬁt-maximizing price, write an expression for proﬁts as a function of Q and S. Profits= 0 50 100 150 .15Q − .10Q − 10S. 200 Quantity Using the solution for Q(S) that you found in the last section, substitute Q(S) for Q to write an expression for proﬁts as a function of S alone. 10S = 222.22S 1/2 Profits =.05(4, 444.44S 1/2 ) − − 10S. (e) If the Daily Calumny charges its proﬁt-maximizing price, and prints the proﬁt-maximizing amount of scandal, how many column inches of scandal should it print? are sold 49,383 123.456 inches. How many copies and what is the amount of proﬁt for the Daily Calumny if it maximizes its proﬁts? 1,234.5. 24.6 (0) In the graph below, use black ink to draw the inverse demand curve, p1 (y) = 200 − y. (a) If the monopolist has zero costs, where on this curve will it choose to operate? At y = 100, p = 100. (b) Now draw another demand curve that passes through the proﬁtmaximizing point and is ﬂatter than the original demand curve. Use a red pen to mark the part of this new demand curve on which the monopolist would choose to operate. (Hint: Remember the idea of revealed preference?) 304 Chapter 25 NAME to maximize its proﬁts Monopoly Behavior $13. How much will those proﬁts be? (b) If McSwill can charge a diﬀerent price in each country and wants to maximize proﬁts, how many copies should it sell in the United States? 23,000. discrimination by monopolists. There are also problems related to spatial markets, where transportation costs are accounted for and we show that lessons learned about spatial models give us a useful way of thinking about competition under product diﬀerentiation in economics and in politics. Remember that a price discriminator wants the marginal revenue in each market to be equal to the marginal cost of production. Since he produces all of his output in one place, his marginal cost of production is the same for both markets and depends on his total output. The trick for solving these problems is to write marginal revenue in each market as a function of quantity sold in that market and to write marginal cost as a function of the sum of quantities sold in the two markets. The proﬁtmaximizing conditions then become two equations that you can solve for the two unknown quantities sold in the two markets. Of course, if marginal cost is constant, your job is even easier, since all you have to do is ﬁnd the quantities in each market for which marginal revenue equals the constant marginal cost. $13.50. 25.1 (0) Ferdinand Sludge has just written a disgusting new book, Orgy in the Piggery. His publisher, Graw McSwill, estimates that the demand for this book in the United States is Q1 = 50, 000 − 2, 000P1 , where P1 is the price in the U.S. measured in U.S. dollars. The demand for Sludge’s opus in England is Q2 = 10, 000 − 500P2 , where P2 is its price in England measured in U. S. dollars. His publisher has a cost function C(Q) = $50, 000 + $2Q, where Q is the total number of copies of Orgy that it produces. (a) If McSwill must charge the same price in both countries, how many What price should it charge What price should it charge in the United States? How many copies should it sell in England? What price should it charge in England? total proﬁts be? $11. 4,500. How much will its $255,000. 25.2 (0) A monopoly faces an inverse demand curve, p(y) = 100 − 2y, and has constant marginal costs of 20. (a) What is its proﬁt-maximizing level of output? (b) What is its proﬁt-maximizing price? Example: A monopolist sells in two markets. The inverse demand curve in market 1 is p1 = 200 − q1 . The inverse demand curve in market 2 is p2 = 300−q2 . The ﬁrm’s total cost function is C(q1 +q2 ) = (q1 +q2 )2 . The ﬁrm is able to price discriminate between the two markets. Let us ﬁnd the prices that it will charge in each market. In market 1, the ﬁrm’s marginal revenue is 200 − 2q1 . In market 2, marginal revenue is 300 − 2q2 . The ﬁrm’s marginal costs are 2(q1 + q2 ). To maximize its proﬁts, the ﬁrm sets marginal revenue in each market equal to marginal cost. This gives us the two equations 200 − 2q1 = 2(q1 + q2 ) and 300 − 2q2 = 2(q1 + q2 ). Solving these two equations in two unknowns for q1 and q2 , we ﬁnd q1 = 16.67 and q2 = 66.67. We can ﬁnd the price charged in each market by plugging these quantities into the demand functions. The price charged in market 1 will be 183.33. The price charged in market 2 will be 233.33. 27,500. (Ch. 25) $252,500. Introduction. Problems in this chapter explore the possibilities of price copies should it sell? MONOPOLY BEHAVIOR 20. $60. (c) What is the socially optimal price for this ﬁrm? $20. (d) What is the socially optimal level of output for this ﬁrm? 40. (e) What is the deadweight loss due to the monopolistic behavior of this ﬁrm? 400. (f ) Suppose this monopolist could operate as a perfectly discriminating monopolist and sell each unit of output at the highest price it would fetch. The deadweight loss in this case would be Calculus 0. 25.3 (1) Banana Computer Company sells Banana computers in both the domestic and foreign markets. Because of diﬀerences in the power supplies, a Banana purchased in one market cannot be used in the other market. The demand and marginal revenue curves associated with the two markets are as follows: Pd = 20, 000 − 20Q M Rd = 20, 000 − 40Q Pf = 25, 000 − 50Q M Rf = 25, 000 − 100Q. Banana’s production process exhibits constant returns to scale and it takes $1,000,000 to produce 100 computers. NAME 305 (a) Banana’s long-run average cost function is AC(Q) = $10,000 and its long-run marginal cost function is M C(Q) = $10,000. (Hint: If there are constant returns to scale, does long-run average cost change as output changes?) Draw the average and marginal cost curves on the graph. (b) Draw the demand curve for the domestic market in black ink and the marginal revenue curve for the domestic market in pencil. Draw the demand curve for the foreign market in red ink and the marginal revenue curve for the foreign market in blue ink. 306 MONOPOLY BEHAVIOR (Ch. 25) (d) At the proﬁt-maximizing price and quantity, what is the price elas- −3. ticity of demand in the domestic market? elasticity of demand in the foreign market? What is the price −2.33. Is demand more or less elastic in the market where the higher price is charged? Less elastic. (e) Suppose that somebody ﬁgures out a wiring trick that allows a Banana computer built for either market to be costlessly converted to work in the other. (Ignore transportation costs.) On the graph below, draw the new inverse demand curve (with blue ink) and marginal revenue curve (with black ink) facing Banana. Dollars (1,000s) 60 Dollars (1,000s) 40 50 30 40 Blue line 20 30 10 LRAC LRMC Black line 20 Black line LRAC LRMC 10 0 100 200 300 400 500 600 Red line Blue line 0 Pencil line 100 200 300 500 600 700 800 should Banana sell? 400. What price will it charge? $15,714. How will Banana’s proﬁts change now that it can no longer practice price discrimination? (c) If Banana is maximizing its proﬁts, it will sell 15,000 in the foreign market at total proﬁts? 250 dollars each and 17,500 computers in 150 computers dollars each. What are Banana’s $2,375,000. NAME 307 30. 30. (a) What is its proﬁt-maximizing level of output? 308 MONOPOLY BEHAVIOR 110. (Ch. 25) What price will be charged to nonstudents? Grand Theater make? What price $90. many nonstudent tickets will be sold? (c) If you wanted to choose a price ceiling for this monopolist so as to maximize consumer plus producer surplus, what price ceiling should you choose? Decrease by $89,284. 25.4 (0) A monopolist has a cost function given by c(y) = y 2 and faces a demand curve given by P (y) = 120 − y. will the monopolist charge? (b) If you put a lump sum tax of $100 on this monopolist, what would its output be? 800 (f ) Given that costs haven’t changed, how many Banana computers 400 Banana Computers the domestic market at 700 Banana Computers 70. $3.50. How How much proﬁt will the $47.50. $80. (d) How much output will the monopolist produce at this price ceiling? 40. (e) Suppose that you put a speciﬁc tax on the monopolist of $20 per unit output. What would its proﬁt-maximizing level of output be? 25. 25.5 (1) The Grand Theater is a movie house in a medium-sized college town. This theater shows unusual ﬁlms and treats early-arriving movie goers to live organ music and Bugs Bunny cartoons. If the theater is open, the owners have to pay a ﬁxed nightly amount of $500 for ﬁlms, ushers, and so on, regardless of how many people come to the movie. For simplicity, assume that if the theater is closed, its costs are zero. The nightly demand for Grand Theater movies by students is QS = 220−40PS , where QS is the number of movie tickets demanded by students at price PS . The nightly demand for nonstudent moviegoers is QN = 140 − 20PN . (a) If the Grand Theater charges a single price, PT , to everybody, then at prices between 0 and $5.50, the aggregate demand function for movie tickets is QT (PT ) = 360 − 60PT . (d) If you know calculus, see if you can do this part. Suppose that the Grand Theater can hold only 150 people and that the manager wants to maximize proﬁts by charging separate prices to students and to nonstudents. If the capacity of the theater is 150 seats and QS tickets are sold to students, what is the maximum number of tickets that can be sold to nonstudents? QN = 150 − QS . Write an expression for the price of nonstudent tickets as a function of the number of student tickets sold. (Hint: First ﬁnd the inverse nonstudent demand function.) PN = −1/2 + QS /20. Write an expression for Grand Theater proﬁts as a function of the number QS only. (Hint: Make substitutions using your previous answers.) QS (11/2 − QS /40) + (−1/2 + 20/QS )(150 − QS ) − 500 = −3Q2S /40 + 27QS /2 − 575. tickets should the Grand sell to maximize proﬁts? How many student 90. What price Over this range of prices, the inverse demand function is then PT (QT ) = 6 − QT /60. is charged to students? sold? 60. $3.25. How many nonstudent tickets are What price is charged to nonstudents? $4. How much (b) What is the proﬁt-maximizing number of tickets for the Grand Theater to sell if it charges one price to everybody? would this number of tickets be sold? the Grand make? 100. $40. To nonstudents? $3. 180. proﬁt does the Grand make under this arrangement? At what price How much proﬁts would How many tickets would be sold to students? 80. (c) Suppose that the cashier can accurately separate the students from the nonstudents at the door by making students show their school ID cards. Students cannot resell their tickets and nonstudents do not have access to student ID cards. Then the Grand can increase its proﬁts by charging students and nonstudents diﬀerent prices. What price will be charged to students? $32.50. $2.75. How many student tickets will be sold? 25.6 (2) The Mall Street Journal is considering oﬀering a new service which will send news articles to readers by email. Their market research indicates that there are two types of potential users, impecunious students and high-level executives. Let x be the number of articles that a user requests per year. The executives have an inverse demand function PE (x) = 100 − x and the students have an inverse demand function PU (x) = 80 − x. (Prices are measured in cents.) The Journal has a zero marginal cost of sending articles via email. Draw these demand functions in the graph below and label them. NAME 309 310 MONOPOLY BEHAVIOR (Ch. 25) (d) What is the the maximum price that the Journal can charge for 100 articles per year if it wants executives to prefer this deal to buying 80 articles a year at the highest price the undergraduates are willing to pay Price 120 for 80 articles? 100 Solve 50 − p = 48 − 32 to find p = $34. 80 (e) Suppose that the Mall Street Journal decides to include only 60 articles in the student package. What is the most it could charge and still P (X) = 100 - X E 60 get student to buy this package? 40 20 P (X) = 80 - X U buying the student package? 20 0 40 60 80 100 120 student package? (a) Suppose that the Journal can identify which users are students and which are executives. It oﬀers each type of user a diﬀerent all or nothing deal. A student can either buy access to 80 articles per year or to none at all. What is the maximum price a student will be willing to pay for $32, which equals the area under the student’s demand curve. (Hint: Recall the lesson on consumer’s surplus and the area under the demand curve.) An executive can either buy access to 100 articles per year or to none at all. What is the maximum price an executive would be willing to pay for $50 which is the area under access to 100 articles? an executive’s demand curve. (b) Suppose that the Journal can’t tell which users are executives and which are undergraduates. Thus it can’t be sure that executives wouldn’t buy the student package if they found it to be a better deal for them. In this case, the Journal can still oﬀer two packages, but it will have to let the users self-select the one that is optimal for them. Suppose that it oﬀers two packages: one that allows up to 80 articles per year the other that allows up to 100 articles per year. What’s the highest price that the undergraduates will pay for the 80-article subscription? $32. (c) What is the total value to the executives of reading 80 articles per year? $48. (Hint: Look at the area under their demand curve and to the right of a vertical line at 80 articles.) NAME 311 (a) Let us assume that MightySoft also oﬀers the products separately as well as bundled. In order to determine how to price the bundle, Bill Barriers asks himself the following questions. In order to sell the bundle to the wordprocessor purchasers, the price would have to be less than 200 + 120 = 320. (b) In order to sell the bundle to the spreadsheet users, the price would have to be less than 250 + 100 = 350. $38. (h) If the number of executives in the population equals the number of students, would the Mall Street Journal make higher proﬁts by oﬀering a student package of 80 articles or a student package of 60 articles? Everyone buys the bundle so 60. 25.7 (2) Bill Barriers, CEO of MightySoft software, is contemplating a new marketing strategy: bundling their best-selling wordprocessor and their spreadsheet together and selling the pair of software products for one price. From the viewpoint of the company, bundling software and selling it at a discounted price has two eﬀects on sales: (1) revenues go up due to to additional sales of the bundle; and (2) revenues go down since there is less of a demand for the individual components of the bundle. The proﬁtability of bundling depends on which of these two eﬀects dominates. Suppose that MightySoft sells the wordprocessor for $200 and the spreadsheet for $250. A marketing survey of 100 people who purchased either of these packages in the last year turned up the following facts: 1) 20 people bought both. 2) 40 people bought only the wordprocessor. They would be willing to spend up to $120 more for the spreadsheet. 3) 40 people bought only the spreadsheet. They would be willing to spend up to $100 more for the wordprocessor. In answering the following questions you may assume the following: 1) New purchasers of MightySoft products will have the same characteristics as this group. 2) There is a zero marginal cost to producing extra copies of either software package. 3) There is a zero marginal cost to creating a bundle. 312 MONOPOLY BEHAVIOR (Ch. 25) (h) Is it more proﬁtable to bundle or not bundle? Bundle. (i) Suppose that MightySoft worries about the reliability of their market survey and decides that they believe that without bundling t of the 100 people will buy both products, and (100−t)/2 will buy the wordprocessor only and (100 − t)/2 will buy the spreadsheet only. Calculate proﬁts as a function of t if there is no bundling. 225 × (100 − t) + 450 × t. (j) What are proﬁts with the bundle? (c) What would MightySoft’s proﬁts be on a group of 100 users if it priced the bundle at $320? $12. (g) What is the most that the Mall Street Journal could charge for a 100article package and expect executives to buy this package rather than the Quantity access to 80 articles? $30. (f ) If the Mall Street Journal oﬀers a “student package” of 60 articles at this price, how much net consumer surplus would executives get from $32,000. (k) At what values of t would it be unproﬁtable to oﬀer the bundle? Solve for the t that equates the two profits are 100 × 320 = $32, 000. profits to find t = 42.22. So if more (d) What would MightySoft’s proﬁts be on a group of 100 users if 42 of the 100 new purchasers would buy both it priced the bundle at $350? both anyway, 20 people would buy 40 people would buy the spreadsheet only and would be willing to buy the bundle, 40 people would buy the wordprocessor, but not the spreadsheet. Total profits would be 20 × 350 + 40 × 350 + 40 × 200 = 29, 000. products anyway, it $320 (l) This analysis so far has been concerned only with customers who would purchase at least one of the programs at the original set of prices. Is there any additional source of demand for the bundle? What does this say about the calculations we have made about the proﬁtability of bundling? Yes, it may be that wordprocessor or $250 for the spreadsheet, but would be willing to bundling would be even (f ) What would proﬁts be without oﬀering the bundle? Without the bundle, profits would be 20×(200+250)+ 40 ∗ 200 + 40 ∗ 250 = 27, 000. 32, 000 100 × 320 = there are some consumers who were not willing to pay pay $320 for the bundle. is the more profitable price. (g) What would be the proﬁts with the bundle? is not profitable to bundle them. $200 for the (e) If MightySoft oﬀers the bundle, what price should it set? than This means that more profitable than the calculations above indicate. 25.8 (0) Colonel Tom Barker is about to open his newest amusement park, Elvis World. Elvis World features a number of exciting attractions: you can ride the rapids in the Blue Suede Chutes, climb the Jailhouse Rock and eat dinner in the Heartburn Hotel. Colonel Tom ﬁgures that Elvis World will attract 1,000 people per day, and each person will take x = 50 − 50p rides, where p is the price of a ride. Everyone who visits Elvis World is pretty much the same and negative rides are not allowed. The marginal cost of a ride is essentially zero. NAME 313 (a) What is each person’s inverse demand function for rides? p(x) = (b) If Colonel Tom sets the price to maximize proﬁt, how many rides will 25. be taken per day by a typical visitor? (c) What will the price of a ride be? (d) Suppose that the bowling alleys at the end of town have a marginal cost of $3 per customer and set their prices to maximize proﬁts. (For the time being assume that these bowling alleys face no competition from $12.50 the other bowling alleys in town.) How many customers will they have? Zero. (f ) If Colonel Tom charged the Pareto eﬃcient price for a ride, how many rides would be purchased? 100 × (15 − p). p = 15 − q/100. 50 cents. (e) What is the Pareto eﬃcient price of a ride? 600. What price will they charge? $9. How far away from the edge of town does their most distant customer live? 50. (g) How much consumers’ surplus would be generated at this price and 25. 6 miles. (e) Now consider the bowling alley in the center of town. If it charges a price of $p, how many customers will it have per week? (h) If Colonel Tom decided to use a two-part tariﬀ, he would set an $25 and charge a price per ride of 0. 2*(15-p). (f ) If the bowling alley in the center of town also has marginal costs of $3 per customer and maximizes its proﬁts, what price will it charge? 25.9 (1) The city of String Valley is squeezed between two mountains and is 36 miles long, running from north to south, and only about 1 block wide. Within the town, the population has a uniform density of 100 people per mile. Because of the rocky terrain, nobody lives outside the city limits on either the north or the south edge of town. Because of strict zoning regulations, the city has only three bowling alleys. One of these is located at the city limits on the north edge of town, one of them is located at the city limits on the south edge of town, and one is located at the exact center of town. Travel costs including time and gasoline are $1 per mile. All of the citizens of the town have the same preferences. They are willing to bowl once a week if the cost of bowling including travel costs and the price charged by the bowling alley does not exceed $15. (a) Consider one of the bowling alleys at either edge of town. If it charges $10 for a night of bowling, how far will a citizen of String Valley be willing to travel to bowl there? Up to 5 miles. How many customers would this bowling alley have per week if it charged $10 per night of bowling? (Ch. 25) (c) Write a formula for this bowling alley’s inverse demand function. (d) What will Colonel Tom’s proﬁts be per person? admission fee of MONOPOLY BEHAVIOR (b) Write a formula for the number of customers that a bowling alley at the edge of town will have if it charges $p per night of bowling. 1 − x/50. quantity? 314 $9. How many customers will it have per week? 1,200. How far away from the center of town will its most distant customers live? 6 miles. (g) Suppose that the city relaxes its zoning restrictions on where the bowling alleys can locate, but continues to issue operating licenses to only 3 bowling alleys. Both of the bowling alleys at the end of town are about to lose their leases and can locate anywhere in town that they like at about the same cost. The bowling alley in the center of town is committed to stay where it is. Would either of the alleys at the edge of town improve its proﬁts by locating next to the existing bowling alley in the center of town? No. What would be a proﬁt-maximizing location for each of these two bowling alleys? One would be located 12 miles north of the town center and one 500. 12 miles south. NAME 315 25.10 (1) In a congressional district somewhere in the U.S. West a new representative is being elected. The voters all have one-dimensional political views that can be neatly arrayed on a left-right spectrum. We can deﬁne the “location” of a citizen’s political views in the following way. The citizen with the most extreme left-wing views is said to be at point 0 and the citizen with the most extreme right-wing views is said to be at point 1. If a citizen has views that are to the right of the views of the fraction x of the state’s population, that citizen’s views are said to be located at the point x. Candidates for oﬃce are forced to publically state their own political position on the zero-one left-right scale. Voters always vote for the candidate whose stated position is nearest to their own views. (If there is a tie for nearest candidate, voters ﬂip a coin to decide which to vote for.) (a) There are two candidates for the congressional seat. Suppose that each candidate cares only about getting as many votes as possible. Is there an equilibrium in which each candidate chooses the best position given the position of the other candidate? If so, describe this equilibrium. The only equilibrium is one in which both candidates choose the same position, and that position is at the point 1/2. 25.11 (2) In the congressional district described by the previous problem, let us investigate what will happen if the two candidates do not care about the number of votes that they get but only about the amount of campaign contributions that they receive. Therefore each candidate chooses his ideological location in such a way as to maximize the amount of campaign contributions he receives, given the position of the other.* Let us deﬁne a left-wing extremist as a voter whose political views lie to the left of the leftmost candidate, a right-wing extremist as a voter whose political views lie to the right of the rightmost candidate, and a moderate voter as one whose political views lie between the positions of the two candidates. Assume that each extremist voter contributes to the candidate whose position is closest to his or her own views and that moderate voters make no campaign contributions. The number of dollars that an extremist voter contributes to his or her favorite candidate is proportional to the distance between the two candidates. Speciﬁcally, we assume that there is some constant C such that if the left-wing candidate is located at x and the right-wing candidate is located at y, then total campaign contributions received by the left-wing candidate will be $Cx(y − x) and total campaign contributions received by the right-wing candidate will be $C(1 − y)(y − x). * This assumption is a bit extreme. Candidates typically spend at least some of their campaign contributions on advertising for votes, and this advertising aﬀects the voting outcomes. 316 MONOPOLY BEHAVIOR (Ch. 25) (a) If the right-wing candidate is located at y, the contributionmaximizing position for the left-wing candidate is x = y/2 If the left-wing candidate is located at x, the contribution-maximizing position for the right-wing candidate is y = derivative and set it equal to zero.) ((1 + x)/2 . (Hint: Take a (b) Solve for the unique pair of ideological positions for the two candidates such that each takes the position that maximizes his campaign contributions given the position of the other. x = 1/3, y = 2/3 (c) Suppose that in addition to collecting contributions from extremists on their side, candidates can also collect campaign contributions from moderates whose views are closer to their position than to that of their rivals’ position. Suppose that moderates, like extremists, contribute to their preferred candidate and that they contribute in proportion to the diﬀerence between their own ideological distance from their less-preferred candidate and their ideological distance from their more-preferred candidate. Show that in this case the unique positions in which the leftand right-wing candidates are each maximizing their campaign contributions, given the position of the other candidate, occurs where x = 1/4 and y = 3/4. Total contributions received by the left wing candidate will be C x(y − x) + (y − x)2 /4 . Total contributions received by the right-wing candidate will be C (1 − y)(y − x) + (y − x)2 /4 . Differentiating the former expression with respect to x and the latter with respect to y and solving the resulting simultaneous equations yields x = 1/4 and y = 3/4. 318 Chapter 26 FACTOR MARKETS (Ch. 26) If the industry were competitive, then the industry would employ the factor up to the point where the value of the marginal product was equal to 4. This gives us the equation NAME Factor Markets p2 = 4, Introduction. In this chapter you will examine the factor demand decision of a monopolist. If a ﬁrm is a monopolist in some industry, it will produce less output than if the industry were competitively organized. Therefore it will in general want to use less inputs than does a competitive ﬁrm. The value marginal product is just the value of the extra output produced by hiring an extra unit of the factor. The ordinary logic of competitive proﬁt maximization implies that a competitive ﬁrm will hire a factor up until the point where the value marginal product equals the price of the factor. The marginal revenue product is the extra revenue produced by hiring an extra unit of a factor. For a competitive ﬁrm, the marginal revenue product is the same as the value of the marginal product, but they diﬀer for monopolist. A monopolist has to take account of the fact that increasing its production will force the price down, so the marginal revenue product of an extra unit of a factor will be less than the value marginal product. Another thing we study in this chapters is monopsony, which is the case of a market dominated by a single buyer of some good. The case of monopsony is very similar to the case of a monopoly: The monopsonist hires less of a factor than a similar competitive ﬁrm because the monopsony recognizes that the price it has to pay for the factor depends on how much it buys. Finally, we consider an interesting example of factor supply, in which a monopolist produces a good that is used by another monopolist. Example: Suppose a monopolist faces a demand curve for output of the form p(y) = 100 − 2y. The production function takes the simple form y = 2x, and the factor costs $4 per unit. How much of the factor of production will the monopolist want to employ? How much of the factor would a competitive industry employ if all the ﬁrms in the industry had the same production function? Answer: The monopolist will employ the factor up to the point where the marginal revenue product equals the price of the factor. Revenue as a function of output is R(y) = p(y)y = (100 − 2y)y. To ﬁnd revenue as a function of the input, we substitute y = 2x: R(x) = (100 − 4x)2x = (200 − 8x)x. The marginal revenue product function will have the form M RPx = 200− 16x. Setting marginal revenue product equal to factor price gives us the equation 200 − 16x = 4. Solving this equation gives us x∗ = 12.25. NAME 319 (b) What is the equilibrium quantity sold by Bubble Up? ∗ D(p ) = 1,000−c . 2 so p∗ = 2. How much output would be demanded at this price? We plug this into the demand function to get the equation 2 = 100 − 2y, which implies y ∗ = 49. Since the production function is y = 2x, we can solve for x∗ = y ∗ /2 = 24.5. 26.1 (0) Gargantuan Enterprises has a monopoly in the production of antimacassars. Its factory is located in the town of Pantagruel. There is no other industry in Pantagruel, and the labor supply equation there is W = 10 + .1L, where W is the daily wage and L is the number of persondays of work performed. Antimacassars are produced with a production function, Q = 10L, where L is daily labor supply and Q is daily output. Q The demand curve for antimacassars is P = 41 − 1,000 , where P is the price and Q is the number of sales per day. (a) Find the proﬁt-maximizing output for Gargantuan. (Hint: Use the production function to ﬁnd the labor input requirements for any level of output. Make substitutions so you can write the ﬁrm’s total costs as a function of its output and then its proﬁt as a function of output. Solve for the proﬁt–maximizing output.) 1,000. (b) How much labor does it use? it pays? 10,000. (c) What is the price of antimacassars? made? What is the wage rate that $110. $31. How much proﬁt is $200,000. 26.2 (0) The residents of Seltzer Springs, Michigan, consume bottles of mineral water according to the demand function D(p) = 1, 000 − p. Here D(p) is the demand per year for bottles of mineral water if the price per bottle is p. The sole distributor of mineral water in Seltzer Springs, Bubble Up, purchases mineral water at c per bottle from their supplier Perry Air. Perry Air is the only supplier of mineral water in the area and behaves as a proﬁt-maximizing monopolist. For simplicity we suppose that it has zero costs of production. (a) What is the equilibrium price charged by the distributor Bubble Up? p∗ = 1,000+c . 2 320 FACTOR MARKETS (Ch. 26) (a) What is the ﬁrst-order condition for proﬁt maximization? For future reference, let x∗ be the proﬁt-maximizing amount produced and p∗ be the price at which it sells. (In this part, assume that tapes cannot be copied.) p(x∗ ) + p (x∗ )x∗ = c. (c) What is the equilibrium price charged by the producer Perry Air? ∗ c = 500. (d) What is the equilibrium quantity sold by Perry Air? D(c∗ ) = 250. (e) What are the proﬁts of Bubble Up? πb = (500−250)(750− 500) = 2502 . (f ) What are the proﬁts of Perry Air? πp = 500 · 250. (g) How much consumer’s surplus is generated in this market? CSe = Now a new kind of consumer digital tape recorder becomes widely available that allows the user to make 1 and only 1 copy of a prerecorded digital tape. The copies are a perfect substitute in consumption value for the original prerecorded tape, and there are no barriers to their use or sale. However, everyone can see the diﬀerence between the copies and the orginals and recognizes that the copies cannot be used to make further copies. Blank tapes cost the consumers c per tape, the same price the monopolist pays. (b) All Moosecake fans take advantage of the opportunity to make a single copy of the tape and sell it on the secondary market. How is the price of an original tape related to the price of a copy? Derive the inverse demand curve for original tapes facing UPUR. (Hint: There are two sources of demand for a new tape: the pleasure of listening to it, and the proﬁts from selling a copy.) If UPUR produces x tapes, 2x tapes reach the market, so UPUR can sell for p(2x) + [p(2x) − c]. 2502 /2. a single tape (h) Suppose that this situation is expected to persist forever and that the interest rate is expected to be constant at 10% per year. What is the minimum lump sum payment that Perry Air would need to pay to Bubble first term is the willingness-to-pay for Up to buy it out? 10 × 2502 . The listening; the second term is profit from selling a copy. (i) Suppose that Perry Air does this. What will be the new price and quantity for mineral water? p∗ = 500 and D(p∗ ) = 500. (j) What are the proﬁts of the new merged ﬁrm? πp = 5002 . (k) What is the total amount of consumers’ surplus generated? How does this compare with the previous level of consumers’ surplus? Calculus CSi = (c) Write an expression for UPUR’s proﬁts if it produces x tapes. [p(2x) + p(2x) − c]x − cx = 2p(2x)x − 2cx. (d) Let x∗∗ be the proﬁt-maximizing level of production by UPUR. How does it compare to the former proﬁt-maximizing level of production? From the two profit functions, one sees 5002 /2 > CSe . that 2x∗∗ = x∗ , so 26.3 (0) Upper Peninsula Underground Recordings (UPUR) has a monopoly on the recordings of the famous rock group Moosecake. Moosecake’s music is only provided on digital tape, and blank digital tapes cost them c per tape. There are no other manufacturing or distribution costs. Let p(x) be the inverse demand function for Moosecake’s music as a function of x, the number of tapes sold. (e) How does the price of a copy of a Moosecake tape compare to the price determined in Part (a)? x∗∗ = x∗ /2. The prices are the same. (f ) If p∗∗ is the price of a copy of a Moosecake tape, how much will a new Moosecake tape sell for? 2p∗∗ − c. 322 Chapter 27 NAME Oligopoly Introduction. In this chapter you will solve problems for ﬁrm and industry outcomes when the ﬁrms engage in Cournot competition, Stackelberg competition, and other sorts of oligopoly behavior. In Cournot competition, each ﬁrm chooses its own output to maximize its proﬁts given the output that it expects the other ﬁrm to produce. The industry price depends on the industry output, say, qA + qB , where A and B are the ﬁrms. To maximize proﬁts, ﬁrm A sets its marginal revenue (which depends on the output of ﬁrm A and the expected output of ﬁrm B since the expected industry price depends on the sum of these outputs) equal to its marginal cost. Solving this equation for ﬁrm A’s output as a function of ﬁrm B’s expected output gives you one reaction function; analogous steps give you ﬁrm B’s reaction function. Solve these two equations simultaneously to get the Cournot equilibrium outputs of the two ﬁrms. Example: In Heifer’s Breath, Wisconsin, there are two bakers, Anderson and Carlson. Anderson’s bread tastes just like Carlson’s—nobody can tell the diﬀerence. Anderson has constant marginal costs of $1 per loaf of bread. Carlson has constant marginal costs of $2 per loaf. Fixed costs are zero for both of them. The inverse demand function for bread in Heifer’s Breath is p(q) = 6 − .01q, where q is the total number of loaves sold per day. Let us ﬁnd Anderson’s Cournot reaction function. If Carlson bakes qC loaves, then if Anderson bakes qA loaves, total output will be qA + qC and price will be 6 − .01(qA + qC ). For Anderson, the total cost of producing qA units of bread is just qA , so his proﬁts are pqA − qA = (6 − .01qA − .01qC )qA − qA 2 = 6qA − .01qA − .01qC qA − qA . Therefore if Carlson is going to bake qC units, then Anderson will choose 2 qA to maximize 6qA − .01qA − .01qC qA − qA . This expression is maximized when 6 − .02qA − .01qC = 1. (You can ﬁnd this out either by setting A’s marginal revenue equal to his marginal cost or directly by setting the derivative of proﬁts with respect to qA equal to zero.) Anderson’s reaction function, RA (qC ) tells us Anderson’s best output if he knows that Carlson is going to bake qC . We solve from the previous equation to ﬁnd RA (qC ) = (5 − .01qC )/.02 = 250 − .5qC . We can ﬁnd Carlson’s reaction function in the same way. If Carlson knows that Anderson is going to produce qA units, then Carlson’s proﬁts will be p(qA + qC ) − 2qC = (6 − .01qA − .01qC )qC − 2qC = 6qC − .01qA qC − 2 .01qC −2qC . Carlson’s proﬁts will be maximized if he chooses qC to satisfy the equation 6 − .01qA − .02qC = 2. Therefore Carlson’s reaction function is RC (qA ) = (4 − .01qA )/.02 = 200 − .5qA . NAME 323 and qS is the number that Simon sells. The cost of producing pumpkins for either farmer is $.50 per pumpkin no matter how many pumpkins he produces. (a) The inverse demand function for pumpkins at the Farmers’ Market is p = a − b(qC + qS ), where a = 2 and b = 1/1, 600. marginal cost of producing a pumpkin for either farmer is The $.50. (b) Every spring, each of the farmers decides how many pumpkins to grow. They both know the local demand function and they each know how many pumpkins were sold by the other farmer last year. In fact, each farmer assumes that the other farmer will sell the same number this year as he sold last year. So, for example, if Simon sold 400 pumpkins last year, Carl believes that Simon will sell 400 pumpkins again this year. If Simon sold 400 pumpkins last year, what does Carl think the price of pumpkins will be if Carl sells 1,200 pumpkins this year? 1. If Simon sold qSt−1 pumpkins in year t − 1, then in the spring of year t, Carl t thinks that if he, Carl, sells qC pumpkins this year, the price of pumpkins this year will be 2 − (qSt−1 + qCt )/1, 600. (c) If Simon sold 400 pumpkins last year, Carl believes that if he sells t qC pumpkins this year then the inverse demand function that he faces is t t p = 2 − 400/1, 600 − qC /1, 600 = 1.75 − qC /1, 600. Therefore if Simon sold 400 pumpkins last year, Carl’s marginal revenue this year will be t 1.75 − qC /800. More generally, if Simon sold qSt−1 pumpkins last year, t then Carl believes that if he, himself, sells qC pumpkins this year, his marginal revenue this year will be 2 − qSt−1/1, 600 − qCt /800. (d) Carl believes that Simon will never change the amount of pumpkins that he produces from the amount qSt−1 that he sold last year. Therefore Carl plants enough pumpkins this year so that he can sell the amount that maximizes his proﬁts this year. To maximize this proﬁt, he chooses the output this year that sets his marginal revenue this year equal to his marginal cost. This means that to ﬁnd Carl’s output this year when Simon’s output last year was qSt−1 , Carl solves the following equation. 2 − qSt−1 /1, 600 − qCt /800 = .5. t (e) Carl’s Cournot reaction function, RC (qSt−1 ), is a function that tells us what Carl’s proﬁt-maximizing output this year would be as a function of Simon’s output last year. Use the equation you wrote in the last answer to t−1 t ﬁnd Carl’s reaction function, RC (qSt−1 ) = 1, 200 − qS /2. (Hint: This is a linear expression of the form a − bqSt−1 . You have to ﬁnd the constants a and b.) OLIGOPOLY (Ch. 27) Let us denote the Cournot equilibrium quantities by q̄A and q̄C . The Cournot equilibrium conditions are that q̄A = RA (q̄C ) and q̄C = RC (q̄A ). Solving these two equations in two unknowns we ﬁnd that q̄A = 200 and q̄C = 100. Now we can also solve for the Cournot equilibrium price and for the proﬁts of each baker. The Cournot equilibrium price is 6 − .01(200 + 100) = $3. Then in Cournot equilibrium, Anderson makes a proﬁt of $2 on each of 200 loaves and Carlson makes $1 on each of 100 loaves. In Stackelberg competition, the follower’s proﬁt-maximizing output choice depends on the amount of output that he expects the leader to produce. His reaction function, RF (qL ), is constructed in the same way as for a Cournot competitor. The leader knows the reaction function of the follower and gets to choose her own output, qL , ﬁrst. So the leader knows that the industry price depends on the sum of her own output and the follower’s output, that is, on qL + RF (qL ). Since the industry price can be expressed as a function of qL only, so can the leader’s marginal revenue. So once you get the follower’s reaction function and substitute it into the inverse demand function, you can write down an expression that depends on just qL and that says marginal revenue equals marginal cost for the leader. You can solve this expression for the leader’s Stackelberg output and plug in to the follower’s reaction function to get the follower’s Stackelberg output. Example: Suppose that one of the bakers of Heifer’s Breath plays the role of Stackelberg leader. Perhaps this is because Carlson always gets up an hour earlier than Anderson and has his bread in the oven before Anderson gets started. If Anderson always ﬁnds out how much bread Carlson has in his oven and if Carlson knows that Anderson knows this, then Carlson can act like a Stackelberg leader. Carlson knows that Anderson’s reaction function is RA (qC ) = 250−.5qc . Therefore Carlson knows that if he bakes qC loaves of bread, then the total amount of bread that will be baked in Heifer’s Breath will be qC + RA (qC ) = qC + 250− .5qC = 250+ .5qC . Since Carlson’s production decision determines total production and hence the price of bread, we can write Carlson’s proﬁt simply as a function of his own output. Carlson will choose the quantity that maximizes this proﬁt. If Carlson bakes qC loaves, the price will be p = 6 − .01(250 + .5qC ) = 3.5−.005qC . Then Carlson’s proﬁts will be pqC −2qC = (3.5−.005qC )qC − 2 . His proﬁts are maximized when qC = 150. (Find 2qC = 1.5qC − .005qC this either by setting marginal revenue equal to marginal cost or directly by setting the derivative of proﬁts to zero and solving for qC .) If Carlson produces 150 loaves, then Anderson will produce 250 − .5 × 150 = 175 loaves. The price of bread will be 6 − .01(175 + 150) = 2.75. Carlson will now make $.75 per loaf on each of 150 loaves and Anderson will make $1.75 on each of 175 loaves. 27.1 (0) Carl and Simon are two rival pumpkin growers who sell their pumpkins at the Farmers’ Market in Lake Witchisit, Minnesota. They are the only sellers of pumpkins at the market, where the demand function for pumpkins is q = 3, 200 − 1, 600p. The total number of pumpkins sold at the market is q = qC + qS , where qC is the number that Carl sells 324 OLIGOPOLY (Ch. 27) (f ) Suppose that Simon makes his decisions in the same way that Carl does. Notice that the problem is completely symmetric in the roles played by Carl and Simon. Therefore without even calculating it, we can guess t−1 t−1 that Simon’s reaction function is RSt (qC ) = 1, 200−qC /2. (Of course, if you don’t like to guess, you could work this out by following similar steps to the ones you used to ﬁnd Carl’s reaction function.) (g) Suppose that in year 1, Carl produced 200 pumpkins and Simon produced 1,000 pumpkins. In year 2, how many would Carl produce? 700. How many would Simon produce? many would Carl produce? 650. 1,100. In year 3, how How many would Simon produce? 850. Use a calculator or pen and paper to work out several more terms in this series. To what level of output does Carl’s output appear to be converging? 800 How about Simon’s? 800. (h) Write down two simultaneous equations that could be solved to ﬁnd outputs qS and qC such that, if Carl is producing qC and Simon is producing qS , then they will both want to produce the same amount in the next period. (Hint: Use the reaction functions.) qs = 1, 200 − qC /2 and qC = 1, 200 − qS /2. (i) Solve the two equations you wrote down in the last part for an equilibrium output for each farmer. Each farmer, in Cournot equilibrium, produces 800 units of output. The total amount of pumpkins brought to the Farmers’ Market in Lake Witchisit is pumpkins in that market is make? $1. 1,600. The price of How much proﬁt does each farmer $400. 27.2 (0) Suppose that the pumpkin market in Lake Witchisit is as we described it in the last problem except for one detail. Every spring, the snow thaws oﬀ of Carl’s pumpkin ﬁeld a week before it thaws oﬀ of Simon’s. Therefore Carl can plant his pumpkins one week earlier than Simon can. Now Simon lives just down the road from Carl, and he can tell by looking at Carl’s ﬁelds how many pumpkins Carl planted and how many Carl will harvest in the fall. (Suppose also that Carl will sell every pumpkin that he produces.) Therefore instead of assuming that Carl will sell the same amount of pumpkins that he did last year, Simon sees how many Carl is actually going to sell this year. Simon has this information before he makes his own decision about how many to plant. NAME 325 326 OLIGOPOLY (Ch. 27) t (a) If Carl plants enough pumpkins to yield qC this year, then Simon knows that the proﬁt-maximizing amount to produce this year is qSt = when neither knows the other’s output this Hint: Remember the reaction functions you found in the last problem. year until after planting time is a Cournot 1, 200 − equilibrium, Carl will want Simon to know qCt /2. (b) When Carl plants his pumpkins, he understands how Simon will make his decision. Therefore Carl knows that the amount that Simon will produce this year will be determined by the amount that Carl produces. t In particular, if Carl’s output is qC , then Simon will produce and sell 1, 200 − qCt /2 1, 200 + qCt /2. and the total output of the two producers will be 27.3 (0) Suppose that Carl and Simon sign a marketing agreement. They decide to determine their total output jointly and to each produce the same number of pumpkins. To maximize their joint proﬁts, how many Therefore Carl knows that if his own output is qC , pumpkins should they produce in toto? 1.25 − the price of pumpkins in the market will be qCt /3, 200. (c) In the last part of the problem, you found how the price of pumpkins this year in the Farmers’ Market is related to the number of pumpkins that Carl produces this year. Now write an expression for Carl’s total t revenue in year t as a function of his own output, qC . (qCt )2 /3, 200. 1.25 − qCt /1, 600. (d) Find the proﬁt-maximizing output for Carl. 600. proﬁt-maximizing output for Simon. 1,200. much proﬁt does Carl make? $225. 600. 1,200. How much does each How much proﬁt does each one of them 450. make? 27.4 (0) The inverse market demand curve for bean sprouts is given by P (Y ) = 100 − 2Y , and the total cost function for any ﬁrm in the industry is given by T C(y) = 4y. Find the $7/8. How $−2. The (b) If the bean-sprout industry were perfectly competitive, the industry 48 , and the industry price would be $4. How much proﬁt does Simon (c) Suppose that two Cournot ﬁrms operated in the market. The reaction function for Firm 1 would be y1 = 24 − y2 /2. (Reminder: Unlike the example in your textbook, the marginal cost is not zero here.) The equilibrium. (e) If he wanted to, it would be possible for Carl to delay his planting until the same time that Simon planted so that neither of them would know the other’s plans for this year when he planted. Would it be in Carl’s interest to do this? Explain. (Hint: What are Carl’s proﬁts in the equilibrium above? How do they compare with his prof- No. its in Cournot equilibrium?) $4. change in price for a one-unit increase in output is equal to output would be $450. An equilibrium of the type we discuss here is known Stackleberg (a) The marginal cost for any ﬁrm in the industry is equal to Find the equilibrium price of pumpkins in the Lake Witchisit Farmers’ Market. make? 1.25qCt − one of them produce? Write an expression for Carl’s marginal revenue in t year t as a function of qC . as a his output. Carl’s profits in Stackleberg equilibrium are reaction function of Firm 2 would be y2 = 24 − y1 /2. If the ﬁrms were operating at the Cournot equilibrium point, industry output would be 32 16 , each ﬁrm would produce , and the market price $36. would be larger than (d) For the Cournot case, draw the two reaction curves and indicate the equilibrium point on the graph below. in Cournot equilibrium. So if the output NAME 327 328 OLIGOPOLY (Ch. 27) (a) Write down an expression for proﬁts as a function of q: Π(q) = (20 − .20q)q − 2q. y2 24 Grinch. 18 (b) What price does Grinch get per gallon of mineral water if he produces Firm 2's reaction function e the proﬁt-maximizing quantity? 12 6 6 12 18 24 y1 (e) If the two ﬁrms decided to collude, industry output would be and the market price would equal 24 $52. (f ) Suppose both of the colluding ﬁrms are producing equal amounts of output. If one of the colluding ﬁrms assumes that the other ﬁrm would not react to a change in industry output, what would happen to a ﬁrm’s own proﬁts if it increased its output by one unit? Profits would increase by $22. written as maxy1 [100 − 2(y1 + 24 − y1 /2)]y1 − 4y1 . Solving this problem results in the leader producing an output of and the follower producing and price of 50/3. What is the price in the Cournot equilibrium? $9.33. 27.6 (1) Albatross Airlines has a monopoly on air travel between Peoria and Dubuque. If Albatross makes one trip in each direction per day, the demand schedule for round trips is q = 160 − 2p, where q is the number of passengers per day. (Assume that nobody makes one-way trips.) There is an “overhead” ﬁxed cost of $2,000 per day that is necessary to ﬂy the airplane regardless of the number of passengers. In addition, there is a marginal cost of $10 per passenger. Thus, total daily costs are $2, 000+10q if the plane ﬂies at all. (a) On the graph below, sketch and label the marginal revenue curve, and the average and marginal cost curves. (g) Suppose one ﬁrm acts as a Stackleberg leader and the other ﬁrm behaves as a follower. The maximization problem for the leader can be 36 How much proﬁt does he make? (c) Suppose, now, that Grinch’s neighbor, Grubb ﬁnds a mineral spring that produces mineral water that is just as good as Grinch’s water, but that it costs Grubb $6 a bottle to get his water out of the ground and bottle it. Total market demand for mineral water remains as before. Suppose that Grinch and Grubb each believe that the other’s quantity decision is independent of his own. What is the Cournot equilibrium out- Firm 1's reaction function put for Grubb? output of $11. $405. 0 24 Find the proﬁt-maximizing choice of q for 45. 12. MR, MC 80 60 This implies an industry $28. 27.5 (0) Grinch is the sole owner of a mineral water spring that costlessly burbles forth as much mineral water as Grinch cares to bottle. It costs Grinch $2 per gallon to bottle this water. The inverse demand curve for Grinch’s mineral water is p = $20 − .20q, where p is the price per gallon and q is the number of gallons sold. mr ac 40 20 0 mc 20 40 60 80 Q NAME 329 330 (b) Calculate the proﬁt-maximizing price and quantity and total daily 45 proﬁts for Albatross Airlines. p = 70 , q = OLIGOPOLY (Ch. 27) she herself will sell, q2 , after taking this information into account. The inverse demand function for kangaroos is P (q1 + q2 ) = 2, 000 − 2(q1 + q2 ). It costs $400 to raise a kangaroo to sell. , π = $450 per day. (a) Alex and Anna are Stackelberg competitors. (c) If the interest rate is 10% per year, how much would someone be willing to pay to own Albatross Airlines’s monopoly on the Dubuque-Peoria route. (Assuming that demand and cost conditions remain unchanged forever.) and Anna revenue be if she herself sells q1 kangaroos? (d) If another ﬁrm with the same costs as Albatross Airlines were to enter the Dubuque-Peoria market and if the industry then became a Cournot (c) What is Anna’s reaction function, R(q2 )? would be about $900 per day. (d) Now if Alex sells q2 kangaroos, what is the total number of kangaroos that will be sold? $60 , π = M R(q2 ) = 1, 200 − 2q2 . $900. sell? $45. 80 , and proﬁts? What will (a) What is the supply curve of one of the competitive ﬁrms? is the monopolist’s proﬁt-maximizing price? 500. What p = 5. (d) How much output will the competitive sector provide at this price? What will be the total amount of output sold in ym + yc = 750. 27.9 (0) Consider a market with one large ﬁrm and many small ﬁrms. The supply curve of the small ﬁrms taken together is S(p) = 100 + p. The demand curve for the product is D(p) = 200 − p. The cost function for the one large ﬁrm is c(y) = 25y. 50p. (b) If the monopolist sets a price p, the amount that it can sell is Dm (p) = 1, 000 − 100p. 331 (c) The monopolist’s proﬁt-maximizing output is ym = y = p. The total supply from the competitive sector at price p is S(p) = $400. NAME 50 × 5 = 250. 200. $800. D(p) = 1, 000 − 50p. 27.7 (0) Alex and Anna are the only sellers of kangaroos in Sydney, Australia. Anna chooses her proﬁt-maximizing number of kangaroos to sell, q1 , based on the number of kangaroos that she expects Alex to sell. Alex knows how Anna will react and chooses the number of kangaroos that this industry? How many kangaroos will Anna sell? No. $40 , quantities, How many kangaroos will Alex 27.8 (0) Consider an industry with the following structure. There are 50 ﬁrms that behave in a competitive manner and have identical cost functions given by c(y) = y 2 /2. There is one monopolist that has 0 marginal costs. The demand curve for the product is given by How much would its proﬁts (g) Suppose that AA stuck with one plane and another ﬁrm entered the market with a plane of its own. If the second ﬁrm has the same cost function as the ﬁrst and if the two ﬁrms act as Cournot oligopolists, what will be the price, 400. the industry price be? If AA could prevent entry by another competitor, would it choose to add a second plane? What will be the market price as P (q2 ) = 1, 200 − q2 . (e) What is Alex’s marginal revenue as a function of q2 only? (f ) Let us assume that the overhead costs per plane are constant regardless of the number of planes. If AA added a second plane with the same costs be? 400 + 1/2q2 . a function of q2 only? $2,000. and capacity as the ﬁrst plane, what price would it charge? R(q2 ) = 400 − 1/2q2 . (e) Suppose that the throbbing night life in Peoria and Dubuque becomes widely known and in consequence the population of both places doubles. As a result, the demand for airplane trips between the two places doubles to become q = 320 − 4p. Suppose that the original airplane had a capacity of 80 passengers. If AA must stick with this single plane and if no other airline enters the market, what price should it charge to maximize its output and how much proﬁt would it make? p = M R(q1 + q2 ) = 2, 000 − 4q1 − 2q2 . No; losses duopoly, would the new entrant make a proﬁt? 140. is the leader (b) If Anna expects Alex to sell q2 kangaroos, what will her own marginal About $1.6 million. How many tickets would it sell? Alex is the follower. 332 Calculus OLIGOPOLY (Ch. 27) 27.10 (2) In a remote area of the American Midwest before the railroads arrived, cast iron cookstoves were much desired, but people lived far apart, roads were poor, and heavy stoves were expensive to transport. Stoves could be shipped by river boat to the town of Bouncing Springs, Missouri. Ben Kinmore was the only stove dealer in Bouncing Springs. He could buy as many stoves as he wished for $20 each, delivered to his store. The only farmers who traded in Bouncing Springs lived along a road that ran east and west through town. Along that road, there was one farm every mile and the cost of hauling a stove was $1 per mile. There were no other stove dealers on the road in either direction. The owners of every farm along the road had a reservation price of $120 for a cast iron cookstove. That is, any of them would be willing to pay up to $120 to have a stove rather than to not have one. Nobody had use for more than one stove. Ben Kinmore charged a base price of $p for stoves and added to the price the cost of delivery. For example, if the base price of stoves was $40 and you lived 45 miles west of Bouncing Springs, you would have to pay $85 to get a stove, $40 base price plus a hauling charge of $45. Since the reservation price of every farmer was $120, it follows that if the base price were $40, any farmer who lived within 80 miles of Bouncing Springs would be willing to pay $40 plus the price of delivery to have a cookstove. Therefore at a base price of $40, Ben could sell 80 cookstoves to the farmers living west of him. Similarly, if his base price is $40, he could sell 80 cookstoves to the farmers living within 80 miles to his east, for a total of 160 cookstoves. (a) Suppose that the large ﬁrm is forced to operate at a zero level of output. What will be the equilibrium price? equilibrium quantity? 50. What will be the 150. (b) Suppose now that the large ﬁrm attempts to exploit its market power and set a proﬁt-maximizing price. In order to model this we assume that customers always go ﬁrst to the competitive ﬁrms and buy as much as they are able to and then go to the large ﬁrm. In this situation, the equilibrium price will be large ﬁrm will be 25. competitive ﬁrms will be $37.50. The quantity supplied by the of the base price, $p, that he charges: 2(120 − p)(p − 20) = 2(140p − p2 − 2, 400). (b) Ben’s proﬁt-maximizing base price is $70. (Hint: You just wrote proﬁts as a function of prices. Now diﬀerentiate this expression for proﬁts with respect to p.) Ben’s most distant customer would be located at a 137.5. $312.50. (d) Finally suppose that the large ﬁrm could force the competitive ﬁrms out of the business and behave as a real monopolist. What will be the 175/2. cookstoves he could sell? 2(120−p). (Remember to count the ones he could sell to his east as well as to his west.) Assume that Ben has no other costs than buying the stoves and delivering them. Then Ben would make a proﬁt of p − 20 per stove. Write Ben’s total proﬁt as a function and the equilibrium quantity supplied by the (c) What will be the large ﬁrm’s proﬁts? equilibrium price? (a) If Ben set a base price of $p for cookstoves where p < 120, and if he charged $1 a mile for delivering them, what would be the total number of 225/2. What will be the equilibrium quantity? What will be the large ﬁrm’s proﬁts? (175/2)2 . distance of 50 miles from him. Ben would sell and make a total proﬁt of 100 cookstoves $5,000. (c) Suppose that instead of setting a single base price and making all buyers pay for the cost of transportation, Ben oﬀers free delivery of cookstoves. He sets a price $p and promises to deliver for free to any farmer who lives within p−20 miles of him. (He won’t deliver to anyone who lives NAME 333 further than that, because it then costs him more than $p to buy a stove and deliver it.) If he is going to price in this way, how high should he set p? $120. How many cookstoves would Ben deliver? 200. How much would his total revenue be? $24,000 How much would his total costs be, including the cost of deliveries and the cost of buying the stoves? $14,000. (Hint: What is the average distance that he has to haul a cookstove?) How much proﬁt would he make? $10,000. Can you explain why it is more proﬁtable for Ben to use this pricing scheme where he pays the cost of delivery himself rather than the scheme where the farmers pay for their own deliveries? If Ben pays between nearby farmers and faraway ones. He charges a higher price, net of transport to pay less net of transport cost. (a) If Ben’s base price is pB and Huey’s is pH , write down an equation that could be solved for the distance x∗ to the west of Bouncing Springs that NAME 335 whose price to him is cheaper by one penny. When the two merchants have this pricing policy, all farmers who live within 20 miles of Ben will buy from Ben and all farmers who live within 20 miles of Huey will buy from Huey. A farmer who lives x miles west of Ben and buys from Ben must pay 59.99 − x dollars to have a cookstove delivered to him. A farmer who lives x miles east of Huey and buys from Huey must pay 59.99 − x for delivery of a stove. On the graph below, use blue ink to graph the cost to Ben of delivering to a farmer who lives x miles west of him. Use red ink to graph the total cost to Huey of delivering a cookstove to a farmer who lives x miles west of Ben. Use pencil to mark the lowest price available to a farmer as a function of how far west he lives from Ben. Dollars 80 40 20 ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; Pencil;;;;;;;;;;;; line ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; Huey's profit ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; Ben's profit ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; Red line ;;;;;;;;;;;; ;;;;;;;;;;;; Blue line 0 10 20 30 40 Miles west of Ben (f ) With the pricing policies you just graphed, which farmers get stoves delivered most cheaply, those who live closest to the merchants or those who live midway between them? Those who live midway between them. On the graph you made, shade in the area representing each merchant’s proﬁts. How much proﬁts does each merchant make? $400. If Ben and Huey are pricing in this way, is there any way for either of them to increase his proﬁts by changing the price he charges to some farmers? No. Ben’s market extends. pB +x∗ = pH +(40−x∗ ). price is pB and Huey’s is pH , then Ben will sell cookstoves and Huey will sell If Ben’s base 20 + (pH − pB )/2 20 + (pB − pH )/2 cookstoves. (b) Recalling that Ben makes a proﬁt of pB − 20 on every cookstove that he sells, Ben’s proﬁts can be expressed as the following function of pB and pH . (20 + (pH − pB )/2)(pB − 20). (c) If Ben thinks that Huey’s price will stay at pH , no matter what price Ben chooses, what choice of pB will maximize Ben’s proﬁts? pB = (Hint: Set the derivative of Ben’s proﬁts with respect to his price equal to zero.) Suppose that Huey thinks that Ben’s price will stay at pB , no matter what price Huey chooses, what choice of pH will maximize Huey’s proﬁts? pH = 30 + pB /2. (Hint: Use the symmetry of the problem and the answer to the last question.) (d) Can you ﬁnd a base price for Ben and a base price for Huey such that each is a proﬁt-maximizing choice given what the other guy is doing? (Hint: Find prices pB and pH that simultaneously solve the last two equations.) 27.11 (2) Perhaps you wondered what Ben Kinmore, who lives oﬀ in the woods quietly collecting his monopoly proﬁts, is doing in this chapter on oligopoly. Well, unfortunately for Ben, before he got around to selling any stoves, the railroad built a track to the town of Deep Furrow, just 40 miles down the road, west of Bouncing Springs. The storekeeper in Deep Furrow, Huey Sunshine, was also able to get cookstoves delivered by train to his store for $20 each. Huey and Ben were the only stove dealers on the road. Let us concentrate our attention on how they would compete for the customers who lived between them. We can do this, because Ben can charge diﬀerent base prices for the cookstoves he ships east and the cookstoves he ships west. So can Huey. Suppose that Ben sets a base price, pB , for stoves he sends west and adds a charge of $1 per mile for delivery. Suppose that Huey sets a base price, pH , for stoves he sends east and adds a charge of $1 per mile for delivery. Farmers who live between Ben and Huey would buy from the seller who is willing to deliver most cheaply to them (so long as the delivered price does not exceed $120). If Ben’s base price is pB and Huey’s base price is pH , somebody who lives x miles west of Ben would have to pay a total of pB + x to have a stove delivered from Ben and pH + (40 − x) to have a stove delivered by Huey. 60 (Ch. 27) and a lower net price to faraway farmers, who are willing Calculus OLIGOPOLY 30 + pH /2. for delivery, he can price-discriminate cost, to nearby farmers 334 pB = pH = 60. to farmers living west of him? these sales? How many cookstoves does Ben sell 20. How much proﬁt does he make on $800. (e) Suppose that Ben and Huey decided to compete for the customers who live between them by price discriminating. Suppose that Ben oﬀers to deliver a stove to a farmer who lives x miles west of him for a price equal to the maximum of Ben’s total cost of delivering a stove to that farmer and Huey’s total cost of delivering to the same farmer less 1 penny. Suppose that Huey oﬀers to deliver a stove to a farmer who lives x miles west of Ben for a price equal to the maximum of Huey’s own total cost of delivering to this farmer and Ben’s total cost of delivering to him less a penny. For example, if a farmer lives 10 miles west of Ben, Ben’s total cost of delivering to him is $30, $20 to get the stove and $10 for hauling it 10 miles west. Huey’s total cost of delivering it to him is $50, $20 to get the stove and $30 to haul it 30 miles east. Ben will charge the maximum of his own cost, which is $30, and Huey’s cost less a penny, which is $49.99. The maximum of these two numbers is $49.99 . Huey will charge the maximum of his own total cost of delivering to this farmer, which is $50, and Ben’s cost less a penny, which is $29.99. Therefore Huey will charge $50.00 336 to deliver to this farmer. This farmer will buy from OLIGOPOLY (Ch. 27) Ben 338 Chapter 28 GAME THEORY Game Theory Introduction. In this introduction we oﬀer two examples of two-person The Battle of the Bismarck Sea games. The ﬁrst game has a dominant strategy equilibrium. The second game is a zero-sum game that has a Nash equilibrium in pure strategies that is not a dominant strategy equilibrium. Example: Albert and Victoria are roommates. Each of them prefers a clean room to a dirty room, but neither likes housecleaning. If both clean the room, they each get a payoﬀ of 5. If one cleans and the other doesn’t clean, the person who does the cleaning has a utility of 2, and the person who doesn’t clean has a utility of 6. If neither cleans, the room stays a mess and each has a utility of 3. The payoﬀs from the strategies “Clean” and “Don’t Clean” are shown in the box below. Clean Room–Dirty Room Clean 5, 5 6, 2 Clean Albert Don’t Clean Victoria Don’t Clean 2, 6 3, 3 In this game, whether or not Victoria chooses to clean, Albert will get a higher payoﬀ if he doesn’t clean than if he does clean. Therefore “Don’t Clean” is a dominant strategy for Albert. Similar reasoning shows that no matter what Albert chooses to do, Victoria is better oﬀ if she chooses “Don’t Clean.” Therefore the outcome where both roommates choose “Don’t Clean” is a dominant strategy equilibrium. This is true despite the fact that both persons would be better oﬀ if they both chose to clean the room. Example: This game is set in the South Paciﬁc in 1943. Admiral Imamura must transport Japanese troops from the port of Rabaul in New Britain, across the Bismarck Sea to New Guinea. The Japanese ﬂeet could either travel north of New Britain, where it is likely to be foggy, or south of New Britain, where the weather is likely to be clear. U.S. Admiral Kenney hopes to bomb the troop ships. Kenney has to choose whether to concentrate his reconnaissance aircraft on the Northern or the Southern route. Once he ﬁnds the convoy, he can bomb it until its arrival in New Guinea. Kenney’s staﬀ has estimated the number of days of bombing NAME 339 (a) If (top, left) is a dominant strategy equilibrium, then we know that a> e d ,b> , c > g, and f > h. (b) If (top, left) is a Nash equilibrium, then which of the above inequalities must be satisﬁed? a > e; b > d. Why? North Kenney South Imamura North South 2, −2 2, −2 1, −1 3, −3 This game does not have a dominant strategy equilibrium, since there is no dominant strategy for Kenney. His best choice depends on what Imamura does. The only Nash equilibrium for this game is where Imamura chooses the northern route and Kenney concentrates his search on the northern route. To check this, notice that if Imamura goes North, then Kenney gets an expected two days of bombing if he (Kenney) chooses North and only one day if he (Kenney) chooses South. Furthermore, if Kenney concentrates on the north, Imamura is indiﬀerent between going north or south, since he can be expected to be bombed for two days either way. Therefore if both choose “North,” then neither has an incentive to act diﬀerently. You can verify that for any other combination of choices, one admiral or the other would want to change. As things actually worked out, Imamura chose the Northern route and Kenney concentrated his search on the North. After about a day’s search the Americans found the Japanese ﬂeet and inﬂicted heavy damage on it.∗ 28.1 (0) This problem is designed to give you practice in reading a game matrix and to check that you understand the deﬁnition of a dominant strategy. Consider the following game matrix. A Game Matrix Player A Top Bottom Player B Left Right a, b c, d e, f g, h ∗ This example is discussed in R. Duncan Luce and Howard Raiﬀa’s Games and Decisions, John Wiley, 1957, or Dover, 1989. We recommend this book to anyone interested in reading more about game theory. 340 GAME THEORY (Ch. 28) (b) If the other player contributes, what is your payoﬀ if you keep? $18. tribute? If the other player contributes, what is your payoﬀ if you con- $16. Yes. (c) Does this game have a dominant strategy equilibrium? (c) If (top, left) is a dominant strategy equilibrium must it be a Nash equilibrium? (Ch. 28) time for each of the outcomes. The payoﬀs to Kenney and Imamura from each outcome are shown in the box below. The game is modeled as a “zero-sum game:” for each outcome, Imamura’s payoﬀ is the negative of Kenney’s payoﬀ. NAME Yes. A dominant strategy equilibrium is always a Nash equilibrium. 28.2 (0) In order to learn how people actually play in game situations, economists and other social scientists frequently conduct experiments in which subjects play games for money. One such game is known as the voluntary public goods game. This game is chosen to represent situations in which individuals can take actions that are costly to themselves but that are beneﬁcial to an entire community. In this problem we will deal with a two-player version of the voluntary public goods game. Two players are put in separate rooms. Each player is given $10. The player can use this money in either of two ways. He can keep it or he can contribute it to a “public fund.” Money that goes into the public fund gets multiplied by 1.6 and then divided equally between the two players. If both contribute their $10, then each gets back $20 × 1.6/2 = $16. If one contributes and the other does not, each gets back $10 × 1.6/2 = $8 from the public fund so that the contributor has $8 at the end of the game and the non-contributor has $18–his original $10 plus $8 back from the public fund. If neither contributes, both have their original $10. The payoﬀ matrix for this game is: so, what is it? If Both keep. 28.3 (1) Let us consider a more general version of the voluntary public goods game described in the previous question. This game has N players, each of whom can contribute either $10 or nothing to the public fund. All money that is contributed to the public fund gets multiplied by some number B > 1 and then divided equally among all players in the game (including those who do not contribute.) Thus if all N players contribute $10 to the fund, the amount of money available to be divided among the N players will be $10BN and each player will get $10BN/N = $10B back from the public fund. (a) If B > 1, which of the following outcomes gives the higher payoﬀ to each player? a) All players contribute their $10 or b) all players keep their $10. All players contribute their $10. (b) Suppose that exactly K of the other players contribute. If you keep your $10, you will have this $10 plus your share of the public fund contributed by others. What will your payoﬀ be in this case? $10 + $10BK/N . If you contribute your $10, what will be the total number of contributors? K+1. What will be your payoﬀ? $10B(K + 1)/N . (c) If B = 3 and N = 5, what is the dominant strategy equilibrium for this Voluntary Public Goods Game game? Player A Contribute Keep Player B Contribute Keep $16, $16 $8, $18 $18, $8 $10, $10 All keep their $10 Explain your answer. If K other players contribute, a player’s payoff will be $10 + 30K/5 if he keeps the $10 and $30(K + 1)/5 if he contributes it. Where B = 3 and N = 5, the difference between (a) If the other player keeps, what is your payoﬀ if you keep? $10. If the other player keeps, what is your payoﬀ if you contribute? $8. these two payoffs is $10 − $6 = $4 > 0, so we see that for all possible K, the payoff from keeping the money is always higher. NAME 341 (d) In general, what relationship between B and N must hold for “Keep” We ask when the to be a dominant strategy? Using the formulas above, we want to know when 10(1+BK/N ) > 10B(K + 1)/N , which reduces to B/N < 1. (e) Sometimes the action that maximizes a player’s absolute payoﬀ, does not maximize his relative payoﬀ. Consider the example of a voluntary public goods game as described above, where B = 6 and N = 5. Suppose that four of the ﬁve players in the group contribute their $10, while the ﬁfth player keeps his $10. What is the payoﬀ of each of the four contributors? $60×4/5 = $48. his $10? GAME THEORY (Ch. 28) always higher. payoff from keeping is larger than the payoff from contributing. 342 28.4 (1) The Stag Hunt game is based on a story told by Jean Jacques Rousseau in his book Discourses on the Origin and Foundation of Inequality Among Men (1754). The story goes something like this: “Two hunters set out to kill a stag. One has agreed to drive the stag through the forest, and the other to post at a place where the stag must pass. If both faithfully perform their assigned stag-hunting tasks, they will surely kill the stag and each will get an equal share of this large animal. During the course of the hunt, each hunter has an opportunity to abandon the stag hunt and to pursue a hare. If a hunter pursues the hare instead of the stag he is certain to catch the hare and the stag is certain to escape. Each hunter would rather share half of a stag than have a hare to himself.” The matrix below shows payoﬀs in a stag hunt game. If both hunters hunt stag, each gets a payoﬀ of 4. If both hunt hare, each gets 3. If one hunts stag and the other hunts hare, the stag hunter gets 0 and the hare hunter gets 3. What is the payoﬀ of the player who keeps $10 + $60 × 4/ = $58. Who has the highest payoﬀ in the group? The player who keeps his $10. What would be the payoﬀ to the ﬁfth player if instead of keeping his $10, he The Stag Hunt Game contributes, so that all ﬁve players contribute. $60 × 5/5 = $60. If the other four players contribute, what should the ﬁfth player to maximize his absolute payoﬀ? Contribute. Hunt Stag Hunter A Hunt Hare What should he do to maximize his payoﬀ relative to that of the other players? Hunter B Hunt Stag Hunt Hare 4, 4 0, 3 3, 0 3, 3 Keep his $10. (a) If you are sure that the other hunter will hunt stag, what is the best (f ) If B = 6 and N = 5, what is the dominant strategy equilibrium for this game? All contribute. Where B = 6 and thing for you to do? (b) If you are sure that the other hunter will hunt hare, what is the best N = 5, if K other players contribute, a player’s payoff will be $10 + 60K/5 if he keeps the $10 and $60(K + 1)/5 if he to take No. If The best action depends on what the other player does. $10 is NAME 343 (d) This game has two pure strategy Nash equilibria. What are they? Both hunters hunt stag. Both hunters hunt hare. 344 GAME THEORY (Ch. 28) (c) Do any of the pure Nash equilibria that you found seem more rea- Although (Stay, sonable than others? Why or why not? Stay) is a Nash equilibrium, it seems (e) Is one Nash equilibrium better for both hunters than the other? Yes Hunt hare. so, what is it? If not explain why not. two payoffs is$10 − $12 = −$2, so the payoff from contributing the thing for you to do? (c) Does either hunter have a dominant strategy in this game? contributes it. For every possible K, the difference between these Hunt stag. Explain your answer. If so, which is the better equilibrium? Both hunt a silly one. If either player stag. will go to the (f ) If a hunter believes that with probability 1/2 the other hunter will hunt stag and with probability 1/2 he will hunt hare, what should this go. hunter do to maximize his expected payoﬀ? Hunt hare. 28.5 (1) Evangeline and Gabriel met at a freshman mixer. They want desperately to meet each other again, but they forgot to exchange names or phone numbers when they met the ﬁrst time. There are two possible strategies available for each of them. These are Go to the Big Party or Stay Home and Study. They will surely meet if they both go to the party, and they will surely not otherwise. The payoﬀ to meeting is 1,000 for each of them. The payoﬀ to not meeting is zero for both of them. The payoﬀs are described by the matrix below. believes that there is any chance that the other party, he or she will also (d) Let us change the game a little bit. Evangeline and Gabriel are still desperate to ﬁnd each other. But now there are two parties that they can go to. There is a little party at which they would be sure to meet if they both went there and a huge party at which they might never see each other. The expected payoﬀ to each of them is 1,000 if they both go to the little party. Since there is only a 50-50 chance that they would ﬁnd each other at the huge party, the expected payoﬀ to each of them is only 500. If they go to diﬀerent parties, the payoﬀ to both of them is zero. The payoﬀ matrix for this game is: More Close Encounters Close Encounters of the Second Kind Evangeline Go to Party Stay Home Gabriel Go to Party Stay Home 1000, 1000 0, 0 0, 0 0, 0 (a) A strategy is said to be a weakly dominant strategy for a player if the payoﬀ from using this strategy is at least as high as the payoﬀ from using any other strategy. Is there any outcome in this game where both players are using weakly dominant strategies? The only one is (Go to party, Go to party). (b) Find all of the pure-strategy Nash equilibria for this game. are two: (Go, Go) and (Stay, Stay). Evangeline Little Party Big Party Gabriel Little Party Big Party 1000, 1000 0, 0 0, 0 500, 500 (e) Does this game have a dominant strategy equilibrium? are the two Nash equilibria in pure strategies? No. What (1) Both go to the little party. (2) Both go to the big party. There (f ) One of the Nash equilibria is Pareto superior to the other. Suppose that each person thought that there was some slight chance that the other would go to the little party. Would that be enough to convince them both to attend the little party? No. Can you think of any reason why the NAME 345 346 GAME THEORY (Ch. 28) Pareto superior equilibrium might emerge if both players understand the game matrix, if both know that the other understands it, and each knows (b) Suppose that Victoria and Albert each believe that the other will try to take the actions that will maximize his or her total payoﬀ for the day. that the other knows that he or she understands the game matrix? Does either believe the other will start a quarrel? No Assuming that each is trying to maximize his or her own payoﬀ, given the actions of the other, what would you expect each of them to do in the ﬁrst stage If both know the game matrix and each Not Clean. knows that the other knows it, then each of the game, clean or not Clean? may predict the other (c) Suppose that Victoria and Albert are governed by emotions that they cannot control. Neither can avoid getting angry if the other does not clean. And if either one is angry, they will quarrel so that the payoﬀ of each is diminished by 2. Given that there is certain to be a quarrel if either does not clean, the payoﬀ matrix for the game between Victoria and Albert becomes: will choose the little party. 28.6 (1) The introduction to this chapter of Workouts, recounted the sad tale of roommates Victoria and Albert and their dirty room. The payoﬀ matrix for their relationship was given as follows. Vengeful Victoria and Angry Albert Domestic Life with Victoria and Albert Albert Clean Don’t Clean Clean 5, 5 6, 2 Clean 5, 5 4, 0 Clean Albert Don’t Clean Victoria Don’t Clean 2, 6 3, 3 Victoria Don’t Clean 0, 4 1, 1 Clean (d) If the other player cleans, is it better to clean or not clean? If the other player does not clean, is it better to clean or not clean. Suppose that we add a second stage to this game in which Victoria and Albert each have a chance to punish the other. Imagine that at the end of the day, Victoria and Albert are each able to see whether the other has done any housecleaning. After seeing what the other has done, each has the option of starting a quarrel. A quarrel hurts both of them, regardless of who started it. Thus we will assume that if either or both of them starts a quarrel, the day’s payoﬀ for each of them is reduced by 2. (For example if Victoria cleans and Albert doesn’t clean and if Victoria, on seeing this result, starts a quarrel, Albert’s payoﬀ will be 6 − 2 = 4 and Victoria’s will be 2 − 2 = 0.) (a) Suppose that it is evening and Victoria sees that Albert has chosen not to clean and she thinks that he will not start a quarrel. Which strategy Clean quarrel whether or not the first player cleans. Given that there will be a quarrel anyway, the second player is better off not cleaning than cleaning. (e) Does this game have a dominant strategy? No. Explain The best response depends on what the other does. NAME 347 (f ) This game has two Nash equilibria. What are they? Both clean the room and both don’t clean the 348 GAME THEORY (Ch. 28) (c) On the following axes, draw the lines that represent each of the two equations you mentioned in Part (a). Label the equilibrium attendance level. y room. 80 (g) Explain how it could happen that Albert and Victoria would both be better oﬀ if both are easy to anger than if they are rational about when to get angry, but it might also happen that they would both be worse 60 oﬀ. Not clean, the other will be angry so there will be a will give her a higher payoﬀ for the whole day, Quarrel or Not Quarrel? Not Quarrel. If one player does not Explain e 40 Y=11+.8X If they both are easy to anger, there are two equilibria, one that is better for both of them and one that is worse for both of them than the equilibrium where neither gets angry and neither cleans. (h) Suppose that Albert and Victoria are both aware that Albert will get angry and start a quarrel if Victoria does not clean, but that Victoria is level-headed and will not start a quarrel. What would be the equilibrium outcome? Albert does not clean and Victoria does clean. Nobody starts a quarrel. 28.7 (1) Maynard’s Cross is a trendy bistro that specializes in carpaccio and other uncooked substances. Most people who come to Maynard’s come to see and be seen by other people of the kind who come to Maynard’s. There is, however, a hard core of 10 customers per evening who come for the carpaccio and don’t care how many other people come. The number of additional customers who appear at Maynard’s depends on how many people they expect to see. In particular, if people expect that the number of customers at Maynard’s in an evening will be X, then the number of people who actually come to Maynard’s is Y = 10 + .8X. In equilibrium, it must be true that the number of people who actually attend the restaurant is equal to the number who are expected to attend. Y=10+.8X 20 X=Y 0 20 40 60 80 x (d) Suppose that one additional carpaccio enthusiast moves to the area. Like the other 10, he eats at Maynard’s every night no matter how many others eat there. Write down the new equations determining attendance at Maynard’s and solve for the new equilibrium number of customers. y = 11 + .8x and y = x, so x = y = 55. (e) Use a diﬀerent color ink to draw a new line representing the equation that changed. How many additional customers did the new steady 4. customer attract (besides himself)? (f ) Suppose that everyone bases expectations about tonight’s attendance on last night’s attendance and that last night’s attendance is public knowledge. Then Xt = Yt−1 , where Xt is expected attendance on day t and Yt−1 is actual attendance on day t − 1. At any time t, Yt = 10 + .8Xt . Suppose that on the ﬁrst night that Maynard’s is open, attendance is 20. What will be attendance on the second night? 26. (g) What will be the attendance on the third night? 30.8. (a) What two simultaneous equations must you solve to ﬁnd the equilibrium attendance at Maynard’s? Y = 10+ .8X and X = Y . (b) What is the equilibrium nightly attendance? 50. (h) Attendance will tend toward some limiting value. What is it? 50. 28.8 (0) Yogi’s Bar and Grill is frequented by unsociable types who hate crowds. If Yogi’s regular customers expect that the crowd at Yogi’s will be X, then the number of people who show up at Yogi’s, Y , will be the larger of the two numbers, 120 − 2X and 0. Thus Y = max{120 − 2X, 0}. NAME 349 350 GAME THEORY (Ch. 28) 352 GAME APPLICATIONS (a) Solve for the equilibrium attendance at Yogi’s. Draw a diagram depicting this equilibrium on the axes below. y 80 Y=120-2X 60 40 e 20 X=Y 0 20 40 60 80 x (b) Suppose that people expect the number of customers on any given night to be the same as the number on the previous night. Suppose that 50 customers show up at Yogi’s on the ﬁrst day of business. How many 20. will show up on the second day? fourth day? 0. 0. The ﬁfth day? The ninety-ninth day? 120. The third day? 120. 80. The The sixth day? The hundredth day? 0. (c) What would you say is wrong with this model if at least some of Yogi’s customers have memory spans of more than a day or two? They’d notice that last night’s attendance is not a good predictor of tonight’s. If attendance is low on odd-numbered days and high on even-numbered days, it would be smart to adjust by coming on odd-numbered days. Chapter 29 Game Applications Introduction. As we have seen, some games do not have a Nash equilibrium in pure strategies. But if we allow for the possibility of Nash equilibrium in mixed strategies, virtually every game of the sort we are interested in will have a Nash equilibrium. The key to solving for such equilibria is to observe that if a player is indiﬀerent between two strategies, then he is also willing to choose randomly between them. This observation will generally give us an equation that determines the equilibrium. Example: In the game of baseball, a pitcher throws a ball toward a batter who tries to hit it. In our simpliﬁed version of the game, the pitcher can pitch high or pitch low, and the batter can swing high or swing low. The ball moves so fast that the batter has to commit to swinging high or swinging low before the ball is released. Let us suppose that if the pitcher throws high and the batter swings low, or the pitcher throws low and the batter swings high, the batter misses the ball, so the pitcher wins. Also suppose that if the pitcher throws high and the batter swings high, the batter always connects, but if the pitcher throws low and the batter swings low, the batter will connect only half the time. This scenario leads us to the following payoﬀ matrix, where if the batter hits the ball he gets a payoﬀ of 1 and the pitcher gets 0, and if the batter misses, the pitcher gets a payoﬀ of 1 and the batter gets 0. Simpliﬁed Baseball Pitch High Pitcher Pitch Low Batter Swing Low Swing High 1, 0 0, 1 .5, .5 1, 0 This game has no Nash equilibrium in pure strategies. There is no combination of actions taken with certainty such that each is making the best response to the other’s action. The batter always wants to swing the same place the pitcher throws, and the pitcher always wants to throw to the opposite place. What we can ﬁnd is a pair of equilibrium mixed strategies. (Ch. 29) In a mixed strategy equilibrium each player’s strategy is chosen at random. The batter will be willing to choose a random strategy only if the expected payoﬀ to swinging high is the same as the expected payoﬀ to swinging low. The payoﬀs from swinging high or swinging low depend on what the pitcher does. Let πP be the probability that the pitcher throws high and 1 − πP be the probability that he throws low. The batter realizes that if he swings high, he will get a payoﬀ of 0 if the pitcher throws low and 1 if the pitcher throws high. The expected payoﬀ to the batter is therefore πP . If the pitcher throws low, then the only way the batter can score is if the pitcher pitches low, which happens with probability 1 − πP . Even then the batter only connects half the time. So the expected payoﬀ to the batter from swinging low is .5(1 − πP ). These two expected payoﬀs are equalized when πP = .5(1 − πP ). If we solve this equation, we ﬁnd πP = 1/3. This has to be the probability that the pitcher throws high in a mixed strategy equilibrium. Now let us ﬁnd the probability that the batter swings low in a mixed strategy equilibrium. In equilibrium, the batter’s probability πB from swinging low must be such that the pitcher gets the same expected payoﬀ from throwing high as from throwing low. The expected payoﬀ to the pitcher is the probability that the batter does not score. If the pitcher throws high, then the batter will not connect if he swings low, but will connect if he swings high, so the expected payoﬀ to the pitcher from pitching high is πB . If the pitcher throws low, then with probability (1 − πB ), the batter will swing swing high, in which case the pitcher gets a payoﬀ of 1. But when the pitcher throws low, the batter will swing low with probability πB and connect half the time, giving a payoﬀ to the pitcher of .5πB . Therefore the expected payoﬀ to the pitcher from throwing low is (1 − πB ) + .5πB = 1 − .5πB . Equalizing the payoﬀ to the pitcher from throwing high and throwing low requires πB = 1 − .5πB . Solving this equation we ﬁnd that in the equilibrium mixed strategy, πB = 2/3. Summing up, the pitcher should throw low two-thirds of the time, and the batter should swing low two-thirds of the time. NAME Calculus 29.1 (2) Two software companies sell competing products. These products are substitutes, so that the number of units that either company sells is a decreasing function of its own price and an increasing function of the other product’s price. Let p1 be the price and x1 the quantity sold of product 1 and let p2 and x2 be the price and quantity sold of product 2. Then x1 = 1000 90 − 12 p1 + 14 p2 and x2 = 1000 90 − 12 p2 + 14 p1 . Each company has incurred a ﬁxed cost for designing their software and writing the programs, but the cost of selling to an extra user is zero. Therefore each company will maximize its proﬁts by choosing the price that maximizes its total revenue. NAME 353 354 GAME APPLICATIONS (Ch. 29) (a) Write an expression for the total revenue of company 1, as a function of the its price p1 and the other company’s price p2 . 1000 90p1 − 1 2 p1 2 + 1 4 p1 p2 Revenue is Battle of the Sexes . Wrestling Roger Opera (b) Company 1’s best response function BR1 (·) is deﬁned so that BR1 (p2 ) is the price for product 1 that maximizes company 1’s revenue given that the price of product 2 is p2 . With the revenue functions we have speciﬁed, the best response function of company 1 is described by the formula BR1 (p2 ) = 90 + p2 /4 . (Hint: Take a derivative of revenue with respect to p1 and solve for the revenue-maximizing p1 given p2 .) Michelle Wrestling Opera 2, 1 0, 0 0, 0 1, 2 (a) Is the sum of the payoﬀs to Michelle and Roger constant over all outcomes? No. Does this game have a dominant strategy equilibrium? No. (c) Use a similar method to solve for company 2’s best response function BR2 (p1 ) = (b) Find two Nash equilibria in pure strategies for this game. 90 + p1 /4. go to opera. (d) Solve for the Nash equilibrium prices p1 = 120 and p2 = 120. (e) Suppose that company 1 sets its price ﬁrst. Company 2 knows the price p1 that company 1 has chosen and it knows that company 1 will not change this price. If company 2 sets its price so as to maximize its revenue given that company 1’s price is p1 , then what price will company 2 choose? p2 = $90 + 3p1 /4 If company 1 is aware of how company 2 will react to its own choice of price, what price will company 1 choose? $128.55 2 choose? Given this price for company 1, what price will company $186.41. 29.2 (1) Here is an example of the “Battle of the Sexes” game discussed in the text. Two people, let us call them Michelle and Roger, although they greatly enjoy each other’s company, have very diﬀerent tastes in entertainment. Roger’s tastes run to ladies’ mud wrestling, while Michelle prefers Italian opera. They are planning their entertainment activities for next Saturday night. For each of them, there are two possible actions, go to the wrestling match or go to the opera. Roger would be happiest if both of them went to see mud wrestling. His second choice would be for both of them to go to the opera. Michelle would prefer if both went to the opera. Her second choice would be that they both went to see the mud wrestling. They both think that the worst outcome would be that they didn’t agree on where to go. If this happened, they would both stay home and sulk. NAME 355 (a) Does this game have a dominant strategy? two Nash equilibria in pure strategies? No. What are the The two outcomes where one teenager swerves and the other (c) Find a Nash equilibrium in mixed strategies. chooses Both Both go to mud wrestling. Michele opera with probability 2/3 and wrestling with probability 1/3. Roger chooses opera with probability 1/3 and mud wrestling with probability 2/3. 29.3 (1) This is an example of the game of “Chicken.” Two teenagers in souped-up cars drive toward each other at great speed. The ﬁrst one to swerve out of the road is “chicken.” The best thing that can happen to you is that the other guy swerves and you don’t. Then you are the hero and the other guy is the chicken. If you both swerve, you are both chickens. If neither swerves, you both end up in the hospital. A payoﬀ matrix for a chicken-type game is the following. Chicken Swerve 1, 1 2, 1 Swerve Joe Bob Don’t Swerve 356 GAME APPLICATIONS Leroy Don’t Swerve 1, 2 0, 0 (Ch. 29) 29.5 (0) Ned and Ruth love to play “Hide and Seek.” It is a simple game, but it continues to amuse. It goes like this. Ruth hides upstairs or downstairs. Ned can look upstairs or downstairs but not in both places. If he ﬁnds Ruth, Ned gets one scoop of ice cream and Ruth gets none. If he does not ﬁnd Ruth, Ruth gets one scoop of ice cream and Ned gets none. Fill in the payoﬀs in the matrix below. does not. (b) Find a Nash equilibrium in mixed strategies for this game. Hide and Seek Play each strategy with probability 1/2. Ned 29.4 (0) I propose the following game: I ﬂip a coin, and while it is in the air, you call either heads or tails. If you call the coin correctly, you get to keep the coin. Suppose that you know that the coin always comes up heads. What is the best strategy for you to pursue? Upstairs Downstairs Upstairs 1, 0 0, 1 Ruth Downstairs 0, 1 1, 0 Always call (a) Is this a zero-sum game? heads. pure strategies? (a) Suppose that the coin is unbalanced and comes up heads 80% of the time and tails 20% of the time. Now what is your best strategy? Yes. What are the Nash equilibria in There are none. (b) Find a Nash equilibrium in mixed strategies for this game. Ruth hides upstairs and Ned searches Always call heads. upstairs with probability (b) What if the coin comes up heads 50% of the time and tails 50% of the downstairs and Ned searches downstairs with time? Then what is your best strategy? It doesn’t matter. You can call heads always, tails always, or randomize your calls. (c) Now, suppose that I am able to choose the type of coin that I will toss (where a coin’s type is the probability that it comes up heads), and that you will know my choice. What type of coin should I choose to minimize my losses? probability 1/2. (c) After years of playing this game, Ned and Ruth think of a way to liven it up a little. Now if Ned ﬁnds Ruth upstairs, he gets two scoops of ice cream, but if he ﬁnds her downstairs, he gets one scoop. If Ned ﬁnds Ruth, she gets no ice cream, but if he doesn’t ﬁnd her she gets one scoop. Fill in the payoﬀs in the graph below. Advanced Hide and Seek A fair coin. (d) What is the Nash mixed strategy equilibrium for this game? (It may help to recognize that a lot of symmetry exists in the game.) I choose a fair coin, and you randomize with 50% heads and 50% 1/2; Ruth hides tails. Upstairs Ned Downstairs Upstairs 2, 0 0, 1 Ruth Downstairs 0, 1 1, 0 NAME 357 (d) Are there any Nash equilibria in pure strategies? strategy equilibrium can you ﬁnd? No. What mixed Ruth hides downstairs 358 GAME APPLICATIONS (Ch. 29) (a) Is there a dominant strategy for the little pig? there a dominant strategy for the big pig? Yes, Wait. Is No. 2/3 of the time. Ned looks downstairs 1/2 (b) Find a Nash equilibrium for this game. Does the game have more than the time. one Nash equilibrium? If both use equilibrium strategies, what fraction of the 1/2. time will Ned ﬁnd Ruth? The only Nash equilibrium is where the little pig waits and the big pig 29.6 (0) Perhaps you have wondered what it could mean that “the meek shall inherit the earth.” Here is an example, based on the discussion in the book. In a famous experiment, two psychologists∗ put two pigs—a little one and a big one—into a pen that had a lever at one end and a trough at the other end. When the lever was pressed, a serving of pigfeed would appear in a trough at the other end of the pen. If the little pig would press the lever, then the big pig would eat all of the pigfeed and keep the little pig from getting any. If the big pig pressed the lever, there would be time for the little pig to get some of the pigfeed before the big pig was able to run to the trough and push him away. Let us represent this situation by a game, in which each pig has two possible strategies. One strategy is Press the Lever. The other strategy is Wait at the Trough. If both pigs wait at the trough, neither gets any feed. If both pigs press the lever, the big pig gets all of the feed and the little pig gets a poke in the ribs. If the little pig presses the lever and the big pig waits at the trough, the big pig gets all of the feed and the little pig has to watch in frustration. If the big pig presses the lever and the little pig waits at the trough, then the little pig is able to eat 2/3 of the feed before the big pig is able to push him away. The payoﬀs are as follows. (These numbers are just made up, but their relative sizes are consistent with the payoﬀs in the Baldwin-Meese experiment.) presses. (Incidentally, while Baldwin and Meese did not interpret this experiment as a game, the result they observed was the result that would be predicted by Nash equilibrium.) (c) Which pig gets more feed in Nash equilibrium? The little pig. 29.7 (1) Let’s have another look at the soccer example that was discussed in the text. But this time, we will generalize the payoﬀ matrix just a little bit. Suppose the payoﬀ matrix is as follows. The Free Kick Jump Left Goalie Jump Right Kicker Kick Left Kick Right 1, 0 0, 1 1-p,p 1, 0 Big Pig–Little Pig Little Pig Press Wait Now the probability that the kicker will score if he kicks to the left and the goalie jumps to the right is p. We will want to see how the equilibrium probabilities change as p changes. Big Pig Press Wait −1, 9 −1, 10 6, 4 0, 0 (a) If the goalie jumps left with probability πG , then if the kicker kicks πG . right, his probability of scoring is (b) If the goalie jumps left with probability πG , then if the kicker kicks ∗ Baldwin and Meese (1979), “Social Behavior in Pigs Studied by Means of Operant Conditioning,” Animal Behavior. NAME 359 left, his probability of scoring is 360 GAME APPLICATIONS (c) Find the probability πG that makes kicking left and kicking right lead to the same probability of scoring for the kicker. (Your answer will be a p function of p.) πG = 1+p . (Ch. 29) The Hawk-Dove Game (d) If the kicker kicks left with probability πK , then if the goalie jumps left, the probability that the kicker will not score is p(1 − πG ). πK . Hawk Animal A Dove Animal Hawk −5, −5 0, 10 B Dove 10, 0 4, 4 (e) If the kicker kicks left with probability πK , then if the goalie jumps right, the probability that the kicker will not score is (1 − p)πK + (1 − πK ). (f ) Find the probability πK that makes the payoﬀ to the goalie equal from jumping left or jumping right. 1 1+p . (g) The variable p tells us how good the kicker is at kicking the ball into the left side of the goal when it is undefended. As p increases, does the equilibrium probability that the kicker kicks to the left increase or decrease? Decreases. Explain why this happens in a way that even a TV sports announcer might understand. The better the kicker’s weak side gets, the less often the goalie defends the kicker’s good side. So teh kicker can kick to the good side more often. 29.8 (1) This problem is an illustration of the Hawk-Dove game described in the text. The game was ﬁrst used by biologist John Maynard Smith to illustrate the uses of game theory in the theory of evolution. Males of a certain species frequently come into conﬂict with other males over the opportunity to mate with females. If a male runs into a situation of conﬂict, he has two alternative strategies. If he plays “Hawk,” he will ﬁght the other male until he either wins or is badly hurt. If he plays “Dove,” he makes a bold display but retreats if his opponent starts to ﬁght. If two Hawk players meet, they are both seriously injured in battle. If a Hawk meets a Dove, the Hawk gets to mate with the female and the Dove slinks oﬀ to celibate contemplation. If a Dove meets another Dove, they both strut their stuﬀ but neither chases the other away. Eventually the female may select one of them at random or may get bored and wander oﬀ. The expected payoﬀs to each male are shown in the box below. (a) Now while wandering through the forest, a male will encounter many conﬂict situations of this type. Suppose that he cannot tell in advance whether another animal that he meets is a Hawk or a Dove. The payoﬀ to adopting either strategy for himself depends on the proportions of Hawks and Doves in the population at large. For example, that there is one Hawk in the forest and all of the other males are Doves. The Hawk would ﬁnd that his rival always retreated and would therefore enjoy a payoﬀ of 10 on every encounter. Given that all other males are Doves, if the remaining male is a Dove, his payoﬀ on each encounter would be 4. (b) If strategies that are more proﬁtable tend to be chosen over strategies that are less proﬁtable, explain why there cannot be an equilibrium in which all males act like Doves. If you know that you are meeting a Dove, it pays to be a Hawk. (c) If all the other males are Hawks, then a male who adopts the Hawk strategy is sure to encounter another Hawk and would get a payoﬀ of −5. If instead, this male adoptled the Dove strategy, he would again be sure to encounter a Hawk, but his payoﬀ would be 0. (d) Explain why there could not be an equilibrium where all of the animals acted like Hawks. If everyone plays Hawk, it would be profitable to play Dove. NAME 361 (e) Since there is not an equilibrium in which everybody chooses the same strategy, we look for an equilibrium in which some fraction of the males are Hawks and the rest are Doves. Suppose that there is a large male population and the fraction p are Hawks. Then the fraction of any player’s encounters that are with Hawks is about p and the fraction that are with Doves is about 1 − p. Therefore with probability p a Hawk meets another Hawk and gets a payoﬀ of −5 and with probability 1 − p he meets a Dove and gets 10. It follows that the payoﬀ to a Hawk when the fraction of Hawks in the population is p, is p × (−5) + (1 − p) × 10 = 10 − 15p. Similar calculations show that the average payoﬀ to being a Dove when the proportion of Hawks in the population is p will be p × 0 + (1 − p) × 4. (f ) Write an equation that states that when the proportion of Hawks in the population is p, the payoﬀ to Hawks is the same as the payoﬀs to Doves. 4 − 4p = 10 − 15p. (g) Solve this equation for the value of p such that at this value Hawks do exactly as well as Doves. This requires that p = 6/11. (h) On the axes below, use blue ink to graph the average payoﬀ to the strategy Dove when the proportion of Hawks in the male population who is p. Use red ink to graph the average payoﬀ to the strategy, Hawk, when the proportion of the male population who are Hawks is p. Label the equilibrium proportion in your diagram by E. Payoff 8 Red Line 6 4 2 e 0 25 Blue Line 50 75 100 Percentage of hawks NAME 363 average nesting episode takes only 4 days for invaders. Suppose that when they meet, half the time the founder of the nest wins the ﬁght and half the time the invader wins. Let D be the number of wasps that dig their own burrows and let I be the number of wasps that invade the burrows of others. The fraction of the digging wasps that are invaded will be I I about 54 D . (Assume for the time being that 54 D < 1.) Half of the diggers who are invaded will win their ﬁght and get to keep their burrows. The fraction of digging wasps who lose their burrows to other wasps is then 15 I 5 I 2 4 D = 8 D . Assume also that all the wasps who are not invaded by other wasps will successfully stock their burrows and lay their eggs. 362 GAME APPLICATIONS (Ch. 29) (i) If the proportion of Hawks is slightly greater than E, which strat- Dove. egy does better? If the proportion of Hawks is slightly less than E, which strategy does better? Hawk. If the more proﬁtable strategy tends to be adopted more frequently in future plays, then if the strategy proportions are out of equilibrium, will changes tend to move the proportions back toward equilibrium or further away from equilibrium? Closer to equilibrium. 29.9 (2) Economic ideas and equilibrium analysis have many fascinating applications in biology. Popular discussions of natural selection and biological ﬁtness often take it for granted that animal traits are selected for the beneﬁt of the species. Modern thinking in biology emphasizes that individuals (or strictly speaking, genes) are the unit of selection. A mutant gene that induces an animal to behave in such a way as to help the species at the expense of the individuals that carry that gene will soon be eliminated, no matter how beneﬁcial that behavior is to the species. A good illustration is a paper in the Journal of Theoretical Biology, 1979, by H. J. Brockmann, A. Grafen, and R. Dawkins, called “Evolutionarily Stable Nesting Strategy in a Digger Wasp.” They maintain that natural selection results in behavioral strategies that maximize an individual animal’s expected rate of reproduction over the course of its lifetime. According to the authors, “Time is the currency which an animal spends.” Females of the digger wasp Sphex ichneumoneus nest in underground burrows. Some of these wasps dig their own burrows. After she has dug her burrow, a wasp goes out to the ﬁelds and hunts katydids. These she stores in her burrow to be used as food for her oﬀspring when they hatch. When she has accumulated several katydids, she lays a single egg in the burrow, closes oﬀ the food chamber, and starts the process over again. But digging burrows and catching katydids is time-consuming. An alternative strategy for a female wasp is to sneak into somebody else’s burrow while she is out hunting katydids. This happens frequently in digger wasp colonies. A wasp will enter a burrow that has been dug by another wasp and partially stocked with katydids. The invader will start catching katydids, herself, to add to the stock. When the founder and the invader ﬁnally meet, they ﬁght. The loser of the ﬁght goes away and never comes back. The winner gets to lay her egg in the nest. Since some wasps dig their own burrows and some invade burrows begun by others, it is likely that we are observing a biological equilibrium in which each strategy is as eﬀective a way for a wasp to use its time for producing oﬀspring as the other. If one strategy were more eﬀective than the other, then we would expect that a gene that led wasps to behave in the more eﬀective way would prosper at the expense of genes that led them to behave in a less eﬀective way. Suppose the average nesting episode takes 5 days for a wasp that digs its own burrow and tries to stock it with katydids. Suppose that the 364 GAME APPLICATIONS (Ch. 29) invaders return toward equilibrium or move further away? Further away. Success Blue line 8-5(I/D) (a) Then the fraction of the digging wasps who do not lose their burrows 5 I is just 1 − . 8D Therefore over a period of 40 days, a wasp who dug her own burrow every time would have 8 nesting episodes. Her expected number of I successes would be 8 − 5 D . Black line e 5 _I D (b) In 40 days, a wasp who chose to invade every time she had a chance would have time for 10 invasions. Assuming that she is successful half the time on average, her expected number of successes would be 5. Write an equation that expresses the condition that wasps who always dig their own burrows do exactly as well as wasps who always invade burrows dug by others. 8 − 5 DI = 5. I (c) The equation you have just written should contain the expression D . I Solve for the numerical value of D that just equates the expected number of successes for diggers and invaders. The answer is 3 5. (e) The authors noticed this likely instability and cast around for possible changes in the model that would lead to stability. They observed that an invading wasp does help to stock the burrow with katydids. This may save the founder some time. If founders win their battles often enough and get enough help with katydids from invaders, it might be that the expected number of eggs that a founder gets to lay is an increasing rather than a decreasing function of the number of invaders. On the axes below, show an equilibrium in which digging one’s own burrow is an increasingly I eﬀective strategy as D increases and in which the payoﬀ to invading is constant over all ratios of I . D Is this equilibrium stable? Yes. Success (d) But there is a problem here: the equilibrium we found doesn’t appear to be stable. On the axes below, use blue ink to graph the expected number of successes in a 40-day period for wasps that dig their own burrows I every time where the number of successes is a function of D . Use black ink to graph the expected number of successes in a 40-day period for inI vaders. Notice that this number is the same for all values of D . Label the point where these two lines cross and notice that this is equilibrium. Just I to the right of the crossing, where D is just a little bit bigger than the equilibrium value, which line is higher, the blue or the black? Black. I At this level of D , which is the more eﬀective strategy for any individual wasp? Invade. Suppose that if one strategy is more eﬀective than the other, the proportion of wasps adopting the more eﬀective one increases. If, after being in equilibrium, the population got joggled just a little to the right of equilibrium, would the proportions of diggers and e _I D 29.10 (1) The Iron Chicken restaurant is located along a busy interstate NAME 365 highway. Most of its customers are just passing through and will never return to the Iron Chicken. But some are truck drivers whose routes take them past the Iron Chicken on a regular basis. Sybil, the nearsighted waitress at the Iron Chicken is unable to distinguish regular customers from one-time customers. Sybil can either give a customer good service or bad service. She knows that if she gives bad service to any customer, then she will get a small tip. If she gives good service to a truck driver, he will give her a large tip in the (futile) hope that she will recognize him the next time he comes, but if she gives good service to a one-time customer, he will still leave a small tip. Suppose that the cost to Sybil of giving a customer good service rather than bad service is $1. The tips given by dissatisﬁed customers and customers just passing through average $.50 per customer. A truck driver who has received good service and plans to come back will leave a tip of $2. Sybil believes that the fraction of her customers who are truck drivers who plan to come back is x. In equilibrium we would expect Sybil to give good service if x is greater than 2/3 and bad service if x is smaller. 366 GAME APPLICATIONS (Ch. 29) (a) Suppose that the story is as before except that Mona will be out of town for three days. The value to Lisa of having the car is again $50 per day and the total cost to Mona of letting Lisa use the car is $20, regardless of how many days Lisa uses it. This time, let us suppose that Lisa makes the ﬁrst oﬀer. Mona can either accept the oﬀer or refuse it and make a counteroﬀer. Lisa, in turn, can either accept Lisa’s counteroﬀer or refuse it and make another counteroﬀer. Each time an oﬀer is rejected and a new oﬀer is made, a day passes and so there is one less day in which the car can be rented. On the evening before the ﬁrst of these three days, Lisa reasons as follows. “If Mona rejects the oﬀer that I make tonight, then there will be two days left and it will be Mona’s turn to make an oﬀer. If this happens, Mona will get a proﬁt of slightly less than $50. (Hint: We found this answer above, for the two-day case.) Since her total costs are $20, Mona will make a proﬁt of $50 if I oﬀer her a price of $70. for the three days rental.” Since three days of car rental is worth $150 to Lisa, Lisa would make a proﬁt of proﬁt of $50. 29.11 (2) Mona is going to be out of town for two days and will not be needing her car during this time. Lisa is visiting relatives and is interested in renting her car. The value to Lisa of having the car during this time is $50 per day. Mona ﬁgures that the total cost to her of letting Lisa use the car is $20, regardless of how many days Lisa uses it. On the evening before the ﬁrst of these two days, Lisa can send a message to Mona, oﬀering a speciﬁed amount of money to rent the car for two days. Lisa can either accept the oﬀer or reject the oﬀer and make a counteroﬀer. The only problem is that it takes a full day for a counteroﬀer to be made and accepted. Let us consider the Rubinstein bargaining solution to this problem. We start by working back from the end. If Lisa rejects the original oﬀer, then the car can only be rented for one day and there will be no time for Mona to make a counteroﬀer. So if Lisa rejects the original oﬀer, she can oﬀer Mona slightly more than $20 to rent the car for the last day and Mona will accept. In this case, Lisa will get a proﬁt of slightly less than $50 − $20 = $30. Mona understands that this is the case. Therefore when Mona makes her original oﬀer, she is aware that Lisa will reject the oﬀer unless it gives Lisa a proﬁt of slightly more than $30. $80. and Mona would make a (b) Now suppose that the story is as before except that Mona will be out of town for four days and suppose that Mona makes the ﬁrst oﬀer. Mona knows that if Lisa rejects her ﬁrst oﬀer, there will be three days left, it will be Lisa’s turn to make an oﬀer and so Lisa can make a proﬁt of $80. (See the previous answer.) Having the car for four days is worth $200 to Lisa, so to the highest price that Mona can expect Lisa to accept for the four days is slightly less than a proﬁt of slightly less than more than $100. $120. In this case, Mona makes and Lisa makes a proﬁt of slightly $80. Mona is aware that the value to Lisa of renting the car for two days is $100. Therefore the highest price for two days car rental that Lisa will accept is slightly less than $70. Since Mona’s total costs for renting the car are $20, Mona would make a net proﬁt of slightly less than Lisa would make a net proﬁt of slightly more than $50. and $30. 368 Chapter 30 NAME Exchange Introduction. The Edgeworth box is a thing of beauty. An amazing amount of information is displayed with a few lines, points and curves. In fact one can use an Edgeworth box to show just about everything there is to say about the case of two traders dealing in two commodities. Economists know that the real world has more than two people and more than two commodities. But it turns out that the insights gained from this model extend nicely to the case of many traders and many commodities. So for the purpose of introducing the subject of exchange equilibrium, the Edgeworth box is exactly the right tool. We will start you out with an example of two gardeners engaged in trade. Example: Alice and Byron consume two goods, camelias and dahlias. Alice has 16 camelias and 4 dahlias. Byron has 8 camelias and 8 dahlias. They consume no other goods, and they trade only with each other. To describe the possible allocations of ﬂowers, we ﬁrst draw a box whose width is the total number of camelias and whose height is the total number of dahlias that Alice and Byron have between them. The width of the box is therefore 16 + 8 = 24 and the height of the box is 4 + 8 = 12. Dahlias Byron 12 6 0 Alice 6 12 18 24 Camelias Any feasible allocation of ﬂowers between Alice and Byron is fully described by a single point in the box. Consider, for example, the allocation where Alice gets the bundle (15, 9) and Byron gets the bundle (9, 3). This allocation is represented by the point A = (15, 9) in the Edgeworth box. The distance 15 from A to the left side of the box is the number of camelias for Alice and the distance 9 from A to the bottom of the box is the number of dahlias for Alice. This point also determines Byron’s consumption of camelias and dahlias. The distance 9 from A to the right side of the box is the total number of camelias consumed by Byron, and the distance from A to the top of the box is the number of dahlias consumed by Byron. Since the width of the box is the total supply of camelias and the height of the box is the total supply of dahlias, these conventions ensure that any point in the box represents a feasible allocation of the total EXCHANGE (Ch. 30) supply of camelias and dahlias. It is useful to mark the initial allocation on the Edgeworth box. In this case, the initial allocation is represented by the point E = (16, 4). Now suppose that Alice’s utility function is U (c, d) = c + 2d and Byron’s utility funtion is U (c, d) = cd. Alice’s indiﬀerence curves will be straight lines with slope −1/2. The indiﬀerence curve that passes through her initial endowment, for example, will be a line that runs from the point (24, 0) to the point (0, 12). Since Byron has Cobb-Douglas utility, his indiﬀerence curves will be rectangular hyperbolas, but since quantities for Byron are measured from the upper right corner of the box, these indiﬀerence curves will be ﬂipped over as in the diagram. The Pareto set or contract curve is the set of points where Alice’s indiﬀerence curves are tangent to Byron’s. There will be tangency if the slopes are the same. The slope of Alice’s indiﬀerence curve at any point is −1/2. The slope of Byron’s indiﬀerence curve depends on his consumption of the two goods. When Byron is consuming the bundle (cB , dB ), the slope of his indiﬀerence curve is equal to his marginal rate of substitution, which is −dB /cB . Therefore Alice’s and Byron’s indiﬀerence curves will nuzzle up in a nice tangency whenever −dB /cB = −1/2. So the Pareto set in this example is just the diagonal of the Edgeworth box. Some problems ask you to ﬁnd a competitive equilibrium. For an economy with two goods, the following procedure is often a good way to calculate equilibrium prices and quantities. • Since demand for either good depends only on the ratio of prices of good 1 to good 2, it is convenient to set the price of good 1 equal to 1 and let p2 be the price of good 2. • With the price of good 1 held at 1, calculate each consumer’s demand for good 2 as a function of p2 . • Write an equation that sets the total amount of good 2 demanded by all consumers equal to the total of all participants’ initial endowments of good 2. • Solve this equation for the value of p2 that makes the demand for good 2 equal to the supply of good 2. (When the supply of good 2 equals the demand of good 2, it must also be true that the supply of good 1 equals the demand for good 1.) • Plug this price into the demand functions to determine quantities. Example: Frank’s utility function is U (x1 , x2 ) = x1 x2 and Maggie’s is U (x1 , x2 ) = min{x1 , x2 }. Frank’s initial endowment is 0 units of good 1 and 10 units of good 2. Maggie’s initial endowment is 20 units of good 1 and 5 units of good 2. Let us ﬁnd a competitive equilibrium for Maggie and Frank. Set p1 = 1 and ﬁnd Frank’s and Maggie’s demand functions for good 2 as a function of p2 . Using the techniques learned in Chapter 6, we ﬁnd that Frank’s demand function for good 2 is m/2p2 , where m is his income. Since Frank’s initial endowment is 0 units of good 1 and 10 units of good 2, his income is 10p2 . Therefore Frank’s demand for good 2 is 10p2 /2p2 = 5. Since goods 1 and 2 are perfect complements for Maggie, she will choose to consume where x1 = x2 . This fact, together with her budget constraint implies that Maggie’s demand function for good 2 is NAME 369 m/(1 + p2 ). Since her endowment is 20 units of good 1 and 5 units of good 2, her income is 20 + 5p2 . Therefore at price p2 , Maggie’s demand is (20 + 5p2 )/(1 + p2 ). Frank’s demand plus Maggie’s demand for good 2 adds up to 5 + (20 + 5p2 )/(1 + p2 ). The total supply of good 2 is Frank’s 10 unit endowment plus Maggie’s 5 unit endowment, which adds to 15 units. Therefore demand equals supply when (20 + 5p2 ) 5+ = 15. (1 + p2 ) 370 EXCHANGE (Ch. 30) (b) At any Pareto optimum, where both people consume some of each good, it must be that their marginal rates of substitution are equal. No matter what he consumes, Morris’s marginal rate of substitution is equal -1. to When Philip consumes the bundle, (bP , wP ), his MRS is −wP /bP . Therefore every Pareto optimal allocation where both wP = consume positive amounts of both goods satisﬁes the equation bP . Solving this equation, one ﬁnds that the equilibrium price is p2 = 2. At the equilibrium price, Frank will demand 5 units of good 2 and Maggie will demand 10 units of good 2. 30.1 (0) Morris Zapp and Philip Swallow consume wine and books. Morris has an initial endowment of 60 books and 10 bottles of wine. Philip has an initial endowment of 20 books and 30 bottles of wine. They have no other assets and make no trades with anyone other than each other. For Morris, a book and a bottle of wine are perfect substitutes. His utility function is U (b, w) = b + w, where b is the number of books he consumes and w is the number of bottles of wine he consumes. Philip’s preferences are more subtle and convex. He has a Cobb-Douglas utility function, U (b, w) = bw. In the Edgeworth box below, Morris’s consumption is measured from the lower left, and Philip’s is measured from the upper right corner of the box. Use black ink on the diagram above to draw the locus of Pareto optimal allocations. (c) At a competitive equilibrium, it will have to be that Morris consumes some books and some wine. But in order for him to do so, it must be that the ratio of the price of wine to the price of books is 1. Therefore we know that if we make books the numeraire, then the price of wine in 1. competitive equilibrium must be (d) At the equilibrium prices you found in the last part of the question, At these prices, Philip will choose to consume 25 books and 25 bottles of wine. If Morris Zapp consumes all of the books and all of the wine that Philip doesn’t consume, he will consume Wine 50. what is the value of Philip Swallow’s initial endowment? Philip 15 40 55 books and bottles of wine. Red curve (e) At the competitive equilibrium prices that you found above, Morris’s 20 Blue curve 0 Morris 20 income is 70. Therefore at these prices, the cost to Morris of consuming all of the books and all of the wine that Philip doesn’t consume Black e line 40 60 is (the same as, more than, less than) the same as his income. At these prices, can Morris aﬀord a bundle that he likes better than the No. bundle (55, 15)? 80 Books (a) On this diagram, mark the initial endowment and label it E. Use red ink to draw Morris Zapp’s indiﬀerence curve that passes through his initial endowment. Use blue ink to draw in Philip Swallow’s indiﬀerence curve that passes through his initial endowment. (Remember that quantities for Philip are measured from the upper right corner, so his indiﬀerence curves are “Phlipped over.”) (f ) Suppose that an economy consisted of 1,000 people just like Morris and 1,000 people just like Philip. Each of the Morris types had the same endowment and the same tastes as Morris. Each of the Philip types had the same endowment and tastes as Philip. Would the prices that you found to be equilibrium prices for Morris and Philip still be competitive equilibrium prices? Yes. If each of the Morris types and each of the Philip types behaved in the same way as Morris and Philip did above, would supply equal demand for both wine and books? NAME 371 30.2 (0) Consider a small exchange economy with two consumers, Astrid and Birger, and two commodities, herring and cheese. Astrid’s initial endowment is 4 units of herring and 1 unit of cheese. Birger’s initial endowment has no herring and 7 units of cheese. Astrid’s utility function is U (HA , CA ) = HA CA . Birger is a more inﬂexible person. His utility function is U (HB , CB ) = min{HB , CB }. (Here HA and CA are the amounts of herring and cheese for Astrid, and HB and CB are amounts of herring and cheese for Birger.) (a) Draw an Edgeworth box, showing the initial allocation and sketching in a few indiﬀerence curves. Measure Astrid’s consumption from the lower left and Birger’s from the upper right. In your Edgeworth box, draw two diﬀerent indiﬀerence curves for each person, using blue ink for Astrid’s and red ink for Birger’s. Herring 4 e Birger 372 EXCHANGE Yes. (Ch. 30) (d) Where the price of cheese is 1 and p is the price of herring, what is 1+4p . the value of Astrid’s initial endoment? Astrid demand at price p? How much herring will 2 + (1/2p) 30.3 (0) Dean Foster Z. Interface and Professor J. Fetid Nightsoil exchange platitudes and bromides. When Dean Interface consumes TI platitudes and BI bromides, his utility is given by UI (BI , TI ) = BI + 2 TI . When Professor Nightsoil consumes TN platitudes and BN bromide, his utility is given by UN (BN , TN ) = BN + 4 TN . Dean Interface’s initial endowment is 12 platitudes and 8 bromides. Professor Nightsoil’s initial endowment is 4 platitudes and 8 bromides. Blue curves Nightsoil Platitudes 16 2 Black line Red curves 0 Astrid 2 e 12 4 6 8 Cheese Red curve 8 Pencil curve (b) Use black ink to show the locus of Pareto optimal allocations. (Hint: Since Birger is kinky, calculus won’t help much here. But notice that because of the rigidity of the proportions in which he demands the two goods, it would be ineﬃcient to give Birger a positive amount of either good if he had less than that amount of the other good. What does that tell you about where the Pareto eﬃcient locus has to be?) 4 3.2 Blue line Pareto 0 Interface efficient allocations lie on the line with slope 1 extending from Birger’s corner of the box. (c) Let cheese be the numeraire (with price 1) and let p denote the price of herring. Write an expression for the amount of herring that Birger will demand at these prices. 7/(p + 1) (Hint: Since Birger initially owns 7 units of cheese and no herring and since cheese is the numeraire, the value of his initial endowment is 7. If the price of herring is p, how many units of herring will he choose to maximize his utility subject to his budget constraint?) 4 8 12 16 Bromides (a) If Dean Interface consumes TI platitudes and BI bromides, his mar−1/2 ginal rate of substitution will be −TI . If Professor Nightsoil consumes TN platitudes and BN bromides, his marginal rate of substitu−1/2 tion will be −2TN . (b) On the contract curve, Dean Interface’s marginal rate of substitution equals Professor Nightsoil’s. Write an equation that states this condition. √ √ TI = TN /2. This equation is especially simple because each person’s marginal rate of substitution depends only on his consumption of platitudes and not on his consumption of bromides. NAME 373 374 EXCHANGE (Ch. 30) (c) From this equation we see that TI /TN = 1/4 at all points on the contract curve. This gives us one equation in the two unknowns TI and TN . Wine 8 (d) But we also know that along the contract curve it must be that TI + 6 Barbie Black line Red curve TN = 16 , since the total consumption of platitudes must equal the total endowment of platitudes. ce 4 (e) Solving these two equations in two unknowns, we ﬁnd that everywhere on the contract curve, TI and TN are constant and equal to and Blue curve TI = 3.2 2 Pareto w efficient points TN = 12.8. (f ) In the Edgeworth box, label the initial endowment with the letter E. Dean Interface has thick gray penciled indiﬀerence curves. Professor Nightsoil has red indiﬀerence curves. Draw a few of these in the Edgeworth box you made. Use blue ink to show the locus of Pareto optimal points. The contract curve is a (vertical, horizontal, diagonal) horizontal line in the Edgeworth box. (g) Find the competitive equilibrium prices and quantities. You know what the prices have to be at competitive equilibrium because you know what the marginal rates of substitution have to be at every Pareto optimum. 30.4 (0) A little exchange economy has just two consumers, named Ken and Barbie, and two commodities, quiche and wine. Ken’s initial endowment is 3 units of quiche and 2 units of wine. Barbie’s initial endowment is 1 unit of quiche and 6 units of wine. Ken and Barbie have identical utility functions. We write Ken’s utility function as, U (QK , WK ) = QK WK and Barbie’s utility function as U (QB , WB ) = QB WB , where QK and WK are the amounts of quiche and wine for Ken and QB and WB are amounts of quiche and wine for Barbie. (a) Draw an Edgeworth box below, to illustrate this situation. Put quiche on the horizontal axis and wine on the vertical axis. Measure goods for Ken from the lower left corner of the box and goods for Barbie from the upper right corner of the box. (Be sure that you make the length of the box equal to the total supply of quiche and the height equal to the total supply of wine.) Locate the initial allocation in your box, and label it W . On the sides of the box, label the quantities of quiche and wine for each of the two consumers in the initial endowment. NAME 375 equilibrium prices. In equilibrium Ken’s income will be the value of his endowment at competitive prices. Knowing his income and the prices, you can compute his demand in competitive equilibrium. Having solved for Ken’s consumption and knowing that total consumption by Ken and Barbie equals the sum of their endowments, it should be easy to ﬁnd Barbie’s consumption.) (g) On the Edgeworth box for Ken and Barbie, draw in the competitive equilibrium allocation and draw Ken’s competitive budget line (with black ink). 30.5 (0) Linus Straight’s utility function is U (a, b) = a + 2b, where a is his consumption of apples and b is his consumption of bananas. Lucy Kink’s utility function is U (a, b) = min{a, 2b}. Lucy initially has 12 apples and no bananas. Linus initially has 12 bananas and no apples. In the Edgeworth box below, goods for Lucy are measured from the upper right corner of the box and goods for Linus are measured from the lower left corner. Label the initial endowment point on the graph with the letter E. Draw two of Lucy’s indiﬀerence curves in red ink and two of Linus’s indiﬀerence curves in blue ink. Use black ink to draw a line through all of the Pareto optimal allocations. 6 6 (b) Use blue ink to draw an indiﬀerence curve for Ken that shows allocations in which his utility is 6. Use red ink to draw an indiﬀerence curve for Barbie that shows allocations in which her utility is 6. (c) At any Pareto optimal allocation where both consume some of each good, Ken’s marginal rate of substitution between quiche and wine must equal Barbie’s. Write an equation that states this condition in terms WB /QB = (d) On your graph, show the locus of points that are Pareto eﬃcient. (Hint: If two people must each consume two goods in the same proportions as each other, and if together they must consume twice as much wine as quiche, what must those proportions be?) (e) In this example, at any Pareto eﬃcient allocation, where both persons consume both goods, the slope of Ken’s indiﬀerence curve will be −2. Therefore, since we know that competitive equilibrium must be Pareto eﬃcient, we know that at a competitive equilibrium, pQ /pW = 2. (f ) In competitive equilibrium, Ken’s consumption bundle must be quiche, 4 wine. 2 How about Barbie’s consumption bundle? 2 quiche, 4 wine. (Hint: You found competitive equilibrium prices above. You know Ken’s initial endowment and you know the 376 EXCHANGE (Ch. 30) consumption. In competitive equilibrium, total consumption of apples equals the total supply of apples and total consumption of bananas equals the total supply of bananas. Therefore Lucy will consume 12 − as apples and 12 −bs bananas. At a competitive equilibrium, Lucy will be consuming at one of her kink points. The kinks occur at bundles where Lucy consumes 2 Therefore we know that apples for every banana that she consumes. 12−as 12−bs 2. = (c) You can solve the two equations that you found above to ﬁnd the quantities of apples and bananas consumed in competitive equilibrium by Linus and Lucy. Linus will consume 9 6 units of bananas. Lucy will consume and 3 units of bananas. units of apples and 6 units of apples Lucy 30.6 (0) Consider a pure exchange economy with two consumers and two goods. At some given Pareto eﬃcient allocation it is known that both consumers are consuming both goods and that consumer A has a marginal rate of substitution between the two goods of −2. What is consumer B’s Black line Blue curves 0 Linus 4 Quiche WK /QK . √1 . 3.2 Bananas 12 e Red curves 2 of the consumptions of each good by each person. TI = 3.2, TN = 12.8, platitude price/bromide price = 0 Ken marginal rate of substitution between these two goods? 12 Apples −2. 30.7 (0) Charlotte loves apples and hates bananas. Her utility function is U (a, b) = a − 14 b2 , where a is the number of apples she consumes and b is the number of bananas she consumes. Wilbur likes both apples and √ bananas. His utility function is U (a, b) = a+2 b. Charlotte has an initial endowment of no apples and 8 bananas. Wilbur has an initial endowment of 16 apples and 8 bananas. (a) In this economy, in competitive equilibrium, the ratio of the price of apples to the price of bananas must be 1/2. (b) Let aS be Linus’s consumption of apples and let bS be his consumption of bananas. At competititive equilibrium, Linus’s consumption will have to satisfy the budget constraint, as + 2 bS = 24. This gives us one equation in two unknowns. To ﬁnd a second equation, consider Lucy’s (a) On the graph below, mark the initial endowment and label it E. Use red ink to draw the indiﬀerence curve for Charlotte that passes through this point. Use blue ink to draw the indiﬀerence curve for Wilbur that passes through this point. NAME 377 EXCHANGE (Ch. 30) (a) Sketch an Edgeworth box for Mutt and Jeﬀ. Use blue ink to show a couple of indiﬀerence curves for each. Use red ink to show the locus of Pareto optimal allocations. (Hint: Look for boundary solutions.) Wilbur Apples 378 16 Black line 12 Red line Jeff Juice 8 8 Blue line 4 Red point 6 e 0 Charlotte 4 8 12 Blue curves (Jeff) 4 16 Bananas Blue curves (Mutt) can Charlotte be consuming at a Pareto optimal allocation? 0. On the graph above, use black ink to mark the locus of Pareto optimal allocations of apples and bananas between Charlotte and Wilbur. 0 Mutt (c) We know that a competitive equilibrium allocation must be Pareto optimal and the total consumption of each good must equal the total supply, so we know that at a competitive equilibrium, Wilbur must be consuming 16 1/4 4 6 8 Milk and his marginal utility of apples will be 1. If apples are the numeraire, then the only price of bananas at which he will want to consume exactly 16 1/4. 2 bananas. If Wilbur is consuming this number of bananas, his marginal utility for bananas will be bananas is Red point 2 (b) If Charlotte hates bananas and Wilbur likes them, how many bananas In competitive equilibrium, for the Charlotte-Wilbur economy, Wilbur will consume 16 0 and Charlotte will consume bananas and bananas and 14 2 apples apples. 30.8 (0) Mutt and Jeﬀ have 8 cups of milk and 8 cups of juice to divide between themselves. Each has the same utility function given by u(m, j) = max{m, j}, where m is the amount of milk and j is the amount of juice that each has. That is, each of them cares only about the larger of the two amounts of liquid that he has and is indiﬀerent to the liquid of which he has the smaller amount. NAME 379 30.9 (1) Remember Tommy Twit from Chapter 3. Tommy is happiest when he has 8 cookies and 4 glasses of milk per day and his indiﬀerence curves are concentric circles centered around (8,4). Tommy’s mother, Mrs. Twit, has strong views on nutrition. She believes that too much of anything is as bad as too little. She believes that the perfect diet for Tommy would be 7 glasses of milk and 2 cookies per day. In her view, a diet is healthier the smaller is the sum of the absolute values of the diﬀerences between the amounts of each food consumed and the ideal amounts. For example, if Tommy eats 6 cookies and drinks 6 glasses of milk, Mrs. Twit believes that he has 4 too many cookies and 1 too few glasses of milk, so the sum of the absolute values of the diﬀerences from her ideal amounts is 5. On the axes below, use blue ink to draw the locus of combinations that Mrs. Twit thinks are exactly as good for Tommy as (6, 6). Also, use red ink to draw the locus of combinations that she thinks is just as good as (8, 4). On the same graph, use red ink to draw an indiﬀerence “curve” representing the locus of combinations that Tommy likes just as well as 7 cookies and 8 glasses of milk. 380 EXCHANGE (Ch. 30) On the planet Drongo there is just one commodity, cake, and two time periods. There are two kinds of creatures, “old” and “young.” Old creatures have an income of I units of cake in period 1 and no income in period 2. Young creatures have no income in period 1 and an income of I ∗ units of cake in period 2. There are N1 old creatures and N2 young creatures. The consumption bundles of interest to creatures are pairs (c1 , c2 ), where c1 is cake in period 1 and c2 is cake in period 2. All creatures, old and young, have identical utility functions, representing preferences over cake in the two periods. This utility function is U (c1 , c2 ) = ca1 c21−a , where a is a number such that 0 ≤ a ≤ 1. Milk 16 14 12 Red curve Blue curve 10 8 4 bundle (c1 , c2 ). Tommy's red curve ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; Black;;;;;;;;;;;; ;;;;;;;;;;;; line;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; ;;;;;;;;;;;; 6 (a) If current cake is taken to be the numeraire, (that is, its price is set at 1), write an expression for the present value of a consumption c1 + c2/(1 + r). of income for old creatures I Write down the present value and for young creatures I ∗ /(1 + r). The budget line for any creature is determined by the condition that the present value of its consumption bundle equals the present value of its income. Write down this budget equation for old c1 + c2/(1 + r) = I creatures: 2 and for young creatures: ∗ c1 + c2/(1 + r) = I /(1 + r). (b) If the interest rate is r, write down an expression for an old creature’s 0 2 4 6 8 10 12 14 16 Cookies (a) On the graph, shade in the area consisting of combinations of cookies and milk that both Tommy and his mother agree are better than 7 cookies and 8 glasses of milk, where “better” for Mrs. Twit means she thinks it is healthier, and where “better” for Tommy means he likes it better. (b) Use black ink to sketch the locus of “Pareto optimal” bundles of cookies and milk for Tommy. In this situation, a bundle is Pareto optimal if any bundle that Tommy prefers to this bundle is a bundle that Mrs. Twit thinks is worse for him. The locus of Pareto optimal points that you just drew should consist of two line segments. These run from the point (8,4) to the point 5,7 and from that point to the point 2,7. 30.10 (2) This problem combines equilibrium analysis with some of the things you learned in the chapter on intertemporal choice. It concerns the economics of saving and the life cycle on an imaginary planet where life is short and simple. In advanced courses in macroeconomics, you would study more-complicated versions of this model that build in more earthly realism. For the present, this simple model gives you a good idea of how the analysis must go. demand for cake in period 1 c1 = aI and in period 2 c2 = (1−a)I(1+r). Write an expression for a young creature’s demand for cake in period 1 c1 = aI ∗ /(1 + r) and in period 2 c2 = ∗ (1 − a)I . (Hint: If its budget line is p1 c1 + p2 c2 = W and its utility function is of the form proposed above, then a creature’s demand function for good 1 is c1 = aW/p and demand for good 2 is c2 = (1 − a)W/p.) If the interest rate is zero, how much cake would a young creature choose in period 1? aI ∗ . For what value of a would it choose the same amount in each period if the interest rate is zero? a = 1/2. If a = .55, what would r have to be in order that young creatures would want to consume the same amount in each period? .22. (c) The total supply of cake in period 1 equals the total cake earnings of all old creatures, since young creatures earn no cake in this period. There are N1 old creatures and each earns I units of cake, so this total is N1 I. Similarly, the total supply of cake in period 2 equals the total amount earned by young creatures. This amount is N2 I ∗ . NAME 381 382 EXCHANGE (Ch. 30) 384 PRODUCTION (d) At the equilibrium interest rate, the total demand of creatures for period-1 cake must equal total supply of period-1 cake, and similarly the demand for period-2 cake must equal supply. If the interest rate is r, then aI the demand for period-1 cake by each old creature is and the ∗ demand for period-1 cake by each young creature is aI /(1 + r). Since there are N1 old creatures and N2 young creatures, the total demand for period-1 cake at interest rate r is N1 aI + N2 aI ∗ /(1 + r). (e) Using the results of the last section, write an equation that sets the demand for period-1 cake equal to the supply. r) = N1 I. N1 aI +N2 aI ∗ /(1+ Write a general expression for the equilibrium value of r, given N1 , N2 , I, and I ∗ . r= N2 I ∗ a − 1. Solve this equation N1 I(1−a) for the special case when N1 = N2 and I = I ∗ and a = 11/21. r = 10%. (f ) In the special case at the end of the last section, show that the interest rate that equalizes supply and demand for period-1 cake will also equalize supply and demand for period-2 cake. (This illustrates Walras’s law.) Supply = demand for period 2 if a)I(1 + r) + N2 (1 − a)I ∗ = N2 I ∗ . and N1 (1 − If N1 = N2 I = I ∗, then (1 − a)(1 + r) + (1 − a) = 1. If a = 11/21, then Chapter 31 r = 10%. (Ch. 31) NAME Production Problems 80 Introduction. In this section we explore economywide production possibility sets. We pay special attention to the principle of comparative advantage. The principle is simply that eﬃciency suggests that people should specialize according to their relative abilities in diﬀerent activities rather than absolute abilities. 60 40 30 Joint 20 18 Example: For simplicity, let us imagine an island with only two people on it, both of them farmers. They do not trade with the outside world. Farmer A has 100 acres and is able to grow two crops, wheat and hay. Each acre of his land that he plants to wheat will give him 50 bushels of wheat. Each acre of his land that he plants to hay will give him 2 tons of hay. Farmer B also has 100 acres, but his land is not so good. Each acre of his land yields only 20 bushels of wheat and only 1 ton of hay. Notice that, although Farmer A’s land is better for both wheat and hay, Farmer B’s land has comparative advantage in the production of hay. This is true because the ratio of tons of hay to bushels of wheat per acre 2/50 = .04 for Farmer A and 1/20 = .05 for Farmer B. Farmer A, on the other hand, has comparative advantage in the production of wheat, since the ratio of bushels of wheat to tons of hay is 50/2 = 25 for Farmer A and 20/1 = 20 for Farmer B. The eﬃcient way to arrange production is to have Farmer A “specialize” in wheat and farmer B “specialize” in hay. If Farmer A devotes all of his land to wheat and Farmer B devotes all of his land to hay, then total wheat production will be 5,000 bushels and total hay production will be 100 tons. Suppose that they decide to produce only 4,000 bushels of wheat. Given that they are going to produce 4,000 bushels of wheat, the most hay they can possibly produce together will be obtained if Farmer A devotes 80 acres to wheat and 20 acres to hay while Farmer B devotes all of his land to hay. Suppose that they decide to produce 6,000 bushels of wheat. Then they will get the most hay possible given that they are producing 6,000 bushels of wheat if Farmer A puts all of his land into wheat and Farmer B puts 50 acres into wheat and the remaining 50 acres into hay. 31.1 (0) Tip and Spot ﬁnally got into college. Tip can write term papers at the rate of 10 pages per hour and solve workbook problems at the rate of 3 per hour. Spot can write term papers at the rate of 6 pages per hour and solve workbook problems at the rate of 2 per hour. Which of 12 Tip Spot 0 20 36 40 60 80 96 100 120 Pages (a) Tip and Spot each work 6 hours a day. They decide to work together and to produce a combination of term papers and workbook problems that lies on their joint production possibility frontier. On the above graph plot their joint production possibility frontier. If they produce less than 60 pages of term papers, then If they produce more than Tip 60 will write all of the term papers. pages of term papers, then will continue to specialize in writing term papers and write some term papers. Spot Tip will also 31.2 (0) Robinson Crusoe has decided that he will spend exactly 8 hours a day gathering food. He can either spend this time gathering coconuts or catching ﬁsh. He can catch 1 ﬁsh per hour and he can gather 2 coconuts per hour. On the graph below, show Robinson’s production possibility frontier between ﬁsh and coconuts per day. Write an equation these two has comparative advantage in solving workbook problems? for the line segment that is Robinson’s production possibility frontier. Spot. F + C/2 = 8. NAME 385 16 (Ch. 31) a ﬁsh. At wages below $ 1 per hour, I would want to produce inﬁnitely many ﬁsh since I would make a proﬁt on every one. So the only possible wage rate that would make me choose to produce a positive Utility of 4 Production possibility frontier ﬁnite amount of ﬁsh is $ 1 per hour. Now what would the price of coconuts have to be to induce me to produce a positive number of coconuts. At the wage rate I just found, the cost of producing a coconut 8 is $.50. At this price and only at this price, would I be willing to produce a ﬁnite positive number of coconuts.” 4 Utility of 8 0 PRODUCTION one ﬁsh or 2 coconuts. At wages above $ 1 per hour, I wouldn’t produce any ﬁsh at all, because it would cost me more than $1 to produce Coconuts 12 386 4 8 12 16 Fish (a) Robinson’s utility function is U (F, C) = F C, where F is his daily ﬁsh consumption and C is his daily coconut consumption. On the graph above, sketch the indiﬀerence curve that gives Robinson a utility of 4, and also sketch the indiﬀerence curve that gives him a utility of 8. How many ﬁsh will Robinson choose to catch per day? 4. How many 31.3 (0) We continue the story of Robinson Crusoe from the previous problem. One day, while walking along the beach, Robinson Crusoe saw a canoe in the water. In the canoe was a native of a nearby island. The native told Robinson that on his island there were 100 people and that they all lived on ﬁsh and coconuts. The native said that on his island, it takes 2 hours to catch a ﬁsh and 1 hour to ﬁnd a coconut. The native said that there was a competitive economy on his island and that ﬁsh were the numeraire. The price of coconuts on the neighboring island must have been $.50. The native oﬀered to trade with Crusoe at these prices. “I will trade you either ﬁsh for coconuts or coconuts for ﬁsh at coconuts will he collect? 8. (Hint: Robinson will choose a bundle that maximizes his utility subject to the constraint that the bundle lies in his production possibility set. But for this technology, his production possibility set looks just like a budget set.) the exchange rate of 2 coconuts for a ﬁsh,” said he. “But you will have to give me 1 ﬁsh as payment for rowing over to your island.” (b) Suppose Robinson is not isolated on an island in the Paciﬁc, but is retired and lives next to a grocery store where he can buy either ﬁsh or coconuts. If ﬁsh cost $1 per ﬁsh, how much would coconuts have to cost in ﬁsh and sell coconuts or vice versa? prices are the same as the order that he would choose to consume twice as many coconuts as ﬁsh? he can transform the two goods, he can gain $.50. nothing by Suppose that a social planner decided that he wanted Robinson to consume 4 ﬁsh and 8 coconuts per day. He could do this by setting the price of ﬁsh equal to $1, the price of coconuts equal to 8 giving Robinson a daily income of $ $.50 and . (c) Back on his island, Robinson has little else to do, so he pretends that he is running a competitive ﬁrm that produces ﬁsh and coconuts. He wonders, “What would the price have to be to make me do just what I am actually doing? Let’s assume that ﬁsh are the numeraire and have a price of $1. And let’s pretend that I have access to a competitive labor market where I can hire as much labor as I want at some given wage. There is a constant returns to scale technology. An hour’s labor produces NAME 387 show the bundles he can aﬀord if he chooses to trade and specializes appropriately. Remember to take away 2 ﬁsh to pay the trader. Would Robinson gain by trading with him? No. If so, would he buy Neither. Since their rate at which trading. (a) Several days later, Robinson saw another canoe in the water on the other side of his island. In this canoe was a native who came from a diﬀerent island. The native reported that on his island, one could catch only 1 ﬁsh for every 4 hours of ﬁshing and that it takes 1 hour to ﬁnd a coconut. This island also had a competitive economy. The native oﬀered to trade with Robinson at the same exchange rate that prevailed on his own island, but said that he would have to have 2 ﬁsh in return for rowing between the islands. If Robinson decides to trade with this island, he chooses to produce only fish and will get his coconuts from the other island. On the graph above, use black ink to draw Robinson’s production possibility frontier if he doesn’t trade and use blue ink to 388 PRODUCTION (Ch. 31) Milk 80 Coconuts 16 12 60 Utility of 4 Production possibility frontier Red line 40 Pencil line 8 20 Blue line Black line 4 Utility of 8 0 0 4 8 12 15 10 20 40 60 80 Wheat 16 Fish (b) Write an equation for Crusoe’s “budget line” if he specializes appropriately and trades with the second trader. If he does this, what bundle will he choose to consume? 3 fish, 12 coconuts. Does he like this bundle better than the bundle he would have if he didn’t trade? Yes. (b) On the diagram below, use black ink to draw the production possibility set for a typical farmer from the south side of the island. Given the equilibrium prices, will this farmer specialize in milk, specialize in wheat, or produce both goods? Specialize in wheat. Use blue ink to draw the budget that he faces in his role as a consumer if he makes the optimal choice of what to produce. Milk 80 31.4 (0) The Isle of Veritas has made it illegal to trade with the outside world. Only two commodities are consumed on this island, milk and wheat. On the north side of the island are 40 farms. Each of these farms can produce any combination of non-negative amounts of milk and wheat that satisﬁes the equation m = 60 − 6w. On the south side of the island are 60 farms. Each of these farms can produce any combination of non-negative amounts of milk and wheat that satisﬁes the equation m = 40 − 2w. The economy is in competitive equilibrium and 1 unit of wheat exchanges for 4 units of milk. (a) On the diagram below, use black ink to draw the production possibility set for a typical farmer from the north side of the island. Given the equilibrium prices, will this farmer specialize in milk, specialize in wheat, or produce both goods? Specialize in milk. Use blue ink to draw the budget that he faces in his role as a consumer if he makes the optimal choice of what to produce. 60 Blue line Red line 40 Black line 20 0 Pencil line 20 40 60 80 Wheat (c) Suppose that peaceful Viking traders discover Veritas and oﬀer to exchange either wheat for milk or milk for wheat at an exchange rate of NAME 389 1 unit of wheat for 3 units of milk. If the Isle of Veritas allows free trade with the Vikings, then this will be the new price ratio on the island. At this price ratio, would either type of farmer change his output? No. 390 PRODUCTION Bubble gum (d) On the ﬁrst of the two graphs above, use red ink to draw the budget for northern farmers if free trade is allowed and the farmers make the right choice of what to produce. On the second of the two graphs, use red ink to draw the budget for southern farmers if free trade is allowed and the farmers make the right choice of what to produce. 20 In favor. 10 5 How will the southerners vote? How is it that you can make a deﬁnite answer to the last two questions without knowing anything about the farmers’s consumption 0 The change strictly enlarges the budget set for northerners and strictly shrinks it for southerners. (f ) Suppose that instead of oﬀering to make exchanges at the rate of 1 unit of wheat for 3 units of milk, the Vikings had oﬀered to trade at the price of 1 unit of wheat for 1 unit of milk and vice versa. Would either type of farmer change his output? Use pencil to sketch the budget line for each kind of farmer at these prices if he makes the right production decision. How will the northerners vote now? In favor. How will the southerners vote now? Depends preferences about consumption. Explain why it is that your answer to one of the last two questions has to be “it depends.” The two alternative budget lines for southerners are not nested. NAME 391 (g) If Harold were spending all of its money of both colors on ambrosia and it decided to purchase a little bit of bubble gum, which currency The blue currency. (h) How much bubble gum could it buy before it ran out of that color money? He could buy 9 units of bubble gum. (i) What would be the slope of this budget line before it ran out of that kind of money? ;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;; 9 5 10 12 15 20 Ambrosia (b) The Free Choice party campaigns on a platform that Mungoans should be allowed to purchase goods at either the blue-money price or the redmoney price, whichever they prefer. We want to construct Harold’s budget set if this reform is instituted. To begin with, how much bubble gum could Harold consume if it spent all of its blue money and its red money on bubble gum? 15 units of bubble gum. (c) How much ambrosia could it consume if it spent all of its blue money and all of its red money on ambrosia? 21 units of ambrosia. (d) If Harold were spending all of its money of both colors on bubble gum and it decided to purchase a little bit of ambrosia, which currency would it use? The red currency. (e) How much ambrosia could it buy before it ran out of that color money? 31.5 (0) Recall our friends the Mungoans of Chapter 2. They have a strange two-currency system consisting of Blue Money and Red Money. Originally, there were two prices for everything, a blue-money price and a red-money price. The blue-money prices are 1 bcu per unit of ambrosia and 1 bcu per unit of bubble gum. The red-money prices are 2 rcu’s per unit of ambrosia and 4 rcu’s per unit of bubble gum. would it use? Part a budget set ;;;;;;;;;;; Yes. Southerners would now switch to specializing in milk. on their (12,9) 9 6 Against. preferences? Part j budget set 15 (e) The council of elders of Veritas will meet to vote on whether to accept the Viking oﬀer. The elders from the north end of the island get 40 votes and the elders from the south end get 60 votes. Assuming that everyone votes in the selﬁsh interest of his end of the island, how will the northerners vote? (Ch. 31) (a) Harold has a blue income of 9 and a red income of 24. If it has to pay in both currencies for any purchase, draw its budget set in the graph below. (Hint: You answered this question a few months ago.) The slope would be −1. (j) Use your answers to the above questions to draw Harold’s budget set in the above graph if it could purchase bubble gum and ambrosia using either currency. 12 units of ambrosia. (f ) What would be the slope of this budget line before it ran out of that kind of money? 392 PRODUCTION The slope would be − 12 . (Ch. 31) 394 Chapter 32 NAME WELFARE (Ch. 32) (a) Is the social preference ordering deﬁned in this way complete? Welfare Yes. Reﬂexive? Transitive? Yes. Yes. Introduction. Here you will look at various ways of determining social (b) If everyone prefers x to y, will the Borda count rank x as socially preferences. You will check to see which of the Arrow axioms for aggregating individual preferences are satisﬁed by these welfare relations. You will also try to ﬁnd optimal allocations for some given social welfare functions. The method for solving these last problems is analogous to solving for a consumer’s optimal bundle given preferences and a budget constraint. Two hints. Remember that for a Pareto optimal allocation inside the Edgeworth box, the consumers’ marginal rates of substitution will be equal. Also, in a “fair allocation,” neither consumer prefers the other consumer’s bundle to his own. preferred to y? Explain your answer. Yes. If everybody ranks x ahead of y, then everyone must give x a higher rank than y. Then the sum of the ranks of x must be larger than the sum of the ranks of y. Example: A social planner has decided √ that √she wants to allocate income between 2 people so as to maximize Y1 + Y2 where Yi is the amount of income that person i gets. Suppose that the planner has a ﬁxed amount of money to allocate and that she can enforce any income distribution such that Y1 + Y2 = W , where W is some ﬁxed amount. This planner would have ordinary convex indiﬀerence curves between Y1 and Y2 and a “budget constraint” where the “price” of income for each person is 1. Therefore the planner would set her marginal rate of substitution between income for the two people equal to the relative price which is 1. When you solve this, you will ﬁnd that she sets Y1 = Y2 = W/2. Suppose instead that it is “more expensive” for the planner to give money to person 1 than to person 2. (Perhaps person 1 is forgetful and loses money, or perhaps person 1 is frequently robbed.) For example, suppose√ that the √ planner’s budget is 2Y1 + Y2 = W . Then the planner maximizes Y1 + Y2 subject to 2Y1 + Y2 = W . Setting her MRS equal to the price ratio, we ﬁnd that √ √Y2 = 2. So Y2 = 4Y1 . Therefore the planner makes Y1 = W/5 and Y1 Y2 = 4W/5. 32.1 (2) One possible method of determining a social preference relation is the Borda count, also known as rank-order voting. Each voter is asked to rank all of the alternatives. If there are 10 alternatives, you give your ﬁrst choice a 1, your second choice a 2, and so on. The voters’ scores for each alternative are then added over all voters. The total score for an alternative is called its Borda count. For any two alternatives, x and y, if the Borda count of x is smaller than or the same as the Borda count for y, then x is “socially at least as good as” y. Suppose that there are a ﬁnite number of alternatives to choose from and that every individual has complete, reﬂexive, and transitive preferences. For the time being, let us also suppose that individuals are never indiﬀerent between any two diﬀerent alternatives but always prefer one to the other. NAME 395 (c) Suppose that there are two voters and three candidates, x, y, and z. Suppose that Voter 1 ranks the candidates, x ﬁrst, z second, and y third. Suppose that Voter 2 ranks the candidates, y ﬁrst, x second, and z third. What is the Borda count for x? 3. For y? 4. For z? 5. Now suppose that it is discovered that candidate z once lifted a beagle by the ears. Voter 1, who has rather large ears himself, is appalled and changes his ranking to x ﬁrst, y second, z third. Voter 2, who picks up his own children by the ears, is favorably impressed and changes his ranking to y ﬁrst, z second, x third. Now what is the Borda count for x? 4. 3. For y? 5. For z? (d) Does the social preference relation deﬁned by the Borda count have the property that social preferences between x and y depend only on how people rank x versus y and not on how they rank other alternatives? Ex- No. In the above example, the ranking plain. of z changed, but nobody changed his mind about whether x was better than y or versa. vice Before the change x beat y, and after the change y beat x. 32.2 (2) Suppose the utility possibility frontier for two individuals is given by UA + 2UB = 200. On the graph below, plot the utility frontier. 396 WELFARE (Ch. 32) √ √ (a) The parent’s utility function is U (a, b) = a + b, where a is the amount of money she gives to A and b is the amount of money she gives UB to B. How will she choose to divide the money? 200 Black line a = b = $500. (b) Suppose that her utility function is U (a, b) = − a1 − 1b . How will she 150 choose to divide the money? a = b = $500. (c) Suppose that her utility function is U (a, b) = log a + log b. How will 100 Red line she choose to divide the money? (d) Suppose that her utility function is U (a, b) = min{a, b}. How will she Blue line 50 choose to divide the money? Utility frontier 0 50 100 200 150 choose to divide the money? 200 UA 66.66 0. and UB equal to and UB equals a = $1, 000, b = 0, or vice versa. (b) If instead we use a Rawlsian criterion, W (UA , UB ) = min{UA , UB }, then the social welfare function is maximized on the above utility possibility frontier where UA equals a = b = $500. (e) Suppose that her utility function is U (a, b) = max{a, b}. How will she (a) In order to maximize a “Nietzschean social welfare function,” W (UA , UB ) = max{UA , UB }, on the utility possibility frontier shown above, one would set UA equal to a = b = $500. 66.66. 1/2 1/2 (c) Suppose that social welfare is given by W (UA , UB ) = UA UB . In this case, with the above utility possibility frontier, social welfare is maximized where UA equals 100 and UB is 50. (Hint: You might want to think about the similarities between this maximization problem and the consumer’s maximization problem with a Cobb-Douglas utility function.) (d) Show the three social maxima on the above graph. Use black ink to draw a Nietzschean isowelfare line through the Nietzschean maximum. Use red ink to draw a Rawlsian isowelfare line through the Rawlsian maximum. Use blue ink to draw a Cobb-Douglas isowelfare line through the Cobb-Douglas maximum. (Hint: In each of the above cases, we notice that the parent’s problem is to maximize U (a, b) subject to the constraint that a + b = 1, 000. This is just like the consumer problems we studied earlier. It must be that the parent sets her marginal rate of substitution between a and b equal to 1 since it costs the same to give money to each child.) (f ) Suppose that her utility function is U (a, b) = a2 + b2 . How will she choose to divide the money between her children? Explain why she doesn’t set her marginal rate of substitution equal to 1 in this case. She gives everything to one child. Her preferences are not convex, indifference curves are quarter circles. 32.4 (2) In the previous problem, suppose that A is a much more eﬃcient shopper than B so that A is able to get twice as much consumption goods as B can for every dollar that he spends. Let a be the amount of consumption goods that A gets and b the amount that B gets. We will measure consumption goods so that one unit of consumption goods costs $1 for A and $2 for B. Thus the parent’s budget constraint is a + 2b = 1, 000. (a) If the mother’s utility function is U (a, b) = a + b, which child will get 32.3 (2) A parent has two children named A and B and she loves both of them equally. She has a total of $1,000 to give to them. more money? A. Which child will consume more goods? A. NAME 397 (b) If the mother’s utility function is U (a, b) = a × b, which child will get They get the same amount of money. more money? A consumes more. Which child will get to consume more? (c) If the mother’s utility function is U (a, b) = − a1 − 1b , which child will get more money? B gets more money. Which child will get They consume the same amount. to consume more? (d) If the mother’s utility function is U (a, b) = max{a, b}, which child A. will get more money? 398 WELFARE (Ch. 32) (c) Derive an equation for the slope of the above utility possibility curve. dUR = −2UN . dUN (d) Both Ralph and Norton believe that the ideal allocation is given by maximizing an appropriate social welfare function. Ralph thinks that UR = 75, UN = 5 is the best distribution of welfare, and presents the maximization solution to a weighted-sum-of-the-utilities social welfare function that conﬁrms this observation. What was Ralph’s social welfare function? (Hint: What is the slope of Ralph’s social welfare function?) W = UR + 10UN . Which child will get to consume more? (e) Norton, on the other hand, believes that UR = 19, UN = 9 is the A. best distribution. What is the social welfare function Norton presents? (e) If the mother’s utility function is U (a, b) = min{a, b}, which child will B. get more money? Which child will get to consume more? They consume the same amount. Calculus 32.5 (1) Norton and Ralph have a utility possibility frontier that is given 2 by the following equation, UR + UN = 100 (where R and N signify Ralph and Norton respectively). (a) If we set Norton’s utility to zero, what is the highest possible utility 100. Ralph can achieve? the best Norton can do? W = UR + 18UN . 32.6 (2) Roger and Gordon have identical utility functions, U (x, y) = x2 + y 2 . There are 10 units of x and 10 units of y to be divided between them. Roger has blue indiﬀerence curves. Gordon has red ones. (a) Draw an Edgeworth box showing some of their indiﬀerence curves and mark the Pareto optimal allocations with black ink. (Hint: Notice that the indiﬀerence curves are nonconvex.) If we set Ralph’s utility to zero, what is Gordon y 10 10. Fair Black lines (b) Plot the utility possibility frontier on the graph below. Red curves Ralph's utility 100 Blue curves 75 Black lines Fair 50 25 0 10 Roger x (b) What are the fair allocations in this case? 0 5 10 32.7 (2) Paul and David consume apples and oranges. Paul’s utility function is UP (AP , OP ) = 2AP + OP and David’s utility function is 15 20 Norton's utility NAME 399 UD (AD , OD ) = AD + 2OD , where AP and AD are apple consumptions for Paul and David, and OP and OD are orange consumptions for Paul and David. There are a total of 12 apples and 12 oranges to divide between Paul and David. Paul has blue indiﬀerence curves. David has red ones. Draw an Edgeworth box showing some of their indiﬀerence curves. Mark the Pareto optimal allocations on your graph. ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; Blue shading ;;;;;;;;;;;;;;;;;;;;;;;;; Pareto ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; optimal Red curves ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; Blue ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; curves ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; Red shading;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; Fair ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;; 0 Pareto optimal Apples (a) Write one inequality that says that Paul likes his own bundle as well as he likes David’s and write another inequality that says that David likes and WELFARE (Ch. 32) also likes Juliet to be happy and he knows that spaghetti makes her happy. Juliet likes spaghetti, but she also likes Romeo to be happy and she knows that spaghetti makes Romeo happy. Romeo’s utility function a 1−a is UR (SR , SJ ) = SR SJ and Juliet’s utility function is UJ (SJ , SR ) = 1−a SJa SR , where SJ and SR are the amount of spaghetti for Romeo and the amount of spaghetti for Juliet respectively. There is a total of 24 units of spaghetti to be divided between Romeo and Juliet. (a) Suppose that a = 2/3. If Romeo got to allocate the 24 units of spaghetti exactly as he wanted to, how much would he give himself? 16. How much would he give Juliet? 8. (Hint: Notice that this problem is formally just like the choice problem for a consumer with a Cobb-Douglas utility function choosing between two goods with a budget constraint. What is the budget constraint?) (b) If Juliet got to allocate the spaghetti exactly as she wanted to, how much would she take for herself? Romeo? 16. How much would she give 8. 12 Paul his own bundle as well as he likes Paul’s. 400 David Oranges 12 See diagram. 2AP +OP ≥ 2AD +OD AD + 2OD ≥ AP + 2OP . a gift from one to the other.) The Pareto optimal allocations are all of the allocations in which each person gets at least 8 units of (b) Use the fact that at feasible allocations, AP +AD = 12 and OP +OD = 12 to eliminate AD and OD from the ﬁrst of these equations. Write the resulting inequality involving only the variables AP and OP . Now in your Edgeworth box, use blue ink to shade in all of the allocations such that Paul prefers his own allocation to David’s. (c) What are the Pareto optimal allocations? (Hint: An allocation will not be Pareto optimal if both persons’ utility will be increased by 2AP + OP ≥ 18. (c) Use a procedure similar to that you used above to ﬁnd the allocations where David prefers his own bundle to Paul’s. Describe these points spaghetti. (d) When we had to allocate two goods between two people, we drew an Edgeworth box with indiﬀerence curves in it. When we have just one good to allocate between two people, all we need is an “Edgeworth line” and instead of indiﬀerence curves, we will just have indiﬀerence dots. Consider the Edgeworth line below. Let the distance from left to right denote spaghetti for Romeo and the distance from right to left denote spaghetti for Juliet. with an inequality and shade them in on your diagram with red ink. AD + 2OD ≥ 18. (d) On your Edgeworth box, mark the fair allocations. 32.8 (3) Romeo loves Juliet and Juliet loves Romeo. Besides love, they consume only one good, spaghetti. Romeo likes spaghetti, but he (e) On the Edgeworth line you drew above, show Romeo’s favorite point and Juliet’s favorite point. NAME 401 (f ) Suppose that a = 1/3. If Romeo got to allocate the spaghetti, how much would he choose for himself? 8. If Juliet got to allocate the spaghetti, how much would she choose for herself? 8. Label the Edgeworth line below, showing the two people’s favorite points and the locus of Pareto optimal points. 402 WELFARE (Ch. 32) (c) If it is possible to throw away some whiskey and they must consume equal amounts of whiskey, how much should they throw away? 3 quarts. (g) When a = 1/3, at the Pareto optimal allocations what do Romeo and Juliet disagree about? Romeo wants to give spaghetti to Juliet, but she doesn’t want to take it. Juliet wants to give spaghetti to Romeo, but he doesn’t want to take it. Both like spaghetti for themselves, but would rather the other had it. 32.9 (2) Hatﬁeld and McCoy hate each other but love corn whiskey. Because they hate for each other to be happy, each wants the other to 2 have less whiskey. Hatﬁeld’s utility function is UH (WH , WM ) = WH −WM 2 and McCoy’s utility function is UM (WM , WH ) = WM − WH , where WM is McCoy’s daily whiskey consumption and WH is Hatﬁeld’s daily whiskey consumption (both measured in quarts). There are 4 quarts of whiskey to be allocated. (a) If McCoy got to allocate all of the whiskey, how would he allocate it? All for himself. If Hatﬁeld got to allocate all of the whiskey, how would he allocate it? All for himself. (b) If each of them gets 2 quarts of whiskey, what will the utility of each of them be? −2. If a bear spilled 2 quarts of their whiskey and they divided the remaining 2 quarts equally between them, what would the utility of each of them be? 0. If it is possible to throw away some of the whiskey, is it Pareto optimal for them each to consume 2 quarts of whiskey? No. 404 Chapter 33 EXTERNALITIES (Ch. 33) NAME AP, MP 12 Externalities 10 8 Introduction. When there are externalities, the outcome from independently chosen actions is typically not Pareto eﬃcient. In these exercises, you explore the consequences of alternative mechanisms and institutional arrangements for dealing with externalities. Example: A large factory pumps its waste into a nearby lake. The lake is also used for recreation by 1,000 people. Let X be the amount of waste that the ﬁrm pumps into the lake. Let Yi be the number of hours per day that person i spends swimming and boating in the lake, and let Ci be the number of dollars that person i spends on consumption goods. If the ﬁrm pumps X units of waste into the lake, its proﬁts will be 1, 200X − 100X 2 . Consumers have identical utility functions, U (Yi , Ci , X) = Ci + 9Yi − Yi2 − XYi , and identical incomes. Suppose that there are no restrictions on pumping waste into the lake and there is no charge to consumers for using the lake. Also, suppose that the factory and the consumers make their decisions independently. The factory will maximize its proﬁts by choosing X = 6. (Set the derivative of proﬁts with respect to X equal to zero.) When X = 6, each consumer maximizes utility by choosing Yi = 1.5. (Set the derivative of utility with respect to Yi equal to zero.) Notice from the utility functions that when each person is spending 1.5 hours a day in the lake, she will be willing to pay 1.5 dollars to reduce X by 1 unit. Since there are 1,000 people, the total amount that people will be willing to pay to reduce the amount of waste by 1 unit is $1,500. If the amount of waste is reduced from 6 to 5 units, the factory’s proﬁts will fall from $3,600 to $3,500. Evidently the consumers could aﬀord to bribe the factory to reduce its waste production by 1 unit. 6 AP MP 4 Cost 2 2 0 4 6 8 1. It costs $2,000 dollars a month to operate a lobster boat. 2. If there are x boats operating in Horsehead Bay, the total revenue from the lobster catch per month will be f (x) = $1, 000(10x − x2 ). x there are zero proﬁts?) 8 boats. (c) What number of boats maximizes total proﬁts? Set M P equal to cost to give 10 − 2x = 2, or x = 4 boats. (d) If Horsehead, Massachusetts, wants to restrict the number of boats to the number that maximizes total proﬁts, how much should it charge per month for a lobstering permit? (Hint: With a license fee of F thousand dollars per month, the marginal cost of operating a boat for a month $4,000 per month. 33.2 (2) Suppose that a honey farm is located next to an apple orchard and each acts as a competitive ﬁrm. Let the amount of apples produced be measured by A and the amount of honey produced be measured by H. The cost functions of the two ﬁrms are cH (H) = H 2 /100 and cA (A) = A2 /100 − H. The price of honey is $2 and the price of apples is $3. (a) If the ﬁrms each operate independently, the equilibrium amount of honey produced will be (a) In the graph below, plot the curves for the average product, AP (x) = f (x)/x, and the marginal product, M P (x) = 10, 000 − 2, 000x. In the same graph, plot the line indicating the cost of operating a boat. 12 (b) If the permits are free of charge, how many boats will trap lobsters in Horsehead, Massachusetts? (Hint: How many boats must enter before would be (2 + F ) thousand dollars per month.) 33.1 (2) The picturesque village of Horsehead, Massachusetts, lies on a bay that is inhabited by the delectable crustacean, homarus americanus, also known as the lobster. The town council of Horsehead issues permits to trap lobsters and is trying to determine how many permits to issue. The economics of the situation is this: 10 produced will be 100 150. and the equilibrium amount of apples NAME 405 (b) Suppose that the honey and apple ﬁrms merged. What would be the proﬁt-maximizing output of honey for the combined ﬁrm? What would be the proﬁt-maximizing amount of apples? 150. 150. EXTERNALITIES $1 per unit. 33.3 (2) In El Carburetor, California, population 1,001, there is not much to do except to drive your car around town. Everybody in town is just like everybody else. While everybody likes to drive, everybody complains about the congestion, noise, and pollution caused by traﬃc. A typical resident’s utility function is U (m, d, h) = m + 16d − d2 − 6h/1, 000, where m is the resident’s daily consumption of Big Macs, d is the number of hours per day that he, himself, drives, and h is the total amount of driving (measured in person-hours per day) done by all other residents of El Carburetor. The price of Big Macs is $1 each. Every person in El Carburetor has an income of $40 per day. To keep calculations simple, suppose it costs nothing to drive a car. 8. $6. 33.4 (3) Tom and Jerry are roommates. They spend a total of 80 hours a week together in their room. Tom likes loud music, even when he sleeps. His utility function is UT (CT , M ) = CT + M , where CT is the number of cookies he eats per week and M is the number of hours of loud music per week that is played while he is in their room. Jerry hates all kinds of music. His utility function is U (CJ , M ) = CJ − M 2 /12. Every week, Tom and Jerry each get two dozen chocolate chip cookies sent from home. They have no other source of cookies. We can describe this situation with a box that looks like an Edgeworth box. The box has cookies on the horizontal axis and hours of music on the vertical axis. Since cookies are private goods, the number of cookies that Tom consumes per week plus the number that Jerry consumes per week must equal 48. But music in their room is a public good. Each must consume the same number of hours of music, whether he likes it or not. In the box, let the height of a point represent the total number of hours of music played in their room per week. Let the distance of the point from the left side of the box be “cookies for Tom” and the distance of the point from the right side of the box be “cookies for Jerry.” (a) If an individual believes that the amount of driving he does won’t affect the amount that others drive, how many hours per day will he choose to drive? (Ch. 33) (f ) The same objective could be achieved with a tax on driving. How much would the tax have to be per hour of driving? (Hint: This price would have to equal an individual’s marginal rate of substitution between driving and Big Macs when he is driving the “right” amount.) (c) What is the socially eﬃcient output of honey? 150. If the ﬁrms stayed separate, how much would honey production have to be subsidized to induce an eﬃcient supply? 406 Jerry Music 80 (Hint: What value of d maximizes U (m, d, h)?) b Blue Line (b) If everybody chooses his best d, then what is the total amount h of driving by other persons? 60 Red Line 8,000. 40 (c) What will be the utility of each resident? 56. Red Line 20 (d) If everybody drives 6 hours a day, what will be the utility level of a Blue Shading typical resident of El Carburetor? 64. a Blue Line (e) Suppose that the residents decided to pass a law restricting the total number of hours that anyone is allowed to drive. How much driving should everyone be allowed if the objective is to maximize the utility of the typical resident? (Hint: Rewrite the utility function, substituting 1, 000d for h, and maximize with respect to d.) 5 hours per day. NAME 407 blue ink to sketch the indiﬀerence curve for Jerry that passes through this point. [Hint: When you draw Jerry’s indiﬀerence curve, remember two things: (1) He hates music, so he prefers lower points on the graph to higher ones. (2) Cookies for Jerry are measured from the right side of the box, so he prefers points that are toward the left side of the box to points that are toward the right.] Use blue ink to shade in the points representing situations that would make both roommates better oﬀ than they are at point A. 0 Tom 12 24 36 48 Cookies (a) Suppose the dorm’s policy is that you must have your roommate’s permission to play music. The initial endowment in this case denotes the situation if Tom and Jerry make no deals. There would be no music, and each person would consume 2 dozen cookies a week. Mark this initial endowment on the box above with the label A. Use red ink to sketch the indiﬀerence curve for Tom that passes through this point, and use 408 EXTERNALITIES (Ch. 33) (b) The extra proﬁt that the jeweler would get from an extra dollar’s worth of advertising by the clothing store is approximately equal to the derivative of the jeweler’s proﬁts with respect to the clothing store’s advertising expenditure. When the two stores are doing the equilibrium amount of advertising that you calculated above, a dollar’s worth of advertising by the clothing store would give the jeweler an extra proﬁt of about $32 and an extra dollar’s worth of advertising by the jeweler would give the clothing store an extra proﬁt of about (b) Suppose, alternatively, that the dorm’s policy is “rock-n-roll is good for the soul.” You don’t need your roommate’s permission to play music. Then the initial endowment is one in which Tom plays music for all of the 80 hours per week that they are in the room together and where each consumes 2 dozen cookies per week. Mark this endowment point in the box above and label it B. Use red ink to sketch the indiﬀerence curve for Tom that passes through this point, and use blue ink to sketch the indiﬀerence curve for Jerry that passes through this point. Given the available resources, can both Tom and Jerry be made better oﬀ than they are at point B? Calculus Yes. 33.5 (0) A clothing store and a jewelry store are located side by side in a small shopping mall. The number of customers who come to the shopping mall intending to shop at either store depends on the amount of money that the store spends on advertising per day. Each store also attracts some customers who came to shop at the neighboring store. If the clothing store spends $xC per day on advertising, and the jeweler spends $xJ on advertising per day, then the total proﬁts per day of the clothing store are ΠC (xC , xJ ) = (60 + xJ )xC − 2x2C , and the total proﬁts per day of the jewelry store are ΠJ (xC , xJ ) = (105 + xC )xJ − 2x2J . (In each case, these are proﬁts net of all costs, including advertising.) (a) If each store believes that the other store’s amount of advertising is independent of its own advertising expenditure, then we can ﬁnd the equilibrium amount of advertising for each store by solving two equations in two unknowns. One of these equations says that the derivative of the clothing store’s proﬁts with respect to its own advertising is zero. The other equation requires that the derivative of the jeweler’s proﬁts with respect to its own advertising is zero. These two equations are written as 60 + xJ − 4xC = 0 and 105 + xC − 4xJ = 0. equilibrium amounts of advertising are Proﬁts of the clothing store are are 2,048. xC = 23 $1,058 and The xJ = 32. and proﬁts of the jeweler $23. (c) Suppose that the owner of the clothing store knows the proﬁt functions of both stores. She reasons to herself as follows. Suppose that I can decide how much advertising I will do before the jeweler decides what he is going to do. When I tell him what I am doing, he will have to adjust his behavior accordingly. I can calculate his reaction function to my choice of xC , by setting the derivative of his proﬁts with respect to his own advertising equal to zero and solving for his amount of advertising as a function of my own advertising. When I do this, I ﬁnd that xJ = 105/4+xC /4. If I substitute this value of xJ into my proﬁt function and then choose xC to maximize my own proﬁts, I will choose xC = choose xJ = 32.41 . and his proﬁts will be 24.64 In this case my proﬁts will be and he will $1,062.72 $2,100.82. (d) Suppose that the clothing store and the jewelry store have the same proﬁt functions as before but are owned by a single ﬁrm that chooses the amounts of advertising so as to maximize the sum of the two stores’ proﬁts. The single ﬁrm would choose xC = $37.50 and xJ = $45. Without calculating actual proﬁts, can you determine whether total proﬁts will be higher, lower, or the same as total proﬁts would be when they made their decisions independently? would be higher. Yes, they How much would the total proﬁts be? $3,487.50. 33.6 (2) The cottagers on the shores of Lake Invidious are an unsavory bunch. There are 100 of them, and they live in a circle around the lake. Each cottager has two neighbors, one on his right and one on his left. There is only one commodity, and they all consume it on their front lawns in full view of their two neighbors. Each cottager likes to consume the commodity but is very envious of consumption by the neighbor on his left. Curiously, nobody cares what the neighbor on his right is doing. In fact every consumer has a utility function U (c, l) = c − l2 , where c is NAME 409 his own consumption and l is consumption by his neighbor on the left. Suppose that each consumer owns 1 unit of the consumption good and consumes it. 410 0. (a) Calculate his utility level. EXTERNALITIES UJ (cJ , x, y) + UT (cT , x, y) = cJ + 500 ln x − 20y + cT + 500 ln y − 10x = cJ + cT + 500 ln x − 10x + 500 ln y − 10y. (b) Suppose that each consumer consumes only 3/4 of a unit. Will all Better off. individuals be better oﬀ or worse oﬀ? (c) What is the best possible consumption if all are to consume the same 1/2. amount? (Ch. 33) (b) Because Jim and Tammy have quasilinear utility functions, their utility possibility frontier includes a straight line segment. Furthermore, this segment can be found by maximizing the sum of their utilities. Notice that But we know from the family budget constraint that cJ +cT = 1, 000, 000− 20x − 100y. Therefore we can write UJ (cJ , x, y) + UT (cT , x, y) = 1, 000, 000 − 20x − 100y + 500 ln x − 10x + 500 ln y − 10y = 1, 000, 000 + 500 ln x + 500 ln y − 30x − 110y. (d) Suppose that everybody around the lake is consuming 1 unit. Can any two people make themselves both better oﬀ either by redistributing No. consumption between them or by throwing something away? Let us now choose x and y so as to maximize UJ (cJ , x, y) + UT (cT , x, y). Setting the partial derivatives with respect to x and y equal to zero, we No. (e) How about a group of three people? (f ) How large is the smallest group that could cooperate to beneﬁt all its described by the equation UJ + UC = 1,001,163.86. (You need a calculator or a log table to ﬁnd this answer.) Along this frontier, the 100. members? ﬁnd the maximum where x = 16.67 and y = 4.54 . If we plug these numbers into the equation UJ (cJ , x, y)+UT (cT , x, y) = 1, 000, 000+ 500 ln x+500 ln y−30x−110y, we ﬁnd that the utility possibility frontier is 33.7 (0) Jim and Tammy are partners in Business and in Life. As is all too common in this imperfect world, each has a little habit that annoys the other. Jim’s habit, we will call activity X, and Tammy’s habit, activity Y . Let x be the amount of activity X that Jim pursues and y be the amount of activity Y that Tammy pursues. Due to a series of unfortunate reverses, Jim and Tammy have a total of only $1,000,000 a year to spend. Jim’s utility function is UJ = cJ + 500 ln x − 10y, where cJ is the money he spends per year on goods other than his habit, x is the number of units of activity X that he consumes per year, and y is the number of units of activity Y that Tammy consumes per year. Tammy’s utility function is UT = cT + 500 ln y − 10x, where cT is the amount of money she spends on goods other than activity Y , y is the number of units of activity Y that she consumes, and x is the number of units of activity X that Jim consumes. Activity X costs $20 per unit. Activity Y costs $100 per unit. (a) Suppose that Jim has a right to half their joint income and Tammy has a right to the other half. Suppose further that they make no bargains with each other about how much activity X and Y they will consume. How much of activity X will Jim choose to consume? 25 units. 5 units. How much of activity Y will Tammy consume? total expenditure on the annoying habits X and Y by Jim and Tammy is 787.34 . The rest of the $1,000,000 is spent on cJ and cT . Each possible way of dividing this expenditure corresponds to a diﬀerent point on the utility possibility frontier. The slope of the utility possibility frontier (a) “Free to Choose with No Bargaining”: Suppose that no bargains can be struck between the airport and the developer and that each can decide on its own level of activity. No matter how many houses the developer builds, the number of planes per day that maximizes proﬁts for the airport 24. is NAME 411 number of houses that maximizes the developer’s proﬁts is proﬁts of the airport will be will be 324. 576 18. Total and total proﬁts of the developer The sum of their proﬁts will be developer. Then no planes will ﬂy. The developer will build 30 900. (c) “Lawyer’s Paradise”: Suppose that a law is passed that makes the airport liable for all damages to the developer’s property values. Since the developer’s proﬁts are 60Y − Y 2 − XY and his proﬁts would be 60Y − Y 2 if no planes were ﬂown, the total amount of damages awarded to the developer will be XY . Therefore if the airport ﬂies X planes and the developer builds Y houses, then the airport’s proﬁts after it has paid damages will be 48X − X 2 − XY . The developer’s proﬁts including the amount he receives in payment of damages will be 60Y −Y 2 −XY +XY = 60Y − Y 2 . To maximize his net proﬁts, the developer will choose to build 30 houses no matter how many planes are ﬂown. To maximize its proﬁts, net of damages, the airport will choose to land Total proﬁts of the developer will be 81. airport will be Calculus 900 9 planes. 981. 33.9 (1) This problem concerns the airport and the developer from the previous problem. (a) “The Conglomerate”: Suppose that a single ﬁrm bought the developer’s land and the airport and managed both to maximize joint proﬁts. Total proﬁts, expressed as a function of X and Y would be when X = 12 and Y = 24. EXTERNALITIES the airport to eliminate? (a) Assuming that no tolls are charged for using either road, in equilibrium how many commuters will use the road through Middle Potato? 2,500. What will be the total number of person-minutes per day spent by commuters traveling from East Potato to West Potato? 45 × 6, 000 = 270, 000. Total proﬁts are maximized (b) Suppose that a social planner controlled access to the road through Middle Potato and set the number of persons permitted to travel this way so as to minimize the total number of person-minutes per day spent by commuters traveling from East Potato to West Potato. Write an expression for the total number of person-minutes per day spent by commuters traveling from East Potato to West Potato as a function of the number N of commuters permitted to travel on the Middle Potato N road. N (20 + 100 ) + (6, 000 − N )45. How many commuters per day would the social planner allow to use the road through Middle Potato? 1,250. In this case, how long would it take com- muters who drove through Middle Potato to get to work? minutes. (b) “Dealing”: Suppose that the airport and the developer remain independent. If the original situation was one of “free to choose,” could the developer increase his net proﬁts by bribing the airport to cut back one ﬂight per day if the developer has to pay for all of the airport’s lost Yes. 12. 33.10 (1) Every morning, 6,000 commuters must travel from East Potato to West Potato. Commuters all try to minimize the time it takes to get to work. There are two ways to make the trip. One way is to drive straight across town, throught the heart of Middle Potato. The other way is to take the Beltline Freeway that circles the Potatoes. The Beltline Freeway is entirely uncongested, but the drive is roundabout and it takes 45 minutes to get from East Potato to West Potato by this means. The road through Middle Potato is much shorter, and if it were uncongested, it would take only 20 minutes to travel from East Potato to West Potato by this means. But this road can get congested. In fact, if the number of commuters who use this road is N , then the number of minutes that it takes to drive from East Potato to West Potato through Middle Potato is 20 + N/100. Total proﬁts are then equal to 1,008. proﬁts? (Ch. 33) ﬂights by paying for all lost proﬁts coming from the reduction of ﬂights. To maximize his own net proﬁts, how many ﬂights per day should he pay and total proﬁts of the The sum of their proﬁts will be 48X − X 2 + 60Y − Y 2 − XY 412 Given that the airport is landing this number of planes, the 900. (b) “Strict Prohibition”: Suppose that a local ordinance makes it illegal to land planes at the airport because they impose an externality on the houses and will have total proﬁts of -1. constructed in this way is 33.8 (0) An airport is located next to a large tract of land owned by a housing developer. The developer would like to build houses on this land, but noise from the airport reduces the value of the land. The more planes that ﬂy, the lower is the amount of proﬁts that the developer makes. Let X be the number of planes that ﬂy per day and let Y be the number of houses that the developer builds. The airport’s total proﬁts are 48X −X 2 , and the developer’s total proﬁts are 60Y − Y 2 − XY . Let us consider the outcome under various assumptions about institutional rules and about bargaining between the airport and the developer. The developer decides to get the airport to reduce its 32.5 What would be the total number of person-minutes per day spent by commuters traveling from East Potato to West Potato? 1, 250 × 32.5 + 4, 750 × 45 = 254, 375 NAME 413 (c) Suppose that commuters value time saved from commuting at $w per minute and that the Greater Potato metropolitan government charges a toll for using the Middle Potato road and divides the revenue from this toll equally among all 6,000 commuters. If the government chooses the toll in such a way as to minimize the total amount of time that people spend commuting from East Potato to West Potato, how high should it set the toll? $12.5w. 414 EXTERNALITIES (Ch. 33) by expanding the capacity of the Middle Potato highway? 0 Do you think people will think that this capacity expansion will be a good use of their tax dollars? No. How much revenue will it collect per day from this toll? $15, 625w. Show that with this policy every commuter is better oﬀ than he or she was without the tolls and evaluate the gain per consumer in dollars. Before the toll was in place, all commuters spent 45 minutes traveling to work. With the toll in place, commuters who travel on the Beltline still spend 45 minutes traveling to work and commuters who travel through Middle Potato are indifferent between spending 45 minutes traveling on the Beltline and paying the toll to go through Middle Potato. Thus nobody would be worse off even if toll revenue were wasted. But everybody gets back about $2.6w per day from the toll revenue, so all are better off. 33.11 (2) Suppose that the Greater Potato metropolitan government rejects the idea of imposing traﬃc tolls and decides instead to rebuild the Middle Potato highway so as to double its capacity. With the doubled capacity, the amount of time it takes to travel from East Potato to West Potato on the Middle Potato highway is given by 20 + N/200, where N is the number of commuters who use the Middle Potato highway. In the new equilibrium, with expanded capacity and no tolls, how many commuters will use the Middle Potato highway? 5,000 How long will it take users of the Middle Potato highway to get to work? minutes 45 How many person-minutes of commuting time will be saved 416 Chapter 34 INFORMATION TECHNOLOGY (Ch. 34) (c) On the same graph, draw and label a separate demand curve for the case where DoorKnobs’s perceived market share s takes on each of the following values: s =1/8, 1/4, 3/4, 7/8, 1. NAME Information Technology Willingness to Pay 256 Introduction. We all recognize that information technology has revolutionized the way we produce and consume. Some think that it is necessary to have a “new economics” to understand this New Economy. We think not. The economic tools that you have learned in this course can oﬀer very powerful insights into the economics of information technology, as we illustrate in this set of problems. 224 S=1 192 S=7/8 160 128 34.1 (2) Bill Barriers, the president of MightySoft software company is about to introduce a new computer operating system called DoorKnobs. Because it is easier to swap ﬁles with people who have the same operating system, the amount people are willing to pay to have DoorKnobs on their computers is greater the larger they believe DoorKnobs’s market share to be. The perceived market share for DoorKnobs is the fraction of all computers that the public believes is using DoorKnobs. When the price of DoorKnobs is p, then its actual market share is the fraction of all computer owners that would be willing to pay at least $p to have DoorKnobs installed on their computers. Market researchers have discovered that if DoorKnobs’s perceived market share is s and the price of DoorKnobs is $p, then its actual market share will be x, where x is related to the price p and perceived market share s by the formula p = 256s(1 − x). (1) 96 S=3/4 64 S=1/2 32 S=1/4 S=1/8 0 2 4 6 8 10 (a) If the perceived market share is s, then the demand curve for DoorKnobs is given by Equation 1. On the graph below, draw the demand curve relating price to actual market share in the case in which DoorKnobs’s perceived market share is s = 1/2. Label this curve s = 1/2. (b) On the demand curve that you just drew with s = 1/2, mark a red dot on the point at which the actual market share of DoorKnobs is 1/2. (This is the point on the demand curve directly above x = 1/2.) What is the price at which half of the computer owners actually want to buy DoorKnobs, given that everybody believes that half of all computer owners want to buy DoorKnobs? $64 14 16 (d) On the demand curve for a perceived market share of s = 1/4, put a red dot on the point at which the actual market share of DoorKnobs is 1/4. (This is the point on this demand curve directly above x = 1/4.) If the perceived market share of DoorKnobs is 1/4, at what price is the actual market share of DoorKnobs also 1/4? In the short run, MightySoft can inﬂuence the perceived market share of DoorKnobs by publicity, advertising, giving liquor and gifts to friendly journalists, and giving away copies in conspicuous ways. In the long run, the truth will emerge, and DoorKnobs’s perceived market share s must equal its actual market share x. 12 Actual Market Share (in sixteenths) $48 (e) Just as you did for s = 1/2 and s = 1/4, make red marks on the demand curves corresponding to s = 1/8, 3/4, 7/8, and 1, showing the price at which the actual market share is s, given that the perceived market share is s. (f ) Let us now draw the long-run demand curve for DoorKnobs, where we assume that computer owners’ perceived market shares s are the same as the actual market shares x. If this is the case, it must be that s = x, so the demand curve is given by p = 256x(1 − x). On the graph above, plot a few points on this curve and sketch in an approximation of the curve. (Hint: Note that the curve you draw must go through all the red points that you have already plotted.) (g) Suppose that MightySoft sets a price of $48 for DoorKnobs and sticks with that price. There are three diﬀerent perceived market shares such that the fraction of consumers who would actually want to buy DoorKnobs for $48 is equal to the perceived market share. One such perceived NAME 417 market share is 0. What are the other two possibilities? s = 1/4 and s = 3/4 (h) Suppose that by using its advertising and media inﬂuence, MightySoft can temporarily set its perceived market share at any number between 0 and 1. If DoorKnobs’s perceived market share is x and if MightySoft charges a price p = 256x(1 − x), the actual market fraction will also be x and the earlier perceptions will be reinforced and maintained. Assuming that MightySoft chooses a perceived market share x and a price that makes the actual market share equal to the perceived market share, what market share x should MightySoft choose in order to maximize its revenue and what price should it charge in order to maintain this market share? (Hint: Revenue is px = 256x2 (1 − x).) Use calculus and show your work. d x = 2/3. The first-order condition is 256(x2 − x3 ) = 0. This implies 2x = 3x2 , dx which implies that x = 2/3 or x = 0. The second order condition is satisfied only when x = 2/3. Price should be $256 × 1/3 × 2/3 = $56.89. (a) If the actual market share in the ﬁrst period was 1/2, ﬁnd the actual market share in the second period .75 , the third period Write down the actual market shares for the next few periods Do they seem to be approaching a limit? .833. .8529, If so, what? .853553. NAME 419 Frisian Press has already made all its ﬁxed cost investments in the book and must pay a constant marginal cost of $c for each copy that it sells. Let p1 be the price charged for new copies sold in the ﬁrst year of publication and let p2 be the price charged for new copies sold in the second year of publication. The publisher and the students who buy the book are aware that there will be an active market for used copies of Microeconomics for the Muddleheaded one year after publication and that used copies of the ﬁrst edition will have zero resale value two years after publication. At the end of the ﬁrst year of publication, students can resell their used textbooks to bookstores for 40% of the second-year price, p2 . The net cost to a student of buying the book in the ﬁrst year, using it for class, and reselling it at the end of the year is p1 − 0.4p2 . The number of copies demanded in the ﬁrst year of publication is given by a demand function, q1 = D1 (p1 − 0.4p2 ). Some of the students who use the book in the ﬁrst year of publication will want to keep their copies for future reference, and some will damage their books so that they cannot be resold. The cost of keeping one’s old copy or of damaging it is the resale price 0.4p2 . The number of books that are either damaged or kept for reference is given by a “keepers” demand function, Dk (0.4p2 ). It follows that the number of used copies available at the end of the ﬁrst year will be D1 (p1 − 0.4p2 ) − D k (0.4p2 ). Students who buy Microeconomics for the Muddleheaded in the second year of publication will not be able to resell their used copies, since a new edition will then be available. These students can, however, buy either a new copy or a used copy of the book. For simplicity of calculations, let us assume that students are indiﬀerent between buying a new copy or a used copy and that used copies cost the same as new copies in the book store. (The results would be the same if students preferred new to used copies, but bookstores priced used copies so that students were indiﬀerent between buying new and used copies.) The total number of copies, new and used, that are purchased in the second year of publication is q2 = D2 (p2 ). (a) Write an expression for the number of new copies that East Frisian Press can sell in the second year after publication if it sets prices p1 in year 1 and p2 in year 2. INFORMATION TECHNOLOGY (Ch. 34) (b) Notice that when price is held constant at p, if DoorKnobs’s market share converges to a constant x̄, it must be that x̄ = 1 − (p/256x̄). Solve this equation for x̄ in the case where p = $32. What do you make of the fact that there are two solutions? This equation implies x2 − x + 1/8 = 0. Solutions x = 0.85355 and x = 0.14645. equilibrium market shares D2 (p2 ) − D1 (p1 − .4p2 ) + Dk (.4p2 ). (b) Write an expression for the total number of new copies of Microeconomics for the Muddleheaded that East Frisian can sell over two years at prices p1 and p2 in years 1 and 2. D1 (p1 − .4p2 ) + D2 (p2 ) − D1 (p1 − .4p2 ) + Dk (.4p2 ) = D2 (p2 ) + Dk (.4p2 ). (c) Would the total number of copies sold over two years increase, decrease, or remain constant if p1 were increased and p2 remained constant? It would remain constant. are Both are with a price of $32. 34.3 (1) A group of 13 consumers are considering whether to connect to a new computer network. Consumer 1 has an initial value of $1 for hooking up to the network, consumer 2 has an initial value of $2, consumer 3 has an iinitial value of $3, and so on up to consumer 13. Each consumer’s willingness to pay to connect to the network depends on the total number of persons who are connected to it. In fact, for each i, consumer i’s willingness to pay to connect to the network is i times the total number of persons connected. Thus if 5 people are connected to the network, consumer 1’s willingness to pay is $5, consumer 2’s willingness to pay is $10 and so on. (a) What is the highest price at which 9 customers could hook up to the $45 market and all of them either make a proﬁt or break even? 34.2 (1) Suppose that demand for DoorKnobs is as given in the previous problem, and assume that the perceived market share in any period is equal to the actual market share in the previous period. Then where xt is the actual market share in period t, the equation p = 256xt−1 (1 − xt ) is satisﬁed. Rearranging this equation, we ﬁnd that xt = 1 − (p/256xt−1 ) whenever p/256xt−1 ≤ 1. If p/256xt−1 ≤ 0, then xt = 0. With this formula, if we know actual market share for any time period, we can calculate market share for the next period. Let us assume that DoorKnobs sets the price at p = $32 and never changes this price. (To answer the following questions, you will ﬁnd a calculator useful.) 8534. 418 (b) Suppose that the industry that supplies the computer network is competitive and that the cost of hooking up each consumer to the network is $45. Suppose that consumers are very conservative and nobody will sign up for the network unless her buyer value will be at least as high as the price she paid as soon as she signs up. How many people will sign up if 0 the price is $45? (c) Suppose that the government oﬀers to subsidize “pioneer users” of the system. The ﬁrst two users are allowed to connect for $10 each. After the ﬁrst two users are hooked up, the government allows the next two to connect for $25. After that, everyone who signs up will have to pay the full cost of $45. Assume that users remain so conservative that will sign up only if their buyer values will be at least equal to the price they are charged when they connect. With the subsidy in place, how many 9 consumers in toto will sign up for the network? 34.4 (2) Professor Kremepuﬀ has written a new, highly simpliﬁed economics text, Microeconomics for the Muddleheaded, which will be published by East Frisian Press. The ﬁrst edition of this book will be in print for two years, at which time it will be replaced by a new edition. East 420 INFORMATION TECHNOLOGY (Ch. 34) (d) Write an expression for the total revenue that East Frisian Press will receive over the next two years if it sets prices p1 and p2 . p1 D1 (p1 − .4p2 )+p2(D2 (p2 )−D1(p1 −.4p2 )+Dk (.4p2 )) = (p1 − p2 )D1 (p1 − .4p2 ) + p2(D2 (p2) + Dk (.4p2 )). (e) To maximize its total proﬁts over the next two years, East Frisian must maximize the diﬀerence between its total revenue and its variable costs. Show that this diﬀerence can be written as (p1 − p2 )D1 (p1 − .4p2 ) + (p2 − c) D2 (p2 ) + D k (.4p2 ) . Variable cost is c(D2(p2 ) + Dk (.4p2 )). Subtract this from previous answer. (f ) Suppose that East Frisian has decided that it must charge the same price for the ﬁrst edition in both years that it is sold. Thus it must set p = p1 = p2 . Write an expression for East Frisian’s revenue net of variable costs over the next two years as a function of p. (p − c)(D2 (p) + Dk (.4p)) 34.5 (2) Suppose that East Frisian Press, discussed in the previous problem, has a constant marginal cost of c = $10 for each copy of Microeconomics for the Muddleheaded that it sells and let the demand functions be D1 (p1 − 0.4p2 ) = 100 (90 − p1 + 0.4p2 ) D2 (p2 ) = 100(90 − p2 ). The number of books that people either damage or keep for reference after the ﬁrst year is Dk (0.4p2 ) = 100(90 − 0.8p2 ). (This assumption is consistent with the assumption that everyone’s willingness to pay for keeping the book is half as great as her willingness to pay to have the book while she is taking the course.) Assume that East Frisian Press is determined to charge the same price in both years, so that p1 = p2 = p. NAME 421 (a) If East Frisian Press charges the same price p for Microeconomics for the Muddleheaded in the ﬁrst and second years, show that the total sales of new copies over the two years are equal to 422 INFORMATION TECHNOLOGY (Ch. 34) (b) Find the price that maximizes total revenue net of variable costs. $50. 18, 000 − 180p. Total sales are D2 (p)+Dk (.4p2 ) = 100(90− p) + 100(90 − .8p)) = 18, 000 − 180p (c) The total number of new books sold in the ﬁrst year would be 4,000 , and the total number of books sold in the second year would 4,000. be (b) Write an expression for East Frisian’s total revenue, net of variable (p − costs, over the ﬁrst two years as a function of the price p. 10)(18, 000 − 180p) = 19, 800p − 180p − 180, 000 (d) East Frisian’s total revenue net of variable costs, if it markets a new $320,000. edition after one year, will be 2 (c) Solve for the price p that maximizes its total revenue net of variable costs over the ﬁrst two years. p = $55. At this price, the net cost to students in the ﬁrst year of buying the text and reselling it is The total number of copies sold in the ﬁrst year will be $33. 5,700. The total number of copies that are resold as used books is 1,100. The total number of copies purchased by students in the second year will be 3,500. (Remember students in the second year know that they cannot resell the book, so they have to pay the full price p for using it.) The total number of new copies purchased by students in the second year will be 2,400. will be $364,500. Total revenue net of variable costs over the two years (e) Would it be more proﬁtable for East Frisian Press to produce a new edition after one year or after two years? After two years. Which would be better for students? (Hint: The answer is not the same for all students.) After two years is better for students who take the course in the first year of publication and plan to sell. After one year is better for the other students. 34.7 (3) Suppose that East Frisian Press publishes a new edition only after two years and that demands and costs are as in the previous problems. Suppose that it sets two diﬀerent prices p1 and p2 in the two periods. (a) Write an expression for the total number of new copies sold at prices 34.6 (2) East Frisian Press is trying to decide whether it would be profitable to produce a new edition of Microeconomics for the Muddleheaded after one year rather than after two years. If it produces a new edition after one year, it will destroy the used book market and all copies that are purchased will be new copies. In this case, the number of new copies that will be demanded in each of the two years will be 100(90 − p), where p is the price charged. The variable cost of each copy sold remains $10. (a) Write an expression for the total number of copies sold over the course of two years if the price is p in each year 200(90-p) . Also, write p1 and p2 and show that this number depends on p2 but not on p1 . 100 ((70 − p1 + .4p2 ) + (140 − 1.8p2 ) − (70 − p1 + .4p2 )) = 100(140 − 1.8p2 ) (b) Show that at prices p1 and p2 , the diﬀerence between revenues and variable costs is equal to 100 90p1 + 108p2 + 1.4p1 p2 − p21 − 2.2p22 − 1, 800 . This difference is 100(p1 − p2 )(90 − p1 + an expression for total revenue net of variable costs as a function of p. .4p2 ) + (p2 − 10)(180 − 1.8p2 ). Expand this 200(p − 10)(90 − p). expression. NAME 423 (c) Calculate the prices p1 and p2 that maximize the diﬀerence between total revenue and variable costs and hence maximize proﬁts. p1 = $80, p2 = $50 424 INFORMATION TECHNOLOGY (Ch. 34) (d) Fasten contemplates raising the price of checks to $22 per period. Will its customers switch? No. The present value of continuing to use Fasten are $220 while the (d) If East Frisian Press chooses its proﬁt-maximizing p1 and p2 , compare the cost of using Microeconomics for the Muddleheaded for a student who buys the book when it is ﬁrst published and resells it at the end of the ﬁrst year with the cost for a student who buys the book at the beginning of the second year and then discards it. The former has a net cost of $80 − .4 × 50 = $60 and the latter present value of switching to Czechwriter is $250. (e) At what price for checks will Fasten’s customers just be indiﬀerent to switching? (Hint: Let x be this amount. Compare the present value of staying with Fasten with the present value of switching to Czechwriter.) Solve the equation x/.10 = 250 to find has a cost of $50. x = 25. 34.8 (2) The Silicon Valley company Intoot produces checkwriting software. The program itself, Fasten, sells for $50 and includes a package of checks. Check reﬁll packets for Fasten cost $20 to produce and Intoot sells the checks at cost. Suppose that a consumer purchases Fasten for $50 in period 1 and spends $20 on checks in each subsequent period. Assume for simplicity that the consumer uses the program for an inﬁnite number of periods. (f ) If it charges the highest price that it can without making its customers switch, what proﬁt does Fasten make on checks from each of its customers per period? $5. What is the present value of the proﬁt per customer that Fasten gets if it sets the price of checks equal to the number determined in the last question? P V = 5/.10 = 50. compare to the customer switching cost? (a) If the interest rate is r = .10 per period, what is the present value of the stream of payments made by the consumer? (Hint: a stream of payments of x starting next period has a present value of x/r.) The total cost of ownership of Fasten is 50+20/.10 = $250. purchasing the program. This means his switching costs are $50 (c) Fasten is contemplating raising the price of checks to $30 per period. If so, will its customers switch to Czechwriter? Explain. Yes, the present value of continuing to use Fasten are $300 while the costs of switching to Czechwriter are $250. (g) Suppose now that the cost of switching also involves several hours of data conversion that the consumer values at $100. The total cost of switching is the cost of the new program plus the data conversion cost which is (b) Fasten’s competitor produces an equally eﬀective product called Czechwriter. Czechwriter can do everything Fasten can do and vice versa except that Fasten cannot use check reﬁll packets that are sold by anyone other than Fasten. Czechwriter also sells for $50 and sells its checks for $20 per period. A Fasten customer can switch to Czechwriter simply by How does this It is the same. $150. (h) Making allowances for the cost of data conversion, what is the highest price that Intoot can charge for its checks? 250 + 100 for x = 35. from this price? costs? $150. Solve x/.10 = What is the present value of proﬁt How does this compare to total switching It is the same. (i) Suppose that someone writes a computer program that eliminates the cost of converting data and makes this program available for free. Suppose that Intoot continues to price its check reﬁll packages at $25. A new customer is contemplating buying Fasten at a price of $50 and paying $25 per period for checks, versus paying $50 for Czechwriter and paying $20 for checks. If the functionality of the software is identical, which will the consumer buy? Czechwriter. NAME 425 (j) Intoot decides to distribute a coupon that oﬀers a discount of $50 oﬀ of the regular purchase price. What price would it have to set to make consumers indiﬀerent between purchasing Fasten and Czechwriter? Solve 50 + 25/.10 − d = 50 + 20/.10 to find 426 INFORMATION TECHNOLOGY (Ch. 34) (e) Sol Microsystems decides that the independent subsidiary system is cumbersome, so it sets up Guava Computing which sells a bundled system consisting of the chip and the language. Let p be the price of the bundle. Guava Computing’s proﬁt function is [120 − p]p that the discount should be $50. (f ) Diﬀerentiate this proﬁt with respect to p and set the resulting expres- (k) Suppose that consumers are shortsighted and only look at the cost of the software itself, neglecting the cost of the checks. Which program sion to zero to determine p = would they buy if Intoot oﬀered this coupon? Czechwriter respond to the Fasten oﬀer? Fasten. How might coupon for $50 and raise the price of its is better for consumers? more proﬁt? checks to $35. 34.9 (2) Sol Microsystems has recently invented a new language, Guava, which runs on a proprietary chip, the Guavachip. The chip can only be used to run Guava, and Guava can only run on the Guavachip. Sol estimates that if it sells the chip for a price pc and the language for a price pg , the demand for the chip-language system will be x = 120 − (pc + pg ). (a) Sol initially sets up two independent subsidiaries, one to produce the chip and one to produce the language. Each of the subsidiaries will price its product so as to maximize its proﬁts, while assuming that a change in its own price will not aﬀect the pricing decision of the other subsidiary. Assume that marginal costs are negligible for each company. If the price of the language is set at pg , the chip company’s proﬁt function (neglecting ﬁxed costs) is . (g) Compare the prices charged by the integrated system and the separate subsidiaries. Which is lower? Issue its own 60 Integrated system. Integrated system. Which makes Integrated system. 34.10 (2) South Belgium Press produces the academic journal Nanoeconomics, which has a loyal following among short microeconomists, and Gigaeconomics, a journal for tall macroeconomists. It oﬀers a license for the electronic version of each journal to university libraries at a subscription cost per journal of $1,000 per year. The 200 top universities all subscribe to both journals, each paying $2,000 per year to South Belgium. By revealed preference, their willingness to pay for each journal is at least $1,000. (a) In an attempt to lower costs, universities decide to form pairs, with one member of each pair subscribing to Nanoeconomics and one member of each pair subscribing to Gigaeconomics. They agree to use interlibrary loan to share the other journal. Since the copies are electronic, there is no incremental cost to doing this. Under this pairing scheme, how many subscriptions of each journal will South Belgium sell? . [120 − pc − pg ]pc Which 100 (b) Diﬀerentiate this proﬁt function with respect to pc and set the result (b) In order to stem the revenue hemorrhage, South Belgium raises the price of each journal. Assuming library preferences and budgets haven’t equal to zero to calculate the optimal choice of pc as a function of pg . changed, how high can they set this price? pc = 120 − 2pg . the price to $2,000, since libraries have (c) Now consider the language subsidiary’s pricing decision. The optimal choice of pg as a function of pc is pg = 120 − 2pc . 40 and pg = , so that pc + pg = 80 NAME 427 (d) If there were a cost of interlibrary loan, how would your an- Assuming they still bought both journals, libraries would be worse off since they would have to pay the transactions that they are willing (c) How does library expenditure and South Belgium’s revenue compare to those of the previous regime? . swer change? raise to pay this much for the pair of journals. (d) Solving these two equations in two unknowns, we ﬁnd that pc = 40 already indicated They can cost for interlibrary loan. 428 INFORMATION TECHNOLOGY They remain the same. (Ch. 34) 430 Chapter 35 PUBLIC GOODS (Ch. 35) have the same utility function. This function is U (Xi , G) = Xi − 100/G, where Xi is the number of bottles of Labatt’s consumed by citizen i and G is the size of the town skating rink, measured in square meters. The price of Labatt’s ale is $1 per bottle and the price of the skating rink is $10 per square meter. Everyone who lives in Muskrat has an income of $1,000 per year. NAME Public Goods Introduction. In previous chapters we studied selﬁsh consumers consuming private goods. A unit of private goods consumed by one person cannot be simultaneously consumed by another. If you eat a ham sandwich, I cannot eat the same ham sandwich. (Of course we can both eat ham sandwiches, but we must eat diﬀerent ones.) Public goods are a different matter. They can be jointly consumed. You and I can both enjoy looking at a beautiful garden or watching ﬁreworks at the same time. The conditions for eﬃcient allocation of public goods are diﬀerent from those for private goods. With private goods, eﬃciency demands that if you and I both consume ham sandwiches and bananas, then our marginal rates of substitution must be equal. If our tastes diﬀer, however, we may consume diﬀerent amounts of the two private goods. If you and I live in the same town, then when the local ﬁreworks show is held, there will be the same amount of ﬁreworks for each of us. Eﬃciency does not require that my marginal rate of substitution between ﬁreworks and ham sandwiches equal yours. Instead, eﬃciency requires that the sum of the amount that viewers are willing to pay for a marginal increase in the amount of ﬁreworks equal the marginal cost of ﬁreworks. This means that the sum of the absolute values of viewers’ marginal rates of substitution between ﬁreworks and private goods must equal the marginal cost of public goods in terms of private goods. Example: A quiet midwestern town has 5,000 people, all of whom are interested only in private consumption and in the quality of the city streets. The utility function of person i is U (Xi , G) = Xi + Ai G − Bi G2 , where Xi is the amount of money that person i has to spend on private goods and G is the amount of money that the town spends on ﬁxing its streets. To ﬁnd the Pareto optimal amount of money for this town to spend on ﬁxing its streets, we must set the sum of the absolute values of marginal rates of substitution between public and private goods equal to the relative prices of public and private goods. In this example we measure both goods in dollar values, so the price ratio is 1. The absolute value of person i’s marginal rate of substitution between public goods and private goods is the ratio of the marginal utility of public goods to the marginal utility of private goods. The marginal utility of private goods is 1 and the marginal utility of public goods for person i is Ai − Bi G. Therefore the absolute of absolute value of person i’s MRS is Ai − B i G and the sum values of marginal rates of substitution is i (A = i Ai − ( Bi )G. i − Bi G) Therefore Pareto eﬃciency requires that A − ( B )G = 1. Solving i i i i this for G, we have G = ( i Ai − 1)/ i Bi . 35.1 (0) Muskrat, Ontario, has 1,000 people. Citizens of Muskrat consume only one private good, Labatt’s ale. There is one public good, the town skating rink. Although they may diﬀer in other respects, inhabitants NAME 431 (a) Write down an expression for the absolute value of the marginal rate of substitution between skating rink and Labatt’s ale for a typical citizen. 100/G2 What is the marginal cost of an extra square meter of skating 10. rink (measured in terms of Labatt’s ale)? (b) Since there are 1,000 people in town, all with the same marginal rate of substitution, you should now be able to write an equation that states the condition that the sum of absolute values of marginal rates of substitution equals marginal cost. Write this equation and solve it for the 100 Pareto eﬃcient amount of G. 1, 000 G2 = 10. So G = 100. (c) Suppose that everyone in town pays an equal share of the cost of the skating rink. Total expenditure by the town on its skating rink will be $10G. Then the tax bill paid by an individual citizen to pay for the skating rink is $10G/1, 000 = $G/100. Every year the citizens of Muskrat vote on how big the skating rink should be. Citizens realize that they will have to pay their share of the cost of the skating rink. Knowing this, a citizen realizes that if the size of the skating rink is G, then the amount of Labatt’s ale that he will be able to aﬀord is 1, 000 − G/100. (d) Therefore we can write a voter’s budget constraint as Xi + G/100 = 1, 000. In order to decide how big a skating rink to vote for, a voter simply solves for the combination of Xi and G that maximizes his utility subject to his budget constraint and votes for that amount of G. How much G is that in our example? G = 100. (e) If the town supplies a skating rink that is the size demanded by the voters will it be larger than, smaller than, or the same size as the Pareto optimal rink? The same. (f ) Suppose that the Ontario cultural commission decides to promote Canadian culture by subsidizing local skating rinks. The provincial government will pay 50% of the cost of skating rinks in all towns. The costs of this subsidy will be shared by all citizens of the province of Ontario. There are hundreds of towns like Muskrat in Ontario. It is true that to pay for this subsidy, taxes paid to the provincial government will have to be increased. But there are hundreds of towns from which this tax 432 PUBLIC GOODS (Ch. 35) is collected, so that the eﬀect of an increase in expenditures in Muskrat on the taxes its citizens have to pay to the state can be safely neglected. Now, approximately how large a skating rink would citizens of Muskrat (a) What is Bob’s reservation price for the sofa? vote for? (Hint: Rewrite the budget constraint for individuals observing that local taxes will be only half as large as before and the cost of increasing the size of the rink only half as much as before. Then solve for the utility-maximizing combination.) (b) What is Ray’s reservation price for the sofa? √ G = 100 2. (g) Does this subsidy promote economic eﬃciency? No. 35.2 (0) Ten people have dinner together at an expensive restaurant and agree that the total bill will be divided equally among them. Solve WB = 2(WB − pB ) to get pB = WB /2. Solve 2WR = 3(WR − pR ), which gives pR = WR /3. (c) If Bob has a total of WB = $100 and Ray has a total of WR = $75 to spend on sofas and other stuﬀ, they could buy the sofa and have a Pareto improvement over not buying it so long as the cost of the sofa is no greater than $75. (a) What is the additional cost to any one of them of ordering an appetizer that costs $20? $2. (b) Explain why this may be an ineﬃcient system. Each pays less than full cost of own meal, so all overindulge. 35.3 (0) Cowﬂop, Wisconsin, has 1,000 people. Every year they have a ﬁreworks show on the Fourth of July. The citizens are interested in only two things—drinking milk and watching ﬁreworks. Fireworks cost 1 gallon of milk per unit. People in Cowﬂop are all pretty much the same. In fact, they have identical utility functions. The utility function of each √ citizen i is Ui (xi , g) = xi + g/20, where xi is the number of gallons of milk per year consumed by citizen i and g is the number of units of ﬁreworks exploded in the town’s Fourth of July extravaganza. (Private use of ﬁreworks is outlawed.) (a) Solve for the absolute value of each citizen’s marginal rate of substitution between ﬁreworks and milk. √ 1/(40 g). (b) Find the Pareto optimal amount of ﬁreworks for Cowﬂop. 625. 35.4 (0) Bob and Ray are two hungry economics majors who are sharing an apartment for the year. In a ﬂea market they spot a 25-year-old sofa that would look great in their living room. Bob’s utility function is uB (S, MB ) = (1 + S)MB , and Ray’s utility function is uR (S, MR ) = (2+S)MR . In these expressions MB and MR are the amounts of money that Bob and Ray have to spend on other goods, S = 1 if they get the sofa, and S = 0 if they don’t get the sofa. Bob has WB dollars to spend, and Ray has WR dollars. 35.5 (0) Bonnie and Clyde are business partners. Whenever they work, they have to work together. Their only source of income is proﬁt from their partnership. Their total proﬁt per year is 50H, where H is the number of hours that they work per year. Since they must work together, they both must work the same number of hours, so the variable “hours of labor” is like a public “bad” for the two person community consisting of Bonnie and Clyde. Bonnie’s utility function is UB (CB , H) = CB − .02H 2 and Clyde’s utility function is UC (CC , H) = CC − .005H 2 , where CB and CC are the annual amounts of money spent on consumption for Bonnie and for Clyde. (a) If the number of hours that they both work is H, what is the ratio of Bonnie’s marginal utility of hours of work to her marginal utility of private goods? −.04H. What is the ratio of Clyde’s marginal utility of hours of work to his marginal utility of private goods? −.01H. (b) If Bonnie and Clyde are both working H hours, then the total amount of money that would be needed to compensate them both for having to work an extra hour is the sum of what is needed to compensate Bonnie and the amount that is needed to compensate Clyde. This amount is approximately equal to the sum of the absolute values of their marginal rates of substitution between work and money. Write an expression for this amount as a function of H. .05H. they make if they work an extra hour? How much extra money will $50. (c) Write an equation that can be solved for the Pareto optimal number of hours for Bonnie and Clyde to work. .05H = 50. Find the Pareto optimal H. H = 1, 000. (Hint: Notice that this model is formally the same as a model with one public good H and one private good, income.) NAME 433 35.6 (0) Lucy and Melvin share an apartment. They spend some of their income on private goods like food and clothing that they consume separately and some of their income on public goods like the refrigerator, the household heating, and the rent, which they share. Lucy’s utility function is 2XL + G and Melvin’s utility function is XM G, where XL and XM are the amounts of money spent on private goods for Lucy and for Melvin and where G is the amount of money that they spend on public goods. Lucy and Melvin have a total of $8,000 per year between them to spend on private goods for each of them and on public goods. (a) What is the absolute value of Lucy’s marginal rate of substitution 1/2. between public and private goods? What is the absolute value XM /G. of Melvin’s? (b) Write an equation that expresses the condition for provision of the 1/2 + XM /G = 1. Pareto eﬃcient quantity of the public good. (c) Suppose that Melvin and Lucy each spend $2,000 on private goods for themselves and they spend the remaining $4,000 on public goods. Is this a Pareto eﬃcient outcome? Yes. (d) Give an example of another Pareto optimal outcome in which Melvin gets more than $2,000 and Lucy gets less than $2,000 worth of private One example: Melvin gets $2,500; Lucy goods. gets $500 and G = $5, 000. (e) Give an example of another Pareto optimum in which Lucy gets more than $2,000. Lucy gets $5,000; Melvin gets $1, 000 and G = $2, 000. 434 PUBLIC GOODS (Ch. 35) 35.7 (0) This problem is set in a fanciful location, but it deals with a very practical issue that concerns residents of this earth. The question is, “In a Democracy, when can we expect that a majority of citizens will favor having the government supply pure private goods publicly?” This problem also deals with the eﬃciency issues raised by public provision of private goods. We leave it to you to see whether you can think of important examples of publicly supplied private goods in modern Western economies. On the planet Jumpo there are two goods, aerobics lessons and bread. The citizens all have Cobb-Douglas utility functions of the form 1/2 1/2 Ui (Ai , Bi ) = Ai Bi , where Ai and Bi are i’s consumptions of aerobics lessons and bread. Although tastes are all the same, there are two diﬀerent income groups, the rich and the poor. Each rich creature on Jumpo has an income of 100 fondas and every poor creature has an income of 50 fondas (the currency unit on Jumpo). There are two million poor creatures and one million rich creatures on Jumpo. Bread is sold in the usual way, but aerobics lessons are provided by the state despite the fact that they are private goods. The state gives the same amount of aerobics lessons to every creature on Jumpo. The price of bread is 1 fonda per loaf. The cost to the state of aerobics lessons is 2 fondas per lesson. This cost of the state-provided lessons is paid for by taxes collected from the citizens of Jumpo. The government has no other expenses than providing aerobics lessons and collects no more or less taxes than the amount needed to pay for them. Jumpo is a democracy, and the amount of aerobics to be supplied will be determined by majority vote. (a) Suppose that the cost of the aerobics lessons provided by the state is paid for by making every creature on Jumpo pay an equal amount of taxes. On planets, such as Jumpo, where every creature has exactly one head, such a tax is known as a “head tax.” If every citizen of Jumpo gets 20 lessons, how much will be total government expenditures on lessons? 120 million fondas. have to pay? 40 fondas. rich creature have left to spend on bread after it has paid its taxes? fondas. (f ) Describe the set of Pareto optimal allocations. The allocations that satisfy the equations XM /G = 1/2 and XL + XM + G = $8, 000. 60 How much will a poor creature have left to spend on bread 10 fondas. after it has paid its taxes? (b) More generally, when everybody pays the same amount of taxes, if x lessons are provided by the government to each creature, the total cost 6 million to the government is (g) The Pareto optima that treat Lucy better and Melvin worse will have (more of, less of, the same amount of) public good as the Pareto optimum that treats them equally. How much taxes will every citizen If 20 lessons are given, how much will a creature has to pay is 2 times x and the taxes that one times x. Less of. NAME 435 (c) Since aerobics lessons are going to be publicly provided with everybody getting the same amount and nobody able to get more lessons from another source, each creature faces a choice problem that is formally the same as that faced by a consumer, i, who is trying to maximize a CobbDouglas utility function subject to the budget constraint 2A + B = I, where I is its income. Explain why this is the case. If A lessons are provided, your taxes are 2A fondas. After taxes, you have I − 2A fondas to 436 PUBLIC GOODS (Ch. 35) There are 3,000,000 people in all. If each gets A lessons and lessons cost 2 fondas, total cost is 6, 000, 000A. (h) With the proportional income tax scheme discussed above, what budget constraint would a rich person consider in deciding how many aerobics lessons to vote for? 3A + B = 100. What is the relevant bud- 1.5A + B = 50. spend on B. get constraint for a poor creature? (d) Suppose that the aerobics lessons are paid for by a head tax and all lessons are provided by the government in equal amounts to everyone. tax rates, how many aerobics lessons per creature would the rich favor? How many lessons would the rich people prefer to have supplied? 25. How many would the poor people prefer to have supplied? 12.5. (Hint: In each case you just have to solve for the Cobb-Douglas demand with an appropriate budget.) 50/3. How many would the poor favor? 50/3. With these What quantity of aerobics lessons per capita would be chosen under majority rule? How much bread would the rich get? poor get? 50. How much bread would the 25. (e) If the outcome is determined by majority rule, how many aerobics lessons will be provided? 75. 12.5. How much bread will the rich get? How much bread will the poor get? 25. √ 833.3. How many will the poor get? rich get? 50. 12.5. How much bread will the How much bread will the poor get? √ 1, 250. Under a proportional income tax. (Hint: In each case, solve for the consumption of bread and the consumption of aerobics lessons that a rich person gets, and plug these into the utility function.) Now calculate the utility of each poor creature under the head tax. stays at 2 fondas per unit. How many aerobics lessons will the rich get? 25. √ 312.5. √ 312.5. Un√ 416.67. (Express these utilities Under privatization. der the proportional income tax. as square roots rather than calculating out the roots.) 25. (j) Is privatization Pareto superior to the head tax? (g) Suppose that aerobics lessons remain publicly supplied but are paid for by a proportional income tax. The tax rate is set so that tax revenue pays for the lessons. If A aerobics lessons are oﬀered to each creature on Jumpo, the tax bill for a rich person will be 3A fondas and the tax bill for a poor person will be 1.5A fondas. If A lessons are given to each creature, show that total tax revenue collected will be the total cost of A lessons. There are 2,000,000 poor and 1,000,000 rich, total revenue is 2, 000, 000 × 1.5A + 1, 000, 000 × 3A = 6, 000, 000A. √ 937.5 (i) Calculate the utility of a rich creature under a head tax. Under privatization. (f ) Suppose that aerobics lessons are “privatized,” so that no lessons are supplied publicly and no taxes are collected. Every creature is allowed to buy as many lessons as it likes and as much bread as it likes. Suppose that the price of bread stays at 1 fonda per unit and the price of lessons 50/3. tional income tax Pareto superior to the head tax? Pareto superior to the proportional income tax? two answers. prefer Yes. No. Is a propor- Is privatization No. Explain the last Rich prefer privatization, proportional income tax. poor 438 Chapter 36 INFORMATION (Ch. 36) (d) Now we change our assumptions about the technology. Suppose that each producer can choose to manufacture either a high-quality or a low-quality pencil-sharpener, with a unit cost of $11.50 for the former and $11 for the latter, what would we expect to happen in equilib- NAME Information rium? Introduction. The economics of information and incentives is a relatively new branch of microeconomics, in which much intriguing work is going on. This chapter shows you a sample of these problems and the way that economists think about them. No trade. Producers would produce the low-quality product since it has a lower production cost. If all producers produce low-quality output, costs will be $11 and 36.1 (0) There are two types of electric pencil-sharpener producers. “High-quality” manufacturers produce very good sharpeners that consumers value at $14. “Low-quality” manufacturers produce less good ones that are valued at $8. At the time of purchase, customers cannot distinguish between a high-quality product and a low-quality product; nor can they identify the manufacturer. However, they can determine the quality of the product after purchase. The consumers are risk neutral; if they have probability q of getting a high-quality product and 1 − q of getting a low-quality product, then they value this prospect at 14q + 8(1 − q). Each type of manufacturer can manufacture the product at a constant unit cost of $11.50. All manufacturers behave competitively. the willingness-to-pay for low quality is $8. (e) Assuming that each producer is able to make the production choice described in the last question, what good would it do if the government banned production of low-quality electric pencil-sharpeners? If there is no ban, there will be no output and no consumers’ surplus. If low-quality (a) Suppose that the sale of low-quality electric pencil-sharpeners is illegal, so that the only items allowed to appear on the market are of high there is output $11.50. quality. What will be the equilibrium price? (b) Suppose that there were no high-quality sellers. How many low-quality sharpeners would you expect to be sold in equilibrium? Sellers won’t sell for less than $11.50, consumers won’t pay that much for low-quality product. So in equilibrium there would be no sales. (c) Could there be an equilibrium in which equal (positive) quantities of the two types of pencil sharpeners appear in the market? No. Average willingness to pay would be $11, which is less than the cost of production. So there would be zero trade. NAME 439 workers all choose not to. The competitive wage for people who take the microeconomics course will be $15 and the wage for people who don’t take the microeconomics course will be $10. (c) If there is a separating equilibrium, with high-productivity workers taking the course and low-productivity workers not taking it, then the net beneﬁts from taking the microeconomics course will be for the high-productivity workers and $−1 workers. Therefore there (will be, won’t be) equilibrium of this type. $2 for the low-productivity will be a separating be $15 and the wage for people who don’t take the microeconomics $10. (e) If there is a separating equilibrium with high-productivity workers taking the course and low-productivity workers not taking it, then the net beneﬁts from taking Kremepuﬀ’s microeconomics course will be for the high-productivity workers and $1 workers. Therefore there (will be, won’t be) equilibrium of this type. $4 for the low-productivity won’t be a separating 36.3 (1) In Enigma, Ohio, there are two kinds of workers, Klutzes whose labor is worth $1,000 per month and Kandos, whose labor is worth $2,500 per month. Enigma has exactly twice as many Klutzes as Kandos. Klutzes look just like Kandos and are accomplished liars. If you ask, they will claim to be Kandos. Kandos always tell the truth. Monitoring individual work accomplishments is too expensive to be worthwhile. In the old days, there was no way to distinguish the two types of labor, so everyone was paid the same wage. If labor markets were competitive, what was this wage? $1,500 and positive consumers’ surplus. 36.2 (0) In West Bend, Indiana, there are exactly two kinds of workers. One kind has a (constant) marginal product worth $10 and the other kind has a (constant) marginal product worth $15. There are equal numbers of workers of each kind. A ﬁrm cannot directly tell the diﬀerence between the two kinds of workers. Even after it has hired them, it won’t be able to monitor their work closely enough to determine which workers are of which type. (a) If the labor market is competitive, workers will be paid the average value of their marginal product. This amount is $12.50. (b) Suppose that the local community college oﬀers a microeconomics course in night school, taught by Professor M. De Sade. The highproductivity workers think that taking this course is just as bad as a $3 wage cut, and the low-productivity workers think it is just as bad as a $6 wage cut. The ﬁrm can observe whether or not an individual takes the microeconomics course. Suppose that the high-productivity workers all choose to take the microeconomics course and the low-productivity 440 INFORMATION (Ch. 36) (a) A professor who loves to talk oﬀered to give a free monthly lecture on macroeconomics and personal hygiene to the employees of one small ﬁrm. These lectures had no eﬀect on productivity, but both Klutzes and Kandos found them to be excruciatingly dull. To a Klutz, each hour’s lecture was as bad as losing $100. To a Kando, each hour’s lecture was as bad as losing $50. Suppose that the ﬁrm gave each of its employees a pay raise of $55 a month but insisted that he attend the professor’s lectures. What would happen to the ﬁrm’s labor force? would All Klutzes leave. Kandos would stay on. More Kandos could be hired at these terms. Klutzes would not accept job. What would (d) Suppose that Professor De Sade is called oﬀ to Washington, to lecture wayward representaatives on the economics of family values. His replacement is Professor Morton Kremepuﬀ. Kremepuﬀ prides himself on his ability to make economics “as easy as political science and as fun as the soaps on TV.” Professor Kremepuﬀ’s claims are exaggerated, but at least students like him better than De Sade. High-productivity workers think that taking Kremepuﬀ’s course is as bad as a $1 wage cut, and low-productivity workers think that taking Kremepuﬀ’s course is as bad as a $4 wage cut. If the high-productivity workers all choose to take the microeconomics course and the low-productivity workers all choose not to, the competitive wage for people who take the microeconomics course will course will be products are banned, then in equilibrium to the average productivity of the ﬁrm’s employees? happen Rise by $1,000--from $1,500 to $2,500. (b) Other ﬁrms noticed that those who had listened to the professor’s lectures were more productive than those who had not. So they tried to bid them away from their original employer. Since all those who agreed to listen to the original lecture series were Kandos, their wage was bid up to $2,500. (c) After observing the “eﬀect of his lectures on labor productivity,” the professor decided to expand his eﬀorts. He found a huge auditorium where he could lecture to all the laborers in Enigma who would listen to him. If employers believed that listening to the professor’s lectures improved productivity by the improvement in productivity in the ﬁrst small ﬁrm and oﬀered bonuses for attending the lectures accordingly, who would attend the lectures? Everybody. Having observed this outcome, how much of a wage premium would ﬁrms pay for those who had attended the professor’s lectures? 0. (d) The professor was disappointed by the results of his big lecture and decided that if he gave more lectures per month, his pupils might “learn more.” So he decided to give a course of lectures for 20 hours a month. Would there now be an equilibrium in which the Kandos all took his course and none of the Klutzes took it and where those who took the course were paid according to their true productivity? Yes. If those who take the course get $2,500 and people who do not get $1,000 a month, then Kandos would take the course, since the NAME 441 pain of 20 hours of lecture costs $1,000, but the wage premium is $1,500. Klutzes would not take the course, since the pain (Ch. 36) buyers be willing to pay for a used car? $1,400. Would owners $1,500. car owners won’t sell. sell. 15 hours 36.4 (1) Old MacDonald produces hay. He has a single employee, Jack. If Jack works for x hours he can produce x bales of hay. Each bale of hay sells for $1. The cost to Jack of working x hours is c(x) = x2 /10. (a) What is the eﬃcient number of bales of hay for Jack to cut? No. of good used cars be willing to sell their used cars at this price? Describe the equilibrium that would take place in Rustbucket. (e) What is the smallest number of hours the professor could lecture and still maintain a separating equilibrium? INFORMATION (a) If all 200 used cars in Rustbucket were for sale, how much would No. Would there be an equilibrium in which all used cars are sold? of lectures costs $2,000 a month and the wage premium is 442 5. Good Lemon owners will Price of a used car will be $300. (b) Suppose that instead of there being 100 cars of each kind, everyone in town is aware that there are 120 good cars and 80 lemons. How much would buyers be willing to pay for a used car? $1,620. Would owners of good used cars be willing to sell their used cars at this price? Yes. Would there be an equilibrium in which all used cars are sold? Yes. Would there be an equilibrium in which only the lemons were (b) If the most that Jack could earn elsewhere is zero, how much would MacDonald have to pay him to get him to work the eﬃcient amount? sold? 2 5 /10 = $2.50. Yes. Describe the possible equilibrium or equilibria that would take place in Rustbucket. (c) What is MacDonald’s net proﬁt? 5 − 2.50 = $2.50. One equilibrium has all cars sold at a price of $1,620. There is (d) Suppose that Jack would receive $1 for passing out leaﬂets, an activity that involves no eﬀort whatsoever. How much would he have to receive also from MacDonald for producing the eﬃcient number of bales of hay? are sold. $3.50. 36.6 (1) Each year, 1,000 citizens of New Crankshaft, Pennsylvania, sell their used cars and buy new cars. The original owners of the old cars have no place to keep second cars and must sell them. These used cars vary a great deal in quality. Their original owners know exactly what is good and what is bad about their cars, but potential buyers can’t tell them apart by looking at them. Lamentably, though they are in other respects model citizens, the used-car owners in New Crankshaft have no scruples about lying about their old jalopies. Each car has a value, V , which a buyer who knew all about its qualities would be willing to pay. There is a very large number of potential buyers, any one of which would be willing to pay $V for a car of value $V. The distribution of values of used cars on the market is quite simply described. In any year, for any V between 0 and $2,000, the number of used cars available for sale that are worth less than $V is V /2. Potential used-car buyers are all risk-neutral. That is if they don’t know the value of a car for certain, they value it at its expected value, given the information they have. (e) Suppose now that the opportunity for passing out leaﬂets is no longer available, but that MacDonald decides to rent his hayﬁeld out to Jack for a ﬂat fee. How much would he rent it for? $2.50. 36.5 (0) In Rustbucket, Michigan, there are 200 people who want to sell their used cars. Everybody knows that 100 of these cars are “lemons” and 100 of these cars are “good.” The problem is that nobody except the original owners know which are which. Owners of lemons will be happy to get rid of their cars for any price greater than $200. Owners of good used cars will be willing to sell them for any price greater than $1,500, but will keep them if they can’t get $1,500. There are a large number of buyers who would be willing to pay $2,500 for a good used car, but would pay only $300 for a lemon. When these buyers are not sure of the quality of the car they buy, they are willing to pay the expected value of the car, given the knowledge they have. NAME 443 Rod’s Garage in New Crankshaft will test out any used car and ﬁnd its true value V . Rod’s Garage is known to be perfectly accurate and perfectly honest in its appraisals. The only problem is that getting an accurate appraisal costs $200. People with terrible cars are not going to want to pay $200 to have Rod tell the world how bad their cars are. But people with very good cars will be willing to pay Rod the $200 to get their cars appraised, so they can sell them for their true values. Let’s try to ﬁgure our exactly how the equilibrium works, which cars get appraised, and what the unappraised cars sell for. 444 an equilibrium where only the lemons INFORMATION (Ch. 36) (f ) In equilibrium, what will be the total net revenue of all owners of used cars, after Rod’s Garage has been paid for its appraisals? $1, 000, 000 − 800 × 200 = 840, 000. all sell for $1,000 for total revenue of 36.7 (2) In Pot Hole, Georgia, 1,000 people want to sell their used cars. These cars vary in quality. Original owners know exactly what their cars are worth. All used cars look the same to potential buyers until they have bought them; then they ﬁnd out the truth. For any number X between 0 and 2,000, the number of cars of quality lower than X is X/2. If a car is of quality X, its original owner will be willing to sell it for any price greater than X. If a buyer knew that a car was of quality X, she would be willing to pay X + 500 for it. When buyers are not sure of the quality of a car, they are willing to pay its expected value, given their knowledge of the distribution of qualities on the market. $1,000,000. (a) Suppose that everybody knows that all the used cars in Pot Hole are (a) If nobody had their car appraised, what would the market price for used cars in North Crankshaft be and what would be the total revenue received by used-car owners for their cars? They’d (b) If all the cars that are worth more than $X are appraised and all the cars that are worth less than $X are sold without appraisal, what will the market price of unappraised used cars be? (Hint: What is the expected value of a random draw from the set of cars worth less than $X?) for sale. What would used cars sell for? cars would appear on the market? $X/2. $1,500. used car owner be willing to sell at this price? No. Would every Which used Those worth less than $1,500. (c) If all the cars that are worth more than $X are appraised and all the cars that are worth less than $X are sold without appraisal, then if your car is worth $X, how much money would you have left if you had it appraised and then sold it for its true value? $X − 200. How (b) Let X ∗ be some number between 0 and 2,000 and suppose that all cars of quality lower than X ∗ are sold, but original owners keep all cars of quality higher than X ∗ . What would buyers be willing to pay for a used car? much money would you get if you sold it without having it appraised? for sale? $X/2. (d) In equilibrium, there will be a car of marginal quality such that all cars better than this car will be appraised and all cars worse than this car will be sold without being appraised. The owner of this car will be just indiﬀerent between selling his car unappraised and having it appraised. What will be the value of this marginal car? Solve X/2 = X − 200 to get X = $400. (e) In equilibrium, how many cars will be sold unappraised and what will they sell for? The worst 200 cars will be unappraised and will sell for $200. X ∗ /2 + 500. At this price, which used cars would be Cars worth less than X ∗ /2 + 500. (c) Write an equation for the equilibrium value of X ∗ , at which the price that buyers are willing to pay is exactly enough to induce all cars of quality less than X ∗ into the market. X ∗ /2 + 500 = X ∗ . this equation for the equilibrium value of X ∗ . Solve X ∗ = $1, 000. NAME 445