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ak_ps03_micro_s08

ak_ps03_micro_s08 - Answer Keys for Problem Set 3...

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Answer Keys for Problem Set 3 Microeconomics Spring 2008 1. A household consumes only apples ( X ) and bananas ( Y ). We denote a consumption bundle consisting of x bags of apples and y bags of bananas is denoted by ( x, y ) . The preferences of the household are given by the utility function U ( x, y ) = Ax α y 1 α where A > 0 and 0 < α < 1 . Suppose that prices for apples and bananas are P X and P Y and let I denote the income of the household. (a) Determine the marginal rate of substitution (MRS) for any consumption bundle ( x, y ) . MRS = MU x MU y = ∂U ∂x ∂U ∂y = Aαx α 1 y 1 α A (1 α ) x α y α = α 1 α y x (b) Describe the consumer’s utility maximaiztion problem using the information provided. max ( x,y ) Ax α y 1 α s.t. P X x + P Y y = I (c) Illustrate graphically where the optimal consumption bundle will lie with appropriately drawn indi ff erence curve and budget line. X Y I/P X I/P Y x * y * 1
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(d) Write out the Lagrangian function for the consumer’s maximization problem. L = Ax α y 1 α + λ ( I P X x P Y y ) (e) What are the fi rst order conditions ?(Hint: There are three.) ∂L ∂x = Aαx α 1 y 1 α P X λ = 0 ∂L ∂y = A (1 α ) x α y α P Y λ = 0 ∂L ∂λ = I P X x P Y y = 0 (f) Use the fi rst order conditions to solve for the optimal quantity of x and y. X ( P X , P Y , I ) = αI P X Y ( P X , P Y , I ) = (1 α ) I P Y (g) If α = 0 . 25 , I = 100 , and both prices are equal to 2 , fi nd the optimal consumption bundle of apples and bananas. x = 0 . 25 (100) 2 = 12 . 5 y = 0 . 75 (100) 2 = 37 . 5 (h) Now assume that the income of the household is $80 and the prices for apples and bananas stay the same. What’s the new demand (utility maximizing amount) for apples and bananas?
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