# Monopolistic competition

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?

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.