NYU UA 10 – Homework Set 2 Preferences and Utility

Intermediate MicroeconomicsHomework Set 2: Preferences and UtilityDue: Wednesday, February 8th at 2 pm via NYU Classes website Exercise 1Consider the following utility functions on the choice set X = {(x1 , x2 ) : x1 , x2 ? 0}:(1) u(x1 , x2 ) = x1 + 2×2 ,0.75(2) u(x1 , x2 ) = 2×0.251 x2 , (3) u(x1 , x2 ) = ? 21 x21 ? x2 ,(4) u(x1 , x2 ) = min{x1 , 2×2 },(5) u(x1 , x2 ) = max{x1 , 2×2 }.For each of (1) to (5), sketch two indifference curves of the utility function. (Please usedifferent graphs for each utility.)Exercise 2Consider the following utility functions on the choice set X = {(x1 , x2 ) : x1 , x2 > 0}:(1) u1 (x1 , x2 ) = x21 x22 ,(2) u2 (x1 , x2 ) = x1 + x2 ,(3) u3 (x1 , x2 ) = log x1 + log x2 ,(4) u4 (x1 , x2 ) = x31 + 3×21 x2 + 3×1 x22 + x33 ,(5) u5 (x1 , x2 ) = x1 x2 .How many distinct preference relations do these utility functions represent? Whichutilities represent the same preferences?Exercise 3Consider a two-period economy with a single commodity (say leisure): x1 is the consumption of leisure in period 1, and x2 is the consumption of leisure in period 2. WhenPeter evaluates consumption streams, he cares only about the best time in his life. Onthe other hand, Christine only cares about the worst time in her life. Provide utilityrepresentation for their preferences.Exercise 41 1/2 1/3 Martina’s preferences are represented by u(x1 , x2 ) = x1 + x2 , where x1 > 0, x2 > 0.(1) What is the marginal utility of good 1 at (x1 , x2 )?(2) What is the marginal utility of good 2 at (x1 , x2 )?(3) What is the marginal rate of substitution of good 2 for good 1 at (x1 , x2 )?(4) Perform the same computations for the preferences represented by v(x1 , x2 ) =1/21/3log(x1 + x2 ). How do they relate to the results for the first utility function?Exercise 5Burt’s utility function is u(x1 , x2 ) = (x1 + 2)(x2 + 6), where x1 is the number of cookiesand x2 is the number of glasses of milk that he consumes.(1) What is the slope of Burt’s indifference curve at the point where he is consuming(4, 6)? Draw a line with this slope through (4, 6).(2) The indifference curve through (4, 6) passes through the points (·, 0), (7, ·) and(2, ·). Give the missing coordinates. What is the equation for Burt’s indifferencecurve through (4, 6)? Sketch it.(3) Burt currently has the bundle (4, 6). Ernie offers to give Burt 9 glasses of milkif Burt will give Ernie 3 cookies. If Burt makes this trade, which bundle would hehave? Burt refuses to trade. Was this a wise decision? Mark the bundle (1, 15) onyour graph.(4) Ernie says to Burt, “Burt, your marginal rate of substitution (MRS) is -2. Thatmeans that an extra cookie is worth only twice as much to you as an extra glass ofmilk. I offered to give you 3 glasses of milk for every cookie you give me. If I offer togive you more than your MRS, then you should want to trade with me.” CompleteBurt’s reply: “Ernie, you are right that my MRS is -2. But. . . ” Would Burt bewilling to give up 1 cookie for 3 glasses of milk? Would Burt object to giving up 2cookies for 6 glasses of milk?Exercise 6Imagine that you are about to purchase a stereo for $125 and a calculator for $15. Thesalesman tells you that one of these two goods is on sale for $5 less at the other branchof the store, located 20 minutes away. Suppose that• You are indifferent between a $5 discount on the calculator and the same discounton the stereo.• You would make the trip to the other store to get the discount on the calculator.• You would not travel to the other store to get the discount on the stereo.Does there exist a complete, transitive preference relation over consumption bundles of(number of stereos, number of calculators, money, travel time) that rationalizes thesechoices? Why or why not? 2