Consumption, investment and healthcare with aging

  • Paolo GuasoniEmail author
  • Yu-Jui Huang


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.


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

Mathematics Subject Classification (2010)

91G80 49L25 

JEL Classification

E21 I12 



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).


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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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