# Monopolistic Competition

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- Additional Contributors: None
Categories: CategoryModel CategoryMicro CategoryIndustrialOrganisation CategoryImperfectCompetition

Related Models: Locational_Product_Differentiation

Created: 2008-07-15

Suggested Citation: See license page.

Contents

## Introduction

Represents a situation of competition between products where products provide **variety**. Specifically, products are not perfect substitutes but consumers have a taste for variety and buy multiple products. This contrasts with Locational_Product_Differentiation models where a given consumer only purchases from a single supplier and prefers, all other things being equal, to purchase from the supplier located closest to them.

## Model

Take a *representative* consumer. Divide goods into the numeraire, good 0, and all other goods $$i = 1, ... n$$ in the sector of interest. Let $$q_{i}, p_{i}$$ denote, respectively, consumption and price of good $$i$$. Normalize price of good 0 to 1. For the differentiated product we assume there is free entry into the industry with fixed and marginal costs respectively $$f, c$$. The consumer's budget constraint is:

$$ \[ q_{0} + \sum_{i=1}^{n} p_{i}q_{i} \leq L \] $$

The utility function is of CES form where to ensure $$U$$ is concave requires $$\rho \lt 1$$:

$$ \[ U = U( q_{0}, ( \sum_{i=1}^{n} q_{i}^\rho )^{\frac{1}{\rho}}) \] $$

**Aside:** this utility function has certain special properties that simplify the analysis which the reader should keep in mind. First it treats all products as symmetric. Second it allows for no location decision. Every product that is introduced is equally differentiated from every other product.

## Solving

For simplicity define $$y = ( \sum_{i=1}^{n} q_{i}^\rho )^{\frac{1}{\rho}}, p = ( \sum_{i=1}^{n} q_{i}^\beta )^{\frac{1}{\beta}}, \beta = \frac{ -\rho }{ 1 - \rho }$$

### Consumers

The first order condition for utility maximization yields:

$$ \[ U_{1}p_{i} = U_{2} q_{i}^{\rho - 1} ( \sum_{j=1}^{n} q_{j}^{\rho} )^{\frac{ 1-\rho }{ \rho }} \] $$

Where $$U_{k}$$ is the partial derivate of $$U$$ with respect to its kth argument.

Working through the algebra we get (see dixit_ea_1977 p. 288-289)

$$ \[ q_{i} = y (\frac{q}{p_{i}})^{1/(1-rho)} \approx \textrm{const} p_{i}^{-1/(1-rho)} \] $$

If $$n$$ is large then the effect on both $$q$$ and $$y$$ of changes in $$q_{i}$$ are neglible thus we have approximately:

$$ \[ \epsilon_{i} = -\frac{ \partial q_{i} }{ \partial p_{i} }= \frac{1}{1-\rho} \] $$

### Production

The producer of good $$i$$ solves:

$$ \[ \textrm{max}_{p_{i}} ( (p_{i} - c)q_{i} - f ) \] $$

This is standard monopolist's problem with solution:

$$ \[ p_{i} = \frac{c}{\rho} \] $$

Imposing symmetry ($$q_{i} = q$$, the zero-profit condition is:

$$ \[ f = q(\frac{c}{\rho} - c) \] $$

Substituting this into the FOC for utility maximization above we can solve for $$n$$ implicitly (to solve explicitly the form of $$U_{h}$$ must be specified).

## Applications and Extensions

Monopolistic competition is regularly used as a key part of other larger models. For example:

It has been heavily used in Endogenous Growth models.

## Bibliography and References

- **Dixit, Avinash and Stiglitz, Joseph E.** Monopolistic Competition and Optimum Product Diversity, American Economic Review, 1977.
- **Ethier, Wilfred** National and International Returns to Scale in the Modern Theory of International Trade, American Economic Review pp. 389-405, 1982-06.
- **Hart, Oliver** Monopolistic Competition in the Spirit of Chamberlin: A General Model, Review of Economic Studies, 1985.
- **Hart, Oliver** Monopolistic Competition in the Spirit of Chamberlin: Special Results, Economic Journal, 1985.
- **Tirole, Jean** The Theory of Industrial Organization, 1988.