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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.

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}

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 ([

Existing studies are unsatisfactory in this respect. Although the possibility of sustainable growth against depopulation is demonstrated by Dalgaard and Kreiner [^{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. [

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.

The economy has a one-sector, overlapping-generations structure and operates over an infinite discrete time horizon,

In perfectly competitive environments, firms generate a final good by employing physical capital and human capital (i.e., labor in efficiency units). Let

where

^{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 [

^{7}In what follows,

A new generation is born at the beginning of each period and lives for three periods. Generation t, born in period

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 ^{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,

where

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^{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

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

In terms of

implying that

Substitution of Equation (6) into Equation (5) reveals that the necessary and sufficient condition for an in- terior solution

noting that

^{8}Given the properties of

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

^{10}This specification is viewed as a discrete counterpart of Equation (9) proposed by Jones [

where ^{8}

The working population in period

where ^{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, ^{10} It follows from Equation (9) that the evolution of technology is

where

Equations (9) and (10) constitute a two-dimensional, first-order autonomous system for

This section explores the joint evolution of technology and aggregate human capital. As will be apparent, the initial condition on

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

Second, Equation (10) reveals that for any

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

^{11}As a result of calibration based on the G-7 data, Strulik et al. ([

^{12}A linear approximation reveals that

^{13}In the case of divergence, the growth rate of technology converges to a certain level if

where ^{11}

Proposition 1 clarifies two sufficient conditions on

Proposition 1. Under Equation (A1),

where

Proof. See Appendix A.

The phase diagram in

Any initial pair ^{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 ^{13} The future path is

analytically ambiguous when the economy launches on the shaded area.

As shown above,

Either a decrease in

This subsection investigates population dynamics underlying the growth process. In line with Equation (17) in Strulik et al. [

where

In view of Equations (4) and (8), the dynamic behavior of

^{14}Applying the implicit function theorem to Equation (7),

^{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 [

^{16}Strulik et al. [

^{17}Equation (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

Thus, working population decreases as long as aggregate human capital decreases, or equivalently, as long as ^{15} On the other hand, it does not necessarily begin growing at the onset of the accumulation of aggregate human capital, which occurs when

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 ^{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.

As mentioned earlier, it is generally not clear whether or not

Given Equation (15), a change in

A quantitative prediction of fertility is depicted in

where ^{17} Second, initial values of population and human capital are normalized to one, that is,

^{18}

^{18}In light of Haveman and Wolf ([

^{19}The total fertility rate reported in the reference is 1.67. Note that

Under these circumstances, the initial level of average fertility is^{19} Per worker output

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.

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).

The properties of

where

Under the second condition in Equation (13),

from Equation (A.1).

From Equations (9) and (10),

where

where

Thus, it follows that

Now one finds that

where the function