# Consumption, Investment, and Healthcare with Aging

## TL;DR

This paper models optimal consumption, investment, and healthcare spending with aging, showing healthcare extends lifespan and alters mortality growth, using a novel mathematical approach to solve the control problem.

## Contribution

It introduces a new solution method for the optimal control problem with aging mortality and derives explicit asymptotic solutions for old-age mortality and healthcare spending.

## Key findings

- Healthcare spending increases with age.
- Optimal mortality growth rate is lower than natural mortality.
- Explicit old-age asymptotic solutions are derived.

## Abstract

This paper solves the problem of optimal dynamic consumption, investment, and healthcare spending with isoelastic utility, when natural mortality grows exponentially to reflect 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.