The Impact of Trade Liberalization
on Industrial Productivity
Paul S. Segerstrom

Yoichi Sugita

Stockholm School of Economics

Stockholm School of Economics

First version: June 22, 2012
Current version: November 19, 2012
Abstract: An empirical finding by Trefler (2004, AER) and others that industrial productivity increases more strongly in liberalized industries than in non-liberalized industries has been widely accepted as evidence for the Melitz (2003, Econometrica) model. This paper shows that a multi-industry
version of the Melitz model does not predict this relationship. Instead, it predicts the opposite relationship that industrial productivity increases more strongly in non-liberalized industries than in liberalized
industries.
JEL classification: F12, F13.
Keywords: Trade liberalization, firm heterogeneity, industrial productivity.
Acknowledgments: We thank Tore Ellingsen, Erik Lindqvist and seminar participants at the Stockholm School of Economics, the Midwest International Trade Meeting and Erasmus University Rotterdam for helpful comments. Financial support from the Wallander Foundation is gratefully acknowledged.
Author: Paul S. Segerstrom, Stockholm School of Economics, Department of Economics, Box 6501,
11383 Stockholm, Sweden (E-mail: paul.segerstrom@hhs.se, Tel: +46-8-7369203, Fax: +46-8-313207).
Author: Yoichi Sugita, Stockholm School of Economics, Department of Economics, Box 6501, 11383
Stockholm, Sweden (E-mail: yoichi.sugita@hhs.se, Tel: +46-8-7369254, Fax: +46-8-313207).

1

Introduction

A central question in the study of international trade is how trade liberalization improves resource
allocation in the liberalizing country. While traditional studies have emphasized reallocation acrossindustries, recent studies have discovered that reallocation occurs even within industries. In the last
decade, the empirical literature has established that trade liberalization improves productivity by shifting resources from less productive to more productive firms within industries.
By comparing industries that experienced different degrees of trade liberalization (e.g. tariff cuts),
several studies found that intra-industry reallocation improves industrial productivity more strongly
in liberalized industries than in non-liberalized industries.1 For instance, by investigating the longrun impact of the Canada-USA free trade agreement on Canadian manufacturing industries, Trefler(2004) found that industrial productivity increased more strongly in liberalized industries than in nonliberalized industries, and that the rise in industrial productivity was mainly due to the shift of resourcesfrom less productive to more productive firms. Other studies also found that the exit of low productive
firms from an industry, which contributes to a rise in industrial productivity, is positively associatedwith the decrease in tariffs and other import costs in the industry (Bernard, Jensen, and Schott, 2006,
for the US; Lilieva, 2008, for Canada; Nataraj, 2011, for India; and Eslava, Haltiwanger, Kugler, andKugler, 2009, for Colombia).2
The seminal model by Melitz (2003) has been accepted as the central model of intra-industry
reallocation due to trade liberalization. By combining the Hopenhayn (1992) model of the entry andexit of heterogeneous firms and the Krugman (1979, 1980) model with fixed trade costs, Melitz (2003)
theoretically demonstrated that trade liberalization improves the aggregate productivity of economies
through resource reallocation toward more productive firms.
It is widely believed that the Melitz model explains the fact that industrial productivity increases
more strongly in liberalized industries than in non-liberalized industries. Virtually all survey papers
recently written by leading scholars cite Trefler (2004) and Bernard, Jensen, and Schott (2006) as
1

Early studies use firm size as a proxy for firm productivity (Head and Ries, 1999; Baggs, 2005) and/or trade volume
as a proxy for trade policy meaures (Pavcnik, 2002). Recent studies use more direct productivity measures and trade policy
measures. Tybout (2003) surveys early studies in this literature.
2
Other studies found positive but statistically insignificant associations between productivity improvement due to intraindustry reallocation and the extent of liberalization (Tybout and Westbrook, 1995, for Mexico; Fernandes, 2007, for Colombia; Harrison, Martin, and Nataraj, 2011 and Sivadasan, 2009, for India).

2evidence in support of the Melitz model (Bernard, Jensen, Redding, and Schott, 2007, 2012; Helpman,
2011; Redding, 2011; Melitz and Trefler, 2012). For instance, Helpman (2011, p.107) writes in his
recent book: “In other words, the Canadian experience conforms to the theoretical analysis.” With thisempirical support, the reallocation mechanism of Melitz (2003) has been applied to various issues andis central to the theoretical trade research of the last decade.3
The purpose of our paper is to revisit the correspondence between empirical studies and the Melitz
model. The type of trade liberalization that typical empirical studies investigated is different fromthe type of trade liberalization that Melitz (2003) theoretically analyzed. The liberalization in the one
industry Melitz model is multilateral and uniform liberalization, in which all countries reduce tariffs
on all goods by the same amount. On the other hand, the above mentioned empirical studies investigate
unilateral and non-uniform liberalization, in which one country reduces tariffs and the extent of tariff
reductions vary across industries.4
The gap between theory and evidence has probably been overlooked because the analysis of unilateral and non-uniform liberalization in a multi-industry Melitz model appears to be complicated andintractable. To overcome this theoretical difficulty, we develop a brand new way of solving the Melitz
model using simple and intuitive diagrams. We show that these new techniques can be used to solve a
multi-industry version of the Melitz model and study what happens when one country reduces tariffs
in some industries but not others.
Our surprising result is that the multi-industry Melitz model does not predict that industrial productivity rises more strongly in liberalized industries than in non-liberalized industries. Instead, it predicts
the opposite relationship that industrial productivity rises more strongly in non-liberalized industries
than in liberalized industries. This result forces us to re-think the match between theory and evidence:an empirical fact that has been widely cited as evidence for the Melitz model is actually evidence
against the Melitz model. Furthermore, when the size of a liberalized industry is small compared tothe size of the liberalizing country, we find a more striking prediction: industrial productivity decreases
in the liberalized industry, while it increases in the non-liberalized industries. This result calls for re3
Applications of the Melitz (2003) mechanism of intra-industry reallocation include too many papers to cite in this limited
space. We encourage readers to consult with the above-mentioned survey papers for applications.
4
Trefler (2004) and Lileeva (2008) controlled for the US tariff changes when they estimate the impact of the Canadiantariff cuts introduced by the Canada-USA free trade agreement (CUFTA). Therefore, although the CUFTA is a bilateral trade
agreement, the effects of Canadian tariff cuts estimated by these two papers should be interpreted as the effects of unilateralliberalization.

3

thinking the model’s implications for industrial promotion policy. If the government of a country isinterested in raising the productivity of a small “target” industry through a resource reallocation fromless productive to more productive firms, the theoretically correct advice based on the Melitz model
is to protect the target industry, not trade liberalization. This is obviously the opposite of what Trefler(2004) and other empirical studies suggest.
As illustrated in Table 1, no previous paper has analyzed unilateral and non-uniform trade liberalization in a Melitz model with multiple Melitz industries. This is required to compare the model withthe empirical facts from cross-industry regressions. Melitz and Ottaviano (2008), Demidova (2008),
Demidova and Rodriguez-Clare (2009, 2011), and Felbermayr, Jung, and Larch (2012) analyze unilateral trade liberalization in models with just one Melitz industry. Bernard, Redding, and Schott (2007)
and Okubo (2009) develop models with multiple Melitz industries but only analyze multilateral tradeliberalization.

One Melitz Industry
Multilateral
Liberalization

Melitz (2003)

Unilateral
Liberalization

Melitz & Ottaviano(08)Demidova(08)Demidova & Rodriguez-Clare (09, 11)Felbermayr, Jung & Larch (12)Multiple Melitz Industries
Bernard, Redding & Schott(07)Okubo (09)Our paper

Table 1: Related Literature
The rest of the paper is organized as follows. In section 2, we present a multi-industry version of
the Melitz (2003) model. In section 3, we solve the model analytically for the effects of unilateral andnon-uniform trade liberalization. We also present an intuitive explanation for our results in subsection
3.2 and solve the model numerically to illustrate the intuition. In section 4, we offer some concludingcomments and there is an Appendix where calculations that we did to solve the model are presented in
more detail.

2

The ModelThis section presents a multi-industry version of the Melitz (2003) model. Our model differs from theoriginal model in five points: (1) our model has two industries and two countries; (2) industries and4countries are asymmetric so that wages differ between countries; (3) trade costs are asymmetric anddepend on the direction of trade; (4) the utility function of consumers has two tiers, the Cobb-Douglas
upper tier and the CES lower tier; and (5) firms draw their productivities from Pareto distributions.
The last two specifications are commonly used in applications of the Melitz model.

2.1

Setting

Consider two countries, 1 and 2, with two differentiated goods sectors (or industries), A and B.
Throughout the paper, subscripts i and j denote countries (i, j ∈ {1, 2}) and subscript s denotes
sectors (s ∈ {A, B}). Though the model has infinitely many periods, there is no means for saving overperiods. We focus on a stationary steady state equilibrium where aggregate variables do not changeover time and omit notation for time periods.
The representative consumers in both countries have an identical two-tier (Cobb-Douglas plus
CES) utility function:
αA αB
U ≡ CA
CB

ˆ
where

Cs ≡

ρs

qs (ω) dω

1/ρs
.

(1)

ω∈Ωs

In equation (1), qs (ω) is the consumer’s quantity consumed of a product variety ω produced in sectors, Ωs is the set of available varieties in sector s and ρs measures the degree of product differentiationin sector s. We assume that products within a sector are closer substitutes than products across sectors,
which implies that a within-sector elasticity of substitution σ s ≡ 1/(1 − ρs ) satisfies σ s > 1. Given
that αA + αB = 1, αs represents the share of consumer expenditure on sector s products.
Country i is endowed with Li unit of labor as the only factor of production. Labor is inelasticallysupplied and workers in country i earn the competitive wage rate wi . We measure all prices relative tothe price of labor in country 2 by setting w2 = 1.
Firms are risk neutral and maximize expected profits. In each time period, let Mise denote themeasure of firms that choose to enter in country i and sector s. Each firm uses Fis units of labor toenter and incurs the fixed entry cost wi Fis . Each firm then independently draws its productivity ϕ froma Pareto distribution. The cumulative distribution function Gis (ϕ) and the density function gis (ϕ) are5

given by
Gis (ϕ) = 1 −

bis
ϕ

θ s
and gis (ϕ) =θs bθissfor ϕ ∈ [bis , ∞),ϕθs +1(2)where θs and bis are the shape and scale parameters of the distribution for country i and sector s. We
assume that θs > σ s − 1 to guarantee that expected profits are finite. A firm with productivity ϕ uses
1/ϕ units of labor to produce one unit of output and has constant marginal cost wi /ϕ in country i.
This firm must use fijs units of labor and incur the fixed “marketing” cost wi fijs to sell in country
j. There are also iceberg trade costs associated with shipping products across countries: a firm thatexports from country i to country j 6= i in sector s needs to ship τ ijs > 1 units of a product in orderfor one unit to arrive at the foreign destination (if j = i, then τ iis = 1).
Because of the fixed marketing costs, there exist productivity cut-off levels ϕ∗ijs such that only
firms with ϕ ≥ ϕ∗ijs sell products from country i to country j in sector s. In each country and sector,
we assume that exporting requires higher fixed costs than local selling (fijs > fiis ). We solve themodel for an equilibrium where both countries produces both goods A and B, and the more productive
firms export (ϕ∗iis < ϕ∗ijs ). In each period, there is an exogenous probability δ is with which actively
operating firms in country i and sector s die and exit. In a stationary steady state equilibrium, the massof actively operating firms Mis and the mass of entrants Mise in country i and sector s satisfy[1 − Gis (ϕ∗iis )] Mise = δ is Mis ,(3)

that is, firm entry in each time period is matched by firm exit.
Let pijs (ϕ) denote the price charged in country j by a firm with productivity ϕ from country iin sector s. Let qijs (ϕ) denote the quantity that consumers in country j buy from this firm and let
rijs (ϕ) ≡ pijs (ϕ) qijs (ϕ) denote the corresponding firm revenue. Also, let Pjs denote the index of
consumer prices in country j and sector s. Since free entry implies that aggregate profit income iszero, in each time period, consumers in country j spend exactly what they earn in wage income wj Lj .
Consumer optimization calculations imply that consumer demand and the corresponding firm revenue
areqijs (ϕ) =pijs (ϕ)−σs αs wj Ljpijs (ϕ)1−σs αs wj Lj
and rijs (ϕ) =
.
1−σ s
1−σ s
Pjs
Pjs

A firm with productivity ϕ from country i earns variable profit π ijs (ϕ) = rijs (ϕ) −6(4)wi τ ijs
ϕ qijs (ϕ) from

selling to country j in sector s. Solving for the profit-maximizing price, we obtain thatpijs (ϕ) =wi τ ijs
,
ρs ϕ

(5)

that is, each firm charges a fixed markup over its marginal cost wi τ ijs /ϕ. Substituting this price backinto the variable profit function yields π ijs (ϕ) = rijs (ϕ)/σ s .

2.2

Sector Equilibrium

We first derive equilibrium conditions for each sector, following the steps in Melitz (2003) and otherprevious studies. Since a firm with cut-off productivity ϕ∗ijs just breaks even from selling to country j,
it follows that ϕ∗ijs is determined by the cut-off productivity condition
rijs ϕ∗ijs
σs= wi fijs .

(6)

A firm from country i needs to have productivity ϕ ≥ ϕ∗ijs to justify paying the fixed marketing cost
wi fijs of serving the country j market in sector s.
From (4), (5) and (6), the cut-off productivity levels of domestic and foreign firms in country j are
related by trade costs and labor costs as follows:ϕ∗ijs = Tijs



wi
wj

1/ρs

ϕ∗jjs ,

(7)

where Tijs ≡ τ ijs (fijs /fjjs )1/(σs −1) captures both variable and fixed trade costs from country i tocountry j relative to the fixed trade cost within country j.
Let µis (ϕ) denote the equilibrium productivity density function for country i and sector s. Since
only firms with productivity ϕ ≥ ϕ∗iis produce in equilibrium and firm exit is uncorrelated with productivity, the equilibrium productivity density function is given byµis (ϕ) ≡gis (ϕ)1−Gis (ϕ∗iis ) 0

if ϕ ≥ ϕ∗iisotherwise.

7(8)

Given (3) and (8), the price index Pjs satisfies
1−σ s
Pjs
=

X Mkse ˆ ∞
k=1,2

δ ks

ϕ∗kjs

pkjs (ϕ)1−σs dGks (ϕ).

(9)

In each time period, there is free entry by firms in each sector s and country i. Let π̄ is denote theaverage profits across all domestic firms in country i and sector s (including the fixed marketing costs).
Pt
Let v̄is ≡ ∞t=0 (1 − δ is ) π̄ is = π̄ is /δ is denote the present value of average profit flows in country iand sector s, taking into account the rate δ is at which firms exit in each time period. Free entry implies
that the probability of successful entry times the expected profits earned from successful entry must
equal the cost of entry, that is, [1 − Gis (ϕ∗iis )]π̄ is /δ is = wi Fis . Calculating the average profits acrossall domestic firms (exporters and non-exporters), we obtain
1 X
δ is

ˆ ∞ 

∗
j=1,2 ϕijsrijs (ϕ)− wi fijs dGis (ϕ) = wi Fis ,
σs

(10)

that is, the expected lifetime profit from entry must be equal to the entry costs. Following Melitz (2003)
and Demidova (2008), equation (10) can be rewritten as an expression of the cut-off productivity levels
using (2), (5), and (6). Doing so yields the free entry conditionXsγ is fijs ϕ∗−θ
ijs = Fis

(11)

j=1,2

where γ 1s ≡ bθ1ss (σ s − 1)/ [δ 1s (θs − σ s + 1)].
For each sector s, four equations [(7) for ij = 12, 21 and (11) for i = 1, 2] determine four cutoff productivity levels [ϕ∗ijs for ij = 11, 12, 21, 22] as functions of w1 and trade costs (T12s , T21s ).
This simple observation highlights a general equilibrium effect of trade liberalization on industrial
productivities: liberalization in one sector affects the cut-off productivity levels in other sectors throughthe factor market.

2.3

General Equilibrium

To analyze the general equilibrium effect linking the two sectors, we solve for the country 1 equilibrium
wage rate w1 directly from the country 1 labor market clearing condition. We are able to do so thanks
8to two convenient properties of the current model with the Cobb-Douglas upper tier utility (1) and thePareto distribution (2).
The first convenient property is that labor demand Lis by all firms in country i and sector s is
proportional to the mass of entrants Mise . We show this in three steps. First, equation (11) implies
that the fixed costs (the entry costs plus the marketing costs) are proportional to the mass of entrants ineach country i and sector s:wi Mise Fis +

X ˆ ∞



∗
j=1,2 ϕijs

fijs Mis µis (ϕ) dϕ = wi Miseθs Fisσs − 1.

(12)Second, equation (10) implies that the fixed costs are equal to the gross profits in each country i and


´∞
P
is
sector s, that is, wi Mise θσssF−1= σ −1s
j=1,2 Rijs where Rijs ≡ ϕ∗ rijs (ϕ)Mis µis (ϕ)dϕ is theijs

total revenue associated with shipments from country i to country j in sector s. Third, free entry
also implies that wage payments to labor equal total revenue in each country i and sector s, that is,P
wi Lis = j=1,2 Rijs . These three steps lead immediately to:
Lis =

1 X
Rijs = Mise Xis ,
wi

(13)

j=1,2

where Xis ≡ θs Fis /ρs is the labor demand per entrant in country i and sector s. Notice that the industrial labor demand Lis depends only on the mass of entrants Mise and not on any cut-off productivity
levels ϕ∗ijs . We will exploit this remarkable simple property to solve the model.
The second convenient property of the model is that we can solve for the mass of entrants M1se
as a function of the wage w1 and trade costs T12s and T21s . Let φijs denote the ratio of the expected
profit of an entrant in country i from selling to country j in sector s to that captured by an entrant incountry j from selling to country j:i´ ∞ h rijs (ϕ)
−
w
f
δ −1
i ijs dGis (ϕ)
is ϕ∗ijs
σshi.
φijs ≡´ ∞ rjjs (ϕ)
δ −1
−
w
f
dG
(ϕ)
∗
j
jjs
js
js ϕ
σs
jjs

9

Using (2), (5), (6) and (7), this relative expected profit simplifies toδ js fijs
φijs =
δ is fjjs



bis
bjs

θs

−θs
Tijs



wi
wj

1−θs /ρs,(14)

so φijs is a function of Tijs and w1 . From (2), (3), (4), (5), (6), (7), (8), (9), and (14), total revenue
Rijs can be written as
Mise φijs.
k=1,2 Mkse φkjs

(15)

φijs
= wi Xis .
k=1,2 Mkse φkjs

(16)

Rijs = αs wj Lj P
From (13) and (15), we obtainX

α s wj L j P

j=1,2

For each sector s, (16) represents a system of linear equations that can be solved using Cramer’s Rulefor Mise . We find that the mass of entrants in country 1 and sector s isM1se = αs

φ21s L2
w1 L1−
w1 X1s − φ12s X2s X2s − φ21s w1 X1s


.

(17)

Given (14), equation (17) defines M1se as a function of w1 , T12s and T21s , and can be written in function form as M1se (w1 , T12s , T21s ). It is straightforward to verify that it has the properties ∂M1se /∂w1 <0, ∂M1se /∂T12s < 0 and ∂M1se /∂T21s > 0. These effects are quite intuitive. Increases in the wage
(w1 ↑) and export barriers (T12s ↑) discourages entry (M1se ↓), while an increase in import barriers(T21s ↑) encourages entry (M1se ↑).
Having already established that labor demand in country 1 is proportional to the mass of entrants(L1s = M1se X1s ), it follows that labor demand in country 1 is a function of w1 , T12s and T21s . Thisfunction can be written in function form as L1s (w1 , T12s , T21s ) and it has the same properties as theM1se (w1 , T12s , T21s ) function: ∂L1s /∂w1 < 0, ∂L1s /∂T12s < 0 and ∂L1s /∂T21s > 0. In particular,
we obtain the nice property that country 1 labor demand in each sector (s = A and s = B) is downward
sloping in the country 1 wage rate w1 . The country 1 labor supply is given by L1 so the requirement10that labor supply equals labor demand
L1 =X

L1s (w1 , T12s , T21s )

(18)

s=A,B

uniquely determines the equilibrium wage rate w1 given the trade costs (T12s , T21s ).
Figure 1 describes the determination of the equilibrium wage from (18) by using a graphical technique commonly used for the specific factors model. The vertical axis represents country 1’s wage rate
w1 and the width of the box is set equal to country 1’s labor endowment L1 . The left bottom corner
represents the origin for sector A, while the right bottom corner represents the origin for sector B. The
labor demand of each sector is drawn as a downward sloping curve relative to its corresponding origin.
The intersection of the two curves determines the equilibrium wage and the allocation of labor acrosssectors.

Figure 1: The labor market equilibrium
Having found the equilibrium wage rate w1 , we can now solve for the equilibrium cut-off productivity levels. From (7) and (11), we obtain the export productivity cut-off ϕ∗12s for country 1 in sectors as a function of the country 1 wage rate w1 :ϕ∗12s =

1/θs
γ 1s f12s (1 − φ12s φ21s ),F2s (φ12s /w1 ) − φ12s φ21s F1s
11

(19)

where φ12s φ21s < 1 from fijs > fiis . Given (14), the export productivity cut-off ϕ∗12s is an unambiguously increasing function of w1 . When the wage rate goes up, firms have higher labor costs and need
to be more productive to justify exporting their products. Following Demidova and Rodriguez-Clare(2011), we refer to equation (19) as the “competitiveness curve” for country 1 and sector s. Figure
2 illustrates the competitiveness curves (C1s curves) for both sectors s in country 1 together with thelabor market diagram.

Figure 2: Determination of the equilibrium
Finally, we show that industrial productivity only depends on the export productivity cut-off level.
We consider three measures of industrial productivity. The first measure of industrial productivity ΦR1s
is defined as the industrial average of firm productivity weighted by each firm’s revenue share in theindustry:
ˆ ∞
ΦR
1s ≡

P
ϕv1s (ϕ) dϕ where v1s (ϕ) ≡0∗
j=1,2 I(ϕ ≥ ϕ1js )r1js (ϕ)M1s µ1s (ϕ)

P

k=1,2 R1ks

.

(20)

In this definition, I(ϕ ≥ ϕ∗1js ) is an indicator function that takes on the value 1 if ϕ ≥ ϕ∗1js
12

and 0 otherwise. The function v1s (ϕ) is a revenue-weighted density function for ϕ and satisfies´∞
R
0 v1s (ϕ)dϕ = 1. We need to assume θ s > σ s so that Φ1s takes a finite value. This measure is
widely used in empirical studies (e.g. Olley and Pakes, 1996) and is a simpler version of the measure that Melitz (2003) used. The second measure of industrial productivity ΦL
1s is industrial laborproductivity defined as the real industrial output per unit of labor:
P


R
/P̃1s
1js
j=1,2

ΦL
1s =

L1s

In this definition, the price deflator P̃1s ≡

.

(21)

´∞

ϕ∗11s p11s (ϕ) µ1s (ϕ)dϕ is the simple average of prices

set by domestic firms at the factory gate and aims to resemble the industrial product price index,
which is used for the calculation of the real industrial output. This measure is also widely used inempirical studies (e.g. Trefler, 2004). The third measure of industrial productivity ΦW
1s is industrial
labor productivity calculated using the theoretically consistent “exact” price index P1s that we derived
earlier in equation (9):
P
ΦW
1s =


R
j=1,2 1js /P1s
L1s

.

(22)

This measure is motivated by thinking about consumer welfare. Consider the representative consumer
in country 1 who supplies one unit of labor. Since her utility U1 satisfies
U1 = αA ΦW
1A

 αA

α B ΦW
1B

 αB

,

(23)

W
ΦW
1A and Φ1B are the productivity measures for industries A and B that are directly relevant for calcu-

lating consumer welfare U1 .
The next lemma shows that, regardless of which measure of industrial productivity we use, we candraw a negative-sloped curve between industrial productivity and the export productivity cut-off, andthis curve does not shift as a result of changes in variable trade costs.
Lemma 1 All three measures of industrial productivity Φk1s (k = R, L, W ) can be expressed as decreasing functions of the export productivity cut-off ϕ∗12s and these functions Φk1s (ϕ∗12s ) do not containany other endogenous variables or variable trade costs.

13

Proof. The first step is to rewrite Φk1s (k = R, L, W ) as functions of ϕ∗11s and ϕ∗12s . From (2), (3), (4),(5), (6) and (8), ΦR
1s is simplified as
ΦR
1s = ξ s

1
1 + zsϕ∗11s +



12s
where ξ s ≡ (θs − σ s + 1) / (θs − σ s ) > 0 and zs ≡ ff11s

zs
1 + zsϕ∗12s

 ∗ θ s
ϕ11s
ϕ∗12s


,

(24)

. From (13), it holds that ΦL1s =w1 /P̃1s and ΦW
1s = w1 /P1s . By calculating P̃1s from (2), (5) and (8), and P1s from (4), (5) and (6),
we obtain
ΦL
1s =



θs + 1θsρs ϕ∗11s

and

ΦW
1s =



αs L1
σ s f11s

1/(σs −1)ρs ϕ∗11s .

(25)

Second, a total differentiation of the free entry condition (11) givesdϕ∗11s= −zsdϕ∗12s



ϕ∗11s
ϕ∗12s


< 0.

(26)

W
∗
From (25) and (26), we immediately see that both ΦL
1s and Φ1s are decreasing functions of ϕ12s . We

∗
R
∗
R
∗
∗
∗
obtain the total derivative dΦR
1s /dϕ12s = ∂Φ1s /∂ϕ12s + ∂Φ1s /∂ϕ11s (dϕ11s /dϕ12s ) from (24) and(26) asdΦR1s= −ξ sdϕ∗12szs
1 + zs





ϕ∗11s
(θs − 1) 1 − ∗
< 0.
ϕ12s

∗
Therefore, ΦR
1s is a decreasing function of ϕ12s . Finally, note that equations (11), (24), and (25) do notcontain any other endogenous variables or variable trade costs.
The intuition behind Lemma 1 is as follows. Suppose the export productivity cut-off ϕ∗12s falls.
This means that exporting becomes more profitable for some firms in country 1 that could not previously afford to pay the exporting fixed cost w1 f12s . Since all exporters face the same demand function
and the same level of trade barriers, exporting must become more profitable for existing exporters also.
As a result, a potential entrant in country 1 sees an increase in the expected profits from entry and more
firms enter the industry in country 1. Some of these new entrants draw sufficiently high productivity tosurvive. This means that the industry becomes more populated with firms and local consumer demandfor each individual firm’s product decreases.5 Thus, all firms earn lower profits from domestic sales
The decrease in local consumer demand can be confirmed as follows. By using ΦW
1s = w1 /P1s in the proof of Lemma 1,
1−σs(4) and (5), local consumer demand for an individual firm can be written as q11s (ϕ) = (ρs ϕ)σs ΦW
αs L1 . Therefore,
1s
local demand q11s (ϕ) falls if and only if productivity ΦWrises.
1s
5

14and the lowest productivity non-exporting firms exit, that is, the domestic productivity cut-off ϕ∗11s
increases. Since expanding exporters are more productive than the exiting non-exporters, resources arereallocated from less to more productive firms, increasing industrial productivity Φk1s .6
Using Lemma 1, we draw the negative relationship between ϕ∗12s and Φk1s for each sector s in thebottom two diagrams in Figure 2. We refer to the Φk1s (ϕ∗12s ) functions as “productivity curves” andlabel them as P1s curves in Figure 2. Factor market clearing determines w1 , then the competitivenesscurves determine ϕ∗12s and then the productivity curves determine Φk1s . This figure clearly shows thatthe productivity of each sector s is connected though the factor market.
Lemma 1 implies that the source of a rise in industrial productivity is higher profits from exporting.
Since the productivity curves Φk1s (ϕ∗12s ) do not include any variable trade costs, trade liberalization(multilateral or unilateral) improves industrial productivity Φk1s only when exporting becomes more
profitable and the export productivity cut-off ϕ∗12s falls.

3

Trade Liberalization

We are now ready to analyze the impact of trade liberalization on industrial productivity. While Melitz(2003) considered only multilateral and uniform liberalization, in which all countries reduce variable
trade costs on all products in a uniform way, we consider unilateral and non-uniform liberalization:
country 1 liberalizes tariffs only for sector A. Following Melitz (2003), import tariffs take the form of
iceberg trade costs. So trade liberalization for us means decreasing τ 21A while holding τ 12A , τ 12B andτ 21B fixed. We call sector A the liberalized industry and sector B the non-liberalized industry.

3.1

Structurally Symmetric Industries

We focus on the impact of trade liberalization when the two industries are structurally symmetric
except for their consumption share in GDP (αA is allowed to differ from αB ).
6

The weighted average productivity measure in Melitz (2003), ϕ̃1s1/(σ s −1)

≡

h´

∞
ϕσs −1 µ1s (ϕ)dϕϕ∗11si1/(σs −1)=[θs / (θs − σ s + 1)]
ϕ∗11s also satisfies Lemma 1. Since ϕ∗11s and ϕ∗12s move in the opposite direction from (11),productivity ϕ̃1s rises if and only if ϕ∗12s falls. Since w1 and τ ijs do not show up in either ϕ̃1s or (11), they affect ϕ̃1s only
through ϕ∗12s .

15Definition 1 The two industries are structurally symmetric if ρA = ρB , θA = θB , δ iA = δ iB , biA =biB , fijA = fijB , FiA = FiB , and τ ijA = τ ijB .
This is a natural benchmark case for the analysis of unilateral and non-uniform trade liberalization.
The Melitz (2003) model only has one industry but requires balanced trade and labor market clearing
as in general equilibrium models. Thus, it is natural to think of the one industry in the Melitz model asa representative industry. Note that Definition 1 requires symmetry only across industries. Countries
can differ in their factor endowments, technologies and trade costs.

Figure 3: Productivity rises more strongly in the non-liberalized industry
The diagrams developed in the previous section greatly simplifies the analysis. Figure 3 shows
the same diagrams we used in Figure 2 for the structurally symmetric industries case. Before tradeliberalization, both industries have symmetric competitiveness curves (the C1A and C1B curves) andsymmetric productivity curves (the P1A and P1B curves), which implies that both industries have thesame productivity Φk1A = Φk1B (k = R, L, W ).7
7

Before liberalization, country 1 always produce positive outputs in both sectors since L1A /αA = L1B /αB holds from(13) and (17).

16The top-center diagram in Figure 3 describes the impact of liberalization on the labor market of theliberalizing country 1. The labor demand curve of the liberalized industry A shifts leftward since themass of entrants drops for a given wage level w1 from (17). Workers move from the liberalized to thenon-liberalized industry and the wage drops in the liberalizing country. The top-right and bottom-right
diagrams in Figure 3 describe how the general equilibrium effect operating through the labor market
affects the productivity of the non-liberalized industry B. The competitiveness curve in industry B
does not shift in response to liberalization since ϕ∗12B in (19) includes no trade costs for industry A.
As country 1’s wage w1 declines, less productive firms start exporting and the export productivity cutoff ϕ∗12B unambiguously falls in the non-liberalized industry B. Then, from the bottom-right diagram,industrial productivity Φk1B unambiguously rises in the non-liberalized industry B.
The liberalized industry A responds to trade liberalization in a more complex way since the competitiveness curve depends on trade costs in the industry. From (14) and (19), the competitiveness0 ) for a given wage level
curve of the liberalized industry A shifts leftward (curve C1A shifts to C1A

w1 .8 This leftward shift of the C1A curve tends to increase the export productivity cut-off ϕ∗12A , while
the declining wage w1 tends to decrease it. Therefore, the net effect on the export productivity cutoff ϕ∗12A in the liberalized industry is ambiguous. Lemma 1 then implies that the net effect on the
productivity of the liberalizing industry is also ambiguous. We have established
Theorem 1 In the multi-industry Melitz model with structurally symmetric industries, unilateral tradeliberalization by country 1 in industry A (τ 21A ↓) leads to a decrease in the country 1 wage rate
(w1 ↓) and an increase in the productivity of the non-liberalized industry (Φk1B ↑). However, whether
productivity rises or falls in the liberalized industry is in general ambiguous (Φk1A ↑ or ↓).
Although trade liberalization has an ambiguous effect on productivity in the liberalized industry,
we can make an unambiguous statement about the difference in the productivity change between theliberalized and the non-liberalized industries. If the C1A curve remained at the pre-liberalization position, then productivity would rise by the same amount in each industry due to the decrease in the equi0 ), productivity increases
librium wage rate w1 . However, since the C1A curve shifts leftward (to C1A

more strongly in industry B than in industry A. As shown in Figure 3, productivity increases from
8
The Appendix spells out the derivation in detail. Demidova and Rodriguez-Clare (2011) show that this leftward shift
occurs in a one industry Melitz model.

17

k
k
Φk1B to Φk0
1B in industry B, while the productivity increase is smaller in industry A, from Φ1A = Φ1Bk0
to Φk0
1A < Φ1B . Thus, productivity rises more strongly in the non-liberalized industry than in the lib-

eralized industry, i.e. ∆Φk1B − ∆Φk1A > 0, k = R, L, W . This “difference-in-difference” prediction
is sufficient for our purpose of matching the model with empirical studies. Because typical empiricalstudies estimate cross-industry regressions with time fixed effects and industry fixed effects (e.g. Trefler, 2004), their estimates only tell us whether trade liberalization increases productivity in liberalized
industries relative to non-liberalized industries. We have established
Theorem 2 In the multi-industry Melitz model with structurally symmetric industries, unilateral tradeliberalization by country 1 in industry A (τ 21A ↓) leads to productivity rising more strongly in the
non-liberalized industry than in the liberalized industry (∆Φk1B > ∆Φk1A for k = R, L, W ).
Theorem 2 is our central result. An empirical finding by Trefler (2004) and others that industrial
productivity increases more strongly in liberalized industries than in non-liberalized industries has
been widely accepted as evidence for the Melitz (2003) model. Theorem 2 shows that a multi-industry
version of the Melitz model does not predict this relationship. Instead, it predicts the opposite relationship that industrial productivity increases more strongly in non-liberalized industries than in liberalized
industries. Theorem 2 forces us to re-think the match between theory and evidence: an empirical factthat has been widely cited as evidence for the Melitz model is actually evidence against the Melitz
model.
Next, we study whether the effects of trade liberalization depend on the size of the industry that
opens up to trade. Does trade liberalization have different effects, depending on whether the liberalized
industry is small or large? Since the parameter αs determines the size of industry s, we analyze how
the response of industrial productivity to trade liberalization depends on αA , the size of the liberalized
industry.
Holding all other parameter values fixed, a change in αA has no effect on the equilibrium wage w1 .
Since employment in the two industries satisfies L1A /αA = L1B /αB from (13) and (17), the labormarket clearing condition (18) can be rewritten as
L1 = L1A + L1B = L1A

18

αA + αB
αA


=

L1A
.
αA

Now L1 = L1A (w1 , T12s , T21s )/αA uniquely determines the equilibrium wage w1 and L1A /αA does
not depend on αA from (17). Thus, the equilibrium wage w1 does not depend on αA .

}Figure 4: How much the wage declines depends on the size of the liberalized industry
As illustrated in Figure 4, the pre-liberalization wage w1 is the same whether αA is small or large.
Trade liberalization causes the labor demand curve L1A to shift leftward, or equivalently, to shift down.
Equation (17) implies that the size of the downward shift in the labor demand curve L1A (“d” in Figure4) does not depends on αA . Equation (17) also implies that as αA increases, the slope of the labordemand curve L1A becomes flatter because the number of entrants in industry A increases in αA .
Similarly, as αA increases, which means that αB = 1 − αA decreases, the slope of the labor demand
curve L1B becomes steeper. Thus, as illustrated in Figure 4, the wage drop due to trade liberalizationis larger when αA is larger.
Figure 3 shows that whether productivity increases in the liberalized industry A depends on the net
effect of two opposite changes: the leftward shift of the competitiveness curve C1A (“the competitiveness effect”) and the drop in the wage rate w1 (“the wage effect”). The competitiveness effect does notdepend on αA since equation (19) does not include αA . However, as we have just shown, the wage
effect is larger when αA is larger. If αA is sufficiently small and the wage effect is sufficiently small,then the competitiveness effect must dominate the wage effect. Figure 5 illustrates this case.
The export productivity cut-off ϕ∗12A rises and productivity Φk1A unambiguously falls in the liberalized industry. If αA is sufficiently large and satisfies αA = 1, then the model reduces to a one industry
19

Figure 5: Productivity falls in the liberalized industry if the liberalized industry is small
Melitz model where Demidova and Rodriguez-Clare (2011) proved that unilateral liberalization raises
industrial productivity. Since the model’s properties are continuous in parameter αA , we obtain thefollowing theorem:
Theorem 3 In the multi-industry Melitz model with structurally symmetric industries, suppose that
there is unilateral trade liberalization by country 1 in industry A (τ 21A ↓). Then there exists a thresholdᾱA ∈ (0, 1) such that productivity Φk1A falls in the liberalized industry if αA < ᾱA and rises if
αA > ᾱA .
Theorem 3 helps us to understand contrasting results in previous studies on unilateral trade liberalization using versions of the Melitz model with one Melitz industry. Melitz and Ottaviano (2008)
and Demidova (2008) find that the productivity of the liberalized industry falls, while Demidova andRodriguez-Clare (2011) and Felbermayr, Jung, and Larch (2012) find that the productivity of the liberalized industry rises. The main difference between these papers is that Melitz and Ottaviano (2008)
and Demidova (2008) consider a class of models with a homogeneous numeraire good that is pro20

duced with constant returns to scale technology and is freely traded under perfect competition, while
Demidova and Rodriguez-Clare (2011) and Felbermayr, Jung, and Larch (2012) consider models witha single Melitz industry. The former with the numeraire good that fixes the wage rate corresponds toour model with a negligible αA , while the latter corresponds to our model with αA = 1.
By combining the results in Theorems 1 and 3, we obtain one more theorem:
Theorem 4 In the multi-industry Melitz model with structurally symmetric industries, suppose that
there is unilateral trade liberalization by country 1 in industry A (τ 21A ↓). If αA is sufficiently small,then productivity falls in the liberalized industry A and rises in the non-liberalized industry B (Φk1A ↓and Φk1B ↑).
Theorem 4 provides a surprising policy implication. If the government of a country is interested inraising the productivity of a small “target” industry through a resource reallocation from less productive
to more productive firms, the theoretically correct advice based on the Melitz model is to protect thetarget industry, not trade liberalization. This is obviously the opposite of the policy recommendationthat is suggested by Trefler (2004) and other empirical studies.

3.2

Intuition

Trade liberalization (decreasing τ 21A ) causes two curves to shift in Figure 3. The shift in the competitiveness curve of the liberalized industry C1A expresses the competitiveness effect of trade liberalization, a partial equilibrium effect that decreases productivity in the liberalized industry. The shift in thelabor demand curve of the liberalized industry L1A expresses the wage effect of trade liberalization,a general equilibrium effect that increases productivity in both industries. To understand the overall
effect of trade liberalization on industrial productivity, we need to understand both the competitivenesseffect and the wage effect.
To understand the competitiveness effect, suppose that country 1 opens up to trade in industry A(τ 21A ↓) and the wage rate w1 is held fixed. Then firms in country 2 earn higher profits from exporting
to country 1 in industry A. Since exports increase not only for existing exporters (the intensive margin)
but also by the entry of less productive firms into exporting (the extensive margin), the export productivity cut-off ϕ∗21A falls. Because exporting becomes more profitable in country 2, more firms enter21and industry A in country 2 becomes more populated with firms, so local consumer demand for eachindividual firm’s product decreases in country 2. Thus all firms earn lower profits from domestic sales
and the lowest productivity non-exporting firms exit. Since expanding exporters are more productive
than exiting non-exporters, resources are reallocated from less to more productive firms, increasing
industry A productivity Φk2A in country 2. This increase in country 2 productivity sets in motion the
opposite chain of events in country 1. Now industry A in country 2 is occupied by more firms and moreproductive firms. This means that it becomes less profitable for country 1 firms to export to country 2.
The export productivity cut-off ϕ∗12A increases and industry A productivity Φk1A decreases in country
1.
To understand the wage effect, it is helpful to think about the balanced trade condition. Let Eijs be
the expenditure of country i on country j goods in sector s. Then the exports in sector s by country 1
P
is j=1,2 R1js − E11s and the imports in sector s by country 1 is E12s . The balanced trade condition
can be written asPhP= w1 L1s and

P

s=A,B

From

P

j=1,2 R1js

j=1,2 R1js − E11s

j=1,2 E1js

i

− E12s = 0.

(27)

= αs w1 L1 , the excess exports of sector s can be

expressed as
P

j=1,2 R1js − E11s
− E12s = w1 αsL1s (w1 , T12s , T21s )− L1 .
αs(28)

By summing up (28) for both industries, we see that the balanced trade condition (27) is equivalent to
the labor market clearing condition (18).
Starting from balanced trade and holding the wage w1 fixed, trade liberalization leads to excess
imports in industry A by the liberalizing country 1. Then (27) and (28) imply that the wage w1 must
drop to increase exports by both industries in the liberalizing country until trade balance is restored.
Since exports increase not only for existing exporters (the intensive margin) but also by the entry ofless productive firms into exporting (the extensive margin), the export productivity cut-offs ϕ∗12s fall inboth industries when w1 falls. Because exporting becomes more profitable, more firms enter and bothindustries become more populated with firms, so local consumer demand for each individual firm’s
product decreases. Thus all firms earn lower profits from domestic sales and the lowest productivity

22

non-exporting firms exit. Since expanding exporters are more productive than exiting non-exporters,
resources are reallocated from less to more productive firms, increasing industrial productivity. Withstructurally symmetric industries, the wage effect by itself contributes to increase productivity equally
in both industries.
Balanced trade also explains why the size of the liberalized industry αA determines the relative
size of the wage effect and the competitiveness effect. Since L1s (·) /αs does not depend on αs from(17), equation (28) shows that the size of excess exports in sector s is proportional to αs and that therise in exports in sector s due to a wage drop increases in αs . Therefore, if αA is sufficiently large,
trade balance after liberalization is mainly restored by an increase in the exports in the liberalized
industry. Since the mass of firms in the liberalized industry falls, the exports increases by the entryof less productive firms into exporting (the extensive margin) as well as for existing exporters (theintensive margin). To achieve this, the wage must drop by a sufficiently large amount so that the wage
effect dominates the competitiveness effect and the export productivity cutoff ϕ∗12s falls. On the other
hand, if αA is small, then liberalization leads to a small trade deficit, so a small wage drop can restore
trade balance by increasing exports in the non-liberalized industry. Therefore, the wage effect is too
small to dominate the competitiveness effect if αA is sufficiently small.
The effects of trade liberalization are summarized in Table 2. The wage effect of trade liberalization tends to increase productivity in both industries symmetrically, while the competitiveness effecttends to decrease productivity in the liberalized industry. As a consequence, industrial productivity
unambiguously rises in the non-liberalized industry but it can rise or fall in the liberalized industry,
depending on the relative size of the wage effect and the competitiveness effect. Figure 3 illustrates
the case where the wage effect of trade liberalization dominates the competitiveness effect, with theconsequence that productivity rises in the liberalized industry. Productivity rises even more in the
non-liberalized industry because there is a positive wage effect and no negative competitiveness effect.
Figure 5 illustrates what happens when a sufficiently small industry is opened up to trade. Then thepositive wage effect of trade liberalization is sufficiently small so that the negative competitivenesseffect dominates. Productivity falls in the liberalized industry and only rises in the non-liberalized
industry.

23

Competitiveness Effect
Wage EffectTotal EffectImpacts on Industrial Productivity
Liberalized (A) Non-liberalized (B)Difference
∆Φk1A
∆Φk1B
∆Φk1A − ∆Φk1B
−
0
−
+
+
0
+ or −+−

Table 2: The effects of trade liberalization

3.3

Numerical Results

As a check that our analytically derived results are correct, we also solve the model numerically.
Looking at a numerical example is helpful for understanding the intuition behind the results.9
For the numerical results reported in Table 3, we assume structurally symmetric industries andcountries. Then there are only ten parameters that need to be chosen. We use the following benchmark
parameter values: σ s = 3.8, δ is = .025, bis = .2, θs = 4.582, Fis = 2, fiis = .043, Li = 1,
αA = .5, τ ijs = 1.3 and fijs = .0588. The first six parameter values come from Balistreri, Hillbery
and Rutherford (2011), where a version of the Melitz model is calibrated to fit trade data. Li = 1 is aconvenient normalization given that an increase in country size Li has no effect on the key endogenous
variables that we are solving for (the relative wage w1 /w2 , productivity cutoff levels ϕ∗ijs and industry
productivity levels ΦRis ). αA = .5 means that both industries are equally large: consumers spend 50
percent of their income on industry A products and 50 percent of their income on industry B products.
τ ijs = 1.3 corresponds to a 30 percent tax on all traded goods. Finally, we chose fijs = .0588 toguarantee that 18 percent of firms export in our benchmark equilibrium, consistent with evidence for
the United States (Bernard et al., 2007).
The first column of numbers in Table 3 shows the benchmark equilibrium (when αA = .5 andτ 21A = 1.30). The second column shows what happens when country 1 unilaterally opens up to trade
in industry A (τ 21A is decreased from 1.30 to 1.15 holding τ 21B = τ 12A = τ 12B = 1.30 fixed).
This leads to productivity rising more strongly in the non-liberalized industry B (ΦR
1B increases from.5564 to .5651) than in the liberalized industry A (ΦR
1A increases from .5564 to .5590), consistent with
Theorem 2. Since productivity rises in the liberalized industry, we are illustrating a case where thewage effect of trade liberalization dominates the competitiveness effect. The third and fourth columns
9

The MATLAB files used to solve the model can be obtained from the authors upon request.

24αA = .5 Case

αA = .3 Caseτ 21A = 1.30

τ 21A = 1.15

τ 21A = 1.30

τ 21A = 1.15

w1 /w2
L1A
L1B
L2A
L2B

1.0000
.5000
.5000
.5000
.5000

.9707
.4221
.5779
.5757
.4243

1.0000
.3000
.7000
.3000
.7000

.9801
.2275
.7725
.3711
.6289

ϕ∗12A
ϕ∗11A
ϕ∗12B
ϕ∗11B

.3257
.2240
.3257
.2240

.3206
.2250
.3092
.2274

.3257
.2240
.3257
.2240

.3273
.2238
.3144
.2262

ϕ∗21A
ϕ∗22A
ϕ∗21B
ϕ∗22B

.3257
.2240
.3257
.2240

.3012
.2296
.3443
.2214

.3257
.2240
.3257
.2240

.2957
.2314
.3380
.2222

ΦR
1A
R
Φ1B
ΦR
2A
ΦR
2B

.5564
.5564
.5564
.5564

.5590
.5651
.5694
.5476

.5564
.5564
.5564
.5564

.5556
.5623
.5724
.5505

U1
U2

.1230
.1230

.1242
.1238

.1376
.1376

.1385
.1381

Table 3: Effects of Trade Liberalization

25

show the effects of the same trade liberalization when industry A is smaller (αA = .3, all otherparameter values unchanged). Then the wage effect of trade liberalization is smaller and is dominatedby the competitiveness effect. Productivity in the liberalized industry decreases (ΦR
1A decreases from.5564 to .5556) and productivity in the non-liberalized industry increases (ΦR
1B increases from .5564
to .5623), consistent with Theorem 4.
To see the intuition behind these results, consider the αA = .3 “small industry” case first andfocus on what happens in industry A. When country 1 opens up to trade in industry A, country 2
firms earn higher profits from exporting. These higher export profits lead to more entry and greater
industrial employment (L2A , which is proportional to the mass of entrants and active firms, increases
from .3000 to .3711). As the industry becomes more populated with firms, the country 2 demand foreach individual firm’s product decreases, so the least productive firms are forced to exit (ϕ∗22A increases
from .2240 to .2314). Even though the increase in labor demand bids up the wage rate in country 2(w1 /w2 decreases from 1.000 to .9801), the wage increase is not large enough to completely offset the
tariff reduction by country 1 and more country 2 firms become exporters (ϕ∗21A decreases from .3257 to.2957). Since expanding exporters are more productive than exiting non-exporters, productivity rises
for country 2 in industry A (ΦR
2A increases from .5564 to .5724). For firms in country 1, the picture
is very different. Now they are competing against more productive firms in their export market, theyearn lower profits from exporting and this sets into motion the opposite effects. Fewer country 1 firms
become exporters (ϕ∗12A increases from .3257 to .3273), entry is discouraged and the mass of firms in
the industry falls (L1A decreases from .3000 to 2275) until the expected profits from domestic sales
increase to offset the loss of expected profits from exporting. The increase in domestic profits allows
less productive firms to survive in the domestic market (ϕ∗11A decreases from .2240 to .2238). Thus,
we get a reallocation of resources from more productive to less productive firms in country 1, lowering
industry productivity (ΦR
1A decreases from .5564 to .5556).
Next, focus on what happens in industry B when country 1 opens up to trade in industry A. Because wages rise in country 2 (w1 /w2 decreases from 1.000 to .9801), it becomes less profitable forcountry 2 firms to export and there is a reallocation of resources from more productive to less productive firms, lowering productivity (ΦR
2B decreases from .5564 to .5505). Because wages fall in country
1 (w1 /w2 decreases from 1.000 to .9801), the export productivity cutoff falls (ϕ∗12B decreases from26

.3257 to .3144) and there is a reallocation of resources from less productive to more productive firms,
raising productivity (ΦR
1B increases from .5564 to .5623).
Finally, turn to the effects of trade liberalization when industry A is larger (αA = .5). We obtain thesame qualitative effects in industry B: because wages rise in country 2 (w1 /w2 decreases from 1.000
to .9707), productivity falls (ΦR
2B decreases from .5564 to .5476) and because wages fall in country 1(w1 /w2 decreases from 1.000 to .9707), productivity rises (ΦR
1B increases from .5564 to .5651). But
the qualitative effects are different for the industry A that opens up to trade because there is a larger fallin country 1 wages. Even though trade liberalization raises productivity in country 2 (ΦR
2A increases
from .5564 to .5694), which by itself makes exporting less attractive for country 1 firms, the larger
fall in country 1 wages now dominates and country 1 productivity in industry A actually rises (ΦR
1A
increases from .5564 to .5590).
Although the impact of trade liberalization on industrial productivity is the main focus of thispaper, we also report the impact of trade liberalization on consumer welfare in the last two rows ofTable 3. U1 and U2 denote the steady-state utility levels of the representative consumer in countries 1
and 2, respectively. In the αA = .5 case, trade liberalization by country 1 raises consumer welfare incountry 2 (U2 increases from .1230 to .1238) and raises even more consumer welfare in country 1 (U1increases from .1230 to .1242). Thus country 2 benefits when country 1 opens up to trade and country
1 benefits even more by unilaterally opening up to trade. Looking at the αA = .3 case, we obtain
qualitatively similar welfare effects.

3.4

Symmetric Multilateral Trade Liberalization

In this subsection, we replicate the analysis of symmetric multilateral trade liberalization in Melitz(2003) using our diagrams. The two countries are assumed to be identical as in Melitz (2003) but
each industry may have different parameters. We analyze multilateral but non-uniform liberalization
by decreasing τ 12A and τ 21A by the same amount while holding τ 12B and τ 21B fixed.
Assuming symmetric countries simplifies the model. First, wages are equalized between the twocountries: w1 = w2 = 1. Second, the notation for describing the model takes a simpler form: Xis =Xs , φijs = φs , Tijs = Ts , Fis = Fs , fiis = fs and fijs = fxs for i 6= j.
Figure 6 describes the impact of liberalization. The employment in sector s becomes L1s =27

αs L1 from (13) and (17), so multilateral trade liberalization in sector A (τ 12A = τ 21A ↓) leads to noequilibrium change in the wage w1 and the labor allocation.10 The top-right and bottom-right diagrams
show that multilateral liberalization does not affect productivity Φk1B in the non-liberalized industry.

Figure 6: Multilateral trade liberalization in industry A
The impact on the liberalized industry is different from the case of unilateral trade liberalization.
Given symmetric countries, the export productivity cut-off in sector A [given by (19)] simplifies toϕ∗12A =

γ 1A fxA
FA


1+

1
φA

1/θA(29)

and (14) implies that φA = (fxA /fA )TA−θA . Thus multilateral trade liberalization leads to a decreasein the export productivity cut-off ϕ∗12A and an increase in productivity Φk1A in the liberalized industry.
We have established
Theorem 5 In the multi-industry Melitz model with symmetric countries, symmetric multilateral trade10

The labor demand curve of the liberalized industry becomes flatter as illustrated in Figure 6. This is shown in the
Appendix.

28

liberalization (τ 12A = τ 21A ↓) increases productivity in the liberalized industry (Φk1A ↑) but not in the
non-liberalized industry (Φk1B unchanged).
A comparison of Theorems 2 and 5 confirms that the source of the rise in industrial productivity
in the Melitz model is the expansion of export opportunity, not the increased import competition fromtrade liberalization.
Notice that by setting αA = 1, the model becomes identical to the original Melitz (2003) modelwith one industry. Therefore, our analysis nests the analysis of multilateral and uniform liberalizationin Melitz (2003). We obtain Melitz’s original result using new diagrams:
Corollary 1 (Melitz, 2003) If there is only one industry (αA = 1) and symmetric countries, thensymmetric multilateral trade liberalization (τ 12A = τ 21A ↓) increases industrial productivity (Φk1A ↑).

4

Conclusion

In order to establish that trade liberalization improves industrial productivity through within-industry
resource reallocation, the empirical literature and the theoretical literature have taken different approaches. On the one hand, empirical studies have typically investigated episodes of unilateral andnon-uniform liberalization in order to utilize the cross-industry variations in trade liberalization. On
the other hand, theoretical studies have typically analyzed multilateral and uniform liberalization inmodels with a single representative industry.
We have demonstrated that this apparently small difference in approach between theoretical andempirical studies is not innocuous. Once the Melitz (2003) model is extended to analyze unilateral andnon-uniform trade liberalization, it does not predict what researchers find empirically, that industrial
productivity increases more strongly in liberalized industries than in non-liberalized industries.
Instead, we find that industrial productivity increases more strongly in non-liberalized industries
than in liberalized industries. When one country opens up to trade in one industry, we find that productivity falls if the industry is sufficiently small. If the industry is larger, so productivity rises, we findthat it rises even more in the industries that did not open up to trade. So, if we take the Melitz model
seriously when looking at the real world, other things being equal, we should observe that productivity29

is high in industries with high tariffs and productivity is low in industries with low tariffs. This is the
opposite of what reseachers find empirically.

References[1] Baggs, Jen (2005) “Firm survival and exit in response to trade liberalization.” Canadian Journalof Economics, 38(4), 1364-1383.
[2] Balisteri, Edward, Russell H. Hillberry, and Thomas F. Rutherford (2011) “Structural estimationand solution of international trade models with heterogeneous firms,” Journal of International
Economics, 83, 95-108.
[3] Bernard, Andrew B., J. Bradford Jensen, and Peter K. Schott (2006) “Trade Costs, Firms andProductivity,” Journal of Monetary Economics, 53, 917-937.
[4] Bernard, Andrew B., J. Bradford Jensen, Stephen J. Redding, and Peter K. Schott (2007), “Firms
in International Trade,” Journal of Economic Perspectives, 21(3), 105-130.
[5] Bernard, Andrew B., J. Bradford Jensen, Stephen J. Redding, and Peter K. Schott (2012) “TheEmpirics of Firm Heterogeneity and International Trade,” Annual Review of Economics, 4, 283313[6] Bernard, Andrew B., Stephen J. Redding, and Peter K. Schott (2007) “Comparative Advantage
and Heterogeneous Firms,” Review of Economic Studies,74 (1), 31-66.
[7] Demidova, Svetlana (2008), “Productivity Improvements and Falling Trade Costs: Boon or
Bane?”, International Economic Review, 49(4), 1437-1462.
[8] Demidova, Svetlana and Andres Rodriguez-Clare (2009), “Trade Policy under Firm-Level Heterogeneity in a Small Economy”, Journal of International Economics, 78(1), 100-112.
[9] Demidova, Svetlana and Andres Rodriguez-Clare (2011), “The Simple Analytics of the Melitz
Model in a Small Open Economy.” NBER working paper 17521.

30

[10] Eslava, Marcela, John C. Haltiwanger, Adriana D. Kugler, and Maurice Kugler (2010) “Trade Reforms and Market Selection: Evidence from Manufacturing Plants in Colombia.” NBER working
paper, No. 14935.
[11] Felbermayr, Gabriel, Benjamin Jung, and Mario Larch (2012) “Optimal tariffs, retaliation, andthe welfare loss from tariff wars in the Melitz model.” Journal of International Economics, inpress.
[12] Fernandes, Ana M. (2007) “Trade policy, trade volumes and plant-level productivity in Colombian manufacturing industries.” Journal of International Economics, 71(1), 52-71.
[13] Harrison, Ann E., Leslie A. Martin, and Shanthi Nataraj (2011) “Learning Versus Stealing: How
Important are Market-Share Reallocations to India’s Productivity Growth?.” NBER working paper w16733.
[14] Head, Keith and John Ries (1999) “Rationalization effects of tariff reductions.” Journal of International Economics, 47(2), 295-320.
[15] Helpman, Elhanan (2011) Understanding Global Trade. The Belknap Press of Harvard University
Press, Cambridge, Massachusetts.
[16] Hopenhayn, Hugo A. (1992) “Entry, Exit, and Firm Dynamics in Long Run Equilibrium,” Econometrica, 60(5), 1127-50.
[17] Krugman, Paul (1979) “Increasing Returns, Monopolistic Competition, and International Trade,”
Journal of International Economics, 9(4), 469-79.
[18] Krugman, Paul (1980) “Scale Economy, Product Differentiation, and the Pattern of Trade,” American Economic Review, 70(5), 950-59.
[19] Lileeva, Alla (2008) “Trade liberalization and productivity dynamics: evidence from Canada,”
Canadian Journal of Economics, 41(2), 360-390.
[20] Melitz, Marc J. (2003) “The Impact of Trade on Intra-Industry Reallocations and Aggregate
Industry Productivity,” Econometrica, 71(6), 1695-1725.

31

[21] Melitz, Marc J. and Gianmarco I.P. Ottaviano (2008) “Market Size, Trade and Productivity,”
Review of Economic Studies, 75, 295-316.
[22] Melitz, Marc J. and Daniel Trefler (2012) “Gains form Trade when Firms Matter,” Journal of
Economic Perspectives, 26(2), 91-118.
[23] Nataraj, Shanthi (2011) “The impact of trade liberalization on productivity: Evidence from India’s formal and informal manufacturing sectors.” Journal of International Economics, 85(2),292-301.
[24] Okubo, Toshihiro (2009) “Firm heterogeneity and Ricardian comparative advantage within andacross sectors.” Economic Theory, 38(3), 533-559.
[25] Olley, C. Steven, and Ariel Pakes (1996) “The dynamics of productivity in the telecommunications equipment industry.” Econometrica, 64(6), 1263-1297.
[26] Pavcnik, Nina (2002) “Trade Liberalization, Exit, and Productivity Improvement: Evidence from
Chilean Plants,” Review of Economic Studies, 69(1), 245-76[27] Redding, Stephen J. (2011) “Theories of Hetrogeneous Firms and Trade,” Annual Review of Economics, 3, 77-105.
[28] Sivadasan, Jagadeesh. (2009) “Barriers to competition and productivity: Evidence from india.”
The BE Journal of Economic Analysis & Policy, 9(1), 1935-1682.
[29] Trefler, Daniel (2004) “The Long and Short of the Canada-U.S. Free Trade Agreement,” American Economic Review, 94(4), 870-895.
[30] Tybout, James R. (2003) “Plant- and Firm-Level Evidence on “New” Trade Theories,” in E. Kwan
Choi and James Harrigan eds. Handbood of International Trade, 388-415. Blackwell Publishing,
Oxford.
[31] Tybout, James R. and M. Daniel Westbrook (1995) “Trade Liberalization and Dimensions of
Efficiency Change in Mexican Manufacturing Industries,” Journal of International Economics,
39, 53-78.

32

Appendix: Solving The Model (Not for Publication)
In this Appendix, calculations done to solve the model are spelled out in more detail.

Consumers
First, we solve the within-sector consumer optimization problemˆmax Cs ≡

ˆ

1/ρs

ρs

qs (ω) dωqs (·)ps (ω)qs (ω) dω = Es

s.t.
ω∈Ωs

ω∈Ωs

where qs (ω) is quantity demanded for variety ω in sector s, ps (ω) is the price of variety ω in sector s
and Es is individual consumer expenditure on sector s products. This problem of maximizing a CES
utility function subject to a budget constraint can be rewritten as the optimal control problemˆqs (ω)ρs dωmax
qs (·)ẏs (ω) = ps (ω)qs (ω), ys (0) = 0, ys (+∞) = Es

s.t.

ω∈Ωs

where ys (ω) is a new state variable and ẏs (ω) is the derivative of ys with respect to ω. The Hamiltonian
function for this optimal control problem is
H = qs (ω)ρs + γ(ω)ps (ω)qs (ω)∂Hwhere γ(ω) is the costate variable. The costate equation ∂y= 0 = −γ̇(ω) implies that γ(ω) iss∂H
constant across ω. ∂q= ρs qs (ω)ρs −1 + γ · ps (ω) = 0 implies thatsqs (ω) =ρs−γ · ps (ω)

1/(1−ρs ).

Substituting this back into the budget constraint yields
ˆ

ˆ

Es =

ps (ω)qs (ω) dω =
ω∈Ωs


=



ρs
−γ

ps (ω)ω∈Ωs1/(1−ρs ) ˆρs−γ · ps (ω)

1−ρs −1

ps (ω) 1−ρs dω.
ω∈Ωs

−ρs
1
s −1
Now σ s ≡ 1−ρ
implies that 1 − σ s = 1−ρ
= 1−ρ
, so
1−ρ
s

s

s

Es
´
=
1−σ s dωω∈Ωs ps (ω)

1



ρs
−γ

1/(1−ρs ).

1/(1−ρs )dωIt immediately follows that the individual consumer demand function isqs (ω) =where Ps ≡h´

1−σ s dωω∈Ωs ps (ω)

i1/(1−σs )ps (ω)−σs Es
Ps1−σsis the price index for sector s. Substituting this consumer

demand function back into the CES utility function yields
ˆ

ρs

Cs =

"ˆ

1/ρs

qs (ω) dω

=

ω∈Ωs

#1/ρs

ρ

ps (ω)−σs ρs Es sω∈Ωs

(1−σ s )ρs

Ps

dω

=

ˆ

Es
Ps1−σs−σ s ρsps (ω)

1/ρs
dω

ω∈Ωs

−ρs
Taking into account that −σ s ρs = 1−ρ
= 1 − σ s , the CES utility can be simplified further to
s

Cs =

ˆ

Es
Ps1−σsps (ω)

1−σ s

1/ρs
dω

=

ω∈Ωs

Es
Ps1−σs 1−σs 1/ρs
=
Ps

Es
−σ s
=
.
1−σ s Ps
Ps
Ps
Es

Thus, we can write the across-sector consumer optimization problem as EA αA EB αB
αA αB
max U ≡ CA
CB =
s.t.
EA + EB = E
EA ,EB
PA
PB
where E is consumer expenditure on products in both sectors combined. The solution to this problem
is EA = αA E and EB = αB E.
In country i, workers earn the wage rate wi and total labor supply is Li , so total wage income thatcan be spent on products produced in both sectors is wi Li . Given free entry, there are no profits earned
from entering markets, so consumers spend exactly what they earn in wage income. Let Eis denotethe expenditure by all consumers in country i on sector s products. It follows that
Eis = αs wi Li .

Firms
Let π ijs (ϕ) denote the gross profits (or variable profits) earned by a firm with productivity ϕ fromcountry i to country j in sector s. It follows thatwi τ ijs
qijs (ϕ)ϕ
pijs (ϕ)1−σs αs wj Lj
wi τ ijs pijs (ϕ)−σs αs wj Lj
−
.
1−σ s
1−σ s
ϕ
Pjs
Pjs

π ijs (ϕ) = rijs (ϕ) −=2.

We obtain the price that maximizes gross profits by solving the first order condition∂π ijs (ϕ)∂pijs (ϕ)==
wτwi τ ijs σ s pijs (ϕ)−σs −1 αs wj Lj(1 − σ s )pijs (ϕ)−σs αs wj Lj
+
1−σ s
1−σ s
Pjs
ϕPjs


pijs (ϕ)−σs αs wj Lj
wi τ ijs σ s
1 − σs +=01−σ sϕpijs (ϕ)
Pjs
σ1−(1−ρs )
ijs s
σs
1
which yields σ s − 1 = ϕpi ijs= ρ1 , we obtain the(ϕ) . Taking into account that σ s −1 = 1−ρ / 1−ρsss

profit-maximizing pricepijs (ϕ) =wi τ ijs
.
ρs ϕ

(5)

Substituting this expression for price back into gross profits, we obtain
π ijs (ϕ) =====wi τ ijs pijs (ϕ)−σs αs wj Ljpijs (ϕ)1−σs αs wj Lj
−
1−σ s
1−σ s
ϕ
Pjs
Pjs


pijs (ϕ)1−σs αs wj Lj
wi τ ijs
1−1−σ s
ϕpijs (ϕ)
Pjswi τ ijs ρs ϕrijs (ϕ) 1 −ϕ wi τ ijs
rijs (ϕ) [1 − ρs ]rijs (ϕ)σs1implies that 1 − ρs = σ1s . A firm from country i and sector s needs to have asince σ s = 1−ρsproductivity ϕ ≥ ϕ∗ijs to justify paying the fixed “marketing” cost wi fijs of serving the country jmarket. Thus ϕ∗ijs is determined by the cut-off productivity condition
rijs (ϕ∗ijs )= wi fijs .
σs(6)Comparing the cut-off productivity levels of domestic firms and foreign firms in country j, we find

3

that
wi fijs
wj fjjs

=

=

=

=

rijs (ϕ∗ijs )/σ s
rjjs (ϕ∗jjs )/σ s
1−σs
α s wj L jpijs (ϕ∗ijs )/Pjs
from (4)1−σspjjs (ϕ∗jjs )/Pjs
α s wj L j1−σswi τ ijs /ρs ϕ∗ijs1−σs from (5)∗wj τ jjs /ρs ϕjjs
!1−σswi τ ijs ϕ∗jjs.
wj ϕ∗ijs

Rearranging terms yields
ϕ∗jjs
ϕ∗ijs

!1−σs

ϕ∗ijs
ϕ∗jjs


wi σ s=wj σs 1/(σs −1)fwi
s −1 ijs= τ σijs
fjjs wjfs −1 ijs
τ σijsfjjs

and letting Tijs ≡ τ ijs (fijs /fjjs )1/(σs −1) , it follows thatϕ∗ijs = Tijs



wi
wj

1/ρs

ϕ∗jjs .

(7)

The Price Index
Next we solve for the value of the price index Pjs for country j and sector s. Given the Pareto distribution function Gis (ϕ) ≡ 1 − (bis /ϕ)θs , let gis (ϕ) ≡ G0is (ϕ) = bθiss θs ϕ−θs −1 denote the corresponding
productivity density function. Let µis (ϕ) denote the equilibrium productivity density function for
country i and sector s. Since only firms with productivity ϕ ≥ ϕ∗iis produce in equilibrium, firm exit isuncorrelated with productivity and ϕ∗iis < ϕ∗ijs , the equilibrium productivity density function is givenbyµis (ϕ) ≡gis (ϕ)1−Gis (ϕ∗iis ) 0

if ϕ ≥ ϕ∗iisotherwise.

4(8)

Using Ps ≡h´

1−σ s dωω∈Ωs ps (ω)

i1/(1−σs )and[1 − Gis (ϕ∗iis )] Mise gis (ϕ)
Misegis (ϕ),=∗δ is [1 − Gis (ϕiis )]δ is

Mis µis (ϕ) =(A.1)

the price index Pis for country i and sector s satisfies
Pis1−σs

ˆ ∞

ˆ ∞
1−σ s

=
ϕ∗iis

=

piis (ϕ)

Mise
δ is

Mis µis (ϕ) dϕ +

ˆ ∞
ϕ∗iis

piis (ϕ)ϕ∗jispjis (ϕ)1−σs Mjs µjs (ϕ) dϕ

MjsedGis (ϕ) +
δ js

1−σ s

ˆ ∞
ϕ∗jis

pjis (ϕ)1−σs dGjs (ϕ).

This expression can be written more conveniently by switching indexes i and j
Mjse
1−σ s
=
Pjs
δ js

ˆ ∞
1−σ s

ϕ∗jjs

pjjs (ϕ)

MisedGjs (ϕ) +
δ is

ˆ ∞
ϕ∗ijs

pijs (ϕ)1−σs dGis (ϕ)

and it follows that the price index Pjs satisfies
1−σ s
Pjs
=

X Mkse ˆ ∞
k=1,2

δ ks

ϕ∗kjs

pkjs (ϕ)1−σs dGks (ϕ).

(9)

Free Entry
Free entry implies that the probability of successful entry times the expected profits earned from successful entry must equal the cost of entry, that is, Prob.(ϕ ≥ ϕ∗iis )v̄is = wi Fis or[1 − Gis (ϕ∗iis )]π̄ is= wi Fis .
δ is

The average profits across all domestic firms (exporters and non-exporters) is given by
(ˆ
)
ˆ ∞
∞
1
π̄ is =
[π iis (ϕ) − wi fiis ]Mis µis (ϕ) dϕ +[π ijs (ϕ) − wi fijs ]Mis µis (ϕ) dϕ
Mis
ϕ∗iis
ϕ∗ijs


ˆ ∞ 
ˆ ∞
rijs (ϕ)
riis (ϕ)
gis (ϕ)gis (ϕ)=− wi fiisdϕ +− wi fijsdϕ.
∗σs
1 − Gis (ϕiis )σs1 − Gis (ϕ∗iis )
ϕ∗iis
ϕ∗ijs
Substituting yields[1 − Gis (ϕ∗iis )] π̄ is =

ˆ ∞
ϕ∗iis



ˆ ∞ 
rijs (ϕ)
riis (ϕ)− wi fiis gis (ϕ) dϕ+− wi fijs gis (ϕ) dϕ = δ is wi Fis .
σs
σs
ϕ∗ijs

5

Thus we obtain
1 X
δ is

ˆ ∞ 

∗
j=1,2 ϕijsrijs (ϕ)− wi fijs dGis (ϕ) = wi Fis .
σs

(10)

To evaluate the integrals, next note that from (4) and (5),1−σ spijs (ϕ)1−σs (αs wj Lj ) /Pjs
rijs (ϕ)=
1−σ s =
rijs (ϕ∗ijs )pijs (ϕ∗ijs )1−σs (αs wj Lj ) /Pjs

pijs (ϕ)
pijs (ϕ∗ijs )

!1−σs


=

wi τ ijs ρs ϕ∗ijs
ρs ϕ wi τ ijs

1−σs
=

ϕ
ϕ∗ijs

Using the cut-off productivity condition, it follows thatrijs (ϕ∗ijs )rijs (ϕ)
=
σs
σs

ϕ
ϕ∗ijs

!σs −1σ s wi fijs
=
σs

ϕ
ϕ∗ijs

!σs −1= wi fijs

ϕϕ∗ijs!σs −1
(A.2)

and
ˆ ∞ 
ϕ∗ijs



!σs −1ˆ ∞rijs (ϕ)ϕ
wi fijs− wi fijs dGis (ϕ) =− wi fijs  dGis (ϕ)
σs
ϕ∗ijs
ϕ∗ijs


!σs −1ˆ ∞ ϕ= wi fijs− 1 dGis (ϕ)
ϕ∗ijs
ϕ∗ijs
= wi fijs Jis (ϕ∗ijs ),(A.3)

where the function Jis (·) is given byˆ ∞  ϕ σs −1
Jis (x) ≡− 1 dGis (ϕ)
xxˆ ∞ 
ϕ σs −1 θs=bis θs ϕ−θs −1 dϕ − [1 − Gis (x)]
x
x
 θs
ˆ ∞
bis
θs
1−σ s
σ s −1−θs −1= bis θs x
ϕ
dϕ −
x
x
 θ s
σ
−1−θ
s
s
x
bis
= bθiss θs x1−σs−θs − σs + 1x θsθs − (θs − σ s + 1) bis=
θs − σs + 1x θ sσs − 1bis=.
θs − σs + 1 x(A.4)

We assume that θs > σ s − 1 to guarantee that expected profits are finite. Making substitutions and6!σs −1.

rearranging terms, it follows thatX ˆ ∞  rijs (ϕ)− wi fijs dGis (ϕ) = δ is wi Fis
∗
σs
ϕ
ijs
j=1,2
X
wi fijs Jis (ϕ∗ijs ) = δ is wi Fisfrom (A.3)

j=1,2

X

fijs Jis (ϕ∗ijs ) = δ is Fis

j=1,2

X
j=1,2

σs − 1fijs
θs − σs + 1

bis
ϕ∗ijs

!θ s
= δ is Fis from (A.4)(A.5)

and using γ is ≡ bθiss (σ s − 1) / [δ is (θs − σ s + 1)], yields the free entry conditionXsγ is fijs ϕ∗−θ
ijs = Fis .

(11)

j=1,2

Labor Demand
Let Lis denote labor demand by all firms in country i and sector s. We use a three step argument tosolve for labor demand.
First, we show that the fixed costs (the entry costs plus the marketing costs) are proportional to themass of entrants in each country i and sector s.




X ˆ ∞
X ˆ ∞
Mise
gis (ϕ) dϕ from (A.1)
wi Mise Fis +fijs Mis µis (ϕ) dϕ = wi Mise Fis +
fijs
∗
∗
δ
is
ϕ
ϕ
ijs
ijs
j=1,2
j=1,2


X
M
ise
fijs [1 − Gis (ϕ∗ijs )]
= wi Mise Fis +
δ is
j=1,2

!θs 
X
M
b
ise
is

= wi Mise Fis +
fijs
δ is
ϕ∗ijs
j=1,2


Mise
θs − σs + 1= wi Mise Fis +δ is Fis
from (A.5)δ isσs − 1σs − 1 + θs − σs + 1= wi Mise Fis
σs − 17

from which it follows thatwi Mise Fis +X ˆ ∞∗j=1,2 ϕijsfijs Mis µis (ϕ) dϕ = wi Miseθs Fisσs − 1.

(12)Second, we show that the fixed costs are equal to the gross profits in each country i and sector s.
From the free entry condition (10), we obtainδ is wi Fis =wi δ is Fis +X

fijs [1 − Gis (ϕ∗ijs )]

=X ˆ ∞  rijs (ϕ)∗
j=1,2 ϕijs

X ˆ ∞ rijs (ϕ)∗
j=1,2 ϕijs

j=1,2




wi Mise Fis +Mise Xδ isfijs [1 − Gis (ϕ∗ijs )] =

j=1,2


wi Miseθs Fisσs − 1


=
=
=

σs

σs


− wi fijs dGis (ϕ)

dGis (ϕ)ˆ ∞rijs (ϕ)
dGis (ϕ)σsj=1,2X ˆ ∞ rijs (ϕ)
MisdGis (ϕ) from (12)∗1 − Gis (ϕ∗iis )σsj=1,2 ϕijs
ˆ
1 X ∞rijs (ϕ)Mis µis (ϕ)dϕ from (A.1)
∗
σs
ϕ
ijs
j=1,2
1 X
Rijs
σs
Mise X
δ is

ϕ∗ijs

j=1,2

where Rijs ≡´∞ϕ∗ijs rijs (ϕ)Mis µis (ϕ)dϕ is the total revenue associated with shipments from country

i to country j in sector s.
Third, we show that the wage payments to labor equals the total revenue in each country i andsector s. Firms use labor for market entry, for the production of goods sold to domestic consumers
and for the production of goods sold to foreign consumers. Taking into account both the marginal and8fixed costs of production, we obtainwi Lis =

=
=
=
=


X ˆ ∞ τ ijs
fijs + qijs (ϕ)
wi Mise Fis + wi
Mis µis (ϕ) dϕ
∗
ϕ
ϕ
ijs
j=1,2


X ˆ ∞
X ˆ ∞
wi τ ijs

qijs (ϕ)
fijs Mis µis (ϕ) dϕ +wi Mise Fis +ρ Mis µis (ϕ) dϕ∗∗
ρs ϕ sj=1,2 ϕijs
j=1,2 ϕijs


X ˆ ∞
θs Fis+ ρsrijs (ϕ)Mis µis (ϕ) dϕ from (4), (5) and (12)wi Mise∗σs − 1j=1,2 ϕijs
X
1 X
Rijs + ρs
Rijs
σs
j=1,2
j=1,2
X
(1 − ρs + ρs )
Rijs
j=1,2

=

X

Rijs .

j=1,2

Thus
1
1 X
wi MiseRijs =
Lis =
wi
wi
j=1,2



θs Fisσs − 1σsand it immediately follows that
Lis =

1 X
Rijs = Mise Xis
wi

(13)

j=1,2

where Xis ≡ θs Fis /ρs is the labor demand per entrant in country i and sector s.

Relative Expected Profit
The expected profit of an entrant in country i from selling to country j in sector s (after the entrant has
paid the entry cost wi Fis ) is



ˆ
ˆ ∞ 
rijs (ϕ)[1 − Gis (ϕ∗iis )] ∞ rijs (ϕ)
gis (ϕ)−1− wi fijs
dϕ = δ is− wi fijs dGis (ϕ).
δ isσs1 − Gis (ϕ∗iis )
σs
ϕ∗ijs
ϕ∗ijs
The expected profit of an entrant in country j from selling to country j in sector s (after the entrant has
paid the entry cost wi Fis ) is
h
i


ˆ ∞ 
1 − Gjs (ϕ∗jjs ) ˆ ∞  r (ϕ)
gjs (ϕ)rjjs (ϕ)
jjs−1− wj fjjs
dϕ = δ js− wj fjjs dGjs (ϕ).
δ jsσs
1 − Gjs (ϕ∗jjs )
σs
ϕ∗jjs
ϕ∗jjs

9

Thus the expected profit of an entrant in country i from selling to country j in sector s relative to that
captured by an entrant in country j from selling to country j (or the relative expected profit) is givenbyi´ ∞ h rijs (ϕ)
δ −1
−
w
f
i ijs dGis (ϕ)
is ϕ∗ijs
σsihφijs ≡´ ∞ rjjs (ϕ)
−
w
f
δ −1
∗
j jjs dGjs (ϕ)
js ϕ
σs
jjs

=

∗
δ −1
is wi fijs Jis (ϕijs )from (A.3)∗
δ −1
js wj fjjs Jjs (ϕjjs )

θ s
bis
s −1
δ js wi fijs θsσ−σ
∗
ϕijs
s +1
=

 from (A.4)
bjs θss −1
δ is wj fjjs θsσ−σ
∗
ϕjjs
s +1
"
 
 −1/ρs #θsδ js wi fijs bis θs
wi
−1
=
Tijs
from (7)δ is wj fjjs bjs
wjorδ js fijs
φijs =
δ is fjjs



bis
bjs

θs

−θs
Tijs



wi
wj

1−θs /ρs.

It follows that
φ12s φ21s =
=
=
=

 
 
δ 2s f12s b1s θs −θs 1−θs /ρs δ 1s f21s b2s θs −θs −1+θs /ρs
T12s w1
T21s w1δ 1s f22s b2s
δ 2s f11s b1s
f12s f21s[T12s T21s ]−θs
f11s f22s
"
#−θs




f12s f21s
f12s 1/(σs −1)
f21s 1/(σs −1)
τ 21s
τ 12s
f11s f22s
f22s
f11s

(θs −σs +1)/(σs −1)
f11s f22s
1<1
θs
f12s f21s(τ 12s τ 21s )since τ 12s > 1, τ 21s > 1, f12s > f11s and f21s > f22s .

10

(A.6)

(14)

Total Revenue
To solve for total revenue Rijs associated with shipments from country i to country j in sector s, wefirst establish three properties:
pijs (ϕ∗ijs ) =wi τ ijs
ρs ϕ∗ijs=
ρs Tijs

wi Tijs 1/ρswi
wj−1/(σ s −1)

=

wi


ϕ∗jjs

fijs
fjjs

σ /(σ s −1) 

wj s

ρs ϕ∗jjs

1/(1−σs )

fijs
fjjs

from (7)1/(1−σs )
(A.7)

s
= σ−1
,
since 1 − ρ1 = (σsσ−1)−σ
s −1
s −1
s


ˆ ∞  ϕ σs −1− 1 dGis (ϕ) + 1 − Gis (x)Jis (x) + 1 − Gis (x) =x
ˆx∞  ϕ σs −1=dGis (ϕ) − [1 − Gis (x)] + 1 − Gis (x)xxˆ ∞ 
ϕ σs −1=dGis (ϕ)
x
x
 θs
bis
σs − 1+ 1 − Gis (x)=θs − σs + 1 x
σs − 1 + θs − σs + 1=[1 − Gis (x)]θs − σs + 1θsθs − σs + 1=
Jis (x)θs − σs + 1 σs − 1θs=Jis (x),σs − 1

11

(A.8)

and
ˆ ∞

ˆ ∞
ϕ∗ijs

pijs (ϕ)1−σs dGis (ϕ) =

ϕ∗ijs

ϕ
ϕ∗ijs

pijs (ϕ∗ijs )1−σs= pijs (ϕ∗ijs )1−σs

ˆ ∞

ϕ
ϕ∗ijs

ϕ∗ijs

!σs −1dGis (ϕ)!σs −1dGis (ϕ)= pijs (ϕ∗ijs )1−σs [Jis (ϕ∗ijs ) + 1 − Gis (ϕ∗ijs )] from (A.4)" −1/(σs −1) σs /(σs −1) 
#1−σs

wi
wj
fijs 1/(1−σs )=[Jis (ϕ∗ijs ) + 1 − Gis (ϕ∗ijs )] from (A.7)ρs ϕ∗jjs
fjjs

wi wj−σs fijs
θs
= 1−σs ∗1−σsJis (ϕ∗ijs ) from (A.8)ρsϕjjs fjjs σ s − 1 −1
wi wj−σs fijs
δ js wj fjjs Jjs (ϕ∗jjs )θsφijs from (A.6)= 1−σs ∗1−σs
δ −1ρs
ϕjjs fjjs σ s − 1
is wi fijs
!1−σs
Jjs (ϕ∗jjs )wjθs=δ is φijs .
(A.9)
σ s − 1 ρs ϕ∗jjs
δ js
Using these properties, we can solve for total revenue
ˆ ∞Rijs ≡
=
=
=
=

=

=

ϕ∗ijs

rijs (ϕ)Mis µis (ϕ) dϕ

Mis
1 − Gis (ϕ∗iis )

ˆ ∞
ϕ∗ijs

rijs (ϕ) dGis (ϕ) from (A.1)ˆ[1 − Gis (ϕ∗iis )]Mise ∞pijs (ϕ)qijs (ϕ) dGis (ϕ)δ is [1 − Gis (ϕ∗iis )] ϕ∗ijs
ˆpijs (ϕ)−σs αs wj Lj
Mise ∞
pijs (ϕ)
dGis (ϕ) from (4)1−σ sδ is ϕ∗ijs
Pjs
ˆαs wj Lj Mise ∞pijs (ϕ)1−σs dGis (ϕ)
1−σ s δ
Pjs
is
ϕ∗ijs
´
Mise ∞
1−σ s dG (ϕ)
isδ isϕ∗ijs pijs (ϕ)αs wj Lj P
from (9)´∞Mkse
1−σ s dG (ϕ)
ks
k=1,2 δ ks
ϕ∗kjs pkjs (ϕ)
1−σs J (ϕ∗ )jswj
Mise θsjjs
δ is φijs
δ is σ s −1 ρs ϕ∗jjs
δ js
αs wj Lj P
from (A.9)
1−σs J (ϕ∗ )jswjMkse θsjjs
δ ks φkjs
k=1,2 δ ks σ s −1 ρ ϕ∗
δ js
s

12

jjs

and it follows that total revenue can be written simply as
Mise φijs.
k=1,2 Mkse φkjs

Rijs = αs wj Lj P

(15)

The Mass of Entrants
We are now in a position to solve for the mass of entrants using the property that labor demand isP
proportional to the mass of entrants. From Lis = w1i j=1,2 Rijs = Mise Xis , we obtain
X

Rijs = wi Mise Xis

j=1,2

X
j=1,2

Mise φijs
αs wj Lj P
k=1,2 Mkse φkjs

= wi Mise Xis from (15)from which it follows thatX

φijs
= wi Xis .
k=1,2 Mkse φkjs

α s wj L j P

j=1,2
δ f

ijs
Now φijs = δjs
is fjjs



bis
bjs

θ s

−θs
Tijs



wi
wj

1−θs /ρsand Tijs ≡ τ ijs



fijs
fjjs

(16)

1/(σs −1)and φiis = 1. Thus equation (16) can be written out as
α s w1 L 1 α s L 2+
φ= w1 X1s
G1s
G2s 12sαs L2α s w1 L 1
φ21s +
= X2s
G1s
G2s
where
G1s ≡ M1se + M2se φ21s
G2s ≡ M1se φ12s + M2se .
Written in matrix form, these systems of linear equations become







1
φ21s

φ12s  



1


αs w1 L1
G1s
αs L2
G2s

13





  w1 X1s 
 


=
 

X2s

imply that Tiis = 1








1
φ12s







φ21s   M1se   G1s 

 


=
.

 

1
M2se
G2s

Solving using Cramer’s Rule yields1
αs w1 L1=
(w1 X1s − φ12s X2s )
G1s
∆s
αs L2
1
=
(X2s − φ21s w1 X1s )
G2s
∆s

(A.9)

where ∆s ≡ 1 − φ12s φ21s > 0 is the common determinant andM1se ==

1(G1s − φ21s G2s )∆s1
αs w1 L1 ∆s
αs L2 ∆s− φ21s.
∆s w1 X1s − φ12s X2sX2s − φ21s w1 X1sThus the mass of entrants is given by
w1 L 1
φ21s L2
M1se = αs−
w1 X1s − φ12s X2s X2s − φ21s w1 X1s
whereδ 2s f12s
φ12s =
δ 1s f22s



b1s
b2s

and
δ 1s f21s
φ21s =
δ 2s f11s



b2s
b1s

θ s

θs

(17)

1−θs /ρs

−θs
w1
T12s

−1+θs /ρs

−θs
w1
T21s

.

Equation (17) defines M1se as a function of w1 , T12s and T21s , and can be written in function
form as M1se (w1 , T12s , T21s ). To determine the properties of this function, we calculate the partial
∂φ12s
∂φ21s
∂φ21s
12s
derivatives. Given θs > σ s − 1 = ρs σ s > ρs , we obtain ∂φ∂w1 < 0, ∂T12s < 0, ∂w1 > 0, ∂T21s < 0,14and it follows that
∂M1se
∂w1

"
= αs

12s
(w1 X1s − φ12s X2s )L1 − w1 L1 (X1s − ∂φ∂w1 X2s )(w1 X1s − φ12s X2s )2
−
"

= αs
∂M1se
∂T12s

∂M1se
∂T21s

∂φ21s
21s
(X2s − φ21s w1 X1s ) ∂φ
∂w1 L2 + φ21s L2 (φ21s + ∂w1 w1 )X1s

(X2s − φ21s w1 X1s )212s−φ12s X2s L1 + w1 L1 ∂φ∂w1 X2s(w1 X1s − φ12s X2s )2
"

#

−

2
21s
X2s ∂φ∂w1 L2 + (φ21s ) L2 X1s(X2s − φ21s w1 X1s )2
#

#
<0

12s
(w1 X1s − φ12s X2s )0 + w1 L1 ∂φ∂T12s X2s

−0
(w1 X1s − φ12s X2s )2"#12s
w1 L1 ∂φ
X
2s
∂T12s
= αs
<0
(w1 X1s − φ12s X2s )2
"
#
∂φ21s
21s
(X2s − φ21s w1 X1s ) ∂φ
∂T21s L2 + φ21s L2 ∂T21s w1 X1s= αs 0 −(X2s − φ21s w1 X1s )2"#
21s
X2s ∂φ∂T21s L2
= −αs
> 0.
(X2s − φ21s w1 X1s )2
= αs

∂M1se
∂M1se
1se
Thus, the function M1se (w1 , T12s , T21s ) has the properties ∂M∂w1 < 0, ∂T12s < 0 and ∂T21s > 0.

Equilibrium Cut-off Productivities
Having found the equilibrium wage rate w1 , we can now solve for the equilibrium cut-off productiviP
ties. Writing out the free entry conditions j=1,2 wi fijs Jis (ϕ∗ijs ) = δ is wi Fis , we obtainw1 f11s
w1 f12sJ1s (ϕ∗11s ) +J1s (ϕ∗12s ) = w1 F1s
δ 1s
δ 1s
f22s
f21s
J2s (ϕ∗21s ) +J2s (ϕ∗22s ) = F2s .
δ 2s
δ 2s
δ js wi fijs Jis (ϕ∗ )Writing out the relative expected profit conditions φijs = δis wj fjjs Jjs (ϕijs∗ ) , we obtainjjs

φ12s =
φ21s =

δ 2s w1 f12s J1s (ϕ∗12s )δ 1s f22s J2s (ϕ∗22s )δ 1s f21s J2s (ϕ∗21s ).
δ 2s w1 f11s J1s (ϕ∗11s )15Thus the free entry conditions can be rewritten as
f21s
w1 f12sJ2s (ϕ∗21s ) +J1s (ϕ∗12s ) = w1 F1s
φ21s δ 2s
δ 1s
f21s
w1 f12sJ2s (ϕ∗21s ) +J1s (ϕ∗12s ) = F2s
δ 2s
φ12s δ 1s
and in matrix form become




 φ1
 21s


1







f21s J2s (ϕ∗21s )
δ 2s

1
1
φ12s

  w1 F1s 
 

=
.
 

∗
1 12s J1s (ϕ12s )
F2s
δ 1s




 w fSolving using Cramer’s Rule yields
w1 f12s J1s (ϕ∗12s )δ 1s

w1 f12s σ s − 1
b1s θsδ 1s θs − σ s + 1 ϕ∗12s
1 − φ12s φ21s
w1 f12s σ s − 1bθsδ 1s θs − σ s + 1 1s F2s φ12s − φ12s φ21s w1 F1s=

F2s
φ21s − w1 F1s
1φ12s φ21s − 1=

F2s φ12s − φ12s φ21s w1 F1s
1 − φ12s φ21s

∗θs.
= ϕ12sLetting γ 1s ≡ bθ1ss (σ s − 1)/[δ 1s (θs − σ s + 1)], we can write the last expression more simply as
ϕ∗12s =

1/θs
γ 1s f12s (1 − φ12s φ21s ).
F2s (φ12s /w1 ) − φ12s φ21s F1s(19)

Equation (19) shows the export productivity cut-off ϕ∗12s for country 1 in sector s as a function of thecountry 1 wage rate w1 . To determine the slope of this function, note that1
φ12s φ21s =(τ 12s τ 21s )θs

f11s f22s
f12s f21s

(θs −σs +1)/(σs −1)

does not depend on w1 and
"
#
"
#
 
 
φ12s
δ 2s f12s b1s θs −θs 1−θs /ρsδ 2s f12s b1s θs −θs
−θ /ρ
−1
=
T12s w1
w1 =T12s w1 s s
w1δ 1s f22s b2s
δ 1s f22s b2s
is decreasing in w1 . Thus the export productivity cut-off ϕ∗12s is an unambiguously increasing functionof w1 .

16

Proof of Lemma 1(Part 1) One measure of industrial productivity ΦR
1s is the industrial average of firm productivity ϕweighted by each firm’s revenue share in the industry and is given byPˆ ∞∗j=1,2 I(ϕ ≥ ϕ1js )r1js (ϕ)M1s µ1s (ϕ)
RP
ϕv1s (ϕ) dϕ where v1s (ϕ) ≡
Φ1s ≡
.
0
k=1,2 R1ks
The function v1s (ϕ) is a revenue-weighted density function for ϕ since
ˆ ∞P
ˆ ∞
∗
j=1,2 I(ϕ ≥ ϕ1js )r1js (ϕ)M1s µ1s (ϕ)
P
v1s (ϕ) dϕ =dϕ00
k=1,2 R1ks
X ˆ ∞1I(ϕ ≥ ϕ∗1js )r1js (ϕ)M1s µ1s (ϕ) dϕ
= P
R
1ks
k=1,2
j=1,2 0X ˆ ∞1
= P
r1js (ϕ)M1s µ1s (ϕ) dϕ
k=1,2 R1ks j=1,2 ϕ∗1js
X
1
= P
R1js
k=1,2 R1ks
j=1,2

= 1.
From (3), (4), (5), (6) and (8), R1js can be written asˆ ∞R1js =

=

=
=

ϕ∗1js

r1js (ϕ) M1s µ1s (ϕ) dϕ[1 − G1s (ϕ∗11s )]M1se
δ 1s
M1se σ s w1 f1js
δ 1s

ˆ ∞

ˆ ∞
ϕ∗1js

M1se σ s w1 f1js θs bθ1ss
s −1
δ 1s
ϕ∗σ
1js

=

M1se σ s w1 f1js θs bθ1ss
s −1
δ 1s
ϕ∗σ
1js

=

M1se σ s w1 bθ1ss θs
δ 1s (θs − σ s + 1)

ϕ∗1js

ϕ
ϕ∗1js
ˆ ∞

r1js ϕ∗1js



s
ϕ∗θ
1js

!σs −1g1s (ϕ)dϕ from (A.1, A.2)
1 − G1s (ϕ∗11s )!σs −1g1s (ϕ) dϕ from (6)ϕσs −1−θs −1 dϕ

ϕ∗1js

f1js

ϕ
ϕ∗1js

∗σ s −1−θs
−ϕ1js

!

σs − 1 − θs!
.

17

By following similar steps, we can rewrite ΦR
1s as
ˆ ∞ P
∗
j=1,2 I(ϕ ≥ ϕ1js )r1js (ϕ)M1s µ1s (ϕ)
R
P
Φ1s =
ϕ
dϕ
0
k=1,2 R1ks
X ˆ ∞1
ϕ r1js (ϕ) M1s µ1s (ϕ) dϕ= P
k=1,2 R1ks j=1,2 ϕ∗1js!σs −1ˆ ∞X
M1se σ s w1 f1js
1
ϕ

= P
ϕ
g1s (ϕ) dϕ
∗
δ 1s
ϕ∗1js
ϕ1js
k=1,2 R1ks j=1,2θs ˆ ∞
X
M1se σ s w1 f1js θs b1s1

ϕσs −θs −1 dϕ
= P∗σ s −1
∗
δ
R
ϕ1js
1s
ϕ1js
k=1,2 1ks j=1,2!
∗σ s −θs −1+1
X M1se σ s w1 f1js θs bθs
−ϕ
1
1js
1s


= P
∗σ s −1Rδσ−θs − 1 + 1
ϕ1ss
k=1,2 1ks j=1,2
1js!−1 !
θs
X
X M1se σ s w1 bθs θs
f1js
f1ks   M1se σ s w1 b1s θs
1s
ϕ∗1js = ∗θs∗θs
δ 1s (θs − σ s + 1) ϕ1ks
δ 1s (θs − σ s )
ϕ1js
j=1,2
k=1,2




∗θs
f1js /ϕ1js
θs − σs + 1  X
 ϕ∗1js =P∗θsθs − σs
f /ϕ
=

j=1,2
k=1,2
∗
∗
ξ s [h11s ϕ11s + h12s ϕ12s ] ,

1ks

1ks

where ξ s ≡ (θs − σ s + 1) / (θs − σ s ) > 0 holds given our assumption that θs > σ s ,h11s ≡

1zs
f12s, h12s ≡and zs ≡1 + zs
1 + zs
f11s



ϕ∗11s
ϕ∗12s

θs
.

Notice that
∂h11s
∂h12s
=− ∗ =−
∗
∂ϕ11s
∂ϕ11s



1
1 + zszs
1 + zs



θs
ϕ∗11s




= −h11s h12s

θs
ϕ∗11s


<0

and
∂h11s
∂h12s
=− ∗ =
∗
∂ϕ12s
∂ϕ12s



1
1 + zszs
1 + zs

18



θs
ϕ∗12s




= h11s h12s

θs
ϕ∗12s


> 0.

Using these results, we can calculate the partial derivatives of ΦR
1s with respect to the domestic productivity cut-off and the export productivity cut-off:∂ΦR∂h11s ∗
∗
1s
=
ξ
h
+
(ϕ
−
ϕ
)
11s
s
12s
∂ϕ∗11s∂ϕ∗11s 11s




θs
∗
∗
(ϕ11s − ϕ12s )= ξ s h11s − h11s h12s
ϕ∗11s
 ∗


ϕ12s
= ξ s h11s 1 + h12s θs
−1
>0
ϕ∗11s
and
∂ΦR
1s
∂ϕ∗12s∂h11s ∗∗= ξs(ϕ − ϕ12s ) + h12s∂ϕ∗12s 11s




θs
∗
∗
(ϕ11s − ϕ12s ) + h12s
= ξ s h11s h12s
ϕ∗12s



 ∗
ϕ11s
−
1
+
1
.
= ξ s h12s h11s θs
ϕ∗12s

The free entry condition
f11s(ϕ∗11s )

+

θs

f12s
θs

(ϕ∗12s )

=

F1s
γ 1s

determines ϕ∗11s as an implicit function of ϕ∗12s . We can solve for its derivative by totally differens −1
s −1
dϕ∗12s = 0 anddϕ∗11s − f12s θs ϕ∗−θ
tiating the free entry condition. This yields −f11s θs ϕ∗−θ
12s
11s

rearranging terms, we obtain the derivativedϕ∗11s
f12s
=−
∗
dϕ12s
f11s



ϕ∗11s
ϕ∗12s

θs +1


= −zs

ϕ∗11s
ϕ∗12s


.

It follows that∗∂ΦR
1s dϕ11s∂ϕ∗11s dϕ∗12s

  ∗ 
ϕ∗12s
ϕ11s= −ξ s h11s 1 + h12s θs− 1 zs
∗
ϕ11s
ϕ∗12s
 ∗


ϕ11s
ϕ∗11s
= −ξ s (1 − h11s ) ∗ + h12s θs 1 − ∗
ϕ12s
ϕ
 ∗

12s
∗
ϕ11s
ϕ
= −ξ s h12s
+ h12s θs 1 − 11s
.
∗
ϕ12s
ϕ∗12s




19

∗
We can now express ΦR
1s as a function of ϕ12s and take its derivative:dΦR
1s
dϕ∗12s∗∂ΦR∂ΦR
1s dϕ11s
1s+∂ϕ∗11s dϕ∗12s ∂ϕ∗12s
 ∗



 ∗


ϕ11s
ϕ∗11s
ϕ11s
+ ξ s h12s h11s θs+ h12s θs 1 − ∗−1 +1= −ξ s h12s
ϕ∗12s
ϕ12s
ϕ∗12s
 ∗



ϕ11s
ϕ∗11s
= −ξ s h12s
−
1
+
(h
θ
+
h
θ
)
1
−
12s s
11s s
ϕ∗12s
ϕ∗12s


ϕ∗
= −ξ s h12s (θs − 1) 1 − ∗11s < 0.
ϕ12s

=

∗
Therefore, ΦR
1s is a decreasing function in ϕ12s .
∗∗Notice that the expression ΦR
1s = ξ s [h11s ϕ11s + h12s ϕ12s ] does not include either variable trade∗
costs τ ijs or w1 . Thus changes in variable trade costs affect ΦR
1s only through the change in ϕ12s .

(Part 2) The second measure of industrial productivity ΦL
1s is industrial labor productivity:Pˆ ∞j=1,2 R1js
L
Φ1s ≡p11s (ϕ) µ1s (ϕ)dϕ.
where P̃1s =P̃1s L1s
ϕ∗11s
P
From w1 L1s = j=1,2 R1js andˆ ∞ P̃1s =
=
=
=

w1
ρs ϕ



g1s (ϕ)dϕ
1 − G1s (ϕ∗11s )
ϕ∗11s
ˆ ∞
θs bθ1ss
w1dϕρs (b1s /ϕ∗11s )θs ϕ∗11s ϕθs +2"#∗−(θ +2)+1∗θs−ϕ11s s
w1 θs ϕ11s
ρs−(θs + 2) + 1


w1
θs
,
ρs ϕ∗11s θs + 1

industrial labor productivity becomesΦL
1s =From the free entry conditionθs + 1θs∗−θs
j=1,2 fijs ϕijsPρs ϕ∗11s .

= Fis /γ is , ϕ∗11s decreases when ϕ∗12s increases.

∗
Therefore, ΦL
1s decreases when ϕ12s increases. Furthermore, a change in variable trade costs only∗
affects industrial productivity ΦL
1s through its influence on ϕ12s since the trade costs τ ijs and the wage

w1 do not appear separately in the above expression for ΦL
1s or the free entry condition.

20

(Part 3) Another measure of industrial productivity ΦW
1s is industrial labor productivity calculatedusing a theoretically consistent “exact” price index:P

j=1,2 R1js

ΦW
1s ≡

P1s L1s.

It is easy to calculate how a change in ϕ∗12s affects this measure of industrial productivity. Startingfrom the cut-off productivity condition (6)r11s (ϕ∗11s )
σs
∗
1−σ
s
p11s (ϕ11s )αs w1 L1
1−σ s
P1s
1−σs

w1 τ 11s
α s w1 L 1ρs ϕ∗11s P1s
w1 1−σs
P1s= w1 f11s= σ s w1 f11s from (4)= σ s w1 f11s from (5)σ s f11s(ρ ϕ∗ )1−σs
αs L1 s 11s
σ s f11s 1/(1−σs )=ρs ϕ∗11s
αs L1

=

w1
P1s
and then using w1 L1s =P

j=1,2 R1js , we obtain

P
ΦW
1s ≡
From the free entry condition

j=1,2 R1js

P1s L1s

w1
=
=
P1s

∗−θs
j=1,2 fijs ϕijs



αs L1
σ s f11s

1/(σs −1)ρs ϕ∗11s .

= Fis /γ is , ϕ∗11s decreases when ϕ∗12s increases.

P∗
Therefore, ΦW
1s decreases when ϕ12s increases. Furthermore, a change in variable trade costs only∗
affects ΦW
1s through its influence on ϕ12s since the trade costs τ ijs and the wage w1 do not appear

separately in the above expression for ΦW
1s or the free entry condition.
Finally, we derive the welfare formula (23) for the representative consumer in country 1 who
supplies one unit of labor. Since her income is w1 , her aggregate consumption over varieties in sector
s is
C1s =

α s w1.
P1s

21

From the utility function (1) and ΦW
1s = w1 /P1s , her utility is written as: 
α A w1 αA α B w1 αB
U =
P1A
P1B

 αB
W αA
= αA Φ1A
α B ΦW.
1B

Footnote 5
Local consumer demand for an individual firm’s product is given by
p11s (ϕ)−σs αs w1 L1
1−σ s
P1s


w1 τ 11s −σs αs w1 L1
=
1−σ s
ρs ϕ
P1s


w1 1−σs= (ρs ϕ)σs
αs L1
P1s
1−σs
αs L1 .
= (ρs ϕ)σs ΦW
1sq11s (ϕ) =Footnote 6
The weighted average productivity measure in Melitz (2003) satisfies"ˆ∞
ϕ̃1s ≡#1/(σs −1)
ϕ

σ s −1

ϕ∗11s

"ˆ
∞

µ1s (ϕ)dϕ#1/(σs −1)
g1s (ϕ)ϕ=dϕ
1 − G1s (ϕ∗11s )ϕ∗11s"ˆ#1/(σs −1)
θs
∞
θ
b
s 1s
ϕσs −1 θs +1
dϕ
=
∗
θ
∗
ϕ
(b
1s /ϕ11s ) s
ϕ11s"#1/(σs −1)
ˆ
=

s
ϕ∗θ
11s θ sσ s −1∞

ϕσs −1−θs −1 dϕ

ϕ∗11s

"

∗σ s −1−θs −1+1
−ϕ11s
s
= ϕ∗θ
θ
s
11s
σs − 1 − θs − 1 + 11/(σs −1)θs∗σ s −1=ϕθs − σ s + 1 11s1/(σs −1)θs=
ϕ∗11s .
θs − σs + 1

22!#1/(σs −1)

How the Competitiveness Curve Shifts Due to Trade Liberalization
We can determine how the competitiveness curve shifts due to unilateral trade liberalization by country
1 in sector A. When τ 21A decreases holding all other parameter values fixed and holding ϕ∗12A fixed,
the free entry condition∗−θA
A
f11A ϕ11A+ f12A ϕ∗−θ
12A = F1A /γ 1A

implies that ϕ∗11A remains fixed. The other free entry condition∗−θA
A
f21A ϕ∗−θ
21A + f22A ϕ22A = F2A /γ 2A

implies that ϕ∗21A and ϕ∗22A move in opposite directions. Because the cut-off productivity levels satisfy



−1/ρ
1/ρ
ϕ∗12A ϕ∗21A = T12A w1 A ϕ∗22A T21A w1 A ϕ∗11A = T12A ϕ∗22A T21A ϕ∗11A ,ϕ∗12A is fixed, T12A is fixed, T21A decreases and ϕ∗11A is fixed, it follows that ϕ∗21A decreases if ϕ∗22Adecreases. But this is impossible since ϕ∗21A and ϕ∗22A move in opposite directions. If ϕ∗22A remains
fixed, then ϕ∗21A decreases. This is also impossible. The only possibility left is that ϕ∗22A increases
and ϕ∗21A decreases. Thus, a decrease in τ 21A holding ϕ∗12A fixed leads to ϕ∗11A remaining fixed, ϕ∗22A
1/ρA

increasing and ϕ∗21A decreasing. But then the cut-off productivity condition ϕ∗12A = T12A w1

ϕ∗22A

implies that w1 must decrease. It follows that when τ 21A decreases holding ϕ∗12A fixed, then the wage
rate w1 must decrease and the competitiveness curve shifts down.

Balanced Trade
FromP

j=1,2 R1js = w1 L1s andP

j=1,2 E1js = E11s + E12s = αs w1 L1 , the excess exports of sectors for country 1 is
P


R
−
E
11s − E12s = w1 L1s (·) − (αs w1 L1 − E12s ) − E12s
j=1,2 1js
L1s (·)= w1 α s− L1 .
αs23

Summing up for both industries, we obtain that the balanced trade condition is equivalent to the labormarket clearing condition:
0=

P

s=A,B





i
P
L1s (·)
R
−
E
−
E
=
w
α
−
L
1 s
11s
12s
1
j=1,2 1js
s=A,B
αs
 



L1A (·)
L1B (·)= w1 α A− L1 + αB
− L1
αA
αB
= w1 [L1A (·) + L1B (·) − L1 ] .

hPMultilateral Trade Liberalization
With symmetric countries and w1 = w2 = 1,
L1s = M1se X1s


w1 L1
φ21s L2= αs−
X1s
w1 X1s − φ12s X2s X2s − φ21s w1 X1s


φs L1
L1
−
Xs
= αs
Xs − φs Xs Xs − φs Xs= αs L1 ,
and
ϕ∗12s
ϕ∗12A

1/θs
γ 1s f12s (1 − φ12s φ21s )=F2s (φ12s /w1 ) − φ12s φ21s F1s
γ 1A fxA (1 − φA φA ) 1/θA=
FA (φA /1) − φA φA FA
γ 1A fxA (1 − φA )(1 + φA ) 1/θA=,
FA φA (1 − φA )
from which it follows thatϕ∗12A =

γ 1A fxAFA

Since
δ js fijs
φijs =
δ is fjjs



bis
bjs



θs

1
1+
φA
−θs
Tijs

1/θA



.

wi
wj

1−θs /ρs

simplifies toφA =fxA −θAT,
fA A

a decrease in TA leads to an increase in φA and a decrease in ϕ∗12A for fixed w1 = 1.

24

(27)

Finally, we show the labor demand curve of industry A, L1A , becomes flatter in response to liberalization of industry A as illustrated in Figure 6. To draw the labor demand curve, we allow w1 can bedifferent from one; therefore φ12s can be different from φ21s . The labor demand by sector s in country
1 is

L1s = M1se Xs

w1 L1
φ21s L1= αs Xs−
w1 Xs − φ12s Xs Xs − φ21s w1 Xs
 


1
φ21s w1
1−.
= αs L1
1 − φ12s /w1
w1 1 − φ21s w1Notice that w1 > φ12s and 1 > φ21s w1 are required for an interior solution from (A.9).
 (θs −σs +1)/(σs −1)
θ /ρ
fs
s
Let $ ≡ w1s s and κs ≡ τ −θ
. Then φ12s and φ21s become
s
fxs
κs
φ12s
=
< 1 and φ21s w1 = κs $ < 1,
w1$
from which it follows that κs < 1. By substituting these into the labor demand, we obtain




$
κs $
1
L1s = αs L1.
−$ − κs$σs /θs 1 − κs $We take its derivative with respect to κs
∂L1s
∂κs




1
1
1
= $αs L1−
$σs /θs (1 − κs $)2($ − κs )2" #
$1−σs /θs αs L1$ − κs 2
σ s /θs=$−.
1 − κs $($ − κs )2

Since ∂κs /∂τ s < 0,
∂L1s
∂τ s
∂L1s
∂τ s
∂L1s
∂τ s

> 0 if LHS($) ≡ $σs /2θs <= 0 if LHS($) = RHS($)< 0 if LHS($) > RHS($).

25$ − κs≡ RHS($)
1 − κs $SincedLHS($)d$2
d LHS($)
d$2
dRHS($)
d$d2 RHS($)
d$2

σ s σs /2θs −1
$
> 0,
2θs


σ s 2θs − σ s$−(2θs −σs )/2θs −1 < 0,
= −
2θs
2θs
1 − κ2s> 0,=(1 − κs $)2
2κs 1 − κ2s> 0,=(1 − κs $)3=

and
LHS($ = 1) = RHS($ = 1)
dLHS($ = 1)σs
1 + κsdRHS($ = 1)
=
<1<
=
,
d$
2θs
1 − κsd$
we have
LHS($) < RHS($) if $ > 1
LHS($) = RHS($) if $ = 1
LHS($) > RHS($) if $ < 1.
θ /ρsSince $ = w1s, we obtain
∂L1s
∂τ s
∂L1s
∂τ s
∂L1s
∂τ s

> 0 for w1 > 1= 0 for w1 = 1< 0 for w1 < 1.

Therefore, a reduction in τ A makes L1A flatter and tilt counterclockwise around point E in Figure 6.

26