2.4 GENERATING SYSTEMS OF INCOME DISTRIBUTIONS

As was already mentioned in the preceding chapter, a huge variety of size

distributions, including almost all of the best known continuous univariate

distributions supported on the positive hal¬‚ine, have been introduced during the

last hundred years. It is therefore of particular interest to have, apart from

classi¬cation systems like those surveyed in the preceding section, generating

systems that yield, starting from a few basic principles, models that should be useful

for the modeling of size phenomena. Not surprisingly, the largest branch of the size

distributions literature, the literature on income distributions, has come up with

several generating systems for the derivation of suitable models. We present the

systems proposed by D™Addario (1949) and Dagum (1980b,c, 1990a, 1996).

D™Addario™s System

Following the idea of transformation functions applied earlier by Edgeworth (1898),

Kapteyn (1903), van Uven (1917), and Frechet (1939), D™Addario (1949) speci¬ed

´

his system by means of the generating function

g( y) ¼ A{b þ exp ( y1=p )}À1 , (2:79)

where p . 0 and b is real, and the transformation function

a

dy

yq x , 1,

,c x0 (2:80)

¼

dx x À c

where a = 0 and q is real. Here x is the income variable and A is a normalizing

constant. The differential equation (2.80) yields

y ¼ h(x) ¼ a(x À c)a , q ¼ À1, a . 0, a = 0, (2:81)

y ¼ h(x) ¼ [(1 þ q){alog(x À c) þ a}]1=(1þq) , q = À1, a = 0, (2:82)

where a is a constant of integration. Equations (2.79) and (2.80) imply that the

transformed variable y ¼ h(x) is a monotonic function of income, taking values on

the interval [x0 , x1 ] if h(x) is increasing and on the interval [x1 , x0 ] if h(x) is

decreasing, where x0 ¼ h( y0 ) and x1 ¼ limx!1 h(x).

The general form of solution of D™Addario™s system is given by

dh(x)

f (x) ¼ A 1=p À1

dx {b þ exp[h(x) ]} , (2:83)

where dh(x)=dx is obtained from (2.81) or (2.82), depending on the value of q.

Table 2.5 presents the income distributions that can be deduced from D™Addario™s

system.

56 GENERAL PRINCIPLES

Table 2.5 D™Addario™s Generating System

Distribution Generating Transformation Support

Function Function

a

b p c q

x0 x , 1

.0

Pareto (I) 0 1 0 0

c , x0 x , 1

.0

Pareto (II) 0 1 =0 0

0 x,1

.0

Lognormal (2 parameters) 0 1/2 0 0

c x,1

.0

Lognormal (3 parameters) 0 1/2 =0 0

c , x0 x , 1

.0 21

Generalized gamma 0 =0 =0

c , x0 x , 1

21 .0 2p 21

Davis =0

It is worth noting that the Davis distribution (see Section 7.2), a distribution

not easily related to any other system of distributions, is a member of

D™Addario™s transformation system. Also, the four-parameter generalized gamma

distribution, introduced by Amoroso (1924 “1925), comprises a host of

distributions as special or limiting cases, including the gamma and inverse

gamma, Weibull and inverse Weibull, chi and chi square, Rayleigh, exponential,

and half-normal distributions. The Amoroso distribution will be discussed in

some detail in Chapter 5.

Dagum™s Generalized Logistic System

The Pearson system is a general-purpose system not necessarily derived from

observed stable regularities in a given area of application. D™Addario™s system is a

translation system with ¬‚exible generating and transformation functions constructed

to encompass as many income distributions as possible. In contrast, the system

speci¬ed by Dagum (1980b,c, 1983, 1990a) starts from the characteristic properties

of empirical income and wealth distributions. He observes that the income elasticity

dlog{F(x)}

h(x, F) ¼

dlog x

of the c.d.f. of income is a decreasing and bounded function of F starting from a

¬nite and positive value as F(x) ! 0 and decreasing toward zero as F(x) ! 1, that

is, for x ! 1. This pattern leads to the speci¬cation of the following generating

system for income and wealth distributions:

dlog{F(x) À d}

x0 , x , 1,

¼ q(x)f(F) k, 0 (2:84)

dlog x

where k . 0, q(x) . 0, f(x) . 0, d , 1, and d{q(x)f(F)}=dx , 0. These

constraints assure that the income elasticity of the c.d.f. is indeed a positive,

decreasing, and bounded function of F and therefore of x. For each speci¬cation of f

57

2.4 GENERATING SYSTEMS OF INCOME DISTRIBUTIONS

Table 2.6 Dagum™s Generating System

q(x) f(F) (d, b)

Distribution Support

x,1

a 0 , x0

Pareto (I) (1 À F)=F (0, 0)

ax

x,1

0 , x0

Pareto (II) (1 À F)=F (0, 0)

xÀc

ax

x,1

bx þ 0 , x0

Pareto (III) (1 À F)=F (0, þ)

xÀc

x,1

0 , x0

Benini 2a log x (1 À F)=F (0, 0)

aÀ1

x,1

bx(x À c)

Weibull (1 À F)=F (0, þ) c