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

International Trade and Industrialization with Negative
Population Growth

Hiroaki Sasaki

Discussion Paper No. E-12-009

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

December 2012

International Trade and Industrialization with
Negative Population Growth
Hiroaki Sasaki∗
December 2012

Abstract
This paper builds a small-open-economy, non-scale-growth model with negative population growth and investigates the relationship between trade patterns and per capita
consumption growth. Under free trade, if the population growth rate is negative and its
absolute value is small, the home country becomes an agricultural country. Then, thelong-run growth rate of per capita consumption is positive and depends on the world
population growth rate. On other hand, if the population growth rate is negative andits absolute value is large, the home country becomes a manufacturing country. Then,the long-run growth rate of per capita consumption is positive and depends on boththe home country and the world population growth rate. Moreover, the home country is
better oﬀ under free trade than under autarky in terms of per capita consumption growthirrespective of whether the population growth is positive or negative.
Keywords: non-scale-growth model; negative population growth; trade patterns; percapita 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 develops a small-open-economy, two-sector (manufacturing and agriculture),
non-scale-growth model in which the growth rate of population is negative and examines
the relationship between trade patterns and the long-run growth rate of per capita consumption.
In many developed countries, population growth is stagnant, and in some countries, population growth is negative.1 The existing economic growth theories assume positive population growth. However, given that population growth can be negative in reality, we need toconsider 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.2 As Ferrara (2011) and Christiaans (2011) show, incorporating
negative population growth in growth models is more complicated than replacing a positive
population growth rate with a negative population growth rate.
For example, Christiaans (2011) shows the importance of negative population growthusing a simple model. Consider a Solow growth model with a production function thatexhibits increasing, but relatively small, returns to scale.3 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, percapita income growth is positive and decreasing in the population growth rate.
In this paper, we focus on the so-called non-scale-growth model in which population
growth plays an important role in determining the growth rates of per capita income and consumption. The non-scale-growth model can overcome the shortcomings of the endogenous
growth model with scale eﬀects. In the endogenous growth model with scale eﬀects, longrun per capita income growth depends positively on the population level (Romer, 1990).
That is, population size positively aﬀects per capita growth. However, this assumption
seems counterfactual. Jones (1995) attempts to remove the scale eﬀects and presents a
non-scale-growth model in which the growth rate of output per capita depends positively on
the population growth rate and not on the population size.4 That is, the higher the population
1

For example, according to the Ministry of Internal Aﬀairs and Communications, as of March 2012, Japan
has experienced its largest-ever decline in population.
2
Ritschel (1985) argues that in the standard Solow growth model, a negative savings rate is necessary forthe existence of a steady state equilibrium with negative population growth. See also Felderer (1998).
3
In the Solow model with a constant returns to scale production function, per capita income growth is zero
when population growth is positive, whereas it is positive when population growth is negative. For details, see
Christiaans (2011).
4
Strulik (2005) introduces human capital accumulation into a growth model with endogenous technicalprogress due to R&D and shows that the growth rate of per capita consumption along the balanced growth path1

growth rate, the faster the country grows.5
Christiaans (2008) is an example of a study that investigates international trade andgrowth using a non-scale-growth model.6 He extends Wong and Yip’s (1999) small-openeconomy endogenous growth model to a non-scale-growth one. Wong and Yip’s model includes scale eﬀects, whereas Christiaans’ model does not. Using a non-scale-growth model,
Christiaans shows that the relationship between the trade patterns and per capita income
growth of the home country is determined by whether population growth is larger in thehome country or in the world.
Based on Christiaans’ (2008) model, we investigate the case where population growth isnegative. From our analysis, we obtain the following results.
First, when population growth is negative and its absolute value is small, the home country asymptotically completely specializes in agriculture in the long run. Per capita consumption growth depends positively on the world population growth rate, does not depend on thehome country population growth rate, and is larger than that under autarky.
Second, when population growth is negative and its absolute value is large, the homecountry completely specializes in manufacturing in the long run. Per capita consumption
growth depends on both the home country and the world population growth rates and islarger than those under autarky and in the above case.
The rest of the paper is organized as follows. Section 2 derives the equilibrium under autarky. Section 3 derives the equilibrium under free trade. In this analysis, we investigate thecase where the home country diversifies and produces both agricultural and manufactured
goods, the case where the home country completely specializes in manufacturing, and thecase where the home country asymptotically completely specializes in agriculture. Section
4 investigates the transitional dynamics with regard to whether the home country approaches
complete specialization in manufacturing or in agriculture. Section 5 concludes the paper.

depends positively or negatively on the growth rate of population.
5
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). In contrast, by estimating the population of the worldfrom one million B.C. to 1990, Kremer (1993) shows that the world population is positively correlated withthe growth rate of per capita income. This empirical fact supports the existence of scale eﬀects.
6
For other studies that examine international trade and economic growth using non-scale-growth models,see Sasaki (2008, 2011a, 2011b, 2012).

2

2 Autarky
Consider an economy with manufacturing and agricultural sectors. Let good 1 denote goods
produced in the manufacturing sector and good 2 denote goods produced in the agricultural
sector. Good 1 is used for both consumption and investment, and good 2 is used only for
consumption. Manufactured goods are produced with labor L1 and capital stock K, whereas
agricultural goods are produced with labor L2 . Both production functions are specified asfollows:
X1 = AK α L11−α ,= K α+β L11−α ,A = Kβ(1)0 < α < 1, 0 < β < 1, α + β < 1,(2)

X2 = L2 ,(3)

where X1 and X2 denote the production of goods 1 and 2, respectively. Here, A = K β captures
the externality due to capital accumulation and implies that manufacturing production has
increasing returns to scale. The assumption α + β < 1 means that the extent of the increasing
returns is not very large, which means our model exhibits non-scale growth. Moreover, theMarshallian externality is imposed so that profit-maximizing firms regard A as exogenously
given. Therefore, increasing returns to scale are consistent with perfect competition.
Suppose that labor supply is equal to the population L, which grows at a constant rate n,
and that labor is fully employed. Therefore, we have L1 + L2 = L. Moreover, we assume thatlabor is free to move across the two sectors. Accordingly, wages in both sectors are equal.
Let the agricultural goods be the numéraire. Moreover, let the wage rate, the profit rate,
and the price of manufactured goods be w, r, and p, respectively. From profit maximization,
we obtain the following equations:∂X1= p(1 − α)K α+β L1−α = 1,
∂L1
∂X1
r=
= αK α+β−1 L11−α .
∂K A:given

w=p

(4)
(5)

As long as agricultural goods are produced, the wage rate is w = 1.
For simplification, we make the classical assumption that wage income wL and capitalincome rpK are entirely devoted to consumption and saving, respectively. In the canonicalone-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 thumbfor consumers with dynamic optimization (Christiaans, 2008). In addition, we assume that3

consumers’ preferences take the Cobb-Douglas form.
max U = C1γC21−γ ,C1 ,C20 < γ < 1,

s.t. pC1 + C2 = wL.

(6)
(7)

Solving the above utility-maximizing problem, we find that a fraction γ of total wage income
is spent on good 1 and the rest 1 − γ on good 2.
pC1 = γL,

(8)

C2 = (1 − γ)L.

(9)

In addition, solving the expenditure-minimizing problem, we find that the consumer priceindex pc is given by pc = pγ .7
We assume that all savings are spent on investments. From our assumption, savings areequal to profits. Then the following equation is satisfied.
K̇
= r.
K

(10)

That is, the rate of capital accumulation is determined by the rate of profit.
The market clearing conditions for both goods are given by
X1 = C1 + I,

(11)

X2 = C2 .

(12)

By Walars’ law, one of these two equations is redundant.
From these equations, the short-run equilibrium is given by
L1 = γL,(13)

L2 = (1 − γ)L,
(γL)α
p=
.
(1 − α)K α+β

(14)
(15)

First, we consider the case where n > 0.
Along the balanced growth path (BGP, hereafter), the main variables grow at constantrates. The growth rate of capital stock is constant when the profit rate is constant. Calculating the growth rate of the profit rate from equation (5) and letting the resulting expression
Strictly speaking, the consumer price index leads to pc = γ−γ (1 − γ)−(1−γ) pγ . However, for simplicity, we
use pc = pγ because the constant term does not aﬀect the growth rate of pc that is needed for later analysis.
74equal zero, we obtain the following equation: (α + β − 1)gK + (1 − α)n = 0. From this, theBGP growth rate of capital stock g∗K is given byg∗K = ϕn > 0,

where ϕ ≡

1−α
> 1.
1−α−β

(16)

In the following analysis, g x = ẋ/x denotes the growth rate of a variable x, and an asterisk“∗” denotes the long-run value of a variable.
Other BGP growth rates are given byg∗X1 = gK = ϕn > 0,

(17)

g∗X2 = n,(18)g∗p = −(ϕ − 1)n < 0,(19)g∗c = g∗w − γg∗p = −γg∗p = γ(ϕ − 1)n > 0.

(20)

The relative price of manufactured goods is decreasing at a constant rate along the BGP.
When deriving gc , we use the fact that per capita real consumption c is equal to the real
wage w/pγ .
Considering the BGP growth rate of capital stock, we introduce a new state variable
k = K/Lϕ . When K grows at its BGP rate, k is constant. From equations (5) and (10), thedynamics of the scale-adjusted capital stock lead tok̇ = αγ1−α kα+β − ϕnk.

(21)

The steady state is where k̇ = 0. When population growth is positive, there exists a k > 0such that k̇ = 0, given by1( 1−α ) 1−α−β
αγ
k∗ =
.
(22)
ϕn
Moreover, this steady state is stable becausedk̇= −k∗ [(1 − α − β)αγ1−α (k∗ )α+β−2 + ϕn] < 0.
dk k=k∗(23)

Next, we consider the case where n < 0.
If n < 0, there never exists a k > 0 such that k̇ = 0, and we have k̇ > 0 for k > 0. That is,
if k > 0, then k diverges to infinity.
In this case, we examine the growth rate of per capita consumption. Considering thatper capita consumption is equal to the real wage measured in terms of the consumer priceindex, we obtaingc = α(α + β)γ2−α kα+β−1 − γαn.
5(24)From this, we have
lim gc = −γαn > 0.

k→+∞

(25)

From the above analysis, we obtain the following proposition.
Proposition 1. Suppose that the population growth rate is negative. Then, g∗c is decreasing
in n, and g∗c > 0 even though n < 0.

3 Free trade
We consider free trade between the home country and the world. Suppose that the homecountry is a small open economy. Suppose also that the world’s parameters are the same as
those of the home country except for the population growth rate and that the world is locatedon its BGP. If we denote nw > 0 as the average population growth rate of the world, then thegrowth rate of the terms of trade determined by the world market is given byg∗p = −(ϕ − 1)nw < 0.

(26)

We investigate the existence of a steady state when the home country diversifies, when it
asymptotically completely specializes in agriculture, and when it completely specializes inmanufacturing.

3.1 Diversification
As long as agricultural goods are produced, we have w = 1. From the goods market clearing
condition, we obtain the growth rate of capital stock.
β
K̇
1−α
1−α
= α(1 − α) α K α p α .
K(27)Along the BGP, we haveg∗K = −1−αg p = ϕnw > 0.
β(28)Considering the BGP growth rate of capital stock, we introduce a new variable k̃ =K p . The dynamic equation of k̃ is given by1−αβ1−α α+βk̃˙ = α(1 − α) α k̃ α − ϕnw k̃.

6

(29)

There exists a steady state value k̃ > 0 such that k̃˙ = 0.
[
] αβ
ϕnw
k̃e =
.
1−α
α(1 − α) α(30)

However, this steady state is unstable because dk̃˙ /dk̃ k=k̃ > 0, and hence, k̃ converges to zeroeor diverges to infinity.

3.2 Specialization in agriculture
As long as agricultural goods are produced, we have w = 1. In the above diversificationcase, if k̃0 < k̃e , then k̃ approaches zero. Then, we can prove that X1 approaches zero whenk̃ approaches zero. Manufacturing production can be expressed by k̃ and p as follows:α+βX1 = (1 − α) α k̃ α p− β .
1−α

1−α

(31)

Here, we investigate limt→+∞ X1 . The rate of change in X1 is given byβϕẊ1 α + β k̃˙ 1 − α ṗ
1−α β
=
−
= (α + β)(1 − α) α k̃ α −
nw .
X1
α k̃
β p
α

(32)

When k̃ → 0, Ẋ1 /X1 approaches −(βϕ/α)nw , which is negative. Accordingly, in the longrun, we obtainlim X1 = 0.

t→+∞

(33)

Therefore, if k̃0 < k̃e , manufacturing production approaches zero, and consequently, thehome country asymptotically completely specializes in agriculture.
In this case, the home country becomes entirely agricultural, and capital stock is notaccumulated. Nevertheless, per capita consumption growth is not zero because consumptionof manufactured goods is met by imports and the relative price of these goods is decreasing at
the constant rate g∗p . Therefore, the BGP growth rate of per capita consumption approachesg∗c = −γg∗p = γ(ϕ − 1)nw > 0.

(34)

3.3 Specialization in manufacturing
When only manufactured goods are produced, the wage rate is given by w = p(1−α)K α+β L−α .
From the goods market clearing condition, the growth rate of capital stock is given byK̇
= αK α+β−1 L1−α .
K
7

(35)

In this case, the dynamics are described by k as in autarky and given byk̇ = αkα+β − ϕnk.

(36)

The same arguments as in autarky are applicable. If n > 0, there exists a steady state valueof k, which is given by() 1
α 1−α−β
kI =
,
ϕn

(37)

and the steady state is stable because dk̇/dk k=k < 0. On the other hand, if n < 0, there neverI
exists a steady state value of k > 0 such that k̇ = 0. Then, we have k̇ > 0 for k > 0.
If the home country continues to completely specialize in manufacturing, the growth rateof per capita consumption in the long run is given byg∗c = g p + (α + β)gK − αn − γg p
= −αn − (1 − γ)(ϕ − 1)nw .

(38)

4 Transitional dynamics
The analysis in the preceding section assumes that when switching from autarky to free
trade, one of the three trade patterns is realized. In this section, by contrast, using a phase
diagram, we investigate how trade patterns evolve through time by assuming that whenswitching from autarky to free trade, the home country diversifies.
As the foregoing analysis shows, the dynamics of our model are described by k̃ and k, so
we investigate the dynamics on the (k, k̃)-plain. Considering the definitions of k and k̃, we
obtain the following relationship between k and k̃.
α−1

k = k̃L−ϕ p β .

(39)

Log-diﬀerentiating both sides, we obtaink̇ k̃˙= + ϕ(nw − n).
k k̃

(40)

If we want to express the dynamics under diversification using k, we can substitute theequation for k̃˙ (i.e., equation (29)) into the right-hand side of equation (40). Similarly, if wewant to express the dynamics under complete specialization in manufacturing by using k̃,
we can substitute the equation for k̇ (i.e., equation (36)) into the left-hand side of equation(40).
8

From the above reasoning, we obtain the dynamic systems for diversification and forcomplete specialization in manufacturing.

1−α β



k̇ = α(1 − α) α k̃ α k − ϕnkDiversification : 
(41)
α+β


k̃˙ = α(1 − α) 1−αα k̃ α − ϕn k̃
w


α+β


k̇ = αk − ϕnkSpecialization in M : (42)k̃˙ = αkα+β−1 k̃ − ϕn k̃
w

Next, we derive the borderline that divides the (k, k̃)-plain into the diversification region
and the complete specialization region. This borderline corresponds to the combination of k
and k̃ such that X2 = 0. Expressing X2 = 0 using k and k̃, we have
[
k̃ =

1

] β1

(1 − α)1−α kα(1−α−β)

.

(43)

This is a downward sloping curve on the (k, k̃)-plain. The region above this curve corresponds to complete specialization in manufacturing, whereas the region below this curve
corresponds to diversification (X2 > 0).
We first consider diversification. When k̇ = 0, from the first part of equation (41), thefollowing relation holds.
β

1−αα(1 − α) α k̃ α − ϕn = 0.

(44)

However, no k̃ > 0 satisfies equation (44). On the other hand, when k̃˙ = 0, we have k̃ = k̃efrom the second part of equation (41).
Next, we consider complete specialization in manufacturing. When k̇ = 0, from the firstpart of equation (42), the following relation holds.
αkα+β−1 − ϕn = 0.

(45)

However, no k > 0 satisfies equation (45). On the other hand, when k̃˙ = 0, from the second
equation of equation (42), the following relation holds.
) 1
α 1−α−β
ke =
.
ϕnw
(

Summarizing the above results, we obtain figure 1.
[Figure 1 around here]

9

(46)

The transitional dynamics that start at point S1 in figure 1 are interesting, and we obtain
two results. First, starting from diversification, the home country then completely specializes in manufacturing, sooner or later begins to diversify, and finally asymptotically completely specializes in agriculture. Second, starting from diversification, the home country
completely specializes in manufacturing and continues to do so for all time.
Which of the two results is realized depends on whether the home country crosses theborderline. The second is obtained if the home country remains in the complete specialization region. For example, we consider point P in figure 2. The home country continues
to move southeast through time, and it either begins to cross the borderline and enter thediversification region or stays in the complete specialization region. For the home country to always stay in the complete specialization region, from equation (43), the following
condition must continue to be satisfied.
] β1
[
1
k̃ >
.
(47)
(1 − α)1−α kα(1−α−β)If we take the growth rates of both sides, we obtaingk̃ > −

α
gk .
ϕ−1

(48)

If complete specialization in manufacturing continues to hold, then from equation (42),
we have
gk = αkα+β−1 − ϕn > 0,(49)

gk̃ = αkα+β−1 − ϕnw .

(50)

Since k diverges to infinity in figure 2, at the limit, we obtainlim gk = −ϕn > 0,(51)

lim gk̃ = −ϕnw < 0.

(52)

k→+∞
k→+∞

Substituting these expressions into equation (48), we obtain−αn − (ϕ − 1)nw > 0 =⇒ n < −ϕ−1
nw ≡ n̄.
α(53)

As long as this condition is satisfied, once the home country specializes in manufacturing, it
continues to do throughout time.
The long-run growth rate of per capita consumption is given by equation (38). Since
0 < γ < 1, equation (38) is positive if equation (53) is satisfied. That is, the long-run growthrate of per capita consumption is positive. Moreover, it depends on both n and nw .
10[Figure 2 around here]
We can summarize the above results as the following propositions.
Proposition 2. Suppose that n̄ < n < 0. Then, the home country asymptotically completely
specializes in agriculture and per capita consumption growth depends on nw . On the other
hand, suppose that n < n̄. Then, the home country completely specializes in manufacturing
and per capita consumption growth depends on both n and nw .
Figure 3 summarizes the relationships between population growth and per capita consumption growth under autarky and free trade. The case where n > 0 under free trade is
obtained from Christiaans (2008).
[Figure 3 around here]
The solid line corresponds to autarky, and the dashed line corresponds to free trade.
When n < n̄ under free trade, the home country completely specializes in manufacturing in
the long run, and the growth rate of per capita consumption under free trade is larger thanthat under autarky. When n̄ ≤ n < nw under free trade, the home country asymptoticallycompletely specializes in agriculture, and the growth rate of per capita consumption underfree trade is larger than that under autarky. When nw ≤ n under free trade, the home country
completely specializes in manufacturing, and the growth rate of per capita consumption
under free trade is equal to that under autarky.8
Summarizing the above results, we obtain the following proposition.
Proposition 3. The growth rate of per capita consumption under free trade is equal to or
larger than that under autarky irrespective of the size and the sign of the growth rate of
population.

5 Conclusions
This paper has built a small-open-economy, two-sector, non-scale-growth model and investigated the relationship between trade patterns and per capita consumption growth. Importantly, we have assumed that the growth rate of population is negative. The main results are
as follows.
Both nw ≤ n and k̃0 > k̃e are the necessary and suﬃcient conditions for the home country to completely
specialize in manufacturing in the long run. Therefore, nw ≤ n is a necessary condition for complete specialization. For details, see Christiaans (2008).
8

11

Under autarky, if the population growth rate is negative, then the long-run growth rate of
per capita consumption is positive and decreasing in the population growth rate.
Under free trade, if the population growth rate is negative and its absolute value is small,then the home country becomes an agricultural country. Then, the long-run growth rate of
per capita consumption is positive and depends on the world population growth rate. On
other hand, if the population growth rate is negative and its absolute value is large, thenthe home country becomes a manufacturing country. Then, the long-run growth rate of
per capita consumption is positive and depends on both the home country and the world
population growth rate. Moreover, the long-run growth rate of per capita consumption islarger in the case where the home country becomes a manufacturing country than in the casewhere it becomes an agricultural country.
In addition, from the results of Christiaans (2008), we know that when population growthis positive, the long-run growth rate of per capita consumption under free trade is larger thanor equal to that under autarky.
Therefore, the home country is better oﬀ under free trade than under autarky in terms ofper capita consumption growth when population growth is both positive and negative.

References
Christiaans, T. (2008) “International trade and industrialization in a non-scale model of
economic growth,” Structural Change and Economic Dynamics 19 (3), pp. 221–236.
Christiaans, T. (2011) “Semi-endogenous growth when population is decreasing,” Economics Bulletin 31 (3), pp. 2667–2673.
Dinopoulos, E. and Thompson, P. (1998) “Schumpeterian growth without scale eﬀects,”
Journal of Economic Growth 3 (4), pp. 313–35.
Felderer, B. (1988) “The existence of a neoclassical steady state when population growth isnegative,” Journal of Economics 48 (4), pp. 413–418.
Ferrara, M. (2011) “An AK Solow model with a non-positive rate of population growth,”
Applied Mathematical Sciences 5 (25), pp. 1241–1244.
Howitt, P. (1999) “Steady endogenous growth with population and R&D inputs growth,”
Journal of Political Economy 107 (4), pp. 715–30.
Jones, C. I. (1995) “R&D-based models of economic growth,” Journal of Political Economy
103 (4), pp. 759–784.

12

Jones, C. I. (1999) “Growth: with or without scale eﬀects?” American Economic Review 89(2), pp. 139–44.
Kortum, S. (1997) “Research, patenting, and technological change,” Econometrica 65 (6),pp. 1389–419.
Kremer, M. (1993) “Population growth and technological change: one million B.C. to1990,” Quarterly Journal of Economics 108 (3), pp. 681–716.
Peretto, P. F. (1998) “Technological change and population growth,” Journal of Economic
Growth 3 (4), pp. 283–311.
Ritschel, A. (1985) “On the stability of the steady state when population is decreasing,”
Journal of Economics 45 (2), pp. 161–170.
Sasaki, H. (2008) “International trade and industrialization with capital accumulation andskill acquisition,” The Manchester School 76 (4), pp 464–486.
Sasaki, H. (2011a) “Trade, non-scale growth, and uneven development,” Metroeconomica
62 (4), pp. 691–711.
Sasaki, H. (2011b) “Population growth and north-south uneven development,” Oxford Economic Papers 63 (2), pp. 307–330.
Sasaki, H. (2012) “Trade patterns and non-scale growth between two countries,” Kyoto
University, Graduate School of Economics Research Project Center Discussion Paper
Series, No. E-12-006.
Segerstrom, P. (1998) “Endogenous growth without scale eﬀects,” American Economic Review 88 (5), pp. 1290–310.
Strulik, H. (2005) “The role of human capital and population growth in R&D-based models
of economic growth,” Review of International Economics 13 (1), pp. 129–145.
Wong, K.-Y., and Yip, C. K. (1999) “Industrialization, economic growth, and international
trade,” Review of International Economics 7 (3), pp. 522–540.
Young, A. (1998) “Growth without scale eﬀects,” Journal of Political Economy 106 (1), pp.
41–63.

13    %$              ! "  #      %Figure 1: Phase diagram when n < 0    

    

Figure 2: Long-run relationship between population growth and trade patterns

14




        "             ! "   








Figure 3: Long-run relationship between population growth and per capita consumption
growth15