<p>Great question! One can make things quite complicated in these models, and their flexibility is one of the reasons they are so prevalent. Below, I try to keep the simplicity of the <span class="math-container">$2$</span> firm case you outlined and highlight how you can think about generalising to the <span class="math-container">$N$</span> firm case.</p>
<p>First, it is informative to think about what are the key strategic elements driving the equilibrium and comparative statics in the <span class="math-container">$2$</span> firm case. Let's consider the profit functions of the <span class="math-container">$2$</span> firms with zero marginal costs.</p>
<p><span class="math-container">$$\begin{align*}\Pi_i(p_i,p_j)=p_i(A_i-a_ip_i+b_ip_j) \end{align*}$$</span></p>
<p>We assume <span class="math-container">$A_i,a_i,b_j&gt;0$</span>. In most of economics, it is the marginals which really matter for strategic considerations. These are, <span class="math-container">$$\begin{align*} \frac{\partial \Pi_i}{\partial p_i}(p_i,p_j)&amp;=A_i-2a_ip_i+b_ip_j \\
\frac{\partial \Pi_i}{\partial p_j}(p_i,p_j)&amp;= b_ip_i\end{align*}$$</span> I emphasise <span class="math-container">$4$</span> important takeaways from these equations. <span class="math-container">$(1)$</span> for any fixed <span class="math-container">$p_j\geq 0$</span> there exists a unique <span class="math-container">$(!)$</span> <span class="math-container">$p_i$</span> such that <span class="math-container">$\frac{\partial \Pi_i}{\partial p_i}(p_i,p_j)=0$</span>. <span class="math-container">$(2)$</span> profit is concave in <span class="math-container">$p_i$</span>, <span class="math-container">$\frac{\partial^2 \Pi_i}{\partial p_i^2}(p_i,p_j)=-2a_i&lt;0$</span>, and <span class="math-container">$(3)$</span> since <span class="math-container">$\frac{\partial^2 \Pi_i}{\partial p_i\partial p_j}(p_i,p_j)=b_i&gt;0$</span>, the marginal benefit of increasing <span class="math-container">$p_i$</span> rises as <span class="math-container">$p_j$</span> increases. In jargon, <span class="math-container">$p_i$</span> and <span class="math-container">$p_j$</span> are <span class="math-container">$\textit{strategic complements}$</span> and the game is super-modular. Finally, we note that <span class="math-container">$(4)$</span> the marginals are affine in <span class="math-container">$p_i$</span> and <span class="math-container">$p_j$</span>.</p>
<p>Points <span class="math-container">$(1)$</span> and <span class="math-container">$(2)$</span> ensure the game has a unique equilibrium and <span class="math-container">$(3)$</span> captures the key strategic component of the game. Point <span class="math-container">$(4)$</span> is a simplifying element effectively enforcing that consumer's valuations for the goods are drawn from a uniform distribution which is what makes quantity demand affine in prices.</p>
<p>When generalising, we would like to preserve each of these key elements. Suppose now there are <span class="math-container">$N$</span> firms each producing differentiated goods and competing on prices.</p>
<p>We shall begin in a general framework and work out what we need to assume to recover our <span class="math-container">$4$</span> observations from the <span class="math-container">$2$</span> firm case. We start with some exogenously given demand curves <span class="math-container">$\{D_i\}_{i=1}^N$</span> which are common knowledge and perfectly observed. Given prices <span class="math-container">$p_1,p_2,\dots , p_N$</span> posted by the firms, the demand for firm <span class="math-container">$i$</span>'s good is <span class="math-container">$D_i(p_1,p_2,\dots , p_N)$</span>. Thus, the profit function of firm <span class="math-container">$i$</span> in this extended model is <span class="math-container">$$\Pi_i(p_1,p_2,\dots, p_N)=p_iD_i(p_1,p_2,\dots , p_N)$$</span> Examining the marginal profits again, <span class="math-container">$$\begin{align*}\frac{\partial \Pi_i}{\partial p_i} &amp;= D_i(p_1,p_2,\dots , p_N)+p_i\frac{\partial D_i}{\partial p_i}(p_1,p_2,\dots , p_N) \\
\frac{\partial \Pi_i}{\partial p_j} &amp;= p_i\frac{\partial D_i}{\partial p_j}(p_1,p_2,\dots , p_N) \end{align*}$$</span> For these equations to be affine for all <span class="math-container">$i=1,2,\dots ,N$</span>, it must be that <span class="math-container">$D_i(p_1,p_2,\dots , p_N)$</span> is affine in prices. Therefore, <span class="math-container">$D_i$</span> has form <span class="math-container">$$D_i(p_1,p_2,\dots , p_N)=c_0^i+c_1^ip_1+c_2^ip_2+\dots + c_N^ip_N$$</span>
and <span class="math-container">$\frac{\partial \Pi_i}{\partial p_i}=c_i^i$</span> and <span class="math-container">$\frac{\partial \Pi_i}{\partial p_j}=c_j^i$</span> are constants.</p>
<p>At prices equal to <span class="math-container">$0$</span>, demand for good <span class="math-container">$i$</span> is <span class="math-container">$c_0^i$</span>, so naturally we shall require <span class="math-container">$c_0^i&gt;0$</span>. Next, super-modularity (point <span class="math-container">$(3)$</span>) requires <span class="math-container">$c_j^i&gt;0$</span> for any <span class="math-container">$j\neq i$</span>. Finally, points <span class="math-container">$(1)$</span> and <span class="math-container">$(2)$</span> require <span class="math-container">$c_i^i&lt;0$</span>. So, based on just our <span class="math-container">$4$</span> observations, we have found that we must require <span class="math-container">$$D_i(p_1,p_2,\dots , p_N)=A_i-a_ip_i +\sum_{j\neq i}b^i_jp_j$$</span> for some constants <span class="math-container">$A_i,a_i,b_j^i&gt;0$</span> for all <span class="math-container">$i,j$</span>.</p>
<p>So, we have shown that if we want to capture the strategic considerations and the simplicity of the <span class="math-container">$2$</span> firm model, we have to assume quantity demanded is of the form <span class="math-container">$$q_i=A_i-a_ip_i +\sum_{j\neq i}b^i_jp_j$$</span></p>
<p>Therefore, in this model, as <span class="math-container">$N$</span> increases we simply add new cross-price coefficients to the demand equation. We cannot say much more as the demand curves and number of firms are exogenously given, so asking how they scale with <span class="math-container">$N$</span> is difficult to answer. We can only compare statically between two specific values of <span class="math-container">$N$</span>, with all of the demand curves defined.</p>
<p>This may seem an unsatisfactory answer to your question <span class="math-container">$2$</span>. If I understand your question <span class="math-container">$2$</span>, you are asking what exactly determines the cross-price coefficients as we scale <span class="math-container">$N$</span>, and whether these should be homogenous or heterogeneous between firms as <span class="math-container">$N$</span> gets large. To properly answer this question, one would have to endogenise the number of firms <span class="math-container">$N$</span> within the market and examine what type of firms enter the market. That is, we would want to know whether firms with high or low <span class="math-container">$b_j^i$</span> coefficients on the firm's already in the market are most likely to enter. That is, whether firm's with more or less similar goods relative to the ones currently in the market are most likely to enter.</p>

CC BY-SA 4.0
2024-12-06T11:25:54.620Z
2024-12-06T11:25:54.620Z
Joseph Basford
<p>I am exploring how the demand equations generalize in a Differentiated Bertrand competition model as the number of firms,

N, increases beyond 2. In the standard two-firm case, the demand equations are defined as:
<span class="math-container">$$
q_1 = A_1 - a_1 p_1 + b_1 p_2
$$</span></p>
<p><span class="math-container">$$
q_2 = A_2 - a_2 p_2 + b_2 p_1
$$</span></p>
<p>Where:</p>
<ul>
<li><span class="math-container">$q_1, q_2$</span>: Quantities demanded for firms 1 and 2, respectively.</li>
<li><span class="math-container">$p_1, p_2$</span>: Prices set by firms 1 and 2.</li>
<li><span class="math-container">$A_1, A_2$</span>: Demand intercepts for each firm.</li>
<li><span class="math-container">$a_1, a_2$</span>: Negative slope coefficients for a firm’s own price.</li>
<li><span class="math-container">$b_1, b_2$</span>: Positive cross-price coefficients showing how a competitor’s price affects a firm’s demand.</li>
</ul>
<p>As the number of firms <span class="math-container">$N$</span> increases:</p>
<ol>
<li>How does <span class="math-container">$q_i$</span> (firm <span class="math-container">$i$</span>'s demand) generalize? (e.g. for 3 or more firms)</li>
<li>How do interactions between the firms (cross-price effects) scale with <span class="math-container">$N$</span>?</li>
</ol>
<p>From the Wikipedia article on <a href="https://en.wikipedia.org/wiki/Differentiated_Bertrand_competition" rel="nofollow noreferrer">Differentiated Bertrand competition</a>, the model resolves the Bertrand paradox by assuming each firm produces a differentiated product, facing downward-sloping demand curves. When a competitor raises its price, a firm can also raise its price, benefiting from an upward shift in demand.</p>
<p>I suspect that as <span class="math-container">$N$</span> increases, the formulation for <span class="math-container">$q_i$</span> might involve summing the cross-price effects of all other firms, but I’m unsure how to account for this mathematically or how <span class="math-container">$b_{ij}$</span> (cross-price coefficients) might vary across firms.</p>
<h3>Question:</h3>
<ol>
<li>How do we mathematically generalize the demand equation for <span class="math-container">$q_i$</span> in the <span class="math-container">$N$</span>-firm case?</li>
<li>Are there specific considerations for cross-price coefficients (e.g., symmetry or heterogeneity among firms)?</li>
</ol>
<p>Any insights, examples, or references to literature addressing this would be appreciated!</p>

<competitive-equilibrium><competition><dynamic-games><equilibrium-price>
How Does the Number of Firms Impact Demand (q) in a Differentiated Bertrand Competition Model?
Laurence_jj
# How Does the Number of Firms Impact Demand (q) in a Differentiated Bertrand Competition Model?

(Asked by: username_0 on 2024-12-05)

I am exploring how the demand equations generalize in a Differentiated Bertrand competition model as the number of firms,
N, increases beyond 2. In the standard two-firm case, the demand equations are defined as:
$$
q\_1 = A\_1 - a\_1 p\_1 + b\_1 p\_2
$$

$$
q\_2 = A\_2 - a\_2 p\_2 + b\_2 p\_1
$$

Where:

* $q\_1, q\_2$: Quantities demanded for firms 1 and 2, respectively.
* $p\_1, p\_2$: Prices set by firms 1 and 2.
* $A\_1, A\_2$: Demand intercepts for each firm.
* $a\_1, a\_2$: Negative slope coefficients for a firm’s own price.
* $b\_1, b\_2$: Positive cross-price coefficients showing how a competitor’s price affects a firm’s demand.

As the number of firms $N$ increases:

1. How does $q\_i$ (firm $i$'s demand) generalize? (e.g. for 3 or more firms)
2. How do interactions between the firms (cross-price effects) scale with $N$?

From the Wikipedia article on [Differentiated Bertrand competition](https://en.wikipedia.org/wiki/Differentiated_Bertrand_competition), the model resolves the Bertrand paradox by assuming each firm produces a differentiated product, facing downward-sloping demand curves. When a competitor raises its price, a firm can also raise its price, benefiting from an upward shift in demand.

I suspect that as $N$ increases, the formulation for $q\_i$ might involve summing the cross-price effects of all other firms, but I’m unsure how to account for this mathematically or how $b\_{ij}$ (cross-price coefficients) might vary across firms.

### Question:

1. How do we mathematically generalize the demand equation for $q\_i$ in the $N$-firm case?
2. Are there specific considerations for cross-price coefficients (e.g., symmetry or heterogeneity among firms)?

Any insights, examples, or references to literature addressing this would be appreciated!

**Tags:** competitive-equilibrium, competition, dynamic-games, equilibrium-price


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## 1 answers

### Answer by username_1 on 2024-12-06. Score: 3 | Accepted answer

Great question! One can make things quite complicated in these models, and their flexibility is one of the reasons they are so prevalent. Below, I try to keep the simplicity of the $2$ firm case you outlined and highlight how you can think about generalising to the $N$ firm case.

First, it is informative to think about what are the key strategic elements driving the equilibrium and comparative statics in the $2$ firm case. Let's consider the profit functions of the $2$ firms with zero marginal costs.

$$\begin{align\*}\Pi\_i(p\_i,p\_j)=p\_i(A\_i-a\_ip\_i+b\_ip\_j) \end{align\*}$$

We assume $A\_i,a\_i,b\_j>0$. In most of economics, it is the marginals which really matter for strategic considerations. These are, $$\begin{align\*} \frac{\partial \Pi\_i}{\partial p\_i}(p\_i,p\_j)&=A\_i-2a\_ip\_i+b\_ip\_j \\
\frac{\partial \Pi\_i}{\partial p\_j}(p\_i,p\_j)&= b\_ip\_i\end{align\*}$$ I emphasise $4$ important takeaways from these equations. $(1)$ for any fixed $p\_j\geq 0$ there exists a unique $(!)$ $p\_i$ such that $\frac{\partial \Pi\_i}{\partial p\_i}(p\_i,p\_j)=0$. $(2)$ profit is concave in $p\_i$, $\frac{\partial^2 \Pi\_i}{\partial p\_i^2}(p\_i,p\_j)=-2a\_i<0$, and $(3)$ since $\frac{\partial^2 \Pi\_i}{\partial p\_i\partial p\_j}(p\_i,p\_j)=b\_i>0$, the marginal benefit of increasing $p\_i$ rises as $p\_j$ increases. In jargon, $p\_i$ and $p\_j$ are $\textit{strategic complements}$ and the game is super-modular. Finally, we note that $(4)$ the marginals are affine in $p\_i$ and $p\_j$.

Points $(1)$ and $(2)$ ensure the game has a unique equilibrium and $(3)$ captures the key strategic component of the game. Point $(4)$ is a simplifying element effectively enforcing that consumer's valuations for the goods are drawn from a uniform distribution which is what makes quantity demand affine in prices.

When generalising, we would like to preserve each of these key elements. Suppose now there are $N$ firms each producing differentiated goods and competing on prices.

We shall begin in a general framework and work out what we need to assume to recover our $4$ observations from the $2$ firm case. We start with some exogenously given demand curves $\{D\_i\}\_{i=1}^N$ which are common knowledge and perfectly observed. Given prices $p\_1,p\_2,\dots , p\_N$ posted by the firms, the demand for firm $i$'s good is $D\_i(p\_1,p\_2,\dots , p\_N)$. Thus, the profit function of firm $i$ in this extended model is $$\Pi\_i(p\_1,p\_2,\dots, p\_N)=p\_iD\_i(p\_1,p\_2,\dots , p\_N)$$ Examining the marginal profits again, $$\begin{align\*}\frac{\partial \Pi\_i}{\partial p\_i} &= D\_i(p\_1,p\_2,\dots , p\_N)+p\_i\frac{\partial D\_i}{\partial p\_i}(p\_1,p\_2,\dots , p\_N) \\
\frac{\partial \Pi\_i}{\partial p\_j} &= p\_i\frac{\partial D\_i}{\partial p\_j}(p\_1,p\_2,\dots , p\_N) \end{align\*}$$ For these equations to be affine for all $i=1,2,\dots ,N$, it must be that $D\_i(p\_1,p\_2,\dots , p\_N)$ is affine in prices. Therefore, $D\_i$ has form $$D\_i(p\_1,p\_2,\dots , p\_N)=c\_0^i+c\_1^ip\_1+c\_2^ip\_2+\dots + c\_N^ip\_N$$
and $\frac{\partial \Pi\_i}{\partial p\_i}=c\_i^i$ and $\frac{\partial \Pi\_i}{\partial p\_j}=c\_j^i$ are constants.

At prices equal to $0$, demand for good $i$ is $c\_0^i$, so naturally we shall require $c\_0^i>0$. Next, super-modularity (point $(3)$) requires $c\_j^i>0$ for any $j\neq i$. Finally, points $(1)$ and $(2)$ require $c\_i^i<0$. So, based on just our $4$ observations, we have found that we must require $$D\_i(p\_1,p\_2,\dots , p\_N)=A\_i-a\_ip\_i +\sum\_{j\neq i}b^i\_jp\_j$$ for some constants $A\_i,a\_i,b\_j^i>0$ for all $i,j$.

So, we have shown that if we want to capture the strategic considerations and the simplicity of the $2$ firm model, we have to assume quantity demanded is of the form $$q\_i=A\_i-a\_ip\_i +\sum\_{j\neq i}b^i\_jp\_j$$

Therefore, in this model, as $N$ increases we simply add new cross-price coefficients to the demand equation. We cannot say much more as the demand curves and number of firms are exogenously given, so asking how they scale with $N$ is difficult to answer. We can only compare statically between two specific values of $N$, with all of the demand curves defined.

This may seem an unsatisfactory answer to your question $2$. If I understand your question $2$, you are asking what exactly determines the cross-price coefficients as we scale $N$, and whether these should be homogenous or heterogeneous between firms as $N$ gets large. To properly answer this question, one would have to endogenise the number of firms $N$ within the market and examine what type of firms enter the market. That is, we would want to know whether firms with high or low $b\_j^i$ coefficients on the firm's already in the market are most likely to enter. That is, whether firm's with more or less similar goods relative to the ones currently in the market are most likely to enter.


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