Consider an oligopoly between two identical firms producing a homogenous good with constant marginal cost where firms face linear market demand. $B_i(q_j)$ denotes firm i’s best response given the output by firm j for i,j=1,2 and $i\not= j$. Let $q_c$ denote the Cournot equilibrium output choice where firms choose output simultaneously. Assume now a Stackelberg setup, where firm 2 observes firm 1’s choice before making its own output choice.
Consider the following pair of strategies in the Stackelberg setup: firm 1 chooses $q_c$. Firm 2 chooses $q_c$ if firm 1 chooses $q_c$ otherwise firm 2 chooses $q^*_2 >>q_c$.
(A) show whether or not the pair of strategies described above constitutes a Nash equilibrium in the game.
(B) show whether or not the pair of strategies described above constitutes a subgame perfect equilibrium in the game.
What I did is
Leaks inverse demand function $p=a-bq$ And $q=q_1+q_2$
And since marginal cost is constant c, then cost function is $cq_i$
First of all, I find Cournot equilibrium outputs $q_c$
I obtain by the profit maximization problem
$$q_c=\frac{a-c}{3b}$$
for both firms.
Secondly, since firm 2 also has a choice $q_2^* >> q_c$
Then let’s assume that $q_2^* = \frac{a-c}{2b}>>\frac{a-c}{3b}=q_c$
Then in order to obtain Stackelberg setup, by backward induction method
Let’s calculate $q^*_1$ which is the firm 1’s quantity when firm 2 chooses $q_2^*=\frac{a-c}{2b}$.
$$\pi_1=(a-bq_1-b(\frac{a-c}{2b}))q_1-cq_1$$
By FOC,
$$q_1^*=\frac{a-c}{4b}$$
And $$q^*_1<< q_c$$
Thirdly I calculated profits for four cases
Case 1: firm1’s quantity and firm 2’s quantity are both $q_c$
Then both firms’ profits are equal and equal to $\frac{(a-c)^2}{9b}$
Case 2: firm1’s quantity is $q_1^*$ firm 2’s quantity is $q_c$
Then,
Firm 1’s profit =$\frac{5(a-c)^2}{48b}$
Firm 2’s profit =$\frac{5(a-c)^2}{36b}$
Case 3: firm1’s quantity is $q_c$ firm 2’s quantity is $q_2^*$
Then,
Firm 1’s profit =$\frac{(a-c)^2}{18b}$
Firm 2’s profit =$\frac{5(a-c)^2}{12b}$
Case 4: firm1’s quantity is $q_1^*$ firm 2’s quantity is $q_2^*$
Then,
Firm 1’s profit =$\frac{(a-c)^2}{16b}$
Firm 2’s profit =$\frac{(a-c)^2}{8b}$
Now let’s look at the part (A)
I construct the following table
So as it is seen, only $(q_c, q_c)$ is Nash equilibrium. But $(q_1^*, q_2^*)$ is not Nash equilibrium.
Next let’s look at the part (B)
I construct tree in this case.
When we look at the table,
First of all, it is for firm 2,
In the left hand side, since the payoff of $q_c$ is greater than the payoff of $q_2^*$, the firm 2 will choose $q-c$.
In the right hand side, since the payoff of $q_2^*$ is greater the payoff of $q_c$ for firm 2, then the firm 2 will choose $q^*_2$.
Now, as for firm 1, if firm 1 choose $q_c$ then it knows that its payoff will be $(a-c)^2/9b$. But if firm 1 choose $q^*_1$ the its payoff will be $(a-c)^2/16b$. So the firm 1 will decide to choose $q_c$.
That’s SPNE is $(q_c,q_cq_c)$.
That is, $(q_1^*,q_2^*)$ is not SPNE.
I solve this question in this way. But I am exactly not sure about my way. Please share your ideas with me. I will be happy if say something abou my solution. Thank you.