Are marginal utilities higher when you are sick or when you are healthy? What this question basically asks is whether or not you enjoy having more income when you are healthy or when you are sick. On the one hand, when you are healthy, you may value expensive items (e.g., travel, flat screen tv, sports car) more than when you are sick. On the other hand, when you are sick, you may value income more in order to pay for prepared meals, or home health care.
A paper by Finkelstein, Luttmer and Notowidigdo (2008) explore how the utility function changes with illness. They derive an interesting model based on earlier work of De Nardi, French and Jones 2006 and Palumbo 1999.
- U(C,S) = γ0S + (1 + γ1S)u(C)
The utility function above is based on the amount of consumption, C, and whether or not the person is sick, S. S is equal to unity if the individual is sick and zero if they are in good health. The γ0 term is negative represents the absolute change in utility which occurs when someone is sick and the γ1 parameter gives the effect of sickness on marginal utility. If γ1 is positive, then being sick increases marginal utility (the benefit from one additional dollar of income) and if γ1 is negative then being sick decreases marginal utility.
The authors use a CRRA utility function as follows:
Individuals can purchase a health insurance benefit, b, to protect them against income risk from becoming sick. The authors find that the optimal health insurance level depends not only on risk aversion, but also on how marginal utility changes between healthy and sick states (i.e., the parameter γ1).
|Optimal Health Insurance, b|
Similarly, risk aversion and state-dependent marginal utilities will also affect an individual’s optimal savings rate.
|Optimal Savings, s|
We see that the optimal levels of health insurance and saving depend significantly on whether or not marginal utility increases or decreases when one becomes sick. If marginal utility increases when one is sick, more health insurance is optimal. If marginal utility decreases when one is sick, less health insurance is optimal. Similarly, increased marginal utility when a person is sick, increases optimal savings rates but the effect is smaller than the effect on optimal health insurance. The question remains: when a person gets sick, does marginal utility increase or decrease empirically?
The authors utilize data from the Health and Retirement Survey. The study uses a self-assessment of subjective well being as a proxy for utility, and estimates how changes in permanent income affect the marginal utility of non-medical consumption when an individual is sick and when they are healthy. The authors results are summarized as follows:
The results suggest that, relative to the standard practice of assuming a state-independent utility function, accounting for our estimate of state dependence lowers the optimal share of medical expenditures reimbursed by health insurance by about 20 to 45 percentage points, and lowers the optimal fraction of earnings saved for retirement by about 1 percentage point.
In summary, people adjust to their new, sick state and adjust their marginal utilities accordingly to cope with the change in their life situation.
- Finkelstein, Luttmer and Notowidigdo (2008) “What Good Is Wealth Without Health? The Effect of Health on the Marginal Utility of Consumption“, NBER WP #14089.