Skip to main content


Log in

50 is the new 30—long-run trends of schooling and retirement explained by human aging

  • Published:
Journal of Economic Growth Aims and scope Submit manuscript


Workers in the US and other developed countries retire no later than a century ago and spend a significantly longer part of their life in school, implying that they stay less years in the work force. The facts of longer schooling and simultaneously shorter working life are seemingly hard to square with the rationality of the standard economic life cycle model. In this paper we propose a novel theory, based on health and aging, that explains these long-run trends. Workers optimally respond to a longer stay in a healthy state of high productivity by obtaining more education and supplying less labor. Better health increases productivity and amplifies the return on education. The health accelerator allows workers to finance educational efforts with less forgone labor supply than in the previous state of shorter healthy life expectancy. When both life-span and healthy life expectancy increase, the health effect is dominating and the working life gets shorter if the intertemporal elasticity of substitution for leisure is sufficiently small or the return on education is sufficiently large. We calibrate the model and show that it is able to predict the historical trends of schooling and retirement.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others


  1.   There is no contradiction between Lee’s (2001) observation of a (mild) increase in age of retirement and the observed decline in labor force participation of the elderly (Costa 1998). The reason is that the labor force participation rate captures individuals who manage to survive until age 65; individuals who retired and died before the age of 65 do not count. During the period of observation improvements in health allowed more people to survive until the age of 65. This implies that the expected age of retirement increased while labor force participation declined.

  2.   There is also a direct link from better health in childhood to performance at school (Bleakley 2007; Bharadwaj et al. 2013) and to parental investment in education (Hazan and Zoabi 2006) and thus to adult health and productivity. Andersen et al. (2015) find a strong impact of eye disease on contemporaneous income across countries and argue that it results partly from the delay of the fertility transition caused by the strong exposure to UV light in some countries and the reduced return on human capital due to the entailed high prevalence of eye disease. The trade-off between fertility and education is not explored in the present paper but it is potentially amplifying the positive association between life-expectancy, healthy life expectancy, and productivity.

  3.   Recent research by Hamermesh (2013) suggests that this is even true for economists. Hamermesh observes a trend towards higher average age of authors publishing in the top five journals and explains this by a trend that made the profession less like pure mathematics (requiring mostly fluid abilities, which decline early in life) and more like a humanistic field (requiring mostly crystallized abilities, which are more persistent).

  4.   The assumption that \(T>\lambda \) is plausible but not necessary for our results, which hold for \(\lambda \rightarrow T\) as well. Moreover, health could decrease non-linearly without affecting our results. The linearity assumption is made to obtain a closed-form solution. The crucial assumption is that there exists a point in life after which health and productivity decline, which allows us to disentangle the effects of increasing life expectancy and increasing healthy life expectancy.

  5. In Appendix F we compare the treatment of depreciation in the standard human capital model vs. our approach in more detail.

  6. Better health also increases the hours worked per time increment because of reduced sickness absence. In our model a reduction of sickness absence would be captured by an increase of supply of human capital per time increment, i.e. an upward shift of the a(t)-curve. In the Appendix we show that such a shift leaves all results unaffected.

  7. For the benchmark calibration of the model we assume a mild increase of \(\lambda \) together with T. The implied \(\beta \) is given by \( [{\lambda (1970)-\lambda (1850)]}/[{T(1970)-T(1850)}]\) = 0.119. Thus \(\alpha > 0.119+0.03=0.22\) is sufficient for life time labor supply to decline. For the benchmark calibration we use \(\alpha =0.84\), based upon the estimate provided with Fig. 3.

  8. Keeping a zero interest rate does not much affect the predicted trends but it implies a too high level of education for all cohorts. The reason is that the opportunity costs of education would be too low when consumption during the schooling period is financed at a zero interest rate.

  9. In the Appendix we extend the model by a constant trend of wage growth (due to technological progress). In this case the best fit of the data is obtained for \(\theta =0.0715\), a value slightly smaller than in the benchmark case.


  • Acemoglu, D., & Johnson, S. (2006). Disease and development: The effect of life expectancy on economic growth. NBER Working Paper 12269.

  • Acemoglu, D., & Johnson, S. (2007). Disease and development: The effect of life expectancy on economic growth. Journal of Political Economy, 115, 925–985.

    Article  Google Scholar 

  • Aghion, P., Howitt, P., & Murtin, F. (2011). The relationship between health and growth: When lucas meets Nelson-Phelps. Review of Economics and Institutions, 2, 1–24.

    Google Scholar 

  • Akerlof, G. A., & Katz, L. F. (1989). Workers’ trust funds and the logic of wage profiles. Quarterly Journal of Economics, 104, 525–536.

    Article  Google Scholar 

  • Andersen, T. B., Dalgaard, C.-J., & Selaya, P. (2015). Climate and the Emergence of Global income difference. Review of Economic Studies, forthcoming.

  • Arking, R. (2006). The biology of aging: Observations and principles. Oxford: Oxford University Press.

    Google Scholar 

  • Ben-Porath, Y. (1967). The production of human capital and the life cycle of earnings. Journal of Political Economy, 75, 352–365.

    Article  Google Scholar 

  • Bharadwaj, P., Loeken, K., & Neilson, C. (2013). Early life health interventions and academic achievement. American Economic Review, 103, 1862–1891.

    Article  Google Scholar 

  • Bleakley, H. (2007). Disease and development: Evidence from hookworm eradication in the American South. Quarterly Journal of Economics, 122, 73–117.

    Article  Google Scholar 

  • Blinder, A. S., & Weiss, Y. (1976). Human capital and labor supply: A synthesis. Journal of Political Economy, 84, 449–72.

    Article  Google Scholar 

  • Bloom, D. E., Canning, D., & Moore, M. (2014a). Optimal retirement with increasing longevity. Scandinavian Journal of Economics, 116(3), 838.

    Article  Google Scholar 

  • Bloom, D., Canning, D., & Fink, G. (2014b). Disease and development revisited. Journal of Political Economy, 122, 1355–1366.

    Article  Google Scholar 

  • Boucekinne, R., de la Croix, D., & Licandro, O. (2002). Vintage human capital, demographic trends, and endogenous growth. Journal of Economic Theory, 104, 340–375.

    Article  Google Scholar 

  • Boucekinne, R., de la Croix, D., & Licandro, O. (2003). Early mortality declines at the dawn of modern growth. Scandinavian Journal of Economics, 105, 401–418.

    Article  Google Scholar 

  • Burtless, G. (2013). The impact of population aging and delayed retirement on workforce productivity. Center for Retirement Research, Working Paper 2013–11.

  • Card, D. (1999). The causal effect of education on earnings. In O. C. Ashenfelter & R. Layard (Eds.), Handbook of labor economics (Vol. 3, pp. 1801–1863). Amsterdam: Elsevier.

    Google Scholar 

  • Case, A. (2010). What’s past is prologue: The impact of early life health and circumstance on health in old age. In D. A. Wise (Ed.), Research findings in the economics of aging (pp. 211–228). Chicago: University of Chicago Press.

    Chapter  Google Scholar 

  • Case, A., & Paxson, C. (2008). Height, health, and cognitive function at older ages. American Economic Review, 98, 463–467.

    Article  Google Scholar 

  • Cervellati, M., & Sunde, U. (2005). Human capital formation, life expectancy, and the process of development. American Economic Review, 95, 1653–1672.

    Article  Google Scholar 

  • Cervellati, M., & Sunde, U. (2011). Life expectancy and economic growth: The role of the demographic transition. Journal of Economic Growth, 16, 99–133.

    Article  Google Scholar 

  • Cervellati, M., & Sunde, U. (2013). Life expectancy, schooling, and lifetime labor supply: Theory and evidence revisited. Econometrica, 81, 2055–2086.

    Article  Google Scholar 

  • Cervellati, M., & Sunde, U. (2015). The economic and demographic transition, mortality, and comparative development. American Economic Journal: Macroeconomics, 7, 189–225.

    Google Scholar 

  • Chetty, R., Guren, A., Manoli, D. S., & Weber, A. (2011). Does indivisible labor explain the difference between micro and macro elasticities? A meta-analysis of extensive margin elasticities. NBER Working Paper 16729.

  • Costa, D. L. (1998). The Evolution of retirement: Summary of a research project. American Economic Review, 88, 232–36.

    Google Scholar 

  • Costa, D. L. (2000). Understanding the twentieth-century decline in chronic conditions among older men. Demography, 37, 53–72.

    Article  Google Scholar 

  • Costa, D. L. (2002). Changing chronic disease rates and long-term declines in functional limitation among older men. Demography, 39, 119–137.

    Article  Google Scholar 

  • Cutler, D. M., Lleras-Muney, A., & Vogl, T. (2011). Socioeconomic status and health: Dimensions and mechanisms. In S. Glied & P. C. Smith (Eds.), The Oxford handbook of health economics (pp. 124–163). Oxford: Oxford University Press.

    Google Scholar 

  • d’Albis, H., Lau, S. P., & Sánchez-Romero, M. (2012). Mortality transition and differential incentives for early retirement. Journal of Economic Theory, 147, 261–283.

    Article  Google Scholar 

  • Dalgaard, C.-J., & Strulik, H. (2012). The genesis of the golden age—Accounting for the rise in health and leisure. University of Copenhagen Discussion Paper 12–10, 2012.

  • Dalgaard, C.-J., & Strulik, H. (2014). Optimal aging and death: Understanding the Preston curve. Journal of the European Economic Association, 12, 672–701.

    Article  Google Scholar 

  • De Grey, A., & Rae, M. (2007). Ending Aging: The rejuvenation breakthroughs that could reverse human aging in our lifetime. Macmillan.

  • Floud, R., Fogel, R. W., Harris, B., & Hong, S. C. (2011). The changing body: Health, nutrition, and human development in the western world since 1700. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Fogel, R. W. (1994). Economic growth, population theory, and physiology: The bearing of long-term processes on the making of economic policy. American Economic Review, 84, 369–395.

    Google Scholar 

  • Fries, J. F. (1980). Aging, natural death, and the compression of morbidity. New England Journal of Medicine, 303, 130–135.

    Article  Google Scholar 

  • Galor, O. (2011). Unified growth theory. Princeton: Princeton University Press.

    Google Scholar 

  • Galor, O., & Weil, D. N. (2000). Population, technology and growth: From Malthusian stagnation to the demographic transition and beyond. American Economic Review, 76, 807–828.

    Google Scholar 

  • Galor, O., & Mountford, A. (2008). Trading population for productivity: Theory and evidence. Review of Economic Studies, 75, 1143–1179.

    Article  Google Scholar 

  • Gavrilov, L. A., & Gavrilova, N. S. (1991). The biology of human life span: A quantitative approach. London: Harwood Academic Publishers.

    Google Scholar 

  • Grossman, M. (2006). Education and nonmarket outcomes. In E. Hanushek & F. Welch (Eds.), Handbook of the economics of education (Vol. 1, pp. 577–633). Amsterdam: Elsevier.

    Google Scholar 

  • Gruber, J., & Wise, D. (1998). Social security and retirement: An international comparison. American Economic Review, 88, 158–63.

    Google Scholar 

  • Hamermesh, D. S. (2013). Six decades of top economics publishing: Who and how? Journal of Economic Literature, 51, 1–11.

    Article  Google Scholar 

  • Hansen, C. W., & Lønstrup, L. (2012). Can higher life expectancy induce more schooling and earlier retirement? Journal of Population Economics, 25(4), 1249–1264.

    Article  Google Scholar 

  • Hansen, C. W., & Strulik, H. (2015). Life expectancy and education: Evidence from the cardiovascular revolution. University of Copenhagen, Discussion Paper 15-01.

  • Hazan, M. (2009). Longevity and lifetime labor supply: Evidence and implications. Econometrica, 77, 1829–1863.

    Article  Google Scholar 

  • Hazan, M., & Zoabi, H. (2006). Does longevity cause growth? A theoretical critique. Journal of Economic Growth, 11, 363–76.

    Article  Google Scholar 

  • Heckman, J. (1976). A life cycle model of earnings, learning, and consumption. Journal of Political Economy, 84, S11–S44.

    Article  Google Scholar 

  • Heijdra, B., & Romp, W. E. (2009). Retirement, pensions, and aging. Journal of Public Economics, 93, 586–604.

    Article  Google Scholar 

  • Heijdra, B., & Reijnders, L. (2012). Human capital accumulation and the macroeconomy in an aging society. CESifo Working Paper 4046.

  • Kalemli-Ozcan, S., Ryder, H. E., & Weil, D. N. (2000). Mortality decline, human capital investment, and economic growth. Journal of Development Economics, 62, 1–23.

    Article  Google Scholar 

  • Kalemli-Ozcan, S., & Weil, D. N. (2010). Mortality change, the uncertainty effect, and retirement. Journal of Economic Growth, 15, 65–91.

    Article  Google Scholar 

  • Kotlikoff, L. J., & Gokhale, J. (1992). Estimating a firm’s age-productivity profile using the present value of workers’ earnings. Quarterly Journal of Economics, 107, 1215–1242.

    Article  Google Scholar 

  • Kuhn, M., Wrzaczek, S., Prskawetz, A., & Feichtinger, G. (2015). Optimal choice of health and retirement in a life-cycle model. Journal of Economic Theory, 158, 186–212.

    Article  Google Scholar 

  • Kurzweil, R., & Grossman, T. (2010). Transcend: Nine steps to living well forever. Emmaus: Rodale Press.

    Google Scholar 

  • Lazear, E. P. (1981). Agency, earnings profiles, productivity, and hours restrictions. American Economic Review, 71, 606–620.

    Google Scholar 

  • Lee, C. (2001). The expected length of male retirement in the United States, 1850–1990. Journal of Population Economics, 14, 641–650.

    Article  Google Scholar 

  • Manton, K. G., Gu, X. L., & Lamb, V. L. (2006). Long-term trends in life expectancy and active life expectancy in the United States. Population and Development Review, 32, 81–105.

    Article  Google Scholar 

  • Mehra, R., & Prescott, E. C. (1985). The equity premium: A puzzle. Journal of Monetary Economics, 15, 145–161.

    Article  Google Scholar 

  • Mincer, J. (1974). Schooling, experience and earnings. New York: Columbia University Press.

    Google Scholar 

  • Mitnitski, A. B., Mogilner, A. J., MacKnight, C., & Rockwood, K. (2002). The accumulation of deficits with age and possible invariants of aging. Scientific World, 2, 1816–1822.

    Article  Google Scholar 

  • Nair, K. S. (2005). Aging muscle. American Journal of Clinical Nutrition, 81, 953–963.

    Google Scholar 

  • OECD. (1998). Work force ageing in OECD countries, OECD employment outlook, Ch. 4 (June) (pp. 123–150). Paris: OECD Publications.

  • Oeppen, J., & Vaupel, J. W. (2002). Broken limits to life expectancy. Science, 296, 1029–1031.

    Article  Google Scholar 

  • Olshansky, S. J., et al. (2005). A potential decline in life expectancy in the United States in the 21st century. New England Journal of Medicine, 352, 1138–1145.

    Article  Google Scholar 

  • Oster, E., Shoulson, I., & Dorsey, E. (2013). Limited life expectancy, human capital and health investments. American Economic Review, 103, 1977–2002.

    Article  Google Scholar 

  • Porter, R. (2001). The Cambridge illustrated history of medicine. Cambridge: Cambridge University Press.

    Google Scholar 

  • Restuccia, D., & Vandenbroucke, G. (2013). A century of human capital and hours. Economic Inquiry, 51, 1849–1866.

    Article  Google Scholar 

  • Salomon, J. A., Wang, H., Freeman, M. K., Vos, T., Flaxman, A. D., Lopez, A. D., et al. (2013). Healthy life expectancy for 187 countries, 1990–2010: A systematic analysis for the global burden disease study 2010. The Lancet, 380, 2144–2162.

    Article  Google Scholar 

  • Schaie, K. W. (1994). The course of adult intellectual development. American Psychologist, 49, 304–313.

    Article  Google Scholar 

  • Skirbekk, V. (2004). Age and individual productivity: A literature survey, Vienna Yearbook of Population Research, pp. 133–153.

  • Soares, R. R. (2005). Mortality reductions, educational attainment, and fertility choice. American Economic Review, 95, 580–601.

    Article  Google Scholar 

  • Strauss, J., & Thomas, D. (1998). Health, nutrition, and economic development. Journal of Economic Literature, 36, 766–817.

    Google Scholar 

  • Strulik, H., & Vollmer, S. (2013). Long-run trends of human aging and longevity. Journal of Population Economics, 26, 1303–1323.

    Article  Google Scholar 

  • Strulik, H., & Werner, K. (2014). Elite education, mass education, and the transition to modern growth. University of Goettingen, Cege Discussion Paper 205.

  • WHO. (2012). Global health observatory data repository. Accessed June, 12, 2015.

Download references


We would like to thank Lothar Banz, Carl Johan Dalgaard, David de la Croix, Paula Gobbi, Moshe Hazan, Ben Heijdra, Gregory Ponthiere, Alexia Prskwetz, and four anonymous referees for helpful comments.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Holger Strulik.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 228 KB)


Appendix A: derivation of the optimal solution

The interior solution of the optimization problem of the household is derived from the first order conditions (5) and (6), i.e.

$$\begin{aligned} 0&\mathop {=}\limits ^{!} \theta + \frac{2(\lambda -\tau )}{R^2-2 \lambda R + 2s(\lambda -\tau )+\tau ^2}, \end{aligned}$$
$$\begin{aligned} 0&\mathop {=}\limits ^{!} \frac{2 T (R-\lambda )}{R^2-2R\lambda +2s(\lambda -\tau )+\tau ^2} - (\lambda -R)^{-\eta } \omega . \end{aligned}$$

Rearranging Eq. (13) we obtain

$$\begin{aligned} \frac{2}{R^2-2R\lambda + 2s(\lambda -\tau ) + \tau ^2} = -\frac{\theta }{\lambda -\tau }. \end{aligned}$$

Inserting this into Eq. (14) we arrive at

$$\begin{aligned} \omega (\lambda -R)^{-\eta } = - \frac{T \theta (R-\lambda )}{\lambda -\tau }\quad \Leftrightarrow \quad R = \lambda - \left( \frac{(\lambda -\tau )\omega }{T\theta } \right) ^{\frac{1}{1+\eta }}. \end{aligned}$$

Optimal schooling follows from inserting the optimal retirement age R from (15) into Eq. (14), i.e.

$$\begin{aligned} s = \frac{\lambda + \tau }{2} - \frac{1}{\theta } - \frac{\left( \frac{(\lambda -\tau )\omega }{T\theta } \right) ^{\frac{2}{1+\eta }}}{2(\lambda -\tau )}. \end{aligned}$$

Therefore, optimal labor supply is, given by \(L=R-s\), is

$$\begin{aligned} L = \frac{1}{\theta } + \frac{1}{2(\lambda -\tau )} \left( \tau -\lambda +\left( \frac{(\lambda -\tau ) \omega }{T \theta } \right) ^{\frac{1}{1+\eta }}\right) ^2. \end{aligned}$$

If the Hessian of U is negative definite at the critical point (sR), then this point is a local maximum. The Hessian of U is given by

$$\begin{aligned} H_U(s,R):= \begin{pmatrix} \frac{\partial ^2 U}{\partial s^2} &{} \frac{\partial ^2U}{\partial s \partial R}\\ \frac{\partial ^2 U}{\partial R \partial s} &{} \frac{\partial ^2 U}{\partial R^2} \end{pmatrix} =: \begin{pmatrix} H_{1,1} &{} H_{1,2} \\ H_{2,1} &{} H_{2,2} \\ \end{pmatrix}, \end{aligned}$$


$$\begin{aligned} H_{1,1}&= \frac{2 T (-R^2 + 2 R \lambda + 2(s-\lambda )\lambda -2s\tau +\tau ^2)}{(R^2-2R\lambda + 2s(\lambda -\tau )+\tau ^2)^2} - \eta \omega (\lambda -R)^{-1-\eta },\\ H_{1,2}&= - \frac{4 T (R-\lambda )(\lambda -\tau )}{(R^2-2R \lambda + 2s(\lambda -\tau )+\tau ^2)^2},\\ H_{2,1}&= H_{1,2},\\ H_{2,2}&= -\frac{4 T (\lambda -\tau )^2}{(R^2-2R\lambda +2s(\lambda -\tau )+\tau ^2)^2}. \end{aligned}$$

The two principal minors of H evaluated in the critical point are

$$\begin{aligned} H_{U,1}&= -\eta \frac{T \theta }{(\lambda -\tau )} - \frac{T \theta \left( \lambda -\tau +\theta \left( \frac{(\lambda -\tau )\omega }{T \theta } \right) ^{\frac{2}{1+\eta }} \right) }{(\lambda -\tau )^2} <0, \end{aligned}$$
$$\begin{aligned} H_{U,2}&= \det H(R,s) = \frac{T^2 \theta ^3}{\lambda -\tau } (1 + \eta ) >0. \end{aligned}$$

Hence, the Hessian is negative definite and the critical point is a maximum.

Appendix B: proofs of the propositions

Proof of Proposition 1

The partial derivatives of Rs and L, c.f. (7) – (9), with respect to T are

$$\begin{aligned} \frac{\partial s}{\partial T}&= \frac{\omega \left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{1-\eta }{1+\eta }}}{T^2 (1+\eta )\theta } >0,\\ \frac{\partial R}{\partial T}&= \frac{\left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{1}{1+\eta }}}{T(1+\eta )} >0,\\ \frac{\partial L}{\partial T}&=-\frac{\omega \left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{-\eta }{1+\eta }}}{T^2(1+\eta )\theta } \left( -\lambda + \tau + \left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{1}{1+\eta }} \right) . \end{aligned}$$

We have \({\partial L}/{\partial T}>0\) because

$$\begin{aligned} \frac{\omega \left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{-\eta }{1+\eta }}}{T^2(1+\eta )\theta } \ge 0 \ \text {and} \ \left( -\lambda + \tau + \left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{1}{1+\eta }} \right) = \tau - R \le 0. \end{aligned}$$

\(\square \)

Proof of Proposition 2

The partial derivative of s, c.f. (8), with respect to \(\tau \) is

$$\begin{aligned} \frac{\partial s}{\partial \tau } = \frac{1}{2} \left( 1+ \frac{(1-\eta )\left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{2}{1+\eta }}}{(1+\eta )(\lambda -\tau )^2} \right) . \end{aligned}$$

For \(\eta \le 1\) it is easy to see that \(\frac{\partial s}{\partial \tau } >0\). If \(\eta >1\), it holds that

$$\begin{aligned} 1+ \frac{(1-\eta )\left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{2}{1+\eta }}}{(1+\eta )(\lambda -\tau )^2} >0 \ \Leftrightarrow \ (1+\eta ) (\lambda -\tau )^2 > (\eta -1)\left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{2}{1+\eta }} \end{aligned}$$

This inequality holds because

$$\begin{aligned} (\eta -1)\left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{2}{1+\eta }} \le (\eta -1)(\lambda -\tau )^2\mathop {<}\limits ^{!}(1+\eta )(\lambda -\tau )^2 \ \Leftrightarrow \ \eta -1<1+\eta . \end{aligned}$$

The partial derivatives of R and L, c.f. (7) and (9), with respect to \(\tau \) are

$$\begin{aligned} \frac{\partial R}{\partial \tau }&= \frac{\left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{1}{1+\eta }}}{(1+\eta )(\lambda -\tau )}>0,\\ \frac{\partial L}{\partial \tau }&= \frac{\left( -\lambda + \tau + \left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{1}{1+\eta }}\right) \left( (1+\eta )(\lambda -\tau ) + (\eta -1)\left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{1}{1+\eta }} \right) }{2(1+\eta )(\lambda -\tau )^2}. \end{aligned}$$

It holds

$$\begin{aligned} \frac{\partial L}{\partial \tau } <0&\Leftrightarrow \ \left( (1+\eta )(\lambda -\tau ) + (\eta -1)\left( \frac{(\lambda -\tau )\omega }{\theta T} \right) ^{\frac{1}{1+\eta }} \right) >0 \\&\Leftrightarrow (1+\eta )(\lambda -\tau ) + (\eta -1)(\lambda -R) >0\\&\Leftrightarrow \frac{\lambda -\tau }{\lambda -R}>\frac{1-\eta }{1+\eta }, \end{aligned}$$

which holds \(\forall \ \eta \ge 0\). \(\square \)

Proof of Proposition 3

An equal increase in \(\tau \) and T is captured by the directional derivative \({\partial L}/{\partial T} + {\partial L}/{\partial \tau }\), i.e.

$$\begin{aligned} \frac{\partial L}{\partial T} + \frac{\partial L}{\partial \tau } = \frac{\left( -\lambda +\tau + \left( \frac{(\lambda -\tau )\omega }{T\theta }\right) ^{\frac{1}{1+\eta }} \right) }{2 T^2(1+\eta )\theta (\lambda -\tau )^2}\cdot \Lambda \end{aligned}$$


$$\begin{aligned} \Lambda :=- \omega \left( \frac{(\lambda -\tau )\omega }{T\theta }\right) ^{\frac{-\eta }{1+\eta }} 2 (\lambda -\tau )^2 + T^2\theta \left( (1+\eta )(\lambda -\tau )+(\eta -1)\left( \frac{(\lambda -\tau )\omega }{T\theta }\right) ^{\frac{1}{1+\eta }} \right) . \end{aligned}$$

For the numerator of the first term of Eq. (21) it holds that

$$\begin{aligned} \left( -\lambda +\tau + \left( \frac{(\lambda -\tau )\omega }{T\theta }\right) ^{\frac{1}{1+\eta }} \right) = \tau -R <0. \end{aligned}$$

The denominator is greater than 0. Therefore, the problem simplifies to

$$\begin{aligned}&- \omega \left( \frac{(\lambda -\tau )\omega }{T\theta }\right) ^{\frac{-\eta }{1+\eta }} 2 (\lambda -\tau )^2 + T^2\theta \left( (1+\eta )(\lambda -\tau )+(\eta -1)\left( \frac{(\lambda -\tau )\omega }{T\theta }\right) ^{\frac{1}{1+\eta }} \right) \mathop {>}\limits ^{!}0 \\&\quad \Leftrightarrow \ \frac{\omega }{T^2 \theta } \left( \frac{(\lambda -\tau )\omega }{T\theta }\right) ^{\frac{-\eta }{1+\eta }} 2(\lambda -\tau )^2 - (\eta -1)\left( \frac{(\lambda -\tau )\omega }{T\theta }\right) ^{\frac{1}{1+\eta }} < (1+\eta )(\lambda -\tau )\\&\quad \Leftrightarrow \ \left( \frac{(\lambda -\tau )\omega }{T\theta }\right) ^{\frac{1}{1+\eta }} \left[ \frac{2(\lambda -\tau )^2 \omega }{T^2\theta } \frac{T^2 \theta }{(\lambda -\tau )\omega }-(\eta -1) \right] < (1+\eta )(\lambda -\tau )\\&\quad \Leftrightarrow \ \left( \frac{(\lambda -\tau )\omega }{T}\right) ^{\frac{1}{1+\eta }} \left[ \frac{2(\lambda -\tau )+T(1-\eta )}{T(1+\eta )(\lambda -\tau )} \right] < \theta ^{\frac{1}{1+\eta }}, \end{aligned}$$

which is always fulfilled for \(\eta \ge \frac{2(\lambda -\tau )}{T}+1\). If \(\eta < \frac{2(\lambda -\tau )}{T}+1\), the last inequality becomes

$$\begin{aligned} \theta > \frac{(\lambda -\tau )\omega }{T} \left[ \frac{2(\lambda -\tau )+T(1-\eta )}{T(1+\eta )(\lambda -\tau )}\right] ^{1+\eta }. \end{aligned}$$

\(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Strulik, H., Werner, K. 50 is the new 30—long-run trends of schooling and retirement explained by human aging. J Econ Growth 21, 165–187 (2016).

Download citation

  • Published:

  • Issue Date:

  • DOI:


JEL Classification