W14 MIC Problem Set 5 Solutions

The long run cost function is c wl rk 14q 8q

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Unformatted text preview: isoquant for Q units of output. The long run cost function is C = wL + rk = (1)(4Q) + (8)(Q) = 12Q. 4) A firm produces output (Q) using labor (L), whose factor price is w, and capital (K), whose factor price is r. The production function is Q = KL + K. Throughout this problem assume there is an interior optimum (with L > 0 and K > 0). a) Find the factor demand equations, L[w,r,Q] and K[w,r,Q]. Hint: You are effectively finding (L*,K*) for any levels of [w,r,Q] that generate an interior solution. b) Show that the firm’s long run total cost function is C[ w, r, Q ] = 2 Qwr − w Answer: 4) a) MRTSL, K = MPL K w = = ⇒ w( L + 1) = rK [Condition #1: Tangency] MPK L + 1 r MPL MPK K w < ⇔ < w r L +1 r (One could imagine using only cheap K inputs when the wage gets very high.) Q Q = KL + K = K ( L + 1) ⇒ K = [Condition #2: Production] L +1 #Q& Qr Qr 2 Combine and solve for L: w( L + 1) = r % ⇒ L +1 = ( ⇒ ( L + 1) = $ L +1' w w NOTE: Bang for the buck logic would generate L = 0 when ⇒ L[ w, r, Q ] = ⇒K= Q = L +1 Qr −1 w Q Qw = = K [ w, r, Q ] r Qr −1+1 w b) Plug the two factor demands into the general cost function and simplify: " Qr % " Qw % C = wL + rK = w $ $ w − 1' + r $ r ' = Qwr − w + Qwr = 2 Qwr − w = C[ w, r, Q ] '$ ' # &# & 5) Now work problem 4 “backwards,” starting with the cost functio...
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This homework help was uploaded on 03/03/2014 for the course ECON 310 taught by Professor Whinston during the Spring '12 term at Northwestern.

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