Stata Technical Bulletin
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Kleiber, C. 1996. Dagum vs. Singh-Maddala income distributions. Economics Letters 53: 265-268.
McDonald, J. B. 1984. Some generalized functions for the size distribution of income. Econometrica 52: 647-663.
Singh, S. K. and G. S. Maddala. 1976. A function for the size distribution of income. Econometrica 44: 963-970.
Wifling, B. and W. Kraemer. 1993. The Lorenz-ordering of Singh-Maddala income distributions. Economics Letters 43: 53-57.
sg107 Generalized Lorenz curves and related graphs
Stephen P. Jenkins, ISER, University of Essex, UK, [email protected]
Philippe Van Kerm, GREBE, University of Namur, Belgium, [email protected]
Generalized Lorenz curves (henceforth GLC’s) are frequently used by economists as a tool for representing and comparing
empirical distributions, typically of income. The GLC of a continuously distributed variable y plots the cumulative total of
y divided by total population size against p = Fɑ/), the cumulative distribution function. Mathematically, point coordinates
[p{y), GL(p(j∕))] of the GLC are given by
P{y) = F{y}, θL(p(y)) = [ xf(x}dx
Jo
with f(x) = dF(ж) ∕dx. If the GLC coordinates are computed using a series of discrete data points y1,..., yw, where observations
have been ordered so that y1 ≤ у2 ≤ ∙ ∙ ∙ ≤ yon one obtains
p(⅜) = θL(p(⅜)) =
and analogously for weighted data.
GLCs of income distributions have attractive properties, related to checks of “welfare dominance” and “poverty dominance.”
For example, if one were to draw the GLCs for two countries A and B, and found that the GLC for A lay above the GLC for B
at each value of p, then one may conclude that welfare is higher and poverty lower in distribution A compared to distribution
B, according to all measures of welfare and poverty satisfying a standard set of desirable axioms. See for example Shorrocks
(1983) or the texts by Cowell (1995) or Lambert (1993) for further details.
A series of graphical instruments are closely related to GLCs, some of them perhaps better known. The most obvious is the
Lorenz curve. The Lorenz curve of a variable у plots the cumulative share of у against p = F(y), the cumulative distribution
function. The LC coordinates for the corresponding discrete case are thus p(yi) = i∕N L(p{yi'}'} = ∑2}=ι ‰7 ∑≠^=ι ¾ The
Lorenz curve of у is simply the GLC of y/ μy where μy is the mean of y. If two Lorenz curves do not intersect, one may
conclude that inequality in the distribution with the higher curve is lower than inequality in the other distribution, according to
all standard inequality indices (e.g., all those in the Atkinson and Generalized Entropy classes, and the Gini coefficient).
Imagine now that one plots the cumulative share of some other variable s (observed jointly with y) against p = F{y),
the cumulative distribution function. The picture obtained is the concentration curve of s against y. Say we observe a set of
pairs (?/i, si),..., sjv) indexed in such a way that y1 ≤ y2 ≤ ∙ ∙ ∙ ≤ yιh, the coordinates of the concentration curve are
p{yi∙,si) = i∕N, C(p(yi,Si}'} = ∑2}=ι ∙sj∕Σ≠!=i ∙sj = ∑}=ι ∙sj∕Ms∕-^, where μs is the mean of s. Concentration curves are
particularly useful for the analysis of taxes, benefits, and income redistribution (see, for example, Lambert 1993).
The so-called TIP (Three I’s of Poverty) curves can also be easily introduced in this framework (Jenkins and Lambert 1997).
Let z be some threshold and define the variables g as g =z — y and a as r =1 — (y∕z) = g∣z. The coordinates of the TIP curve
are
г___, ,
P⅛)i,z) = T7,TIP(p(⅜,z)) =
whereas the coordinates of the TIP of normalized poverty gaps are
Σj=l r^j
г___, ,
p{yi-,z) = τ7,TIPn⅛(‰z)) =