Stochastic dominance and decomposable measures of inequality and poverty

• Published date: 01 April 2021
• Author: Buhong Zheng
• Date: 01 April 2021
• DOI: http://doi.org/10.1111/jpet.12496

J Public Econ Theory. 2021;23:228–247.wileyonlinelibrary.com/journal/jpet228

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© 2020 Wiley Periodicals LLC

Received: 26 September 2019

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Revised: 24 November 2020

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Accepted: 29 November 2020

DOI: 10.1111/jpet.12496

ORIGINAL ARTICLE

Stochastic dominance and decomposable

measures of inequality and poverty

Buhong Zheng

Department of Economics, University of

Colorado Denver, Denver, Colorado, USA

Correspondence

Buhong Zheng, Department of Economics,

University of Colorado Denver, Denver

80217‐3364, CO, USA.

Email: buhong.zheng@ucdenver.edu

Abstract

In this paper, we characterize some new links be-

tween stochastic dominance and the measurement of

inequality and poverty. We show that: for second‐

degree normalized stochastic dominance (NSD), the

weighted area between the NSD curve of a distribu-

tion and that of the equalized distribution is a de-

composable inequality measure; for first‐degree and

second‐degree censored stochastic dominance (CSD),

the weighted area between the CSD curve of a dis-

tribution and that of the zero‐poverty distribution is a

decomposable poverty measure. These characteriza-

tions provide graphical representations for decom-

posable inequality and poverty measures in the same

manner as Lorenz curve does for the Gini index. The

extensions of the links to higher degrees of stochastic

dominance are also investigated.

KEYWORDS

censored stochastic dominance, decomposable inequality measure,

decomposable poverty measure, normalized stochastic dominance,

stochastic dominance

JEL CLASSIFICATION

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1|INTRODUCTION

A milestone in the history of income inequality measurement research is the establishment of

the close connection among Lorenz curve, summary measures of inequality, and Pigou‐Dalton's

principle of transfers. Atkinson (1970), building upon Rothchild and Stiglitz's (1970) seminal

work on second‐degree stochastic dominance (SSD), proved that the dominance of Lorenz

curves is tantamount to the unanimous ranking by all relative inequality measures that satisfy

Pigou‐Dalton's principle of transfers. The equivalence between Lorenz dominance and SDD (for

equal‐mean distributions) also makes SSD a dual inequality ordering condition. By normalizing

all income distributions to have equal mean, the application of SSD would yield the same

inequality rankings as Lorenz dominance. Although not as frequently used as Lorenz dom-

inance, this dual inequality ordering condition has been first noted in Foster and Sen (1997) and

is generally referred to as “normalized stochastic dominance (NSD)”(Zheng et al., 2000)to

distinguish it from the original dominance condition.

The close connection that Atkinson uncovered also allows us to define and characterize

Lorenz‐consistent inequality measures by extracting and summarizing the information con-

tained in a Lorenz curve. The best‐known example is the Gini index which is twice the area

enclosed by the Lorenz curve and the 45‐degree line. Another well‐known example is the

Schutz coefficient which is the maximum distance between the Lorenz curve and the 45‐degree

line. Generalizing the Gini index which is an unweighted area, Shorrocks and Slottje (2002)

defined a class of “Gini type inequality measures”by weighting the area between the Lorenz

curve and the 45‐degree line. The Gini type measures, as noted by Shorrocks and Slottje (2002),

are closely related to the “generalized Gini”measures examined in Donaldson and Weymark

(1980), Weymark (1981), and Bossert (1990), and the linear measures by Mehran (1976).

Aaberge (2000), drawing upon the similarity between Lorenz curve and the cumulative prob-

ability distribution function, characterized the moments of a Lorenz curve as inequality mea-

sures. Since Aaberge's “Lorenz family of inequality measures”essentially employs a specific

weighting function (the power function) in weighting the area, the family is also included in the

class that Shorrocks and Slottje (2002) defined.

1

None of the inequality measures characterized via the Lorenz curve is decomposable. In

fact, all Gini type measures are the so‐called rank‐based measures since the relative position of

each income matters in determining the level of inequality for society. For a decomposable

inequality measure, in contrast, the relative rank does not matter and the overall inequality can

be decomposed as a sum of the “within‐group inequality”and the “between‐group inequality.”

Given the increasing importance of decomposable measures in income inequality analysis, it

would be helpful to provide them with a geometric representation which can also better

enunciate the difference among the various measures.

2

In this paper, we show that the dual inequality ordering condition—NSD—is able to

characterize the entire class of decomposable inequality measures. Specifically, we show

that the weighted area between the second‐degree NSD curve of a distribution and that of

its equalized distribution is a decomposable inequality measure; different weighting

functions give rise to different inequality measures. Formby et al. (1999)demonstrated

through a direct calculation that the area between the two second‐degree NSD curves is

one‐half of the squared coefficient of variation which is a member of the generalized

entropy (GE) family (Shorrocks, 1980,1984). This paper generalizes the connection to all

decomposable inequality measures. The characterization of decomposable inequality

1

Zheng (2017) extended the Shorrocks–Slottje approach to poverty measurement and characterizes a class of gen-

eralized Sen poverty measures.

2

Magdalou (2018) recently provided a geometric interpretation for decomposable inequality measures in his attempt to

explain some puzzles in inequality measurement. His interpretation is utility‐curve‐based and is different from what we

do in this paper.

ZHENG

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