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Consumption, investment and healthcare with aging

  • Paolo GuasoniEmail author
  • Yu-Jui Huang
Article
  • 15 Downloads

Abstract

This paper solves the problem of optimal dynamic investment, consumption and healthcare spending with isoelastic utility, when natural mortality grows exponentially to reflect the Gompertz law and investment opportunities are constant. Healthcare slows the natural growth of mortality, indirectly increasing utility from consumption through longer lifetimes. Optimal consumption and healthcare imply an endogenous mortality law that is asymptotically exponential in the old-age limit, with lower growth rate than natural mortality. Healthcare spending steadily increases with age, both in absolute terms and relative to total spending. The optimal stochastic control problem reduces to a nonlinear ordinary differential equation with a unique solution, which has an explicit expression in the old-age limit. The main results are obtained through a novel version of Perron’s method.

Keywords

Healthcare Consumption–investment Gompertz’ law Viscosity solutions Perron’s method 

Mathematics Subject Classification (2010)

91G80 49L25 

JEL Classification

E21 I12 

Notes

Acknowledgements

We thank for helpful comments from seminar participants at Collegio Carlo Alberto, ETH Zürich, University of Limerick, Alfred Rényi Institute, National Central University in Taiwan, the QMF conference at UTS Sydney, the Congress of the Bachelier Finance Society, the University of Colorado at Boulder, and National Center for Theoretical Sciences in Taiwan. This paper is dedicated to Nicomede Guasoni (1939–2012).

References

  1. 1.
    Bayraktar, E., Zhang, Y.: Minimizing the probability of lifetime ruin under ambiguity aversion. SIAM J. Control Optim. 53, 58–90 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beeler, J., Campbell, J.Y.: The long-run risks model and aggregate asset prices: an empirical assessment. Crit. Finance Rev. 1, 141–182 (2012) CrossRefGoogle Scholar
  3. 3.
    Bommier, A.: Portfolio choice under uncertain lifetime. J. Public Econ. Theory 12, 57–73 (2010) CrossRefGoogle Scholar
  4. 4.
    Bommier, A., Rochet, J.-C.: Risk aversion and planning horizons. J. Eur. Econ. Assoc. 4, 708–734 (2006) CrossRefGoogle Scholar
  5. 5.
    Chetty, R., Stepner, M., Abraham, S., Lin, S., Scuderi, B., Turner, N., Bergeron, A., Cutler, D.: The association between income and life expectancy in the United States, 2001–2014. JAMA 315, 1750–1766 (2016) CrossRefGoogle Scholar
  6. 6.
    Cohen, S.N., Elliott, R.J.: Stochastic Calculus and Applications, 2nd edn. Springer, Cham (2015) CrossRefzbMATHGoogle Scholar
  7. 7.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cutler, D., Deaton, A., Lleras-Muney, A.: The determinants of mortality. J. Econ. Perspect. 20(3), 97–120 (2006) CrossRefGoogle Scholar
  9. 9.
    Ehrlich, I.: Uncertain lifetime, life protection, and the value of life saving. J. Health Econ. 19, 341–367 (2000) CrossRefGoogle Scholar
  10. 10.
    Ehrlich, I., Chuma, H.: A model of the demand for longevity and the value of life extension. J. Polit. Econ. 98, 761–782 (1990) CrossRefGoogle Scholar
  11. 11.
    Gompertz, B.: On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos. Trans. R. Soc. Lond. 115, 513–583 (1825) CrossRefGoogle Scholar
  12. 12.
    Grossman, M.: On the concept of health capital and the demand for health. J. Polit. Econ. 80, 223–255 (1972) CrossRefGoogle Scholar
  13. 13.
    Hall, R.E., Jones, C.I.: The value of life and the rise in health spending. Q. J. Econ. 122, 39–72 (2007) CrossRefGoogle Scholar
  14. 14.
    Harrison, G.W., Lau, M.I., Rutström, E.E.: Estimating risk attitudes in Denmark: a field experiment. Scand. J. Econ. 109, 341–368 (2007) CrossRefGoogle Scholar
  15. 15.
    Hartman, M., Catlin, A., Lassman, D., Cylus, J., Heffler, S.: US health spending by age, selected years through 2004. Health Aff. 27(1), w1–w12 (2008) CrossRefGoogle Scholar
  16. 16.
    Huang, H., Milevsky, M.A., Salisbury, T.S.: Optimal retirement consumption with a stochastic force of mortality. Insur. Math. Econ. 51, 282–291 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hugonnier, J., Pelgrin, F., St-Amour, P.: Health and (other) asset holdings. Rev. Econ. Stud. 80, 663–710 (2013) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Janeček, K., Sîrbu, M.: Optimal investment with high-watermark performance fee. SIAM J. Control Optim. 50, 790–819 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Richard, S.F.: Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. J. Financ. Econ. 2, 187–203 (1975) CrossRefGoogle Scholar
  20. 20.
    Rosen, S.: The value of changes in life expectancy. J. Risk Uncertain. 1, 285–304 (1988) CrossRefGoogle Scholar
  21. 21.
    Shepard, D.S., Zeckhauser, R.J.: Survival versus consumption. Manag. Sci. 30, 423–439 (1984) CrossRefGoogle Scholar
  22. 22.
    Smith, J.P.: Healthy bodies and thick wallets: the dual relation between health and economic status. J. Econ. Perspect. 13(2), 145–166 (1999) CrossRefGoogle Scholar
  23. 23.
    Smith, J.P.: The impact of socioeconomic status on health over the life-course. J. Hum. Resour. 42, 739–764 (2007) CrossRefGoogle Scholar
  24. 24.
    Yaari, M.E.: Uncertain lifetime, life insurance, and the theory of the consumer. Rev. Econ. Stud. 32, 137–150 (1965) CrossRefGoogle Scholar
  25. 25.
    Yogo, M.: Portfolio choice in retirement: health risk and the demand for annuities, housing, and risky assets. J. Monet. Econ. 80, 17–34 (2016) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  2. 2.School of Mathematical SciencesDublin City UniversityGlasnevinIreland
  3. 3.Department of Applied MathematicsUniversity of ColoradoBoulderUSA

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