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368 PART • Market Structure and Competitive Strategy additional units would exceed the cost Because marginal costs must be the same at each plant, and because marginal revenue must equal marginal cost, we see that profit is maximized when marginal revenue equals marginal cost at each plant We can also derive this result algebraically Let Q1 and C1 be the output and cost of production for Plant 1, Q2 and C2 be the output and cost of production for Plant 2, and QT = Q1 + Q2 be total output Then profit is p = PQT - C1(Q1) - C2(Q2) The firm should increase output from each plant until the incremental profit from the last unit produced is zero Start by setting incremental profit from output at Plant to zero: ⌬(PQT) ⌬C1 ⌬p = = ⌬Q1 ⌬Q1 ⌬Q1 Here ⌬(PQT)/⌬Q1 is the revenue from producing and selling one more unit— i.e., marginal revenue, MR, for all of the firm’s output The next term, ⌬C1/⌬Q1, is marginal cost at Plant 1, MC1 We thus have MR - MC1 ϭ 0, or MR = MC1 Similarly, we can set incremental profit from output at Plant to zero, MR = MC2 Putting these relations together, we see that the firm should produce so that MR = MC1 = MC2 Note the similarity to the way we obtained a competitive industry’s supply curve in §8.5 by horizontally summing the marginal cost curves of the individual firms (10.3) Figure 10.6 illustrates this principle for a firm with two plants MC1 and MC2 are the marginal cost curves for the two plants (Note that Plant has higher marginal costs than Plant 2.) Also shown is a curve labeled MCT This is the firm’s total marginal cost and is obtained by horizontally summing MC1 and MC2 Now we can find the profit-maximizing output levels Q1, Q2, and QT First, find the intersection of MCT with MR; that point determines total output QT Next, draw a horizontal line from that point on the marginal revenue curve to the vertical axis; point MR* determines the firm’s marginal revenue The intersections of the marginal revenue line with MC1 and MC2 give the outputs Q1 and Q2 for the two plants, as in equation (10.3) Note that total output QT determines the firm’s marginal revenue (and hence its price P*) Q1 and Q2, however, determine marginal costs at each of the two plants Because MCT was found by horizontally summing MC1 and MC2, we know that Q1 + Q2 = QT Thus these output levels satisfy the condition that MR = MC1 = MC2 10.2 Monopoly Power Pure monopoly is rare Markets in which several firms compete with one another are much more common We say more about the forms that this competition can take in Chapters 12 and 13 But we should explain here why each firm in a
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