generalized beta (or transformed beta) distribution given by

axapÀ1

, x . 0:

f (x) ¼ (1:14)

bap B( p, q)[1 þ (x=b)a ] pþq

It is also known in the statistical literature as the generalized F (see, e.g., Kalb¬‚eisch

and Prentice, 1980) and was rediscovered in a slightly different parameterization by

Majumder and Chakravarty (1990) a few years later. This family includes numerous

models used as income and size distributions, in particular the Singh and Maddala

(1976) model, the Dagum type I model (Dagum, 1977), the Fisk model, and

evidently the beta distribution of the second kind. In actuarial science the Singh “

Maddala and Dagum models are usually referred to as the Burr and inverse Burr

distributions, respectively, since they are members of the Burr (1942) system of

distributions.

We also mention the natural generalization of the Pareto distribution proposed by

Stoppa in 1990b,c. It is given by

Àa !u

x

0 , x0

F(x) ¼ 1 À , x: (1:15)

x0

This book is devoted to a detailed study of the distributions surveyed in this

section and their interrelations. The literature is immense and omissions are

unavoidable although we tried to utilize all the references collected during a

six-month extensive search. Due to the rather sporadic developments in that area,

only some isolated multivariate distributions are included.

1.4 STOCHASTIC PROCESS MODELS FOR

SIZE DISTRIBUTIONS

Interestingly enough, income and wealth distributions of various types can be

obtained as steady-state solutions of stochastic processes.

The ¬rst example is Gibrat™s (1931) model leading to the lognormal distribution.

He views income dynamics as a multiplicative random process in which the product

of a large number of individual random variables tends to the lognormal distribution.

This multiplicative central limit theorem leads to a simple Markov model of the “law

15

1.4 STOCHASTIC PROCESS MODELS FOR SIZE DISTRIBUTIONS

of proportionate effect.” Let Xt denote the income in period t. It is generated by a

¬rst-order Markov process, depending only on XtÀ1 and a stochastic in¬‚uence

Xt ¼ Rt XtÀ1 :

Here {Rt } is a sequence of independent and identically distributed random variables

that are independent of XtÀ1 as well. X0 is the income in the initial period.

Substituting backward, we see that

Xt ¼ X0 Á R0 Á R1 Á R2 Á . . . RtÀ1 ,

and as t increases, the distribution of Xt tends to a lognormal distribution provided

var( log Rt ) , 1.

In the Gibrat model we assume the independence of Rt , which may not be

realistic. Moreover, the variance of log Xt is an increasing function of t and this often

contradicts the data. Kalecki (1945), in a paper already mentioned, modi¬ed the

model by introducing a negative correlation between XtÀ1 and Rt that prevents

var( log Xt ) from growing. Economically, it means that the probability that income

will rise by a given percentage is lower for the rich than for the poor. (The

modi¬cation is an example of an ingenious but possibly ad hoc assumption.)

Champernowne (1953) demonstrated that under certain assumptions the

stationary income distribution will approximate the Pareto distribution irrespectively

of the initial distribution. He also viewed income determination as a Markov process

(income for the current period depends only on one™s income for the last period and

random in¬‚uence). He subdivided the income into a ¬nite number of classes and

de¬ned pij as the probability of being in class j at time t þ 1 given that one was in

class i at time t. The income intervals de¬ning each class are assumed (1) to form a

geometric (not arithmetic) progression. The limits of class j are higher than those of

class j À 1 by a certain percentage rather than a certain absolute amount of income

and the transitional probabilities pij depend only on the differences j À i. (2) Income

cannot move up more than one interval nor down more than n intervals in any one

period; (3) there is a lowest interval beneath which no income can fall, and (4) the

average number of intervals shifted in a period is negative in each income bracket.

Under these assumptions, Champernowne proved that the distribution eventually

behaves like the Pareto law.

The assumptions of the Champernowne model can be relaxed by allowing for

groups of people (classi¬ed by age, occupation, etc.) and permitting movement from

one group to another. However, constancy of the transition matrix is essential;

otherwise, no stationary distribution will emerge from the Markov process.

Moreover, probabilities of advancing or declining ought to be independent of the

amount of income. Many would doubt the existence of a society whose institutional

framework is so static, noting that such phenomena as “inherited privilege,” and

cycles of poverty or prosperity are part and parcel of all viable societies.

To complicate the matter with the applicability of Champernowne™s model, it was

shown by Aitchison and Brown (1954) that if the transition probabilities pij depend

16 INTRODUCTION

on j=i (rather than j À i, as is the case in Champernowne™s model) and further that

the income brackets form an arithmetic (rather than geometric) progression, then the

limiting distribution is lognormal rather than Pareto. In our opinion the dependence

on j À i may seem to be more natural, but it is a matter of subjective opinion.

It should also be noted that the Champernowne and Gibrat models and some

others require long durations of time until the approach to stationarity is obtained.

This point has been emphasized by Shorrocks (1975).

Rutherford (1955) incorporated birth “ death considerations into a Markov model.

His assumptions were as follows:

The supply of new entrants grows at a constant rate.

.

These people enter the labor force with a lognormal distribution of income.

.

The number of survivors in each cohort declines exponentially with age.

.

Under these assumptions, the data eventually approximate the Gram “ Charlier type

A distribution, which often provides a better ¬t than the lognormal. In Rutherford™s

model the overall variance remains constant over time.

Mandelbrot (1961) constructed a Markov model that approximates the Pareto

distribution similarly to Champernowne™s model, but does not require the strict law

of proportionate effect (a random walk in logarithms).

Wold and Whittle (1957) offered a rather general continuous-time model that also

generates the Pareto distribution: It is applied to stocks of wealth that grow at a

compound interest rate during the lifetime of a wealth-holder and are then divided

among his heirs. Deaths occur randomly with a known mortality rate per unit time.

Applying the model to wealth above a certain minimum (this is necessary because

the Pareto distribution only applies above some positive minimum wealth), Wold and

Whittle derived the Pareto law and expressed the exponent a as a function of (1) the

number of heirs per person, (2) the growth rate of wealth, and (3) the mortality rate

of the wealth owners.

The most complicated model known to us seems to be due to Sargan (1957). It is

a continuous-time Markov process: The ways in which transitions occur are

explicitly spelled out. His approach is quite general; it accommodates

Setting of new households and dissolving of old ones

.

Gifts between households

.

Savings and capital gains

.

Inheritance and death

.

It is its generality that makes it unwieldy and unintelligible.

As an alternative to the use of ergodic Markov processes, one can also explain

wealth or income distributions by means of branching processes. Steindl (1972),

building on the model of Wold and Whittle (1957) mentioned above, showed in this

way that the distribution of wealth can be regarded as a certain transformation of an

age distribution. Shorrocks (1975) explained wealth accumulations using the theory

17

1.4 STOCHASTIC PROCESS MODELS FOR SIZE DISTRIBUTIONS