<p>I assume that you found Firm 3's best response to be 
\begin{equation}
q_3^*(q_1,q_2)=\frac12(16-q_1-q_2).
\end{equation}</p>

<p>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 
\begin{equation}
\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.
\end{equation}
Solve this problem, you should get Firm 2's best response as a function of $q_1$ only. Denote this best response $q_2^*(q_1)$.</p>

<p>Then, you solve Firm 1's profit maximization, with Firm 1 correctly anticipating the responses of the subsequent two firms: 
\begin{equation}
\max_{q_1}\; \bigl(16-q_1-q_2^*(q_1)-q_3^*(q_1,q_2)\bigr)q_1.
\end{equation}</p>

CC BY-SA 4.0
2018-05-24T17:43:29.450Z
Yes; upon solving the Best Response functions and Maximization problem, I got my answer. Thank You for confirming!
S.Rana
CC BY-SA 4.0
2018-05-24T17:37:25.523Z
2018-05-24T17:37:25.523Z
Herr K.
<p>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 <em>Ci(qi)=0</em> for <em>qi≥0</em> for each <em>i∈{1,2,3}</em>;  the price at which output is sold when the total output is Q is <em>Pd(Q)=max{16−Q,0}</em>, where <em>Q=q1+q2+q3.</em> </p>

<p>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 <em>q1,q2,q3</em></p>

<p>The answer has been stated as (8,4,2) </p>

<p>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! </p>

CC BY-SA 4.0
2018-05-24T17:24:32.930Z
Please consider formatting the mathematical content of your post with [MathJax](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference).
Herr K.
<game-theory><oligopoly>
3 Firm Stackelberg's Oligopoly Game
S.Rana
# 3 Firm Stackelberg's Oligopoly Game

(Asked by: username_0 on 2018-05-24)

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!

**Tags:** game-theory, oligopoly

> **Comments:**
> **username_1**: Please consider formatting the mathematical content of your post with [MathJax](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference).  

---

## 1 answers

### Answer by username_1 on 2018-05-24. Score: 2 | Accepted answer

I assume that you found Firm 3's best response to be
\begin{equation}
q\_3^\*(q\_1,q\_2)=\frac12(16-q\_1-q\_2).
\end{equation}

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
\begin{equation}
\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.
\end{equation}
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:
\begin{equation}
\max\_{q\_1}\; \bigl(16-q\_1-q\_2^\*(q\_1)-q\_3^\*(q\_1,q\_2)\bigr)q\_1.
\end{equation}

> **Comments:**
> **username_0**: Yes; upon solving the Best Response functions and Maximization problem, I got my answer. Thank You for confirming!  

---

