In a duopoly market, the market demand function is $y(p) = 120-p$ where $y=y_1+y_2$

Total function of firm 1 is $C_1(y_1)=20y_1$ while the reaction function of firm 2 is $y_2= 60-0.5y_1$. Assume that the firm 1 is a Stackelberg leader and firm 2 is a follower. What is the equilibrium output of firm 1?

I have the above question. And I have solved as follows:

Step 1: Firm 2's problem for total cost (TC) and total revenue (TR).

$$\pi_2(y) = TR_2 - TC_2$$

F.O.C. $$\frac{\partial \pi_2}{\partial y_2} = MR_2 - MC_2 = 0$$

Setting $MR_2 = MC_2 $, I can get the reaction function of firm 2 which is

$$y_2= 60-0.5y_1$$

Step 2: Firm 1's problem

$$\pi_1(y) = [120-y_1-y_2]y_1 - 20y_1$$

$$\pi_1(y) = [120-y_1-60+0.5y_1]y_1 - 20y_1$$

$$= [60- 0.5y_1]y_1 - 20y_1$$

F.O.C. $$\frac{\partial \pi_1}{\partial y_1} = 60- y_1 - 20 =0$$


Is this solution correct? I don't know exactly what the reaction function means? Thus, I am confused while solving it. Please share your opinions with me. Thank you.

  • $\begingroup$ You have the correct solution. The reaction function just gives the best response of firm $2$ given firm 1's output. Note that firm 2's optimal response when $y_1\geq 120$ is to produce nothing. Is firm 2's marginal cost zero? That seems to be the case given the reaction function you have derived. Your notation is a bit off, as you have used $y$ for three different things: the demand function, the total output of the two firms, and the argument of the profit functions (which should be $(y_1,y_2)$). $\endgroup$
    – smcc
    Commented Sep 14, 2023 at 6:34


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.