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

Positive and Negative Population Growth andLong-Run Trade Patterns: A Non-Scale Growth Model
Hiroaki Sasaki

Discussion Paper No. E-13-004

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

October 2013

Positive and Negative Population Growth andLong-Run Trade Patterns: A Non-Scale Growth
Model∗
Hiroaki Sasaki†

Abstract
This paper builds a two-country, two-sector, non-scale growth model and investigates
the relationship between trade patterns and the growth rate of per capita real consumption. We consider negative population growth as well as positive population growth.
We show that, as long as the population growth rates of the two countries are diﬀerent, if the country that accumulates capital stock has negative population growth, no
trade patterns are sustainable in the long run. This is true irrespective of the population
growth rate of the other country. Moreover, we show that, if the country that accumulates capital stock has positive population growth, two trade patterns are sustainable in
the long run. In this case, either each country’s per capita growth is determined by the
population growth of the capital-accumulating country or the population growth of bothcountries, depending on which of the two trade patterns is realized.
Keywords: positive/negative population growth; trade patterns; non-scale growth model
JEL Classification: F10; F43; O11; O41∗

I am grateful to Grants-in-Aid for Scientific Research of Japan Society for the Promotion of Science(KAKENHI 25380295) for their financial support of this study.
†
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

1 Introduction
This study investigates the relationships between population growth, trade patterns, and per
capita consumption growth. We consider negative population growth as well as positive
population growth. For this purpose, we present a two-country, two-sector, two-factor, nonscale growth model.
In scale-growth models, the long-run growth rate of per capita income is proportional
to the scale of the population, which seems counterfactual. On the other hand, in nonscale growth models, the long-run growth rate of per capita income is proportional to thegrowth rate of the population. A pioneering work on non-scale growth models was thatof Jones (1995). Jones (1995) attempted to remove the scale eﬀects, and presented a nonscale growth model in which the growth rate of output per capita depends positively on
the population growth rate, and not on the size of the population. That is, the higher the
population growth rate, the faster the country grows.1)
We use the non-scale growth model for two reasons. First, we can obtain sustainable
income per capita growth, even when population growth is strictly positive. Second, we do
not need to impose knife-edge conditions on the parameters of the model.
Other studies analyze the relationship between trade patterns and growth.2) Two good
examples are the studies by Kaneko (2003) and Felbermayr (2007).
Kaneko (2003) builds a two-country, two-sector, AK growth model and endogenizes theterms of trade.3) He finds that, if an autarkic country with a lower growth rate than its trade
partner 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 in which 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 in which 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 with the de1) For a systematic exposition of scale eﬀects and non-scale growth, see Jones (1999), Jones (2005), Aghion
and Howitt (2005), and Dinopoulos and Sener (2007). For more sophisticated non-scale growth models, seealso Kortum (1997), Dinopoulos and Thompson (1998), Peretto (1998), Segerstrom (1998), Young (1998),
Howitt (1999), and Dinopoulos and Syropoulos (2007).
2) Wong and Yip (1999) present a small, open-economy, two-sector model of endogenous growth, including capital accumulation and learning by doing. They analyze the relationship between economic growth,industrialization, and international trade.
3) Kaneko (2000) builds a growth model with human capital accumulation and shows that the relationshipbetween the terms of trade and growth depends on whether the country specializes in the consumption goods
sector or the investment goods sector. However, Kaneko’s (2000) model is a small, open-economy model,
which means the terms of trade are exogenous.

2

creasing returns to scale, the South can grow at the same rate as the North. Therefore, theSouth can eliminate the growth gap by opening trade.
However, these are both scale growth models, and hence are subject to the aforementioned problems specific to these models.
In contrast, examples of studies that investigate the relationship between trade patterns
and growth rates using non-scale growth models include Sasaki (2011a) and Sasaki (2012).
Sasaki (2011a) derives a model to investigate which trade pattern is realized in the long run
when both countries’ population growth rates are equal. Then, Sasaki (2012) extends thismodel to include the case in which the countries’ population growth rates are diﬀerent.4)
However, both studies consider only positive population growth. Accordingly, we consider
cases in which population growth is both negative and positive.
Many developed countries have stagnant population growth, and in some cases, a negative growth rate.5) Existing economic growth theories assume positive population growth.
However, given that population growth can be negative, we need to consider this case as
well.
At first, it may seem easy to include negative population growth in economic growth theory, but this is not the case.6) As Ferrara (2011) and Christiaans (2011) show, incorporating
negative population growth in growth models is more complicated than simply replacing a
positive value with a negative value.
Christiaans (2011) shows the importance of negative population growth using a simple
model. Consider a Solow growth model with a production function that exhibits increasing,
but relatively small returns to scale.7) When the population growth rate is positive, per capita
income growth is positive and increasing in the population growth rate. On the other hand,when the population growth rate is negative, contrary to expectations, per capita income
growth remains positive, but is decreasing in the population growth rate.
Using a two-country model of international trade, we investigate the relationships between the size of exogenously given population growth rates, the sustainability of tradepatterns, and the long-run growth rate of per capita consumption in each country, given that
4) In this respect, Sasaki (2011b) builds a non-scale growth, North-South economic development model,
and shows that both countries grow at the same rate along the balanced growth path. However, their per
capita incomes grow at diﬀerent rates because the diﬀerences in population growth. Nevertheless, since theproduction pattern is fixed and given exogenously in this model, we cannot know whether the given tradepattern is sustainable over time.
5) For example, according to Japan’s Ministry of Internal Aﬀairs and Communications, as of March 2012,
Japan has experienced its largest-ever decline in population.
6) Ritschel (1985) argues that, in the standard Solow growth model, a negative savings rate is necessary forthe existence of a steady-state equilibrium with a negative population growth rate. See also Felderer (1988).
7) In the Solow model with a constant returns to scale production function, per capita income growth is zero
when the population growth is positive, but positive when the population growth is negative. For details, see
Christiaans (2011).

3

a specific trade pattern is sustainable.
The main results are as follows. We show that, as long as the population growth ratesof the two countries are diﬀerent, if the country that accumulates capital stock has negative population growth, no trade patterns are sustainable in the long run, irrespective of the
population growth rate of the other country. Moreover, we show that, if the country that accumulates capital stock has positive population growth, two trade patterns are sustainable in
the long run. Here, either each country’s growth rate is determined by the population growthof the capital-accumulating country or the population growth of both countries, dependingon which of the two trade patterns is realized.
The rest of the paper is organized as follows. Section 2 presents our model and determines the equilibrium and long-run growth rate of per capita consumption under autarky.
Section 3 investigates the equilibrium under free trade. Section 4 investigates the long-run
growth rate of per capita consumption under free trade. We compare the growth rates underfree trade and autarky, and then compare our results to those of related studies. Finally,
Section 5 concludes the paper.

2 The model
Consider a world that contains 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 , andproduce agricultural goods, XiA , with only labor input, LiA . Here, i = 1 and i = 2 denote
Home and 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 introduced by Arrow (1962). Substituting Ai
into equation (1), we obtain equation (2), which shows that manufactured goods production

4

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.
We further suppose that the labor supply is equal to the population and that the population is fully employed. Moreover, 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. Note that population
growth can be negative.
Let pi denote the price of manufactured goods relative to agricultural goods. Then, theprofits of the 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 paid to produce agricultural
goods and ri 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.8) In the canonical one-sector
Solow model, under the golden rule steady state in which per capita consumption is maximized, consumption is equal to the total real wage and total capital income is saved andinvested. Hence, our assumption has some rationality and can be interpreted as a simple ruleof thumb for consumers with dynamic optimization (Christiaans, 2008). We define real consumption per capita, 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,

8) The same assumption is used in Uzawa (1961), which considers a two-sector growth model, and Krugman(1981), which considers a two-country, two-sector, North-South trade and development model. If dynamic
optimization is used, then the Euler equation for consumption appears and the number of diﬀerential equations
increases, significantly complicating the analysis. Therefore, an analysis with dynamic optimization will be left
for future research. In addition, it is true that consumption smoothing with dynamic optimization is a standard
tool in macroeconomics, although Mankiw (2000) states that, in reality, consumption behavior deviates fromconsumption smoothing.

5

1 − γ, is spent on CiA .
piCiM = γwi Li ,(6)

CiA = (1 − γ)wi Li .

(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:
g Ki ≡

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).

2.3 Equilibrium under autarky and per capita consumption growth
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)First, we derive the balanced-growth path (BGP) under autarky when the population
growth rate is positive, that is, ni > 0. Along the BGP, the rate of capital accumulationis constant and equal to the rental rate of capital, which is given in equation (5) as ri =αKiα+β−1 (γLi )1−α . With ġKi /gKi = (α + β − 1)gKi + (1 − α)ni = 0, 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

Consumption is defined as wages only, and hence, the growth rate of per capita realconsumption is equal to the growth rate of the real wage.9)
gci = gwi − γg pi .

(12)Here, the real wage is deflated by the consumer price index, pγi .10) To obtain gci , we must
know gwi and g pi . Note that, as long as agricultural goods are produced, the nominal wage isequal to unity, that is, wi = 1, which means that gwi = 0. Accordingly, we obtain the growthrate of per capita consumption under autarky as follows:
gcATi =

γβni > 0,
1−α−β

(13)

where “AT” denotes autarky. Therefore, gcATi is increasing in ni .
Considering the BGP growth rate of capital stock, we introduce a new variable, scale1−αadjusted capital stock: ki ≡ Ki /Liϕ , where ϕ ≡ 1−α−β. The dynamics of the scale-adjusted
capital stock are given byk̇i = αγ1−α kiα+β − ϕni ki .

(14)

In the steady state, k̇i = 0, from which we obtain() 1
1−α 1−α−β
αγ
ki∗ =
.
ϕni

(15)

The steady state is stable because dk̇i /ki |ki =ki∗ = −ki∗ [(1 − α − β)αγ1−α (ki∗ )α+β−2 + ϕni ] < 0.
Next, we derive the long-run equilibrium under autarky when the population growth rateis negative, that is, ni < 0. If ni < 0, from equation (14), there never exists a ki > 0 suchthat k̇i = 0, and we have k̇i > 0 for ki > 0. That is, if the initial value of the capital stock isstrictly greater than zero, ki (0) > 0, then ki diverges to infinity. However, even in this case,
we can examine the long-run growth rate of per capita consumption. Considering that percapita consumption is equal to the real wage measured in terms of the consumer price index,

9) In every case in our model, the long-run growth rate of per capita real consumption is equal to that of percapita real income. Therefore, we use the growth rate of per capita real consumption.
10) Let pc denote the consumer price index that is consistent with the expenditure minimization problem ofconsumers. Then, pc = γ−γ (1 − γ)−(1−γ) pγi , and by definition, pc ci = wi . Strictly speaking, the consumer priceindex is given by γ−γ (1 − γ)−(1−γ) pγi . However, we use pγi because the constant terms have no eﬀect on the
results.

7

we havegcATi = −γg pi = −γ[αni − (α + β)(gki + ϕni )]= α(α + β)γ2−α kiα+β−1 − γαni .

(16)Since α + β − 1 < 0 by assumption, kiα+β−1 approaches zero when ki approaches infinity.
From this, we have
lim gcATi = −γαni > 0.

ki →+∞(17)

Therefore, gcATi is positive, even when ni < 0 and is decreasing in ni .

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)without loss of generality.
It is suﬃcient for our purpose to consider the following four trade patterns from theviewpoint of Home:
Pattern 1: Both countries produce both goods; that is, both countries diversify.
Pattern 2: Home diversifies and Foreign completely specializes in agriculture.
Pattern 3: Home completely specializes in manufacturing and Foreign completely specializes in agriculture.
Pattern 4: Home completely specializes in manufacturing and Foreign diversifies.
In addition, we consider the following eight cases according to the size of the population
growth: Case 1, 0 < n1 = n2 ;11) Case 2, n1 = n2 < 0; Case 3, 0 < n2 < n1 ; Case 4,
0 < n1 < n2 ; Case 5, n2 < 0 < n1 ; Case 6, n1 < 0 < n2 ; Case 7, n2 < n1 < 0; and Case 8,
n1 < n2 < 0.

11) Long-run growth rates and transitional dynamics in the case where 0 < n1 = n2 are analyzed in detail in
Sasaki (2011a). When n1 , n2 , with regard to the state variable ki , the locus of k̇i = 0 moves over time, which
complicates the analysis. The analysis of transitional dynamics, including a numerical analysis when n1 , n2 ,is left for future research.

83.1 Equilibrium when both countries diversify: Pattern 1
The market-clearing conditions for both goods are given by
X1M + X2M = C1M + C2M + I1 + I2 ,(18)

X1A + X2A = C1A + C2A .

(19)

From these, we obtainp=[γ(L1 + L2 )]α
( α+β
) .
α+β α
α
α
(1 − α) K1 + K2

(20)

Each 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(21)

The rates of capital accumulation in both countries are given by
gK2 = αK2α+β−1 (θ2M L2 )1−α .

gK1 = αK1α+β−1 (θ1M L1 )1−α ,(22)First, if n1 = n2 , so that L1 = L2 ,12) then, after enough time has passed, we obtainlim θ1M = 2γ,

lim θ2M = 0,

t→+∞

t→+∞

(23)

where γ < 1/2 is needed. Then, the manufacturing employment share in Foreign goes tozero, and Foreign asymptotically completely specializes in agriculture.13) Hence, Pattern 1
is not sustainable when n1 = n2 .
Second, if n1 > n2 , then after enough time has passed, we obtainlim θ1M = γ,lim θ2M = 0.

t→+∞

t→+∞

(24)

In this case too, the manufacturing employment share in Foreign goes to zero, and Foreign
asymptotically completely specializes in agriculture. Hence, Pattern 1 also is unsustainable12) In addition, if K1 (0) = K2 (0), the manufacturing employment share in each country is given by θiM = γ,
which is constant. Pattern 1 is only sustainable in this case. However, the relative prices in both countries
under autarky are equal, and therefore trade does not occur.
13) In our model, the agricultural output approaches zero, but it never vanishes because we assume that Foreign’s capital stock is strictly positive. Therefore, the phrase “asymptotically” completely specializes in agriculture is more appropriate. For more information, see Christiaans (2008).

9

when n1 > n2 .
Third, if n1 < n2 , then after enough time has passed, we obtainlim θ1M = +∞,lim θ2M = 0.

t→+∞

t→+∞

(25)

In this case, the manufacturing employment share in Home exceeds unity, the manufacturing employment share in Foreign goes to zero, and Foreign asymptotically completely
specializes in agriculture. Hence, Pattern 1 also is unsustainable when n1 < n2 .
Summarizing the above results, we obtain the following proposition.
Proposition 1. Pattern 1, in which both Home and Foreign diversify, is unsustainable in thelong run in every case.

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

(26)
(27)

Hence, we obtainp=[γ(L1 + L2 )]α(1 − α)K1α+β

.

(28)

The manufacturing employment share in Home is given by(θ1M = γ

)
L2
1+
.
L1

(29)

First, if n1 = n2 , the manufacturing employment share in Home becomesθ1M = 2γ.

(30)

In this case, we need γ < 1/2 for Pattern 2 to hold. Second, if n1 > n2 , we obtainlim θ1M = γ.

t→+∞

(31)

Here, the manufacturing employment share of Home converges to γ, and thus Pattern 2 is10

sustainable. Third, if n1 < n2 , then θ1M continues to increase, becomes more than unity, andapproaches infinity.
lim θ1M = +∞.

(32)

t→+∞

In this case, Pattern 2 is again unsustainable.
The growth rate of capital stock is given by
gK1 = αγ1−α (L1 + L2 )1−α K1α+β−1 .

(33)Note that, in this case, we obtain c1 = c2 because w1 = w2 = 1 as long as agricultural
goods are produced and both countries face the same relative price, p. Accordingly, in
Pattern 2, the long-run growth rates of per capita consumption in Home and Foreign are, where “FT” denotes free trade.
equalized; that is, gcFT
= gcFT
1
2
We now examine in detail the conditions under which Pattern 2 holds. Following Wongand Yip (1999), we investigate whether the trade pattern is sustainable by comparing thesize of the terms of trade and the size of the marginal rate of transformation (MRT) of
the production possibilities frontier (PPF) at the corner point where a country completely
specializes in manufacturing.
[Figure 1 around here]
The size of the MRT of the PPF in Home is given by[K α+β (θ1M L1 )1−α ] 1−α
dX1A
= 1
− M
.
α+β
dX1
1−α
(1 − α)K
α

(34)

1

Substituting θ1M = 1 into 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)

For Pattern 2 to be sustainable over time, we need p < χ̄1 ; that is, from equations (28)and (35),[γ(L1 + L2 )]αL1α(1 − α)K1(1 − α)K1α+β

<
α+β

11

.

(36)

Rearranging this condition, we obtain
L2 <

1−γ
L1 .
γ

(37)

From this, if n1 = n2 , Pattern 2 is sustainable if γ < 1/2. However, if n1 > n2 , Pattern 2 issustainable in the long run irrespective of the size of γ. In contrast, if n1 < n2 , Pattern 2 isunsustainable in the long run.
If n1 > 0, then from equation (33), the growth rate of gK1 is given by
(
)
ġK1
L2
L1
= (1 − α)n1 +n2 + (α + β − 1)gK1 .
g K1
L1 + L2
L1 + L2(38)

When n1 ≥ n2 , this leads toġK1= (1 − α)n1 + (α + β − 1)gK1 .
g K1

(39)

With ġK1 /gK1 = 0, the growth rates of capital stock, terms of trade, and per capita consumption are respectively given by1−αn1 > 0,1−α−ββgp = −
n1 < 0,
1−α−β
γβ
n1 .
gcFT
= gcFT=1
2
1−α−βg K1 =

(40)
(41)
(42)

Accordingly, both countries’ growth rates are increasing in n1 .
On the other hand, if n1 < 0, the growth rate of k1 = K1 /L1ϕ is given byk̇1= αγ1−α k1α+β−1 − ϕn1 > 0.
k1

(43)

Here, ki continues to increase and k̇1 /k1 approaches −ϕn1 > 0. In this case, the growth rateof capital stock, the terms of trade, and per capita consumption are respectively given by
gK1 = gk1 − ϕn1 = 0,(44)g p = αn1 < 0,

(45)

gcFT
= gcFT= −γαn1 > 0.
1
2

(46)

Accordingly, both countries’ growth rates are decreasing in n1 .
12However, we must also consider another condition to ascertain the sustainability of Pattern 2, that is, the relationship between the MRT of the PPF in Foreign and the terms of trade.
The size of the MRT at the point where Foreign completely specializes in manufacturing isgiven byχ̄2 =L2α(1 − α)K2α+β

.

(47)

For Pattern 2 to be sustainable in the long run, it is necessary that p < χ̄2 .
There exist five cases such that n1 ≥ n2 : 0 < n1 = n2 (Case 1); n1 = n2 < 0 (Case 2);
0 < n2 < n1 (Case 3); n2 < 0 < n1 (Case 5); and n2 < n1 < 0 (Case 7).
Case 1: If 0 < n1 = n2 , we find that p < χ̄2 holds over time because, in the long run,βn1 < 0 and gχ̄2 = αn1 > 0. Therefore, Pattern 2 is sustainable.
g p = − 1−α−βCase 2: If n1 = n2 < 0, we find that p < χ̄2 holds over time because, in the long run,g p = gχ̄2 = αn1 . Therefore, Pattern 2 is sustainable.
βCase 3: If 0 < n2 < n1 , we find that p < χ̄2 holds over time because g p = − 1−α−βn1 < 0 andgχ̄2 = αn2 > 0. Therefore, Pattern 2 is sustainable.
βCase 5: If n2 < 0 < n1 , we have g p = − 1−α−βn1 < 0 and gχ̄2 = αn2 < 0. For the conditionp < χ̄2 to hold in the long run, we need |g p | > |gχ̄2 |.
|g p | − |gχ̄2 | =βn1 + αn2 .
1−α−β

(48)

β
β
Accordingly, if 1−α−βn1 + αn2 > 0, Pattern 2 is sustainable. In contrast, if 1−α−βn1 + αn2 < 0,
Pattern 2 is unsustainable.
Case 7: If n2 < n1 < 0, we have g p = αn1 < 0 and gχ̄2 = αn2 < 0, which imply that|g p | − |gχ̄2 | = α(n2 − n1 ) < 0.

(49)This contradicts |g p | > |gχ̄2 |, and so Pattern 2 is unsustainable.
Summarizing the above results, we obtain the following propositions.
Proposition 2. Pattern 2 (Home diversifies while Foreign asymptotically completely specializes in agriculture) is sustainable in the long run in Cases 1, 2, 3, and 5.
Proposition 3. If Pattern 2 is sustainable in the long run, both countries grow at the same
per capita rate under free trade, irrespective of whether their population growth is positiveor negative.
βNote that Cases 1 and 2 require γ ≤ 1/2, while Case 5 requires 1−α−βn1 + αn2 > 0.

13

The reason why this trade pattern is unsustainable in the long run when n1 < n2 is thatdemand for manufactured goods in Foreign grows faster than the supply from Home. As aresult, Home cannot meet the world demand for manufactured goods on its own.

3.3 Equilibrium when Home specializes in manufacturing and Foreign
specializes in agriculture: Pattern 3
The market-clearing conditions for both goods are given by
X1M = C1M + C2M + I1 ,(50)

X2A = C1A + C2A .

(51)

With L1M = L1 , because of complete specialization in manufacturing, we obtainp=γL2(1 − α)(1 − γ)K1α+β L11−α

.

(52)

The growth rate of capital stock is given by
gK1 = αK1α+β−1 L11−α .

(53)Pattern 3 is sustainable if p > χ̄1 , which, from equations (35) and (52), can be rewrittenasγL2(1 − α)(1 − γ)K1α+β L11−α>

L1α(1 − α)K1α+β

.

(54)

From this, we obtain
L2 >

1−γ
L1 .
γ

(55)

Therefore, if n1 = n2 , Pattern 3 is sustainable as long as γ > 1/2. If n2 > n1 , Pattern 3
is sustainable in the long run, irrespective of the size of γ. On the other hand, if n2 < n1 ,Pattern 3 is unsustainable in the long run. In summary, n2 ≥ n1 is necessary for Pattern 3 tobe sustainable.
Note that, in this case, we obtain c1 > c2 because w1 = [γ/(1 − γ)] · (L2 /L1 ) > 1 > w2 = 1
when n2 > n1 , and both countries face the same relative price, p.
If n1 > 0, the growth rates of the capital stock, terms of trade, and per capita consumption

14

are given by1−αn1 > 0,1−α−β1−αg p = n2 −
n1 ,
1−α−β
(
FT
gc1 = (n2 − n1 ) − γ n2 −
gK1 =

=
gcFT
2

(56)
(57)
)
1−α
β − (1 − γ)(1 − α)n1 =n1 + (1 − γ)n2 ,
1−α−β
1−α−β

γ(1 − α)n1 − γn2 .
1−α−β

(58)
(59)

is increasing in n1 if β > (1−γ)(1−α), decreasing in n1 if β < (1−γ)(1−α),Accordingly, gcFT
1
and increasing in n2 . In addition, gcFT
is increasing in n1 and decreasing in n2 . In standard
2
non-scale growth models, the growth rate of per capita consumption (income) is increasingin the growth rate of population. However, in our model, Home’s per capita consumption
growth can be increasing or decreasing in its population growth, and Foreign’s per capita
consumption growth is decreasing in its population growth.14)
On the other hand, if n1 < 0, the growth rate of k1 is given byk̇1= αk1α+β−1 − ϕn1 > 0.
k1

(60)

Accordingly, k1 continues to increase over time. When k1 increases, k̇1 /k1 approaches−ϕn1 > 0. From this, we obtain
gK1 = 0,(61)g p = n2 − (1 − α)n1 ,

(62)

gcFT
= (n2 − n1 ) − γ[n2 − (1 − α)n1 ] = −[1 − γ(1 − α)]n1 + (1 − γ)n2 ,
1

(63)

gcFT
= −γ[n2 − (1 − α)n1 ] = γ(1 − α)n1 − γn2 .
2(64)Accordingly, gcFT
is decreasing in n1 and increasing in n2 . In addition, gcFT
is increasing in
12
n1 and decreasing in n2 .
However, as in Pattern 2, we must also consider another condition. For Pattern 3 to besustainable in the long run, we need p < χ̄2 .
There exist five cases such that n1 ≤ n2 : 0 < n1 = n2 (Case 1); n1 = n2 < 0 (Case 2);
0 < n1 < n2 (Case 4); n1 < 0 < n2 (Case 6); and n1 < n2 < 0 (Case 8).
14) Sasaki (2011b) presents empirical evidence indicating that, in developed countries, the correlation between
per capita income growth and population growth is ambiguous, whereas in developing countries, the correlationis negative.

15

βCase 1: If 0 < n1 = n2 , we obtain g p = − 1−α−βn1 < 0 and gχ̄2 = αn1 > 0, which show thatp < χ̄2 holds over time, and hence, Pattern 3 is sustainable.
Case 2: If n1 = n2 < 0, we obtain g p = αn1 < 0 and gχ̄2 = αn1 < 0, which show that p < χ̄2holds over time, and hence, Pattern 3 is sustainable.
1−αCase 4: If 0 < n1 < n2 , we have g p = n2 − 1−α−βn1 and gχ̄2 = αn2 > 0. In this case, we
obtain()
1|gχ̄2 | − |g p | = −(1 − α) n2 −
n1 .
(65)
1−α−β
1−α
1−α
n1 , that is, g p < 0, Pattern 3 is sustainable. If n2 > 1−α−βn1 , that is, g p > 0, we
If n2 < 1−α−β1
1−α1need n2 < 1−α−β n1 for |gχ̄2 | > |g p | to hold. Therefore, if 1−α−β n1 < n2 < 1−α−βn1 , Pattern 3 is1sustainable. On the other hand, if 1−α−β n1 < n2 , Pattern 3 is unsustainable. In summary, if
1n2 < 1−α−βn1 , Pattern 3 is sustainable.
Case 6: If n1 < 0 < n2 , we have g p = n2 − (1 − α)n1 > 0 and gχ̄2 = αn2 > 0. For Pattern 3 tobe sustainable, we need |gχ̄2 | > |g p |. However, we obtain|gχ̄2 | − |g p | = −(1 − α)(n2 − n1 ) < 0.

(66)

Accordingly, Patterns 3 is unsustainable in this case.
Case 8: If n1 < n2 < 0, we have g p = n2 − (1 − α)n1 and gχ̄2 = αn2 < 0. If n2 − (1 − α)n1 > 0,that is, g p > 0, Pattern 3 is clearly unsustainable. If n2 − (1 − α)n1 < 0, that is, g p < 0, we
need |g p | > |gχ̄2 | for Pattern 3 to be sustainable. However, we obtain|g p | − |gχ̄2 | = −(1 − α)(n2 − n1 ) < 0.

(67)

Therefore, Pattern 3 is unsustainable in this case.
Summarizing the above results, we obtain the following propositions.
Proposition 4. Pattern 3 (Home completely specializes in manufacturing while Foreign
asymptotically completely specializes in agriculture) is sustainable in the long run in Cases
1, 2, and 4.
Proposition 5. If Pattern 3 is sustainable in the long run and if both countries’ population
growth rates are diﬀerent, both countries grow at diﬀerent per capita rates under free trade.
1
Note that Cases 1 and 2 require 1/2 < γ, while Case 4 requires n2 < 1−α−βn1 .
The reason why this trade pattern is unsustainable in the long run when n1 > n2 is thatdemand for agricultural goods in Home grows faster than they are supplied by Foreign.
Hence, Foreign cannot meet the world demand for agricultural goods on its own.

16

3.4 Equilibrium when Home specializes in manufacturing and Foreign
diversifies: Pattern 4
The market-clearing conditions for both goods are given by
X1M + X2M = C1M + C2M + I1 + I2 ,(68)

X2A = C1A + C2A .

(69)

From equations (68) and (69), 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

(70)

Here, p is implicitly and uniquely determined from equation (70), and hence, p is a functionof K1 , K2 , L1 , and L2 : p = p(K1 , K2 , L1 , L2 ).15)
The growth rate of the capital stock in each country is given by
gK1 = αK1α+β−1 L11−α ,

1−α

1−α

β

gK2 = α(1 − α) α p α K2α ,(71)

where p is endogenously determined by equation (70).
The employment share of manufacturing in Foreign is given by
1

1

α+β

(1 − α) α p α K2 α
θ2M =
.
L2

(72)

In this case, analytical solutions are diﬃcult to obtain, so we conduct numerical simulations.16) Using equation (70), we obtain the time derivative of p as follows:
1

ṗ =

β

(1 − α) α K2α K̇2
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
α
α+β

1
1−α
1
(1 − α) α p α K2 αα+ (1 − α)(1 − γ)K1α+β L11−α(73)

Substituting L̇1 = n1 L1 , L̇2 = n2 L2 , and the two equations from (71) into equation (73), we
obtain the diﬀerential equation of p.
Using the initial conditions K1 (0), K2 (0), L1 (0), L2 (0), as well as the parameters, we can15) The left-hand side of equation (70) is an increasing function of p, whereas the right-hand side of thisequation is a decreasing function of p for given values of K1 , K2 , L1 , and L2 . Plotting both functions, we findthat their intersection is unique and gives an instantaneous equilibrium value of p.
16) We use the following values for the parameters and initial capital stocks: α = 0.3, β = 0.2, γ = 0.6,
K1 (0) = 1.2, K2 (0) = 1.

17.

obtain the initial value of the terms of trade, p(0), using equation (70). Using this initialvalue, p(0), and equation (73), we obtain the time path of p(t).
From the numerical simulation, we find that, regardless of whether n1 ⋛ n2 , the manufacturing employment share in Foreign tends to zero in finite time; that is, θ2M → 0. Therefore,
Pattern 4 is unsustainable in the long run.
Proposition 6. Pattern 4 (Home completely specializes in manufacturing while Foreign diversifies) is unsustainable in the long run in every case.

4 Per capita consumption growth under free trade
From the above analysis, we find that the sustainable trade patterns are Patterns 2 and 3. In
this section, we summarize the long-run growth rate of per capita consumption under free
trade according to the rate of the population growth.

4.1 Case 1: 0 < n1 = n2
If 0 < γ < 1/2, only Pattern 2 is sustainable, and if 1/2 < γ < 1, only Pattern 3 is sustainable.
In both cases, the BGP growth rates of per capita consumption are given by
gcFT= gcFT=12

γβn1 > 0.
1−α−β

(74)

4.2 Case 2: n1 = n2 < 0
If 0 < γ < 1/2, only Pattern 2 is sustainable, and if 1/2 < γ < 1, only Pattern 3 issustainable. In both cases, balanced growth is impossible, but the long-run growth rates ofper capita consumption are given by
gcFT= gcFT= −γαn1 > 0.
1
2

(75)

4.3 Case 3: 0 < n2 < n1
Only Pattern 2 is sustainable, and the BGP growth rates of per capita consumption are given
bygcFT= gcFT=12

γβn1 > 0.
1−α−β

18

(76)

4.4 Case 4: 0 < n1 < n2
Only Pattern 3 is sustainable. The BGP growth rates of per capita consumption are given by(
gcFT= (n2 − n1 ) − γ
1
gcFT
=
2

)
1−α
n2 −
n1 ,
1−α−β

γ(1 − α)n1 − γn2 .
1−α−β

(77)
(78)

1
Note that we need the condition that n2 < 1−α−β
n1 .
1−α
1−α
1
< 0.17) If n2 < 1−α−β> 0, and gcFT
If 1−α−β n1 < n2 < 1−α−β n1 , we have g p > 0, gcFT
n1 ,
2
1
FT
FT
we have g p < 0, gc1 > 0, and gc2 > 0.
In either case, we find that

= n2 − n1 > 0,
− gcFT
gcFT
2
1

(79)

from which we obtain gcFT
> gcFT.
1
2
This case is realistic because we can regard Home and Foreign as a developed country and a developing country, respectively: (1) the rate of population growth in developed
countries is lower than that in developing countries; (2) the per capita income growth indeveloped countries is higher than that in developing countries; and (3) developed countries
are industrialized countries while developing countries are agricultural countries. Chamonand Kremer (2009) also point out the importance of relative population growth for the development of developing countries. Population growth in developing countries is considered a
problem, although since it is declining, it may not be an obstacle to development. Nevertheless, if the population growth in developed countries declines more rapidly than that indeveloping countries, the size of the relative population growth also declines. This decline
will be an obstacle for developing countries.
1−α1Then, as stated above, when 1−α−βn1 ≤ n2 ≤ 1−α−βn1 , the per capita consumption growth
FTof the developing country is negative (gc2 ≤ 0). To obtain positive growth, other things
being equal, a developing country needs to decrease its population growth and satisfy the
1−αcondition that n2 < 1−α−βn1 . However, since the population growth in developed countries
is currently decreasing (i.e., n1 is decreasing), even though some policies aim to decrease n2 ,the above inequality will not be satisfied as long as n1 decreases. Therefore, a decrease in17) The necessary and suﬃcient condition for gcFT
> 0 is given by β−(1−γ)(1−α)
n1 + (1 − γ)n2 > 0, which can be
1−α−β
1
[
]
β
1−α
− (1−γ)(1−α)
rewritten as n2 > 1−α−βn1 . Note that the coeﬃcient of n1 is less than unity. Then, if 0 < n1 < n2 ,
[
]
β
1−α
the condition n2 > 1−α−β− (1−γ)(1−α)
n1 is always satisfied. Therefore, if 0 < n1 < n2 , we necessarily haveFT
gc1 > 0.

19

the population growth in developing countries does not necessarily narrow the growth gap
between them and developed countries.

4.5 Case 5: n2 < 0 < n1
Only Pattern 2 is sustainable, and the BGP growth rates of per capita consumption are givenby== gcFTgcFT
2
1

γβn1 > 0.
1−α−β

(80)

β
Note that, for Pattern 2 to be sustainable, we need the condition that 1−α−βn1 + αn2 > 0.

4.6 Case 6: n1 < 0 < n2
Only Pattern 3 is sustainable. The long-run growth rates of per capita consumption are givenbygcFT= (n2 − n1 ) − γ[n2 − (1 − α)n1 ] = (1 − γ)n2 − [1 − γ(1 − α)]n1 > 0,
1

(81)

gcFT
= −γ[n2 − (1 − α)n1 ] < 0.
2(82)Hence, we have gcFT
> gcFT. Note that, in this case, Pattern 3 is sustainable for a while, but
1
2
is unsustainable in the long run.

4.7 Case 7: n2 < n1 < 0
In this case, the long-run growth rates of per capita consumption are given by
gcFT= gcFT= −γαn1 > 0.
1
2

(83)

Here, Pattern 2 is sustainable for a while, but unsustainable in the long run.

4.8 Case 8: n1 < n2 < 0
When n2 > (1 − α)n1 , the long-run growth rates of per capita consumption are given by
gcFT= (1 − γ)n2 − [1 − γ(1 − α)]n1 ,
1

(84)

gcFT
= −γ[n2 − (1 − α)n1 ] < 0.
2

(85)

20

To obtain gcFT
> 0, we must have1n2 <1 − γ(1 − α)
n1 .
1−γ

(86)

The coeﬃcient of n1 is less than unity, and accordingly, there exist combinations of n1 and> 0 is possible.
n2 that simultaneously satisfy n1 < n2 < 0 and equation (86). Therefore, gcFT
1
In addition, we have gcFT
> gcFT, irrespective of whether gcFT
> 0 or gcFT
< 0.
1
2
1
1
When n2 < (1 − α)n1 , the long-run growth rates of per capita consumption are given by= (1 − γ)n2 − [1 − γ(1 − α)]n1 = (n2 − n1 ) − γ[n2 − (1 − α)n1 ] > 0,
gcFT
1

(87)

= −γ[n2 − (1 − α)n1 ] > 0.
gcFT
2

(88)

.
Therefore, we clearly have gcFT
> gcFT
1
2
Note that Pattern 3 is sustainable for a while, but unsustainable in the long run.

4.9 Comparisons of autarky and free trade growth rates
In this section, we compare gcAT1 to gcFT
and gcAT2 to gcFT
. The results are summarized in Table
1
2
1.18)
[Table 1 around here]
With regard to per capita consumption growth, we obtain the following propositions.
Proposition 7. In Cases 1, 2, 3, and 5, both countries’ per capita consumption growthunder free trade can be equal to or more than per capita consumption growth under autarky(gcATi ≤ gcFT
, i = 1, 2). Other than in Cases 1 and 2, which are special cases, Pattern 2 isirealized in Cases 3 and 5.
Proposition 8. In Cases 3 and 5, the growth gap between Home and Foreign under autarky
narrows, and both countries’ growth rates are equalized by switching from autarky to free
trade (gcAT1 > gcAT2 to gcFT
= gcFT). In these cases, Pattern 2 holds.
1
2
Proposition 9. In Case 4, the growth gap between Home and Foreign under autarky reverses(from gcAT1 < gcAT2 to gcFT
> gcFT). In this case, Pattern 3 holds. In addition, in Case 4,
12
switching from autarky to free trade decreases per capita consumption growth (gcAT2 > gcFT).
218) For derivation, see Appendix A.

21

5 Conclusions
In this study, we built a two-country, two-sector, non-scale growth model and investigated
the relationship between trade patterns and per capita consumption growth. In addition,
we considered both negative and positive population growth. Our analysis yielded someinteresting results with regard to trade patterns and per capita consumption growth.
First, when the population growth of Home is higher than that of Foreign, a trade patternsuch that Home diversifies while Foreign specializes in agriculture is sustainable in the longrun. In this case, after switching from autarky to free trade, the growth gap between the
countries disappears, and both countries grow at the same per capita rate.
Second, when the population growth of Home is lower than that of Foreign, a tradepattern such that Home specializes in manufacturing while Foreign specializes in agriculture
is sustainable in the long run. In this case, after switching from autarky to free trade, thegrowth gap between the two countries reverses, and Home grows faster than Foreign in percapita terms.
Third, no trade patterns are unsustainable when the population growth of a country that
produces manufactured goods is negative. This is true irrespective of whether the population
growth of the other country, which does not produce manufactured goods, is positive or
negative.
Finally, under autarky, even if the population growth of Home is negative, the per capita
growth rate is positive in the long run. However, under free trade, if the population growthof Home is negative, it can neither specialize in manufacturing nor diversify. Therefore, we
can say that whether the economy is sustainable in the long run when the population growthis negative depends on trade openness.

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24

A Appendix: Comparison between autarky and free tradeCases 1 and 2: We have gcAT1 = gcFT
and gcAT2 = gcFT
.
1
2
AT
FT
FT
AT
Case 3: We have gc1 = gc1 and gc2 < gc2 .
Case 4: We havegcAT1 − gcFT
= (1 − γ)(n1 − n2 ) < 0,1γ(1 − α)=−gcAT2 − gcFT
(n1 − n2 ) > 0.
2
1−α−β
.
and gcAT2 > gcFT
Therefore, we obtain gcAT1 < gcFT
2
1
Case 5: We have
(
)
β
AT
FT
gc2 − gc2 = −γ
n1 + αn2 .
1−α−β

(A-1)
(A-2)

(A-3)

β
n1 + αn2 > 0.
Therefore, we obtain gcAT2 < gcFT
if 1−α−β2Case 6: We havegcAT1 − gcFT
= (1 − γ)(n1 − n2 ) < 0,
1
(
)
1
AT
FT
gc2 − gc2 = −γ(1 − α) n1 −n2 > 0.
1−α−β

(A-4)
(A-5)

Therefore, we obtain gcAT1 < gcFT
and gcAT2 > gcFT.
1
2
Case 7: We havegcAT2 − gcFT
= γα(n1 − n2 ) > 0.
2

(A-6)

Therefore, we obtain gcAT2 > gcFT.
2
Case 8: We havegcAT1 − gcFT
= −(1 − γ)(n2 − n1 ) < 0.
1
Therefore, we obtain gcAT1 < gcFT
.
1

25

(A-7)

X1A

X2A
p
¯ 1

O

X1M O

¯ 2

Figure 1: Production possibility frontiers in Home and Foreign

26

p

X2M

27

Case 10 < n1 = n22 and 3†
gcAT1 = gcAT2
gcFT= gcFT1
2
AT
FT
g c1 = g c1
gcAT2 = gcFT
2

Case 2n1 = n2 < 0
2 and 3†
gcAT1 = gcAT2
gcFT= gcFT
1
2
AT
FT
gc1 = gc1
gcAT2 = gcFT
2

Case 30 < n2 < n1
2
gcAT1 > gcAT2
gcFT= gcFT1
2
AT
FT
g c1 = g c1
gcAT2 < gcFT
2

gcFT
< 0♠
2

Case 4♢0 < n1 < n2
3
gcAT1 < gcAT2
gcFT> gcFT1
2
AT
FT
gc1 < gc1
gcAT2 > gcFT2

n/a

Case 5n2 < 0 < n1
2
gcAT1 > gcAT2 •gcAT1 < gcAT2 ••gcFT= gcFT1
2
gcAT1 = gcFT1•
gcAT2 < gcFT2••
gcAT2 > gcFT
2
gcFT
<0
2

Case 6*n1 < 0 < n2
3
gcAT1 > gcAT2 ⋆gcAT1 < gcAT2 ⋆⋆gcFT> gcFT1
2
gcAT1 < gcFT1
gcAT2 > gcFT
2

Case 7*n2 < n1 < 0
2
gcAT1 < gcAT2
gcFT= gcFT1
2
AT
FT
g c1 = g c1
gcAT2 > gcFT
2

gcFT
<0
1
FT
g c2 < 0Case 8*n1 < n2 < 0
3
gcAT1 > gcAT2
gcFT> gcFT1
2
AT
FT
gc1 < gc1
gcAT2 > gcFT
2

1n2 < 1−α−βn1 is imposed.
*
These cases are unsustainable in the long run.
**
Pattern 2 is the case in which Home diversifies and Foreign specializes in agriculture. Pattern 3 is the case in which Home andForeign specialize in manufacturing and agriculture, respectively.
†
Pattern 2 is obtained if γ ≤ 1/2, while Pattern 3 is obtained if 1/2 < γ.
1
♠ 1−αn < n2 < 1−α−β
n1 .
1−α−β 1β•n + αn2 > 0.
1−α−β 1β••n + αn2 < 0. However, this case is unsustainable in the long run.
1−α−β 1β⋆
αn1 + 1−α−βn2 < 0.
β⋆⋆
αn1 + 1−α−β n2 < 0♢

Negative growth

Case
Population growth
Trade pattern**
Relationshipbetween g1 and g2

Table 1: Comparisons of autarky and free trade growth rates