Trade, Unemployment, and Reallocation
with Search Frictions∗
Ryo Makioka†
Hitotsubashi University
First Draft: April 2011
This Version: November 2011

Abstract
The impact of international trade on unemployment can be aﬀected by intersectoral labor reallocation as well as intra-sectoral one. To see the eﬀect of trade liberalization on unemployment with both intra-sectoral and inter-sectoral labor movements, we construct a two-country, two-sector model of international trade where
one sector, called a diﬀerentiated-good sector, is characterized by heterogeneousﬁrms and the search frictional labor market. The analysis shows that trade liberalization increases both wages and the labor market tightness in a diﬀerentiated-good
sector through a decrease in hiring costs. These labor market reactions result in job
creation within a diﬀerentiated-good sector which has a positive eﬀect on economywide employment, and labor inﬂow from the outside sector which has a negativeeﬀect. These opposite eﬀects can be totally cancelled out each other. The net
impact of international trade on an economy-wide unemployment rate depends onthe country’s characteristics of the labor market such as matching quality betweenﬁrms and workers and the skill distribution of workers.

JEL Classiﬁcation : F12; F16; J64
Keywords : International trade, Heterogeneous ﬁrms, Labor search, Inter-sectoral labor
reallocation∗

I am very grateful to Prof. Jota Ishikawa and Taiji Furusawa for their considerate guidance. I also
thank Prof. Daisuke Oyama and Keisuke Kawata for valuable comments and suggestions. Financial
support of the Global COE program and the Research Unit for Statistical and Empirical Analysis in
Social Sciences at Hitotsubashi University is gratefully acknowledged.
†
Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi-shi, Tokyo, Japan.
Email: ed112005@g.hit-u.ac.jp

1

Introduction

The impact of trade liberalization on unemployment is a controversial issue in academics
as well as in public debates. Although this issue has attracted a great deal of attention, no
concensus has yet emerged among academics, among public statements, and as a whole.
Articles such as Krugman (1993) and Mussa (1993) claim that trade does not aﬀect therate of unemployment. When we look at empirics, while Attanasio et al. (2004) reveal
no evidence of relationship between trade and the likelihood of unemployment, manystudies show an increase in worker displacement after trade liberalization (Goldberg andPavcnik (2003), Revenga (1997), Menezes-Filho and Muendler (2007)).1 These mixed
insights naturally give rise to the following questions. What is the impact of tradeliberalization on unemployment? How and through which does trade liberalization aﬀect
unemployment? Providing a rational explanation for these questions is important inorder for policy makers not to make a wrong decision based on false perceptions.
Toward these questions, many recent studies are based on the Melitz (2003) type heterogeneous ﬁrm trade model. By introducing ﬁrm heterogeneity, these can explain employment expansion or shrinkage through intra-sectoral labor reallocation without labor
movement across sectors (Egger and Krieckmeier 2009, Felbermayr, Prat, and Schmerer
2008, 2011, Janiak 2006). However, trade liberalization aﬀects inter-sectoral labor reallocation as well as intra-sectoral labor reallocation, which is supported by Kambourov(2009). In other words, one sector analyses of the impact of trade liberalization on unemployment seem to be inadequate.
Based on these background, we construct a two-country, two-sector model of international trade where one sector is characterized by heterogeneous ﬁrms and the search
frictional labor market. Because of our two-sector setting where labor markets are also
separated like the Harris-Todaro model, trade liberalization leads to both intra-sectoral
1

Public debate also has both pros and cons of trade liberalization. For instance, a report publishedby a NGO association Policy Matters Ohio shows the NAFTA resulted in 52,265 job losses in Ohio state(http://www.policymattersohio.org/trade.htm, accessed on 12 January 2011) and disagrees with tradeliberalization. On the other hand, Egypt’s Ambassador has an optimistic viewof the impact of trade on
unemployment, saying: “The question is why all of a sudden, when third world labor has proved to becompetitive, why do industrial countries start feeling concerned about our workers?”2and inter-sectoral labor reallocations. In other words, our model can analyze the eﬀectof trade liberalization on unemployment by taking into account both worker compositions across sectors (how many workers are in the search frictional sector) and sectoral
unemployment rates. Under these settings, we obtain the following results.
First, trade liberalization through a reduction of transport costs increases both wages
and the labor market tightness in the diﬀerentiated-good sector. Although this result
itself is the same as Felbermayr et al. (2008, 2011), the mechanism is diﬀerent. In theirpapers, trade liberalization heightens the marginal revenue of hiring additional workers,
encouraging remaining ﬁrms to employ more workers.2 This causes an increase in wages
and the labor market tightness. In our model, on the other hand, the eﬀect on themarginal revenue is cancelled out by an opposite force generated by ﬁercer competitionin the product market. Therefore the marginal revenue of hiring a worker does not change.
Instead, trade liberalization reduces the marginal cost of hiring workers which leads toincreases in wages and the labor market tightness.
Second, trade liberalization does not cause a clear reaction of unemployment, as
the net eﬀect on unemployment depends on the country’s characteristics such as thematching quality between ﬁrms and workers and the skill distribution of workers. This
ambiguity is derived from two competing forces. While trade liberalization creates job in
the diﬀerentiated-good sector (job creation eﬀect), which reﬂects an increase in the labormarket tightness, this job expansion encourages some workers in outside sector to change
their occupations (labor inﬂow eﬀect). These two factors may cancell out. This result isgenerated because of the two-sector framework, and so is in contrast to Felbermayr et al.
(2008, 2011) where higher productivity always creates jobs in one and only one sector.
Within the literature, our model is closely related to the following three papers. Larchand Lechthaler (2009) extend Felbermayr et al. (2008) by incorporating the second sector
which also has frictional labor market and monopolistically competitive product market.
It has two factors (skilled and unskilled workers) according to Bernard, Redding andSchott (2007) in order to examine the eﬀect of trade liberalization on distribution of gains
2

Technically speaking, this is due to their assumption that the price index P is set to one.

3

from trade as well as skill speciﬁc-unemployment rate. While these ample ingredients may
enable them to derive many results, these complications seem to prevent us from intuitive
understanding. Moreover, their calibration methodology also makes it diﬃcult for us totrace the inﬂuence path of trade liberalization on unemployment.
Coşar, Guner and Tybout (2010) also construct a two-sector heterogeneous ﬁrm
model, where one sector has a frictional labor market and the other has a friction-lesslabor market. Although their framework looks like ours, their main focus is on a ﬁrm’s
response to productivity shocks as well as the impact of trade liberalization on the labormarket. In addition, their results are obtained by a calibration method, which makes
it diﬃcult for us to follow the logical relation. On the other hand, our model employs
a simpler two-sector model with analytical solutions, which enables us to keep up with
the inﬂuence path, therefore, contributing to better understanding on the labor market
outcomes.
Helpman and Itskhoki (2007, 2009, 2010) use a similar framework to study the eﬀectof trade liberalization on unemployment. Among these, Helpman and Itskhoki (2007)
has the closest setting: the diﬀerentiated-good sector with a frictional labor market andthe homogeneous product sector with a frictionless labor market.3 An economy-wide
unemployment rate is determined by the product of (1) a sectoral unemployment rate in
the diﬀerentiated sector, and (2) the fraction of job seekers in that sector.
Even though this component of unemployment is the same as ours, the obtained
results of trade liberalization are diﬀerent due to the following points. On the one handin Helpman and Itskhoki (2007), the sectoral labor market tightness is pinned down only
by domestic labor market parameters such as matching quality, costs of vacancy, andmarket frictions, because hiring costs are paid in the unit of homogeneous good. With
this structure, trade liberalization changes only composition of workers across two sectors,
but does not change a sectoral unemployment rate. Since trade liberalization in a more
frictional sector encourages workers to shift into that sector, it expands an economy-wide3

Helpman and Itskihoki (2010) give a similar analysis to their paper in 2007 except for a homogeneous
product sector having frictional labor market. Their paper in 2009 extends the model in 2010 into a
dynamic analysis. Despite of these diﬀerences, all of these obtain the same implications.

4unemployment rate.
On the contrary in our model, hiring costs are paid in the unit of ﬁnal good in thediﬀerentiated-good sector. Hence, trade liberalization has an impact on a sectoral unemplorment rate through a change of the labor market tightness, as well as the proportion ofworkers in the diﬀerentiated-good sector. Since these two eﬀects change an economy-wide
unemployment rate to the opposite directions, the net result is ambiguous depending on
the country’s characteristics. In that sense, it can be said that our paper is testing therobustness of Helpman and Itskhoki (2010).4
The remainder of this paper is as follows: Section 2 gives the setup of the model.
Section 3 also provides the basic structure of sector Y and occupational choice and labormarket equilibrium. After product market equilibrium is exhibited in Section 4, the eﬀectof trade liberalization is supplied in Section 5. The last section concludes.

2

The setup

The present model is based on Felbermayr et al. (2008, 2010) framework, where a sector
has monopolistically competitive product market with ﬁrms diﬀerentiated with respect toproductivity and labor market with search friction (hereafter, sector X). The only thingthat is notably diﬀerent from their work is to include a perfectly competitive homogeneous
good sector (hereafter, sector Y ), and therefore, workers choose their occupations in
consultation with their skills. In addition, we suppose the world is composed of twosymmetric countries, home and foreign, across which intermediate inputs in sector X andﬁnal good Y are traded. Symmetry enables us to focus on home country with omitting
the subscript of the country.

4

Concerning the relationship between international trade and unemployment in the dual sector setting,our model is also related to extensions of Harris and Todaro (1970).

5

2.1

Production in the sector X

The preference of a representative consumer is given by the following quasi-linear form:

U = µlnCX + CY ,µ ∈ (0, 1),(1)where CX is the total amount of consumption of good X, CY is that of good Y . We
assume that the representative consumer has enough income to consume both goods.
While good Y is produced only by labor, good X is produced from continuously
diﬀerentiated intermediate goods, according to the following Blanchard and Giavazzi(2003) type production function:·Z− σ1
QX = (M )[x(j)]

σ−1
σ

σ
¸ σ−1

dj

,

σ>1

(2)

j∈J

where x(j) is the quantity of intermediate variety j, the measure of the set J is the mass
M of available intermediate inputs, and σ denotes the elasticity of substitution between
each variety. This Blanchard and Giavazzi type speciﬁcation enables us to eliminate
scale expansion eﬀect (love of variety eﬀect) due to trade liberalization on production,and hence to focus our attention only on trade-induced-ﬁrm-reallocation-eﬀect developedby Melitz (2003). This can be seen in a symmetric demand case: x(j) = Q/M where Q
denotes an aggregate input demand. By substituting it, we obtain QX = Q, which meansthat the total output is independent of the number of ﬁrms M .5
1
h
i 1−σ
R
−1
1−σ
With the price index of good X deﬁned P ≡ (M )[p(j)] dj, where p(j)j∈Jis the price of a variety j, we obtain the demand for a varietyj:µxH (j) =P
pH (j)

¶σ ³

µ ´
,
PM

(3)

where xH (j) indicates the demand for an intermediate variety j by a domestic ﬁnal good
producer. Similarly, if an intermediate ﬁrm can engage in exporting, it encounters the5This speciﬁcation is also used in Egger and Kreickemeier (2009), Ebell and Haefke (2009), andFelbermayr et al.(2010). The detailed chacracteristics of this fucntion is dealt in these papers.

6following demand by a foreign ﬁnal good producer:µxF (j) =P
pF (j)

¶σ ³

τµ ´
,
PM

(4)

where τ ≥ 1 is the iceberg transport cost to ship one unit of good internationally. In
addition to this variable costs, a ﬁrm serving domestic market has to incur ﬁxed costs
fH per period in the unit of homogeneous good, and a ﬁrm engaging in export activity
incurs additionally fF per preriod in the same way. These ﬁxed costs are supposed tofF ≥ fH so as to focus on the case where within domestically active ﬁrms only a fractionof ﬁrms engage in exporting like Melitz (2003).6
The production function of an intermediate good j simply depends on the amount oflabor inputs h(j) and ﬁrm’s productivity ϕ(j), x(j) = h(j)ϕ(j). As each ﬁrm produces adiﬀerent variety from the others and ﬁrms with the same productivity behave similarly,
we, hereafter, denote each ﬁrm as its productivity ϕ. If a ﬁrm can aﬀord to export its goodto the foreign market, the total output of that ﬁrm has to be optimally allocated so as
to equalize the marginal revenue from home and foreign, which requires pH (ϕ)τ = pF (ϕ)
clearly shown later (in equation (26)).
Given these demand functions, now we derive the revenue function of a ﬁrm:

R(h; ϕ) = P

σ−1σ³ µ ´ σ1 £
¤1
σ−1
σ−1
1 + I(ϕ)τ 1−σ σ (ϕh(ϕ)) σ ≡ D(ϕ)(ϕh(ϕ)) σ ,M(5)where h(ϕ) = hH (ϕ) + I(ϕ)hF (ϕ) is the total amount of employment for a ﬁrm withproductivity ϕ, I(ϕ) is the indicator function which takes one if a ﬁrm engages in export
activity or otherwise zero, and D(ϕ) controls for aggregate variables.
According to Melitz (2003), average productivity ϕ̃ is such that xH (ϕ̃) = QMX : output
of a ﬁrm with average productivity is equal to average output in this economy.

6

More exactly, to focus on the separation case, we have to impose τ σ−1 fF ≥ fH .

7

2.2

Labor search in the sector X

The labor market in sector X is modeled by following Pissarides (2000), being imperfectlycompetitive because of search and matching friction. This frictional matching between àlabor and a ﬁrm is summarized by the constant returns to scale and concave matching
function. By using these properties, the rate of ﬁlling a vacancy for a ﬁrm is denotedby q(θ) where θ denotes the labor market tightness. It is worth noting that the rate isdecreasing in θ, meaning that if the number of vacancies relative to that of unemployedworkers is larger, it is more diﬃcult for a ﬁrm to ﬁnd a worker. On the contrary, the rateof ﬁnding a job for a worker is denoted θq(θ), which is increasing in θ.
In addition, these rates also have the following feaures : limθ→0 q(θ) = limθ→∞ θq(θ) =∞ , and limθ→∞ q(θ) = limθ→0 θq(θ) = 0. These properties mean that as the labor market
tightness is close to zero, matching is quite diﬃcult for a worker, and easy for a ﬁrm. In
the case that the labor market tightenss is close to inﬁnity, the situation is the opposite.
Because of this frictional matching process, ﬁrms are forced to incur hiring costs pervacancy cv in the unit of ﬁnal good X.7 Taking the matching probability into consideracv Ption, a ﬁrm has to pay total hiring costs q(θ)
to ﬁll a vacancy.

In each period, an employed worker encounters job destruction at the probability s,
which consists of two elements. One is match-speciﬁc job destruction, denoted by χ, andthe other is the job destruction as a result of a ﬁrm itself being disrupted, denoted by δ.
The reason of existing δ is that without this kind of ﬁrm destruction, every ﬁrm could
easily pay the entry ﬁxed costs in installments over the inﬁnite periods. Since these two
probabilities are independent, the total job destruction rate for a worker in each period
is deﬁned by s ≡ δ + χ − δχ.

7Justiﬁcation for this assumption is as follows. An intermediate-good ﬁrm faces resource allocation
problem between production and recruiting activity, and if it uses additional resourse for recruiting
activity, it has to give up one unit of production. In that sense, the cost of hiring additional workers is
the price of intermediate-good. In our setting, this cost is paid by the weighted average of the prices ofeach intermediate-good, that is, the price index. By doing so, hiring cost per vacancy does not dependon productivity of a ﬁrm.

8

2.3

Maximization of an intermediate ﬁrm

Knowing or expecting the demand, costs and wages, a ﬁrm maximizes its expected discount value with respect to the number of vacancies posted.
J(h; ϕ) =maxv( ϕ)1[R(h; ϕ) − w(h; ϕ)h(ϕ) − cv Pv(ϕ) − fH − I(ϕ)fF + (1 − δ)J 0 (h0 ; ϕ)] ,1+r
σ−1

s.t. R(h; ϕ) = D(ϕ)(ϕh(ϕ)) σ ,h0 (ϕ) = (1 − χ)h(ϕ) + q(θ)v(ϕ),(6)
where prime denotes the level of a variable in the next period, and w is the wage level
determined, as we will see, by bargaining game.
The ﬁrst order condition for the vacancy posting v is
cv P
∂J 0 (h0 ; ϕ)
= (1 − δ),q(θ)∂h0 (ϕ)(7)

which indicates that a ﬁrm determines the amount of job vacancies so as to equalize
the marginal cost of posting a vacancy in this period with the marginal beneﬁt obtainedin the next period. In other words, if the right hand side is greater than the left handside, a ﬁrm would be better oﬀ by posting more vacancies. Also, by envelop theorem,
diﬀerentiating with respect to h(ϕ) yields·¸∂R(h; ϕ)∂J(h; ϕ)1∂w(h; ϕ)∂J 0 (h0 ; ϕ)=− w(h; ϕ) −h(ϕ) + (1 − δ)(1 − χ)
.
∂h(ϕ)
1+r
∂h(ϕ)
∂h(ϕ)
∂h0 (ϕ)(8)
By combining these two equations, we obtain the optimal employment condition for theintermediate producer,∂w(h; ϕ)cv P
∂R(h; ϕ)= w(h; ϕ) +h(ϕ) + ∆
,
∂h(ϕ)
∂h(ϕ)
q(θ)

(9)

r+s
where ∆ ≡ 1−δ
is a parameter that controls job destruction rates. Clearly, a ﬁrm opti-

9mally equalizes the marginal revenue of hiring a worker with the marginal costs. Diﬀerences from the optimal condition in the usual frictionless labor market are the existence of
second and third term in the right hand side. The former indicates the eﬀect of marginal
employment on the wage level of all employed workers and the latter reﬂects the hiringcosts due to the search and matching friction.
This equation also can be interpreted as the Euler equation of a ﬁrm, which is readily
understood by rewriting as follows:·¸
cv P
1 − δ ∂R(h; ϕ)
∂w(ϕ)
cv P
=
− w(h; ϕ) −h(ϕ) + (1 − χ)
.
q(θ)
1+r
∂h(ϕ)
∂h(ϕ)
q(θ)
The right hand side represents costs of hiring in this period. The left hand side represents
discounted beneﬁt of hiring in this period.

2.4

Wage bargaining

Before introducing wage bargaining structure, we deﬁne the value of a worker in eachstatus. An employed worker has the following value function,

E(h; ϕ) =1[w(ϕ) + sU + (1 − s)E(h; ϕ)] ,
1+r

(10)

where E is the expected discount value of an employed worker, and U is that of an
unemployed worker. Since a wage is assumed to pay at the end of each period, the wageof the current period is also discounted. As we saw, a worker is faced with job destruction
at the rate of s, resulting in being unemployed in the next period. There is no on-the-job
search. The counterpart for an unemployed worker is,U=

1[θq(θ)Ẽ + (1 − θq(θ))U ],
1+r

(11)

where an unemployed worker earns no income in the current period. In the next period,
an unemployed worker obtaining a job at the rate of θq(θ), can earn the expected value

10

¡ R∞
¢
Ẽ = 0 E(ϕ)dG(ϕ) , where G(ϕ) is the productivity distribution deﬁned later.
After matching a ﬁrm with a worker or at the begining of every period while thematching remains, they bargain only about wages according to the Stole and Zwiebel(1996) solution to the bargaining problem. In other words, they determine wage rates so
as to maximize the following equation:

E(h; ϕ) − U =∂J(h; ϕ)
,
∂h(ϕ)

(12)

where E − U is the surplus of the matching for a worker, and ∂J
is the surplus for a ﬁrm∂hby ﬁring the marginal worker. For simplicity, we assume that the bargaining power of aworker is 0.5.8
By solving diﬀerential equation, we can obtain the following wage equation and job
creation equation respectively:9
cv P
cv P
+θ
,
q(θ)
1−δ
cv P
2(σ − 1)ϕ̃P − ∆.
Job creation curve w =2σ − 1q(θ)
Wage curve

w=∆

(13)

The wage equation, increasing in w − θ plane, indicates worker’s and ﬁrm’s optimal wage
setting condition: if the labor market tightness is large, the ﬁrm is willing to pay highcv P
wages above the outside option in order to save high hiring costs q(θ)
which has to be paid

if the ﬁrm could not have been succeed in the bargaining. Without any friction in thelabor market (cv = 0), the wage curve is equalized to zero which is the outside option ofan unemployed worker who commits to this sector. Clearly, the intercept of wage curve
does not change in any case.
On the other hand, the job creation curve, decreasing in w − θ plane, shows a ﬁrm’s
optimal hiring behavior in the frictional labor market, which is a counterpart of thelabor demand curve in the frictionless case. In other words, an increase of the labor
8

Abowd and Allain (1996) examine that worker’s bargaining power in an individual bargaining caseis close to 0.5.
9Derivation of these equations is given in appendix.

11w
2(σ − 1) ~ϕP
2σ − 1

Wage Curve
Job Creation CurveOθ

Figure 2.1: Labor market equilibrium
market tightness leads to a dominance of the marginal costs of hiring over the marginal
revenues, and hence forces ﬁrms to decrease wages so as to satisfy the optimal employment
condition. In the case where there is no search friction, ﬁrms set their wages as w =³´
2(σ−1)
ϕ̃P for any θ.
2σ−1
It is also worth noting that these two equations do not depend on respective ﬁrm’s
productivity ϕ but only on average productivity ϕ̃. Hence, by the intersection of these two
curves, labor market equilibrium (w∗ , θ∗ ) is determined given ϕ̃, as shown in Figure2.1.
Lemma 1 : Given the average productivity ϕ̃, the labor market equilibrium (w∗ , θ∗ )uniequely exists.
Proof. See appendix. ¤Now, we show the eﬀect of the change in average productivity ϕ̃ on the labor market
equilibrium. First, while an increase in average productivity directly leads to an increasein the marginal revenue of hiring a worker, it also indirectly causes a decrease in themarginal revenue through a decrease of the price index P. This latter eﬀect captures
ﬁercer competition in the labor market caused by an increase in average productivity

12

in the product market.10 These two eﬀects are cancelled out in our model because ofa property in the Melitz model, so the intercept of job creation curve does not shift.11
Second, an increase of average productivity leads to a fall of total hiring costs to ﬁll avacancy, because it is paid in the unit of ﬁnal good. This ﬂattens both wage curve andjob creation curve.
With these respective eﬀects, the net eﬀect of average productivity on the labor market
equilibrium is summarized in the following proposition.
Proposition 1 : An increase of average productivity ϕ̃ leads to an increase of the labormarket tightness θ. Moreover, if the elasticity of ﬁrm’s macthing rate with respect
to the labor market tightness is negative but greater than -1, wages also increases.
Proof. See appendix. ¤As shown in the appendix, the restriction concerning the elasticity of ﬁrm’s matchingrate is satisﬁed in the case of the matching function being Cobb-Douglas form.
Even though Felbermayr et al. (2008, 2011) obtain the same results as ours, theirresults rely on the assumption that P = 1. By assuming it, their job creation curves
move upward, slopes of both job creation curve and wage curve being remained, so wages
and the labor market tightness are raised. On the other hand, in our model, an increaseof average productivity leads to a decrease in hiring costs, resulting in increases of wages
and the labor market tightness. Therefore, our mechanism is diﬀerent from Felbermayr
et al. (2008, 2011).

Felbermayr et al. (2008, 2011) set P = 1, and hence they do not capture this ﬁercer competitioneﬀect.
11
In the Melitz model, the price index can be written as P = constant × ϕ̃1 .
10

13

w
2(σ − 1) ~ϕP
2σ − 1OθFigure 2.2: The impact of price index decreased

3

Sector Y , occupational choice, and the labor market equilibrium

As mentioned in the above, sector Y is assumed to have a perfectly competitive productmarket and labor market, and each ﬁrm produces homogeneous good Y directly using
only labor. This production process is done according to constant returns to scale andnormalized as unit labor requirement being one. These simpliﬁcations make it appropriate
to set home’s good Y as numeraire, and therefore wage rate for one unit of labor as asector Y worker in home country is also one. In addition, as good Y is freely traded
across countries, the price of good Y and wage rate is equalized across countries.
This economy is composed of L families, and each familiy has a continuum of workers.
Each worker within a family has the same one unit of labor as a sector X worker anddiﬀerentiated α unit of labor as a sector Y worker. The latter skill is distributed witha cummulative distribution function F (α) (α ∈ [αmin , ∞)). Within these heterogeneous
workers, while a worker with low α chooses to engage in sector X, one with high α chooses
sector Y , which reﬂects a kind of a comparative advantage.12 In other words, a worker
12

Gibbson et al.(2005) empirically shows that the occupational choice process depends on the compar-

14optimally chooses his occupation by comparing his expected values and then the cutoﬀ
skill level is determined at such a point that makes discounted lifetime values indiﬀerent
between occupations.
On the one hand, the discounted value for a worker in sector Y is,
1+r
α,
r

(14)

where α is per period wages for a worker with α unit of labor. On the other hand, thediscounted expected value for a sector X worker is
·
¸
·
¸
1
θq(θ)w
θ cv PU==.
r r + s + θq(θ)
r 1−δ

(15)

This equation explicitly shows that the expected value for unemployed workers is increasing in wages and the probability of matching. Second equality is obtained by substituting
the wage curve, in which the value of a worker depends on average productivity and thelabor market tightness. While an increase of average productivity has a positive eﬀect
itself on the worker’s value, it also has an indirect eﬀect through the change of the labormarket tightness.
In equilibrium, the indiﬀerence condition between two values determines the cutoﬀ
skill level denoted by Ã(θ, P),
·
¸
cv P
θ
Ã(θ, P) =.
1+r 1−δ

(16)

As the skill as a sector Y worker α is distributed with c.d.f. F (α), the fraction of F (Ã)
engages in sector X, and 1 − F (Ã) in sector Y .
The following lemma shows the movement of this cutoﬀ skill level with respect toaverage productivity ϕ̃.
Lemma 2 : The cutoﬀ skill level Ã(θ, P) increases as a result of an increase of average
ative advantage.

15

productivity ϕ̃.
Proof. See appendix. ¤Since an increase in average productivity leads to a rise of wages and the labor market
tightness in sector X, an expected value of engaging in that sector goes up, resulting ininﬂow of some workers into the sector.
In addition to the division of workers into each sector, workers sorted into sector X
have a possibility to be unemployed because of search and matching friction in sector X
labor market. In other words, all workers sorted into sector X are divided into unemployedor employed workers, and some of whom in each category move toward the other category
by a given probability in every period. While a fraction of unemployed workers LU θq(θ)
in a family changes their status into employed workers by matching with ﬁrms, that of
employed workers [F (Ã) − LU ]s becomes unemployed workers, where LU denotes thenumber of unemployed workers in each family.13 In stationary equilibrium, the inﬂow
into unemployed pool has to be equalized to the outﬂow from that pool,

LU (θ, Ã) =F (Ã)s.
θq(θ) + s

(17)

We summarize the determination of occupational choice equilibrium as,
Deﬁnition 1 : Given average productivity ϕ̃, the occupational choice equilibrium ( Ã∗ ,L∗U ) is pinned down by equation (16), and (17).

4

Product market equilibrium

Now we move on to the ﬁrm’s entry and exit process, and the product market equilibrium
in sector X, both of which are similar to Melitz (2003).
The entry process of a ﬁrm is mainly divided into two steps. First, among an inﬁnitely
large number of potential entrants, ﬁrms which incur the entry sunk cost FE in the unit
13

LU can be also interpreted as an economy-wide unemployment rate.

16

of homogeneous good, is allowed to enter the market. Second, after ﬁrms having known
their productivity ϕ which is distributed with c.d.f. G(ϕ), and p.d.f. g(ϕ) (ϕ ∈ (0, ∞)),
only ﬁrms having suﬃciently high productivity to earn positive proﬁts, can actually start
the production by incurring per period ﬁxed costs fH in the unit of homogeneous good,or otherwise exit from the market without starting the production. This cutoﬀ point isdenoted by ϕ∗H , pinned down by the product market equilibrium conditions we will see
later. In addition, a highest productivity fraction of ﬁrms can engage in exporting activity
by incurring additional per period export ﬁxed costs fF in the unit of homogeneous
good. The cutoﬀ productivity between a ﬁrm engaging only in domestic activity and aﬁrm engaging both domestic and exporting activities, is denoted by ϕ∗F , which is also
determined by conditions deﬁned later. The rate of ﬁrms engaging in exporting activity
1−G(ϕ∗ )conditioned by that engaging in domestic activity is deﬁned by ρ = 1−G(ϕ∗F ) .
H

Following these entry processes, the average productivity of active ﬁrms ϕ̃T in homecountry is deﬁned by the following weighted average,
·

¸ 1
¡
¢ σ−1
1
σ−1
−1
σ−1
,
ϕ̃T =
MH ϕ̃H + ρMH (τ ϕ̃F )MH + ρMH(18)

where MH is mass of domestic ﬁrms active in home market. Taking into account thatcountries are symmetric (MH = MF ), MH + ρMH can be interpreted as the total mass ofcompeting ﬁrms in home country, and ϕ̃T also can be regarded as the average productivityof all ﬁrms engaging in home country (all domestic ﬁrms plus some foreign ones).
In equation (18), two average productivities ϕ̃H (ϕ∗H ) and ϕ̃F (ϕ∗F ) are deﬁned as follows:
"

1
ϕ̃H =
1 − G(ϕ∗H )

Z ∞

1
# σ−1

ϕσ−1 g(ϕ)dϕ

"

, and

ϕ∗H

1
ϕ̃F =
1 − G(ϕ∗F )

Z ∞

1
# σ−1

ϕσ−1 g(ϕ)dϕ

,

ϕ∗F

(19)
each of which is strictly increasing function of each cutoﬀ productivity. Therefore to
determine the average productivity ϕ̃T , the two cutoﬀ productivities have to be pinned
down by product market equilibrium conditions.
17

The expected operating proﬁt of a domestic ﬁrm is,ΠH (ϕ) = (1 − δ)∞X

(1 − r − δ)t πH (ϕ) −

t=0

cv P
hH (ϕ) − fH ,
q(θ)

(20)

where the ﬂow proﬁt in each period is

πH (ϕ) = pH (ϕ)xH (ϕ) − whH (ϕ) −cv P
χhH (ϕ) − fH .
q(θ)

(21)

The equation (20) suggests that in the initial period, an entering ﬁrm only pays ﬁxed
costs and hiring costs for the proﬁt from the next period and some ﬁrms are forced to exitby facing constant shocks δ without reaping any proﬁt. Moreover, these two equations
imply that after a ﬁrm employs the optimal number of workers hH at the initial period, it
recruits, from the next period, the exact number of workers it lost by the match speciﬁc
job destruction χ in order to keep their hiring levels being optimal.
Let us consider the determination of domestic cutoﬀ productivity ϕ∗H . At this productivity level, the operating expected proﬁt has to be equalized to zero in order to admit
only proﬁtable ﬁrms in the market. This feature leads to,r + δ cv P+
πH (ϕ̃H ) = hH (ϕ̃H )1 − δ q(θ)

"µ

ϕ̃H
ϕ∗H

¶σ−1

#
1+r
− 1 fH ,1−δ(22)which is called as the domestic zero cutoﬀ proﬁt condition, whose derivation is given in
the appendix. This condition is almost the same as that in Melitz (2003) except for theﬁrst term in the right hand side, which exists due to search frictions in the labor market.
In the same manner, at the exporting cutoﬀ productivity ϕ∗F , we can derive the exporting
zero cutoﬀ proﬁt condition as follows,r + δ cv P+
πF (ϕ̃F ) = hF (ϕ̃F )1 − δ q(θ)

"µ

ϕ̃F
ϕ∗F

¶σ−1

#
1+r
− 1 fF ,1−δ(23)which means that a ﬁrm with this productivity is indiﬀerent between engaging in exporting activity or not.
18

In addition to these ex-post entry equilibrium conditions, ex-ante entry process iscontrolled by the free entry condition,
·
¸
FE
1−δ
cv P
1−δ
cv P
=
πH (ϕ̃H ) −hH (ϕ̃H ) − fH + ρ
πF (ϕ̃F ) −hF (ϕ̃F ) − fF .
1 − G(ϕ∗H )
r+δ
q(θ)
r+δ
q(θ)
(24)
That is, in equilibrium, entry sunk costs FE is equalized to the sum of the expected
proﬁt of engaging only in domestic market (ﬁrst term in the right hand side) plus thatof exporting activity (second term). In other words, if the sum of the expected proﬁts isgreater than the entry sunk costs, more ﬁrms are willing to enter the market until theequality is achieved.
By combining these three conditions, we obtain an equation concerning the relation
between ϕ∗H and ϕ∗F ,"µFE = (1 − G(ϕ∗H ))

ϕ̃H
ϕ∗H

¶σ−1

#

1+r
−1
fH + (1 − G(ϕ∗F ))
r+δ

"µ

ϕ̃F
ϕ∗F

¶σ−1

#
−1

1+r
fF .
r+δ
(25)

With the other equation on the relation between ϕ∗H and ϕ∗F ,
µ
τ

1−σ

ϕ∗F
ϕ∗H

¶σ−1
fH = fF ,(26)

the equilibrium in the product market is determined.14
In the sector X product market, the mass of ﬁrms is determined so as to clear thelabor market,
M[hH (ϕ̃H ) + ρhF (ϕ̃F )] = L(F (Ã) − LU ),
1+ρ

(27)

where M = MH + ρMH is the total mass of eﬀective intermediate ﬁrms in this country.
Even though in the usual heterogeneous ﬁrm model, the mass of ﬁrms plays an importantrole in determining aggregate variables, it is not the case in the current setting becauseof the modeling of the production function.
Deﬁnition 2 : The product market equilibrium (ϕ∗H , ϕ∗F , MH∗ ) is determined by equation
14

Derivation of this equation is given in appendix.

19

(25), (26), and (27).
Here, cutoﬀ productivities ϕ∗H and ϕ∗F are not aﬀected by labor market variables. This
property is called separavility in Felbermayr et al. (2011), and enables us to analyze theeﬀect of trade liberalization on economy-wide unemployment in a simple manner. It is
obtained because revenue, wage payments, and hiring costs for each ﬁrm can be written
in linear form in terms of the amount of employment for each ﬁrm.

5

The eﬀect of trade liberalization

Here, we show the eﬀect of trade liberalization on the labor market outcomes and theoccupational choice outcomes. The scenario of trade liberalization supposes to be thedecrease of iceberg transport costs τ . By Proposition 1 and Lemma 2, it suﬃces toconsider the eﬀect of decreasing τ on the average productivity ϕ̃T .15
To identify the direction of equilibrium average productivity movement after tradeliberalization, it is rewritten as follows,·ϕ̃T = ϕ∗H

1
1+ρ

µ

FE /fH r + δ
fF
+1+ρ
∗
1 − G(ϕH ) 1 + r
fH

¶¸
.

(28)

Then, the following proposition can be obtained.
Proposition 2 : A decrease of iceberg transport cost τ leads to an increase of the average
productivity ϕ̃T , under the pertition condition τ σ−1 fF ≥ fH being satisﬁed.
Proof. See appendix. ¤The rise of average productivity due to trade liberalization reﬂects exits of low productivity ﬁrms out of the product market and a change of exporting strategies for high15

Because of the modeling of production function (Blanchard and Giavazzi type), the price index does
not depend on the mass of ﬁrms, but only on the average productivity. This characteristic is shown in
the appendix.

20

productive ﬁrms. Then as seen in the proposition 1, this change caused by trade liberalization leads to an increase in wages and the labor market tightness. This means that eventhough both hiring expansion by high productivity exporting ﬁrms and hiring shrinkage
by low productive domestic ﬁrms and exiting ﬁrms occur, the former eﬀect dominates as
cv Pa result of hiring costs reduction ( q(θ)↓ ). In other words, labor is successfully reallocatedfrom low productivity ﬁrms to high producitivty ones.
Even though the within-sector labor market reaction mentioned above is intriguing, itthen also leads to a change in the occupational choice decision in the current two-sector
economy. As can be seen in Lamma 2, a rise of wages and the labor market tightness
causes an increase in the cutoﬀ skill level Ã(θ, P), which means that more workers inﬂow
into the sector X due to higher wages and higher probability of matching with a ﬁrm.
Thus, the implication on unemployment after trade liberalization is clearly diﬀerent fromwhat one-sector analysis suggests. In other words, while one sector models only capture
the change of unemployment rates due to job creations or losses within a sector, two
sector models can additionally take into account the change of labor composition acrosssectors.
As deﬁned in the previous section, the rate of unemployment in this economy depends
on how many workers shift to sector X, as determined by the cutoﬀ skill level Ã(θ, P) as
well as on the extent of job creation captured by the labor market tightness θ. As a result,in some cases higher labor market tightness ceteris paribus decreases unemployment insector X (job creation eﬀect), induced expansion of job creation and higher wages leadto some workers to moving from sector Y to sector X (labor inﬂow eﬀect), resulting in
the net unemployment outcome depending on the scale of these two eﬀects. This can beseen in a following equation:h ∂F (Ã) ∂ Ã∂LUs
F (Ã) ∂θq(θ) i
=
−
,
∂ ϕ̃
θq(θ) + s
∂ ϕ̃
θq(θ) + s ∂ ϕ̃∂Ã
| {z }|{z}labor inﬂow eﬀect(29)

job creation eﬀect

where the ﬁrst term in the bracket indicates labor inﬂow eﬀect which leads to an increase21

Hiring costs

P↓
τ↓

ϕ~ ↑
T

Product Market

w↑

and

θ↑

Labor inflow effect

A~ ↑

LU

Job creation effectLabor Market

Figure 5.1: The eﬀect of trade liberalization on unemploymentof unemployment and the second term in the bracket is job creation eﬀect causing a
reduction of unemployment. In addition, we notice from the equation that matchingquality captured by the function q(·) and the skill distribution F (·) should aﬀect the
net eﬀect of trade liberalization on unemployment. For instance, if the workers’ skill
distribution around the cutoﬀ skill level is thick in a country, workers’ movement fromsector Y to sector X is large, and so this labor inﬂow eﬀect should dominate job creationeﬀect in sector X, causing a rise of unemployment rates.
To analyze the net impact of trade liberalization on unemployment, the skill distribution of workers is assumed to be uniformly distributed and the matching function is
Cobb-Douglas form: q(θ) = m̄θ−α1 .
The eﬀect of trade liberalization on unemployment is summarized in the following
proposition, and Figure5.1 illustrates the inﬂuence path for each case.
Proposition 3 : a) Trade liberalization has an impact on economy-wide unemployment
through two channels: (1) labor inﬂow eﬀect, and (2) job creation eﬀect, whose effects operate opposite directions. Net eﬀect depends on the countries characteristics
such as matching quality and worker distribuction.
b) Under the assumption that the matching function is Cobb-Douglas and the skill
distribution is Pareto, economy-wide unemployment decreases if
µ³
¶³
´k
´
s
1
Ã
m̄θ1−α1
1 + m̄θ1−α1 − k− 1 1 + 2(r+s) is negative, and vice versa.
αminProof. See appendix. ¤Our distinctive result is caused by introduction of the outside sector and worker
heterogeneiry. Despite the job creation after trade liberalization, a rise of matching22

probability for a worker leads to the inﬂow of workers into sector X, and thus both eﬀects
attenuate each other. In the case of the Cobb-Douglas matching function and the Pareto
skill distribution, the net impact of trade liberalization on economy-wide unemployment
depends on the eﬃciency of the matching parameter m̄ and α1 , job destruction rate s andr, skill distribution parameter k and αmin , and endogenous variable Ã and θ. In otherwords, if the quality of matching process and the labor market tightness is relatively
high, trade liberalization reduces an economy-wide unemployment rate, whareas if job
destruction rates are high, it increases unemployment rates.
This result is in contrast with Helpman and Itkhoki (2010) where an economy-wide
unemployment rate is determined only by labor inﬂow eﬀect. In their model, a sectoral unemployment rate depend only on the labor market parameters which do notchange even after trade liberalzation and hence an economy-wide unemployment rate isdetermined by how many workers are in the more frictional diﬀerentiated-good sector.
Trade liberalization in a diﬀerentiated-good sector motivates workers to move from low
frictional outside sector to a diﬀerentiated-good sector, resulting in an increase in an
economy-wide unemployment rate. In contrast, the economy-wide unemployment rate in
our model takes into account both labor movements and the change of the sectoral unemployment rate, resulting in ambiguous eﬀect. Moreover, the net outcome is inﬂuenced
by shape of skill distribution in our heterogenous workers setting.

6

Conclusions

The purpose of this paper is to examine the impact of trade liberalization on unemployment in the two-sector situation where one sector has the Melitz type product market
with the search-matching frictional labor market, and the other has perfectly competitivemarkets. The results demonstrate that trade liberalization has generally ambiguous eﬀect
on the rate of economy-wide unemployment, for job creation eﬀect and labor inﬂow eﬀect
operate in the opposite direction. In the case of the Cobb-Douglas matching function
and Pareto skill distribution of workers, we show a suﬃcient condition under which trade23liberalization decreases the economy-wide unemployment rate. That condition is mainlycomposed of labor market parameters and skill distributional parameters.
Although the within-sectoral labor market outcomes are similar to Felbermayr etal. (2008, 2011), our results are obtained by the diﬀerent mechanism. That is, tradeliberalization does not lead to a change of the marginal revenue of hiring, but leads toa reduction of the marginal costs of hiring, resulting in an increase in both wages andthe labor market tightness. In addition, because of our two-sector setting, economy-wide
outcome is aﬀected by labor inﬂow eﬀect. Moreover, our result is also diﬀerent from
Helpman and Itskhoki (2010) because our model takes into account both labor inﬂow
eﬀect and within-sectoral job creation eﬀect in equilibrium.
Despite these results, there are some limitations to our study. First, our results dependon the assumption that hiring costs are paid in the unit of ﬁnal good X. Even though it
is partly justiﬁed by an explanation, we have to endogenize this mechanism. Second, an
economy in our model is composed of two extreme types of sectors in that only one sector
has labor market frictions and the other has no friction. This artiﬁcial situation can bejustiﬁed by the situation of developing countries. Another justiﬁcation is that the model
considers the extreme case of two arbitrary sectors where one sector has a relatively highfrictional labor market and the other a low frictional labor market.
Finally, even though within-sectoral frictions (search and matching friction) in sector
X are highlighted in this model, between-sectoral frictions are not considered. In reality, the latter friction is greater than the former: a worker must encounter uncertainty
and other costs resulting from sector changes which is greater than these in the case ofoccupational changes within a sector. Therefore in the future work, the eﬀect of tradeliberalization on labor movement across sectors might be considered with friction.

24

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25

[12] Fonseca, Raquel, Paloma Lopez-Garcia, and Christpher Pissarides (2001) “Entrepreneurship, Start-up Costs and Employment,” European Economic Review 45
692-705.
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27

Appendix• Derivation of wage curve and job creation curve
By substituting the modiﬁed value function of an employed worker and the ﬁrm’s
envelop condition into Stole and Zweibel bargaining equation, we obtain
∂w
w rU
1 ∂R
= −2 +
+
.
∂h
h
h
h ∂h
Now, we use the method of variation of constants. After solving the homogeneous
equation, we assume the constant term C as C(h):w = C(h)h−2 .

Substituting this into the former equation,
C 0 (h) = rU h−1 +

∂R −1
h .
∂h

Noticing that ∂R= σ−1
D(ϕ)ϕ σ h(ϕ)− σ , C(h), which is obtained by the inte∂h
σ
σ−1

1

0

grateion of C (h), is substituted into w:
1
σ ∂Rw = rU +.
2
2σ − 1 ∂h(A1)Based on the above, we ﬁrst derive the wage equation. By (A1) and ﬁrm’s optimal
employment condition (9), some calculations leads to

w = rU + ∆

cv P
.
q(θ)

By the value function of both unemployed and employed workers, we obtain thewage equation:
w=∆

cv P
cv P
+θ
.
q(θ)
1−δ

28

, a bargaining condiAs for the derivation of job creation curve, substitutions of ∂R∂htion, and pH (ϕ̃) = P into the equation (A1) lead tow=

2(σ − 1)cv P
ϕ̃P − ∆.
2σ − 1q(θ)

This is the job creation curve.
• Proof for Lemma 1
As be mentioned, the wage curve is increasing and the job creation curve decreasing
in θ:
∂w ¯¯
cv P
cv P 0−∆q (θ) > 0,
¯ =
∂θ W
1−δ
[q(θ)]2∂w ¯¯
cv P 0q (θ) < 0.
¯ =∆
∂θ JC
[q(θ)]2
In order to obtain the intersection of these curves, the intercept of wage curve
has to lower than that of job creation curve. While the former is zero, the latter2(σ−1)
ϕ̃P > 0, desirable result being obtained.
2σ−1¤• Proof for Proposition 1
Tottaly diﬀerentiation of the wage curve and the job creation curve generates thefollowing two equation:
·
¸
·
¸
dw
cv
cv
dP
cv P 0cv P dθ= ∆+θ− ∆q (θ) +,dϕ̃q(θ)
1 − δ dϕ̃[q(θ)]2
1 − δ dϕ̃
cv dP
cv P 0 dθ
dw
= −∆
+∆
q (θ) .
dϕ̃q(θ) dϕ̃[q(θ)]2dϕ̃
From these equation, we can obtain,
h

cv
cv θ
2∆ q(θ)
+ 1−δ

i

dP

dϕ̃
dθ
=
> 0.
cv P
c
vP
dϕ̃
2∆ [q(θ)]2 q 0 (θ) − 1−δ

29

(A2)

= −1. Since the second order diﬀerential of matching function
where we use Pϕ̃ dPdϕ̃
is negative and dP
is also negative, the sign of ddθϕ̃ is positive.
dϕ̃
Similarly, as for dw, we conduct total diﬀerentiation,
dϕ̃
·
¸
dw
θ
1
cv dP
=
1+
.
2
0
dϕ̃
2 − q(θ) /[(r + s)q (θ)]²q(θ) 1 − δ dϕ̃where,²q(θ) =

θ dq(θ).
q(θ) dθTherefore, if −1 < ²q(θ) < 0, then ∂w
is positive. In the case of Cobb-Douglas∂ ϕ̃matching function where q(θ) = m̄θ−α1 , this inequality is satisﬁed.
¤• Proof for Lemma 2
Diﬀerentiation of Ã(θ, ϕ̃) with respect to ϕ̃ generates the following equation,iÃ(θ, ϕ̃) h ϕ̃ ∂θ∂ Ã(θ, ϕ̃)
=
−1 .
∂ ϕ̃
ϕ̃
θ ∂ ϕ̃
So, the sign of ∂∂Ãϕ̃ depends on whether ϕ̃θ ∂∂θϕ̃ is greater than one or not.
Now, by equation (A2), and the assumption that the matching function is CobbDouglas form,
h

cv P
cv P
2∆ q(θ)+ θ 1−δ

i

ϕ̃ dP

P dϕ̃
ϕ̃ ∂θ
²θ ≡
=
,
cv P
cv P
θ ∂ ϕ̃
2∆ [q(θ)]2 θq 0 (θ) − θ 1−δ

=

cv P
cv P
+ θ 1−δ2∆ q(θ)
cv P
cv P
0
θ 1−δ
− 2∆ [q(θ)]
2 θq (θ)(A3).

The second equality uses Pϕ̃ dP= −1. Since the matching function is concavedϕ̃
θ dq(θ)(− q(θ)≤ 1), the numerator is greater than the denominator, so ϕ̃θ ∂∂θϕ̃ is greaterdθthan one.
¤30

• Derivation of the zero cutoﬀ proﬁt conditions
By the per period ﬂow proﬁt, we can derive
πH (ϕ̃H ) + fH
hH (ϕ̃H )=.
∗πH (ϕH ) + fH
hH (ϕ∗H )(A4)

From the production function of an intermediate-good ﬁrm and the demand function,hH (ϕ̃H )=hH (ϕ∗H )

µ

ϕ̃H
ϕ∗H

¶σ−1
.

(A5)

Now, at the domestic cutoﬀ productivity ϕ∗H , the operating proﬁt is,¸
cv P∗
hH (ϕH ) + fH .
1 − δ qH (θ)

r+δπH (ϕ∗H ) =·Combining these equations,r + δ cv P
πH (ϕ̃H ) = hH (ϕ̃H )+1 − δ q(θ)

"µ

ϕ̃H
ϕ∗H

¶σ−1

#
1+r
− 1 fH .
1−δ

By the similar way, the exporting zero cutoﬀ proﬁt condition is also obtained.
¤• Derivation of equation (26)
By operating proﬁt from domestic market,
πH (ϕ∗F ) + fH=
πH (ϕ∗H ) + fH

µ

ϕ∗F
ϕ∗H

¶σ−1
,

and, this can be rewritten as follow,µ¶r + δ cv P
hH (ϕ∗F )1 − δ q(θ)¸µ ∗ ¶σ−1 µ
¶·
1−δ
ϕF
cv P
r+δ
1−δ
∗
∗
=
πH (ϕH ) +fH −hH (ϕH ) .
ϕ∗H
1−δ
r+δ
r+δ
q(θ)

πH (ϕ∗H ) + fH −

31

(A6)

Then, substituting the domestic zero cutoﬀ proﬁt condition results in,µπH (ϕ∗H ) + fH −

r+δ
1−δ

¶

cv P
hH (ϕ∗F ) =
q(θ)

µ

ϕ∗F
ϕ∗H

¶σ−1

1+r
fH .
1−δ

(ϕ)
Now, since hhHF (ϕ)= τ σ−1 , we can derive,µπH (ϕ∗F )+fH −

r+δ
1−δ

¶

¶
·
µ
¸
cv P
r + δ cv P
∗
σ−1
∗
∗
hH (ϕF ) = τπF (ϕF ) + fF −hF (ϕF ) .
q(θ)
1 − δ q(θ)

Then, by substituting this into equation (A6), and some calculations lead to,
µ
τ

σ−1

ϕ∗F
ϕ∗H

¶σ−1
fH = fF .

¤• The dependence of price index only on the average productivity
As deﬁned in the above, the price index is,·Z−1
P = (M )

1−σ

p(j)

1
¸ 1−σ

dj

,

j∈J

which can be rewritten as,
1
·
¸ 1−σ
Z ∞
−1
1−σ
P = (M )p(ϕ) µ(ϕ)M dϕ
,

0
g(ϕ)
σ 1
∗
where µ is 1−G(ϕ∗ ) if ϕ ≥ ϕH , and otherwise zero. By substituting p(ϕ) = σ−1 ϕ M C
H

∂w
cv P
where M C = w + ∂h(ϕ)h(ϕ) + ∆ q(θ), we obtain,P=

where ϕ̃ ≡

£R ∞
0

σ 1
M C = p(ϕ̃),σ − 1 ϕ̃¤ 1
ϕσ−1 µ(ϕ)dϕ σ−1 . This clearly depends only on the average produc-

tivity.
¤32• Proof for Proposition 2First, we want to show how the cutoﬀ productivities ϕ∗H and ϕ∗F are changed by
the movement of τ . By diﬀerentiating equation (25) with respect to τ ,dG(ϕ∗H )1+rdk(ϕ∗H ) 1 + rk(ϕ∗H )
fH + (1 − G(ϕ∗H ))
fH
dτ
r+δ
dτ r + δdG(ϕ∗F )1+rdk(ϕ∗F ) 1 + r−k(ϕ∗F )
fF + (1 − G(ϕ∗F ))
fF .
dτ
r+δ
dτ r + δ

0=−Now, sincedG(ϕ∗H )ϕ∗= g(ϕ∗H ) H ,dτ·dτ ∗µ ∗¶ ∗ ¸dG(ϕ∗F )ϕFϕF dϕH∗= g(ϕF )
+
,
dτ
τ
ϕ∗H
dτ
"µ ¶
#
σ−1
ϕ̃
k(ϕ) =−1 ,ϕ∂k(ϕ)g(ϕ)k(ϕ) (σ − 1)[k(ϕ) + 1]=−,
∂ϕ
1 − G(ϕ)ϕ
we obtain,ϕ∗fF τF Ξ(ϕ∗F )
∂ϕ∗H
³ ∗´
< 0,
=−
ϕF
∂τ
∗
Ξ(ϕ
)
f Ξ(ϕ∗ ) + f
∗
H

F

H

ϕH

F

where
Ξ(ϕ∗i ) ≡ (1 − G(ϕ∗i ))k 0 (ϕ∗i ) − g(ϕ∗i )k(ϕ∗i ),(i = H, F ).

Next, the derivative of ϕ∗F with respect to τ is
∂ϕ∗X
∂τ





=

ϕ∗F 
τ

1− ³

ϕH
ϕ∗F

´

1
fH Ξ(ϕ∗H )+1
fF Ξ(ϕ∗F ) > 0.

Given the above results, we can conclude that as long as fF ≥ fH is satisﬁed, ϕ̃T isdecreasing in τ .
¤• Proof for Proposition 3
U
We calculate ∂Lin the case that the skill distribution is Pareto and the matching
∂ ϕ̃

33

function is Cobb-Douglas: F (α) = 1 −

¡α

min

α

¢k

(k > 1) and q(θ) = m̄θ−α1 . Since∂ Ãcv= (1+r)(1−δ)(P ∂∂θϕ̃ + θ Pϕ̃ ), Pϕ̃ ∂P= −1, and ²θ ≡ ϕ̃θ ∂∂θϕ̃ ,
∂ ϕ̃
∂ ϕ̃

sign

n ∂L oU∂ ϕ̃n= sign ²θÃkÃ−k (1 − α1 )m̄θ1−α1
1 − αmin1−km̄θ1−α1
Ã−k
kαmin

!

o
−1 ,

By using (A3), we know thatnsign ²θÃk
1 − αminÃ−k (1 − α1 )m̄θ1−α11−km̄θ1−α1
kαmin
Ã−k

!

o
−1

n ² − 1 1 − αk Ã−ko
1 − α1
θ
min
=sign
−
,
k
²θ
kαmin
Ã−k 1 + s/(m̄θ1−α1 )¶n .µk
1 − αminÃ−k . ³s ´om̄θ1−α1=sign 1−
1+.
1+
k2(r + s)m̄θ1−α1kαmin
Ã−kTherefore,signn ∂L oU∂ ϕ̃n= sign 1 +

Ã

!k
¶
µ
Ã
s
1
m̄θ1−α1 o

.
−
−1
1+
m̄θ1−α1k
αmin
2(r + s)¤34