<p>(B) Define the set of firms as <span class="math-container">$F= \{1,2,3,....N \}$</span></p>
<p>Demand function given as <span class="math-container">$P=a -bQ$</span></p>
<p>Let the leader firm's quantity produced be <span class="math-container">$q_l $</span> for some <span class="math-container">$l \in F$</span></p>
<p>Since it's a extensive game (assuming perfect information is available to all) of choosing quantity and the Leader being the first mover can anticipate the choice by the rest of the firms (i.e. their best response to the leader's quantity choice)</p>
<p>We can try to use backward induction to arrive at Subgame Perfect Nash Equilibrium</p>
<p>Let's try to figure out the best response of each of the remaining firms to the leader's quantity choice in the initial period <span class="math-container">$t_0$</span> and to the other firm's choice (simultaneous game)</p>
<p>Apart from the leader, each firm <span class="math-container">$j$</span> tries to maximise their profit given by:-</p>
<p><span class="math-container">$$\underset{q_j\geq0}{max} \: \left(a-b(q_l + q_j + \sum_{k\neq \{j, l\}} q_k)\right) q_j - cq_j$$</span></p>
<p><span class="math-container">$ \: \: \: \forall \: k \in F - \{ q_l , q_j\}$</span></p>
<p>This yields</p>
<p><span class="math-container">$BR_j (q_l,\sum_{k\neq \{j, l\}} q_k)$</span> = <span class="math-container">$\left\{\begin{matrix}
\frac{a-bq_l - b\sum_{k \neq \{j, l\}} q_k -c}{2b}&amp; if \:  \frac{a-c}{b} \geq q_l + \sum_{k\neq \{j, l\}} q_k \\ 
 0&amp;  otherwise
\end{matrix}\right.$</span></p>
<p>Since it is symmetric for all <span class="math-container">$j$</span></p>
<p><span class="math-container">$ Q_{-l}  = \sum_{j \neq l}q_j$</span></p>
<p><span class="math-container">$ = \left\{\begin{matrix}
\frac{N-1}{bN}(a-bq_l -c) &amp; if \:   \frac{a-c}{b} \geq q_l  \\ 
0 &amp;  otherwise
\end{matrix}\right. $</span></p>
<p>Now, Leader takes the above information into account while taking the decision</p>
<p><span class="math-container">$\underset{q_l\geq0}{max} (a - b(q_l + Q_{-i}))q_l -cq_l $</span></p>
<p><span class="math-container">$s.t. Q_{-i} =  \left\{\begin{matrix}
\frac{N-1}{bN}(a-bq_l -c) &amp; if \:   \frac{a-c}{b} \geq q_l  \\ 
0 &amp;  otherwise
\end{matrix}\right. $</span></p>
<p>This would result in SPNE outcome such that
<span class="math-container">$q_l^* = \frac{a-c}{2b}$</span> and<br />
<span class="math-container">$q_{i \neq l}^* = \frac{a-c}{2bN}$</span> for all firms other than l (leader)</p>

CC BY-SA 4.0
2023-10-17T19:20:20.217Z
Thanks for the solution! Could you explain how you got ![Q-l](https://latex.codecogs.com/svg.image?Q_{-l}) value after you found BR, I assume it is (N-1)*BR, since other firms are equal and we exclude leader, but I do not come to your  value of ![Q-l](https://latex.codecogs.com/svg.image?Q_{-l}) when solving like that.
Super_Clown
CC BY-SA 4.0
2023-10-18T02:08:47.310Z
If all firms except leader simultaneously choose the strategy $q_j = BR_j$ (and satisfying non-negativity constraint), then we can find $\sum q_j$ for all j. So, we have $ Q_{-l} = \frac{(N-1)a - (N-1)b q_l - (N-2) b Q_{-l} - (N-1)c}{2b}$  . Now we can solve for $Q_{-l}$ and get the result.
Divyam Verma
CC BY-SA 4.0
2023-10-17T12:13:34.560Z
2023-10-17T12:13:34.560Z
Divyam Verma
<p>Consider a market, where <strong>N</strong> firms with average cost <strong>c</strong> compete in quantities. Assume linear demand P=a-bQ. Assume that <strong>N</strong> is arbitrary.</p>
<p>a)Find the equilibrium of the game, where these <strong>N</strong> firms compete a la Cournot.</p>
<p>b)Assume now that one firm (out of these <strong>N</strong> firms) acts as a “leader” by setting quantity first
and the other firms move second (simultaneously) knowing the leader’s choice. Find the new
equilibrium.</p>
<p>Here is my solution for point a)</p>
<p><img src="https://latex.codecogs.com/svg.image?Q=q_i+Q_%7B-i%7D" alt="Q" /></p>
<p><img src="https://latex.codecogs.com/svg.image?P=a-b(q_i+Q_%7B-i%7D)" alt="C" /></p>
<p>Then profit will be:</p>
<p><img src="https://latex.codecogs.com/svg.image?%5Cpi&amp;space;_i=(a-b(q_i+Q_%7B-i%7D))q_i-cq_i%5Cto&amp;space;max&amp;space;q_i" alt="F" /></p>
<p><img src="https://latex.codecogs.com/svg.image?q_i=(a-c-bQ_%7B-i%7D)/2b" alt="E" /></p>
<p>imposing symmetry: <img src="https://latex.codecogs.com/svg.image?q_1=q_2=q_3=...=q_n" alt="V" /></p>
<p>Therefor <img src="https://latex.codecogs.com/svg.image?Q_%7B-i%7D=(n-1)q" alt="p" /></p>
<p><img src="https://latex.codecogs.com/svg.image?q=(a-c-bQ_%7B-i%7D)/2b=(a-c-bq(n-1))/2b" alt="E" /></p>
<p><img src="https://latex.codecogs.com/svg.image?q%5E%7B%5Cast%7D=(a-c)/(b(n+1))" alt="w" /></p>
<p>How do I apporach point b), I'm a bit lost? Also extra question: what if all firms sequentially chose quantities?</p>

<microeconomics><game-theory><cournot>
Stackelberg equilibrium with arbitrary number of firms
Super_Clown
# Stackelberg equilibrium with arbitrary number of firms

(Asked by: username_0 on 2023-10-15)

Consider a market, where **N** firms with average cost **c** compete in quantities. Assume linear demand P=a-bQ. Assume that **N** is arbitrary.

a)Find the equilibrium of the game, where these **N** firms compete a la Cournot.

b)Assume now that one firm (out of these **N** firms) acts as a “leader” by setting quantity first
and the other firms move second (simultaneously) knowing the leader’s choice. Find the new
equilibrium.

Here is my solution for point a)

![Q](https://latex.codecogs.com/svg.image?Q=q_i+Q_%7B-i%7D)

![C](https://latex.codecogs.com/svg.image?P=a-b(q_i+Q_%7B-i%7D))

Then profit will be:

![F](https://latex.codecogs.com/svg.image?%5Cpi&space;_i=(a-b(q_i+Q_%7B-i%7D))q_i-cq_i%5Cto&space;max&space;q_i)

![E](https://latex.codecogs.com/svg.image?q_i=(a-c-bQ_%7B-i%7D)/2b)

imposing symmetry: ![V](https://latex.codecogs.com/svg.image?q_1=q_2=q_3=...=q_n)

Therefor ![p](https://latex.codecogs.com/svg.image?Q_%7B-i%7D=(n-1)q)

![E](https://latex.codecogs.com/svg.image?q=(a-c-bQ_%7B-i%7D)/2b=(a-c-bq(n-1))/2b)

![w](https://latex.codecogs.com/svg.image?q%5E%7B%5Cast%7D=(a-c)/(b(n+1)))

How do I apporach point b), I'm a bit lost? Also extra question: what if all firms sequentially chose quantities?

**Tags:** microeconomics, game-theory, cournot


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## 1 answers

### Answer by username_1 on 2023-10-17. Score: 3 | Accepted answer

(B) Define the set of firms as $F= \{1,2,3,....N \}$

Demand function given as $P=a -bQ$

Let the leader firm's quantity produced be $q\_l $ for some $l \in F$

Since it's a extensive game (assuming perfect information is available to all) of choosing quantity and the Leader being the first mover can anticipate the choice by the rest of the firms (i.e. their best response to the leader's quantity choice)

We can try to use backward induction to arrive at Subgame Perfect Nash Equilibrium

Let's try to figure out the best response of each of the remaining firms to the leader's quantity choice in the initial period $t\_0$ and to the other firm's choice (simultaneous game)

Apart from the leader, each firm $j$ tries to maximise their profit given by:-

$$\underset{q\_j\geq0}{max} \: \left(a-b(q\_l + q\_j + \sum\_{k\neq \{j, l\}} q\_k)\right) q\_j - cq\_j$$

$ \: \: \: \forall \: k \in F - \{ q\_l , q\_j\}$

This yields

$BR\_j (q\_l,\sum\_{k\neq \{j, l\}} q\_k)$ = $\left\{\begin{matrix}
\frac{a-bq\_l - b\sum\_{k \neq \{j, l\}} q\_k -c}{2b}& if \: \frac{a-c}{b} \geq q\_l + \sum\_{k\neq \{j, l\}} q\_k \\
0& otherwise
\end{matrix}\right.$

Since it is symmetric for all $j$

$ Q\_{-l} = \sum\_{j \neq l}q\_j$

$ = \left\{\begin{matrix}
\frac{N-1}{bN}(a-bq\_l -c) & if \: \frac{a-c}{b} \geq q\_l \\
0 & otherwise
\end{matrix}\right. $

Now, Leader takes the above information into account while taking the decision

$\underset{q\_l\geq0}{max} (a - b(q\_l + Q\_{-i}))q\_l -cq\_l $

$s.t. Q\_{-i} = \left\{\begin{matrix}
\frac{N-1}{bN}(a-bq\_l -c) & if \: \frac{a-c}{b} \geq q\_l \\
0 & otherwise
\end{matrix}\right. $

This would result in SPNE outcome such that
$q\_l^\* = \frac{a-c}{2b}$ and  
$q\_{i \neq l}^\* = \frac{a-c}{2bN}$ for all firms other than l (leader)

> **Comments:**
> **username_0**: Thanks for the solution! Could you explain how you got ![Q-l](https://latex.codecogs.com/svg.image?Q_{-l}) value after you found BR, I assume it is (N-1)*BR, since other firms are equal and we exclude leader, but I do not come to your  value of ![Q-l](https://latex.codecogs.com/svg.image?Q_{-l}) when solving like that.  
> **username_1**: If all firms except leader simultaneously choose the strategy $q_j = BR_j$ (and satisfying non-negativity constraint), then we can find $\sum q_j$ for all j. So, we have $ Q_{-l} = \frac{(N-1)a - (N-1)b q_l - (N-2) b Q_{-l} - (N-1)c}{2b}$  . Now we can solve for $Q_{-l}$ and get the result.  

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