Altoona High

• ECN GAME THEOR

• 1. Free-entry Consider Industry X, where firms compete `a la Cournot and produce a homogeneous good. In the industry, n identical firms operate and their cost function is T C(qi) = 20qi +F, where qi denotes the quantity produced by each firm. Observe that, in addition to marginal costs, firms bear fixed entry costs F amounting to 25. Market demand is expressed by the function P(Q) = 200 − 4Q, where Q denotes total output produced by firms in the industry. • Analytically determine the quantity produced by each firm, total output, price and individual profits at the post-entry equilibrium, as functions of n. The profit function of Firm i is πi = (200−4Q−20)qi where Q = Pn j=1 qj . The F.O.C. is 200 − 4 Xn j=1 qj − 20 − 4qi = 0 (1) By symmetry among firms, q ∗ = 45 n + 1 Q ∗ = 45n n + 1 p ∗ = 20n + 200 n + 1 π ∗ = 4(45)2 (n + 1)2 (2) • Determine the number of firms that will be able to compete in the industry in the long-run equilibrium. Given F = 25, the number of firms that compete in the industry in the long-run equilibrium can be obtained through the following: 4(45)2 (n + 1)2 = 25 ⇒ n ∗ = 17 (3) • What will happen to the number of firms n if fixed costs F considerably increase, shifting from 25 to 100? Explain. If fixed costs shift to 100, the number of firms that compete in the industry in the long run equilibrium will be obtained through the following: 4(45)2 (n + 1)2 = 100 ⇒ n ∗ = 8 (4) • Now assume that fixed entry costs are again at their st