ECON6001/6701 Microeconomic Analysis 1 Problem Set 3

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Problem Set 3

ECON6001/6701 Microeconomic Analysis 1

Exercise 1. Consider the following utility functions:

U (x1 ; x2)  =  min{x1 ; x2g                                                                       (2)

U (x1 ; x2)  =   x1(a) x1 ¡ a)          where 0 < a < 1.                                          (3)

Derive the   a) Marshallian Demand, b) Indirect Utility function, and where appropriate, verify Roy's identity.

Exercise. The following two functions were estimated by observing a consumers purchases for her consumption of three goods at various income levels y (sufficiently high) and prices p = (p1 ; p2 ; p3).

x1   =  100 ¡ 5  + p  + 6

x2   =  100 + p  + V  + 6 :

The Greek letters are (as yet undetermined) constants.

1) If this is a utility maximizing locally non-satiated consumer, what must be the demand for good 3?

2) Are these functions homogeneous of degree zero in income and prices?

3) What further restrictions do we need on p ; V and 6 to conclude that the above functions are generated by a utility maximizing consumer?

Hint: Look up Theorem 1 on (See Lec 3 Slides!) You have already verified two of the required three in the previous parts. Also remember, even though there are 3 goods here, the demand for good 3 is not independently given, so, for all purposes it is a 2 good economy.