Consider a utility function of two goods x and y: U (x,y) = A (ax +by') where A >0, a>0, b>0, r € (-∞,0)U(0, 1) are constants. This utility function is called a "constant elasticity of substitution (CES)" function and is frequently used in Macroeconom- ics. (a) Prove that when a+b = 1, this utility function converges to a Cobb-Douglas utility function as r→0. Hint: apply l'Hopital's rule to lim In ) = limm(ar +by') (b) Calculate the slope of the indifference curves of U. Based on your answer, are good x and y perfect/imperfect substitutes/complements when r → 1? When r → -0?

Consider a utility function of two goods x and y: U (x,y) = A (ax +by') where A >0, a>0, b>0, r € (-∞,0)U(0, 1) are constants. This utility function is called a "constant elasticity of substitution (CES)" function and is frequently used in Macroeconom- ics. (a) Prove that when a+b = 1, this utility function converges to a Cobb-Douglas utility function as r→0. Hint: apply l'Hopital's rule to lim In ) = limm(ar +by') (b) Calculate the slope of the indifference curves of U. Based on your answer, are good x and y perfect/imperfect substitutes/complements when r → 1? When r → -0?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.13P

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