1、Problem set # 1 (Preferences and Utility)ECON 3113Multiple choice1. The assumption that preferences are complete requires the consumer(a) to rank any two baskets.(b) to say that basket C is preferred to basket A if basket B is preferred to basket A andbasket C is preferred to basket B.(c) to rank a
2、basket with more units of all goods higher than a basket with fewer units ofall goods.(d) to have a diminishing marginal rate of substitution.2. Assume that a consumer prefers watching the World Series to watching the SuperBowl and that this same consumer prefers watching the Super Bowl to watching
3、the StanleyCup championship game. Further assume that this same consumer states, “I would prefer towatch the Stanley Cup championship game to the World Series.” This consumers preferencesviolate which of the following key assumptions?(a) Completeness.(b) Transitivity.(c) More is better.(d) Both comp
4、leteness and transitivity.3. Identify the truthfulness of the following statements.I. Ordinal utility gives us information about which basket the consumer prefers and themagnitude of the preference.II. Cardinal utility gives us information about which basket the consumer prefers but notabout the int
5、ensity of those preferences.(a) Both I and II are true.(b) Both I and II are false.(c) I is true; II is false.1(d) I is false; II is true.4. Marginal utility(a) is the ratio of total utility to total consumption.(b) is the rate at which total utility changes as the level of consumption rises.(c) wil
6、l always be equal to the products price.(d) tells us nothing; were only concerned with total utility.5. Suppose that a consumer has the utility function u(A,B) = 3A+ 4B. The MRS is(a) 0.75(b) 1.25(c) 1.00 since A and B are perfect substitutes(d) cannot be determined since the values of A and B are u
7、nknown6. Suppose that a consumer has the utility function u(A,B) = 3A+4B. If A is measuredon the horizontal axis,(a) The indifference curves will be vertical.(b) The indifference curves will be horizontal.(c) The indifference curves will be straight lines with slope -0.75.(d) The indifference curves
8、 will be straight lines with slope -1.25.7. Consider the following three market baskets.Basket Good x Good yA 2 6B 6 4C 4 5If Basket A and Basket B are on the same indifference curve, preferences satisfy the usualassumptions, and the indifference curves have a diminishing marginal rate of substituti
9、on,(a) Basket C is preferred to Basket A(b) Basket A is preferred to Basket C(c) The consumer is indifferent between Basket A and Basket C2(d) There is not enough information to determine how the consumer would rank BasketA relative to Basket C.8. Consider the utility function u = min2x,y. If x is o
10、n the horizontal axis, theindifference curves for this utility function will be(a) vertical(b) horizontal(c) upward sloping(d) L-shapedProblems1. Coach Steroid likes his players to be big, fast, and obedient. If player A is better thanplayer B in two of these three characteristics, Steroid will pref
11、er A to B. Three players tryout for quarterback. Wilbur Westinghouse (W) weighs 320 pounds, runs very slowly, and isquite obedient. Harold Hotpoint (H) weighs 240 pounds, runs extremely fast, and is extremelydisobedient. Jerry Jacuzzi (J) weighs 150 pounds, runs at average speed, and is extremelyobe
12、dient. Does Coach Steroid have transitive preferences? Explain your answer.2. Ikes utility function is U(x,y) = 25xy. He has 12 units of good x and 8 units of goody. Bens utility function for the same two goods is U(x,y) = 4x + 4y. Ben has 9 units of xand 13 units of y. Does Ike prefer his own baske
13、t or Bens? How about Ben?3. Consider the following utility function: u(x1,x2) = 2x1 + 3x2.a) Is the preference represented by this utility function monotonic, i.e., more is better?b) Draw a typical indifference curve and label a weakly preferred set.c) Calculate the MRS. Does this preference exhibit
14、 diminishing MRS?4. Consider the following utility function: u(x1,x2) = x21x32.a) Is the preference represented by this utility function monotonic?b) Draw a typical indifference curve and label the weakly preferred set.3c) Calculate the marginal utilities and the MRS. Does this preference exhibit di
15、minishingMRS?5. Consider the following utility function: u(x1,x2) = min2x1,3x2.a) Is the preference represented by this utility function monotonic?b) Draw a typical indifference curve and label a weakly preferred set.c) calculate the MRS. Does this preference exhibit diminishing MRS?4Solution to Pro
16、blem Set # 1 (Preferences and Utility)ECON 3113Multiple choice1. (a).2. (b).3. (b).4. (b).5. (a). The utility function u(A,B) = 3A+ 4B.MUA = 3,MUB = 4, MRS = MUAMUB= 34 = 0.75.6. (c)7. (a). Note that C is exactly half of A + B. Therefore, if we draw a line, these threebaskets will be on the same lin
17、e. Now we know that the preference exhibits diminishing MRS,which implies that the indifference curve is convex. Drawing an indifference curve that goesthrough A and B (they are on the same indifference curve), then C must be outside this curvesince the curve is convex.8. (d).Problems1. Steroid pref
18、ers W to H (W follows H) because W is heavier and more obedient. He prefersH to J (H follows J) because H is heavier and faster. But he prefers J to W (J follows W) becauseJ is more obedient and faster than W. Since his preferences have a cycle, they cannot betransitive.2. Ikes basket is (x,y) = (12
19、,8) and Bens basket is (9,13). Ike would prefer Bens basketsince913 128 U(9,13) U(12,8).However, Ben prefers his own basket since,49 + 413 412 + 48 U(9,13) U(12,8).13. a) Yes, the preference is monotonic, since the utility goes up when either x1 or x2 goesup. Alternatively, we can say that the prefe
20、rence is monotonic since MU1 = ux1 = 2 0 andMU2 = ux2 = 3 0.b) The equation for the indifference curve with utility level k is 2x1 + 3x2 = k. We plotthe graph for k = 6 in Figure 1.Figure 1: Indifference curve: 2x1 + 3x2 = 6.Pick any basket on this indifference curve, say (3/2,1). The set weakly pre
21、ferred to (3/2,1)is the whole area to the up and right of this indifference curve, including the curve itself. InFigure 1, its the area above the red line.c) Using the marginal utilities we calculated in a), we can obtain MRS = MU1MU2 = 23. Sincethe MRS is constant, we say the preference does not ex
22、hibit (strictly) diminishing MRS.4. a) The preference is monotonic, since MU1 = 2x1x32 0 and MU2 = 3x21x22 0.b) A typical indifference curve with utility level k satisfies the equation x21x32 = k. We plotthe indifference curve with k = 9 in Figure 2. The set weakly preferred to any basket on thisind
23、ifference curve is the area above this curve, including the curve itself.c) MRS = MU1MU2 = 2x1x323x21x22= 2x23x1. As x1 increases and x2 decreases, MRS decreases. So thepreference exhibits diminishing MRS.5. a) Yes, the preference is monotonic, since u(x1 +c,x2) u(x1,x2) and u(x1,x2 +c) u(x1,x2) for
24、 any c 0. That is, more of either good makes the consumer (weakly) better off.2Figure 2: Indifference curve: x21x32 = 9.Note that the utility function is not differentiable everywhere, we cant use marginal utilityto answer this question.b) A typical indifference curve with utility level k has the eq
25、uation min2x1,3x2 = k. Theindifference curve is L-shaped, with the corner point being (k2, k3). Two sample indifferencecurves for k = 12 and k = 18 respectively are plotted in Figure 3. The set weakly preferred toany basket on an indifference curve is the area above the L-shaped indifference curve,
26、includingthe curve itself.c) We can see that MRS = 0 on the horizontal line part and MRS = on the verticalline part of the indifference curve. This preference does not exhibit diminishing MRS sinceMRS is a constant (zero) on the horizontal part of the indifference curve.3Problem set # 3 (Optimal Cho
27、ice)ECON 3113Multiple choice1. Economists describe consumer choice as a constrained optimization problem. Whatis the consumer trying to do?a) Maximize income subject to the budget constraint.b) Maximize the budget constraint.c) Maximize utility subject to the budget constraint.d) Maximize market bas
28、kets.2. At a consumers interior optimum solution, which of the following will not neces-sarily hold true?a) MUx = MUyb) MUxMUy = pxpyc) MRS = MUxMUyd) MRS = pxpy3. Suppose the price of A is $4, the price of B is $2, and that the consumer is currentlyspending all available income and is consuming bot
29、h goods. At the consumers currentconsumption basket the marginal utility of A is 5 and the marginal utility of B is 3.a) The consumer is currently maximizing utility.b) The consumer could increase utility by consuming more of good B and less of goodA.c) The consumer could increase utility by consumi
30、ng more of good A and less of goodB.d) Nothing can be said about the consumers utility because we do not know theconsumers income or utility function.4. Suppose the price of good x is $2 and the price of good y is $3. Also, supposeMUx = y and MUy = x. Which of the following baskets could be an inter
31、ior optimum?a) x = 10,y = 151b) x = 7.5,y = 5c) x = 2,y = 4d) x = 6,y = 85. The “same bang for the buck” idea means that the consumer is equatinga) the marginal utilities of all of the goods purchased.b) the prices of all the goods purchased.c) the marginal utilities of the last dollar spent on each
32、 good purchased.d) the ratios of the last dollar spent on each good purchased.6. Suppose that a consumer considers coffee and tea to be perfect substitutes, but herequires two cups of tea to give up one cup of coffee. This consumers budget constraintcan be written as 2.5C +T = 10. What should the co
33、nsumer buy?a) 2 cups of tea and no coffee.b) 10 cups of tea and no coffeec) 2.5 cups of coffee and no tea.d) 4 cups of coffee and no tea.Problems1. Consider the following utility function: u(x1,x2) = x1x2. The price of good 1 isp1 = 1. The price of good 2 is p2 = 2 and consumer has an income of I =
34、20. Derive theconsumers optimal choice for both goods.2. Consider the following utility function: u(x,y) = x2/3y1/3. Suppose that Px = 4,Py = 2 and the income is I = 30. Derive the optimal choice for both goods.3. Suppose that John is always willing to substitute 3 bottles of Budweiser with 2bottles
35、 of Heineken. A utility function representing this preference is u(x1,x2) = 2x1+3x2,where x1 is the bottles of Budweiser and x2 is the bottles of Heineken(a) Find his optimal consumption bundle if the price of one bottle of Budweiser is 2and the price of Heineken is 4. His income is 12.(b) Find his
36、optimal consumption bundle if the price of one bottle of Budweiser is 3and of Heineken is 4. His income is still 12.2To simplify the problem assume that John can buy and consume any fraction of onebottle of beer.4. Suppose that John always consumes 5 tablespoons of coffee together with 2 table-spoon
37、s of sugar.(a) Write down his utility function.(b) Find his optimal consumption choice if the price of a tablespoon of coffee is p1 = 1and of sugar is p2 = 3 and his income is I = 44. Assume that John can buy and consumeany fraction of a tablespoon.5. Kristen takes two exams in her Public Economics
38、course. Her overall grade forthe course is the minimum of her scores on these two exams. She would like to spendno more than 2400 minutes studying for the two exams and she wants to maximize heroverall course grade. If she spends m1 minutes studying for the first exam, her score onthis exam will be
39、m120 . For each point in the second exam, she needs 10 minutes studying.(1) Write down a utility function that represents her preference.(2) Write down the “budget equation”.(3) Derive her optimal choice, i.e., how much time to study for each exam, the resultingexam scores and her overall grade.3Sol
40、utions to problem set # 3 (Optimal Choice)ECON 3113Multiple choice1. c).2. a).3. b). The marginal utility to price ratio for the two goods arerA = MUApA= 54,rB = MUBpB= 32.We can find that rA r2, implying that good 1 (Budweiser) delivers more bang for the samebuck. So the optimal choice involves no
41、consumption of Heineken, i.e.,x2 = 0 .Plug this into the equation for budget line, we havep1x1 +p2x2 = I 2x1 + 40 = 12 x1 = 6 .(b) Now p1 = 3. The two ratios are,r1 = MU1p1= 23, r2 = MU2p2= 34.It can be showed that r1 r1, implying that only good 2 (clothing) should be consumed and x1 = 0.Now plug x1
42、 = 0 use the budget equation:p1x1 + p2x2 = I 0 + 1x2 = I x2 = I .The Engel curve of clothing is drawn in Figure 6. If we draw a similar graph for food,then its Engel curve will simply be the vertical axis in that graph.Figure 6: Engel curves for clothing: x2 = I.Clothing is a normal good. Food is NO
43、T a normal good, or an inferior good, since itsdemand does not change when income changes.(2) Now we are given p2 = 2, I = 30. The marginal utility to price ratios for both goodsare,r1 = MU1p1= 3p1, r2 = MU2p2= 22 = 1.When p1 3, r1 r2, implying that only good 1 (food) should be consumed. Thusx2 = 0.
44、 Plug it into the budget equation:p1x1 + p2x2 = I p1x1 + 0 = 30 x1 = 30p1 .When p1 = 3, r1 = r2, the consumer is indifferent between any combinations of food orclothing. If all income is spent on food, we have x1 = Ip1 = 303 = 10. Thus the demand offood is anywhere between 0 and 30 when p1 = 3. Comb
45、ine all these three cases, the demandcurve for food is drawn in Figure 7.6Figure 7: Demand curve when the preference is perfect substitutes.Good 1 is ordinary (demand goes down when price goes up) when p1 3. , it is not ordinary overall.4. The inverse demand function is p = 302b, where p is the pric
46、e of burger, and b isthe number of burgers demanded.(1) The current price of burger is $10. Plug it into the inverse demand function, andsolve for demand, we havep = 302b 10 = 302bb = 10That is, he is consuming 10 burgers. The green lines illustrate the current price andconsumption. See Figure 8.Fig
47、ure 8: Inverse demand function: p = 302b.(2) His total willingness to pay (WTP) is the area below the demand curve, but to theleft of the vertical green line, which is difference between the areas of two triangles. That is,7WTP(b = 10) = 12 3015 12 10(1510) = 22525 = 200.(3) While his WTP is $200 fo
48、r 10 burgers at a price of $10 each, he is only payingpb = 1010 = $100. The consumer surplus he is enjoying is the different between WTPand the amount he is paying, i.e., the consumer surplus isCS = 200100 = $100.(4) When price increases to $14, using the inverse demand function, we havep = 302b 14
49、= 302b 2b = 3014 b = 8.The relevant lines for the price and consumption now are the blue lines. Calculate theWTP and CS the same way we did when p = $10, we can obtain the WTP and CS whenp = $14.WTP(b = 8) = 12 3015 12 14(158) = 22549 = 176.The total amount he pays is pb = 148 = 112.His consumer surplus is the difference, 176112 = $64.The change in consumer surplus is the difference in two consumer surplus under differentprices, 64100 = $36. Therefore, when price of burger increases from $10 to $14, Mr
