Abstract
In this paper, we show a simple correction for the aggregation effect when testing the relationship between income inequality and life expectancy using aggregated data. While there is evidence for a negative correlation between income inequality and a population’s average life expectancy, it is not clear whether this is due to an aggregation effect based on a nonlinear relationship between income and life expectancy or to income inequality being a health hazard in itself. The proposed correction method is general and independent of measures of income inequality, functional form assumptions of the health production function, and assumptions on the income distribution. We apply it to data from the Human Development Report and find that the relationship between income inequality and life expectancy can be explained entirely by the aggregation effect. Hence, there is no evidence that income inequality itself is a health hazard.
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Notes
Wagstaff and van Doorslaer (2000) use the term income inequality hypothesis instead. In their work, the relative income hypothesis is defined by health depending “on the deviation of the individual’s income from the population mean income” (ibid., p. 547). Because of the ambiguous use of the terminology in the literature and the popularity of the term relative income hypothesis, we employ the latter.
We are grateful to Hugh Gravelle who proposed this exposition on the following two pages.
In our applications below, this will basically be a test on assumptions of the income distribution and the health production function. In principle, the question of the appropriate assumed income distribution can be settled before estimating Eq. 3 and the test of equality of the parameters is mainly on the functional form assumptions of the life expectancy production function.
Apart from the finding of Pinkovski and SalaiMartin that the lognormal distribution provides a good approximation to income data, this possibility of deriving s from the Gini index (which is reported in official data) is the other important reason for using this distribution. Chotikapanich et al. (2007) assume a generalised beta distribution which, however, requires more than one parameter to retrieve the standard deviation of log income from the Gini index.
Examples from other fields like migration economics, where the Gini index is employed, are, e.g. Chojnicki et al. (2011).
Both lines are predicted values after the two regressions, where we set the Gini index to its mean value of 40.66 and insert different values of income between 0 and 60,000.
According to the results in column (5), income is positive as long as 1. 378 × Income / 1, 000 − 0. 020 × (Income / 1, 000)^{2} > 0 which holds for Income / 1, 000 > 68. 9. The marginal effect of income is positive as long as 1. 378 − 0. 020 × 2 × Income / 1, 000 > 0 which holds for Income / 1, 000 > 34. 45.
Note that in our model, the shape of the lognormal distribution depends on two scale parameters: average log income and the standard deviation of log income/the Gini index. Since both parameters differ between countries, the shape of the lognormal distributions will also vary.
In a recent study, Hupfeld (2011) finds that the relationship between income and life expectancy is nonmonotonic (convex) for pensioners in the public pension system in Germany. However, the data used have at least two drawbacks. The first is that one cannot identify subjects who were selfemployed or civil servants for some time, where lowpension claims might be due not to low income but to other labor conditions. Higher life expectancy in lower income deciles may therefore be an artefact. The second drawback is that the highest income decile is rightcensored. Hence, higher life expectancy in the highest decile could also be due to an artefact. Both drawbacks can lead to a rather more convex than concave relationship between income and life expectancy.
The same holds for share of GDP spent on health care or education.
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Acknowledgments
We are grateful to Hugh Gravelle and an anonymous referee who greatly helped to improve the paper. We also thank Stefan Felder, Miriam Krieger, as well as the participants of the international conference “Health, Happiness, Inequality” (2010) in Darmstadt, Germany, the international conference “Distributive Justice in the Health System—Theory and Empirics” (2009) in Halle, Germany, and the 2009 Annual Meeting of the German Health Economics Association (dggö) in Hanover, Germany for the very useful comments. All remaining errors are our own.
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Appendices
Appendix 1
Derivation of the correction factors
Let m and s be the mean and standard deviation of the natural logarithm of y _{ i k } and let y _{ i k } follow a lognormal distribution. Then, log(y _{ i k }) follows a normal distribution.It is wellknown that:

\(E[\text {log}(y_{ik})] = m\)

\(\text {Var}[\text {log}(y_{ik})] = s^{2}\)

\(E[y_{ik}] = e^{m+\frac {s^{2}}{2}}\)

\(\text {Var}[y_{ik}] = e^{2m+s^{2}}\left (e^{s^{2}}1\right )\).
Hence, in the logarithmic case:
In the quadratic case:
according to the computational formula for the variance. To compute m, we use \(E[y_{ik}] = e^{m+\frac {s^{2}}{2}} \Leftrightarrow m = \text {log}(E[y_{ik}]) \frac {s^{2}}{2}\) and the log of the mean income for \(\text {log}(E[y_{ik}])\).
Appendix 2
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Mayrhofer, T., Schmitz, H. Testing the relationship between income inequality and life expectancy: a simple correction for the aggregation effect when using aggregated data. J Popul Econ 27, 841–856 (2014). https://doi.org/10.1007/s0014801304837
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DOI: https://doi.org/10.1007/s0014801304837
Keywords
 Income inequality
 Life expectancy
 Aggregation effect
JEL Classification
 D31
 I10
 O15