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Unformatted text preview: Chapter 1 NAME The Market
Introduction. The problems in this chapter examine some variations on
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. 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.) Person Price =A = 40 B 25 CD 30 35 E 10 F 18 G 15 H 5 (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.) 2 THE MARKET (Ch. 1) Price 60 50 40 30 20 10 0 1 2 3 4 5 67 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 units? $18. $15. A, B, C, D. $10 to $15. (c) What is the lowest price that would make the market demand equal to 5 units? (d) With a supply of 4 apartments, which of the people A–H end up getting apartments? (e) What if the supply of apartments increases to 6 units. What is the range of equilibrium prices? 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 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. (a) Write down your budget equation. 10x1 + 5x2 = 40. (b) If you spent all your income on commodity 1, how much could you buy? 4. 8.
Use blue ink to draw your budget line in the graph (c) If you spent all of your income on commodity 2, how much could you buy? below. x2 8 6 4 2 ;;;;;; ;;;;;; Line Blue ;;;;;; ;;;;;; ;;;;;; Red Line ;;;;;; ;;;;;; ;;;;;;Black Shading ;;;;;; ;;;;;; ;;;;;; ;;;;;;;;;;;;; ;;;;;; ;;;;;;;;;;;;; Black Line ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;; Blue ;;;;;;;;;;;;; ;;;;;;;;;;;;; Shading ;;;;;;;;;;;;; ;;;;;;;;;;;;; ;;;;;;;;;;;;;
2 4 6 8 x1 0 (d) Suppose that the price of commodity 1 falls to $5 while everything else stays the same. Write down your new budget equation. 5x1 +5x2 = 40. 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. Use black ink to draw this budget (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). 2.2 (0) On the graph below, draw a budget line for each case. 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.) (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?)
x2 20 15 Blue Line = Pencil Line Black Line
10 5 Red Line 0 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 12 8 4 0 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? 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 price of x is 1. x + 2y = 16. 3x + 6y = 48. (f ) Write another budget equation that gives you the same budget line, but where the price of x is 3. 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. (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 ? $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. Comic books 32 24 16 8 0 8 12 16 24 32 Candy bars 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 of jars of jam consumed by Gunnar in a year. 2P + 4M + 6J = 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 the price of meatballs? 2 crowns. 3 (c) In terms of the new currency, what is the price of jam? crowns.
(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. P + 2M + 3J = (e) Write down Gunnar’s new budget equation. 180. No. Is Gunnar’s budget set any diﬀerent than it was before the change? 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. (a) If Edmund accepts zero sacks of garbage, how many video cassettes can he buy? 0. NAME 11 (b) If he accepts 15 sacks of garbage, how many video cassettes can he buy? 5. 6C − 2G = 0. (c) Write down an equation for his budget line. (d) Draw Edmund’s budget line and shade in his budget set.
Garbage 20 15 10 5 ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;;Budget Line ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; Set Budget ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;; ;;;;;;;;;
5 10 15 20 Video cassettes 0 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.
Administrator speeches 100 75 50 25 0 25 50 75 100 Politician speeches 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 twodimensional 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. 2.9 (0) Jonathan Livingstone Yuppie is a prosperous lawyer. He has, in his own words, “outgrown those conﬁning twocommodity 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 threedimensional diagram to show his budget set. Label the intersections of the budget set with each axis.
M 8 5 20 T W (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 restaurant meals and whiskey that he could aﬀord? 20W +50M = 320.
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 (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 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.) (b) How many pages of economics could she read if she decided to spend all of her time reading economics? 50 pages. 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 10,000 recent M.B.A.’s and by 3,000 lawyers with hot tubs. (b) If he spends his entire advertising budget on the consumer publication, his ad will be read by lawyers with hot tubs. 6,000 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 4,000 (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 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. 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/x2 . (If you don’t know calculus, you will have to take our A word for this.) Find Charlie’s marginal rate of substitution at the point, (10, 10). −1. −4. (l) What is his marginal rate of substitution at the point (5, 20)? (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). (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. (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 (d) What is the slope of his indiﬀerence curve at the point (4, 12)? −1. Berries 40 30 20 10 ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Pencil Shading ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Red Curve ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Red ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Blue Curve ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; Shading ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;
10 20 30 40 Nuts ; 0 (e) What is the slope of his indiﬀerence curve at the point (9, 12)? at the point (4, 16)? −2/3 −1. (f ) Do the indiﬀerence curves you have drawn for Ambrose exhibit diminishing marginal rate of substitution? Yes. (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 16ounce cans and 8ounce cans of beer. Use blue ink to draw these indiﬀerence curves. 22 PREFERENCES (Ch. 3) 8ounce 8 6 Blue Lines 4 Red Lines 2 0 2 4 6 8 16ounce (b) Lorraine Quiche likes to have a beer while she watches “Masterpiece Theatre.” She only allows herself an 8ounce 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. 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. NAME 23 Dimes 8 6 4 2 ;;;;;;;;;;;;; ; ; ;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; Blue ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; Red ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; shading ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; shading ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ; ;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ;;;;;; ;;;;;;;;;;;; ; ; ; ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;; ; ; ; ;;;;;;;;;;;;;;;;;; Black ;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;; ; ; ; ;;;;;;;;;;;; ;;;;;;;;;;;; ; ;;;;;;;;;;;;;;;;;; ; lines ;;;;;;;;;;;;;;;;;; ; ; ; ;;;;;;;;;;;;;;;;;; ; ; ;;;;;;;;;;;;;;;;;; ;
2 4 6 8 Quarters 0 (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.
(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. (c) Does Elmo have convex preferences between dimes and quarters? Yes.
(d) Does Elmo always prefer more of both kinds of money to less? No. (e) Does Elmo have a bliss point? 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 24 PREFERENCES (Ch. 3) as if Elmo’s fractional quarters and fractional dimes are accepted at the corresponding fraction of their value.) Describe these new indiﬀerence curves in words. Line segments with slope −2.5. 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. (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. Steeper. (b) Do Randy’s indiﬀerence curves get steeper or ﬂatter as you move from left to right along one of them? Hours studying history 8 6 Preference direction 4 2 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 31 (h) Are Coach Steroid’s new preferences reﬂexive? 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? Grubs. (c) What order should Baby Bear suggest if he wants to get his favorite food for dinner? 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 sixcup 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’ sixcup 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 . (b) B ∼ A. True. True. 32 PREFERENCES (Ch. 3) (c) B ∼ C . (d) A ∼ C . (e) C ∼ A. (f ) A B. True. False. False. True. True. True. False. True. False. False. False. False. True.
, transitive? (g) B A. (h) B C. (i) A C. (j) C A. (k) A B. (l) B A. (m) B C. (n) A C. (o) C A. (p) Is Olson’s “atleastasgoodas” relation, No. No. (q) Is Olson’s “can’ttellthediﬀerence” relation, ∼, transitive? (r) is Olson’s “betterthan” relation, , transitive. Yes. NAME 35 u(x1 , x2 ) 2x1 + 3x2 4x1 + 6x2 ax1 + bx2 √ 2 x1 + x2 ln x1 + x2 v (x1 ) + x2 x1 x2 xa xb 12 (x1 + 2)(x2 + 1) (x1 + a)(x2 + b) xa + xa 1 2 M U1 (x1 , x2 ) M U2 (x1 , x2 ) M RS (x1 , x2 ) 2 4 a
√1 x1 3 6 b 1 1 1 x1 bxaxb−1 12 x1 + 2 x1 + a axa−1 2 − − − −2/3 −2/3 −a/b − √1 1 x −1/x1 − v (x 1 ) −x2 /x1
2 − ax1 bx 1/x1 v (x 1 ) x2 axa−1 xb 2 1 x2 + 1 x2 + b axa−1 1 x2 +1 x1 +2 x2 +b x1 +a a−1 x1 x2 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 . (a) Charlie has 40 apples and 5 bananas. Charlie’s utility for the bundle (40, 5) is U (40, 5) = 200. The indiﬀerence curve through (40, 5) includes all commodity bundles (xA , xB ) such that xA xB = 200. So 200 . On the indiﬀerence curve through (40, 5) has the equation xB = 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 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 welloﬀ 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 39 “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, you see.” Would Burt be willing to give up 1 cookie for 3 glasses of milk? 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, 10. (c) Now use red ink to sketch in the indiﬀerence curve along which U (x, y ) = 10. Does Phil have convex preferences?
y 20 No. 15 Blue lines 10 5 2y=10 Red indifference curve 5 x=10 10 15 20 x 0 40 UTILITY (Ch. 4) 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. (a) Write a utility function that represents Nancy’s preferences over alternative combinations of test scores x1 and x2 on tests 1 and 2 respectively. U (x1 , x2 ) = min{x1 , x2 }, or any monotonic transformation.
4.6 (0) Remember Shirley Sixpack and Lorraine Quiche from the last chapter? Shirley thinks a 16ounce can of beer is just as good as two 8ounce cans. Lorraine only drinks 8 ounces at a time and hates stale beer, so she thinks a 16ounce can is no better or worse than an 8ounce can. (a) Write a utility function that represents Shirley’s preferences between commodity bundles comprised of 8ounce cans and 16ounce cans of beer. Let X stand for the number of 8ounce cans and Y stand for the number of 16ounce cans. 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? Yes. Would the utility function U (x, y ) = (5X + 10Y )2 represent her preferences? Yes. Would the utility function U (x, y ) = X + 3Y represent her preferences? 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 second bundle better than the ﬁrst bundle. Shirley prefers (0,2) to (3,0). Lorraine disagrees. 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. (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 43 calculus users: A diﬀerentiable function f (u) is an increasing function of u if its derivative is positive.) (a) f (u) = 3.141592u. Yes. No. Yes. (b) f (u) = 5, 000 − 23u. (c) f (u) = u − 100, 000. (d) f (u) = log10 u. (e) f (u) = −e−u . Yes. Yes. (f ) f (u) = 1/u. No. Yes. (g) f (u) = −1/u. 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. (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 8 6 (6,4) 4 2 (8,2) 0 2 4 6 8 Animal crackers 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 (6,4) 4 2 (8,2) 0 2 4 6 8 Animal crackers (c) Are Martha’s preferences convex? Yes. Are Bertha’s? Yes. (d) What can you say about the diﬀerence between the indiﬀerence curves you drew for Bertha and those you drew for Martha? 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 by a monotonic transformation.
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 ) = x2 + x2 . 1 2 (a) Draw a few of his indiﬀerence curves. What kind of geometric ﬁgure are they? Quarter circles centered at the
Does Willy have convex preferences? origin. No. True or false 1. F 2. F 3. F 4. F 5. F 6. T Multiple choice 1. e 2. b 3. b 4. e 5. a 6. c 1 ...
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 Fall '08
 Hansen
 Microeconomics

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