Kyoto University,
Graduate School of Economics
Research Project Center Discussion Paper Series

Trade Patterns and Non-Scale Growth between Two Countries

Hiroaki Sasaki

Discussion Paper No. E-12-006

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

August 2012

Trade Patterns and Non-Scale Growth between
Two Countries
Hiroaki Sasaki∗

Abstract
This paper builds a two-country, two-sector (manufacturing and agriculture), non-scale
growth model and investigates the relationship between trade patterns and the growthrate of per capita real consumption. If the population growth rate of the home country ishigher than that of the foreign country, the following results are obtained. (1) Under autarky, the growth rate of per capita real consumption is higher in the home country thanin the foreign country. (2) Under free trade, if the home country completely specializes in manufacturing and the foreign country asymptotically completely specializes inagriculture, then the growth rate of the foreign country is higher than that of the homecountry, though this trade pattern is not sustainable in the long run. (3) Under free
trade, if the home country produces both goods and the foreign country asymptoticallycompletely specializes in agriculture, then the growth rates of the home country and theforeign country are equal, and this trade pattern is sustainable in the long run.
Keywords: non-scale growth model; trade patterns; population growth; per capita growth
JEL Classification: F10; F43; O11; O41∗

Graduate School of Economics, Kyoto University. Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan.
E-mail: sasaki@econ.kyoto-u.ac.jp. Phone: +81-(0)75-753-3446

1 Introduction
This paper builds a two-country, two-sector, non-scale growth model and investigates therelationship between trade patterns and economic growth. We investigate how the per capita
growth rate of a country changes depending on the sector in which the country specializes.
Some other studies analyze the relationship between trade patterns and growth.1 Kaneko(2000) builds a growth model with human capital accumulation and shows that the relationship between the terms of trade and growth depends on whether the country specializes in
the consumption goods sector or the investment goods sector. If the home country specializes in the investment goods sector, its growth rate does not depend on the terms of trade. On
the other hand, if the home country specializes in the consumption goods sector, its growthrate does depend on the terms of trade and increases as the terms of trade improve. However,Kaneko’s (2000) model is a small-open-economy model, and hence, the terms of trade are
given exogenously.
Kaneko (2003) builds a two-country, two-sector, AK growth model and endogenizes theterms of trade. He finds that if a country with a lower growth rate than its trade partner underautarky has a comparative advantage in the consumption goods sector, then the country cannarrow or even reverse the growth gap by opening trade.
Felbermayr (2007) describes a situation where a capital-abundant North and a capitalscarce South trade with each other. In his model, the trade pattern is endogenously determined, and he analyzes the situation where the North produces investment goods and theSouth produces consumption goods. The production technology of investment goods is AK
and that of consumption goods is decreasing returns to scale. Along the balanced growthpath, the Southern terms of trade are continuously improving such that even the decreasingreturns-to-scale South can grow at the same rate as the North. Therefore, the South caneliminate the growth gap by opening trade.
The above studies use scale-growth models. That is, population size positively aﬀects
per capita growth. This assumption, however, seems counterfactual. Jones (1995) attempts
to remove the scale eﬀects and presents a non-scale growth model in which the growth rateof output per capita depends positively on the population growth rate and not on the size ofthe population. That is, the higher the population growth rate, the faster the country grows.2
1Wong and Yip (1999) present a small-open-economy, two-sector model of endogenous growth with capital
accumulation and learning by doing and analyze the relationship between economic growth, industrialization,and international trade.
2
For a systematic exposition of scale eﬀects and non-scale growth, see Jones (1999). For more sophisticated non-scale growth models, see also Kortum (1997), Dinopoulos and Thompson (1998), Peretto (1998),
Segerstrom (1998), Young (1998), and Howitt (1999).

1

In this paper, we build a two-country, two-sector, non-scale growth model in which manufacturing has increasing returns to scale and agriculture has constant returns to scale. We
investigate the relationship between trade patterns and the growth and income gaps between
the two countries under free trade in the long run.3
We use the non-scale growth model for two reasons. First, we can obtain sustainable
income per capita growth even though population growth is strictly positive, and second,
we do not need to impose knife-edge conditions on the parameters of the model. To ourknowledge, this model diﬀers from almost all other models in that we explicitly consider
population growth. In addition, existing models belong to the AK class of models, and assuch, impose knife-edge conditions on the production functions.
According to the analysis, we find that the diﬀerence between the population growth
rates of the two countries aﬀects the trade patterns and the relationship between the per capita
growth of the home country (Home, hereafter) and that of the foreign country (Foreign,
hereafter). In this paper, without loss of generality, we assume that the population growthrate of Home is greater than or equal to that of Foreign.
We first discuss the case where Home and Foreign have equal population growth rates.
Under autarky, the per capita consumption growth rates of the two countries are equal. Under
free trade, two trade patterns may be realized in the long run. First, Home diversifies while
Foreign asymptotically completely specializes in agriculture along the balanced growth path
(BGP, hereafter).4 This trade pattern is sustainable, which means that if an economy is located on the BGP, then it continues on the BGP over time. Second, Home completely specializes in manufacturing while Foreign asymptotically completely specializes in agriculture
along the BGP. This trade pattern is also sustainable. If population growth is equal in the twocountries, then under both autarky and free trade, per capita consumption growth is equal.
Next, we discuss the case where the population growth rate is larger in Home than in Foreign. Under autarky, per capita consumption in Home is larger than in Foreign. Under free
trade, two trade patterns may be realized. First, Home diversifies while Foreign asymptotically completely specializes in agriculture along the BGP. This trade pattern is sustainable.
In this case, per capita consumption growth is equal in the two countries and is the sameas per capita consumption growth in Home under autarky. Therefore, Foreign’s growth rate
under free trade is higher than under autarky, and the growth gap under autarky vanishes.
3

Our model is based on the work of Christiaans (2008). He extends Wong and Yip’s (1999) model anddevelops a small-open-economy, non-scale growth model in which agriculture has constant returns to scale
and manufacturing has increasing returns to scale and examines the dynamics as the economy moves toward
the long-run equilibrium.
4
The word “asymptotically” means that the agricultural output converges to zero, but it never vanishes
because we assume that Foreign’s capital stock is strictly positive. See also Christiaans (2008).

2

Second, Home completely specializes in manufacturing while Foreign asymptoticallycompletely specializes in agriculture along the BGP. However, this trade pattern is not sustainable. While Foreign is on the BGP, its per capita consumption growth exceeds that ofHome. Therefore, the growth gap under autarky is reversed. However, as stated above, thistrade pattern eventually becomes unsustainable and shifts to the first trade pattern. At thispoint, the growth rates of per capita consumption in both countries are equal.
The rest of the paper is organized as follows. Section 2 presents the framework of themodel and analyzes the equilibrium under autarky. Section 3 presents a free trade equilibrium corresponding to each trade pattern and investigates whether each trade pattern issustainable. Section 4 compares the growth rates of per capita real consumption under autarky and free trade in both countries. Section 5 concludes the paper.

2 The model
Consider a world that consists of two countries: Home and Foreign. Both countries produce
homogeneous manufactured and agricultural goods. The manufactured good is used for bothconsumption and investment whereas the agricultural good is used only for consumption.

2.1 Production
Firms produce manufactured goods XiM with labor input LiM and capital stock Ki and produce agricultural goods XiA with only labor input LiA . Here, i = 1 and i = 2 denote Homeand Foreign, respectively. Both countries have the same production functions, which are
specified as follows:
XiM = Ai Kiα (LiM )1−α ,where Ai = Kiβ(1)= Kiα+β (LiM )1−α ,XiA = LiA .

0 < α < 1, 0 < β < 1, α + β < 1,

(2)
(3)

Here, Ai in equation (1) represents an externality associated with capital accumulation,
which captures the learning-by-doing eﬀect à la Arrow (1962). Substituting Ai into equation(1), we obtain equation (2), which shows that manufactured goods production has increasing
returns to scale with β corresponding to the extent of the increasing returns. Equation (3)shows that agricultural goods production has constant returns to scale.
Suppose that labor supply is equal to the population and that the population is fully
employed. Moreover, suppose that the population grows at a constant rate ni and the initial
population is unity in each country: Li (t) = LiM (t) + LiA (t) = eni t , ni > 0.
3

Let pi denote the price of manufactured goods relative to agricultural goods. Then, theprofits of manufacturing and agricultural firms are given by πiM = pi XiM − wi LiM − pi ri Ki and
πiA = XiA − wi LiA , respectively, where wi denotes the wage in terms of agricultural goods andri denotes the rental rate of capital.
From the profit-maximizing conditions, we obtain the following relations:pi∂XiM= wi = 1,∂LiM∂XiM= ri with Ai given.
∂Ki

(4)
(5)

From equation (4), we find that the wage is unity as long as agricultural production is positive. We assume a Marshallian externality in deriving equation (5); profit-maximizing firms
regard Ai as exogenously given. Accordingly, firms do not internalize the eﬀect of Ai .

2.2 Consumption
For simplification, we make the classical assumption that wage income and capital income
are entirely devoted to consumption and saving, respectively.5 In the canonical one-sector
Solow model, under the golden rule steady state where per capita consumption is maximized, consumption is equal to the real wage and all profits are saved and invested. Hence,
our assumption has some rationality and can be interpreted as a simple rule of thumb for
consumers with dynamic optimization (Christiaans, 2008). We define real consumption percapita ci as ci = Ci /Li = (CiM )γ (CiA )1−γ /Li , where Ci denotes economy-wide real consumption. In this case, a fraction γ of wage income is spent on CiM and the rest 1 − γ is spent on
CiA .
piCiM = γwi Li ,CiA = (1 − γ)wi Li .

(6)
(7)

Moreover, the following relationship between real investment Ii and saving holds: pi Ii =pi ri Ki . From this equation, we obtain the rate of capital accumulation:
K̇i= ri .
Ki(8)

That is, the rate of capital accumulation is equal to the rental rate of capital. A dot over avariable denotes the time derivative of the variable (e.g., K̇i ≡ dKi /dt).
5

The same assumption is also used in Krugman (1981), which considers a two-country, two-sector, NorthSouth trade and development model.

42.3 Equilibrium under autarky
Under autarky, both goods have to be produced. The market-clearing conditions are as follows: XiM = CiM + Ii and XiA = CiA . Note that wi = 1 under autarky. From the market-clearing
condition for manufactured goods, we obtain pi , which is used to derive each sector’s employment share: LiM /Li = γ and LiA /Li = 1 − γ. Therefore, under autarky, each sector’s
employment share is constant.
Under autarky, the relative price of manufactured goods is given bypi =(γLi )α(1 − α)Kiα+β.

(9)

We now derive the BGP under autarky. Along the BGP, the rate of capital accumulationis constant and equal to the rental rate of capital, which is given from equation (5) by ri =αKiα+β−1 (γLi )1−α . From this, the BGP growth rates of Ki and pi are, respectively, given by
1−αni > 0,1−α−ββg∗pi = −ni < 0,1−α−βg∗Ki =(10)(11)

where g x ≡ ẋ/x denotes the growth rate of a variable x and an asterisk “∗” denotes a BGP
value. The rate of capital accumulation is positive and proportionate to population growth,and the relative price of manufactured goods is decreasing at a constant rate.6

3 Equilibrium under free trade
Suppose that Home and Foreign engage in free trade at time zero. If K1 (0) > K2 (0), thenfrom equation (9), p1 (0) < p2 (0) because L1 (0) = L2 (0) = 1. Thus, if K1 (0) > K2 (0),Home has a comparative advantage in manufactured goods and Foreign has a comparativeadvantage in agricultural goods. In the following analysis, we assume that K1 (0) > K2 (0).
6Considering the BGP growth rate of capital stock, we introduce a new variable, scale-adjusted capitalstock: ki ≡ Ki /Liϕ . The dynamics of the scale-adjusted capital stock are given byk̇i = αγ1−α kiα+β − ϕni ki ,where ϕ ≡

1−α
.
1−α−β

In the steady state, k̇i = 0, from which we obtainki∗ = {[αγ1−α /(ϕni )]} 1−α−β .
1The steady state is stable because dk̇i /ki |ki =ki∗ = −ki∗ [(1 − α − β)αγ1−α (ki∗ )α+β−2 + ϕni ] < 0.

5

It is suﬃcient for our purpose to consider the following four trade patterns from theviewpoint of Home:7
Case 1 : Both countries produce both goods, that is, both countries diversify.
Case 2 : Home diversifies and Foreign completely specializes in agriculture.
Case 3 : Home completely specializes in manufacturing and Foreign completely specializes
in agriculture.
Case 4 : Home completely specializes in manufacturing and Foreign diversifies.

3.1 Equilibrium when both countries diversify—Case 1
The market-clearing conditions for both goods are given by
X1M + X2M = C1M + C2M + I1 + I2 ,(12)

X1A + X2A = C1A + C2A .

(13)

From these, we obtain1

pα =γ(L1 + L2 )
( α+β
).
α+β
1
(1 − α) α K1 α + K2 αEach country’s employment share of the manufacturing sector θiM is given by
(
)
L2
M
γ
1
+
L
L1
,
θ1M ≡ 1 =
α+β
)
(
L1
K2 α
1 + K1
)
(
L1
M
γ
1
+
L
L2
θ2M ≡ 2 =
.
α+β
(
)
L2
K1 α
1 + K2

(14)

(15)

(16)

The rates of capital accumulation in both countries are given byK̇1= αK1α+β−1 (θ1M L1 )1−α ,K1
K̇2= αK2α+β−1 (θ2M L2 )1−α .
K2

7

(17)
(18)

In Sasaki (2011a), a detailed analysis with the use of phase diagrams is conducted under the assumptionthat n1 = n2 = n.

6

First, if n1 = n2 , so that L1 = L2 ,8 then, after enough time has passed, we obtainlim θ1M = 2γ,(19)lim θ2M = 0,

(20)

t→+∞

t→+∞

where γ < 1/2. The manufacturing employment share in Foreign goes to zero, and Foreign
asymptotically completely specializes in agriculture. Hence, Case 1 is not sustainable whenn1 = n2 .
Next, if n1 > n2 , after enough time has passed, we obtainlim θ1M = γ,(21)

lim θ2M = 0.

(22)

t→+∞
t→+∞

In this case too, the manufacturing employment share in Foreign goes to zero, and Foreign
asymptotically completely specializes in agriculture. Hence, Case 1 also is unsustainable
when n1 > n2 .
Figures 1 to 6 show a numerical example in which α = 0.3, β = 0.2, γ = 0.6, K1 (0) = 1.2,K2 (0) = 1, n1 = 0.02, and n2 = 0.01.
[Figures 1 to 6 around here]
The above analysis shows that Case 1 is unsustainable irrespective of the size of population growth.

3.2 Equilibrium when Home diversifies and Foreign specializes in agriculture—
Case 2The market-clearing conditions for both goods are given by
X1M = C1M + C2M + I1 ,X1A + X2A = C1A + C2A .

(23)
(24)

Hence, we obtain1

pα =γ(L1 + L2 )1α+β

(1 − α) α K1 α

.

(25)

In addition, if K1 (0) = K2 (0), the manufacturing employment share in each country is given by θiM = γ,
which is constant. Case 1 is only sustainable in this case. However, the relative prices in both countries underautarky are equal, and therefore trade does not occur.
87The manufacturing employment share in Home is given by()
L2M
θ1 = γ 1 +
.
L1

(26)

If n1 = n2 = n, the manufacturing employment share in Home becomesθ1M = 2γ.

(27)

In this case, we need γ < 1/2 for Case 2 to hold.
On the other hand, if n1 > n2 , we obtainlim θ1M = γ.

(28)

t→+∞

The manufacturing employment share of Home converges to γ.
The growth rate of capital stock is given byK̇1= αγ1−α (L1 + L2 )1−α K1α+β−1 .
K1The rate of change of capital stock growth rate is given by
(1 − α)n + (α + β − 1)gK1
ġK1 

(
)
=
L
L

1
2

gK1 
n1 +
n2 + (α + β − 1)gK1
(1 − α)L1 + L2
L1 + L2(29)

if n1 = n2 = n,if n1 > n2 .

(30)

When n1 > n2 we obtain L1 /(L1 + L2 ) → 1 and L2 /(L1 + L2 ) → 0 after enough time has
passed, and hence,ġK1= (1 − α)n1 + (α + β − 1)gK1 .
g K1

(31)

From this, the BGP growth rate of capital stock and the terms of trade become
1−αn1 > 0,1−α−ββg∗p = −
n1 < 0.
1−α−βg∗K1 =(32)(33)

Along the BGP, the terms of trade continue to decrease at a constant rate.
We now examine the conditions under which Case 2 holds in detail. Following Wongand Yip (1999), we investigate whether or not the trade pattern is sustainable by comparing
the size of the terms of trade and that of the marginal rate of transformation (MRT) of
the production possibilities frontier (PPF) at the corner point where a country completely
specializes in manufacturing.
8

First, the size of the MRT of PPF in Home is given by[K1α+β (θ1M L1 )1−α ] 1−α
dX1A
− M =
.
α+β
dX1
(1 − α)K 1−α
α

(34)

1

Second, substituting θ1M = 1 in equation (34), the size of the MRT at the point where
Home completely specializes in manufacturing is given by
χ̄1 =

L1α
(1 − α)K1α+β

.

(35)

[Figure 7 around here]
For Case 2 to be sustainable, we need p < χ̄1 .9 Rearranging this condition, we obtain
L2 <

1−γ
L1 .
γ

(36)

If n1 = n2 , that is, L1 = L2 , this condition simplifies to γ < 1/2. Therefore, if theexpenditure share of manufactured goods is less than 1/2, Case 2 is sustainable.
If n1 > n2 , then L1 > L2 . If γ < 1/2, we obtain (1 − γ)/γ > 1, and hence, condition (36)holds. If γ > 1/2, we obtain (1 − γ)/γ < 1. In this case, if L1 is suﬃciently large, condition(36) holds. In fact, with n1 > n2 , L1 becomes larger than L2 after some time has passed.
Therefore, if n1 > n2 , Case 2 is sustainable regardless of the size of γ.
Note that in this case, we obtain c1 = c2 because w1 = w2 = 1 and both countries face
the same relative price p.

3.3 Equilibrium when Home specializes in manufacturing and Foreign
specializes in agriculture—Case 3
The market-clearing conditions for both goods are given by
X1M = C1M + C2M + I1 ,(37)

X2A = C1A + C2A .

(38)

With L1M = L1 , we obtain
p=

9

γL2
(1 − α)(1 − γ)K1α+β L11−α.

Appendix A explains why Foreign cannot completely specialize in manufacturing.

9

(39)

The growth rate of capital stock is given byK̇1= αK1α+β−1 L11−α .
(40)
K1The BGP growth rate of Home’s capital stock becomes
1−αg∗K1 =
n1 > 0.
(41)
1−α−βIn addition, from equation (39), the growth rate of the terms of trade is given by

β



n<0
if n1 = n2 = n,−
 1−α−β∗gp = (42)
1−α(1 − α)(n1 − n2 ) + βn2



n
−
n
=
−
<
0
if
n
>
n
.
1
1
2
 2
1−α−β
1−α−β
Hence, along the BGP, the relative price of manufactured goods continues to decrease at aconstant rate.
Case 3 is sustainable if χ̄1 < p, which can be rewritten as
γ
L1 <
L2 .
(43)
1−γ
First, if n1 = n2 , that is, L1 = L2 , this condition simplifies to γ > 1/2. Therefore, if
the population growth rates in both countries are equal, Case 3 is sustainable as long as theexpenditure share of manufactured goods is more than 1/2.
Second, we consider the case where n1 > n2 , that is, L1 > L2 . If γ < 1/2, we obtain0 < γ/(1 − γ) < 1, and hence, condition (43) is not satisfied. If γ > 1/2, we obtainγ/(1 − γ) > 1, and hence, condition (43) can be satisfied even though L1 > L2 . However, as
the time passes, L1 becomes suﬃciently larger than L2 and condition (43) will be violated.
Thus, if the population growth rate is larger in Home than in Foreign, Case 3 is sustainable
as long as the expenditure share for manufactured goods is greater than 1/2, but it will be
unsustainable in the long run. Once Case 3 becomes unsustainable, it will switch to Case
2.10
Note that in this case, we obtain c1 > c2 because w1 = [γ/(1 − γ)] · (L2 /L2 ) > 1 > w2 = 1
and both countries face the same relative price p.

3.4 Equilibrium when Home specializes in manufacturing and Foreign
diversifies—Case 4
The market-clearing conditions for both goods are given by
X1M + X2M = C1M + C2M + I1 + I2 ,X2A = C1A + C2A .
10

Appendix B explains why Case 3 switches to Case 2.

10

(44)
(45)

From these equations, we find that the terms of trade satisfy the following equation:α+β(1 − α) α p α K2 α = γL2 − (1 − α)(1 − γ)pK1α+β L11−α .
1

1

(46)

From this equation, p is implicitly and uniquely determined, and hence, p is a function ofK1 , K2 , L1 , and L2 : p = p(K1 , K2 , L1 , L2 ).11
The growth rate of the capital stock in each country is given byK̇1= αK1α+β−1 L11−α ,
K1
β
K̇2
1−α
1−α
= α(1 − α) α p α K2α ,
K2

(47)
(48)

where p is endogenously determined by equation (46).
The employment share of manufacturing in Foreign is given by
1

1

α+β

(1 − α) α p α K2 α
.
θ2M =
L2

(49)

In this case, analytical solutions are hard to obtain, so we conduct numerical simulations.
Using equation (46), we obtain the time derivative of p as follows:
1

ṗ =

β

γ L̇2 − (1 − α)2 (1 − γ)pK1α+β L1−α L̇1 − (1 − α)(1 − γ)(α + β)pK1α+β−1 L11−α K̇1 − α+β(1 − α) α K2α K̇2
α
α+β

1−α
1
1
(1 − α) α p α K2 αα+ (1 − α)(1 − γ)K1α+β L11−α(50)

Substituting equations (47) and (48) into equation (50), we obtain the diﬀerential equationof p.
With initial conditions K1 (0), K2 (0), L1 (0), L2 (0), and the parameters, we can obtain theinitial value of the terms of trade, p(0), using equation (46). Using this initial value p(0) andequation (50), we obtain the time path of p(t).
From the numerical simulation, we find that regardless of whether n1 > n2 or n1 = n2 ,the manufacturing employment share in Foreign tends to zero in finite time, that is, θ2M → 0(see Figures 7 to 10). Therefore, Case 4 is unsustainable in the long run.
[Figures 7 to 10 around here]
If p is extremely large when switching from autarky to free trade, Home completely specializes in manufacturing while Foreign diversifies. However, since p continues to decreaseover time, Foreign asymptotically completely specializes in agriculture (Case 3).
11

The left-hand side of equation (46) is an increasing function of p, whereas the right-hand side of thisequation is a decreasing function of p. Plotting both functions, we find that the intersection of the functions is
unique and gives an instantaneous equilibrium value of p.

11.

4 Per capita consumption growth rates
Consumption is defined as wages only, and hence, the growth rate of per capita real consumption is equal to the growth rate of the real wage.
gci = gwi − γg pi .

(51)Here, the real wage is deflated by the consumer price index pγi .12
To obtain gci , we must determine the growth rate of the nominal wage and that of theterms of trade. Note that as long as agricultural goods are produced, the nominal wage isequal to unity, that is, wi = 1, which means that gwi = 0 as long as XiA > 0. However,when a country completely specializes in manufacturing under free trade, we obtain wi =[γ/(1 − γ)] · (L2 /L1 ) > 1, which means that gwi = n2 − n1 < 0.

4.1 Autarky
Under autarky, the growth rate of per capita consumption is given by
γβ
gcATi =
ni > 0,
1−α−β

(52)

where AT denotes “autarky.” From this, if n1 = n2 , we obtain gcAT1 = gcAT2 , and if n1 > n2 , weobtain gcAT1 > gcAT2 .

4.2 Case 2
When Home diversifies and Foreign asymptotically completely specializes in agriculture,
the growth rates of per capita consumption are equal and are given by

γβ



n > 0 if n1 = n2 = n,



1−α−β
FT 2
2
(53)
gcFT
=
g
=

c2

1
γβ



n
>
0
if
n
>
n
,
1
1
2

1−α−β
where FT 2 denotes “free trade” in Case 2.

4.3 Case 3
When both countries completely specialize, if n1 = n2 = n, both countries’ growth rates are
given byγβ33n > 0.
(54)
gcFT= gcFT=
1
2
1−α−βStrictly speaking, the consumer price index is given by γ−γ (1 − γ)−(1−γ) pγi . However, we use pγi because
the constant terms have no eﬀect on the results.
1212

On the other hand, if n1 > n2 , both countries’ growth rate are given byβ − (1 − γ)(1 − α)n1 + (1 − γ)n2 ,1−α−βγ(1 − α)3gcFT=
n1 − γn2 > 0.
2
1−α−β

3
gcFT
=
1

(55)
(56)

3
3
gcFT
> 0 necessarily holds because the condition under which gcFT
> 0 becomes
2
2

n2 <

1−α
n1 ,
1−α−β

(57)

which necessarily holds because n1 > n2 .
3
The coeﬃcient on n1 in gcFT
can be positive or negative. Indeed, under the restriction of1
α + β < 1, the term β − (1 − γ)(1 − α) can be positive or negative. Therefore, an increase in
the population growth in Home can either increase or decrease the growth rate of per capita
consumption in Home.
3> 0 holds. If the coeﬃcient on n1 is
We consider the conditions under which gcFT1
FT 3
positive, we always have gc1 > 0. Even if the coeﬃcient on n1 is negative, there exists a
combination of n1 and n2 that satisfies n1 > n2 and gcFT3 > 0:1n2 >(1 − γ)(1 − α) − β
n1 .
(1 − γ)(1 − α − β)(58)These results are similar to those obtained by Sasaki (2011b). He builds a non-scale
growth, North-South economic development model and shows that along the BGP, bothcountries grow at the same rate but per capita incomes grow at diﬀerent rates because of
the diﬀerences in population growth. In Sasaki (2011b), the growth rate of per capita consumption in the North may be increasing or decreasing in Northern population growth but isincreasing in Southern population growth, and the growth rate of per capita consumption in
the South is decreasing in Southern population growth but is increasing in Northern population growth. We obtain similar results with the North corresponding to Home and the Southcorresponding to Foreign.
However, this paper does diﬀer from Sasaki (2011b) in two ways. First, in Sasaki(2011b), the production pattern is fixed and given exogenously, and population growth in the
North is lower than that in the South, which corresponds to n1 < n2 in this paper. The lowpopulation-growth North produces only manufactured goods, whereas the high-populationgrowth South produces only agricultural goods. However, in our model, the trade patterns
are endogenously determined, so that the country with high population growth specializes
in manufacturing and the country with low population growth specializes in agriculture in
the long run.
13

Moreover, in Sasaki (2011b), if n1 < n2 , we obtain gc1 > gc2 . That is, the growth rateof per capita consumption in the low-population-growth manufacturing country is higher
than that of the high-population-growth agricultural country. In this paper, by contrast, if
33n1 > n2 , we obtain gcFT
< gcFT. That is, the per capita growth rate of the high-population1
2
growth manufacturing country is lower than that of the low-population-growth agriculturalcountry.

4.4 Comparison between autarky and free trade
In this subsection, we assume n1 > n2 .
Under autarky, we have gcAT1 − gcAT2 = n1 − n2 > 0, and hence, there exists a growth gap.
Proposition 1. Suppose that population growth in Home is higher than in Foreign. Then,
under autarky, the BGP growth rate of per capita consumption in Home is higher than in
Foreign.
When Home diversifies and Foreign asymptotically completely specializes in agriculture22= 0, and hence, the growth gap disappears.
− gcFT(i.e., Case 2), we have gcFT
2
1
Proposition 2. Suppose that when switching from autarky to free trade, both countries diversify. Then, in the long run, Home diversifies and Foreign asymptotically completely specializes in agriculture. This situation is sustainable. Moreover, the growth rates and levelsof per capita consumption in both countries are equal.
When Home completely specializes in manufacturing and Foreign asymptotically com33
= −(n1 − n2 ) < 0, and− gcFT
pletely specializes in agriculture (i.e., Case 3), we have gcFT
2
1
hence, there exists a growth gap.
Proposition 3. Suppose that when switching from autarky to free trade, Home completely
specializes in manufacturing and Foreign asymptotically completely specializes in agriculture. This situation is sustainable for a while. Moreover, the growth rate of per capita
consumption in Foreign is higher than that in Home, but the level of per capita consumption
in Foreign is lower than that in Home.
We now compare the growth rates under autarky with those under free trade. When
Home completely specializes in manufacturing and Foreign asymptotically completely specializes in agriculture, we obtain the following relations:
3
gcFT− gcAT1 = −(1 − γ)(n1 − n2 ) < 0,1γ(1 − α)3
gcFT
− gcAT2 =(n1 − n2 ) > 0.
2
1−α−β

14

(59)
(60)

3
3
From these, we have gcFT
< gcAT1 and gcFT
> gcAT2 . Next, we obtain
1
2

3
gcFT
− gcAT2 =1

β − (1 − γ)(1 − α)(n1 − n2 ),
1−α−β

3
gcFT
− gcAT1 = γ(n1 − n2 ) > 0.
2

(61)
(62)

3
3
Accordingly, we have gcFT
> gcAT1 . Moreover, if β − (1 − γ)(1 − α) > 0, we have gcFT
> gcAT2 ,2
1
3< gcAT2 .
and if β − (1 − γ)(1 − α) < 0, we have gcFT
1
From the above analysis, we obtain the following three sets of inequalities:2
2
Case 2 : gcAT2 < gcAT1 = gcFT
= gcFT
,
1
2

(63)

3a
3a
,
< gcAT1 < gcFT
Case 3a : gcAT2 < gcFT
1
2

where β − (1 − γ)(1 − α) > 0,

(64)

3b
3b
,
Case 3b : gcFT
< gcAT2 < gcAT1 < gcFT2
1

where β − (1 − γ)(1 − α) < 0.

(65)

Therefore, we obtain the following two propositions:
Proposition 4. Suppose that under free trade, in the long run, Home diversifies and Foreign
asymptotically completely specializes in agriculture. The growth rate of per capita consumption in Home under free trade is equal to that under autarky, whereas the growth rateof per capita consumption in Foreign under free trade is higher than that under autarky.
Proposition 5. Suppose that under free trade, in the long run, Home completely specializes
in manufacturing and Foreign asymptotically completely specializes in agriculture. The
growth rate of per capita consumption in Home under free trade is lower than that underautarky, whereas the growth rate of per capita consumption in Foreign under free trade ishigher than that under autarky. In addition, the growth rate of Home under free trade may
be higher or lower than that of Foreign under autarky.
In Case 2, agricultural production is positive and hence, wi = 1 in both countries. In
addition, the terms of trade are common to both countries. Therefore, the growth rate of per2
2.
= gcFT
capita consumption is equal in both countries, that is, gcFT
1
2
In Case 3, the growth rate of the nominal wage is diﬀerent in each country. In Home,
FT 3
gw1 = −(n1 − n2 ) < 0, whereas in Foreign, gwFT2 3 = 0. Therefore, the growth rate of per capita
3
3
consumption in Home is less than that in Foreign, that is, gcFT
< gcFT.
1
2

5 Conclusions
This paper has built a two-country, two-sector, non-scale growth model and has investigated
the relationship between trade patterns and per capita growth.
15

If population growth is greater in Home than in Foreign, Home becomes a manufacturing country and Foreign becomes an agricultural country when switching from autarky tofree trade. However, based only on this specialization pattern, we cannot say that Foreign
becomes worse oﬀ because the per capita growth rate under free trade exceeds that underautarky.
In the case where Home completely specializes in manufacturing and Foreign asymptotically completely specializes in agriculture, the per capita growth rate of Foreign exceedsthat of Home, although this trade pattern is unsustainable in the long run.
Therefore, in both cases, Foreign can be better oﬀ under free trade than under autarky.

Acknowledgments
I am grateful to Grants-in-Aid for Scientific Research of Japan Society for the Promotion of
Science (KAKENHI 23730234) for financial support. The usual disclaimer applies.

Appendix
A Why Foreign’s complete specialization in manufacturing is unsustainable
The slope of the PPF at the corner point where Foreign completely specializes in manufacturing is given by
χ̄2 =

L2α
(1 − α)K2α+β

.

(A-1)

When Foreign specializes in agriculture, the growth rate of χ̄2 is given bygχ̄2 = αn2 > 0.

(A-2)

Hence, the slope of the PPF at the corner point continues to become steeper, which meansthat Foreign cannot completely specialize in manufacturing.

B Why Case 3 switches to Case 2 with the passage of time
As stated above, even if the BGP in Case 3 exists, the economy cannot remain on this pathfor long. Even if χ̄1 < p holds at first, it will become χ¯1 ≤ p after some time. Then, Case16

3 is unsustainable and switches to Case 2. At the very moment when Home switches from
Case 3 to Case 2, the absolute value of the rate of change of the terms of trade and that of
the MRT are given byK̇1− α[σ(t)n1 + (1 − σ)n2 ],K1
K̇1|gχ̄1 | = (α + β) − αn1 ,K1|g p | = (α + β)

(B-3)
(B-4)

where σ(t) ≡ L1 (t)/[L1 (t) + L2 (t)]. From these, we obtain|g p | − |gχ̄1 | = α(n1 − n2 )[1 − σ(t)] ≥ 0.

(B-5)

When n1 > n2 , σ(t) converges to unity after enough time has passed. Then, we have |g p | =|χ̄1 |, and both p and χ̄1 decrease at the same rate.

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18

Figures












Figure 1: Capital stock in Home in Case 1














Figure 2: Capital stock in Foreign in Case 1

19





      



 

 





 


         Figure 3: Growth rate of capital stock in Home in Case 1

      



 

 

 





Figure 4: Growth rate of capital stock in Foreign in Case 1

20



 
 
           













Figure 5: Manufacturing employment share in Home in Case 1











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







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











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Figure 6: Manufacturing employment share in Foreign in Case 1

21





 




 



   

Figure 7: Production possibility frontiers in Home and Foreign




            





 

Figure 8: Capital stock in Home in Case 4

22





 
 
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 




 





Figure 9: Capital stock in Foreign in Case 4



















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

















Figure 10: Terms of trade in Case 4

23













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

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Figure 11: Manufacturing employment share in Foreign in Case 424