ï»¿ WPS6449
Policy Research Working Paper 6449
The Inequality Possibility Frontier
Extensions and New Applications
Branko Milanovic
The World Bank
Development Research Group
Poverty and Inequality Team
May 2013
Policy Research Working Paper 6449
Abstract
This paper extends the Inequality Possibility Frontier post-1960 civil conflict around the world. The duration
approach in two methodological directions. It allows of conflict and the casualty rate are positively associated
the social minimum to increase with the average income with the inequality extraction ratio, that is, with the
of a society, and it derives all the Inequality Possibility extent to which elite pushes the actual inequality
Frontier statistics for two other inequality measures closer to its maximum level. Inequality, albeit slightly
besides the Gini. Finally, it applies the framework reformulated, is thus shown to play a role in explaining
to contemporary data, showing that the inequality civil conflict.
extraction ratio can be used in the empirical analysis of
This paper is a product of the Poverty and Inequality Team, Development Research Group. It is part of a larger effort by
the World Bank to provide open access to its research and make a contribution to development policy discussions around
the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be
contacted at bmilanovic@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
The inequality possibility frontier: Extensions and new applications
Branko Milanovic 1
Key words: inequality, inequality possibility frontier, civil war
JEL classification: D31, N3, O1, Q3
Number of words: About 7,700
Sector board: Social protection
1
World Bank, Development Research. Earlier versions of the paper was presented at the conference
â€œMeasuring inequality in economic historyâ€? held at the Universidad de la Republica in Montevideo in August
2012, economic history seminar at the University of Lund in September 2012, and Poverty and Inequality
Measurement and Analysis Practice group seminar at the World Bank, Washington in April 2013. I am grateful
to the participants, and in particular to Jorge Alvarez, Luis Bertola, Cecilia Castelnovo, Esteban Nicolini, Javier
Rodriguez Weber and Henry Willebald in Montevideo, Ewout Frankema, Christer Gunnarsson and Debin Ma in
Lund, and Peter Lanjouw in Washington for very valuable comments. It is written as a part of the KCP-funded
research project â€œChageable inequalities: facts, perceptions and policiesâ€? TF012968. The opinions expressed in
the paper are authorâ€™s and should not be attributed to the World Bank or its affiliated organizations.
1. Introduction
The purpose of the inequality extraction ratio (Milanovic, Lindert and Williamson
2007, 2011, in further text MLW; first defined in Milanovic, 2006) is to measure how close is
measured inequality to the maximum inequality that can exist in a given society (called
maximum feasible inequality). The maximum feasible inequality was defined under a special
condition such that all but an infinitesimal minority of people (Îµ) live at the physiological
subsistence (s). Then, if we define Y=total income and n=number of people, the surplus (S)
over the subsistence will be
í µí±† = í µí±Œ âˆ’ í µí±›(1 âˆ’ í µí¼€ )í µí±
The surplus will be, under conditions of maximum feasible inequality, received by an
infinitesimally small percentage of people, in the extreme case by one person. It should be
apparent that the derivation of the maximum feasible inequality follows exactly the same
method as used when defining the maximum values of the Gini coefficient or other
inequality statistics. The only difference is that here the floor is physiological subsistence (s)
rather than income of 0. The conventional maximum Gini of 1 is defined as the situation
when all individuals but one have zero incomes, and the one, rich, individual appropriates
the entire income of a community. It should also be clear that the conventional maximum
Gini is a special case of the maximum feasible Gini when s=0.
This can be readily seen from the formal definition of the maximum feasible Gini. As
mentioned, the society consists of two groups of people: Îµn with income (Y-n(1-Îµ)s)/Îµ and
n(1-Îµ) with income of s. The Gini coefficient (G*) is by definition
í µí±› í µí±›
1
í µí°º = ï¿½ ï¿½(í µí±¦í µí±— âˆ’ í µí±¦í µí±– )í µí±?í µí±– í µí±?í µí±—
í µí¼‡
í µí±–=1 í µí±—>í µí±–
where m=mean income, yi=income of individual or group i, and pi=share of iâ€™s in total
population. With only two groups and average incomes as given, the Gini coefficient
reduces to
1 í µí±›í µí±šâˆ’í µí±›(1âˆ’í µí¼€ )í µí±
í µí°º âˆ— = í µí¼‡ ( í µí±›í µí¼€
âˆ’ í µí± )(í µí¼€)(1 âˆ’ í µí¼€ )) (1)
Rewriting (1), we get
2
1 1 1âˆ’í µí¼€
í µí°º âˆ— = í µí±š (í µí¼€ (í µí±š âˆ’ (1 âˆ’ í µí¼€ )í µí± ) âˆ’ í µí± )(í µí¼€)(1 âˆ’ í µí¼€)) = í µí±š
(í µí±š âˆ’ í µí± ) (2)
Obviously, when Îµâ†’0, the expression simplifies to
1 í µí±
í µí°º âˆ— = í µí±š (í µí±š âˆ’ í µí± ) = 1 âˆ’ í µí±š (3)
If we express the mean income in terms of the physiological subsidence (which we
shall find particularly useful when dealing with pre-industrial economies), and denote this as
Î±=m/s, then (3) becomes
1 í µí»¼âˆ’1
í µí°º âˆ— = 1 âˆ’ í µí»¼ = í µí»¼
(4)
Equation (4) is our final expression for the maximum Gini under the condition of all
but one person receiving the physiological minimum. Obviously, G* depends on how rich
the society is: the richer it is, the more inequality it can theoretically accommodate because
the surplus will be greater, and if it is, by assumption, appropriated by one person,
inequality will be greater as well. To fix ideas, suppose that Î±=2; then G*=0.5; if Î±=3, then
G*=0.66 etc. For very high values of Î±, as in todayâ€™s advanced economies, where Î±>100, the
maximum feasible Gini will approach 1. In other words, in very rich societies, the maximum
feasible Gini (G*) will not differ much from the conventional Gini derived under the
condition that s=0.
The G*â€™s that are charted as Î± increases define the locus of maximum Ginis which we
call the inequality possibility frontier (IPF). As shown in Figure 1, the Gini inequality
possibility frontier is concave. This can also be checked from equation (4) whose second
derivative is negative.
3
Figure 1. Inequality Possibility Frontier
1
.8
maximum feasible Gini
.4 .2
0 .6
1 5 20 50 100
alpha log scale
Consider the situation when the minimum is zero: then, the second term in (3)
becomes 0, and the maximum Gini is simply the maximum value of the â€œstandardâ€? Gini
index, that is 1 regardless of the average income of society. Then, the IPF is a straight line
fixed at G=1 throughout the range of Î±â€™s.
The inequality extraction ratio (IER) is defined as the ratio between the recorded
(measured) Gini and the maximum feasible Gini (G*):
í µí°º
í µí°¼í µí°¸í µí±… = í µí°º âˆ— (5)
IER gives an estimate of how close a society is to its inequality possibility frontier. It
also implies that the same recorded Gini in two societies that differ in terms of their mean
incomes will have very different implications. For a poor society, the measured Gini can be
quite close to the maximum feasible Gini, and the IER will be high. For a rich society, whose
G* is much greater, the G/G* ratio will be lower: the extraction ratio will be less. The ratio
will be thus representative both of the level of development of a society and of the ability of
the elite to extract the surplus. Thus, to illustrate the role of the average income we may
take a contemporary example of Tanzania and Malaysia. Tanzania has a lower Gini than
Malaysia (0.38 vs. 0.47 in 2008), but its IER is greater (51 percent vs. 48 percent) simply
because Malaysiaâ€™s income is much higher. And to illustrate the role of predatory elites, we
4
may take the finding from Milanovic, Lindert and Williamson (2011). The authors find
particularly high IERs to have obtained in colonies: inequality there was pushed almost to its
maximum, with IERs approaching 100%. 2 Figure 2 shows the actual Ginis in pre-industrial
societies calculated from the social tables drawn against the Inequality Possibility Frontier.
The data include new results for the United States in 1774 and 1860 and Russia in 1904 that
had become available after the publication of the MLW paper. 3 It is easy to notice that the
dots representing most of the colonies in the sample (Moghul India 1750 and British India
1938, Nueva EspaÃ±a 1790, Maghreb 1880, and Kenya 1912 and 1927) lie around or slightly
above the frontier. 4
Figure 2. Estimated Gini coefficients and the Inequality Possibility Frontier (pre-industrial economies)
80
IPF
70
Nueva EspaÃ±a 1790 Chile 1861
Holland 1732
60 Maghreb 1880
Holland 1561
India 1938
Netherlands. 1808 USA1860
Old Castille 1752 France 1788
50 India 1750
Siam 1929 USA 1774 England 1801
Kenya 1927 Brazil 1872
Florence 1427
Gini index
Peru 1876 Engl1759
Engl 1688
40 Byzant 1000
Java 1880
Rome 14 Levant 1596 Russia 1904
Engl 1290 Japan 1886
Kenya 1914 Bihar 1807
30 Java 1924
Naples 1811
China 1880
20
Serbia 1455
10
0
0 300 600 900 1200 1500 1800 2100 2400
GDI per capita (in 1990 $PPP)
Source: Updated from MLW (2011).
2
The social minimum was assumed to be 300 international 1990 dollars (these are the same dollars that
underlie Maddisonâ€™s tables). This amount is somewhat lower than the World Bank absolute poverty line (see
discussion in MLW, 2011, p. 262, fn. 16). In the empirical work, we find that there are people with incomes
even lower than the World Bank absolute poverty line, but their distance from the poverty line is typically low.
Moreover, if one thought, based on this evidence, that 300 international dollars is greater than what is needed
for subsistence, all relations given here can be recalculated with an even lower s. The key point is that the
subsistence minimum must be greater than 0.
3
US estimates were published by Lindert and Williamson (2011), Russia estimates by Nafziger and Lindert
(2012).
4
The observations for two countries with at least three data points (Holland/Netherland and England) are
linked by the dashed lines.
5
So far the derivation of the IER has been done using two important simplifying
assumptions. First, it was assumed that the subsistence is an unalterable physiological
minimum s. But what would happen if the subsistence itself changes in function of the
average income of society, s=s(Î±), so that it is no longer an absolute minimum but a socially-
influenced or socio-cultural minimum? How would our formulas for the maximum Gini and
the IER change?
Second, inequality was assumed to be measured by the Gini and the IER was defined
as the ratio of the two (actual and maximum feasible) Ginis. The question can be asked: Can
the Inequality Possibility Frontier and the IER be derived for other inequality measures and
how would they compare with IPF and IER derived for the Gini? Will countriesâ€™ extraction
ratios move the same way whether we measure inequality using one or another measure?
In other words, we need to show that the IER is not â€œa prisoner of the Giniâ€?. It is to these
two questions, both extensions of the original IER concept, that we turn next.
2. When physiological minimum becomes social minimum
There are three sources of evidence that the social minimum tends to rise with
affluence of a community. We use the term â€œsocial minimumâ€? to indicate that the â€œfloor
incomeâ€? may rise as mean income increases, while the term â€œsubsistenceâ€? or â€œphysiological
minimumâ€? is reserved for an â€œabsolute income floorâ€? below which life is not sustainable. To
differentiate between the two, we use notation Ïƒ for social minimum and s for the
physiological minimum.
The first type of empirical evidence comes from the countriesâ€™ official poverty lines.
They may be thought similar to social minimum, and they tend to increase as the mean
income of a society goes up (see Chen and Ravallion, 2013, in particular Figure 1, p. 6). Thus,
they gradually diverge from the physiological minimum which remains a poverty threshold
only in the poorest countries. Chen and Ravallion (2013, p. 9) find that the elasticity of the
official poverty lines with respect to mean income is around 0.33, once we move away from
the poorest 15 to 20 countries where the elasticity is zero (i.e., their poverty lines are equal
6
to subsistence). 5 Converting this cross-sectional regularity into historical terms implies that
the physiological minimum was much more likely to have been a â€œreasonableâ€? minimum in
poor, pre-industrial societies than in todayâ€™s more affluent ones.
The second type of evidence comes from the studies on subjective poverty. It
emerges there that what people consider to be the â€œminimum income necessary to make
ends meetâ€? increases in the respondentâ€™s family per capita income. Most research has
yielded the elasticity values between 0.4 and 0.7 (see, e.g. Flik and van Praag, 1991, p. 325;
van Praag and Flik, 1992, p. 10). This also accords well with our intuitive perception that as
people get richer they set the necessary minimum higher, but do not raise it (in percentage
terms) as much as their income increases.
Third, classical economists, most famously Adam Smith, later reprised by Amartya
Sen, argued that poverty is not solely a physiological attribute (inability to satisfy some basic
minimum needs or functions), but to operate â€œwithout shameâ€? in a society. In words of
Adam Smith: â€œBy necessaries, I understand, not only the commodities which are
indispensably necessary for the support of life, but whatever the custom of the country
renders it indecent for creditable people, even of the lowest order, to be withoutâ€?. 6 Thus,
again, a more affluent society would require a higher social minimum. The same argument
was used recently in Ravallion (2012) to argue in favor of a â€œweakly relative poverty lineâ€? for
the developing world, such that in addition to the absolute (subsistence) component it
would also make allowance for the â€œsocial inclusionâ€? needs that rise with the average
income of society.
Let the social minimum (Ïƒ) be related to the average income normalized by s (=Î±) as
in (6)
í µí¼Ž = í µí± í µí»¼ í µí±? (6)
where b represents the elasticity with which social minimum increases as mean income
(normalized by s) of a society goes up. Clearly, if the mean income is at the subsistence itself
5
When estimated across all poverty lines and mean consumptions, the estimated elasticity is 0.65 (Chen and
Ravallion, 2013, p. 8).
6
The Wealth of Nations, book 5, ii.
7
(Î¼/s=Î±=1), then Ïƒ must also be equal to s regardless of the elasticity. As Î¼ increases, Ïƒ will
diverge from s, depending on the elasticity b. By writing the new expression for Ïƒ into (3),
we obtain
1 1 í µí¼‡ 1 1
í µí°º âˆ— (í µí±?, í µí»¼ ) = í µí¼‡ (í µí¼‡ âˆ’ í µí¼Ž) = í µí¼‡ ï¿½í µí¼‡ âˆ’ í µí± ( í µí± )í µí±? ï¿½ = í µí¼‡ (í µí¼‡ âˆ’ í µí± (í µí»¼)í µí±? ) = 1 âˆ’ í µí»¼ í µí»¼ í µí±? (7)
Relationship (7) is the general expression linking the maximum feasible Gini, average
income, and elasticity of the minimum with respect to average income. If elasticity b=0, the
relationship reduces to (3). There is no social element in the minimum and the minimum is
purely physiologically determined. At the other extreme, if b=1, so that the social minimum
increases pari passu with the average income, G* becomes 0. In other words, there cannot
be any surplus if it is assumed that all members of a community have to be guaranteed a
social minimum equal to the mean income. Then, obviously, everybody has the average
income and Gini is equal to zero. Formula (7) is a general case of the maximum feasible Gini
which allows subsistence to vary between 0; the physiological minimum s; or to increase in
function of average income growth of a country up to being equal to mean income.
Empirically, b will lie between the two extremes (0 and 1). For simplicity and also
based on the subjective poverty literature as well on the observations of how poverty lines
rise with the average income (GDP per capita), a reasonable approximation of b may be
0.5. 7 But higher or lower elasticities are possible. Figure 3 shows the shape of the inequality
possibility frontier for several values of b while Î± ranges from 1 to 48, and then (at the last
point on the horizontal axis) attains the value of Î±=100. The curve on the top labeled
â€œminimum= subsistenceâ€?, drawn for b=0, is the same IPF as the one which we already drew
in Figure 1. It is the IPF constructed under the assumption that the subsistence minimum
does not include any â€œsocially-influencedâ€? part and is purely physiological. As b increases,
and the social minimum begins to (increasingly) respond to the rise in the average income,
the IPF shifts downward. This should be easy to understand intuitively. As we require that
all members of a community have at least an income that rises in proportion to the mean,
the â€œsurplusâ€? to be divided among an infinitesimally small elite, will be less. That surplus will
7 1
Notice also that in that case, equation (7) for the maximum feasible Gini conveniently simplifies to 1 - . For
âˆší µí»¼
example, if Î±=100, as in todayâ€™s rich economies, G*=0.9.
8
obviously be less the greater the elasticity of the social minimum with respect to the mean.
In consequence, maximum feasible inequality must be less and its locus, IPF must shift
downward (for any given Î±). 8
Figure 3. Inequality possibility frontier for different values of the social minimum
1.0 minimum=subsistence
0.9 elasticity=0.5
0.8 elasticity=0.6
0.7
0.6
Gini
0.5
0.4
elasticity=0.9
0.3
0.2
0.1
0.0
1 6 11 16 21 26 31 36 41 46
Alpha
The introduction of a social, rather than a merely physiological minimum, which in
turn shifts downward the IPF, has a straightforward implication for the inequality extraction
ratio. Since it is the ratio between measured and maximum feasible Gini, and the latter is
now lower, the IER increases.
Table 1 and Figure 4 show historical IER for England and Wales (or United Kingdom)
and the United States under two assumptions: (a) that the subsistence minimum is constant
and (b) that it increases in proportion to the mean income with elasticity of 0.5. Figure 4
shows the inequality extraction ratios over a long historical period covering more than 300
years for England/UK and 200 years for the United States. The usefulness of the IER ratio
appears most obviously in such examples because of the big variations in real incomes over
such long periods. For the UK, real per capita income between 1688 (the first year for which
8
The concave shape of IPF (for any given b) remains.
9
we have inequality estimates) and 2010 increased by more than 16 times. 9 The figure
shows the large difference between the IERs calculated with subsistence only, and under
the assumption that the social minimum increases as the mean income goes up. Under the
first assumption, the IER in England/UK oscillated around 60 percent throughout 18th and
19th century. But if we assume that the social minimum had risen in proportion with real
income (b=0.5), the IERs attained 80 percent. The numbers for the United States are very
similar.
The political implications of b=0 and b=0.5 are very different. If the elites had to
acquiesce to a rise of the minimum living standard for the masses, then an 80% inequality
extraction ratio shows that they were able or willing to push overall inequality rather close
to the maximum. If, on the other hand, we believe that there was no social pressure to
increase the social minimum, then the elitesâ€™ appetites may be thought to have been
relatively modest.
In both countries, it is only in the 20th century, as illustrated more clearly in Figure 5,
that the IER, calculated under the more stringent conditions of a changing social minimum,
began its downward slide, to be arrested and reversed in the last quarter of the 20th century
and early 21st. But because real incomes have by then risen to very high levels (exceeding,
in the United States, 100 times the subsistence minimum), the inequality possibility frontier
was close to 1 whether we use b=0.5 or b=0 assumption, and the inequality extraction ratio
was similar under both scenarios. 10 Today, actual inequality in the UK reaches the level of
some 40 percent of the maximum feasible inequality while in the United States it is just over
50 percent. This is a significant improvement compared to the situation some 150 years ago
when the IER ratios in both countries were around 80 percent.
While the changes in the IER are more dramatic and probably more informative
when calculated over longer periods of time, the results can be very interesting for the
contemporary period as well particularly when we are dealing with countries that register
9
GDP per capita data, expressed in 1990 Geary-Khamis PPPs, are from Maddison (2007). The physiological
subsistence minimum in the same prices is assumed to be $PPP 300 (for discussion see MLW 2011, p. 262).
10
The maximum feasible Gini is bounded from above (at 1), and the gap between the maximum feasible Ginis
calculated for different bâ€™s is decreasing in Î±, as can be seen in Figure 3. To see this, define, using expression
(3), the â€œgap functionâ€? between the two maximum Ginis for two different bâ€™s as, G* (b1,Î±) â€“ G*(b0, Î±) where
b1>b0 and differentiate it with respect to Î±. It can be easily verified that the â€œgap functionâ€? is decreasing in Î±.
10
high income increases or declines. Recently, perhaps the most startling is the comparison
between China and Russia.
Figure 6 shows their diverging patterns in the 1990s and in the first decade of the
21st century (assuming in both cases that the subsistence is fixed at the absolute level). In
both Russia and China, as shown in the left-hand panel, Gini increased substantially. In the
case of China this happened against the background of an almost seven-fold increase in GDP
per capita while in Russia it happened while GDP per capita at first went steeply down and
then recovered, reaching in 2009 the same level as 21 years earlier. This had very different
implication for the inequality extraction ratio (see right-hand side panel). In China, higher
GDP per capita offset higher inequality and the IER remained at 50 percent. In Russia, the
extraction ratio almost doubled going up from 21 percent to 41 percent. These patterns
would be somewhat less striking if we assumed that the social minimum changes with
mean income: both Chinese gains and Russian loses would be less dramatic.
11
Table 1. Historical inequality extraction ratios
for England/United Kingdom and the United States, selected years
England/United Kingdom
Year Estimated Estimated Î± Maximum Gini Inequality extraction
Gini (GDP per ratio (in percent)
capita/s)
With s only with b=0.5 With s with
only b=0.5
1209 36.7 2.1 53.1 31.5 69 117
1688 45.0 4.7 78.8 54.0 57 83
1759 45.9 5.9 82.9 58.7 55 78
1801 51.5 6.7 85.0 61.3 61 84
1867 53.0 9.9 89.9 68.2 59 78
1964 33.6 31.9 96.9 82.2 35 41
1979 28.8 43.9 97.7 84.9 29 34
1991 35.9 53.9 98.1 86.4 37 42
2002 37.4 69.8 98.6 88.0 38 42
2007 37.1 78.8 98.7 88.7 38 42
2010 37.4 76.2 98.7 88.5 38 42
United States
Country/year Estimated Estimated Î± Maximum Gini Inequality extraction
Gini (GDP per ratio (in percent)
capita/s)
With s only with b=0.5 With s with
only b=0.5
1774 45.6 5.3 81.1 56.5 56 81
1860 53.0 7.3 86.2 62.9 62 84
1929 48.1 23.0 95.7 79.1 50 61
1935 47.2 18.2 94.5 76.6 50 62
1950 40.5 31.9 96.9 82.3 42 49
1967 39.7 47.8 98.0 85.5 41 46
1979 40.4 62.6 98.4 87.4 41 46
1991 42.8 76.2 98.7 88.5 43 48
2002 46.2 95.3 99.0 89.8 47 51
2007 46.3 104.5 99.0 90.2 47 51
2009 46.8 99.4 99.0 89.7 47 52
Sources: GDP per capita for both countries from Maddison (2007), expressed in 1990 PPPs. The subsistence minimum
assumed $PPP 300 in 1990 international prices.
UK Ginis: 1209-1801 from Milanovic, Lindert and Williamson (2011) based respectively on social tables drawn by Campbell
(2007) for year 1209, Gregory King for 1688 [1696], Joseph Massie for 1759 [1760] and Patrick Colquhoun for 1801-3
[1806]. Data for 1867 calculated from Lindert and Williamson (1983) which are in turn based on social tables produced by
Dudley Baxter [1869]. UK Ginis from 1967 onward from Luxembourg Income Study (income concept: disposable household
per capita income across individuals).
US Ginis: 1774 and 1860 from Lindert and Williamson (2011). Period 1929-50 from Goldsmith et al (1954, fn. 4, p. 7). Gini
after 1967 from US Census Bureau (2010). The US concept is gross household income across households.
12
Figure 4. Inequality extraction ratios with b=0 and b=0.5
Inequality extraction ratio, Inequality extraction ratio, USA
England/UK 1688-2010 1774-2009
100 100
Inequality extraction ratio
Inequality extraction ratio
90 90
80 With s=300 and b=0.5
80 With s=300 and b=0.5
70 70
With s=300 and b=0
60
60
50
50 With s=300 and b=0
40
40
30
30
20
20 10
10 0
0 1750 1800 1850 1900 1950 2000 2050
1600 1700 1800 1900 2000 2100
Year
Year
Source: See notes to Table 1.
13
Figure 5. UK and US historical inequality extraction ratios
(elasticity of the social minimum with respect to mean income = 0.5)
90
80
England/UK
70
USA
60
50
40
30
20
10
0
1650 1700 1750 1800 1850 1900 1950 2000 2050
Years
Source: See notes to Table 1.
Figure 6. Inequality possibility frontier (left) and the inequality extraction ratios (right panel)
for China and Russia
100
100
IPF
70
70
Inequality extraction ratio
60
60
China
Gini
50
50
China 2011
2011 1990
2009
40
40
2009
1990
30
30
Russia Russia
1988
20
20
1988
0
0
300 10002000 5000 20000 300 1000 2000 5000 20000
GDI per capita GDI per capita
14
Another way to look at the behavior of the inequality extraction ratio with a
changing social minimum is to take the current actual social minima, find out what they
implicitly imply about b, and calculate the inequality extraction ratio based on such â€œrealâ€?
data (that is, without a prior assumption of a given elasticity). Take the United States in
2012. Its GDP per capita was $43,000 and the federal poverty line for a four-member
household was $23,050. On a per capita basis, this gives a poverty line of $5,762 per year.
Assuming as before a physiological subsistence of $300 per year, and plugging these actual
values into (6), enables us to calculate the implicit elasticity b. It works out as 0.59. Figure 7
shows the inequality extraction ratio for the United States over the period 1965-2012 using
this implicit observed elasticity of the social minimum with respect to real income. The
pattern of the increase is the same as with b=0.5, but the level of the extraction ratio is
higher. In 2012, it is about 3 points higher: 51 percent instead of 48 percent. Of course, in
either case, the rise of the extraction ratio during the past 45 years was substantial.
Figure 7. Inequality extraction ratio in the United States, 1967-2012 (calculated using the
observed elasticity between the social minimum and GDP per capita, b=0.59)
0.52
0.50
Inequality extraction ratio
0.48
0.46
0.44
0.42
0.40
0.38
1967 1972 1977 1982 1987 1992 1997 2002 2007 2012
Year
15
3. The Inequality Possibility Frontier with measures other than Gini
So far the analysis has been conducted exclusively in terms of the Gini coefficient. It
is around the Gini coefficient that we have created the inequality possibility frontier and the
inequality extraction ratio, In other words, we had assumed throughout that inequality =
Gini. But that of course is not true. Inequality can be measured by many other measures.
The question can then be asked: Would our results remain if instead of Gini we used
another measure of inequality? That implies re-expressing the entire framework (i.e.,
deriving IPF and IER) in terms of other inequality measures. We do this here for three
measures: the two Theil indices, and the standard deviation of logs of incomes.
The first step is the derivation of the maxima for each of the measures and for
different Î±â€™s, under the assumption that the subsistence is fixed in absolute amounts.
Annex 1 shows the derivations of the inequality possibility frontier for Theil (0), Theil (1) or
Theilâ€™s entropy measure, and the standard deviation of log of incomes. 11 Table 2 shows the
formulas, the range of each measure, and the final expression for the inequality possibility
frontier. For both Theil (0) and the standard deviation of logs, the maximum feasible
inequality is equal to Î±, that is, is directly proportional to the mean income (expressed in
multiples of subsistence). For Theil (1), there is no upper bound to the maximum and hence
the Inequality Possibility Frontier cannot be defined. 12
Figure 8 shows the Inequality Possibility Frontiers and the calculated Gini and Theil
(0) coefficients for the same sample of pre-industrial economies. Two things stand out. First,
results with Theil generally show colonies (full dots) at a greater distance from the
Inequality Possibility Frontier than when we use Gini. The inequality extraction ratio,
calculated across nine colonies in the sample is 90 percent with Gini and 55 percent with
Theil. In other words, within the Theil framework, it would seem that the elites in colonies,
while exploitative, have not nearly exhausted the entire surplus as implied by the Gini
framework. Part of the reason may lie in the difference in sensitivities to various parts of
11
Note that this measure of inequality (standard deviation of logs of incomes) is also a key functional
parameter if incomes are lognormally distributed.
12
This is an interesting reversal: the range of Theil (0) coefficient is [0, âˆž) but the maximum Theil (0) and
hence the IPF are well defined. On the other hand, the range of Theil (1) is [0, ln n] where is n=number of
observations, but the maximum Theil (1) is not bounded from above and hence the IPF cannot be defined.
16
the income distribution exhibited by the two measures. As is well-known, Theil is much
more sensitive to extreme values, while Gini is most sensitive to the values around the
mode. Since our data for pre-industrial economies are calculated from the social tables with
a limited number of social classes, the top of the distribution is truncated, or more exactly
â€œsqueezedâ€?. The top is represented by the mean income of the richest class, not by the
individual incomes of the richest individuals. Hence, Theil may tend to show much lower
values than if we had more finely-grained (individual-level) data. The gap between the
measured and maximum Theil may thus be overestimated.
Second, for high values of Î± the distance of the calculated (actual) Theils from the
Inequality Possibility Frontier is much greater than the analogous distance of the Gini. The
reason is that the IPF within the Theil framework increases linearly in logs while within the
Gini framework IPF is concave. Thus, for example, for countries with Î±>4, the average IER
using Gini is 66 percent; using Theil, it is less than half (30 percent). 13 But while the levels of
IER calculated with Gini and with Theil differ, the correlation between the two IER measures
is quite high: both the linear and rank correlations are 0.89.
In conclusion, Gini and Theil seem to rank countries according to their inequality
extraction ratios very much alike: more â€œexploitativeâ€? countries will be deemed such by
both approaches. But the implied levels of IER are significantly lower with Theil than with
Gini. This, in turn, has implications on our view regarding how close to the inequality
frontier were different societies. It also seems that, given the nature of social tables that are
used to estimate pre-industrial inequalities, Gini would be more likely to capture inequality
well, and in this case at least, it would be reasonable to prefer it to Theil.
13
There are 7 countries with Î±>4.
17
Table 2. Definitions of inequality measures and inequality possibility frontier for
Theil (0), Theil (1) and standard deviation of logs
Theil (0) or mean log Theil (1) or entropy Standard deviation of
deviation index logs
Formula for 1 1/ n 1 yi yi
inequality T (0) = âˆ‘
n i
ln
yi / mn
T (1) = âˆ‘ ln
n i m m
1
í µí¼Ž = ï¿½ ï¿½(í µí±™í µí±›í µí±¦ âˆ’ í µí±™í µí±›í µí±š)2
measure í µí±›
í µí±–
Range of the 0 to infinity 0 to ln n 0 to infinity
measure
Inequality ln Î± Infinity ln Î±
possibility
frontier (with
s=given)
Figure 8. Actual (measured) inequality and inequality possibility frontier within Gini and
Theil (0) frameworks
100
80
200
Chile
Nueva EspaÃ±a
Hol1732
60
150
Maghreb Hol1561
France Netherlands Engl1867
Theil
Old Castiille
Gini
Eng1801
USA1860
India-British
India-Moghul
Siam
Kenya Florence
USA1774 Eng1759
Eng1688
Brazil
Peru
Byzantium
40
Java1880
Roman Japan
Empire
100
Eng1290
Bihar
Kenya Java1924
Nueva EspaÃ±a
Kingdom of Naples Chile
Hol1732
China Maghreb
Engl1867
20
Netherlands
50
France
Old Castiille Eng1801
India-British
Kenya
Byzantium Siam USA1774
Roman FlorenceEng1688
Peru Empire Eng1759
India-Moghul
Kenya Java1880
Brazil
China
Eng1290 of
BiharKingdom Naples
Java1924
0
0
1000 2000 3000 4000 1000 2000 3000 4000
GDI per capita in 1990 PPP dollars GDI per capita in 1990 PPP dollars
Note: Augmented sample of pre-industrial economies from Milanovic, Lindert and Williamson (2010).
18
Another way to check whether IERs calculated from Gini and Theil (0) behave
similarly is to use the regression results from Milanovic, Lindert and Williamson (2007)
derived within Gini-based framework and check if they â€œsurviveâ€? when we use IERs based on
Theil. MLW (2007) find, on the sample of 28 pre-industrial societies, that two variables are
strongly associated with the level of inequality extraction: being a colony increases the IER
by some 25 Gini points; countries that are more densely populated are associated with
lower IER (an increase of 10 people per km2 reduces IER by 1.9 points). 14 As shown in Table
3, both variables remain highly statistically significant when the IER is calculated within the
Theil framework. 15 The absolute values of the coefficients are slightly different because the
values of Theil indexes and Theil-based IERs differ from those of Gini and Gini-based IERs.
But both approaches yield strikingly similar results. In other words, and this is very
important, IER results (in this case at least) do not depend on whether we situate ourselves
within the Gini or Theil framework.
Table 3. Explaining the Inequality Extraction Ratio:
dependent variable = measured inequality (Gini or Theil) / maximum feasible inequality
(Gini or Theil)
Gini-framework Theil-framework
GDP per capita (in ln) 128.22 151.26
(0.34) (0.49)
Squared (ln) GDP per capita -9.81 -11.46
(0.32) (0.48)
Percent of urban population 0.31 0.21
(0.24) (0.65)
Population density per km2 -0.19** -0.20**
(0.000) (0.001)
Colony dummy 26.61** 24.18**
(0.000) (0.002)
Constant -344.46** -455.26
(0.46) (0.54)
Adjusted R2 (F) 0.67 (12.1) 0.53 (6.6)
No. of observations 28 28
Note: ** Indicates statistical significance at less than 1%. p-values between brackets.
14
The finding regarding the role of population density is probably the most interesting because it lends itself to
several interpretations (see MLW, 2007). The effect of colonies is rather expected.
15
In both cases, IERs are calculated under the assumption that the subsistence minimum is fixed in real terms
at $PPP 300 (in 1990 international dollars).
19
4. A contemporary application: Explaining conflict
One of the ways to look at the contemporary relevance of the inequality extraction
ratio is to study its potential role in one of the areas where, on an a priori grounds, it should
matter, namely in the analysis of civil war and within-national conflict. As is well known,
simple interpersonal inequality reflected in a Gini coefficient is seldom found to be a
statistically significant determinant of conflict (see Collier and Hoeffler 2004; Collier,
Hoeffler and Rohner 2008, p. 16; and review in Sambanis 2004) . This leads to a somewhat
bizarre and counter-intuitive conclusion that inequality is not associated with conflict. More
recently, other types of inequalities, most notably horizontal, that is, inequality in average
incomes between groups, have gained prominence (Stewart 2000; Ã˜stby 2008; Cederman,
Weidmann and Gleditsch, 2011). While in principle such inequality may be thought related
to conflicts in multi-ethnic or multi-religious societies, it cannot capture inequality in
societies that are homogeneous along these dimensions. For example, the civil war waged
by the Khmer Rouge in Cambodia was dominantly politically-driven, not ethnic. So were the
Great Leap Forward, the first civil war in Angola between the government and UNITA and a
number of conflicts in Argentina and Colombia.
The inequality extraction ratio, by capturing how close to the frontier is actual
inequality, conveys the information about the relative â€œrapaciousnessâ€? of the elite and
combines in its formulation two aspects that are often found important for the explanation
of civil conflict: the average level of development of a country (its GDP per capita) and its
income distribution. To check the role of IER, we use a very detailed database on civil war
created by Nicholas Sambanis (Sambanis and Schulhofer-Wohl, 2009). The database covers
151 conflicts in 70 countries over the period 1945-2002. This is probably the most complete
data base of civil conflict: it treats as distinct the conflicts that might have overlapped in a
given country over the same time period but were motivated by different reasons and had
different actors; it includes the data on the duration of each conflict and its estimated
casualties. Here we consider the role of IER in â€œexplainingâ€? the number of years of conflict
per country and the overall casualty rate in the period 1960-2002. We exclude the pre-
1960s conflicts that were often driven by the decolonization movement and whose
determinants differed from those of the post-1960s conflicts.
20
The results are shown in Table 4. The regression is run across countries with the
dependent variable summing or averaging the conflict outcomes for the period 1960-2002.
The years of war variable shows the total number of years of civil war (with a year coded as
being a â€œcivil war yearâ€? even if conflict lasted only one month). The variable has,
unsurprisingly, a mode at 0 with 116 countries (out of the sample of 185 countries) not
experiencing civil conflict. The maximum value is 41 years for the Philippines and Colombia.
The other dependent variable, the casualty rate, is the ratio between total estimated
casualties from civil war(s) divided by the average population size during the 1960-2002
period. The casualty rate is obviously 0 for countries without civil wars, and its maximum
value is 23 percent in Cambodia followed by 8.4 percent in Angola and 8 percent in
Afghanistan. Even when we exclude countries with a zero casualty rate, the density
function is heavily skewed to the right and is strongly â€œbunchedâ€? around very small (less
than 1 percent) values (see Figure 9).
The distribution of the inequality extraction ratio in countries with no civil conflict
and those with at least one conflict is shown in Figure 10. The latter distribution is shifted to
the right, with a higher mean and median. In countries without conflict, the mean extraction
ratio is 41.3 percent and the median 36.9 percent; in countries with civil conflict, the mean
is 50.6 percent and the median 48.1 percent. 16 Equality of the means and medians is
rejected at less than 1% level. So is the equality of the two distributions by the Smirnov-
Kolmogorov test.
16 2
The standard deviations however are very close: 14.4 and 14.8 percent. Ï‡ test accepts their equality.
21
Figure 9. Distribution of the casualty rate from civil conflict over the period 1960-2002
Density function of casualty rate
.6
.4
Density
.2
0
0 5 10 15 20 25
Casualty rate
kernel = epanechnikov, bandwidth = 0.1385
Note: Casualty rate = total number of dead over 1960-2002 divided by the average population over
the same period (in percent).
Figure 10. Distribution of the inequality extraction ratio in countries
with no civil conflict and in countries with a positive number of civil conflicts
no conflict
.03
conflict
.02
density
.01
0
20 40 60 80 100
extraction ratio in %
22
As shown in Table 4, the introduction of IER to replace both GDP per capita and Gini
does not reduce the explanatory power of the models. 17 The inequality extraction ratio is
positively related to the civil war variables, and for the number of years of civil war, it is
statistically highly significant. If we compare IER with Gini only, in both instances, IER
outperforms Gini. While in the explanation of the casualty rate, the IER is not statistically
significant, its sign is positive while Gini is marginally negative. For the duration of conflict,
the difference is small: both are significant, but IER is significant at a lower p-level. In
interpreting the role of IER, we conclude that a ten percentage point increase in IER is
associated with, on average, an additional 0.2 years (2.4 months) of civil war, and with 0.12
percent increase in the overall casualty rate. Finally, we note that, as expected, ethno-
linguistic fractionalization is strongly correlated with both duration of civil wars and casualty
rates, while the effect of democracy is rather ambiguous: it is negatively associated with
casualty rates but positively with years of conflict.
Table 4. Regressing civil war related variables
Number of years of civil war Overall casualty rate (ln)
GDP per capita (ln) -0.665** -0.322
(0.000) (0.073)
Gini (in %) 0.010* -0.0001
(0.013) (0.994)
Inequality extraction 0.020** 0.012
ratio (in %) (0.000) (0.192)
Democracy (Polity 0.148** 0.027* -0.772 -0.133**
measure) (0.000) (0.036) (0.176) (0.002)
Ethno-linguistic 3.661** 3.398** 8.193** 8.262**
fractionalization (0.000) (0.000) (0.000) (0.000)
Constant 5.595** 0.180 -0.794 -3.822**
(0.000) (0.228) (0.633) (0.000)
R2 or pseudo R2 0.32 0.25 0.38 0.38
Number of 143 143 143 143
observations
Note: For number of years of civil war we use Poisson regressions. When transforming the casualty
rate into logs, 0 casualty rate is treated as 0.01. IER is expressed in percentage points (G/G* times 100); so is
Gini. Democracy is measured by the democracy variable from PolityIV database. It ranges from 0 (absence of
democracy) to 10 (full democracy). All explanatory variables are 1960-2002 country averages. ** (*) Indicates
statistical significance at less than 1% (5%). p-values between brackets.
17
Both Gini and the IER are expressed in percentages which facilitates the comparison between the two.
23
5. Summary and conclusions
The paper had two key objectives: to extend the methodological work done on the
inequality possibility frontier by allowing for a flexible social minimum that changes with the
increase in the average income of a society, and to derive the inequality possibility frontier
and the inequality extraction ratio for other inequality measures than Gini.
The inclusion of an income-flexible social minimum makes the surplus, at any
average income level, less than if the subsistence is entirely physiologically determined. The
IPF thus shifts downward, and the IER becomes greater. The difference between the two
IERs is particularly marked in the past when the average incomes were much lower. Thus,
over the 18th and 19th centuries, the inequality extraction ratios in the UK and the United
States, calculated using a social minimum that increases modestly with the average income
(b=0.5), in percent, were in the 80â€™s. They were much lower, in the 60â€™s, when calculated
with a simple physiological minimum. The political implication of the finding is that, once
the elite had to concede an increase in living standards of the poor as the economy got
richer, it was quite efficient or ruthless in maximizing inequality to a very high degree. This is
however different from the present-day situation. Using the observed elasticity of the social
minimum with respect to the mean income in the United States of 0.59, the IER, despite its
recent increase, turns out to be just above 50 percent.
The use of other inequality measures, like Theil, in the IPF framework is both feasible
(although Theilâ€™s entropy measure cannot be used since its maximum is not bounded from
above) and consistent with the results obtained within the Gini framework. We thus find
that using either Theil (0) or Gini does not make any difference in our conclusion that,
among pre-industrial economies, colonies were significantly more â€œexploitativeâ€? while
densely populated countries were significantly less. This finding has political implications
too. If more densely populated countries were less unequal, was it because they somehow
escaped the Malthusian trap, and then the populace by its sheer multitude presented
enough of a threat to those in power to deter them from a more exploitative behavior? It is
a question that needs further research, perhaps leading to results that may prove important
for explaining countriesâ€™ historical inequality and growth trajectories.
24
The usefulness of the IPF approach was also illustrated on the contemporary
example of factors associated with civil conflict. We find that IER can successfully replace
GDP per capita and Gini in such explanations without reducing the explanatory power of the
models. In other words, instead of an elusive role of the Gini coefficient in explaining civil
conflict, we argue that the extraction ratio brings inequality (albeit formulated somewhat
differently) back to the center stage.
For the IPF framework to prove valuable, further work will have to assess its
empirical relevance. We need to know much more about whether the inequality extraction
ratio can be usefully deployed to answer historical questions and to address the role of
inequality today, particularly in poorer societies, where the Inequality Possibility Frontier is
more binding.
25
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26
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27
Annex 1. Derivation of the Inequality Possibility Frontier for three other measures of inequality
A. Derivation of the maximum Theil (Theil 0, or mean log deviation) for a given Î±
1 1/ n
(1) T * (0) = âˆ‘
n i
ln
yi / mn
Where n = total population, m = mean income, yi = income of i-th individual. There are two groups of
people: n(1-Îµ) people with y=s, another group of Îµ people with income yh
nm âˆ’ n(1 âˆ’ Îµ ) s nm âˆ’ ns + nÎµs m âˆ’ s + Îµs
(2) y h = = =
Îµn Îµn Îµ
Substitute (2) into (1)
n(1 âˆ’ Îµ ) 1 / n Îµn 1/ n mn mnÎµ
T* = ln + ln = (1 âˆ’ Îµ ) ln + Îµ ln
n s / mn n m âˆ’ s + Îµs sn n(m âˆ’ s + Îµs )
Îµ
mn
mÎµ Î± sÎµ Î±Îµ
T * = (1 âˆ’ Îµ ) ln Î± + Îµ ln = (1 âˆ’ Îµ ) ln Î± + Îµ ln = (1 âˆ’ Îµ ) ln Î± + Îµ ln
m âˆ’ s + Îµs Î± s âˆ’ s + Îµs Î± âˆ’1+ Îµ
Î±Îµ
limí µí¼€â†’0 í µí±‡ âˆ— = ln Î± + Îµ ln
Î± âˆ’1+ Îµ
The latter expression tends to 0 x -âˆž. Transform so that it can be solved by Lâ€™HÃ´pitalâ€™s rule
Î±Îµ Î± Î±Îµ Î± âˆ’1+ Îµ Î±A âˆ’ Î±Îµ A
ln { âˆ’ }
Î± âˆ’1+ Îµ = Î± âˆ’1+ Îµ [Î± âˆ’ 1 + Îµ )] Î±Îµ A2 Î±Îµ = âˆ’ ( A âˆ’ Îµ )Îµ
2
(3) =
1 âˆ’1 âˆ’1 A
Îµ Îµ 2
Îµ 2
where A = Î± âˆ’ 1 + Îµ
When Îµâ†’0, Aâ†’ Î±-1.
âˆ’ (Î± âˆ’ 1)Îµ
When Îµâ†’0, then the whole expression tends to = âˆ’Îµ = 0
(Î± âˆ’ 1)
Consequently, the maximum Theil (0) for a given alpha is T* = ln(Î±).
Note that when Î±=1, the maximum T*=0.
28
B. Derivation of the maximum Theil (Theil 1, or Theil entropy index) for a given Î±
1 yi yi
(1) T * (1) = âˆ‘ ln
n i m m
n(1-Îµ) people with y=s.
nÎµ people with y=yh. As before,
nm âˆ’ n(1 âˆ’ Îµ ) s nm âˆ’ ns + nÎµs m âˆ’ s + Îµs
(2) y h = = =
Îµn Îµn Îµ
Substitute (2) and the rest into (1)
n(1 âˆ’ Îµ ) s s nÎµ m âˆ’ s + Îµs m âˆ’ s + Îµs
T * (1) = ln + ln =
n m m n Îµm Îµm
1 1 m âˆ’ s + Îµs m âˆ’ s + Îµs 1 1 Î±s âˆ’ s + Îµs Î±s âˆ’ s + Îµs
= (1 âˆ’ Îµ ) ln +Îµ ln = (1 âˆ’ Îµ ) ln + Îµ ln =
Î± Î± Îµm Îµm Î± Î± ÎµÎ±s ÎµÎ±s
1 1 Î± âˆ’1+ Îµ Î± âˆ’1+ Îµ
= (1 âˆ’ Îµ ) ln + ln
Î± Î± Î± ÎµÎ±
lim T* when Îµâ†’0,
1 1 Î± âˆ’1 Î± âˆ’1 1 1 Î± âˆ’1
T * (1) = ln + ln = ln + ln âˆ?
Î± Î± Î± ÎµÎ± Î± Î± Î±
So T* (1) tends to infinity. Note also that the first term will be negative because 1/Î±<1, and thus ln
(1/Î±) < 0.
Consider several numerical examples.
Let Î±=2,
1 1 1 1
T * (1) = ln + ln = A+ B
2 2 2 2Îµ
When Îµ=0.01, then T* = A + 1.96. When Îµ is smaller (0.001), T* = A + 3.11, and if Îµ = 0.0001, then T*
= A + 4.26. So as Îµ decreases, T* diverges rather than converges. Thatâ€™s the problem.
Let Î±=5. With Îµ=0.01, T* = A + 3.51. When Îµ is smaller (0.001), T* = A + 5.35 etc. Again, it diverges.
Thus, the maximum feasible Theil (1) as a function of Î± diverges.
Let Î±=10. With Îµ=0.01, T* = A + 4.05; with Îµ=0.001, T*= A + 6.12 etc.
As can be seen in Table A1, the maximum feasible Theil (1) diverges as Îµ becomes smaller and tends
toward 0. Thus, the inequality possibility frontier will be âˆž.
29
Table A1. Maximum T*(1) for different Î± and Îµ
Î±\Îµ 1/100 1/1000 1/10000
2 1.96 3.11 4.26
5 3.51 5.35 7.19
10 4.05 6.12 8.19
C. Derivation of the maximum standard deviation of logs for a given Î±
1 1
í µí¼Ž âˆ—= ï¿½ ï¿½(í µí±™í µí±›í µí±¦í µí±– âˆ’ í µí±™í µí±›í µí¼‡)2 = ( ï¿½(í µí±™í µí±›í µí±¦í µí±– âˆ’ í µí±™í µí±›í µí¼‡)2 )1/2
í µí±› í µí±›
We know as before that the income of the elite is:
nm âˆ’ n(1 âˆ’ Îµ ) s nm âˆ’ ns + nÎµs m âˆ’ s + Îµs
yh = = =
Îµn Îµn Îµ
1/2
1 1
í µí¼Ž âˆ—= ï¿½ í µí±›(1 âˆ’ í µí¼€ ) (ln í µí± âˆ’ í µí±™í µí±›í µí¼‡)2 + í µí¼€í µí±›(í µí±™í µí±›í µí±¦â„Ž âˆ’ í µí±™í µí±›í µí¼‡)2 ï¿½
í µí±› í µí±›
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2
2
í µí¼‡ âˆ’ í µí± + í µí¼€í µí±
í µí¼Ž âˆ—= ï¿½(1 âˆ’ í µí¼€ ) (ln í µí± âˆ’ í µí±™í µí±›í µí± âˆ’ í µí±™í µí±›í µí»¼ ) + í µí¼€ ï¿½í µí±™í µí±› âˆ’ í µí±™í µí±›í µí± âˆ’ í µí±™í µí±›í µí»¼ï¿½ ï¿½
í µí¼€
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í µí¼Ž âˆ—= ï¿½(1 âˆ’ í µí¼€ ) ( âˆ’í µí±™í µí±›í µí»¼ )2 + í µí¼€ [ln(í µí¼‡ âˆ’ í µí± + í µí¼€í µí± ) âˆ’ í µí±™í µí±›í µí¼€ âˆ’ í µí±™í µí±›í µí± âˆ’ í µí±™í µí±›í µí»¼ ]2 ï¿½
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í µí¼Ž âˆ—= ï¿½(1 âˆ’ í µí¼€ ) ( í µí±™í µí±›í µí»¼ )2 + í µí¼€ [ln(í µí¼‡ âˆ’ í µí± + í µí¼€í µí± ) âˆ’ ln(í µí¼€í µí± í µí»¼)]2 ï¿½
when Îµâ†’0, the second term in the previous expression tends to 0[ln(Î¼-s)+âˆž]2 = 0 x âˆž.
Thus we have to find the limit of
(í µí¼€ (ln(í µí¼‡ âˆ’ í µí± + í µí¼€í µí± ) âˆ’ ln(í µí¼€í µí± í µí»¼))2 )
when Îµâ†’0.
í µí¼€ 1 í µí±™í µí±›í µí°´ âˆ’ í µí±™í µí±›í µí¼€í µí± í µí»¼ âˆž
= = =
1 1 1 1 1
âˆ’2(í µí±™í µí±›í µí°´ âˆ’ í µí±™í µí±›í µí¼€í µí± í µí»¼ )âˆ’1 ( í µí°´â€² âˆ’ âˆ’2(í µí°´ í µí°´â€² âˆ’ ) âˆž
(í µí±™í µí±›í µí°´ âˆ’ í µí±™í µí±›í µí¼€í µí± í µí»¼)2 í µí°´ í µí¼€í µí± í µí»¼ í µí± í µí»¼) í µí¼€
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where A = Î¼-s+Îµs and lnA = constant if Îµâ†’âˆž, also A'/A constant if Îµâ†’âˆž. Also A'=s.
Continue with L'HÃ´spital's rule when Îµâ†’0
í µí°´â€² í µí± í µí»¼ 1
í µí±™í µí±›í µí°´ âˆ’ í µí±™í µí±›í µí¼€í µí± í µí»¼ âˆ’ í µí¼€í µí± í µí»¼ (í µí± í µí¼€ âˆ’ í µí°´) í µí°´í µí¼€(í µí± í µí¼€ âˆ’ í µí°´) 0
= í µí°´ = í µí°´í µí¼€ = = =0
1 â€² 1 1 2 1 2 2 í µí¼€ 2 âˆ’ í µí°´2 ) 2(í µí± í µí¼€ âˆ’ í µí°´ ) âˆ’2í µí°´2
2 2 2
âˆ’2(í µí°´ í µí°´ âˆ’ í µí¼€ ) âˆ’2(âˆ’ 2 í µí± + 2 ) (í µí±
í µí°´ í µí¼€ í µí°´2 í µí¼€ 2
Therefore,
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lim(í µí¼€ â†’ âˆž) í µí¼Ž âˆ—= ï¿½(1 âˆ’ í µí¼€ ) ( í µí±™í µí±›í µí»¼ )2 ï¿½ = ln í µí»¼.
Thus, the maximum value taken by the standard deviation of logs when average income = Î±, is lnÎ±.
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