Monopolistic Competition, As You Like it Paolo Bertoletti and …repec.dems.unimib.it/repec/pdf/mibwpaper454.pdf · 2021. 1. 25. · Monopolistic Competition, As You Like It Paolo

Department of Economics, Management and Statistics University of Milano – Bicocca

Piazza Ateneo Nuovo 1 – 2016 Milan, Italy http://dems.unimib.it/

DEMS WORKING PAPER SERIES

Monopolistic Competition, As You Like it

Paolo Bertoletti and Federico Etro

No. 454 – November 2020

http://dems.unimib.it/

Monopolistic Competition, As You Like It

Paolo Bertoletti and Federico Etro1

University of Milan-Bicocca and University of Florence

November 2020

Keywords: Monopolistic competition, Asymmetric preferences, Heterogeneousrms, Generalized separability, Variable markups

JEL Codes: D11, D43, L11

Abstract

We study monopolistic competition with asymmetric preferences over a va-riety of goods provided by heterogeneous rms, and show how to compute equi-libria (which approximate Cournot and Bertrand equilibria when market sharesare negligible) through the Morishima measures of substitution. Further resultsconcerning pricing and entry emerge under homotheticity and when demandsdepend on a common aggregator, as with GAS preferences. Under additivitywe can determine which goods are going to be provided under free entry, andthe selection e¤ects associated with changes in market size (i.e. opening upmarkets), consumersincome (i.e. demand shocks), aggregate productivity (i.e.supply shocks or technological growth) and preference parameters.

1We thank Lilia Cavallari, Avinash Dixit, Mordecai Kurz, James Heckman, FlorencioLopez de Silanes, Mario Maggi, Peter Neary and seminar participants at Oxford Univer-sity, EIEF (Rome), University of Pavia and the National University of Singapore. Corre-spondence. Paolo Bertoletti: Dept. of Economics, Management and Statistics, University ofMilan-Bicocca, Piazza dellAteneo Nuovo 1, 20126 Milan, Italy. Tel: +390264483152, email:[email protected]. Federico Etro: Florence School of Economics and Management,Via delle Pandette 32, Florence, 50127. Phone: 055-2759603. Email: [email protected].

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Which products and at which prices will be provided by markets whereheterogeneous rms sell di¤erentiated goods? This is a core question of mod-ern economic theories that depart from the perfectly competitive paradigm byadopting the monopolistic competition set up pioneered by Chamberlin (1933).Most of these theories rely on symmetric, Constant Elasticity of Substitution(CES) preferences based on Dixit and Stiglitz (1977: Section I), which deliv-ers constant markups, either across countries and among rms in trade models(Krugman, 1980; Melitz, 2003) or over time in macroeconomic applications withexible prices (see Woodford, 2003 and Barro and Sala-i-Martin, 2004). Fewapplications use more general but still symmetric preferences (Dixit and Stiglitz,1977: Section II; Bertoletti and Etro, 2016), even when considering variable pro-ductivity across rms (as in Melitz and Ottaviano, 2008, Parenti et al., 2017,Arkolakis et al., 2019) and over time (as in Kimball, 1995, or Bilbiie et al., 2012).In an attempt to capture the features of monopolistic competition in the spiritof Chamberlin,2 we study a large industry with heterogeneous rms supplyinggenuinely di¤erentiated commodities, and develop methodologies to characterizemonopolistic competition in such a setting.3 This allows for markups variableacross markets and goods of di¤erent quality, possibly depending on aggregatevariables, and for some progress concerning the way markets select not only howmany but also which goods are going to be provided.4

Consider demand systems derived from asymmetric preferences over a va-riety of di¤erent commodities that can be represented by well-behaved utilityfunctions. Each commodity is produced with idiosyncratic marginal and xedcosts. Our basic question is simply which strategies are adopted by rms in sucha market. The starting point is the analysis of Cournot and Bertrand equilib-ria in which rms choose either their quantities or their prices taking as giventhe strategies of competitors and demand systems. We generalize the familiarmonopoly pricing conditions by expressing the equilibrium markup of rms interms of their market shares and of the substitutability of their own productswith those sold by competitors. Substitutability is measured by (the averageof) the Morishima Elasticities of Substitution, as rediscovered and formalizedby Blackorby and Russell (1981).5

Competition among a large number of rms with negligible market shares(Spence, 1976; Dixit and Stiglitz, 1977, 1993) corresponds to the concept of mo-

2Chamberlin (1933) dened monopolistic competition with reference to factors a¤ectingthe shape of the demand curve, and certainly did not intend to limit his analysis to the case ofsymmetric goods. He saw no discontinuity between its own market theory and the theory ofmonopoly as familiarly conceived, claiming inter alia that monopolistic competition embracesthe whole theory of monopoly. But it also looks beyond, and considers the interrelations,wherever they exist, between monopolists who are in some appreciable degree of competitionwith each other. (Chamberlin, 1937, p. 571-2).

3See also the seminal work of Spence (1976), who explicitly deals with the problem ofproduct selection, focusing on quasi-linear preferences.

4 Important empirical works on entry in function of the market size include Bresnahan andReiss (1987) and Campbell and Hopenhayn (2005). See also Cosman and Schi¤ (2019) on amarket closer to monopolistic competition such as restaurants in New York City.

5The Morishima Elasticity of Substitution was originally proposed by Morishima (1967).We also employ the related concept of the Morishima Elasticity of Complementarity.

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nopolistic competition. Here, we dene it as the market structure in which rmsperceive the demand elasticity as given by the average Morishima elasticity(which approximately coincides with the actual one when market shares are in-deed small enough). We show that in this case, under the regularity conditionthat the second derivatives of the utility functions are bounded, the equilibriumprices are approximately the same under both quantity and price competition(as in the CES case). Introducing free entry we can also ask which products areprovided by the market, and what kind of selection is associated with changesin market size (for instance due to opening up markets), expenditure (due to ademand shock), aggregate productivity (due to a supply shock or technologicalgrowth) as well as preference parameters. The answers are simple under (asym-metric) CES preferences, because the set of goods provided by the market isnot a¤ected by changes of aggregate productivity, and an increase of expendi-ture or market size delivers new goods but without a¤ecting the entry sequence(as implicit in endogenous growth models à la Romer, 1990).6 This is not thecase in general, but we will show that the irrelevance of (common) productivityshocks is preserved under homothetic preferences, the neutrality of expenditureunder directly additive preferences, and the neutrality of the market size underindirectly additive preferences.Since typical demand systems depend on simple aggregators of rm choices,

we study in further depth monopolistic competition for the Generalized Addi-tively Separable (GAS) preferences introduced by Pollak (1972) and Gorman(1970), which deliver demand systems depending on one aggregator. Analyzingmonopolistic competition with GAS preferences, intuition suggests that to takethe common aggregator as given should be approximately correct (i.e., protmaximizing) when market shares are negligible. We show that this is indeedthe case, in the sense that when shares are negligible the impact of the singlerm on the aggregator is negligible too, and the perceived demand elasticitiesare approximately equal to the average Morishima measures (which in turn areclose to the actual ones). In addition, the equilibrium strategies do not dependon whether prices or quantities are chosen by rms, implying that imperfectlycompetitive choices do actually convergeto those of monopolistic competition.This approach provides a simple way to solve for asymmetric equilibria, and canbe extended to other demand functions that depend on multiple aggregators.Free-entry equilibria can be naturally dened as those where rms make both

entry and pricing decisions anticipating the value of the aggregator and takingit as given. Under additivity of preferences we can actually show uniqueness ofthe free entry equilibrium in spite of asymmetries between goods. For the classof directly additive preferences we show that an increase in the market size fa-vors the entry of rms producing goods with a less elastic demand, which enjoythe largest unit protability. At the same time, these rms are the less favoredby an increase in (aggregate) productivity, while changes in expenditure areneutral on the entry sequence. For the class of indirectly additive preferences,equilibrium pricing is independent across rms and the price of each rm only

6For an extension of endogenous growth models to

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depends on its marginal cost, product substitutability and consumersexpendi-ture. Moreover, we show that an increase of either expenditure or productivitya¤ects proportionally more the rms that face the most elastic demands, whichmake the best of them in terms of their survival ability, while an increase ofmarket size has a proportional impact on all rms and is neutral on pricing aswell as on the entry sequence.This work is related to di¤erent literatures. We generalize the analysis of

imperfect competition with di¤erentiated products (usually studied under qua-silinear preferences: see Vives, 1999) by reframing it in terms of the Morishimaelasticities. After the seminal contribution of Spence (1976), only few papershave analyzed monopolistic competition with asymmetric preferences. The workof Dixit and Stiglitz (1977: Section III) only dealt with a specic example withintersectoral perfect substitutability. The earliest treatement we are aware ofis in a work of Pascoa (1997), mainly focused on an example with Stone-Gearypreferences and a continuum of goods. More recently, DAspremont and DosSantos Ferreira (2016, 2017) have provided a related analysis of asymmetricpreferences with an outside good adopting an alternative equilibrium concept(but their monopolistic competition limit is consistent with ours). The tradeliterature with heterogeneous rms, started by Melitz (2003) and Melitz andOttaviano (2008), has usually considered monopolistic competition with sym-metric preferences; only a few works have added asymmetries to model qualitydi¤erentials among goods (for instance Baldwin and Harrigan, 2012, Crozetet al., 2012, and Feenstra and Romalis, 2014), but retaining a CES structure.Heterogeneity in demand and costs is instead at the basis of the empirical litera-ture of industrial organization on the impact of market size on entry (Bresnahanand Reiss, 1987, Campbell and Hopenhayn, 2005). We build a bridge betweenthese distant literatures considering asymmetric preferences that generate dif-ferent markups among goods a¤ecting the determinants of market selection.7

Our companion paper (Bertoletti and Etro, 2018) analyzes in further detailmonopolistic competition with GAS preferences under heterogeneous rms, butretaining the symmetry of preferences.The work is organized as follows. Section 1 presents alternative equilibria

of imperfect competition for the same demand microfoundation. Section 2 and3 study monopolistic competition respectively when preferences are homotheticand when the demand system depends on an aggregator. Section 4 concludes.All proofs are in the Appendix.

1 The Model

We consider L identical consumers with preferences over a nite number n ofcommodities represented by well-behaved direct and indirect utility functions:

U = U (x) and V = V (s) ; (1)

7See also Mrázová and Neary (2019) on selection e¤ects with heterogeneous rms andHottman et al. (2016) for an empirical approach based on a nested-CES utility system.

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where x is the n-dimensional vector of quantities and s = p=E is the corre-sponding vector of prices normalized by expenditure E. We assume that theutility maximizing choices are unique, interior (x;p > 0) and characterized bythe rst-order conditions for utility maximization. Therefore, the inverse anddirect demand systems are delivered by Hotelling-Wolds and Roys identities:

si(x) =Ui (x)e� (x) , xi(s) = Vi (s)� (s) ; (2)

where e� (x) = nXj=1

Uj (x)xj , � (s) =nXj=1

Vj (s) sj (3)

and Ui and Vi denote marginal utilities, i = 1; ::; n. Here e� is the marginal utilityof income times the expenditure level, and it holds that j� (s)j = e� (x(s)),as can be veried by adding up the market shares bj = sjxj . As a simpleexample we will occasionally refer to the asymmetric CES preferences, that canbe represented by:

U =

nXj=1

eqjx1��j and V = nXj=1

qjs1�"j ; (4)

where qj = eq�j > 0 can be interpreted as an idiosyncratic quality index for goodj, and � = 1=" 2 [0; 1) is the parameter that governs substitutability amonggoods.Firm i produces good i at the marginal cost ci = eci=A > 0, where the

common parameter A > 0 represents aggregate productivity: the variable protsof rm i are then given by:

�i = (pi � ci)xiL: (5)

We begin by studying market equilibria in which rms correctly perceive thedemand system and choose their prot-maximizing strategies. In the traditionof industrial organization we have to consider two cases, with each rm choosingeither its production level (Cournot competition) or its price (Bertrand compe-tition). Throughout this work we assume that the rst-order condition for protmaximization characterizes rm behaviour. Of course, to behave well marketequilibria may also require that the demand system satises other regularityconditions (for a related discussion see Vives, 1999, Ch. 6). We assume thatthese equilibria are well dened (but see our results on existence and unique-ness under additivity in Section 3), and use them to study a generalized formof monopolistic competition and to discuss free entry conditions.

1.1 Cournot competition

Let us consider rms that choose their quantities on the basis of the inversedemand functions si(x) in (2). Correctly anticipating the quantities produced

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by competitors, each rm i chooses xi to equate its marginal revenue to itsmarginal cost ci. The relevant (per-consumer) marginal revenue of rm i isMRi = @ (pixi) =@xi, where pi(x) = si(x)E. It can be written as:

MRi(x) =[Ui(x) + Uii(x)xi] e� (x)� Ui(x)xi hUi(x) +Pnj=1 Uji(x)xjie� (x)2 E

= pi(x)

241� si(x)xi � nXj=1

�ij(x)sj(x)xj

35 ;where we have dened the (gross) Morishima Elasticity of Complementarity,henceforth MEC, between varieties i and j as follows:8

�ij(x) = �@ ln fsi(x)=sj(x)g

@ lnxi=Uji(x)xiUj(x)

� Uii(x)xiUi(x)

: (6)

Notice that this inverse measure of substitutability depends on preferences andnot on the specic utility function which is chosen to represent them. Sincesubstitutability can di¤er among goods, let us compute the weighted average ofthe MECs for good i with respect to all the other goods j, with weights basedon the expenditure shares bj(x) = sj(x)xj , namely:

�i(x) =nXj 6=i

�ij(x)bj(x)

1� bi(x): (7)

It is then immediate to verify that the marginal revenue above can be rewrittenas MRi = pi(1� bi)(1� �i), and that the Cournot equilibrium quantities satisfythe system:

pi(x) =ci

1� �Ci (x)for i = 1; 2; :::; n, (8)

where the left hand side comes from the inverse demand given in (2) and theright hand side depends on:

�Ci (x) = bi(x) + [1� bi(x)]�i(x): (9)

Here �Ci is an increasing function of the market share of rm i and of its averageMorishima elasticity �i (which we assume to be smaller than unity). Intuitively,a rms markup is higher when it supplies a good that is on average less sub-stitutable with the other goods (high �i), and its market share is larger (highbi). In the CES example (4) � corresponds to the common and constant MEC,and the market shares depend on the idiosyncratic quality and cost parametersqj and cj , but closed form equilibrium solutions can be obtained only in simplecases.

8See Blackorby and Russell (1981) on the corresponding net measure which applies tocompensated demands. The larger is �ij the smaller is the possibility of good j to substitutefor good i. Notice that �ii = 0 and that in general �ij 6= �ji for i 6= j.

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1.2 Bertrand competition

Consider now rms that choose their prices on the basis of the direct demandxi(s) in (2), while correctly anticipating the prices of the competitors. Theelasticity of the Marshallian direct demand of rm i can be computed as:

����@ lnxi (s)@ ln pi���� = � sixi (s)

Vii(s)� (s)� Vi(s)hVi(s) +

Pnj=1 Vji(s)sj

i� (s)

2 :

Let us consider the (gross) Morishima Elasticity of Substitution, or MES, be-tween goods i and j:9

"ij (s) = �@ ln fxi(s)=xj(s)g

@ ln si=siVji(s)

Vj(s)� siVii(s)

Vi(s); (10)

which again depends on preferences and not on their specic utility representa-tion, and compute the weighted average:

�"i (s) �nXj 6=i

"ij (s)bj (s)

1� bi (s)(11)

which is assumed larger than unity, and where, with a little abuse of notation,bj (s) = sjxj(s) is now the expenditure share of rm i as a function of normalizedprices. We can now rewrite demand elasticity j@ lnxi=@ ln pij as:

"Bi (s) = bi (s) + [1� bi (s)]�"i (s) ; (12)

to dene the Bertrand equilibrium through the following system:

pi ="Bi (s) ci"Bi (s)� 1

for i = 1; 2; ::; n: (13)

Firms set higher markups if their goods are on average less substitutable thanthose of competitors (low �"i) and their market shares larger (high bi). In theCES example (4) the parameter " is the constant and common MES and is thereciprocal of the common MEC.

1.3 Generalized monopolistic competition

The remainder of this work is dedicated to analyze large markets of monopolisticcompetition under asymmetric preferences. There are alternative ways to makesense of this concept but, in the spirit of Dixit and Stiglitzs (1993) reply to Yangand Heidra (1993), we interpret monopolistic competition as the result of havingrms that correctly perceive market shares as negligible. In fact, what Dixit and

9See Blackorby and Russell (1981) and Blackorby et al. (2007). The higher is "ij thegreater is the possibility of good j to substitute for good i. Notice that "ii = 0 and that ingeneral "ij 6= "ji for i 6= j.

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Stiglitz (1977) did in their symmetric setting amounts to neglect any term oforder 1=n in the demand elasticities, where n was a number of rms assumedsu¢ ciently large to make the omitted terms small. Similarly, in our setting,when there are many goods we expect consumers to spread their expenditure ifpreferences are well behaved and not too asymmetric, so that the market sharesshould be small for all goods.10 On this basis, our previous results suggest toapproximate the relevant demand elasticities with the corresponding averagesof the Morishima measures.Accordingly, we dene as monopolistically competitive an environment where

market shares are negligible, that is bi � 0 for any i = 1; ::; n, and where rms,correctly anticipating the value of actual demands, perceivethe relevant elas-ticities as given by the average Morishima elasticities. This approach actuallyleads to two approximations according to whether we refer either to quantityor to price competition. In the rst case we can approximate (8) by using thefollowing system of pricing rules:

pi(x) =ci

1� ��i(x)for i = 1; 2; ::; n: (14)

In the second case we can approximate (13) with the pricing rules:

pi =�"i (p=E) ci�"i (p=E)� 1

for i = 1; 2; ::; n: (15)

Before analyzing the close relation between these two approaches, which wewill show to be approximately equivalent when the market shares are negli-gible, we can learn something more by considering the relevant cross demandelasticities. They can be computed as:

@ ln pj (x)

@ lnxi=

Uji (x)xiUj (x)

�nXh=1

Uhi(x)xiUh(x)

bh(x)

= �ij (x)� ��i(x) + bi(x)��i(x); (16)

@ lnxj (s)

@ ln si= "ij (s)�

����@ lnxi (s)@ ln pi����

= "ij (s)� �"i (s)� bi (s) (1� �"i (s)) : (17)

When shares are indeed negligible the cross e¤ects should be perceived as negli-gible too whenever the di¤erences �ij���i and "ij��"i are small and the perceivedown elasticities are not very large. Apparently, this is the case that Dixit andStiglitz (1993) had in mind, and we expect it to apply to the typical monop-olistic competition equilibrium with positive markups. Notice that the formercondition is satised in any equilibrium of a symmetric environment. However,both conditions might be violated in our asymmetric setting: in similar cases

10Su¢ cient conditions on preferences to deliver this result are studied in Vives (1987).

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the perceived cross demand elasticities can be large, and associated to a largeown demand elasticity and therefore to small equilibrium markups. In otherwords, it can happen that goods are perceived as highly substitutable and thatmonopolistic competition pricing approximates marginal cost pricing as in aperfectly competitive setting.11 We will exemplify this possibility in the nextsection for the case of translog preferences, and in Appendix for the case ofrestricted AIDS preferences (Deaton and Muellbauer, 1980).In the CES case (4) the conditions (14) and (15) exactly characterize the

same monopolistic competition solution:

bpi = ci1� � =

"ci"� 1 ; (18)

and in such a case the cross e¤ects (16) and (17) actually vanish when marketshares become negligible. More generally, under the regularity condition thatthe second derivatives of the utility functions (1) are bounded (analogous con-ditions are used in Vives, 1987), we can prove that the prices generated by thesystems of pricing rules (14) and (15) approximately coincide when the marketshares are neglibile (see Appendix A):

Proposition 1. When the second derivatives of the utility functions arebounded and market shares are negligible each average MES is approximatelyequal to the reciprocal of the corresponding average MEC, i.e. �"i (p=E) ���i (x (p=E))

�1.

The systems (14) and (15) need to be solved to derive the prices and quanti-ties which arise in a monopolistic competition equilibrium (that ought to implynegligible market shares). Once we depart from symmetry this may still be aformidable task, but in next sections we will consider methodologies that allowone to obtain explicit solutions for some classes of asymmetric preferences.

1.4 Entry

Which set of goods will be provided in a monopolistic competition equilibrium,and how is the latter a¤ected by market fundamentals? In this section we intro-duce these questions by considering free entry equilibria when the production ofeach good requires a positive xed cost. It may be useful to remind the readerthat without xed costs a perfectly competitive market would provide all thesuitable goods by pricing them at marginal cost: the question of which goods areactually introduced becomes relevant under xed costs, imperfect competitionand asymmetries between goods (with symmetry it simplies to the question ofwhich number of goods should be provided, already explored elsewhere). The

11Notice that, in general, the value of these cross demand elasticities need not be negligible ina strategic setting. In fact, if they were null there would be no reason for strategic interactionand we could think of those producers as isolated monopolists.

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analysis is important for both industrial organization applications and generalequilibrium applications.12

Let us assume that preferences are dened over a large but nite set of Ndi¤erent commodities, and that each good i 2 can be produced by a singlerm only after paying a xed entry cost Fi > 0. In the spirit of Chamberlin(1933), one can think of rms entering the market as long as they can price abovethe average cost.13 Namely, in a monopolistic competition equilibrium with freeentry there are n � N active rms which all get non-negative prots: the otherN � n rms would not obtain a positive prot by entering the market. Theprices of the goods produced by the former rms are set at their monopolisticcompetition levels, say bp, and the prices of the goods of the latter rms shouldbe set above their choke levels (if any), or equivalently at 1. The variableprots of an active rm i = 1; :::; n can be written as �i =

pi�cipi

biEL. By usingequilibrium pricing condition (15) and dening �i � �@ lnV=@ ln si as the priceelasticity of utility of commodity i, with average � � 1n

Pnj=1 �j , we can express

equilibrium variable prots as:

b�i = �i(bs)EL�(bs)�"i(bs)n (19)

(a corresponding formula can be obtained from the dual representation of pref-erences through the average MEC). Since EL=n are common to all rms, thisimplies that active rms with a lower average MES �"i and a higher ratio �i=�have higher variable prots because they can set higher markups and conquerlarger market shares (these elasticities determine the intensive and extensiveprot margins). In a free entry equilibrium only rms covering xed costs withtheir variable prots can be active.14

12For these applications, one can also add to our basic setting a good representing theoutside economy. This is particularly relevant for trade applications with a competitive sectorand for macroeconomic applications with labor supply. Pricing within the monopolisticallycompetitive sector carries on unchanged after imposing independent pricing for the outsidegood and taking this into account in the computation of the Morishima elasticities.13One can also consider an entry process à la Melitz (2003) that exhausts expected prots:

given an ex ante probability distribution over parameters indexing the goods, rms would enterthe market until they expect prots to cover the entry cost. This would leave unchanged thecompetition stage whenever costs and market size attract a number of rms large enough tojustify the assumption of negligible market shares.14A social planner maximizing utility under a resource constraint EL =

Pnj=1(cjxjL+Fj)

would set a common markup on all goods: this implies that a market equilibrium tends toprovide too much of the goods with a low average MES. Without loss of generality, the optimalprices can be set at the marginal costs when the xed costs are directly paid out of individualexpenditure. Accordingly, the social planner chooses the set �� of goods to be provided tosolve:

max��

V

sii2�

=ci

E �Pi2� Fi=L

; sii=2�

=1!:

As long as unproduced goods become less costly (or either E or L or A increases) they canenter the set of optimally provided goods. However, there is no general reason why themarket should be expected to either provide the optimal set of goods or to introduce them inthe optimal order (see Spence, 1976, on some special cases).

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At the present level of generality, we cannot exclude a multiplicity of marketequilibria. However, in the next sections we will make some progress (and proveexistence and uniqueness) under further assumptions on the preferences. Here,let us reconsider the CES example (4) as a benchmark, and let b� � be a setof goods provided in a free-entry equilibrium at prices (18).15 We can directlycompute prots (19) for a given market size as:

b�i = qiec1�"i EL"P

j2b� qjec1�"j ; (20)which is independent from aggregate productivity A (an increase of productivityreduces prices and unit costs while increasing proportionally demand so thatprots, and thus b�, remain unchanged), and linear with respect to the totalmarket size EL (for a given set of rms). Thus, the condition of a non-negativeprot for good i,

qiec1�"iFi

�"P

j2b� qjec1�"jEL

; (21)

uniquely denes an order among rms based on the value of the left-hand sideof (21): it is natural to think of it as establishing the sequence of market in-troduction. Thus, as we will prove formally in Section 3, the asymmetric CESpreferences generate a free entry equilibrium such that the identity of the goodsintroduced is independent from aggregate productivity A and it is determinedby the total market size EL, while the sequence of introduction is una¤ectedfrom either expenditure E or market size L. We will see that some of thespecial properties of the CES example extend to more general classes of pref-erences. In particular, the irrelevance of productivity shocks will be preservedunder homothetic preferences, the neutrality of expenditure under directly addi-tive preferences, and the neutrality of the market size under indirectly additivepreferences.

2 Monopolistic competition with homotheticity

Monopolistic competition with symmetric homothetic preferences has been stud-ied by Benassy (1996) and others.16 Here we are concerned with the moregeneral case of asymmetric homothetic preferences, because they are crucialfor representative agent models and provide an interesting application of ourproposed equilibria. Let us normalize the indirect utility function to be:

V =E

P (p); (22)

15The set b� is actually unique under additive (encompassing CES) preferences, as we willprove in Section 3.16See Feenstra (2003) on translog preferences, and Feenstra (2018) for their generalization to

the case of the so-called quadratic mean of order r(QMOR) preferences with heterogeneousrms.

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where P (p) is homogeneous of degree 1 and represents a fully-edged price

index. For instance P =hPn

j=1 qjp1�"j

i 11�"

in the CES case (4). The Roys

identity delivers direct demands xi = Pi (s) =P (s) and market shares bi =siPi (s) =P (s), which are homogeneous respectively of degree �1 and 0. Thisallows us to compute the MES:

"ij(s) =siPji(s)

Pj(s)� siPii(s)

Pi(s);

which is homogeneous of degree 0, being the di¤erence of two functions thatare both homogeneous of that degree. Therefore also the average MES �"i(s)is homogeneous of degree zero, which implies immediately that pricing is inde-pendent from the expenditure level (for a given set of rms).17 Similar resultscan be derived starting from the direct utility (which can be written as a con-sumption index) and using the inverse demand system and the average MEC tostudy quantity competition.

2.1 Examples

We now consider equilibrium pricing for two specications of homothetic pref-erences.

Translog preferences As a rst example, let us consider the homothetictranslog preferences (Christensen et al., 1975) represented by the following priceindex:

P (s) = exp

24ln�0 +Xi

�i ln si +1

2

Xi

Xj

�ij ln si ln sj

35 ; (23)where we assume without loss of generality �ij = �ji, and we need

Pi �i = 1

andP

j �ij = 0 to satisfy the linear homogeneity of P (a symmetric version ofthese preferences is used by Feenstra, 2003). The direct demand for good i is:

xi(s) =�i +

Pj �ij ln sj

si;

which delivers the market share bi = �i +P

j �ij ln sj . Maximization of protsprovides the Bertrand equilibrium conditions:

pi = ci

�1 +

bi�i

�; (24)

where the positiveness of �i � ��ii is necessary to ensure "Bi = 1 + �i=bi > 1.17When preferences are homothetic and symmetric, this also implies that Morishima elas-

ticities and markups in a symmetric equilibrium can be at most a function of the number ofgoods. While this result has been used elsewhere (for instance in Bilbiie et al., 2012), we arenot aware of a formal proof (we are thankful to Mordecai Kurz for pointing this out).

12

We can obtain the same result, as well as the monopolistic competitionequilibrium, by deriving the Morishima elasticity between goods i and j as:

"ij = 1 +�ibi+�jibj;

so that the average MES is:

�"i =nXj 6=i

"ijbj

1� bi= 1 +

�i(1� bi) bi

:

This allows one to get (24) from (13), and to obtain the monopolistic competitionprices:

pi = ci

�1 +

(1� bi) bi�i

�(25)

from (15). Notice that these prices of monopolistic competition are below theBertrand prices (24) for given market shares, and that, when market shares arenegligible (bi � 0), the average MES is large, goods are highly substitutable andprices must be close to the marginal costs (i.e., bpi � ci), approaching the caseof perfect competition.

Generalized linear preferences Let us now consider an example of ho-mothetic preferences due to Diewert (1971). Suppose that preferences can berepresented by the following direct utility/consumption index:

U =px0Apx =

Xi

Xj

pxiaij

pxj (26)

where, without loss of generality, we can take the matrix A to be symmetric.To satisfy the standard regularity conditions we assume that aij � 0 for any i; j(notice that parameters aii, i = 1; ::; n have no impact on the Hessian D2U).Here we obtain Ui =

Pj aij

pxj=pxi and e� = U , with market shares bi =

(pxiP

j aijpxj)=U . Since the MECs can be computed as:

�ij =1

2

�1 +

aijpxiP

h ajhpxh� aii

pxiP

h aihpxh

�;

we obtain the average MEC:

�i =1

2

�1� aii

pxiP

h aihpxh+bi � aiixi=U (x)

1� bi

�;

which allows us to determine the equilibrium conditions.18 Here ��i is strictlypositive for every good, implying positive markups, unless aij = 0 for any i 6= j(in which case commodities would be perfect substitutes).

18Notice that in the special, fully symmetric case with aij = a > 0 and xi = x for i; j =1; ::; n, one gets �ij = 1=2.

13

A simple case emerges when aii = 0 for any i, which implies �i = 1= [2 (1� bi)].This allows us to express Cournot prices as:

pi =2ci

1� 2bi; (27)

and monopolistic competition prices as:

pi =2 (1� bi) ci1� 2bi

: (28)

With these preferences markups do not vanish when market shares are negligible,but rather approach to twice the marginal cost: indeed we get bpi � 2ci whenbi � 0.

2.2 Entry

As discussed in Section 1.4, in general changes in market size, individual expen-diture and productivity a¤ect the set of active rms. However, under homo-theticity the equilibrium variable prot (19) can be computed as:

b�i = bpiPi (bp)EL" (bp)P (bp) ; (29)

where bpi = "(bp)ci"(bp)�1 for i 2 b� (with innite prices for i =2 b�), and one can verifythat is independent from the productivity component A, and linear with respectto EL for a given set b�. Thus changes in aggregate productivity do not a¤ectb�, while increases in market size and individual expenditures exert the sameexpansionary e¤ect on it. We summarize these facts as follows (see the proof inAppendix B):

Proposition 2. When preferences are homothetic, the identity of the goodsprovided in a free entry equilibrium does not depend on aggregate productivity,and is symmetrically a¤ected by expenditure and market size.

In practice, general purpose technological progress or aggregate shocks re-ducing marginal costs do not expand the set of goods provided by the market(unless they also a¤ects xed costs) and do not a¤ect markups. Instead, anincrease of the aggregate market size is likely to generate the provision of newgoods (and, possibly, the replacement of some), independently from whether itssource is higher spending or more consumers, as in endogenous growth mod-els à la Romer (1990), and this entry process might a¤ect markups as well asprotability.19 As we will see now, the impact of supply shocks, spending andpopulation on entry is radically di¤erent under di¤erent preferences.

19For an application to endogenous growth models à la Romer (1990) departing from CESproduction function and allowing for general technologies see Etro (2020).

14

3 Monopolistic competition with an aggregator

Although well-behaved demands can depend on prices in a general way, thedemand systems adopted in usual theoretical and empirical applications aresimpler and depend on price aggregators or quantity indices (as in the CEScase). For these cases we can study an alternative approach to monopolisticcompetition and to verify its consistency with our previous proposal. In thisSection we mainly explore preferences that generate direct demand functionsthat depend on the own price and one common aggregator of all prices or,equivalently, inverse demand functions that depend on the own quantity andone common aggregator of all quantities (but in Section 3.3 we discuss how thesame approach can be used with more aggregators). Pollak (1972) has denedGeneralized Additively Separable (GAS) preferences as those exhibiting demandfunctions that can be written as:

si = si(xi; �(x)) and xi = xi(si; �(s)); (30)

where @si=@xi, @xi=@pi < 0 and �(x) and �(s) are common functions (ag-gregators) of respectively quantities and prices. Notice that we can writesi = x

�1i (xi; �(x)), so that si (�) is the partial inverse of xi (�) with respect

to its rst argument, and one can also write �(x) = �(s (x)).Gorman (1970, 1987) has shown that GAS preferences encompass an exten-

sion of additive preferences that we call Gorman-Pollak preferences.20 Theycan be represented by the utility functions:

U =

nXj=1

uj (xj�)� � (�) and V =nXj=1

vj(sj�)� � (�) ; (31)

where � and � are implicitly dened by �0 (�) �Pn

j=1 u0j(xj�)xj and �

0 (�) �Pnj=1 v

0j(sj�)sj , under suitable restrictions on the good-specic sub-utilities uj

and vj and the common functions � and �.21

GAS preferences provide an ideal setting to study monopolistic competition,since we can naturally dene it as the environment in which each rm correctlyanticipates the value of the aggregators � and �, but takes (perceives) themas given while choosing its strategy to maximize prots:

�i = (siE � ci)xi (si; �)L = (si(xi; �)E � ci)xiL: (32)

It is important to stress that in this case the price and quantity equilibria ofmonopolistic competition do coincide. Since the perceivedinverse demand of

20GAS preferences also include the class of implicit CES preferences (Hanoch, 1975, andBlackorby and Russell, 1981). See Bertoletti and Etro (2018) for a discussion.21These preferences are homothetic when � (�) = ln � and � (�) = � ln �, a case which covers

the GAS demand systems investigated in Matsuyama and Ushchev (2017). They are directlyadditive when � (�) = �� (so that � = 1). Finally, they are indirectly additive when � (�) = �(so that � = 1). Obviously, the functional forms have to satisfy the usual regularity conditions(explored in Fally, 2018).

15

a commodity is just the inverse of the perceived direct demand, the corre-sponding elasticities

�i =@ ln si(xi; �)

@ lnxiand "i =

@ lnxi(si; �)

@ ln si(33)

are simply related by the exact condition "i = 1=�i, as in a monopoly. For

instance, with preferences (31) these elasticities are given by �i = �xiu00i (xi�)

u0i(xi�)

and "i = � siv00i (si�)

v0i(si�). This approach is entirely consistent with that adopted by

Dixit and Stiglitz (1977) who suggested to neglect the impact of an individualrm on marginal utility of income (the relevant aggregator in their setting),provided that this is su¢ ciently small to make this behaviour approximatelycorrect (i.e., prot maximizing). In fact, we can prove that, provided thatthe market shares are negligible, the impact of a single rm on the aggregatoris negligible too, and to take the aggregator as given is approximately protmaximizing for rms, since the perceived demand elasticity is approximatelyequal to the average Morishima measure. Formally, we have (see Appendix Cfor a proof):

Proposition 3. When preferences are of the GAS type and market sharesbecome negligible, the impact of a single rm on the aggregator vanishes and theperceived demand elasticity approximates the average Morishima elasticity.

Accordingly, a monopolistic competition equilibrium where rms take aggre-gators as given approximates the imperfect competition equilibria of Section 1,which in this sense do converge, when market shares become negligible. Theconditions for prot maximization of (32) taking as given either � or � dene asystem of pricing or production rules as:

pi = pi(ci; �) and xi = xi(ci; �): (34)

These rules, together with the budget constraintP

j pjxj = E and the assump-tion that rms correctly anticipate the actual demands, can be used to derivethe equilibrium value of the aggregators as a function of the marginal cost vec-tor c and of expenditure E, and therefore the equilibrium prices bpi(c; E) andquantities bxi(c; E).Moreover, under GAS preferences entry decisions can be studied by assum-

ing that in a free entry equilibrium each rm decides to enter taking as giventhe relevant aggregator, which is approximatively correct when market shareare negligible (due to Proposition 3). We can make substantial progress in theanalysis of monopolistic competiton with free entry under additive preferences,therefore below we focus on both directly additive and indirectly additive pref-erences. In particular, we will be able to characterize which goods are going tobe provided under free entry, and the selection e¤ects associated to changes inmarket size (i.e., opening up to free trade), expenditure (i.e., a demand shock),aggregate productivity (i.e., technological growth) and preference parameters.

16

3.1 Directly Additive preferences

Directely Additive (DA) preferences can be represented by a direct utility thatis additive as in:

U =nXj=1

uj(xj); (35)

where the sub-utility functions uj are potentially all di¤erent but always in-creasing and concave. The inverse demand system is given by

si(xi; �(x)) =u0i(xi)

�(x);

where � = e� = Pj xju0j and xi(si; �) = u0�1i (si�). These preferences clearlybelong to the GAS type, and were originally used by Dixit and Stiglitz (1977:Section II) in the symmetric version with uj(x) = u(x) for all j.22 We canexpress the variable prots of rm i as:

�i =

�u0i(xi)E

�� ci

�xiL: (36)

The prot-maximizing condition with respect to xi, taking � as given, canbe rearranged in the pricing conditions:

pi (xi) =ci

1� �i(xi); i = 1; 2; :::; n; (37)

where pi (xi) = u0i(xi)E=� and we dene the elasticity of the marginal subutil-ity �i(x) � �xu00i (x)=u0i(x), which corresponds to the elasticity of the inversedemand si(x; �) for given �. In this case �i is also the MEC �ij between good iand any other good j 6= i, therefore it coincides also with the average MEC �idiscussed in Section 1. In general, the markups can either increase or decreasein the consumption, depending on whether �i(x) is increasing or decreasing.Given a set of n active rms, a monopolistic competition equilibrium is a

vector (x; �) that satises the n + 1 equations u0i(xi)E = �ci= [1� �i(xi)] foreach i = 1; ::; n and � =

Pj xju

0j (xj). Asymmetries of preferences and costs

complicate its derivation because the quantity of each good depends on thequantities of all the other goods through the inverse demand system. However,under assumptions that guarantee that the prot-maximization problem is welldened for all rms and any value of ci�=E (essentially, assuming that all mar-ginal revenues u0i(xi)+u

00i (xi)xi are positive and decreasing), in Appendix D we

show that it exists a unique equilibrium. Formally, we have:

Proposition 4. Assume that preferences are DA and that r0i (x) > 0 >r00i (x), where ri (x) � xu0i (x), with limx!0 r0i (x) = 1 and limx!1 r0i (x) = 022For a further analysis of symmetric DA preferences see Zhelobodko et al. (2012), as well

as Bertoletti and Epifani (2014) and Arkolakis et al. (2019) for applications to trade, andLatezer et al. (2020) and Cavallari and Etro (2020) for applications to macroeconomics.

17

for i = 1; 2; :::; n. Then for that set of rms it exists a unique equilibrium ofmonopolistic competition.

We can easily study the comparative statics of this equilibrium. In partic-ular, an increase in the expenditure level E increases all quantities, and raisesthe markup of rm i if and only if �0i(x) > 0: this allows one to obtain di¤er-ent forms of pricing to market for di¤erent goods depending on their MECfunctions. A rise of the marginal cost ci decreases the quantity xi, inducingan incomplete pass-through on the price of rm i if and only if its MEC isincreasing. Also the indirect e¤ect on the markups of the other rms (takingplace through the change of the aggregator) depends on whether their MECs areincreasing or decreasing. Finally, when a new good is introduced in the marketthrough entry of an additional rm (for given E and L), the production of allother commodities decreases and therefore the markup of a rm decreases if andonly if its MEC is increasing.

3.1.1 Examples

We now present some examples of asymmetric DA preferences for which we canexplicitly solve for the monopolistic competition prices.

Power sub-utility A simple case of DA preferences is based on the sub-utilitypower function:

ui(xi) = ~qix1��ii ; (38)

where both the MEC �i 2 [0; 1) and the shift parameters ~qi > 0 can di¤er amonggoods. These preferences are a special instance of the direct addilogprefer-ences discussed by Houthakker (1960). They are neither CES nor homotheticunless the exponents are all identical.23 Under monopolistic competition, sincethe MECs are constant, markups are also constant and di¤erent across rms,and the equilibrium prices are:

bpi = ci1� �i

; (39)

which shows a full pass-through of changes in the marginal cost and indepen-dence from the pricing behavior of competitors and the expenditure level. Theequilibrium quantities bxi depend on the equilibrium value b�.Stone-Geary sub-utility Consider a simple version of the well-known Stone-Geary preferences (see Geary, 1950-51 and Stone, 1954) where:

ui(xi) = log(xi + �xi); (40)

23They have been often used in applications with perfect competition. Dhrymes and Kurz(1964) is an early example of these functional forms as production technologies. More recently,Fieler (2011) has used them as utility functions in a trade model.

18

with every �xi positive but small enough to insure a positive demand.24 Solvingfor the elasticity of the perceived inverse demand we get �i(x) = x=(x+ �xi) andthen the pricing condition:

pi (xi) = ci

�1 +

xi�xi

�:

The right-hand side is decreasing in �xi because a higher value of it increases de-mand elasticity. However, the equilibrium price of each rm cannot be derivedindependently from the behavior of competitors: the interdependence betweenrms created by demand conditions requires the following, fully-edged equilib-rium analysis. By the Hotelling-Wold identity we have:

si(xi; �) =1

(xi + �xi)�;

where � =P

j xj= (xj + �xj). Combining this with the pricing condition we can

compute the quantity xi =p�xiE=(ci�) � �xi and the (normalized) price rules

si =pci=(�xiE�) for rm i. Dening =

Pnj=1

p�xjcj and using the adding up

constraint we obtain the condition n=� �=pE� = 1, which can be solved for

the equilibrium value of the aggregator:

b� = �p2 + 4nE ��24E

:

Replacing b� in the price rule we nally get the closed-form solution for themonopolistic competition price of any rm i:

bpi = 2Eq

ci�xip

2 + 4nE �: (41)

In this example the price of each rm i increases less than proportionally in itsmarginal cost ci and decreases in the preference parameter �xi (which reducesthe relevant MEC). Moreover, an increase in expenditure raises the markupof each good less than proportionally. Note that each price is increasing in ,therefore an increase of the marginal cost cj of a competitor or an increase of thepreference parameter �xj (which reduces the associated marginal utility) induce,albeit indirectly, a small increase in the markup of rm i.

3.1.2 Entry

Let us now move to the question of which set of goods is actually introduced inthe market under monopolistic competition, and of its comparative statistics.

24Simonovska (2015) has recently used a symmetric version of these preferences to studymonopolistic competition among heterogeneous rms.

19

Firm i can survive in a monopolistic competition equilibrium only if �i � Fi or,using (36), if:

Eb� � cibxi + Fi=Lbxiu0i (bxi) :In Appendix E, extending the approach of Spence (1976) we (inversely) rankthe rms according to their survival coe¢ cient:

Si �Minxi

�cixi + Fi=L

xiu0i (xi)

�; (42)

and show that the free-entry equilibrium is unique.25 The survival coe¢ cient iscomputed at the survival quantity level that minimizes the ratio between aver-age cost and average revenue, and therefore it is only a function of exogenousparameters concerning technology (ci and Fi) and preferences (depending onthe subutility) as well as of market size (L), and captures the ability to survivein a free market, allowing us to identify the equilibrium active rms. Only rmswith the smallest coe¢ cients can survive in a market equilibrium. We can thenthink of the survival ranking as determining the actual sequence of market intro-duction, which allows us to study how entry is a¤ected by a change of marketsize, expenditure, aggregate productivity and also di¤erent preferences para-meters. The main result concerning the free-entry equilibrium of monopolisticcompetition is the following:

Proposition 5. When preferences are DA, the identity of the goods providedin the free entry equilibrium is uniquely determined, and an increase of themarket size or a fall of productivity favor rms with the largest values of theMEC (computed at survival quantity), while a change of expenditure is neutralon the survival ranking.

With the expression to favor we refer to improvements of the survivalranking (which apply to the marginal rm selected by the market to be active atthe equilibrium), while the referred numerical values of the MECs are computedat the quantities which dene the survival coe¢ cients (therefore again dependingonly on exogenous parameters).The rationale for these results is simple. An increase of the market size

(for given individual quantities) increases protability more for rms that havehigh markups due to a less elastic demand, which makes them relatively morelikely to enter. Analogously, and increase of aggregate productivity increasesproportionally more the prots of the rms that start with lower markups be-cause they face a more elastic demand, while a fall of productivity favours rmsfacing a less elastic demand. Finally, the neutrality of expenditure relies on thefact that this has a proportional impact on the revenue of all rms: therefore

25As mentioned in Section 1, in this paper we assume that the market equilibrium exists:generally speaking, the exact conditions required for existence are likely to be case dependent.

20

an expansion of demand attracts new rms in the market, but without alteringtheir survival coe¢ cients.26

Our examples provide an illustration of these results and of the impact ofexogenous preference parameters. The power sub-utility delivers the survivalcoe¢ cients:

Si =

�FiL

��i (1� �i)�i�2~qic

�i�1i �

�ii

and we remind that only the rms with the smallest value of this coe¢ cient canbe active in the market. A larger market size promotes entry (by decreasingall survival coe¢ cients), but favors the introduction of goods with a less elasticdemand, that is those with a high �i (since the impact of market size is stronger).In the Stone-Geary case the survival coe¢ cient is:

Si =

rFiL+p�xici

!2

as dened at exi =pxiFi=ciL, with �i (exi) = [1 +pxiciL=Fi]�1. Here, it is thecombination of preference and cost parameters xici=Fi that a¤ects the survivorindex and the relevant elasticity. In particular, a larger market size favors entryof goods with the lowest value of �xici=Fi, which are the goods produced withhigh xed costs and low marginal costs and facing a less elastic demand (due toa low �xi).

3.2 Indirectly Additive preferences

Preferences with an indirect utility that is additive (IA preferences) can berepresented by:

V =nXj=1

vj(sj); (43)

with sub-utilities vj decreasing and convex (Houthakker, 1960). Elsewhere wehave used the symmetric version of these preferences for the analysis of monop-olistic competition.27 The direct demand function is given by:

xi(si; � (s)) =�v0i(si)� (s)

;

26This provides a rationale for the results on the selection e¤ects of globalization derived byZhelobodko et al. (2012) and Bertoletti and Epifani (2014) in a setting with symmetric goodsand rms with heterogeneous marginal costs. They show that a market size increase has animpact on e¢ ciency which depends on whether the MEC is increasing or decreasing. In fact,in that setting the rms facing the largest MECs are either the most or the least e¢ cient rmsaccording to the MEC being either decreasing or increasing in consumption.27Also see Bertoletti et al. (2018) and Macedoni and Weinberger (2018) for applications to

trade and Boucekkine et al. (2017) and Anderson et al. (2018) for applications to macroeco-nomics.

21

with � = j�j = �Pn

j=1 sjv0j and si (xi; �) = v

0�1i (�xi�), which conrms that

these preferences belong to the GAS type. Here we can express the variableprots of rm i as:

�i =(pi � ci) [�v0i(pi=E)]L

�: (44)

For a given value of price aggregator � the elasticity of perceived demandxi(si; �) is given by "i(s) = �sv00i (s)=v0i(s), which is also the MES "ij betweengoods i and j (i 6= j) and thus coincides with the average MES "i of Section 1.The prot-maximizing price for each rm is then given by the solution to theprice condition:

pi ="i(pi=E)ci"i(pi=E)� 1

; i = 1; 2; :::; n (45)

(which requires "i > 1). Under weak conditions a solution to (45) exists, and itis unique under the assumption that 2"i > �siv000i =v00i for any i, which ensuresthat the second-order condition for prot maximization is satised.Remarkably, each condition (45) is now su¢ cient to determine the monop-

olistic competition price of each rm in function of its own marginal cost andconsumersexpenditure. This means that for the entire class of IA preferenceseach rm i can choose its price bpi(ci; E) independently from the behavior ofcompetitors, as well as from their cost conditions or from parameters concerningtheir goods (e.g., from their qualities). An increase of expenditure increasesthe price of a good, and changes in its marginal cost are undershifted on theprice if and only if the MES is increasing, which means that the demand isperceived as less elastic when expenditure is higher. These prices, for a set ofn rms, together with the corresponding value of the aggregator, provide anequilibrium of monopolistic competition. Formally, this is a vector (s; �) thatsatises the n+ 1 equations siE = "i(si)ci= ["i(si)� 1] for each i = 1; ::; n and� = �

Pj sjv

0j (vj). In Appendix E we prove the following:

Proposition 6. Assume that preferences are IA, that 2"i(s) > �sv000i (s)=v00i (s) and that a solution to the prot maximization problem exists for eachmember of a given set of rms i = 1; 2; :::; n and for any value of ci and E. Thenfor that set of rms it exists a unique equilibrium of monopolistic competition.

All the equilibrium quantities (and the other rm-level variables, such assales and prots) as well as welfare measures can then be recovered from thedirect demand functions. For this reason, this class of preferences can be nat-urally employed in applied industrial organization (and trade) models, whereasthe e¤ects of di¤erential (trade) costs, qualities and demand elasticities can beempirically assessed. A natural outcome of this environment is that goods ofhigher quality or lower substitutability generate higher revenues in a given mar-ket and therefore are more likely to be sold in more distant countries. SimilarAlchian-Allen e¤ects (shipping the good apples out) have been explored inrecent works by Baldwin and Harrigan (2011), Crozet et al. (2012), Feenstraand Romalis (2014) and others, but always retaining a CES structure (4) thatgenerates identical markups across goods. The IA class allows us to move easily

22

beyond the case of common markups, and to endogenize quality di¤erentiationacross rms and across destinations within rms, whose empirical relevance hasbeen pointed out in Manova and Zhang (2012).

3.2.1 Examples

As for the other classes of preferences considered above, we briey discuss ex-amples where one can compute explicitly the equilibrium prices.

Power sub-utility Consider a power sub-utility as:

vi(si) = qis1�"ii ; (46)

where heterogeneity derives from the shift parameter qi > 0 and the constantMES parameter "i > 1, implying that preferences are neither CES nor homo-thetic (unless all the exponents are identical).28 The pricing of rm i undermonopolistic competition is immediately derived as:

bpi = "ici"i � 1

; (47)

which implies again full pass-through of changes of the marginal cost. It isstraightforward to derive the equilibrium quantity (for a given set of activerms):

bxi = qih("i�1)Eci"i

i"iPn

j=1 qj

h("j�1)E"jcj

i"j�1 ;and consequently sales and prots. Clearly, qi is a shift parameter capturingthe quality of good i, that leaves unchanged the price but increases prot byincreasing sales. The relative productions, sales and prots of rms depend onthe relative quality of their goods, on their cost e¢ ciency and demand elasticity,and on the level of expenditure in simple ways that can be exploited in empiricalwork. We can also solve for equilibrium welfare (for a given set of rms) as:

bV = nXj=1

qj ("jcj)1�"j E"j�1

("j � 1)2�"j;

which allows one to analyze the welfare impact of any parameter change.

Translated power sub-utility Consider the following sub-utility:

vi(s) =(ai � s)1+i1 + i

; (48)

28This generalization of the CES case is a special instance of the indirect addilogprefer-ences of Houthakker (1960) and di¤ers from the one based on DA power sub-utilities (presentedin Section 3.1.1).

23

with quality (willingness-to-pay) parameter ai > 0 (such that vi(s) = 0 if s > ai)and i > 0. It delivers simple perceived demand functions, including the caseof a linear demand (for i = 1) and the limit cases of a perfectly rigid demand(i ! 0) and a perfectly elastic demand (i ! 1). These preferences havebeen recently applied by Bertoletti et al. (2018) and Macedoni and Weinberger(2018) to study the welfare impact of trade liberalization and quality regulationin multicountry models with heterogeneous rms. Since the MES for good i is"i (s) = is=(ai � s), the price of rm i can be computed as:

bpi = aiE + ici1 + i

; (49)

which shows incomplete pass-through of marginal cost changes (parametrizedby the rm-specic parameter i) and markups increasing in the intensity ofpreference for each good (as captured by willingness-to-pay ai) and in the ex-penditure level.

3.2.2 Entry

Contrary to what happens under alternative preferences, under IA the entryof a new rm does not change the prices of the pre-existing goods, but justreduces their production (through an increase of �). Firm i can survive in amonopolistic competition equilibrium only if �i � Fi, i.e., from (44) only if:

L

�� Fi(pi � ci) [�v0i(pi=E)]

:

This allows us to characterize the unique free entry equilibrium (that we assumeto exist) in terms of a survival ranking, which is simply given by an index ofrelative (total) protability:

eSi =Minpi

�Fi

v0i(pi=E) (ci � pi)

�: (50)

which is again a function of exogenous technological and preference parametersas well as income: in this case the survival coe¢ cient eSi is dened at the equilib-rium price of monopolistic competition bpi which is the relevant one to survive inthe market. Accordingly, the introduction of commodities follows this ranking,in the sense that goods with a lower value of eSi are always more protable andare introduced before others. In Appendix E we prove:

Proposition 7. When preferences are IA, the identity of the goods providedin the free entry equilibrium is uniquely determined, and an increase of theexpenditure level or a rise of productivity favors rms with the largest values ofthe MES (computed at equilibrium prices), while an increase of market size isneutral on the survival ranking.

The intuition is simple. An increase of expenditure exerts its impact (forgiven prices) through an expansion of the demand size, which is stronger for

24

rms facing a more elastic demand, which therefore manage to expand moretheir gross prots and are more likely to cover the entry costs. Analogously,an increase of aggregate productivity has an impact on unit protability (forgiven prices and given demand) that is larger for rms starting with lowermarkups and therefore facing a more elastic demand, which again induces alarger expansion of the gross prots. Finally, an increase of market size doesnot a¤ect prices or individual demand and has a proportional impact on all rmswithout altering the sequence of entry.29 These results should be contrasted withthose obtained under direct additivity (the neutrality of expenditure and marketsize are inverted and the qualitative impact of productivity is the opposite),stressing the crucial role of preferences in driving entry patterns.Again we can use our examples to illustrate the fact that a positive shock to

the expenditure level favors rms facing relatively more elastic demands. Thepower sub-utility delivers the survivor coe¢ cients:

eSi = Fi �"iE

�"i c"i�1iqi

("i � 1)�"i ;

and we remind that a lower coe¢ cient makes it easier to survive. In this case,a higher expenditure favors rms with the higher parameters "i, that is facinga more elastic demand. The translated power sub-utility provides:

eSi = Fi�E

i

�i � 1 + iaiE � ci

�1+i;

and higher expenditure favors goods with the largest values of the equilibriumMES, which can be computed here as "i (bsi) = aiE+iciaiE�ci and depends on all theexogenous preference parameters and income. Since this value is increasing inci and i, and decreasing with respect to ai, an income expansion favors entryof goods produced with a high marginal cost, characterized by a relatively lowwillingness to pay of consumers, and facing a highly elastic demand.

3.3 Multiple aggregators

We conclude by noting that in principle the approach to monopolistic compe-tition that we have explored when preferences are separable can be extendedto other cases in which each demand function depends on multiple aggregators,generalizing (30) as in:

si = si (xi; � (x)) and xi = xi (si;� (s)) ;

where � and � are now vectors. In fact, the associated procedure to determinethe equilibrium can be applied to any system of well dened perceived de-mands as soon as the alledged behavioral rules (based on the perceived demand29This provides a rationale for results derived by Bertoletti and Etro (2017) in a setting with

symmetric goods and rms with heterogeneous marginal costs. They show that an increaseof expenditure has an impact on e¢ ciency which depends on whether the MES is increasingor decreasing. In fact, the rms facing the largest MESs are in that setting either the least orthe most e¢ cient rms according to the MES being increasing or decreasing.

25

elasticities) are consistent with the demand system, so that rms can be seen ascorrectly anticipating the actual demands. However, to argue that taking all ag-gregators as given is approximately prot-maximizing for rms, one has to verifythat when the market shares become negligible the perceived demand elasticitydoes converge to the relevant Morishima measure (this basically requires thatthe impact of each rm on the aggregators vanishes). An example with twoaggregators is given by the implicitly additive preferences of Hanoch (1975), towhich we can extend the convergence result of Proposition 3.30 Other examplesinclude preferences for which the marginal utility of each good is separable, asin a generalization of the preferences of Melitz and Ottaviano (2008) analyzedin Appendix F, and a restricted version of the AIDS preferences (Deaton andMuellbauer, 1980) presented in Appendix G.

4 Conclusion

We have analyzed imperfect competition when consumers have asymmetric pref-erences over many di¤erentiated commodities and rms are heterogeneous incosts. Dening monopolistic competition as the market structure which ariseswhen market shares are negligible, we have been able to obtain a well-denedand workable characterization of monopolistic competition pricing. Moreover,we have presented a simple and consistent approach to the functioning of a mar-ket with monopolistic competition when demand functions depend on commonaggregators.Under general microfoundations, changes of aggregate productivity due to

technological shocks or growth, changes of expenditure due to demand shocksand changes of the market size across di¤erent market conditions do a¤ect notonly the markups of the goods provided under a free-entry equilibrium, but eventhe same identity of the goods that can be provided. For the case of additivepreferences, we have been able to determine the exact direction of these selec-tion e¤ects, which can inform future empirical research. Our approach can beusefully employed in industrial organization, trade and macroeconomic applica-tions. The research on the impact of market size on entry (Bresnahan and Reiss,1987, Campbell and Hopenhayn, 2005) usually accounts for heterogeneity on thedemand and cost size but without providing a theoretical microfoundation (forinstance of the di¤erential impact of income and numerosity of consumers) thatshould drive empirical research. Instead, most of the recent research on tradewith heterogeneous rms is actually based on symmetric preferences (Melitz,2003), which is hardly realistic, especially to analyze empirical issues. And themacroeconomic applications of monopolistic competition have mostly focusedon symmetric homothetic aggregators (Woodford, 2003). Building on more gen-eral microfoundations of imperfect competition should allow to examine markup

30The implicitly additive preferences include the implicit CES preferences (whose optimalityproperties have been explored in Bertoletti and Etro, 2018) that also belong to the GAS type,and others employed in applications by Kimball (1995), Feenstra and Romalis (2014) andMatsuyama (2019).

26

variability among goods both across markets and over time and its inuence onwelfare in industrial organization analysis as well as along the business cycleand across countries.

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AppendixA: Monopolistic competition with general preferences.Proof of Proposition 1. We want to prove that "i(s) � �i(x (s))�1 when

bi � 0 for all i if the second derivatives of the utility functions are bounded. Byusing the denition of �i(x) in (7) and of "i(s) in (10) one gets:

[1� bi(x)] �i(x) =nXj 6=i

Uji(x)xiUj(x)

bj(x)� [1� bi(x)]Uii(x)xiUi(x)

;

[1� bi(s)] "i(s) =nXj 6=i

Vji(s)siVj(x)

bj(s)� [1� bi(s)]Vii (s) siVi(s)

:

Thus, when the second (cross) derivatives of the utility functions are boundedabove and bi � 0 for all i:

�i(x) � �Uii(x)xiUi(x)

and "i(s) � �Vii (s) siVi(s)

:

30

Notice that, by using Roy and Hotelling-Wald identities, "i(s) � �i(x (s))�1 isthen equivalent to

�Vii(s)Uii (x(s)) � e� (x(s))2 ;where V and U have been normalized to be such that V (s) = U(x(s)). Considernow the scaleelasticities:

�j(x) =@ lnUj(�x)

@ ln�j�=1=

Xi

@ lnUj(x)

@ lnxi;

�i(s) =@ lnVi(�s)

@ ln�j�=1=

Xj

@ lnVi(s)

@ ln sj:

By di¤erentiating si � Ui (x (s)) =�(x (s)) one gets:

1 =Xj

�Uij(x (s))xj (s)

Ui (x (s))��1 + �j(x (s))

�bj(x (s))

��Vji(s)siVj(s)

� [1 + �i(s)] bi(s)�

which, in the case in which bi � 0 for all i, becomes:

1 �Xj

�Uij(x (s))xj (s)

Ui (x (s))

Vji(s)siVj(s)

�=

Pj Uij(x (s))Vji(s)

� (x(s))2 ;

under the assumption that �i(s) and �j(x) do not diverge (which is granted bythe assumption that the second derivative Uij and Vji are bounded). Accord-ingly we obtain Viisi=Vi � [Uiixi=Ui]�1 under the condition that:X

j 6=iUij(x (s))Vji(s) = �(s)

2Xj 6=i

Uij(x (s))

Uj(x (s))

Vji(s)

Vj(s)bj(s)

vanishes when market shares are negligible. A su¢ cient condition for the latteris again that the second (cross) derivative of the utility functions are boundedabove. �

B: Monopolistic competition with Homothetic preferences.Proof of Proposition 2. Due to the homogenenity of degree zero of both the

average MES " (p) and the market share bi = piPi (p)EL=P (p), changes inaggregate productivity A do not alter the monopolistic competition prots of agiven set of rms, given by (29). Since A a¤ects neither the equlibrium protsof the active rms, nor the prot that each other rm may get by entering themarket, it does not a¤ect the free-entry equilibrum set b�, which on the contrarypossibly depends on EL. �

C: Monopolistic competition with GAS preferences.Proof of Proposition 3. Assume that preferences belong to the GAS type.

Taking as given the relevant aggregator, in a monopolistic competition equi-librium rms compute the perceived (inverse) demand elasticity according to

31

�i = �@ ln si (xi; �) =@ lnxi in (33). We now show that, when market shares arenegligible, to take the aggregator � as given approximately coincides with usingthe average Morishima measures as the relevant demand elasticities, and is thusapproximately prot maximizing. Let us start by computing the MEC betweencommodities i and j (i 6= j):

�ij = �@ ln fsi (x) =sj (x)g

@ lnxi

=

�@ ln sj (xj ; � (x))

@ ln �� @ ln si (xi; � (x))

@ ln �

�@ ln � (x)

@ lnxi� @ ln si (xi; � (x))

@ lnxi:

This implies that the average MEC is:

�i =

24Xj 6=i

@ ln sj (xj ; � (x))

@ ln �

bj(x)

1� bi(x)� @ ln si (xi; � (x))

@ ln �

35 @ ln � (x)@ lnxi

�@ ln si (xi; � (x))@ lnxi

:

By di¤erentiating the identityP

j si (xj ; �)xj = 1 we can also compute:

@ ln � (x)

@ lnxi= �

@ ln si(xi;�(x))@ ln xi

+ 1Pnj=1

@ ln sj(xj ;�(x))@ ln xj

bi(x)

� (x)2 :

This shows that when bi � 0 then @ ln �=@ lnxi � 0: accordingly we have�i � �i � �ij when bi � 0.31 Notice that �i = �i = �ij even when shares are notnegligible if both preferences and the consumption bundle (and then the pricevector) are symmetric.32 Analogously, one can derive the MES and show thatwith GAS preferences small market shares imply "i � "i � "ij . Thus to takethe aggregator � as given while choosing the own price is approximately correctwhen market shares are indeed negligible. �

Self-dual addilog preferences As an example, the family of self-dual ad-dilogpreferences introduced by Houthakker (1965) and investigated by Pollak(1972) belongs to the Gorman-Pollak class (31). For this family of preferencesthe direct demand system is given by:

xi(s) = qis�"ii

� (s)"i+

��1�

;

31This formally assumes that not all the demand own elasticities and the quantity aggregatorare too small.32Since (h 6= i 6= j)

�ij � �ih =�@ ln sj (xj ; � (x))

@ ln �� @ ln sh (xi; � (x))

@ ln �

�@ ln � (x)

@ lnxi;

from (16) cross demand e¤ects are approximately zero when market shares are negligible,unless the own demand elasticities are indeed large.

32

where qi > 0 is a shift parameter reecting the quality of good i, "i > 1 governsthe perceived elasticity of demand and � (s) is implicitly dened by the conditionPn

i=1 qis1�"ii �

1��� �"i = 1. We assume � 2 (0; 1), and "i 6= "j for some i and j

(otherwise preferences are CES). Moreover, the inverse demand system is givenby:

si(x) = eqi x��ii� (x)

�i+e��1e�

;

where � (x) is implicitly dened by the conditionPn

i=1 eqix1��ii � 1�e�e� ��i = 1, with"i =

1�i> 0, qi = eq�ii and � = 1 � e�. Pollak (1972) showed that the underlying

preferences can be represented for � 6= 1=2 by:

U =

nXj=1

eqj (xj�)1��j1� �j

�e�� 2e��1e�2e� � 1 and V =

nXj=1

qj (sj�)1�"j

"j � 1+��

2��1�

2� � 1 :

In the special case with � = 1=2 preferences are homothetic and � and � take alogarithmic form with respect to the corresponding aggregators.Given the inverse and direct demand systems, when rms maximize prots

taking as given the aggregators, we immediately obtain the following pricesunder monopolistic competition:

bpi = ci1� �i

="ici"i � 1

;

where the idiosyncratic markups are constant as in our additive, power sub-utility examples (see Sections 3.1.1 and 3.2.1). In fact, we can also derive theequilibrium quantities as:

bxi = qi("i � 1)"iE"ic"ii "

"ii � (bs) ��1� +"i :

These results make this family the natural extension of the power additive prefer-ences. The availability of a homothetic version, with the associated well-denedprice and consumption indexes, and the exibility of the general specicationprovide interesting advantages for applications that depart from the CES para-digm.

D: Monopolistic competition with DA preferences.Proof of Proposition 4. It follows immediately from the rst-order condition

for prot maximization (the second-order condition is satised) rewritten as:

r0i(xi)E

�= ci

that for any ci�=E it exists a unique prot-maximizing quantity xi > 0 for anyrm i = 1; 2; :::; n, and that this quantity decreases with respect to �. Moreover,

33

it also follows from (37) that:

pixi =u0i (xi)xi

�E =

ciri(xi)

r0i(xi):

The right-hand side of latter expression implies that the prot-maximizing rev-enue of a monopolistically competitive rm is increasing with respect to its equi-librium quantity, rising form zero to innite. Since total revenue must be equalto the expenditure level E, and in such a case it holds that � =

Pj xju

0j (xj), it

follows that it exists a single value of � which characterizes a unique equilibriumfor a given set of rms (a level of expenditue E and a vector of marginal costc).�Proof of Proposition 5. Consider free entry under DA of preferences. By

using (36) we can write the condition of a non-negative prot as:

E

�� cixi + Fi=L

xiu0i (xi):

Following Spence (1976), let us rank rms increasingly according to their sur-vival coe¢ cient (SN � SN�1 � ::: � S1):

Si =Minxi

�cixi + Fi=L

xiu0i (xi)

�:

Notice that the ranking is independent from the values of expenditure and �.The entry equilibrium can be described as follows: for a given E=�, any activerm maximizes prot by setting its Lerner index equal to the MEC �i, inde-pendently from Fi=L. This determines the whole set of quantities for the activerms, and then the aggregator �. After any entry, the value of aggregator �must increase to reduce the expenditure in the incumbent commodities, makingroom for the entrant and survival more di¢ cult for all rms. In a free-entryequilibrium, all active rms get non-negative prots, and their quantities areconsistent with the value of the aggregator �. All the other rms do not expecta positive prot if entering the market, taking as given the equilibrium value ofthe aggregator �. It is then the case that Si � E=� for all rms i 2 b� = f1; ::; ng,and Si � E=� for i =2 b�. Notice that rm k cannot belong to the equilibriumset b� if rm j < k does not. Di¤erentiating Si and using the envelope theorem,we have:

@ lnSi@ lnL

= ��i (exi) , @ lnSi@ lnE

= 0 and@ lnSi@ lnA

= �i (exi)� 1;where �i is evaluated at the quantity exi = FiciL 1��i(exi)�i(exi) which denes Si. Accord-ingly, an increase of market size (which has a positive impact on all rms) or ina common component of the marginal cost (a reduction of productivity) altersthe survival ranking favoring rms producing varieties with the largest MECs(thus facing steeper perceived demand functions), while expenditure is neutral.�

34

E: Monopolistic competition with IA preferences.Proof