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

$$y_1=40$$

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.

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  • $\begingroup$ 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

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