This note provides simple tests for first-order bi-polarization orderings of distributions of living standards. In doing so, the paper also offers an ethical basis and an interpretation for the common use of some simple measures of distances from the median. Illustrations using Luxembourg Income Study data show that several countries can be ranked by such tests of bi-polarization.Polarization, inequality, stochastic dominance, relative poverty

Damien Echevin

Jean-Yves Duclos

English

Research Papers in Economics

 
 
 
CIRPÉE 
Centre interuniversitaire sur le risque, les politiques économiques et l’emploi 
 
 
 
 
 
Cahier de recherche/Working Paper 03-41 
 
 
 
 
 
Bi-Polarization Comparisons 
 
 
 
 
Jean-Yves Duclos 
Damien Échevin 
 
 
 
 
 
 
 
Octobre/October 2003 
 
 
 
 
 
 
 
 
_______________________ 
Duclos: CIRPÉE and Département d’économique, Pavillon J.-A.-DeSève, Université Laval, Québec, Canada, 
G1K 7P4; Phone: 1-418-656-7096; Fax: 1-418-656-7798 
jyves@ecn.ulaval.ca 
Échevin: Direction de la Prévision – Ministère de l’Économie, des Finances et de l’Industrie – Télédoc 673, 139 
rue de Bercy, 75572 Paris Cedex 12; Phone: +33-1-53-18-56-40; Fax: +33-1-53-18-95-64 
damien.echevin@dp.finances.gouv.fr 
 
This research was funded in part by Canada’s SSHRC, Québec’s FQRSC and the PEP program of IDRC. 
Échevin also thanks CIRPÉE for financial support during the post-doctoral research year he spent at Université 
Laval, and during which this paper started. Abstract:  This note provides simple tests for first-order bi-polarization orderings of 
distributions of living standards. In doing so, the paper also offers an ethical basis 
and an interpretation for the common use of some simple measures of distances 
from the median. Illustrations using Luxembourg Income Study data show that 
several countries can be ranked by such tests of bi-polarization 
 
Keywords:  Polarization, inequality, stochastic dominance, relative poverty 
 
JEL Classification: D31, D63, I32 
 1 Introduction
The last decade has seen a surge of interest in the theoretical and empirical
measurement of polarization – loosely speaking, the clustering of incomes around
localpoles.1 Therecentliteraturehasalsoprovidedtoolsfororderingdistributions
over classes of bi-polarization indices, that is, over classes of indices that exhibit
an ethical ”preference for the middle”2. This note builds on that literature by
deriving simple tests for ﬁrst-order bi-polarization orderings of distributions of
living standards (Section 2). This differs from the earlier and recent work that
concentrated on second-order bi-polarization orderings. The paper (Section 3)
also offers an ethical basis for the common use of simple measures of distances
from the median, thus allowing a re-interpretation of some of the simple indicators
that could otherwise be described by (see for instance, Wolfson (1994), p.354) as
”unsatisfactory” and ”incoherent”.
Focussing on ﬁrst-order (as opposed to second-order) bi-polarization order-
ings has the advantage of greater generality regarding the ethical properties of
the bi-polarization indices, thus enabling searches for ”more unanimous” order-
ings. Such a focus can, however, limit the ordering power of the resulting tests.
Illustrations (Section 4) using cross-country Luxembourg Income Study data nev-
ertheless show that the ﬁrst-order tests proposed by this paper seem empirically
quite powerful in ordering many countries in terms of bi-polarization.
The proofs are found in the Appendix along with a sketch of the sampling
distribution of the statistics used in the dominance tests.
2 Polarization Dominance
2.1 Measuring bi-polarization
Let x = (x1;x2;:::;xn) 2 <n
++ be an n-dimensional vector of positive in-
comes, ordered in increasing values such that x1 · x2 · ::: · xn, and letting xi
be the income of the ith person. We assume that n is even – we will see later using
the population-replication axiom 2 that this assumption is without consequence
here. Median income is thus mx = xn=2. Let dx = (dx(1);dx(2);:::;dx(n)) 2
1See, among many others, Davis and Huston (1992), Foster and Wolfson (1992), Esteban and
Ray (1994), Wolfson (1994), Jenkins (1995), Esteban and Ray (1999), Gradin (2000), Chakravarty
and Majumder (2001), and Zhang and Kanbur (2001).
2See in particular Foster and Wolfson (1992), Wolfson (1994) and Wang and Tsui (2000).
1<n
++, with dx(i) = j1 ¡ xi=mxj. dx(i) is thus the proportional ”spread” of i’s
income from the median. Then:
Deﬁnition 1 A bi-polarization index P (dx) : <n
++ ! < is a function of the
differences dx (i); i = 1;:::;n of xi from the median income mx.
Axiom 1 (Homogeneity) The index P is homogeneous of degree zero in dx, viz,
for any ° > 0, we have
P (dx) = P (°dx): (1)
Axiom 2 (Population invariance) Adding a replication of a distribution x to that
same distribution has no impact on P.
We can therefore suppose, for the sake of expositional simplicity, that all income
vectors x are of the same dimension n.
Axiom 3 (Monotonicity) For a given n and a given mx, the index P (dx) is mono-
tonically increasing in the distance dx. In other words, for any x and y in <n
++
such that mx = my, with dx(i) ¸ dy(i) for all i = 1;:::;n and with dx(i) > dy(i)
for some i = 1;:::;n, then P (dx) ¸ P (dy).
A simple index which obeys all of the above axioms is given by
Qx(¸) = n
¡1
n X
i=1
I(dx(i) ¸ ¸) (2)
where I(¢) is an indicator function that takes the value 1 if its argument is true and
0 otherwise. Qx(¸) shows the proportion of the population whose proportional
distance from the median exceeds ¸ – loosely speaking, an index of ”bipolarity”.3
2.2 First-order bi-polarization dominance
Now deﬁne the class C(A1;A2;A3) as the class of all polarization indices
P (¢) which obey Axioms 1, 2 and 3. Then:
Theorem 1 (First-order bi-polarization dominance)
P (dx) ¸ P (dy); 8P(¢) 2 C(A1;A2;A3) (3)
iff dx(i) ¸ dy(i); 8i = 1;:::;n: (4)
3See, for instance, Morris, Bernhardt, and Handcock (1994).
2Proof. See appendix.
Let d¤
x be the vector dx rearranged in increasing values of the dx(i). A con-
dition that is regularly implicitly imposed on bi-polarization indices derives from
the following symmetry axiom:
Axiom 4 (Symmetry) P (dx) = P (d¤
x) for any x 2 <n
++:
Axiom 4 says that ignoring whether distances from the median occur on the
right or on the left of the median should not matter for the measurement of bi-
polarization. We then have:
Theorem 2 (First-order symmetric polarization dominance)
P (dx) ¸ P (dy); 8P(¢) 2 C(A1;A2;A3;A4) (5)
iff d¤
x(i) ¸ d¤
y(i); 8i = 1;:::;n: (6)
Proof. The proof follows from the same arguments as for Theorem 1, this
time using d¤
x and d¤
y as opposed to dx and dy.
It can be seen by inspection that (6) is a weaker condition than (4). It is
easier to order bi-polarization indices that consider distances from the median
symmetrically than over those that do not.
3 Discussion
Consider the following two income distributions expressed as a proportion of
the median mx:
x = (2mx=3;2mx=3;2mx=3;mx;mx;mx;mx;2mx;2mx;2mx)
y = (0;0;0;0;mx;mx;2:5mx;2:5mx;2:5mx;2:5mx):
Moving from distribution x to distribution y illustrates an increased ”spreads” ef-
fect: both the poor and the rich are getting more distant from the median. Figure 1
displays this. Individual i appears on the horizontal axis at percentile p = i=n and
the proportional distance from the median is shown on the vertical axis. The rela-
tive differences from the median are smaller in x than in y whatever the percentiles
considered.
Inverting the distance curves provides an equivalent test that is interpretable in
terms of a ”bi-polarity” criterion. For this, let dx¡ = (dx(1);dx(2);:::;dx(n=2))
3and dx+ = (dx(n=2 + 1);dx(n=2 + 2);:::;dx(n)). It can be checked that condi-
tion (4) is then equivalent to
Qx¡(¸) ¸ Qy¡(¸); 8¸ > 0 (7)
and Qx+(¸) ¸ Qy+(¸); 8¸ > 0: (8)
Note that Qx¡(¸) is the proportion of the population which lies below the median
by a proportional distance ¸ or greater. Alternatively, Qx¡(1 ¡ ³) is the propor-
tion of the population whose income is lower than ³ times the median – this is a
frequently-used relative poverty measure. Note that (7) can then be equivalently
written as
Qx¡(1 ¡ ³) ¸ Qy¡(1 ¡ ³); 8³ 2 [0;1]: (9)
Hence, one of the necessary conditions for ﬁrst-order bi-polarization dominance
is that relative poverty be uniformly higher in x than in y for all proportions of the
median between 0 and 1.
Another simple polarization index is the share of the population within a cer-
tain sub-interval of income spread symmetrically on each side of the median in-
come. This is given by 1¡Qx(¸), where ¸ is that symmetric spread expressed as a
proportion of the median – a popular measure of relative dispersion. 1¡Qx(1), for
instance, is the proportion of the population located within one median distance
of the median.
Note ﬁnally that condition (6) can also be rewritten as
Qx(¸) ¸ Qy(¸); 8¸ > 0; (10)
or as
1 ¡ Qx(¸) · 1 ¡ Qy(¸); 8¸ > 0: (11)
A symmetric bi-polarization ordering is then obtained when the share of the pop-
ulation away from the median is greater in x than in y, whatever the proportional
spread considered, or when the share of the population within the median is lower
in x than in y, whatever the spread considered4.
4Note that the popular interquartile range (expressed as a proportion of the median) is given as
dx¡(n=4)+dx+(n=4) whenever n=4 is an integer. But neither condition (4) nor condition (6) can
be rewritten in terms of that range.
44 Illustration
We use Luxembourg Income Study (LIS) data to illustrate the above tools5.
Figure 2 shows the distance curves d(i = np) across percentiles p and Figures
3 and 4 display polarization indices Q¡(1 ¡ ³) and Q(¸), all computed for the
UnitedStates(2000)-US00-theUnitedKingdom(1999)-UC99-Canada(1998)
- CN98 - the Netherlands (1994) - NL94 - Mexico (1998) - MX98 - and France
(1994) - FR94. Figure 2 shows that many pairs of countries can be distinguished –
inter alia, many of those involving Mexico, the US, France, Canada and the UK.
Figure 3 shows why a statistical test of relative poverty can rank unambiguously
all possible pairs of countries that do not include the Netherlands – except for
Canada’s curve which crosses the UK’s and which a conventional test of size 5%
cannot distinguish statistically for larger values of ³. Note also that Mexico has
the highest level of relative poverty of all 6 countries. Finally, once we impose
the symmetry axiom, most pairs of countries are ranked with statistical signiﬁ-
cance (see Figure 4) and that, in particular, the Netherlands has lower ﬁrst-order
polarization than any one of the other countries (except possibly France).
5 Conclusion
It is well known that ”inequalities can diverge” (e.g., Wolfson (1994) and Es-
teban and Ray (1994)), namely that polarization and inequality can evolve in op-
posite directions. The distinction between inequality and bi-polarization is even
sharper in this paper. This is mainly because of the monotonicity axiom 3 by
which movements of income away from the median must increase bi-polarization,
regardless of their impact on the mean – a crucial element in accounting for the
movements of the usual relative inequality indices. This focus on ﬁrst-order eth-
ical properties has the advantage of making the bi-polarization comparisons po-
tentially more general. Interestingly, illustrations using cross-country data suggest
that a number of important bi-polarization orderings can still be made in spite of
this increased ethical generality.
5 http://lissy.ceps.lu for detailed information on the structure of these data. Living
standards are measured by household disposable income divided by the squared root of household
size. Observations with negative incomes are removed as well as those with incomes exceeding
50 times the median. Household observations are weighted by the LIS sample weights times the
number of persons in the household.
5A Proof of Theorem 1
Note ﬁrst that Axiom 1 implies that
P (dx) = P (dx=mx) (12)
for any x 2 <n
++. We can therefore suppose that, for all of the comparisons of
bi-polarization that need to be consistent with Axiom 1, all incomes have been
pre-normalized by the median of their distribution. We then check in turn the
necessity and sufﬁciency of condition 4:
1. Necessity: P (dx) ¸ P (dy), for all P 2 C(A1;A2;A3), implies dx(i) ¸
dy(i), for all i = 1;:::;n.
Suppose we have dx(i) · dy(i) for all i = 1;:::;n and dx(i) < dy(i) for
some i = 1;:::;n. Note that
Qx(dy(i)) < Qy(dy(i)) (13)
ButQ(dy(i))doesbelongtoC(A1;A2;A3). Hence, wecannothavedx(i) <
dy(i) anywhere if P (dx) ¸ P (dy)8P(¢) 2 C(A1;A2;A3).
2. Sufﬁciency: dx(i) ¸ dy(i), for all i = 1;:::;n, implies P (dx) ¸ P (dy), for
all P 2 C(A1;A2;A3).
This is straightforward since, by Axiom 3 and whenever dx(i) ¸ dy(i)8i =
1;:::;n, we can write:
P(dx(1);dx(2);:::;dx(n)) ¸ P(dy(1);dx(2);:::;dx(n)) (14)
¸ ::: (15)
¸ P(dy(1);dy(2);:::;dy(n)) (16)
for all of the bi-polarization indices that belong to C(A1;A2;A3).
B The sampling distribution of the polarization es-
timators
First, considerthe”spreads”dx(np), wherepissomepercentile. Thesespreads
are functions of p-quantiles and of the median, the sampling distribution of which
was derived by Bahadur (1966). Suppose that a population is characterized by
6a twice differentiable distribution function F. Then, if the p¡quantile of F is
denoted by X(p), and the sample p¡quantile from a sample of n independent
drawings yi from F by ˆ X(p), we have
ˆ X(p) ¡ X(p) = ¡
1
nf
¡
X(p)
¢
n X
i=1
³
I
¡
xi < X(p)
¢
¡ p
´
+ O
¡
n
¡3=4(logn)
3=4¢
;
(17)
where f = F 0 is the density. The asymptotic sampling distribution of dx(np) can
then be obtained by applying Rao (1973)’s ”delta” method.
Second, consider the following estimator ˆ Q(¸) of the bi-polarity indicator
Q(¸):
ˆ Q(¸) =
Z ˆ m(1¡¸)
0
d ˆ F(y) +
Z 1
ˆ m(1+¸)
d ˆ F(y); (18)
where ˆ F is the empirical distribution function. ˆ Q(¸) can be expressed asymptoti-
cally as
ˆ Q(¸) ' (ˆ m ¡ m)[(1 ¡ ¸)f (m(1 ¡ ¸)) + (1 + ¸)f (m(1 + ¸))]
+ ˆ F (m(1 ¡ ¸)) + 1 ¡ ˆ F (m(1 + ¸)):
Everything above can be expressed as sums of iid variables: for ˆ m, see (17), and
note that ˆ F(y) = 1
n
Pn
i=1 I(xi · y). The asymptotic sampling distribution of
ˆ Q(¸) then follows by simple computation.
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Bi-Polarization Comparisons

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