Altoona High

• ECN GAME THEOR

30283 - BIEF - Markets, Organizations and Incentives Chiara Fumagalli Problem set # 3: SOLUTIONS 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 starting level, i.e. F = 25, but grow as the market size grows. What type of costs is this? Explain what happens to the structure of the industry in case of demand increases and provide a reason for this (no calculations are required). In such a framework, fixed costs are endogenous: as market size grows, equilibrium sunk costs borne by firms increase. As a consequence, the decreasing trend in market concentration weakens. 2. Free-entry (2) Consider a new industry where entry by firms is sequential and competition takes place only after all the potential entrants decided whether to enter the market or not. Firms sell homogeneous goods and compete in prices. In such a situation, how will the equilibrium number of firms in the industry vary as sunk costs of entry K > 0 increase? In such a framework, only one firm can opera