Solutions_to_PS8 - ECON W3211.002 Intermediate...

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ECON W3211.002 Intermediate Microeconomics Problem Set 8 Due Monday, November 22, 2010 1. Answer the following questions: (a) The monopolist faces a demand curve given by Q = 100 - 2 p . The firm’s cost function is C ( Q ) = 2 Q . What is the optimal level of output and price? The inverse demand curve is p ( Q ) = 50 - Q/ 2 . The firm’s optimization problem is max Q (50 - Q/ 2) Q - 2 Q the first-order-condition (FOC) is 50 - Q - 2 = 0 , and hence Q * = 48 and p * = 26 . (Observe that MR ( Q ) = 50 - Q and MC = 2 .) (b) How does your answer change if Q = 10 p - 3 The inverse demand curve is p ( Q ) = ( Q 10 ) - 1 3 . The firm’s optimization problem is max Q ( Q 10 ) - 1 3 Q - 2 Q FOC is ( 1 10 ) - 1 3 2 3 Q - 1 3 - 2 = 0 Therefore Q * = 10 × 3 - 3 and p * = 3 . There is a shortcut to solve this problem. Notice that the demand curve has a constant elasticity of - 3 . Using the formula p [1 + 1 ] = MC , we substitute to get p [1 - 1 / 3] = 2 . Therefore p * = 3 and Q * = 10 × 3 - 3 as before. (c) If Q = 100 /p and C ( Q ) = Q 2 , what is the optimal level of output of the monopolist? : The demand curve has a constant elasticity of - 1 . Thus marginal revenue is zero for all levels of output (Observe that revenue is Q × p = 100 .) Hence it can never be equal to marginal cost, where MC = 2 Q .
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