Homework 2.pdf - Intermediate Microeconomics Homework Set...

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Intermediate MicroeconomicsHomework Set 2: Preferences and UtilityDue: Wednesday, February 8th at 2 pm via NYU Classes websiteExercise1Consider the following utility functions on the choice setX={(x1, x2) :x1, x20}:(1)u(x1, x2) =x1+ 2x2,(2)u(x1, x2) = 2x0.251x0.752,(3)u(x1, x2) =-12x21-x2,(4)u(x1, x2) = min{x1,2x2},(5)u(x1, x2) = max{x1,2x2}.For each of (1) to (5), sketch two indierence curves of the utility function. (Please usedierent graphs for each utility.)Exercise2Consider the following utility functions on the choice setX={(x1, x2) :x1, x2>0}:(1)u1(x1, x2) =x21x22,(2)u2(x1, x2) =x1+x2,(3)u3(x1, x2) = logx1+ logx2,(4)u4(x1, x2) =x31+ 3x21x2+ 3x1x22+x33,(5)u5(x1, x2) =x1x2.How many distinct preference relations do these utility functions represent?Whichutilities represent the same preferences?Exercise3Consider a two-period economy with a single commodity (say leisure):x1is the con-sumption of leisure in period 1, andx2is 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.Exercise41
Martina’s preferences are represented byu(x1, x2) =x1/21+x1/32, wherex1>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 byv(x1, x2) =log(x1/21+x1/32). How do they relate to the results for the first utility function?Exercise5Burt’s utility function isu(x1, x2) = (x1+ 2)(x2+ 6), wherex1is the number of cookiesandx2is the number of glasses of milk that he consumes.(1) What is the slope of Burt’s indierence curve at the point where he is consuming(4,6)? Draw a line with this slope through (4,6).(2) The indierence curve through (4,6) passes through the points (·,0), (7,·) and(2,·).Give the missing coordinates.

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Term
Spring
Professor
Azc
Tags
Utility, X1, Would Burt

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