export growth and factor market competition
TRANSCRIPT
Export growth and factor market competition∗
Julian Emami Namini†, Giovanni Facchini‡, Ricardo A. Lopez§
June 6, 2012
Abstract
Empirical evidence suggests that exporters are significantly more skilled labor intensive thannon–exporters. In this paper we analyze how this source of heterogeneity impacts the firmselection induced by trade liberalization. We develop a two–factor framework in which firms canchoose their input mix. First, we show that export growth increases competition for the factorintensively used by exporters, harming them while benefitting non–exporters. Furthermore, ifthe firms’ choices of input mix are sufficiently different, part of the exporters will actually die.Second, we extend our analysis to incorporate heterogeneity in total factor productivity (TFP).We show that while trade liberalization does increase aggregate TFP, this increase is smallerthe larger are the differences in the choice of input mix. This finding provides a rationale forrecent evidence, suggesting that the effect of trade liberalization on aggregate TFP might beonly moderate.
JEL classification numbers: F12, F16, L11.Keywords: Firm dynamics, two–factor trade model, firm heterogeneity in factor intensities.
∗The authors gratefully acknowledge financial assistance from the Efige project (No. 225551) funded by the Euro-pean Commission under the Seventh Framework Programme. We would also like to thank participants at the NorthAmerican Summer Meeting of the Econometric Society in St. Louis, EEA Annual Congress in Glasgow, ERWIT–CEPR conference in Nottingham, German Economic Association meeting in Kiel, Midwest International EconomicsMeetings in Madison and the SAET conference in Faro for very useful comments. A previous version of this paperwas circulated under the title “Export growth and factor market competition: Theory and evidence”.
†Department of Economics, Erasmus University Rotterdam, P.O. Box 1738, 3000DR Rotterdam, the Netherlands;Centro Studi Luca d’Agliano, Italy; email: [email protected].
‡Corresponding author: Department of Economics, Erasmus University Rotterdam and Tinbergen Institute, P.O.Box 1738, 3000DR Rotterdam, the Netherlands; Universita degli Studi di Milano, Centro Studi Luca d’Agliano, Italy;CEPR and CES–Ifo; email: [email protected].
§International Business School, Brandeis University, Waltham, MA, USA; email: [email protected].
1
1 Introduction
Ever since detailed firm level trade data have become available, many studies have highlighted that
exporting and non–exporting firms differ from each other and have investigated how this heterogeneity
affects the impact of trade liberalization. Early empirical work has shown that exporters tend to
exhibit higher total factor productivity (TFP) than non–exporters.1 More recent studies have also
emphasized the existence of other important sources of heterogeneity between exporters and non–
exporters, a particularly salient one being their skilled labor intensity. A large number of studies
has emphasized that exporting firms are more skilled labor intensive than non–exporting ones (see
Bernard and Jensen (1995) for the United States, Munch and Skaksen (2008) for Denmark, Klein,
Moser, and Urban (2010) for Germany, Martins and Opromolla (2011) for Portugal, Alvarez and
Lopez (2005) for Chile among many others). This is true both when looking at the entire economy,
and also when considering narrowly defined sectors (see for instance the recent evidence by Wagner
(2010) for the case of Germany).
What are the effects of differences in factor input choices between exporters and non–exporters
on the firm selection induced by trade liberalization? The purpose of this paper is to develop what
is, to the best of our knowledge, the first theoretical model to study the impact of trade liberaliza-
tion on factor market competition in a setting in which firms can choose their factor intensities in
production. As exporters and non–exporters differ in their input choices, the selection mechanism we
identify works through factor markets, and complements the mechanism identified by Melitz (2003).
Importantly, our model might help rationalizing some of the recent empirical evidence suggesting
that the effects of trade liberalization on sector–wide TFP might only be moderate (Lawless and
Whelan 2008, Chen, Imbs, and Scott 2009).
We cast our discussion in a general equilibrium setting with one monopolistically competitive
sector in each country. Each firm produces a unique variety of a differentiated final good using skilled
and unskilled labor. Upon market entry, firms choose the factor share parameter characterizing their
CES production function, and in general, they find it optimal to adopt different technologies to limit
competition in factor markets.
We start by characterizing the autarkic equilibrium. We show that it is unique in the sense that
the mass of firms which choose a specific factor intensity in production is uniquely determined by the
parameters of the model. Next, we study the trade equilibrium arising in a symmetric two–country
world. In a setting with fixed export costs, and in which only skilled labor intensive firms can afford
to serve the foreign market, we characterize the firm selection induced by trade liberalization. We
highlight three different effects. First, trade liberalization provides additional profit opportunities
for exporting firms. Second, it decreases the domestic market share for both exporters and non–
exporters. Third, it increases factor market competition because of the additional production needed
1For an overview of this literature, see Bernard, Jensen, Redding, and Schott (2012).
1
to serve the export market. In particular, since exporters are skilled labor intensive, the increase
in factor market competition increases (decreases) the relative price of skilled (unskilled) labor and
negatively affects exporters, while positively affecting non–exporters. We also show that this effect
becomes stronger, the larger is the difference in factor intensities between the two types of firms. As
a result, the burden of increased factor market competition falls entirely on skilled labor intensive
exporters and some of them might be forced to cease production. Extending the model to multiple
trading countries strengthens this result.
Much of the existing literature has emphasized the role of productivity differences among firms.
How do differences in factor input choices interact with heterogeneity in TFP in shaping the firm
selection process? To answer this question we extend our model and assume that within a group of
firms with identical factor intensities, firms differ with respect to TFP. The firm selection induced by
trade liberalization works then along two dimensions. First, within each group of firms we observe se-
lection against the non–exporters, as in Melitz (2003). Second, because of factor market competition,
selection works against the skilled labor intensive firms. While the first process increases sector–wide
TFP, the second one has a priori an ambiguous effect. Still, under some mild assumptions, we can
show that the larger is the difference in factor intensities between firms, the smaller is the increase
in sector–wide TFP induced by a trade expansion. Thus, factor market competition dampens the
positive effect of trade on sector–wide TFP, which has been highlighted by Melitz (2003). This allows
our model to provide a rationale for some recent findings in the literature, suggesting that the effects
of trade liberalization on sector–wide TFP might only be moderate (Lawless and Whelan 2008).2
Our paper contributes to the literature on trade with firm heterogeneity, which has been pioneered
by Bernard, Eaton, Kortum, and Jensen (2003) and Melitz (2003). Bernard et al. (2007) extend the
Melitz (2003) setup by considering two factors of production and, additionally, two monopolistically
competitive sectors with different capital–labor ratios in production. In their model – differently
from ours – within each sector firms are homogeneous with respect to factor intensities, while they
still differ with respect to TFP. Bernard et al. (2007) thus are able to provide important insights
into the inter–industry and intra–industry factor relocations induced by trade liberalization. By
construction, though, they do not analyze how firm heterogeneity in factor intensities interacts with
globalization. This is because, within sectors, a firm’s export status only depends on its TFP. In
Yeaple (2005), on the other hand, firms choose their technology upon market entry. Labor is the
only factor of production, but workers differ with respect to their skills. The author assumes that
for each technology, a higher skill level leads to higher revenues per worker, and similarly a more
advanced technology also leads to higher revenues for any given skill level of the employee. Because
of this monotone relationship, trade liberalization leads to the same type of firm selection as in Melitz
(2003): the relative mass of exporters increases, whereas the relative mass of non–exporters decreases.
2For a recent alternative explanations see Chen, Imbs, and Scott (2009), Atkeson and Burstein (2010) or Raff andWagner (2010).
2
In our setup, on the other hand, firms produce with standard CES technologies with two inputs, and,
as a result, we do not have a monotone relationship between factor intensities and profits. While
the paper by Yeaple (2005) provides important insights into how trade liberalization affects workers’
skill–premia, it does not consider firm heterogeneity in factor intensities and thus it cannot explain
those stylized facts about trade liberalization, which refer to factor market competition.
The papers that come closest to ours are Emami Namini (2009), Emami Namini (2011), Crozet
and Trionfetti (2011) and Furusawa and Sato (2008). All of these contributions develop models of
trade in which firms differ in factor intensities. Emami Namini (2009, 2011) considers a setting in
which the factor intensity parameter is randomly assigned to firms and studies the impact of trade
liberalization on welfare and growth. Because of the randomness of the technology assignment, the
relative mass of firms with different factor intensities is given exogenously, whereas the effect of
trade liberalization on firm selection is the focus of the analysis carried out in this paper. Crozet
and Trionfetti (2011) also consider a model with random factor intensities and study how a firm’s
technology and a country’s relative factor endowment interact to determine a firm’s sales volume.
Furusawa and Sato (2008) assume instead a random TFP parameter like in Melitz (2003) and a
technology in which a continuum of intermediates, which differ in their factor intensities, is used to
produce a final good. Their focus is on the effects of trade liberalization on the adoption of a new
technology for the intermediate good. Also in this paper, the authors do not consider the effect of
the heterogeneity in factor intensities on firm selection.
The remainder of the paper is organized as follows. Section 2 lies out our model, while in section
3 we characterize the autarkic equilibrium. In section 4 we consider the effects of trade liberalization
in a symmetric two–country setting. Section 5 extends the model to N countries, whereas section 6
combines our setting with the standard Melitz–type heterogeneity in TFP. Section 7 concludes the
paper.
2 Model setup
Home’s economy is characterized by a representative consumer and a single monopolistically com-
petitive industry. We start by describing the demand side, and proceed then to consider production,
focusing first on the technologies available to the firms and then on market entry.
The preferences of the representative consumer are given by a CES utility function of the type
U =
[∫υ∈Υ
q(υ)σ−1σ dυ
] σσ−1
. (1)
The parameter σ > 1 is the elasticity of substitution between different varieties, and Υ is the set of
available goods, indexed by υ. The representative consumer is endowed with fixed amounts of skilled
and unskilled labor, respectively denoted by S and L, which we assume to be perfectly mobile within
3
a country, but immobile between countries. In other words, we are considering a short run setting in
which international migration is restricted. The consumer’s overall income is given by
I = wSS + wLL,
where wS and wL are respectively the wages paid to skilled and unskilled labor. Utility maximization
subject to the budget constraint leads to the demand for each individual variety, which is given by
q(υ) = IP σ−1p(υ)−σ, (2)
where P =[∫
υ∈Υ p(υ)1−σdυ] 1
1−σ is the price index, which is dual to the utility function.
Turning to the supply side of the economy, there is a continuum of firms, each of which produces
a different variety of the same good, combining skilled and unskilled labor S and L according to the
following CES production function:
q(ϕ) =[ϕ1−αSα + (1− ϕ)1−αLα
]1/α, 0 < α < 1 (3)
where q(ϕ) is the firm’s output and ϕ ∈ [0, 1] a factor share parameter characterizing the technology.
In the remainder of the paper we index firms by ϕ. The elasticity of substitution between inputs is
given by ς = 11−α
.3
An active firm faces a fixed production cost and a constant marginal cost c(ϕ). The latter is given
by
c(ϕ) =[ϕw1−ς
S + (1− ϕ)w1−ςL
]1/(1−ς).
Clearly, as long as wS = wL, firms choosing different values of ϕ face different marginal costs.
Production requires a fixed cost which takes the following form:
F (ϕ) = c(ϕ)f(ϕ). (4)
This structure of fixed costs is common in two–factor trade models (e.g., Markusen and Venables
2000) and implies that firms have to pay for the fixed input requirement f(ϕ) in terms of their final
output.4 We assume that f(ϕi) > f(ϕj) if ϕi > ϕj, i.e. the more skilled labor intensive is the
technology, the higher is the fixed input requirement.5
3Notice that in the representative consumer’s utility function (equation 1) each variety receives an identical weight,regardless of its factor intensity in production. While we could assume that, e.g., a higher skilled labor intensityimplies a higher quality level, i.e. varieties with a higher skilled labor intensity get a larger weight in utility (e.g.,Haruyama and Zhao 2008), this would not add to our analysis of the factor market effect of trade liberalization, whilecomplicating the algebra.
4Alternatively, we could assume that firms have to pay for f(ϕ) in terms of skilled labor, i.e. F (ϕ) = wSf(ϕ), or interms of unskilled labor, i.e. F (ϕ) = wLf(ϕ). Our results are robust to these alternative specifications of F (ϕ).
5Long, Raff, and Stahler (2011) also assume that a more advanced technology leads to higher fixed costs, as reflected
4
The market entry process takes the following form. Ex–ante, all firms are identical. Market entry
is costless. After entry, firms have the choice between two different technologies: a skilled labor
intensive technology, characterized by ϕS, and an unskilled labor intensive technology, characterized
by ϕL, with ϕS > ϕL.6 Firms maximize profits, which are given by
π(ϕi) =Ip(ϕi)
−σ
P 1−σ[p(ϕi)− c(ϕi)]− c(ϕi)f(ϕi), i = S, L, (5)
and the resulting output price is given by p(ϕi) =σ
σ−1c(ϕi).
3 Autarkic equilibrium
We choose unskilled labor as the numeraire and set wL = 1. As a result, wS denotes the relative price
of skilled labor. In equilibrium factor markets clear. Applying Shephard’s Lemma to the marginal
cost functions, this implies:
L =∑i=L,S
aLi [q(ϕi) + fi] ηi (6)
S =∑i=L,S
aSi [q(ϕi) + fi] ηi. (7)
ηi denotes the mass of firms of type i active in the market, whereas the terms aSi ≡ ϕiw−ςS c(ϕi)
ς and
aLi ≡ (1− ϕi) c(ϕi)ς are, respectively, the unit skilled and unskilled labor requirements for variety i.
Furthermore, let f(ϕi) ≡ fi in order to save on notation.
Since market entry is costless, in the autarkic equilibrium each firm’s profits are driven to zero.
As firms can choose among two different technologies, a zero profit condition has to be formulated
for each of them separately, and is given by
IP σ−1p(ϕi)−σ = q(ϕi) = (σ − 1) fi, with i = S, L. (8)
Notice that since fS > fL, equation 8 implies that, in equilibrium, q(ϕS) > q(ϕL), i.e. a skilled labor
intensive firm produces a larger amount of output.
Equations 6, 7 and 8 can be used to perform some comparative statics exercises. For this purpose
and for the remainder of our paper we will assume ς = σ to simplify the algebra, without affecting
the thrust of our results.7 Thus, σ will denote both the elasticity of substitution between inputs in
by a sunk R&D investment.6For simplicity we assume a firm to retain the same technology throughout its life. Notice though that since entry
is costless, firms of one type could exit the market and re–enter it with a different technology choice and this wouldnot affect our results.
7See for instance the proof of Lemma 1.
5
production and between varieties in consumption. We start by considering the relationship between
firms’ production and factor prices:
Lemma 1 An exogenous increase in the aggregate production of the skilled (unskilled) labor intensive
firms increases (decreases) the relative price of skilled labor wS.
Proof. Dividing equations 6 and 7 by each other, considering that, in general equilibrium, q(ϕi)σ−1
= fi
and some simple transformations lead to:
L
S=
aLS + aLLq(ϕL)ηLq(ϕS)ηS
aSS + aSLq(ϕL)ηLq(ϕS)ηS︸ ︷︷ ︸
≡Θ
. (9)
The term Θ denotes relative demand for unskilled labor in the economy. An increase in aggregate
production of skill intensive firms (q(ϕS)ηS) has the following impact on Θ:
∂Θ
∂[q(ϕS)ηS]=
q(ϕL)ηL(aLSaSL − aLLaSS)
[aSSq(ϕS)ηS + aSLq(ϕL)ηL]2. (10)
∂Θ∂[q(ϕS)ηS ]
< 0 since aLSaSL−aLLaSS < 0 due to ϕS > ϕL. Thus, wS has to adjust such that Θ becomes
larger again. We can show that an increase in wS has two positive effects on Θ. First, it increases
aLi, while it decreases aSi, i = S, L. Second, it increases q(ϕL)q(ϕS)
= c(ϕL)−σ
c(ϕS)−σ , which has a positive impact
on Θ since aLSaSL − aLLaSS < 0. Thus, wS has to increase so that factor markets clear again after
the increase in q(ϕS)ηS. However, if we assume σ = ς, equation 9 simplifies to:
L
S=
1− ϕS + (1− ϕL)ηLηS
ϕS + ϕLηLηS
wσS, (11)
which illustrates the negative relationship between relative skilled labor demand and wS in a more
straightforward way.
A second comparative statics result is given by lemma 2:
Lemma 2 If, in the initial equilibrium, both types of firms realize zero profits, an increase in the
relative price of skilled labor leads to strictly positive (negative) profits for unskilled (skilled) labor
intensive firms.
Proof. Substituting the terms for I, P and p(ϕS) into equation 5, calculating the partial derivative
of π(ϕS) with respect to wS and, afterwards, considering equation 8 leads to:
∂π(ϕS)
∂wS
∣∣∣∣π(ϕS)=0
=S(1− ϕS)− LϕSw
−σS
σP 1−σ+
(L+ wSS
)(1− σ)w−σ
S ηL (ϕS − ϕL)
σP 2−2σ< 0 (12)
6
∂π(ϕS)∂wS
∣∣∣∣π(ϕS)=0
is negative since SL
< ϕS
1−ϕSw−σ
S , which follows from equation 11, ϕS > ϕL and σ > 1. It
can be shown along the same lines that the profits of unskilled labor intensive firms become positive
with an increase in wS.
The intuition for lemma 2 is as follows. An increase in the relative price of skilled labor ceteris
paribus increases the relative price of the skilled labor intensive goods. This shifts demand away from
skilled labor intensive goods and towards unskilled labor intensive ones, leading to positive (negative)
profits for the unskilled (skilled) labor intensive firms.
Using lemma 1 and 2, we are now ready to establish our first proposition.
Proposition 1 If ϕS
ϕL>(
fSfL
)(σ−1)/σ
there exists a unique and stable autarkic equilibrium with both
skilled and unskilled labor intensive firms active in the market.
Proof. The proof proceeds in two steps. First, equation 11 shows that the relationship between wS
and ηLηS
as it results from the factor market clearing condition is negative. Second, taking the ratio
of the zero profit conditions for skilled and unskilled labor intensive firms (equation 8) we have
q (ϕS)
q (ϕL)=
(ϕSw
1−σS + 1− ϕS
)−σ/(1−σ)(ϕLw
1−σS + 1− ϕL
)−σ/(1−σ)=
fSfL
. (13)
Equation 13 can be solved to determine the relative price of skilled labor wS in the autarkic equilib-
rium (superscript a):
waS =
[Ψa(1− ϕL)− (1− ϕS)
ϕS −ΨaϕL
]1/(1−σ)
, (14)
with Ψa =(
fSfL
)(σ−1)/σ
. Notice that waS does not depend on ηL and ηS since these affect only the
price index P which enters identically the zero profit conditions for both types of firms (see equation
8). Furthermore, as long as ϕS
ϕL>(
fSfL
)(σ−1)/σ
, waS is well defined and both types of firms are active in
equilibrium. Finally notice that waS < 1 since fS > fL. Thus, substituting equation 14 into equation
11 we can solve for ηLηS. Once wa
S and ηLηS
are known, we can determine all the remaining variables of
the model.
In the remainder of the paper, we will refer to equation 11 as the relative factor market clearing
condition (FMC). Equation 14 determines instead the relative price of skilled labor, given that both
types of firms are active, and we will refer to it as the price of skilled labor condition (PS). In the left
panel of Figure 1, we depict the two curves. Their intersection establishes the relative price of skilled
labor wS and the relative mass of unskilled labor intensive firms ηLηS
in the autarkic equilibrium. OnceηLηS
has been determined, we can also obtain the absolute number of active firms by using one of the
two zero profit conditions. This is done in the right panel of the figure.
7
6
-ηLηS
wS
PSa
FMCa
Ea
-
6ηL
ηS
Ea
(ηLηS
)aZero profitcondition
Figure 1: Autarkic equilibrium
Finally, it is useful to calculate the average skilled labor share parameter over all active firms,
which we define by ϕa:
ϕa =ϕS + ϕL
ηLηS
1 + ηLηS
Notice that, as shown in appendix A, an industry with ηL + ηS homogenous firms producing with
the skilled labor share parameter ϕa leads to the same aggregate outcomes as an industry with ηL
and ηS firms, each producing with the skilled labor share parameters ϕL and ϕS, respectively.
4 Free trade equilibrium
In this section, we extend our analysis to a two–country setting to study the effect of a bilateral trade
liberalization. To keep our analysis simple, we compare the autarkic and the free trade equilibria,
and assume zero transport costs.
We study the firm selection in each country, which is due to increased competition on goods and
factor markets. The former is induced by the inflow of foreign varieties. The latter is instead the
result of increased production by exporters. To provide intuition for our results, we consider first
the impact of increased competition on goods markets, and then turn to increased competition on
factor markets. Thus, we first focus on how the inflow of foreign varieties influences the mass of the
two types of firms, holding factor prices fixed. We then consider the full general equilibrium effects
with endogenously determined factor prices. Throughout our discussion we assume that production
factors are immobile across countries.
Home and Foreign are assumed to be completely symmetric. Utility maximization in Foreign
8
results in the following demand function for a variety produced in Home:
qF (ϕ) = IFPσ−1F p(ϕ)−σ,
where the subscript F denotes variables of Foreign. In order to export, we assume that a domestic
firm needs to set up a distribution network, which leads to a fixed export cost given by:
FX = c(ϕ)fX , (15)
where fS ≥ fX > fL. This assumption implies that only skilled labor intensive firms will earn non–
negative profits by serving the foreign market, whereas no unskilled labor intensive firm will find it
optimal to export.8 Thus, this assumption is crucial for our model to replicate the stylized fact that
only some domestic firms serve the foreign market.
Total demand for a domestically produced skilled labor intensive variety increases to
q (ϕS) + qF (ϕS) = 2IP σ−1p (ϕS)−σ (16)
and the aggregate price index decreases to
P =[2ηSp (ϕS)
1−σ + ηLp (ϕL)1−σ]1/(1−σ)
(17)
following trade liberalization. For unskilled labor intensive firms, trade liberalization ceteris paribus
does not affect the supply decision and the zero profit condition is still given by equation 8. On the
other hand, trade liberalization affects the supply decision of skilled labor intensive firms and their
zero profit condition becomes:
2q (ϕS) = (σ − 1)(fS + fX). (18)
Dividing equations 18 and 8 by each other and remembering that q(ϕi) = IP σ−1p(ϕi)−σ, we can solve
for wS in the free trade equilibrium (superscript ft):
wftS =
[Ψft(1− ϕL)− (1− ϕS)
ϕS −ΨftϕL
]1/(1−σ)
, (19)
with Ψft =(
fS+fX2fL
)(σ−1)/σ
. Notice that Ψft < Ψa, implying that wftS > wa
S, i.e. the PS–curve shifts
upward with trade liberalization. We will refer to equation 19 as the PS–equation in the free trade
equilibrium.
8Remember that the two types firms earn zero profits from serving the domestic market. Since both countries arecompletely symmetric, the assumption fS ≥ fX > fL implies that unskilled labor intensive firms would make negativeprofits from serving the foreign market, whereas skilled labor intensive firms will earn non–negative profits.
9
Finally, considering also the additional factor demand due to production for exports leads to the
following FMC condition with free trade:
L
Sw−σ
S =2(1− ϕS) + (1− ϕL)
ηLηS
2ϕS + ϕLηLηS
. (20)
The additional factor 2 on the right hand side of equation 20 implies that, for a given magnitude of
wS,ηLηS
increases, implying that the FMC–curve shifts rightward due to trade liberalization.
Summarizing our results so far we obtain:
Lemma 3 Compared to autarky, a bilateral trade liberalization has the following consequences:
i) the aggregate price index P decreases in each country due to the availability of additional varieties
from abroad; the decrease in P ceteris paribus decreases the profits of exporting and non–exporting
firms and reflects an increase in goods market competition;
ii) skilled labor intensive firms increase their production due to additional profit opportunities abroad;
iii) the relative price of skilled labor wS increases due to additional production by skilled labor inten-
sive exporters; the increase in wS ceteris paribus decreases the profits of skilled labor intensive
firms and increases the profits of unskilled labor intensive firms.
Proof. Parts i) and ii) follow from equations 16 and 17. Part iii) follows from lemma 1 and lemma
2.
Notice that it is a priori ambiguous whether trade liberalization leads to a firm selection in favor
of or against either type of firms, i.e. whether ηLηS
increases or decreases. The additional availability of
foreign varieties affects both skilled and unskilled labor intensive firms negatively, and ceteris paribus
drives both types of firms out of the market. At the same time, the increased profit opportunities
abroad affect skilled labor intensive firms positively, ceteris paribus leading to additional entry of
this type of firms. Finally, the increased competition on factor markets, which is reflected by the
rightward shift of the FMC–curve, affects skilled labor intensive firms negatively and unskilled labor
intensive firms positively, ceteris paribus leading to exit of skilled labor intensive firms and entry of
unskilled labor intensive firms.
The net effect of trade liberalization on the two types of firms crucially depends on the difference
in skilled labor share parameters ϕS − ϕL. In the following, we will refer to ϕS − ϕL as the factor
intensity gap between exporters and non–exporters. The factor intensity gap determines (i) the
extent to which wS increases with trade liberalization and (ii) the extent to which firms are affected
by the increase in wS. Its role is characterized in the following:
Proposition 2 There exists a threshold value Φ for the factor intensity gap such that bilateral trade
liberalization leads to the following pattern of firm selection:
10
6
-Φ
ϕS − ϕL
1(ϕS − ϕL)min
(ηLηS
)ft−(
ηLηS
)a
Figure 2: The role of the factor intensity gap
i) if ϕS − ϕL > Φ, ηLηS
increases, i.e. the relative mass of non–exporters increases;
ii) if ϕS − ϕL < Φ, ηLηS
decreases, i.e. the relative mass of non–exporters decreases.
In general, the larger is ϕS − ϕL, the more detrimental (beneficial) is trade liberalization for a single
exporting (non–exporting) firm.
Proof. See appendix B.
Figure 2 illustrates the firm selection with trade liberalization. (ηLηS)ft stands for the relative mass
of unskilled labor intensive firms in the free trade equilibrium, while (ηLηS)a stands for the relative
mass of unskilled labor intensive firms in the autarkic equilibrium. The minimum factor intensity
gap, which is denoted by (ϕS − ϕL)min, is defined as that difference ϕS−ϕL, which leads to (ηS)a = 0.
In appendix B we prove that the relationship between (ηLηS)ft − (ηL
ηS)a and ϕS − ϕL is as illustrated by
figure 2.
The intuition behind proposition 2 is as follows. First, the increase in wS brought about by
trade liberalization is larger, the larger is the difference ϕS − ϕL. Second, for a given increase in the
relative price of skilled labor wS the losses (gains) for the skilled (unskilled) labor intensive firms are
larger, the larger is ϕS − ϕL. Thus, we can conclude that unskilled (skilled) labor intensive firms
will unambiguously gain (lose) from trade liberalization and firms of this type will enter (exit) the
market if the factor intensity gap is sufficiently large.9
Figure 3 illustrates the effect of trade liberalization on the mass of firms active in equilibrium.
The left panel shows that, starting from the autarkic equilibrium Ea, trade liberalization shifts the
PS–curve upward. This results from new profit opportunities abroad for skilled labor intensive firms,
9Furthermore, notice that the threshold value Φ is smaller for larger values of fX . The reason for this as fXincreases, profits from exporting will decline. Thus, if fX becomes larger, a smaller factor intensity gap, i.e. a smallerincrease in wS is sufficient to decrease ηS relative to ηL.
11
6
-ηLηS
wS
PSa
FMCa
Ea
-
6ηL
ηS
Ea
(ηLηS
)aEft
(ηLηS
)ft..................................................................
Eft
PSft
FMCft
Figure 3: Trade liberalization
which require an increase in the relative price of skilled labor wS for the free entry conditions to hold
again. Trade liberalization also leads to increased competition in factor markets, which shifts the
FMC–curve rightward. In fact, if relative demand for skilled labor increases and if wS is given by
the PS–curve, ηLηS
has to increase to re–establish factor market clearing. The free trade equilibrium
is illustrated by point Eft. Notice that we have drawn the curves for a “large” factor intensity gap
such that ηLηS
increases with trade liberalization.
The right panel of the same figure captures also the role played by the increased availability of
foreign varieties. We keep factor prices constant for the moment in order to separate the effects of
increased factor market competition from those of increased goods market competition. Starting from
the autarkic equilibrium Ea, increased availability of foreign varieties and new profit opportunities
abroad make the line illustrating the zero profit conditions for skilled labor intensive firms shift
inward and become steeper (dotted line). Allowing factor prices to adjust (wS increases) flattens the
curve and makes it shift inward. The new equilibrium point is indicated by Eft. In general, the mass
of skilled labor intensive firms ηS decreases, whereas ηL can increase or decrease.10
Importantly, notice that in our model a skilled labor intensive firm will never react to the in-
crease in wS by exiting the foreign market, while still serving the domestic one.11 If factor market
competition is sufficiently strong so that some exporting firms leave the foreign market, these firms
will cease production completely, i.e. they will also exit the domestic market. The reason for this
result lies in our market entry procedure. Since firms do not have any uncertainty about their tech-
nology and market entry is free, each firm realizes zero profits on the domestic market in the autarkic
equilibrium, i.e. π(ϕ) = 0 (see equation 8).
10Appendix C formally derives the shifts of the zero profit condition in the right panel.11This result follows, of course, from our focus on steady states. It is well–known that, in the short run, firms also
enter and exit the export market, without necessarily dying, as it is pointed out by Schroder and Sørensen (2011).
12
This finding is in contrast with the standard results in the literature (see Melitz 2003, among
others). In these models an increase in sector–wide exports increases the wage rate, which decreases
profits of all firms proportionately and leads the least productive firms to exit the market, whereas
the marginal exporting firms become non–exporters.
It is interesting to determine the effect of trade liberalization on the industry–wide average skilled
labor share parameter ϕ. This is done in the following:
Proposition 3 Compared to autarky, trade liberalization leads to an increase in the average industry–
wide skilled labor share parameter.
Proof. See appendix D.
5 The N country case
We now extend our analysis to the case of N ≥ 2 symmetric countries, which are freely trading among
each other.12 We focus on a trade liberalization experiment that involves all countries simultaneously.
Compared to the two–country case, the aggregate output of a skilled labor intensive firm now
increases to
Nq(ϕS) = NIP σ−1 [p(ϕS)]−σ , N ≥ 2,
with trade liberalization. The zero profit condition for a skilled labor intensive firm is now given by
q(ϕS) =(σ − 1)[fS + (N − 1)fX ]
N, (21)
whereas the corresponding condition for unskilled labor intensive firms is still given by equation 8.
Dividing equation 21 by equation 8 and solving for the relative price of skilled labor, we obtain the
N country version of the free trade PS–curve:
wftS =
[Ξ(1− ϕL)− (1− ϕS)
ϕS − ΞϕL
]1/(1−σ)
,
with Ξ =[fS+(N−1)fX
fL
1N
](σ−1)/σ
. The relative factor market clearing condition (FMC) for the N
country case can be solved directly for ηLηS:
ηLηS
=1− ϕS − w−σ
SLSϕS
w−σS
LSϕL − (1− ϕL)
N. (22)
We can now study the effect of an increase in N on the firm selection induced by trade liberalization,
starting from the initial equilibrium E1 (see figure 4). Consider the PS–curve. It is straightforward to
12Notice that increasing the size of the single trading partner, which we have considered before, would lead to similarresults.
13
6
-
E1
E2
1
wS
ηLηS
FMC
PS
-
6ηL
ηS
E2
E1
(ηLηS
)(
ηLηS
)′
Figure 4: The effect of an increase in N
show that as N increases it shifts upward (the thicker black line). Intuitively, since N ceteris paribus
increases the profits of skilled labor intensive firms, wS has to increase as well for the zero profit
condition of skilled labor intensive firms to hold. Remember from lemma 2 that the skilled labor
intensive firms’ profits become negative as wS increases. Furthermore, in the limit, as N approaches
infinity, wS converges to the following value
wSft =
(
fXfL
)(σ−1)/σ
(1− ϕL)− (1− ϕS)
ϕS −(
fXfL
)(σ−1)/σ
ϕL
1/(1−σ)
,
and wSft < 1 since fX > fL.
Turning now to the FMC–curve, as N becomes larger, the curve shifts rightward, i.e. ηLηS
increases
for a given wS (see equation 22). This is because as the number of trading partners becomes larger,
aggregate relative demand for skilled labor ceteris paribus increases. Thus, the new equilibrium is
given by E2. Importantly, in equilibrium the relationship between ηLηS
and N is linear. Thus, if N
goes to infinity, ηLηS
goes to infinity as well.13
Consider now the right panel of figure 4. An increase in the number of trading partners N shifts
the zero profit condition further to the left and the curve becomes steeper. Thus, we can summarize
our main finding for the N country case in the following proposition.
Proposition 4 As the number N of trading partners becomes sufficiently large, trade liberalization
always leads to an increase in the relative mass of non–exporting firms ηLηS.
Proof. See appendix E.
13Notice that wS is bounded from above by 1. Thus, if N is sufficiently large, trade liberalization always increasesηL
ηS, irrespective of the magnitude of the factor intensity gap ϕS − ϕL. In other words, an increase in N shifts the
concave curve in figure 2 downward.
14
6 Adding heterogeneity in TFP
A large empirical literature has documented the existence of substantial firm heterogeneity in TFP
within a narrowly defined sector (Bernard and Jensen 1995 and Alvarez and Lopez 2005, among
others), and thus it is important to study how heterogeneity in factor intensities interacts with
heterogeneity in TFP in shaping firm selection with trade liberalization. To keep the analysis general,
we focus on the case of N ≥ 2 symmetric trading partners in the free trade situation.
To incorporate firm heterogeneity in TFP we follow Melitz (2003) and modify the market entry
procedure. In particular, after having chosen the technology parameter ϕL or ϕS and to actually
enter the market, firms have to pay a sunk market entry fee fE. Payment of this allows firms to draw
their TFP parameter A from a common and exogenously given probability distribution with support
[1,∞), density g(A) and cumulative density G(A).14 Since the random TFP parameter reflects a
firm’s uncertainty about, e.g., how well workers perform or how consumers evaluate a variety, it is
reasonable to assume that a firm learns its TFP after it has chosen its skilled labor share parameter.
The production function of a firm with skilled labor share parameter ϕ is now given by:
q(ϕ,A) = A[ϕ1−αSα + (1− ϕ)1−α Lα
]1/α, 0 < α < 1.
The corresponding marginal cost function c(ϕ,A) results as:
c (ϕ,A) =1
A
(ϕw1−σ
S + 1− ϕ)1/(1−σ)
, σ > 1.
As in Melitz (2003), we assume that TFP does not influence the fixed production cost fPi , i = S, L.
We therefore choose the following specification
F P (ϕ) = Ac(ϕi, A)fPi =
(ϕiw
1−σS + 1− ϕi
)1/(1−σ)fPi , i = S, L.
A firm’s current period profits can then be expressed as
π (ϕi, A) =Ip (ϕi, A)
1−σ
P 1−σσ− Ac (ϕi, A) f
Pi .
Again, we assume that firms pay for fE with their final output, so that the sunk market entry costs
for a firm with skilled labor share parameter ϕi, i = S, L, are given by:
FE (ϕi) = Ac (ϕi, A) fE =
(ϕiw
1−σS + 1− ϕi
)1/(1−σ)fE.
Notice that TFP does not affect the sunk entry cost either. We can now define the minimum
14Notice that without a sunk entry fee firms could enter and exit the market costlessly, and thus draw their produc-tivity parameter repeatedly until they obtain the highest possible productivity level.
15
productivity level A∗i , such that a firm starts producing. A∗
i is determined by the following zero
cutoff profit condition:
IP σ−1p (ϕi, A∗i )
−σ = q (A∗i , ϕi) = A∗
i (σ − 1)fPi , i = S, L. (23)
Given the threshold TFP parameter A∗i , free entry implies that the ex–ante expected profits from
market entry are equal to zero. Thus, the free entry condition can be written as follows:
[1−G(A∗i )]
∫ ∞
A∗i
π(ϕi, A)µ(A)dA = FE(ϕi), i = S, L, (24)
where µ(A) = g(A)1−G(A∗
i ). The first term on the left hand side of equation 24 represents the probability
that a firm of type i starts producing after entry. The second term describes the average profits of
active firms. The term on the right hand side represents the sunk entry cost.
The following lemma characterizes the threshold TFP parameter in the autarkic equilibrium:
Lemma 4 The threshold TFP parameter A∗a,i in the autarkic equilibrium (subscript a) is given by
the solution to the following equation:
[1−G
(A∗
a,i
)] (Aa,i
A∗a,i
)σ−1
− 1
=fE
fPi
, i = S, L, (25)
where(
1
Aa,i
)1−σ
≡∫∞A∗
a,i
(1A
)1−σµ(A)dA.
Proof. See appendix F.
Notice that A∗a,i depends only on σ, fE, fP
i and g(A). Thus, A∗a,i is independent from A∗
a,j, i = j.
To determine the autarkic equilibrium, we proceed as in section 3, and construct the modified version
of the price of skilled labor curve (PS) and the factor market clearing condition (FMC). To derive
the autarkic PS–curve under the presence of firm heterogeneity in TFP, we take the ratio of the zero
cutoff profit conditions for the two types of firms (equation 23), and solve this for waS:
waS =
(
fPS
fPL
)(σ−1)/σ (A∗a,S
A∗a,L
)(1−σ)2/−σ
(1− ϕL)− (1− ϕS)
ϕS −(
fPS
fPL
)(σ−1)/σ (A∗a,S
A∗a,L
)(1−σ)2/−σ
ϕL
1/(1−σ)
. (26)
To derive the FMC condition, we need to consider that, compared to the baseline model, an increase
in productivity decreases the unit factor requirements, whereas it increases aggregate output since
16
the price of each variety declines. The modified FMC condition becomes:
L
S=
(1− ϕS) Aσ−1a,S + (1− ϕL) A
σ−1a,L
ηLηS
ϕSw−σS Aσ−1
a,S + ϕLw−σS Aσ−1
a,LηLηS
. (27)
and we refer the reader to appendix G for the derivation. Combining the PS and the FMC conditions
we can determine the autarkic equilibrium, which is characterized in the following
Proposition 5 There exists a unique, stable autarkic equilibrium with firm heterogeneity in factor
intensities and TFP.
Proof. See appendix G.
We are now ready to determine the industry–wide average skilled labor share parameter ϕ and
the industry–wide average TFP parameter A in the autarkic equilibrium:
ϕa =ϕSA
σ−1a,S + ϕLA
σ−1a,L
ηLηS
Aσ−1a,S + Aσ−1
a,LηLηS
, Aa =
[Aσ−1
a,S + Aσ−1a,L
ηLηS
1 + ηLηS
]1/(σ−1)
.
Notice that an industry with ηL+ηS homogeneous firms, each producing with technology parameters
ϕa and Aa, leads to the same aggregate outcome as an industry with ηL and ηS firms, each producing
with parameters ϕL, Aa,L, ϕS and Aa,S, respectively (see appendix H).
Now, we consider the effects of a multilateral trade liberalization (i.e. a movement from autarky
to free trade) among N countries. We assume that the fixed exporting cost fX is such that fX ≥fPS > fP
L .15 If fX is strictly larger than fP
S , only some of the skilled labor intensive firms export
after trade liberalization. We can now determine a second threshold value A∗Xi, which represents the
minimum productivity level that enables a firm to serve the N foreign markets after liberalization.
This threshold is determined by the following zero cutoff profit condition:
IFPσ−1F p(ϕi, A
∗Xi)
−σ = q(ϕi, A∗Xi) = A∗
Xi(σ − 1)fX . (28)
Equation 28 implies the following. First, A∗Xi ≥ A∗
i since fX ≥ fPi . Thus, not all firms which serve
the domestic market do also export. Second, A∗XL > A∗
XS, i.e. unskilled labor intensive firms need
a higher productivity level in order to be able to export, compared to skilled labor intensive firms.
This follows from the fact that p(ϕS, A) < p(ϕL, A) for any given TFP parameter A. Thus, a higher
TFP parameter has to compensate for the otherwise higher marginal costs of unskilled labor intensive
15Notice that more recent contributions also consider firm heterogeneity in fX (e.g., Jørgensen and Schroder 2008).Since heterogeneity in fX would not add to our analysis of the factor market effect of trade liberalization, we keep fXidentical across firms.
17
firms. Finally, dividing equations 23 and 28 by each other and solving for A∗Xi leads to:
A∗Xi = A∗
i
(fPi
fX
)1/(1−σ)
. (29)
The free entry condition has to be modified to account for the additional ex–ante expected export
profits, and becomes
[1−G(A∗i )]
∫ ∞
A∗i
π(ϕi, A)µ(A)dA+ (N − 1) [1−G(A∗Xi)]
∫ ∞
A∗Xi
πX(ϕi, A)µX(A)dA = FE(ϕi),
where µX(A) =g(A)
1−G(A∗Xi)
. The term 1−G(A∗Xi) stands for the probability that a firm of type i will be
exporting after market entry. The term∫∞A∗
Xiπ(ϕi, A)µX(A)dA stands for the average export profits
over all exporting firms. Notice that 1 − G(A∗XS) > 1 − G(A∗
XL) since A∗XL > A∗
XS. Thus, if a firm
has chosen a skilled labor intensive technology, it is more likely to become an exporter than if it had
chosen an unskilled labor intensive technology.
Lemma 5 characterizes the threshold TFP parameter for firms of type i, i = S, L, in the free
trade equilibrium and the impact of trade liberalization on the threshold TFP parameter:
Lemma 5 The threshold TFP parameter A∗ft,i, i = S, L, in the free trade equilibrium (subscript ft)
is given by the solution to the following equation:
[1−G
(A∗
ft,i
)](Aft,i
A∗ft,i
)σ−1
− 1
+ (N − 1) [1−G (A∗Xi)]
(AXi
A∗Xi
)σ−1
− 1
fXfPi
=fE
fPi
, (30)
where(
1
AXi
)1−σ
≡∫∞A∗
Xi
(1A
)1−σµX(A)dA. Trade liberalization increases A∗
ft,i.
Proof. See appendix I.
Furthermore, in the following we will assume that the TFP parameter follows a Pareto–distribution
with density g(A) = kAk+1 and shape parameter k ≥ σ − 1.16 We can then formulate lemma 6:
Lemma 6 The increase in A∗S due to trade liberalization is larger than the increase in A∗
L. Further-
more, the increase in AS due to trade liberalization is larger than the increases in AL.
Proof. See appendix J.
To understand the intuition behind lemma 5 and 6, notice that trade liberalization increases
ex–ante expected profits and thus triggers additional entry of skilled and unskilled labor intensive
firms. Competition becomes stronger, which implies that only the more productive firms of either
16Notice that the assumption k ≥ σ − 1 guarantees that the average TFP parameters Aσ−1i and Aσ−1
Xi are defined.Axtell (2001) and Luttmer (2007), among others, have shown that a Pareto–distribution describes appropriately thedistribution of TFP across firms in manufacturing.
18
type will survive. Since the share of exporters among skilled labor intensive firms is larger, new
entry of skilled labor intensive firms exceeds new entry of unskilled labor intensive firms. Thus, the
TFP improvement among skilled labor intensive firms is larger than the TFP improvement among
unskilled labor intensive firms.
Notice though that the PS–curve is only indirectly affected by trade liberalization since it has
been derived from the zero cutoff profit condition for the supply to the domestic market. The increase
in the ratio of TFP thresholdsA∗
S
A∗Ldue to trade liberalization will shift the PS–curve upward.17 As
for the relative factor market clearing condition, following trade liberalization it takes the following
form:
L
Sw−σ
S =
ηS(1− ϕS) + ηL(1− ϕL)(
Aft,L
Aft,S
)σ−1 1+
(AXLAft,L
)σ−1
(N−1)sL
1+
(AXSAft,S
)σ−1
(N−1)sS
ηSϕS + ηLϕL
(Aft,L
Aft,S
)σ−1 1+
(AXLAft,L
)σ−1
(N−1)sL
1+
(AXSAft,S
)σ−1
(N−1)sS
, (31)
with si ≡1−G(A∗
Xi)
1−G(A∗i )
denoting the share of exporters among firms of type i. Notice that trade liberal-
ization decreases Aσ−1L relative to Aσ−1
S , while the term1+
(AXLAft,L
)σ−1
(N−1)sL
1+
(AXSAft,S
)σ−1
(N−1)sS
, which adds due to trade
liberalization, is smaller than 1.18 Thus, relative demand for skilled labor ceteris paribus increases,
which shifts the FMC–curve to the right.
The extent of the factor reallocation between skilled and unskilled labor intensive firms depends,
as in section 5, on the factor intensity gap ϕS − ϕL between these two types of firms. We can show
that ηLηS
increases (decreases) with trade liberalization if ϕS −ϕL is at its maximum (minimum) level.
Furthermore, trade liberalization is more detrimental (beneficial) for the skilled (unskilled) labor
intensive firms, the larger is ϕS − ϕL (see appendix K).
Finally, the industry–wide average technology parameters in the free trade equilibrium are given
17This follows from equation 26.
18This follows from the fact that, if A follows a Pareto–distribution, then si =(
A∗i
A∗Xi
)k=(
fPi
fX
) kσ−1
. Furthermore,(AXi
Ai
)σ−1
=(
A∗Xi
A∗i
)σ−1
=(
fPi
fX
)−1
. Thus, si
(AXi
Ai
)σ−1
=(
fPi
fX
) k−σ+1σ−1
. Notice that we assume k−σ+1 > 0 since this
guarantees that the average TFP parameters Aσ−1i and Aσ−1
Xi are defined. This implies(
AXS
Aft,S
)σ−1
sS >(
AXL
Aft,L
)σ−1
sL,
i.e. the term1+
(AXLAft,L
)σ−1
(N−1)sL
1+
(AXSAft,S
)σ−1
(N−1)sS
is smaller than 1.
19
by (see appendix L):
ϕft =
ϕSAσ−1ft,S
[1 +
(AXS
Aft,S
)σ−1
(N − 1)sS
]+ ϕLA
σ−1ft,L
ηLηS
[1 +
(AXL
Aft,L
)σ−1
(N − 1)sL
]Aσ−1
ft,S
[1 +
(AXS
AS
)σ−1
(N − 1)sS
]+ Aσ−1
LηLηS
[1 +
(AXL
AL
)σ−1
(N − 1)sL
] ,
Aft =
Aσ−1
ft,S
[1 +
(AXS
Aft,S
)σ−1
(N − 1)sS
]+ Aσ−1
ft,LηLηS
[1 +
(AXL
Aft,L
)σ−1
(N − 1)sL
][1 + (N − 1)sS] +
ηLηS
[1 + (N − 1)sL]
1
σ−1
.
Comparing Aa with Aft leads to proposition 6
Proposition 6 Trade liberalization increases the sector–wide average TFP parameter A. The in-
crease in A is smaller, the larger is the factor intensity gap ϕS − ϕL.
Proof. See appendix M.
The intuition for proposition 6 is as follows. In our setting with firm heterogeneity in TFP and in
factor input choices, the firm selection induced by trade liberalization works along two dimensions.
On the one hand, since firms differ in TFP, the sector experiences a firm selection as described
by Melitz (2003): within each group of firms (skilled and unskilled labor intensive ones) factors
relocate from less productive non–exporters towards more productive exporters. This type of firm
selection is described by lemma 5, and it increases the average TFP of both skilled and unskilled
labor intensive firms. On the other hand, due to heterogeneity in the firms’ input choice, trade
liberalization also impacts ηLηS. If ηL
ηSincreases, those firms which experience a smaller average TFP
gain become relatively more frequent (see Lemma 6). Thus, if ηLηS
increases, the positive impact on
sector–wide TFP is dampened. If, in contrast, ηLηS
decreases, those firms which experience a larger
increase in average TFP become relatively more frequent, which implies that the positive impact on
sector–wide TFP is strengthened. Finally, the factor intensity gap ϕS−ϕL determines how ηLηS
adjusts
due to trade liberalization. If the gap is large, the increased factor market competition (rightward
shift of the FMC–curve in Figure 3) dominates the impact of increased profit opportunities abroad
(upward shift of PS–curve in Figure 3), so that ηL increases relative to ηS. On the other hand, if
the factor intensity gap is “small”, the increased profit opportunities abroad dominate the impact of
increased factor market competition, i.e. ηS increases relative to ηL. Thus, the more heterogeneous
are firms in their input choices, the smaller will be the gain in TFP induced by trade liberalization.
The theoretical models that have built upon Melitz’s (2003) pioneering contribution have empha-
sized the positive TFP effect of trade liberalization. At the same time, recent empirical evidence
(Lawless and Whelan 2008, Chen, Imbs, and Scott 2009) has suggested that these effects might be
only moderate. Our analysis suggests that, in the presence of substantial heterogeneity in factor
intensities, the increase in factor market competition actually dampens the increase in average TFP
20
brought about by trade liberalization, by forcing some of the skilled labor intensive firms out of the
market. Looking at factor markets is thus crucial to gain a more nuanced understanding of the firm
selection process.
7 Conclusions
A large empirical literature has shown that exporting and non–exporting firms differ in the mix of
inputs used in production even within narrowly defined sectors. In this paper, we have developed a
new theoretical framework to analyze how this type of heterogeneity affects the firm selection process
brought about by trade liberalization.
Under the realistic assumption that only skilled labor intensive firms can serve the foreign market,
we have shown that an increase in sector–wide exports increases competition for skilled labor, raising
its relative price. This effect is stronger, the bigger is the difference in factor intensities between
firms. Thus, in our model factor market competition plays against exporters, inducing some of them
to exit the market.
Much of the existing theoretical and empirical literature has highlighted how exporting firms tend
to be more productive. To take this into account, we have extended our analysis to allow also for
differences in TFP. In this richer environment we have shown that both skilled and unskilled labor
intensive firms can export, but that skilled labor intensive firms are more likely to do so. Furthermore,
we have argued that the firm selection induced by trade growth works along two dimensions. First,
within each group of (skilled and unskilled labor intensive) firms we observe selection against the
non–exporters, as in Melitz (2003). Second, because of factor market competition, selection works
against the skilled labor intensive firms. Combining these two effects, we have shown that the greater
is the difference in factor intensities between firms, the smaller is the increase in sector wide TFP
induced by trade liberalization. Our analysis has also broader policy implications. Competition
in factor markets is thus likely to have a moderating effect on the growth in average productivity
brought about by trade liberalization.
Our analysis can be extended to tackle several additional important questions. First, introducing
heterogeneity in factor input choices among firms suggests that even intra–industry trade might
bring about income distribution effects. How big are these effects? What are the implications for
the existence of gains from trade? A second possible extension involves considering multiple sectors,
with differences in factor intensities both within and across industries and Heckscher–Ohlin trade. In
this context one could study whether reallocations within sectors can dominate reallocations between
sectors, so that even the unskilled labor abundant country might experience an increase in country
wide skilled labor intensity following a trade liberalization. While both these questions are important,
they are left for future research.
21
Appendix
A Average skilled labor share parameter in autarky
If the economy were populated by η = ηL + ηS average firms, each of which producing with the
skilled labor share parameter ϕa =ϕS+ϕL
ηLηS
1+ηLηS
, the relative factor market clearing (FMC) equation
would become:L
Sw−σ
S =(1− ϕa)η
ϕaη. (32)
Substituting the terms for ϕa and η into equation 32 leads to equation 11 in the main part of thetext.
B Proof of proposition 2
The proof proceeds in four steps. First, we show that wftS ≥ wa
S. Let Ψa ≡(
fSfL
)(σ−1)/σ
and
Ψft ≡(
fS+fX2fL
)(σ−1)/σ
. The ratiowft
S
waSis then given by:
wftS
waS
=
{[Ψft(1− ϕL)− (1− ϕS)
][ϕS −ΨaϕL]
[Ψa(1− ϕL)− (1− ϕS)] [ϕS −ΨftϕL]
}1/(1−σ)
.
wftS
waS≥ 1 since Ψft ≤ Ψa. The latter follows from our assumption that fS ≥ fX .
Second, we show that the upward shift of the PS–curve becomes smaller, the larger is ϕS and thesmaller is ϕL, i.e. the larger is the factor intensity gap between skilled and unskilled labor intensive
firms. The upward shift of the PS–curve is reflected bywft
S
waSand the partial derivatives with respect
to ϕS and ϕL result as follows:
∂(
wftS
waS
)∂ϕS
=
(ϕS −ΨftϕL
)(ϕS −ΨaϕL)
[ϕL(w
ftS )1−σ(wa
S)1−σ + 1− ϕL
](
waS
wftS
)σ1−σ
Ψa−Ψft {[Ψa(1− ϕL)− (1− ϕS)] [ϕS −ΨftϕL]}2.
∂(wftS /wa
S)∂ϕS
< 0 since Ψa ≥ Ψft and ϕS − ΨmϕL > 0, m = a, ft, if the two types of firms are active in
general equilibrium (see proposition 1). Furthermore:
∂(
wftS
waS
)∂ϕL
=
(ϕS −ΨftϕL
)(ϕS −ΨaϕL)
[ϕS(w
ftS )1−σ(wa
S)1−σ + 1− ϕS
](
waS
wftS
)σ1−σ
Ψft−Ψa {[Ψa (1− ϕL)− (1− ϕS)] [ϕS −ΨftϕL]}2.
∂(wftS /wa
S)∂ϕL
> 0 since, again, Ψa ≥ Ψft and ϕS −ΨmϕL > 0, m = a, ft. Since the relationship betweenwft
S
waSand ϕi, i = S, L is monotonic, we assume for the remainder of appendix B that ϕS = 1−ϕL and,
thus, ϕS > 0.5 > ϕL.Third, we can show that the rightward shift of the FMC–curve with trade liberalization does not
depend on the factor intensity gap ϕS − ϕL. Solving the FMC conditions (equations 11 and 20) for
22
(ηLηS)a and (ηL
ηS)ft, and taking their ratio results in:
(ηL/ηS)ft
(ηL/ηS)a =
2(1−ϕS)−2LS(wft
S )−σϕS
LS(wft
S )−σ(1−ϕS)−ϕS
1−ϕS−LS(wa
S)−σϕS
LS(wa
S)−σ(1−ϕS)−ϕS
.
Thus, for each constant level of wS = wftS = wa
S we get (ηL/ηS)ft
(ηL/ηS)a = 2, i.e. the relative mass of unskilled
labor intensive firms ceteris paribus doubles with trade liberalization. Therefore, the rightward shiftof the FMC–curve does not depend on the factor intensity gap.
Fourth, we can show that ηLηS
decreases (increases) with trade liberalization if the factor intensity
gap is at its minimum (maximum) level. The maximum value of the factor intensity gap is 1 since ϕS
and ϕL are restricted by the interval [0, 1]. We define the minimum level of the factor intensity gapas that level which leads to ( ηS
ηL)a = 0. ( ηS
ηL)a is given by (remember that we assume ϕS = 1− ϕL):(
ηSηL
)a
=LS(wa
S)−σ(1− ϕS)− ϕS
1− ϕS − LS(wa
S)−σϕS
(33)
Thus, ( ηSηL)a = 0 if L
S(wa
S)−σ(1 − ϕS) − ϕS = 0. We define that value of ϕS which leads to ( ηS
ηL)a =
0 as ϕminS . In order to prove that ϕmin
S is uniquely defined, we substitute the term for waS into
LS(wa
S)−σ(1− ϕmin
S )− ϕminS = 0. Rearranging terms then leads to:(
ηSηL
)a
= 0 ⇐⇒[ϕS (Ψ
a + 1)− 1
ϕS (Ψa + 1)−Ψa
]σ/(σ−1)1− ϕS
ϕS︸ ︷︷ ︸≡Π
=S
L.
Notice that the squared bracket is defined since we have restricted the technology space so thatskilled and unskilled labor intensive technologies coexist in equilibrium (see proposition 1). We arenow able to determine the following partial derivative:
∂Π
∂ϕS
=σ
σ − 1
[ϕS (Ψ
a + 1)− 1
ϕS (Ψa + 1)−Ψa
] 11−σ (Ψa + 1) (1−Ψa)
[ϕS (Ψa + 1)−Ψa]2−[ϕS (Ψ
a + 1)− 1
ϕS (Ψa + 1)−Ψa
] σσ−1 1
ϕ2S
. (34)
Equation 34 shows that ∂Π∂ϕS
< 0 for all values of ϕS since Ψa > 1. Thus, ( ηSηL)a = 0 only if ϕS = ϕmin
S .
Furthermore, notice that ( ηSηL)a > 0 if the numerator in the term for ( ηS
ηL)a (equation 33) is negative
since the denominator is already negative due to 1−ϕS
w−σS ϕS
< LS. Thus, if ϕS > ϕmin
S we get ( ηSηL)a > 0.
Therefore, since LS(wa
S)−σ(1 − ϕmin
S ) − ϕminS = 0 and since wS increases with trade liberalization,
it follows immediately that ηSηL
increases with trade liberalization if the factor intensity gap is at itsminimum level. Finally, if the factor intensity gap between exporters and non–exporters is at its
maximum, i.e. ϕS = 1 and ϕL = 0, we get ( ηSηL)a = SfL
LfS> ( ηS
ηL)ft = SfL
L(fX+fS), i.e. ηS
ηLdecreases with
trade liberalization.
C The zero profit condition in the right panel of figure 3
In this appendix the superscript a denotes variables in the autarkic equilibrium, ft1 variables in thefree trade equilibrium before any adjustment of relative factor prices and ft2 variables in the free
23
trade equilibrium after the adjustment of relative factor prices. Considering equations 2, 8 and 18, wecan derive the axis–intercepts of the skilled labor intensive firms’ zero profit condition in the autarkicequilibrium and after trade liberalization. In the autarkic equilibrium the axis–intercepts are givenby:
ηaS =
[I
p(ϕS)
]a1
(σ − 1)fSand ηaL =
[Ip(ϕS)
−σ
p(ϕL)1−σ
]a1
(σ − 1)fS.
After trade liberalization and before any adjustment of relative factor prices, the axis intercepts aregiven by:
ηft1S =
[I
p(ϕS)
]ft11
(σ − 1)(fS + fX)and ηft1L =
[Ip(ϕS)
−σ
p(ϕL)1−σ
]ft12
(σ − 1)(fS + fX).
Since[
Ip(ϕS)
]a=[
Ip(ϕS)
]ft1and
[Ip(ϕS)
−σ
p(ϕL)1−σ
]a=[Ip(ϕS)
−σ
p(ϕL)1−σ
]ft1, we get the following result:
ηft1S
ηaS= fS
fS+fX<
1 andηft1L
ηaL= 2fS
fS+fX≥ 1. Thus, trade liberalization without any adjustment of relative factor prices
shifts the skilled labor intensive firms’ zero profit condition inward and makes it steeper.In order to determine how the increase in wS affects the ηS–axis intercept, we have to consider
the following partial derivative:
∂[I/p(ϕS)]
∂wS
= −w−σS c(ϕS)
σ−2SϕS
(L
S− 1− ϕS
ϕSw−σS
)< 0.
Thus,[
Ip(ϕS)
]ft2<[
Ip(ϕS)
]ft1, which implies ηft2S < ηft1S .
In order to determine how the increase in wS affects the ηL–axis intercept, we first have to considerthat the increase in wS makes the zero profit condition ceteris paribus flatter since its slope is given
by dηLdηS
= −2[p(ϕS)p(ϕL)
]1−σ
and ∂[p(ϕS)/p(ϕL)]∂wS
> 0. Second, we have to consider that a division of the zero
profit conditions of the two types of firms leads to[p(ϕS)p(ϕL)
]−σ
= fSfL
in the autarkic equilibrium (see
equation 8) and to[p(ϕS)p(ϕL)
]−σ
= fS+fX2fL
in the free trade equilibrium (see equation 18). Thus, we can
rewrite the expressions for the ηL–axis intercepts the following way:
ηaL =
[I
p(ϕL)
]a1
(σ − 1)fLand ηft2L =
[I
p(ϕL)
]ft21
(σ − 1)fL.
It follows immediately that ηaL < ηft2L since ∂[I/p(ϕL)]∂wS
> 0. Third, since the zero profit condition
becomes flatter with the increase in wS and since ηft2S < ηft1S , we can conclude that ηft2L < ηft1L .
D Proof of proposition 3
Remember that, in the autarkic equilibrium ϕa =ϕS+ϕL
(ηLηS
)a
1+(
ηLηS
)a . Since the production of each skilled
labor intensive firm ceteris paribus doubles with trade liberalization, the average sector–wide skilled
labor share parameter in the free trade equilibrium is given by ϕft =2ϕS+ϕL
(ηLηS
)ft
2+(
ηLηS
)ft . Thus, ϕft > ϕa if
24
and only if (ϕS − ϕL)
[2(
ηLηS
)a−(
ηLηS
)ft]≥ 0. This condition always holds since 2
(ηLηS
)a=(
ηLηS
)ftif fS = fX and 2
(ηLηS
)a>(
ηLηS
)ftif fS > fX , for any factor intensity gap ϕS − ϕL.
E Proof of proposition 4
If we define ΨN ≡[fS+(N−1)fX
NfL
]σ−1σ, the PS–condition can be written as follows:
wftS =
[ΨN(1− ϕL)− (1− ϕS)
ϕS −ΨNϕL
] 11−σ
.
The partial derivative of wftS with respect to the number of trading partners N results as follows:
∂wftS
∂N=
1
1− σ(wft
S )σϕS − ϕL
(ϕS −ΨNϕL)2
∂ΨN
∂N, with
∂ΨN
∂N=
σ − 1
σ(ΨN)1/(1−σ)fL (fX − fS)
(NfL)2 .
Since fS ≥ fX and ϕS > ϕL, it follows that∂wft
S
∂N> 0, i.e. the PS–curve shifts upward with an
increase in N . Notice also that limN→∞ ΨN =(
fXfL
)σ−1σ. Thus, if N → ∞, we get wft
S > waS if
fS > fX and wftS = wa
S if fS = fX . Furthermore, it is straightforward to check that limN→∞ wftS < 1
since(
fXfL
)(σ−1)/σ
> 1, i.e. wftS is always strictly smaller than 1, even if N → ∞. The FMC–curve,
instead, establishes a linear relationship between ηLηS
and N . Thus, if N is sufficiently large, ηLηS
alwaysincreases with trade liberalization, irrespective of the magnitude of the factor intensity gap ϕS − ϕL.
Turning now to the panel on the right of figure 4, the ηL–axis intercept of the zero profit conditionfor skilled labor intensive firms is given by:
ηftL =NIp (ϕS)
−σ
p (ϕL)1−σ
1
[fS + (N − 1)fX ] (σ − 1).
The partial derivative with respect to N results as:
∂ηftL∂N
=Ip (ϕS)
−σ
p (ϕL)1−σ
fS − fX
[fS + (N − 1) fX ]2 (σ − 1)
≥ 0.
Furthermore, limN→∞ ηftL = Ip(ϕS)−σ
fXp(ϕL)1−σ(σ−1)
≥ ηaL = Ip(ϕS)−σ
fSp(ϕL)1−σ(σ−1)
. Finally, the ηS–axis intercept of
that zero profit condition is given by:
ηftS =I
p (ϕS)
1
[fS + (N − 1) fX ] (σ − 1).
Thus, if N → ∞ we get ηftS → 0 and ηftL > 0.
F Proof of lemma 4
Using our definition of AS, equation 24 can be rewritten as follows:
25
[1
σ − 1
1
AS
q(AS, ϕS
)− fP
S
]=
fE
[1−G (A∗S)]
. (35)
Sinceq(A∗
S ,ϕS)q(AS ,ϕS)
=(
AS
A∗S
)−σ
, the zero cutoff profit condition (equation 23) can be transformed to:
q(Ai, ϕi
)=
(Ai
A∗i
)σ
A∗i (σ − 1)fP
i . (36)
Substituting equation 36 into equation 35 and simplification leads to equation 25 in the main part.
G Proof of proposition 5
In this section we first derive the FMC condition under heterogeneity in TFP and then establishproposition 5.
With heterogeneity in TFP we get aSS = Aσ−1S ϕSw
−σS c (AS, ϕS)
σ and aLS = Aσ−1S (1−ϕS)c (AS, ϕS)
σ.
Notice that ∂aSS
∂AS< 0 and ∂aLS
∂AS< 0, since c (AS, ϕS)
σ =[
1AS
(ϕSw
1−σS + 1− ϕS
)1/(1−σ)]σ. Fixed costs
are not influenced by the TFP parameter, and are still given by equation 4. Let fi ≡ fE
1−G(A∗i )
+ fPi ,
i.e. fi stands for the total fixed costs of a single firm of type i in general equilibrium. The FMCcondition can be rewritten as:
L
S=
∑i=L,S (1− ϕi)
[(ϕiw
1−σS + 1− ϕi
)1/(1−σ)]σ [
1
Aiq(Ai, ϕi
)+ fi
]ηi∑
i=L,S ϕiw−σS
[(ϕiw
1−σS + 1− ϕi
)1/(1−σ)]σ [
1
Aiq(Ai, ϕi
)+ fi
]ηi
. (37)
Finally, remembering that q(Ai, ϕi
)= IP σ−1
[σ
σ−11
Ai
(ϕiw
1−σS + 1− ϕi
)1/(1−σ)]−σ
and that free entry
impliesq(Ai,ϕi)Ai(σ−1)
= fi =fE
1−G(A∗i )
+ fPi , equation 37 can be simplified to:
L
S=
(1− ϕS)Aσ−1S + (1− ϕL)A
σ−1L
ηLηS
ϕSw−σS Aσ−1
S + ϕLw−σS Aσ−1
LηLηS
(38)
To establish proposition 5, notice that from lemma 4 we know that A∗a,S and A∗
a,L only depend
on the parameters fE and fPi and the distribution of A. Therefore, Aσ−1
a,S and Aσ−1a,L are determined
from equation 25 alone. Thus, equations 26 and 27 can be solved for wS and ηLηS
like in the autarkicequilibrium without firm heterogeneity in TFP. Finally, notice that the right hand side of equation38 still depends positively on ηL
ηS, i.e. equation 38 is still represented by a negatively sloping curve.
H Aggregation under TFP heterogeneity — autarky
If the economy were populated by η = ηL + ηS average firms, each of which producing with the
technology parameters ϕa =ϕSA
σ−1a,S +ϕLA
σ−1a,L
ηLηS
Aσ−1a,S +Aσ−1
a,LηLηS
and Aa =
[Aσ−1
a,S +Aσ−1a,L
ηLηS
1+ηLηS
] 1σ−1
, the relative factor market
26
clearing (FMC) equation would become:
L
Sw−σ
S =(1− ϕa)Aσ−1
a η
ϕaAσ−1a η
. (39)
Substituting the terms for ϕa, Aa and η into equation 39 leads to equation 27 in the main part of thetext.
I Proof of lemma 5
Since A∗Xi is a function of A∗
i (see equation 29), equation 30 can be solved for A∗ft,i. In order to prove
that A∗i increases with trade liberalization, we show that the term (1−G (A∗
i ))
[(Ai
A∗i
)σ−1
− 1
]≡ Λ
depends negatively on A∗i . Remember that Ai =
[∫∞A∗
iAσ−1µ(A)dA
]1/(σ−1)
is also a function of A∗i .
Using Leibniz’s rule to calculate ∂Ai
∂A∗i, we obtain:
∂Λ
∂A∗i
= −1−G (A∗i )
A∗i
(σ − 1)
(Ai
A∗i
)σ−1
< 0.
Since trade liberalization adds the ex–ante expected profits from serving N − 1 foreign markets tothe left hand side of the free entry condition (see equation 30), the threshold TFP–parameter A∗
i hasto increase so that Λ decreases and the free entry condition in the free trade situation holds again.
J Proof of lemma 6
Assuming that A is distributed on [1,∞) according to a Pareto distribution with density g(A) = kAk+1
and k > σ − 1, we get the following:
1−G(A) =
(1
A
)k
andAi
A∗i
=
(k
1 + k − σ
)1/(σ−1)
.
Thus, in the autarkic equilibrium the free entry condition is given by:(1
A∗a,i
)k
fPi
(k
1 + k − σ− 1
)= fE.
Solving for A∗a,i yields:
A∗a,i =
(fPi
fE
σ − 1
1 + k − σ
)1/k
.
In the free trade equilibrium the free entry condition results as:
(1
A∗ft,i
)k
fPi
σ − 1
1 + k − σ+
1
A∗ft,i
[fPi
(N−1)fX
]1/(1−σ)
k
fXσ − 1
1 + k − σ= fE.
27
Solving for A∗ft,i leads us to:
A∗ft,i =
fPi +
[fPi
(N−1)fX
]k/(σ−1)
fE
σ − 1
1 + k − σ
1/k
.
Thus, we can determine the following ratio:
A∗ft,i
A∗a,i
=[1 + (fP
i )(1+k−σ)/(σ−1)(NfX)
k/(1−σ)]1/k
. (40)
Equation 40 shows that a larger value for fPi leads to a larger increase in A∗
i with trade liberalization.Thus, since fP
S > fPL , the increase in A∗
S due to trade liberalization exceeds the increase in A∗L. Since
Ai
A∗i=(
k1+k−σ
)1/(σ−1), it follows immediately that the increase in AS also exceeds the increase in AL
due to trade liberalization.
K Firm selection with trade liberalization and TFP heterogeneity
We can illustrate the relationship between the factor intensity gap and firm selection with tradeliberalization again by the upward shift of the PS–curve and the rightward shift of the FMC–curve.
The upward shift of the PS–curve is reflected by the ratiowft
S
waS, whereas the rightward shift of the
FMC–curve is reflected by the ratio (ηL/ηS)ft
(ηL/ηS)a .
If we define Ψa ≡(
fPS
fPL
)(σ−1)/σ (A∗a,S
A∗a,L
)(1−σ)2/−σ
and Ψft ≡(
fPS
fPL
)(σ−1)/σ (A∗ft,S
A∗ft,L
)(1−σ)2/−σ
, the ratio
wftS
waSis given by:
wftS
waS
=
{[Ψft(1− ϕL)− (1− ϕS)
][ϕS −ΨaϕL]
[Ψa(1− ϕL)− (1− ϕS)] [ϕS −ΨftϕL]
}1/(1−σ)
.
Notice that Ψft ≤ Ψa since A∗S increases with trade liberalization. Thus,
wftS
waS> 1.
Furthermore, the ratio (ηL/ηS)ft
(ηL/ηS)a results as follows:
(ηL/ηS)ft
(ηL/ηS)a =
(1−ϕS)−LS(wft
S )−σϕS
LS(wft
S )−σ(1−ϕS)−ϕS
1−ϕS−LS(wa
S)−σϕS
LS(wa
S)−σ(1−ϕS)−ϕS
(Aa,L
Aa,S
)σ−1[1 +
(AXS
Aft,S
)σ−1
(N − 1)sS
](
Aft,L
Aft,S
)σ−1[1 +
(AXL
Aft,L
)σ−1
(N − 1)sL
] . (41)
Compared to the case without firm heterogeneity in TFP, the term with the average TFP parametersadds to the right hand side of equation 41. Since the average TFP parameters do not depend on the
factor intensity gap ϕS − ϕL, bothwft
S
waSand (ηL/ηS)
ft
(ηL/ηS)a react to a change in the factor intensity gap the
same way as we have described in appendix B for the case without firm heterogeneity in TFP.
28
L Aggregation under TFP heterogeneity — free trade
If η = ηS [1 + (N − 1)sS] + ηL [1 + (N − 1)sL] average firms were active in the market, each of which
producing with the technology parameters ϕft and Aft, as defined in section 6, the relative factormarket clearing condition would become:
L
Sw−σ
S =(1− ϕft)Aσ−1
ft η
ϕftAσ−1ft η
. (42)
Substituting the terms for ϕft, Aft and η into equation 42 leads to equation 31 in the main part ofthe text.
M Proof of proposition 6
If A follows a Pareto distribution, then(
AXi
Ai
)σ−1
=(
A∗Xi
A∗i
)σ−1
. SinceA∗
Xi
A∗i=(
fPi
fX
)1/(1−σ)
, we can thus
conclude:(
AXi
Ai
)σ−1
=(
fPi
fX
)−1
. Furthermore, si =(
A∗i
A∗Xi
)k=(
fPi
fX
) kσ−1
. Thus, the term(
AXi
Aft,i
)σ−1
si
can be simplified to:(
fPi
fX
) k−σ+1σ−1
. Thus, we can rewrite the term for Aσ−1ft the following way:
Aσ−1ft =
Aσ−1ft,S
[1 +
(fPS
fX
) k−σ+1σ−1
(N − 1)
][1 +
(fPS /fX)
kσ−1
(N−1)−1
]+ ηL
ηS
[1 +
(fPL /fX)
kσ−1
(N−1)−1
] +
Aσ−1ft,L
[1 +
(fPL
fX
) k−σ+1σ−1
(N − 1)
]ηSηL
[1 +
(fPS /fX)
kσ−1
(N−1)−1
]+
[1 +
(fPL /fX)
kσ−1
(N−1)−1
](43)
Equation 43 has the following implications. Due to factor market competition, which is reflectedby a rightward shift of the FMC–curve, ηL
ηSceteris paribus increases, which decreases the weighting
factor for Aσ−1ft,S, while it increases the weighting factor for Aσ−1
ft,L. Thus, the increase in factor marketcompetition due to trade liberalization shifts resources to those firms, which experience a smallerincrease in TFP due to trade liberalization. Notice that, if all firms would produce with identicalfactor intensities, ηL
ηSwould stay constant across equilibria.
Furthermore, we can show that Aσ−1ft > Aσ−1
a even if trade liberalization leads to the maximumpossible increase of ηL
ηS. We have shown previously that a factor intensity gap ϕS − ϕL = 1 leads to
the maximum possible increase of ηLηS
with trade liberalization. If ϕS = 1 and ϕL = 0, the FMC andthe PS conditions reduce, respectively, to:
L
S(wft
S )−σ =1 +
(AXL
Aft,L
)σ−1
(N − 1)sL
1 +(
AXS
Aft,S
)σ−1
(N − 1)sS
Aσ−1ft,L
Aσ−1ft,S
(ηLηS
)ft
and wftS =
(fPS
fPL
)−1/σ(A∗
ft,L
A∗ft,S
)(1−σ)/σ
.
Notice that the limiting case of N = 1 would represent the autarkic equilibrium. Combining these
two conditions and considering that(
AXi
Aft,i
)σ−1
si =(
fPi
fX
) k−σ+1σ−1
and
Aft,LA∗ft,L
Aft,SA∗ft,S
= 1, as A follows a Pareto
29
distribution, leads to:
L
S
fPS
fPL
=1 +
(fPL
fX
) k−σ+1σ−1
(N − 1)
1 +(
fPS
fX
) k−σ+1σ−1
(N − 1)
(ηLηS
)ft
. (44)
Thus, if countries liberalize trade, i.e. N increases to a value N ≥ 2, ηLηS
has to increase such that the
term1+
(fPLfX
) k−σ+1σ−1
(N−1)
1+
(fPS
fX
) k−σ+1σ−1
(N−1)
(ηLηS)ft stays constant across equilibria.
Rewriting equation 43 and considering (ηLηS)ft = L
S
fPS
fPL
1+
(fPSfX
) k−σ+1σ−1
(N−1)
1+
(fPL
fX
) k−σ+1σ−1
(N−1)
, as it follows from equation
44, leads to:
Aσ−1ft =
Aσ−1ft,S + Aσ−1
ft,L(ηLηS)ft
1+
(fPLfX
) k−σ+1σ−1
(N−1)
1+
(fPS
fX
) k−σ+1σ−1
(N−1)
1+
(fPS
fX
) kσ−1
(N−1)
1+
(fPS
fX
) k−σ+1σ−1
(N−1)
+ LS
fPS
fPL
1+
(fPL
fX
) kσ−1
(N−1)
1+
(fPL
fX
) k−σ+1σ−1
(N−1)
. (45)
Notice, first, that the numerator of equation 45 increases with trade liberalization since both Aσ−1ft,S
and Aσ−1ft,L increase, while the term (ηL
ηS)ft
1+
(fPLfX
) k−σ+1σ−1
(N−1)
1+
(fPS
fX
) k−σ+1σ−1
(N−1)
in the numerator remains constant across
equilibria. Second, the denominator decreases with an increase in N . This results from the followingpartial derivative:
∂
[1+(fP
i /fX)k
σ−1 (N−1)
1+(fPi /fX)
k−σ+1σ−1 (N−1)
]∂N
=
(fPi
fX
) kσ−1(1− fX
fPi
)[1 +
(fPi
fX
) k−σ+1σ−1
(N − 1)
]2 < 0. (46)
Thus, Aσ−1 increases with trade liberalization, also with the maximum possible increase in ηLηS.
30
References
Alvarez, R. and R. Lopez (2005). Exporting and performance: evidence from Chilean plants.Canadian Journal of Economics 38, 1384–1400.
Atkeson, A. and A. Burstein (2010). Innovation, firm dynamics and international trade. Journalof Political Economy 118, 433–484.
Axtell, R. L. (2001). Zipf distribution of U.S. firm sizes. Science 293, 1818–1820.
Bernard, A. B., J. Eaton, S. Kortum, and J. B. Jensen (2003). Plants and productivity in inter-national trade. American Economic Review 93, 1268–1290.
Bernard, A. B. and J. B. Jensen (1995). Exporters, jobs, and wages in U.S. manufacturing: 1976–1987. Brookings Papers on Economic Activity: Microeconomics 1995, 67–119.
Bernard, A. B., J. B. Jensen, S. J. Redding, and P. K. Schott (2012). The empirics of firm hetero-geneneity and international trade. Annual Review of Economics . forthcoming.
Bernard, A. B., S. Redding, and P. K. Schott (2007). Comparative advantage and heterogeneousfirms. Review of Economic Studies 74, 31–66.
Chen, N., J. Imbs, and A. Scott (2009). The dynamics of trade and competition. Journal ofInternational Economics 77, 50–62.
Crozet, M. and F. Trionfetti (2011). Comparative advantage and within–industry firms perfor-mance. CEPII WP 2011–1.
Emami Namini, J. (2009). The short and long run impact of globalization if firms differ in factorshares. Tinbergen Institute WP 2009–020.
Emami Namini, J. (2011). Trade and the gains from trade when firms differ in factor input ratios.Mimeo, Erasmus University Rotterdam.
Furusawa, T. and H. Sato (2008). A factor–proportions theory of endogenous firm heterogeneity.Mimeo, Hitotsubashi University.
Haruyama, T. and L. Zhao (2008). Trade and firm heterogeneity in a quality–ladder model ofgrowth. Mimeo, Kobe University.
Jørgensen, J. G. and P. J. H. Schroder (2008). Fixed export cost heterogeneity, trade and welfare.European Economic Review 52, 1256–1274.
Klein, M. W., C. Moser, and D. M. Urban (2010). The contribution of trade to wage inequality:The role of skill, gender and nationality. NBER WP 15985.
Lawless, M. and K. Whelan (2008). Where do firms export, how much, and why? Mimeo, UCDublin.
Long, N. V., H. Raff, and F. Stahler (2011). Innovation and trade with heterogeneous firms. Journalof International Economics 84, 149–159.
Luttmer, E. G. J. (2007). Selection, growth, and the size distribution of firms. Quarterly Journalof Economics 122, 1103–1144.
Markusen, J. R. and A. J. Venables (2000). The theory of endowment, intra–industry and multi–national trade. Journal of International Economics 52, 209–234.
Martins, P. S. and L. D. Opromolla (2011). Why ex(im)porters pay more: Evidence from matchedfirm–worker panels. Mimeo, Queen Mary University London.
31
Melitz, M. (2003). The impact of trade on intra–industry reallocations and aggregate industryproductivity. Econometrica 71, 1695–1725.
Munch, J. and J. R. Skaksen (2008). Human capital and wages in exporting firms. Journal ofInternational Economics 75, 363–372.
Raff, H. and J. Wagner (2010). Intra–industry adjustment to import competition: Theory andapplication to the german clothing industry. The World Economy 33, 1006–1002.
Schroder, P. J. H. and A. Sørensen (2011). Firm exit, technological progress and trade. EuropeanEconomic Review . forthcoming.
Wagner, J. (2010). Exports and firm characteristics in German manufacturing industries. IZA WP5244.
Yeaple, S. R. (2005). A simple model of firm heterogeneity, international trade, and wages. Journalof International Economics 65, 1–20.
32