اثر رشد در مقابل اثر سطحی تغییرات جمعیت بر توسعه اقتصادی : بازرسی در مکانیسم های مربوط به سرمایه انسانی

This paper studies the different mechanisms and the dynamics through which demography is channelled to the economy. We analyze the role of demographic changes in the economic development process by studying the transitional and the long-run impact of both the rate of population growth and the initial population size on the levels of per capita human capital and income. We do that in an enlarged Lucas–Uzawa model with intergenerational altruism. In contrast to the existing theoretical literature, the long-run level effects of demographic changes, i.e. their impact on the levels of the variables along the balanced growth path, are deeply characterized in addition to the more standard long-run growth effects. We prove that the level effect of the population rate of growth is non-negative (positive in the empirically most relevant case) for the average level of human capital, but a priori ambiguous for the level of per capita income due to the interaction of three transmission mechanisms of demographic shocks, a standard one (dilution) and two non-standard (altruism and human capital accumulation). Overall, the sign of the level effects of population growth depend on preference and technology parameters, but numerically we show that the joint negative effect of dilution and altruism is always stronger than the induced positive human capital effect. The growth effect of population growth depends basically on the attitude to intergenerational altruism and intertemporal substitution. Moreover, we also prove that the long-run level effects of population size on per capita human capital and income may be negative, nil, or positive, depending on the relationship between preferences and technology, while its growth effect is zero. Finally, we show that the model is able to replicate complicated time relationships between economic and demographic changes. In particular, it entails a negative effect of population growth on per capita income, which dominates in the initial periods, and a positive effect which restores a positive correlation between population growth and economic performance in the long term.

The relationship between demographic change and economic development is an important topic which has suggested a huge empirical and theoretical literature in both demography and economics. While correlations between certain economic and demographic variables may sound as obvious at first glance, a general conclusion from most of the empirical studies performed is that such correlations are far from compelling, which has opened an ongoing intense population debate. For example, Kelley and Schmidt (2001) (see also Kelley, 1988 and Kelley and Schmidt, 1995) report “a general lack of correlation between the growth rates of population and per capita output”, documented in more than two dozen studies. The same conclusion was reached by the demographer Lee (1983) two decades ago, who particularly pointed out the “inconclusivity” of cross-national studies. As mentioned by Kelley and Schmidt (2001), simple correlations between demographic and economic variables would be anyway difficult to interpret “…plagued as they are by failure to adequately account for reverse causation between economic and demographic change complicated timing relationships associated with the Demographic Transition, …, complexity of economic–demographic linkages that are poorly modeled, …, and data of dubious quality”. Other problems come from the data limitations which have led to simplified specifications of the relationship between demographic and economic change. For example, while the levels of physical and human capital stocks are a priori key variables in the analysis of the latter relationship, they are quite difficult to construct, specially for developing countries. Usually, proxies of their respective growth rates are incorporated in modified relationships in terms of variables’ growth rates. Even worse, a key variable like human capital, which sounds as the major variable connecting demographic and economic trends, is difficult to compile, be it in level or in growth rates. This paper is a theoretical contribution to the population debate outlined above. Concretely, we study the impact of population change on human capital and income in a traditional setting where growth is endogenously generated by human capital accumulation in line with the Lucas–Uzawa two-sector model. In doing so, we abstract from the very well-known reverse causation highlighted by Kelley and Schmidt (2001). As in standard endogenous growth models with infinite-lived representative agents, we keep demographics exogenous, summarized in two parameters, population size (N)(N) and population growth rate (n)(n). There are some quite popular models studying the relationship between population, human capital, and growth under the assumption of endogenous fertility, mostly based on the well-known quality–quantity of children trade-off. An overwhelming part of the latter literature uses overlapping-generations models (see for example Nerlove et al., 1985). Here, we choose to investigate the demographic–economic link in a standard endogenous growth model with infinite-lived agents, and as most demographers we do not incorporate any form of the traditional quality–quantity trade-off into the analysis; that is we keep fertility exogenous. As we shall see throughout the paper, our framework with exogenous demographics is already extremely complicated. The reference model is the Lucas (1988) two-sector model of endogenous growth with physical and human capital stocks, which distinguishes between the number of individuals (population) and the quality of individuals (human capital). Such a model endogenizes quality but leaves the number to follow an exogenous process. Human capital may be considered under different perspectives as knowledge, education, or experience and on the job training. Hence, knowledge and skills embodied in people are the cause of advances in technological and scientific knowledge, which in turn fosters economic development. We focus on the relationship between population, human capital, and growth by studying the impact of population growth and size on the long-run level (level effect)1 and the rate of growth (growth effect)2 of human capital and income per capita. We also study the transitional dynamics in an attempt to distinguish between the short-run and long-run effects of population growth and size on economic performance, and to uncover part of the “complicated timing relationships” pointed out by Kelley and Schmidt (2001). Moreover, the basic model has been enlarged to include the Benthamite principle of maximizing total utility (classical utilitarianism) and the Millian principle of maximizing per capita utility (average utilitarianism) as the two polar cases of social welfare criteria, in line with Palivos and Yip (1993) and Razin and Yuen (1995). Our contribution is therefore threefold. In first place, we do not restrict our analysis to the relationship between demographic and economic growth rates as it is the case in the related theoretical literature and in the vast majority of the empirical work. We also study the relationship between income and human capital levels and demographic parameters. Mankiw et al. (1992) do consider a two-sector growth model with physical and human capital and do estimate the shape of the relationship between the level of income per capita and the population growth rate. However, this was done in an exogenous growth setting with exogenous saving rates. When one turns to optimization-based endogenous growth theory with infinite-lived agents, our paper is the first which goes beyond the typical analysis of the link between demographic and economic growth rates. Strulik (2005) and Bucci (2008), among others, do study the latter link in proper endogenous growth frameworks but they do not account for any level effect. Investigating the impact of demographic change on the level of income per capita seems however a necessary task, especially if one is concerned with development policies in developing countries where level measures are generally much more meaningful than growth rate indicators, as argued by Parente and Prescott (1993). The main reason why this task has not been undertaken so far in the class of endogenous growth models is technical: long-run levels are undetermined along balanced growth paths, only growth rates and ratios of variables are identifiable along these paths (see for example, chapter 5 of Barro and Sala-i-Martin’s textbook, Barro and Sala-i-Martin (1995), devoted to the Lucas–Uzawa model). Typically, those long-run levels depend on initial conditions, therefore implying that uncovering the long-run levels requires the characterization of transitional dynamics, a daunting task for non-AK endogenous growth models. In our enlarged Lucas model, we rely on Boucekkine and Ruiz-Tamarit (2008) who produced analytical solutions to the Lucas–Uzawa model, to extract the closed-form expressions corresponding to the optimal paths of all variables in level. The analytical solutions make use of a specific class of special functions, the so-called Gaussian hypergeometric functions, which naturally result from the resolution of the dynamic system formed by the first-order conditions. Because of the omnipresent special functions, comparative statics with respect to the demographic parameters, while possible, are very complicated to handle analytically. Second, we show that population size, that is the scale of the economy, is also an important determinant of economic performance. Precisely, we show that the size of the population affects the levels of income and human capital but not the long-run rate of growth of the economy, which in contrast depends on the population growth rate. As outlined by Kelley and Schmidt (2001), “…curiously, even though studies in the economic–demographic tradition have long harkened the importance of population size and density, these influences have been strikingly missing in empirical growth in recent decades”. The quite thin related literature points at a generally significant impact of population size and density on economic performance although it varies a lot across places and over time (Kelley and Schmidt, 2001). Our theoretical analysis identifies a nonzero level effect of population size while its growth effect is shown to be nil, which is to our knowledge the first characterization in the related theoretical literature. Last but not least, using the closed-form solution paths we are able, along with Kelley and Schmidt, 1995 and Kelley and Schmidt, 1996, to distinguish between short and long term effects of population growth and size on economic performance. This is a valuable exercise if one has in mind the ongoing population debate. Pessimistic theories of population growth would emphasize its short term adverse impacts given the apparent fixity of resources and diminishing returns. Optimistic theories would rather take a long term perspective where the short-run costs of population growth are counterbalanced by benefits. Therefore, having the possibility to compare short and long term level and growth effects of a demographic change is extremely worthwhile to deliver a global picture of the demographic–economic nexus. Again this contribution is quite original since most endogenous growth theories only focus on the long-run results. Empirical literature on the relationship between human capital and population, and the population scale effect The interaction between population and human capital has been quantitatively studied at family and country levels. At family level, it has been shown that beyond a fixed family size, extra children are associated with lower average educational attainments, worse nutritional standards, and a lower spending on health services (King, 1985 and Birdsall, 1977). Kelley (1996) reviews the available evidence from empirical studies and suggests that additional children reduce the years of schooling completed by other children in the household, although the size of this effect is usually small. In fact, the negative effect of larger families on the quantity of human capital is not always found, or may it not be statistically significant. For example, Mueller (1984) presents evidence from Botswana and Sierra Leone that children from larger families achieve higher average levels of schooling, controlling other pertinent variables. However, Birdsall (1977) points out that children from large families do less well in test intelligence, that mothers’ health is negatively affected by pregnancies, especially among poor women, and that large families adjust to economic constraints transferring the burden on the children in the form of a declining quantity and quality of food and medical cares. At the aggregate level, the empirical evidence also shows an uncertain effect of demographic change on human capital accumulation measured by enrollment rates, years of school attainment of adults, school dropout rates, the student–teacher ratio, and scores on international examinations. For example, Schultz (1987) and Kelley (1996) find that rapid population growth is relatively unimportant in explaining the increased quantity of education (enrollment and attainment rates); however, it seems that it reduces the quality of the education provided, as it increases the student–teacher ratio and decreases the government expenditures per school-age child, mainly at the secondary level and during the sixties and the seventies.3 Concerning the population scale effect one would conclude, from the literature quoted above on the impact of family size on human capital level, that it is not less disputed. As mentioned by Kelley and Schmidt (2001), population size has been traditionally viewed as a positive factor of long-run growth in countries with abundant resources, strong institutions, and relatively low population densities. However, the latter conditions are seldom met, notably in developing countries. The empirical literature is rather thin on this question. Most of the existing related papers focus on the agricultural sector (see for example Pingali and Binswanger, 1987) where the impact of larger population densities on the efficiency of transportation and irrigation can be more directly apprehended. Still the available studies show a great variability in their conclusions (see again Kelley and Schmidt, 2001). More recently, some authors have studied whether “small states” have specific properties in terms of the development pattern. Among them, Easterly and Kraay (2000) have found that, controlling for location, smaller states are actually richer than other states in per capita GDP. That is, there exists a negative correlation between population size and level of income per capita. However, they have also found that small states do not have different per capita growth rates, therefore concluding that population size looks uncorrelated with per capita growth rates. We shall show that our enlarged Lucas model displays a similar picture. Relation to the theoretical literature We now briefly review the related theoretical literature. As mentioned above, an early contribution is due to Mankiw et al. (1992) who consider a two-sector exogenous growth model, which by definition cannot give rise to a growth effect. According to this model, population size does not affect the long-run levels of both per capita human capital and income. But the model predicts a negative level effect of the population growth rate on the long-run level of per capita income due to the effect of dilution experienced by both human and physical capital. Since the corresponding investment rates are exogenous, both capital stocks cannot increase in proportion to population growth, resulting in decreasing stocks in per capita terms. In the case of the endogenous growth models, Jones (1999) has comprehensively evaluated the “demographic” properties of most of R & D based models of endogenous growth with no human capital accumulation, which may be classified as scale effect growth models, semi-endogenous growth models, and fully endogenous growth models.4 Our paper is more closely related to Dalgaard and Kreiner (2001), Strulik (2005), and Bucci (2008) as these authors analyze the impact of population on economic growth in endogenous growth models with human capital accumulation, although they also consider endogenous R&D activities. They all focus on the relationship between population, human capital, and output per capita stressing the role played by the agents’ degree of altruism. However, for the technical reasons mentioned above, none of these papers analyze separately the effect of population growth and size on the rate of growth of per capita income (growth effect) as well as on the long-run level of income per capita (level effect). As outlined above, our novel study of level effects is notably relevant for the design of theories primary concerned with policies which raise income levels and not growth rates (Parente and Prescott, 1993). Even more importantly, a substantial part of the empirical literature relies on direct level measures of human capital accumulation like enrollment rates or years of schooling. Hence, providing an explicit theory of how demographic change affects human capital in level is therefore not only theoretically challenging, it might be also illuminating from the empirical point of view. Main findings Four findings should be emphasized. 1.A first decisive outcome of our work is the separation of level vs. growth effects of demographic change on economic development. In particular, our paper is the first which does the job in the class of endogenous growth models considered. While the study of growth effects of population change is standard, and usually displays the property that population growth has a non-negative impact on economic growth in the long term, the mere inspection of its level effects is novel and, therefore, provides new insight into how demographic change influences economic development. Essentially, we have identified three causation mechanisms from population growth to the long-run income level. The first one is associated with the ratio of physical to human capital which originates in the standard effect of physical capital dilution: a larger population growth increases the magnitude of dilution, which is detrimental to the per capita income level. As outlined above, this sole effect explains the negative level effect obtained by Mankiw et al. (1992). A second more original mechanism is connected with the fraction of non-leisure time devoted to goods production and, consequently, with preference parameters. This effect is nonzero if and only if economic agents are not selfish, and we therefore refer to it from now as the effect of altruism utility. As non-leisure time devoted to production of the final good is shown to be a decreasing function of the population growth rate, provided that economic agents are not selfish, this effect also generates a negative correlation between population growth and the level of per capita income.5 Last but not least, a third causation line induced by the average level of human capital arises, therefore representing the effect of human capital. Unfortunately, the third effect has a non-trivial sign. Consistently with the empirical literature on the link between the level of human capital and population growth, the relationship between these two variables is highly complex and depends nonlinearly on preference and production parameters, and on the initial conditions. Consequently, the total impact of population growth on the level of income per capita is ambiguous, which is again consistent with the empirical literature. This departs sharply from the simple comparative statics usually performed to study the impact of demographics on the long-run economic rate of growth: the level effects of population change are by far trickier. 2.Deeply inspecting the sources of ambiguity, we show that when the inverse of the intertemporal elasticity of substitution is equal to the value of the capital share in the final good sector, population growth rate has no effect on the long-run level of human capital, that is the effect of human capital mentioned above is nil. Consequently, the level effect on the long-run levels of per capita income and output is negative. However, when the inverse of the intertemporal elasticity of substitution is no longer equal to, say bigger than, the value of the capital share in the final good sector, things are substantially different. Considering the initial position of the economy with respect to its long-run equilibrium in terms of the ratio physical to human capital, we analytically show that if the economy starts from below or is exactly equal to the long-run value of the latter ratio, then population growth has a positive effect on the long-run level of human capital; that is, the effect of the human capital mechanism is positive. In such a case, the total level effect of population growth is ambiguous: the physical capital dilution and altruism impact negatively while the effect of human capital is positive. Resorting to numerical investigation we find that, for all the empirically relevant cases, population growth positively affects the (detrended) long-run level of human capital, and negatively affects the (detrended) long-run levels of per capita income and broad output. The sign of these effects are invariant to the configuration chosen for initial conditions and to the assumed degree of altruism. 3.We also investigate the growth and level effects of population size, or scale effects. We find that there is no growth effect due to population size. The common long-run rate of growth of average human capital stock, per capita broad output, and per capita income does not depend on population size. In contrast, the level effect of the population size is nonzero. Here results also depend on the relationship between the inverse of the intertemporal elasticity of substitution in consumption and the physical capital share in goods production. For example, we find that in the normal case when the former parameter is bigger than the latter, a larger initial population size leads to lower long-run detrended levels of per capita income, per capita broad output, and average human capital independently of the initial conditions and of the degree of altruism of economic agents. This roughly illustrates a negative level effect of population size, just like the level effect of population growth rate is generally found to be, although population growth does raise the level of human capital in the most relevant parametric configuration of the model. The non-positive growth and level effects of population size obtained here may seem opposite to the corresponding empirical literature. Notice however that the largest part of such literature has been more concerned with growth effects of the scale economies, and even more concerned with the agricultural sector in developed countries. We believe that our results on the level effect of population size in a human-capital-based growing economy are truly original. On the other hand, they are clearly consistent with the recent empirical work of Easterly and Kraay (2000) highlighted above, one of the very few papers separating growth and level effects. 4.Last but not least, we study the effects of population change over time by depicting the optimal transition paths. In doing so, we get the optimal paths accounting for both the level and growth effects together. The results are highly interesting if one has in mind the population debate. In particular, we find that the effect of a higher population growth rate on per capita income is generally negative in the short-run, reflecting the negative level effect outlined above, while this effect is positive in the long-run through the positive growth effect also mentioned above. As such, our theory neatly explains why the relationship between population change and economic development depends on time. The distinction between level and growth effects of population change allows to give a simple and powerful explanation to this complicated time relationship. The paper is organized as follows. Section 2 is devoted to briefly present an enlarged version of the Lucas–Uzawa model which includes an altruism parameter. Section 3 examines the balanced growth path and exposes the closed-form solution for the variables involved in the relationship between population, human capital, and growth. Section 4 analyzes the growth effect of population size and growth. Sections 5 and 6 analyze the level effect of population growth and population size, respectively. Section 7 studies the impact of different demographic shocks on the optimal transition paths of the more significant variables. Section 8 concludes.

In this paper we have analytically studied the short- and long-run impact of two demographic variables (population size and the rate of growth of population) on two kind of economic variables (the rate of growth of the economy and the level of the essential economic indicators) in a growth model based on the accumulation of human capital. In comparison with the related existing literature, three breakthroughs have been achieved: a separate analysis of the level effects of demographic change, an inspection into the level and growth effects of population size in the context of a growing economy driven by human capital accumulation, and the study of the possible “complicated time relationships” between economic development and demographic change through the analysis of transitional dynamics. It goes without saying that many research lines are still open. One is the inclusion of feedback effects from economic growth to population change, which ultimately requires endogenizing demographics. There are several ways to undertake such a task (see for example Boucekkine and Fabbri (2013) for a quite general one-sector model). However, it is very likely that such a step will destroy the closed-form solutions developed in this paper. Without the latter, the exercise will turn fully computational, disabling any analytical decomposition of the mechanisms at work. A second more valuable line of research is empirical and concerns the development of tools in order to identify level vs. growth effects in the data. Our paper shows that such a distinction is highly relevant.