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Unformatted text preview: 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 firms 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 firms optimization problem is max Q (50 Q/ 2) Q 2 Q the firstordercondition (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 firms 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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This note was uploaded on 10/03/2011 for the course ECON W3211 taught by Professor Elmes during the Fall '09 term at Columbia.
 Fall '09
 Elmes
 Microeconomics

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