ECOS3005 Problem set 2

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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 xed costs. Assume we have the same rationing rule as the Bertrand model. That is, the lower priced rm captures the entire market; if both rms set the same price, they share the market equally. The rms play sequentially. First, rm 1 sets its price. Firm 1 is then committed to this price. Firm 2 then observes rm 1’s price and chooses a price of its own. That is, rst rm 1 sets its price, then rm 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 rms competing by simultaneously choosing price. In this problem, we will examine what happens if only one rm is capacity constrained. Consider the same basic information. There are two rms. Market demand is given by

Q(p) = 1000 · 1000p5 Q = q1 +q2 ·

Each rm has constant marginal costs and no xed costs: C(qi) = 0 · 3qi , i = 15 2. Firm 1 has a capacity of 340. Firm 2 has no capacity constraints. The rms 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 rm 2 might set.

(b) Find the reaction function for rm 2.

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

3. In this problem, we want to examine the sustainability of a cartel in a market in which rms 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 rms operate in this market, and each rm has constant marginal costs of 20, and no xed costs. The rms play an in nitely 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 pro ts of each rm. That is, solve the Cournot model in which rms compete by simultaneously choosing quantities in a single period. We will label the Cournot output qN , and the Cournot pro ts 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 rm shares the market equally, how much does each rm produce (call this qC), and what are the pro ts of each rm (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 rm 1’s optimal output in response ] Call this output qD , and calculate the corresponding pro ts for rm 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 rst 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 identi ed 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.