Lecture 2: Stochastic Dominance, Inequality Decompositions and Inequality of Opportunity

LECTURE 2: STOCHASTIC DOMINANCE, INEQUALITY DECOMPOSITIONS AND INEQUALITY OF OPPORTUNITY

Francisco H. G. Ferreira

Poverty and Inequality Analysis Course 2011

Module 5: Inequality and Pro-Poor Growth

OUTLINE

1. Stochastic Dominance and Rank Robustness

2. The Determinants of Inequality: a conceptual overview

3. Inequality Decompositions By Population Subgroup

The Classic Decomposition The ELMO modification

By Income Source Generalizing Oaxaca-Blinder

4. An application: Measuring inequality of opportunity

STOCHASTIC DOMINANCE AND RANK ROBUSTNESS

Welfare: First or Second Order Stochastic Dominance

Poverty: Mixed Poverty Dominance

Inequality: Lorenz Dominance

Welfare Dominance: First Order

Figure 1 F(y)

1

B A

y

yyFyF BA , and yyFyF BA ,

Distribution A displays first-order stochastic dominance over distribution B if its cumulative distribution function FA(y) lies nowhere above and at least somewhere below that of B, FB(y). For any income level y, fewer people earn less than it in A than in B. For any income level y, fewer people earn less than it in A than in B. If that is the case, a theorem due to Saposnik (1981) establishes that any social welfare function which is increasing in income will record higher levels of welfare in A than in B.

Welfare Dominance: Second Order

Figure 2 G (yk) B A yk

kkBkA yyGyG , and kkBkA yyGyG ,

Distribution A displays second order stochastic dominance over B if its deficit function (the integral of the distribution function

G y F y dyk

yk

0

) lies nowhere above (and somewhere below) that of

B. It is a weaker concept than its first order analogue, and is in fact implied by it.

Shorrocks (1983) has shown that if it holds, any social welfare function that is increasing and concave in income will record higher levels of social welfare in A than in B.

Lorenz Dominance Figure 3 L(p)

A

B

0 p

ppLpL BA , and ppLpL BA , Distribution A displays mean-normalized second-order stochastic dominance (also known as Lorenz dominance) over distribution B, if the Lorenz curve associated with it lies nowhere below, and at least somewhere above that associated with B. A Lorenz curve, such as those depicted in the figure above, is a mean-normalized integral of the inverse of a distribution

function: L py

F dp

1 1

0 . In other words, it plots the share of income

accruing to the bottom p% of the population, against p. For a Lorenz curve (A) to lie everywhere above another (B) means that in A, the poorest p% of the population receive a greater share of the income than in B, for every p.

Atkinson (1970) has shown that if it holds, inequality in A is lower than in B according to any inequality measure that satisfies the Pigou-Dalton transfer axiom.

2. THE DETERMINANTS OF INEQUALITY:

A CONCEPTUAL OVERVIEW

Inequality measures dispersion in a distribution. Its determinants are thus the determinants of that distribution. In a market economy, that’s nothing short of the full general equilibrium of that economy.

One could think schematically in terms of: y = a.r

This suggests a scheme based on assets and returns: Asset accumulation Asset allocation / Use Determination of returns Demographics Redistribution



Box 1: Schematic Representation of Household Income Determination I (Z, w)

Investment in Human Capital P (X, Z, w) V(J) The Matching Function

D( p(X, Z, J), X, Z, J, w) Remuneration in the Labor Market G(, w) Household Formation

F(y) Redistribution H(y+t)



Modeling these processes in an empirically testable way is quite challenging. Though there are G.E. models of wealth and income

distribution dynamics

Historically, empirical researchers have used ‘shortcuts’, such as: decomposing inequality measures by population

subgroups, and attributing “explanatory power” to those variables which had large “between” components;

Decomposing inequality by income sources, to understand which contributed most to inequality, and why;

Decomposing changes in inequality into changes in group composition, group mean and group inequality.

11

3. INEQUALITY DECOMPOSITIONS:POPULATION SUBGROUPS

The significance of group differences in well-being is thus often at the center of the study of inequality.

Techniques for the decomposition of inequality into a “between-group” and a “within-group” component have become a workhorse in the inequality literature.

Much of the methodological development occurred in the 1970s and early 1980s: Bourguignon (1979), Cowell (1980), Shorrocks (1980)

proposed a class of sub-group decomposable inequality measures

Pyatt (1975), Yitsaki (various) have explored the decomposability of the Gini coefficient.


n

i y

iy

nyE

12

111

);(

n

i iy

y

nE

1

log1

)0(

n

i

ii

y

y

y

y

nE

1

log1

)1(

n

ii yy

ynE

1

2

22

1)2(

Not all inequality measures are decomposable, in the sense that I = IW + IB. The Generalized Entropy class is.

Examples includeTheil – L

Theil – T

0.5 CV2


Let Π (k) be a partition of the population into k subgroups, indexed by j. Similarly index means, n, and subgroup inequality measures. Then if we define:

n

i y

jjB fyE

12

11

);(

k

jjjW yEwyE

1

;;

where 1

jjj fvw

n

nv jj

j n

nf j

j

Then, E = EB + EW.

14


Given a partition and functional we can summarize between-group inequality as:

Moving from any partition to a finer sub-partition cannot decrease.

AN EXAMPLE FROM BRAZIL

Source: Ferreira, Leite and Litchfield, 2008

16

TWO CONCERNS

Concern #1: Between-group shares in practical applications are usually quite small:

Anand (1983) decomposes Malaysian inequality and finds a between-ethnic group contribution of only about 15%

Cowell and Jenkins (1995) decompose U.S. inequality by groups defined in terms of age, sex, race and earner status of the household head, and finds that most inequality remains “unexplained”

Elbers, Lanjouw, and Lanjouw (2003) use poverty maps to show that between-community inequality (across many hundreds of communities) is still vastly outweighed by within-community inequality.

17

TWO CONCERNS

Such findings have left some observers worried: Kanbur (2000) states that the use of such

decompositions “…assists the easy slide into a neglect of inter-group inequality in the current literature”

He argues that social stability and racial harmony can (and does) break down once the average differences between groups go beyond a certain threshold.

Concern #2: It is difficult to compare decompositions across settings Over time Across settings

18

CONCERN 2, CONT. AN EXAMPLE FROM THREE COUNTRIES The shares of income inequality attributable to

differences between racial groups in Brazil, and South Africa are 16%, and 38%, respectively. In the U.S. the between-race inequality share is only 8%

In each country, the mean income of the non-white groups is much below that of the white group, but the non-white groups form the majority in South Africa (80%), half of the population in Brazil (50%), and a minority in the U.S. (28%).

The standard decomposition, is sensitive to differences in relative mean incomes across groups, but also to the numbers of groups, their population shares, and their “internal” inequality. Does it capture the “salience” of horizontal inequality as we might wish?

19

3. INEQUALITY DECOMPOSITIONS:THE ELMO MODIFICATION

Elbers, Lanjouw, Mistiaen, and Ozler (2007) propose comparing IB against a benchmark of maximum between-group inequality holding the number and relative sizes of groups constant:

J groups in partition of size j(n).

20

PROPERTIES cannot be smaller than

However, may not rise with finer sub-partitioning. for both the numerator and denominator

change as a result of finer partitioning.

For any finer partitioning of an original partition, the rate of change of is lower than or equal to that of .

21

CALCULATING We calculate in the usual way. Maximum between-group inequality:

For a maximum, groups must occupy non-overlapping intervals (Shorrocks and Wan, 2004).

In the case of n sub-groups in the partition we take a particular permutation of sub-groups {g(1),….g(n)} allocate lowest incomes to g(1), then to g(2), etc.

Calculate the corresponding between group inequality.

Repeat this for all n! permutations of sub-groups. Select the highest resulting between-group

inequality.

Calculate the ratio of the two

22

AN EXAMPLE (ELMO, 2007)

3. INEQUALITY DECOMPOSITIONS:BY INCOME SOURCES

Shorrocks A.F. (1982): “Inequality Decomposition by Factor Components, Econometrica, 50, pp.193-211.

Noted that could be written as:

n

ii yy

ynE

1

2

22

1)2(

2

1

22)2( ff

ff EEE

Correlation of income source with total income

Share of income source

Internal inequality of the source

3. INEQUALITY DECOMPOSITIONS:BY INCOME SOURCES

Source: Ferreira, Leite and Litchfield, 2008.

ff

f

f

ff

Total Household Income per

Capita

Total Earnings from

Employment*

Total Income from Self-

Employment**

Total Employer Income***

Total Social Insurance

Transfers #

All Other Incomes ##

Mean 393.88 196.06 60.76 44.12 76.82 16.11E(2) 1.618 2.101 6.801 43.301 6.925 23.090Correlation with household income ( f)

1 0.569 0.310 0.598 0.443 0.299

Relative mean ( f) 1 0.498 0.154 0.112 0.195 0.041Absolute factor contribution (S f)

1.618 0.522 0.158 0.561 0.289 0.088

Proportionate factor contribution (sf)

1 0.323 0.098 0.347 0.179 0.054

E(2), yf>0 1.618 1.365 1.991 2.115 1.923 6.567Pop share with yf>0 1 0.717 0.341 0.060 0.326 0.300

Table 4: The Contribution of Income Sources to Total Household Income Inequality in 1981, 1993 and 2004.

2004

f

f

3. INEQUALITY DECOMPOSITIONS:

DYNAMICS FOR SCALAR MEASURES

Mookherjee, D. and A. Shorrocks (1982): "A Decomposition Analysis of the Trend in UK Income Inequality", Economic Journal, 92, pp.886-902.

))y( ( )f - v( +

f )( - + f G(0) +

)G(0f

= G(0)

jjj

k

j=1

jjj

k

j=1jj

k

j=1

jj

k

j=1

log

log

Pure inequality

Group Size

Relative means

THE (OBLIGATORY) EXAMPLE FROM BRAZIL…

Observed Proportional change in E(0)

a b c d a b c d a b c d

Age 0.112 -0.003 0.000 0.002 -0.139 -0.002 0.000 0.017 -0.044 -0.003 0.000 0.019

Education 0.110 0.000 0.043 -0.035 -0.089 0.001 0.019 -0.053 0.011 0.001 0.088 -0.136

Family Type 0.120 -0.005 0.015 -0.004 -0.138 -0.005 0.022 0.005 -0.039 -0.004 0.040 -0.032

Gender 0.116 -0.005 0.000 0.000 -0.120 -0.004 0.000 0.000 -0.018 -0.009 0.000 -0.001

Race n.a. n.a. n.a. n.a. -0.101 -0.003 0.001 -0.021 n.a. n.a. n.a. n.a.

Region 0.141 -0.003 -0.003 -0.024 -0.118 -0.001 -0.001 -0.005 0.012 -0.005 -0.004 -0.028

Urban/rural 0.178 0.005 -0.032 -0.040 -0.104 0.002 -0.014 -0.009 0.054 0.017 -0.048 -0.049

Table 5. A Decomposition of Changes in Inequality by Population Subgroups.

1981-1993 1993-2004 1981-2004

-0.035

Note: Term a is the pure inequality effect; terms b and c are the allocation effect; term d is the income effect.Source: Authors’ calculations from PNAD 1981, 1993 and 2004.

0.107 -0.128

Source: Ferreira, Leite and Litchfield, 2008.

3. INEQUALITY DECOMPOSITIONS: DYNAMICS FOR THE WHOLE

DISTRIBUTION

In practice, decompositions of changes in scalar measures suffer from serious shortcomings: Informationally inefficient, as information on entire

distribution is “collapsed” into single number. Decompositions do not ‘control’ for one another. Can not separate asset redistribution from changes in

returns.

With increasing data availability and computational power, studies that decompose entire distributions have become more common. Juhn, Murphy and Pierce, JPE 1993 DiNardo, Fortin and Lemieux, Econometrica, 1996

3. INEQUALITY DECOMPOSITIONS: THE OAXACA-BLINDER DECOMPOSITION

These approaches draw on the standard Oaxaca-Blinder Decompositions (Oaxaca, 1973; Blinder, 1973)

Let there be two groups denoted by r = w, b.

Then and

So that

Or:

Caveats: (i) means only; (ii) path-dependence; (iii) statistical decomposition; not suitable for GE interpretation.

irririr Xy

wiwyw X bibyb X

bibiwbwiwybyw XXX

wibiwbwibybyw XXX

“returns component” “characteristics component”

3. INEQUALITY DECOMPOSITIONS: JUHN, MURPHY AND PIERCE (1993)

irririr Xy irirrir XF 1

001

010' iiii XFXy

where

Define:

001

110" iiii XFXy

Then: 0' ii yFIyFI

ii yFIyFI '"

ii yFIyFI "1 Observed charac. Component.

Returns component

Unobserved charac. component

3. INEQUALITY DECOMPOSITIONS: BOURGUIGNON, FERREIRA AND LUSTIG (2005)

Figure 15a: A Complete Decomposition

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

Percentiles

Dif

fere

nces

of

log

inco

mes

alphas and betas 1996-1976

Source: "Pesquisa Nacional por Amostra de Domicilios" (PNAD), 1976 and 1996.


Figure 15b: A Complete Decomposition

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

Percentiles

Dif

fere

nces

of

log

inco

mes

alphas and betas alphas, betas, gammas 1996-1976



Figure 15: A Complete Decomposition

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

Percentiles

Dif

fere

nces

of

log

inco

mes

alphas and betas alphas, betas, gammas

mu(d), alphas, betas, gammas mu(d), mu(e), alphas, betas, gammas

1996-1976


4. MEASURING INEQUALITY OF OPPORTUNITY

MOTIVATION Amartya Sen’s Tanner Lectures (1980) question:

“Equality of what?” Modern theories of social justice want to move beyond

the distribution-neutral, sum-based approach of utilitarianism.

Desire to place some value on “equality”. But are outcomes, such as incomes, the appropriate

space? What role for individual effort and responsibility? Are all inequalities unjust?

“We know that equality of individual ability has never existed and never will, but we do insist that equality of opportunity still must be sought”

(Franklin D. Roosevelt, second inaugural address.)

Equality of opportunity is a normatively appealing concept. Many philosophers (and politicians) increasingly see it as the appropriate “currency of egalitarian justice”.

Dworkin (1981): What is Equality? Part 1: Equality of Welfare; Part 2: Equality of Resources”, Philos. Public Affairs, 10, pp.185-246; 283-345.

Arneson (1989): “Equality of Opportunity for Welfare”, Philosophical Studies, 56, pp.77-93.

Cohen (1989): “On the Currency of Egalitarian Justice”, Ethics, 99, pp.906-944.

Roemer (1998): Equality of Opportunity, (Cambridge, MA: Harvard University Press)

Sen (1985): Commodities and Capabilities, (Amsterdam: North Holland)


MOTIVATION

Economists have also become interested. Van de Gaer (1993) and John Roemer (1993, 1998) suggested an influential definition, based on the distinction between “circumstances” and “efforts” among the determinants of individual advantage. Circumstances are morally-irrelevant, pre-determined

factors over which individuals have no control. Equality of opportunity is attained when advantage is

distributed independently of circumstances.

“According to the opportunity egalitarian ethics, economic inequalities due to factors beyond the individual responsibility are inequitable and [should] be

compensated by society, whereas inequalities due to personal responsibility are equitable, and not to be compensated”

(Peragine, 2004, p.11)

yFCyF


MOTIVATION

Roemer’s definition of equality of opportunity:“Leveling the playing field means guaranteeing that those who

apply equal degrees of effort end up with equal achievement, regardless of their circumstances. The centile of the effort distribution of one’s type provides a meaningful intertype comparison of the degree of effort expended in the sense that the level of effort does not” (Roemer, 1998, p.12)

Inverting the quantile function yields:

Test for equality of conditional distributions across types: Lefranc, Pistolesi and Trannoy (2008).

lklk TTyy ,;1,0,

lklk TTklyFyF ,,,


DOMINANCE APPROACH

An example of :0

.2.4

.6.8

1

-2 -1 0 1 2consumption

none incomplete

primary complete

Colombia

0.2

.4.6

.81


none incomplete

primary complete

Ecuador

0.2

.4.6

.81


none incomplete

primary complete

Guatemala

0.2

.4.6

.81


none incomplete

primary complete

Panama

0.2

.4.6

.81


none incomplete

primary complete

Peru

Distribution of p.c.h. consumption conditional on mother’s education

yFyF lk


DOMINANCE APPROACH

Partition population into circumstance-homogeneous groups: types.

Consider inequality in the value of opportunity sets faced by people with different exogenous circumstances. Compute between-type inequality: IOL:

IOR:

Standard inequality decomposition, interpreted as a lower-bound on inequality of opportunity.

Can be computed non-parametrically or parametrically Bourguignon, Ferreira and Menendez (2007) Checchi and Peragine (2010) Ferreira and Gignoux (2011)


CARDINAL INDICES: THE EX-POST APPROACH

kia I

yI

I ki

r

Partition population into effort-homogeneous groups: tranches

Consider inequality among those people who exert same degree of effort

Compute within-tranche inequality. Checchi and Peragine (2010)

The two approaches do not yield identical solutions.

Related to the debate between Roemer’s “Mean of mins”

Van de Gaer’s “Min of Means”


CARDINAL INDICES: THE EX-ANTE APPROACH

1

0

,minmaxarg*

dy k

k

kk

k

k

kVDG ydy

minmaxarg,minmaxarg*

1

0


ILLUSTRATION FOR LATIN AMERICA

Per capita household consumption

Total inequality and levels of inequality of opportunity

0,000

0,050

0,100

0,150

0,200

0,250

0,300

0,350

0,400

0,450

0,500

COL ECU GUA PAN PER

E(0

) in

dic

es

Total inequality

Inequality of opportunity(difference between non-parametric and parametricestimates)Inequality of opportunity(parametric estimate)

In Latin America, (lower-bound ex-post) inequality of economic opportunity:

• ranges from 23% to 35% for income per capita.• ranges from 24% to 50% for consumption per capita.


OPPORTUNITY-DEPRIVATION PROFILES

“The rate of economic development should be taken to be the rate at which the mean advantage level of the worst-off types grows over time. […] I look forward to a future number of the WDR that

carries out the computation, across countries, of this new definition of economic development” (p.243).

Roemer, John E. (2006): “Review Essay, ‘The 2006 world development report: Equity and development”, Journal of Economic Inequality (4): 233-

244

Define an opportunity profile:

And an opportunity-deprivation profile:

KTTT ,...,,* 21 K ...21

Jj TTTT ,...,,...,, 21* | J ...21 ; JkkJ , ; and

J

jj

J

jj NNN

1

1

1


OPPORTUNITY-DEPRIVATION PROFILES

The Brazilian profile, by income per capita

Type Ethnicity Father's occupation

Father's education

Mother's education Place of birth

Estimated population

Share of national population

Mean advantage (HPCY)

Ratio of overall mean

1 black and mix-raced agricultural

worker none or unknown none or unknown Nordeste or

North 2,276,662 0.06776 105.9 0.261

2 black and mix-raced agricultural worker

Upper primary (5) or more

none or unknown Sao Paulo or Federal District

1,417 0.00004 116.5 0.287


none or unknown lower primary Nordeste or North

313,664 0.00934 136.6 0.337


Lower primary none or unknown Nordeste or North

352,729 0.01050 136.9 0.338


Upper primary (5) or more

none or unknown Nordeste or North

7,564 0.00023 144.2 0.355

6 black and mix-raced Other none or unknown none or unknown Nordeste or North

2,063,415 0.06141 144.5 0.356

Brazil’s “opportunity-deprivation profile” in 1996: six poorest “social types” (adding up to 10% of the population), defined by pre-determined background

characteristics.

Documents

Lecture 2: Stochastic Dominance, Inequality Decompositions and Inequality of Opportunity