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«Oded Stark, Yong Wang Overlapping Number ZEF – Discussion Papers on Development Policy Bonn, August 2002 The CENTER FOR DEVELOPMENT RESEARCH (ZEF) ...»

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ZEF Bonn

Zentrum für Entwicklungsforschung

Center for Development Research

Universität Bonn

Oded Stark, Yong Wang



ZEF – Discussion Papers on Development Policy

Bonn, August 2002

The CENTER FOR DEVELOPMENT RESEARCH (ZEF) was established in 1997 as an international,

interdisciplinary research institute at the University of Bonn. Research and teaching at ZEF aims

to contribute to resolving political, economic and ecological development problems. ZEF closely

cooperates with national and international partners in research and development organizations.

For information, see: http://www.zef.de.

ZEF – DISCUSSION PAPERS ON DEVELOPMENT POLICY are intended to stimulate discussion among researchers, practitioners and policy makers on current and emerging development issues. Each paper has been exposed to an internal discussion within the Center for Development Research (ZEF) and an external review. The papers mostly reflect work in progress.

Oded Stark, Yong Wang: Overlapping, ZEF – Discussion Papers On Development Policy No. 50, Center for Development Research, Bonn, August 2002, pp. 17.

ISSN: 1436-9931

Published by:

Zentrum für Entwicklungsforschung (ZEF) Center for Development Research Walter-Flex-Strasse 3 D – 53113 Bonn Germany Phone: +49-228-73-1861 Fax: +49-228-73-1869 E-Mail: zef@uni-bonn.de http://www.zef.de

The authors:

Oded Stark, Center for Development Research, University of Bonn Yong Wang, City University of Hong Kong Overlapping Contents Acknowledgements


1 Kurzfassung 1 1 The idea 2 2 The analytical framework 4 3 The effect of an extended overlapping on human capital formation 7 4 The welfare effect of an extended overlapping 12 5 Complementary reflections 14 Appendix 16 References 17 ZEF Discussion Papers on Development Policy 50 List of Figures Figure 1 8 Figure 2 8 Figure 3 8 Figure 4 10 Figure 5 11 Figure 6 11 Overlapping

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Abstract We propose a new microeconomic explanation for the divergent experiences of economies in forming human capital. We suggest that the positive effect of a longer life expectancy on human capital formation arises from two separate effects: a life expectancy effect and a prolonged intergenerational overlap effect. We argue that the duration of the overlap between generations and the associated parental support can affect the marginal cost of human capital formation and hence its level: parental support is cheaper than market financing. We thus attribute the strong correlation between the formation of human capital and life expectancy not merely to a higher marginal benefit arising from a longer payback period but also to a lower marginal cost arising from a prolonged intergenerational overlap. We provide conditions under which a longer overlap results in a higher level of per capita output.

Kurzfassung Wir schlagen eine neue mikroökonomische Erklärung für die voneinander abweichenden Erfahrungen von Wirtschaftssystemen bei der Bildung von Humankapital vor. Die positive Wirkung einer längeren Lebenserwartung deutet darauf hin, dass Humankapitalbildung durch zwei getrennte Wirkungen entsteht: eine Lebenserwartungswirkung und eine anhaltende Wirkung der Intergenerationsüberlappung. Wir argumentieren, dass die Zeitdauer der Überlappung zwischen Generationen und die damit verbundene elterliche Unterstützung die

Grenzkosten der Bildung von Humankapital beeinflussen können und somit auch deren Höhe:

Elterliche Unterstützung ist billiger als Marktfinanzierung. Wir schreiben daher die hohe Korrelation zwischen der Bildung von Humankapital und der Lebenserwartung nicht nur einem höheren Grenznutzen zu, der durch eine längere Rückzahlungsdauer entsteht, sondern auch den niedrigeren Grenzkosten, die durch eine verlängerte Intergenerationsüberlappung entstehen. Wir liefern Bedingungen, unter denen eine längere Überlappung ein höheres Ertragsniveau pro Kopf ergibt.

ZEF Discussion Papers on Development Policy 50

1 The idea It has been long recognized that the stock of human capital affects the level of per capita output in an economy. Whether the effect arises because human capital is an ordinary input in the economy’s production function or because the effect manifests itself through enhancement of total factor productivity (in that it leads to the creation, adoption, implementation, and diffusion of new technologies) are largely empirical issues. The notion that an economy that forms a large quantity of human capital will have a higher per capita output than an economy that forms a small quantity of human capital can safely be taken as given, requiring little, if any, additional inquiry. But why is it that one economy has, or forms, abundant per capita human capital, while another has, or forms, little per capita human capital? Why does the per capita human capital gap between economies not close? Much – though not all – of the human capital in an economy is the result of decisions made by individuals. Clearly, several factors are involved and one of them is life expectancy: a longer life expectancy entails a longer payback period that in turn encourages larger investments in human capital. An economy consisting of individuals with a long life expectancy will then form more human capital than an economy consisting of individuals with a short life span.

The impact of a lengthened life expectancy comes from the returns side of the human capital investment calculus: the marginal benefit is higher. We argue, however, that typically, imbedded in a lengthened life expectancy is a lowered marginal cost of forming human capital.

We seek to unearth this effect and study its role in accounting for the divergent experiences of economies in the formation of human capital. We suggest that the lowered marginal cost effect arises from a correlate of extended life expectancy: prolonged duration of the overlap between generations. Suppose that as long as they are alive, parents support the human capital formation of their children, and that the parental support is cheaper than market financing. An extended life expectancy that results in a prolonged overlap entails more parental support, which in turn can foster the formation of more human capital. An example will serve to illustrate.

Suppose that life expectancy is 45. An individual gives birth to one child when the individual is 20 years of age. The child is cared for in his infancy and for as long as he engages in acquiring human capital, conditional on the individual being alive. The age at which the child makes the human capital formation decision is 15. At this age, if the child were to engage in human capital formation, the child could expect parental support for up to 10 years. If the child finds it optimal to devote more than 10 years to human capital formation, he can do so by borrowing at a fixed market interest rate. When the child reaches the age of 20, he gives birth to a child whom he, in turn, will support in the same manner in which he was supported. Suppose that the child finds it optimal to acquire human capital for a little more than 10 years, say for τ

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years in excess of 10. During these years the child has to bear the entire cost of forming human capital, which includes the market rate of interest.

Suppose now that life expectancy is 55. Retaining all other assumptions as before, the child can now expect parental support for up to 20 years. To see the implications of this assumption for human capital formation, consider the case 0 τ 10. All of the years of human capital formation previously financed by commercial loans now become parentally supported, interest-free years. Since the marginal cost of forming human capital goes down, more human capital will be formed. This effect is separate from the returns to human capital, a marginal benefit that arises from the addition of years during which returns to the human capital investment can be reaped.

In section 2 we present our analytical framework. In section 3 we investigate formally the effect of extended overlapping on the formation of human capital by optimizing individuals.

To this end we decompose the “gross” life expectancy effect into a “net” life expectancy effect and an overlapping effect. In section 4 we trace the welfare implication of extended overlapping for an economy that is subjected to such a change. In section 5 we further explain the rationale underlying our idea and offer a suggestion as to how to differentiate empirically between the overlapping model of human capital formation and the received model of human capital formation.

ZEF Discussion Papers on Development Policy 50

2 The analytical framework Consider an overlapping-generations economy. In every period t a generation is born. A generation consists of a continuum of individuals of measure N. Each member of generation t has a single parent in generation t-1, and each parent of generation t-1 has a single offspring in generation t. The economy consists, therefore, of a continuum of dynasties of measure N.

Individuals live for two periods. In the first period of their lives, individuals work and form human capital. In the second and last period of their lives individuals work and procreate.

Let the duration of the first period be normalized at 1, and let the duration of the second period be 0 l 1. Thus, the individual’s lifespan is 1+l. An individual gives birth to a child after l c (≥0) of l has elapsed. Thus, l p ≡ l − l c measures the duration of the overlap between the individual and his child.1 Let st represent the proportion of the first period that an individual chooses to allocate to human capital formation. Hence (1 − s t ) of the first period is allocated to work. The first period earnings of a member of generation t thus become (1 − s t ) wt where wt is the prevailing wage at time t. Investment in human capital is costly. Let the cost be a proportion λ of the individual’s wage. The cost of forming human capital is born by the individual’s parent as long as the parent is alive, and by the individual himself through borrowing at the market interest rate if additional human capital is formed past the parent’s death. When the child reaches the point l c of the second period of his life he has a child. That child too is faced with a choice of allocating the first period of his life between work and human capital formation, drawing on his parent’s support in a manner akin to that described above, that is, up to a duration of l p. The amount of human capital (measured in efficiency units of labor) that is available in the second period of the individual’s life, generated by investment of the time proportion st in human capital formation in the first period, is given by ϕ( s t ) where ϕ( 0) = 1 ; ϕ( s t ) 1, ϕ′( s t ) 0, ϕ′′( s t ) 0 for all st ∈ (0,1); lim ϕ′( st ) = ∞, and lim ϕ′( s t ) = 0.2 st → 0 s t →1 Alternatively, it can be assumed that the individual gives birth to a child during the first period of his life and that the child reaches the human capital formation age only at a point in time that is 1+lc into the individual’s life. The years prior to that point in time are immaterial since they do not affect the child’s human capital formation decision.

The assumption that the human capital investment has a lagged effect on productivity, affecting the efficiency units of labor in the second period of the individual’s lifetime, is expositionally convenient but is not essential.

Our analysis also applies in the case in which the human capital investment affects the individual’s productivity in the first period of his lifetime.


Let ct+1 be consumption in the second period of the individual’s life. The individual’s preferences are represented by the utility function u(ct+1 ) where u ′(ct +1 ) 0 and u ′′(ct +1 ) ≤ 0 for all c t ≥ 0; u ( 0) −∞. Thus, u(ct+1 ) is strictly increasing, concave, and bounded from below.

For simplicity’s sake we have assumed a preference for consumption only in the second period of life.

Given rt, wt, wt+1, the human capital formation decision of the individual’s child, and recalling ϕ( st ), the individual chooses the proportion of time allocated to human capital formation so as to maximize his utility. Namely,

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