Disparities in socio-economic outcomes: some positive propositions and their normative implications

See the Scanlan’s Rule page of www.​jpscanlan.​com.

As a particular scenario, Scanlan posits an ‘across the board change’, or social improvement, such that everyone between a previously set poverty line and 50 % of that poverty line, escapes poverty. Following that decline, the disparity in the black/white rates of experiencing poverty increases given Scanlan’s data.

Other descriptors could include economic benefit, means, good, or ‘economic advantage’. In this paper we reserve the descriptor ‘advantaged’ for one of two groups relative to the other.

For the reader’s convenience and ease of transcription, \(\prec \) is denoted \(\le _{lr} \) in this book, and the random variables and distributions corresponding to our \(Y_{D},\,Y_{A}\) and \(F_{D}(y), F_{A}(y) \) are denoted \(X, Y\) and \(F(t), G(t)\) respectively.

The condition can equally be expressed in terms of \(F_A (y)\) and \(F_D (y)\). For \(u\in [a,b],\,{Y}_{\!\!D} \prec {Y}_{\!\!A}\) says that \(\frac{F_D (b)-F_D (u)}{F_D (b)-F_D (a)}\le \frac{F_A (b)-F_A (u)}{F_A (b)-F_A (a)}\).

Notice, however, that if one allowed for the possibility of \({\varphi }'(y)\ge 0\forall y\), then inter-group disparity could be altogether eliminated by having \(\phi (y)\) be perfectly horizontal at \(\bar{{y}}\), the mean. Provided that society is not destitute, i.e. provided that the poverty line is below the mean (Cowell 1988, p.159), no poor person would pay tax in this case.

In fact, from Definition 3, \({Y}_{\!\!D} \prec {Y}_{\!\!A} \) is equivalent to \(\left\{ {Y_D \left| {a\le Y_D \le b} \right. } \right\} \prec _{fsd} \left\{ {Y_A \left| {a\le Y_A \le b} \right. } \right\} \) whenever \(a\le b\), which is clearly significantly stronger than \(F_D \prec _{fsd} F_A \).

If \(X\) is uniform on {1,2,3,4} and \(Y\) is defined by P(\(Y\) = 1) = 0.1, P(\(Y\) = 2) = 0.3, P(\(Y\) = 3) = 0.2, P(\(Y\) = 4) = 0.4 then, as Shaked and Shanthikumar (2007, p. 43) note, \( X\prec _{hr} Y \quad \& \quad X\prec _{rh} Y\) (and of course \(X\prec _{fsd} Y)\) but it is not the case that \(X \prec Y\).

A referee suggested that HRX might be identified by means of a carefully formulated geometric condition on the discrimination curve, namely, a condition whose stringency is intermediate between that required for the likelihood ratio ordering and that required for first-degree stochastic dominance. We thank the referee for this challenging suggestion, which remains for future investigation. It is interesting to note that several empirical studies on wage discrimination document cases in which a ‘disadvantaged’ group performs better than a supposed ‘advantaged’ one, at least on some limited wage intervals. The likelihood ratio ordering would not hold in such a case: the curve of Fig. 1 could cross the \(45^{\circ }\) line. The possibility that HRX could hold in such a circumstance also remains to be investigated in future work.

Its ingredients, which are the relative frequency curves \(f_D (y)\) and \(f_A (y)\), feature centrally in the class of segregation measures proposed by Duncan and Duncan (1955), and have been used since in modified form to capture occupational discrimination by gender and by race. In that literature, both the differences in the distributions of the respective groups, and the differences in mean earnings between the respective groups, are highly relevant; see (Wolff 1976, p. 152).

There are noteworthy examples of such measures, for example, the reservation of quotas in education and employment for ethnic, religious, or caste minorities, and supplementary nutrition targeted to the poor, children, or lactating mothers.