Unformatted text preview: ponents sum to 1. There are constant
returns to scale. You can either just state this fact or do a lambda proof:
1/ 3 Q1 = ( K1) 1/ 3 1/ 3 ( L1) ( M 1) . Let K 2 = λK1 (and similarly for the other inputs)
1/ 3
1/ 3
1/ 3
1/ 3
1/ 3
1/ 3
1/ 3
1/ 3
1/ 3
⇒ Q2 = ( K 2 ) ( L2 ) ( M 2 ) = ( λK1) ( λL1) ( λM 1) = λ1( K1 ) ( L1) ( M 1) = λQ1
€ b) We will have an interior optimum because this is a CobbDouglas production function. We
could also observe that the optimum will have to be interior because the firm always needs
positive amounts of all inputs to produce any output. [To see this, try plugging in zero for any of
the inputs.] Thus, at a long run optimum, three conditions must be satisfied:
1/3 1/3 1/3
i) Q = K L M (the firm must produce with the production function)
ii) MPL/MPK = w/r, or K/L = 1/27 (This says we have tangency for inputs L and K)
1/3 2/3 1/3
1/3 1/3
2/3
1/3 1/3
2/3
(MPL = (1/3) K L M = (1/3) K M / L . Similarly, MPK = (1/3) L M / K .) So MRTSL,K
= MPL/MPK = K/L.)
iii) MPL/MPM = w/m, or M/L = 1 (We have tangency for inputs L and M)
Together (ii) and (...
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 Spring '12
 Whinston
 Economics, Economics of production, total cost function

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