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Econ501aH1ansS07

# Econ501aH1ansS07 - The Ohio State University Department of...

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The Ohio State University Department of Economics Econ 501.02—Prof. James Peck Homework #1 Answers 1. Suppose we have the utility function, U = x 2 y 3 . (a) Find the function for the marginal rate of substitution. (b) Show that this utility function satis fi es the fi rst 3 axioms of consumer preference. Answer: (a) The MRS is given by ∂u ∂x ∂u ∂y = 2 xy 3 3 x 2 y 2 = 2 y 3 x . (b) Suppose preferences are based on this utility function, and denote bun- dle A by ( x A , y A ) , bundle B by ( x B , y B ) , and bundle C by ( x C , y C ) . Then preferences are complete, because either ( x A ) 2 ( y A ) 3 > ( x B ) 2 ( y B ) 3 (in which case A P B ), or ( x A ) 2 ( y A ) 3 < ( x B ) 2 ( y B ) 3 (in which case B P A ), or ( x A ) 2 ( y A ) 3 = ( x B ) 2 ( y B ) 3 (in which case A I B ). Preferences are re fl exive, because ( x A ) 2 ( y A ) 3 = ( x A ) 2 ( y A ) 3 . Preferences are transitive, because if ( x A ) 2 ( y A ) 3 > ( x B ) 2 ( y B ) 3 and ( x B ) 2 ( y B ) 3 > ( x C ) 2 ( y C ) 3 hold, then ( x A ) 2 ( y A ) 3 > ( x C ) 2 ( y C ) 3 must hold. In other words, the " > " relation on real numbers is complete, re fl exive, and tran- sitive. 2. Suppose we have the utility function, U = 2 xy + x + y. (a) Find the function for the marginal rate of substitution. (b) If prices are p x = \$2 and p y = \$4 , and if income is M = \$17 , fi nd the utility maximizing consumption bundle.

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Econ501aH1ansS07 - The Ohio State University Department of...

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