International Trade and Income Inequality∗
Taiji Furusawa†

Hideo Konishi‡

Hitotsubashi University

Boston College

October 12, 2012

Abstract
We propose a theory that explains why international trade can widen a wage gap
between top income-earners and others. We consider two symmetric countries in which
individuals with diﬀerent abilities work either as knowledge workers, who develop products produced in a diﬀerentiated-good sector, or as production workers, who engage
in actual production processes. In equilibrium, ex ante symmetric ﬁrms post diﬀerent
wages for knowledge workers and hence attract workers with diﬀerent abilities fromother ﬁrms, creating the diﬀerence in quality of their products. International trade
will beneﬁt ﬁrms that produce high-quality products and harm ﬁrms that produce
low-quality products. The relative wage gap between individuals with high ability andthose with low ability expands as a result. Indeed, we show that international trade
increases the real wages for those with lowest and highest abilities but decreases thereal wages for those with intermediate abilities. We also extend the basic model to theone with asymmetric countries and show that the wage gap created by international
trade is severer in the smaller country than the larger country.
Preliminary and incomplete.

∗

Work in progress. We are grateful to James Anderson, Pol Antras, Arnaud Costinot, Elhanan Helpman,Oleg Itskhoki, Tomoya Mori, and Esteban Rossi-Hansberg for helpful comments. We also thank Laura
Bonacorsi for her research assitance.
†
Graduate School of Economics, Hitotsubashi University. (E-mail) furusawa@econ.hit-u.ac.jp‡
Department of Economics, Boston College. Email: hideo.konishi@bc.edu

1Introduction

It has been widely documented that wage gap increases signiﬁcantly in many countries,
especially in the developed countries. In Britain, for example, “chief executives can expect
to receive average compensation in excess of £4.5m ($ 6.9m) this year. Pay at the top grewby over 300% between 1998 and 2010. At the same time, the median British worker’s real
wage has been pretty stagnant. These trends mean the ratio of executive to average pay
at FTSE 100 ﬁrms jumped from 47 to 120 times in 12 years.” (The Economist, January
14th-20th, 2012, p. 11) It has also been documented that job polarization has occurred inmany developed countries, including the United States and some European countries, suchthat the shares of employment in high-skilled occupations and low-skilled occupations growwhile that of middle-wage occupation declines (Acemoglu and Autor, 2011).
Technological changes and globalization are often argued to be the causes of the job polarization and expansion of income inequality. Machines have replaced middle-skilled workers
who engage in routine tasks. Oﬀshoring routine tasks to low-wage countries reduces demands
for middle-skilled workers in high-wage developed countries. Oﬀshoring is unambiguously an
important aspect of globalization. But we argue here that international trade in goods, which
is more fundamental part of globalization, alone can cause some sort of job polarization andexpand income inequality among diﬀerent skill groups of workers. In particular, we relate
the recent trend of market concentration in manufacture industries, i.e., the winner-take-all
market trend, to the job polarization and the expansion of the income inequality. Thanks
to globalization (i.e., an increase in the market size and wider information transmission), a
diﬀerence in quality becomes an important source of diﬀerential proﬁtability within industries: ﬁrms that sells high-quality products command disproportional market shares. This,in turn, gives workers who work as knowledge workers in the winning ﬁrms disproportional
shares of income. Knowledge workers in such ﬁrms are winners because they are the sourcesof the ﬁrms’ success. But that implies that the war for talent arises and large portion of
the ﬁrms’ operating proﬁts go to the knowledge workers as the rents for their talents. What
determines the product quality is the talent of knowledge workers such as managers and1

R&D workers hired in ﬁrms. Firms that hire talented knowledge workers enjoy beneﬁts fromthe employment, which in turn beneﬁts such knowledge workers in the form of high wages.
To show this phenomenon, which becomes more and more important in recent years, webuild a two-country model in which ex ante symmetric ﬁrms in the diﬀerentiated good (manufacture) sector hire knowledge workers with diﬀerent abilities, thereby producing productswith diﬀerent qualities. In equilibrium, knowledge workers are assortatively allocated in
diﬀerent ﬁrms: ﬁrms that hire a group of highly-talented knowledge workers produce highquality products, while those hire mediocre knowledge workers produce relatively low-quality
products. Firms also hire production workers to produce their products. We assume thatworkers are homogeneous in their productivities when hired as production workers despite
the diﬀerence in their abilities. The wage gap may arise between knowledge workers andproduction workers even within a ﬁrm; the wage gap within a ﬁrm is particularly serious in
proﬁtable ﬁrms that produce high-quality products.
International trade aﬀects ﬁrms’ proﬁtability diﬀerently across ﬁrms. Top-tier ﬁrms that
produce high-quality products are the winners of opening to trade; getting access to an
additional market give them large beneﬁts. Medium-tier and Lowest-tier ﬁrms, on the other
hand, loose from opening to trade. They suﬀer from foreign top-tier ﬁrms’ penetration into
their own market. Although the medium-tier ﬁrms sell their products to the foreign market
as well as their own market, the additional export proﬁts after the subtraction of ﬁxed export
costs are not enough to oﬀset the loss that they incur in their domestic market. Consequently,
top income earners who are best talented and work in the top-tier exporting ﬁrms beneﬁt
from opening to trade; they are the winners of globalization. On the contrary, workers in the
middle-income class, who (used to) work as knowledge workers in the middle-tier ﬁrms, arelikely to suﬀer from opening to trade. Their nominal incomes drop because the proﬁts forthe ﬁrms in which they are working fall after trade liberalization. Although the real wages
may still rise thanks to the increased varieties of products in their consumption after tradeliberalization, we show that under a relatively mild condition their real wages decline by
opening to trade. Indeed, workers in the middle-income class are the only losers from trade2liberalization. Thanks to the increased varieties in consumption, the real wages for the least
talented workers, who work as production workers, increase by opening to trade.
We also extend the basic model to the one with two countries of asymmetric size. We
ﬁnd that production workers earn a higher wage in the larger country than the smaller one.
The export cutoﬀ quality is higher in the larger country due to the higher wage than in
the smaller country, while the entry cutoﬀ quality is lower in the large country. The wage
gap created by international trade can be said to be smaller in the larger country since thewages for production workers increase while the range of qualities for products sold only
domestically expands in the larger country and contracts in the smaller country by opening
to trade. We also ﬁnd that as the variable trade costs increase, the wage rate for production
workers in the larger country relative to the one in the smaller country increases.
We are not the ﬁrst to theoretically predict that international trade widens wage gap
across diﬀerent income groups. Manasse and Turrini (2001), Yeaple (2005), Costinot andVogel (2010), Helpman et al. (2010), Helpman et al. (2011), Helpman et al. (forthcoming),
and Blanchard and Willmann (2011), among others, show in their models that international
trade in goods widens wage gap at least in some trading countries.
Among these studies, Manasse and Turrini (2001) and Yeaple (2005) are the closest to
our paper. Manasse and Turrini (2001) employ the same basic model structure as ours;
ex ante symmetric ﬁrms produce products of diﬀerent qualities because they are run byentrepreneurs with diﬀerent abilities. In their analysis, however, the number of ﬁrms inthe diﬀerentiated good industry is ﬁxed and it is not aﬀected by opening to trade. As aconsequence, they cannot analyze how international trade aﬀects proﬁtability of ﬁrms andhence the wage distribution through a change in the number of operating ﬁrms. This channel
is important because the winner-take-all phenomenon is prevalent in the global markets andthis reduces the number of ﬁrms in each industry, thereby reducing the jobs for workers in
the middle-income class. Yeaple (2005) derives similar predictions to ours in a similar model
environment. In his model, ﬁrms choose both their individual production technologies andtypes of workers. A distinguishing feature of his model is the complementarity between the3technology and skills of labor; high productivity technology is matched with high-skilledworkers. Beside the fact that the average talent of knowledge workers is the only source ofﬁrm heterogeneity in our model, our model is diﬀerent from his in that we separate labor
into two endogenously allocated categories: knowledge workers and production workers. Thisdistinction is a key to our analysis of the impact of trade on income redistribution. First, we
can discuss diﬀerential eﬀects of trade on workers within ﬁrms, which is an important wage
gap as well as the wage gap within sectors and within occupations. Second and more broadly,
separating knowledge workers from production workers is important to understand the eﬀectof globalization on labor market. Knowledge can be embedded into products so that it isduplicated unboundedly with the help of capital and production workers to possibly earn afortune in the global market. Globalization does not necessarily increase the demands forknowledge workers. It only increases the demands for talent. Knowledge created by a limited
number of knowledge workers is embedded in the products and travels over the world.

2

The Model

We consider two countries, 1 and 2, in which a diﬀerentiated good with many varieties is
produced and consumed. In each country i = 1, 2, there is a continuum of workers withthe mass Li . The diﬀerentiated good consists of a continuum of varieties each of which(denoted by ω ∈ Ω) is produced under monopolistic competition by a ﬁrm in a continuum
pool of ﬁrms. Let x(ω) denote the consumption level of a variety ω of the diﬀerentiatedgood. Quality of the diﬀerentiated good may vary across varieties, which is represented byα(ω). Then, a representative consumer’s preferences are represented by the utility function:[∫1σα(ω) x(ω)

u=

σ−1
σ

σ
] σ−1

dω

.

(1)

Ω

The utility levels derived from the consumption of the varieties of the good are aggregatedwith the use of CES aggregator where σ > 1 denotes the elasticity of substitution. The
higher the α(ω), the higher the utility a consumer derives from the consumption of variety
ω.
4

Labor is the only production factor in this economy. But workers are employed either as
knowledge workers in the diﬀerentiated good sector to develop a product or as production
workers to produce the diﬀerentiated goods. We choose labor provided by production workers
as the numeraire. Workers are heterogeneous in their abilities (or abilities) that only matter
when they are hired as knowledge workers. Thus, they are heterogeneous as knowledge
workers, but homogenous as production workers. In the basic model, ability is measured bya ∈ R+ , the distribution of which in country i is described by the cumulative probability
distribution Gi with probability density function gi ; the mass of workers with their abilities
less than or equal to a is, therefore, given by LGi (a).
In the basic model, we assume that countries 1 and 2 are symmetric: that is, L1 = L2 = L
and G1 (a) = G2 (a) = G(a) for all a ∈ R+ . Until Section 5, we do not put subscripts on L
and G to simplify the notations.
In the production of the diﬀerentiated good, each ﬁrm needs to hire ls knowledge workers
to develop a product. The average ability level of these workers determines the quality ofthe product; we simply assume that the quality of the product α(ω) is equal to the averagetalent level of the knowledge workers employed in the ﬁrm. The diﬀerentiated good market
is under the monopolistic competition with free entry and exit. To enter the market, ﬁrms
need to develop the product by hiring ls knowledge workers, which serves as entry costs.
Production of the product itself requires only production workers; 1 unit of labor produces
1 unit of the good.
Because the average ability level of knowledge workers determines the product quality,ﬁrms in the diﬀerentiated good sector compete for talent. They post wages for knowledge
workers, and workers apply for those positions. Then each ﬁrm chooses ls workers from those
who have applied for the ﬁrm’s position. There is no friction in the labor markets, both forknowledge workers and production workers, nor does there exist information asymmetry
between workers and ﬁrms on individual workers’ ability levels.
In equilibrium, ﬁrms post diﬀerent wages for knowledge workers attracting workers with
diﬀerent ability levels. Workers with highest abilities are hired by the ﬁrms that post highest

5wages, such that workers are sorted according to their ability levels; knowledge workers withthe same or similar abilities work together in equilibrium. Thanks to the competition amongﬁrms, the entire operating proﬁts for each ﬁrm are given as the rents to the knowledge
workers. We assume that all knowledge workers, possibly with diﬀerent ability level, hiredin the same ﬁrm obtain the same wage. Letting w(ω) denote the knowledge worker’s wage
in the ﬁrm that produces the variety ω and π̃(ω) denote the ﬁrm’s operating proﬁts, wetherefore havew(ω)ls = π̃(ω).

(2)

Firms produce varieties of diﬀerent quality. Since the average ability level of knowledge
workers determines the product quality, the ﬁrm distribution with respect to their productquality is determined from the ability distribution. Let f (α) be the density of the ﬁrms that
produce a variety of quality α and F be the corresponding cumulative (non-probabilistic)∫αfunction such that F (α) = 0 f (α′ )dα′ . Since the ﬁrms that produce varieties of qualitythat is greater or equal to α hire all workers whose ability levels are greater or equal to α(i.e., a ≥ α), we have
l

s

∫ ∞

f (α′ )dα′ = L[1 − G(α)].

α

Taking the derivative of this identity with respect to α, we obtain the density of ﬁrms that
produce a variety of quality α asf (α) =

Lg(α)
.
ls

(3)

As this relationship shows, each ﬁrm hires ls knowledge workers of the same ability level;
the equilibrium allocation of knowledge workers are perfectly assortative.
Workers who are not hired as knowledge workers will work as production workers. In
equilibrium, there will be a cutoﬀ ability level α∗ such that all workers with a ≥ α∗ work as
knowledge workers, while all workers with a < α∗ work as production workers.
Once α∗ is given, together with (3), the distribution of operating ﬁrms is completely
determined.

6

3

Autarkic Equilibrium

This section derives the autarkic equilibrium and shows that knowledge workers obtain higher
wages than production workers and that knowledge workers’ wages increase proportionately
with their ability levels.
Thanks to the symmetry assumed in the basic model, we need only consider a country toderive an autarkic equilibrium. Let I denote the aggregate income of a country, and let w be
the wage rate of production worker. Since the wage of production workers can be normalized
to 1 in one country model, each ﬁrm optimally selects the price of p(α) = σ/(σ − 1), theconstant mark-up price over the marginal cost of 1, regardless of its product quality α, so
the ﬁrm that produces a variety of quality α sells
αp(α)−σ(σ − 1)Iα
I = ∫∞ ′
′
1−σ
′
′
σ
αp(α ) dF (α )α dF (α )α∗α∗x(α) = ∫ ∞(4)units of the good. A ﬁrm’s production level is higher if the quality of its product is higher and∫∞if the quality index, α∗ α′ dF (α′ ), is lower. The operating proﬁts for the ﬁrm that produce
a product with quality α are given byπ̃(α) = ∫ ∞
α∗

α

I

α′ dF (α′ ) σ.

(5)Letting w(α) denote the wage for a knowledge worker hired by the ﬁrm that produces the
product with quality α with slight abuse of notation, we can write the proﬁts for the ﬁrm asπ(α) = π̃(α) − w(α)ls .
If π(α) is greater than 0 for some ﬁrm with α, an entrant posts a slightly higher wage thanw(α) and get all the knowledge workers from such a ﬁrm to proﬁtably operate. Therefore,
π(α) = 0 in equilibrium, so that the knowledge workers’ wage schedule is given by w(α) =π̃(α)/ls as (2) indicates.
The equilibrium is characterized by the two conditions: free entry (FE) condition andlabor market clearing (LM) condition. Free entry condition expresses that the cutoﬀ ﬁrm
with α∗ earns zero proﬁts. The knowledge workers in the cutoﬀ ﬁrm earn the wage of 1, i.e.,7w(α∗ ) = 1, in equilibrium, since if w(α∗ ) > 1 proﬁtable entry by a ﬁrm that posts the wageof 1, for example, would occur. Thus, the free entry condition can be written asα∗I= ls .
σ
αdF
(α)
α∗

∫∞

(6)

The labor market clearing condition, on the other hand, expresses that total labor demands,
demands for knowledge workers and those for production workers, must equal the labor∫∞
supply L. Total demands for knowledge workers are given by ls α∗ f (α)dα. Total demands
for production workers equal (σ − 1)I/σ. Thus, the labor market clearing condition can be
written as
l

s

∫ ∞
f (α)dα +α∗σ−1
I = L.
σ(7)

Figure 1 depicts the relationships between α∗ and I that express the free entry and labormarket clearing conditions. The free entry condition can be expressed by a negatively-slopedschedule F E, since the left-hand side of (6) increases with both α∗ and I. The labor market
clearing condition, on the other hand, can be expressed by a positively-sloped schedule LM ,since the left-hand side of (7) decreases with α∗ but increases with I. The intersection of∗these two schedules gives us the equilibrium values of α∗ and I, which we call αA
and IA .

Once the equilibrium threshold of α∗ is determined, the equilibrium wage schedule readily
obtains. As Figure 2 shows, wages are ﬂat at 1 for all workers with their ability levels smaller∗
than αA. Their wages are 1 because they work as production workers. Those who have∗abilities greater than αA, on the other hand, work as knowledge workers. Their wages are

the rents for their abilities and are greater than 1 except for the knowledge workers whose∗
ability levels are exactly equal to αA. Note that the ratio of wages for knowledge workers

with any two diﬀerent levels of abilities is equal to the ratio of their abilities itself, as we cansee from (5):π̃(α)α
wA (α)== ∗.
∗∗wA (αA )π̃(αA )
αA∗∗∗
Since wA (αA) = 1, we have wA (α) = α/αA
for all α ∈ [αA, ∞), and wA (α) = 1 for all∗α ∈ [0, αA].

The equilibrium real wages can also be readily derived. It is easy to infer from (4) that8

′

the worker with ability level of α consumes ∫ ∞∗ α′′αdF (α′′ ) (σ−1)w(α)
units of the variety of quality
σ
α

A

∗
α′ for any α′ ≥ αA
. Then we substitute them into (1) to obtain the worker’s indirect utility:u(α) =

( σ

) [∫

σ−1

w(α)

1 .
] 1−σ

∞ ′
α dF (α′ )α∗A

To see the impact of international trade on each worker’s well-being, we deﬁne the real wage
for workers with ability level α byv(α) = [∫

w(α)

] 1
∞ ′
α dF (α′ ) 1−σ
α∗

,

(8)

which is proportional to the indirect utility u(α). The real wage in the autarkic equilibrium∗
is given by vA (α), which is nothing but v(α) deﬁned in (8) with α∗ being equal to αA.

4

Trade equilibrium

Let us turn to the analysis of the impact of international trade on the ﬁrm distribution andwages. We suppose that ﬁrms in the diﬀerentiated good sector need to incur the ﬁxed costof exporting in terms of fX units of labor. Exporting ﬁrms also incur iceberg trade cost suchthat they need to ship τ units of the good to supply 1 unit in the foreign market.
We consider the realistic case in which only a fraction of the ﬁrms export their products.
Let αX denote the threshold quality such that the products are exported (as well as supplied
domestically) if and only if their individual qualities are higher or equal to αX . It is easy to
see that the operating proﬁts from the domestic sales and foreign sales are given byπ̃d (α) = ∫ ∞

α

∫∞

I

,

(9)

ατ 1−σ
I
∫∞
,
′
′
1−σ
′
′
α dF (α ) + τ
α dF (α ) σ
α∗
αX

(10)

α∗

α′ dF (α′ ) + τ 1−σαX

α′ dF (α′ ) σπ̃X (α) = ∫ ∞respectively. Then, the proﬁts for the ﬁrm that produces a product of quality α are equal to{π(α) =π̃d (α) − w(α)lsπ̃d (α) + π̃X (α) − w(α)ls − fX

if α∗ ≤ α < αX
if α ≥ αX(11)

Since the values of the quality index are the same between the two countries due to thesymmetry, it is easy to compare the operating proﬁts from exporting and those from domestic
9

sales. In particular, we compare the operating proﬁts from exporting for the threshold ﬁrm
with αX , i.e., π̃X (αX ) = fX , and those for the other threshold ﬁrm with α∗ , i.e., π̃d (α∗ ).
Then, it follows from (9) and (10) that the relationship between αX and α∗ can be expressedby the function:
αX (α∗ ) =τ σ−1 fX ∗α .
ls

We assume that τ σ−1 fX > ls so that only a fraction of the ﬁrms export their products.
The free-entry condition and the labor market clearing condition can be written asα∗I= ls ,1−σσ
αdF (α) + τ
αdF (α)α∗αX (α∗ )
∫ ∞
∫ ∞
σ−1
f (α)dα + fX
f (α)dα +I = L.
lsσα∗αX (α∗ )
∫∞

∫∞

(12)
(13)

By comparing the free-entry condition (12) with the autarky counterpart (6), we ﬁnd that
the FE schedule shifts up as Figure 3 indicates. International trade intensiﬁes the domestic
competition. In order for the same threshold producer to be break-even, the total incomemust increase. Similarly, by comparing (13) with (7), we see that the LM schedule shifts
down. The labor market becomes tighter because demands for labor used for exportingare created by opening to trade. Thus, the income must decrease so that these increased
demands are oﬀset by the decreased demands for production workers. Trade equilibriumincome denoted by IT may be greater or smaller than IA , i.e., the impact of trade on nominal
income measured by the numeraire good is ambiguous. However, trade will unambiguously
raise the threshold quality αT∗ . International trade intensiﬁes competition in individual
domestic markets, which lowers the proﬁtability of ﬁrms that only serve their individual
domestic markets. In addition, the labor market becomes tighter due to the demands forlabor for exporting. These two eﬀects work as factors to increase the bar for entry into thediﬀerentiated good sector.
Proposition 1 International trade raises the threshold quality of the diﬀerentiated good∗αA
< αT∗ and hence raises the average quality of the good.

10

The equilibrium wage schedule is described by a piecewise linear function: (i) for α ∈[0, αT∗ ],wT (α) = 1(ii) for α ∈ [αT∗ , αX ],wT (α) =ααT∗and (iii) for α ∈ [αX , ∞),wT (α) =ατ 1−σ IT] × (α − αX ).
[
+
∫∞
∫∞
αT∗
′
′
1−σ
σ
α′ dF (α′ ) ℓS
∗ α dF (α ) + ταXαTFigure 4 shows how wage schedule changes as a result of opening to trade. The wage schedulein trade equilibrium shows that there are more production workers who obtain the wage of 1
than before the opening to trade due to the reduction of the mass of ﬁrms in the diﬀerentiated∗good sector. Note in particular that workers whose ability α is between αAand αT∗ used towork as knowledge workers but now work as production workers after opening to trade. In
trade equilibrium, workers whose ability is between αT∗ and αX work as knowledge workers
in the ﬁrms that serve only their own domestic market. Their wages are lower in tradeequilibrium than in autarky because their ﬁrms suﬀer from an increased competition in theirdomestic market. Workers whose ability is greater than αX are the knowledge workers who
work in ﬁrms that export their products. Proﬁts for a ﬁrm that barely meets the criterion
for exporting are smaller than those in autarky. This is because proﬁts from exporting after
paying the ﬁxed costs of exporting are not suﬃcient to oﬀset the losses in the domestic sales
caused by the import penetration. Wages for knowledge workers who work in such ﬁrms
decline as a result of opening to trade. However, the wage schedule beyond αX is steeper
than that in autarky between αT∗ and αX due to the existence of the ﬁxed cost of exporting
paid by the numeraire good, as (11) shows. Consequently, the wages for knowledge workers
whose ability is higher than a certain threshold increase as a result of opening trade. Trade
increases nominal wages (measured by labor provided by the production workers) only for
those who are highly talented.

11

Proposition 2 International trade raises the nominal income (measured by labor provided
by the production workers) only for those who are most talented.
What is the impact of trade on real wages? It is easy to see that workers who work as
production workers both before and after trade beneﬁt from trade. Their real wages increase
because the price index falls due to the increased varieties in consumption while the nominal
wages are unaﬀected. What about the impact on the middle income class? To derive aclear-cut answer to this question, we assume now that the ability is distributed according to
the Pareto distribution with its cumulative distribution function G(a) = 1 − (a0 /a)k , wherea0 > 0 and k > 1. Then, it can be shown that under a mild restriction, international tradedecreases the real wages for the middle income class as Figure 5 shows.
Proposition 3 Suppose that there are two symmetric countries in which workers’ ability
distribution follows Pareto distribution. Then, the lowest income earners who work as production workers as well as the highest income earners who work as knowledge workers are
better oﬀ by international trade. Those who belong to the middle income class, however, experience a decrease in real wages by opening to trade, if and only if σ > 2 holds. All knowledge
workers who work in ﬁrms that only serve their individual domestic markets belong to such
middle income class.
We use two measures of social welfare to evaluate the eﬀect of international trade below.
If the ability is distributed according to the Pareto distribution, we also have an unambiguous result regarding the impact of trade on social welfare that is a simple aggregation ofindividuals’ utilities (or real wages).
Proposition 4 Suppose that there are two symmetric countries in which workers’ ability
distribution follows a Pareto distribution. Then, international trade unambiguously improves
utilitarian social welfare for individual countries.
The other measure of social welfare is the Lorenze domination, which is a measure toevaluate the equality of income distribution. We deﬁne the Lorenz function by∫aw(a′ )dG(a′ )L(a) = ∫ 0∞,w(a′ )dG(a′ )012the fraction of total income earned by those who have the ability a or less. We say that theincome distribution characterized by the Lorenz function LA is Lorenz-dominated by theone characterized by LB , if LA (a) ≤ LB (a) for any a with strict inequality for some a.
Proposition 5 Suppose that there are two symmetric countries in which workers’ ability
distribution follows a Pareto distribution. Then, the income distribution under international
trade is Lorenz-dominated by the one under autarky.

5

Asymmetric Countries

In this section, we extend the basic model to the one in which countries are asymmetric interms of population. We will show that the impact of trade on income inequality is severer
in a smaller country than a larger country.
Let us suppose that country 1 is larger than country 2, i.e., L1 > L2 . We continue toassume that the ability distributions are the same between the two countries, so the ﬁrm
distribution in country i is given by the density function similar to the one in (3):fi (α) =Li g(α).
ls(14)

We choose the labor provided by the production workers in country 2 as a numeraire, so thewage rate for the production workers in country 2 equals 1. We let w denote the wage rate
for the production workers in country 1.
To derive equilibrium conditions, we ﬁrst note that the total income for each country
equals the total revenue for all the domestic ﬁrms. We can write the total income forcountry 1, for example, as∫∞∫∞
w1−σ τ 1−σ αX αdF1 (α)
w1−σ α∗ αdF1 (α)
11∫∫I + ∫∞
I.
I1 = 1−σ ∫ ∞1−σ ∞ αdF (α) 11−σ τ 1−σ ∞ αdF (α) 2
αdF
(α)
+
τ
αdF
(α)
+
w
w
2
1
1
2
αX
αX
α∗
α∗
1

2

1

2

Using (14), we can rewrite this equality as∫∞∫∞
w1−σ τ 1−σ L1 αX αdG(α)
w1−σ L1 α∗ αdG(α)
1
1
∫∞
∫∞
I1 + ∫ ∞I2 .
I1 = 1−σ ∫ ∞w L1 α∗ αdG(α) + τ 1−σ L2 αX αdG(α)
L2 α∗ αdG(α) + w1−σ τ 1−σ L1 αX αdG(α)
1

2

2

13

1

Similarly, we have∫∞
L2 α∗ αdG(α)∫∞τ 1−σ L2 αX αdG(α)
2
2
∫∞
∫∞
∫∞
I2 + 1−σ ∫ ∞
I1 ,
I2 =
L2 α∗ αdG(α) + w1−σ τ 1−σ L1 αX αdG(α)
w L1 α∗ αdG(α) + τ 1−σ L2 αX αdG(α)
2

1

1

2

for country 2. Although they look diﬀerent, it is easy to show that these two equations areequivalent and can be written as∫∞∫∞
w1−σ τ 1−σ L1 αX αdG(α)τ 1−σ L2 αX αdG(α)
1
2
∫∞
∫∞
∫∞
I2 = 1−σ ∫ ∞I1 .
L2 α∗ αdG(α) + w1−σ τ 1−σ L1 αX αdG(α)
w L1 α∗ αdG(α) + τ 1−σ L2 αX αdG(α)
2
1
1
2
(15)
The left-hand side of this equation represents country 1’s export volume while the right-handside represents country 2’s export volume. Thus, this equation shows that the trade should
balance between the two countries, which is the ﬁrst equilibrium condition of this extended
model.
The second set of equilibrium conditions are the free-entry conditions for countries 1 and
2. Using (14), we can write the conditions as
w1−σ α1∗ I1[] = w,∫∞∫∞
σ w1−σ L1 α∗ αdG(α) + τ 1−σ L2 αX αdG(α)
1

(16)

2

α2∗ I2[ ∫] = 1.
∫∞∞σ L2 α∗ αdG(α) + w1−σ τ 1−σ L1 αX αdG(α)
2

(17)

1

Whereas these conditions determine the entry cutoﬀs for the two countries, the next two
conditions determine the cutoﬀ qualities for exporting ﬁrms:
ls w−1σ τ 1−σ α1X I2] = wfX ,∫∞∫∞σ L2 α∗ αdG(α) + w1−σ τ 1−σ L1 αX αdG(α)
[

2

σ

1

s 1−σ

[
w1−σ L1

(18)

∫∞
α∗1

lτ

α2X I1αdG(α) + τ 1−σ L] = fX ,2 αX αdG(α)
∫∞

(19)

2

for countries 1 and 2, respectively.
The last two conditions are the labor market clearing conditions:∫∞∫ ∞∫w1−σ L1 α∗ αdG(α)
fX L1 ∞σ−11∫∞dG(α) +dG(α) + 1−σ ∫ ∞
L1
I1
s
1−σ L
l
σ
w
L
X αdG(α)∗ αdG(α) + τ
2
1
α∗1
αX
α
α
1
2
1
∫∞
1−σ 1−σw τ L1 αX αdG(α)
σ−1
1
∫∞
∫∞
+
I2 = L1 ,(20)
L2 α∗ αdG(α) + w1−σ τ 1−σ L1 αX αdG(α) σ
1

2

14

∫ ∞
L2

fX L2
dG(α) +lsα∗2

∫ ∞dG(α) +

L2
∫∞

αX
2

∫∞
L2 α∗ αdG(α)∫∞

2αdG(α) + w1−σ τ 1−σ L1
α∗2

σ−1
I2
αdG(α) σαX∫∞1τ 1−σ L2 αX αdG(α)
σ−1
2
∫∞
∫∞
+
I1 = L2 ,1−σ
1−σw L1 α∗ αdG(α) + τ L2 αX αdG(α) σ
1

(21)

2

for countries 1 and 2, respectively.
We have 7 equations, (15)–(21), and 7 unknowns (α1∗ , α1X , α2∗ , α2X , w, I1 , I2 ). Naturally, it
is rather diﬃcult to analytically solve this simultaneous equation system. Thus, we conduct
some simulations to derive the eﬀects of international trade on income inequality when thetwo countries are asymmetric. In the ﬁrst simulation, we ﬁx the trade cost, τ , and change thepopulation size of country 2. Then we ﬁx the population of the two individual countries andchange the trade cost to see the impact of trade on income inequality within each country.
In both simulations, we select a0 = 1 and k = 5 as the Pareto-distribution parameter values
for both countries. We also select σ = 4, ls = 1, fX = 3, and L1 = 100.
The ﬁrst simulation examines the eﬀect of the diﬀerence in population between the twocountries. We ﬁx τ = 1.2 and change L2 from 0 to 100. We ﬁnd that as L2 decreases from
L1 = 100, α1∗ decreases while α1X increases in country 1, and α2∗ increases while α2X decreases
in country 2. This is intuitive. As the size of country 2 decreases relative to country 1, theper-capita mass of exporting ﬁrms in country 1 decreases and that in country 2 increases.
But this implies that country 2’s market becomes more competitive relative to country 1’s,so the per-capita mass of ﬁrms in country 1 increases while that in country 2 decreases.
We also observe that the wage rate for production workers in country 1 relative to that incountry 2 increases as the diﬀerence in country size expands. The average income in country
1, I1 /L1 , increases while that in country 2, I2 /L2 , remains the same as L2 decreases. We
conﬁrm a typical result in the ﬁeld of the new economic geography that the larger country
has a higher per-capita income in the existence of trade costs.
In the next simulation, we ﬁx L2 = 50 and change τ from 1 to 2. We ﬁnd that as trade
costs decreases from τ = 2 to τ = 1, both α1∗ and α2∗ increase while α1X and α2X decrease
as expected. In addition, both w and I1 /L1 decrease while I2 /L2 remains the same as τ
decreases. Although the larger country still enjoys a higher per-capita income than the15

smaller country, the diﬀerence diminishes as the trade costs falls (when the trade costs are
suﬃciently small).
Figure 6 shows the (nominal) wage schedule of the two countries of asymmetric size
when they trade goods with trade costs. As illustrated, country 2, the smaller country, hasa smaller range of middle-income class. But the top income earners are richer in country
2. Trade beneﬁts top income earners disproportionately especially in country 2 that exports
goods to the larger country. Income inequality caused by trade is greater in the small country
than the large country.

6

Conclusion

We have built a two-country trade model in which the average ability of knowledge workers
hired in a ﬁrm determines the quality of the product that the ﬁrm produces, in order toexamine the impact of international trade on income inequality across workers with diﬀerent
abilities. Knowledge workers are assortatively allocated across ﬁrms in equilibrium, which
entails ﬁrm heterogeneity in terms of product quality. We ﬁnd that international trade will
beneﬁt the ﬁrms that produce the highest qualities, while trade decreases the proﬁts for thosethat barely export their products and those that serve only their individual domestic markets.
Consequently, income inequality within knowledge workers, who earn higher income than
production workers, expands. International trade increases the real wages for top income
earners while decreases those for the middle-income class. The real wages for those in the
low-income class who work as production workers increase thanks to the reduced price index
as a result of increased varieties of products consumed.
In the basic model, we focus on a single dimensional ability (or talent). In reality, however,there is a variety of abilities–the mathematical ability and artistic ability for example–andworkers’ abilities in these diﬀerent dimensions are not perfectly correlated. In the presenceof multiple dimensional abilities, globalization can aﬀect diﬀerent workers diﬀerently. Tradeliberalization raises the relative price for the export good. If a worker has a great talentthat is valued highly in the comparative disadvantage sector, she is likely to be hired in16the comparative disadvantage sector and receive relatively wages. But trade liberalization
reduces her wages, while she would obtain much higher wages if she migrate to the foreigncountry that has a comparative advantage in that sector. Thus, globalization can change
industrial structure signiﬁcantly, and aﬀect workers quite diﬀerently based on their ability
proﬁles.

Appendix
This appendix collects the proofs of Propositions 3, 4 and 5.
Proposition 3. Suppose that there are two symmetric countries in which workers’ ability
distribution follows a Pareto distribution. Then, the lowest income earners who work as
production workers as well as the highest income earners who work as knowledge workers
are better oﬀ by international trade. Those who belong to the middle income class, however,experience a decrease in real wages by opening to trade, if and only if σ > 2 holds. All
knowledge workers who work in ﬁrms that only serve their individual domestic markets
belong to such middle income class.
Proof. Before trade, product quality index (8) is written as
∫ ∞

Lkαk
α dF (α ) = S 0
ℓ
α∗A
′

′

∫ ∞

α−k dα =

α∗A

Lkα0k
∗ k−1
(k − 1)ℓS (αA),

and after trade product quality index is written as
∫ ∞

′

′

α dF (α ) + τ

1−σ

α∗T

=

Lkα0k
(k − 1)ℓS (αT∗ )k−1

∫ ∞
τ σ−1 f X ∗αTℓS

α′ dF (α′ )( X )1−k ]f.
1 + τ k(1−σ)ℓS[∗
Now, we will calculate αAand αT∗ by utilizing FE and LM equations, (6) and (7). First,

17

we start with autarkic equilibrium. Substituting autarkic index into condition FE, we obtain
IA

σℓS
=
∗
αA

∫ ∞

α′ dF (α′ )

α∗A

σℓS
Lkα0k
×
∗
∗ k−1
αA
(k − 1)ℓS (αA
)
( )k
σLk
α0
=
.
∗
(k − 1) αA=

With Pareto distribution, condition LM is written as:
L =
=
=
=
=
=

)
σ−1
ℓ
dF (α ) +IAσα∗A( )k (
)
L α0
σ−1
S
ℓ × S
+
IA
∗
ℓ
αA
σ
( )k ()( )kL α0
σ − 1 σLk α0
S
ℓ × S
+
∗
∗
ℓ
αA
σ
k − 1 αA( )k [
(
)
]
α0
σ−1
σk
L
1+
∗
αA
σ
k−1
( )k []α0kL
1 + (σ − 1)∗αA
k−1( )k ()
α0
σk − 1
L
.
∗
αA
k−1
S

∫ ∞

(

′

Thus, we have

(
∗
αA
= α0

σk − 1
k−1

) k1
.

∗
This implies that αA> α0 holds, since the contents of the parenthesis is positive by noting

σ > 1.
Let us turn to trade equilibrium. Substituting the after trade index into condition FE

18

(12), we obtain
IT

σℓS
=
αT∗

[∫
∞

α′ dF (α′ ) + τ 1−σ

α∗T

∫ ∞

]
α′ dF (α′ )αX[( X )1−k ]
σℓS
Lkα0k
f
=
×
1 + τ k(1−σ)
∗
k−1
∗
αT
ℓS
(k − 1)ℓS (αT )( X )1−k ]( )k [σLkf
α0=
1 + τ k(1−σ)∗(k − 1) αTℓS( )k [( S )k−1 ]
σLk
α0
ℓ
.
=
1 + τ k(1−σ)∗(k − 1) αT
fX

With Pareto distribution, condition LM (13) is written as:
L =
=
=

=
=

)
σ−1
IT
ℓ
dF (α ) + f
dF (α ) +
τ σ−1 f X ∗
σ
α∗T
αT
S
ℓ
( )k
( )k (
)k (
)
L α0
ℓS
σ−1
L α0
S
X
ℓ × S
+f × S
+
IT
ℓ
αT∗
ℓ
αT∗
τ σ−1 f X
σ
( )k
( )k
( S )k−1α0ℓ
L α0k(1−σ)
S
+
L
τ
ℓ × S
ℓ
αT∗
αT∗
fX
(
)
( )k [( S )k−1 ]
σ−1
σLk
α0
ℓ
+
1 + τ k(1−σ)∗σ(k − 1) αT
fX
)
(
(
( )k
( S )k−1 [
)
]
σ−1
α0
ℓ
σk
k(1−σ)
1+
L
1+τ
αT∗
fX
σ
(k − 1)( )k (( S )k−1 ) ()α0
σk − 1ℓk(1−σ)
L
1
+
τ
.
αT∗
fX
k−1
S

∫ ∞

′

Thus, we have(αT∗ = α0

X

(

∫ ∞

′

( S )k−1 ) k1 ()1ℓσk − 1 kk(1−σ)
1+τ
.
fX
k−1

The above calculations show:
∫ ∞
α∗A

Lkα0k

α′ dF (α′ ) =∗ k−1(k − 1)ℓS (αA
)
Lkα0k
=
(
) k−1
k
(k − 1)ℓS α0k−1 σk−1
k−1

19

and
∫ ∞

′

′

α dF (α ) + τ

1−σ

α∗T

=

[

Lkα0k
(k − 1)ℓS (αT∗ )k−1=
(k − 1)ℓS α0k−1∫ ∞τ σ−1 f X ∗αTℓS

α′ dF (α′ )( S )k−1 ]ℓ
1 + τ k(1−σ)
fX
Lkα0k
(

(
1 + τ k(1−σ)ℓSfX)k−1 ) k−1
k (( S )k−1 ]ℓ
1 + τ k(1−σ)
fX

[
σk−1
k−1

) k−1
k

[
( S )k−1 ] k1
Lkα0k
ℓ
=
1 + τ k(1−σ)
k−1
(
)
fX
k
(k − 1)ℓS α0k−1 σk−1
k−1[
( S )k−1 ] k1
∫ ∞
ℓ
.
=
α′ dF (α′ ) × 1 + τ k(1−σ)
fXα∗T
By these explicit formulae, we can ﬁrst conclude
∫ ∞
∫ ∞
∫ ∞
′
′
1−σ
′
′
α dF (α ) + τ
α dF (α ) >α′ dF (α′ ).
σ−1 X
α∗T

τ

f

ℓS

α∗T

α∗A

This implies that international trade increases the real income of consumers whose grosswage rates stay the same. Since unskilled workers’ wages stay the same, the poorest workers
are better oﬀ by international trade.
Finally we can check whether or not the condition for the existence of suﬀering middle
class by trade holds. This condition1
 σ−1
∫ ∞ ′
∫
′
1−σ ∞
′
′
τ σ−1 f X ∗ α dF (α )∗ α dF (α ) + τ
∗
α
αT
αT
T
ℓS

∫∞
>
∗
′
′
αA
α
dF
(α
)
∗
α
A

is satisﬁed if and only if we have(
 1( S )k−1 ) k1
( S )k−1 ) k1 σ−1ℓℓ
1 + τ k(1−σ)>  1 + τ k(1−σ)
.
X
f
fX

(

1
The last inequality is satisﬁed if and only if σ−1< 1, or σ > 2.

Proposition 4. Suppose that there are two symmetric countries in which workers’ ability
distribution follows a Pareto distribution. Then, international trade unambiguously improves
utilitarian social welfare for individual countries.
20

Proof. Since our utility function is homogeneous of degree one, aggregate income adjusted
by quality index describes the country’s utilitarian social welfare:
SW = (

) [∫
σ
σ−1

I
α′ dF (α′ )

∝ [∫
1
] 1−σ

I

] 1 .
α′ dF (α′ ) 1−σ

Thus, trade improves utilitarian social welfare if and only if
IT
<[
1
1 ,
] σ−1
] σ−1
∫∞
∫∞
′
′
α dF (α )
′
′
1−σ
′
′
α∗
τ σ−1 f X ∗ α dF (α )∗ α dF (α ) + τ

[∫
∞

IA

αT

A

or

∫ ∞
α∗T

IT
>
IA

IT
=
IA

αT

 1
∫∞
α′ dF (α′ ) + τ 1−σ τ σ−1 f X α∗ α′ dF (α′ ) σ−1
T
ℓS

∫∞
′
′
α
dF
(α
)
∗
α
A

We know
σLk
k−1

ℓS

(

α0
α∗T

)k [

1 + τ k(1−σ)
( )k

σLk
k−1

( S )k−1 ]
ℓ
fX

α0
α∗A

( S )k−1 ]ℓ=
1 + τ k(1−σ)
fX(( S )k−1 )−1 [( S )k−1 ]ℓℓ=
1 + τ k(1−σ)
1 + τ k(1−σ)
=1
X
f
fX
(

∗
αA
αT∗

)k [

and
∫ ∞


α∗T

 1
∫∞
1
[
( S )k−1 ] k1 × σ−1
α′ dF (α′ ) + τ 1−σ τ σ−1 f X α∗ α′ dF (α′ ) σ−1
ℓ
T
ℓS

∫∞
= 1 + τ k(1−σ).
′′
fX
α dF (α )α∗A

1Since k1 × σ−1> 0, the above inequality is always satisﬁed.

Proposition 5. Suppose that there are two symmetric countries in which workers’ ability
distribution follows a Pareto distribution. Then, the income distribution under international
trade is Lorenz-dominated by the one under autarky.
Proof. In autarkic equilibrium, the wage schedule satisﬁes∗]
1. wA (α) = 1 for all α ∈ [0, αA

21

∗
2. wA (α) = αα∗ for all α ∈ [αA, ∞).
A

In contrast, the after-trade wage schedule satisﬁes
1. wT (α) = 1 for α ∈ [0, αT∗ ],2. wT (α) = αα∗ for α ∈ [αT∗ , αX ],T

3. wT (α) = αα∗ +T[σ

τ 1−σ IT]× (α − αX ) for α ∈ [αX , ∞).
∫′ dF (α′ )+τ 1−σ ∞ α′ dF (α′ ) ℓS
α
α∗
αX

∫∞

T

∗
< αT∗ < αX , there must be ᾱ > αX such that (i)From Proposition 1 we have αAwT (ᾱ) = wA (ᾱ), (ii) wT (α) ≤ wA (α) for all α ≤ ᾱ, and (iii) wT (α) > wA (α) for all α > ᾱ.
∗This implies that the Lorenz function LA (a) = LT (a) for all a ≤ αA
, and LA (a) > LT (a)
∫
∫
∞
∞
∗
for all a > αA
, since 0 wA (a)dG(a) = 0 wT (a)dG(a) due to IT = IA from the proof of

Proposition 4. Thus, the income distribution under international trade is Lorenz-dominatedby the one under autarky.

References[1] Acemoglu, Daron and David Autor (2011), “Skills, Tasks and Technologies: Implications for Employment and Earnings,” Handbook of Labor Economics 4(B), 1043-1171,
Elsevier.
[2] Azevedo, E., and J. Leshno, 2011, A Supply and Demand Framework for Two-Sided
Matching Markets, Working Paper, Harvard University.
[3] Blanchard, Emily and Gerald Willmann (2011), “Trade, Education, and the Shrinking
Middle Class,” unpublished manuscript, Dartmouth College.
[4] Burnstein, A.T., and A. Monge-Naranjo, 2009, Foreign Know-How, Firm Control, andthe Income of Developing Countries, Quarterly Journal of Economics 124, 149-195.
[5] Caliendo, L., and E. Rossi-Hansberg, 2011, The Impact of Trade on Organization and
Productivity, forthcoming in Quarterly Journal of Economics.
22

[6] Costinot, Arnaud and Jonathan Vogel (2010), “Matching and Inequality in the World
Economy,” Journal of Political Economy, 118, 747-786.
[7] The Economist, January 14th-20th 2012.
[8] Helpman, Elehana, Oleg Itskhoki, and Stephen Redding (2010), “Inequality and Unemployment in a Global Economy,” Econometrica, 78, 1239-1283.
[9] Helpman, Elehana, Oleg Itskhoki, Marc-Andreas Muendler, and Stephen Redding(2011), “Trade and Inequality: From Theory to Estimation,” unpublished manuscript,
Harvard University.
[10] Helpman, Elehana, Oleg Itskhoki, and Stephen Redding (forthcoming), “Unequal Effects of Trade on Workers with Diﬀerent Abilities,” Journal of European Economic
Association, Papers and Proceedings.
[11] Krussell, P., L. Ohanian, J.-V. Rios-Rull, and G. Violante, 2000, Capital-Skill Complementarity and Inequality, Econometrica 68, 1026-1053.
[12] Lucas, R.E. Jr., 1978, On the Size Distribution of Business Firms, Bell Journal of
Economics 9, 508-523.
[13] Mas-Colell, A., 1984, ”A Theorem of Schmeidler”, Journal of Mathematical Economics
13, 201-206.
[14] Manasse, Paolo and Alessandro Turrini (2001), “Trade, Wages, and ‘Superstars’,” Journal of International Economics, 54, 97-117.
[15] Melitz, M., 2003, “The Impact of Trade on Intra-Industry Reallocations and Aggregate
Industry Productivity”, Econometrica, 71, 1695-1725.
[16] Yeaple, Stephen Ross (2005), “A Simple Model of Firm Heterogeneity, International
Trade, and Wages”, Journal of International Economics, 65, 1-20.

23

Figure 1. Autarkic equilibrium

Figure 2. Autarkic equilibrium wage scheduleFigure 3. Trade equilibrium

Figure 4. Impact of trade on nominal wages

Figure 5. Impact of trade on real wages

Figure 6. Impact of trade on nominal wages when countries are difference in size