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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 shortrun cost function is: SC ( v,w,k,q ) = vk + wqk k q : q > , k > q ; vk : q = 0 (4) b) By definition of longrun 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 shortrun cost function SR yields the longrun 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 longrun cost: C ( v,w,q ) = min k SC ( v,w,k,q ) SC ( v,w,k,q ) . (8) Equality holds whenever k in the shortrun 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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 '08
 StephenE.Morris

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