trade, competition, and efficiency

Journal of International Economics 87 (2012) 1–17

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Journal of International Economics

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Trade, competition, and efficiency

Kristian Behrens a,b,c,⁎, Yasusada Murata d,e

a Canada Research Chair, Department of Economics, Université du Québec à Montréal (UQAM), Case postale 8888, Succ. Centre-Ville, Montréal, QC, Canada H3C 3P8b CIRPÉE ESG-UQAM, Canadac CEPR, UKd Advanced Research Institute for the Sciences and Humanities (ARISH), Nihon University, 12-5, Goban-cho, Chiyoda-ku, Tokyo 102-8251, Japane Nihon University Population Research Institute (NUPRI), Japan

⁎ Corresponding author at: Canada Research Chair, Dversité du Québec à Montréal (UQAM), Case postale 888H3C 3P8, QC, Canada. Tel.: +1 514 987 3000x6513; fax

E-mail addresses: [email protected] (K. [email protected] (Y. Murata).

1 Dixit (2004, p.128) summarizes the gains from tradetion as follows: (i) availability of greater variety; (ii) beof scale; and (iii) greater degree of competition, drivingMore recently, the World Trade Organization (2008, pp.4sification: (i) gains from increased variety; (ii) gains fro(iii) gains from increased economies of scale.

2 For instance, the World Trade Organization (2008, pgains from intra-industry trade are concerned, most studof the variety, scale or pro-competitive (price) effects o

0022-1996/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.jinteco.2011.11.005

a b s t r a c t
a r t i c l e i n f o
Article history:Received 17 July 2010Received in revised form 24 October 2011Accepted 5 November 2011Available online 19 November 2011

JEL classification:D43D51F12

Keywords:Pro-competitive effectsCompetitive limitExcess entryTrade and efficiencyMonopolistic competition

We present a general equilibrium model of monopolistic competition featuring pro-competitive effects and acompetitive limit, and investigate the impact of trade on welfare and efficiency. Contrary to the constant elas-ticity case, in which all gains from trade are due to product diversity, our model allows for a welfare decom-position between gains from product diversity and gains from pro-competitive effects. We show that themarket outcome is not efficient because too many firms operate at an inefficiently small scale by chargingtoo high markups. We further illustrate that trade raises efficiency by narrowing the gap between the equi-librium utility and the optimal utility. As the population gets arbitrarily large in the integrated economy, theequilibrium utility converges to the optimal utility because of the competitive limit. We finally extend thevariable elasticity model to a multi-sector setting, and show that intersectoral distortions are eliminated inthe limit. The multi-sector model allows us to illustrate some new aspects arising from intersectoral andintrasectoral allocations, namely that trade leads to structural convergence, rather than sectoral specializa-tion, and that trade induces domestic exit in the nontraded sector.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Few trade theorists would disagree with the statement that prod-uct diversity, scale economies, and pro-competitive effects are centralto any discussion about gains from trade and efficiency under monop-olistic competition. Yet, it is fair to say that these questions have notbeen fully and jointly explored within a simple and solvable generalequilibrium model. This is largely due to the fact that the workhorsemodel, namely the constant elasticity of substitution (henceforth,CES) framework, displays two peculiar features. First, it does notallow for pro-competitive effects so that “there is no effect of trade

epartment of Economics, Uni-8, Succ. Centre-Ville, Montréal,: +1 514 987 8494.rens),

under monopolistic competi-tter exploitation of economiesprices closer to marginal costs.8–50) provides a similar clas-m increased competition; and

.48) states that “(a)s far as theies have focused on either onef trade opening”.

rights reserved.

on the scale of production, and the gains from trade come solelythrough increased product diversity” (Krugman, 1980, p.953).Second, the equilibrium in the CES model is usually constrained(second-best) optimal, i.e., the market provides the socially desirablenumber of varieties at an efficient scale (Dixit and Stiglitz, 1977).Consequently, trade is not efficiency enhancing in the CES model be-cause it does not correct the only existing market failure, pricingabove marginal cost.

In order to more fully explore gains from trade and efficiencyunder monopolistic competition, we must depart from the standardCES model. Doing so, however, has long been difficult since the vari-able elasticity of substitution (henceforth, VES) model in Krugman(1979) has “not proved tractable, and from Dixit and Norman(1980) and Krugman (1980) onwards, most writers have used theCES specification […] with its unsatisfactory implications that firmsize is fixed by tastes and technology, and all adjustments in industrysize (due to changes in trade policy, for example) come aboutthrough changes in the number of firms” (Neary, 2004, p.177).Building on the new VES specification by Behrens and Murata(2007), which satisfies the properties of the utility function inKrugman (1979), we present a simple general equilibrium model ofinternational trade featuring pro-competitive effects (i.e., profit-maximizing prices are decreasing in the mass of competing firms)and a competitive limit (i.e., profit-maximizing prices converge to

http://dx.doi.org/10.1016/j.jinteco.2011.11.005

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http://dx.doi.org/10.1016/j.jinteco.2011.11.005

http://www.sciencedirect.com/science/journal/00221996

4 See Behrens et al. (2009) for an example using such general equilibrium conditionswhen estimating a gravity equation.

5 See Epifani and Gancia (2011) for a recent multi-sector analysis on the class of util-ity functions to which two-stage budgeting is applicable.

6 This is reminiscent of what Feenstra and Weinstein (2010) call the “domestic exit

2 K. Behrens, Y. Murata / Journal of International Economics 87 (2012) 1–17

marginal costs when the mass of competing firms becomes arbitrarilylarge). Within this framework, where varieties, markups, and firm-level scale economies are endogenous, we investigate the impact oftrade on welfare and efficiency.

Our results can be summarized as follows. First, unlike in the stan-dard CES model, the market outcome is not efficient because toomany firms operate at an inefficiently small scale due to the negativeexternality each firm imposes on the others through markups. Notethat in a more general CES model presented in Benassy (1996),where market power and taste for variety are disentangled, the equi-librium mass of firms can also be larger (or smaller) than the optimalone. However, that inefficiency is not due to pro-competitive effectsas markups are constant because of the CES specification. According-ly, entry restriction (or promotion), if any, would not affect price-costmargins in Benassy (1996), whereas it would in our model sincemarkups depend on the mass of firms competing in the market.

Second, due to pro-competitive effects, autarky markups are nolonger the same across countries of different sizes. It is therefore notobvious that free trade leads to the equalization of price-cost marginsand to product and factor price equalization when country sizes dif-fer, labor markets are segmented, and products are differentiated.For instance, the seminal paper by Krugman (1979, p.476) states,after pointing out that “countries have identical tastes and technolo-gies”, that “(s)ymmetry will ensure that wage rates in the two coun-tries will be equal and that the price of any good produced in eithercountry will be the same”, even in the presence of country-size asym-metry. As we are not aware of any formal proof of this assertion, weprovide one in this paper.3

Third, contrary to the CES case, in which all gains from trade aredue to greater product diversity, our model allows for a welfare de-composition between gains from product diversity and gains frompro-competitive effects. It is worth emphasizing that such a welfaredecomposition is necessary for understanding each channel throughwhich gains from trade materialize. Recently, Feenstra andWeinstein (2010) compared the estimated gains from trade in a VESmodel based on Feenstra (2003) with those in the CES model byBroda and Weinstein (2006). Interestingly, although the overallgains are roughly the same between the two specifications, the un-derlying mechanism is quite different: the CES model ascribes allgains to new import varieties, whereas in the VES model increasedproduct diversity explains only two-thirds of the overall gains withthe remaining one-third being driven by pro-competitive effects. Ig-noring endogenous markups may thus overstate gains from new im-port varieties.

Fourth, we illustrate that trade raises efficiency by narrowing thegap between the equilibrium utility and the optimal utility. In ourmodel, product diversity is greater in equilibrium than in optimum.The associated equilibrium gains approach zero as the populationgets arbitrarily large in the integrated economy. By contrast, whilemarkups are too high, the associated equilibrium losses also vanishin the limit as prices converge to marginal costs. Hence, we obtainthe overall efficiency result.

Our approach is closely related to that of Feenstra (2003) in thatboth the mass of varieties and markups are made endogenous with-out relying on an additively separable numeraire good. However,there are several important differences. For instance, to solve forprices, Feenstra (2003) uses an approximation that applies to thecase where markups are sufficiently small. In our framework, exactprices are obtained. Furthermore, in Feenstra (2003) there is noclosed form solution for the direct utility function, although it ishomothetic. By contrast, we use a class of non-homothetic

3 The absence of a formal proof in Krugman (1979) is not an exception. On the con-trary, most monopolistic competition models of trade assume, rather than prove, thatproduct price equalization holds under free trade. See Helpman (1981) for anotherrepresentative example.

preferences that admits a workable direct utility function. Finally,our model is tractable enough for obtaining several analytical resultsand for incorporating labor market clearing and zero profit conditionsthat must be satisfied in general equilibrium.4 Hence, our approach iscomplementary to that of Feenstra (2003).

Finally, we extend our framework to include two monopolisticallycompetitive sectors. Extending Krugman (1979) to a multi-sector set-ting has been difficult since preferences over varieties in each sectorare non-homothetic. As is well known and as pointed out by Dixitand Stiglitz (1977, p.302) in the context of monopolistic competition,under such non-homothetic preferences, two-stage budgeting is notapplicable.5 Our specification, however, allows for closed form solu-tions for all equilibrium expressions, even in the two-sector case.This enables us to explore some new aspects arising from intersec-toral and intrasectoral allocations.

The main contribution of the two-sector analysis is threefold. First,despite intersectoral heterogeneity, we can establish the efficiency re-sult, namely that intersectoral distortions are eliminated as the popula-tion gets arbitrarily large in the integrated economy, while losses fromintrasectoral distortions vanish in the limit as in the single-sector case.Second, when both sectors are freely traded, we can establish structuralconvergence, i.e., countries having different sectoral compositionsunder autarky converge to the same industry structure under freetrade. This is in sharp contrast to the prediction of Ricardian andHeckscher–Ohlin models that trade causes sectoral specialization. Fur-thermore, unlike the new trade theory that emphasizes intra-industrytrade between similar countries, with similarity giving rise to moretrade, we show that countries become more similar due to trade, thussuggesting circular causation between similarity and intra-industrytrade. Finally, when either of the two sectors is nontraded, we canshow that trade induces domestic exit, or a variety loss in the nontradedsector.6 Unlike in the single-sector case with a traded good, such a vari-ety loss in the nontraded sector is not compensated by import varieties.Given that the observed share of nontraded goods is not negligible, thissuggests an important welfare implication: monopolistic competitionmodels that abstract from nontraded varieties may overestimate gainsfrom trade.7

The remainder of the paper is organized as follows. Section 2 de-velops a single-sector model, and Section 3 focuses on the autarkycase. Section 4 analyzes the trade equilibrium, decomposes thegains from trade, and shows that trade enhances efficiency.Section 5 extends the single-sector model to a multi-sector setting.Section 6 concludes.

2. Model

We first analyze the single-sector case. Consider a world with twocountries, labeled r and s. Variables associated with each country willbe subscripted accordingly. There is a mass Lr of workers/consumersin country r, and each worker supplies inelastically one unit oflabor. Thus, Lr also stands for the total amount of labor available incountry r. We assume that labor is internationally immobile andthat it is the only factor of production.

effect” in the traded sector, i.e., a decrease in the mass of varieties produced in eachcountry that is illustrated in Krugman (1979) and Feenstra (2004, Ch. 5). One notabledifference is that, in our extended model with a nontraded good, trade induces domes-tic exit in the nontraded sector.

7 Dotsey and Duarte (2008), for instance, state that consumption of nontraded goodsaccounts for about 40% of GDP in the United States.

9 It is well known that price and quantity competition yield the same outcome inmonopolistic competition models with a continuum of firms (Vives, 1999, p.168).We thus focus on prices as the only choice variable.10 As shown by Roberts and Sonnenschein (1977), the existence of (price) equilibriais usually problematic in monopolistic competition models, since firms' reaction func-tions may be badly behaved. Because our model relies on a continuum of firms, whichare individually negligible, we do not face this problem. In a similar spirit, Neary(2003) uses a general equilibriummodel of oligopolistic competition with a continuumof sectors, in which firms are ‘large’ in their own markets but ‘negligible’ in the wholeeconomy. This also allows the restoration of equilibrium since firms cannot directly in-

3K. Behrens, Y. Murata / Journal of International Economics 87 (2012) 1–17

2.1. Preferences

There is a single monopolistically competitive industry producinga horizontally differentiated consumption good with a continuum ofvarieties. Let Ωr (resp., Ωs) be the set of varieties produced in countryr (resp., s), of measure nr (resp., ns). Hence, N≡nr+ns stands for theendogenously determined mass of available varieties in the globaleconomy. International markets are assumed to be integrated, sothat each firm in each country sets a unique free-on-board price forconsumers in both countries. We assume that preferences are addi-tively separable over varieties as in Krugman (1979), and that thesub-utility functions are of the ‘constant absolute risk aversion’(CARA) type as in Behrens and Murata (2007). A representative con-sumer in country r solves the following utility maximization problem:

maxqrrðiÞ; qsrðjÞ

Ur≡∫Ωr1−e−αqrr ið Þh i

diþ ∫Ωs1−e−αqsr jð Þh i

dj

s:t:∫Ωrpr ið Þqrr ið Þdiþ ∫Ωs

ps jð Þqsr jð Þdj ¼ Er;ð1Þ

where α>0 is a utility parameter; Er stands for the expenditure; pr(i)denotes the price of variety i, produced in country r; and qsr(j) standsfor the per-capita consumption of variety j, produced in country s andsold in country r.

The demand functions for country-r consumers are given by (seeAppendix A.1):

qrr ið Þ ¼ − 1αln pr ið Þ þ Er

Pþ 1αHP

ð2Þ

qsr jð Þ ¼ − 1αln ps jð Þ þ Er

Pþ 1αHP; ð3Þ

where P≡∫Ωrpr(i)di+∫Ωs

ps(j)dj and H≡∫Ωrpr(i)ln pr(i)di+∫Ωs

ps(j)ln ps(j)dj are the sum of prices and a measure of price dispersion, re-spectively. Mirror expressions hold for country-s consumers. Becauseof the continuum assumption firms are negligible, and thus take P andH as given. The own-price derivatives of the demand functions arethen as follows:

∂qrr ið Þ∂pr ið Þ ¼ − 1

αpr ið Þ∂qsr jð Þ∂ps jð Þ ¼ − 1

αps jð Þ ; ð4Þ

which yields the variable demand elasticities εrr(i)=[αqrr(i)]−1 andεsr(j)=[αqsr(j)]−1.8 Mirror expressions hold again for country-sconsumers.

2.2. Technology

All firms have access to the same increasing returns to scale technol-ogy. To produceQ(i) units of any variety requires l(i)=cQ(i)+F units oflabor, where F is the fixed and c is the marginal labor requirement. Weassume that firms can costlessly differentiate their products and thatthere are no scope economies. Thus, there is a one-to-one correspon-dence between firms and varieties, so that the mass of varieties N alsostands for the mass of firms operating in the global economy. There isfree entry and exit in each country, which implies that nr and ns are en-dogenously determined by the zero profit conditions. Consequently, theexpenditure Er equals the wagewr. Under integratedmarkets, the profitof firm i∈Ωr is then as follows:

Πr ið Þ ¼ pr ið Þ−cwr½ �Qr ið Þ−Fwr ; ð5Þ

where Qr(i)≡Lrqrr(i)+Lsqrs(i) stands for its total output.

8 Our utility function Ur thus has the same properties as those in Krugman (1979)because it is additively separable across varieties, the sub-utility is increasing and con-cave in q, and the elasticity of demand ε decreases with q.

2.3. Equilibrium

Country-r (resp., country-s) firms maximize their profit (5) withrespect to pr(i) (resp., ps(j)), taking the vectors (nr,ns) and (wr,ws)of firm distribution and wages as given.9 This yields the followingfirst-order conditions:

∂Πr ið Þ∂pr ið Þ ¼ Qr ið Þ þ pr ið Þ−cwr½ � Lr

∂qrr ið Þ∂pr ið Þ þ Ls

∂qrs ið Þ∂pr ið Þ

� �¼ 0; ð6Þ

∂Πs jð Þ∂ps jð Þ ¼ Qs jð Þ þ ps jð Þ−cws½ � Ls

∂qss jð Þ∂ps jð Þ þ Lr

∂qsr jð Þ∂ps jð Þ

� �¼ 0: ð7Þ

We define a price equilibrium as a distribution of prices satisfying (6)and (7) for all i∈Ωr and j∈Ωs. Wewill discuss its existence, uniqueness,and some other properties in the following sections.10 An equilibrium is aprice equilibriumand vectors (nr,ns) and (wr,ws) offirmdistribution andwages such that national labormarkets clear, trade is balanced, andfirmsearn zero profits. Formally, an equilibrium is a price equilibrium satisfy-ing (6) and (7), and a solution to the following three conditions:

∫ΩrcQr ið Þ þ F½ �di ¼ Lr ; ð8Þ

∫ΩscQs jð Þ þ F½ �dj ¼ Ls; ð9Þ

Ls∫Ωrpr ið Þqrs ið Þdi ¼ Lr∫Ωs

ps jð Þqsr jð Þdj; ð10Þ

where all quantities are evaluated at a price equilibrium. It is readily ver-ified that firms earn zero profits when conditions (8)–(10) hold. Onemay set eitherwr orws as the numeraire. However, we need not choosea numeraire since the model is fully determined in real terms.11

3. Autarky

Assuming that the two countries can initially not trade with eachother, we first characterize the equilibrium and the optimum in theclosed economy, and show that there are too many firms operatingat an inefficiently small scale in equilibrium. Without loss of general-ity, we consider country r in what follows.

3.1. Equilibrium

Inserting (2) and (4) into (6), and letting qrs(i)=∂qrs(i)/∂pr(i)=0, one can show that the price equilibrium is symmetric and unique,and given by (see Appendix A.2 for the derivation):

par ¼ 1þ αcna

r

� �cwa

r ; ð11Þ

where an a-superscript henceforth denotes autarky values. At thesymmetric price equilibrium, the profit of each firm is given byΠr

a=Lrqrra (pra−cwr

a)−Fwra. Using the consumer's budget constraint

wra=nr

apraqrr

a , the above expression can be rewritten as Πra=pr

aqrra

fluence aggregates of the whole economy.11 The choice of the numeraire is immaterial in our monopolistic competition frame-work. This is an important departure from general equilibrium oligopoly models,where the choice of the numeraire is usually not neutral (Gabszewicz and Vial, 1972).

13 As−e−1b−e−1−αF/(cLr)b0, there is another possible real value ofW(−e−1−αF/(cLr))satisfying −1bW(−e−1−αF/(cLr))b0. However, it leads to nr

ob0 and must, therefore, beruled out.14 This can be seen from Dixit and Stiglitz (1977, p.301), when letting s=1 and θ=0in their Eqs. (20) and (21), since there is no homogeneous good in our setting. Further-more, as shown by Bilbiie et al. (2008, abstract), the market outcome is first-best effi-cient “if and only if symmetric, homothetic preferences are of the CES form”. This iswhy we must depart from the CES specification to study trade and efficiency. See


[Lr(1−cnraqrr

a )−Fnra]. Zero profits then imply that the quantities must

be such that

qarr ¼1c

1nar− F

Lr

� �; ð12Þ

which are positive because nraFbLr must hold from the resource con-

straint when nrafirms operate. Utility is then given by

U nar

� � ¼ nar 1−e

−αc

1nar− F

Lr

� " #: ð13Þ

Note that (12) and (13) hold whenever prices are symmetric andfirms earn zero profit.

Inserting qrra =wr

a/(nrapra) into the labor market clearing condition(8), we get:

nar ¼

LrF

1− cwar

par

� �: ð14Þ

The equilibriummass of firms can then be found by using (11) and(14), which yields12:

nar ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcFLr þ αFð Þ2

q−αF

2cF≡ν α; Lrð Þ > 0: ð15Þ

The function ν will be useful to make notation compact whenextending the model to a two-sector setting. The output per firm isgiven by Qr

a≡Lrqrra =(Lr/nra)(wr

a/pra)=Lr/(cnra+α), where we use(11). Plugging (15) into the last expression, we have

Qar ¼

2FLrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcFLr þ αFð Þ2

qþ αF

¼ Fαν α; Lrð Þ: ð16Þ

Finally, inserting (15) into (13), the equilibrium utility in autarkyis given by

U Lrð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcFLr þ αFð Þ2

q−αF

2cF1−e

− 2αFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcFLrþ αFð Þ2

pþαF

" #> 0; ð17Þ

which is a strictly increasing and strictly concave function of the pop-ulation size Lr for all admissible parameter values, i.e., α>0, c>0,F>0, and Lr>0. Alternatively, the equilibrium utility can beexpressed in terms of ν as U(Lr)=ν(α,Lr)[1−e− Fν(α, Lr)/Lr].

3.2. Optimum

We now analyze the first-best problem. The planner maximizesthe utility, as given by (1), subject to the technology and resourceconstraint (8). The first-order conditions of this problem with respectto qrr(i) show that the quantities must be symmetric. This, togetherwith (8), implies that:

qrr ¼Qr

Lr¼ 1

c1nr

− FLr

� �: ð18Þ

Hence, the planner maximizes

U nor

� � ¼ nor 1−e

−αc

1nor− F

Lr

� " #; ð19Þ

12 The other root is negative and must, therefore, be ruled out.

with respect to the mass of varieties nro, where an o-superscript

henceforth denotes the first-best values. Utility maximization re-quires the following first-order condition to hold:

cnor

α þ cnor¼ e

−αc

1nor− F

Lr

� ⇒− 1þ α

cnor

� �e− 1þ α

cnor

� ¼ −e−1−αF

cLr : ð20Þ

Using the Lambert W function, which is defined as the inverse ofthe function x↦xex (e.g., Corless et al., 1996; Hayes, 2005), the lattercan be rewritten as:

− 1þ αcno

r

� �¼ W −e−1−αF

cLr

� :

Solving this equation for nro yields a unique optimal mass of firms

nor ¼ − α

c 1þW−1 −e−1−αFcLr

� h i > 0; ð21Þ

where W−1 is the real branch of the Lambert W function satisfyingW(−e−1−αF/(cLr))≤−1 (Corless et al., 1996, pp.330–331; Hayes,2005).13 Note that W−1 is increasing in Lr, and that −∞bW−1b−1for 0bLrb∞.

Furthermore, letting Qro be the optimal output per firm given by nr

o

(cQro+F)=Lr, we can establish the following proposition.

Proposition 1. There are too many firms operating at an inefficientlysmall scale in equilibrium, i.e., nr

a>nro and Qr

abQro.

Proof. See Appendix B. □

Note that excess entry arises because of pro-competitive effects(∂pra/∂nrab0 by (11)). The negative externality each firm imposes onthe other firms gives rise to the ‘business-stealing effect’ (Mankiwand Whinston, 1986, p.49), i.e., the equilibrium output per firm de-clines as the number of firms grows (∂Qr

a/∂nra=∂(Lrqrra )/∂nrab0 by(12)).

Interestingly, this result contrasts starkly with the constant elas-ticity case, where the equilibrium mass of varieties is (second-best)optimal.14 Stated differently, the basic CES model does not accountfor the tendency that too many firms produce at an inefficientlysmall scale in autarky (the so-called ‘Eastman–Stykolt hypothesis’;Eastman and Stykolt, 1967), an argument often used to criticizeimport-substituting industrialization policies (Krugman et al., 2012,pp.292–293) or tariff barriers (Horstmann and Markusen, 1986) onefficiency grounds.

Combining (19) and (20) yields Uo(nro)=αnro/(α+cnro). Inserting

(21) into this expression, the optimal utility is given by

Uo Lrð Þ ¼ − α

cW−1 −e−1−αFcLr

� > 0; ð22Þ

which is a strictly increasing function of the population size Lr for alladmissible parameter values, i.e., α>0, c>0, F>0, and Lr>0.

Behrens and Murata (2009a) and Dhingra and Morrow (2011) for related issues inan urban context and in heterogeneous firm models, respectively.

15 It is readily verified that expressions (15) and (25) yield claim (i), which, togetherwith (11) and (23), implies (ii). Claim (ii) and expressions (14) and (24) then yield(iii). Finally, from (16) and (26), we obtain claim (iv).16 In the CES model by Lawrence and Spiller (1983, Proposition 7), (Proposition 7),trade leads to a redistribution of existing firms between the two countries while the to-tal mass of firms remains unchanged. This result is driven by changes in relative factorprices and, as pointed out by the authors, need not hold under variable markups.17 See Redding (2011) for a recent state-of-the-art survey on theories of heteroge-neous firms and trade.


4. Free trade

We now analyze the impacts of trade on welfare and efficiency in aworld with pro-competitive effects and a competitive limit. Section 4.1analyzes the equilibrium. Section 4.2 then shows the existence of gainsfrom trade and decomposes them into gains from product diversity andgains from pro-competitive effects. Section 4.3 finally illustrates thattrade narrows the gap between the equilibrium utility and the optimalutility by driving prices closer to marginal costs.

4.1. Equilibrium

We have shown that the profit-maximizing price under autarky isgiven by (11), where nr

a is evaluated at (15). Accordingly, markups inautarky are no longer the same across countries of different sizes. It istherefore not obvious that free trade leads to the equalization ofprice-cost margins and to product and factor price equalizationwhen country sizes differ, labor markets are segmented, and productsare differentiated. Assume that both countries can trade freely. Theprofits and the first-order conditions are still given by (5)–(7),respectively. Using these expressions, we establish the followingresult.

Proposition 2. Free trade leads to product and factor price equalization,i.e., pr(i)=ps(j)=p for all i∈Ωr and for all j∈Ωs, and wr=ws=w. Theprice equilibrium is then given by

p ¼ 1þ αcN

� cw; ð23Þ

where markups are equalized across varieties and countries.

Proof. See Appendix C. □

Note that (23) is an extension of the autarky case (11). Sinceprices and wages are equalized, qrr=qsr=qss=qrs=w/(Np) musthold by (2) and (3). Accordingly, all firms sell the same quantityQ=(Lr+Ls)q. Labor market clearing then implies that nr/ns=Lr/Ls,which, together with q=w/(Np), yields

nr ¼LrF

1− cwp

� �: ð24Þ

Plugging (23) into (24) and the analogous expression for countrys, we obtain two equations with two unknowns nr and ns. Solving forthe equilibrium masses of firms, we get

nr ¼Lr

Lr þ Ls

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcF Lr þ Lsð Þ þ αFð Þ2

q−αF

2cF

ns ¼Ls

Lr þ Ls

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcF Lr þ Lsð Þ þ αFð Þ2

q−αF

2cF:

Thus, the equilibriummass of firms in the global economy is givenby

N ¼ nr þ ns ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcF Lr þ Lsð Þ þ αFð Þ2

q−αF

2cF¼ ν α; Lr þ Lsð Þ; ð25Þ

which is an extension of the autarky expression (15). The output perfirm is then given by Q=(Lr+Ls)q=[(Lr+Ls)/N](w/p)=(Lr+Ls)/(cN+α), where we use (23). Plugging (25) into the last expression,we have

Q ¼ 2F Lr þ Lsð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcF Lr þ Lsð Þ þ αFð Þ2

qþ αF

¼ Fαν α; Lr þ Lsð Þ: ð26Þ

The impacts of trade on product diversity, markups, and outputper firm are the same as those in Krugman (1979) and Feenstra(2004), except that we obtain the closed form solution for each vari-able that is useful for many applications such as efficiency and multi-ple sectors. When compared with autarky, free trade leads in eachcountry to: (i) greater product diversity, an increase in the mass ofvarieties consumed, N>max{nra,nsa}; (ii) pro-competitive effects, adecrease in markups, p/(cw)bmin{pra/(cwr

a),psa/(cwsa)}; (iii) domestic

exit effects, a decrease in the mass of varieties produced, max{nr,ns}bnra; and (iv) better exploitation of scale economies, an increasein output per firm, Q>max{Qr

a,Qsa}.15

These results capture the relationship among product diversity,markups, and output per firm. First, the mass of varieties consumedincreases due to new import varieties by property (i). This intensifiescompetition and reduces markups by property (ii), thus driving somefirms out of each domestic market by property (iii).16 Labor marketclearing then makes sure that output per firm expands by property(iv), as labor is reallocated from the fixed requirements of closingfirms to the marginal requirements of surviving firms. This is an im-portant departure from the standard CES model, in which only chan-nel (i) operates. Recall that the equalization of markups inProposition 2 holds regardless of country size. In autarky, a smallercountry has a smaller mass of firms, which implies higher markups.Therefore, markups in a smaller country decrease more than thosein a larger country under free trade. Similarly, a smaller country expe-riences a greater increase in product diversity and output per firm.

Our approach is closely related to that of Feenstra (2003) in thatboth the mass of varieties and markups are made endogenous with-out relying on a quasi-linear specification. However, there are severalimportant differences. For instance, to solve for prices, Feenstra(2003) uses an approximation that applies to the case wheremarkups are sufficiently small. In our framework, exact prices areobtained. Furthermore, in Feenstra (2003) there is no closed form so-lution for the direct utility function, although it is homothetic. By con-trast, we use a class of non-homothetic preferences that admits aworkable direct utility function. Our model is also tractable enoughfor obtaining several analytical results and for incorporating labormarket clearing and zero profit conditions that must be satisfied ingeneral equilibrium. Note that although there is a growing literatureon firm heterogeneity in international trade (e.g., Melitz, 2003), theprice-cost margin for each firm is usually assumed to be constant inthose models because of the CES specification.17

One notable exception is Melitz and Ottaviano (2008) who illus-trate pro-competitive effects in a quasi-linear framework with firmheterogeneity. Both their and our models predict that the marketsize is the crucial determinant of markups. However, the extent ofthe market in question differs, and so does the mechanism that drivesthe markup reduction. In Melitz and Ottaviano (2008), the local mar-ket size matters for the (average) prices and markups, so that the sizeof the trading partner has no impact on the domestic utility level, aswell as the number of firms selling in the home country (see their ex-pressions (23)–(25)). In contrast, in our model, it is the globalmarketsize (Lr+Ls) that affects the prices and markups, as well as the massof varieties consumed, utility, and efficiency. This difference, whichgives rise to quite different policy implications regarding the choiceof trading partners, arises due to income effects. Indeed, we canshow that if the numeraire good were added to our model, the prices


and markups would be exogenously fixed by preferences (α) andtechnology (c), and thus independent of the global market size. Asdiscussed in detail by Melitz and Ottaviano (2008, Sections 3.5 and3.6), firm heterogeneity models, and more generally, monopolisticcompetition models, typically display either increased factor marketcompetition as in Melitz (2003) or increased product market compe-tition as in Melitz and Ottaviano (2008). We allow for both factor andproduct market competition by incorporating income effects as inMelitz (2003) and pro-competitive effects as in Melitz andOttaviano (2008). Our framework is thus useful especially when ana-lyzing how trading partners of different size affect domestic con-sumption diversity and markups, as well as welfare and efficiency.18

4.2. Welfare decomposition and gains from trade

Contrary to the CES case, in which all gains from trade are due toincreased import varieties, our model allows for both gains fromproduct diversity and gains from pro-competitive effects. In order tofocus on each channel through which gains from trade materialize,we now decompose welfare as in Krugman (1981).19 Since varietiesare symmetric under both free trade and autarky, the utility differ-ence is given by:

Ur−Uar ¼ N 1−e−

αwNp

� −na

r 1−e−αwa

rnar p

ar

!:

Adding and subtracting nrae−αw/(nr

ap), and rearranging the resultingterms, we obtain the following welfare decomposition:

Ur−Uar ¼ N 1−e−

αwNp

� −na

r 1−e−αw

nar p

� |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Product diversity

þnar e

−αwar

nar par −e

−αwnar p

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Pro−competitive effects

: ð27Þ

We now examine the role and the sign of each component in ex-pression (27) in more details, both from a theoretical and an empiri-cal point of view.

4.2.1. Product diversityThe first term in (27) captures the beneficial effects of increased

product diversity, given the wage-price ratio under free trade, w/p.As shown before, trade expands the mass of varieties consumed, de-spite the exit of some domestic producers. This raises utility, holdingw/p constant, as we have

Ur ¼ N 1−e−αwNp

� ;

∂Ur

∂N ¼ 1−e−αwNp 1þ αw

Np

� �> 0 ∀N:

To obtain the last inequality, let z≡αw/(Np) andh(z)≡1−e−z(1+z).Clearly, h(0)=0 and h′(z)>0 for all z>0,which shows that for any givenwage-price ratio w/p, utility increases with the mass of varietiesconsumed.

Despite its central role in new trade theory, little is known aboutthe empirical importance of gains from product diversity (Feenstra,

18 See Behrens et al. (2009) for an example in the context of Canada–US interregionaltrade, where the sizes of trading partners matter in general equilibrium. Note that suchan analysis is infeasible in the quasi-linear framework by Melitz and Ottaviano (2008),and as pointed out by Feenstra (2010, p.20), “its zero income elasticities suggest that inempirical application it is best suited for partial equilibrium analysis.”19 Krugman (1981) illustrates a similar decomposition in a model where two types ofsector-specific workers earn different real wages. However, because of the CES specifi-cation, there are no pro-competitive effects.

1995). Yet, there is an emerging literature on measuring gains fromvarieties. Using extremely disaggregated data and the method devel-oped by Feenstra (1994), Broda andWeinstein (2006) document thatthe number of varieties in US imports rose by 212% between 1972 and2001, and according to their estimates this maps into US welfaregains of about 2.6% of GDP. A more recent study by Feenstra andWeinstein (2010), however, points out that the CES specificationused in Broda and Weinstein (2006) ignores endogenous markupsand thus may overstate gains from import varieties. Indeed,Feenstra and Weinstein (2010) compare the estimated gains fromtrade in a VES model based on Feenstra (2003), with those in Brodaand Weinstein (2006). Interestingly, although the overall gains areroughly the same between the two specifications, the underlyingmechanism is quite different: the CES model ascribes all gains tonew import varieties, whereas in the VES model increased product di-versity explains only two-thirds of the overall gains with the remain-ing one-third being driven by pro-competitive effects.

4.2.2. Pro-competitive effectsThe second term in (27) captures the beneficial effects of intensi-

fied competition, given the mass of firms under autarky, nra. As shownbefore, trade reduces markups, which ceteris paribus raises utility. Itis worth emphasizing that the reduction in markups is a social gain:lower markups in a zero-profit equilibrium indicates a smallerwedge between firm's marginal and average costs, which implieslarger output per firm and greater economies of scale. Note that wr

a/pra=w/p would hold in the CES case, i.e., there would be no gains

from trade due to pro-competitive effects.It is well known from various industrial organization studies that

prices in many imperfectly competitive industries are increasingfunctions of producer concentration (see Schmalensee, 1989,pp.987–988, for a survey). In our symmetric equilibrium, theHerfindahl-index of concentration, defined as the sum of squaredmarket shares, reduces to H=N(1/N)2=1/N. Since the mass offirms is increasing in market size in our model, markups are lowerin larger markets (see Campbell and Hopenhayn, 2005, for empiricalevidence). Similarly, by increasing the number of competitors ineach market, import competition decreases concentration, whichmaps into lower consumer prices. Several case studies confirm this‘imports-as-market-discipline hypothesis’ (e.g., Levinsohn, 1993;Harrison, 1994; Tybout, 2003). More recently, Badinger (2007) findssolid evidence that the Single Market Programme of the EU has re-duced markups by 26% in aggregate manufacturing of 10 memberstates.

Let us summarize our results as follows:

Proposition 3. Free trade raises welfare both by increasing the mass ofvarieties consumed and by reducing markups.

Proof. To prove our claim, it is sufficient to examine the sign of thetwo components in (27). As shown above, they are both positive,which ensures gains from trade. □

4.3. Entry, markups, and efficiency

We now compare the equilibrium and optimal allocations in theglobal economy. Since in our model free trade amounts to increasingthe population size, the result on excess entry established inSection 3.2 continues to hold, even under free trade. Stated different-ly, there is a unique optimal mass of firms NobN satisfying the first-order condition (20) under free trade. Hence, there are too manyfirms operating at an inefficiently small scale and the market outcomeis not efficient. Furthermore, it can be verified that

limL→0

NNo ¼ 1 and lim

L→∞

NNo ¼

ffiffiffi2

p


hold regardless of parameter values.20 By continuity, for a sufficientlysmall population size, excess entry tends to be small, whereas it getslarger when the population gets arbitrarily large.

Turning to pro-competitive effects, expressions (23) and (25) yield

pcw

¼ 1þ 2αFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcFLþ αFð Þ2

q−αF

and∂∂L

pcw

� b0:

It is then readily verified that our model exhibits a competitive limit

limL→0

pcw

¼ ∞ and limL→∞

pcw

¼ 1;

so that prices converge to marginal costs as the population getsarbitrarily large. Despite the fact that there remains excess entry evenwhen the population gets arbitrarily large in the integrated economy,the associated gains from excessive varieties are shown to approachzero in the limit. As the gap between prices and marginal costs eventu-ally vanishes, we obtain the overall efficiency result as follows.21

Proposition 4. When the population gets arbitrarily large in the inte-grated economy, the equilibrium utility converges to the optimal utility,i.e.,

limL→∞

U Lð Þ ¼ limL→∞

Uo Lð Þ ¼ αc:

Proof. Applying l'Hospital's rule to (17), it is readily verified thatlimL→∞U(L)=α/c. Furthermore, by definition of the Lambert W func-tion, limL→∞W−1(−e−1−αF/(cL))=−1 holds. Hence, taking the limitof expression (22) yields limL→∞U

o(L)=α/c.22 □

Proposition 4 uses the optimum given technology and the re-source constraint that has been analyzed in Section 3.2. Alternatively,we can confirm the efficiency result by implementing the first-bestallocation via marginal cost pricing po=cwo. This requires lump-sum transfers as each firm earns negative profits −Fwo. When thereis a mass No of operating firms, a lump-sum tax (NoFwo)/L is leviedon each consumer's income. Accordingly, the income net of this taxis given by Eo=wo(1−NoF/L). The consumer's budget constraintqo=Eo/(Nopo), together with po=cwo, then immediately yieldsqo=(1/c)(1/No−F/L), which is the same as (18). Hence, the plannerfaces the same utility (19) to maximize, thus achieving the optimalmass of varieties given by (21).

This alternative way of implementing the first-best allocationallows for a welfare decomposition in terms of product diversityand pro-competitive effects as follows.

U−Uo ¼ N 1−e−αwNp

� −No 1−e−

αwo

Nopo

� �¼ N 1−e−

αwNp

� −No 1−e−

αwNop

� |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

excess entry

þNo e−αwo

Nopo−e−αwNop

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

too high markups

;

where we add and subtract No[1−e−αw/(Nop)], and rearrange theresulting terms. The first term is positive due to the excessive massof varieties, N>No, given the equilibrium wage–price ratio. By

20 When taking the limit of expressions involving the Lambert W function, we useMathematica, where W−1(·) can be computed by ProductLog[−1,·]. Note thatwe will use ProductLog[·] for the principal branch of W(·) that we will encounterin the subsequent analysis.21 This result is reminiscent of Mankiw and Whinston (1986, Proposition 3) who es-tablish conditions for the equilibrium utility in a partial equilibrium closed-economymodel to converge to the optimal utility when excess entry gets large.22 See Behrens and Murata (2006, Appendix E) for an alternative proof that does notuse the Lambert W function.

definition of the optimal utility, the second term must be negative,reflecting too high equilibrium markups, p/(cw)>po/(cwo)=1,given the optimal mass of varieties. As UbUo, the second term mustalways dominate the first term in any economy of finite size. Yet, bytaking the limit of each term, we have

limL→∞

N 1−e−αwNp

� −No 1−e−

αwNop

� h i¼ 0

limL→∞

No e−αwo

Nopo−e−αwNop

� �¼ 0;

which shows that both equilibrium gains from excessive varieties andequilibrium losses from too high markups vanish in the limit, therebyyielding the overall efficiency result.

The limit result established in Proposition 4 may be extended to afinite economy by investigating whether

maxU Lrð ÞUo Lrð Þ ;

U Lsð ÞUo Lsð Þ

� bU Lð ÞUo Lð Þb1;

where the last inequality comes from the definition of the optimum.Whether U/Uo monotonically increases in L is not a trivial questionsince both the equilibrium and the optimal utility increase in L, ascan be seen from (17) and (22). Fig. 1 shows that N/No is increasingin L, meaning that excess entry gets larger as the population in-creases.23 At the same time, Fig. 2 illustrates that [p/(cw)]/[po/(cwo)]=p/(cw) is decreasing in L, implying that the gap betweenthe equilibrium prices and marginal costs gets smaller as the popula-tion increases. The overall effect is depicted in Fig. 3. As expected, U/Uob1. The market outcome thus remains inefficient for finite popula-tion sizes. At the same time, however, Fig. 3 illustrates that U/Uo in-creases monotonically with L. In such a case, trade between largercountries yields higher efficiency than trade between smallercountries.

5. Two-sector model

We now extend our basic model to include two sectors, each ofwhich produces a differentiated good. Doing so allows us to shedlight on various issues arising from intersectoral and intrasectoral al-locations. In particular, we establish the following three results. First,when both sectors are freely traded, countries having different sec-toral compositions under autarky converge to the same industrystructure under free trade. This is in sharp contrast to the predictionof Ricardian and Heckscher–Ohlin models that trade causes sectoralspecialization. Furthermore, unlike the new trade theory, wheremore similar countries engage in more intra-industry trade, weshow that countries become more similar due to trade, thus suggest-ing circular causation between similarity and intra-industry trade.Second, despite intersectoral heterogeneity, we establish the efficien-cy result, namely that intersectoral distortions are eliminated as thepopulation gets arbitrarily large in the integrated economy, whilelosses from intrasectoral distortions vanish in the limit as in thesingle-sector case. Finally, when either of the two sectors is non-traded, trade induces domestic exit, or a variety loss in the nontradedsector. Since the variety loss in the nontraded sector is not compen-sated by import varieties, this result suggests an important welfareimplication: monopolistic competition models that abstract fromnontraded varieties may overestimate gains from trade.

23 The parameter values for Figs. 1–3 are as follows: α={0.1,1,5};c=0.5; and F=1.Other admissible parameter values yield qualitatively similar figures, thus suggestingthat the underlying property is robust.

0 10 20 30 40 501.0

1.1

1.2

1.3

1.4N

/No

smaller α

L

Fig. 1. N/No as a function of L.

0 10 20 30 40 501.0

1.5

2.0

2.5

[p/(

cw)]

/[po /

(cw

o )]

L

smaller α

Fig. 2. [p/(cw)]/[po/(cwo)] as a function of L.

0 10 20 30 40 500.6

0.7

0.8

0.9

1.0

U/U

o

L

smaller α

Fig. 3. U/Uo as a function of L.

24 Such heterogeneity in preference parameters is also useful for empirically explain-ing observed markup differences within traded sectors and between traded and non-traded sectors (e.g., Badinger, 2007).25 Note that W is different from W−1 that we have introduced in Section 3.2.


5.1. Autarky

Assume that there are two sectors, denoted by 1 and 2, each pro-ducing a horizontally differentiated consumption good with a contin-uum of varieties. Without loss of generality, we consider country r inthis subsection. Let q1r and q2r denote the distribution of demands forthe varieties of goods 1 and 2, respectively. We assume that prefer-ences are (weakly) separable across the two goods so that the utilitymaximization problem can be expressed as follows:

maxqr1 ;q2r

Ur≡U U1r q1rð Þ;U2r q2rð Þð Þ s:t: ∫Ω1rp1r ið Þq1r ið Þdi

þ ∫Ω2rp2r jð Þq2r jð Þdj ¼ Er ; ð28Þ

where U has standard properties. As in the single-sector case, U1r andU2r are given by

U1r≡∫Ω1r1−e−α1q1r ið Þh i

di and U2r≡∫Ω2r1−e−α2q2r jð Þh i

dj;

where we assume, following the tradition of the new trade theory(e.g., Helpman and Krugman, 1985, Chs. 6 and 9), that the two sec-tors differ in preference parameters that govern the price elastici-ties of demand.24 Since U1r and U2r are not homothetic, two-stagebudgeting is not applicable as pointed out by Dixit and Stiglitz(1977, p.302). However, as shown in Appendix A.3, the demandfunction for each variety in each sector can be expressed compact-ly as follows:

q1r ið Þ ¼ 1α1

lnp̃1rp1r ið Þ� �

q2r jð Þ ¼ 1α2

lnp̃2rp2r jð Þ� �

; ð29Þ

where p̃1r and p̃2r are: (i) common to all firms within a sector; (ii)taken as given by each firm because of the continuum assumption;yet (iii) endogenously determined in equilibrium (see AppendixA.3 for their expressions). Note that p̃1r and p̃2r can be interpretedas reservation prices since q1r(i)>0 if and only if p1r ið Þb p̃1r andq2r(j)>0 if and only if p2r jð Þb p̃2r . Given the demand functions, aswell as p̃1r and p̃2r , firms in each sector maximize profits

Π1r ið Þ ¼ p1r ið Þ−cwr½ � Lrα1


−Fwr ð30Þ

Π2r jð Þ ¼ p2r jð Þ−cwr½ � Lrα2


−Fwr: ð31Þ

5.1.1. EquilibriumAs in the single-sector case, we first analyze the price equilibri-

um. Let W denote the principal branch of the Lambert W func-tion.25 From the first-order conditions for profit-maximization,we obtain the prices, quantities, and operating profits under autar-ky as follows (see Appendix A.4 for the derivations and the prop-erties of W):

pa1r ¼cwa

r

Wa1r

qa1r ¼1α1

1−Wa1r

� �πa1r ¼

Lrcwar

α1

1Wa

1rþWa

1r−2� �

ð32Þ

pa2r ¼cwa

r

Wa2r

qa2r ¼1α2

1−Wa2r

� �πa2r ¼

Lrcwar

α2

1Wa

2rþWa

2r−2� �

;

ð33Þ

where

Wa1r≡W e

cwar

p̃a1r

� �and Wa

2r≡W ecwa

r

p̃a2r

� �: ð34Þ

Since p̃a1r and p̃a2r are common to all firms within the same sector,the price equilibrium and the associated quantity and operatingprofits in each sector are symmetric. We thus drop the firm indices iand j.

The equilibrium is characterized by the price equilibrium, zeroprofits in each sector, labor market clearing, and the relationshipbetween p̃a1r and p̃a2r that results from the consumer's optimization


problem. First, plugging the profit-maximizing prices and quantitiesinto (30) and (31), the zero profit conditions are given by26

Π1r ¼Lrcw

ar

α1

1Wa

1rþWa

1r−2� �

−Fwar ¼ 0

Π2r ¼Lrcw

ar

α2

1Wa

2rþWa

2r−2� �

−Fwar ¼ 0;

which can be uniquely solved for W1ra and W2r

a as follows:27

Wa1r ¼ 1−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4α1cFLr þ α1Fð Þ2

q−α1F

2cLr∈ 0;1ð Þ ð35Þ

Wa2r ¼ 1−


q−α2F

2cLr∈ 0;1ð Þ: ð36Þ

Expressions (35) and (36), together with (32) and (33), implythat the profit-maximizing prices, quantities, and operating profitsin one sector do not depend on the characteristics of the other sector.This is due to the fact that preferences are (weakly) separable acrossgoods: given consumers' budget allocation across sectors, firms careonly about what happens in their own sector when maximizingprofits.

Using (32), (33), (35), and (36), and noting that W1ra and W2r

a areincreasing in Lr, we know that, in autarky, equilibrium prices andquantities, p1ra , p2ra , q1ra , and q2r

a , are smaller in larger countries. Equilib-rium outputs, Q1r

a and Q2ra , defined as Q1r

a ≡Lrq1ra and Q2r

a ≡Lrq2ra are,

however, larger in larger counties. We can also establish the followingresult.

Proposition 5. Suppose that α1>α2. Then, the firms in sector 1 chargehigher markups and produce smaller output in each country, i.e.,p1ra >p2r

a and Q1ra bQ2r

a . Furthermore, larger countries have a smallerprice ratio, p1r

a /p2ra , and a greater output ratio, Q1ra /Q2r

a .

Proof. Noting that W1ra bW2r

a when α1>α2, and that Lr(1−W1ra )/α1

and Lr(1−W2ra )/α2 are decreasing in α1 and α2, respectively, we ob-

tain the first claim. Furthermore, noting that the price ratio is givenby p1r

a /p2ra =W2ra /W1r

a , it is readily verified that

∂ln pa1r=pa2r

� �∂lnLr

¼ffiffiffiffiffiffiffiffiffiα2F

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4cLr þ α2F

p −ffiffiffiffiffiffiffiffiffiα1F


p :

Sinceffiffiffiffiffiffiffiffiffiα2F

p=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4cLr þ α2F

pand

ffiffiffiffiffiffiffiffiffiα1F

p=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4cLr þ α1F

pare increasing in

α2 and α1, respectively, we obtain ∂ ln(p1ra /p2ra )/∂ lnLrb0 when α1>α2.Finally, differentiating the output ratio Q1r

a /Q2ra =(α2/α1)[(1−W1r

a )/(1−W2r

a )], we have

∂ln Qa1r=Q

a2r

� �∂lnLr

¼ 12

ffiffiffiffiffiffiffiffiffiα1F


p −ffiffiffiffiffiffiffiffiffiα2F


p !

;

which, using the same argument as for the price ratio, yields the sec-ond claim. □

26 The same solution technique can be applied to the single-sector model (see Appen-dix A.5 for details).27 Note that the other root is greater than one, which is infeasible as it implies pricesbelow marginal costs.

We now turn to the equilibrium sectoral labor allocation L1ra and

L2ra and the equilibriummasses of firms n1ra and n2r

a . This requires mak-ing use of the remaining two equilibrium conditions – labor marketclearing and the relationship between p̃a1r and p̃a2r – which are givenas follows (see Appendix A.3 for the latter derivation):

Lr ¼ La1r þ La2r ; where La‘r≡na‘r

Lrcα‘

1−Wa‘r

� �þ F� �

ð37Þ

p̃a1r

p̃a2r

¼ α1

α2

∂Ur=∂U1r

∂Ur=∂U2r: ð38Þ

The left-hand side of (38) is obtained by plugging W1ra and W2r

a in(35) and (36) into the left-hand side of (34). Indeed, we can solveuniquely for p̃a

1r and p̃a2r from (34) as follows:

p̃a1r ¼ 1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4α1cFLr þ α1Fð Þ2

qþ α1F

2cLr

24

35e F

Lrν α1 ;Lrð Þcwa

r > pa1r

p̃a2r ¼ 1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4α2cFLr þ α2Fð Þ2

qþ α2F

2cLr

24

35e F

Lrν α2 ;Lrð Þcwa

r > pa2r ;

so that the reservation prices are higher than the market prices inequilibrium.

The right-hand side of (38), in turn, can be obtained as follows. Wehave so far made no explicit assumption regarding Ur. We assume inwhat follows that Ur≡β1lnU1r+β2lnU2r, with β1,β2>0 and β1+β2=1.In that case, we have

∂Ur=∂U1r

∂Ur=∂U2r¼ β1U2r

β2U1r¼

β1na2r 1− pa2r

p̃a2r

� β2n

a1r 1− pa1r

p̃a1r

� ¼β1n

a2r 1− cwa

r

Wa2r p̃

a2r

� �β2n

a1r 1− cwa

r

Wa1r p̃

a1r

� � : ð39Þ

Plugging (39) into (38) and using the expressions forW1ra ,W2r

a , p̃a1r

and p̃a2r that we have obtained above, the two equilibrium conditions

(37) and (38) depend on n1ra and n2r

a only. Letting

κa1r≡e

FLrν α1 ;Lrð Þ−1 and κa

2r≡eFLrν α2 ;Lrð Þ−1; ð40Þ

the two equations can be solved for the equilibrium masses of firms:

na1r ¼

La1rLr

ν α1; Lrð Þ and na2r ¼

La2rLr

ν α2; Lrð Þ; ð41Þ

where

La1rLr

¼β1κa1rν α1; Lrð Þ

β1κa1rν α1; Lrð Þ þ β2

κa2rν α2; Lrð Þ

andLa2rLr

¼β2κa2rν α2; Lrð Þ



ð42Þ

are sectoral labor shares. We can show that, in equilibrium, less laboris allocated to the sector with the lower elasticity of demand (largerα) or the smaller weight on utility (smaller β).


Proposition 6. Suppose that α1>α2 and β1=β2, or that α1=α2 andβ1bβ2. Then, less labor is allocated to sector 1 than to sector 2 in equilib-rium, i.e., L1r

a bL2ra .

Proof. Taking thedifference between L1ra and L2r

a , andusing (40),wehave

sign La1r−La2r� � ¼ sign β1

ν α1; Lrð Þe

FLrν α1 ;Lrð Þ−1

−β2ν α2; Lrð Þ

eFLrν α2 ;Lrð Þ−1

� :

Since ν(⋅,Lr)/[eFν(⋅, Lr)/Lr−1] is decreasing in ν and ν is increasingin its first argument, α1>α2 and β1=β2 imply L1r

a bL2ra . When

α1=α2, we have ν(α1,Lr)/[eFν(α1, Lr)/Lr−1]=ν(α2,Lr)/[eFν(α2, Lr)/Lr−1].Then, the relationship reduces to sign{L1ra −L2r

a }=sign{β1−β2}, thuscompleting the proof. □

Since the quantities within each sector are symmetric in equilibri-um and given by q1r

a =Fν(α1,Lr)/(α1Lr) and q2ra =Fν(α2,Lr)/(α2Lr), the

equilibrium U1ra and U2r

a are written as

Ua1r ¼ na

1r 1−e−α1qa1r

� ¼ La1r

Lrν α1; Lrð Þ 1−e−

FLrν α1 ;Lrð Þh i

ð43Þ

Ua2r ¼ na

2r 1−e−α2qa2r

� ¼ La2r

Lrν α2; Lrð Þ 1−e−

FLrν α2 ;Lrð Þh i

; ð44Þ

which finally yields the equilibrium utility

Uar ¼ β1ln

La1rLr

ν α1; Lrð Þ 1−e−FLrν α1;Lrð Þh i�

þ β2lnLa2rLr

ν α2; Lrð Þ 1−e−FLrν α2 ;Lrð Þh i�

:

The foregoing expression is an extension of the equilibrium utilityin the single-sector case.

5.1.2. OptimumWe now investigate the optimal allocation. The planner maxi-

mizes utility subject to technology and the resource constraint inthe two-sector economy. As firms are symmetric within each sector,the first-best problem is given by

maxq1r ;q2r ;n1r ;n2r

Ur ¼ U U1r q1r ;n1rð Þ;U2r q2r ;n2rð Þð Þ

s:t: Lr ¼ n1r Lrcq1r þ Fð Þ þ n2r Lrcq2r þ Fð Þ:

As in the foregoing, we assume that Ur≡β1lnU1r+β2lnU2r with β1

+β2=1. Letting λ be the Lagrange multiplier, the first-order condi-tions are then given by

β1

n1r

α1e−α1q1r

1−e−α1q1r¼ λLrc ð45Þ

β2

n2r

α2e−α2q2r

1−e−α2q2r¼ λLrc ð46Þ

β1

n1r¼ λ Lrcq1r þ Fð Þ ð47Þ

β2

n2r¼ λ Lrcq2r þ Fð Þ: ð48Þ

Expressions (47) and (48), together with the resource constraint,yield

Lr ¼ n1r Lrcq1r þ Fð Þ þ n2r Lrcq2r þ Fð Þ ¼ β1

λþ β2

λ¼ 1

λ:

This result, together with (47) and (48), yields the optimal sector-al labor allocation as follows

Lo1r ¼ n1r Lrcq1r þ Fð Þ ¼ β1Lr and Lo2r ¼ n2r Lrcq2r þ Fð Þ ¼ β2Lr ; ð49Þ

which implies L1ro /L2ro =β1/β2. Using the equilibrium and optimal

labor allocation, (42) and (49), we can establish the followingproposition.

Proposition 7. Suppose that α1>α2. Then, the equilibrium labor alloca-tion in sector 1 is insufficient, whereas that in sector 2 is excessive, i.e.,L1ra bL1r

o and L2ra >L2r

o .

Proof. From expressions (42) and (49), the difference between theequilibrium and optimal labor allocation is given by

La1r−Lo1r ¼β1β2Lr



ν α1; Lrð Þκa1r

−ν α2; Lrð Þκa2r

� �:

Using (40), we thus have

sign La1r−Lo1r� � ¼ sign

ν α1; Lrð Þe


− ν α2; Lrð Þe


� :

Since ν(⋅,Lr)/[eFν(⋅, Lr)/Lr−1] is decreasing in ν and ν is increasingin its first argument, α1>α2 implies L1r

a bL1ro . As L1r

a +L2ra =L1r

o +L2ro =Lr, we then also have L2r

a >L2ro . □

Using (45)–(48), we show in Appendix D that the optimal mass offirms in each sector and the optimal utility are given by

no1r ¼ − α1β1

c 1þW−1 −e−1−α1FcLr

� �� > 0; no2r ¼ − α2β2

c 1þW−1 −e−1−α2FcLr

� �� > 0

ð50Þ

and

Uor ¼ β1ln − α1β1

cW−1 −e−1−α1FcLr

� �2664

3775þ β2ln − α2β2


� �2664

3775; ð51Þ

whereW−1(⋅) is the real branch of the LambertW function satisfyingW(−e−1−αF/(cLr))≤−1. Note that expressions (50) and (51) arestraightforward extensions of the single sector case.


We now consider whether or not there is excess entry. Since theequilibrium mass of firms in each sector (e.g., n1ra ) depends on thecharacteristics of both sectors (e.g., both α1 and α2), the proof ofProposition 1 is not applicable. However, noting that the equilibriumquantity in each sector (e.g., q1ra ) does not depend on the characteris-tics of the other sector (e.g., α2), we can show that the firms in eachsector operate at an inefficiently small scale in equilibrium. It isthen readily verified that at least one sector displays excess entry byusing sectoral labor misallocations established in Proposition 7.

Proposition 8. The firms in both sectors operate at an inefficiently smallscale in equilibrium, i.e., Q1r

a bQ1ro and Q2r

a bQ2ro . Furthermore, there is

excess entry in at least one sector, i.e., when α1>α2, we have n2ra >n2r

o ,whereas we have n1r

a >n1ro when α1bα2.

Proof. From (45) and (47), the optimal quantity in sector 1 mustsatisfy

LHS qo1r;α1� � ¼ e−α1q

o1r

1−e−α1qo1r¼ Lrc

α1 Lrcqo1r þ F

� � ¼ RHS qo1r ;α1� �

;

where both the LHS and RHS are decreasing in q1ro . Furthermore, we

have limq1ro →0LHS=∞, limq1r

o →0RHS=Lrc/(α1F), and limq1ro →∞LHS=

limq1ro →∞RHS=0. Since q1r

o is uniquely determined by (49) and (50),the RHS cuts the LHS only once from below. We now evaluate theLHS and RHS at the equilibrium value, q1ra =(1−W1r

a )/α1, and showthat LHS(q1ra ,α1)>RHS(q1ra ,α1). Using (35) and the definition of ν,we can show that

LHS qa1r ;α1� �

−RHS qa1r ;α1� � ¼ 1

e

FLrν α1; Lrð Þ

−1

− cα1

ν α1; Lrð Þ

¼ cα1

1

e

FLrν α1; Lrð Þ

−1

α1

c−ν α1; Lrð Þ e

FLrν α1; Lrð Þ

−1

264

375

8><>:

9>=>;:

We can derive exactly the same relationship in the single sectorcase, except that α1 is replaced with α. Since we know byProposition 1 that qrabqro holds in the single sector case, regardless ofparameter values, LHS(qra,α)>RHS(qra,α) must hold for any α. Wethus have LHS(q1ra ,α1)>RHS(q1ra ,α1). This establishes that q1r

a bq1ro

and Q1ra bQ1r

o even in the two-sector case. Similarly, we can provethat Q2r

a bQ2ro . Finally, when α1>α2, we know by Proposition 7 that

n2ra (cQ2r

a +F)=L2ra >L2r

o =n2ro (cQ2r

o +F). Noting that Q2ra bQ2r

o , we haven2ra >n2r

o . Similarly, when α1bα2, we have n1ra >n1r

o . □

Unlike in the single-sector case, Proposition 8 states that at leastone sector displays excess entry. In fact, we can construct numericalexamples of insufficient entry in one sector when the other sector dis-plays excess entry. Such insufficient entry arises due to intersectorallabor misallocations between the two differentiated sectors. For in-stance, when α1>α2 we know by Proposition 7 that n1ra (cQ1r

a +F)=L1ra bL1r

o =n1ro (cQ1r

o +F), i.e., there is insufficient labor allocated to sec-tor 1 in equilibrium. It is then possible to have insufficient entry insector 1, n1ra bn1r

o , even though the firms in sector 1 operate at an inef-ficiently small scale in equilibrium, Q1r

a bQ1ro .28

5.2. Trade

We turn to the open economy version of our two-sector model.We first analyze the properties of equilibrium when both goods are

28 In the two-sector case, pro-competitive effects in each sector do not necessarilymap into excess entry in both sectors. Hence, the general tendency toward excess en-try in partial equilibrium models (Vives, 1999, Ch. 6) does not carry over to generalequilibrium models with multiple sectors.

freely traded. We then investigate an intermediate case where onegood is freely traded, while the other is nontraded.

5.2.1. Free tradeThe model involves a mixture of Sections 4 and 5.1. We denote

sectors by subscripts 1 and 2, and countries by subscripts r and s.We assume that trade is free, and that markets are integrated. Sincethe price of each variety is the same in the two countries under theintegrated market assumption, we can show that PPE and FPE holdby using the same technique as in the single-sector open economycase (see Appendix C). To alleviate notation, when there is no confu-sion, we provide expressions for country r only, with mirror expres-sions holding for country s. Starting with profits, we have:

Π1r ið Þ ¼ p1r ið Þ−cwr½ � Lrq1r ið Þ þ Lsq1s ið Þ½ �−FwrΠ2r ið Þ ¼ p2r ið Þ−cwr½ � Lrq2r ið Þ þ Lsq2s ið Þ½ �−Fwr ;

where the expressions for quantities are analogous to those in (29).Noting that PPE and FPE hold under the integrated market assump-tion, and that p̃1r ¼ p̃1s ¼ p̃1 and p̃2r ¼ p̃2s ¼p̃2, we can aggregate de-mand across countries. Hence, the price equilibrium is alsoanalogous to (32) and (33) in Section 5.1. We further know from(34) that W1 and W2, and thus all prices and quantities are no longercountry specific. Accordingly, the zero profit conditions are given by

Π1 ¼ Lr þ Lsð Þcwα1

W−11 þW1−2

� −Fw ¼ 0

Π2 ¼ Lr þ Lsð Þcwα2

W−12 þW2−2

� −Fw ¼ 0;

which, as in the foregoing, can be solved for W1 and W2 to yield:

W1 ¼ 1−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4α1cF Lr þ Lsð Þ þ α1Fð Þ2

q−α1F

2c Lr þ Lsð Þ ∈ 0;1ð Þ

W2 ¼ 1−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4α2cF Lr þ Lsð Þ þ α2Fð Þ2

q−α2F

2c Lr þ Lsð Þ ∈ 0;1ð Þ:

By comparing the free trade and autarky values, we see that 1/W1bmin{1/W1r

a ,1/W1sa } and 1/W2bmin{1/W2r

a ,1/W2sa }. In words, trade

reduces markups in both sectors in both countries due to pro-competitive effects.

The trade equilibrium is, in turn, characterized by the followingfour conditions. First, the labor market in each country must clear,which requires that

Lr ¼ n1rLr þ Lsð Þc

α11−W1ð Þ þ F

� �þ n2r

Lr þ Lsð Þcα2

1−W2ð Þ þ F� �

ð52Þ

Ls ¼ n1sLr þ Lsð Þc

α11−W1ð Þ þ F

� �þ n2s

Lr þ Lsð Þcα2

1−W2ð Þ þ F� �

: ð53Þ

Second, the relationship between p̃1 and p̃2 is given as before by(38). The construction of the left-hand side of (38) under free tradeis analogous to that in Section 5.1. The right-hand side of (38) isobtained by assuming again that U≡β1lnU1+β2lnU2. We then have(∂U/∂U1)/(∂U/∂U2)=β1U2/(β2U1), where the ratio is given by

β1U2

β2U1¼

β1 n2r þ n2sð Þ 1− p2p̃2

� �β2 n1r þ n1sð Þ 1− p1

p̃1

� � ¼β1 n2r þ n2sð Þ 1− cw

W2 p̃2

� �

β2 n1r þ n1sð Þ 1− cwW1 p̃1

� � :


Finally, in the open economy, trade must balance, which requiresthat:

Lr n1sW−1

1 −1α1

þ n2sW−1

2 −1α2

" #¼ Ls n1r

W−11 −1α1

þ n2rW−1

2 −1α2

" #: ð54Þ

The four equilibrium conditions (38) and (52)–(54) yield {n1r,n2r,n1s,n2s} as follows:

n1r ¼Lr

Lr þ Ls

L1rLr

ν α1; Lr þ Lsð Þ n1s ¼Ls

Lr þ Ls

L1sLs

ν α1; Lr þ Lsð Þ ð55Þ

n2r ¼Lr

Lr þ Ls

L2rLr

ν α2; Lr þ Lsð Þ n2s ¼Ls

Lr þ Ls

L2sLs

ν α2; Lr þ Lsð Þ; ð56Þ

where

L1rLr

¼ L1sLs

¼β1

κ1ν α1; Lr þ Lsð Þ

β1

κ1ν α1; Lr þ Lsð Þ þ β2


ð57Þ

L2rLr

¼ L2sLs

¼β2


β1

κ1ν α1; Lr þ Lsð Þ þ β2


; ð58Þ

and

κ1 ¼ eF

LrþLsν α1 ;LrþLsð Þ−1 and κ2 ¼ e

FLrþLs

ν α2 ;LrþLsð Þ−1: ð59Þ

Hence, the population distribution between the two countries(e.g., Lr/(Lr+Ls)) and the sectoral labor allocation (e.g., L1r/Lr) are cru-cial for the mass of firms in each sector in each country. Holding thesectoral labor allocation constant, one can readily verify that there isdomestic exit in each sector due to trade, as in the single-sectorcase. However, trade alters the sectoral labor allocation. Thisstrengthens domestic exit in the shrinking sector, but weakens thetendency toward domestic exit in the other sector. We still find do-mestic exit and variety expansion in both sectors in both countries.

Having extended our single-sector model to the two-sector set-ting, we now address various intersectoral issues. As shown inSection 5.1, the sectoral variables under autarky such as the relativemass of firms, n1ra /n2ra , the relative mass of workers, L1ra /L2ra , the relativeoutput, Q1r

a /Q2ra , as well as the relative price, p1ra /p2ra , depend on the

country size Lr. However, we can show that trade eliminates thesecross-country differences as follows.

Proposition 9. Trade leads to structural convergence regardless of pop-ulation sizes, Lr and Ls. The relative price, the relative output, the relativemass of firms, and the relative mass of workers are equalized betweenthe two countries, i.e., p1r/p2r=p1s/p2s, Q1r/Q2r=Q1s/Q2s, n1r/n2r=n1s/n2s, and L1r/L2r=L1s/L2s hold.

Proof. The equalization of the relative price and the relative outputcan be obtained by noting that W1 and W2 depend only on theworld population. Using expressions (55) to (58), it is readily verifiedthat

n1r

n2r¼ n1s

n2s¼ β1=κ1

β2=κ2

ν α1; Lr þ Lsð Þν α2; Lr þ Lsð Þ� �2

;

and that

L1rL2r

¼ L1sL2s

¼ β1=κ1

β2=κ2

ν α1; Lr þ Lsð Þν α2; Lr þ Lsð Þ ;

where κ1 and κ2 are given by (59). Both expressions depend only onthe world population Lr+Ls, thus completing the proof. □

Proposition 9 states that countries having different sectoral com-positions under autarky converge to the same industry structureunder free trade. This is in sharp contrast to the prediction of Ricar-dian and Heckscher–Ohlin models that trade causes sectoral speciali-zation. Furthermore, unlike the new trade theory that emphasizesintra-industry trade between similar countries, with similarity givingrise to more trade, Proposition 9 shows that countries become moresimilar due to trade, thus suggesting circular causation between sim-ilarity and intra-industry trade. Finally, an immediate corollary to thisproposition is that, under free trade, the smaller country is a small-scale replica of the larger country, e.g., n1r/n1s=n2r/n2s=Lr/Ls. Itshould be noted, however, that this almost never holds underautarky, unlike in the CES case. The only exception that we observen1r/n1s=n2r/n2s=Lr/Ls even under autarky is the case of symmetriccountries Lr=Ls.

5.2.2. EfficiencyWe now analyze whether trade enhances efficiency in the two-

sector model. As in the single-sector case, free trade amounts to in-creasing the population size. Yet, unlike in that case, we can addresshow trade affects the difference between the equilibrium and opti-mum through both intersectoral and intrasectoral allocations.

Concerning intersectoral allocations, we can show that intersec-toral distortions are eliminated in the limit. In particular, when thepopulation gets arbitrarily large in the integrated economy, intersec-toral output, variety, and labor distortions vanish, i.e.,

limL→∞

Q1

Q2¼ lim

L→∞

Qo1

Qo2¼

ffiffiffiffiffiffiα2

α1

rlimL→∞

n1

n2¼ lim

L→∞

no1

no2¼

ffiffiffiffiffiffiα1

α2

rβ1

β2

limL→∞

L1L2

¼ limL→∞

Lo1Lo2

¼ β1

β2:

Turning to intrasectoral distortions, it is verified that limL→0(n1/n1o)= limL→0(n2/n2o)=1, and that limL→∞ n1=no

1

� � ¼ limL→∞ n2=no2

� � ¼ffiffiffi2

p, as in the single-sector case. By continuity, for a sufficiently

small population size, excess entry tends to be small, whereas itgets larger when the population gets arbitrarily large. Despite excessentry in the limit in both sectors, we can show the following overallefficiency result.

Proposition 10. When the population gets arbitrarily large, the equilib-rium utility converges to the optimal utility, i.e.,

limL→∞

U ¼ limL→∞

Uo ¼ β1lnα1β1

cþ β2ln

α2β2

c:

Proof. In this context, U1 and U2 in equilibrium can be expressed as in(43) and (44). Since we already know by Proposition 4 that limL→∞ν(α1,L)[1−e−Fν(α1,L)/L]=α1/c and limL→∞ν(α2,L)[1−e−Fν(α2,L)/L]=α2/c, wenow establish the limit of L1/L and L2/L. They are given by

limL→∞

L1L¼ β1 and lim

L→∞

L2L¼ β2:

Hence, it is readily verified that limL→∞U1=α1β1/c andlimL→∞U2=α2β2/c, so that, using the chain rule for the limits of con-tinuous functions, we have:

limL→∞

U ¼ β1lnα1β1

cþ β2ln

α2β2

c:


Turning to the optimum, since W−1(−e−1)=−1, it is readilyverified that limL→∞U1

o=α1β1/c and limL→∞U2o=α2β2/c, so that

limL→∞

Uo ¼ β1lnα1β1

cþ β2ln

α2β2

c:

Therefore, the equilibrium is efficient in the limit, which provesour claim. □

Thus, our overall efficiency result in the single-sector case carries overto the two-sector case as intersectoral distortions vanish in the limit.

5.2.3. Nontraded goodAssume now that sector 1 produces a nontraded good, whereas

sector 2 produces a freely traded good. Each good is differentiatedas before. The objective of this subsection is to show that trade in sec-tor 2 induces domestic exit in the nontraded sector 1. Given this ob-jective, it is sufficient to focus on the simple case where the twocountries are symmetric (Lr=Ls=L). Then, we can show again thatPPE and FPE hold. Since the expressions for the autarky case are thesame as those in Section 5.1, we only derive the expressions for theopen economy. The price equilibrium is analogous to the previouscases, which yields the zero profit conditions:

Π1 ¼ Lcwα1

W−11 þW1−2

� −Fw ¼ 0

Π2 ¼ 2Lcwα2

W−12 þW2−2

� −Fw ¼ 0:

From those conditions, we obtain

W1 ¼ 1−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4α1cFLþ α1Fð Þ2

q−α1F

2cL∈ 0;1ð Þ

W2 ¼ 1−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8α2cFLþ α2Fð Þ2

q−α2F

4cL∈ 0;1ð Þ:

Observe that the market size for sector 2 is doubled, whereas thatfor sector 1 is unchanged.

The trade equilibrium is characterized by the following conditions.First, the labor market must clear in each country. Letting n1r=n1s=n1 and n2r=n2s=n2 by symmetry, this requires that

L ¼ n1cLα1

1−W1ð Þ þ F� �

þ n22cLα2

1−W2ð Þ þ F� �

:

Second, the relationship between the two reservation prices is againgiven by (38). The construction of the left-hand side of (38) in the openeconomy is analogous to that in Section 5.1. The right-hand side of (38)is obtained by assuming as before that U≡β1lnU1+β2lnU2. We thenhave (∂U/∂U1)/(∂U/∂U2)=β1U2/(β2U1), where the ratio is given by

β1U2

β2U1¼

2β1n2 1− p2p̃2

� �

β2n1 1− p1p̃1

� � ¼2β1n2 1− cw

W2p̃2

� �

β2n1 1− cwW1p̃1

� � :

Finally, trade is balanced because Lr=Ls=L, n2s=n2r=n2, andFPE and PPE hold. From these conditions we obtain {n1,n2} as follows:

n1 ¼ L1Lν α1; Lð Þ ¼

2β1

κa1ν α1; Lð Þ

2β1

κa1ν α1; Lð Þ þ β2

κ2ν α2;2Lð Þ

ν α1; Lð Þ

n2 ¼ 12L2Lν α2;2Lð Þ ¼

β2

κ2ν α2;2Lð Þ

2β1


κ2ν α2;2Lð Þ

ν α2;2Lð Þ:

We can then establish the following results.

Proposition 11. Trade induces domestic exit in the nontraded goodsector, i.e., n1bn1

a. Put differently, there is a consumption variety loss inthe nontraded good sector.

Proof. The mass of firms in the nontraded good sector under autarkyis given by

na1 ¼

β1

κa1ν α1; Lð Þ

β1


κa2ν α2; Lð Þ

ν α1; Lð Þ

¼β1

κa1

ν α1; Lð ÞL

β1

κa1

ν α1; Lð ÞL

þ β2

κa2

ν α2; Lð ÞL

ν α1; Lð Þ;

whereas in the open economy it becomes

n1 ¼2β1

κa1ν α1; Lð Þ

2β1


κ2ν α2;2Lð Þ

ν α1; Lð Þ

¼β1

κa1

ν α1; Lð ÞL

β1

κa1

ν α1; Lð ÞL

þ β2

κ2

ν α2;2Lð Þ2L

ν α1; Lð Þ:

The only change between autarky and free trade in sector 2 ap-pears in the second term of the denominator. Let ω≡ν(α2,2L)/(2L).Noting that

1κ2

ν α2;2Lð Þ2L

¼ 1

eFν α2 ;2Lð Þ

2L −1

ν α2;2Lð Þ2L

¼ ωeFω−1

decreases withω, and thatω decreases with the market size for sector2, we get n1bn1a. □

Proposition 12. The mass of varieties consumed in the traded goodsector increases, i.e., 2n2>n2

a. Furthermore, domestic entry occurs inthe traded good sector, i.e., the mass of varieties produced in the tradedsector in each country increases.

Proof. The mass of varieties consumed in the traded good sectorunder autarky is given by

na2 ¼

β2

κa2ν α2; Lð Þ

β1


κa2ν α2; Lð Þ

ν α2; Lð Þ

¼β2

κa2

ν α2; Lð ÞL

β1

κa1

ν α1; Lð ÞL

þ β2

κa2

ν α2; Lð ÞL

ν α2; Lð Þ;

whereas in the open economy it becomes

2n2 ¼ 2

β2

κ2ν α2;2Lð Þ

2β1


κ2ν α2;2Lð Þ

ν α2;2Lð Þ

¼β2

κ2

ν α2;2Lð Þ2L

β1

κa1

ν α1; Lð ÞL

þ β2

κ2

ν α2;2Lð Þ2L

2ν α2;2Lð Þ½ �:


Using the same notation as in the previous proposition, and notingthat 2ν(α2,2L)>ν(α2,L), we obtain the first claim. Turning to domes-tic entry, we have

n2 ¼β2

κ2ν α2;2Lð Þ

2β1


κ2ν α2;2Lð Þ

ν α2;2Lð Þ

¼β2

κ2

ν α2;2Lð Þ2L

β1

κa1

ν α1; Lð ÞL

þ β2

κ2

ν α2;2Lð Þ2L

ν α2;2Lð Þ:

Again, using the same notation as in the previous proposition, wecan show that n2>n2

a. □

These two propositions have important implications for assessinggains from trade in the presence of nontraded goods. Although tradeexpands the range of varieties that consumers face in the traded goodsector, it induces domestic exit, or a variety loss in the nontraded sec-tor. Unlike the single sector case with a traded good, such a varietyloss in the nontraded sector is not compensated by import varieties.Accordingly, monopolistic competition models that abstract fromnontraded varieties may overestimate gains from trade.

Furthermore, our results suggest that, contrary to general belief,VES models per se may not explain domestic exit in the traded goodsector when there are nontraded varieties. This reconfirms the impor-tance of other factors such as firm heterogeneity in explaining domes-tic exit in the traded good sector. Extending the two-sector CES modelwith firm heterogeneity by Bernard et al. (2007) to a VES setting orextending the one-sector VES model with firm heterogeneity byDhingra and Morrow (2011) to a multi-sector setting seems to be apromising step toward a better understanding of this issue.

6. Conclusions

We have developed a VES model of international trade displayingpro-competitive effects and a competitive limit, and have investigat-ed the impacts of trade on welfare and efficiency. Unlike the standardCES model, our framework allows us to capture the impacts of tradeon varieties, markups, and exploitation of scale economies withoutresorting to an additively separable numeraire good. The welfare de-composition helps us to understand the relative contribution of prod-uct diversity and pro-competitive effects to gains from trade, which isbecoming increasingly more important given the recent empirical ev-idence such as Broda and Weinstein (2006) and Feenstra andWeinstein (2010). We have also explored whether or not trade is ul-timately efficiency enhancing, and have shown that this is indeed thecase.

The basic framework presented in this paper is flexible enough toallow for many extensions. In this paper, we have provided one,namely a multi-sector setting.29 In our model, trade tends to reducewithin-sector efficiency losses, which disappear at the competitivelimit. However, as is well known, markup heterogeneity across sec-tors is a source of between-sector distortions (Bilbiie et al., 2008;Epifani and Gancia, 2011), which may become more important withfreer trade. Yet, we have established that such intersectoral distor-tions also vanish when the population gets arbitrarily large in the in-tegrated economy. Our multi-sector analysis further illustrates somenew aspects arising from the interaction between intersectoral andintrasectoral allocations, namely structural convergence, rather thansectoral specialization, and domestic exit in the nontraded good sec-tor induced by trade in the other sector.

29 Other extensions include, for instance, heterogeneous firms and trade costs(Behrens et al., 2009) and heterogeneous consumers (Behrens and Murata, 2009b).

Finally, as we have obtained the closed form solutions for theequilibrium utility and the optimal utility, our model can also be ex-tended to a multi-region setting in a spatial economy. Doing sosheds new light on whether or not larger cities are more efficient. Insuch a setting, a larger population not only allows for greater diversitybut also exacerbates congestion in cities while achieving prices closerto marginal costs. Although we have established efficiency gains fromtrade in this paper, it is not obvious whether or not our efficiency re-sult carries over to a spatial economy with urban congestion. Explor-ing this formally is left for future research.

Acknowledgments

Wewould like to thank Robert Staiger and two anonymous refereesfor useful comments and suggestions that helped us to significantly im-prove the paper. Behrens is holder of the Canada Research Chair in Re-gional Impacts of Globalization. Financial support from the CRCProgram of the Social Sciences and Humanities Research Council(SSHRC) of Canada is gratefully acknowledged. Behrens also gratefullyacknowledges financial support from UQAM, and from FQRSC Québec(grant NP-127178). YasusadaMurata gratefully acknowledges financialsupport from Japan Society for the Promotion of Science (17730165)and MEXT Academic Frontier (2006–2010). This research was partiallysupported by the Ministry of Education, Culture, Sports, Science andTechnology, Grant-in-Aid for 21st century COE Program.

Appendix A. Computations for equilibrium

A.1. Demand functions

A representative consumer in country r solves problem (1). Let-ting λ stand for the Lagrange multiplier, the first-order conditionsfor an interior solution are given by:

αe−αqrr ið Þ ¼ λpr ið Þ;∀i∈Ωr ðA:1Þ

αe−αqsr jð Þ ¼ λps jð Þ;∀j∈Ωs ðA:2Þ

and the budget constraint

∫Ωrpr ið Þqrr ið Þdiþ ∫Ωs

ps jð Þqsr jð Þdj ¼ Er : ðA:3Þ

Taking the ratio of (A.1) with respect to i and j, we obtain

e−α qrr ið Þ−qrr jð Þ½ � ¼ pr ið Þpr jð Þ⇒qrr ið Þ ¼ qrr jð Þ þ 1

αln

pr jð Þpr ið Þ� �

∀i; j∈Ωr :

Multiplying the last expression by pr(j) and integrating with re-spect to j∈Ωr we obtain:

qrr ið Þ∫Ωrpr jð Þdj ¼ ∫Ωr

pr jð Þqrr jð Þdjþ 1α∫Ωr

lnpr jð Þpr ið Þ� �

pr jð Þdj: ðA:4Þ

Analogously, taking the ratio of (A.1) and (A.2) with respect to iand j, we get:

e−α qrr ið Þ−qsr jð Þ½ � ¼ pr ið Þps jð Þ⇒qrr ið Þ ¼ qsr jð Þ þ 1

αln

ps jð Þpr ið Þ� �

∀i∈Ωr ;∀j∈Ωs:

Multiplying the last expression by ps(j) and integrating with re-spect to j∈Ωs we obtain:

qrr ið Þ∫Ωsps jð Þdj ¼ ∫Ωs

ps jð Þqsr jð Þdjþ 1α∫Ωs

lnps jð Þpr ið Þ� �

ps jð Þdj: ðA:5Þ


Summing (A.4) and (A.5), and using the budget constraint (A.3)yield

qrr ið Þ ¼Er− 1

α ∫Ωrln

pr ið Þpr jð Þ� �

pr jð Þdj− 1α ∫Ωs

lnpr ið Þps jð Þ� �

ps jð Þdj

∫Ωrpr jð Þdjþ ∫Ωs

ps jð Þdj:

Finally, noting the definitions of P and H given in the main text, weobtain the demands (2). The derivations of the demands (3) areanalogous.

A.2. Price equilibrium

Behrens and Murata (2007, Proposition 2) show that the priceequilibrium is symmetric and unique. Given that result, expression(11) is obtained as follows. Plugging Qr(i)≡Lrqrr(i)+Lsqrs(i) into (6),and setting qrs(i)=∂qrs(i)/∂pr(i)=0, we have

∂Πr ið Þ∂pr ið Þ ¼ Lrqrr ið Þ þ pr ið Þ−cwr½ �Lr

∂qrr ið Þ∂pr ið Þ ¼ 0: ðA:6Þ

Noting that ∂qrr(i)/∂pr(i)=−1/[αpr(i)] by (4), imposing symme-try on (A.6) implies

pr−cwr

αpr¼ qrr ¼ − 1

αlnpr þ

ErPþ 1αHP;

where we use (2) to get the last equality. Since P=nrpr and H=nrprlnprbecause of symmetry, it can be readily verified that

pr ¼ cwr þαErnr

:

Noting that Er=wr in equilibrium as profits are zero due to freeentry, we obtain (11).

A.3. Two-sector demand functions and reservation prices

The first-order conditions for an interior solution of the utilitymaximization problem (28) are given by:

∂Ur

∂U1rα1e

−α1q1r ið Þ ¼ λp1r ið Þ;∀i∈Ω1r

∂Ur

∂U2rα2e

−α2q2r jð Þ ¼ λp2r jð Þ;∀j∈Ω2r;

where λ is the Lagrange multiplier. Applying the same technique asthat we used for the single-sector case in Appendix A.1, we readilyobtain the demands as follows:

q1r ið Þ ¼ − 1α1

lnp1r ið Þ þ ErPr

þ 1α1

Hr

Pr− 1

α2Prln

α2

α1

∂Ur=∂U2r

∂Ur=∂U1r

� �∫Ω2r

p2r jð Þdj ðA:7Þ

q2r jð Þ ¼ − 1α2

lnp2r jð Þ þ α1

α2

ErPr

þ 1α2

Hr

Pr

þ 1α2Pr

lnα2

α1

∂Ur=∂U2r

∂Ur=∂U1r

� �∫Ω1r

p1r ið Þdi; ðA:8Þ

where

Pr≡∫Ω1rp1r ið Þdiþ α1

α2∫Ω2r

p2r jð Þdj

Hr≡∫Ω1rp1r ið Þlnp1r ið Þdiþ α1

α2∫Ω2r

p2r jð Þlnp2r jð Þdj

are the sum of prices and a measure of price dispersion in the two-sector economy. The demands (A.7) and (A.8) can then be rewritten as

q1r ið Þ ¼ 1α1


; where p̃1r≡eα1ErPr þHr

Pr − α1α2Pr

lnα2α1

∂Ur =∂U2r∂Ur =∂U1r

� ∫Ω2r

p2r jð Þdj

q2r jð Þ ¼ 1α2


; where p̃2r≡eα1ErPr þHr

Pr þ 1Pr ln

α2α1


� ∫Ω1r

p1r ið Þdi:

Observe that p̃1r and p̃2r are common to all firms within eachsector, and taken as given by each firm because of the continuum as-sumption. Taking the ratio of p̃1r and p̃2r , we obtain

p̃1rp̃2r

¼ e−α1

α21Pr

lnα2α1


� ∫Ω2r

p2r jð Þdj− 1Pr

lnα2α1

∂Ur=∂U2r∂Ur=∂U1r

� ∫Ω1r

p1r ið Þdi ¼ α1

α2

∂Ur=∂U1r

∂Ur=∂U2r:

A.4. Profit-maximization and properties of W

The first-order conditions ∂Π1r/∂p1r(i)=0 are given by


¼ 1− cwr

p1r ið Þ :

Taking the exponential of both sides and rearranging terms, we have

ecwr

p̃1r¼ cwr

p1r ið Þ ecwrp1r ið Þ:

Noting that the Lambert W function is defined as φ=W(φ)eW(φ)

and setting φ ¼ e cwr=p̃1r , we obtain W e cwr=p̃1r� � ¼ cwr=p1r ið Þ,

which implies the expression of p1ra as given in (32). Combining thefirst-order conditions and demand functions, quantities are given byq1r(i)=(1/α1)[1−cwr/p1r(i)]. Plugging p1r(i)=p1r

a , we have theexpression for q1r

a . The expression for operating profits is given byπ1r(i)=[p1r(i)−cwr]Lrq1r(i), which together with p1r

a and q1ra , yields

π1ra . Mirror expressions hold for sector 2.Turning to the properties of the Lambert W function, φ=W(φ)eW(φ)

implies that W(φ)≥0 for all φ≥0. Taking the logarithm of both sidesand differentiating yield

W ′ φð Þ ¼ W φð Þφ W φð Þ þ 1½ � > 0;∀φ > 0:

Finally, 0=W(0)eW(0) and e=W(e)eW(e) imply W(0)=0 andW(e)=1, respectively.

A.5. Application of Lambert W to the single-sector model

Take the benchmark case of a single sector (say sector 1), as devel-oped in Sections 2 and 3. Then, solving the zero profit conditionπ1r=Fwr for W1r yields

W1r ¼ 1−


q−α1F

2cLr:

Substituting this expression into the labor market clearing condi-tion

n1r Lrcq1r þ Fð Þ ¼ Lr⇒n1rLrcα1

1−W1rð Þ þ F� �

¼ Lr

Yr

1

f

g

0

g (0)

Fig. A1. f and g as a function of Yr.

n0 nor na

r

1

f

g

g(nar)

f(nar)

Fig. A2. Excess entry.


yields

n1r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4α1cFLr þ α1Fð Þ2

q−α1F

2cF;

which is exactly the same as (15) in the single sector case.

Appendix B. Proof of Proposition 1

We compare the equilibrium nra with the optimum nr

o defined asthe solution of the equation:

f nrð Þ ¼ g nrð Þ; where f nrð Þ≡ cnr

α þ cnrand g nrð Þ≡e−α

c1nr

− FLr

� �: ðB:1Þ

Note first that f is strictly increasing in nr, taking values from 0 to 1,and that g is also strictly increasing, taking values from 0 to eαF/(cLr)

>1. Some standard calculations show that there is a unique intersec-tion since: (i) both functions are continuous; (ii) f is concave, whereasg is convex for nr sufficiently small; (iii) the slope of f is strictly greaterthan that of g for nr sufficiently small30; and (iv) g admits a singlevalue for which its second-order derivative is equal to zero.

We next show that nra>nro. To prove our claim, we use a convexity

argument. The equilibriummass of varieties is given by (15), whereasthe optimal mass of varieties is the unique solution to (B.1). First,evaluate f at nra, which yields

f nar

� � ¼ cnar

α þ cnar¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcFLr þ αFð Þ2

q−αFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4αcFLr þ αFð Þ2q

þ αF¼ Xr−2αF

Xr; ðB:2Þ

where Xr≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4αcFLr þ αFð Þ2

qþ αF. Second, evaluate g at nra to get

g nar

� � ¼ e−α

c1nar

− FLr

� ¼ e−

2αFXr : ðB:3Þ

Let Yr≡(2αF)/Xrb1 and g(Yr)=e−Yr. Note that (B.2) can then beexpressed as f(Yr)=1−Yr, which is tangent to (B.3) at Yr=0:

1−Yr ¼ g 0ð Þ þ g ′ 0ð Þ Yr−0ð Þ:

Since (B.3) is strictly convex, it lies strictly above its tangent. Put dif-ferently, f(Yr)=1−Yrbe−Yr=g(Yr) holds for all Yr>0 (see Fig. A1).Hence, the right-hand side of (B.1) exceeds the left-hand side of (B.1)at the equilibrium mass of firms nra. By uniqueness of the optimal massof firms, and since the right-hand side of (B.1) exceeds the left-handside if and only if nr>nr

o, we may conclude that nra>nro (see Fig. A2). Ex-

pression (8) then implies that nra(cQr

a+F)=nro(cQr

o+F)=Lr, whichyields Qr

abQro.

Appendix C. Proof of Proposition 2

Conditions (6) and (7) must hold for both country-r and country-sfirms at every price equilibrium which, using (2)–(4), yields

∂Πr ið Þ∂pr ið Þ −

∂Πs jð Þ∂ps jð Þ ¼ 0⇔c

wr

pr ið Þ−ws

ps jð Þ� �

¼ lnpr ið Þps jð Þ� �

: ðC:1Þ

30 To check this, note that limnr→0 f′(nr)=c/α> limnr→0g′(nr)=0. The last equality is

obtained as follows. Noting that lng′ nrð Þ ¼ − 2nr

ln nrð Þ1=nr

þ α2c

� �þ ln

αc

� þ αFcLr

; and that

limnr→0lnnr/(1/nr)=0 by l'Hospital's rule, we have limnr→0

lng′ nrð Þ ¼ − limnr→0

2nr

�limnr→0

ln nrð Þ1=nr

þ α2c

� �þ ln

αc

� þ αFcLr

¼ −∞; which, by continuity of the logarithmic func-

tion, implies limnr→0g′(nr)=0.

It is also readily verified that

Qr ið ÞfQs jð Þ ⇔ − Lr þ Lsα

lnpr ið Þps jð Þ� �

f 0: ðC:2Þ

Furthermore, an equilibrium is such that firms earn zero profit, i.e.,

Πr ið Þ ¼ wrpr ið Þwr

−c� �

Qr ið Þ−F�

¼ 0

Πs jð Þ ¼ wsps jð Þws

−c� �

Qs jð Þ−F�

¼ 0:

Assume that there exists i∈Ωr and j∈Ωs such that pr(i)>ps(j).Then condition (C.1) implies that

wr

pr ið Þ >ws

ps jð Þ⇒pr ið Þwr

bps jð Þws

;

whereas condition (C.2) implies that Qr(i)bQs(j). Hence,Πr(i)bΠs(j),which is incompatible with an equilibrium. We may hence concludethat pr(i)=ps(j) must hold for all i∈Ωr and j∈Ωs, which shows thatproduct prices are equalized. Condition (C.1) then shows thatwr=ws, i.e., factor prices are equalized whenever product prices areequalized. Finally, since pr(i)=ps(j)=p and wr=ws=w, we can re-write (2)–(4) as follows:

qrr ¼ qsr ¼ qss ¼ qrs ¼wNp

ðC:3Þ


and

∂qrr∂pr

¼ ∂qsr∂ps

¼ ∂qss∂ps

¼ ∂qrs∂pr

¼ − 1αp

: ðC:4Þ

Inserting (C.3) and (C.4) into the first-order condition (6), weobtain the price equilibrium, in which markups are equalized acrossvarieties and countries.

Appendix D. The optimal masses of firms and the optimal utility inthe two-sector case

As in the single sector case, we first derive the optimal mass offirms in each sector. Expression (45) becomes

β1

n1r

α1e−α1q1r

1−e−α1q1r¼ c⇒e−α1q1r ¼ cn1r

α1β1 þ cn1r: ðD:1Þ

Expression (47), in turn, becomes

β1

n1r¼ Lrcq1r þ F

Lr⇒q1r ¼

1c

β1

n1r− F

Lr

� �: ðD:2Þ

Plugging (D.2) into (D.1), we have

e−α1

cβ1n1r

− FLr

� ¼ cn1r

α1β1 þ cn1r⇒− 1þ α1β1

cn1r

� �e− 1þα1β1

cn1r

� ¼ −e−1−α1F

cLr :

We thus obtain

− 1þ α1β1

cn1r

� �¼ W −e−1−α1F

cLr

� �;

which can be solved for n1ro as shown in (50). The derivation of n2ro isanalogous. Finally, turning to the optimal utility, expressions (50) and(D.1) yield

Uo1r ¼ no

1r 1−e−α1qo1r

� ¼ no

1rα1β1

α1β1 þ cno1r

¼ − α1β1


� � :

Again, the derivation of U2ro is analogous. Hence, the optimal utility

is given by (51).

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