## Abstract

This paper examines contemporaneous choices of saving and health prevention under a two-argument utility of wealth and health. Unlike the traditional approach to modelling the cost of prevention as a decline in wealth, health prevention here is assumed to mainly reduce current health level in exchange for a lower probability of contracting a disease in the future. We show that the optimal levels of the two instruments of risk management can move either in the same direction or in opposite directions. One key element in distinguishing these two cases is whether a decision maker is correlation averse or correlation prone. Together with the relative importance of substitution effect over income effect on saving, the sign of correlation attitude is also relevant in determining the reaction of optimal saving and health prevention when the interest rate changes. Lastly, several combinations of preferences in different aspects are provided, which lead to unambiguous effects of a background risk in wealth on the two optimal choices.

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## Notes

- 1.
Empirical studies such as Kantor and Fishback (1996) demonstrate that the introduction of social insurance programs such as workers’ compensation in the US, which reduces the uncertainty of future income, leads to a decline in households’ precautionary saving.

- 2.
Eeckhoudt et al. (2007) refer to this kind of health investment as “tertiary prevention”.

- 3.
Self-insurance reduces the negative effect of a future bad event leaving probabilities unchanged, whereas prevention merely reduces the probability of incurring the bad event.

- 4.
Indeed, health prevention is a form of self-protection, while health investment can be related to self-insurance.

- 5.
We use the following notation for derivatives: \( \frac{\partial u}{\partial W}=u_{1}\), \(\frac{\partial u}{ \partial H}=u_{2}\), \(\frac{\partial ^{2}u}{\partial W^{2}} =u_{1}{}_{1}\), \(\frac{\partial ^{2}u}{\partial H^{2}} =u_{2}{}_{2}\), etc.

- 6.
See Finkelstein et al. (2009) for a detailed review.

- 7.
Also notice that the use of a two-argument utility of wealth and health allows us to model this feature, which is novel to the prevention literature. This determines a clear difference with previous models of financial prevention and saving.

- 8.
We use the following notation for derivatives: \(\frac{\partial U}{\partial s} =U_{s}\), \(\frac{\partial U}{\partial e}=U_{e}\), \( \frac{\partial ^{2}U}{\partial s\partial e}=U_{s}{}_{e}\), \( \frac{\partial ^{2}U}{\partial s^{2}}=U_{s}{}_{s}\)\(\frac{ \partial ^{2}U}{\partial e^{2}}=U_{e}{}_{e}\), etc.

- 9.
Note that, in this section where we allow for multiple joint optima, we consider the case where condition (9) holds locally. In next sections we will instead assume that the condition holds for every

*e*and*s*. - 10.
This result may also be related to one finding by Lee (2005), who studies a model without saving where prevention is the unique instrument for the DM and shows that, in this context, an increase in initial wealth increases prevention when wealth and health are complements.

- 11.
Note that the comparative statics is also unambiguous for changes in \(h_{1}^{G}\), but this is not our main interest.

- 12.
For the traditional analysis of substitution and income effects see, for instance, Hicks (1946).

- 13.
Note that the index \(-sR\frac{u_{11}}{u_{1}}\) is the index of relative risk aversion when the the baseline wealth in the second period is null while it is the index of partial relative risk aversion when the the baseline wealth in the second period is positive.

- 14.
In fact, an increase in the interest rate generates in the optimum an increase in the wealth of second period by \(s^*\).

- 15.
To see this it is sufficient to note that for

*R*tending to zero, \(U_{{ sw}_1}\) and thus \(U_{{ sw}_1}U_{{ ee}}\) tend to zero too while \(U_{{ se}}U_{{ ew}_1}\) does not. - 16.
For this reason, we assume that the range of possible realizations of \(\tilde{ \epsilon }\) is bounded from below by \(-w_{1}\).

- 17.
- 18.
For instance, using a formulation similar to that in Baiardi et al. (2016, p. 443), we have \(u(W,H)=\frac{W^{1-\gamma }H^{\theta (1-\gamma )}-1}{1-\gamma }\) implying correlation loving, prudence in wealth and cross-imprudence in health (as required in Proposition 3 (c) for \(\theta >0\) and \(0<\gamma <1\).

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## Acknowledgements

Desu Liu gratefully acknowledges financial support of the China National Social Science Fund under Grant No. 15BJL093.

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Liu, D., Menegatti, M. Optimal saving and health prevention.
*J Econ* **128, **177–191 (2019). https://doi.org/10.1007/s00712-018-00652-6

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### Keywords

- Saving
- Health prevention
- Correlation aversion/loving
- Background risk

### JEL Classification

- D81
- I12
- D15