# Dixit Stiglitz (1977) - Monopolistic Competition And Optimum Product Diversity

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Has article title | Monopolistic Competition And Optimum Product Diversity |

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## Contents

## Reference(s)

- Dixit, A. and J. Stiglitz (1977), "Monopolistic competition and optimum product diversity", American Economic Review 67, 297-308. pdf (Class Slides)

@article{dixit1977monopolistic, title={Monopolistic competition and optimum product diversity}, author={Dixit, A.K. and Stiglitz, J.E.}, journal={The American Economic Review}, volume={67}, number={3}, pages={297--308}, year={1977}, publisher={JSTOR} }

## Abstract

The basic issue concerning production in welfare economics is whether a market solution will yield the socially optimum kinds and quantities of commodities. It is well known that problems can arise for three broad reasons: distributive justice; external effects; and scale economies. This paper is concerned with the last of these.

## Summary

A model of product differentiation driven by sheer taste for variety (not risk diversification or distance).

## The Model

The model involves consumer optimization, firm optimization of production scale and entry decisions, solving for number of firms and a comparison to a planner's solution.

The assumptions of the model are:

- [math]n\,[/math] (large) firms producing differentiated goods [math]x_1,\ldots,x_n\,[/math] which sell for [math]p_1,\ldots,p_n\,[/math].
- [math]x_0\,[/math] is a numeraire good (i.e. money)
- Free entry

Consumers have utility:

- [math]U = u(x_0,(\sum_{i=1}^{n} x^\rho )^{\frac{1}{\rho}})\,[/math]

and budgets:

- [math]B = x_0 + \sum_{i=1}^{n} p_i x_i\,[/math]

### Consumer optimization

Subbing in for [math]x_0\,[/math] from the budget contraint gives a constrained utility:

- [math]U = u(B - \sum_{i=1}^{n} p_i x_i,(\sum_{i=1}^{n} x^\rho )^{\frac{1}{\rho}})\,[/math]

The FOC wrt [math]x_i\,[/math] gives:

- [math]-p_i u_{x_0} + u_y \frac{1}{\rho} \left ( \sum_{i=1}^{n} x_i^\rho \right)^{\frac{1}{\rho} - 1} \cdot \rho x_i^{\rho - 1} = 0 \quad \forall i\,[/math]

Rearranging gives:

[math]x_i = \left(\frac{1}{p_i}\right)^{\frac{1}{1-\rho}} \cdot \frac{1}{q^{1-\rho} y} \,[/math]

where [math]y = (\sum_{i=1}^{n} x^\rho )^{\frac{1}{\rho}}\;[/math] and [math]q = \left(\sum_{i=1}^{n}p_i^{-1}{\frac{1-\rho}{\rho}}\right)^{-\frac{1-\rho}{\rho}}\,[/math]

### Market Behaviour

The firm's problem is that changing [math]p_i\,[/math] will not only affect demand for its own good, but for also affect all other firms (i.e. elements [math]q\,[/math] and [math]y\,[/math]).

However, if we can consider [math]q\,[/math] to be invariant in the firm's decisions, demand elasticity is easier to characterize. This assumption is realistic (i.e [math]\frac{dq}{dp_i} \approx 0\,[/math] and [math]\frac{dq}{dp_i} \approx 0\,[/math]) if the number of firms is very large. It is in fact checkable in a symmetric equilibrium that these partials go to zero as [math]n \to \infty\,[/math].

Then:

- [math]\frac{dx_i}{dp_i} = -\frac{1}{1-\rho} (\frac{q}{p_i})^{\frac{1}{1-\rho}} \frac{y}{p_i}\,[/math]

and elasticity of demand is:

- [math]\frac{dx_i p_i}{dp_i x_i} = -\frac{1}{1-\rho}\,[/math]

We can therefore consider demand to be of the form:

- [math]x_i = k p_i^{\frac{-1}{1-\rho}}\,[/math]

Firms therefore solve:

- [math]\max_{p_i} (p_i - c) k p_i^{\frac{-1}{1-\rho}} - f\,[/math]

which gives (using the FOC):

- [math]p_i^* = \frac{c}{\rho}\,[/math]

With a free entry condition profits are zero, so:

- [math](p_i - c) x - f = 0 \quad \therefore \; x^* = \frac{f\rho}{c(1-\rho)}\,[/math]

#### Solving for the number of firms

The FOC of the consumer was:

- [math]-p_i^* u_{x_0} + u_y \frac{1}{\rho} \left ( \sum_{i=1}^{n} x_i^\rho \right)^{\frac{1}{\rho} - 1} \cdot \rho x_i^{\rho - 1} = 0 \,[/math]

Using a symmetric equilibrium and plugging in [math]p_i^*\,[/math] we can rewrite this as:

- [math]n^* = \left( \frac{c u_{x_0}}{\rho u_y} \right)^{\rho}{1-\rho}\,[/math]

### A Planner's solution

A planner would use lump sum taxation (so as not to distort incentives) to cover fixed costs, and then marginal cost pricing to get efficient production.

Therefore a planner would solve:

- [math]\max_x u(B - nf - ncx, xn^{\frac{1}{\rho}}\,[/math]

and then optimize by the number of firms. However, by the envelope theorem, both optimizations can be performed simultaneously. Therefore FOCs wrt [math]x\,[/math] and [math]n\,[/math] give (respectively):

- [math]-ncu_{x_0} + n^{\frac{1}{\rho}}u_y = 0\,[/math]

- [math](-f -cx)u_{x_0} + \frac{1}{\rho} x n^{\frac{1}{\rho} - 1} u_y = 0\,[/math]

This gives planner solutions [math]x^p\,[/math] and [math]n^p\,[/math].

### Comparing solutions

To make the comparison we need to plug in for [math]u(\cdot)\,[/math] into the market solution and eliminate it (by substitution) from the planner's solution (see handout. Doing this gives

- [math]n^* = \frac{B}{\frac{f}{1-\rho} (1+\alpha)}\,[/math]

- [math]n^p = \frac{B}{(\frac{\rho}{1-\rho}(1+\alpha) + 1) f}\,[/math]

On comparison is turns out that the market solution **does not create too much entry**. The business stealing effect is mitigated by difficulties in appropriating consumer surplus.