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HWIII.08

# HWIII.08 - HOMEWORK ECONOMICS 200A III DEMAND THEORY AND...

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HOMEWORK ECONOMICS 200A III. DEMAND THEORY AND DUALITY R. R. RUSSELL FALL 2008 1. Consider the following three utility functions: U ( x 1 , x 2 ) = ( x 1 ) 1 / 2 + x 2 , U ( x 1 , x 2 ) = x 1 + ln x 2 U ( x 1 , x 2 ) = ( x 1 - γ ) α x 2 , α > 0 . (a) Restricting prices to those that yield interior solutions, find the first-order conditions for maximizing these utility functions subject to a budget constraint, p · x = y , and check to see that second-order conditions are satisfied. (b) Without restricting prices to those that yield interior solutions, find the first-order conditions for maximizing these utility functions subject to the budget constraint. (Be careful here to consider different signs of γ and to restrict choices to those where the utility function is well defined.) (c) Derive the demand functions for each utility function and verify the requisite ho- mogeneity condition. (d). Derive the indirect utility function dual to each utility function and verify Roy’s Identity.

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