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.
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).