**Theoretical Economics Letters**

Vol.05 No.02(2015), Article ID:55137,9
pages

10.4236/tel.2015.52021

Notes on Economic Growth with Scale Effects: Is Depopulation Compatible with Growth?

Masao Nakagawa^{1}, Asuka Oura^{2}, Yoshiaki Sugimoto^{3*}

^{1}Graduate School of Social Sciences, Hiroshima University, Higashi Hiroshima,
Hiroshima, Japan

^{2}Graduate School of Economics, Osaka University, Toyonaka, Osaka, Japan

^{3}Faculty of Economics, Kansai University, Suita, Osaka, Japan

Email: ^{*}sugimoto@kansai-u.ac.jp

Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 2 February 2015; accepted 18 March 2015; published 27 March 2015

ABSTRACT

This research develops a simple theory to analyze the compatibility of depopulation and sustain- able growth. By introducing the scale effect of aggregate rather than average human capital, it shows that the economy may enter a sustainable growth path with fertility recovery, keeping away from a non-Malthusian poverty trap.

**Keywords:**

Scale Effect, Depopulation, Human Capital, Growth

1. Introduction

Over the last few decades, there has been a strand of theoretical literature studying
the role of population in economic growth. One of its main arguments is that population
expansion improves total factor productivity through various channels, including
the increase in potential inventors (Kremer [1] ) and R&D activities (Romer
[2] , Aghion and Howitt [3] ). The size of population is proposed as a key determinant
for the level of output per worker, although its effect on the growth rate may vanish
in the long run.^{1}

Such theories typically build on the condition that population increases in size
over time. Certainly, they can incorporate population aging as long as fertility
rates are above replacement level.^{2} However, this restriction will not
be appropriate for some advanced economies in the future. According to the United
Nations ([6] , p. 11), the world between 2005 and 2010 has as many as 75 countries
or areas whose total fertility levels are below replacement level. Germany, Italy,
and Japan will enter the phase of population decline over the decades up to 2050
(ibid., pp. 61-62). Because depopulation would exert adverse effects on the growth
process, it is not theoretically apparent whether these economies will sustain growth
in the long run.

Existing studies are unsatisfactory in this respect. Although the possibility of
sustainable growth against depopulation is demonstrated by Dalgaard and Kreiner
[7] and Strulik [8] , they treat demographic factors exogenously and thus are silent
about the potential interaction between the fertility decision and the living stand-
ard. Taking this approach neglects the feedback effect of economic growth on population
and fails to examine the possibility of population recovery.^{3}

Equally important, it is questionable whether their result-steady growth in output per worker accompanied by permanent depopulation-should be interpreted as economic growth. These issues are not addressed even in a seminal work by Strulik et al. [10] , which would be the most relevant to the present paper. They aim to account for the past and future dynamics of an economy in which fertility decisions and private R&D activities are en- dogenous. Although the fertility rate they calibrate exhibits an upward recovery to replacement level, it is gene- rated by an exogenous parameter change in the development process. Furthermore, they omit the possibility of stagnation resulting from depopulation by taking the entrance to a persistent growth path as historically inde- pendent. The present paper, in contrast, conducts a global dynamic analysis to present the existence of a non- Malthusian, catastrophic poverty trap, toward which population keeps shrinking and productivity goes into a stall.

Motivated by these observations, this research offers a theoretical framework for
analyzing the compatibility of economic growth with depopulation. It develops a
simple and tractable dynamic model of an economy that exhibits the following features.^{4}
First, households face a quantity-quality trade-off in child rearing. Fertility
decline results from a rise in education investment or, alternatively, from a decline
in parental income. Second, the amount of new technology is assumed to depend on
the level of aggregate rather than average human capital. This formulation, along
with the quantity-quality trade-off, implies that education investment does not
nece- ssarily accelerate technological progress. Third, a rise in the technology
level in turn augments skills by pro- viding new knowledge/ideas to the young generation.^{5}

Under such circumstances, average human capital increases over time and thus aggregate human capital exhibits faster growth than working population. Consequently the initially depopulating economy may sustain growth away from the non-Malthusian poverty trap, depending on the initial conditions on technology and human capital. The associated income growth may ultimately push the average fertility above the replacement level. The possibility of the fertility recovery is confirmed by a numerical analysis.

2. The Model

The economy has a one-sector, overlapping-generations structure and operates over an infinite discrete time horizon,. It is small in size and open to global capital markets, where the interest rate is stationary at.

2.1. Firms

In perfectly competitive environments, firms generate a final good by employing physical capital and human capital (i.e., labor in efficiency units). Let and, respectively, be the aggregate levels of these factor inputs in period t. Further, let be the level of labor-augmenting technology in period t. Their relation- ship with the level of aggregate output in period t, , is expressed by a neoclassical production function F such that

(1)

where and. The price of the final good is normalized to unity. As a result of profit maximization by price-taking firms, and maximize the aggregate profit , where and denote the rental price of physical capital and the wage rate per unit of efficiency labor, respectively, in period t. For simplicity, physical capital is assumed not to depreciate, so that the rental price equals the global interest rate r through arbitration. Then, it follows that where.

2.2. Households

A new generation is born at the beginning of each period and lives for three periods. Generation t, born in period, comprises a continuum of identical individuals existing on the interval.

Consider the lifetime of an individual of generation t. In the first period, the
individual engages in skill acqui- sition possibly with parental support. In the
second period, he/she acquires
efficiency units of labor and supplies all of them through the labor market. The
earned wage income
is allocated between saving and child rearing. The adult individual raises
units of identical children. Raising a child incurs
units of real expenditure, where
and
are the basic and education cost, respectively, in terms of effici- ency units of
labor.^{6} Capital and interest are used for consumption in the post-retirement
period, such that no bequests are left to the offspring. To summarize, the budget
constraint is

(2)

The utility of an individual of generation t, , depends not only on consumption in elderhood but also on aggregate income of his/her children. Each of these children acquires efficiency units of labor in period. Taking these into account, the utility function is formulated as

(3)

where measures the degree of parental altruism.

The level of efficiency units of labor hinges on two factors: the levels of education and technology. Human capital is augmented by technology, which embodies knowledge and ideas, on the grounds that their availability improves the efficiency of education. This is referred to as the skill-augmenting effect. The formation of human capital is formulated as

(4)

where,
and
^{7}
The function h exhibits diminishing marginal productivity with respect to education
and is increasing in technology. The positive cross derivative indicates their complemen-
tarity in skill formation. The second-last property (i.e., the first Inada condition)
precludes the existence of a corner solution at
This preclusion would be plausible considering the present paper’s focus on ad-
vanced stages of economic development.

As price takers, parents maximize their own utility by allocating resources between consumption, the quantity of children, and education for them. Substituting Equations (2) and (4) into Equation (3), the maximization problem faced by a member of generation t becomes

(5)

subject to. The logarithmic form implies that.

In terms of, the objective function is strictly concave and the first-order condition yields

(6)

implying that is the resource allocated to child rearing. Childbirth is encouraged by a rise in parental skill (i.e., the income effect) and is discouraged by a rise in the education cost (i.e., the substitution effect).

Substitution of Equation (6) into Equation (5) reveals that the necessary and sufficient condition for an in- terior solution which is unique, is given by

(7)

noting that Since the existence of the solution is ensured by the Inada condi- tions on, the implicit function theorem establishes that the optimal education choice is given by a continuously differentiable, single-valued function

(8)

where
^{8}

2.3. Aggregate Human Capital and Technological Progress

The working population in period is given by. Thus, substituting Equation (8) into Equations (4) and (6), the level of aggregate human capital in period can be expressed as

(9)

where
is historically given and, for any
and.
The last pro- perty,
reflects
only the direct effect of knowledge/ideas on human capital (i.e., the aforementioned
skill-augmenting effect) because the effect of any marginal change in
on skill formation is neutralized by its opposing effect on fertility.^{9}

As mentioned in the introduction, the present model abstracts from microeconomic
foundations that account for the innovation process. Suppose that the creation of
new technology is a by-product of manufacturing final output and depends on the
level of aggregate rather than average human capital, on the ground that more skilled
labor would come up with more ideas. Specifically,
where
the parameter
measures the degree of learning by doing.^{10} It follows from Equation
(9) that the evolution of technology is

(10)

where is historically determined.

Equations (9) and (10) constitute a two-dimensional, first-order autonomous system for and. Its notable feature is a dynamic interaction between technology and aggregate human capital. The level of technology in period t, , affects not only but also aggregate human capital in the subsequent period, through the skill-augmenting effect. The resulting amount of in turn determines through learning by doing.

3. The Dynamical System

This section explores the joint evolution of technology and aggregate human capital. As will be apparent, the initial condition on nails down the future path of, and thus, those of the other en- dogenous variables.

First, as follows from Equations (9) and (10),

(11)

Second, Equation (10) reveals that for any

Third and finally, Equation (9) reveals that for any

(12)

where
is a unique value such that
To ensure the existence of
suppose that the growth factor of
varies with
so remarkably that^{11}

(A1)

Proposition 1 clarifies two sufficient conditions on, one for convergence and the other for diver- gence.

Proposition 1. Under Equation (A1),

(13)

where, is a fixed value, and

Proof. See Appendix A.

The phase diagram in Figure 1 graphically represents the properties of the dynamical system derived thus far. The upper and lower boundaries of the shaded area are the sets of for which the first and second con- ditions of Proposition 1 are satisfied with equality, respectively. A stationary-state equilibrium occurs at any point on the horizontal line, where and In particular, the subset is referred to as the non-Malthusian poverty trap.

Any initial pair
below the shaded area converges to one of the stationary-state equilibria in the
trap.^{12} Note that the fall into the trap is not due to the Malthusian
mechanism: the saturation of population under the resource constraint. A large population
size is rather growth-promoting in the developed stages considered herein. The fall
is caused by scarce initial endowments, as they restrain the aforementioned dynamic
interaction between technology and aggregate human capital. As long as
is given above the shaded area,
eventually
exceeds
and then the evolution of
displays a U-shaped turnaround.^{13} The future path is

Figure 1. Global Dynamics of Technology and Human Capital. Notes: The diagram depicts the dynamical system on of the A-H plane. The horizontal line is a set of stationary-state equilibria. The economy may fall into the subset Non-Malthusian Poverty Trap or enter an explosive path, depending on whether the initial pair lies below or above the shaded area.

analytically ambiguous when the economy launches on the shaded area.

4. Analyses

4.1. The Hurdle to Sustainable Growth

As shown above, is the critical value that triggers the accumulation of aggregate human capital. In this sense, is the hurdle to prosperous, sustainable growth. Since by definition, it follows from Equa- tion (9) that depends on two structural parameters: and which respectively identity the degree of intergenerational altruism and the fixed cost of child rearing.

Either a decrease in or an increase in lowers the growth factor of aggregate human capital for a given. The resulting rise in makes it more difficult for the economy to enter on the explosive path. In light of the first condition of Equation (13), a rise in expands the domain of on which the economy falls into the non-Malthusian poverty trap in Figure 1. Even if the economy is initially on the ex- plosive path, its growth process may be averted by a permanent change in those parameters. For instance, the degree of parental altruism might be affected by the growth of nuclear families and a shift in social norms through international exchanges. The fixed cost of child rearing would be higher as a result of natural disasters and deterioration in the quality of public services.

4.2. Growth and Depopulation

This subsection investigates population dynamics underlying the growth process. In line with Equation (17) in Strulik et al. [10] , aggregate human capital is decomposed into average human capital and working population when expressing the growth factor:

(14)

where and Noting Equation (10), we can see that at least either or needs to grow in the long run to sustain productivity growth at a positive rate.

In view of Equations (4) and (8), the dynamic behavior of depends on how education investment reacts to technological progress. The analysis below considers a strong complementarity between education and tech- nology in skill formation. Formally,

(A2)

The complementary relationship in Equation (A2) generates a stimulative effect of
technological progress on education investment; that is to say,^{14}

(15)

These results, along with Equation (11) showing monotonic technological progress, reveal that and, therefore,

(16)

Thus, working population decreases as long as aggregate human capital decreases,
or equivalently, as long as
is below
in Figure 1.^{15} On the other hand, it
does not necessarily begin growing at the onset of the accumulation of aggregate
human capital, which occurs when
exceeds.
These indicate that initial de- population does not necessarily block the way to
sustainable growth.

While Equation (16) indicates the possibility of initial depopulation on an explosive
path, it is not apparent whether or not such a demographic trend continues. One
certain fact from Equation (14) is that the onset of population expansion is inevitable
if average human capital
converges toward a certain level on the path.^{16} This case is brought
about by a bounded production function of human capital satisfying Equation (A1).
In the long-run, depopulation is compatible with productivity growth only if average
human capital keeps increasing.

4.3. The Dynamic Trend in Fertility

As mentioned earlier, it is generally not clear whether or not reaches the replacement level, and it is analytically difficult to characterize its dynamic behavior. The difficulty stems from two counteracting effects of technological change on fertility. In light of Equations (4), (6), and (8),

(17)

Given Equation (15), a change in enhances both directly and indirectly, generating the income effect on. On the other hand, an increase in shifts resources from the quantity to the quality of children (i.e., the substitution effect). These forces would make the dynamic trend in fertility, analytically ambiguous.

A quantitative prediction of fertility is depicted in Figure 2, where one time period equals 30 years. The observed recovery to replacement level occurs under the following conditions. First, the production function of human capital is specified as

(18)

where
and
is interpreted as the level of basic skills any individual acquires in adulthood.
Note that the function exhibits the property in Equation (A2): strong complementarity
between education and technology in skill formation.^{17} Second, initial
values of population and human capital are normalized to one, that is,
and
Third, the structural parameters and initial condition on technology are such that

Figure 2. A Quantitative Prediction of Fertility.

and,
which are calibrated to match the em- pirical evidence in developed countries.^{18}

Under these circumstances, the initial level of average fertility is, which is consistent
with the experience of more developed regions from 1990 to 1995 (United Nations
[6] , p. 12).^{19} Per worker output
increases twofold in 30 years, implying an annual growth rate of about 2.34%.

5. Concluding Remarks

The growth theory developed above has demonstrated that in the presence of a scale effect of human capital, the initial conditions on technology, population, and human capital determine whether an economy undergoing depopulation enters a steady-growth path, along which population growth may ultimately turn positive. The possibility of falling into the poverty trap is explained by initial depopulation, which depresses the scale effect on productivity growth. In addition to the main result, a permanent decline in parental altruism or in the fixed cost of child rearing raises the hurdle to sustainable growth and may thereby divert the economy from the pro- sperous path.

Without taking a unified-approach, the present paper focuses on the developed economy whose initial fertility rate is below replacement level. As such, the unsolved questions are how the initial conditions are determined and why they are different among advanced countries. In order to answer them, it is necessary to extend the model and consider the process of fertility decline from a longer-term perspective. This theme is left for future research.

Acknowledgements

The authors are grateful to Koichi Futagami, Ken Tabata, and the participants of the 8th Conference of Macroeconomics for Young Professionals (2014, Osaka), the 2014 Asian Meeting of the Econometric Society, and the 2014 Australia Meeting of the Econometric Society, for their useful comments and encouragement. This research benefited from a Grant-in-Aid for JSPS Fellows (26-3695).

References

- Kremer, M. (1993) Population Growth and Technological Change: One Million B.C. to 1990. Quarterly Journal of Economics, 108, 681-716. http://dx.doi.org/10.2307/2118405
- Romer, P.M. (1990) Endogenous Technological Change. Journal of Political Economy, 98, S71-S102. http://dx.doi.org/10.1086/261725
- Aghion, P. and Howitt, P. (1992) A Model of Growth through Creative Destruction. Econometrica, 60, 323-351. http://dx.doi.org/10.2307/2951599
- Jones, C.I. (1999) Growth: With or Without Scale Effects? American Economic Review: Papers and Proceedings, 89, 139-144. http://dx.doi.org/10.1257/aer.89.2.139
- Prettner, K. (2013) Population Aging and Endogenous Economic Growth. Journal of Population Economics, 26, 811- 834. http://dx.doi.org/10.1007/s00148-012-0441-9
- Department of Economic and Social Affairs, Population Division, United Nations (2013) World Population Prospects: The 2012 Revision, Volume I: Comprehensive Tables ST/ESA/SER.A/336. http://esa.un.org/wpp/Documentation/pdf/WPP2012_Volume-I_Comprehensive-Tables.pdf
- Dalgaard, C.J. and Kreiner, C.T. (2001) Is Declining Productivity Inevitable? Journal of Economic Growth, 6, 187-203. http://dx.doi.org/10.1023/A:1011343715594
- Strulik, H. (2005) The Role of Human Capital and Population Growth in R & D-Based Models of Economic Growth. Review of International Economics, 13, 129-145. http://dx.doi.org/10.1111/j.1467-9396.2005.00495.x
- Department of Economic and Social Affairs, Population Division, United Nations (2014) World Population Prospects: The 2012 Revision, Methodology of the United Nations Population Estimates and Projections. Working Paper No. ESA/P/WP.235. http://esa.un.org/wpp/Documentation/pdf/WPP2012_Methodology.pdf
- Strulik, H., Prettner, K. and Prskawetz, A. (2013) The Past and Future of Knowledge-Based Growth. Journal of Economic Growth, 18, 411-437. http://dx.doi.org/10.1007/s10887-013-9098-9
- Galor, O. and Weil, D.N. (2000) Population, Technology, and Growth: From Malthusian Stagnation to the Demographic Transition and Beyond. American Economic Review, 90, 806-828. http://dx.doi.org/10.1257/aer.90.4.806
- Jones, C.I. (1995) R & D-Based Models of Economic Growth. Journal of Political Economy, 103, 759-784. http://dx.doi.org/10.1086/262002
- Romer, P.M. (1986) Increasing Returns and Long-Run Growth. Journal of Political Economy, 94, 1002-1037. http://dx.doi.org/10.1086/261420
- Maddison, A. (2006) The World Economy, Volume 1: A Millennial Perspective. The OECD Development Centre, Paris (Originally Published in 2001). http://dx.doi.org/10.1787/9789264022621-en
- Haveman, R. and Wolfe, B. (1995) The Determinants of Children’s Attainments: Findings and Review of Methods. Jour- nal of Economic Literature, 33, 1829-1878.
- de la Croix, D. and Doepke, M. (2003) Inequality and Growth: Why Differential Fertility Matters. American Economic Review, 93, 1091-1113. http://dx.doi.org/10.1257/000282803769206214

Appendix

A. Proof of Proposition 1

The properties of in Equations (9) and (A1) ensure the existence of a unique value such that. Since is the stock of inventions up to period t, Equations (9)-(11) reveal that for

(A.1)

where and Then, it follows that if in period then

where Thus, given the first condition in Equation (13),

and then This implies that; otherwise, occurs in a period t.

Under the second condition in Equation (13), because

from Equation (A.1).

B. Local Analysis of the Poverty Trap

From Equations (9) and (10),

where This dynamical system has a steady-state equilibrium where A linear approximation in the neighborhood yields

where and

Thus, it follows that and

Now one finds that and

where the function is from Equation (13). Therefore, converges to provided that .

NOTES

^{*}Corresponding author.

^{1}Jones [4] categorizes major R&D-based growth models, including the
pioneer works referred in the text, by the effectiveness of scale effects on the
growth rate of per capita income. He also argues that scale effects on the level
of per capita income are at work in all of those models.

^{2}By utilizing the existing endogenous growth models, Prettner [5] investigates
the effect of population aging on long-run growth performance. However, population
aging is not accompanied by a population decline in his model. The model developed
below makes no distinction between these two types of demographic changes.

^{3}The rise in fertility is consistent with a forecast of the United Nations,
which is made by a time series model based on national experience (United Nations
[9] , p. 17). In “more developed” regions, including Europe, Northern America, Australia,
New Zealand, and Japan, total fertility turns upward around the year 2000, approaching
to replacement level (United Nations [6] , p. viii and p. 12). While Strulik et
al. ([10] , p. 432) demonstrate fertility recovery by controlling a weight parameter
for children in utility, in the model developed below, the future trend in fertility
depends on the initial conditions.

^{4}While this paper extends the growth theory developed by Galor and Weil
[11] , it does not take their unified approach. The focus here is on the modern
era in which advanced economies have already experienced a demographic transition.
Thus, the poverty trap presented in this paper does not describe the Malthusian
stagnation, which results from population expansion under resource constraints.

^{5}The scale effect on the growth rate of technology becomes negligible
in the long run provided that the skill-augmenting effect is bounded above (cf.
Footnote 13). The dissipation of the scale effect is in line with Jones [12] , whose
model is more empirically plausible in this aspect than previous R&D-based growth
theories.

^{6}Measuring the child rearing costs in labor, rather than in time, is
one of the crucial deviations from the model of Galor and Weil [11] . Such an extension
generates an income effect on fertility, which is the potential force of population
recovery in developed stages.

^{7}In what follows,
denotes
the partial derivative of f with respect to x. Similarly,
and
denote the second and cross derivatives, respectively.

^{8}Given the properties of
in Equation (4),
and
where the inequality holds because the difference in the square brackets is negative
by concavity and strictly decreasing in.

^{9}Recall that Equation (7) is the condition for

^{10}This specification is viewed as a discrete counterpart of Equation
(9) proposed by Jones [12] , in which R&D exhibits no externalities. Taking
into account the long time interval of the OLG economy, it would be plausible to
consider that
depends on
rather than.
The resulting dependence of
on
yields increasing returns to scale in line with Romer [13] .

^{11}As a result of calibration based on the G-7 data, Strulik et al. ([10]
, pp. 421 and 429) impose a similar restriction to the second inequality in Equation
(A1). Unlike in the present model, however, their restriction makes the transition
to the prosperous path automatic regardless of the initial conditions.

^{12}A linear approximation reveals that
converges to a stationary state-equilibrium
where
provided that
See Appendix B for the proof.

^{13}In the case of divergence, the growth rate of technology converges
to a certain level if
Since Equations (9) and (10) yield
where,
it follows that
and.
This outcome is consistent with the dissipation of scale effect asserted by Jones
[12] .

^{14}Applying the implicit function theorem to Equation (7),
where
Equations (7) and (A2) imply that the numerator is negative, whereas

^{15}Negative population growth with low technology appears to be inconsistent
with the historical experience of most economies, whose population has been expanding
(cf. Maddison [14] , p. 241). As mentioned in the introduction, however, the focus
here is on the contemporary period in which advanced economies begin aging, and
encompassing the demographic patterns over the past millennia is beyond the scope
of the present research. For this reason, the initial technology level
is considered to be above a certain level, even though
may be smaller than.

^{16}Strulik et al. [10] preclude such a case by setting the productivity
parameter for education investment so large that
(cf. Footnote 11). This condition is crucial, but not sufficient, for the accumulation
of aggregate human capital against depopulation in the long run.

^{17}Equation (18) yields

^{18}In light of Haveman and Wolf ([15] , Table
1) and de la Croix and Doepke ([16] , p. 1099), we choose the US as a representative
developed country and assume that parents initially spend 7.25% of their potential
income on the fixed cost and 7.3% on child education.

^{19}The total fertility rate reported in the reference is 1.67. Note that
should be half of the total fertility rate in the real world because all adult individuals
give birth in this economy.