(In)Equality and (In)Justice

Jasso, Guillermina

doi:10.15448/1984-7289.2016.2.23118

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Dossier: Inequalities, Stratification and Social Justice • Civitas, Rev. Ciênc. Soc. 16 (2) • Apr-Jun 2016 • https://doi.org/10.15448/1984-7289.2016.2.23118 copy

(In)Equality and (In)Justice

(Des)Igualdade e (In)Justiça

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

Understanding the connection between inequality and justice is important because justice is classically regarded as the first line of defense against self-interest and inequality. Absent a strong and clear link between inequality and justice, the sense of justice would not awaken to exert its moral suasion, no matter how great the inequality or how fast its increase. We obtain exact links between economic inequality and three parameters of the justice evaluation distribution -the mean, median, and variance-across a comprehensive set of inequality measures and a substantial starter set of just reward scenarios. This work shows that there is no general necessary connection between inequality and justice. There is, however, a striking pattern in some situations: as economic inequality increases, the average of the justice evaluations moves deeper into the territory of unjust underreward, and the distribution stretches outward, increasing the gulf between underrewarded and overrewarded and hollowing out the middle class.

Keywords:
Inequality; Justice; Fairness; Probability distributions; Measures of inequality and of justice and fairness

X Distribution	Average of the Distribution of Justice Evaluations	Effect of Economic Inequality
	$E (J) = E [In (X)] - E [In (X^{*})]$	$\frac{\partial E (J)}{\partial (\cdot)}$	$\frac{\partial E (J)}{\partial I (X)}$
A. Inequality in X represented by Atkinson's measure A
General	$ln [1 - A (X) + ln (E (X)] - E [In (X^{*})]$	$\frac{1}{1 - A (X)}$	< 0
B. Inequality in X represented by Theil ’ s MLD
General	$- M L D (X) + In [E (X)] - E [In (X^{*})]$	-1	< 0
C. Inequality in X represented by general inequality parameter c in continuous univariate two-parameter probability distributions
Lognormal (c > 0)	$- \frac{c^{2}}{2} + \ln [E (X)] - E [\ln (X^{*})]$	$- c < 0$	<0
Pareto (c > 1)	$\frac{1}{c} + \ln (\frac{c - 1}{c}) + \ln [E (X)] - E [\ln (X^{*})]$	$\frac{1}{c^{2} (c - 1)} > 0$	< 0
Power-Function (c > 0)	$- \frac{1}{c} + \ln (\frac{c + 1}{c}) + \ln [E (X)] - E [\ln (X^{*})]$	$\frac{1}{c^{2} (c + 1)} > 0$	< 0

X Distribution	Median of the Distribution of Justice Evaluations	Effect of Economic Inequality
	$M (J) = Q_{J} (.5)$	$\frac{\partial M (J)}{\partial c}$	$\frac{\partial M (J)}{\partial I (X)}$
A. Scenario 2: X* is a positive constant Thus $Q_{J} (.5) = Q_{In (X)} (.5) - In (X^{*})$
Lognormal (c > 0)	$- \frac{c^{2}}{2} + \ln [E (X)] - \ln (X^{*})$	−c < 0	< 0
Pareto (c > 1)	$\frac{\ln (2)}{c} + \ln (\frac{c - 1}{c}) + \ln [E (X)] - \ln (X^{*})$	$\frac{1}{c (c - 1)} - \frac{\ln (2)}{c^{2}} > 0$	< 0
Power-Function (c > 0)	$- \frac{\ln (2)}{c} + \ln (\frac{c + 1}{c}) + \ln [E (X)] - \ln (X^{*})$	$\frac{\ln (2)}{c^{2}} - \frac{1}{c (c + 1)} > 0$	< 0
B. Scenario 2.1: X* E (X) Thus $Q_{J} (.5) = Q_{In (X)} (.5) - In [E (X)]$
Lognormal (c > 0)	$- \frac{c^{2}}{2}$	−c < 0	< 0
Pareto (c > 1)	$\frac{\ln (2)}{c} + \ln (\frac{c - 1}{c})$	$\frac{1}{c (c - 1)} - \frac{\ln (2)}{c^{2}} > 0$	< 0
Power-Function (c > 0)	$- \frac{\ln (2)}{c} + \ln (\frac{c + 1}{c})$	$\frac{\ln (2)}{c^{2}} - \frac{1}{c (c + 1)} > 0$	< 0
C. Scenario 2.2: X* = G (X) Thus, $Q_{J} (.5) = Q_{In (X)} (.5) - In [G (X)]$
Lognormal (c > 0)	0	0	0
Pareto (c > 1)	$\frac{\ln (2) - 1}{c}$	$\frac{1 - \ln (2)}{c^{2}} > 0$	< 0
Power-Function (c > 0)	$\frac{1 - \ln (2)}{c}$	$\frac{l n (2) - 1}{c^{2}} < 0$	> 0

X Distribution	Variance of the Distribution of Justice Evaluations	Effect of Economic Inequality
	$V a r (J) = V a r [In (X)] + V a r [In (X^{})] - 2 C o v [In (X), In (X^{})]$	$\frac{\partial V a r (J)}{\partial c}$	$\frac{\partial V a r (J)}{\partial I (X)}$
Scenario 2: X* is a constant. Thus, $V a r {(J)}^{*} = V a r [In (X)]$
Lognormal (c > 0)	c ²	2c > 0	> 0
Pareto (c > 1)	$\frac{1}{c^{2}}$	$- \frac{1}{c^{3}} < 0$	> 0
Power-Function (c > 0)	$\frac{1}{c^{2}}$	$- \frac{1}{c^{3}} < 0$	> 0

	Parameters of the Justice Evaluation Distribution
	E(J)	M(J)	V ar(J)
Scenario 1	− 0	− 0 +	0 +
Scenario 2	− 0	− 0 +	+
Scenario 2.1	–	–	+
Scenario 2.2	0	− 0 +	+
Scenario 3	0	0	0
Scenario 3.1	0	0	0
Scenario 3.2	0	0	0

X Distribution	Mean of ln(X)	Variance of ln(X)
	$E [In (X)] = In [G (X)]$	$V a r [In (X)]$
Lognormal (c > 0)	$- \frac{c^{2}}{2} + \ln [E (X)]$	c ²
Pareto (c > 1)	$\frac{1}{c} + \ln (\frac{c - 1}{c}) + \ln [E (X)]$	$\frac{1}{c^{2}}$
Power-Function (c > 0)	$- \frac{1}{c} + \ln (\frac{c + 1}{c}) + \ln [E (X)]$	$\frac{1}{c^{2}}$

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E-mail: civitas@pucrs.br

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[1] Corresponding Author: Guillermina Jasso, New York University, Department of Sociology, 295 Lafayette Street, 4^th Floor, New York, NY 10012-9605

Scenario 1	X* can assume any positive value
Scenario 2	X* is a positive constant k
Scenario 2.1	X* is the arithmetic mean of X (comparison to equality)
Scenario 2.2	X* is the geometric mean of X (comparison to all)
Scenario 3	X* equals kX, k positive
Scenario 3.1	X^:* equals X (k positive and equal to one), so that J= 0
Scenario 3.2	X^:* equals kX, k positive and not equal to one. so that J = -ln(k)