A monopoly faces the following demand function: Q = 100 - p + sqrt(A ), where A equals the dollar amount spent on advertising. If the cost function is A + 10 + 2Q, what are the profit-maximizing levels of price, output, and advertising? Compare this outcome to the case where the firm does not advertise at all
What will be an ideal response?
(1 ) ? = (100 - Q + sqrt(A ))Q - 10 - 2Q - A
(2 ) ??/?Q = 100 - 2Q + sqrt(A ) - 2 = 0
(3 ) ??/?A = Q/2sqrt(A ) - 1 = 0
From (3 ), sqrt(A ) = Q/2. Substituting into (2 ) and rearranging yields Q = 65.33. So, A = 1067.32 and p = 67.33. Total revenue is (65.33 ? 67.33 ) = 4398.67. Total costs = (10 + 2(65.33 ) + 1067.32 ) = 1207.98. Profit equals 3190.69.
Without advertising, set A = 0 in equation (2). This yields Q = 49 and p = 51. Total revenue equals 2499; total cost equals 108. Profit equals 2391. The firm is better off advertising.
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