10fB_ps04solutions

10fB_ps04solutions - Microeconomics ECON 100A Problem Set 4...

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Unformatted text preview: Microeconomics ECON 100A Problem Set 4 Solutions [THIS MATERIAL IS NOT ON THE MIDTERM] Due October 20, 2010 (Solutions Posted October 22, 2010) 1. Now, let’s see if you can do the general perfect complements problem. Matt gets utility from X and Z. Matt always consumes α X with β Z. a. What type of utility function represents Matt’s preferences? Write down an expression for Matt’s utility. U=min{( 1 / α) X, ( 1 / β) Z} b. What is Matt’s utility maximizing combination of X*(P x , P z , Y) and Z*(P x , P z , Y)? Use the facts that (1) that there will be no excess X or Z (so 1 / α X*= 1 / β Z*) and (2) the utility maximizing bundle must be on the budget constraint (PxX*+PzZ* =Y). Substitute the expression for Z*= β / α X* from (1) into the budget constraint in (2) to get: Px(X*)+ Pz( β / α X*)=Y X*(Px+ β / α Pz) = Y and then solve for X*= z x p p Y α β + Finally, substitute this expression for X* into Z*= β / α X* to solve for Z*: Z*= ) ( z x p p Y α β α β + c. What is the Matt’s indirect utility function, i.e., V(P x , P z , Y)=U(X*(P x , P z , Y), Z*(P x , P z , Y))? U*(px, pz, Y)=min{ 1 / α X*, 1 / β Z* }=min{ ) ( , ) ( z x z x p p Y p p Y α β α α β α + + } d. Show that Roy’s identity holds for good x. Show that Roy’s identity holds for good x....
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This note was uploaded on 11/30/2011 for the course ECON 311 taught by Professor Zambrano during the Fall '08 term at Cal Poly.

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10fB_ps04solutions - Microeconomics ECON 100A Problem Set 4...

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