by Rasche et al. (1980), cf. (2.17), and a proposal due to Ortega et al. (1991).

The resulting family is baptized the Pareto family; its members are listed in Table 2.1.

Sarabia, Castillo, and Slottje obtained excellent results when ¬tting this family to

Swedish and Brazilian data given in Shorrocks (1983) and concluded that the Pareto

distribution, which has the disadvantage not to ¬t the entire income range, does

much better when serving as a generator of parametric Lorenz curves.

The process may be repeated using other Lorenz curves as the generating function

L0 . Speci¬cally, for the model proposed by Chotikapanich (1993),

eku À 1

, 0 , u , 1,

L0 (u) ¼ k (2:18)

e À1

where k . 0, one obtains a family that is called the exponential family of Lorenz

curves by Sarabia, Castillo, and Slottje (2001), whereas for

L0 (u) ¼ ubuÀ1 , 0 , u , 1, (2:19)

where b . 0, a speci¬cation proposed by Gupta (1984), one obtains the Rao “Tam

(1987) curve as the L1 -type curve, namely

L1 (u) ¼ ua buÀ1 , 0 , u , 1: (2:20)

A further method for generating new parametric Lorenz curves from previously

considered ones has been suggested by Ogwang and Rao (2000) who employed

convex combinations”their “additive model””and weighted products”their

“multiplicative model””of two constituent Lorenz curves. Employing combinations

Table 2.1 Parametric Lorenz Curves: The Pareto Family

Lorenz Curve Gini Coef¬cient Source

1Àk

L0 (u) ¼ 1 À (1 À u)k

1þk

L1 (u) ¼ ua [1 À (1 À u)k ] 1 À 2[B(a þ 1, 1) Ortega et al. (1991)

À B(a þ 1, k þ 1)]

2

L2 (u) ¼ [1 À (1 À u)k ]g 1 À B(k À1 , g þ 1) Rasche et al. (1980)

k

P G(i À g)

1À2 1

L3 (u) ¼ ua [1 À (1 À u)k ]g Sarabia, Castillo, and

i¼0

G(i þ 1)G(Àg)

Slottje (1999)

‚B(a þ 1, ki þ 1)

29

2.1 SOME CONCEPTS FROM ECONOMICS

of the Ortega et al. (1991) and Chotikapanich (1993) as well as the Rao “Tam (1987)

and Chotikapanich (1993) models, they concluded that the additive models perform

distinctly better than either constituent model and moreover yield a satisfactory ¬t

over the entire range of income.

Among the many further proposals, we should mention the work of Maddala and

Singh (1977b) who suggested expressing the Lorenz curve as a sum of powers of u

and 1 À u. Holm (1993) proposed a maximum entropy approach using side

conditions on the Gini coef¬cient and the distance of the mean income from the

minimum income. For the resulting maximum entropy Lorenz curve, the ¬t is often

excellent. Ryu and Slottje (1996) considered nonparametric series estimators based

on Bernstein or exponential polynomials and Sarabia (1997) took one of the few

distributions parameterized in terms of their associated quantile function, the Tukey™s

lambda distribution, as the starting point.

For any given parametric Lorenz curve, it is natural to inquire about the form of

the implied p.d.f. of the income distribution. Not surprisingly, the resulting

expressions are often rather involved, yielding, for example, distributions with

bounded support”this being the case, for instance, for the Chotikapanich (1993)

model”or a severely restricted behavior in the upper tail. For example, the elliptical

model of Villasen and Arnold (1989) implies that f (x) $ xÀ3 , for large x,

˜or

irrespective of the parameters. This does however not diminish the usefulness

of these models for approximating the Lorenz curve. It is also interesting that a sub-

class of the Rasche et al. curve corresponds to a subclass of Lorenz curves implied

by the Singh “ Maddala income distribution (cf. Section 6.2). See Chotikapanich

(1994) for a discussion of the general form of the p.d.f. implied by the Rasche et al.

curves.

All the speci¬cations considered are usually estimated by a nonlinear

(generalized) least-squares procedure, possibly after a logarithmic transformation.

In a comprehensive study ¬tting 13 parametric Lorenz curves to 16 data sets

describing the disposable household incomes in the Federal Republic of Germany

for several nonconsecutive years between 1950 and 1988, Schader and Schmid

(1994) concluded that one- and two-parameter models are often inappropriate for

this purpose. (The investigation also include several curves obtained from parametric

models of the income density, namely, the lognormal, Singh “ Maddala, and Dagum

type I models.) Their criterion is a comparison with Gastwirth™s (1972) non-

parametric bounds for the Gini coef¬cient that are violated by a number of curves. In

particular, the Kakwani “Podder (1973) and Gupta (1984) models perform rather

poorly, whereas the Kakwani “ Podder (1976) model does very well.

A popular benchmark data set in this line of research consists of the deciles of the

income distributions of 19 countries derived from Jain (1975) and later published by

Shorrocks (1983).

2.1.3 Inequality Measures

Inequality measures are an immensely popular and favorite topic in the modern

statistical and econometric literature, especially among “progressive” researchers.

30 GENERAL PRINCIPLES

Oceans of ink and tons of computer software have been used to analyze this

somewhat controversial and touchy topic, and numerous books, theses, pamphlets,

technical and research reports, memoranda, etc. are devoted to this subject matter

(see, e.g., Chakravarty, 1990; Sen, 1997; or Cowell, 1995, 2000). It is not our aim to

analyze these works and we shall take it for granted that the reader is familiar with

the structure of the basic time-honored inequality measure, the Gini coef¬cient.

However, in the authors™ opinion the overemphasis”bordering on obsession”on

the Gini coef¬cient as the measure of income inequality that permeates the relevant

publications of research staffs and their consultants in the IMF and World Bank is an

unhealthy and possibly misleading development.

One of the numerous de¬nitions of the Gini index is given as twice the area

between the Lorenz curve and the “equality line”:

°1 °1

G¼2 [u À L(u)] du ¼ 1 À 2 L(u) du: (2:21)

0 0

Clearly, the Gini coef¬cient satis¬es the Lorenz order; in economic parlance: It is

“Lorenz consistent.” Alternative representations are too numerous to mention here;

see Giorgi (1990) for a partial bibliography with 385 mainly Italian sources. We do

however require a formula in terms of the expectations of order statistics

Ð1

[1 À F(x)]2 dx

E(X1:2 ) 0

G ¼1À ¼1À , (2:22)

E(X ) E(X )

which is presumably due to Arnold and Laguna (1977), at least in the non-Italian

literature. In the economics literature it has been independently rediscovered by

Dorfman (1979).

It should not come as a surprise that various generalizations of the Gini

coef¬cient have also been suggested. Kakwani (1980a), Donaldson and Weymark

(1980, 1983), and Yitzhaki (1983) proposed a one-parameter family of generalized

Gini indices by introducing different weighting functions for the area under the

Lorenz curve

°1

L(u)(1 À u)nÀ2 du,

Gn ¼ 1 À n(n À 1) (2:23)

0

where n . 1. Muliere and Scarsini (1989) observed that

E(X1:n )

:

Gn ¼ 1 À (2:24)

E(X )