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ECOS3005

Problem set 2

1. Suppose that two rms compete by choosing prices. Both rms produce identical products and have identical constant marginal costs, and no fixed costs.  Assume we have the same rationing rule as the Bertrand model. That is, the lower priced firm captures the entire market; if both firms set the same price, they share the market equally. The firms play sequentially. First, firm 1 sets its price. Firm 1 is then committed to this price. Firm 2 then observes firm 1’s price and chooses a price of its own. That is, first firm 1 sets its price, then firm 2 sets it price. Identify any subgame perfect Nash equilibria to this game. Explain your reasoning.

2. The Edgeworth model we studied in class involved two capacity-constrained firms competing by simultaneously choosing price.  In this problem, we will examine what happens if only one firm is capacity constrained. Consider the same basic information. There are two firms. Market demand is given by

Q(p) = 1000 · 1000p5                            Q = q1 +q2 ·

Each rm has constant marginal costs and no fixed costs: C(qi) = 0 · 3qi , i = 15 2. Firm 1 has a capacity of 340. Firm 2 has no capacity constraints. The firms interact only once and set prices simultaneously.

(a) Find the reaction function for rm 1.  That is, identify rm 1’s best response to any

possible price that firm 2 might set.

(b) Find the reaction function for firm 2.

(c) Draw the reaction functions for both rms. Can you identify any Nash equilibria to this game? Briefly explain the intuition behind your result.

3. In this problem, we want to examine the sustainability of a cartel in a market in which firms engage in quantity competition. Demand is given by the relationship P = 140 · 0 · 1Q, where P is the market price, and Q is market output. Two firms operate in this market, and each firm has constant marginal costs of 20, and no fixed costs. The firms play an infinitely repeated game in which they simultaneously choose quantities each period.

 

(a) Calculate the Cournot Nash equilibrium output for each rm, and the corresponding market price and profits of each firm. That is, solve the Cournot model in which firms compete by simultaneously choosing quantities in a single period.  We will label the Cournot output qN , and the Cournot profits mN .

(b) Calculate the optimal output for an unconstrained cartel (a cartel unconcerned with cheating, detection, and other considerations).  [Hint: the cartel collectively behaves like a monopolist.]  If each firm shares the market equally, how much does each firm produce (call this qC), and what are the profits of each firm (call this mC)?

(c) If rm 2 were to produce the cartel output, qC , what is rm 1’s single period best response? [That is, if the game were played for a single period, what is firm 1’s optimal output in response?]  Call this output qD , and calculate the corresponding profits for firm 1, mD .

(d) We want to examine the conditions under which the grim trigger strategy is sustainable. Consider the following strategy:

●  set q = qC in the rst period of the game or if both rms produced qC last period and in every previous period;

●  set q = qN otherwise - that is, if either rm has chosen any output other than qC in any previous period.

For what discount factors (patience levels), 8, is this strategy sustainable (ie constitutes a subgame perfect Nash equilibrium)?

(e) The grim-trigger strategy involves a fairly extreme level of punishment. Let’s consider whether a more moderate punishment system could sustain collusion in this market. Consider the following variant of a ‘tit-for-tat’ strategy:

set q = qC in the first period of the game or if both rms produced qC last period, or if both rms produced qN last period;

●  set q = qN otherwise.

That is, both rms choose the cooperative output if they have cooperated in the past. If either rm deviates from this, both rms punish for one period by choosing the Cournot- Nash output, and then reverting to the cooperative output.  For what discount factors (patience levels), 8, is this strategy sustainable?

(f) Now, suppose that, instead of the cartel output you identified in part (3b), the cartel decides to set output qC  = 350.  For what discount factors (patience levels), 8, is the tit-for-tat strategy of part (3e) sustainable now? Explain.