# Stochastic accumulation of human capital and welfare in the Uzawa–Lucas model: an analytical characterization

- 104 Downloads

## Abstract

Stochastic growth models are often solved numerically, because they are not tractable in general. However, recent several studies find the closed-form solution to the stochastic Uzawa–Lucas model in which technological progress or population dynamics follow a Brownian motion process with one or two parameter restriction(s). However, they assume that the return on the accumulation of human capital is deterministic, which is inconsistent with empirical evidence. Therefore, I develop the Uzawa–Lucas model in which the accumulation of human capital follows a mixture of a Brownian motion process and many Poisson jump processes, and obtain the closed-form solution. Moreover, I use it to examine the nexus between human capital uncertainty, technological progress, expected growth rate of human capital, and welfare.

## Keywords

Human capital Welfare Endogenous growth Uncertainty## JEL Classification

C61 J24 O33 O41## Notes

### Acknowledgements

I thank Prof. Noritsugu Nakanishi, Assoc. Prof. Quoc Hung Nguyen, Prof. Hiroyuki Nishiyama, Prof. Masao Oda, Prof. Yoshifumi Okawa and seminar participants at the summer 2017 JSIE Kansai Branch Meeting for their many extensive and thoughtful comments. I would particularly like to thank Assoc. Prof. Shiro Kuwahara for his encouragement, pointing out troublesome typos, and constructive comments, “from the cradle to completion” of this paper. I am especially grateful to Prof. Yoichi Gokan, my discussant, and two anonymous referees of this journal for their detailed and exceptionally helpful comments that lead to the unthinkably substantial improvement of the earlier version of the manuscript. All remaining mistakes are my own. This paper was accepted to the *J Econ* under the guidance of Prof. Giacomo Corneo (Editor). Figure 2 is created with Eviews 9.5 Student Version, while the others are with MATLAB R2016b (Version 9.1, MATLAB and Simulink Student Suite). This research does not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

## Supplementary material

## References

- Aghion P, Howitt P (1992) A model of growth through creative destruction. Econometrica 60(2):323–352CrossRefGoogle Scholar
- Ahn S, Kaplan G, Moll B, Winberry T, Wolf C (2017) When inequality matters for macro and macro matters for inequality. In: Eichenbaum MS, Parker J (eds) NBER macroeconomics annual 2017, vol 32. University of Chicago Press, ChicagoGoogle Scholar
- Barro RJ, Sala-i-Martin X (2004) Economic growth, 2nd edn. MIT Press, CambridgeGoogle Scholar
- Bilkic N, Gries T, Pilichowski M (2012) Stay in school or start working? The human capital investment decision under uncertainty and irreversibility. Lab Econ 19:706–717CrossRefGoogle Scholar
- Brock WA, Mirman LJ (1972) Optimal economic growth and uncertainty: the discounted case. J Econ Theory 4(3):479–513CrossRefGoogle Scholar
- Brunnermeier MK, Sannikov Y (2014) A macroeconomic model with a financial sector. Am Econ Rev 104(2):379–421CrossRefGoogle Scholar
- Bucci A, Colapinto C, Forster M, La Torre D (2011) Stochastic technology shocks in an extended Uzawa–Lucas model: closed-form solution and long-run dynamics. J Econ 103(1):83–99CrossRefGoogle Scholar
- Caballé J, Santos MS (1993) On endogenous growth with physical and human capital. J Polit Econ 101(6):1042–1067CrossRefGoogle Scholar
- Chang F-R (2004) Stochastic optimization in continuous time. Cambridge University Press, New YorkCrossRefGoogle Scholar
- Chaudhry A, Naz R (2018) Closed-form solutions for the Lucas–Uzawa growth model with logarithmic utility preferences via the partial hamiltonian approach. Discret Contin Dyn Syst Ser S 11(4):643–654Google Scholar
- Cho J, Cooley TF, Kim SH (2015) Business cycle uncertainty and economic welfare. Rev Econ Dyn 18(2):185–200CrossRefGoogle Scholar
- Cinnirella F, Streb J (2017) The role of human capital and innovation in economic development: evidence from post-Malthusian Prussia. J Econ Growth 22(2):193–227CrossRefGoogle Scholar
- Duffie D, Epstein LG (1992) Stochastic differential utility. Econometrica 60(2):353–394CrossRefGoogle Scholar
- Eaton J (1981) Fiscal policy, inflation and the accumulation of risky capital. Rev Econ Stud 48(3):435–445CrossRefGoogle Scholar
- Hartog J, Ophem HV, Bajdechi SM (2007) Simulating the risk of investment in human capital. Educ Econ 15(3):259–275CrossRefGoogle Scholar
- Higham DJ (2001) An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev 43(3):525–546CrossRefGoogle Scholar
- Hiraguchi R (2009) A note on the closed-form solution to the Lucas–Uzawa model with externality. J Econ Dyn Control 33:1757–1760CrossRefGoogle Scholar
- Hiraguchi R (2013) On a closed-form solution to the stochastic Lucas–Uzawa model. J Econ 108(2):131–144CrossRefGoogle Scholar
- Hiraguchi R (2014) A note on the analytical solution to the neoclassical growth model with leisure. Macroecon Dyn 18(2):473–479CrossRefGoogle Scholar
- Hojo M (2003) An indirect effect of education on growth. Econ Lett 80:31–34CrossRefGoogle Scholar
- Karabarbounis L, Neiman B (2014) The global decline of the labor share. Q J Econ 129(1):61–103CrossRefGoogle Scholar
- Krebs T (2003) Human capital risk and economic growth. Q J Econ 118(3):709–744CrossRefGoogle Scholar
- Kuwahara S (2017) Multiple steady states and indeterminacy in the Uzawa–Lucas model with educational externalities. J Econ 122(2):173–190CrossRefGoogle Scholar
- Ladrón-De-Guevara A, Ortigueira S, Santos MS (1999) A two-sector model of endogenous growth with leisure. Rev Econ Stud 66(3):609–631CrossRefGoogle Scholar
- Lee JW, Lee H (2016) Human capital in the long run. J Dev Econ 122:147–169CrossRefGoogle Scholar
- Lester R, Pries M, Sims E (2014) Volatility and welfare. J Econ Dyn Control 38:17–36CrossRefGoogle Scholar
- Levhari D, Weiss Y (1974) The effect of risk on the investment in human capital. Am Econ Rev 64(6):950–963Google Scholar
- Lucas RE (1988) On the mechanics of economic development. J Monet Econ 22(1):3–42CrossRefGoogle Scholar
- Lucas RE (2003) Macroeconomic priorities. Am Econ Rev 93(1):1–14CrossRefGoogle Scholar
- Madsen JB (2014) Human capital and the world technology frontier. Rev Econ Stat 96(4):676–692CrossRefGoogle Scholar
- Mankiw NG, Romer D, Weil DN (1992) A contribution to the empirics of economic growth. Q J Econ 107(2):407–437CrossRefGoogle Scholar
- Marsiglio S, La Torre D (2012a) A note on demographic shocks in a multi-sector growth model. Econ Bull 32(3):2293–2299Google Scholar
- Marsiglio S, La Torre D (2012b) Population dynamics and utilitarian criteria in the Lucas–Uzawa model. Econ Model 29(4):1197–1204CrossRefGoogle Scholar
- Moll B (2014) Productivity losses from financial frictions: can self-financing undo capital misallocation? Am Econ Rev 104(10):3186–3221CrossRefGoogle Scholar
- Naz R, Chaudhry A, Mahomed FM (2016) Closed-form solutions for the Lucas–Uzawa model of economic growth via the partial Hamiltonian approach. Commun Nonlinear Sci Numer Simul 30:299–306CrossRefGoogle Scholar
- Rebelo S, Xie D (1999) On the optimality of interest rate smoothing. J Monet Econ 43(2):263–282CrossRefGoogle Scholar
- Sennewald K (2007) Controlled stochastic differential equations under Poisson uncertainty and with unbounded utility. J Econ Dyn Control 31(4):1106–1131CrossRefGoogle Scholar
- Sennewald K, Wälde K (2006) “Itô’s Lemma” and the Bellman equation for Poisson processes: an applied view. J Econ 89(1):1–36CrossRefGoogle Scholar
- Smith WT (2007) Inspecting the mechanism exactly: a closed-form solution to a stochastic growth model. BE J Macroecon 7(1):1–31Google Scholar
- Steger TM (2005) Stochastic growth under Wiener and Poisson uncertainty. Econ Lett 86:311–316CrossRefGoogle Scholar
- Turnovsky SJ (1997) International macroeconomic dynamics. MIT Press, CambridgeGoogle Scholar
- Turnovsky SJ (2000) Methods of macroeconomic dynamics, 2nd edn. MIT Press, CambridgeGoogle Scholar
- Uzawa H (1965) Optimum technical change in an aggregative model of economic growth. Int Econ Rev 6(1):18–31CrossRefGoogle Scholar
- Wälde K (2011a) Production technologies in stochastic continuous time models. J Econ Dyn Control 35(4):616–622CrossRefGoogle Scholar
- Wälde K (2011b) Applied intertemporal optimization. Mainz University Gutenberg Press, MainzGoogle Scholar
- Xie D (1991) Increasing returns and increasing rates of growth. J Polit Econ 99(2):429–435CrossRefGoogle Scholar
- Xie D (1994) Divergence in economic performance: transitional dynamics with multiple equilibria. J Econ Theory 63(1):97–112CrossRefGoogle Scholar
- Xu S (2017) Volatility risk and economic welfare. J Econ Dyn Control 80:17–33CrossRefGoogle Scholar