Final solution Q1 to Q4

# Final solution Q1 to Q4 - ECO 310 Fall 2008 Microeconomic...

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Unformatted text preview: ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach Final Exam 01/14/09 - Answer Key 0.1 Question 1 a) The Marshallian demand functions are: x ( P,I ) = ln(1 /P ) : P ≤ 1 ,- e- I/P ≤ I + P (ln(1 /P )- 1); I/P : P ≤ 1 ,- e- I/P ≥ I + P (ln(1 /P )- 1); : P ≥ 1 , (1) and: y ( P,I ) = I + P ln(1 /P ) : P ≤ 1 ,- e- I/P ≤ I + P (ln(1 /P )- 1); : P ≤ 1 ,- e- I/P ≥ I + P (ln(1 /P )- 1); I : P ≥ 1 , (2) b) The indirect utility function V ( P,I ) is found by substituting the above Marshallian demands in the utility function. Hence: V ( P,I ) = I + P (ln(1 /P )- 1) : P ≤ 1 ,- e- I/P ≤ I + P (ln(1 /P )- 1);- e- I/P : P ≤ 1 ,- e- I/P ≥ I + P (ln(1 /P )- 1); I : P ≥ 1 , (3) c) Straightforward. 11 pts were awarded for writing down the interior solution in part a) and answering correctly the remaining parts; similarly, 17 pts, for the interior solution and either corner solution; 25 pts, for the interior solution and both corner solutions. E.g.: if one answered ” x = ln(1 /P ) and y = I + P ln(1 /P ) ” in part a); ” V = I + P (ln(1 /P )- 1) ” in part b), and verified the Slutsky equation solely for x = ln(1 /P ) , then he received 11 pts for the whole Question 1. The breakdown for part a) was 7 pts for the interior solution, and 4 pts for each corner solution. Question 2 a) To produce q > 0 units of output with k > q given units of capital requires l units of labor, where l satisfies q = kl/ ( k + l ). That is, l = qk/ ( k- q ). Else, for any given q , k , such that k ≤ q , there is no nonnegative value for l that satisfies q = kl/ ( k + l ). Otherwise, if q = 0, then l = 0 suffices. So the short-run cost function is: SC ( v,w,k,q ) = vk + wqk k- q : q > , k > q ; vk : q = 0 (4) b) By definition of long-run cost: C ( v,w,q ) = min k SC ( v,w,k,q ) ≤ SC ( v,w,k,q ) . (5) The optimal k that minimizes the RHS is: k ( q ) = q (1 + p v/w ) . (6) Substituting the above conditional input demand k in the short-run cost function SR yields the long-run cost function: C ( v,w,q ) = SR ( v,w,k ( q ) ,q ) = q ( √ v + √ w ) 2 . (7) 1 c) For q > 0 and all k > q , by definition of long-run cost: C ( v,w,q ) = min k SC ( v,w,k,q ) ≤ SC ( v,w,k,q ) . (8) Equality holds whenever k in the short-run cost function on the RHS takes the k value that minimizes that function. From part b), that value for k is k = q (1 + p v/w )....
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Final solution Q1 to Q4 - ECO 310 Fall 2008 Microeconomic...

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