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Dominant Firm and Competitive Fringe The behavior of a dominant firm with a competitive fringe can be analyzed using calculus. This appendix illustrates such an analysis with the no-entry model with n fringe firms, using more general demand and cost functions than were (implicitly) assumed in Figure 4.6 in your textbook and concentrating on a long-run analysis in which average variable costs and average costs are equal. If the cost function of a fringe firm is Cf(qf), then its average cost is AC = Cf(qf)/qf and its marginal cost is MC = C'f(qf), where the prime indicates differentiation. The fringe firm's objective is to maximize its profits, f, through its choice of its output level, qf: (Equation 1) max qf f = pqf - Cf(qf), where pqf is its total revenues. This firm believes it is a price-taker that can sell as much as it wants at the going price and that it cannot affect the price through its own actions. The first-order condition for profit maximization for a fringe firm is: (Equation 2) p = C'f(qf). That is, the firm sets its output where price (the firm's marginal revenue) equals its marginal cost. The second-order condition is C"f(qf) > 0; that is, the marginal cost curve must be upward sloping at the equilibrium quantity for profits to be maximized. [The theory of the competitive firm requires that, in addition to meeting the first- and second-order conditions, a firm must make sure that its profits are positive (or else it should go out of business). Profits are positive if the market price is above the minimum average cost ( in Figure 4.6a).] The combined output of the fringe (Qf = nqf) and the dominant firm (Qd) determines the market price: p(Q) = p(Qf + Qd). Thus, all else the same, as the dominant firm increases its quantity, the price falls. As the price falls, each fringe firm chooses to produce less (because its marginal cost is increasing in qf and it sets price equal to marginal cost from Equation 2). We can show formally that the fringe supply falls as Qd rises. First, rewrite Equation 2 to reflect how price varies with Qd: (Equation 2') p(nqf + Qd) = C'f(qf). Then, totally differentiate Equation 2' to show that p'ndqf + p'dQd = C"fdqf or (rearranging) (Equation 3) dqf - p' ------ = ----------- < 0, dQd np' - C"f where the inequality follows because np' < 0 and -C"f < 0 (by the second-order condition). That is, the quantity supplied by a fringe firm falls as Qd rises. As Qd rises, all else the same, price must fall, and as price falls, the quantity supplied by a fringe firm falls. We can write Qf(Qd) to show that Qf is a function of Qd. From Equation 3, we know that dQf/dQd = ndqf/dQd = - np'/(np' - C"f) < 0, which says Qf falls as Qd rises. The dominant firm takes the relationship (2') into account when trying to maximize its profits through its choice of output level: (Equation 4) max p(Qd + Qf(Qd))Qd - Cd(Qd), Qd where Cd(Qd) is the dominant firm's cost function. The first-order condition for a profit maximization is (Equation 5) dQf p(Qd + Qf) + p'(Qd + Qf)Qd[1 + ------ ] = C'd(Qd). dQd According to Equation 5, profits are maximized if the dominant firm sets its output so that its marginal revenue conditional on the response of the competitive fringe, the left-hand side of the equation, equals its marginal cost, the right-hand side of the equation. From Equation 3, dQf/dQd = - np'/(np' - C"f), so the term in brackets in Equation 5 can be rewritten as- C"f/(np' - C"f). This ratio is positive but less than 1. If Qf 0 and dQf/dQd 0, the dominant firm is a monopoly. Then Equation 5 is the monopoly's profit maximization condition: Marginal revenue (corresponding to the market demand curve) equals marginal cost. The monopoly's p is a function of only the monopoly's output, and Qdp'(Qd) is multiplied by 1; whereas in the dominant firm model, price is a function of the dominant firm's and the competitive fringe's output, and Qdp'(Qd + Qf) is multiplied by a term that is less than 1. We can also express the effect of the fringe's supply on the dominant firm using elasticities. The fringe's supply affects the elasticity of demand that the dominant firm faces and hence helps determine the dominant firm's price. Using slightly different notation, the dominant firm's residual demand, Qd = Dd(p), can be written as the market demand, D(p), minus the supply, S(p), of the fringe: (Equation 6) Dd(p) = D(p) - S(p). The dominant firm's marginal revenue corresponding to this residual demand curve is obtained by differentiating Equation 6 with respect to p: (Equation 7) dDd = dD - dS . ----dp ------ -----dp dp Equation 7 can be expressed in terms of elasticities by multiplying both sides of the equation by p/Q, multiplying the left-side by Qd/Qd, and multiplying the last term on the right side byQf/Qf: (Equation 7') Qd ( ---- ) Q d = Qf - ( ---- ) f, Q where d = [( Dd/ p)(p/Qd)] is the residual demand elasticity, is the elasticity of the market demand curve, f is the fringe's supply elasticity, Qd/Q is the dominant firm's share of output, and Qf/Q is the fringe's share. This expression may be rewritten as (Equation 7'') Q = ----d Qd Qf - ( ----- ) f, Qd where Q/Qd is the ratio of total industry output to that of the dominant firm and Qf/Qd is the ratio of the fringe's output to that of the dominant firm. Thus, all else the same, the absolute value of the elasticity of the residual demand facing the dominant firm is higher (and hence the lower the price it charges), the higher is the supply elasticity of the fringe, the higher is the fringe's relative share of the market Qf/Qd, and the higher is the absolute value of the industry elasticity of demand. If the fringe does not exist (n = 0), the dominant firm's residual demand elasticity equals the industry demand elasticity, and it charges the monopoly price.