1
On Bi-Pola iza ion and he Middle Class in La in Ame ica:
A Look a he Fi s Decade o he Twen y Fi s Cen u y.
Joseph Deu sch
Depa men o Economics, Ba -Ilan Uni e si y
52900 Rama -Gan, Is ael
Email: [email p o ec ed]
Jacques Silbe
Depa men o Economics, Ba -Ilan Uni e si y
52900 Rama -Gan, Is ael
Email: [email p o ec ed]
and
Gas on Yalone zky
Leeds Uni e si y Business School (LUBS)
Uni e si y o Leeds, Leeds, Uni ed Kingdom.
Email: [email p o ec ed]
Oc obe 2012
P elimina y d a . No o be quo ed wi hou he au ho s' pe mission.
2
Abs ac
This pape p oposes a new index and g aphical ep esen a ion o he change in bi-pola iza ion
and in he ela i e impo ance o he middle class ha ook place in a gi en coun y du ing a
gi en pe iod. These ools ex end in ac he concep s o in e -dis ibu ion income inequali y
and Lo enz cu es by making a dis inc ion be ween o e all, “pu e g ow h based” and “shape
ela ed” dis ibu ional changes.
The empi ical illus a ion is based on da a co e ing 17 La in Ame ican coun ies in 2000 and
2009, ob ained om he La inoba óme o su eys o hese yea s. The s anda d o li ing o
indi iduals was de i ed on he basis o co espondence analysis. I appea s ha hese new
ools help unde s anding he changes ha ook place in he dis ibu ion o s anda ds o li ing
du ing he pe iod analysed.
Key Wo ds: bipola iza ion, gene alized Lo enz dominance, in e dis ibu ional income
inequali y, La in Ame ica, La inoba óme o.
J. E. L. Classi ica ion: D31 – O15 - O54
3
1. In oduc ion
In a e y ecen pape C. Kenny (2011) w i es ha “i is ha d o ind a se o cha ac e is ics o
alues ha a e consis en ly and uniquely middle class ac oss coun ies and ime”. The e is
indeed in he li e a u e a long lis o ai s ha a e supposed o help iden i ying indi iduals
who belong o he middle class. The li e a u e seems o ha e s essed he ollowing
cha ac e is ics o indi iduals belonging o he middle class. They a e supposed o be middle
aged people, who in es in hei educa ion, ha e a ela i ely small numbe o child en ( wo o
h ee?), spend mo e on heal h ca e, pay axes, end o own hei own house o apa men as
well as one o wo ca s and, as a consequence, ha e a signi ican amoun o deb . They a e
also supposed o ha e some en ep eneu ial spi i , o wo k in speci ic occupa ions, o ha e
s able jobs and he means o a oid po e y e en when acing an unexpec ed shock such as
becoming unemployed. They a e also expec ed o hold common alues such as a belie in he
i ue o democ acy ( ee elec ions and ee speech) and o ole ance (e.g. owa ds mino i ies)
Finally i is also said ha hey a e op imis ic abou he u u e, belie e ha hey a e doing
be e han hei pa en s, belong o he middle spec um o he dis ibu ion o incomes, no
ma e how such a middle ange is de ined and a e supposed o be one o he main engines o
economic g ow h.
Needless o say, including all hese cha ac e is ics, assuming hey a e all ele an , o
de e mine who belongs o he middle class is an impossible ask, among o he easons
because o he sca ci y o da a sou ces ha would encompass all he po en ially ele an
a iables men ioned p e iously. In addi ion he e is no ag eemen , among hose who ha e
a emp ed o de ine he middle class, abou he mos impo an ea u es o he middle class,
al hough he amoun o income a ailable is almos always men ioned.
4
Bu e en when he main ocus is on he income le el, he e is no consensus conce ning he
c i ical h esholds, hose ha dis inguish he middle class om he poo and om he ich.
The e is hence a need o be e y ca e ul when a emp ing o assess he size o he middle class
in a gi en coun y, o ind ou whe he i s impo ance g ew o e ime, o de ec i s main
cha ac e is ics o o de e mine whe he he iden i y o hose belonging o he middle class
does no change o a ies a lo o e ime.
The impo ance o he middle class is clea ly ela ed o he concep o bipola iza ion. Fos e
and Wol son (1992; 2010) ecommended making a dis inc ion be ween ou s ages when
a emp ing o measu e he ela i e impo ance o he middle class:
- choose a “space” (indi idual/ amily/household income, sala y, expendi u e e c. in an
income- o people-space)
- de ine he "middle" (e.g., he median o he mean income)
- ix a ange a ound he middle (iden i y he middle class by de e mining a pe cen age
in e al abo e and below he median o he mean)
- agg ega e he da a.
Va ious de ini ions based on he “income space” ha e been p oposed. Thu ow (1984)
assumed ha he middle class includes he households whose income anges om 75% o
125% o he median household income. Blackbu n and Bloom (1985) ecommended using a
wide ange (60% o 225% o he median). O he anges ha e been p oposed: 50% o 150%
(Da is and Hus on, 1992) and wo- hi ds o ou - hi ds o men’s median weekly ea nings
(Law ence, 1984). Bi dsall e al. (2007) sugges ed including in he middle class hose
indi iduals abo e he equi alen o $10 day in 2005 and a o below he 90 h pe cen ile o he
income dis ibu ion in hei own coun y. In all hese cases one compu es which sha e o he
o al popula ion he middle class includes.
5
O he s ha e p e e ed o use de ini ions based on he people space. Fo Le y (1987), o
example, he middle class anges om he 20 h o he 80 h pe cen ile. Wha e e he de ini ion
adop ed when using such an app oach, one compu es he e he sha e in o al income o hose
belonging o he selec ed popula ion deciles. G aphical ep esen a ions o ep esen a ions in
bo h he income and he people space ha e been p oposed by Fos e and Wol son (1992;
2010). The p esen pape p oposes a he a g aphical ep esen a ion o he change in bi-
pola iza ion which is de i ed om he concep o in e -dis ibu ional change.
2. Measu ing Changes in Bi-pola iza ion
2.1. The Concep o O e all Dis ibu ional Change and In e -dis ibu ional Inequali y
and Lo enz cu es
The concep s o In e -dis ibu ional inequali y and Lo enz Cu es we e in oduced in he
li e a u e by Bu le and McDonald (1987) and may be summa ized as ollows. Assume wo
di e en densi y unc ions
)(x
and
)(xh
desc ibing he dis ibu ion o income
x
in a gi en
coun y a wo di e en pe iods 0 and 1. Le
)(xF
and
)(xH
be he wo dis ibu ions unc ions
co esponding o he wo densi y unc ions
)(x
and
)(xh
. These wo dis ibu ion unc ions
)(xF
and
)(xH
will now be plo ed espec i ely on he ho izon al and e ical axis o a 1 by 1
squa e. In o he wo ds o each income
x
we plo he pe cen age o indi iduals wi h an
income lowe han o equal o
x
obse ed in he dis ibu ions
)(xF
and
)(xH
. I he
“dis ibu ional change cu e” ob ained happens o be comple ely below he diagonal, we can
ce ainly conclude ha he dis ibu ion i s o de s ochas ically domina es he
dis ibu ion Mo e gene ally i mos o he cu e lies below he diagonal we can
conclude ha he popula ion wi h he income dis ibu ion has an economic ad an age
o e he popula ion wi h an income dis ibu ion (see, Bishop e al., 2011). I howe e

6
mos o he cu e lies abo e he diagonal we would conclude ha he popula ion wi h he
income dis ibu ion has an economic ad an age o e he popula ion wi h an income
dis ibu ion .
2.2. Dis ibu ional Change in he Case o Pu e G ow h
Le us now call
x
m
and
h
x
m
he median incomes co esponding o he dis ibu ions and
and le us, o example, assume ha
x
h
xmm 
. Le now
)(xk
be he densi y unc ion
ob ained when he densi y unc ion
)(x
is ho izon ally ansla ed by an amoun
)(
x
h
xmm 
.
Finally le
)(xK
be he dis ibu ion unc ions co esponding o he densi y unc ion
)(xk
. A
plo o on he e ical axis e sus ha o on he ho izon al axis would hen gi e us
a “dis ibu ional change cu e” ha would only be a ec ed by g ow h (assuming ha
e e s o ime and o ime ) since was de i ed om by a
ansla ion. Assuming he e was posi i e g ow h (since we pos ula ed ha
x
h
xmm 
), he
dis ibu ion change cu e ob ained will hen s a a some poin A on he ho izon al axis (see,
Figu e 1). The segmen OA would hen ep esen he p opo ion o indi iduals who a ime
(co esponding o dis ibu ion ) had an income lowe han he lowes income a ime +1
(co esponding o dis ibu ion ). Such a dis ibu ional cu e would also end a poin B
and he segmen BC would ep esen he sha e o he popula ion who a ime +1 had an
income highe han he highes income a ime (see Figu e 1). In he pa icula and
excep ional case whe e he lowes income a ime +1 would be highe han he highes
income a ime , he dis ibu ional change cu e would become iden ical o he b oken cu e
OFC. In such a case we know ha he e would be no o e lap be ween he dis ibu ions
)(x
and
)(xk
and he index o dis ibu ional change , de ined as being equal o wice he
a ea be ween he dis ibu ional cu e and he diagonal, would e iden ly be equal o 1. The
7
complemen o one o he dis ibu ional change index may hence be conside ed as a measu e
o he deg ee o o e lap be ween he dis ibu ions and in he case o posi i e
g ow h o e ime.
Figu e 1: Dis ibu ional Change Cu e
in he Case o Pu e Posi i e G ow h
O
O
A
O
B
O
O
O
O
O
C
O
O
O
O
O
F
8
Con e sely i he e was nega i e g ow h so ha he median o he dis ibu ion K(x) is smalle
han ha o he dis ibu ion F(x), we would ge a cu e ha would gene ally s a a some
poin E on he e ical axis. OE would hen ep esen he p opo ion o indi iduals who a
ime 1 had an income smalle han he smalles income a ime 0. The cu e would end on he
uppe ho izon al axis a some poin D and DC would ep esen he p opo ion o indi iduals
who a ime 0 had an income highe han he highes income a ime 1.
In he pa icula and excep ional case whe e he highes income a ime +1 would be lowe
han he lowes income a ime , he dis ibu ional change cu e would become iden ical o
he b oken cu e OGC. In such a case we know ha he e would be no o e lap be ween he
dis ibu ions (x) and k(x) and he index o dis ibu ional change de ined p e iously would
e iden ly end owa ds -1. The complemen o -1 o such an index would hence be a measu e
o he deg ee o o e lap in he case o nega i e g ow h.
9
Figu e 2: Dis ibu ional Change Cu e
in he Case o Pu e Nega i e G ow h
O
O
C
E
D
G
16
Figu e 3: O e all dis ibu ional change, by coun y.
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
A gen ina
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Boli ia

17
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
B azil
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Colombia
18
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Cos a Rica
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Chile
19
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Ecuado
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
El Sal ado
20
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Gua emala
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Hondu as
21
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Mexico
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Nica agua

22
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Panama
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Pa aguay
23
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Pe u
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
U uguay
24
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Venezuela
25
“Pu e g ow h ela ed” dis ibu ional change
The esul s conce ning he “pu e g ow h ela ed” dis ibu ional change a e also gi en in Table 1.
I appea s ha as a whole g ow h was highes in Chile, A gen ina, El Sal ado , U uguay, Cos a
Rica and B azil. On he o he side he coun ies whe e as a whole g ow h was nega i e we e
Gua emala, Hondu as, Nica agua, Venezuela and Pa aguay.
Table 2 gi es in o ma ion on he leng h o he “non-o e lapping” segmen s while Table 3a gi es
he a ia ion be ween 2000 and 2009 in he median s anda d o li ing o he di e en coun ies.
Table 3b compa es he anking o he coun ies acco ding o he a e age g ow h a e du ing he
pe iod 2001-2010 and o he alue o he “pu e g ow h ela ed” index o dis ibu ional change.
The index o ank co ela ion be ween hese wo measu es u ns ou o be equal o 0.47.
We hen show in Figu e 4 he “pu e g ow h ela ed” dis ibu ional change cu es o he a ious
coun ies.
32
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
Ecuado
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
El Sal ado

33
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
Gua emala
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
Hondu as
34
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
Mexico
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
Nica agua
35
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
Panama
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
Pa aguay
36
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
Pe u
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
U uguay
37
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 ( i)
CDF o La en a iable 2000 (pi)
Dis ibu ional Change Cu e - Pu e G ow h
Venezuela

38
“Shape ela ed” dis ibu ional change
We now u n o he analysis o “shape ela ed” dis ibu ional change. Table 1 indica es ha his
ype o dis ibu ional change was highes (and posi i e) o Gua emala, U uguay, Nica agua and
Venezuela and lowes (and nega i e) o A gen ina, Chile, Ecuado , Panama, El Sal ado and
Cos a Rica.
Figu e 5 p esen s hese “shape ela ed” dis ibu ional change cu es o he a ious coun ies and
allows one o make a mo e de ailed analysis o he obse ed change.
Le us ake a look o example a he case o Gua emala. One obse es ha om a pu e change in
shape poin o iew almos e e yone in 2009 would ha e had a highe s anda d o li ing han in
2000. The e was hus be ween 2000 and 2009 among he “poo ” a shi o he obse a ions
owa ds he median. We also obse e ha among he “ ich” he e was a igh wa d shi o he
obse a ions owa d highe s anda ds o li ing.
Ano he in e es ing case is Ecuado whe e he “shape ela ed” dis ibu ional change was almos
nil o hose ha ing a s anda d o li ing abo e he median whe eas o hose wi h in 2000 a
s anda d o li ing below he median, he e was a downa d shi owa ds he lowes le els o
s anda d o li ing.
39
Figu e 5: “Shape E ec ” Dis ibu ional Change Cu es
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 ( i)
Dis ibu ional Change Cu e - Shape E ec
A gen ina
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 ( i)
Dis ibu ional Change Cu e - Shape E ec
Boli ia
40
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 ( i)
Dis ibu ional Change Cu e - Shape E ec
B azil
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 ( i)
Dis ibu ional Change Cu e - Shape E ec
Colombia
41
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 ( i)
Dis ibu ional Change Cu e - Shape E ec
Cos a Rica
020 40 60 80 100
020 40 60 80 100
x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 ( i)
Dis ibu ional Change Cu e - Shape E ec
Chile
48
dis ibu ional cu e co esponding o he ich is mos ly below he diagonal, since his implies
ha he ich ha e become iche .
Table 4: Value o he “Change in Bi-Pola iza ion” Index o he Va ious Coun ies.
“Change in
Bi-
Pola iza ion”
Index
Rank
A gen ina
0.0142
4
Boli ia
0.0584
14
B azil
0.0370
10
Colombia
0.0276
7
Cos a_Rica
0.0155
5
Chile
0.0371
11
Ecuado
0.0806
17
El_Sal ado
0.0801
16
Gua emala
0.0204
6
Hondu as
0.0080
3
Mexico
0.0288
8
Nica agua
-0.0042
2
Panama
0.0746
15
Pa aguay
0.0541
13
Pe u
0.0509
12
U uguay
0.0332
9
Venezuela
-0.0473
1
Table 4 hen indica es ha bipola iza ion inc eased he mos in Ecuado and El Sal ado whe eas
i dec eased only in Venezuela and Nica agua. Figu e 4 indica es ha in he case o Venezuela
wha happened was essen ially an imp o emen in he s anda ds o li ing o he poo , he same
being ue o Nica agua hough he changes we e less impo an he e.
In Ecuado and El Sal ado , on he con a y, Figu e 4 shows ha he e was a clea de e io a ion
in he s anda ds o li ing o he poo .

49
The Case o he Middle Class
To ha e a be e iew o wha happened o he middle class we will now unca e he
dis ibu ional change cu e and igno e he lowes 20% and highes 10% o he dis ibu ions o
s anda d o li ing in 2000 and 2009. The co esponding cu es a e gi en in Figu e C-1 in
Appendix C. We will limi ou analysis o some coun ies o which he o e all dis ibu ional
change cu e co e ing he whole popula ion was pa ly abo e and pa ly below he diagonal.
Le us ake he case o B azil, o example. The co esponding cu e shows now clea ly ha he
lowe middle class (mo e o less he lowe 30% o wha we de ined as middle class) we e in a
wo se si ua ion in 2009 han in 2000. This was also ue o he e y uppe middle class ( he
uppe 25% o wha was de ined as middle class). Bu o hose loca ed in he middle o his
unca ed dis ibu ion he s anda d o li ing was clea ly highe in 2009 han in 2000.
Simila conclusions may be d awn when looking a he g aph o Cos a Rica in Figu e C-1.
Fo Chile he pic u e is less clea -cu . The lowe middle class (lowe 15% o he middle class)
and he uppe middle class (uppe 30% o he middle class) we e wo se o in 2009 so ha as a
whole one canno say ha hose we de ined as belonging o he middle class we e be e o in
2009.
Finally in A gen ina we obse e ha e e yone in he middle class, bu he uppe 22-23%, was
be e o in 2009.
50
4. The link wi h adi ional bi-pola iza ion cu es
Le and be wo dis ibu ions o a con inuous a iable and we de ine [ ]
o dis ibu ion and, likewise,
o . I s ands o he median hen
we can s anda dize he dis ibu ions by di iding each alue by he median. Such di ision yields
he a iable and we will ha e, e.g. , and
.
De ine now he sp ead om he median: | |. Following Fos e and Wol son
(2009) le us compa e he ollowing wo ( i s -o de ) change-in-pola iza ion indices. The one
based on he i s -o de pola iza ion cu e appea s in he middle g aph in Figu e 6 and is de ined
as:
∫[ ]
∫ [ ]
{ }
∫ [ ]
{ }
The one based on dis ibu ional change cu es is gi en in he igh g aph in Figu e 6 and is
de ined as:
∫[ ]
∫[ ]
51
Ne posi i e alues mean ha exhibi s ela i ely less pola iza ion han (al hough he e may
be compensa ion e ec s ope a ing a di e en pe cen iles o he dis ibu ions). Ne nega i e
alues mean ha exhibi s ela i ely mo e pola iza ion.
We may no e he p opo ionali y ela ionship be ween and . When he sp ead di e ence
[ ] inc eases, he igh -hand side o has o inc ease as well and ha can only be
accomplished by he widening o some o he pe cen ile gaps, below he median
and/o abo e he median. Hence , which is a unc ion o bo h se s o gaps, also
inc eases when a sp ead di e ence inc eases. Bo h and can be exp essed as unc ions o
he pe cen ile gaps, bu while he i s index is a weigh ed sum o hese gaps in which he
weigh s a e , he second index is a weigh ed sum o pe cen ile gaps in which he weigh s a e
.
No e also ha a Pigou-Dal on ans e ac oss he median should educe pola iza ion and inc ease
and i ep esen s he p e- ans e dis ibu ion and ep esen s he pos - ans e
dis ibu ion.
52
Figu e 6: Fi s -o de pola iza ion cu es and dis ibu ional change cu es
𝑧
𝑞
𝑞𝐵
𝑞𝐴
𝐵
𝐴
𝐴
𝐵
𝑆
𝐴
𝐵
𝑙
𝑙
𝐹
𝐴 𝑙
𝐹𝐵 𝑙
𝐹
𝐴 𝑙
𝐹
𝐴 𝑙
𝐹𝐵 𝑙
𝑞
53
Fos e and Wol son (2009) discuss also wha happens i a Pigou-Dal on ans e akes place on
one side o he median. I he ans e p ese es anks, hen i is easy o show ha is
insensi i e o i , as i is only measu ing pola iza ion wi h espec o he median. Howe e such
ans e should inc ease bi-pola iza ion as i concen a es he dis ibu ion on he side o he
median whe e he ans e ook place. An index ha is sensi i e o hese ans e s and hus
measu es changes in bipola iza ion can be cons uc ed using second-o de pola iza ion cu es.
Le us de ine he cumula i e sp ead om he median as
|∫
| |∫| |
|.
Le us now compa e he ollowing wo ( i s -o de ) change-in-pola iza ion indices.
The i s one is based on he second-o de pola iza ion cu e (middle g aph in Figu e 2)and is
de ined as:
∫[ ]
∫ [ ]
{ }
∫ [ ]
{ }
The second one is based on he concep o cumula i e ela i e dis ibu ion ( igh g aph in Figu e
1) and may be de ined as:
∫[ ]
∫[ ]

54
He e also we may no e he p opo ionali y ela ionship be ween and . I should also be
s essed ha he indices a ach mo e weigh o sp eads close o he median. The e o e i a Pigou-
Dal on ans e occu s on one side o he median, ep esen ing he p e- ans e dis ibu ion and
he pos - ans e dis ibu ion, hen he indices will ake a nega i e alue he eby showing an
inc ease in bipola iza ion.
The empi ical illus a ion gi en in Sec ion 3 deal only wi h a “change in i s o de
bipola iza ion”. We plan in he nea u u e o comple e his illus a ion by looking also a
“second o de changes in bi-pola iza ion”.
55
Figu e 2: Second-o de pola iza ion cu es and cumula i e ela i e dis ibu ions
𝑆
𝑞
𝑞𝐵
𝑞𝐴
𝐵
𝐴
𝐴
𝐵
𝐶
𝐴
𝐵
𝑘
𝐹
𝐴 𝑘
𝐹
𝐴 𝑘
𝐹𝐵 𝑘
𝑞
𝐴
𝐵
56
5. Concluding Commen s
This pape p oposed a new index and g aphical ep esen a ion o he change in bi-pola iza ion
and in he ela i e impo ance o he middle class ha ook place in a gi en coun y du ing a
gi en pe iod. These ools ex end in ac he concep s o in e -dis ibu ion income inequali y and
Lo enz cu es by making a dis inc ion be ween o e all, “pu e g ow h based” and “shape ela ed”
dis ibu ional changes.
The empi ical illus a ion was based on da a co e ing 17 La in Ame ican coun ies in 2000 and
2009, ob ained om he La inoba óme o su eys o hese yea s. The s anda d o li ing o
indi iduals was de i ed on he basis o co espondence analysis. I seems ha he new ools
p oposed in his pape help unde s anding he changes ha ook place in he dis ibu ion o
s anda ds o li ing in La in Ame ica du ing he pe iod analysed. They also sugges a new way o
de e mining wha happened he e o he middle class be ween 2000 and 2009.
This empi ical analysis was limi ed o he case o a “ i s o de change in bi-pola iza ion”. In
u u e wo k we plan o ex end his analysis o he case o a “second o de change in bi-
pola iza ion”.
57
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Cos a Rica
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CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Chile

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x1
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CDF o La en a iable 2000 (pi)
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Ecuado
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x1
CDF o La en a iable 2009 (wi)
CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
El Sal ado
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CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Gua emala
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x1
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CDF o La en a iable 2000 (pi)
O e all Dis ibu ional Change Cu e
Hondu as
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CDF o La en a iable 2000 (pi)
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Mexico
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x1
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Nica agua
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Panama
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x1
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Pa aguay
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Pe u
020 40 60 80 100
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x1
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U uguay

70
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Venezuela