## Abstract

This paper applies stochastic control theory to the Grossman model of investment in health to characterize the case of a serious illness, i.e., one that permanently reduces the individual’s stock of health capital. Health itself is modelled as a stochastic variable, whose variation over time is determined partly by a deterministic factor and partly by a random factor with a Poisson distribution. After setting out the equations for the deterministic and stochastic approaches, phase diagrams illustrate how the introduction of uncertainty alters the model. The framework is also used to consider the effect of the introduction of a vaccine.

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## Notes

Recognition of the importance of uncertainty has, of course, been a key element in health economics since Arrow (1963).

Longer periods of illness could be thought of as the result of autocorrelation in the drawings from the distribution of germ exposure.

This is a simplification that is common to the literature, see Cropper (1977) and Grossman (1972) for example. In principal, it would be fairly straightforward to make income in any period depend on the stock of health capital in that period. But this would add a degree of mathematical complexity without adding to the analysis of the question of interest here, which is the impact of uncertainty on the investment decision. One obvious extension of the present model would be to make the impact of health capital on income a random variable, giving us a two-stochastic state variable problem.

For an extension of the Grossman model to the case including financial assets, see Muurinen (1982).

Thus, sale of organs is excluded from our model.

If the optimal trajectory were to reach the horizontal axis before the end of the planning period, there would be an interval at the end of the individual’s life in which she made no deliberate attempt to affect her stock of health capital, instead allowing it to decline at the natural rate of depreciation,

*δ*. In terms of the phase diagram, she would be sliding to the left along the horizontal axis until the end of the planning horizon. This would affect our later analysis only in the case where a serious illness struck after*I*reached zero. The drop in her health capital resulting from an illness in this range might take*H*far enough below her optimal level to prompt her to raise*I*above zero again, for some period of time.We assume her postillness stock of health capital is so low that an individual who was born healthy would never run her health capital down to that level as part of an optimal plan.

We are, remember, assuming that her optimally chosen terminal health stock at the end of the first period will be greater than the level to which it has fallen at the beginning of the second period. Otherwise, it would appear that the illness could actually increase her stock of health capital.

For an autonomous equation, this term will not be present.

One obvious extension is to make T stochastic.

See Ferguson and Lim (1998).

The Ito derivative need only be used to replace the time derivative of a stochastic variable. The remaining derivatives in a model, which do not involve time

*directly*, can still be taken in the usual way. In the finite horizon case, the individual would initially be following a trajectory slightly below the stable branch to the equilibrium because optimality requires*I*reaching 0 by the end of the (finite) horizon.In a finite horizon problem she would jump to a trajectory below the stable branch because those are the only trajectories open to her which will reach

*I*=0 in finite time. The shorter her remaining planning horizon (i.e., the later in life the illness strikes), the less the upward jump in*I*because the older she is the less time she has left and the sooner the optimal trajectory has to reach the horizontal axis.For another model analysing the effects of the introduction of a vaccine in a Grossman model, see Auld (2003).

For example, an announcement today to the effect that a vaccine has been developed and will be available a year from today.

What matters is the magnitude of the loss of utility that results from the loss of health, which could be very large if

*H*was initially small.The equations for the continuous time stochastic optimization problem of investing in financial assets are set out by Merton (1971).

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## Acknowledgements

We thank two anonymous referees and Paul Grootendorst for their helpful comments and suggestions. The paper also benefited from the discussions with the participants at the Canadian Health Economics Study Group meeting in Banff Alberta.

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*Responsible editor*: Alessandro Cigno

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### Cite this article

Laporte, A., Ferguson, B.S. Investment in health when health is stochastic.
*J Popul Econ* **20**, 423–444 (2007). https://doi.org/10.1007/s00148-006-0068-9

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DOI: https://doi.org/10.1007/s00148-006-0068-9

### Keywords

- Investment in health
- Stochastic
- Grossman model

### JEL Classification

- I12
- D80