This research develops a simple theory to analyze the compatibility of depopulation and sustainable 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.
Scale Effect Depopulation Human Capital Growth1. 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
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
6Measuring 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.
7In what follows, denotes the partial derivative of f with respect to x. Similarly, and denote the second and cross derivatives, respectively.
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
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
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
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
subject to. The logarithmic form implies that.
In terms of, the objective function is strictly concave and the first-order condition yields
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
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
8Given 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.
9Recall that Equation (7) is the condition for
10This 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] .
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
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
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),
Second, Equation (10) reveals that for any
Third and finally, Equation (9) reveals that for any
11As 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.
12A linear approximation reveals that converges to a stationary state-equilibrium where provided that See Appendix B for the proof.
13In 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] .
where is a unique value such that To ensure the existence of suppose that the growth factor of varies with so remarkably that11
Proposition 1 clarifies two sufficient conditions on, one for convergence and the other for diver- gence.
Proposition 1. Under Equation (A1),
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
Global Dynamics of Technology and Human Capital. Notes: The diagram depicts the dynamical system on <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-1500686x134.png" xlink:type="simple"/></inline-formula> 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 <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-1500686x134.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-1500686x135.png" xlink:type="simple"/></inline-formula> lies below or above the shaded area
analytically ambiguous when the economy launches on the shaded area.
4. Analyses4.1. 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:
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,
14Applying the implicit function theorem to Equation (7), where Equations (7) and (A2) imply that the numerator is negative, whereas
15Negative 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.
16Strulik 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.
17Equation (18) yields
The complementary relationship in Equation (A2) generates a stimulative effect of technological progress on education investment; that is to say,14
These results, along with Equation (11) showing monotonic technological progress, reveal that and, therefore,
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),
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
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
A Quantitative Prediction of Fertility
and, which are calibrated to match the em- pirical evidence in developed countries.18
18In 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.
19The 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.
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).
AppendixA. 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
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 .
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