by

Fred E. Foldvary

Department
of Economics, Santa Clara University, CA 95053

ffoldvary@scu.edu

JEL Code: C10

**Abstract**

The Lorenz curve and Gini coefficient are typically used to measure inequality. A different way to measure inequality is introduced here: I = CN, the product of concentration and number of units. The resultant index can be interpreted with reference to an inequality base where one unit owns all and the rest nothing. This inequality index also integrates the measurement of inequality, concentration, and diversification into one system, where diversification is measured as the inverse of concentration. I = CN accommodates various measures of concentration, including the Herfindahl-Hirschman and Tideman-Hall indexes. The Tideman-Hall concentration index also provides indexes of concentration, diversification and inequality as functions of Gini. As one application, the inequality index can be used to provide an index of economic development.

**1. The meaning of inequality**

Inequality can be ordinal or cardinal. An ordinal ranking of distributions in order of inequality implies some cardinal method of measuring or judging each distribution in relation to the others. A cardinal ranking can be relational, defined with reference to other variables, or monadic, its magnitude being defined without reference to other variables (Wright, 1987). This paper is concerned with the measurement of monadic, cardinal inequality.

Inequality is the degree to which the units of a distribution have shares of some attribute which are unequal in quantity. This meaning of inequality is completely general for any type of distribution. In order to measure inequality, one needs to determine the meaning of the degree of being unequal.

Let
S be a set of N elements (also called units or members). There is some
attribute A (such as income) of which the elements have relative shares a_{i},
each a_{i} being a fraction of A. A distribution is designated by (a_{1},
a_{2}, ... , a_{N}) and by (j(a_{i})), which designates
j units of share a_{i} each. The degree of inequality among the a_{i}
is a function of the concentration of A among a_{i} and of the number
of elements, N. The greater the concentration of a_{i}, given some N,
the more unequal is the distribution. The greater N is, given some
concentration, the more unequal the distribution.

Berrebi and Silber (1985) note that inequality indices can be expressed as a income-weighted (more generally, attribute-weighted) sum of individual "deprivation" coefficients, the differences among the indices depending on the way the deprivation is defined. It has been recognized since Dalton (1920) that a measurement of income or wealth inequality implies some concept of social welfare.

It would seem reasonable to posit that, given N, if the concentration doubles, then inequality is also doubled, hence inequality is linearly proportionate to concentration. It would also seem reasonable to posit that, given some concentration C, if N doubles, then inequality also doubles, so that inequality is linearly proportionate to the population size. Putting both variables together, inequality is the product of concentration and number:

(1)

The concentration C of a distribution is often measured by the Herfindahl-Hirschman (HH) index (an alternative, the Tideman-Hall index (12), is discussed below),

(2)

Applying (1), the inequality index specific to the Herfindahl-Hirschman concentration index would then be the Herfindahl-Hirshman-Foldvary inequality index,

(3)

The
HHF index has the desired property that a transfer of a share from a richer to
a poorer member reduces the value of I_{H}, thus reducing inequality.

Hall and Tideman (1967, p. 165)
state that with HH, "the relative sizes of firms are more important than
the absolute number of firms in determining concentration in an industry."
The index weighs each firm by its relative share. It should be noted, however,
that given some accepted inequality index I, that N is as important in
computing C as I, so the problem, if any, in C_{H} would lie in how it
takes into account the inequality of a distribution, the weight increasing
perhaps too much with size, rather than any neglect of the number of units.

As
an example of the effect of changing concentration, consider the distribution
(.5, .5), for which C_{H} = .5. If the distribution is changed to (.4,
.6), C_{H} = .52. The distribution is less equal and has a higher
concentration index. For (.3, .7), (.2, .8), and (.1, .9), C_{H}= .58,
.68, and .82. It is evident that inequality increases with concentration. At
the extreme, the distribution of "monopossession" (1, 0) has C_{H}
= 1, twice the concentration of (.5, .5). Applying equation (1), the inequality
index I_{H} would equal 1 for (.5, .5) and 2 for (1, 0). I_{H}
always equals 1 for complete equality and N for complete inequality among N
elements. The doubling of I_{H} has an intuitive interpretation: as the
share of the first unit doubles from .5 to 1, the inequality also doubles.

For
an example of the effect of population size, again consider the
"monopossessive" distribution (1, 0), where N = 2 and one member has
all of A (i.e. monopolizes income or wealth) and the other has nothing. If the
number of elements increases to 4 with the distribution (1, 0, 0, 0), C_{H}
is unchanged at 1. However, applying (1), I_{H} doubles from 2 to 4.
The second distribution is more unequal, though equally concentrated, because
in (1, 0), 1/2 the population owns all, while in (1, 0, 0, 0), 1/4 of the
population owns all. With complete equality, each would own .25, hence when one
member owns all, he has four times the equal share, whereas with (.5, .5), he
only has twice the equal share. The inequality index I_{H} is thus the
multiple over pure equality when 1 member owns all of A.

Let
I_{F} designate the class of inequality indexes based on (1), with the
concentration index having a range (1/N, 1), 1 being the case of
monopossession. For exactly equal distributions, I_{F} = 1. I_{F}
for any distribution can be interpreted as the equivalent in inequality as that
of a monopossessive distribution with N = I_{F}. For example, if I_{F}
= 3, then the inequality is the same as that with N = 3 and one member owning
everything. The index I_{F} thus has a meaning relative to the
benchmark of a monopossessive-equivalent distribution. This is consistent with
the criterion set by Dalton (1920, p. 349), who stated that "the
inequality of any given distribution may conveniently be defined as the ratio
of the total economic welfare attainable under an equal distribution to the
total economic welfare attained under the given distribution. This ratio is
equal to unity for an equal distribution and is greater than unity for all
unequal distributions."

Equation (1) thus encompasses both the concentration and population aspects of inequality by multiplying them together, since each by itself increases inequality proportionately. It not only measures inequality but implements the very meaning of inequality in its calculation.

** **

**2. Putting the Gini back in the
bottle**

A common way that inequality is measured is with a "size distribution," in which X percent of the population has Y percent of the attribute such as income. The population and attribute are typically divided into quintiles or deciles. One can also measure the inequality by choosing a ratio between two points in the distribution, such as the percentage of income received by the bottom 10% of the population by the percentage received by the top 10%. But these measurements only provide information about points in the distribution rather than reflecting the entire distribution.

The Lorenz curve, devised by the American statistician Conrad Lorenz in 1905, is widely used to show the relationship between population and shares of income. The cumulative percentage of population is drawn on the horizontal axis, and the cumulative share of income received is plotted on the vertical axis. The diagonal line represents strict equality, with inequality reflected by the amount by which the Lorenz curve deviates from it. Although useful as a visual representation of the inequality of a distribution, it is useful also to have a number that captures the degree of inequality, the Gini coefficient.

Formulated by the Italian statistician Corrado Gini in 1912, the coefficient is the ratio of the area between a Lorenz curve and the 45-degree line to the area of the triangle below the 45-degree line. Its formula is

(4)

where a_{i} is the amount
owned by each member in decreasing order of size and ~ is the mean of the a_{i}. The Gini
index is thus a weighted sum of the shares, with the weights determined by rank
order position. As noted by Maasoumi (1995), Gini does not provide for
aggregation consistency or full additive decomposability. Also, Gini places
more weight to transfers affecting the middle of a distribution than at the
tails. However, a function of Gini, such as (14) below, corrects this latter
feature.

As half the relative mean difference,

(5)

G has a maximum value of unity with
extreme inequality and a minimum of zero at pure equality. However, for
distributions with a small N, G is less than one. For example, with (1, 0), G =
.5, representing half the triangle. Since the axes represents relative
proportions of population, the distribution (.5, .5, 0, 0) also has G = .5, and
I_{H} index for (.5, .5, 0, 0) is 2 as it is with (1, 0), so I_{H}
and G agree with respect to relative inequality where the population increases
but there is equality among the "haves".

However,
G does not increase proportionate to number when concentration is held
constant. For the distribution (1, 0, 0, 0), G = .75, an increase of 50% from
(1, 0), in contrast with the doubling of I_{H} and a decrease of (1-G)
of 50%. The monopossessive distribution (1, 7(0)) yields G = .875 and I_{H}
= 8, an increase of 1/6 over (1, 0, 0, 0), while (1-G) again decreases 50%. As
N doubles, G decreases in (1-G) by 50% while I_{H} doubles. At an
extreme, a change from (1, 999(0)) to (1, 99999(0)) yields a change in I_{H}
from 1000 to 100,000 while G changes from .999 to .99999, with (1-G) reduced
one thousandfold from .001 to .000001. Since G is varying directly with (1-G)
rather than with N, G measures little change in inequality if a large N
increases while C is constant, while I_{F} increases proportional to
population. With inequality defined as a function of both concentration and
population, raw G does not fit (1).

With respect to changes in
concentration, the distribution (4(.25)) has C_{H} = .25, I_{H}
= 1, and G = 0. If we change it to (2(.5),2(0)), then C_{H} = .5, I_{H}
= 2, and G = .5. For (1, 3(0)), C_{H} = 1, I_{H} = 4, G = .75.
Here again, I_{H} has doubled with each doubling in concentration, but
G is decreasing (1-G) by half each time, hence increasing proportionately less
with increasing concentration, given some population size.

These traits of G are
overcome with an inverse-reverse function 1/(1-G) as in the Tideman-Hall-Foldvary
inequality index I_{T} presented below in (14), which yields I_{F}
characteristics for I = CN similar to those of I_{H}.

** **

**3. Inequality, Concentration, and
Diversification**

Equation (1) not only measures inequality but demonstrates the relationship between inequality, population size, and concentration. Since I = CN, we can also formulate C as

(7) C = I / N

so that concentration equals inequality divided by population. Given some N, the more unequal the distribution, the more concentrated, and given some I, the greater the population, the less concentrated the distribution. Also,

(8) N = I / C

with population size able to be calculated if the inequality and concentration of a distribution are compatibly measured.

Equation (7) provides a general measurement of concentration, given some measurement of inequality, for any type of distribution. The Herfindahl index and the Tideman-Hall index take both N and I into account with weighted sums of the units.

The diversification of a distribution of items, such as the products of a firm, reflects both the number of items and the relative amount of each item. The diversification of an economy reflects the number of industries and relative size of each, and the diversification of assets in a portfolio reflects the number and relative size of the assets. Diversification is thus similar to concentration in being a function of the number and inequality of items, except that as concentration increases, diversification decreases; diversification is an increasing function of number and decreasing function of inequality. Diversification (D) is thus the inverse of concentration:

(9) D = 1 / C

The index D is equal to N when the units are all equal, since

(10) D = N / I

diversification being defined and calculated as the number of items divided by their inequality.

The inequality index I = CN hence also provides a method for measuring diversification. Diversification, concentration, and inequality are thus measured by one common system, e.g. using the Herfindahl-Tideman-Hall index for C.

The
first three criteria specified by Hall and Tideman (1967) for concentration
also apply to inequality. The index I_{H}
satisfies the first Hall and Tideman criterion as an unambiguous
one-dimensional measure. It satisfies
their second criterion as being a function of all the elements of a distribution
and their relative shares. The third
criteria is that a change in relative shares changes the index, which is
satisfied. Since C_{H}
satisfies all their criteria for C, I using C_{H} satisfies those that
are relevant to inequality.

** **

**4.
Concentration based on Gini**

Suppose that one uses the Gini
coefficient G for inequality. At pure
equality, G = 0, and at pure inequality, G = 1. Then by (1), G = C_{G}N, and we have a Gini-based index
of concentration:

(11)
C_{G} = G / N

With
pure equality, C_{G} = 0 regardless of N, hence the application of Gini
to concentration provides for a deficient index of concentration. With less than pure equality, C_{G}
will vary with N but still be small for very low G. C_{G} varies from zero to unity, as does H, satisfying
the Hall and Tideman (1967) sixth criterion for the range of concentration.

For diversification, (10) using G
provides D_{G} = N / G, and when G is zero, the diversification is
infinite. Also, with a very high degree
of inequality, G is close to 1 and D_{G} almost equals N, so D_{G}
would increase with N even if one element monopolizes the distribution. Using I_{F}, a very high degree of
inequality would yield a high I, which would provide a low degree of diversification
unless N were correspondingly very high.
With G in the range of zero to one, and I_{F} ranging from one
to infinite, I_{F} provides a more general and meaningful basis for
computing diversification and concentration than D_{G}. I_{F} thus provides for greater
consistency in the measurement of C and D than does G. However, this deficiency in G and its
derivatives can be remedied with a function of G.

The
Tideman-Hall (TH) concentration index (Hall and Tideman, 1967, p. 166),
designated here by C_{T}, is

(12)

the
*i*th largest firm receiving weight i, thus weighing each share by its
rank rather than its relative share.
Like C_{H}, C_{T} = 1/N when the distribution consists
of N equal units, so I_{T} = 1 for an equal equality with C_{T}
as with C_{H}. For (.9, .1), C_{T}
= .83 versus .82 for C_{H}, and for (.6, .4), C_{T} = .56
versus C_{H} = .52, so both measures are fairly close, and C_{T}
can serve as well as if not better than C_{H} to provide a
Tideman-Hall-Foldvary index I_{T} of inequality:

(13)

With TH and
Gini both rank-weighted, I_{T} is a function of G:

(14)
I_{T} = 1 / (1-G)

Hence, with C = I / N,

(15) C_{T} = 1/((1-G)N)

I_{T} as a function of G,
with qualities similar to I_{H}, is in the class of indexes I_{F},
with a range of 1 for equality to N for maximum inequality. A diversification index using C_{T}
is

(16) D_{T} = 1/C_{T} =
N/I_{T} = N(1-G)

**5. Using other measures of inequality**

If
other measurements of concentration are used besides C_{H} and C_{T},
equations (1) and (9) may be used to create the corresponding inequality and
diversification indexes. Equations (7)
and (10) can also be applied to formulae for inequality, so long as inequality
increases with increasing values of the index.

For example, the standard deviation of logarithms measures inequality, taking differences from the mean:

(17)

The Theil (1967) entropy-related index is:

(18)

where a_{i} is the fraction of
attribute A as above, the entropy of a distribution being

(19)

which provides a measurement of
diversification.^{1}

An equal distribution yields I_{E}
= 0, while a^{0} monopossessive distribution, with complete
inequality, yields I_{E} = log N.
As Sen (1972) notes, the Theil index is rather arbitrary, but, as Shalit
and Yitzhaki (1984) observe, there is an interesting similarity between
inequality and decision-making under uncertainty.

Shalit (1985) notes that Gini's mean difference can be used as a measure of dispersion:

(20)

The relationship between diversification and inequality formulated in (10) provides an inverse relationship, with I = N / D a measure of inequality if diversification is independently measured.

Atkinson (1970) proposes a inequality/welfare measure that is invariant with respect to linear transformations, an "equally distributed equivalent income" that would give the same level of social welfare as the actual distribution:

(21)

The index proposed by Berrebi and Silber (1985) is:

(22)

where S_{j} is the share of
the j'th rank, S_{1} $ S_{2}
$ ... $ S_{n}.

These various inequality measures, along with the many others that have been proposed, yield different inequality rankings, and, applying (7), yield diverse measurements of concentration. The literature on measurements of inequality thus also provides ways of measuring concentration and diversification.

**6. Application to measuring economic
development**

Michael Todaro (1994) has noted that the concept of economic development has come to be defined as a function of inequality as well as of per-capita GDP. If two countries A and B have an equal per-capita GDP, but A has a much more unequal distribution of income, due to a larger less-developed sector, then it can be regarded as overall less developed than B. Hence, an increase in GDP in a small sector of an economy would not, by this criterion, count as much as a broader-based increase, even if the average increase were the same for both.

Todaro
(1994) proposes a "distributive share index" or a
"poverty-weighted index" as an alternative to the plain GDP or GNP,
and the U.N. Human Development Report calculates "human development
indicators." The inequality index
I_{F}, such as calculated in (3) or (13), can also be used to measure
the degree of economic development Y_{D} as a function of GDP,
population, and inequality:

(23)
Y_{D} = Y_{G} / NI

where Y_{G} is GDP. For a given per-capita GDP, the degree of
economic development declines with increasing inequality.

If
I_{T} is used,

(24) Y_{D} = (Y_{G} / N)(1-G)

As an example, in 1991 Mexico and
Brazil had per-capita incomes of $2870 and $2930, with Gini coefficients .50
and .57, respectively (Schnitzer, 1994, pp. 259, 262). Hence, using I_{T}, Y_{D}
for Mexico and Brazil are 1435 and 1260, Mexico having a higher development
index despite its lower GDP due to the greater inequality in Brazil.

**7. Conclusion**

The equations I = CN and D = 1/C provide an integrated method of computing indexes for inequality, concentration, and diversification, given some distribution with N items. If C is an independently computed variable, then its use to compute inequality places a constraint on C, since certain intuitive constraints on I need to be met. Given N, I should increase with increasing concentration, and given C, I should increase with increasing N. For equal-share distributions, increasing N should decrease C proportionately so as to leave I equal. The Herfindahl-Hirschman and Tideman-Hall indexes satisfy these criteria. If I is to be calculated independently and C is the dependent variable, then the Tideman-Hall-Foldvary inequality index as a function of the Gini coefficient (14) has the desirable qualities needed for I = CN. As functions of Gini, the Tideman-Hall-Foldvary indexes for concentration (15), inequality (14) and diversification (16) may have many useful applications.

An
inequality index I_{F} calculated from C_{H} or C_{T}
in (1) or from G in (14) has some more consistent properties in relationship to
N and C than straight Gini, so it merits empirical investigation as to its
usefulness in computing inequality in income and wealth as a perhaps superior
substitute for the raw Gini Coefficient. I don't disagree, however, with Hall
and Tideman's (1967) statement that no best measurement may exist and that the
measure should be suited to the use. The indexes presented here provide options
worthy of investigation.

1. See Maasoumi (1995) for a discussion of the Theil and Generalized Entropy indices and their applications.

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