SIPA U6400 – Consider consumer Alex with preferences represented by the utility function

Microeconomic Analysis for International and Public Affairs ISIPA U6400. Spring 2016Problem set 1Due: Tuesday 02/09 before the beginning of the lecture to the DRA in charge of your section.1. Consider consumer Alex with preferences represented by the utility function U (x 1 , x 2 ) = (x 1 )2 (x 2 )2and consumer Alessandro with preferences represented by the utility function V (x 1 , x 2 ) = (x 1 ) (x 2 ).(a) Calculate the marginal rate of substitution (of good 1 for good 2) for Alex and for Alessandro.(b) Are the preferences of Alex different from the preferences of Alessandro? That is, can youfind two baskets A and B that Alex and Alessandro rank differently?(c) If Alex and Alessandro face the same prices p 1 and p 2 and have the same income, I willthey choose the same basket? Explain(d) Find the demand of good 1, for Alex and Alessandro.(e) Now calculate the marginal utility of good 1 for Alex and Alessandro. Do they have thesame marginal utility of good 1?2. Consider the following 3 consumers: consumer 1 has preferences represented by the utilityfunction U (x 1 , x 2 ) = (x 1 )(x 2 )6 , consumer 2 has preferences represented by the utility functionU (x 1 , x 2 ) = (x 1 )(x 2 ) and consumer 3 has preferences represented by the utility function U (x 1 , x 2 ) =(x 1 )6 (x 2 )(a) For each consumer calculate the marginal rate of substitution of good 1 for good 2 (M RS (1,2) ).(b) What is the marginal rate of substitution of the 3 consumers calculated at the basket (3, 3)?Assume that they all consume the same basket (3, 3): which consumer likes more good 1?Explain.Now assume that these 3 consumers face the same prices, p 1 = 10 and p 2 = 5. Also assume that consumer 1 has income equal to I 1 = 70, consumer 2 has income I 2 = 300 andconsumer 3 has income I 3 = 700.(c) Find the optimal basket for the 3 consumers.(d) What is the marginal rate of substitution for the 3 consumers at their optimal basket?3. A consumer, Asli, has an income I = 10 and faces prices p 1 = 1 and p 2 = 2. Her preferences arerepresented by the following utility function:U (x 1 , x 2 ) = (x 1 )(x 2 ) + 10x 1Find the optimal choice for Asli.