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Solutions_to_PS_6 - ECON W3211.002 Intermediate...

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ECON W3211.002 Intermediate Microeconomics Problem Set 6 Due Wednesday, November 10, 2010 1. Perloff, Chapter 7: 28, 29, 31, 32 28. The production function is decreasing returns to scale; therefore the long-run cost func- tion will increase at an increasing rate. This means the long-run average cost will rise and marginal cost will be above average cost. Mathematical Derivation: The cost-minimization problem is min K,L wL + rK such that Q = AL α K β Notice that the firm’s decision variable is K and L , where wL + rK is the objective function and Q = AL α K β is a constraint. Since Q = AL α K β , K = Q A L - α 1 β Substituting this into the objective function, we get min L wL + r Q A L - α 1 β We can now differentiate with respect to L and set resulting derivative equal to zero. That is, w + r Q A 1 β - α β L - α β - 1 = 0 The above equation can be solved to get L as a function of w , r , and Q . L ( w, r, Q ) = α β β α + β w - β α + β r β α + β Q A 1 α + β Substituting this into K = ( Q A L - α ) 1 β , we get K ( w, r, Q ) = α β - α α + β w α α + β r - α α +
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