Problem 1 -
Consider an oligopoly with n firms that produce homogeneous goods and compete a la Cournot. Inverse demand is given by P(Q) with P'(Q) < 0, and each firm i has a cost function Ci(qi) with C'i(qi) > 0 and C"i(qi) ≥ 0. Assume that the profit function is twice differentiable.
(a) Compute the first- and second order condition of firm i. Under which (sufficient) conditions is the profit function of firm i, πi, strictly concave? Provide an example for such a profit function.
(b) Compute the slope of the best-response function of firm i, R'i(Q-i) ≡ ∂qi/∂Q-i. In which interval is this slope?
Hint: A sufficient condition for uniqueness of a Cournot equilibrium is
∂2πi/∂qi2 + (n-1)|∂2πi/∂qi∂Q-i|< 0.
Assume this holds and make use of Tirole (1999).
(c) Suppose that P(Q) = a - b i-1∑nqi and Ci(qi) = cqi, for all i ∈ 1, ..., n. Does there mist a unique equilibrium for some n?
Problem 2 -
Consider the following homogeneous Cournot model: The product market is characterized by a set of / active firms, inverse demand is given by P(X) = a - bX, with a, b > 0.
The output of an individual firm i is denoted by xi and X = ∑ixi is the aggregate output. Marginal costs are firm specific and denoted by ci > 0; average marginal costs are denoted by c- = i∑Ici/i.
Assume that, for all i ∈ {1, 2, . . . , I},
(a+Ic-)/(I+1) -ci ≥ 0.
(a) Show that the following statements hold: Equilibrium outputs are
xi = 1/b ((a+Ic-)/(I+1) -ci).
The equilibrium price is
p = (a+Ic-)/(I+1).
Profits are
πi = π (ci, c-) = 1/b ((a+Ic-)/(I+1) -ci)2.
(b) Suppose the competition authority is considering a merger between two firms in a market with three firms. The authority's assessment criterion is consumer surplus. Based on the toolbox, analyze whether the merger should be allowed. Distinguish the following cases:
(i) The firms initially have the same marginal costs, and the merger has no effect on marginal costs.
(ii) The firms initially have different marginal costs, and the merged entity works with the marginal costs of the more efficient party to the merger.
(iii) The firms initially have the same marginal costs, and the merger reduces the marginal costs by some amount.
(c) Discuss how your findings fit with the results of the general analysis in part II ("Price Effects of Horizontal Mergers") of Farrell and Shapiro's 1990 paper.
Problem 3 -
Use your results front the Cournot model in problem 2. Now set b = 1 and a = 1 and suppose the firms can invest into cost reduction before the output game. Suppose the investment cost function is
K(yi) = (yi)2
Assume that initially all firms have the same marginal costs c. For parts (a)-(c) suppose that only one firm can invest in cost reduction, but that there are perfect spillovers. That is, after investment, all firms have marginal cost c - yi, where i is the firm that can invest.
(a) Calculate the equilibrium investment level y*.
(b) Interpret I as a competition parameter. Show that competition reduces the investment level.
(c) Interpret the result of (b) in terms of the four transmission channels.
(d) Suppose that still only one firm can invest into cost reduction but that there are no spillovers. What "animal strategy" does that firm follow?
(e) Suppose now that all firms can invest and that there are no spillovers, i.e., after investment marginal costs become (c - yi) for each firm i. Calculate the symmetric equilibrium investment levels. How do they compare to the investment level in part (a). Discuss your answer. As in (c), interpret I as competition parameter and discuss your results in terms of the four transmission channels.
Hint: Equilibrium outputs are given by
xi = 1/b ((a-c + Iyi -∑i≠j yj)/I+1), with a, b = 1.
Problem 4 -
Consider a Bertrand duopoly problem where demand is given by D(p) = 3 - p, if p is the lower of the two prices charged by the firms. Both firms have constant marginal costs c1 = c2 = 1.
Assume firm 1 can spend a fixed cost F to raise firm 2's marginal cost to c'2 = 7/5 before firm 2 chooses its price (observable investment).
(a) Draw the extensive form (game tree) of the game.
(b) Derive the (subgame-perfect) Nash equilibrium of the game. (You can make the appropriate assumption on tie-braking when p1 = p2.) What is the level of F above which it is not profitable for firm 1 to spend this fixed cost? Discuss your answer.
(c) Suppose now that firms compete in quantities (Cournot) instead of prices. Derive the (subgame-perfect) Nash equilibrium of the game. What is the level of above which it is not profitable for firm 1 to spend this fixed cost? Is it larger or smaller than the maximal fixed cost in part (b)? Discuss your answer.
(d) Returning to Bertrand competition. Assume now, in contrast to (a) and (b), that firm 2 does not observe whether firm 1 spends F, so that firm 2 has to decide on its price level before learning whether its marginal costs will be c2 = 1 Or c'2 = 7/5. What is the (pure-strategy) equilibrium set for F > 0? And for F = 0? Make sure to argue that you have found all pure-strategy equilibria of the game.
Problem 5 -
Consider a market with horizontal product differentiation. A consumer of type θ has the willingness to pay U(q, θ) = a- (x - θ)2
for a product of type x. The parameter θ is uniformly distributed among consumers on the interval [0, 1]. Suppose that a is sufficiently large so that in equilibrium each consumer buys one unit of the offered products. The variable per unit costs are zero for all x ∈ [0, 1].
(a) There are two firms i = 1, 2 in the market, offering goods x1 = 0 and x2 = 1, respectively. A third firm can enter the market and offer good x3 = ½ if it spends a fixed cost F. Determine the Nash equilibrium in prices.
(b) What condition on F must be satisfied such that firm 3 will enter the market?
(c) Suppose F is such that firm 3 will enter the market. Is the entry socially efficient?