GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.198

Cost-Reducing R&D Investment, Occupational
Choice, and Trade

Tadashi Morita

May 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Cost-Reducing R&D Investment, Occupational
Choice, and Trade ∗Tadashi Morita †Graduate School of Economics, Osaka University
May 12, 2011

Abstract
In this paper, I construct a two-country general equilibrium model in which
oligopolistic ﬁrms export goods and undertake cost-reducing R&D investment. Eachcountry imposes tariﬀs. When the cost of education is suﬃciently high, an increasein the tariﬀ rate decreases the level of R&D investment. However, when the costof education is suﬃciently small, an increase in the tariﬀ rate increases the level of
R&D investment.
Keywords: R&D, Occupational Choice, Trade, Oligopoly
JEL classiﬁcation: F12, F13∗

I am very grateful to Koichi Futagami for his valuable help. I would also like to thank Xiao Chen,
Masahisa Fujita, Taiji Furusawa, Takeo Hori, Jota Ishikawa, Yoshinori Kurokawa, Toshihiro Matsumura,Tomoya Mori, Se-il Mun, Takayuki Ogawa, Rhoji Ohdoi, Ryosuke Okamoto, Makoto Okamura, Yoshifumi Okawa, Yoshiyasu Ono, Tsuyoshi Toshimitsu, and seminar participants at Chuo University, Kwansei
Gakuin University, Kyoto University, Osaka University, and Osaka City University, Osaka Prefecture University, at the 23rd Annual Meeting of the Applied Regional Science Conference at Yamagata University,
the 4th Conference of Macroeconomics for Young Professionals, the 2010 Japanese Economic Association
Spring Meeting at Chiba University, Asia Paciﬁc Trade Seminar (2010), and Hitotsubashi COE Trade
Workshop for Young Researchers. I acknowledge ﬁnancial support from the Research Fellowships for
Young Scientists of the Japan Society for the Promotion of Science.
†
Graduate School of Economics, Osaka University, 1.7 Machikaneyama, Toyonaka, Osaka 560-0043,
JAPAN e-mail: gge013mt@mail2.econ.osaka-u.ac.jp1

1

Introduction

Trade liberalization has been occurring in recent decades. Wacziarg and Welch (2008)
showed that the number of countries having open trade policy increased from 22% of all
countries in 1960 to 46% in 2000. 1 However, over the last two decades, R&D investment
has increased sharply and the wage gap between skilled and unskilled workers has widened
sharply. Using data for the US, according to Braun (2008), the ratio of industrial R&D
expenditures to GDP increased from about 1% in 1979 to 1.43% in 1990 and 1.7% in
2004. Moreover, the wage gap between skilled and unskilled workers has increased inmany countries recently. Acemoglu (2002) pointed out that percentage of US workers
with a college education increased sharply from 6% in 1939 to 28% in 1996. He also
pointed out that in the US, the college premium increased from about 0.4 in 1980 toabout 0.6 in 1995.
On the basis of the above data, this paper has three objectives. First, this paper investigates the relationship between the level of cost-reducing R&D investment and tradeliberalization. When trade liberalization occurs, do ﬁrms increase the level of R&D investment? Few papers investigate the relationship between R&D investment and tradeliberalization empirically. Funk (2003) concluded that US manufacturing ﬁrms that sell
their product to the US market decrease their R&D investment when trade liberalization
occurs. However, US manufacturing ﬁrms with foreign sales increase their R&D investment. Scherer and Huh (1992) showed that average US high-tech ﬁrms reduce their R&D
investment in the short run when trade liberalization occurs.
The second and third objectives of this paper are to consider whether the number ofskilled workers and the wage gap between skilled and unskilled workers increases or not
when trade liberalization occurs. Many researchers have investigated the relationship between the wage gap and trade liberalization. Wood (1994), Leamer (1996) and Kurokawa(2010) argued that there is positive relationship between the wage gap and trade liberalization. However, when trade liberalization occurs, does the wage gap between skilled
and unskilled workers widen and does the number of skilled workers increase?
In this paper, I construct a two-country general equilibrium model in which oligopolistic ﬁrms export goods and undertake cost-reducing R&D investment. The governmentsof the countries impose tariﬀs on imported goods. The ability of individuals is heterogeneous. Individuals choose to become skilled workers by paying the cost of education orremain unskilled workers, which involves no cost. In Braun (2008) and Morita (2009),
there are two types of workers: skilled workers and unskilled workers. However, the numbers of skilled and unskilled workers are exogenously given. In this paper, the number ofskilled workers is determined endogenously through the individual’s choice.
1

A deﬁnition of open trade policy is provided by Sachs and Warner (1995).

2

In this paper, I obtain two results regarding the relationship between the tariﬀ rate
and the level of R&D investment. One is that a decrease in the tariﬀ rate reduces the levelof R&D investment when the cost of education is suﬃciently high. However, a decrease in
the tariﬀ rate raises the level of R&D investment when the cost of education is suﬃciently
low. This paper also shows that a decrease in the tariﬀ rate increases the wage gap between
skilled and unskilled workers when the cost of education is suﬃciently high. When thecost of education is suﬃciently low, the eﬀect of decreasing the tariﬀ rate on the wage gap
is ambiguous. Furthermore, this paper shows that a decrease in the tariﬀ rate increases
the number of skilled workers when the cost of education is suﬃciently high. When thecost of education is suﬃciently low, the relationship between trade liberalization and thenumber of skilled workers is ambiguous.
Many papers have investigated the relationship between trade liberalization and costreducing R&D investment. Braun (2008) and Haaland and Kind (2008) constructed asimple model of international oligopoly. In their papers, consumers are homogeneous
agents. They do not consider the labor market for simpliﬁcation. The result of these
papers illustrates that trade liberalization increases R&D investment. In contrast withthese papers, Morita (2009) constructed a general equilibrium model by incorporating
the labor market into the model of Braun (2008) and Haaland and Kind (2008). I showthat trade liberalization decreases R&D investment. In this paper, the cost of education
determines the eﬀects of trade liberalization on the level of R&D investment. Then, thispaper summarizes the results of these papers.
The remainder of this paper is organized into four sections. The next section presents
the basic structure of the model. Section 3 obtains the equilibrium condition of thismodel. I conclude in Section 4.

2

The model

There are two countries, Home and Foreign, indexed by l ∈ {H, F } and these countries
are symmetric. The population size in each country is equal to L. There are two types
of workers; skilled and unskilled. Individuals choose to become either a skilled or an
unskilled worker. This determines the number of skilled and unskilled workers. There are
two types of goods, X and Y. Good X is chosen to be the numeraire. Good X and good
Y can be produced in both countries. The ﬁrm producing good Y in the Home country
is named Firm H. The ﬁrm producing good Y in the Foreign country is named Firm F. I
assume that Firm H and Firm F compete strategically by using their product quantities,
that is, they engage in Cournot competition. The governments of both countries levy
tariﬀs on their imports of good Y and the tariﬀ rate is denoted by τ .

3

2.1

Individual

The utility function of individual i in each country is given byb 2
U i,l (qi,l , xi,l ) = xi,l + aqi,l − qi,l, l ∈ {H, F } , i ∈ {0, 1} ,2

(1)where xi,l is consumption of good X in country l by individual i, qi,l is consumption ofgood Y, and a and b are positive parameters. The budget constraint of consumer i incountry l is as follows:
xi,l + pl qi,l = Ei,l , l ∈ {H, F } ,(2)where pl is the price of good Y, and Ei,l is expenditure in country l by consumer i. From
the ﬁrst-order condition of the individual, I obtain the following inverse demand function:
pl = a − bqi,l= a − bql ,(3)

where ql denotes the average consumption level of good Y in country l. Therefore, theinverse demand function in country l is as follows:
pl = a − b

Ql
,
L

(4)

where Ql denotes the aggregate consumption level of good Y in country l.

2.2

Occupational choice

I assume that individuals can choose their occupation; skilled or unskilled worker. ildenotes the ability of individual il in country l and il ∈ [0, 1]. Ability is distributed
uniformly over the unit interval. When individual il ∈ [0, γ] becomes a skilled worker, heor she does not have to pay good X. When individual il ∈ [γ, 1] decides to become a skilledworker, he or she has to pay D(il − γ) unit of good X. However, when individual il wants
to become an unskilled worker, he or she pays nothing. Each individual has one unit oflabor and supplies one unit of labor inelasticity. When individual il > γ becomes a skilledworker, his or her income becomes wl − D(il − γ), where wl denotes the wage rate of askilled worker in country l. However, when individual il > γ becomes an unskilled worker,
his or her income is the wage rate of an unskilled worker in country l, wU,l . I assume thatan individual îl is indiﬀerent between becoming a skilled worker and an unskilled worker.
Then, the threshold of ability îl becomes:
wl − D(îl − γ) = wU,l , l ∈ {H, F } .

4(5)

Thus, îl L individuals become skilled workers and (1 − îl )L individuals become unskilledworkers. When the cost of education D is zero, the economy is similar to those of Braun(2008) and Haaland and Kind (2008). On the contrary, when the cost of education D is
inﬁnity, the proportion of skilled workers is γ. Then, this economy is the same as thoseof Morita (2009).

2.3

Production

2.3.1

Good X sector

Production of one unit of good X requires one unit of unskilled workers in both countries.
I assume that perfect competition prevails in the good X market and good X can be
traded freely. Thus, the wage rate for unskilled workers in both countries equals to unity,that is, wU,H = wU,F = 1.
2.3.2

Good Y sector

Each ﬁrm produces good Y and conducts cost-reducing R&D investment to decrease their
marginal cost of production. Production of good Y requires both skilled and unskilledworkers. Production of one unit of good Y requires θ units of skilled workers and α(kl ) ∈[0, ᾱ] units of unskilled workers in country l. kl denotes the number of skilled workers thatis allocated to cost-reducing R&D investment in country l. I assume that ∂α(kl )/∂kl < 0and ∂ 2 α(kl )/∂kl2 ≥ 0. The proﬁt of Firm H is then given by
πH = pH yHH + (pF − τ )yHF − (yHH + yHF )(α(kH ) + θwH ) − wH kH ,(6)

where yt,s denotes the output of ﬁrm t that is sold in country s. Hence, the good Y market
clearing conditions in both countries is as follows:
yHH + yF H = QH ,(7)

yHF + yF F = QF .

(8)

The left hand side of these equations represents the supply of good Y and the right-handside of these equations represents the demand for good Y. Substituting the inverse demand
function, (4), (7), and (8) into the proﬁt function (6), I rewrite the proﬁt function of Firm
H as follows:(
πH =yHH + yF H
a−b
L

)

(

yHF + yF F−τ
yHH + a − b
L

)
yHF

− (yHH + yHF )(θwH + α(kH )) − wH kH . (9)5Firms maximize their proﬁts by simultaneously choosing the quantity of good Y in the twomarkets and the level of cost-reducing R&D investment. Then, the proﬁt maximizationof the ﬁrms leads to the following levels of output and R&D investment:yF H
L[a − θwH − α(kH )] −
,
2b
2

(10)

L
yF F
[a − τ − θwH − α(kH )] −
,
2b
2

(11)

yHH =
yHF =

0

wH = −(yHH + yHF )α (kH ).

(12)

I assume that the ﬁrms take the wage rate of skilled workers and the wage rate of unskilledworkers as being constant. In the same way, the output levels of Firm F are as follows:yF F =

L
yHF[a − θwF − α(kF )] −
,
2b
2

(13)

L
yHH
[a − τ − θwF − α(kF )] −
.
(14)
2b
2
Because I assume that Home and Foreign countries are symmetric and that both ﬁrms
have the same unit cost function, Firms H and F produce the same output level. Thus, thelevel of R&D investment, the wage rate for skilled workers, and the proportion of skilledworkers are the same in both countries: k1 = k2 ≡ k, wH = wS ≡ w, and îH = îF = î.
From (10), (11), (13), and (14), the output levels of Firm H and Firm F are given by
yF H =

L[a + τ − θw − α(k)] ,3b
LyHF = yF H = [a − 2τ − θw − α(k)] .
3byHH = yF F =

(15)
(16)

I assume that the parameter a is suﬃciently large in order that the output levels of theﬁrms have positive values. Because the purpose of this paper is to investigate the eﬀects
of tariﬀs, I focus on the case in which positive amounts of good Y are traded between the
countries.

2.4

Labor market equilibrium conditions

The demand for skilled workers is derived from R&D investment and production of goodY. The demand for unskilled workers comes from production of good X and good Y.
Because the supply of skilled workers is îL and that of unskilled workers is (1 − î)L, the6labor market equilibrium conditions in country H are given by:îL = kH + θ(yHH + yHF ),(17)(1 − î)L = xPH + α(kH )(yHH + yHF ),(18)

where xPH denotes the labor demand for the good X sector in country H.

3

Equilibrium

From (12), (15), and (16), I can obtain the wage rate for skilled workers as follows:0−α (k)L [2a − τ − 2α(k)]w=.
3b − 2θα0 (k)L(19)

From (19), I obtain the output level of Firm H as follows:
yHH + yHF =LA(k, τ )
,
B(k)

(20)
0

where A(k, τ ) ≡ 2a − τ − 2α(k) > 0 and B(k) ≡ 3b − 2θα (k)L > 0.
From (5) and (19), I can obtain the threshold of ability î as follows:0−α (k)LA(k, τ )
1
î =
+γ− .
B(k)D
D

(21)

Inserting (19), (20) and (21) into the skilled worker equilibrium condition (17), I canobtain the excess labor demand function as follows:θL(2a − τ − 2θw − 2α(k))
− îL
3b
0
LA(k, τ )(θD + α (k)L)
Dγ − 1
=k+
− L(
).
B(k)D
D

H(k, τ ) = k +(22)When H(k, τ ) = 0, I can obtain the optimal level of R&D investment. Hereafter, Iθassume that α(k) = ᾱe−k and γ = ᾱLfor simplicity. At k = 0, there is positive excess
labor demand when the cost of skilled labor is relatively high, that is D > ᾱL, and τ < τ1
θ
where
3b + 2θᾱLτ1 = 2(a − ᾱ) −
.
(23)
ᾱL
However, at k = 0, there is negative excess labor demand when the cost of skilled labor
is relatively low, that is D < ᾱL, and τ < τ1 . In addition, the slope of the excess laborθ

7demand function at k = 0 has a negative slope when τ > τ2 where:τ2 = 2(a − ᾱ) +(3b + 2θᾱL) [D(3b + 2θᾱL) + 2ᾱL(θD − ᾱL)].
ᾱL2 (3b + 2θ2 D)(24)

The stability condition of this equilibrium is τ < τ̄ whereτ̄ = 2(a − ᾱ) −2ᾱ(3b + 2θᾱL)
.
3b

(25)

Comparing τ1 with τ2 and τ̄ , I can obtain τ1 < τ̄ < τ2 when 3b > 2ᾱ2 L holds.2 Hereafter,
I focus on τ < τ̄ . Then, I can obtain the following proposition (see Appendix for the
proof).
PROPOSITION 1. Suppose that 3b > 2ᾱ2 L. Then, there exists an unique and a positive
level of R&D investment when D > ᾱLand τ1 < τ < τ̄ and when D < ᾱLand τ < τ1 < τ̄ .
θθThe excess labor demand function of H(k, τ ) can be depicted in Figures 1(a) and 1(b). When D > ᾱLand τ < τ1 in Figure 1(a), the intercept of H(k, τ ) has awhen D > ᾱLθθpositive value. However, when D > ᾱLand τ1 < τ < τ̄ in Figure 1(b), the intercept ofθ
H(k, τ ) has a negative value. When D > ᾱLand τ < τ̄ < τ2 in Figures 1(a) and 1(b), theθslope of H(k, τ ) at k = 0 has a positive value. Therefore, when D > ᾱLand τ1 < τ < τ̄ ,
θthere exists a unique and positive level of R&D investment in Figure 1(b).

H(k, τ)H(k, τ)

k



k



(a) τ < τ1(b) τ1 < τ < τ̄

Figure 1: When D > ᾱLθWhen the cost of education, D, is suﬃciently small, the excess labor demand functionof H(k, τ ) can be depicted in Figures 2(a) and 2(b). When D < ᾱLand τ < τ1 in Figureθ2(a), the intercept of H(k, τ ) has a negative value. However, when D < ᾱLand τ1 < τ < τ̄θᾱLin Figure 2(b), the intercept of H(k, τ ) has a positive value. When D < θ and τ < τ̄ < τ2
in Figures 2(a) and 2(b), the slope of H(k, τ ) at k = 0 has a positive value. Therefore,
2

When this inequality holds, τ1 < τ̄ < τ2 . When this inequality does not hold, τ̄ < τ1 < τ2 . I canobtain the same result whether this inequality holds or not.

8

when D < ᾱLand τ < τ1 , there exists a unique and positive level of R&D investment inθFigure 2(a).

H(k, τ)H(k, τ)kk(a) τ < τ1(b) τ1 < τ < τ̄

Figure 2: D < ᾱLθThe relationship between the tariﬀ rate and the level of R&D investment is given byfollowing proposition (see Appendix for the proof).
PROPOSITION 2. When D > ᾱLand τ > τ1 , a decrease in the tariﬀ rate decreasesθ
R&D investment. When D < ᾱLand τ < τ1 , a decrease in the tariﬀ rate increases R&Dθ
investment.
Figure 3 describes the case when the cost of education is suﬃciently high, that is,
D > ᾱL. Then, a decrease in the tariﬀ rate roatates the excess labor demand function ofθᾱL< 0 in a counterclockwise direction. Therefore, a decrease in the
H(k, τ ) around k = ln θD
tariﬀ rate decreases the level of R&D investment. However, Figure 4 describes the casewhen the cost of education is suﬃciently low. Then, a decrease in the tariﬀ rate roatatesᾱLthe excess labor demand function of H(k, τ ) around k = ln θD> 0 in a counterclockwise
direction. Therefore, a decrease in the tariﬀ rate increases the level of R&D investment.

H(k, τ)
τ↓
_

ln __
αL

θD

k




τ↓

H(k, τ)Figure 3: D > ᾱL
θ

_

ln __
αL

k

θD

Figure 4: D < ᾱLθThe eﬀects of the tariﬀ rate can be divided into three eﬀects: trade eﬀect, wage eﬀect,
and occupational choice eﬀect. Firstly, the trade eﬀect is where a decrease in the tariﬀ rate increases cost-reducing R&D investment. When the tariﬀ rate decreases, both9

ﬁrms increase their output levels and the marginal beneﬁt of R&D investment increases.
Therefore, both ﬁrms increase their level of R&D investment and the labor demand forskilled workers increases. Secondly, the wage eﬀect is where a decrease in the tariﬀ rate
reduces cost-reducing R&D investment. When the tariﬀ rate decreases, both ﬁrms increase their output. Then, the demand for skilled workers increases and the wage rate ofskilled workers increases. An increase in the wage rate of skilled workers raises the cost of
R&D and decreases the level of R&D investment. Finally, the occupational choice eﬀect
is where a decrease in the tariﬀ rate increases cost-reducing R&D investment. When the
tariﬀ rate decreases, both ﬁrms increase their output levels. Then, the demand for skilledworkers increases and the wage rate of skilled workers increases. Consequently, the income of skilled workers increases and the number of skilled workers increases. Then, thenumber of skilled workers hired in R&D activities increases and the demand for skilledlabor increases.
Intuitively, when the cost of education is suﬃciently high, individuals are less likely
to become skilled workers. Then, the number of skilled workers is scarce and the wageeﬀect is suﬃciently high. The wage eﬀect overcomes the sum of the trade eﬀect and theoccupational eﬀect. Therefore, when the cost of education is suﬃciently high, a decreasein the tariﬀ rate decreases the level of R&D investment. When the cost of education
is suﬃciently low, individuals easily becomes skilled workers and the supply of skilledworkers is abundant. Then, the wage eﬀect is suﬃciently small. Therefore, the wageeﬀect is smaller than the sum of the trade eﬀect and the occupational eﬀect. Hence, when
the cost of education is suﬃciently small, a decrease in the tariﬀ rate increases the levelof R&D investment.
The next proposition shows the relationship between the tariﬀ rate and the output
level of good Y (see Appendix for the proof).
PROPOSITION 3. When D < ᾱL, a decrease in the tariﬀ rate increases the outputθlevel.
When the tariﬀ rate decreases, there is a direct eﬀect and an indirect eﬀect. The
direct eﬀect is that when the tariﬀ rate decreases, the cost of exports decreases andthe ﬁrms increase their exports and output levels. The indirect eﬀect is that when the
tariﬀ rate decreases, the level of R&D investment changes. An increase in the level of
R&D investment raises productivity and the output level. A decrease in the level of
R&D investment reduces productivity and the output level. Therefore, when the costof education is suﬃciently low, a decrease in the tariﬀ rate increases the level of R&D
investment as shown in Proposition 2 and increases the output level. However, when thecost of education is suﬃciently high, a decrease in the tariﬀ rate decreases the level of
R&D investment. Then, the direct eﬀect is opposite to the indirect eﬀect. The relationship10between the tariﬀ rate and the output level is ambiguous.
When the tariﬀ rate decreases, does the wage rate of skilled workers increase and thenumber of skilled workers increase? As for the relationship between the tariﬀ rate andthe number of skilled workers, I can obtain the following proposition.
PROPOSITION 4. When D > ᾱL, a decrease in the tariﬀ rate increases the wage gap
θbetween skilled and unskilled workers and increases the number of skilled workers.
Proof. The eﬀect of the tariﬀ rate on the wage rate of skilled workers can be divided into
two parts: direct eﬀect and indirect eﬀect. Diﬀerentiating (19) with respect to τ , I obtainthe following equation:∂w∂w ∂w ∂kᾱe−k L ∂w ∂k
=
+
=−
+
.
∂τ
∂τ
∂k ∂τ
B(k)
∂k ∂τ

(26)

< 0. WhenThe ﬁrst term of (26) has a negative value. From the stability condition, ∂w
∂k
ᾱL
∂k
D > θ holds, ∂τ > 0 as shown in Proposition 2. Then, the second term of (26) has anegative value and ∂w< 0. Therefore, when the cost of education is suﬃciently high, a∂τ
holds,
decrease in the tariﬀ rate increases the wage rate of skilled workers. When D < ᾱLθ∂k< 0 as shown in Proposition 2. Then, the second term of (26) has a positive value and∂τthe sign of ∂w
is ambiguous. Therefore, when the cost of education is suﬃciently low, the∂τrelationship between the tariﬀ rate and the wage rate of skilled workers is ambiguous.
I explain the above proposition intuitively. There is a direct eﬀect and an indirect
eﬀect. The ﬁrst term of (26) represents the direct eﬀect and the second term represents
the indirect eﬀect. The direct eﬀect is when the tariﬀ rate decreases given the level of
R&D investment, the cost of exports decreases, and both ﬁrms increase their volume ofexports and increase their output level. Then, the demand for skilled workers increases.
Hence, the direct eﬀect has a positive eﬀect on the wage rate of skilled workers. However,
the indirect eﬀect is that the level of R&D investment aﬀects the price of good Y . Whenthe cost of education is suﬃciently high, a decrease in the tariﬀ rate decreases the levelof R&D investment. Then, the cost of good Y increases and the relative price of good Y
increases. Then, the demand for skilled workers increases relatively. Then, when the costof education is suﬃciently high, the indirect eﬀect also has a positive eﬀect. Therefore,
when the cost of education is suﬃciently high, a decrease in the tariﬀ rate increases thewage rate of skilled workers. However, when the cost of education is suﬃciently small, adecrease in the tariﬀ rate increases the level of R&D investment and decreases the cost ofgood Y . Then, the relative price of good Y decreases, the demand for unskilled workers
increases, and the wage rate of skilled workers decreases relatively. Therefore, when thecost of education is suﬃciently small, the indirect eﬀect has a negative eﬀect. Then, the11—D>αL/θ∂k/∂τ
∂(output)/∂τ

∂w/∂τ

—
D<αL/θ

τ<τ

—
τ<τ<τ


τ<τ



+

-

?

-

-

?

—
τ<τ<τ




-

Figure 5: The results of proposition 2, 3, and 4.

relationship between the tariﬀ rate and the wage rate of skilled workers is ambiguouswhen the cost of education is suﬃciently small.

4

Conclusion

In this paper, I constructed a two-country general equilibrium model in which the abilityof individuals is heterogeneous and oligopolistic ﬁrms produce goods and undertake costreducing R&D investment. There are two main results. The ﬁrst result is the relationshipbetween trade liberalization and the wage gap. Trade liberalization increases the wage gap
between skilled workers and unskilled workers when the cost of education is suﬃciently
high. When the cost of education is suﬃciently low, the relationship between tradeliberalization and the wage rate of skilled workers is ambiguous.
The second result is the relationship between trade liberalization and the level of R&D
investment. This paper investigated the eﬀects of trade liberalization on R&D investment.
There are three eﬀects: trade eﬀect, wage eﬀect, and occupational choice eﬀect. First, thetrade eﬀect is that a decrease in the tariﬀ rate increases R&D investment. This eﬀect
is focused on by Braun (2008) and Haaland and Kind (2008). They concluded that adecrease in the tariﬀ rate increases R&D investment. Second, the wage eﬀect is that adecrease in the tariﬀ rate decreases R&D investment. When the tariﬀ rate decreases, bothﬁrms increase their output level. Morita (2009) considered these two eﬀects: trade eﬀect
and wage eﬀect. He constructed a general equilibrium model by incorporating the labormarket into the model of Braun (2008) and Haaland and Kind (2008). He concluded thatthe wage eﬀect dominates the trade eﬀect for any tariﬀ rate.
The result of this paper is separated into two cases. First, when the cost of education is
suﬃciently low, the trade eﬀect plus the occupational choice eﬀect dominate the wage eﬀect
and a decrease in the tariﬀ rate increases cost-reducing R&D investment. This case is12similar to Braun (2008) and Haaland and Kind (2008). Second, when the cost of education
is suﬃciently high, the wage eﬀect dominate the trade eﬀect plus the occupational choiceeﬀect and a decrease in the tariﬀ rate decreases cost-reducing R&D investment. This caseis similar to Morita (2009). Therefore, the cost of education determines the eﬀects oftrade liberalization on the level of R&D investment.
Comparing this paper and Morita (2009), this paper provided a long-term analysis
and Morita (2009) provided a short-term analysis. In the short term, it is diﬃcult forworkers to acquire skills. Therefore, in the short term, the ratio of skilled workers tounskilled workers is constant. However, the ratio of skilled workers to unskilled workers is
endogenous. This paper and Morita (2009) concluded that trade liberalization decreases
the level of R&D investment in the short term and increases the level of R&D investmentin the long term.

A

Appendix

A.1

Proof of Proposition 1

This proof can be solved in four steps. In the ﬁrst step, I analyze the value of H(k, τ ) at
k = 0. The Second step investigates the value of H(k, τ ) when k approaches to inﬁnity.
The third step examines the gradient of H(k, τ ) at k = 0. Finaly, the fourth step showsthat the equilibrium is stable when τ ≤ τ3 . In addition, I show that τ1 < τ̄ < τ2 .
First step: Investigating the value of H(0, τ ), I obtain the following lemma.
LEMMA 1. If τ > τ1 and D > ᾱL, there is excess labor supply when k = 0. However,θ
if τ > τ1 and D < ᾱL, there is excess labor demand when k = 0.
θθ
Proof. Remember that I assumed that α(k) = ᾱe−k and γ = ᾱL. The intercept of H(k, τ )is given by:

H(0, τ ) =θD − ᾱL[ᾱL(2a − τ − 2ᾱ) − 3b − 2θᾱL] .
ᾱD(3b + 2θᾱL)(A.1)

When τ > τ1 and D > ᾱL, I obtain:θH(0, τ ) <θD − ᾱL[ᾱL(2a − τ1 − 2ᾱ) − 3b − 2θᾱL] = 0.
ᾱD(3b + 2θᾱL)(A.2)

However, when τ > τ1 and D < ᾱL, the intercept of H(0, τ ) has a positive value.
θ
Second step: The second step investigates the value of H(k, τ ) when k approaches

13

inﬁnity. When k approaches inﬁnity, the limiting value of H(k, τ ) is as follows:()
0
LA(k, τ )(θD + α (k)L)
Dγ − 1
lim H(k, τ ) = lim k +
− L(
)
k→∞
k→∞
B(k)D
D
[
]
L(2a − τ )θD
Dγ − 1= lim k +− L(
) = ∞.
k→∞
3bD
D
Therefore, when R&D investment of k approaches inﬁnity, the excess labor demand increases without bound.
Third step: In the third step, I examine the shape of the excess demand function,
H(k, τ ). Then, I obtain the following lemma.
LEMMA 2. When τ > τ2 , the slope of H(k, τ ) at k = 0 is negative.
Proof. Diﬀerentiating H(k, τ ) with respect to k, I can obtain the following:∂H(k, τ )ᾱe−k Lg(k, τ )=1+,∂kDB(k)2(A.3)

g(k, τ ) ≡ 2(θD − ᾱe−k L)B(k) + LA(k, τ )(3b + 2θ2 D).

(A.4)

where

Diﬀerentiating g(k, τ ) with respect to k, I can obtain the following:∂g(k, τ )= 2ᾱe−k LB(k) − 4(θD − ᾱe−k L)θᾱe−k L + 2L(3b + 2θ2 D)ᾱe−k∂k= 4ᾱe−k L(3b + 2θᾱe−k L) > 0.

(A.5)
(A.6)

Therefore, g(k, τ ) is increasing function of k. Then, when k is suﬃciently small, the signof (A.3) is negative. However, when k is suﬃciently large, the sign of (A.3) is positive.
Then, the sign of (A.3) changes once.
When k = 0 and τ > τ2 , the slope of the excess demand function is given by[DB(0)2 + ᾱL {2(θD − ᾱL)B(0) + L(2a − τ − 2ᾱ)(3b + 2θ2 D)}]
∂H(k, τ )|k=0 =∂k
DB(0)2
2[DB(0) + ᾱL {2(θD − ᾱL)B(0) + L(2a − τ2 − 2ᾱ)(3b + 2θ2 D)}]
= 0.
<
DB(0)2(A.7))|k=0 has a negative value.
Thus, when τ > τ2 , the value of ∂H(k,τ∂kFourth step: In this step, I show that the equilibrium is stable when τ ≤ τ̄ andτ1 < τ̄ < τ2 . To analyze the stability condition, I diﬀerentiate the wage rate with respect
14

to the level of R&D investment.
}
]
∂w
1 [{
−k
−k
−k
−k
−k
−ᾱe
LA(k,
τ
)
+
2ᾱe
Lᾱe
B(k)
−
ᾱe
LA(k,
τ
)(−2θ
ᾱe
L)
=
∂k
B(k)2
ᾱe−k L=−C(k, τ ),
(A.8)
B(k)2
where
C(k, τ ) ≡ 3bA(k, τ ) − 2ᾱe−k B(k).

(A.9)

When the equilibrium is stable, the excess labor demand curve is downward sloping against
the wage rate of skilled workers. Then, because the excess labor demand function is an< 0, that is C(k, τ ) > 0. When theincreasing function, the stability condition is ∂w∂kstability condition is satisﬁed and there exists excess labor demand (supply), the wage
rate of skilled labor increases (decreases). Then, the level of R&D investment decreases(increases). Investigating the sign of C(k, τ ), I obtain the following lemma.
LEMMA 3. When τ ≤ τ̄ , the equilibrium is stable.
Proof. The stability condition is ∂w< 0, that is C(k, τ ) > 0. Then,∂k
C(k, τ ) = 3b(2a − τ 2ᾱe−k ) − 2ᾱe−k (3b + 2θᾱe−k L)≥ 3b(2a − τ − 2ᾱ) − 2ᾱ(3b + 2θᾱL)≥ 3b(2a − τ̄ − 2ᾱ) − 2ᾱ(3b + 2θᾱL) = 0.

(A.10)

Then, when τ < τ̄ , C(k, τ ) has a positive value.
has a negative value and the equilibrium is stable. Next, I
Therefore, when τ < τ̄ , ∂w∂kcompare τ̄ to τ1 and to τ2 .
2ᾱ(3b + 2θᾱL)
3b + 2θᾱL− 2(a − ᾱ) +3bᾱL]3b + 2θᾱL [=3b − 2ᾱ2 L > 0,3bᾱLτ̄ − τ1 = 2(a − ᾱ) −(A.11)

because 3b > 2ᾱ2 L. Then, τ̄ is larger than τ1 .
3b + 2θᾱL(3b + 2θᾱL) [D(3b + 2θᾱL) + 2ᾱL(θD − ᾱL)]
−
2(a
−
ᾱ)
+
ᾱL2 (3b + 2θ2 D)
ᾱL[]
3b + 2θᾱL=
3bD(3b + 2θᾱL) + 6bᾱLθD + 4θᾱ3 L3 > 0.
(A.12)
3bᾱL2 (3b + 2θ2 D)

τ2 − τ̄ = 2(a − ᾱ) +Then, τ2 is larger than τ̄ . Therefore, I can obtain τ1 < τ̄ < τ2 .
Summarizing the above four steps, there exists a unique and positive level of R&D
investment when D > ᾱLand τ1 < τ < τ̄ , and when D < ᾱLand τ < τ1 < τ̄ .
θ
θ
15

A.2

Proof of Proposition 2

Diﬀerentiating H(k, τ ) with respect to τ , I can obtain∂H(k, τ )L(θD − ᾱe−k L) ∂A(k, τ )L(θD − ᾱe−k L)
=
=−
.
∂τ
B(k)D
∂τ
B(k)D

(A.13))ᾱL
ᾱL), ∂H(k,τ
has a negative value. However, when k < log( θD),Then, when k > log( θD∂τ∂H(k,τ )ᾱL
has a positive value. Therefore, when D > θ , a decrease in the tariﬀ rate rotates∂τᾱLthe excess labor demand function around k = ln θD< 0 in a counterclockwise direction.
ᾱL
However, when D < θ , a decrease in the tariﬀ rate rotates the excess labor demandᾱL> 0 in a counterclockwise direction.
function around k = ln θD

A.3

Proof of Proposition 3

Diﬀerentiating (20) with respect to τ , I can obtain:∂(yHH + yHF )2ᾱe−k L∂kL=[B(k) + θLA(k, τ )]
−
.
2
∂τ
B(k)
∂τ
B(k)

(A.14)The second term represents the direct eﬀect and the ﬁrst term represents the indirect
eﬀect when the tariﬀ rate changes. The second term of this derivative has a negativevalue for all values of education cost, D. The value of the ﬁrst term in parentheses has a∂kpositive value because A(k, τ ) > 0 and B(k) > 0. When D < ᾱL, ∂τ
is negative as shownθ∂(yHH +yHF )in Proposition 2. Therefore, the sign ofhas a negative value. Hence, a decrease∂τ
in the tariﬀ rate increases the output level when the cost of education is suﬃciently low.
∂kHowever, when D > ᾱL, ∂τ
is positive from Proposition 2. Then, the ﬁrst term has aθpositive value. Therefore, the sign of ∂(yHH∂τ+yHF ) is ambiguous. Hence, when the cost of
education is suﬃciently high, the relationship between the tariﬀ rate and the output levelis ambiguous.

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16

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