400ps1-11-sol - ECON 400A Winter 2011 SOLUTIONS FOR PROBLEM...

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E C O N 4 0 0 A H a r t m a n Winter 2011 SOLUTIONS FOR PROBLEM SET I 1.a. We need a and b to be positive for a downward sloping demand curve and k to be positive for positive costs. 1.b. The monopolist profit is 2 p xk x t x  where p and x satisfy the demand function. Solve the demand function for p in terms of x to get ax p bb , and substitute this into the expression for profit to get 2 2 ax x kx tx . The first order condition for choosing x to maximize profit is 2 20 dax kx t dx b b    and the (sufficient) second order condition is 2 2 2 d k dx b   which is satisfied since b and k are positive. Solve the first order condition to get the profit-maximizing x in terms of t : * () x xt where * 2(1 ) ab t xt bk . An increase in t reduces the profit-maximizing output because * 0 ) dx t b dt bk  since b and k are both positive. 1.c. The (sufficient) second order condition for profit maximization requires that 1 k b     which is weaker than the conditions we imposed in part a above. 1.d. Multiply through the first order condition from part b by b and rearrange to get ) t b kx   . Substitute * ) t bk from part b for x to see that ) ) t t b k t bk     which holds identically for all values of t .
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400ps1-11-sol - ECON 400A Winter 2011 SOLUTIONS FOR PROBLEM...

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