CSIS Discussion Paper No. 85

Trade and gains from trade at the extensive andat the intensive margins∗
Kristian Behrens†

Takaaki Takahashi‡

July, 2007

Abstract
We present a general equilibrium trade model that combines monopolistic competition à la Dixit-Stiglitz-Krugman with Cournot competition à la MarkusenBrander. Firms are free to either compete for the same existing varieties, or torelax competition by introducing new diﬀerentiated varieties at some cost. We
show that both the gains from, and the distributional impacts of, trade depend onwhether ﬁrms compete for the same varieties (‘local competition’ at the intensive
margin) or for diﬀerent varieties (‘global competition’ at the extensive margin). In
particular, when countries produce the same set of varieties in autarky and whenfactors are owned equally, trade may neither expand consumers’ choice sets nor
yield any gains.
Keywords: monopolistic competition; Cournot oligopoly; gains from trade; product diversity; distributional impacts of trade
JEL Classiﬁcation: D43; D51; F12∗

We thank José Sempere-Monerris for useful comments and suggestions. Kristian Behrens gratefully
acknowledges ﬁnancial support from CORE and from the European Commission under the Marie Curie
Fellowship MEIF-CT-2005-024266. Part of this paper was written while Takaaki Takahashi was visiting
CORE, Université catholique de Louvain (Belgium) under the fellowship of the sending-researchers-tospeciﬁc-countries program funded by the Japan Society for the Promotion of Science, whose support is
gratefully acknowledged. The usual disclaimer applies.
†
Center for Operations Research and Econometrics (CORE), Université catholique de Louvain, Belgium. E-mail: behrens@core.ucl.ac.be‡Center for Spatial Information Science (CSIS), University of Tokyo, Japan; and CORE, Université
catholique de Louvain, Belgium. E-mail: takaaki-t@csis.u-tokyo.ac.jp1

1

Introduction

Intra-industry trade in similar products accounts for a substantial and increasing partof world commodity trade. Since the late 1970s, this observation has attracted a lotof attention and has stimulated a huge amount of theoretical and empirical research.1
From a theoretical perspective, countries engage in intra-industry trade for two main
reasons: product diﬀerentiation (Krugman, 1979, 1980; Helpman, 1981) and oligopolistic
ﬁrm behavior (Markusen, 1981; Brander, 1981; Brander and Krugman, 1983). Whilethese two reasons both stem from imperfect competition in product markets and are by
no means mutually exclusive, they have been dealt with rather separately until now. On
the one hand, the general equilibrium new trade literature examines the impact of productdiﬀerentiation focusing on monopolistic competition by treating ﬁrms as negligible anda-strategic players. Doing so allows one to deal with the various interrelations between
product and factor markets, yet comes usually at the price of quite stringent simplifying
assumptions like constant demand elasticity.2 The ‘strategic trade policy’ literature has,on the other hand, extensively dealt with oligopolistic competition in what is essentiallya partial equilibrium framework. Consequently, its analysis of factor markets and theirinterrelations with product markets is limited, and normative conclusions need to bereached and extrapolated cautiously.
The ﬁrst line of research, namely, the general equilibrium new trade literature, has
focused almost exclusively on the highly tractable Dixit-Stiglitz-Krugman (henceforth,DSK) market structure (Dixit and Stiglitz, 1977; Krugman, 1980), which is the de facto
benchmark for establishing both positive and normative results. Despite its analyticaltractability and the deep insights it oﬀers, the DSK framework suﬀers from several wellknown drawbacks. First, there is a one-to-one relationship between ﬁrms and varieties.
A by-product of this rather stringent assumption is that, when trade occurs, all countries
become fully diversiﬁed in terms of the varieties they produce. Because there is no (ﬁnite)
reservation price, trade then mechanically leads to a quite signiﬁcant expansion of consumers’ choice sets, whereas the amount of each variety produced remains constant. In
other words, trade aﬀects the extensive margin of production (i.e., product diversity) but
not its intensive margin (i.e., production scale). Recent empirical evidence by Hummels
and Klenow (2005) shows that this is unrealistic and that the DSK framework starkly over1

Well-known early contributions include Krugman (1979, 1980, 1981), Helpman (1981), Lawrence andSpiller (1983) and Helpman and Krugman (1985). Although ‘new trade theory’ initially emerged to deal
with the empirical shortcomings of classical trade theory, the empirics of new trade theory are more
recent and still at an early stage (see, e.g., Head and Mayer, 2004, for a recent survey).
2
Ottaviano et al. (2002) and Melitz and Ottaviano (2005) present monopolistic competition models
with variable demand elasticity. Yet, their quasi-linear speciﬁcation rules out income eﬀects and reduces
the role of factor markets in the analysis. See also Behrens and Murata (2006) for a monopolisticcompetition model that allows to concisely isolate the variety eﬀect and the pro-competitive eﬀect. Yet,this model has only a single production factor and cannot deal with the redistributive aspects of trade.

2

predicts the relationship between market size and product diversity, particularly when it
comes to exports. Second, the main normative implication of the DSK framework is that,
contrary to the classical Ricardian or Heckscher-Ohlin predictions, all agents in all countries engaged in intra-industry trade may gain. This is because every consumer receives
the beneﬁts of trade that arise from the joint interaction of increasing product diversityand ‘love of variety’. Put diﬀerently, the increase in product diversity may more thanoﬀset the redistributive aspects of international trade (Krugman, 1981). Since this result
crucially hinges on the fact that trade increases consumers’ choice sets ‘suﬃciently’, one
may question its general validity in alternative modelling frameworks.
Our main objectives in this paper are twofold. First, we propose a model that allows
to embed an oligopolistic market structure into a general equilibrium framework with differentiated products, thereby tying more closely together the strands of monopolistic andoligopolistic competitions. To the best of our knowledge, only few contributions deal withboth product diﬀerentiation and oligopolistic ﬁrm behavior in general equilibrium until
now.3 This is largely explained by the fact that general equilibrium models of oligopolistic competition are plagued by serious conceptual and technical problems, including thesensitivity of equilibrium to the choice of numéraire (Gabszewicz and Vial, 1972) andthe bad behavior of reaction functions (Roberts and Sonnenschein, 1977). In such cases,
the existence of equilibrium is not assured because some agents are ‘large enough’ in theaggregate economy to be able to inﬂuence macroeconomic variables. As pointed out by
Neary (2003a,b), a simple solution to this problem consists in eliminating all ‘large’ agentsby assuming that each one is negligible to the aggregate economy.4 Our model is, therefore, made tractable by considering ﬁrms which are large in their own market but small
in the economy as a whole. Because of such a modelling strategy, we can analyze bothproduct diﬀerentiation and oligopolistic ﬁrm behaviors in general equilibrium, which is
the ﬁrst contribution of this paper to the existing literature.
Second, our model concisely captures the fact that trade aﬀects not only the extensive margin but also the intensive margins of production. Indeed, trade usually expands
consumers’ choice sets, by oﬀering access to more variety, and at the same time increases
output of individual varieties and reduces ﬁrms’ market power. These changes jointly map
into welfare gains, although only the ﬁrst one arises in the DSK model. To capture these
two eﬀects in a full general equilibrium framework, we consider the case of imperfectly
3

Neary (2003a,b) presents a general equilibrium oligopoly model of international trade, yet there is noentry at either the intensive or the extensive margin. Ishikawa et al. (2006) develop a general equilibrium
model in which oligopolistic ﬁrms interact with a free entry competitive fringe. Yet, they do not consider
trade issues and there is only a single production factor.
4This is typically done in continuum models of monopolistic competition. Formally, models with acontinuum of agents are closely related to nonatomic games where “the single player has no inﬂuence on
the situation but the aggregate behavior of “large” sets of players can change the payoﬀs” (Schmeidler,
1973, p.295).

3competitive ﬁrms producing a continuum of horizontally diﬀerentiated varieties underﬁrm-level scale economies using two production factors. We depart from the standard
new trade models by assuming that there is not necessarily a one-to-one relationship between ﬁrms and varieties, as several ﬁrms may produce the same variety in an oligopolistic
fashion. Thus, a ﬁrm competes against the others producing the same variety (strategic‘local competition’ ), as well as against those producing diﬀerent imperfectly substitutablevarieties (non-strategic ‘global competition’ ). This allows for markups that decrease with
the number of ﬁrms competing for each variety, while maintaining a tractable general equilibrium interaction framework between diﬀerent varieties. Firms’ decisions as to whether
to create new varieties or to remain competing on existing varieties are endogenously
driven by the trade-oﬀ between incurring additional sunk costs for creating a new variety(e.g., R&D and advertising costs), and enjoying less price competition as the result of aswitch from ﬁercer ‘local’ to softer ‘global’ competition. Stated diﬀerently, we account for
the fact that ﬁrms can relax competition by diﬀerentiating their products (Shaked andSutton, 1982).5 We investigate what the impacts of trade liberalization are on the extensive and on the intensive margins in such a setting. The basic idea is that trade increases
the competitive pressure on existing varieties since more ﬁrms compete for them, which
may lead some ﬁrms to diﬀerentiate their varieties in order to relax competition. Consequently, trade may lead to a non-trivial expansion of consumers’ choice sets by providingﬁrms with additional incentives to introduce new product varieties, as highlighted in the
trade-and-growth literature (e.g., Rivera-Batitz and Romer, 1991).
Our key results may be summarized as follows. First, we show that there is usually alarge number of autarky equilibria that can be sustained in each country. Furthermore, therelative ranking of equilibria is such that all agents always prefer more product diversity.
Focusing on symmetric equilibria only, we then investigate the impacts of free trade on
both countries and both production factors under two alternative assumptions. (i) thecase of ‘full cross-country product diﬀerentiation’ (henceforth, FCCPD), i.e., countries
produce disjoint subsets of varieties in autarky; and (ii) the case of ‘no cross-country
product diﬀerentiation’ (henceforth, NCCPD), i.e. countries produce identical subsets ofvarieties in autarky. In the case of FCCPD, we show that there are always gains fromtrade whether all agents own production factors equally, or some own only the variablefactors while the others own only the ﬁxed factors. In the case of NCCPD, the results
are drastically modiﬁed. Indeed, we show that trade only leads to gains when there is an
expansion in product variety, which will be the case if national markets are ‘suﬃciently
competitive’ before the trade opening. Furthermore, it is the variable factor that now
usually gains from trade, whereas the ﬁxed factor loses when there is no variety creation.
5

In Shaked and Sutton (1982), ﬁrms may relax price competition by diﬀerentiating their products
according to quality. Hence, they deal with vertical but not with horizontal product diﬀerentiation.
Weitzman (1994) proposes a model of localized competition in which the degree of (horizontal) productdiﬀerentiation is one of the ﬁrm’s strategic variables.

4

Only the endogenous introduction of a ‘suﬃcient’ mass of new varieties will lead to gains
from trade for all agents in all countries, thus revealing that the extreme assumptionof fully disjoint sets of varieties under autarky is far from innocuous for the normative
results.
The remainder of the paper is organized as follows. In Section 2, we present themodel and analyze the closed economy equilibrium. Section 3 then extends the model toa general equilibrium ‘reciprocal dumping’ model of international trade. Sections 4 and5 deal with the case of FCCPD and the case of NCCPD, respectively. Section 6 oﬀers
concluding remarks and points towards future research directions.

2

The model

Our model combines a standard monopolistic competition framework à la Dixit-StiglitzKrugman with a model of Cournot oligopoly à la Markusen-Brander.6 This allows for asetting in which ﬁrms are large in their own market but small in the aggregate economy.

2.1

Preferences

Consider an economy with a ﬁxed mass of L identical workers/consumers. The preferencesof a representative consumer over a set N of horizontally diﬀerentiated varieties (withmeasure n) are of the Dixit and Stiglitz (1977) ‘constant elasticity of substitution’ type.
He thus solves the following consumption problem:
(∫

σ−1
σ

U≡
x(v) dvN∫s.t.
p(v)x(v)dv = y,

max

σ
) σ−1

x(v), v∈N

N

where σ > 1 is the elasticity of substitution between any two varieties, and y is the consumer’s income.7 Because preferences are homothetic, the aggregate demand for variety
v is given by:()p(v)−σX(v) ≡ Lx(v) = g p(v) = ∫Y,
1−σp(z) dzwith Y = Ly.

(1)

N
6

In this respect, our contribution is closely related to recent work by Neary (2003a,b), who combinesa ‘quadratic’ monopolistic competition model with a Cournot duopoly. Yet, contrary to Neary, we allow
the mass of varieties to be endogenously determined and consider that more than two ﬁrms may compete
for each existing variety. This allows for both a richer industry structure and the creation of new varieties
in response to competitive trade pressures.
7
See Lawrence and Spiller (1983) for a closely related model. The main diﬀerence between theirapproach and ours is that they have a competitive outside sector but no oligopolistic ﬁrms, whereas thereverse holds true in our case.

5

Since there is a continuum of varieties, changes in any individual price p(v) have no impact
on the price aggregate so that()X(v)
g 0 p(v) = −σ
.
p(v)

2.2

(2)

Technology and market structure

There are two factors of production, both of which are supplied inelastically. The ﬁrst
factor (which we henceforth refer to as ‘labor’) enters the ﬁrm’s variable costs only, whereas
the second factor (which we henceforth refer to as ‘capital’) enters the ﬁrm’s ﬁxed costs
only. Each agent is endowed with one unit of labor and K/L units of capital, so thatthe aggregate labor and capital supply is given by L and K, respectively. We will relax
the rather stringent assumption of equal claims to production factors when discussing
the distributional impacts of trade, yet we may keep it for now. Let c and F stand forthe (constant) marginal labor requirement and the ﬁxed capital requirement of each ﬁrm,
respectively.
Each variety v ∈ N is produced by m(v) ∈ N ﬁrms, which are Cournot competitors
for that variety. Let qk (v) stand for ﬁrm-k’s supply of variety v. Denote furthermore by
w and r the wage rate and the rental rate of capital, respectively. Each variety-v ﬁrm
choses the quantity to maximize its proﬁtΠk (v) = [p(v) − cw] qk (v) − rF,(3)

subject to the product market clearing constraint qk (v) + Q−k (v) = X(v), where we deﬁne∑Q−k (v) ≡ l6=k ql (v). Note that since ﬁrms are negligible, they are ‘aggregate income’takers, which is both an empirically plausible and a technically convenient assumption(e.g., Bonanno, 1990). Using (2), this yields the following ﬁrst-order conditions:∂Πk (v)p(v)= [p(v) − cw] −qk (v) = 0.
∂qk (v)σX(v)In what follows, we assume that all ﬁrms producing the same variety v are identical and,
therefore, have the same market share: qk (v) ≡ X(v)/m(v). Straightforward calculationshows that the equilibrium price for variety v is then given byp(v) =

σm(v)cw,
σm(v) − 1(4)which is decreasing in the number m(v) of ﬁrms competing for variety v. It is furthermore
decreasing in σ: the closer substitutes the diﬀerent varieties are, the lower the equilibrium
prices for all varieties. The ﬁrst eﬀect is linked to local competition (i.e., competition
among the ﬁrms producing the same variety), whereas the second eﬀect is related to
global competition (i.e., competition among ﬁrms producing diﬀerent varieties).
6

Entry and exit of ﬁrms for each existing variety is assumed to be free. Consequently,
for any given value of the rental rate r, incumbent ﬁrms earn non-negative proﬁts:Yp(v)−σ∫Πk (v) = [p(v) − cw]
− rF
m(v)
p(j)1−σ djNY [σm(v) − 1]σ−1
− rF
=
]1−σ
∫ [
σm(j)
σ
σ+1
dj
σ m(v)N σm(j) − 1()≡ π m(v) ≥ 0,
for any v ∈ N ;

(5)

(6)

whereas a potential entrant will earn negative proﬁt:()
π m(v) + 1 < 0,for any v ∈ N.

(7)

Note that because of the integer constraint, incumbent ﬁrms may a priori earn strictly
positive proﬁts from selling any variety v. Yet, since capital is a ﬁxed factor that is perfectly mobile across varieties, in equilibrium, the operating proﬁts of the least proﬁtableindustry are wholly absorbed by the equilibrium rental rate r. Since industries are sym()
metric and ∂π m(v) /∂m(v) < 0, the least proﬁtable industry is obviously the one withthe largest number of competitors. Building on this observation, we ﬁrst prove that inequilibrium the same number of ﬁrms compete for each variety v.
Lemma 1 (symmetric entry) At any equilibrium, m(v) = m for all v ∈ N .
()
Proof. Let vmax denote the least proﬁtable variety, i.e., the one such that π m(vmax ) = 0
at the equilibrium rental rate r. Then,m(vmax ) ≥ m(v),for any v ∈ N

(8)

(
)
because ∂π m(v) /∂m(v) < 0. Suppose furthermore that there exists a variety v 0 suchthat m(v 0 ) < m(vmax ). Since both m(v 0 ) and m(vmax ) are integers, (8) implies thatm(v 0 ) + 1 ≤ m(vmax ).
()
However, ∂π m(v) /∂m(v) < 0 implies that()()
π m(v 0 ) + 1 ≥ π m(vmax ) = 0,
which contradicts (7). Hence, m(v) ≥ m(vmax ) holds for all v and, together with (8),consequently, we derive m(v) = m(vmax ) for all v.
Note that an important by-product of the symmetry result of Lemma 1 is that all ﬁrms
will earn zero proﬁts at the equilibrium rental rate r.
7In our setting, ﬁrms are allowed to diﬀerentiate their products and to introduce newvarieties in order to relax competition for existing brands. To do so, the ﬁrm has to incur
once an additional sunk cost R for innovation (think, e.g., of R&D). Formally, a ﬁrm will
choose to introduce a new variety if and only if it can reap enough monopoly proﬁts tocover its ﬁxed costs and the additional costs for creating this new variety. Let vnew denotethe new variety. The no-entry condition is then given by:
Y [σ − 1]σ−1Π1 (vnew ) = π(m(vnew )) = π(1) = ∫ [− rF ≤ rR.
]1−σ
σm(j)
dj
σσ
N σm(j) − 1(9)Note that m(vnew ) = 1 because the ﬁrm expects to be a monopolist on the new variety.
Note, furthermore, that the eﬀect of the new variety on the price index is negligible
because of the continuum assumption.
In what follows, we denote by m the distribution of the numbers of ﬁrms across thediﬀerent product varieties v. Furthermore, we refer to a pair (m, N ), which describes how
ﬁrms are distributed across a set N of varieties, as an industry structure. Note that both
the distribution of ﬁrms across varieties (i.e., m) and the range of varieties (i.e., the set
N and its measure n) are variable in our model.

2.3

Equilibrium in the closed economy

Factor market clearings for capital and labor, along with the aggregate income constraint,
imply that:∫∫
K = F m(v)dv,
L = c X(v)dv,
(10)
N

and

N

∫
Y = Lw + Kr +m(v)Π(v)dv,

(11)

N

where Π(v) stands for the proﬁt of a ﬁrm producing variety v. For notational convenience,
we omit the subscript because all ﬁrms producing the same variety earn an equal amount of
proﬁt by symmetry. Moreover, since all ﬁrms are negligible by the continuum assumption,
the creation of a new variety has a negligible impact on the capital market. Hence, we
can neglect it in condition (10). By Lemma 1, m(v) = m for all v. This symmetry implies
that prices and quantities are also the same for all varieties and allows us to suppress
the variety index v in what follows. Some straightforward computations show that theequilibrium conditions are then given as follows:8
8

In what follows, we express all nominal variables in terms of r, i.e., we choose the capital rental as ournuméraire. It can be readily veriﬁed that the choice of numéraire is immaterial in this model, contrary to
standard oligopoly models (Gabszewicz and Vial, 1972). This is both due to the existence of a continuumof ﬁrms and because ﬁrms make zero proﬁts.

8• free entry and the equilibrium rental r equate the proﬁts (6) to zero for all ﬁrms
Y− rF = 0;
σm2 n

(12)

• the factor market clearing and aggregate income conditions (10) and (11) reduce to
K = F nm,L = cnX = c

Y
Y (σm − 1)
=
,
p
wσm

Y = wL + rK;(13)• the no-entry condition (9) holds so that no ﬁrm can proﬁtably introduce a new variety
[
]σ−1
σm
Y [σ − 1]σ−1− rF ≤ rR.
(14)
σσ n
σm − 1The set of conditions (12) and (13) can be solved to yield:w
K(σm − 1)
,
=
r
L

Y
= Kmσ
r

and

n=

K
,
Fm

(15)

which, together with the no-deviation condition (14), characterizes the set of equilibria.
Two comments are in order. First, as can be seen from (15), there is an inverse
relationship between m and n. This is due to the fact that all ﬁrms compete for the sameﬁxed capital stock. Hence, expanding the range of varieties necessarily reduces the numberof ﬁrms producing existing varieties and vice versa. Second, as can be seen from (15), thewage-rental ratio w/r is increasing in the number of oligopolistic ﬁrms per industry. The
reason for this is that when there are more oligopolistic ﬁrms, competition on existing
varieties increases, which reduces prices and expands ﬁrms’ production scales. This inturn raises labor demand and, therefore, wages. Since the total mass of ﬁrms nm = K/F
is constant, capital demand and the rental rate r are unchanged, which implies an increase
in w/r. If, on the contrary, ﬁrms relax competition by diﬀerentiating their products, prices
rise and outputs fall. Since demand for labor also falls, the wage decreases with respect
to the rental rate. This result shows that changes in the industry structure from theintensive to the extensive margin shift the returns from the variable to the ﬁxed factor of
production. Depending on the strength of this eﬀect, the classical redistributive problems
of international trade reappear due to changes in industry structure, even in new trademodels with a single sector.
Substituting Y /r and n, as given by (15), in the no-deviation condition (14), the set E
of feasible values for m (which we henceforth call the range of thickness of each industry),is characterized as follows:
{
}
(
)σ−1
F
σ−1
E = m ∈ N,
mσ+1 ≤ 1 .
(16)
R + F σm − 1
Note that the inequality in (16) is independent of the factor endowment ratio K/L. Put
diﬀerently, changes in relative factor endowments do not change the range of thickness.
9

Furthermore, it is readily veriﬁed that the left-hand side of the inequality is strictlyincreasing in m. Hence, there exists a unique threshold m̂ such that the no-deviation
condition is violated for all m > m̂. Let us start with the following result.
Proposition 1 (existence) The equilibrium set E is non-empty for all admissible parameter values since the monopolistically competitive industry structure m∗ = 1 andn∗ = K/F can always be supported as an equilibrium.
Proof. Letting m = 1 and using F/(R + F ) ≤ 1, we obtain the result immediately fromthe deﬁnition (16) of E.
Proposition 2 (monopolistic competition) When R = 0, the equilibrium industry
structure is uniquely determined and given by the monopolistically competitive structure.
Proof. When R = 0, the feasible values of m are such that
(
)σ−1
σ−1
σ+1
≤ 1.
m
σm − 1The left-hand side is strictly increasing for all σ > 1 and m ≥ 1, whereas the equality
holds for m = 1. Since m ∈ N, the result follows.
Proposition 1 shows that monopolistic competition is always an equilibrium, whereas
Proposition 2 highlights the traditional result that it is the only equilibrium when ﬁrms
can costlessly diﬀerentiate their products. This is because it is always more proﬁtable tobe a monopolist on a new variety than to compete for an already existing one. Finally,
as expected, the degree of indeterminacy rises with the value of R&D costs.
Proposition 3 (indeterminacy) The higher R, the larger the range of industry thickness that can be sustained in equilibrium.
Note, in particular, that when R goes to inﬁnity, any conﬁguration (from monopolisticcompetition to a very competitive setting) can be sustained once it has somehow been
established. In other words, there is much structural inertia and the set of possible
equilibria gets very large as the value of R increases.
Proposition 3 raises the classical selection problem. Indeed, there is no a priori satisfying way to choose between the diﬀerent possible equilibria. In what follows, we willthus discuss the properties and the desirability of the diﬀerent equilibria that may besustained. A ﬁrst important question is whether a ranking of the industry structures interms of their social desirability is possible. It is easy to establish the following result.
Proposition 4 (welfare ranking) In a closed economy where agents own all factors
equally, welfare is strictly increasing in n and, therefore, strictly decreasing in m.
10

Proof. Substituting the equilibrium price, quantity, and wage-rental ratio into the utility
function we obtain:
1
1 1
1 yL
= n σ−1 ,V = n σ−1mσc r Kcwhich is strictly increasing in n and, since m = K/(F n), strictly decreasing in m.
Proposition 4 shows that welfare depends only on the mass of varieties n consumed when
agents equally own both labor and capital. Since by equation (13) this mass is inversely related to the number of Cournot competitors for each variety, the monopolistic competition
outcome leads to the highest prices but also the largest choice of variety. This outcome isalways the preferred one, thus showing that consumers’ ‘love for variety’ dominates thecompetition (price) eﬀects.
How does Proposition 4 depend on the assumptions of factor distribution? To see thismost clearly, consider the polar case where there are two types of agents who each own
only one production factor: L workers and K capitalists. In such a setting, aggregate
income remains unchanged, but a worker’s income is w, whereas a capitalist’s income is r.
Subscripting variables pertaining to workers by L and to capitalists by K, we then have1

VL = n σ−11 σm − 1w= n σ−1
p
cσm

and

1

VK = n σ−11r
1 L= n σ−1pcσm K

which, after substitution of the equilibrium expressions and using n = K/(F m), are bothdecreasing functions of m. Hence, there is no ‘class conﬂict’ when it comes to the social
ranking of industry structures: any shift of factor prices in favor of capital, due to adecrease in m and a corresponding increase in n, is more than oﬀset by the consumption
variety gains for workers (see Krugman, 1981, for related results).
To sum up, we may conclude that the monopolistic competition outcome provides
the upper bound to the utility that can be achieved in the closed economy. The procompetitive gains of entry for existing varieties do not allow to compensate for foregone
variety losses.9

3

Equilibrium in the open economy

Assume now that two countries of the kind described above, say 1 and 2, may engage
in international trade. Variables associated with each country will be subscripted accordingly. To simplify the analysis, we assume that the two countries are perfect mirror
images in terms of factor endowments and size, i.e., L1 = L2 = L and K1 = K2 = K.
We further assume that each country produces a subset of the world variety set NW in
9

Although this result is likely to depend on the chosen modelling framework, it suggests that theregulation of entry is a complicated issue once general equilibrium resource constraints and factor prices
are taken into account.

11

autarky. Let N1 (resp., N2 ) stand for the set of varieties produced in country 1 (resp., 2),with measure n1 (resp., n2 ). By deﬁnition NW = N1 ∪ N2 , with measure nW ≤ n1 + n2 .
In principle, an inﬁnity of cases need to be distinguished, depending on which subsets
of world varieties the two countries produce in autarky. In what follows, we deal, forsimplicity, with two simple polar cases only:
1. both countries produce only diﬀerent varieties (i.e., N1 ∩ N2 = ∅), which we call thefull cross-country product diﬀerentiation case (FCCPD);
2. both countries produce only the same varieties (i.e., N1 = N2 ), which we call theno cross-country product diﬀerentiation case (NCCPD).
Note that the latter case contrasts starkly with the standard monopolistic competition
models of international trade, which always assume that there is FCCPD. Analyzing the
NCCPD case allows us to circumvent the problem that trade mechanically expands therange of product variety in most new trade models and yields several new insights.10
We now extend the model presented in the foregoing to both cases of FCCPD and
NCCPD within a single framework. To do so, we modify our model along the linesof ‘reciprocal dumping’ à la Brander (1981) and Brander and Krugman (1983), whichassumes that ﬁrms are Cournot competitors and that markets are spatially segmented.11
Contrary to the standard reciprocal dumping models, our speciﬁcation allows us to deal
with general, rather than partial, equilibrium.
Denote by Xii (v) and Xji (v) the demands in country i = 1, 2, j 6= i, for variety v
satisﬁed by home ﬁrms and by foreign ﬁrms, respectively. Reverting to the setting inwhich all agents have identical factor endowments, we haveXii (v) + Xji (v) = Lxi (v) = ∫pi (v)−σ1−σpi (z)

Yi ,with Yi = Lyi ,

(17)

dz

NW

where pi (v) denotes the price in country i. Solving for the inverse demands, taking theprice aggregates as given, it is readily veriﬁed that∂pi (v)pi (v)∂pi (v)
1==−.
∂Xii (v)∂Xji (v)σ Xii (v) + Xji (v)Since markets are segmented and marginal cost is constant, markets are strategicallyseparable so that ﬁrms choose their quantities for each market independently. Firm k’s
10

Behrens and Murata (2006) use an alternative monopolistic competition model that allows for procompetitive eﬀects, and show that trade always reduces the range of products produced in the global
economy, but expands the range of products consumed. These results again rely on the assumption of a
one-to-one relationship between ﬁrms and varieties.
11
There is a substantial amount of empirical evidence suggesting that international markets are segmented and that ﬁrms’ pricing behavior is discriminatory (e.g., Verboven, 1996; Haskel and Wolf, 2001).

12

proﬁt, when it produces variety v in country i = 1, 2, is then given byΠki (v) = [pi (v) − cwi ] qiik (v) + [pj (v) − cwi ] qijk (v) − rF.

(18)

Note that the wages w1 and w2 may a priori diﬀer across countries (local labor markets),
whereas the rental rates r1 = r2 = r are equalized under the assumption that capital isinternationally mobile.
The ﬁrst-order conditions for proﬁt maximization by country-i ﬁrms are given by1
pi (v)q k (v)σ Xii (v) + Xji (v) ii
pj (v)
1=
q k (v),σ Xjj (v) + Xij (v) ij

pi (v) − cwi =pj (v) − cwiwith mirror expressions holding for country-j ﬁrms. Focusing on the symmetic caseqiik (v) =Xii (v)mi (v)

qijk (v) =andXij (v),mi (v)

for all k

we then have:sii (v)pi (v)σmi (v)sij (v)
pj (v),=σmi (v)pi (v) − cwi =pj (v) − cwi
wheresii (v) ≡Xii (v)Xii (v) + Xji (v)andsij (v) ≡Xij (v)
Xjj (v) + Xij (v)denote country-i ﬁrms’ market shares in both countries. Using the ﬁrst-order conditions,as well as sii + sji = 1, we then readily obtain the following proﬁt maximizing prices:][cσ m1 (v)w1 + m2 (v)w2]
p1 (v) = p2 (v) = [(19)
σ m1 (v) + m2 (v) − 1
as well as the ﬁrms’ export market shares:[]
mi (v) wi + mj (v)σ(wj − wi ),sij (v) =m1 (v)w1 + m2 (v)w2

for i 6= j.

(20)

Note that ﬁrms producing variety v can proﬁtably export from country j to country i 6= j
if and only if the ‘no cut-oﬀ’ condition[()]
c 1 + mi (v)σ wwji − 1
pi (v) > cwj
⇐⇒
>0
(21)
mi (v)σ − 113holds, which is endogenously determined in the full equilibrium since it depends on boththe numbers mi of ﬁrms and on the wage ratio wi /wj .12 In what follows, we will focus oncases in which condition (21) always holds.
To determine the equilibrium, we ﬁnally need the factor market clearing and nodeviation conditions as in the closed economy case.
Factor market clearing conditions: Factor market clearings for each national labormarket and the global capital market require that
∫
[
]
L =
(22)
cX11 (v) + cX12 (v) dv,
N1
∫
[
]
L =
cX22 (v) + cX21 (v) dv,
(23)
N2
∫
∫
2K = F(24)
m1 (v)dv + F
m2 (v)dv
N1

N2

where, using (17) and (20), we haveXii (v) = sii (v) ∫pi (v)−σYi

1−σpi (z)andXij (v) = sij (v) ∫pj (v)−σ1−σdz

pj (z)

NW

Yj

(25)

dz

NW

for country-i ﬁrms. Mirror expressions hold for country-j ﬁrms. Finally, we have as always
the aggregate income constraint Yi = Lwi + Kr for i = 1, 2.
Free entry and no-deviation conditions: Free entry for each variety occurs, andthe rental rate r adjusts, until no ﬁrm can enter to earn a strictly positive proﬁt. Stateddiﬀerently,
Πki (v) = [pi (v) − cwi ]()
Xii (v)Xij (v)+ [pj (v) − cwi ]− rF ≡ πi mi (v) ≥ 0,
mi (v)mi (v)

for all active ﬁrms and varieties in both countries i = 1, 2, and()πi mi (v) + 1 < 0.

12

The endogenous determination of the range of exports is a neglected issue in the monopolistic competition trade literature. Neary (2003a, p.7, emphasis in the original) points out that “[. . .] with Dixit-Stiglitz
preferences, the consumer always demands all goods even if some are much more expensive than others.
This matters in the context of trying to explain intra-industry trade. Dixit-Stiglitz preferences comeclose to assuming that intra-industry trade will take place.” Melitz (2003) extends the CES frameworkby using heterogenous ﬁrms which face ﬁxed costs of exporting. In such a context only a subset ofﬁrms exports, but this subset is determined by other considerations than price (or quantity) competition.
Finally, Ottaviano and Melitz (2005) use a quasi-linear quadratic framework to endogenize the set ofexporting ﬁrms as a function of trade costs.

14

Furthermore, no ﬁrm can proﬁtably deviate by starting to produce a new variety incountry i, which is the case if()
Π1i (vnew ) = πi mi (vnew )= [pi (vnew ) − cwi ] qii (vnew ) + [pj (vnew ) − cwi ] qij (vnew ) − rF](
)σ−1 [
σ
−
1
Y
Y
i
j
∫
+∫
− rF ≤ rR,= σ −σ1−σ dvcwip(v)
p (v)1−σ dv
i
NW
NW j
where

σ
cwi ,
σ−1
are the prices of the new monopolist (mi (vnew ) = 1, mj (vnew ) = 0) for variety vnew ; andwhere the pi (v)’s are the prices for the remaining varieties in country i = 1, 2.
pi (vnew ) = pj (vnew ) =4

Trade with full cross-country product diﬀerentiation (FCCPD)

In almost all new trade models with product diﬀerentiation, countries are strictly speaking
not fully symmetric: indeed, they all produce a diﬀerent set of varieties in autarky.
We also start with this special case where countries are symmetric in terms of factor
endowments, technology, preferences, the number (size of the set) of industries operating,and the number of ﬁrms in each industry; whereas they produce completely disjoint sets
of varieties in autarky. The next section then tackles the, in our opinion, more interestingquestion of what happens in a trading world where the FCCPD assumption is not satisﬁed.
In autarky, the two countries produce the sets N1 and N2 (with N1 ∩ N2 = ∅ under
FCCPD) of varieties, respectively. The masses of varieties are assumed to be the same:n1 = n2 ≡ nA , where autarky values are subscripted with A. Since we focus on the casewhere all varieties are produced by the same number of ﬁrms in each country, furthermore,
the capital market clearing condition for each country implies that the number of ﬁrms
producing each variety in each country is the same: m1 = m2 ≡ mA . Thus, under autarky,
all the variables (including the range of industry thickness) in each country are still givenby (15) and (16), with nA and mA replacing n and m, respectively.
Now, suppose that countries are allowed to trade with (21) being satisﬁed. This mayalter the industry structures in the two countries. As has been suggested, however, there
are multiple equilibria and, for the industry structures, we have many cases to consider.
Here, instead of characterizing all the cases, which would not give us many insights, we
rather focus on a speciﬁc situation in which the industry structures in the autarky arepreserved in the following sense: In country 1, the set N1 of varieties continues to be
produced and each variety is still produced by mA ﬁrms. Similarly, in country 2, the set
N2 of varieties continues to be produced and each variety is still produced by mA ﬁrms.
15

Then, the set of varieties available in each country becomes N1 ∪ N2 and, therefore, themass of available varieties expands to nF ≡ 2nA , where subscript F refers to world market
variables with trade under FCCPD. On the other hand, since each variety is produced bymA ﬁrms, m1 = m2 = mA ≡ mF . Note that such industry structures satisfy the capital
market clearing condition (24):
2K
nF =
.
F mF
Therefore, it is supported by the equilibrium.
Since countries are perfect mirror-images, from the labor market clearing conditions
and the income constraint, with w1 = w2 = w and Y1 = Y2 = Y , we obtain13
w
K(σmF − 1)
=
r
L

and

Y
= KσmF .
r(26)

Because mF = mA , the wage-rental ratio and the income-rental ratio remain unchanged
after the trade opening. Furthermore, the no-deviation condition can be expressed asfollows:
(
)σ−1
σ−1
F
(mF )σ+1 ≤ 1.
(27)
F + R σmF − 1
Since mF = mA , this condition is satisﬁed if and only if the parallel condition is satisﬁed
in autarky. Therefore, trade does not lead to any expansion in the set of varieties producedin the global economy, i.e., there is no variety creation.
Proposition 5 (no variety creation by free trade under FCCPD) Suppose that eachvariety is produced by more than one ﬁrm under autarky. Then, under FCCPD, no newvarieties are created by a switch from autarky to free trade.
Note, ﬁnally, that under FCCPD consumers necessarily gain from trade because themass of available varieties increases. This is the standard result of symmetric new trademodels with product diﬀerentiation.
Proposition 6 (gains from free trade under FCCPD) Under FCCPD, all agents
gain from free trade when they equally own production factors.
Proof. The equilibrium indirect utility is given by
VF =

1
1
(nF ) σ−1 ,c
11which obviously exceeds the indirect utility under autarky (nA ) σ−1 c−1 = (nF /2) σ−1 c−1 .
This establishes the result.
Consider a variety produced in country i, that is, v ∈ Ni (i = 1, 2). For such a variety, we havemi (v) = mF and mj (v) = 0 (j 6= i). Then, (20) implies that sij (v) = 1 and sji = 0. This is followed bysii (v) = 1 and sjj (v) = 0 by construction. We can use these results to compute (25).
13

16

So far, we have assumed that aggregate income is equally distributed across consumers
who own both labor and capital. If we, however, consider diﬀerent types of consumers(workers and capitalists), the existence of gains from trade will again depend on theincome distribution. In that case, the incomes of each type of consumers are w (forworkers) and r (for capitalists). Some straightforward computations yield the following
indirect utilities:
1
1
(
) σ−1
) σ−1
(
2K
σm
−
1
L
2K
F
VFL =
.
,
and
VFK =
F mF
cσmF
F mF
cσmF K
It is readily veriﬁed that V FL and V FK are decreasing in mF . Stated diﬀerently, the monopolistic competition outcome is the preferred one and welfare of both types of agentsincreases with product diversity.
The indirect utility diﬀerential between autarky and free trade is given as follows:
∆VFL
∆VFK

1
) ( K ) σ−1
σmA − 1
= 2
−1
>0
F mA
cσmA
1
) ( K ) σ−1
( 1
L
> 0,
= 2 σ−1 − 1
F mA
cσmA K(1
σ−1which allows us to establish the following result.
Proposition 7 (no class conﬂict under FCCPD) Consider the economy with FCCPD where workers and capitalists exist. Then, all the agents gain from free trade.

5

Trade with no cross-country product diﬀerentiation (NCCPD)

In the preceding section, we have exclusively examined the case where countries produce
completely diﬀerent sets of varieties in autarky. We have shown that there are automatically gains from trade due to an expansion of consumers’ choice sets. This ﬁnding
contrasts starkly with that of classical trade models where there is, on the contrary, no
expansion of consumers’ choice sets due to trade and all gains from trade materialize
through eﬃciency gains driven by the sectoral reallocation of resources. While this latteraspect is clearly important, recent empirical ﬁndings suggest that product variety may
be an even more important, yet still neglected, part of gains from trade (see, e.g., Feenstra, 1994, 1995; Hummels and Klenow, 2005; Broda and Weinstein, 2006; Broda et al.,2006).14
14

Felbermayr and Kohler (2006) propose an empirical speciﬁcation in which the extensive margin is
not interpreted as the creation of new varieties among existing trading partners, but as the establishmentof previously unexploited trading relationships. Since in our model the occurence of trade is endogenous,
an interesting extension to this type of question is left for future research.

17

In this section, we therefore analyze the case where an expansion of varieties is notassumed a priori but is brought about by trade. Put diﬀerently, we show that international
trade may itself be a cause of product variety, instead of a simple consequence. To
make our point most clearly, we consider a setting in which countries are symmetricin terms of factor endowments, technology, preferences, the number (size of the set) ofindustries operating, and the number of ﬁrms in each industry; in addition, they also
produce exactly the same set of goods in autarky. Stated diﬀerently, there is no crosscountry product diﬀerentiation (NCCPD) in autarky, so that countries are really a priori
completely symmetric in all respects.
In autarky, the sets of varieties produced in the two countries are congruent: N1 = N2 .
This implies that the masses of varieties are equal, namely, n1 = n2 = nA . As in theprevious section, we focus on the case where all varieties are produced by the same numberof ﬁrms in each country, i.e., m1 = m2 = mA .
Now, let the two countries engage in trade. As in the case of FCCPD, we focus on thesituation where the autarky industry structure continues to hold: nN = nA varieties areproduced in the world and mN = 2mA ﬁrms compete for each existing variety in the world
market, where subscript N refers to world market variables with trade under NCCPD.
Again, this structure satisﬁes the capital market clearing condition,
nN =

2K
F mN

and, consequently, is supported by equilibrium.
First of all, substituting m1 = m2 = mN /2 and w1 = w2 = w into (19) gives thefollowing price markup rate:
p1 (v)
p2 (v)
cσmN
2cσmA
=
=
=
.
w
w
σmN − 12σmA − 1This is lower than the rate at the autarky, which is equal to cσmA /(σmA − 1). Thus, theprice markup shrinks as a result of the trade opening. This is because under NCCPD,
twice as large number of ﬁrms end up competing in each industry, which makes thecompetition in each industry ﬁercer. It is worth pointing out that such a result, namelythat international trade may actually intensify competition in each industry, does not arisein the traditional monopolistic competition model à la Krugman (1980). In that type ofmodel, markups are constant and do not depend on the number of ﬁrms competing foreach variety (as each ﬁrm is a monopolist on its own variety).
Furthermore, solving the labor market clearing conditions and the income constraint
for each country, with m1 = m2 = mN /2, w1 = w2 = w, and Y1 = Y2 = Y , we thenobtain:w
KYand= (mN σ − 1),= KmN σ.
rr
L
Notice that the wage-rental ratio rises as a result of the trade opening, because mN = 2mA .
This is explained as follows. As has been explained, the competition in each industry
18becomes ﬁercer by the trade opening. This induces each ﬁrm to expand its output level
and, as a result, its labor demand. On the other hand, the total number of ﬁrms in
the world does not change (mN nN = 2mA nA ). Therefore, the demand for capital also
remains the same, and this asymmetrical change in factor demands leads to the rise in
the wage-rental ratio. By the same token, the income-rental ratio Y /r also rises by thetrade opening.
Because competition among ﬁrms producing an existing variety is severer at the NCCPD trade equilibrium than at the autarky, one might conjecture that ﬁrms have a greater
incentive to deviate from the existing industry and create a new variety in the trade regime
than at the autarky. This conjecture turns out to be right, at least as long as the competition has been suﬃciently ﬁerce at the autarky. To see this, note that the no-deviation
condition can be expressed as follows:
(
)σ−1
F
σ−1
≤ 1.
(28)
mσ+1
N
F + R mN σ − 1
Since this condition is the same as the counterpart of the closed economy model (16),we can pay an attention to m̂, the maximum number of oligopolists that satisﬁes (28).
Recalling that 2mA ﬁrms competing for each variety after trade opening, we have thefollowing result:
Proposition 8 (variety creation under NCCPD) Under NCCPD with free trade, whenmA > m̂/2, any trading equilibrium is such thatmN < 2mA

andnN > nA .

In other words, there will be variety creation in the global economy when the autarky
thickness of industry is suﬃciently large.
Proof. Assume that mA > m̂/2. After the initial trade opening, 2mA > m̂ ﬁrms produce each variety. Yet, this is incompatible with the no-deviation condition, so that someﬁrms relax competition by switching into new varieties. Therefore, mN < 2mA will holdin the new trading equilibrium and there is exit of ﬁrms for each variety. Consequently,
we have
nN mN =

2K
< 2mA nN
F=⇒

K
= nA < nN ,F mA

which establishes the result.
We next show that the existence of gains from trade under NCCPD crucially depends
on how the industry structure changes once trade is allowed for. This will highlight theexistence of a trade-oﬀ between pro-competitive eﬀects and product diversity.

19

Proposition 9 (gains from free trade under NCCPD) Assume that agents equally
own all factors. When trade does not expand product variety, there are no gains from free
trade; whereas all agents gain from free trade when trade does expand variety.
Proof. Some straightforward computations show that the equilibrium indirect utility inboth countries under trade is given by1

VN = nNσ−1 c−1 .
Welfare under free trade with NCCPD therefore exceeds its autarky level if and only if
nN > nA .
The intuition underlying this result is as follows. If free trade does not expand productvariety, all gains from trade materialize through resource reallocation and the impliedchanges in factor and product prices. Yet, when agents own all factors equally, thechanges in product prices, in wages, and in the rental rate of capital just oﬀset eachother, thus leaving all agents at the same utility level. Proposition 9, therefore, clearly
hinges on the assumption that each agent has claims to both labor and capital income.
To see what happens when that assumption is removed, let us assume, as before, thatagents may be divided into workers and capitalists. When there is no variety creation, on
the one hand, we have:
(
VNL =
(
VNK

=

2K
F mN
2K
F mN

1
) σ−1

1
) σ−1

σmN − 1
cσmN
L
.
KcσmN

It can be again readily veriﬁed that both VNL and VNK are decreasing in mN .
On the one hand, consider the case with no variety creation. The relationship mN =2mA yields the following indirect utility diﬀerential between autarky and free trade:(∆VNL =

K
F mA

1
) σ−1

1
> 0,
2cσmA

(
∆VNK = −

K
F mA

1
) σ−1

L
< 0.
2cσmA K

Thus, we have established the following result.
Proposition 10 (class conﬂict under NCCPD) Assume that consumers consist ofboth workers and capitalists. Under NCCPD, free trade always makes labor better andcapital worse oﬀ when there is no variety creation.
As in the foregoing section, this result is due to the fact that trade intensiﬁes competition
among ﬁrms, thus leading to lower prices and operating proﬁts. The latter aspect maps
into lower capital rentals and makes capitalists worse oﬀ.
20

On the other hand, when trade leads to the creation of new varieties, i.e., mN < 2mA
and nN > nA . Since
[(
1
1 (
1 (
) σ−1
) (
) σ−1
)]
( ) σ−1
2
1
K
1
1
1
−
1−
1−
∆VNL =
F
c
mN
σmN
mA
σmA
]
[
1
(
) 1
) 1
(
( ) σ−1
2 σ−1 1
1 σ−1 1
K
1 K
K
.
−
∆VN =
F
cσ L
mN
mN
mA
mA
It is easy to show that there are ranges of variety creation for which both factors gainfrom free trade.
Proposition 11 (Pareto improvement under NCCPD) Assume that consumers consist of both workers and capitalists. There exists, under the free trade with NCCPD, anequilibrium industry structure that makes both the workers and capitalists better oﬀ thanat the autarky.
Proof. To see that this possibility may arise, consider the special case where mN =mA , i.e., ﬁrms diﬀerentiate after the trade opening such that the free trade mass ofoligopolists in each industry is equal to the autarky mass. Clearly, both ∆VLN and ∆VKN
are then strictly positive, which implies gains from trade for every consumer.
Proposition 11 shows that who gains and who looses crucially depends on how product variety reacts to international trade, which is in the end an empirical question that deserves
some more attention.

6

Conclusions

As is well known, gains from trade under imperfect competition mainly accrue from the
expansion of consumers’ choice sets (product diversity) and reductions in ﬁrms’ markups(pro-competitive eﬀects). Although the monopolistic competition literature has been paying little attention to pro-competitive eﬀects and has mainly focused on product diversity,
recent empirical evidence points to the importance of both eﬀects (Hummels and Klenow,
2005). This paper is one of the ﬁrst attempts to present a simple general equilibrium
model dealing with both of these aspects of trade.
Using that model, we have shown that the assumption of a one-to-one correspondence
between ﬁrms and varieties, a typical feature of most monopolistic competition models,
is not innocuous when it comes to assessing the welfare impacts of trade. The existenceof gains from trade indeed crucially hinges on how trade aﬀects consumers’ choice sets,
which itself depends on how similar the two countries are in terms of the varieties they
produce before trade. Furthermore, factor prices and ownership of production factors areequally important aspects to take into consideration.
21

We have shown that when countries’ sets of varieties are disjoint under autarky, free
trade expands consumers’ choice sets in each country and, therefore, brings about gains
from trade, at least as long as agents own all factors equally. However, when countries’
sets of varieties are identical under autarky, trade need not yield gains. The reason is
two-fold: First, it does not expand consumers’ choice sets; and second, although markups
should fall due to more competition in the production of each variety, consumer prices
rise because the variable factor becomes more expensive. When combined with the fall incapital rentals, agents’ utilities are left unaﬀected by trade.

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