b1 ¼ 2 Á CV: (5:40)

The mode is at b( p À 1), for p . 1, and at zero otherwise.

A basic property of gamma variables is their closure under addition: Suppose that

Xi $ Ga( pi b), i ¼ 1, 2, are independent. Then

X1 þ X2 $ Ga( p1 þ p2 , b): (5:41)

The mean excess function is not available in simple closed form; we have the

expansion

p À 1 ( p À 1)( p À 2)

þ O(xÀ3 ),

e(x) ¼ 1 þ (5:42)

þ 2

x x

162 GAMMA-TYPE SIZE DISTRIBUTIONS

which shows that e(x) eventually decreases for large x for p . 1 and increases

for p , 1.

Similarly, the hazard rate is

& 'À1

p À 1 ( p À 1)( p À 2)

þ O(xÀ3 )

r(x) ¼ 1 þ , (5:43)

þ

x2

x

which is an increasing function for large x for p . 1 and a decreasing function

for p , 1.

5.2.2 The Angle Process

In a series of papers in the sociological literature, Angle (1986a,b, 1990, 1992,

1993a, 2000) discussed a stochastic model whose long-term wealth distribution

appears to be well approximated by a gamma distribution.

His generating mechanism is motivated by the surplus theory of social

strati¬cation from sociology and anthropology (e.g., Lenski, 1966) that encompasses

two concepts of wealth: (1) subsistence wealth, which is the wealth necessary to

keep producers of wealth alive and to cover the long-term costs of production, and

(2) surplus wealth, which is the difference between subsistence wealth and total

wealth (net product). The central concept of this theory, inequality resulting from

contagious competition, is perhaps as old as the adage “the rich get richer, the poor

get poorer.”

We con¬ne ourselves here to the perhaps most elementary version of the surplus

theory that may be described by the following propositions:

1. When people are able to produce a surplus, some of it will become fugitive

and leave the possession of its producers.

2. Wealth confers, on those who possess it, the ability to extract wealth from

others. Each person™s ability to do this in a general competition for surplus

wealth depends on his or her own surplus wealth; speci¬cally, the rich tend to

take the surplus away from the poor.

Angle™s inequality process formalizes these propositions in what is known in

mathematical physics as an interacting particle system with binary interactions.

(Binary interactions are employed because no others are speci¬ed by the surplus

theory.) The process describes a competition between random pairs of individuals

for each other™s wealth in which the richer individual has a ¬xed probability p of

winning (:5 , p , 1). Also, the loser in a random encounter loses a ¬xed proportion

v of wealth (0 , v , 1). v and p are the parameters of the process.

The transition equations for the wealth of a random pair of individuals i, j are

Xi,t ¼ Xi,tÀ1 þ Dt vXj,tÀ1 À (1 À Dt )vXi,tÀ1 , (5:44)

Xj,t ¼ Xj,tÀ1 þ (1 À Dt )vXi,tÀ1 À Dt vXj,tÀ1 , (5:45)

163

5.2 GAMMA DISTRIBUTION

where Xi,t is i™s surplus wealth after an encounter with j, Xi,tÀ1 is i™s surplus wealth

before the encounter, and Dt is a sequence of identically distributed Bernoulli

variables with P(Dt ¼ 1) ¼ p.

Proposition 1 states only that surplus wealth will be fugitive in encounters

between members of a population and is implemented in (5.44) and (5.45).

Proposition 2 states that people with greater surplus wealth will tend to win

encounters. This can be implemented by specifying Dt to be

&

1 with probability p, if Xi,tÀ1 ! Xj,tÀ1 ,

Dt ¼ (5:46)

if Xi,tÀ1 , Xj,tÀ1 ,

0 with probability 1 À p,

with :5 , p , 1.

There are several generalizations of this basic mechanism: Angle (1990, 1992)

introduced coalitions among the wealth holders, whereas in 1999 he allowed for random

shares of lost wealth v. Angle found, by means of extensive simulation studies, that all

these variants of the inequality process generate income or wealth distributions that are

well approximated by gamma distributions, although a rigorous proof of this fact

appears to be unavailable at present. He further conjectured (1999) that the shape

parameter p of this approximating gamma distribution is related to the parameters of the

process as p ¼ (1 À v)=v. No doubt the Angle process deserves further scrutiny.

5.2.3 Characterizations

A characterization of the gamma distributions in terms of maximum entropy among

all distributions supported on [0, 1) is as follows: If both the arithmetic and

geometric means are prescribed, the maximum entropy p.d.f. is the gamma density

(Peterson and von Foerster, 1971; see also Kapur, 1989, pp. 56“ 57).

Another useful characterization occurs by the following property: If X1 and X2 are

independent positive random variables and

X1

X1 þ X2 and

X1 þ X2

are also independent, then each Xi is gamma with a common b but possibly different

p. This characterization is due to Lukacs (1965) and has been extended by Marsaglia

(1989) relaxing the positivity condition.

In Chapter 2 we saw that the coef¬cient of variation is, up to a monotonic

transformation, a member of the family of generalized entropy measures of

inequality. A characterization of the gamma distribution in terms of the (sample)

coef¬cient of variation should therefore be of special interest in our context. Hwang

and Hu (1999) showed that if X1 , . . . , Xn (n ! 3) are i.i.d. random variables

possessing a density, the independence of the sample mean X n and the sample

coef¬cient of variation CVn ¼ Sn =X n characterizes the gamma distribution. This

result can be re¬ned by replacing the sample standard deviation Sn in the numerator

of CVn with other measures of dispersion. Speci¬cally, we may use Gini™s mean

164 GAMMA-TYPE SIZE DISTRIBUTIONS

P Pn

difference Dn ¼ n

j¼1 jXi À Xj j=[n(n À 1)] and obtain that independence of X n

i¼1

and Dn =X n characterizes the gamma distribution under the previously stated

assumptions (Hwang and Hu, 2000). In our context the latter result is probably best

remembered as stating that independence of the sample mean and the sample Gini

index Gn characterizes the gamma distribution, since one of the many

representations of this inequality measure is Gn ¼ Dn =(2X n ).

5.2.4 Lorenz Curve and Inequality Measures

Specializing from (5.17), we see that the kth moment distribution is given by

F(k) (x; b, p) ¼ F(x; b, p þ k), x . 0: (5:47)

Hence the gamma distribution provides a further example of a distribution that is

closed with respect to the formation of moment distributions. This yields the

parametric expression for the Lorenz curve

{[u, L(u)]} ¼ {[F(x; b, p), F(x; b, p þ 1)]jx [ (0, 1)}: (5:48)

From (5.19) moreover we get