<p>In a symmetric <span class="math-container">$n$</span>-firm Cournot competition with linear inverse demand <span class="math-container">$p=a-bQ$</span> and constant marginal cost <span class="math-container">$c$</span>, each firm's equilibrium profit is <span class="math-container">$$\pi_i = \frac{1}{(n+1)^2}\frac{(a-c)^2}{b}$$</span>
Ensuring that each firm in the market earns nonnegative profit, we can derive the bound on the maximum number of firms the market can accommodate:
<span class="math-container">\begin{equation}
\frac{1}{(n+1)^2}\frac{(a-c)^2}{b} \ge 0 
\quad \Rightarrow\quad 
n\le \left\lfloor \frac{a-c}{\sqrt b} -1 \right\rfloor
\end{equation}</span>
The bound is attained if the market has free entry/exit. Thus, as demand increases (either <span class="math-container">$a\uparrow$</span> or <span class="math-container">$b\downarrow$</span>), the number of firms would increase.</p>

CC BY-SA 4.0
2024-08-13T08:07:48.817Z
Thanks a lot! This is very good.
ExcitedSnail
CC BY-SA 4.0
2024-08-05T17:58:08.340Z
2024-08-05T22:08:06.090Z
2024-08-05T22:08:06.090Z
Herr K.
<p>I want to build a model to capture the following thing: when there is higher demand for fish, this attract more producers to enter and produce fish and there will be a higher number of fish producers in equilibrium. That is, I want a model that can produce a monotonically increasing relationship between the demand for a product and the equilibrium number of operating firms: the equilibrium number of firms rise or fall as demand rise or fall. This seems to me like a classic micro problem. Are there any simple micro model that could produce such a relationship? It would be great if you could provide a concrete model and prove that it indeed implies this relationship. A classnote, textbook or paper reference that does this would also be great. Thanks!</p>

CC BY-SA 4.0
2024-08-05T08:32:28.503Z
You may be interested in this post: https://economics.stackexchange.com/questions/58911/till-what-point-do-firms-enter-a-market/58931#58931 in that model the number of firms $N^\star = (1-\alpha)I/F$ where $I$ measures the size of demand.
Jesper Hybel
CC BY-SA 4.0
2024-08-05T09:01:47.680Z
@jesperhybel Thank you very much! This is great! It would be even better if there are some strategic component in it, i.e., modeled using game theoretic models.
ExcitedSnail
CC BY-SA 4.0
2024-08-05T09:42:12.920Z
Well, there are the obvious candidates of Cournot competition with N players or Bertrand with heterogeneous marginal costs and N players.
Jesper Hybel
CC BY-SA 4.0
2024-08-05T18:01:14.477Z
@JesperHybel: Bertrand with heterogeneous MCs need not deliver the result, since only the lowest cost firm gets to produce in equilibrium. However, Bertrand with heterogeneous products (aka product differentiation) will probably work, though I haven't verified.
Herr K.
CC BY-SA 4.0
2024-08-05T18:57:37.030Z
@Herr K. That is true. I just thought surely someone must have made a Bertrand model with capacity constraints - but I don't really know if that could work.
Jesper Hybel
CC BY-SA 4.0
2024-08-05T20:00:54.697Z
To be more specific I was thinking about: "Peak load pricing with a diverse technology" Michael A. Crew and Paul R. Kleindorfer". Firms with different fixed and constant variable costs compete in several periods. The firms with low fixed costs and high variable costs are peaker plants that are only activated when demand is high. The firms with high fixed costs and low variable costs are base load plants operating in all the periods. There's a long literature on peak load pricing models. It reminds me of a Bertrand game with capacity constraints where the choice of capacity is also explained.
Jesper Hybel
<game-theory><supply-and-demand><demand><industrial-organisation><oligopoly>
Any classic micro models in which the (equilibrium) number of producers of a product increases monotonically with demand?
ExcitedSnail
# Any classic micro models in which the (equilibrium) number of producers of a product increases monotonically with demand?

(Asked by: username_0 on 2024-08-05)

I want to build a model to capture the following thing: when there is higher demand for fish, this attract more producers to enter and produce fish and there will be a higher number of fish producers in equilibrium. That is, I want a model that can produce a monotonically increasing relationship between the demand for a product and the equilibrium number of operating firms: the equilibrium number of firms rise or fall as demand rise or fall. This seems to me like a classic micro problem. Are there any simple micro model that could produce such a relationship? It would be great if you could provide a concrete model and prove that it indeed implies this relationship. A classnote, textbook or paper reference that does this would also be great. Thanks!

**Tags:** game-theory, supply-and-demand, demand, industrial-organisation, oligopoly

> **Comments:**
> **username_1**: You may be interested in this post: https://economics.stackexchange.com/questions/58911/till-what-point-do-firms-enter-a-market/58931#58931 in that model the number of firms $N^\star = (1-\alpha)I/F$ where $I$ measures the size of demand.  
> **username_0**: @jesperhybel Thank you very much! This is great! It would be even better if there are some strategic component in it, i.e., modeled using game theoretic models.  
> **username_1**: Well, there are the obvious candidates of Cournot competition with N players or Bertrand with heterogeneous marginal costs and N players.  
> **username_2**: @JesperHybel: Bertrand with heterogeneous MCs need not deliver the result, since only the lowest cost firm gets to produce in equilibrium. However, Bertrand with heterogeneous products (aka product differentiation) will probably work, though I haven't verified.  
> **username_1**: @username_2 That is true. I just thought surely someone must have made a Bertrand model with capacity constraints - but I don't really know if that could work.  
> **username_1**: To be more specific I was thinking about: "Peak load pricing with a diverse technology" Michael A. Crew and Paul R. Kleindorfer". Firms with different fixed and constant variable costs compete in several periods. The firms with low fixed costs and high variable costs are peaker plants that are only activated when demand is high. The firms with high fixed costs and low variable costs are base load plants operating in all the periods. There's a long literature on peak load pricing models. It reminds me of a Bertrand game with capacity constraints where the choice of capacity is also explained.  

---

## 1 answers

### Answer by username_2 on 2024-08-05. Score: 5 | Accepted answer

In a symmetric $n$-firm Cournot competition with linear inverse demand $p=a-bQ$ and constant marginal cost $c$, each firm's equilibrium profit is $$\pi\_i = \frac{1}{(n+1)^2}\frac{(a-c)^2}{b}$$
Ensuring that each firm in the market earns nonnegative profit, we can derive the bound on the maximum number of firms the market can accommodate:
\begin{equation}
\frac{1}{(n+1)^2}\frac{(a-c)^2}{b} \ge 0
\quad \Rightarrow\quad
n\le \left\lfloor \frac{a-c}{\sqrt b} -1 \right\rfloor
\end{equation}
The bound is attained if the market has free entry/exit. Thus, as demand increases (either $a\uparrow$ or $b\downarrow$), the number of firms would increase.

> **Comments:**
> **username_0**: Thanks a lot! This is very good.  

---

