# 3 Firm Stackelberg's Oligopoly Game

Consider a market in which there are three firms, all producing the same good. Firm i's cost of producing qi units of the good is Ci(qi)=0 for qi≥0 for each i∈{1,2,3}; the price at which output is sold when the total output is Q is Pd(Q)=max{16−Q,0}, where Q=q1+q2+q3.

Each firm's strategic variable is output and the firms make their decisions sequentially: initially firm 1 chooses its output, then firm 2 does so, knowing the output chosen by the firm 1, and finally, firm 3 chooses its output, knowing the output chosen by firms 1 and 2. Find the equilibrium and outcome of Stackelberg's oligopoly game. List q1,q2,q3

The answer has been stated as (8,4,2)

I understand that we're supposed to use the Best Response functions of the other firms and then move sequentially by first solving the subgame of length 1. I was able to find the Best Response of Firm 3 in terms of q1 and q2, but I do not understand how we're supposed to move forward with the question using the given Best Response. For a 2 Firm Stackelberg's oligopoly game, we can simply use the Best Response function of the follower firm and use it in the profit maximization function of leader's firm. However, when I repeat the procedure with a 3 firm Stackelberg's game, the results seem inconclusive. Please help me with this problem!

• Please consider formatting the mathematical content of your post with MathJax. – Herr K. May 24 '18 at 17:24

I assume that you found Firm 3's best response to be $$q_3^*(q_1,q_2)=\frac12(16-q_1-q_2).$$
The next step would be to solve for Firm 2's best response. Since Firm 2 observes Firm 1's output and correctly anticipates Firm 3's best response, its profit maximization problem is $$\max_{q_2}\;(16-q_1-q_2-q_3^*(q_1,q_2))q_2= \max_{q_2}\;\left(16-q_1-q_2-\frac12(16-q_1-q_2)\right)q_2.$$ Solve this problem, you should get Firm 2's best response as a function of $q_1$ only. Denote this best response $q_2^*(q_1)$.
Then, you solve Firm 1's profit maximization, with Firm 1 correctly anticipating the responses of the subsequent two firms: $$\max_{q_1}\; \bigl(16-q_1-q_2^*(q_1)-q_3^*(q_1,q_2)\bigr)q_1.$$