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W14 MIC Problem Set 5 Solutions

# There are constant returns to scale you can either

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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 Cobb-Douglas 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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