Popular Support for Social Evaluation Functions
Rafael Salas
Universidad Complutense de Madrid, Campus de Somosaguas, 28223 Madrid.
Tel: +34 91 3942512. E-mail: [email protected]
East Carolina University (Visiting Fellow), Greenville, NC 27858-4353 USA
Juan Gabriel Rodríguez
Universidad Complutense de Madrid, Campus de Somosaguas, 28223 Madrid.
Tel: +34 91 3942515. E-mail: [email protected]
ABSTRACT
This paper provides sufficient conditions under which the preferences of a social
decision maker accord with majority voting. We show that an additive and concave
utilitarian social evaluation function is consistent with the outcomes of majority voting
for the class of income distributions that are symmetric under a strictly increasing
transformation. An example is the lognormal distribution. The required symmetry
condition is generally accepted employing panel data for 116 countries from the World
Bank’s POVCAL database. In this manner, the proposed methodology provides the
consistent degree of inequality aversion and shows that median income is a good proxy
for welfare.
JEL Classification: D31, D63, H30, P16.
Key Words: inequality aversion; majority voting; social welfare; symmetric
distribution.
1. Introduction
Arrow’s impossibility theorem demonstrates that if the decision-making body has at
least two members and at least three options to decide among, then it is impossible to
design a social welfare function that satisfies unrestricted domain (universality),
nondictatorship, Pareto efficiency, and the independence of irrelevant alternatives
(Arrow, 1950).1 To deal with Arrow’s paradox, we must therefore eliminate or weaken
one of these criteria. Among others, the extant literature explores two main proposals:
majority voting with restricted domain and social evaluation functions.
Majority voting breaks up with universality by imposing a restricted domain of
preferences among voters. For example, if preferences are single peaked, the majority
rule meets Arrow’s remaining axioms, and society commits to the median voter’s
preference (Black, 1948).2 This result has proven useful in many fields. In public
economics, for example, the median voter theorem has been applied to the analysis of
the demand for redistribution. Romer (1975), Roberts (1977), and Meltzer and Richards
(1981), for instance, sought the conditions for progressive taxation as a voting
equilibrium outcome. More particularly, they applied the median voter theorem to linear
tax schedules.
Subsequently, Gouveia and Oliver (1996) generalized the analysis to two-bracket,
piecewise linear tax functions; Cukierman and Meltzer (1991), Roemer (1999) and De
Donder and Hindriks (2003) to quadratic tax functions; and Carbonell-Nicolau and Klor
(2003) to all piecewise linear taxes. In addition, Marhuenda and Ortuño (1995) showed
that if the median voter lies below the mean, then any progressive proposal prevails
over a regressive proposal. In the income inequality and growth literature, Alesina and
Rodrick (1994) and Persson and Tabellini (1994) justified the negative relationship
between growth and income inequality on the grounds of the median voter theorem.
1
The original criteria proposed by Arrow were unrestricted domain, nondictatorship, monotonicity,
nonimposition, and independence of irrelevant alternatives. The most recent version is stronger—that is,
it has weaker conditions—as nonimposition and Pareto efficiency, and the independence of irrelevant
alternatives together do not imply monotonicity.
2
In Schofield (2007) the apparent disparity between the convergence of party leaders towards the centre
of the electoral distribution and the perception of political divergence is explained by the non-policy basis
of political judgement made by the electorate concerning the quality of political contenders.
2
This literature relies on the feature that income distributions are typically right-skewed
so more voters lie to the left of the mean. Consequently, these voters typically vote for
redistribution, which is assumed to be negatively related to growth.
A different strategy to aggregate individual preferences is to assume a social evaluation
function. Here, we abandon the axiom governing the independence of irrelevant
alternatives. The adoption of a particular social evaluation function relies on a set of
generally accepted ethical principles. In this framework, the concavity of the social
evaluation function ensures that an egalitarian principle, the so-called principle of
progressive transfers, applies and the degree of concavity defines the degree of
inequality aversion in a society.
In principle, the two alternatives—majority voting and social evaluation functions—are
rather different. On the one hand, majority voting represents the real-world aggregation
of individual preferences. In addition, majority voting constitutes an ordinal approach
that permits only the partial comparability of social states. On the other hand, a social
evaluation function is a normative methodology based on a set of desirable assumptions
to aggregate the information of individual preferences by a social decision maker
(SDM) (Lambert, 2001). Moreover, a social evaluation function constitutes an ordinal
or cardinal approach that allows for full comparability between social states. It could
therefore be fruitful to study the sufficient conditions under which both alternatives are
equivalent. Among other advantages, to obtain the sufficient conditions under which
both alternatives are equivalent would provide a scenario in which there exists an SDM
whose preferences accord with majority voting. The consistency between the outcome
of majority voting and a social evaluation function is the aim of this paper.
For this result, we first assume the class of concave utilitarian social welfare functions,
probably the most widely used class of welfare functions in the income distribution
3
literature (see, among others, Kolm, 1969, Atkinson, 1970, d’Aspremont and Gevers,
2002, and Blackorby et al., 2002). We then show that for a given set of symmetric
income distributions under the same strictly increasing and concave transformation—
such as the family of lognormal income distributions or the class of distributions that
are symmetric under the Box–Cox transformation more generally—there is a social
evaluation function with a particular degree of aversion to income inequality that is an
ordinal equivalent to median income. That is, given a set of income distributions that
are co-symmetrizable by a strictly increasing and concave transformation, an SDM with
a given level of inequality aversion will prefer the income distribution with the highest
median income. This result is because the equally distributed equivalent (EDE) income
equals the median income for income distributions that are co-symmetrizable.3
Subsequently, we assume that people vote over distributions. In this manner, we can
view the changes in distribution as the result of a political process (see Grandmont,
2006). We then prove that majority voting over distributions that are symmetric under
the same strictly increasing transformation align with the median voter’s preferences. It
then suffices to connect these results to show that the welfarist and majority-voting
approaches are ordinal equivalents.
In the final part of the paper, we test our main assumption, i.e., income distributions are
symmetric under the same strictly increasing transformation. Assuming general power
concave transformations, we test the co-symmetrizability hypothesis for 116 countries
over several years using the World Bank’s POVCAL database. The results confirm that
the required co-symmetrizability condition is generally accepted. Moreover, our
empirical application allows us to provide a consistent aversion parameter of relative
inequality for this set of countries. We also show that median rather than mean income
3
The EDE income is the level of income that if distributed equally to all individuals would generate the
same welfare as the existing distribution X. Note that the EDE value is the counterpart of the certain
equivalent in the theory of choice under uncertainty.
4
is a good proxy for social welfare. This finding should be of great interest to other
fields, such as macroeconomics, where economists usually apply the mean income to
represent an income distribution, but by doing so only consider the size (not the social
welfare) of the distribution.
The organization of the paper is as follows. In Section 2, we link the social evaluation
function of a distribution that is symmetric under a strictly increasing and concave
transformation with the corresponding median income. In Section 3, we show that the
median voter’s preferences drive the solution for majority voting under the cosymmetrizability condition. We also deal with the consistency between majority voting
and the class of social evaluation functions. Section 4 studies as an example the family
of power transformations, while Section 5 provides an empirical illustration. Section 6
discusses future avenues for research.
2. A social decision maker whose preferences accord with median income
We begin our analysis with some notation and definitions. Assume an odd finite number
of individuals, n, who decide over income distributions described by the profile:
X  ( x1 , x2 ,..., xn )  ℝ n  ,
where
xi
is
the
income
of
individual
i,
assumed
positive.4
Let
𝑇 = {𝑋 ∈ ℝ𝑛++ ∕ 0 < 𝑥1 ≤ 𝑥2 ≤ ⋯ ≤ 𝑥𝑛 } be the set of all ordered profiles with
increasing order. Moreover, the mean and median values of X  T are x  and xM ,
respectively.
4
We assume, without lost of generality, that n is odd to avoid discussion on exactly how to define the
median.
5
Assume that M 
(n  1)
is the position of the median income. Then, a profile X  T is
2
said to be symmetric if it satisfies the following property:
𝑥𝑟+𝑀 − 𝑥𝑀 = 𝑥𝑀 − 𝑥𝑀−𝑟 ,
for all 𝑟 ∈ {1,2, … , 𝑀 − 1}. For example, a normal distribution of incomes is symmetric.
We assume 𝕊 to be the set of all symmetric profiles.
Let
𝑓: ℝ ⟶ ℝ
be
a
strictly
increasing
function,
and
define
𝑓(𝑋) =
(𝑓(𝑥1 ), 𝑓(𝑥2 ), … , 𝑓(𝑥𝑛 )). The distribution X is symmetrizable if 𝑓(𝑋) is symmetric for
some choice of f. For example, a lognormal distribution is symmetrizable (with f = log).
We then say that two distributions X and Y are co-symmetrizable if there exists one
strictly increasing function f such that both 𝑓(𝑋) and 𝑓(𝑌) are symmetric. For example,
two different lognormal distributions are co-symmetrizable.
When comparing two distributions, 𝑋 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ 𝑇 and 𝑌 = (𝑦1 , 𝑦2 , … , 𝑦𝑛 ) ∈
𝑇, we define the individual gain function of passing from X to Y as 𝑔𝑖 (𝑋, 𝑌) = 𝑦𝑖 − 𝑥𝑖 ,
for all 𝑖 = 1, 2, … , 𝑛. We assume that there is no reranking among individuals between
X and Y. For example, if X is pre-tax income and Y is post-tax income, we guarantee—
as do real-world statutory tax policies—that the ranking of taxpayers is identical.
Similarly, we define the individual voting function of passing from X to Y as follows:
1 𝑔𝑖 > 0
𝑣𝑖 (𝑋, 𝑌) = { 0 𝑔𝑖 = 0
−1 𝑔𝑖 < 0
for all 𝑖 = 1, 2, … , 𝑛.
Consequently, Y is weakly preferred to X under the majority voting rule, 𝑌 ≿ 𝑋, if and
only if ∑𝑛𝑖=1 𝑣𝑖 ≥ 0. Alternatively, Y is strictly preferred to X under the majority voting
6
rule, 𝑌 ≻ 𝑋 if and only if ∑𝑛𝑖=1 𝑣𝑖 > 0. Finally, Y is indifferent to X under the majority
voting rule, 𝑋 ∼ 𝑌 if and only if ∑𝑛𝑖=1 𝑣𝑖 = 0.
Finally, we assume an SDM with the following evaluation function:
1
𝑊(𝑋) = 𝑛 ∑𝑛𝑖=1 𝑈(𝑥𝑖 ),
where the social utility-of-income function U(·) represents social preferences that are
disinterested or impersonal. Thus, the SDM evaluates utility in society as the average
utility. That is, social welfare is utilitarian (see Kolm, 1969, Atkinson, 1970,
d’Aspremont and Gevers, 2002, and Blackorby et al., 2002).5 For our results below, we
do not need to assume that the utility function U (·) is strictly concave, but we do so for
convenience. We want to ensure that the principle of progressive transfers is fulfilled.
Now, bearing these assumptions in mind, and assuming the class W of utilitarian social
evaluation functions, we find the following result.
Theorem 1
Let 𝑋1 , 𝑋 2 , … , 𝑋 𝐾 ∈ 𝑇 be K possible income distribution vectors, and assume all are cosymmetrizable by a strictly increasing and concave utility function 𝑈(·). Assume an
SDM with a social evaluation function W  W . Then, the SDM will prefer among
𝑋1 , 𝑋 2 , … , 𝑋 𝐾 the distribution with the highest median income.
5
The veil of ignorance (see Harsanyi, 1953) gives another interpretation of the last expression in terms of
risk. The adopted evaluation function would represent the way impartial observers evaluate overall
welfare according to the expected utility.
7
Proof: For any two given income distributions 𝑋 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ 𝑇 and
𝑌 = (𝑦1 , 𝑦2 , … , 𝑦𝑛 ) ∈ 𝑇 that are co-symmetrizable by a strictly increasing and concave
utility function 𝑈(·), we need to prove that the social evaluation function W  W is
fully characterized by the median income; i.e.:
𝑊(𝑋) = 𝑊(𝑌) ⟺ 𝑥𝑀 = 𝑦𝑀 ,
𝑊(𝑋) > 𝑊(𝑌) ⟺ 𝑥𝑀 > 𝑦𝑀 ,
𝑊(𝑋) < 𝑊(𝑌) ⟺ 𝑥𝑀 < 𝑦𝑀 .
By assumption, the utility function 𝑈(·) simultaneously generates 𝑋′, 𝑌′ ∈ 𝕊 that are
symmetric, where 𝑥𝑖′ = 𝑈(𝑥𝑖 ) and 𝑦𝑖′ = 𝑈(𝑦𝑖 ) for all 𝑖 = 1, 2, … , 𝑛. By construction we
have that 𝑥´𝑀 = 𝑈(𝑥𝑀 ). Moreover, we have 𝑥´𝑀 = 𝑥´𝜇 because the distribution 𝑋′ is
symmetric. Noting that 𝑊(𝑋) = 𝑥´𝜇 , we arrive at the following result:
𝑊(𝑋) = 𝑈(𝑥𝑀 ).
The social evaluation function is a strictly increasing transformation of median income.
Therefore, the social evaluation function is ordinal equivalent to median income.
To fully understand this result, it must be noted that the median income of an income
distribution that is symmetric under a strictly increasing transformation is equal to the
equally distributed equivalent income xede. In this manner, social welfare and median
income will be ordinal equivalent because social welfare, W(X), is by definition average
utility, which in turn is U(xede). In other words, we find sufficient conditions under
which the equally distributed equivalent income xede is equal to median income.
8
3. A result on majority voting
After presenting in the previous section the result of an SDM whose preferences accord
with median income, we now show that majority voting over distributions that are
symmetric under the same strictly increasing transformation yields the median voter’s
preferences.
Theorem 26
Let 𝑋1 , 𝑋 2 , … , 𝑋 𝐾 ∈ 𝑇 be K possible income distribution vectors, and assume they are
all co-symmetrizable. Then, the Condorcet winner among 𝑋1 , 𝑋 2 , … , 𝑋 𝐾 is the
distribution with the highest median income.
Proof: A Condorcet winner is an alternative which can beat every other alternative in a
two-way
vote.
Accordingly,
for
any
given
two
income
distributions
X  ( x1 , x2 , ..., xn )  T and Y  ( y1 , y2 , ..., yn )  T that are co-symmetrizable, we need to
prove that
X  Y  xM  yM ,
X  Y  xM  yM
X  Y  xM  yM .
Assume a strictly increasing transformation
f (·) that simultaneously generates
X '  ( x1 ' , x2 ' , ..., xn ' )  S and Y '  ( y1 ' , y2 ' , ..., yn ' ) S, which are symmetric, where
6
This result could be connected with the literature on voting over taxes by assuming that profiles X1,
X ,…,XK are post-tax income distributions from different monotonous tax systems. However, this line of
research goes beyond the scope of the present paper.
2
9
xi '  f ( xi ) and yi '  f ( yi ) for all i  1,2, ..., n . Then, we have xi  yi  xi '  yi ' for all
i  1,2, ..., n . Thus, a majority of voters prefers X to Y if and only if a majority of voters
prefers 𝑋′ to 𝑌′.
Without loss of generality, assume that xM  yM . We must show that a majority of
voters prefers X to Y. But for all i  [1,2, ..., M ] , we have
x'M i  y'M i   x'M i   y'M i  x'M  x'M i  y'M  y'M i
because x'M  y'M . Furthermore, we know that 𝑋′ to 𝑌′ are symmetric by hypothesis so
it is true that
x'M  x'M i  y'M  y'M i  x'M i  x'M  y'M i  y'M .
This implies that x'M i  y'M i because x'M  y'M . Thus, any voter below the median
who prefers Y over X is matched by some voter above the median who prefers X over Y.
By similar logic, any voter below the median who prefers X is matched by some voter
above the median who prefers Y. Consequently, the deciding vote is held by the median
voter. If xM  yM , then the median voter will vote for X; hence X will get majority. 
An illustrative example of this result can be provided for the class of lognormal
distributions. The lognormal distribution is a general function used traditionally to
represent the distribution of income in the economics literature (see, among others,
Aitchison and Brown, 1957 and Cowell, 1995).7 Accordingly, if we assume that the
7
Two reasons justify the general use of the lognormal distribution. First, the product of independent
normal distributions converges asymptotically to a lognormal distribution (see Gibrat, 1957).
Accordingly, we can view the income generation as the product of multiple factors over time. Second,
10
income distribution vectors 𝑋1 , 𝑋 2 , … , 𝑋 𝐾 belong to the family of lognormal
distributions, the Condorcet winner among them will be the distribution with the highest
median income which is xM  exp[ ln X ] .
A possible prediction arising from the result in Theorem 2 is that society will prefer the
income distribution with the highest median income. In point of fact, and despite this
prediction, we do not observe many serious attempts to raise median income. In fact,
median income is usually well below mean income. There are several possible
explanations for this empirical observation. First, voters may base their votes not only
on their anticipated income but also on other factors (e.g., they could prefer greater
economic freedom, or believe that effort should be rewarded, or care about social values
like the extent of individual liberties and the level of national security). Second, income
redistribution may tend to reduce economic growth, and voters anticipating this effect
may then attempt to curb their desire for a redistributive scheme.8 Third, majority voting
in a real-world situation could potentially yield not the median voter’s preferences but
the pivotal voter’s preferences, where the pivotal voter ranks above the median voter.
We find two complementary reasons for this result: (i) certain groups of the population
traditionally do not take part in elections (e.g., those suffering from extreme poverty),
and (ii) certain groups of voters may influence the results of voting (e.g., lobby groups
or coalitions9). Fourth, preferences for redistribution may depend not only on the
median voter’s current position in the income distribution but also on the median voter’s
expectations of future income mobility (see Hirschman and Rothschild, 1973, Ravallion
and Lokshin, 2000, and Bénabou and Ok, 2001). Fifth, the threat of redistribution in
lognormal distributions capture reasonably well the negative skewness that characterizes income
distributions in practice.
8
See Saint Paul and Verdier (1996) for a challenge to the conventional view point that redistribution is
harmful for growth.
9
See Acemoglu et al. (2008).
11
objectively unequal societies may cause the wealthy to alter their consumption patterns
over time such that the median voter no longer supports redistribution (see Kula and
Millimet, 1999). Finally, the co-symmetrizability assumption in Theorems 1 and 2
could be too restrictive. To test the first five possible explanations is clearly beyond the
scope of this paper. However, we can contest the final assertion.
The main assumption in Theorems 1 and 2 requires that the profiles over which society
has to vote should be symmetric under the same strictly increasing transformation. Let f
be a strictly increasing function, and let ℋ𝑓 be the set of all distributions that f maps into
a symmetric distribution. It is obvious that the larger the class ℋ𝑓 , the less restrictive the
assumption in Theorems 1 and 2. Accordingly, we need to measure the range of the
class ℋ𝑓 . For this task, we test the symmetry of more than 500 real distributions for a
particular class of strictly positive (and concave) transformations (the power
transformation) in Section 5. The conclusion is that the assumption is not very
restrictive in practice.
After presenting the results in Theorem 1 and Theorem 2, we obtain the main result of
the paper by combining both. Namely, a particular additive and concave utilitarian
social evaluation function is consistent with the outcome of majority voting if the
income distributions are symmetric under the same strictly increasing and concave
transformation. In principle, majority voting and social welfare are different alternatives
to aggregate individual preferences. The results in Theorems 1 and 2 together provide
sufficient conditions under which both approaches are consistent for a particular level of
inequality aversion. To advocate this proposal, we provide in the next section an
illustrative example.
12
A final remark is worth noting. In this context, the median income can be used as a
proxy for social welfare. Typically, real national income comparisons are based on real
per capita income (𝜇). In this manner, these comparisons explicitly omit distributional
considerations. On the contrary, Sen (1976) has proposed the use of 𝜇(1 − 𝐺), where G
is the Gini coefficient, to permit a welfare interpretation of real income comparisons.10
We propose instead the use of real median income because it is ordinal equivalent to the
social welfare that is consistent with majority voting (Theorems 1 and 2). We contrast
this proposal in the empirical exercise in Section 5. Our results show that real median
income tracks welfare significantly better than real mean income and 𝜇(1 − 𝐺).11
4. An example: the family of power transformations
In statistics and econometrics, the use of the Box–Cox transformation (Box and Cox,
1964) is usually advocated to stabilize variance and make the data more normal
distribution-like.12 In fact, the class of power transformations, which is an affine
transformation of the Box–Cox family, has been used traditionally to evaluate income
distributions. For example, Schwartz (1985) examined the full family of power
transformations using several years of US income data. He found that a transformation
intermediate between the log transformation and no transformation, say 𝜀 = 2/3, most
closely approximated income distributions in the US. Here, we use this family of
transformations to make the data more symmetric distribution-like.
10
Actually, the 𝜇(1 − 𝐺) proposal is a particular case of a more general framework developed by Sen
(1976) for real income comparisons.
11
Note that Newbery (1970) proved that no SDM exists whose preferences belonging to the class W
accord with the ranking of distributions by 𝜇(1 − 𝐺).
12
The Box–Cox transformation (Box and Cox, 1964) is as follows:
 x i  1

0   1
g ( xi )   
 ln x i
 1
13
Assume an initial distribution of incomes 𝑋 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ 𝑇 and the following
family of power transformations:
𝑥𝑖1−𝜀
𝑓(𝑥𝑖 ) = {1 − 𝜀
𝑙𝑛 𝑥𝑖
0<𝜀≠1
𝜀=1
for all 𝑖 = 1, 2, … , 𝑛. These transformations are strictly increasing and concave, where
the parameter  is positive to ensure strict concavity. Moreover, this family corresponds
to the utility function used in Kolm (1969) and Atkinson (1970).13 In this framework,
the parameter  is constant and represents the relative aversion to inequality (see Pratt,
1964, and Arrow, 1965), and the utilitarian social evaluation function is:
𝑛
1
𝑥𝑖1−𝜀
∑
𝑛
1−𝜀
𝑊(𝑋) =
0<𝜀≠1
𝑖=1
𝑛
1
∑ 𝑙𝑛 𝑥𝑖
{𝑛
𝜀 = 1.
𝑖=1
The parameter  that transforms the income distribution 𝑋 into a symmetric distribution
is then the relative aversion to inequality of the Kolm–Atkinson utility function. In this
manner, inequality aversion is country specific as in Lambert et al. (2003) and Millimet
et al. (2008), among others. Then, social welfare will be an increasing transformation of
median income (see Theorem 1). Because of the symmetry of 𝑋 ′ ∈ 𝕊 , we have:
𝑥𝑀 = 𝑓 −1 [𝑥′𝜇 ],
1
where 𝑥′𝜇 = 𝑛 ∑𝑛𝑖=1 𝑓(𝑥𝑖 ). Moreover, we know:
1
𝑓
−1
(𝑥𝑖 ) = {[(1 − 𝜀)𝑥𝑖 ]1−𝜀
exp(𝑥𝑖 )
0<𝜀≠1
𝜀 = 1.
Therefore, the median income 𝑥𝑀 is as follows:
13
By assuming this utility function, both authors imposed homotheticity on the social evaluation function.
14
𝑛
𝑥𝑀 =
1
[ ∑ 𝑥𝑖1−𝜀 ]
𝑛
𝑖=1
1
1−𝜀
0<𝜀≠1
𝑛
1
𝑒𝑥𝑝 [ ∑ 𝑙𝑛(𝑥𝑖 )]
𝑛
{
𝜀 = 1.
𝑖=1
It is clear from the above that the social welfare function à la Kolm–Atkinson is an
increasing transformation of median income. We also observe that the median income
equals the equally distributed equivalent income, xede. Note that for  = 1 we attain the
expression for the lognormal distribution, xM  exp[ ln X ] . Therefore, we can conclude
that under majority voting, society will vote for the income distribution that maximizes
income for the median voter, which in turn is the income distribution that provides
greater social welfare (à la Kolm–Atkinson) for a particular parameter .
5. Empirical exercise
We illustrate these results with data drawn from the World Bank’s POVCAL database.14
This database provides data on after-tax income, either household disposable income (I)
or consumption (C) per person for 116 countries over several years (see Table 1).
Income values are in purchasing power parity (PPP)-corrected monthly US dollars. In
addition, the distributions are population weighted. The main advantage of this database
is the large number of cases that it includes (509) which will allow us to test the main
assumption in our model, i.e., the co-symmetrizability assumption, with a high degree
of accuracy. Moreover, testing for symmetry in poor countries is more difficult than in
rich countries (i.e., the income distribution of a rich country is smoother). Consequently,
14
See http://iresearch.worldbank.org/PovcalNet/povcalSvy.html for detailed information on this database.
15
using the POVCAL database will allow us to be more confident about our results,
because we are testing for symmetry (under a strictly increasing transformation) in the
worst possible case.
To start with, we apply the power transformation specified in Section 3 to these data.
For this transformation, we consider   [0, 3] within two decimal places of accuracy.
We then formally test each transformed distribution for symmetry using the consistent
nonparametric kernel-based test developed by Ahmad and Li (1997). This intuitively
appealing test deals directly with the symmetry issue over the entire domain of the
relevant density function. The procedure used tests the hypothesis that a distribution is
symmetric about the median. Suppose we have a random sample of n observations of
income Xi, i = 1,…, n, drawn from the distribution X and ordered such that
X 1  X 2    X n . We know from Ahmad and Li (1997) that n h Iˆ2 n converges to a
normal distribution with mean 0 and variance 4 2 , where h is the smoothing parameter
and the statistic Iˆ2 n is as follows:
1 n n   Xi  X j
Iˆ2 n  2   K 
n h i 1 j  i  
h

X Xj
  K  i
h




 ,
where K(·) is the kernel function; in our case, the Gaussian density. We estimate the
variance  2 according to the following term:
ˆ 2 
1
1
2  n2h
n
n
 Xi  X j 
.

h


 K 
i 1 j 1
The chosen smoothing parameter is h  sn

1

, where s denotes the standard deviation of
the sample data. In simple density estimation,  = 5, but for the above, Ahmad and Li
16
(1997) suggest a larger value. We provide the results for  = 6. This test is one sided as
the alternative hypothesis states that the statistic Iˆ2 n is positive. Therefore, assuming a
5% level of significance, the critical value is 1.645.
For each distribution, we compute for the inequality aversion parameter  the interval
where symmetry is not rejected (see columns min and max in Table 1). In Table 1, we
reject symmetry when the values min and max are unspecified. We can see that
symmetry is not rejected for 92.14% of cases (469 of the 509 cases), so we can state
with little margin of error that the symmetry condition is generally accepted. Note that
we could reject the symmetry condition in even fewer cases if a more general class than
the power transformation had been used.
After testing for symmetry, we check the range of the class of distributions Hf (see
Section 3). Recall that the larger the class Hf, the less restrictive is our assumption of cosymmetrizability in Theorems 1 and 2. For this task we look for the range [ 1, 2] that is
contained in the majority of intervals [min, max]. In particular, we compute the number
of intervals [min, max] that contain a particular aversion parameter . Graph 1 presents
the results in relative terms. Considering the number of distributions that are symmetric
under an increasing and concave transformation (469), we find that the range [0.95,
1.06] is contained by 80% or more of intervals [min, max]. In the same manner, the
range [0.89, 1.14] is contained by 70% or more of intervals [min, max]. This means that
any of the aversion parameters in the range [0.95, 1.06] allows us to rank at least 80%
of the cases in our sample. In other words, a utilitarian social decision maker with an
aversion parameter in the range [0.95, 1.06] could make a decision that is consistent
with the median voter result over more than 80% of distributions. It is worth noting that
17
the value  = 1 is inside 81.02% of intervals [min, max]. That is, 380 distributions out of
469 could be ranked by the aversion parameter  = 1.
In addition, we carry out the following statistical contrast. For every  in the range
[0.89, 1.14], we compare those distributions with an estimated interval [min, max] that
includes the parameter . For these one-to-one comparisons, we consider the rankings
according to median income and majority voting where people vote for higher incomes.
It is comforting to see that both rankings agree 99.80% of the time or better (see Graph
A1 in the Appendix). Accordingly, we can reinforce that the co-symmetrizability
assumption is not very restrictive in practice.
Graph 1. Relative frequency of the  value contained in the intervals [min, max].
0.85
Relative frequency
0.80
0.75
0.70
0.65
0.60
0.55
1.20
1.18
1.16
1.14
1.12
1.10
1.08
1.06
1.04
1.02
1.00
0.98
0.96
0.94
0.92
0.90
0.88
0.86
0.84
0.82
0.80
0.50
ε value
Now we check the use of median income as a proxy for social welfare. For this task, we
first compute for each interval [min, max] the “optimal” value * as the most probable 
18
for which symmetry is not rejected.15 In Graph 2, we show that the optimal parameter
generally ranges from 0.8 to 1.2. Then, we compute the level of welfare (W) for such an
optimal inequality aversion parameter, the median income (xM), the mean income (),
the Gini coefficient (G) and the value of  1  G  (see Table 1). We observe in Table 1
and Graphs 3a, 3b, and 3c that the median income tracks welfare (consistent with
majority voting) very well, while the mean income and  1  G  exaggerate and shorten
welfare, respectively. The coefficient of determination R2 is 0.999 for median income,
while it is 0.956 and 0.969 for mean income and the term  1  G  , respectively.16
Moreover, the slope of the regression is almost one (0.998) for the median income,
while it is larger than one (1.164) for the mean income and lower than one (0.791) for
the term  1  G  . In sum, the use of the median value could be of great interest to other
fields like macroeconomics, where academics usually apply the mean income to
represent an income distribution, and by doing so, only consider the size of the
distribution.
Though we show the parameter * for all countries, the empirical exercises that follow in this section
are carried out only for those countries whose income distribution is symmetric under a power
transformation.
16
Theoretically we should obtain a perfect correlation (R2=1) for median income. However, the
symmetry obtained is the result of applying a statistical contrast. Despite this, our result is still very close
to 1, which it is another indirect test of the fulfillment of the symmetry assumption in practice.
15
19
Graph 2. Distribution of the optimal aversion parameter *.
Relative frequency
0.3
0.25
0.2
0.15
0.1
0.05
0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
ε value
Once we have computed for each country the “optimal” inequality aversion parameter,
we can contrast the relationship between * and the national level of income. In this
respect, Frisch (1959) argued that we should expect higher values of * in poorer
countries, while Atkinson (1970), Lambert et al. (2003) and Millimet et al. (2008),
among others, remarked that people become more concerned about inequality when the
general level of income rises. In Graph 4, we show the regression between * and mean
income for the POVCAL database. We observe that the inequality aversion parameter
* is uncorrelated with the income level of a country (R2 = 0.052 and the slope is 0).
According to these results, the aversion to inequality is neither a normal good nor an
inferior good. Note also that the ordinate of the regression is close to one (1.074).
20
Graph 3a. Correlation between median income and welfare.
700
Median Income
600
y = 0.998x - 0.199
R² = 0.999
500
400
300
200
100
0
0
200
400
600
800
Welfare
Mean Income
Graph 3b. Correlation between mean income and welfare.
800
700
600
500
400
300
200
100
0
y = 1.164x + 16.23
R² = 0.956
0
200
400
600
800
Welfare
Graph 3c. Correlation between  (1-G) and welfare.
600
y = 0.791x + 0.237
R² = 0.969
μ (1-G)
500
400
300
200
100
0
0
200
400
Welfare
21
600
800
Graph 4. Correlation between * and .
1.8
y = -0.000x + 1.074
R² = 0.052
ε*
1.3
0.8
0.3
-0.2 0
200
400
600
800
μ
Another interesting issue that can be analyzed from the results in Table 1 is the existing
relationship between the propensity to redistribute income (measured by an inequality
aversion parameter) and the level of objective inequality. In this respect, Saint-Paul and
Verdier (1993), Persson and Tabellini (1994) and others consider that greater inequality
increases redistribution, while Perotti (1996), Lambert et al. (2003) and Millimet et al.
(2008) among others conclude that countries with greater levels of objective inequality
do not increase their propensity to redistribute. In Graph 5, we show the regression
between * and objective inequality measured by the Gini coefficient. We observe that
the correlation is negative though barely significant. Accordingly, our analysis suggests
that countries with higher levels of inequality are slightly less averse to inequality.
With respect to the last two results, we must take into account that these relationships
may depend on the level of development of the countries included in the database. Thus,
the POVCAL database only considers low-income countries. Moreover, no controls
have been introduced in the regressions. However, while a more accurate statistical
22
analysis would consider a proper set of explicative variables, this would be beyond the
scope of this paper.
Graph 5. Correlation between * and G.
2.0
y = -0.348x + 1.146
R² = 0.024
ε*
1.5
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
G
Finally, we test in Table 1 for normality of the transformed distributions at the optimal
value of the parameter . In particular, we use the Jarque–Bera statistic to measure the
difference in the skewness and kurtosis of the series relative to the normal distribution.
H = 1 means that normality is rejected, while H = 0 means that normality is not rejected.
Note that the rejection of symmetry implies the rejection of normality but not vice
versa. We do not generally accept normality for our transformed data as we reject the
null hypothesis of normality for 83.50% of cases (425 of the 509 cases). Therefore, we
conclude that the normality assumption is much more restrictive than the symmetry
assumption.
23
6. Concluding remarks
The issue of ranking distributions is implicit in the essence of the political economy
literature. In fact, we can view changes in income distributions as the result of a
political process, likely a majority voting mechanism. However, income distributions
traditionally have been ranked according to a set of “desirable” assumptions, as
represented by a social evaluation function, and this constitutes the essence of
welfarism.
This paper attempts to provide a scenario in which there exists a social decision maker
whose preferences accord with majority voting. More specifically, we propose a set of
sufficient conditions under which a particular additive and concave utilitarian social
evaluation function is consistent with the outcome of majority voting. In this respect, we
restrict our attention to the class of income distributions that are symmetric under
strictly increasing and concave transformations, where lognormal distributions are a
particular case. In fact, and as shown in the paper, symmetry, monotonicity and
concavity are substantially more general assumptions than lognormality.
Among other advantages, this consistency result may help us to understand the apparent
stability of tax schedules in democratic societies. Tax schemes in democratic economies
are commonly viewed as the outcome of a political process—say, majority voting. We
also observe that tax schedules are stable. Our main result states that the outcome of
majority voting is consistent with the maximization of a utilitarian social evaluation
function. This evaluation function depends on an inequality aversion parameter.
Therefore, the stability of a tax system will eventually depend on the stability of the
corresponding inequality aversion parameter. It appears reasonable to assume that the
inequality aversion parameter in a society is stable throughout time (see Li et al., 1998).
In this respect, we observe in Table 2 that this stability requirement is fulfilled.
24
Consequently, the consistency result provides an explanation for the stability of tax
schedules in democratic societies. Nevertheless, there is an obvious need for more
research before we can make any unequivocal statement on the matter.
Furthermore, an alternative method for computing the inequality aversion parameter in a
society is also provided. One approach in the literature to identifying the inequality
aversion parameter has been to measure the elasticity of the marginal social utility of
income (see, among others, Atkinson, 1980 and Amiel et al., 1999). Using the leakybucket method and a large group of student respondents, Amiel et al. (1999) find an
inequality aversion  value between 0.1 and 0.2. In the analysis of income distributions,
these values are significantly lower than any typically employed. Similarly, Pirttila and
Uusitalo (2010) specify a social welfare function à la Kolm−Atkinson for a
representative survey of Finns and obtain an inequality aversion parameter less than 0.5.
In contrast, when respondents are obliged to choose between different income
distributions in a hypothetical society, the estimated  value is generally much larger.
For instance, Carlsson et al. (2005) find that the median inequality aversion lies between
one and two, while Pirttila and Uusitalo (2010) obtain an  value greater than three.
Approaches that derive values of  from observed governmental policy by fitting the
equal-sacrifice model also often find values between one and two. For instance, in the
UK for fiscal year 1973/4, Stern (1977) obtains an  value of 1.97, while Young (1990)
finds  values between 1.52 and 1.72 for the US between 1957 and 1977. By
undertaking a similar exercise, Gouveia and Strauss (1994) provide  values between
1.72 and 1.94 for the US between 1979 and 1989. Lambert et al. (2003) have identified
such a parameter as the one that equalizes subjective inequality across countries to the
so-called “natural rate of subjective inequality”.
25
In this paper, we proposed the use of an aversion to income inequality that is consistent
with a majority voting process, which effectively measures the departure from
symmetry (or skewness). The inequality aversion values found in our empirical exercise
for a panel of 116 countries depend largely on the country and year under consideration,
although the estimated parameters generally range from 0.8 to 1.2. Accordingly, we can
see that our estimates are neither as low as the measures of inequality aversion obtained
with leaky-bucket experiments nor as high as those that result from questioning the
preferred income distribution or the fitting of an equal-sacrifice model.
In addition, we propose the use of median income as a proxy for social welfare. The
advantage of applying median income instead of mean income is that not only
efficiency but also “implicit equity” is considered.
A final comment for future research is worth noting. Thus far, we have used the family
of power transformations that link social evaluation functions à la Kolm–Atkinson with
the majority voting procedure. However, different classes of transformations could be
used, for example, the set of transformations that do not assume constancy for the
inequality aversion parameter. Perhaps, more interestingly, we could search for the set
of transformations that link Yaari’s (1988) rank-dependent social evaluation function to
the majority voting procedure. By doing this, we will be able to find the inequality
aversion parameter endorsed by the family of extended Gini coefficients. However, this
research is clearly beyond the scope of this paper.
26
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30
Country
Year
H
εmin
C
Albania
1996
1
0.74
C
Albania
2002
1
0.71
C
Albania
2005
0
0.77
C
Algeria
1988
1
0.85
C
Algeria
1995
1
0.72
C
Angola
2000
1
I
Argentina-Urban
1986
1
0.74
I
Argentina-Urban
1992
1
0.78
I
Argentina-Urban
1996
1
0.77
I
Argentina-Urban
1998
1
0.79
I
Argentina-Urban
2002
1
I
Argentina-Urban
2005
1
0.73
I
Armenia
1996
0
0.86
C
Armenia
1998
1
0.89
C
Armenia
2002
1
0.97
C
Armenia
2003
1
1.03
C
Azerbaijan
1995
1
0.63
C
Azerbaijan
2001
0
0.91
C
Azerbaijan
2005
1
0.95
C
Bangladesh
1991
1
0.78
C
Bangladesh
1995
1
1.02
C
Bangladesh
2000
0
1.05
C
Bangladesh
2005
0
1.13
I
Belarus
1988
1
0.47
I
Belarus
1993
0
0.61
I
Belarus
1997
1
0.65
I
Belarus
1998
1
0.64
C
Belarus
2000
1
0.71
C
Belarus
2002
1
0.62
C
Belarus
2005
1
0.55
C
Benin
2003
0
0.92
C
Bhutan
2003
1
1.03
I
Bolivia
1997
1
0.7
I
Bolivia
1999
1
I
Bolivia
2002
1
0.68
I
Bolivia
2005
1
0.68
C
Bosnia and Herz.
2001
1
0.67
C
Bosnia and Herz.
2004
1
0.73
C
Botswana
1985
1
C
Botswana
1993
1
1.00
I
Brazil
1981
1
I
Brazil
1984
1
I
Brazil
1987
1
I
Brazil
1990
1
I
Brazil
1993
1
I
Brazil
1996
1
I
Brazil
1999
1
I
Brazil
2002
1
I
Brazil
2005
1
0.86
C
Bulgaria
1989
1
0.57
C
Bulgaria
1994
1
0.58
C
Bulgaria
1997
1
0.82
C
Bulgaria
2001
1
0.52
C
Bulgaria
2003
1
0.61
I: Income; C: Consumption; Herz.: Herzegovina.
Table 1
εmax
ε*
1.23
1.28
1.30
1.44
1.19
1.14
1.06
1.04
0.97
0.98
1.29
1.52
1.62
1.74
1.20
1.44
2.21
1.56
1.73
1.65
1.73
1.29
1.32
1.19
1.32
1.34
1.28
1.21
1.41
1.27
0.98
0.97
0.89
1.39
1.21
1.22
1.03
1.39
1.50
1.64
1.09
1.39
0.98
1.00
1.04
1.15
0.95
0.83
0.94
0.92
0.90
0.88
0.86
0.86
1.08
1.21
1.30
1.40
0.92
1.18
1.60
1.17
1.39
1.36
1.44
0.88
0.96
0.92
0.98
1.03
0.95
0.88
1.17
1.15
0.84
0.70
0.82
0.78
1.03
0.97
1.00
1.11
0.93
0.96
0.91
0.87
0.89
0.87
0.90
0.91
0.95
0.98
1.04
1.24
0.80
1.00
31
W
xM
μ
G
μ (1-G)
132.0
119.3
134.3
92.6
98.1
35.3
385.6
277.5
248.5
230.9
156.2
222.4
75.1
61.5
63.2
65.4
71.7
86.9
125.7
31.4
33.9
34.7
38.4
282.8
189.1
89.5
154.7
176.0
231.6
276.9
39.7
62.8
113.5
102.9
103.5
117.4
307.7
281.6
56.0
62.2
110.3
95.7
119.9
133.4
139.2
155.1
146.5
152.3
158.7
472.5
268.2
134.9
175.7
178.0
131.5
119.0
134.0
93.0
97.6
34.5
383.9
275.0
245.5
227.9
153.4
220.0
75.0
61.8
63.5
65.8
71.8
86.9
125.8
31.4
33.9
34.7
38.4
282.7
188.7
89.3
155.2
176.0
231.7
276.6
39.6
62.4
112.4
100.5
102.5
115.0
308.0
280.7
55.3
61.7
108.2
93.8
117.4
130.1
135.9
151.0
143.6
149.4
156.8
472.5
269.4
135.2
175.5
178.7
150.7
135.2
160.7
122.7
119.0
60.4
526.0
381.2
357.1
337.5
237.8
325.0
104.9
78.0
80.5
82.1
86.4
110.4
134.6
35.6
40.9
41.9
46.8
304.4
203.9
98.8
179.2
204.9
265.5
309.8
51.7
92.1
192.9
166.5
179.2
195.2
350.2
344.4
92.3
115.1
188.1
167.4
211.3
239.2
243.6
268.8
252.9
261.2
264.8
515.1
296.8
154.1
206.1
204.6
0.2782
0.3246
0.2886
0.3836
0.3483
0.5702
0.5126
0.4888
0.4332
0.4435
0.4734
0.4867
0.3414
0.3224
0.4288
0.3468
0.3540
0.1652
0.3422
0.2989
0.3005
0.2568
0.2957
0.2989
0.2924
0.2760
0.2251
0.2215
0.2580
0.2968
0.3736
0.4506
0.5715
0.5646
0.5616
0.5676
0.2752
0.3513
0.5286
0.5652
0.5597
0.5840
0.5716
0.5709
0.5639
0.5408
0.5545
0.5613
0.5711
0.3361
0.2850
0.2316
0.2392
0.2580
108.8
91.3
114.3
75.6
77.6
25.9
256.4
194.9
202.4
187.8
125.3
166.8
69.1
52.9
46.0
53.7
55.8
92.1
88.5
24.9
28.6
31.2
33.0
213.4
144.3
71.6
138.9
159.5
197.0
217.8
32.4
50.6
82.7
72.5
78.5
84.4
253.8
223.4
43.5
50.0
82.8
69.6
90.5
102.6
106.2
123.4
112.7
114.6
113.6
342.0
212.2
118.4
156.8
151.8
Country
Year
Burkina Faso
1994
Burkina Faso
1998
Burkina Faso
2003
Burundi
1992
Burundi
1998
Burundi
2006
Cambodia
1994
Cambodia
2004
Cameroon
1996
Cameroon
2001
Cape Verde
2001
Central Africa Rep.
1993
Central Africa Rep.
2003
Chad
2002
Chile
1987
Chile
1990
Chile
1994
Chile
1996
Chile
1998
Chile
2000
Chile
2003
China-Rural
1981
China-Rural
1984
China-Rural
1987
China-Rural
1990
China-Rural
1993
China-Rural
1996
China-Rural
1999
China-Rural
2002
China-Rural
2005
China-Urban
1981
China-Urban
1984
China-Urban
1987
China-Urban
1990
China-Urban
1993
China-Urban
1996
China-Urban
1999
China-Urban
2002
China-Urban
2005
Colombia
1995
Colombia
1996
Colombia
1999
Colombia
2000
Colombia
2003
Colombia-Urban
1980
Colombia-Urban
1988
Colombia-Urban
1989
Colombia-Urban
1991
Comoros
2004
Congo Dem. Rep.
2005
Congo Rep.
2005
Costa Rica
1981
Costa Rica
1986
Costa Rica
1990
Costa Rica
1993
I: Income; C: Consumption.
C
C
C
C
C
C
C
C
C
C
C
C
C
C
I
I
I
I
I
I
I
I
I
I
C
C
C
C
C
C
I
I
I
C
C
C
C
C
C
I
I
I
I
I
I
I
I
I
C
C
C
I
I
I
I
H
εmin
εmax
ε*
W
xM
μ
G
μ (1-G)
1
1
0
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1.06
1.06
0.93
0.84
0.67
1.35
1.22
1.10
1.06
0.93
0.94
0.84
0.81
0.95
0.89
0.89
0.93
0.91
0.89
0.89
0.53
0.62
0.63
0.90
0.91
0.92
0.92
0.96
0.90
0.33
0.43
0.33
0.59
0.69
0.71
0.70
0.75
0.75
0.81
0.76
0.77
0.75
0.77
0.76
0.74
0.80
0.74
0.91
0.01
0.01
0.62
0.29
0.63
0.66
1.36
1.58
1.45
1.38
1.21
1.63
1.76
1.52
1.42
1.29
1.22
1.09
1.23
1.14
1.17
1.17
1.14
1.13
1.20
1.24
1.23
1.27
1.14
1.53
1.51
1.44
1.41
1.41
1.38
1.41
1.52
1.38
1.31
1.37
1.34
1.28
1.33
1.32
1.19
1.12
1.12
1.07
1.03
1.07
1.04
1.12
1.04
1.24
0.38
0.31
0.88
0.71
1.00
1.01
1.22
1.33
1.20
1.11
0.94
1.49
1.50
1.32
1.25
1.12
1.09
0.89
0.97
1.02
1.04
1.03
1.03
1.04
1.02
1.05
1.07
0.88
0.95
0.88
1.22
1.21
1.18
1.17
1.19
1.14
0.87
0.97
0.85
0.95
1.03
1.03
1.00
1.04
1.04
1.01
0.94
0.94
0.91
0.90
0.92
0.89
0.96
0.89
1.08
0.17
0.12
0.75
0.49
0.80
0.84
24.6
26.5
34.5
21.3
18.1
22.6
38.7
45.1
37.4
53.4
76.5
12.8
30.4
31.4
125.1
170.5
195.3
221.8
234.3
236.2
234.6
19.0
25.8
31.1
28.3
30.3
38.5
37.8
42.1
56.1
39.7
45.5
54.7
53.2
64.0
74.4
84.7
107.8
130.8
119.3
117.0
104.3
109.6
126.3
116.8
141.5
146.4
159.2
41.7
48.8
56.8
89.5
116.5
143.1
150.1
24.6
26.6
34.6
21.2
18.1
22.6
38.8
45.0
37.4
53.2
76.1
12.5
30.1
31.2
123.6
169.0
193.6
219.9
231.6
234.7
234.1
18.9
25.8
31.0
28.3
30.3
38.4
37.7
42.0
55.9
39.8
45.5
54.8
53.2
63.9
74.3
84.6
107.8
130.9
120.6
117.1
104.3
109.0
124.6
116.5
140.2
145.3
157.8
41.8
48.7
56.7
88.5
116.0
142.0
149.6
38.6
39.2
45.7
25.8
23.9
28.5
51.8
62.4
55.4
75.1
117.7
23.8
41.1
40.6
212.1
282.2
321.8
366.5
389.7
390.9
386.2
20.7
28.8
35.3
33.6
36.3
47.3
47.4
54.7
70.3
41.7
47.8
57.9
58.8
73.1
85.4
99.3
129.7
159.9
204.4
193.4
178.5
184.5
218.0
204.8
219.2
232.3
239.2
81.9
50.2
58.0
122.3
128.9
192.3
206.2
0.3806
0.4351
0.4783
0.3232
0.4068
0.3269
0.3990
0.4299
0.4833
0.4267
0.5957
0.3888
0.5231
0.5172
0.5260
0.5241
0.5244
0.5302
0.5365
0.3708
0.3509
0.2457
0.2662
0.2958
0.2962
0.3048
0.3293
0.3464
0.3273
0.3398
0.1825
0.1759
0.1993
0.2531
0.2802
0.2866
0.3106
0.4440
0.5469
0.5617
0.5146
0.5623
0.5132
0.4976
0.5373
0.5324
0.5476
0.5887
0.3038
0.4843
0.4829
0.4600
0.4691
0.4557
0.4691
23.9
22.2
23.9
17.4
14.2
19.2
31.1
35.6
28.6
43.1
47.6
14.6
19.6
19.6
100.5
134.3
153.0
172.2
180.6
246.0
250.7
15.6
21.2
24.9
23.6
25.2
31.7
31.0
36.8
46.4
34.1
39.4
46.4
43.9
52.6
60.9
68.5
72.1
72.5
89.6
93.9
78.1
89.8
109.5
94.8
102.5
105.1
98.4
57.0
25.9
30.0
66.0
68.4
104.6
109.5
32
Country
Year
Costa Rica
1996
Costa Rica
1998
Costa Rica
2000
Costa Rica
2001
Costa Rica
2003
Costa Rica
2005
Côte d'Ivoire
1985
Côte d'Ivoire
1987
Côte d'Ivoire
1988
Côte d'Ivoire
1993
Côte d'Ivoire
1995
Côte d'Ivoire
1998
Côte d'Ivoire
2002
Croatia
1988
Croatia
1998
Croatia
1999
Croatia
2001
Croatia
2005
Czech Republic
1988
Czech Republic
1993
Czech Republic
1996
Djibouti
1996
Djibouti
2002
Dominican Rep.
1986
Dominican Rep.
1989
Dominican Rep.
1992
Dominican Rep.
1996
Dominican Rep.
2000
Dominican Rep.
2003
Dominican Rep.
2005
Ecuador
1987
Ecuador
1994
Ecuador
1998
Ecuador
2003
Ecuador
2005
Egypt
1990
Egypt
1995
Egypt
1999
Egypt
2004
El Salvador
1989
El Salvador
1995
El Salvador
1996
El Salvador
1998
El Salvador
2000
El Salvador
2002
El Salvador
2003
Estonia
1988
Estonia
1993
Estonia
1995
Estonia
1998
Estonia
2000
Estonia
2002
Estonia
2004
Ethiopia
1981
Ethiopia
1995
I: Income; C: Consumption.
I
I
I
I
I
I
C
C
C
C
C
C
C
I
C
C
C
C
I
I
I
C
C
I
I
I
I
I
I
I
I
I
I
I
I
C
C
C
C
I
I
I
I
I
I
I
I
I
C
C
C
C
C
C
C
H
εmin
εmax
ε*
W
xM
μ
G
μ (1-G)
1
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
0
1
0
1
1
1
1
1
1
0.67
0.73
0.70
0.75
0.66
0.75
0.83
0.94
0.79
0.86
0.86
0.90
0.92
0.52
0.59
0.64
0.72
0.65
0.66
1.08
0.77
0.01
0.01
0.69
0.90
0.88
0.79
0.80
0.83
0.85
0.01
0.01
0.01
0.01
0.01
0.96
1.08
1.10
1.03
0.55
0.71
0.73
0.65
0.64
0.62
0.30
0.80
0.57
0.86
0.71
0.72
0.70
0.95
1.02
1.01
1.04
1.06
1.07
1.01
1.06
1.07
1.27
1.21
1.32
1.31
1.36
1.36
1.33
1.27
1.42
1.31
1.29
1.46
1.89
1.72
0.59
0.50
1.04
1.17
1.30
1.12
1.09
1.17
1.11
0.30
0.24
0.24
0.28
0.25
1.54
1.80
1.81
1.76
0.94
1.09
1.07
0.97
0.95
1.00
0.98
1.34
1.09
1.29
1.22
1.21
1.23
1.68
1.64
0.84
0.89
0.88
0.91
0.83
0.91
0.95
1.11
1.00
1.09
1.09
1.13
1.15
0.92
0.93
1.03
1.02
0.97
1.06
1.50
1.25
0.27
0.23
0.87
1.03
1.09
0.95
0.95
1.00
0.98
0.12
0.09
0.00
0.11
0.09
1.25
1.45
1.47
1.41
0.73
0.90
0.89
0.89
0.81
0.79
0.80
0.63
1.07
0.83
1.08
0.96
0.96
0.97
1.33
1.34
164.0
198.3
185.5
210.8
216.1
214.4
105.7
92.4
78.8
67.9
63.6
61.8
64.4
472.7
458.3
361.3
366.4
604.0
458.6
363.7
430.1
158.8
98.7
97.0
105.5
138.6
152.1
189.6
148.6
158.4
200.1
176.8
190.5
263.8
239.6
82.2
79.9
88.1
89.6
130.6
113.1
110.1
124.2
141.8
138.2
120.2
410.9
205.6
198.3
225.0
218.5
213.6
248.6
30.7
32.4
162.8
196.6
184.6
208.8
214.2
212.8
104.8
92.0
78.5
67.6
63.4
61.7
64.6
472.3
458.0
362.8
366.0
603.7
457.7
364.4
433.4
157.3
98.0
96.8
104.5
139.4
150.9
187.9
147.8
156.9
200.0
177.0
190.5
263.4
239.5
82.1
80.0
88.5
90.0
129.5
112.8
108.9
122.3
141.0
137.2
119.4
410.2
206.6
197.7
224.0
218.3
212.8
248.8
30.9
32.6
227.5
282.5
258.5
310.5
311.5
302.4
137.7
122.4
98.1
85.7
79.9
85.9
96.4
510.9
510.8
411.1
429.1
688.9
487.8
423.9
490.7
165.7
102.4
137.4
160.7
216.2
221.3
292.4
230.1
236.7
204.1
179.4
193.1
268.7
243.3
99.7
96.4
109.8
110.4
175.8
166.1
167.8
189.7
211.1
206.3
169.8
433.6
268.3
224.3
285.7
271.0
264.0
305.4
37.8
43.7
0.3389
0.4468
0.4530
0.4605
0.4585
0.4052
0.3939
0.3623
0.3611
0.3588
0.4197
0.3065
0.2858
0.2713
0.2270
0.2655
0.1921
0.2577
0.2504
0.2998
0.2962
0.5027
0.4967
0.4829
0.4860
0.4717
0.4658
0.4852
0.3107
0.3107
0.3083
0.3103
0.3078
0.3080
0.2905
0.3126
0.3122
0.5036
0.5089
0.4760
0.4598
0.4805
0.5041
0.5089
0.3624
0.3610
0.3527
0.2281
0.3819
0.2984
0.3674
0.2903
0.2880
0.3126
0.3766
150.4
156.3
141.4
167.5
168.7
179.9
83.5
78.1
62.7
55.0
46.4
59.6
68.8
372.3
394.8
301.9
346.7
511.4
365.6
296.8
345.4
82.4
51.5
71.0
82.6
114.2
118.2
150.5
158.6
163.2
141.2
123.8
133.7
185.9
172.6
68.5
66.3
54.5
54.2
92.1
89.7
87.2
94.0
103.7
131.6
108.5
280.6
207.1
138.7
200.5
171.5
187.4
217.4
26.0
27.2
33
Country
Year
Ethiopia
1999
Ethiopia
2005
Gabon
2005
Gambia
1998
Gambia
2003
Georgia
1996
Georgia
1999
Georgia
2002
Georgia
2005
Ghana
1987
Ghana
1988
Ghana
1991
Ghana
1998
Ghana
2005
Guatemala
1987
Guatemala
1989
Guatemala
1998
Guatemala
2000
Guatemala
2002
Guatemala
2006
Guinea
1991
Guinea
1994
Guinea
2003
Guinea-Bissau
1991
Guinea-Bissau
1993
Guinea-Bissau
2002
Guyana
1992
Guyana
1998
Haiti
2001
Honduras
1990
Honduras
1992
Honduras
1994
Honduras
1997
Honduras
1999
Honduras
2003
Honduras
2005
Honduras-Urban
1986
Hungary
1987
Hungary
1989
Hungary
1993
Hungary
1998
Hungary
1999
Hungary
2002
Hungary
2004
India-Rural
1977
India-Rural
1993
India-Rural
2004
India-Urban
1977
India-Urban
1983
India-Urban
1987
India-Urban
1993
India-Urban
2004
Indonesia-Rural
1984
Indonesia-Rural
1987
Indonesia-Rural
1990
I: Income; C: Consumption.
C
C
C
C
C
C
C
C
C
C
C
C
C
C
I
I
I
I
I
I
C
C
C
C
C
C
I
I
I
I
I
I
I
I
I
I
I
I
I
I
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
H
εmin
εmax
ε*
W
xM
μ
G
μ (1-G)
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
0
0
0
0
1
0
1
0.88
0.96
0.88
0.86
0.59
0.64
0.67
0.65
0.71
0.76
0.81
0.77
0.73
0.82
0.77
0.78
0.81
0.76
0.83
0.66
0.91
0.88
0.65
0.94
0.78
0.83
0.63
0.76
0.81
0.79
0.78
0.74
0.72
0.88
0.71
0.89
0.58
0.81
0.84
0.57
0.75
0.70
0.70
1.02
0.92
1.05
0.87
0.90
1.00
0.94
0.97
0.77
0.93
0.90
1.65
1.72
1.31
1.17
1.13
1.11
1.15
1.11
1.24
1.29
1.29
1.02
1.15
1.09
0.96
1.16
1.19
1.06
1.13
0.74
1.34
1.29
0.95
1.40
1.32
1.36
1.14
1.05
1.03
0.97
1.05
1.11
1.05
1.04
0.97
1.03
1.60
1.57
1.71
1.36
1.55
1.43
1.37
1.76
1.68
1.80
1.45
1.48
1.48
1.44
1.41
1.46
1.66
1.38
1.27
1.35
1.10
0.95
1.02
0.86
0.87
0.91
0.88
0.98
1.03
1.05
0.90
0.94
0.95
0.86
0.97
1.01
0.91
0.98
0.70
1.13
1.09
0.79
1.18
1.05
1.10
0.88
0.90
0.92
0.88
0.92
0.92
0.88
0.96
0.84
0.96
1.10
1.19
1.29
0.97
1.16
1.06
1.03
1.40
1.31
1.43
1.16
1.20
1.24
1.20
1.19
1.12
1.30
1.14
35.4
42.4
109.6
26.6
54.2
134.1
103.7
81.7
89.5
37.6
38.4
37.6
48.3
57.3
36.4
53.9
100.2
106.0
106.4
119.9
11.1
47.9
26.2
49.7
35.9
38.7
127.1
132.5
34.5
46.8
62.0
70.3
103.6
112.9
95.7
95.6
120.8
432.1
415.2
294.4
230.4
242.4
292.9
329.8
28.9
36.9
40.6
35.4
38.9
39.6
43.7
47.6
31.3
29.6
35.5
35.6
42.6
109.3
26.3
53.9
134.3
103.3
81.6
89.4
37.6
38.4
37.5
47.9
57.1
36.0
52.7
100.0
106.8
105.5
119.0
10.9
47.8
26.1
49.1
36.1
38.7
129.5
132.6
34.2
46.1
61.0
69.6
103.0
112.2
94.3
94.5
119.2
432.9
414.7
296.5
230.6
243.4
292.8
329.9
29.2
37.0
40.7
35.5
38.9
39.6
43.7
47.5
31.4
29.6
35.3
42.2
50.7
146.9
39.9
78.3
164.1
128.1
104.8
114.9
45.9
47.5
47.9
62.0
76.3
62.7
93.9
164.5
173.9
172.4
191.0
14.8
63.6
36.1
79.2
53.4
47.8
196.3
177.6
60.3
79.0
98.4
114.0
161.0
170.1
152.7
156.9
197.7
466.8
466.0
343.7
254.0
279.0
330.6
382.8
36.6
43.3
49.1
44.3
47.7
50.1
54.2
61.4
36.4
34.4
40.3
0.4015
0.4904
0.4579
0.3935
0.3988
0.3638
0.3736
0.4172
0.3471
0.3524
0.3719
0.4016
0.3910
0.5192
0.5312
0.5140
0.5263
0.5544
0.5734
0.3471
0.4191
0.5411
0.4510
0.4660
0.4256
0.4728
0.5657
0.5084
0.4989
0.5193
0.5479
0.5312
0.5498
0.5285
0.5311
0.2641
0.2949
0.2061
0.2464
0.2702
0.2462
0.2709
0.2930
0.3651
0.3258
0.2778
0.3349
0.3444
0.3243
0.3455
0.2704
0.3208
0.2551
0.3372
0.2882
25.2
25.8
79.7
24.2
47.1
104.4
80.3
61.1
75.0
29.7
29.9
28.7
37.7
36.7
29.4
45.7
77.9
77.5
73.6
124.7
8.6
29.2
19.8
42.3
30.7
25.2
85.3
87.3
30.2
38.0
44.5
53.5
72.5
80.2
71.6
115.5
139.4
370.5
351.1
250.8
191.5
203.4
233.7
243.0
24.6
31.3
32.7
29.1
32.2
32.8
39.5
41.7
27.1
22.8
28.7
34
Country
Year
Indonesia-Rural
1993
Indonesia-Rural
1996
Indonesia-Rural
1999
Indonesia-Rural
2002
Indonesia-Rural
2005
Indonesia-Urban
1984
Indonesia-Urban
1987
Indonesia-Urban
1990
Indonesia-Urban
1993
Indonesia-Urban
1996
Indonesia-Urban
1999
Indonesia-Urban
2002
Indonesia-Urban
2005
Iran
1986
Iran
1990
Iran
1994
Iran
1998
Iran
2005
Jamaica
1988
Jamaica
1990
Jamaica
1993
Jamaica
1996
Jamaica
1999
Jamaica
2002
Jamaica
2004
Jordan
1986
Jordan
1992
Jordan
1997
Jordan
2002
Jordan
2006
Kazakhstan
1988
Kazakhstan
1993
Kazakhstan
1996
Kazakhstan
2001
Kazakhstan
2002
Kazakhstan
2003
Kenya
1992
Kenya
1994
Kenya
1997
Kenya
2005
Kyrgyz Rep.
1988
Kyrgyz Rep.
1993
Kyrgyz Rep.
1998
Kyrgyz Rep.
1999
Kyrgyz Rep.
2002
Kyrgyz Rep.
2004
Lao PDR
1992
Lao PDR
1997
Lao PDR
2002
Latvia
1988
Latvia
1993
Latvia
1996
Latvia
1998
Latvia
2002
Latvia
2004
I: Income; C: Consumption.
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
I
I
C
C
C
C
C
C
C
C
I
C
C
C
C
C
C
C
C
I
I
I
C
C
C
H
εmin
εmax
ε*
W
xM
μ
G
μ (1-G)
0
0
1
0
1
1
0
1
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
0
1
1
1
1
1
0.87
0.90
0.76
0.95
0.86
0.79
0.87
0.92
0.98
0.98
1.03
1.02
0.98
0.82
0.77
0.82
0.82
0.77
0.77
0.92
0.66
0.85
0.83
0.93
0.91
0.94
0.93
0.91
0.88
0.94
0.58
0.75
0.68
0.74
0.76
0.74
0.89
0.76
0.96
0.83
0.66
0.84
0.79
0.83
0.87
0.98
0.96
0.97
0.41
0.83
0.48
0.63
0.71
0.69
1.64
1.64
1.59
1.71
1.55
1.39
1.44
1.37
1.45
1.43
1.64
1.52
1.46
1.17
1.18
1.21
1.16
1.24
1.23
1.21
1.19
1.38
1.27
1.17
1.22
1.28
1.38
1.47
1.30
1.47
1.18
0.95
1.09
1.21
1.22
1.19
1.26
1.24
1.32
1.25
0.93
1.22
1.32
1.29
1.21
1.64
1.58
1.56
1.18
1.32
1.32
1.24
1.28
1.21
1.26
1.28
1.18
1.34
1.21
1.09
1.16
1.15
1.22
1.21
1.34
1.27
1.23
1.00
0.97
1.02
1.00
1.01
1.01
1.06
0.92
1.12
1.05
1.05
1.07
1.11
1.16
1.19
1.09
1.21
0.88
0.85
0.88
0.98
1.00
0.97
1.08
1.01
1.14
1.04
1.52
0.79
1.03
1.06
1.06
1.05
1.32
1.28
1.27
0.79
1.08
0.90
0.94
1.00
0.95
34.8
39.4
36.6
45.5
52.9
35.0
32.2
39.9
40.0
46.4
43.8
55.8
65.1
152.5
146.2
168.8
179.6
153.5
137.6
161.7
121.6
141.4
181.9
182.4
188.5
174.9
121.6
117.3
133.7
158.9
302.5
109.4
113.4
131.4
101.1
110.9
49.3
56.9
69.4
74.4
167.0
109.5
52.1
68.9
48.6
60.8
35.7
38.2
41.2
571.1
172.8
194.3
177.8
246.6
284.4
34.7
39.4
36.7
45.5
53.0
35.0
32.2
39.7
39.9
46.2
43.8
55.7
65.0
152.0
145.6
168.0
179.0
153.3
138.1
160.5
121.6
141.9
181.6
180.9
187.4
173.9
121.5
117.3
133.0
159.0
301.6
108.8
112.7
130.9
100.8
110.5
49.3
56.9
69.1
74.4
167.2
108.1
51.9
68.8
48.4
60.5
35.7
38.4
41.2
570.2
172.2
195.2
178.2
246.7
284.3
39.6
45.7
41.0
52.1
62.2
42.3
39.1
49.3
50.3
59.9
55.7
70.0
86.9
220.3
198.2
228.7
246.7
194.9
187.1
217.9
147.8
187.8
251.4
269.3
267.0
219.0
169.0
148.8
172.7
205.7
332.2
127.8
135.9
153.2
123.0
132.7
85.1
76.0
95.1
108.4
193.7
167.7
64.4
84.2
57.4
72.6
42.8
48.1
50.4
608.9
195.5
224.9
211.8
303.8
347.2
0.3844
0.2868
0.3258
0.2670
0.3389
0.2546
0.3441
0.2694
0.3643
0.2430
0.3392
0.3745
0.4596
0.4249
0.4189
0.4307
0.4668
0.4404
0.4115
0.3899
0.4187
0.3503
0.4259
0.3643
0.3544
0.4165
0.3522
0.3787
0.3439
0.3339
0.2560
0.3320
0.3488
0.3093
0.4114
0.4573
0.5362
0.4073
0.3616
0.3120
0.3240
0.2557
0.3535
0.5238
0.3379
0.3171
0.2959
0.3363
0.3502
0.3507
0.2229
0.2683
0.3004
0.3286
0.5126
24.4
32.6
27.6
38.2
41.1
31.6
25.6
36.0
32.0
45.3
36.8
43.8
47.0
126.7
115.1
130.2
131.6
109.1
110.1
132.9
85.9
122.0
144.3
171.2
172.4
127.8
109.5
92.5
113.3
137.0
247.1
85.4
88.5
105.8
72.4
72.0
39.5
45.0
60.7
74.6
131.0
124.8
41.6
40.1
38.0
49.6
30.1
31.9
32.7
395.4
151.9
164.6
148.2
204.0
169.2
35
Country
Year
Lesotho
1986
Lesotho
1993
Lesotho
1995
Lesotho
2002
Liberia
2007
Lithuania
1988
Lithuania
1993
Lithuania
1996
Lithuania
1998
Lithuania
2002
Lithuania
2004
Macedonia, FYR
1998
Macedonia, FYR
2000
Macedonia, FYR
2002
Macedonia, FYR
2003
Madagascar
1980
Madagascar
1993
Madagascar
1999
Madagascar
2001
Madagascar
2005
Malawi
1997
Malawi
2004
Malaysia
1984
Malaysia
1987
Malaysia
1989
Malaysia
1992
Malaysia
1995
Malaysia
1997
Malaysia
2004
Mali
1994
Mali
2001
Mali
2006
Mauritania
1987
Mauritania
1993
Mauritania
1995
Mauritania
2000
Mexico
1984
Mexico
1989
Mexico
1992
Mexico
1994
Mexico
1996
Mexico
1998
Mexico
2000
Mexico
2002
Mexico
2004
Mexico
2006
Moldova Rep.
1988
Moldova Rep.
1992
Moldova Rep.
1997
Moldova Rep.
1999
Moldova Rep.
2002
Moldova Rep.
2004
Mongolia
1995
Mongolia
1998
Mongolia
2002
I: Income; C: Consumption.
C
C
C
C
C
I
I
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
I
I
I
I
I
I
I
C
C
C
C
C
C
C
C
I
C
C
C
C
C
C
C
C
I
I
C
C
C
C
C
C
C
H
εmin
εmax
ε*
W
xM
μ
G
μ (1-G)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
0.86
0.74
0.70
0.46
0.76
0.66
0.62
0.67
0.69
0.59
0.56
0.72
0.74
0.01
0.86
1.07
0.94
0.97
0.86
0.88
0.89
0.90
0.88
0.87
0.79
1.00
0.85
0.82
0.60
1.01
0.67
0.81
0.88
0.81
0.87
0.92
0.82
0.79
0.82
0.85
0.71
0.87
0.51
0.73
0.67
0.74
0.80
0.81
0.67
0.47
0.66
0.97
0.92
1.31
1.26
1.52
1.32
1.22
1.24
1.20
1.02
1.10
1.15
1.17
0.46
1.17
1.74
1.41
1.47
1.21
1.21
1.24
1.15
1.16
1.14
1.14
1.24
1.14
1.24
0.98
1.50
1.15
1.12
1.00
1.14
1.12
1.16
1.15
1.09
1.18
1.18
1.15
1.22
1.25
0.89
1.18
1.24
1.21
1.35
1.10
1.06
1.11
0.91
0.92
0.79
0.83
1.01
0.86
1.14
1.00
0.92
0.95
0.94
0.80
0.83
0.93
0.95
1.34
0.21
1.01
1.03
1.42
1.18
1.22
1.04
1.05
1.07
1.03
1.02
1.01
0.96
1.12
1.00
1.03
0.78
1.26
0.91
0.96
0.93
0.97
1.00
1.04
1.00
0.94
1.01
1.02
0.93
1.05
0.88
0.81
0.93
1.00
1.01
1.08
0.88
0.76
0.89
46.0
34.1
46.9
46.6
21.1
295.6
100.1
205.2
208.9
204.6
250.1
172.4
143.0
222.7
215.9
14.8
38.3
19.5
21.2
27.6
17.9
25.3
157.5
156.2
154.5
168.5
172.9
213.8
161.7
14.7
31.7
37.8
46.2
42.6
63.1
68.9
103.1
153.8
161.5
163.1
132.8
138.1
160.8
159.4
215.6
217.4
60.8
73.1
76.1
41.7
72.2
84.9
68.3
47.8
73.1
45.1
33.5
45.6
45.8
21.3
295.4
101.2
205.9
208.6
204.3
249.7
172.5
142.7
221.5
214.7
14.9
38.1
19.3
21.0
28.3
18.1
25.3
156.9
155.4
153.7
167.3
171.6
212.3
160.5
14.6
31.5
37.6
45.9
43.0
62.9
68.3
101.9
152.7
160.0
161.7
132.4
136.8
160.6
158.4
215.6
216.5
60.8
72.7
76.1
41.8
72.0
84.8
68.0
47.7
72.8
75.7
59.8
89.4
70.4
26.8
317.0
122.8
243.1
239.5
240.5
304.9
191.6
168.8
280.4
273.4
23.6
39.7
25.9
30.9
40.9
27.7
33.3
233.1
226.1
221.2
245.8
255.0
317.8
202.2
22.8
41.1
48.4
60.3
65.9
77.8
87.3
143.7
250.9
246.9
254.6
195.1
201.7
249.6
238.7
300.0
319.1
66.1
86.2
93.9
52.0
90.0
105.3
80.1
53.7
85.5
0.5418
0.5669
0.6169
0.3668
0.3183
0.3517
0.2224
0.3223
0.3156
0.2981
0.3391
0.3805
0.3821
0.2761
0.4618
0.4166
0.4084
0.4681
0.3011
0.3760
0.4703
0.3729
0.4680
0.4550
0.4462
0.4627
0.4699
0.4754
0.3928
0.3805
0.4840
0.4304
0.4628
0.3838
0.3663
0.4776
0.4439
0.4624
0.4522
0.5259
0.4955
0.4920
0.4971
0.4692
0.4755
0.3622
0.2397
0.3477
0.3620
0.3605
0.3473
0.3248
0.3265
0.2996
0.3285
34.7
25.9
34.3
44.6
18.3
205.5
95.5
164.7
163.9
168.8
201.5
118.7
104.3
203.0
147.1
13.8
23.5
13.8
21.6
25.5
14.7
20.9
124.0
123.2
122.5
132.1
135.2
166.7
122.8
14.1
21.2
27.6
32.4
40.6
49.3
45.6
79.9
134.9
135.3
120.7
98.4
102.5
125.5
126.7
157.3
203.5
50.3
56.2
59.9
33.2
58.7
71.1
53.9
37.6
57.4
36
Country
Year
Mongolia
2005
Morocco
1984
Morocco
1990
Morocco
1998
Morocco
2000
Morocco
2007
Mozambique
1996
Mozambique
2002
Namibia
1993
Nepal
1995
Nepal
2003
Nepal-Rural
1984
Nepal-Urban
1984
Nicaragua
1993
Nicaragua
1998
Nicaragua
2001
Nicaragua
2005
Niger
1992
Niger
1994
Niger
2005
Nigeria
1985
Nigeria
1992
Nigeria
1996
Nigeria
2003
Pakistan
1987
Pakistan
1990
Pakistan
1992
Pakistan
1996
Pakistan
1998
Pakistan
2001
Pakistan
2004
Panama
1979
Panama
1991
Panama
1995
Panama
1996
Panama
1997
Panama
2000
Panama
2002
Panama
2004
Paraguay
1990
Paraguay
1995
Paraguay
1997
Paraguay
1999
Paraguay
2002
Paraguay
2005
Peru
1985
Peru
1990
Peru
1994
Peru
1996
Peru
2002
Peru
2005
Philippines
1985
Philippines
1988
Philippines
1991
Philippines
1994
I: Income; C: Consumption.
C
C
C
C
C
C
C
C
I
C
C
I
I
I
I
I
I
C
C
C
C
C
C
C
C
C
C
C
C
C
C
I
I
I
I
C
I
I
I
I
I
I
I
I
I
C
I
C
I
I
I
C
C
C
C
H
εmin
εmax
ε*
W
xM
μ
G
μ (1-G)
1
1
0
1
1
1
1
1
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.60
0.84
0.91
0.87
0.93
0.93
0.88
0.94
1.03
1.19
0.88
0.72
0.75
0.79
0.81
0.84
0.85
0.91
0.84
0.70
0.83
0.73
0.92
0.86
1.13
1.10
1.03
1.05
0.01
0.73
0.71
0.72
0.71
0.75
0.76
0.77
0.71
0.69
0.70
0.73
0.80
0.85
0.83
0.71
0.79
0.84
1.00
1.04
1.01
1.10
1.44
1.34
1.28
1.32
1.44
1.37
1.45
1.55
1.58
1.49
1.00
1.14
1.16
1.20
1.48
1.27
1.40
1.00
0.79
1.29
1.10
1.56
1.46
1.83
1.86
1.74
1.78
0.88
0.81
0.93
0.89
0.92
0.96
1.11
1.00
0.83
1.00
1.00
1.06
1.19
1.34
1.10
1.12
1.10
1.12
1.30
1.24
1.20
0.85
1.15
1.13
1.08
1.13
1.19
1.13
1.20
1.05
1.29
1.39
1.19
1.28
0.86
0.94
0.97
1.01
1.17
1.06
1.16
0.92
0.74
1.06
0.91
1.24
1.16
1.50
1.50
1.40
1.43
0.40
0.89
0.77
0.82
0.81
0.81
0.85
0.86
0.84
0.94
0.89
0.77
0.83
0.85
0.90
1.00
1.10
0.97
0.91
0.94
0.98
1.15
1.15
1.14
1.11
62.1
84.2
116.2
98.1
98.4
116.6
20.2
23.5
44.0
28.7
35.2
25.0
38.0
63.9
80.2
82.2
91.6
26.7
22.3
28.6
36.1
40.6
26.4
29.3
30.2
31.1
51.4
38.7
48.9
44.5
68.7
154.9
137.0
174.5
164.5
188.4
169.8
166.2
179.9
118.7
153.5
117.5
142.5
119.5
156.1
216.2
180.3
135.9
125.1
123.2
139.1
50.2
55.1
56.4
59.5
61.9
84.7
115.9
97.7
98.0
116.8
20.2
23.8
43.5
28.7
35.2
25.0
37.9
63.1
80.0
81.9
91.9
26.9
22.2
28.7
35.9
40.0
26.5
29.1
30.2
31.0
51.5
38.8
49.1
44.7
67.8
152.5
133.7
171.9
161.6
185.6
166.7
163.1
176.2
118.0
151.6
114.7
141.0
118.4
155.2
215.5
181.3
134.9
125.0
122.2
137.8
50.0
54.7
56.1
59.2
72.4
110.2
152.5
127.6
131.0
157.0
28.4
34.7
118.9
37.5
53.1
29.1
48.6
104.5
126.7
123.4
143.8
33.8
29.8
40.0
45.3
53.0
38.0
38.8
37.2
37.8
62.4
46.1
60.8
53.8
72.8
223.6
220.7
287.0
265.7
264.8
278.3
273.0
284.7
151.7
267.4
188.3
234.0
201.5
246.3
304.0
250.8
187.4
174.6
198.3
215.3
67.5
73.7
79.0
81.8
0.3945
0.3920
0.3744
0.3842
0.3846
0.4250
0.4422
0.6901
0.3620
0.4422
0.2809
0.3525
0.4204
0.4834
0.4990
0.5439
0.5123
0.4187
0.3480
0.4033
0.3829
0.4446
0.4453
0.2929
0.2786
0.2920
0.2768
0.3162
0.3230
0.3239
0.5480
0.5347
0.5479
0.4791
0.5545
0.5530
0.5476
0.4765
0.5564
0.5187
0.5516
0.5482
0.3913
0.5669
0.5239
0.5007
0.4436
0.4203
0.4382
0.4487
0.4464
0.4343
0.4305
0.3989
0.3969
43.9
67.0
95.4
78.6
80.6
90.3
15.8
10.8
75.9
20.9
38.2
18.8
28.2
54.0
63.5
56.3
70.1
19.6
19.4
23.9
28.0
29.4
21.1
27.5
26.8
26.8
45.1
31.5
41.1
36.4
32.9
104.0
99.8
149.5
118.4
118.4
125.9
142.9
126.3
73.0
119.9
85.0
142.4
87.3
117.2
151.8
139.5
108.6
98.1
109.3
119.2
38.2
42.0
47.5
49.3
37
Country
Year
Philippines
1996
Philippines
1997
Philippines
2000
Philippines
2003
Philippines
2006
Poland
1985
Poland
1987
Poland
1989
Poland
1992
Poland
1996
Poland
1999
Poland
2002
Poland
2005
Romania
1989
Romania
1992
Romania
1994
Romania
1998
Romania
2000
Romania
2002
Romania
2005
Russia
1988
Russia
1993
Russia
1996
Russia
1999
Russia
2002
Russia
2005
Rwanda
1984
Rwanda
2000
Senegal
1991
Senegal
1994
Senegal
2001
Senegal
2005
Sierra Leone
2003
Slovak Rep.
1988
Slovak Rep.
1992
Slovak Rep.
1996
Slovenia
1987
Slovenia
1993
Slovenia
1998
Slovenia
2002
Slovenia
2004
South Africa
1993
South Africa
1995
South Africa
2000
Sri Lanka
1985
Sri Lanka
1990
Sri Lanka
1995
Sri Lanka
2002
St. Lucia
1995
Suriname
1999
Swaziland
1994
Swaziland
2000
Tajikistan
1999
Tajikistan
2003
Tajikistan
2004
I: Income; C: Consumption.
C
C
C
C
H
εmin
εmax
ε*
W
xM
μ
G
μ (1-G)
1
1
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
0
1
1
1
1
1
0
0
1
1
0
1
1
0
1
1
1
1
0
1
1
1
1
1
0
1
0
0
1
1
1
1
1
0
0
1.03
1.06
0.56
0.58
0.33
0.85
0.79
0.78
0.79
0.32
0.39
0.59
0.01
0.60
0.62
0.72
0.44
0.77
0.69
0.68
0.72
0.75
1.04
0.92
0.85
0.93
1.00
0.76
0.97
0.65
0.48
0.20
0.64
0.86
0.69
0.65
0.71
0.84
1.00
1.01
1.02
0.66
0.73
0.83
0.89
0.65
0.76
0.80
1.22
1.18
1.35
1.27
1.00
1.46
1.33
1.31
1.29
1.15
1.11
1.30
0.85
1.22
1.24
1.38
1.18
1.21
1.15
1.09
1.14
1.16
1.66
1.35
1.07
1.46
1.45
1.20
1.20
1.55
1.63
1.17
1.42
1.59
1.31
1.27
1.32
1.42
1.67
1.51
1.49
1.07
1.00
1.12
1.21
1.32
1.29
1.32
1.07
1.13
1.12
1.06
1.07
0.96
0.33
0.93
0.65
1.16
1.06
1.05
1.04
0.73
0.74
0.94
0.38
0.91
0.93
1.05
0.81
1.00
0.92
0.89
0.93
0.95
1.36
1.14
0.96
1.21
1.22
0.98
1.09
1.10
1.06
0.67
1.03
1.23
1.01
0.96
1.01
0.96
1.03
0.95
1.13
1.34
1.27
1.26
0.86
0.86
0.98
1.05
1.00
1.03
1.06
69.9
70.3
69.4
71.2
69.9
301.6
360.6
383.7
177.8
112.8
246.2
242.3
247.4
323.8
213.2
87.1
139.3
102.3
114.3
158.8
133.8
198.4
201.6
151.3
189.5
239.5
32.6
22.3
26.8
35.4
41.9
51.6
37.0
401.9
377.1
325.0
466.7
468.9
545.4
571.5
582.6
90.0
87.0
83.9
59.9
61.1
64.5
71.8
74.5
117.7
17.5
29.6
40.8
46.7
60.5
69.2
69.9
68.9
70.7
69.2
301.9
350.4
383.5
177.5
112.8
245.6
241.8
246.5
324.5
213.1
87.3
137.2
102.3
114.5
159.3
133.6
200.0
202.0
150.6
188.6
238.1
32.5
22.3
26.5
35.6
41.8
51.4
36.7
401.3
378.1
324.7
466.2
469.4
544.9
570.7
582.0
88.9
85.9
82.9
59.7
61.2
64.4
71.7
74.2
116.2
17.3
29.4
40.9
46.6
60.3
97.1
101.8
100.3
99.5
97.1
332.7
377.7
427.7
199.0
136.4
295.5
293.8
302.3
346.3
230.9
98.6
147.3
117.2
133.1
187.8
144.0
290.8
280.3
186.3
229.9
297.7
38.5
32.4
43.0
48.2
57.0
65.9
50.2
429.3
402.6
349.8
510.8
550.1
620.7
651.5
681.4
164.9
150.9
147.6
72.0
75.1
81.6
97.3
97.2
180.2
31.6
45.2
47.9
55.5
73.0
0.4251
0.4176
0.4463
0.4305
0.2713
0.3102
0.3181
0.3240
0.3340
0.3419
0.2479
0.2653
0.3073
0.2978
0.3091
0.3082
0.2312
0.2527
0.2808
0.2781
0.2356
0.4439
0.3519
0.3687
0.4622
0.3695
0.4450
0.2825
0.5622
0.5466
0.5763
0.3967
0.3836
0.3959
0.5206
0.4132
0.1940
0.1916
0.2468
0.2871
0.3057
0.2326
0.2843
0.2799
0.3907
0.3179
0.3132
0.3434
0.4154
0.5128
0.4840
0.5700
0.3207
0.3295
0.3080
55.8
59.3
55.5
56.7
70.8
229.5
257.6
289.1
132.5
89.8
222.3
215.9
209.4
243.2
159.5
68.2
113.2
87.6
95.7
135.6
110.1
161.7
181.6
117.6
123.6
187.7
21.4
23.3
18.8
21.9
24.1
39.8
31.0
259.3
193.0
205.3
411.7
444.7
467.5
464.5
473.1
126.6
108.0
106.3
43.9
51.2
56.1
63.9
56.8
87.8
16.3
19.4
32.5
37.2
50.5
38
Year
H
εmin
εmax
ε*
W
xM
μ
G
μ (1-G)
Tanzania
1991
Tanzania
2000
Thailand
1981
Thailand
1988
Thailand
1992
Thailand
1996
Thailand
1999
Thailand
2002
Thailand
2004
Timor-Leste
2001
Togo
2006
Trinidad and Tobago
1988
Trinidad and Tobago
1992
Tunisia
1985
Tunisia
1990
Tunisia
1995
Tunisia
2000
Turkey
1987
Turkey
1994
Turkey
2002
Turkey
2005
Turkmenistan
1988
Turkmenistan
1993
Turkmenistan
1998
Uganda
1989
Uganda
1992
Uganda
1996
Uganda
1999
Uganda
2002
Uganda
2005
Ukraine
1988
Ukraine
1992
Ukraine
1996
Ukraine
1999
Ukraine
2002
Ukraine
2005
Uruguay
1981
Uruguay
1989
Uruguay-Urban
1992
Uruguay-Urban
1996
Uruguay-Urban
1998
Uruguay-Urban
2000
Uruguay-Urban
2001
Uruguay-Urban
2005
Uzbekistan
1988
Uzbekistan
1998
Uzbekistan
2002
Uzbekistan
2003
Venezuela
1981
Venezuela
1987
Venezuela
1989
Venezuela
1993
Venezuela
1996
Venezuela
1998
Venezuela
2003
I: Income; C: Consumption.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.67
0.74
0.94
1.07
0.00
1.04
1.04
1.01
0.97
0.85
0.68
0.85
0.75
0.79
0.82
0.92
0.80
0.83
0.74
0.86
0.76
0.89
0.92
0.89
0.97
0.93
0.49
0.53
0.71
0.67
0.64
0.67
0.71
0.74
0.69
0.71
0.70
0.77
0.76
0.80
0.59
0.83
0.81
0.71
0.76
0.69
0.60
0.59
1.24
1.25
1.20
1.34
0.00
1.30
1.32
1.32
1.29
1.30
1.09
1.27
1.19
1.18
1.23
1.40
1.25
1.27
1.18
1.25
1.12
1.43
1.41
1.38
1.43
1.32
1.29
1.09
1.29
1.33
1.30
1.30
1.15
1.21
1.12
1.08
1.05
1.10
1.03
1.05
1.03
1.49
1.40
1.11
1.16
1.06
0.97
0.97
0.96
1.00
1.07
1.21
1.24
1.17
1.15
1.18
1.17
1.14
1.08
0.95
0.88
1.06
0.97
1.00
1.03
1.16
1.03
1.05
0.96
1.54
0.89
1.06
0.94
1.17
1.17
1.14
1.20
1.13
0.89
0.81
1.01
1.01
0.98
1.00
0.93
0.98
0.90
0.90
0.88
0.93
0.90
0.92
1.67
0.80
1.16
1.11
0.98
0.94
0.91
0.96
0.87
0.78
0.77
27.4
18.4
70.5
73.3
100.1
118.4
112.9
122.2
135.1
36.8
45.5
192.9
144.8
100.0
115.3
114.3
136.1
150.1
150.4
152.8
170.0
60.0
31.1
62.0
26.5
26.6
30.6
31.6
33.3
37.5
94.7
221.0
138.0
105.8
149.1
219.3
295.3
298.9
321.0
289.7
317.5
294.3
330.4
248.5
145.0
57.3
41.9
39.9
172.3
162.1
185.8
155.3
104.6
129.7
98.5
27.4
18.4
69.9
73.0
99.7
117.8
112.3
121.6
134.6
36.6
45.4
193.1
144.0
99.7
114.9
114.1
135.6
150.3
150.2
152.6
170.6
60.2
30.9
61.8
26.3
26.8
30.5
31.7
33.3
37.4
94.7
220.3
138.7
105.8
149.1
219.2
295.0
298.6
320.2
288.2
315.5
291.6
327.6
246.0
146.0
57.1
42.2
40.1
169.3
159.7
185.2
154.7
104.3
129.1
97.7
32.8
22.3
99.4
103.6
148.3
165.2
157.6
167.2
186.0
48.4
55.6
263.5
184.0
137.2
149.0
151.5
179.2
209.0
199.8
207.2
229.7
70.1
37.5
82.0
36.1
36.6
39.1
43.6
48.1
51.4
102.4
240.8
169.2
121.1
169.2
248.9
397.9
398.6
421.1
387.7
432.0
400.9
449.6
339.3
170.1
76.2
52.0
50.5
287.4
257.9
251.0
204.2
150.3
181.9
135.2
0.3399
0.3316
0.4067
0.4111
0.4386
0.4227
0.4466
0.4197
0.4231
0.3851
0.3379
0.4346
0.3951
0.3986
0.4209
0.3928
0.4072
0.4139
0.4165
0.4033
0.4203
0.2596
0.3979
0.3553
0.4344
0.4058
0.3607
0.4130
0.4118
0.4337
0.2789
0.2784
0.2304
0.2554
0.3433
0.2847
0.4408
0.4123
0.4281
0.4402
0.4250
0.4420
0.4362
0.4113
0.4364
0.3340
0.3553
0.2539
0.4640
0.4592
0.5378
0.5212
0.4313
0.4072
0.4733
21.6
14.9
59.0
61.0
83.3
95.4
87.2
97.0
107.3
29.7
36.8
149.0
111.3
82.5
86.3
92.0
106.2
122.5
116.6
123.6
133.2
51.9
22.6
52.9
20.4
21.8
25.0
25.6
28.3
29.1
73.8
173.8
130.2
90.2
111.1
178.1
222.5
234.2
240.8
217.1
248.4
223.7
253.5
199.7
95.9
50.8
33.5
37.7
154.0
139.4
116.0
97.8
85.5
107.8
71.2
Country
39
Year
H
εmin
εmax
ε*
W
xM
μ
G
μ (1-G)
Venezuela-Urban
2005
Vietnam
1992
Vietnam
1998
Vietnam
2002
Vietnam
2004
Vietnam
2006
Yemen Rep.
1992
Yemen Rep.
1998
Yemen Rep.
2005
Zambia
1991
Zambia
1993
Zambia
1996
Zambia
1998
Zambia
2002
Zambia
2004
I: Income; C: Consumption.
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0.64
1.01
1.01
1.05
1.04
0.93
0.73
0.69
0.88
0.62
0.81
0.82
0.76
0.89
0.74
1.05
1.47
1.51
1.37
1.36
1.29
1.25
1.22
1.49
0.75
0.86
1.18
1.11
1.37
1.06
0.84
1.24
1.26
1.22
1.20
1.11
1.00
0.96
1.19
0.69
0.84
1.01
0.94
1.13
0.90
134.6
31.1
38.7
45.5
60.0
63.9
120.6
75.2
63.5
28.1
26.9
29.6
33.7
29.4
28.0
134.2
31.1
38.7
45.3
59.7
63.5
120.8
75.2
63.9
27.9
26.5
29.5
33.5
29.4
27.8
187.7
39.4
49.0
58.8
79.0
81.6
155.2
89.6
82.1
47.3
41.0
44.4
53.1
40.0
41.8
0.4746
0.3691
0.3475
0.3654
0.3800
0.3440
0.3621
0.3831
0.3291
0.5121
0.4049
0.4924
0.5144
0.4790
0.5792
98.6
24.9
32.0
37.3
49.0
53.6
99.0
55.3
55.1
23.1
24.4
22.5
25.8
20.8
17.6
Country
Graph A1. Percentage of ranking agreements.
1.000
% of agreement
0.999
0.998
0.997
0.996
0.995
0.89
0.94
0.99
1.04
ε value
40
1.09
1.14