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Substitutability and Competition in the Dixit-Stiglitz Model by Winfried Koeniger* Omar Licandro** DOCUMENTO DE TRABAJO 2004-06

March 2004

* IZA. Corresponding author: Winfried Koeniger, koeniger@iza.org, http://www.iza.org, IZA, P.O. Box 7240, 53072 Bonn, Germany. ** European University Institute and FEDEA.

Substitutability and Competition in the Dixit-Stiglitz Model∗

Winfried Koeniger

Omar Licandro

IZA

European University Institute and FEDEA

January 2004

Abstract

The efects of competition on growth are analyzed in the recent literature by comparing economies with the same market structure but diferent degrees of substitutability. In this note, we show that in a general equilibrium model with monopolistic competition à la Dixit-Stiglitz the efect of substitutability on the allocation of resources is independent of the associated change in competition. Higher substitutability increases welfare, output and productivity because resources shift towards the most productive sectors. However, since markups are equal across sectors, changes in market power do not afect the relative price of consumption goods, implying that the induced changes in market power do not have any direct efect on equilibrium allocations.

∗We thank Vincenzo Denicolò and Alexandru Voicu for helpful comments. Corresponding author: Winfried Koeniger, koeniger@iza.org, http://www.iza.org, IZA, P.O. Box 7240, 53072 Bonn, Germany.

1 Introduction

The efects of product market competition on economic growth and welfare have been substantially analyzed in recent years. See, for example, Aghion et al. (2001). Since the study of exit and entry becomes quite intricate, especially in macroeconomic models, the literature has used changes in the degree of substitution between goods as an indicator of changes in competition, holding the number of market participants constant.1 This seems attractive because the elasticity of substitution is negatively related to the markup, a standard measure of competition.

In this note, we point out some limitations of this approach. A change in substitutability has other implications, apart from afecting the degree of competition in oligopolistic markets. In particular, the environment of the economy is modified after a change in the degree of substitution, what may shift the allocation of resources even under perfect competition. For this motive, we claim that a correct measurement of the efects on growth of changes in competition through market power and substitutability needs a careful analysis.

In this note, we solve a general equilibrium model with monopolistic competition à la Dixit-Stiglitz. We show that the decentralized equilibrium attains an optimal allocation, since relative prices are not distorted by imperfect competition, when the markup is equal across sectors; and the distortion of the price of goods in terms of labor has no efect as long as labor is inelastically supplied. Consequently, the benchmark result in this case is that changes in substitutability only afect the decentralized equilibrium through an optimal reallocation of resources. The associated change in market power and competition has no direct efect on the equilibrium. The nature of our results is related to Denicolò and Zanchettin (2003) in that changes in substitutability have an eficiency and a price efect. However, in Denicolò and Zanchettin (2003) these efects are the result of a switch from Cournot to Bertrand competition in a dynamic Neo-Schumpetarian growth model.

1Alternatively, Syverson (2003) examines the efects of substitutability on productivity through exit and entry.

2 The model

We build on a static Dixit-Stiglitz framework without taste for variety, where only labor is employed in the production of goods. There is a mass of consumer-workers of measure one, each of them endowed with one unit of time. Let us call the marginal productivity of labor in a particular sector and let be a continuous cumulative distribution function representing the distribution of sectors across in the support . The mean of is normalized to one.

The representative consumer derives utility from the consumption of a continuum of goods according to the following utility function:

\[\left(\int_ {\Gamma} c (\rho) ^ {\alpha} \mathrm{dF}\right) ^ {\frac {1}{\alpha}},\tag{1}\]

where is consumption of the good produced with productivity and The elasticity of substitution is . The industry with productivity employs the technology

\[c (\rho) = \rho l (\rho),\tag{2}\]

where is labor.

2.1 The social planner

A social planner maximizes (1) subject to the feasibility constraint

\[\int_ {\Gamma} \frac {c (\rho)}{\rho} \mathrm{dF} = 1.\tag{3}\]

Restriction (3) represents the allocation of the labor endowment across industries, after replacing from (2). From the first order condition and some algebra, we get

\[c (\rho) = \frac {\rho^ {\frac {1}{1 - \alpha}}}{\int_ {\Gamma} \rho^ {\frac {\alpha}{1 - \alpha}} \mathrm{dF}}.\tag{4}\]

Given that , sectorial consumption depends positively on relative eficiency.

Equations (1) and (4) imply

\[V _ {p} = \left(\int_ {\Gamma} \rho^ {\frac {\alpha}{1 - \alpha}} \mathrm{dF}\right) ^ {\frac {1 - \alpha}{\alpha}},\tag{5}\]

where is the optimal utility level for a given value of α. is a true index of output and, given that the labor endowment is equal to unity, it also measures average labor productivity. From (5), average productivity is a weighted average of sectorial productivity. In a symmetric economy, for all industries, implying that , for any [. As soon as industries have diferent productivity, the average labor productivity depends on the degree of substitution, parametrized by α. As shown in Proposition 1 below, an increase in the degree of substitution moves resources to the most eficient sectors, increasing output and productivity.

Proposition 1 is monotonically increasing in α, for

Proof. Let us define where , and rewrite (5) as

\[\left(V _ {p}\right) ^ {\eta} = \mathrm{E} \left(\rho^ {\eta}\right).\tag{6}\]

By diferentiating (6), we get the implicit derivative of w.r.t. η:

\[\frac {\mathrm{d} V _ {p}}{\mathrm{d} \eta} = \frac {\operatorname{E} \left[ \ln (\rho) \rho^ {\eta} \right] - \frac {1}{\eta} \operatorname{E} [ \rho^ {\eta} ] \ln (\operatorname{E} [ \rho^ {\eta} ])}{\eta (V _ {p}) ^ {\eta - 1}}.\]

The denominator of the r.h.s. is strictly positive. Let us introduce the following variable change . The numerator becomes

\[\frac {1}{\eta} \left\{\mathrm{E} [ \ln (z) z ] - \mathrm{E} [ z ] \ln (\mathrm{E} [ z ]) \right\} > 0\]

by the Jensen’s inequality, since the function ln is strictly convex for , which completes the proof.

2.2 The decentralized economy

The representative agent in the decentralized economy maximizes (1) subject to

\[\int_ {\Gamma} p (\rho) c (\rho) \mathrm{dF} = 1 + \Pi ,\tag{7}\]

where aggregate income equals the sum of aggregate profits, Π, plus the value of the labor endowment, which is normalized to unity. Since leisure is not in the utility function, the representative agent ofers inelastically one unit of the labor endowment. Additionally, we take the labor endowment as numeraire. Consequently, prices are measured in units of the labor endowment.

Monopolistic producers exploit their market power and sell goods at the monopolistic price

\[p (\rho) = (\alpha \rho) ^ {- 1},\tag{8}\]

where measures the markup. At equilibrium, it is easy to show that , since profits measured in labor units are equal to . Additionally, welfare at equilibrium is equal to the social optimum . In a general equilibrium economy with monopolistic competition à la Dixit-Stiglitz, all monopolies fix the same markup, which implies that relative prices of consumption goods are equal at equilibrium. The only relative price afected by market power is the price of the labor endowment. But the labor supply is infinitely inelastic, which implies that this distortion does not afect the allocation of labor. Notice that, under homothetic preferences, this result is invariant to changes in the distribution of wealth. Finally, this result would also apply to any imperfectly competitive economy where markups are equal across sectors, or change proportionally to changes in substitutability.

2.3 Conclusions

We have shown that higher substitutability increases welfare, output and productivity because resources shift towards the most productive sectors (see Proposition 1). Moreover, since markups are equal across sectors, changes in market power do not afect the relative price of consumption goods, implying that the decentralized equilibrium is optimal. Consequently, in the decentralized equilibrium changes in substitutability only afect output and productivity through their efect on the optimal allocation of resources. It implies that the induced changes in market power do not have any direct efect on equilibrium allocations. Thus, the degree of substitutability does not measure the efect of competition on output allocation. Of course, the assumption of completely inelastic labor supply and symmetric mark-ups is far from realistic. It is true that these assumptions are necessary to derive the extreme result that changes in substitutability between goods only afect the equilibrium through changes in the optimal allocation of resources. Although relaxing these assumptions would allow for additional efects on the equilibrium through changes in market power, we claim that our result is still important for the recent literature on competition and growth as long as the efect on growth of changes in substitutability cannot be completely assigned to changes in competition and, at least partially, is due to an optimal reallocation of resources.

References

  1. [1] Aghion, Philippe, Christopher Harris, Peter Howitt and John Vickers (2001): “Competition, Imitation and Growth with Step-by-Step Innovation,” Review of Economic Studies, 68, 467-92.
  2. [2] Denicolò, Vincenzo and Piercarlo Zanchettin (2003): “Competition and Growth in a Neo-Schumpeterian Model,” Università di Bologna, mimeo.
  3. [3] Dixit, Avinash K. and Joseph E. Stiglitz (1977): “Monopolistic Competition and Optimum Product Diversity,” American Economic Review, 67, 297-308.
  4. [4] Syverson, Chad (2003): “Product Substitutability and Productivity Dispersion,” NBER Working Paper No. 10049.

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