Trade Liberalization, Division of Labor and Welfare under Oligopoly

Kenji Fujiwara i
Keita Kamei ii
Research Group of Economics and Management
No. 2016-E01
2016.8School of Economics, Kwansei Gakuin University
Faculty of Literature and Social Sciences, Yamagata University.
E-mail: kamei@human.kj.yamagata-u.ac.jpiii

Trade Liberalization, Division of Labor andWelfare under Oligopoly
Kenji Fujiwara
School of Economics, Kwansei Gakuin University
Keita Kamei∗
Faculty of Literature and Social Sciences, Yamagata University
July 29, 2016

Abstract
Incorporating explicitly division of labor into a two-country general oligopolistic equilibrium model, we examine the firm productivity
eﬀect of trade liberalization and its welfare implication. We show thata tariﬀ reduction increases the firm productivity of the trading industries but decreases that of the non-trading industries. An expansionof the trading industries, in contrast, decreases the firm productivityof both the trading and non-trading industries. We then find that atariﬀ reduction necessarily reduces welfare while the welfare eﬀect of
expansion of trading industries is ambiguous.
Keywords: General oligopolistic equilibrium, division of labor, firm
productivity, welfare.
JEL Classifications: F10, F12, L25.

∗

Corresponding author: Faculty of Literature and Social Sciences, Yamagata University. E-mail: kamei@human.kj.yamagata-u.ac.jp1

1

Introduction

It is well-recognized in international economics that trade liberalization improves welfare. Traditional theory of comparative advantage tells that the
opening of trade leads to higher welfare by making the world resource allocation more eﬃcient. In addition, the presence of imperfect competition and/or
economies of scale allows us to discover new sources of trade gains. Amongothers, consumers enjoy a larger variety of products under monopolistic competition, and international trade promotes competition in an oligopoly, bothof which yield welfare improvements.1
The developments of theory and empirics of international trade in thiscentury have uncovered further gains from trade. The large body of literature
on firm heterogeneity finds that international trade is welfare-improving because it increases the share of the more eﬃcient firms.2 Besides, Chaney andOssa (2013) oﬀer a new explanation of gains from trade. By introducing division of labor in a vertical production process into Krugman’s (1979) modelof monopolistic competition, they demonstrate that international trade modelled by an increase in market size is gainful by deepening division of labor.3
While Chaney and Ossa (2013) assume away firm heterogeneity for simplicity,
it is naturally expected that the reallocation eﬀect and the firm productivity
eﬀect jointly enhance welfare. Recent empirical studies find that exporting
firms become more productive once they start to export, which the literature
calls ‘learning-by-exporting.’ De Loecker (2007) uses the micro data of manufacturing firms in Slovenia, finding that exporting entrants improve firm
productivity. Using the Japanese manufacturing data, Yashiro and Hirano(2009) show that the firm productivity growth of large trading firms is higher
1See, for example, Feenstra (2015) for the latest account of classical and new trade
theories of gains from trade, and their empirical tests.
2
Chaney and Ossa (2013, p. 177) refer to this eﬀect as a ‘reallocation eﬀect.’ Melitz(2003) is undoubtedly the most influential work in this literature; see Melitz and Redding(2014) for a comprehensive survey of the firm heterogeneity literature.
3
This eﬀect is called a ‘firm productivity eﬀect’ in Chaney and Ossa (2013, p.177).

2than that of non-trading firms. Wagner (2012) surveys the recent empiricalstudies that international trade improves the firm productivity of exporting
firms.
This paper is closely related to the above direction of researches, but we
employ an approach that is quite diﬀerent from the previous works. In order
to seek the eﬀect of trade liberalization on firm productivity from anotherperspective, we combine the formulation of division of labor by Chaney andOssa (2013) with a general oligopolistic equilibrium (GOLE) model developed by Neary (2016).4 Kamei (2014) develops a closed economy version ofthis model, showing welfare losses from a competition policy, i.e. an increasein the number of oligopolistic firms. We extend his model to a two-country
reciprocal market model of Brander and Krugman (1983), and examine twoforms of trade liberalization. The first form is trade liberalization at theintensive margin, which is modelled by a reduction in import tariﬀ. The
second form is trade liberalization at the extensive margin, i.e. an expansion of the share of trading industries (shrink of the share of non-tradingindustries). We show that trade liberalization at the intensive margin raises
the firm productivity of the trading industries, but lowers that of the nontrading industries. We then find that trade liberalization at the extensive
margin lowers the firm productivity of all industries. After demonstratingthese results, we establish the following results on the welfare eﬀect of tradeliberalization. A tariﬀ reduction necessarily reduces welfare, and the welfare
eﬀect of an expansion of the trading industries is ambiguous. While both ofthese welfare eﬀects seem counter-intuitive in view of the existing literature,
we carefully interpret them, and consider why our results are so diﬀerent
from the previous ones. To this end, we compare our results with those of
Bastos and Straume (2012) and Kreickemeier and Meland (2013), both addressing the welfare eﬀect of the above two kinds of trade liberalization in
4

The first version of Neary (2016) was released in 2002. Colacicco (2015) reviews thefundamental working of the GOLE model and some of its applications.

3the segmented market GOLE model.
This paper is organized as follows. Section 2 presents a model. Sections 3
and 4 consider the eﬀect of trade liberalization on the firm productivity andwelfare, respectively. Section 5 concludes. The proof of a few propositions isleft in Appendices.

2

Model

This section presents the model. Suppose two identical countries and a continuum of oligopolistic industries in a unit interval [0, 1]. Firms in industryz ∈ [0, ze] domestically serve and export while firms in industry z ∈ [ze, 1] justsupply domestically. In what follows, we call the former industry a tradingindustry and the latter a non-trading industry. Both countries impose an
ad-valorem tariﬀ t ≥ 0 on their imports of trading goods.
The representative consumer in the Home country solves the following
utility maximization problem.
∫ 1

max

{x(z)}∫ 1

ln x(z)dz

subject to00p(z)x(z) ≤ I,

(1)

where x(z) and p(z) are the consumption and price of Good z, and I is (nominal) national income. Then, the first-order condition for utility maximization is 1/x(z) = λp(z), where λ is a Lagrangean multiplier, which represents
marginal utility of income.
Following Neary (2016), we assume that all oligopolistic firms are ‘large’in their product market, but ‘small’ in the economy as a whole. In otherwords, oligopolistic firms exercise market power in choosing output, but takeeconomy-wide variables such as the wage rate w and national income I as
given. This assumption allows us to set λ = 1 without loss of generality andexpress the demand function as x(z) = 1/p(z). Substituting this into thedirect utility function in (1), indirect utility or welfare W is measured by
W =−

∫ 1
0

ln p(z)dz = −

∫ e
z
0

4

ln p(z)dz −

∫ 1
e
z

ln p(z)dz.

(2)

The formulation of the production side is the same as that of Chaney andOssa (2013) and Kamei (2014). We suppose division of labor under verticalspecialization. In producing one unit of Good z, a sequence of tasks areneeded. The set of such tasks is given by a closed interval [0, 2], and if taskω ∈ [0, ω1 ] is completed, input ω1 is obtained. Then, input ω1 is transformed
into another input ω2 (ω2 > ω1 ) by completing tasks ω ∈ [ω1 , ω2 ]. In order
for a production team to complete each task in [ω1 , ω2 ], the following amountof labor has to be employed.
1 ∫ ω2l(ω1 , ω2 ) =|c − ω|γ dω,2 ω1

γ > 0,

where c ∈ [0, 2] is a core competency. Furthermore, fixed labor f > 0 isneeded to launch each team. A team with a smaller range of tasks leads
to a smaller labor requirement to produce a final good. Thus, if firms arerational, the core competences of teams are placed at equal intervals on the
product line. Summarizing these assumptions, total cost of producing y unitsof output is derived as
(

∫ 1
T

wT f + y
0

)

(

)

yT −γ
ω dω = w T f +
,
1+γ
γ

(3)

where w is the wage rate, and T is the number of teams. From Eq. (3),the firm cannot reduce production cost by organizing an infinite number ofteams because there is a team fixed cost, f , for organizing a team. If thenumber of teams is determined so as to minimize total cost (3), the optimal
number of teams is obtained as
[

γy
T =
(1 + γ)f

] 1

1+γ

.

(4)

Substituting (4) into (3), total cost T C becomes a function of total output
as follows.

[(1 + γ)f
TC = w
γ
5

] γ

1+γ

1

y 1+γ .

(5)

Note here that increasing returns to scale exhibit in all industries from theassumption that γ > 0.
Having formalized the consumption and production sides, we now derive
the equilibrium in the present model. In each oligopolistic industry, n > 1
firms play a quantity-setting Cournot game. Since the demand function ofGood z is x(z) = 1/p(z), the profit of firm i in the trading industry πi (z) isdefined by
πi (z) = p(z)yi (z) +p(z)yi∗ (z)− T C,
1+t

where yi (z) is supply for the domestic market, yi∗ (z) is supply for the exporting market, and the inverse demand and total cost are given by1
p(z) = ∑n,∑n∗i=1 yi (z) +i=1 yi (z)[(1 + γ)f
TC = w
γ

] γ

1+γ

1

[yi (z) + yi∗ (z)] 1+γ .
(6)Similarly, the profit of the non-trading industry is
πi (z) = p(z)yi (z) − T C,
where inverse demand and total cost are[(1 + γ)f
TC = wγ

1,p(z) = ∑ni=1 yi (z)

] γ

1+γ

1

[yi (z)] 1+γ .

(7)Profit maximization gives the familiar first-order condition that marginal
revenue is equal to marginal cost. Given the symmetry among n firms andthe assumption of identical countries, this condition takes the formsγγ(n − 1)y(z) + ny ∗ (z)w[y(z) + y ∗ (z)]− 1+γ f 1+γ=
γ
1n2 [y(z) + y ∗ (z)]2
γ 1+γ (1 + γ) 1+γγ

γ(1 + t)w[y(z) + y ∗ (z)]− 1+γ f 1+γ
ny(z) + (n − 1)y ∗ (z)=,γ
1n2 [y(z) + y ∗ (z)]2
γ 1+γ (1 + γ) 1+γ
for each trading industry. Summing these equations up and solving for total
output y(z) + y ∗ (z), we havey(z) + y ∗ (z) =

γ

1

1+γ

(2n − 1)γ 1+γ (1 + γ) 1+γ γn2 (2 + t)wf 1+γ
6

.

(8)

It follows from (7) that aggregate labor demand in the trading industries
becomes∫ ez[(1 + γ)f
n
γ
0

] γ

|

1+γ

1

[y(z) + y ∗ (z)] 1+γ dz ={z}ze(2n − 1)(1 + γ).
n(2 + t)w

(9)

labor demand per firm
Making a parallel manipulation, the equilibrium output and aggregate
labor demand in the non-trading industries are computed as follows.
y(z) =∫ 1[(1 + γ)f
n
γ
e
z
|

] γ

1+γ

1

[y(z)] 1+γ dz =
{z

}

γ

1

1+γ

(n − 1)γ 1+γ (1 + γ) 1+γ γ

n2 wf 1+γ
(1 − ze) (n − 1)(1 + γ)
.
nw

(10)
(11)

labor demand per firm
Accordingly, denoting the labor endowment by L, the labor marketclearing condition isze(2n − 1)(1 + γ) (1 − ze) (n − 1)(1 + γ)+
= L,n(2 + t)w
nw
which leads to the equilibrium wage rate:w=(1 + γ) [ze(2n − 1) + (1 − ze) (n − 1)(2 + t)].
nL(2 + t)(12)

Utilizing Eqs. (8), (10) and (12), the optimal number of teams in the trading
and non-trading industries is respectively derived as(2n − 1)γLn(1 + γ)f [ze(2n − 1) + (1 − ze) (n − 1)(2 + t)](n − 1)(2 + t)γL,=n(1 + γ)f [ze(2n − 1) + (1 − ze) (n − 1)(2 + t)]

T1 =

(13)

T2

(14)

where T1 and T2 are the optimal number of teams in each trading and nontrading industry, respectively. These expressions of the optimal number offirms, which serve as a proxy of firm productivity, will prove useful in understanding the eﬀect of trade liberalization. This completes the description of
the model. The subsequent sections address the eﬀects of trade liberalizationon the firm productivity and welfare.
7

3

Trade Liberalization and Firm Productivity

This section investigates the eﬀect of trade liberalization on the firm productivity, namely, the optimal number of teams derived just above.5 As
already noted in Introduction, we consider two types of trade liberalization.6
The first is a reduction in import tariﬀ which we call trade liberalization at
the intensive margin. The second is an increase in ze, namely, an exogenous
expansion of the share of the trading industries, and we call it trade liberalization at the extensive margin.7 Before addressing the firm productivity
eﬀect of trade liberalization, we briefly examine the eﬀect of these trade liberalization policies on the equilibrium wage rate.
Proposition 1. Trade liberalization at the intensive margin and the extensive margin raises the equilibrium wage rate.
Proof. Diﬀerentiating (12) with respect to t and ze, we have∂wze(2n − 1)(1 + γ)
=−
< 0,
∂t
nL(2 + t)2∂w(1 + γ)[1 − (n − 1)t]=.
∂ zenL(2 + t)

As proved in Appendix 1, the prohibitive tariﬀ is t = 1/(n − 1), and hence∂w/∂ ze > 0 for any t ∈ [0, 1/(n − 1)]. These signs establish the proposition.
||
The intuition behind this result is straightforward. From Eq. (8), a
tariﬀ reduction induces an increase in total output and labor demand in the
trading industries, thereby raising the wage rate. By contrast, the eﬀect of
5This definition of firm productivity follows Chaney and Ossa (2013) and Kamei (2014).
Bastos and Straume (2012) and Kreickemeier and Meland (2013) also analyze theimpacts of these two scenarios of trade liberalization.
7
These terminologies of trade liberalization follow those of Kreickemeier and Meland(2013). Bastos and Straume (2012) refer to the former as ‘product market integration’and the latter to ‘increased trade openness,’ respectively.
68

increasing ze on the wage rate seems ambiguous because labor demand in the
trading industries increases but labor demand in the non-trading industries
decreases. However, this type of trade liberalization also raises the wage
rate since an increase in labor demand in the trading industries outweighsa decrease in labor demand in the non-trading industries.8 It is instructive
to mention that the same result as Proposition 1 holds even in the absenceof increasing returns to scale; see Propositions 2 and 4 in Kreickemeier andMeland (2013).
At this stage, one may guess that the eﬀect of two types of trade liberalization on the firm productivity is qualitatively the same. But, we will showthat this is not the case. The following result concerns the eﬀect of tariﬀ
reduction on the firm productivity.
Proposition 2. Trade liberalization at the intensive margin raises the firm
productivity in the trading industries, but lowers that in the non-trading industries.
Proof. Diﬀerentiating (13) and (14) with respect to t, we havedT1
(1 − ze) (2n − 1)(n − 1)γL
= −
<0
dt
n(1 + γ)f [ze(2n − 1) + (1 − ze) (n − 1)(2 + t)]2dT2ze(2n − 1)(n − 1)γL
> 0.
=
dt
n(1 + γ)f [ze(2n − 1) + (1 − ze) (n − 1)(2 + t)]2
These inequalities imply the proposition. ||The eﬀect of tariﬀ reduction on the equilibrium wage plays a crucial role
for this result. As shown in Proposition 1, the wage rate rises as a result of
tariﬀ reduction. The resultant higher wage rate gives rise to a higher marginal
cost in the non-trading industries. Consequently, total output of each firm
8This is algebraically confirmed by noting that total output of the trading industries(Eq. (8)) is larger than that of the non-trading industries (Eq. (10)).

9

decreases in the non-trading industries, and so does the firm productivity.
However, in the trading industries, the first-order eﬀect of tariﬀ reduction(the term 2 + t in the right-hand side of Eq. (8)) dominates the second-ordereﬀect through the wage increase. Thus, firms in the trading industries will
expand total output, which, in turn, raises the firm productivity.
While the above result provides a theoretical rationale for the well-known
evidence that trade liberalization improves the productivity the trading industries, the same is not true of trade liberalization at the extensive margin.
This is stated in:
Proposition 3. Trade liberalization at the extensive margin lowers the firm
productivity in both the trading and the non-trading industries.
Proof. Diﬀerentiating (13) and (14) with respect to ze yields
dT1(2n − 1)γL[1 − (n − 1)t]
<0
= −
dze
n(1 + γ)f [ze(2n − 1) + (1 − ze) (n − 1)(2 + t)]2dT2(n − 1)(2 + t)γL[1 − (n − 1)t]
< 0,
= −
dze
n(1 + γ)f [ze(2n − 1) + (1 − ze) (n − 1)(2 + t)]2
which proves the proposition. ||We know from Proposition 1 that an increase in ze leads to a higher
wage rate. Since total output and the firm productivity in all industries arenegatively related to w, this liberalization policy reduces total output andthe firm productivity in all industries. Moreover, Eqs. (13) and (14) allow us
to see that the number of teams decreases more in trading industries than in
non-trading industries. When the increased wage raises the production costand price, consumers do not decrease the consumption of all goods equally.
The reason is that consumers minimize the reduction of consumption of nontrading goods to maintain the utility level.99This is because marginal utility of trading goods is lower than that of non-trading10

We close this section by comparing our results on the firm productivity
eﬀect of trade liberalization with those in Chaney and Ossa (2013) who incorporate the division of labor under vertical specialization into monopolisticcompetition of Krugman (1979).10 They show the market size expansion due
to economic integration reduces all the prices, and raises the firm productivity.11 In contrast, Proposition 2 above suggests that trade liberalization in
the form of tariﬀ reduction necessarily decreases the prices of traded goods,
but increases those of non-traded goods. This asymmetric impacts on product prices lead to an increase in the firm productivity of trading industries
but a decrease in that of non-trading industries. Furthermore, Proposition 3
tells that an expansion of the proportion of trading industries increases theprices of all goods, and decreases the firm productivity of the all industries.

4

WelfareThis section turns to the welfare eﬀects of trade liberalization. Let us begin
by addressing the welfare eﬀect of trade liberalization at the intensive margin, i.e. tariﬀ reduction. This is formally stated in:
Proposition 4. Trade liberalization at the intensive margin necessarily reduces welfare.
Proof. See Appendix 2. ||This result, which has also been shown in Bastos and Straume (2012) andKreickemeier and Meland (2013), is somewhat surprising in view of the gainsfrom-trade result in a partial equilibrium model of Brander and Krugman
goods.
10
Note that all firms sell domestically and export in Chaney and Ossa (2013) since
Krugman (1979) assumes so.
11
The same survives our model; an exogenous increase in L improves the firm productivity of all industries by lowering the wage rate and marginal costs.

11

(1983). The reason for this seemingly counter-intuitive result is as follows.
When a tariﬀ is reduced, marginal cost and the product price of traded goods
fall since the first-order eﬀect of tariﬀ reduction is stronger than the secondorder eﬀect of wage increase. By contrast, the wage increase induced by tariﬀ
reduction leads to higher marginal cost in the non-trading industries. Hence,
tariﬀ reduction expands the price variability across goods, which results inwelfare losses.
In contrast to the above case, it is regrettably impossible to obtain a clear
result on the welfare eﬀect of an increase in ze. That is,
Proposition 5. It is ambiguous whether trade liberalization at the extensive
margin improves welfare.
Proof. See Appendix 3. ||Proposition 3 convinces us that an increase in z̃ decreases the firm productivity of all industries, which has a negative eﬀect on welfare. In addition
to this eﬀect, the price variability among goods plays a crucial role for thewelfare eﬀect. One of the sources of the price variability is the proportion of
trading industries z̃ as Kreickemeier and Meland (2013) highlight. When z̃
is suﬃciently high, incremental z̃ reduces the price variability among goods,
and positively aﬀects welfare. This is confirmed as follows. Relating Proposition 3 to the product prices, an increase in z̃ raises the prices of all goods,
but the change in prices is smaller as z̃ is large. Therefore, if z̃ is suﬃciently large, trade liberalization at the extensive margin reduces the pricevariability across goods. If this eﬀect dominates the firm productivity eﬀect
explained earlier, increased ze turns out welfare-improving.

12

5Concluding Remarks

This paper has explored the eﬀect of trade liberalization on the firm productivity and welfare by allowing for division of labor in a two-country GOLE
framework. We have shown that trade liberalization at the intensive margin,
i.e. a tariﬀ reduction, improves the trading industries’ productivity but lowers the non-trading industries’ productivity. Furthermore, the productivityof both trading and non-trading industries declines as a result of trade liberalization at the extensive margin which is modelled by an expansion of tradingindustries. We have then demonstrated that trade liberalization at the intensive margin necessarily becomes welfare-reducing whereas it is ambiguouswhether trade liberalization at the extensive margin improves welfare.
Our results hopefully provide a new insight on the eﬀect of trade liberalization, but further research is needed. First, we guess that our lossesfrom-trade result is sensitive to the functional forms we assume: logarithmic
utility function and exponential form of total cost. It is conjectured that
other functional forms, e.g. a quadratic utility function, may dampen orreverse our conclusion of losses from trade. Second, following Chaney andOssa (2013), we have assumed away firm heterogeneity a la Melitz (2003). If
the reallocation eﬀect in Melitz (2013) as well as the firm productivity eﬀect
is allowed, the prediction of this paper may be modified. Third, an empirical
study is called for so as to qualify our theoretical results. These agenda areleft as a future direction of researches.

Appendix 1: Derivation of the Prohibitive Tariﬀ
Solving the system of first-order conditions for profit maximization, theCournot equilibrium outputs are solved asy(z) =(1 + nt)A,
2n − 1y ∗ (z) =13[1 − (n − 1)t]A
2n − 1γ

A≡γ

w [y(z) + y ∗ (z)]− 1+γ f 1+γ
γ

1,

γ 1+γ (1 + γ) 1+γ

where y(z) + y ∗ (z) in A is a function of primitive parameters only; see Eq.
(8). Thus, the prohibitive tariﬀ becomes t = 1/(n − 1) by setting y ∗ (z) = 0.

Appendix 2: Proof of Proposition 4
Noting that the product price and output in all the trading and non-tradingindustries become the same, welfare defined in (2) can be simplified to
W = −ze ln p1 − (1 − ze) ln p2 ,(15)

where p1 and p2 are the price of each trading and non-trading good, and are
given as follows.


p1

γ

1

−(1+γ)(2n − 1)γ 1+γ (1 + γ) 1+γ = n−1 γn2 (2 + t)wf 1+γ


p2 = nγ1−(1+γ)1+γ (1 + γ) 1+γ−1  (n − 1)γγ

n2 wf 1+γ

(16)

,

(17)

where w is defined in (12). By making lengthy manipulations, a small increase
in t aﬀects W as follows.
dW
dt

d ln p1
d ln p2− (1 − ze)dtdt
dp1 /dt
dp2 /dt− (1 − ze)
= −ze
p1
p2
ze (1 − ze) (1 + γ)(n − 1)
(1 − ze) ze(1 + γ)(2n − 1)= −+ze(2n − 1) + (1 − ze) (n − 1)(2 + t) (2 + t) [ze(2n − 1) + (1 − ze) (n − 1)(2 + t)]ze (1 − ze) (1 + γ)[1 − (n − 1)t]
> 0.
=
(2 + t) [ze(2n − 1) + (1 − ze) (n − 1)(2 + t)]= −ze

The last inequality uses the assumption of 0 ≤ t ≤ 1/(n − 1). Thus, any
tariﬀ reduction worsens welfare.

14Appendix 3: Proof of Proposition 5
Diﬀerentiating (15) with respect to ze and using the comparative statics resultin Proposition 1 yielddW
d ln p1
d ln p2= − ln p1 + ln p2 − ze
− (1 − ze)
dze
dze
dze
( )
e
e
p2
dp1 /dz
dp2 /dz= ln− ze− (1 − ze)
p1
p1
p2
[
]
2n − 1(1 + γ)[1 − (n − 1)t]= (1 + γ) ln− ze(n − 1)(2 + t)ze(2n − 1) + (1 − ze) (n − 1)(2 + t)
(1 + γ)[1 − (n − 1)t]− (1 − ze)ze(2n − 1) + (1 − ze) (n − 1)(2 + t)[]
2n − 1(1 + γ)[1 − (n − 1)t]−= (1 + γ) ln.
e(n − 1)(2 + t)z (2n − 1) + (1 − ze) (n − 1)(2 + t)
The sign of the right-hand side is unclear because the first term is positivefor 0 < t < 1/(n − 1) but the second term is negative.

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16