<p>Let <span class="math-container">$n$</span> be the number of firms.</p>
<p>In my answer to your other question I got:</p>
<ul>
<li><span class="math-container">$n$</span> as a function of <span class="math-container">$F$</span>.</li>
</ul>
<p><span class="math-container">$$n = \frac{140(5 + \sqrt{F + 375})}{F+350} - 1$$</span></p>
<ul>
<li>The optimal quantity of each firm as a function of <span class="math-container">$n$</span>.</li>
</ul>
<p><span class="math-container">$$q_i = \frac{140}{n+1}$$</span></p>
<ul>
<li>The total quantity produced by all firms in aggregate:</li>
</ul>
<p><span class="math-container">$$Q = \frac{140 n}{n+1}$$</span></p>
<ul>
<li>The market price as a function of <span class="math-container">$n$</span>.</li>
</ul>
<p><span class="math-container">$$P = \frac{10 (n+15)}{n+1}$$</span></p>
<ul>
<li>The profit function of each firm as a function of <span class="math-container">$n$</span>.</li>
</ul>
<p><span class="math-container">\begin{align*}
\Pi_i &amp;= 10 \cdot \frac{n+15}{n+1} \cdot \frac{140}{n+1} - 10 \cdot 35 - F \\
&amp;= 10 \left(\frac{140 (n+15)}{(n+1)^2} - 35\right) - F
\end{align*}</span></p>
<p>Since in the long run, firms enter the market up to the point where <span class="math-container">$\Pi_i = 0$</span>, then the aggregate profit by firms is</p>
<p><span class="math-container">$$\Pi = 0$$</span></p>
<p>On the other hand, the consumer surplus is given by the area under the demand curve and above the market price.</p>
<p><a href="https://i.sstatic.net/I9X5a.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/I9X5a.jpg" alt="enter image description here" /></a></p>
<p>Since this is the area of a triangle, we get</p>
<p><span class="math-container">\begin{align*}
CS &amp;= \frac{1}{2} Q (150-P) \\
 &amp;= \frac{1}{2} Q^2 \\
 &amp;= \frac{19600 n^2}{2 (n+1)^2} \\
 &amp;= \frac{9800 n^2}{(n+1)^2} \\
\end{align*}</span></p>
<p>With this, the welfare is given by:</p>
<p><span class="math-container">\begin{align*}
W &amp;= \Pi + CS \\
&amp;= \frac{9800 n^2}{(n+1)^2}
\end{align*}</span></p>
<p>It can be easily checked that</p>
<p><span class="math-container">$$\frac{dW}{dn} = \frac{19600 n}{(n+1)^3} &gt; 0, \hspace{2mm} \forall n$$</span></p>
<p>Since the derivative of welfare is always positive, <span class="math-container">$W$</span> is increasing in <span class="math-container">$n$</span>.</p>
<p>Therefore, there is no optimal number of firms. As <span class="math-container">$n \to \infty$</span>, the welfare <span class="math-container">$W$</span> keeps increasing.</p>
<p>This implies we want to pick the level of <span class="math-container">$F$</span> that gives the highest possible value of <span class="math-container">$n$</span>.</p>
<p>Since <span class="math-container">$F$</span> decreases the firms’ profits, the higher the <span class="math-container">$F$</span> value, the lower the number of firms.</p>
<p>Therefore, the optimal policy is to set <span class="math-container">$F = 0$</span>.</p>
<p>Recall that for a given level of <span class="math-container">$F$</span>, the number of firms <span class="math-container">$n$</span> is given by:</p>
<p><span class="math-container">$$n = \frac{140(5 + \sqrt{F + 375})}{F+350} - 1$$</span></p>
<p>Since <span class="math-container">$F = 0$</span>, the number of firms that would enter the market is</p>
<p><span class="math-container">$$n = \frac{140(5+\sqrt{375})}{350} - 1 \approx 8.75$$</span></p>
<p>Since there can’t be fractional firms, the actual number of firms would be <span class="math-container">$n = 8$</span>.</p>

CC BY-SA 4.0
2023-04-13T19:17:42.910Z
2024-12-21T05:42:52.233Z
2024-12-21T05:42:52.233Z
Nicolas Torres
<p>There exist multiple identical firms with a large supply of products available and each firm must decide how much to provide to the market. Each firm has the same cost function, given by <span class="math-container">$C(q) = 10q$</span> and market demand is given by <span class="math-container">$Q = 150 - P$</span>.</p>
<p>Entering the market entails a fixed cost <span class="math-container">$F$</span> to be incurred by each firm in the entry stage. Suppose there is one period of Cournot competition after entry.</p>
<ul>
<li><p>What number of firms maximizes total surplus if entry can be restricted or promoted through manipulation of <span class="math-container">$F$</span>? Note that prices or quantities cannot be regulated; only <span class="math-container">$F$</span> can be altered for the sake of maximizing welfare (defined as the sum of total profits and consumer surplus).</p>
</li>
<li><p>If both the price and number of firms could be regulated in order to maximize total welfare, what price and how many firms will be allowed to enter?</p>
</li>
</ul>

CC BY-SA 4.0
2023-04-14T08:27:10.593Z
Since $F$ can be manipulated, I believe it is some kind of entry fee charged by the regulator. I want to ask if we consider regulator's revenue from this fee as part of the total surplus we aim to maximize?
Amit
<microeconomics><self-study><industrial-organisation><cournot>
How to calculate the maximizing number of firms in a Cournot competition?
Lucas
# How to calculate the maximizing number of firms in a Cournot competition?

(Asked by: username_0 on 2023-04-13)

There exist multiple identical firms with a large supply of products available and each firm must decide how much to provide to the market. Each firm has the same cost function, given by $C(q) = 10q$ and market demand is given by $Q = 150 - P$.

Entering the market entails a fixed cost $F$ to be incurred by each firm in the entry stage. Suppose there is one period of Cournot competition after entry.

* What number of firms maximizes total surplus if entry can be restricted or promoted through manipulation of $F$? Note that prices or quantities cannot be regulated; only $F$ can be altered for the sake of maximizing welfare (defined as the sum of total profits and consumer surplus).
* If both the price and number of firms could be regulated in order to maximize total welfare, what price and how many firms will be allowed to enter?

**Tags:** microeconomics, self-study, industrial-organisation, cournot

> **Comments:**
> **username_1**: Since $F$ can be manipulated, I believe it is some kind of entry fee charged by the regulator. I want to ask if we consider regulator's revenue from this fee as part of the total surplus we aim to maximize?  

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## 1 answers

### Answer by username_2 on 2023-04-13. Score: 0

Let $n$ be the number of firms.

In my answer to your other question I got:

* $n$ as a function of $F$.

$$n = \frac{140(5 + \sqrt{F + 375})}{F+350} - 1$$

* The optimal quantity of each firm as a function of $n$.

$$q\_i = \frac{140}{n+1}$$

* The total quantity produced by all firms in aggregate:

$$Q = \frac{140 n}{n+1}$$

* The market price as a function of $n$.

$$P = \frac{10 (n+15)}{n+1}$$

* The profit function of each firm as a function of $n$.

\begin{align\*}
\Pi\_i &= 10 \cdot \frac{n+15}{n+1} \cdot \frac{140}{n+1} - 10 \cdot 35 - F \\
&= 10 \left(\frac{140 (n+15)}{(n+1)^2} - 35\right) - F
\end{align\*}

Since in the long run, firms enter the market up to the point where $\Pi\_i = 0$, then the aggregate profit by firms is

$$\Pi = 0$$

On the other hand, the consumer surplus is given by the area under the demand curve and above the market price.

[![enter image description here](https://i.sstatic.net/I9X5a.jpg)](https://i.sstatic.net/I9X5a.jpg)

Since this is the area of a triangle, we get

\begin{align\*}
CS &= \frac{1}{2} Q (150-P) \\
&= \frac{1}{2} Q^2 \\
&= \frac{19600 n^2}{2 (n+1)^2} \\
&= \frac{9800 n^2}{(n+1)^2} \\
\end{align\*}

With this, the welfare is given by:

\begin{align\*}
W &= \Pi + CS \\
&= \frac{9800 n^2}{(n+1)^2}
\end{align\*}

It can be easily checked that

$$\frac{dW}{dn} = \frac{19600 n}{(n+1)^3} > 0, \hspace{2mm} \forall n$$

Since the derivative of welfare is always positive, $W$ is increasing in $n$.

Therefore, there is no optimal number of firms. As $n \to \infty$, the welfare $W$ keeps increasing.

This implies we want to pick the level of $F$ that gives the highest possible value of $n$.

Since $F$ decreases the firms’ profits, the higher the $F$ value, the lower the number of firms.

Therefore, the optimal policy is to set $F = 0$.

Recall that for a given level of $F$, the number of firms $n$ is given by:

$$n = \frac{140(5 + \sqrt{F + 375})}{F+350} - 1$$

Since $F = 0$, the number of firms that would enter the market is

$$n = \frac{140(5+\sqrt{375})}{350} - 1 \approx 8.75$$

Since there can’t be fractional firms, the actual number of firms would be $n = 8$.


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