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Consumption, investment and healthcare with aging. (English) Zbl 1411.91365

Summary: 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.

MSC:

91B42 Consumer behavior, demand theory
93E20 Optimal stochastic control
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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