Econ Essay

990 Words4 Pages
Examples and exercises on a profit-maximizing monopolist that sets a single price Procedure * Find the output(s) for which MC(y*) = MR(y*). * For each output you find, check to see whether the condition MC'(y*) MR'(y*) is satisfied. * For each output that satisfies the first two conditions, check to see if profit is nonnegative. * If there are any outputs satisfing these three conditions, the one that has the highest profit is the optimal output for the monopolist. (Most probably there is at most one output that satisfies the three conditions.) If there is no output that satisfies the three conditions the the optimal output for the monopolist is 0. Example A monopolist has the cost function TC(y) = 200y + 15y2 and faces the demand function given by p = 1200 10y. What output maximizes its profit? What is the profit-maximizing price? What is its maximal profit? * We have TR(y) = (1200 10y)y = 1200y 10y2, so MR(y) = 1200 20y. Also MC(y) = 200 + 30y. Thus any output at which MR is equal to MC satisfies 1200 20y = 200 + 30y, or 50y = 1000, or y = 20. * We have MR'(y) = 20 and MC'(y) = 30, so MC'(20) MR'(20). * The price associated with y = 20 is p = 1200 (10)(20) = 1000, so the firm's profit is (1000)(20) 200(20) 15(20)2 = 20000 4000 6000 = 10000. Since this profit is positive, the optimal output for the monopolist is the output we have found, namely y* = 20. The price is 1000 and the monopolist's profit is 10000. Example (A more complicated example to show the possibility of two outputs at which MR is equal to MC.) A monopolist's cost function is TC(y) = (y/2500)(y 100)2 + y, so that MC(y) = 3y2/2500 4y/25 + 5. It faces the inverse demand function P(y) = 4 4y/100. Find its output, the associated price, and its profit. * For MR = MC we need 3y2/2500 4y/25 + 5 = 4 8y/100, or 3y2/2500 8y/100 + 1 = 0, or
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