امید به زندگی, سرمایه انسانی, امنیت اجتماعی و رشد

We analyze the effects of changes in the mortality rate upon life expectancy, education, retirement age, human capital and growth in the presence of social security. We build a vintage growth, overlapping generations model in which individuals choose the length of education and the age of retirement, and where unfunded social security pensions depend on workers' past contributions. Social security has a positive effect on education, but pension benefits favor reductions in retirement age. The net effect is that starting from a benchmark case, higher life expectancies give rise to lower per capita GDP growth in the presence of social security as the share of the active population is reduced. In addition, higher social security contribution rates reduce the growth rate of per capita GDP.

The relationship between life expectancy at birth and per capita GDP growth has been studied both empirically and theoretically. Regarding the empirical evidence, the hypothesis that reductions in the mortality rate has a non monotonic relationship with per capita GDP growth is mostly supported. By using time series data, Rodriguez and Sachs (1999) find a positive effect from life expectancy to per capita GDP growth in Venezuela between 1970 and 1990. However, Malmberg (1994) finds a negative relationship in Sweden between 1950 and 1989. Analysis of cross section data also shows that this relationship is not monotonic. Preliminary data from Latin American and Caribbean countries indicate that the GDP growth rate is positively associated with life expectancy. [See World Health Organization (1999), Box 1.2, p. 9.] Barro and Sala i Martín (1995), using a sample of 97 countries, estimate that an increase in life expectancy of 13 years would increase the per capita growth rate by 1.4% per year. Zhang and Zhang (2005) show a clear, positive relationship, but at a diminishing rate. Still, other studies have found mixed evidence: increases in life expectancy have followed higher growth rates when life expectancies are low, but have followed lower growth rates when life expectancies are high. [See Zhang et al. (2003) and references therein]. Theoretical work mostly assumes that human capital accumulation is the engine of growth. Some studies conclude that the relationship is always positive, whereas others obtain an inverted U pattern. Among the former are Ehrlich and Lui (1991) and Hu (1999). In these models, higher life expectancy increases the length of time in which the return to human capital investment is received, thus allowing for higher rates of return, and, as a consequence, higher investment and per capita GDP growth rates. 1 Still, other works have obtained an inverted U pattern between life expectancy and per capita GDP growth, which is consistent with the mixed empirical evidence mentioned above (both from historical and cross-section data). De la Croix and Licandro (1999) posit an economy where the effect of a reduction in the mortality rate upon the duration of education is such that the per capita GDP growth rate becomes higher for high mortality rates (as in underdeveloped countries), but lower for low mortality rates (as in industrialized countries). The same result is obtained in Boucekkine et al. (2002) under a setting in which there is an uncertain lifetime horizon and endogenous retirement age. In both papers, labor is the unique input in production, and the intuition behind the negatively sloped part is that the average human capital of the labor force becomes more obsolete as life expectancy increases. 2 Zhang and Zhang (2003) and Zhang et al. (2003) also obtain this result but by a different channel: not through own education time, but through expenditure on children's education. Assuming exogenous growth, a second line of research has produced a number of articles dealing with the connections between population aging, a pay-as-you-go social security and the retirement age. One recurring topic in this literature is the effect of social security upon workers' voluntary retirement age. Along these lines, the available empirical evidence suggests that, at least for the US economy, social security is relevant for retirement age issues, even though there is not total agreement on the effect of changes in the payout from the social security program. [See, e.g., Diamond and Gruber (1997) and Coile and Gruber (2000).] In this article, we study to what extent introducing unfunded social security affects the relationship between life expectancy and per capita GDP growth, taking into account the social security impact on education and retirement age incentives. Our starting point is Boucekkine et al. (2002). Boucekkine et al. used an overlapping generations model with uncertain, finite lifetime horizon. Fertility and mortality are exogenous, and individuals choose their optimal length of education and optimal retirement age, thereby influencing average human capital and the economy's growth rate in a vintage way. Our model extends this structure to include an unfunded social security system whose pension benefits depend on the contributions made by workers during their working period. According to this design, social security will influence not only individual decisions (namely, years of education and retirement age), but also aggregate variables such as economic growth. Why might the inclusion of social security be of interest? In such a setup the return to human capital investment is not constrained to labor income while working, but also extends to pensions during retirement, which are in turn related to past wage earnings. Therefore, when individuals choose the optimal length of their education, they take into account not only the effect on future labor earnings, but also on future pension benefits. Additionally, voluntary retirement age will also depend on the incentives that the public pension system embeds. As a consequence, we find that social security will affect the size of the working population and the size of the aggregate human capital in the economy. This means that social security will influence the response of the economy's growth rate to changes in the mortality rate and the corresponding to changes in life expectancy. This article is divided into two parts. In part one we solve analytically the individual problem (individuals and firms), the steady state per capita GDP growth rate and the social security budget balance. We also characterize the parameter space which determines the type of solution for the individuals' problem and prove the existence and the uniqueness of that solution. Furthermore, we prove the existence and uniqueness of the steady state per capita growth rate and social security balance for the case in which there are interior solutions for education and retirement age. In part two we numerically compare two scenarios: with and without unfunded social security. We are able to replicate the observed inverted U relationship between life expectancy and per capita growth in our scenario with social security. Our main finding is that introducing social security affects the incentives for education time and early retirement in such a way that the major force driving the negatively sloped part of that locus is the fall in the share of the working population (i.e., workers), not the obsolescence of human capital among workers, as is the case when there is no social security. The rest of the article is organized as follows. Section 2 introduces the economy: the demographic setup, the individual problem, the aggregate technology for production, the optimal length of education and the optimal retirement age, the economic aggregates, the social security balance and the balanced growth path. The numerical example and numerical results are contained in Section 3. Section 4 concludes. A mathematical Appendix contains the formal proofs.

In this article we have studied the relationship between life expectancy and per capita GDP growth rate. We have used a vintage growth model with a pay-as-you-go social security system where individuals choose education time and retirement age and where pensions depend on the contributions made by workers during their active lives. This way the flow of income during the retirement period also depends on the education time investment. The results obtained in the first part of the article are analytical. We characterized the individual's parameter space which establishes the type of solution for length of education and retirement age. We also proved the existence of, at most, one steady state per capita growth rate and of one unique steady state budget balance for social security. In the second part we compared the responses of the economy to exogenous changes in life expectancy under two regimes (with and without social security), obtaining numerical results. In our model the engine of growth is given by the change in the average human capital of the economy. Average human capital depends on, first, individual decisions such as optimal schooling and optimal age of retirement which affect their own productivity and the share of active workers in the population; and, second, demographic characteristics, namely the survival rate distribution. We saw that increases in life expectancy imply both a behavior effect (through changes in schooling and retirement age) and a composition effect (through changes in the age distribution of workers and in the range of active workers) which in turn imply changes in the growth rate. We found that in an economy with no social security the vintage characteristic seems to play a relevant role as the proportion of agents whose schooling took place a long time ago becomes higher with higher levels of life expectancy. However, in an economy with social security, the vintage description of the economy does not play such an important role in explaining the decreasing part of the life expectancy-per capita growth locus. In this case, the decrease in the share of workers as life expectancy goes up is the main factor. Finally, we studied the relationship between the size of social security and the per capita GDP growth rate. We found that such a relationship is mostly negative, except for very low values for the social security contribution rate. The explanation lies in the discouraging effect that social security imposes on education and, in particular, retirement age, which causes a fall in the share of the working population in the economy. We believe that this line of research requires further empirical work, especially in western economies in which life expectancy has reached significantly high levels and where there is strong debate about the sustainability of current unfunded social security systems and the convenience of postponing retirement age.