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Dixit Stiglitz (1977) - Monopolistic Competition And Optimum Product Diversity (view source)
Revision as of 18:51, 19 June 2010
, 18:51, 19 June 2010no edit summary
*This page is referenced in [[BPP Field Exam Papers]]
==Reference(s)==
*Dixit, A. and J. Stiglitz (1977), "Monopolistic competition and optimum product diversity", American Economic Review 67, 297-308. [http://www.edegan.com/pdfs/Dixit%20Stiglitz%20(1977)%20-%20Monopolistic%20competition%20and%20optimum%20product%20diversity.pdf pdf] [http://www.edegan.com/repository/Dixit%20Stiglitz%20(1977)%20-%20Class%20Slides.pdf (Class Slides)]
==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}} 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 [http://en.wikipedia.org/wiki/Envelope_theorem 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 [http://www.edegan.com/repository/PHDBA279C-DalBo-Lecture2.pdf 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.
