If we use a suitable central limit theorem (CLT) it follows that Xn are asymptotically

lognormally distributed. In the original derivation the et were assumed to be i.i.d. so

109

4.2 HISTORY AND GENESIS

Table 4.1 Gibrat™s Fittings

Variable Location Date

Food expenditures England 1918

Income England 1893“ 1894, 1911“ 1912

Oldenburg 1890

France 1919“ 1927

Prussia 1852, 1876, 1881, 1886, 1892,

1893, 1894, 1901, 1911

Saxony 1878, 1880, 1886, 1888, 1904,

1910

Austria 1898, 1899, 1903, 1910

Japan 1904

Wealth tax Netherlands 1893“ 1910

Wealth Basel 1454, 1887

Old-age state France 1922“ 1927

pensions

Postal check France 1921“ 1922

accounts

Legacies France 1902“ 1913, 1925“ 1927

England 1913“ 1914

Italy 1910“ 1912

9 French 1912, 1926

departements

´

Rents Paris 1878, 1888, 1913

Real estate (tilled) France 1899“ 1900

Real estate Belgium 1850

Real estate sales France 1914

Security holdings France 1896

Wages United Kingdom 1906“ 1907

Italy 1906

Denmark 1906

Bavaria 1909

Dividends Germany 1886, 1896

Firm pro¬ts France 1927

Firm sizes France 1896, 1901, 1906, 1921, 1926

Alsace-Lorraine 1907, 1921, 1926

European countries around 1907

and the United States

Turkey 1925

Germany 1907, 1925

City sizes France 1866, 1906, 1911, 1921, 1926

Europe 1850, 1910, 1920, 1927

United States 1900, 1920, 1927

Family sizes France 1926

Source: Gibrat (1931).

110 LOGNORMAL DISTRIBUTIONS

that the Lindeberg“ Levy CLT is suf¬cient to handle the problem. The same limiting

´

result may, of course, be obtained using more general CLTs if the et are

heterogeneous; see Hart (1973, Appendix B) for a discussion.

It should be noted that Gibrat™s approach is very close to the one exhibited in

Edgeworth™s (1898) method of translations, by means of which a exponential

transform of a normal variable is carried out. Gibrat followed Kapteyn™s (1903)

generic approach (the so-called Kapteyn™s engine) to construct a binomial process

and provided a modi¬cation of the independence assumption inherent in this

process, trying to justify its application in economics. fMuch earlier Galton (1879)

and McAlister (1879) suggested the exponential of a normal variable, inspired by the

Weber “ Fechner law in psychophysics [details are presented in a later publication of

Fechner (1897): Kollektivmasslehre].g Being a French engineer trained to think

geometrically, Gibrat did not use “sophisticated” mathematical tools (such as least-

squares or Cauchy methods). Gibrat argued in favor of his model as compared with

its classical earlier competitor, the Pareto distribution, for a number of cases in

various economic areas. See Table 4.1 for a listing of the data to which Gibrat

successfully ¬t lognormal distributions.

An innovative feature of Gibrat™s model is its potential application in all economic

domains. The Pareto law seemed to be restricted solely to income distributions,

whereas the application of the Gibrat law of proportional effect appeared to be

more extensive. As indicated in Table 4.1, in his 1931 dissertation Gibrat applied

it to other areas of economics such as fortunes and estates, ¬rm pro¬ts, ¬rm sizes

(by number of workers), number of city inhabitants, and family sizes.

Gibrat™s law of proportionate effect was subsequently modi¬ed, for example, to

prevent the variance of log X to grow without bound (see Section 1.4). However, for

more than 70 years it has proved to be a useful ¬rst approximation of ¬rm size

dynamics against which alternatives have to be tested; see, for example, Mans¬eld

(1962), Evans (1987a,b), Hall (1987), or Dunne and Hughes (1994). Sutton (1997)

provided a recent survey of Gibrat™s law in connection with the size distribution

of ¬rms.

4.3 MOMENTS AND OTHER BASIC PROPERTIES

In view of the close relationship with the normal distribution, many properties of the

lognormal distribution follow directly from corresponding results for the normal

distribution. For example, random samples from a lognormal population are

commonly generated from normal random numbers via exponentiation. The main

difference is the new role of the parameters: exp(m) becomes a scale parameter,

whereas s is now a shape parameter. Figure 4.1 illustrates the effect of s; showing

the well-known fact that for small s; a LN(m,s 2 ) distribution can be approximated

by a N(exp m, s 2 ) distribution.

One of the main attractions of the normal distribution is its stability properties

under summation, namely, that sums of independent normal random variables are

111

4.3 MOMENTS AND OTHER BASIC PROPERTIES

Figure 1 Lognormal densities: m ¼ 0 and s ¼ 0:25, 0:5, 1:0, 1:5 (modes from right to left).

also normally distributed. This translates into the following multiplicative

stability property of the lognormal distribution: For independent Xi $ LN(mi , si2 ),

i ¼ 1, 2;

2 2

X1 X2 $ LN(m1 þ m2 , s1 þ s2 ), (4:3)

in particular for the sample geometric mean of n i.i.d. lognormal random variables

!1=n

Y

n

s2

$ LN m, :

Xi (4:4)

n

i¼1

However, sums of lognormal random variables are not very tractable. Unfortu-

nately, as noted by Mandelbrot (1997), “dollars and ¬rm sizes do not multiply, they

add and subtract . . . [hence] the lognormal has invariance properties but not very

useful ones.” In his view this constitutes a strong case against the lognormal

distribution.

112 LOGNORMAL DISTRIBUTIONS

The structure of the lognormal distribution makes it convenient to express its

moments. They are obtained in terms of the moment-generating function mY ( Á ) of

the normal distribution