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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:

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And with some algebra it's equivalent to :

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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.

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

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