Happiness Inequality: How Much is Reasonable?

Abstract

We compute the Gini indexes for income, happiness and various simulated utility levels. Due to decreasing marginal utility of income, happiness inequality should be lower than income inequality. We find that happiness inequality is about half that of income inequality. To compute the utility levels we need to assume values for a key parameter that can be interpreted as a measure of relative risk aversion. If this coefficient is above one, as many economists believe, then a large part of happiness inequality is not related to pecuniary dimensions of life.

Appendix: Relative Risk Aversion

The first and second derivatives of the utility function are:

$$ u'(y) = y^{ - \rho } $$

(2)

$$ u''(y) = - \rho y^{ - \rho - 1} . $$

(3)

A risk neutral individual is indifferent between receiving a payment of $x and a lottery that pays either $x + $z or $x−$z with a probability of 0.5 for each outcome. A concave utility function represents risk averse individuals that strictly prefer the payment of $x over participating in the lottery. A commonly used measure of risk aversion is the Arrow–Pratt coefficient of relative risk aversion, r _R .

$$ r_{R} = y\frac{u''(y)}{u'(y)}. $$

(4)

Substituting (2) and (3) in (4) we get:

$$ r_{R} = \rho . $$

(5)

By now there have been almost 30 years of applied research in risk aversion. Surprisingly, there is not yet a commonly accepted estimate of the coefficient ρ. Although many economists probably believe that the coefficient of relative risk aversion is between 1 and 2, there is a wide range of measures for this coefficient (Table 5).

Fig. 1

Happiness inequality versus income inequality

Table 5 Gini income

The following list is not an exhaustive survey of the literature on risk aversion, instead representing only a small portion of the research efforts in this area. Friend and Blume (1975), studying the demand for risky assets, find that relative risk aversion generally exceeds 1 and probably is above 2. Weber (1975), using expenditure data, and Szpiro (1986) using data on property insurance, estimate relative risk aversion to be in the range between 1.3 and 1.8. Using consumption data, Hansen and Singleton (1983) report lower estimates, between 0.68 and 0.97. Also using data on consumption, Mankiw (1985) finds much larger estimates in the range of 2.44–5.26. Halek and Eisenhauer (2001), using data on life insurance, estimate demographic differences in risk attitudes. They find an average relative risk aversion coefficient of 3.75, but a much lower median risk aversion coefficient of 0.9. Bartunek and Chowdhury (1997) use data from index option prices and estimate low risk aversion coefficients in the range of 0.2–0.3. The authors go into great pains to explain why their results are so different from the rest of the literature. The reasons provided suggest that their results are biased downwards.