<p>The main issue seems to be that you assume that under Bertrand competition a firm is free to set a price and a quantity as well. But under Bertrand competition firms set prices and then have to meet the demand whatever level it takes. Below is a detailed discussion.</p>
<hr />
<p>I will assume that cost functions are the same for all firms and also that the marginal cost function is non-decreasing. (You do not explicitly state these conditions but seem to assume them as well.) I will also assume that demand is non-increasing in price.</p>
<p>In this case, your candidate for the equilibrium is when all firms set the price <span class="math-container">$p^*$</span>, where all firms produce
<span class="math-container">$$
q^* = \frac{D(p^*)}{5}
$$</span>
and where
<span class="math-container">$$
p^* = MC(q^*).
$$</span>
Is this really an equilibrium, or could a firm profit by deviating to a slightly lower price <span class="math-container">$p'$</span>?</p>
<p>If say firm 1 were to deviate to <span class="math-container">$p' &lt; p^*$</span> while the other firms still charge <span class="math-container">$p^*$</span> then all consumers would seek out firm 1. Since <span class="math-container">$p' &lt; p^*$</span> and demand is non-increasing in price
<span class="math-container">$$
D(p') &gt; D(p^*).
$$</span>
Firm 1 has to fulfill this alone, so
<span class="math-container">$$
q_1 = D(p') &gt; D(p^*) = 5 \cdot q^*.
$$</span>
Because marginal cost is non-decreasing you also have
<span class="math-container">$$
MC(q_1) &gt; MC(q^*).
$$</span>
So now firm 1 charges a price lower than <span class="math-container">$p^* = MC(q^*)$</span> but it has to produce so much that the marginal cost is larger than <span class="math-container">$p^*$</span>, hence it earns negative profits, hence deviation from <span class="math-container">$p^*$</span> was not profitable.</p>

CC BY-SA 3.0
2016-08-14T02:34:40.817Z
Thank you, what do you think also about the following argument to compliment yours? If a firm decreases price and only produces the corresponding quantity on the MC curve, then there will be unmet demand for competitors to fulfill using the candidate equilibrium ,as you called it. In the end the deviant makes less per unit on less units and his competitors make more per unit on more units.
Antecedent
CC BY-SA 3.0
2016-08-14T07:10:23.603Z
@Jaco This is not the case. As I wrote, you cannot set both price and quantity. If you set a lower price, all consumers will come to you and you have to fulfill their demands. There isn't unmet demand for your competitors if they set a higher price. (There is another model, the capacitiy constrained Bertrand competition, in which this can be the case.)
Giskard
CC BY-SA 4.0
2016-08-13T09:48:11.343Z
2022-02-23T05:52:28.440Z
2022-02-23T05:52:28.440Z
Giskard
<p>I know that Bertrand Oligopolies will charge a price equal to marginal cost. But if marginal cost is, say, </p>

<blockquote>
  <p>2Q or 4Q^2</p>
</blockquote>

<p>(i.e. not constant) then how does one determine where, on the MC curve, the equilibrium lies?</p>

<p>If there are 5 Bertrand firms, then do I just find where MC intersects the Demand Curve and divide the quantity by 5, and then set price equal to MC at that quantity?</p>

<p>Conceptually this sounds reasonable, but I'm not sure that this is correct. If these firms charge this price then it seems to me that one of them will under produce to charge a lower price until (if MC is 2Q for instance) price and quantity = 0. So that no firms will produce because if they do then someone will charge a cent less till we reach 0 again.</p>

<microeconomics>
How to solve Bertrand Equilibrium with a non-constant MC?
Antecedent
# How to solve Bertrand Equilibrium with a non-constant MC?

(Asked by: username_0 on 2016-08-12)

I know that Bertrand Oligopolies will charge a price equal to marginal cost. But if marginal cost is, say,

> 2Q or 4Q^2

(i.e. not constant) then how does one determine where, on the MC curve, the equilibrium lies?

If there are 5 Bertrand firms, then do I just find where MC intersects the Demand Curve and divide the quantity by 5, and then set price equal to MC at that quantity?

Conceptually this sounds reasonable, but I'm not sure that this is correct. If these firms charge this price then it seems to me that one of them will under produce to charge a lower price until (if MC is 2Q for instance) price and quantity = 0. So that no firms will produce because if they do then someone will charge a cent less till we reach 0 again.

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2016-08-13. Score: 1 | Accepted answer

The main issue seems to be that you assume that under Bertrand competition a firm is free to set a price and a quantity as well. But under Bertrand competition firms set prices and then have to meet the demand whatever level it takes. Below is a detailed discussion.

---

I will assume that cost functions are the same for all firms and also that the marginal cost function is non-decreasing. (You do not explicitly state these conditions but seem to assume them as well.) I will also assume that demand is non-increasing in price.

In this case, your candidate for the equilibrium is when all firms set the price $p^\*$, where all firms produce
$$
q^\* = \frac{D(p^\*)}{5}
$$
and where
$$
p^\* = MC(q^\*).
$$
Is this really an equilibrium, or could a firm profit by deviating to a slightly lower price $p'$?

If say firm 1 were to deviate to $p' < p^\*$ while the other firms still charge $p^\*$ then all consumers would seek out firm 1. Since $p' < p^\*$ and demand is non-increasing in price
$$
D(p') > D(p^\*).
$$
Firm 1 has to fulfill this alone, so
$$
q\_1 = D(p') > D(p^\*) = 5 \cdot q^\*.
$$
Because marginal cost is non-decreasing you also have
$$
MC(q\_1) > MC(q^\*).
$$
So now firm 1 charges a price lower than $p^\* = MC(q^\*)$ but it has to produce so much that the marginal cost is larger than $p^\*$, hence it earns negative profits, hence deviation from $p^\*$ was not profitable.

> **Comments:**
> **username_0**: Thank you, what do you think also about the following argument to compliment yours? If a firm decreases price and only produces the corresponding quantity on the MC curve, then there will be unmet demand for competitors to fulfill using the candidate equilibrium ,as you called it. In the end the deviant makes less per unit on less units and his competitors make more per unit on more units.  
> **username_1**: @Jaco This is not the case. As I wrote, you cannot set both price and quantity. If you set a lower price, all consumers will come to you and you have to fulfill their demands. There isn't unmet demand for your competitors if they set a higher price. (There is another model, the capacitiy constrained Bertrand competition, in which this can be the case.)  

---

