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LECTURE 2: STOCHASTIC DOMINANCE, INEQUALITY DECOMPOSITIONS AND INEQUALITY OF OPPORTUNITY Francisco H. G. Ferreira Poverty and Inequality Analysis Course 2012 Module 5: Inequality and Pro-Poor Growth
Transcript

LECTURE 2: STOCHASTIC DOMINANCE,

INEQUALITY DECOMPOSITIONS AND

INEQUALITY OF OPPORTUNITY

Francisco H. G. Ferreira

Poverty and Inequality

Analysis Course 2012

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 2

STOCHASTIC DOMINANCE AND RANK

ROBUSTNESS

Welfare: First or Second Order Stochastic Dominance

Poverty: Mixed Poverty Dominance

Inequality: Lorenz Dominance

3

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.

4

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.

5

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 d

p

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.

6

7

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 8

2. THE DETERMINANTS OF INEQUALITY:

A CONCEPTUAL OVERVIEW

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)

9

2. THE DETERMINANTS OF INEQUALITY:

A CONCEPTUAL OVERVIEW

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.

10

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), Yitzhaki (various) have explored the decomposability of the Gini coefficient.

3. INEQUALITY DECOMPOSITIONS:

POPULATION SUBGROUPS

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

i

i yyyn

E1

2

22

1)2(

Not all inequality measures are decomposable, in the sense

that I = IW + IB.

The Generalized Entropy class is.

Examples include

Theil – L

Theil – T

0.5 CV2

12

3. INEQUALITY DECOMPOSITIONS:

POPULATION SUBGROUPS

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

j

jB fyE1

21

1);(

k

j

jjW yEwyE1

;;

where 1

jjj fvw

n

nv

jj

j n

nf

j

j

Then, E = EB + EW. 13

14

3. INEQUALITY DECOMPOSITIONS:

POPULATION SUBGROUPS

Given a partition and a specific index, 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 15

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

i

i yyyn

E1

2

22

1)2(

21

22)2( f

f

f

f EEE

Correlation of

income source with

total income

Share of

income

source

Internal

inequality of

the source 23

3. INEQUALITY DECOMPOSITIONS: BY INCOME SOURCES

Source: Ferreira, Leite and Litchfield, 2008.

f

f

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 16.11

E(2) 1.618 2.101 6.801 43.301 6.925 23.090

Correlation 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 0.041

Absolute 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 6.567

Pop 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

24

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=1

jj

k

j=1

jj

k

j=1

log

log

Pure inequality

Group Size

Relative means 25

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.

26

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

27

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”

28

3. INEQUALITY DECOMPOSITIONS:

JUHN, MURPHY AND PIERCE (1993)

irririr Xy irirrir XF 1

00

1

010' iiii XFXy

where

Define:

00

1

110" iiii XFXy

Then: 0' ii yFIyFI

ii yFIyFI '"

ii yFIyFI "1 Observed charac. component.

Returns component

Unobserved charac. component

29

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

nce

s of

log

in

com

es

alphas and betas 1996-1976

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

30

Ferreira and Paes de Barros (1999, 2005)

3. INEQUALITY DECOMPOSITIONS:

BOURGUIGNON, FERREIRA AND LUSTIG (2005)

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

nce

s o

f lo

g i

nco

mes

alphas and betas alphas, betas, gammas 1996-1976

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

31

Ferreira and Paes de Barros (1999, 2005)

3. INEQUALITY DECOMPOSITIONS:

BOURGUIGNON, FERREIRA AND LUSTIG (2005)

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

nce

s o

f lo

g i

nco

mes

alphas and betas alphas, betas, gammas

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

1996-1976

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

32

Ferreira and Paes de Barros (1999, 2005)

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.)

33

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)

4. MEASURING INEQUALITY OF OPPORTUNITY

MOTIVATION

34

Economists have also become interested. Dirk van de Gaer (1993)

and John Roemer (1993, 1998) have 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

4. MEASURING INEQUALITY OF OPPORTUNITY

MOTIVATION

35

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).

lk

lk TTyy ,;1,0,

lk

lk TTklyFyF ,,,

4. MEASURING INEQUALITY OF OPPORTUNITY

DOMINANCE APPROACH

36

An example of :

0.2

.4.6

.81

-2 -1 0 1 2consumption

none incomplete

primary complete

Colombia

0.2

.4.6

.81

-2 -1 0 1 2consumption

none incomplete

primary complete

Ecuador

0.2

.4.6

.81

-2 -1 0 1 2consumption

none incomplete

primary complete

Guatemala

0.2

.4.6

.81

-2 -1 0 1 2consumption

none incomplete

primary complete

Panama

0.2

.4.6

.81

-2 -1 0 1 2consumption

none incomplete

primary complete

Peru

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

yFyF lk

4. MEASURING INEQUALITY OF OPPORTUNITY

DOMINANCE APPROACH

37

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

4. MEASURING INEQUALITY OF OPPORTUNITY

CARDINAL INDICES: THE EX-POST APPROACH

k

ia I

yI

I k

i

r

38 uvCECfy ,,,

,Cy

Cyln

ˆexp~ii C

yI

I iP

r

~

4. MEASURING INEQUALITY OF OPPORTUNITY

ILLUSTRATION FOR LATIN AMERICA

39

Ferreira and Gignoux, forthcoming

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.

4. MEASURING INEQUALITY OF OPPORTUNITY

CARDINAL INDICES: THE EX-ANTE APPROACH

40

4. MEASURING INEQUALITY OF OPPORTUNITY

EX-ANTE VS. EX-POST APPROACH

41 eGk

ky

1

2

1

3

4. MEASURING INEQUALITY OF OPPORTUNITY

EX-ANTE VS. EX-POST APPROACH

42 eGk

ky

1

2

1

3

4. MEASURING INEQUALITY OF OPPORTUNITY

EX-ANTE VS. EX-POST APPROACH

43 eGk

ky

1

The measurement distinction between the ex-ante and ex-post approached is related to the conceptual distinction between the “equal opportunity policies” of Roemer and van de Gaer:

Roemer‟s “Mean of mins”

Van de Gaer‟s “Min of Means”

Bourguignon, Ferreira and Walton (2007)

4. MEASURING INEQUALITY OF OPPORTUNITY

EX-ANTE VS. EX-POST APPROACH

1

0

,minmaxarg*

dy k

k

kk

k

k

kVDG ydy

minmaxarg,minmaxarg*

1

0

44 dse T

s

t

st

Tt

minmax

subject to tTiuu t

iT

t ,,

4. MEASURING INEQUALITY OF OPPORTUNITY

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

Jj TTTT ,...,,...,, 21

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

J

j

j

J

j

j NNN1

1

1

45

4. MEASURING INEQUALITY OF OPPORTUNITY

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

3 black and mix-raced agricultural worker

none or unknown lower primary Nordeste or North

313,664 0.00934 136.6 0.337

4 black and mix-raced agricultural worker

Lower primary none or unknown Nordeste or North

352,729 0.01050 136.9 0.338

5 black and mix-raced agricultural worker

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.

46

4. MEASURING INEQUALITY OF OPPORTUNITY

SELECTED READINGS (ON THE EMPIRICS)

Bourguignon, François, Francisco H.G. Ferreira and Michael

Walton, "Equity, Efficiency and Inequality Traps: A research

agenda," Journal of Economic Inequality, 5 (2), 235-256, August

2007.

Checchi, Daniele and Vito Peragine, "Inequality of Opportunity in

Italy," Journal of Economic Inequality, Series 8 (4), 429-450,

December 2010.

Ferreira, Francisco H.G. and Jérémie Gignoux, "The

Measurement of Inequality of Opportunity: Theory and an

application to Latin America," Review of Income and Wealth,

forthcoming.

Lefranc, Arnaud, Nicolas Pistolesi and Alain Trannoy, "Inequality

of Opportunities vs. Inequality of Outcomes: Are Western

Societies All Alike?," Review of Income and Wealth, Series 54 (4),

513-546, December 2008.

47


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