À1=l

64 PARETO DISTRIBUTIONS

Note that the upper limit is a function of x. This expression is not very tractable

without introducing further assumptions. However, if P(v, x)=M (v, x) is separable in

the form f(x)c(v), this integral simpli¬es to

° (x=m0 À1)=l

c(v)F (v)

lxf(x) dv ¼: lxf(x) Á x(x): (3:17)

(1 þ lv)2

À1=l

Rhodes then considered the special case where x(x) is approximately constant for

large x, which implies F (x) / c Á xf(x), for some c . 0. It remains to determine the

form of f(x). To this end we introduce w(z) :¼ M (v, x)=P(v, x), yielding w Á fc ¼ 1

under the separability assumption. Taking logarithms and differentiating after x, we

obtain

f0 (x)

w0 (z) dz

:

¼À

f(x)

w(z) dx

On the other hand, M (v, x)(dz=dx)(1 þ lv) ¼ 1, whereby

f0 (x)

w0 (z) 1

,

¼À

w(z) M (v, x)(1 þ lv) f(x)

or, using M (v, x) ¼ h(z) Á w(z), we get

f0 (x)

w0 (z) 1

:

¼À

w2 (z)h(z) 1 þ lv f(x)

Since Àf0 (x)=f(x) does not depend on v, a further differentiation with respect to v

yields

!

w0 (z) 0 dz 1 w0 (z) l

¼ 0:

À2

w (z)h(z) (1 þ lv)2

2 (z)h(z) dv 1 þ lv

w

In view of (3.13) and (3.14), this can be rearranged in the form

!0 0 !

w0 (z) w0 (z) M (v, x)

:

¼À

w2 (z)h(z) w2 (z)h(z) m(z)

All the functions in this equation are functions of z (talent) alone. An integration

yields

w0 (z) a

¼

w2 (z)h(z) m(z)

65

3.2 HISTORY AND GENESIS

for some constant a. If we use M (v, x) ¼ w(z)h(z) again, this equals

w0 (z) aM (v, x)

:

¼

w(z) m(z)

Hence, w(z) ¼ c Á m(z)a , for some constant c. Also,

f0 (x)

w0 (z) 1

¼À

w2 (z)h(z) 1 þ lv f(x)

yields

f0 (x) a a

¼À :

¼À

f(x) m(z)(1 þ lv) x

Hence, f(x) ¼ k Á xÀa , for some constant k. From the de¬nition wfc ¼ 1 we further

obtain c(v) Á c Á m(z)a Á k Á xÀa ¼ 1; therefore,

!

xa1

1

¼ (1 þ lv)a ,

c(v) ¼

ck m(z) ck

a function of v alone. Thus, (3.17) is reduced to

° (x=m0 À1)=l

1

F (v)(1 þ lv)ÀaÀ2 dv,

x(x) ¼

ck À1=l

and if this expression is approximately constant for large x, we ¬nally get

F (x) % xf(x) ¼ k Á xÀaÀ1 : (3:18)

Thus, the distribution of income is (approximately) a Pareto distribution.

More recently, the idea of explaining size by (unobserved) aptitudes has also been

used in the literature on the size distribution of ¬rms. Lucas (1978) presented a

model postulating that the observed size distribution is a solution to the problem of

how to allocate productive factors among managers of differing abilities so as to

maximize output. If “managerial talent” follows a Pareto distribution, the implied

size distribution is also of this form in his model.

3.2.3 Markov Processes Leading to the Pareto Distribution

Champernowne (1953) demonstrated that under certain assumptions the stationary

income distribution of an appropriately de¬ned Markov process will approximate the

Pareto distribution irrespectively of the initial distribution.

Champernowne viewed income determination as a discrete-time Markov chain:

Income for the current period”the state of the Markov chain”depends only on

66 PARETO DISTRIBUTIONS

one™s income for the last period and a random in¬‚uence. He assumed that there is

some minimum income x0 and that the income intervals de¬ning each state form a

geometric (not arithmetic) progression (the limits of class j are higher than limits of

class j À 1 by a certain factor, c, say, rather than a certain absolute amount of

income). Thus, a person is in class j if his or her income is between x0 cjÀ1 and x0 cj .

The transition probabilities pij are de¬ned as the probability of being in class j at

time t þ 1 given that one was in class i at time t.

Champernowne required the assumption that the probability of a jump from one

income class to another depends only on the width of the jump, but not on the

position from which one starts (a form of the law of proportionate effect). In other

words, the (time invariant) transition probability pij is a function of j À i ¼ k only,

which is independent of i. If xj (t) is the number of income earners in the income

class j in period t, the process evolves according to

X

j

xjÀk (t)pk :