international trade and firms' heterogeneity under monopolistic competition

Open economies review 13: 291–311, 2002c© 2002 Kluwer Academic Publishers. Printed in The Netherlands.

International Trade and Firms’ Heterogeneityunder Monopolistic Competition

SEBASTIEN JEAN [email protected] d’Etudes Prospectives et d’Informations Internationales, 9 rue Georges-Pitard,75015 Paris, France

Key words: international trade, firms’ heterogeneity, product differentiation, monopolisticcompetition, productive efficiency

JEL Classification Numbers: F12, L11

Abstract

This article studies the interactions between international trade and firms’ heterogeneity by propos-ing a tractable model consistent with the stylised facts. The model describes, in a general equilib-rium framework, two economies producing and trading two goods, one homogeneous and the otherdifferentiated. In the differentiated-good sector, firms are heterogeneous by their marginal cost, ina context of monopolistic competition with free-entry and exit. They incur a fixed production cost,but also a fixed cost if they choose to export. We show that trade in differentiated goods increasesindustry-wide efficiency, through two different logics: one defensive and import-driven; the otheroffensive and export-driven.

The core theories of international trade are based upon the hypothesis of ho-mogenous firms. This is a natural simplifying assumption, but it is strongly ques-tionable, especially when only a fifth of the firms actually export anything, aswas the case in the United States in 1992 (Bernard et al., 2000, p. 4). This may bemisleading, as well: Tybout and Westbrook (1995) show that as a result of tradeliberalisation in Mexico, the scale efficiency gains were minor, while market-share effects were generally important as production shifted toward the moreefficient plants. They suggest that the gains associated with increasing inter-nal returns to scale have probably been largely overdone, while the effects ofreshuffling of output shares among firms have been underestimated. Hetero-geneity among firms may also be important in order to understand the linkbetween international trade and labor skill; Bernard and Jensen (1997a) showthat “increases in employment at exporting plants contribute heavily to the ob-served increase in relative demand for skilled labor in manufacturing” (Bernardand Jensen, 1997a, p. 3).

A growing empirical evidence emphasises and documents the importance offirms’ heterogeneity1 as a determinant both of trade and of its consequences.

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Several robust results emerge from the recent literature, which hold for variouscountries:

i. Only a fraction of firms export anything, and exporters are significantly dif-ferent from nonexporters. They are generally in the minority, and they arelarger and more productive than nonexporters (See Bernard et al., 2000).Furthermore, exporters tend to be more atypical in terms of productivity,and less in terms of size, in industries where the economy has a compara-tive disadvantage (Abd-El-Rahman, 1991).

ii. This specificity of exporters does not seem to be the result of any learning-by-exporting. The causality is the other way around; good firms becomeexporters, and this has a positive influence on their output, but not on theirproductivity (Clerides, Lach, and Tybout, 1998; Bernard and Jensen, 1999aand b).

iii. International trade can induce a significant reshuffling of output-sharesamong firms within a given industry, thus giving rise to important share ef-fects. This appears as an important aspect of the rationalising effect inducedby trade liberalisation in developing countries (Tybout and Westbrook, 1995;Roberts and Tybout, 1997; Clerides, Lach, and Tybout, 1998), but even inthe United States exporting is associated with an important reallocation ofresources from less efficient to more efficient firms (Bernard and Jensen,1999b).

iv. Previous export experience is important in setting the export-status of firms,suggesting that there is a significant fixed cost of exporting in developingcountries (Roberts and Tybout, 1997; Clerides, Lach, and Tybout, 1998), butalso in industrialised countries (Bernard and Jensen, 1997b; Bernard andWagner, 1997).

From a theoretical point of view, however, few studies take into account theheterogeneity of firms as a determinant of international trade. In an informalway, Abd-El-Rahman (1991) proposes hypotheses about the role of firm-specificadvantages, depending on the comparative advantage of the country in thecorresponding industry. He argues that the diversity in firms’ performancesexplains the existence of minor trade flows, whose direction is opposed to thatpredicted by comparative advantages. The assessment of trade policy leadsseveral authors to consider specifically the heterogeneity of unit costs amongfirms (a recent example is van Long and Soubeyran, 1997). But Abd-El-Rahmanand van Long and Soubeyran studies are limited to the case of homogeneousgoods, and they do not provide any explicit description of the selection of firms.

This article aims at shedding light on the interactions between internationaltrade and firms’ heterogeneity by proposing a tractable model consistent withthe stylised facts mentioned previously. While doing this, we will pay specialattention to the way firms’ heterogeneity influences the nature of trade and,reciprocally, to the impact of international trade on the population of firms andits consequences in terms of industry-wide efficiency.

INTERNATIONAL TRADE AND FIRMS’ HETEROGENEITY 293

These objectives require building a trade model in which firms are hetero-geneous, and describing explicitly the selection process of firms by marketentry-exit. In order to do this, we build on the autarkic partial equilibrium modelproposed by Montagna (1995).2 It describes a market where firms produc-ing horizontally differentiated products are in monopolistic competition. Themarginal production cost, randomly drawn, is firm-specific. Given the existenceof a fixed production cost, only firms with a sufficiently low marginal cost areable to survive. Firms can freely enter or quit the market, given that the potentialentrants do not know their marginal production cost until they begin to produce.The dynamic firm-selection process through free entry-exit leads to a uniquesteady-state equilibrium, which can be characterised analytically.

This model completes the description of the determinants of firms hetero-geneity by assuming that incumbent firms (once they know their marginal cost)can freely choose to export or not, given the existence of a fixed cost of ex-porting and of an ad valorem tariff barrier. This description of firms’ behavior isembedded in a general equilibrium framework, with two economies producingand trading two goods, one homogeneous and the other differentiated.

Depending on the values of the parameters, and in particular on the mag-nitude of trade costs, of country-wide efficiency differences, and of firms het-erogeneity, we describe how different trade patterns may emerge. In all cases,we find that trade openness increases the average productive efficiency in thedifferentiated-good sector, but that the impact of trade stems from two differentlogics; one defensive and import-driven, the other offensive and export-driven.

The article is organised as follows: the setup of the model is presented inSection 2; Section 3 discusses the nature of trade, depending on the param-eters; Section 4 studies the implications for the population of firms and forindustry-wide productive efficiency; and Section 5 concludes.

2. The setup

Consider countries A and B producing and trading an homogeneous good Hand a differentiated good D, with one production factor, labor, whose endow-ment is assumed to be exogenously given for each country.

2.1. The demand side

A Cobb-Douglas utility function characterises consumer preferences betweenthe two goods:

Uc = H 1−αc Dα

c , α ∈ ]0; 1[ (1)

Where the index c = A or B refers to the country, U is the utility, H is the con-sumption of homogeneous good, D the composite index for consumption of

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the differentiated good:

Dc =( ∑

i∈Ic ∪ Ic,exp

D(σ−1)/σ

ic

)σ/(σ−1)

. (2)

Where the index c denotes the complement of c in {A, B} (the “other country”),and the index i refers to the variety produced by firm i . Dic is the consumptionof variety i in country c, the set Ic groups the indices of the varieties producedin country c, Ic,exp groups the indices of the varieties produced in country c andexported to country c. Imported varieties are thus treated in the same way asdomestic ones; for an equal price, they will be addressed the same demand. σ

is the elasticity of substitution between varieties (σ > 1).The consumer seeks to maximise his utility under the budget constraint. As-

suming trade to be balanced, this constraint is:

Mc = Hc + Pc Dc = Hc +∑

i∈Ic ∪ Ic,exp

Pic Dic. (3)

Where the price of the homogeneous good is chosen as the numeraire (thisgood is homogeneous and assumed to be tradable without cost, its price is thesame in both countries), Mc is the income of country c, Pic is the price paid forone unity of variety i in country c, Dc is the composite index of consumptionof the differentiated good in country c, and the price index Pc for the bundle ofdifferentiated good in country c is defined by:

Pc =( ∑

i∈Ic ∪ Ic,exp

P1−σic

)1/(1−σ)

. (4)

As a result, the demand for the two types of goods in country c is:

Hc = (1 − α)Mc, Dc = αMc/Pc. (5)

And the demand addressed, in country c, for variety i is:

Dic = αMc Pσ−1c P−σ

ic . (6)

That is, the demand addressed to a given variety depends on the number ofsupplied varieties, on the consumption price index of the homogeneous goodin the country, and on its own price.

2.2. The supply side

The production in the homogeneous-good sector is supposed to exhibit con-stant returns to scale, with a productivity equal to one, so that the labor input


Lc,H required to produce Qc,H is:

Lc,H = Qc,H . (7)

As a consequence, as soon as each country has a non-zero production in thisgood (which we will assume to be the case), the nominal wage in both countriesis equal to one (wc = 1).

For the production of the differentiated good, the cost function of firm i is:

Ci = βi Qi + K (8)

where Qi is the output, βi is the (constant) marginal cost of production, and Kis the fixed cost (costs correspond to labor inputs, as this is the only productionfactor). The existence of a fixed production cost induces a natural monopoly inthe production of each variety and we will assume that each firm produces onlyone variety. There is consequently a one-to-one correspondence between firmsand varieties. The fixed cost is assumed to be identical for all firms, and fromone country to another. The heterogeneity of firms stems from their marginalcost βi , which is firm-specific. The differences in marginal cost can be linkedto differences in organization, management, production techniques, the rate ofdefault, as well as the quality of workers.3

Competition is modelled a la Bertrand, assuming that conjectural variationsare null (for every couple of varieties (i, j), ∂Pj/∂Pi = 0) and that each firm consid-ers itself small with respect to the market (∂P/∂Pi = 0). For sales in the domesticmarket, profit maximization then leads firm i to set its price as follows:

Pi,c = σ

σ − 1βi , ∀i ∈ Ic. (9)

Under these hypotheses of monopolistic competition, the mark-up ratio be-tween the price and the marginal cost is constant and does not depend on themarket share of the firm. The heterogeneity of firms marginal costs is thereforereflected proportionally in prices, and defines an asymmetrical equilibrium interms of market share and profit with the most efficient firms having a highermarket share and higher profits.

The firms which do not export thus earn a profit:

i = ϕ α Mc Pσ−1c β1−σ

i − K , ∀i ∈ Ic,non-exp. (10)

Where we have noted ϕ = (σ −1)σ−1σ−σ , i denotes the profit of firm i, Ic,non-exp isthe set of country c’s firms which do not export (therefore, Ic = Ic,exp ∪ Ic,non-exp).

Exporting firms incur a fixed cost, Kexp, which is a consequence of the costs ofaccess to information, marketing, management, or of technical and commercialadaptation, which is inherent to exporting. As for the fixed production cost, wewill assume that this has to be paid at the beginning of each period.4 Exporting

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firms also incur a tariff barrier ad valorem, with a rate t . This tariff t introduces agap between the price effectively received by the firm and the price paid by theconsumer, Pconsumer. Profit maximization leads the exporters to apply the samemark-up ratio as in their domestic market, between their marginal cost and theprice they receive. However, the consumer must, in addition, pay for the tariff.As a consequence the firm receives the same unit price Pi,c (defined in (9)) as itdoes for sales in its domestic market, but the unit price paid by the consumeris:

Pconsumeri c = (1 + t)

σ

σ − 1βi , ∀i ∈ Ic,exp. (11)

The profit of exporting firms is thus:

i = ϕαMc Pσ−1c β1−σ

i − K︸︷︷︸profit on the domestic market ( i, dom)

+ ϕαMc Pσ−1c (1 + t)−σ β1−σ

i − Kexp︸︷︷︸profit on the foreign market ( i, exp)

, ∀i ∈ Ic,exp

(12)

In this expression, two terms are distinguished for the sake of clarity. The firstone corresponds to the profit on the domestic market, in the sense that it isthe profit the firm would earn, would it sell only in its domestic market; ex-porting involves an additional profit, which we call profit on the foreign market(or export-profit), and which corresponds to the second term.

The tariff revenue is assumed to be redistributed through a lump-sum transfer.For each country, the income is then

Mc = Lc +∑i∈Ic

i + t

1 + t

∑i∈Ic,exp

Pi,c Di,c (13)

where Lc is the exogenous endowment of country c in labor.

2.3. Equilibrium for a given population of firms

For each variety of the differentiated good,

Qi = Di,c + Di,c, ∀i ∈ Ic (14)

where the second term is zero for the non-exported varieties. The equilibriumon the labor market requires

Lc = Lc,H + Nc K + Nc,exp Kexp +∑i∈Ic

βi Qi . (15)

For a given population of firms, Equations (2) to (15) enable the general equilib-rium of the two economies to be characterised (the equilibrium in thehomogeneous-good market does not need to be written explicitly because


of the Walras’ law). Note in particular that for given parameters and factorendowments, trade flows in the differentiated-good sector depend on the num-ber and the characteristics (level of marginal cost and export status) of firmsproducing the differentiated good in each country. These trade flows, if any, canbe either two-way or one way (if one country has no exporting firm). Trade inhomogeneous good, in contrast, is always one-way, and compensates exactlythe net balance of trade in the differentiated-good sector.

2.4. Firms behavior: entry, exit, and export status

Let us turn to the process of entry and exit of firms in the differentiated-goodsector. As in Montagna (1995), both entry and exit are free, but potential entrantsdo not know their marginal cost; they only know the distribution from which itwill be drawn if they decide to produce.

Concretely, the firm-specific marginal cost is randomly drawn from a continu-ous uniform distribution defined over a country-specific interval, [βc,inf; βc,sup] =[φc(1−δ); φc(1+δ)] (with φc > 0 and 0 ≤ δ < 1). The larger this interval is, the moreheterogeneous the firms are likely to be within the industry. When the intervalreduces to a single point (δ = 0), firms are homogeneous. The two country-specific intervals are thus assumed to be homothetic, that is: βA,sup/βB,sup =βA,inf/βB,inf = φA/φB . The standard deviation compared to the average is thena priori identical for both distributions.5 This means that the ex ante degree ofuncertainty of the production process is the same in both countries. Thereby,we assume that the uncertainty in marginal cost is linked to the disparity in in-dividual performances, while international differences in the expected marginalcost are related to country-wide conditions. φA < φB would, for instance, meanthat these country-wide conditions are more favourable in country A.

If a firm chooses to enter the market, i.e., to produce, it incurs a fixed produc-tion cost. It will then (and only then) know the level of its marginal cost. Eachfirm can also freely choose whether to export or not, but the decision to exportis taken once the marginal cost is known.

Each firm has to pay the fixed production cost at the beginning of each pro-duction period. This periodical repetition is justified by the depreciation or thetechnical evolution of material investments (or by the fact that, when the invest-ments are reversible, firms only pay for their use-cost), and by the necessaryadaptation of immaterial investments (like training, R&D, marketing, etc.) to mar-ket evolutions. The fixed export cost is also incured at each period as a resultof the necessity to maintain the capacity to export.6

Each incumbent is free to quit the market at each period: each will do so if it’sprofit is negative. Each firm can also change its export status at each period.The decision with respect to this will be taken on the basis of the sign of theexport profit for the current period.

The behavior of firms can then be summarised as follows. Potential entrantsare assumed to be risk-neutral and to form static expectations about their future

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profits, taking as given the number of firms in each market. As a consequence,some will decide to enter the market if the expected profit on their domesticmarket is non-negative. Among those who entered the market, only firms whosemarginal cost is low enough to enable the export receipts to compensate the as-sociated fixed cost, will export. At each period, only those whose marginal costis sufficiently low enough to be able to make non-negative profits will survive.

2.5. Steady-state equilibrium market structure

We assume that if a steady-state equilibrium is reached, no firm can exportwithout making non-negative profits in the domestic market, i.e., that a firmmaking negative profits in its domestic market cannot export profitably. Thishypothesis is made for the sake of coherence with the hypothesis of monopo-listic competition.7

The population and export status of firms will reach an equilibrium when threeconditions are met for each country:

There is no exit because all incumbents make non-negative profits on theirdomestic market:

∀i ∈ Ic, βi ≤ β∗c / c(β

∗c ) = ϕαMc Pσ−1

c β∗1−σc − K = 0. (16)

There is no shift in export-status as each exporter makes non-negative export-profit, and each nonexporter could not earn positive profit by exporting:

∀i ∈ Ic,exp, ∀J ∈ Ic,non-exp, βi ≤ β∗c,exp and β j ≥ β∗

c,exp/ c,exp(β∗c,exp)

= ϕαMc Pσ−1c (1 + t)−σ β∗ 1−σ

c,exp − Kexp = 0. (17)

There is no more entry in the markets because the expected profits of potentialentrants is non-positive:

E( c) =∫ βc,sup

βc,inf

[ϕαMc Pσ−1

c β1−σ − K]

fc(β) dβ

+∫ β∗

c,exp

βc inf

[ϕαMc Pσ−1

c (1 + t)−σ β1−σ − Kexp]

fc(β) dβ ≤ 0 (18)

where fc(β) = 1/(βc,sup − βc,inf) = 1/(2δφc) is the distribution density function ofrandom variable β in country c. c,exp denotes export-profit, β∗

c is the maximummarginal cost required for a firm to survive in country c, β∗

c,exp is the maximummarginal cost required for a country c’s firm to export (if exporting is not prof-itable even for a firm with the lowest marginal cost possible in country c, βc,inf,then we will set β∗

c,exp = βc,inf). In Equation (18), the first integral corresponds tothe expected domestic profit, which is calculated over all of the possible valuesof β.8 The second integral corresponds to the export profit. As the choice of


the export status is made only once the marginal cost is known, this secondintegral only takes into account the cases where the marginal cost is sufficientlylow enough to enable the export profit of the firm to be positive; its superiorbound is not βc,inf, but β∗

c,exp.Given our hypothesis that a firm making negative profits in its domestic market

cannot export profitably, these three equations, together with Equations (2) to(15), define the free entry-exit steady-state equilibrium of the two economies.These equilibrium conditions for entry and exit are stable. If a firm does notmeet condition (16) or (17), then it will either exit or change its export status.If Equation (18) is not true, then the expected profit for potential entrants ispositive; this would induce some firms to enter the market, thus lowering itsprofitability (because they lower the price index of the industry as soon as theyare able to stay on the market), and therefore lowering the expected profit forall potential entrants until Equation (18) is met.

Note that in spite of the free entry-exit, profits are not necessarily zero inthe steady-state, as potential entrants face uncertainty about the value of theirmarginal costs (See Montagna, 1995).

As soon as at least one country c’s firm export in the steady-state, then con-dition (17), reflecting the stability of firms’ export status, shows that exportershave a lower marginal production cost than nonexporters. Consistently with thestylised facts, exporters are bigger and more efficient than nonexporters in thismodel, and more profitable.9

3. The nature of trade

Depending on the parameters and population of incumbent firms, the set-upexplained previously can lead to different kinds of trade flows.

3.1. Case of prohibitive trade costs

Trade costs are prohibitive when, in both countries, no firm can export. Moreprecisely,

c,exp(βc,inf) ≤ 0 ⇒ Ic,exp = ∅. (19)

In this case, no trade takes place, and both countries are in autarky. The marketstructure is then determined by equilibrium conditions (16) and (18). As shownby Montagna (1995), the maximum marginal cost βc,aut required to be profitableis then10:

βc,aut =( ∫ βc,sup

βc,inf

β1−σ fc(β) dβ

)1/(1−σ)

= φc

( ∫ 1+δ

1−δ

β1−σ

2δdβ

)1/(1−σ)

= φcβaut

(20)

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where we have noted for the sake of simplicity, βaut = (∫ 1+δ

1−δ

β1−σ

2δdβ)1/(1−σ). Note

that βaut only depends on the parameters δ and σ , and can thus be consideredas a parameter here. As (βc,aut) = 0, proposition (19) can be re-written:

(1 + t)σ Kexp

K≥

(φc

φc

βaut

1 − δ

)σ−1

⇒ Ic,exp = ∅. (21)

The term of the inequality on the left measures the importance of trade costscompared to the fixed production cost. Indirectly, it makes it possible to com-pare the profitability of sales in the foreign market with respect to the profitabilityof sales in the domestic market. If this term exceeds the one as the right handside, it means that export costs are so high that even the most efficient firm ina country cannot export profitably. Thus, trade costs are prohibitive, and thetwo countries do not trade at all.

3.2. Case of “semi-prohibitive” trade costs

The level for which trade costs turn to be prohibitive depends on the relativea priori technological levels of the two countries. Let us assume that country-wide production conditions for the differentiated good are more favorable incountry A than in B (φA < φB), and suppose that

(φB

φA

βaut

1 − δ

)σ−1

>(1 + t)σ Kexp

K≥

(φA

φB

βaut

1 − δ

)σ−1

. (22)

In this case, trade costs are too high (compared to the fixed production cost)to make exports profitable for any firm in country B. In contrast, at least withthe autarkic market conditions in B, exporting is profitable for a firm whosemarginal cost is equal to the minimum bound of the distribution in country A. Asa consequence, if at least one firm in country A has a marginal cost sufficientlyclose to the lower bound of the distribution (which we can assume to be thecase, if the number of firms is sufficiently large), then it will export. A purelyinter-industry trade takes place where country A exports differentiated goodsto country B, and imports homogeneous goods from there (trade between bothcountries remains balanced, by assumption).

3.3. Case of non-prohibitive trade costs

When trade costs are sufficiently low they are not prohibitive, even for the econ-omy where country-wide production conditions for the differentiated good areless favorable (let us still assume that this country is B: φA ≤ φB ). However, weassumed that exporting is not profitable for a firm that cannot earn non-negative


profits on its domestic market alone. This hypothesis implies that,

(1 + t)σ Kexp

K≥ 1. (23)

This simply means that when this inequality11 is not met, the model is not well-suited to describe the situation, because the differentiated-good industry hasan oligopolistic structure.

Given this inferior bound on trade costs, it can be shown that there existsϕ0 ∈] 1−δ

βaut; 1[ such that:{

∃TC(φA/φB) ∈]

1;(

βaut

1 − δ

φA

φB

)σ−1[/{(1 + t)σ Kexp

K< TC(φA/φB)

⇒{

βA,exp > βA,inf

βB,exp > βB,inf

}}⇔ φA

φB∈ [ϕ0; 1] (24)

Proof: See Appendix 1. ✷

This means that if the ratio of expected productive efficiencies between the twocountries exceeds a given level (ϕ0), then sufficiently low trade costs lead toa complete specialisation.12 The international efficiency difference is so high,in this case, that an increased competition from country A’s exporters inducesthe cessation of country’s B production in the differentiated-good sector.

In contrast, if the ratio of expected productive efficiencies does not exceedϕ0, then sufficiently low trade costs enable a two-way trade to take place in thedifferentiated goods sector. This trade needs not be balanced by itself, as tradein the homogeneous sector then just balances the net flow in the differentiated-good sector.

3.4. Synthesis

The nature of trade can finally be summarised in Figure 1, where the size of thevarious areas depend on a firms heterogeneity. The case of prohibitive tradecosts corresponds to area (a), where no trade occurs. For what we called above“semi-prohibitive” trade costs (area (b)), trade is purely inter-industry; the a priorimost efficient country for the production of the differentiated-good productexports this good, and it imports only homogeneous goods.

When trade costs are inferior to a given threshold (dependant on the degree offirms heterogeneity) and the difference between country-wide production costlevels in the differentiated good is high, then trade remains purely interindustry.Here there is complete specialisation, as the less efficient country has no moreproduction in the differentiated-good sector. If country-wide production costlevels in the differentiated good are sufficiently similar, then two-way trade inthe differentiated-good may occur, together with a one-way trade flow in thehomogeneous good product.

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Figure 1. The nature of trade.

The nature of the influence of the degree of homogeneity on the shape ofthese areas can be somewhat precise. It can be shown that for any givenσ > 1, [βaut/(1 − δ)] there is an increasing function of δ over the interval [0;1[(See proof in Appendix 2). This shows that, ceteris paribus, as the degree ofhomogeneity among firms increases, the surface of area (a) decreases, whilethe surface of (c) increases. The effect on (b) is uncertain a priori. While it can-not be proved rigourously, (d) is most likely to be reduced when the degree ofheterogeneity increases.

4. Implications for market structure and productive efficiency

We have just emphasized that a firm’s heterogeneity is one of the determinantsof trade. Conversely, trade has an impact on the nature of the population of firms.In particular we will focus on the influence induced on average productive effi-ciency. For the sake of simplicity, we will study separately purely inter-industrytrade and a case intra-industry trade.

4.1. Implications in a context of purely inter-industry trade

Let us first assume that trade costs are lowered from a prohibitive level toa “semi-prohibitive” level. What are the consequences for the least-efficient


country (the one whose a priori expected marginal production cost of thedifferentiated good is the highest, say country B)? Since they catch part of thedemand, imports lower the profitability of the domestic market in this country.The upsurge of imports thus causes the bankruptcy of the least-efficient incum-bent firms in country B. Moreover, in the absence of export opportunities, theexpected profit of a potential entrant is also reduced, and turns out to be strictlynegative; no firms enter the market since the reduction in the number of firmsis not sufficient to restore a positive expected profit for potential entrants. Theleast-efficient firms are crowded out, without creating any entry opportunityfor new firms. The impact on country B’s average productive efficiency in thedifferentiated-good sector is obviously positive, but no domestic firm sees itsprofit increase. Formally, Equations (17) and (18) are inactive as equilibriumconditions for country B. From the initial firms population, only those whosemarginal cost is inferior to β∗

B , as defined by (16), survive.In contrast, in the exporting country (country A), Equations (16) to (18) are all

active in setting the steady-state equilibrium market structure. Equation (18) ex-presses the zero-expected-profit condition for potential entrants. As in autarky,the expected profit in the domestic market is taken into account; but here it issummed up with the expected export-profit. The latter is positive as the exportdecision is taken only once the firm’s marginal cost is known. Under free-trade,potential entrants take into account the additional profit they will realize if theyare sufficiently efficient to export profitably. Consequently, they may be proneto enter the market even when the profitability of the domestic market at a givenlevel of efficiency is inferior to its autarkic level, i.e., when the expected profit inthis single market is negative. The perspective of new profits through exportsattracts new producers, whose entry exacerbates competition.

Formally, using the expression of βA,aut in (20), the zero-expected-profit con-dition for potential entrants expressed through Equation (18) implies that:

(βA,aut) = ϕαMA Pσ−1A β1−σ

A,aut − K

≤ −∫ β∗

A,exp

βA,inf

[ϕαMB Pσ−1

B (1 + t)−σ β1−σ − Kexp]

f A(β) dβ < 0. (25)

This means that the profit of a firm with a marginal cost βA,aut is strictly negative.As a consequence, the maximum marginal cost required to survive in countryA (defined by Equation (16)) is inferior to the one required in autarky. Or, toput it in other words, the efficiency level required to survive is higher. In addi-tion, the most efficient firms, which benefit from export possibilities, increasetheir market shares with respect to autarky. Thus, by the conjunction of thesetwo effects, exporting increases the expected average efficiency level in thedifferentiated-good sector.

As the minimal efficiency level required is higher, the probability of failure fora new entrant is also higher in the free-trade case. On the other hand, the mostefficient firms have the opportunity, through exports, to enlarge their markets,

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and thereby their profits. The market of the exporting country thus becomesmore risky, but more profitable for the “winners.” There is a low probability ofearning a large amount. Moreover, the disparities in the sizes and profits offirms increase, since the most efficient (that is the biggest in terms of size andprofit) are those which most increased their production and their profits thanksto exports.

4.2. Implications in a context of intra-industry trade

When two-way trade in differentiated products takes place, the mechanisms arebasically the same as one-way trade: imports lower the profitability of the do-mestic market, while exports increase the expected profit for potential entrants.The implications are clear-cut for the most efficient country; as previously, theperspective of new profits through exports attracts new producers, whose entryexacerbates competition. The zero-expected-profit condition for potential en-trants (Equation (18)) is then active in setting the equilibrium. As stated throughEquation (25), this implies that the marginal cost required to survive is inferiorto its autarkic level.

For the least efficient country (country B), two cases have to be distinguished:

i. Trade opening does not induce any new entry in the domestic market, be-cause the profit opportunities carried out by exports are not sufficient to bal-ance the negative effect of imports on the expected profit for new entrants.In this context, the mechanisms are similar to those mentioned previouslyfor the country importing differentiated goods and exporting homogeneousgoods; the zero-expected-profit condition for potential entrants is not activein setting the equilibrium, and the least-efficient firms have to quit the mar-ket. The only difference with the case of purely inter-industry trade is thatcountry B’s most efficient firms can export. This also contributes to increas-ing the average productive efficiency in the differentiated-good sector, as itincreases the output of the most efficient firms.

ii. The profit opportunities carried out by exports attract new producers, in-cluding in the least efficient country. This is the case as soon as the dif-ference in average efficiency level is sufficiently weak while trade costs aresufficiently low. In this case, even the least efficient country experimentswith mechanisms similar to those described in the previous section for thecountry exporting differentiated goods and importing homogeneous goods.The zero-expected-profit condition for potential entrants is active in settingthe equilibrium, and the entry of new producers induces a new selectionamong firms. In this context, it can be shown that the probability for a newentrant to survive is superior in the least efficient country, while the proba-bility for a new entrant to be able to export profitably is superior in the mostefficient country (See proof in Appendix 3). This is coherent with the empiricalresult of Abd-El-Rahman (1991), which shows that exporting firms are more


atypical, in terms of productivity, in industry with comparative disadvantage(i.e., the differentiated-good sector in the least efficient country) than in thoseindustries enjoying a comparative advantage (the differentiated-good sectorin the most efficient country). According to Abd-El-Rahman, the specific ad-vantage required for a firm in order to export is more important in an industrysuffering a comparative disadvantage.

4.3. The impact of international trade: defensive logic vs. offensive logic

Within the study of the impact of trade on firms and on average productiveefficiency in the differentiated-good sector, two different types of mechanismsemerge; defensive logic, and offensive logic:

i. The defensive logic is import-driven; it is at work when a country has eitherno exporting firm in the differentiated-good sector, or when the profit op-portunities carried out by exports are not sufficient to balance the negativeeffect of imports on the expected profit for new entrants. In this case, tradeopening does not induce any new entry, and the only firms whose profit mayincrease are the most efficient ones, if they are able to export profitably. In-ternational trade does improve the average productive efficiency level in thedifferentiated-good industry because of the exit of the least efficient firms,and because of the increased output of the most efficient firms when theyare able to export.

ii. The offensive logic is export-driven; it is at work when the profit opportunitiescarried out by exports attract new producers, compared to autarky. Theirentry then exacerbates competition in the sector, reducing the profitabilityof the domestic market for incumbent firms. In addition, the probability offailure for potential entrants is increased compared to autarky. As before,average productive efficiency in the sector is increased as a result both ofthe exit of the least efficient firms and of the increased output of the mostefficient ones. Finally, when trade costs and the international difference inaverage productive efficiency are sufficiently low, this offensive logic dom-inates the defensive one; the maximum marginal cost required to surviveis proportionally more reduced, compared to autarky, in the most efficientcountry where the export intensity and the proportion of exporting firms inthe differentiated-good sector is the highest.

5. Conclusion

The model presented in this article aims at enlightening the interactions be-tween international trade and firms’ heterogeneity, in a context of monopolisticcompetition. Based on the autarkic model proposed by Montagna (1995), thecoexistence of firms with different marginal costs is the consequence of theuncertainty faced by potential entrants about the level of their marginal cost.

306 JEAN

The differences in export status then stem from the existence of a fixed cost ofexporting.

This model fits well with the stylized facts described in the introduction; itis coherent with the fact that only a fraction of firms export anything, that ex-porters are larger and more productive than nonexporters, that exporters aremore atypical in this last respect when the country suffers from a compara-tive disadvantage in the sector, and that export-driven cross-firm reallocations(“share effects”) significantly increase average productive efficiency. In addition,it suggests that the positive effect of import penetration rate on industry-wideproductive efficiency (as described by Cortes and Jean, 1997, or by Hine andWright, 1998) may also be partly the result of cross-firm reallocations. It alsosheds some light on how trade costs, country-wide efficiency differences, andfirms heterogeneity interact to determine the nature of trade.

Finally, we showed that the impact of trade stems from two different logics;one defensive and import-driven, the other offensive and export-driven. Thisresult emphasises that trade is not only a threat that eliminates the least efficientfirms. The trade-induced increase in competitive pressure may also be the resultof the entry of new producers, attracted by the new profit opportunities carriedout by exports. Furthermore, as soon as country-wide efficiency differencesand trade costs are sufficiently low, this offensive logic dominates the defensivelogic in shaping the impact of trade.

Appendix 1

Proposition (21) implies that

IfφA

φB≤ 1 − δ

βaut, then IB,exp = ∅. (A.1)

In other words, if the lowest marginal cost possible in country B is superiorto the maximum marginal cost required in A to survive in the autarkic equilib-rium, then no firms in country B will be able to export profitably to country A,whatever the trade costs. In addition, no firm can survive in country B when(1 + t)σ Kexp/K = 1 (the maximum marginal cost required to survive is then thesame as the maximum marginal cost required to export profitably). By continu-ity (and given that the profitability in country B’s market decreases when tradecosts decrease), there is a critical value for the ratio (1+ t)σ Kexp/K of trade costto the fixed production cost, under which there is no more production of thedifferentiated good, in B.

Now, if both countries have the same expected productive efficiency in thesector (φA/φB = 1), then (1 + t)σ Kexp/K < (βaut/(1 − δ))σ−1 is a sufficient condi-tion to make sure that two-way trade in differentiated goods is likely to takeplace (in both countries, the maximum marginal cost required to export issuperior to the inferior bound of the random distribution of marginal costs:βc,exp > βc,inf).


In addition, ceteris paribus, the export-profitability is a decreasing functionof (1 + t)σ Kexp/K . By continuity, this makes it possible to establish that there isa value ϕ0 such that:

{∃TC(φA/φB) > 1

/{(1 + t)σ Kexp

K< TC(φA/φB) ⇒

{βA,exp > βA,inf

βB,exp > βB,inf

}}

⇔ φA

φB∈ [ϕ0; 1]. (A.2)

Given proposition (A.1), necessarily ϕ0 > (1 − δ)/βaut. In addition, following propo-sition (21), (1 + t)σ Kexp/K = (βaut/(1−δ)φA/φB)σ−1 implies that no firm can exportprofitably in country B. As a consequence, the upper bound of trade costs com-patible with two-way trade TC(φA/φB), is inferior to (βaut/(1 − δ)φA/φB)σ−1. Thisis sufficient to prove that proposition (24) is true.

Appendix 2

Given that βaut = (∫ 1+δ

1−δ

β1−σ

2δdβ)1/(1−σ), it comes: (

βaut

1−δ)1−σ = ∫ 1+δ

1−δ

(β

1−δ)1−σ

2δdβ.

Changing the variable β under the integral for θ = β/(1 − δ), we obtain:

(βaut

1 − δ

)1−σ

= 1 − δ

2δ

∫ 1+δ1−δ

1θ1−σ dθ.

The derivative of this last expression with respect to δ is:

d

dδ

[(βaut

1 − δ

)1−σ]= 1 − δ

2δ

(1 + δ

1 − δ

)1−σ 2

(1 − δ)2− 1

4δ2

∫ 1+δ1−δ

1θ1−σ dθ

≤(

1 + δ

1 − δ

)−σ[1 − 1 + δ

4δ2(1 − δ)

]≤ 0, ∀δ ∈ [0; 1[.

This proves this announced property: for any given σ > 1, [βaut/(1 − δ)] is anincreasing function of δ over the interval [0; 1[.

Appendix 3

This Appendix aims at characterising the populations of firms when the zero-expected-profit condition for potential entrants is active in setting the equi-librium for both countries, that is when the offensive logic is at work both incountry A and B. When this is the case, the steady-state population of firms ischaracterised through Equations (16) to (18). In addition, we will assume thatthe expected profit for potential entrants is exactly zero, instead of assuming it

308 JEAN

to be non-positive. As long as the entry of firms is incremental, and for a largenumber of firms as it is assumed here, this is an acceptable approximation.

From Equation (17), defining the maximum marginal cost required to exportprofitably, it comes:

ϕαMc Pσ−1c = (1 + t)σ Kexpβ

∗ σ−1c,exp . (A.3)

This expression enables Equation (18), expressing the zero-expected-profitcondition for potential entrants, to be written in the following way (using alsothe expression of βaut in (20)):

K

[(1 + t)σ Kexp

K

(β∗

c,exp

φcβaut

)σ−1

− 1

]+ Kexp

∫ β∗c,exp

βc,inf

[(β∗

c,exp

β

)σ−1

− 1

]fc(β) dβ = 0.

(A.4)

This implies that

(β∗

c,exp

φc

)σ−1

= J

[(β∗

c,exp

φc

)σ−1]

(A.5)

Where the function J associates to every positive, real x (for the sake of sim-plicity, we assume here that σ �= 2):

J (x) = βσ−1aut

(1 + t)σ

[K

Kexp+ 1

2δ(σ − 2)

[x1/(σ−1) − (1 − δ)2−σ x

+ (σ − 2)(x1/(σ−1) − (1 − δ)

)]]. (A.6)

Its derivative is:

J ′(x) = βσ−1aut

(1 + t)σ1

2δ(σ − 2)

[x (2−σ)/(σ−1) − (1 − δ)2−σ

]. (A.7)

It is then straightforward to show that

∀x ∈ [(1 − δ)σ−1; (1 + δ)σ−1], −1 ≤ J ′(x) ≤ 0. (A.8)

In addition, J ((1−δ)σ−1) = K(1+t)σ Kexp

βσ−1aut (note: as trade costs are not prohibitive,

this implies that J ((1 − δ)σ−1) > (1 − δ)σ−1).As, (

β∗c,exp

φc)σ−1 ∈ [(1 − δ)σ−1; (1 + δ)σ−1], then Equation (A.5) implies that

(β∗

c,exp

φc)σ−1 < J ((1 − δ)σ−1).


On the other hand, from Equation (16) for country c and (17) for country c, itcomes:(

β∗c,exp

φc

)σ−1

= (1 + t)σ Kexp

K

(β∗

c

φc

)σ−1

≥ (1 − δ)σ−1. (A.9)

As a consequence, (β∗

c,exp

φc)σ−1 ∈ [(1 − δ)σ−1; (1 + δ)σ−1], ∀c = A, B.

Now, given (A.8), (J ◦ J )′ = J ′ ◦ J × J ′ implies that:

(J ◦ J )′((

β∗c,exp

φc

)σ−1)

= J ′((

β∗c,exp

φc

)σ−1)

× J ′((

β∗c,exp

φc

)σ−1)

∈ [0; 1],

∀c = A, B. (A.10)

As J is continuous and decreasing over the interval [(1 − δ)σ−1; (1 + δ)σ−1], thefunction J ◦ J is continuous and derivable over the interval [MINc=A,B(

β∗c,exp

φc)σ−1;

MAXc=A,B(β∗

c,exp

φc)σ−1], and:

∀x ∈[

MINc=A,B

(β∗

c,exp

φc

)σ−1

; MAXc=A,B

(β∗

c,exp

φc

)σ−1], 0 ≤ (J ◦ J )′(x) ≤ 1 (A.11)

Now, assuming that B is the least efficient country (i.e., φA ≤ φB), (A.5) impliesthat

(β∗

A,exp

φA

)σ−1

= φB

φAJ

(φA

φB

(β∗

A,exp

φA

)σ−1)

≥ (J ◦ J )

((β∗

A,exp

φA

)σ−1)

(β∗

B,exp

φB

)σ−1

= φA

φBJ

(φB

φA

(β∗

B,exp

φB

)σ−1)

≤ (J ◦ J )

((β∗

B,exp

φB

)σ−1) (A.12)

Therefore:

(J ◦ J )

((β∗

B,exp

φB

)σ−1)

− (J ◦ J )

((β∗

A,exp

φA

)σ−1)

≥(

β∗B,exp

φB

)σ−1

−(

β∗A,exp

φA

)σ−1

(A.13)

Given (A.11), this implies that (β∗

B,exp

φB) ≤ (

β∗A,exp

φA). This means that the probability

for a new entrant to export is superior in the most efficient country.The first equality in (A.9) then implies that (

β∗A

φA) ≤ (

β∗B

φB): the probability to survive

for a new entrant is inferior in the most efficient country. Or, in other words, theminimum required efficiency has increased, compared to autarky, in the mostefficient country than in the least efficient country. This proves the announcedproperty.

310 JEAN

Acknowledgments

I am grateful to Antoine Bouet, Lionel Fontagne, Michel Fouquin, Jean-LouisGuerin, Catia Montagna and Adrian Wood for helpful comments and sugges-tions on an earlier version of this paper. I also benefited from comments by theparticipants at the ERWIT, CEPR-EPRU, Copenhagen, where this paper waspresented. Usual disclaimers apply.

Notes

1. Many of these results actually deal with heterogeneity among plants, as individual data is avail-able for plants in some countries, and for firms in some others. We will use the term “firm” inthe following, and for the sake of simplicity we will not deal with the problem of multi-plantfirms. However, Clerides, Lach, and Tybout (1998, p. 914) reports that in semi-industrialisedcountries where the calculation is possible, 95 percent of plants are owned by single-plantfirms.

2. Montagna (2001) also proposed an extension of this model to a general equilibrium framework,in order to study the effect of economic integration. But this approach is different as it focuseson the study of the integrated economy. It does not deal with nature of trade, and it is notcoherent with the stylized facts described above.

3. Although it is not strictly compatible with the setup of the model, differences in quality of thegoods produced could also be interpreted as differences in cost per “standardised”unit.

4. There are also sunk entry costs to the export market, which are no longer paid afterwards. Forthe sake of simplicity, we do not take them into account. The fixed cost considered here canalso be understood as the uniform flow whose discounted value is equal to the entry cost, plusthe cost of maintaining the capacity to export.

5. This is not the case when the second interval is obtained from the first one by translation, asassumed by Montagna (2001).

6. This hypothesis is supported by the finding of Roberts and Tybout (1997, p. 559), that the benefitof export experience depreciates very quickly.

7. When trade costs are very weak compared to the fixed production cost, exporting can beprofitable, even for a firm that cannot sell profitably in its domestic market. In this case, allfirms export. This is not unrealistic, if we think for instance about the aeronautics, automotive,pharmaceutical or computer industries. These industries have indeed the specificity of havingvery high fixed costs which, in the context of liberalized trade and low transportation cost, canonly be made profitable through exporting. However, the strategic interaction of firms is crucialin these type of highly concentrated industries. The model presented here, with monopolisticcompetition, therefore does not seem very adequate in this case.

8. This calculation only concerns one period. The implicit assumption is that an entrepreneur canonly raise the required capital if his expected profit in the following period is non-negative. Notethat even the firms with the lowest productivity will produce during one period, because theirvariable profit is positive even though it cannot compensate for the entry cost. I thank MarcMelitz for drawing my attention to these points, which are treated differently in the paper hewrote (Melitz, 1999).

9. As a consequence, differences in productivity among firms are not wholly counterbalancedby differences in wages. This is important, as emphasised by Bernard et al. (2000, p. 2). Aslong as differences in productivity are completely reflected in prices, apparent productivityas measured through value added per worker is the same across firms, regardless of theirdifferences in productive efficiency. In the present case, exporters’ productivity is higher, evenwhen it is measured through value added per worker.


10. This expression is obtained assuming the expected profit to be exactly zero, instead of assumingit to be non-positive. As long as the entry of firms is incremental, and for a large number offirms as it is assumed here, this is an acceptable approximation.

11. The corresponding level for trade cost, compared to the fixed production cost, may seem veryhigh. This is due to the fact that the demand addressed to foreign variety is assumed to bethe same, for the same price, as the demand addressed to domestically-produced varieties. Inpractice, the demand is usually superior for the domestic variety, for the same price; in theseconditions, the inferior limit for ratio (23) is inferior to unity.

12. We assume that the share of the differentiated good in consumption is sufficiently low, so thatthe most efficient country has enough labor to produce sufficiently enough to satisfy the wholeconsumption of this good.

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international trade and firms' heterogeneity under monopolistic competition

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