l ecture 2: s tochastic d ominance, i nequality d ecompositions and i nequality of o pportunity...
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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
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)
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.
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.
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
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
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
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
3. INEQUALITY DECOMPOSITIONS:POPULATION SUBGROUPS
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
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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
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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.
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
nces
of
log
inco
mes
alphas and betas alphas, betas, gammas 1996-1976
Source: "Pesquisa Nacional por Amostra de Domicilios" (PNAD), 1976 and 1996.
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
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
Source: "Pesquisa Nacional por Amostra de Domicilios" (PNAD), 1976 and 1996.
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)
4. MEASURING INEQUALITY OF OPPORTUNITY
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
4. MEASURING INEQUALITY OF OPPORTUNITY
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 ,,,
4. MEASURING INEQUALITY OF OPPORTUNITY
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
-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
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)
4. MEASURING INEQUALITY OF OPPORTUNITY
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”
4. MEASURING INEQUALITY OF OPPORTUNITY
CARDINAL INDICES: THE EX-ANTE APPROACH
1
0
,minmaxarg*
dy k
k
kk
k
k
kVDG ydy
minmaxarg,minmaxarg*
1
0
4. MEASURING INEQUALITY OF OPPORTUNITY
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.
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 ,...,,* 21 K ...21
Jj TTTT ,...,,...,, 21* | J ...21 ; JkkJ , ; and
J
jj
J
jj NNN
1
1
1
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.