# (p.135) A.3.2 Generalized Lorenz Dominance

# (p.135) A.3.2 Generalized Lorenz Dominance

The fixed‐mean comparisons addressed by Atkinson's Theorem are not the only comparisons over which these welfare functions can agree. Extending this line of analysis, a complete characterization of ‘unanimity of welfare’ quasi‐orderings can be obtained with the help of Shorrocks' (1983) *generalized Lorenz* curve, *GL*, defined as the Lorenz curve *L* scaled by the mean μ (i.e., *GL*(*p*) = μ *L*(*p*) for each population share *p*).^{34} Generalized Lorenz dominance is then defined analogous to Lorenz dominance: **x** dominates **y** by the generalized Lorenz criterion, written **x** *GL* **y**, if *GL* _{x} lies above *GL* _{y} (or at least above somewhere and not below anywhere). Diagram A3.1 illustrates comparisons of generalized Lorenz curves.

Shorrocks (1983) shows that **x** *GL* **y** is equivalent to *W*(**x**) > *W*(**y**) for all welfare functions *W* satisfying the requirements

^{35}Consequently, for this class of welfare functions,

*GL*is the appropriate indicator of unanimously higher welfare when means differ. In the special case in which means are the same,

**x**

*GL*

**y**coincides with

**x**

*L*

**y**, which leads us back to the Lorenz theorems of Atkinson and related results (as in

*OEI‐1973*, Chapter 3).

Even though generalized Lorenz rankings extend welfare comparisons quite radically by removing the requirement of fixed means, they too are incomplete (in the way the entire Lorenz approach is). For example, if **x** has the higher mean, while **y** has the higher of the respective smallest incomes, then **x** and **y** cannot be ranked by *GL*. However, Shorrocks and others have provided many empirical examples for which *GL* applies and welfare functions agree, and this extension is of much practical importance indeed.

The Shorrocks (1983) result suggests an alternative characterization of the welfare functions satisfying the required properties, to wit, *GL‐consistent* (since they agree with the generalized Lorenz ranking when it applies). The approach also suggests one specific *GL*‐consistent welfare function that concentrates on the area below the Lorenz curve, analogously to the Lorenz interpretation of the Gini coefficient. And that can then be linked to the Gini‐based ‘corrections’ for inequality in ‘distribution‐adjusted real national income’ μ (1 − *G*) as proposed in Sen (1976a).^{36} Let *W* be twice the area below the generalized Lorenz curve. *W* ranges between 0 (approximated
(p.137)
when *GL* is near the horizontal axis) and μ (approximated when *GL* is near the diagonal of complete equality), and is clearly *GL*‐consistent. It is easy to verify that *W* is μ (1 − *G*), corresponding exactly to the Gini‐based social welfare criterion used in Sen (1976a).^{37} Its simple graphical representation as well as its interpretation as the mean income modified downward by the Gini inequality adds to its attraction as an intuitive and usable welfare indicator.

## Notes:

(34) While Shorrocks was the first to identify the exact conditions and to establish precisely how they operate, there were earlier discussions of this general issue, particularly by Blackorby and Donaldson (1977).

(35) Actually, the Shorrocks result concerns the weak definition of generalized Lorenz dominance; hence the welfare dominance he obtains has a weak inequality. See also Marshall and Olkin (1979, p. 109).

(36)
In fact, as was discussed earlier (in section A.1.2, μ (1 − *G*) is not the welfare function itself, but represents a supporting hyperplane that bounds from below all the superior points in the multicommodity characterization analysed in Sen (1976a). However, μ (1 − *G*) can be used as a welfare function itself, consistently with that analysis, for the special case of a one‐commodity world (or *as if* one‐commodity world with fixed substitution rates) and linear interpersonal weights. It is in that simpler form that μ (1 − *G*) has been most used in actual empirical work for intercountry comparisons; for example, in the United Nations' *Human Development Report 1990* (UNDP 1990, pp. 11–13). See also Sen (1973b) and Kakwani (1980a, 1981, 1984b, 1986) for uses of this and related measures.

(37)
Note that even though μ (1 − *G*) is fully *GL*‐consistent (and indeed corresponds to twice the area under the generalized Lorenz curve), not all of the welfare functions of the form μ (1 − *I*), using other measures of inequality *I*, are *GL*‐consistent. See Blackorby and Donaldson (1978) for examples of violation.