# Tacit Collusion : infinitely repeated Cournot duopoly

This exercise is from Steve Tadelis An Introduction To Game Theory.

(10.8) Tacit Collusion: Two firms, which have zero marginal cost and no fixed cost, produce some good, each producing $$q_i ≥ 0, i ∈ {1, 2}$$. The demand for this good is given by $$p = 200 − Q$$, where $$Q = q1 + q2$$.

a) First consider the case of Cournot competition, in which each form chooses $$q_i$$ and this game is infinitely repeated with a discount factor $$δ < 1$$. Solve for the static stage-game Cournot-Nash equilibrium.

b) For which values of $$δ$$ can you support the firms’ equally splitting monopoly profits in each period as a subgame-perfect equilibrium that uses grim-trigger strategies (i.e., after one deviates from the proposed split, they resort to the static Cournot-Nash equilibrium thereafter)? (Note: Be careful in defining the strategies of the firms!)

I thought that (a) is easy enough to not include it as a separate question, but I would still like to check if my working is correct:

a) Firm $$i\in \{0,1\}$$ maximizes profits when:

$$(200-q_i-q_j)q_i$$ reaches maximum. So,

$$\frac{\partial (200-q_i-q_j)q_i}{\partial q_i}=0 \Leftrightarrow \frac{200-q_j}{2}=0$$.

Nash equilibrium is reached when $$q_i=\frac{200-q_j}{2} \Leftrightarrow q_i=200/3$$

b)

If they split into equal qunatities, then they will maximize profits when both quantities will equal $$50$$.

So, by a formula in my textbook, to see for what values of $$\delta$$ the firms will not deviate from the strategies, we calculate:

And with some algebra it's equivalent to :

If a firm i had to unilaterally deviate from the collusive agreement, it would choose to produce an output $$q^d_i$$ that maximizes its profit, given that firm $$j$$ produces $$q^c_j=50$$. That is

$$q^d_i=\text{ arg max }_{q_i} (200-50-q_i)q_i=(150-q_i)q_i$$

So, $$q_i^d=75$$ Hence $$v^d_i=(200-75-50)75=5625$$

For the current payoff if defect $$v^*_i=\left(200-2\left(\frac{200}{3}\right)\right)(200/3)=4444$$

The collusive outcome, obtained when firms form a cartel that maximizes joint profit, is such that $$Q^c=\text{ arg max }_QP(Q)Q$$ and $$q^c_i=\frac{1}{2}Q^c.$$ Here, $$Q^c=100$$ so $$q_1^c=q_2^c=50$$.

So, $$v^c_i=(200-100)50=5000$$

So, value for $$\delta$$ is :

$$\delta\geq \frac{5625-5000}{5625-4444}=625/1181=0.529$$

So, for $$\delta\in[0.529,1)$$, the firms will not deviate.

• $\frac{200-q_j}{2}=0$ should be $\frac{200-q_j}{2}=q_i$. – VARulle Jun 14 '20 at 0:41