New results linking inequality and justice

ABSTRACT

Recent work revives the classic idea that the sense of justice is the first line of defense against inequality and shows that the link depends, in part, on ideas of the just reward. This paper extends earlier work, making three new contributions: (1) it expands the just reward scenarios from the micro scenarios (which start with the individual's idea of the just reward) to six new macro scenarios (which start with the distribution of everyone's ideas of the just reward); (2) it expands the set of justice measures from three to four, including now the proportion underrewarded; and (3) it obtains results for all the new cases, identifying the conditions that lead to radically different connections between inequality and justice.

KEYWORDS:

• Inequality and justice
• inequality measures and justice measures
• probability distributions
• just rewards; justice evaluations

Acknowledgments

Earlier versions of portions of this article were presented at the 17th Biennial Conference of the International Society for Justice Research (ISJR), Emory University, Atlanta, GA, July 2018; and at the annual meeting of the American Sociological Association (ASA), Philadelphia, PA, August 2018. I thank conference participants, anonymous referees, and the Editor for valuable comments. I also gratefully acknowledge the intellectual and financial support of New York University.

Notes

¹ Formulas for four of the inequality measures — in both observed distributions and mathematically specified distributions — are collected in Jasso and Kotz (2008, p. 37–41, especially Tables 2 and 3).

² Thus, two sets of terms are used interchangeably: (1) ARD, JRD, and JED; and (2) X distribution, X* distribution, and J distribution.

³ Formulas and graphs for the three main associated functions — the cumulative distribution function (CDF), the probability density function (PDF), and the quantile function  (QF) — appear in Jasso and Kotz (2008, p. 36-37).

⁴ For comprehensiveness, the micro scenarios are included in panel A of Table 1.

⁵ For comprehensive information on all these distributional forms, see Johnson et al. (1994, 1995), Kleiber and Kotz (2003), and Kotz, Kozubowski, and Podgórski (2001).

⁶ This paper focuses on the effect of inequality in the X distribution, and does not examine the effects of the three other terms — the means of the X and X* distributions and the dispersion in the X* distribution. But it is worth noting that, as visible in Table 2, in Macro Scenarios 4–6 (when X and X* are different), E(J) increases with the mean of X and the dispersion in X* and decreases with the mean of X*.

⁷ A correlation of −1 is not reached by all perfectly negatively associated pairs of identical distributions, because a correlation of −1 requires perfect linear dependence. When the two identical and perfectly negatively associated distributions are symmetric, their dependence is linear, and their correlation equals −1. In the distributions studied in this paper, those arising from lognormal X and X* are symmetric (they are normal).

⁸ The cells for average J in Macro Scenarios 4–6 show both a zero and a negative effect of inequality. While the inequality effect is negative in the distributions investigated above (Table 2), a zero effect occurs when the X and X* distributions have identical geometric means.

⁹ Figure 1–3 also provide visualization of several symmetries due to the combinations of the c values of the X and X* distributions. Across all three figures, the graphs of panels b and d are mirror images of each other, as are the graphs of panels c and g and the graphs of panels f and h.