during dt has a systematic drift component m dt and a stochastic diffusion

component s dBt : This is essentially a continuous-time version of Gibrat™s law of

proportionate effect.

141

4.11 AN ASYMMETRIC LOG-LAPLACE DISTRIBUTION

The novelty of Reed™s approach lies in the assumption that the time of observation

T is not ¬xed”this would lead to the well-known lognormal case”but random.

An economic interpretation is that even if the evolution of each individual income

follows a geometric Brownian motion (Gibrat™s law of proportionate effect)

when observing the income distribution at a ¬xed point in time, we may not

know for how long a person has lived. If different age groups are mixed and the

distribution of time in the workforce of any individual follows an exponential

distribution, the observed distribution should be that of the state of the geometric

Brownian motion stopped after an exponentially distributed time. What does this

distribution look like?

For a geometric Brownian motion with a ¬xed initial state x0 ; the conditional state

at time T is lognormally distributed

!

s2 2

Y jT :¼ log (X jT ) $ N x0 þ m À T, s T :

2

Hence, Y jT possesses the m.g.f.

& '

s2 s 2 t2

mY jT (t) ¼ exp x0 t þ m À :

tþ

2 2

Assuming that T itself follows an exponential distribution, T $ Exp(l); with m.g.f.

l

mT (t) ¼ ,

lÀt

we obtain for the m.g.f. of the unconditional state variable Y

lex0 t

mY (t) ¼ E(eYt ) ¼ ET [E(eY jTÁt jT )] ¼ , (4:62)

l þ (m À s 2 =2)t À s 2 t 2

which may be rewritten in the form

le x0 t ab

mY (t) ¼ , (4:63)

(a À t)(b þ t)

where a, Àb are the roots of the quadratic equation de¬ned by the denominator of

(4.62). These parameters are therefore functions of the drift and diffusion constants

of the underlying geometric Brownian motion and of the scale parameter l of the

exponentially distributed random time T : Expression (4.63) is recognized as the

m.g.f. of an asymmetric Laplace distribution (see, e.g., Kotz, Kozubowski, and

142 LOGNORMAL DISTRIBUTIONS

Podgorski, 2001), and so the unconditional distribution of the size variable X ¼

´

exp(Y ) randomly stopped at T is given by

8

> x0 ab x bÀ1

>

>

> x , x0 ,

,

< a þ b x0

f (x) ¼ (4:64)

ÀaÀ1

>

> ab x

>

> , x ! x0 ,

:

x0 (a þ b) x0

a density that exhibits power-law behavior in both tails. This is noteworthy, since the

underlying geometric Brownian motion is essentially a multiplicative generative

model and hence in view of the law of proportionate effect a lognormal distribution

would be expected (see Section 4.2). Thus, a seemingly minor modi¬cation”

introducing a random observational time”yields a power-law behavior. This type of

effect was noticed some 20 years earlier by Montroll and Schlesinger (1982, 1983)

who showed that a mixture of lognormal distributions with a geometric weighting

distribution would have essentially a lognormal main part but a Pareto-type

distribution in the upper tail.

Because of the power-law behavior in both tails, Reed referred to (4.64) as a

double-Pareto distribution; in view of its genesis, it could also be called a “log-

asymmetric Laplace distribution.”

A generalization of the above model assumes that the initial state X0 is also

random, following a lognormal distribution. This yields an unconditional

distribution to which Reed (2001b) referred as the double-Pareto-lognormal

distribution. He estimated this four-parameter model for U.S. household incomes

of 1997, Canadian personal earnings in 1996, 6-month household incomes in

Sri Lanka for 1981, and Bohemian personal incomes in 1933 (considered earlier

by Champernowne, 1952), for all of which the ¬t is excellent.

It is worth mentioning that the distribution (4.64) appears in a model of

underreported income discussed by Hartley and Revankar (1974); see also Hinkley

and Revankar (1977). In an underreporting model the goal is to make an inference

about the distribution of the true income XÃ when only a random sample from

observable income X is available. It is therefore necessary to relate the p.d.f. of X to

the parameters of the p.d.f. of XÃ : Suppose the true but unobservable incomes XÃ

follow a Pareto type I distribution (3.2)

a xÃ ÀaÀ1

xÃ ;

f (xÃ ) ¼ , x0

x0 x0

and assume that observable income X is given by

X ¼ XÃ À U ,

143

4.12 RELATED DISTRIBUTIONS

where U is the underreporting factor. It is natural to assume that 0 , U , XÃ :

Hartley and Revankar (1974) postulated that the proportion of XÃ that is

underreported, denoted by

U

WÃ ¼ ,

XÃ

is distributed independently of XÃ with the p.d.f.

f (w) ¼ b(1 À w)bÀ1 , 0 1, b . 0,

w

a special case of the beta distribution. It is not dif¬cult to show that the observable

income X has the p.d.f. (4.64).

4.12 RELATED DISTRIBUTIONS

Since the t distribution can be viewed as a generalization of the normal distribution,

it is not surprising that the log -t distribution has also been suggested as a model for

the size distribution of incomes (Kloek and van Dijk, 1977) or of insurance losses

(Hogg and Klugman, 1983). Its p.d.f. is

!À(nþ1)=2

nn=2 (log x À log m)

Á nþ x . 0,

f (x) ¼ , (4:65)

s2

B(1=2, n=2)x