The market demand for a good is described by the inverse demand function $P(Q) = 120 – Q $ where $Q$ is total quantity demanded and $P(Q)$ the market price. Two firms $i =1,2$ have identical cost functions $c_i(q_i) =0.5 q_i^2$ where $q_i$ denotes firm i's output quantity hence $Q=q_1+q_2$.
a) Find the Cournot Nash Equilibrium.
b) Suppose that both firms cooperate by making the arrangement to produce joint profit maximizing quantity $Q^M$ together (i.e. every firm has to produce $Q^M/2$). Compute that quantity. What is firm 1's best response if firm 2 indeed produces $Q^M/2$ ? does this arrangement constitute a Nash equilibrium?
My Answer:
Part(a) is straightforward. So I found that $q_1^*=q_2^*=30$
Part(b)
For firm i $\forall i =1,2$
$$\pi_i (q_1,q_2)=(120-Q^M)(Q^M/2)-0.5 (Q^M)^2/4$$
$$d \pi_i/ d Q^M = 120-Q^M-(1/4)Q^M=0$$
$$120 = 5/4 Q^M $$
$$Q^M = 96$$
so $$q_1 = q_2 = 96/2 = 48$$
since $48 \not= 30$, this result is not Nash equilibrium.
Does this answer make sense? I am not sure about that. Please share your ideas and corrections with me. Thank you.
Best Answer
It makes perfect sense, since in case they agree, they prefer to produce the monopolistic quantity $(48)$ which is below the (total) quantity they would produce if they were competing (that is $30 \cdot 2 =60$).
Note that indeed, they would maximize their profits, but the result is not a Nash equilibrium since every player has "an incentive to deviate" from the "half-monopolistic production" (that is $30$), since if they have a conjecture that the opponent will produce a quantity of $30$, then their best quantity to produce is different (to compute this, just write $q_2$ as fixed in the profit function of player 1, and derive its best action).