<p>Your answer for the case of two firms is wrong as the expression for the equilibrium prices must involve <span class="math-container">$t$</span>. (Or perhaps you have set <span class="math-container">$t$</span> to some value in your answer?)</p>
<hr />
<p>For the three firm case, assume firm <span class="math-container">$1$</span> is located at <span class="math-container">$0$</span> (or <span class="math-container">$1$</span>) and that firm <span class="math-container">$2$</span> is located at <span class="math-container">$1/2$</span>. Let <span class="math-container">$x_{12}\in(1/2,1)$</span>, <span class="math-container">$x_{13}\in(0,1/4)$</span>, and <span class="math-container">$x_{23}\in(1/4,1/2)$</span> denote the locations of the consumers indifferent between <span class="math-container">$1$</span> and <span class="math-container">$2$</span>, between <span class="math-container">$1$</span> and <span class="math-container">$3$</span>, and between <span class="math-container">$2$</span> and <span class="math-container">$3$</span>.</p>
<p>[If there is a pure strategy Nash equilibrium in prices (added in edit after Giskard's comment), then they must all exist: If not, then at least one firm would have no demand and could do better by lowering its price, or a firm is pricing below zero and so could do better by raising its price.]</p>
<p>We have:
<span class="math-container">$$\begin{align*}
t(1-x_{12})+p_1&amp;=t(x_{12}-1/2)+p_2\\
tx_{13}+p_1&amp;=t(1/4-x_{13})+p_3\\
t(1/2-x_{23})+p_2&amp;=t(x_{13}-1/4)+p_3\\
\end{align*}$$</span></p>
<p>Thus:</p>
<p><span class="math-container">$$\begin{align*}
x_{12}&amp;=\frac{3}{4}+\frac{p_1-p_2}{2t}\\
x_{13}&amp;=\frac{1}{8}+\frac{p_3-p_1}{2t}\\ 
x_{23}&amp;=\frac{3}{8}+\frac{p_2-p_3}{2t}
 \end{align*}$$</span></p>
<p>The profits of each firm are:</p>
<p><span class="math-container">$$\begin{align*}
\pi_1(p)&amp;=p_1(1-x_{12}+x_{13})\\
\pi_2(p)&amp;=p_2(x_{12}-x_{23})\\
\pi_3(p)&amp;=p_3(x_{23}-x_{13})
 \end{align*}$$</span></p>
<p>and</p>
<p><span class="math-container">$$\begin{align*}
\frac{\partial \pi_1}{\partial p_1}&amp;=(1-x_{12}+x_{13})-\frac{p_1}{t}=\frac{3}{8}+\frac{p_2+p_3-4p_1}{2t}\\
\frac{\partial \pi_2}{\partial p_2}&amp;=(x_{12}-x_{23})-\frac{p_2}{t}=\frac{3}{8}+\frac{p_1+p_3-4p_2}{2t}\\
\frac{\partial \pi_3}{\partial p_3}&amp;=(x_{23}-x_{13})-\frac{p_3}{t}=\frac{1}{4}+\frac{p_1+p_2-4p_3}{2t}
 \end{align*}$$</span></p>
<p>Setting each equation to zero and solving gives <span class="math-container">$p_1^*=p_2^*=7t/20$</span> and <span class="math-container">$p_3^*=3t/10$</span>. In equilibrium we have <span class="math-container">$x_{12}^*=3/4$</span>, <span class="math-container">$x_{13}^*=1/10$</span>, <span class="math-container">$x_{23}^*=2/5$</span> and profits are:</p>
<p><span class="math-container">$$\begin{align*}
\pi_1^*=\pi_2^*&amp;=\frac{49t}{400}\\
\pi_3^*&amp;=\frac{9t}{100}
 \end{align*}$$</span></p>
<p>This is not the Nash equilibrium for the reason pointed about by Giskard in the first comment, so it seems there is no pure strategy Nash equilibrium.</p>
<hr />
<p>The rest of this answer is based on the incorrect solution above.</p>
<hr />
<p>Firm <span class="math-container">$3$</span> prices lower than firms <span class="math-container">$1$</span> and <span class="math-container">$2$</span>. The intuition is that firms <span class="math-container">$1$</span> and <span class="math-container">$2$</span> face less price competition between each other than with firm <span class="math-container">$3$</span> because <span class="math-container">$1$</span> and <span class="math-container">$2$</span> are further apart from each other (distance <span class="math-container">$1/2$</span>) than from <span class="math-container">$3$</span> (distance <span class="math-container">$1/4$</span>). Firm <span class="math-container">$3$</span> faces the same level of price competition with both firms <span class="math-container">$1$</span> and <span class="math-container">$2$</span> as it is closer to both (at a distance of <span class="math-container">$1/4$</span> from both).</p>
<p>Mathematically, notice that for any firm <span class="math-container">$i$</span> the profit/revenue is</p>
<p><span class="math-container">$$\pi_i(p)=p_iD_i(p)$$</span></p>
<p>where <span class="math-container">$D_i$</span> is the demand function of firm <span class="math-container">$i$</span>. The marginal revenue for firm <span class="math-container">$i$</span> is</p>
<p><span class="math-container">$$\frac{\partial \pi_i}{\partial p_i}=D_i(p)+p_i\frac{\partial D_i}{\partial p_i}\tag{1}$$</span></p>
<p>Every firm has has <span class="math-container">$\partial D_i/\partial p_i=-1/t$</span>. If all firms were to have the same price, then:</p>
<ul>
<li>The second term in <span class="math-container">$(1)$</span> would be the same for all firms, i.e. a marginal increase in price would cause the same marginal reduction in revenue (through reduction in demand).</li>
<li>The first term (the quantity demanded) in <span class="math-container">$(1)$</span> would be larger (<span class="math-container">$3/8$</span>) for firms <span class="math-container">$1$</span> and <span class="math-container">$2$</span> than for firm <span class="math-container">$3$</span> (<span class="math-container">$1/4$</span>). (This is why firms <span class="math-container">$1$</span> and <span class="math-container">$2$</span> face softer price competition: they are competing over more consumers.)</li>
<li>This cannot be an equilibrium. Firms <span class="math-container">$1$</span> and <span class="math-container">$2$</span> would have a marginal revenue that is larger than firm <span class="math-container">$3$</span> and so either firms <span class="math-container">$1$</span> and <span class="math-container">$2$</span> would want to raise their price or firm <span class="math-container">$3$</span> would want to lower its price.</li>
</ul>
<hr />
<p>For the last part, any fixed cost <span class="math-container">$f&gt;\pi^*_3=9t/100$</span>.</p>

CC BY-SA 4.0
2024-05-10T17:24:22.640Z
I think the spectre of D'Aspremont et al. is looming over your solution, as you are only considering the marginal optimality of $3$'s pricing (e.g. you assume an indifferent buyer $x_{23}$ exists). But firm $3$ could totally undercut the others by pricing at $p_3 = 0.099t$, and attain a higher profit ($0.099t$ vs $0.09t$). Caveat emptor: I only calculated this in my head.
Giskard
CC BY-SA 4.0
2024-05-10T19:38:20.147Z
True. I had assumed because this looked like a homework or textbook problem that it would be straightforward, so hadn't bothered to check that possibility. I suppose there must be no pure strategy equilibrium then?
smcc
CC BY-SA 4.0
2024-05-11T07:22:00.117Z
I am just guessing, but I also don't think that 1) there is a pure strategy equilirium 2) the professor who assigned this exercise thought of this problem.
Giskard
CC BY-SA 4.0
2024-05-11T07:23:36.200Z
I don't understand how the Q is getting more votes than your A? Here, have one.
Giskard
CC BY-SA 4.0
2024-05-10T16:34:02.253Z
2024-05-11T14:20:20.123Z
2024-05-11T14:20:20.123Z
smcc
<p>In a circular city model a la Salop consumers have a linear transportation cost t for each unit they
have to travel. The length of the full circle is normalized to 1. Two firms are active in the market,
have no production cost and are located at 0 and ½ and set prices. The equilibrium level of prices can be calculated easily, and if I´m not mistaken it will equal 1. However, they also ask the following:  Suppose then that a third firm is contemplating to enter. Firm 1 and 2 cannot change their locations anymore and firm 3 thinks about entering at ¼. There is a fixed cost f of entry.
b. If entry takes place what are the equilibrium prices in the market. Which firm charges the lowest market price and provide an economic explanation for that result? (Hint: note that firm 1 and 2 are still symmetric to each other, but firm 3 is not).
I´m not sure how to solve it as I have never seen a Salop linear model with asymmetrical firms
c. For which level of fixed cost f does a third firm not want to enter the market?</p>

<industrial-organisation>
Salop linear Model - Asymmetric
Mario
# Salop linear Model - Asymmetric

(Asked by: username_0 on 2024-05-07)

In a circular city model a la Salop consumers have a linear transportation cost t for each unit they
have to travel. The length of the full circle is normalized to 1. Two firms are active in the market,
have no production cost and are located at 0 and ½ and set prices. The equilibrium level of prices can be calculated easily, and if I´m not mistaken it will equal 1. However, they also ask the following: Suppose then that a third firm is contemplating to enter. Firm 1 and 2 cannot change their locations anymore and firm 3 thinks about entering at ¼. There is a fixed cost f of entry.
b. If entry takes place what are the equilibrium prices in the market. Which firm charges the lowest market price and provide an economic explanation for that result? (Hint: note that firm 1 and 2 are still symmetric to each other, but firm 3 is not).
I´m not sure how to solve it as I have never seen a Salop linear model with asymmetrical firms
c. For which level of fixed cost f does a third firm not want to enter the market?

**Tags:** industrial-organisation


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## 1 answers

### Answer by username_1 on 2024-05-10. Score: 2

Your answer for the case of two firms is wrong as the expression for the equilibrium prices must involve $t$. (Or perhaps you have set $t$ to some value in your answer?)

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For the three firm case, assume firm $1$ is located at $0$ (or $1$) and that firm $2$ is located at $1/2$. Let $x\_{12}\in(1/2,1)$, $x\_{13}\in(0,1/4)$, and $x\_{23}\in(1/4,1/2)$ denote the locations of the consumers indifferent between $1$ and $2$, between $1$ and $3$, and between $2$ and $3$.

[If there is a pure strategy Nash equilibrium in prices (added in edit after username_2's comment), then they must all exist: If not, then at least one firm would have no demand and could do better by lowering its price, or a firm is pricing below zero and so could do better by raising its price.]

We have:
$$\begin{align\*}
t(1-x\_{12})+p\_1&=t(x\_{12}-1/2)+p\_2\\
tx\_{13}+p\_1&=t(1/4-x\_{13})+p\_3\\
t(1/2-x\_{23})+p\_2&=t(x\_{13}-1/4)+p\_3\\
\end{align\*}$$

Thus:

$$\begin{align\*}
x\_{12}&=\frac{3}{4}+\frac{p\_1-p\_2}{2t}\\
x\_{13}&=\frac{1}{8}+\frac{p\_3-p\_1}{2t}\\
x\_{23}&=\frac{3}{8}+\frac{p\_2-p\_3}{2t}
\end{align\*}$$

The profits of each firm are:

$$\begin{align\*}
\pi\_1(p)&=p\_1(1-x\_{12}+x\_{13})\\
\pi\_2(p)&=p\_2(x\_{12}-x\_{23})\\
\pi\_3(p)&=p\_3(x\_{23}-x\_{13})
\end{align\*}$$

and

$$\begin{align\*}
\frac{\partial \pi\_1}{\partial p\_1}&=(1-x\_{12}+x\_{13})-\frac{p\_1}{t}=\frac{3}{8}+\frac{p\_2+p\_3-4p\_1}{2t}\\
\frac{\partial \pi\_2}{\partial p\_2}&=(x\_{12}-x\_{23})-\frac{p\_2}{t}=\frac{3}{8}+\frac{p\_1+p\_3-4p\_2}{2t}\\
\frac{\partial \pi\_3}{\partial p\_3}&=(x\_{23}-x\_{13})-\frac{p\_3}{t}=\frac{1}{4}+\frac{p\_1+p\_2-4p\_3}{2t}
\end{align\*}$$

Setting each equation to zero and solving gives $p\_1^\*=p\_2^\*=7t/20$ and $p\_3^\*=3t/10$. In equilibrium we have $x\_{12}^\*=3/4$, $x\_{13}^\*=1/10$, $x\_{23}^\*=2/5$ and profits are:

$$\begin{align\*}
\pi\_1^\*=\pi\_2^\*&=\frac{49t}{400}\\
\pi\_3^\*&=\frac{9t}{100}
\end{align\*}$$

This is not the Nash equilibrium for the reason pointed about by username_2 in the first comment, so it seems there is no pure strategy Nash equilibrium.

---

The rest of this answer is based on the incorrect solution above.

---

Firm $3$ prices lower than firms $1$ and $2$. The intuition is that firms $1$ and $2$ face less price competition between each other than with firm $3$ because $1$ and $2$ are further apart from each other (distance $1/2$) than from $3$ (distance $1/4$). Firm $3$ faces the same level of price competition with both firms $1$ and $2$ as it is closer to both (at a distance of $1/4$ from both).

Mathematically, notice that for any firm $i$ the profit/revenue is

$$\pi\_i(p)=p\_iD\_i(p)$$

where $D\_i$ is the demand function of firm $i$. The marginal revenue for firm $i$ is

$$\frac{\partial \pi\_i}{\partial p\_i}=D\_i(p)+p\_i\frac{\partial D\_i}{\partial p\_i}\tag{1}$$

Every firm has has $\partial D\_i/\partial p\_i=-1/t$. If all firms were to have the same price, then:

* The second term in $(1)$ would be the same for all firms, i.e. a marginal increase in price would cause the same marginal reduction in revenue (through reduction in demand).
* The first term (the quantity demanded) in $(1)$ would be larger ($3/8$) for firms $1$ and $2$ than for firm $3$ ($1/4$). (This is why firms $1$ and $2$ face softer price competition: they are competing over more consumers.)
* This cannot be an equilibrium. Firms $1$ and $2$ would have a marginal revenue that is larger than firm $3$ and so either firms $1$ and $2$ would want to raise their price or firm $3$ would want to lower its price.

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For the last part, any fixed cost $f>\pi^\*\_3=9t/100$.

> **Comments:**
> **username_2**: I think the spectre of D'Aspremont et al. is looming over your solution, as you are only considering the marginal optimality of $3$'s pricing (e.g. you assume an indifferent buyer $x_{23}$ exists). But firm $3$ could totally undercut the others by pricing at $p_3 = 0.099t$, and attain a higher profit ($0.099t$ vs $0.09t$). Caveat emptor: I only calculated this in my head.  
> **username_1**: True. I had assumed because this looked like a homework or textbook problem that it would be straightforward, so hadn't bothered to check that possibility. I suppose there must be no pure strategy equilibrium then?  
> **username_2**: I am just guessing, but I also don't think that 1) there is a pure strategy equilirium 2) the professor who assigned this exercise thought of this problem.  
> **username_2**: I don't understand how the Q is getting more votes than your A? Here, have one.  

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