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:

• [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.