ECO670......

1. "Deriving the Envelope Thoerem: Consider the more general problem M(o, ) = max, f(x, a, 8) subject to g(r, a, 8) = 0. Show that: dM(a, B) _ Of(x', a, b) ag(x*, a, b) da da da 2. For each of the following, derive x(p,m), h(p,u), v(p,m), h(p,") using the standard budget constraint PIT, + Pal, = m: (a) u(x1, 12) = max(11, 12) (b) u(x1, 12) = min(11, 12) (c) u(x1, 12) = 2:1 + 12 1/2 1/3 (d) u(r1, 12) = 1 12 (e) u(1, 12) = Inn + ; Inx2 (f) What happens if we replace u by e" in part (e)? Compare this to part (d). Can you work out an easy way to derive the hicksian demand functions of a function when you make monotonic transformations of the original function? 3. *A besotted mathematician named Donic consumes either gin or tonic. His preferences are rare 'cause he thinks in the square in a way that is almost sardonic. Specifically, Donic prefers larger drinks to smaller drinks but requires that the square of the amount of lime in a drink equal the sum of the squares of the amounts of gin and tonic. Find a utility function that represents Donic's preferences. Find Donic's Marshalian demand functions for lime, tonic, and gin. 4. (From Midterm 2005) Consider a consumer with a utility function u(11, 12) = e(film(32))1/3 (a) What properties about utility functions will make this problem easier to solve? (b) Which of the non negativity input demand constraints will bind for small m? (c) Derive for the marshallian (uncompensated) demand functions and the indirect utility function. (d) Derive the expenditure function in terms of original utils u. 5. Consider the indirect utility function given by: m v(P1. pz, m) = - P1 + P2 (a) What are the demand functions (b) What is the expenditure function? (c) What is the direct utility function? 6. "Consider the utility function: u(X1, 12) = min(2r1 + 12, X1 + 2.x2) (a) Draw the indiference curve for u(x1, 12) = 20. Shade the area where u(x1, 12) 2 20 (b) For what values of & will the unique optimum be x, = 0 (c) For what values of A will the unique optimum be x2 = 0 (d) If neither x and x2 is equal to zero, and the optimum is unique, what must be the value of #?5. Consider the indirect utility function given by: v(p1, p2, m) = m Pi + p2 (a) What are the demand functions (b) What is the expenditure function? (c) What is the direct utility function? 6. 'Consider the utility function: u(x1, 12) = min(2r1 + 12, $1 + 212) (a) Draw the indiference curve for u(x1, 12) = 20. Shade the area where u(x1, 12) 2 20 (b) For what values of 2 will the unique optimum be x1 = 0 (c) For what values of 2 will the unique optimum be x2 = 0 (d) If neither x, and x, is equal to zero, and the optimum is unique, what must be the value of 37 7. Assume that there is a consumer with weakly monotonic, convex preferences and who is a utility maximizer. For each of the following pairs of bundles, specify if bundle 1 is , 3, or uncomparable to bundle 2. (a) Suppose you have no data: 1. Bundle 1: x1 = 3,x2 = 3, Bundle 2: 21 = 6,12 = 2.5 2. Bundle 1: x1 = 3, 12 = 3, Bundle 2: 21 = 2.5,12 = 2.5 (b) Suppose that you observe that when p1 = 1, p2 = 1, m = 10 the consumer chooses 11 = 2,12 = 8 1. Bundle 1: x1 = 4, x2 = 1, Bundle 2: x1 = 3, x2 = 6 2. Bundle 1 x1 = 6, 12 = 4, Bundle 2: x1 = 3, x2 = 8 (c) Suppose that we have two observations. When p1 = 1,p2 = 1, m = 10 the consumer chooses 21 = 2, x2 = 8. When p, = 1, p2 = 3, m = 15 the consumer chooses x1 = 15, 12 = 0 1. Bundle ]: x1 = 5,12 = 2, Bundle 2: x1 = .0, 12 = 2.5 2. Bundle 1: x1 = 5,12 = 2, Bundle 2: x1 = 6.5, 12 = 05. Consider the indirect utility function given by: v(p1, p2, m) = m Pi + p2 (a) What are the demand functions (b) What is the expenditure function? (c) What is the direct utility function? 6. 'Consider the utility function: u(x1, 12) = min(2r1 + 12, $1 + 212) (a) Draw the indiference curve for u(x1, 12) = 20. Shade the area where u(x1, 12) 2 20 (b) For what values of 2 will the unique optimum be x1 = 0 (c) For what values of 2 will the unique optimum be x2 = 0 (d) If neither x, and x, is equal to zero, and the optimum is unique, what must be the value of 37 7. Assume that there is a consumer with weakly monotonic, convex preferences and who is a utility maximizer. For each of the following pairs of bundles, specify if bundle 1 is , 3, or uncomparable to bundle 2. (a) Suppose you have no data: 1. Bundle 1: x1 = 3,x2 = 3, Bundle 2: 21 = 6,12 = 2.5 2. Bundle 1: x1 = 3, 12 = 3, Bundle 2: 21 = 2.5,12 = 2.5 (b) Suppose that you observe that when p1 = 1, p2 = 1, m = 10 the consumer chooses 11 = 2,12 = 8 1. Bundle 1: x1 = 4, x2 = 1, Bundle 2: x1 = 3, x2 = 6 2. Bundle 1 x1 = 6, 12 = 4, Bundle 2: x1 = 3, x2 = 8 (c) Suppose that we have two observations. When p1 = 1,p2 = 1, m = 10 the consumer chooses 21 = 2, x2 = 8. When p, = 1, p2 = 3, m = 15 the consumer chooses x1 = 15, 12 = 0 1. Bundle ]: x1 = 5,12 = 2, Bundle 2: x1 = .0, 12 = 2.5 2. Bundle 1: x1 = 5,12 = 2, Bundle 2: x1 = 6.5, 12 = 0Problem Set 2 1. (10 points) Annie and David are painting their apartment. At the paint store, David says he prefers Canary Yellow to Bumblebee Yellow, Lime Yellow, and Crayola Yellow. Annie finds new paint samples and asks David to compare Canary Yellow to School Bus Yellow and to Sunrise Yellow. David prefers Sunrise Yellow to Canary Yellow, and prefers School Bus Yellow to Canary Yellow. He also prefers Sunrise Yellow to School Bus Yellow. The store is out of Sunrise Yellow, so they buy School Bus Yellow and paint their apartment with it. David then insists that they go back, buy Lime Yellow, and repaint the apartment. True/False/Uncertain: David has rational preferences (as we define them). Problem 1 courtesy of William Wheaton. Used with permission. 2. (20 points) In each of the following examples, a consumer purchases just two goods: I and y. Based on the information in each of the following parts, sketch a plausible set of indifference curves (that is, draw at least two curves on a set of labeled axes, and indicate the direction of higher utility). Also, write down a utility function u(z, y) consistent with your graph. Note that although all these preferences should be assumed to be complete and transitive (as required for utility representation), not all will be monotone. (a) (4 points) Jessica enjoys bagels r and coffee y, and consuming more of one makes consuming the other more enjoyable. (b) (4 points) Plamen loves mocha swirl ice cream z, but he hates mushrooms y. (c) (4 points) Jennifer likes Cheerios r, and neither likes nor dislikes Frosted Flakes y. (d) (4 points) Edward always buys three white tank tops r for every pair of jeans y. (e) (4 points) Nancy likes both peanut butter I and jelly y, and always gets the same additional satisfaction from an ounce of peanut butter as she does from two ounces of jelly. 3. (20 points) A consumer's preferences are representable by the following utility function: u(x, y) = x3 + y. (a) (10 points) Obtain the MRS of the consumer at an arbitrary point (5. 7), where = > 0 and y > 0. (b) (10 points) Suppose the price of the second good (y) is 1, and the price of the first good (r) is denoted by p > 0. If the consumer's income is m > 0, obtain the optimal consumption bundle of the consumer (in terms of m and p). [Caution: make sure you cover cases in which m is relatively low, as well as cases in which m is relatively high.] 4. (25 points) It is exactly 24 hours before Lauren's physics final. She has an economics final directly after the physics final and has no time to study in between. Lauren wants to be a physicist, so she places more weight on her physics test score. Her utility function is given by u(p. e) = 0.6 1n(p) + 0.4In(e) where p is the score on the physics final and e is the score on the economics final. Although she cares more about physics, she is better at economics; for each hour spent studying economics she will increase her score by 3 points, but her physics score will only increase by 2 points for every hour spent studying physics. Studying zero hours results in a score of zero on both subjects (although In(0) is not defined, assume her utility for a score of zero is negative infinity). (a) (5 points) What constraints does Lauren face in her test score maximization problem? (b) (5 points) How many hours should Lauren optimally spend studying physics? How many hours studying economics? (hours are divisible) (c) (5 points) What economics and physics test scores will she achieve (ie. what are c' and p*)? (d) (5 points) What utility level will she achieve? (e) (5 points) Suppose Lauren can get an economics tutor. If she goes to the tutor, she will increase her economics test score by 5 points for every hour spent studying instead of 3 points, but will lose 4 hours of study time by going to the tutor. She cannot study while at the tutor, and going to the tutor does not directly improve her test score. Should Lauren go to the tutor? Problem 4 courtesy of William Wheaton. Used with permission.Problem Set 3 1. (12 points) Assume the government has two policy options, a cash grant of $200 and providing food stamps worth $200. (a) (3 points) Draw the budget constraint faced by the consumer in each of these two situations. (b) (3 points) Now, draw a set of indifference curves that corresponds to an individual who would be indifferent between these two policies. (c) (3 points) Next, draw a set of indifference curves corresponding to an individual who would prefer to have a cash grant rather than food stamps. (d) (3 points) Is there a consumer who strictly prefers food stamps to cash grants (i.e., is better off with food stamps rather than a cash grant?) Why does the government use food stamps? Problem 1 by MIT OpenCourse Ware. 2. (22 points) Suppose there are exactly two consumers (Albie and Bubbie) who demand strawberries. Suppose that Albie's demand for strawberries is given by q. (p) = p" fu(la) and Bubble's demand is given by qu(p) = p full) where I. and It are Albie and Bubbie's incomes, and fa(.) and fo(.) are two unknown functions. (a) (5 points) Find Albie and Bubbie's (own-price) elasticities of demand, q. p and equip. Use the sign convention that eye = 24. (b) (5 points) Suppose that a > 0 > 8. Are strawberries a Giffen good for Albie? Are strawberries a Giffen good for Bubbie? (c) (12 points) Are strawberries an inferior good for Albie? Are strawberries an inferior good for Bub- bie? Assume that these demands arise from utility maximization, given linear budget constraints. Hint: This question should not require much/any algebra beyond (b). 3. (24 points) Joe considers beer and soda to be perfect substitutes at a rate of 1:2, that is, he always receives the same utility if he has one beer or two sodas to drink. He spends $12 a day on drinks, and beers cost $3 while sodas cost $1 each. However, one day the price of beers decreases to $2; there is no change in his budget. (a) (6 points) How does consumption change when the price of beers changes? What is Joe's new level of utility? (b) (6 points) Show with the aid of a graph what happens to the optimal allocation and the level of utility when the price of beer changes. (c) (6 points) How much must Joe's budget decrease to return him to the original utility level? (d) (6 points) Now assume the price of beers returns to the original price of $3. Describe the demand curve for soda holding the price of beer and income constant; a graphic representation is optional. Problem 3 by MIT OpenCourse Ware. 4. (42 points) Xiaoyu spends all her income on statistical software (S) and clothes (C). Her preferences can be represented by the utility function: U(S, C) = 41(S) + 61(C). (a) (6 points) Compute the marginal rate of substitution of software for clothes. Is the MRS increasing or decreasing in S? How do we interpret this? (b) (6 points) Find Xiaoyu's demand functions for software and clothes, Qs(ps,pc. I) and Qc(ps, pc. I). in terms of the price of software (ps), the price of clothes (pc), and Xiaoyu's income (/). (c) (6 points) Draw the Engel curve for software. (d) (6 points) Suppose that the price of software is ps = 2, the price of clothes is pc = 3, and Xiaoyu's income is / = 10. What bundle of software and clothes (S, C) maximizes Xiaoyu's utility? (e) (6 points) Suppose the price of software increases to ps = 4. What bundle of software and clothes does Xiaoyu demand now? (f) (6 points) Given the price increase, how much income does Xiaoyu need to remain as happy (have the same utility) as she was before the price change? What bundle of software and clothes would Xiaoyu consume if she had that additional income, given the new prices? (g) (6 points) Going back to the situation in part (e) (ps = 4 and / = 10), decompose the total change of software and clothes demanded into substitution and income effects. In a clearly-labeled diagram with software on the horizontal axis, show the income and substitution effects of the increase in the price of software. Problem 4 courtesy of William Wheaton. Used with permission.1. Consider a homeowner with Von-Neumann and Morgenstern utility function u, where u () = 1 - e" for wealth level r, measured in million US dollars. His entire wealth is his house. The value of a house is 1 (million US dollars), but the house can be destroyed by a flood, reducing its value to 0, with probability # 6 (0, 1). (a) What is the largest premium P is the homeowner is willing to pay for a full insurance? (He pays the premium P and gets back 1 in case of a flood, making his wealth 1 - P regardless of the flood.) The homeowner's utility for getting 1 - P always is u(1 - P) =1-e-(1-P) his utility in the outside option is TU(0) + (1 - #)u(1) = (1-7)(1-e-! ) The largest premium Phe is willing to pay is the P that makes him indifferent between buying and not buying insurance. 1 - e-(1-P) = (1 - *)(1 -e-' ) 1 - P = -In(1 - (1 -#)(1 -e-1)) P = 1+In(1 - (1 -#)(1-e-')) (b) Suppose there is a local insurance company who has insured n houses, all in his neigh- borhood, for premium P. Suppose also that with probability # there can be flood in the neighborhood destroying all houses (i.e., either all houses are destroyed or none of them is destroyed). Suppose finally that P is small enough that the homeowner has insured is house. Having insured his house, what is the largest Q that he is willing to pay to get the 1 share of the company? (The value of the company is the total premium it collects minus the payments to the insured homeowners in case of a flood.) The company's value is nP with probability (1-7) and nP -n with probability 7. His utility from buying insurance and not buying stock is u(1 - P) = 1 -e-(1-P) And his utility from buying stock and insurance is (1 - #)u(1 - P - Q + P) + mi(1 - P - Q+ P-1) = (1 - )(1 - exp(-1 + Q)) + #(1 - exp(Q)) We find the Q that makes him indifferent between buying and not buying: 1 - e-(1-P) = (1 - w)(1 - exp(-1 + Q)) + #(1 - exp(Q)) exp(-Q) exp(P) = (1 - #) exp(0) + # exp(1) Q = P - In(1 - * + me) We saw before that it must be true that exp(P)