<p>so the UMP with CES preferences, <span class="math-container">$L$</span> goods produced each by a different firm and weights normalized to 1 is:</p>
<p><span class="math-container">\begin{align}
\max_{x_l} \left[\sum_{l = 1}^{L} x_l^{\frac{\sigma - 1}{\sigma}}\right]^\frac{\sigma}{\sigma -1}\\
s.t. \ \ \sum_{l = 1}^{L} p_l x_l = w \\
\ \ \ \ \ \tilde x =\left[\sum_{l = 1}^{L} x_l^{\frac{\sigma - 1}{\sigma}}\right]^\frac{\sigma}{\sigma -1} \\
 \tilde p \tilde x^*= w
\end{align}</span></p>
<p>Where <span class="math-container">$ \tilde x^* $</span> is the net indirect utility and <span class="math-container">$\tilde p$</span> is the price index, which is taken as given by each firm.
\
Solving the problem using the usual techniques (partial derivatives, ratio of FOCS and plug <span class="math-container">$x_l(x_k)$</span> in the first budget constraint, you find the demand for good <span class="math-container">$L$</span>:</p>
<p><span class="math-container">$$ x_l(\textbf{p},w) = \frac{p_l^{-\sigma}}{\sum_{k=1}^{L}p_k^{-(\sigma -1)}}w $$</span></p>
<p>Use this and the second and the third constraint (it's a bit more involved, but I'm sure you can do it), you get the price index:</p>
<p><span class="math-container">$$\tilde p = \left[ \sum_{k}^{L}p_k^{-(\sigma -1)}\right]^{-\frac{1}{\sigma-1}}$$</span></p>
<p>Then, you can rewrite <span class="math-container">$x_l(\textbf{p},w) = p_l^{-\sigma}\tilde p^{\sigma -1}w$</span>.<br />
As I wrote before, the price index is taken as given by the firms. Since firms are local monopolists that produce a single good, the cross price elasticity is zero
Hence, the own price elasticity is</p>
<p><span class="math-container">$$\epsilon_l(p_l)  =  - d\ln x_l(\textbf{p},w)/d \ln p_l = \sigma$$</span></p>
<p>Using the Lerner Index Condition, the mark up of each firm are, assuming symmetric and constant marginal costs (from the FOCs of the problem of the monopolist)</p>
<p><span class="math-container">$$ (p_l - c) / p_l = 1/\sigma $$</span>, hence</p>
<p><span class="math-container">$p_l = (\sigma/(\sigma-1))c$</span></p>
<p>For a symmetric price <span class="math-container">$p$</span>, you find</p>
<p><span class="math-container">$x = \frac{\sigma-1}{\sigma}w/cL$</span></p>
<p>Suppose now that there are <span class="math-container">$N$</span> identical consumers.</p>
<p>The profits of each firm are gross of fixed costs <span class="math-container">$F$</span> are.</p>
<p><span class="math-container">$$ \pi = (N/L)(w/\sigma)$$</span></p>
<p>In the Long Run, for  the equilibrium number of firms, <span class="math-container">$ L^* $</span>, will be given by the free entry condition <span class="math-container">$ \pi - F = 0$</span>, thus:</p>
<p><span class="math-container">$$ L^* = (N/F)(w/\sigma) $$</span></p>
<p>Clearly, lower fixed costs entail a higher equibrium number of participants, same for a bigger market (bigger <span class="math-container">$N$</span>) .A higher elasticity of substitution entails lower mark-ups, hence, since individual demand of each firm decreases as the number of players increases, the number of players that can be supported in the market is lower.</p>

CC BY-SA 4.0
2022-10-14T19:13:37.347Z
2022-10-14T19:13:37.347Z
Matteo Bulgarelli
<p>I’ve got some questions on economic intuition according to firm behavior in monopolistic competition. Kinda hard to find explanation in the Internet.</p>
<p>What is the effect of the fixed costs of production on the number of firms in monopolistic competition?</p>
<p>What effect the number of consumers has on the number of firms?</p>
<p>And what is the effect of the elasticity of substitution on number of firms in the market?</p>

<microeconomics><monopolistic-competition>
Monopolistic competition
Laney
# Monopolistic competition

(Asked by: username_0 on 2022-10-13)

I’ve got some questions on economic intuition according to firm behavior in monopolistic competition. Kinda hard to find explanation in the Internet.

What is the effect of the fixed costs of production on the number of firms in monopolistic competition?

What effect the number of consumers has on the number of firms?

And what is the effect of the elasticity of substitution on number of firms in the market?

**Tags:** microeconomics, monopolistic-competition

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## 1 answers

### Answer by username_1 on 2022-10-14. Score: 1

so the UMP with CES preferences, $L$ goods produced each by a different firm and weights normalized to 1 is:

\begin{align}
\max\_{x\_l} \left[\sum\_{l = 1}^{L} x\_l^{\frac{\sigma - 1}{\sigma}}\right]^\frac{\sigma}{\sigma -1}\\
s.t. \ \ \sum\_{l = 1}^{L} p\_l x\_l = w \\
\ \ \ \ \ \tilde x =\left[\sum\_{l = 1}^{L} x\_l^{\frac{\sigma - 1}{\sigma}}\right]^\frac{\sigma}{\sigma -1} \\
\tilde p \tilde x^\*= w
\end{align}

Where $ \tilde x^\* $ is the net indirect utility and $\tilde p$ is the price index, which is taken as given by each firm.
\
Solving the problem using the usual techniques (partial derivatives, ratio of FOCS and plug $x\_l(x\_k)$ in the first budget constraint, you find the demand for good $L$:

$$ x\_l(\textbf{p},w) = \frac{p\_l^{-\sigma}}{\sum\_{k=1}^{L}p\_k^{-(\sigma -1)}}w $$

Use this and the second and the third constraint (it's a bit more involved, but I'm sure you can do it), you get the price index:

$$\tilde p = \left[ \sum\_{k}^{L}p\_k^{-(\sigma -1)}\right]^{-\frac{1}{\sigma-1}}$$

Then, you can rewrite $x\_l(\textbf{p},w) = p\_l^{-\sigma}\tilde p^{\sigma -1}w$.  
As I wrote before, the price index is taken as given by the firms. Since firms are local monopolists that produce a single good, the cross price elasticity is zero
Hence, the own price elasticity is

$$\epsilon\_l(p\_l) = - d\ln x\_l(\textbf{p},w)/d \ln p\_l = \sigma$$

Using the Lerner Index Condition, the mark up of each firm are, assuming symmetric and constant marginal costs (from the FOCs of the problem of the monopolist)

$$ (p\_l - c) / p\_l = 1/\sigma $$, hence

$p\_l = (\sigma/(\sigma-1))c$

For a symmetric price $p$, you find

$x = \frac{\sigma-1}{\sigma}w/cL$

Suppose now that there are $N$ identical consumers.

The profits of each firm are gross of fixed costs $F$ are.

$$ \pi = (N/L)(w/\sigma)$$

In the Long Run, for the equilibrium number of firms, $ L^\* $, will be given by the free entry condition $ \pi - F = 0$, thus:

$$ L^\* = (N/F)(w/\sigma) $$

Clearly, lower fixed costs entail a higher equibrium number of participants, same for a bigger market (bigger $N$) .A higher elasticity of substitution entails lower mark-ups, hence, since individual demand of each firm decreases as the number of players increases, the number of players that can be supported in the market is lower.


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