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

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## 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:

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

Consumers have utility:

$U = u(x_0,(\sum_{i=1}^{n} x^\rho )^{\frac{1}{\rho}})\,$

and budgets:

$B = x_0 + \sum_{i=1}^{n} p_i x_i\,$

### Consumer optimization

Subbing in for $x_0\,$ from the budget contraint gives a constrained utility:

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

The FOC wrt $x_i\,$ gives:

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

Rearranging gives:

$x_i = \left(\frac{1}{p_i}\right)^{\frac{1}{1-\rho}} \cdot \frac{1}{q^{1-\rho} y} \,$

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

### Market Behaviour

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

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

Then:

$\frac{dx_i}{dp_i} = -\frac{1}{1-\rho} (\frac{q}{p_i})^{\frac{1}{1-\rho}} \frac{y}{p_i}\,$

and elasticity of demand is:

$\frac{dx_i p_i}{dp_i x_i} = -\frac{1}{1-\rho}\,$

We can therefore consider demand to be of the form:

$x_i = k p_i^{\frac{-1}{1-\rho}}\,$

Firms therefore solve:

$\max_{p_i} (p_i - c) k p_i^{\frac{-1}{1-\rho}} - f\,$

which gives (using the FOC):

$p_i^* = \frac{c}{\rho}\,$

With a free entry condition profits are zero, so:

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

#### Solving for the number of firms

The FOC of the consumer was:

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

Using a symmetric equilibrium and plugging in $p_i^*\,$ we can rewrite this as:

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

### 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:

$\max_x u(B - nf - ncx, xn^{\frac{1}{\rho}}\,$

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

$-ncu_{x_0} + n^{\frac{1}{\rho}}u_y = 0\,$
$(-f -cx)u_{x_0} + \frac{1}{\rho} x n^{\frac{1}{\rho} - 1} u_y = 0\,$

This gives planner solutions $x^p\,$ and $n^p\,$.

### Comparing solutions

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

$n^* = \frac{B}{\frac{f}{1-\rho} (1+\alpha)}\,$
$n^p = \frac{B}{(\frac{\rho}{1-\rho}(1+\alpha) + 1) f}\,$

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.