Stable Paretian Random Functions and the Multiplicative Variation of Income

Abstract

a new family of stable non-gaussian random functions, U(t), and it is intimately connected with random walks of log U(t). Its principal feature is, however, that no use is made of the principle of proportionate effect, the model being rather based upon the fact that there exist certain limits for sums of random functions, upon which the effect of chance in time is multiplicative. (Actually, the result is more general.) This feature provides a new type of motivation for the widespread, convenient, and frequently fruitful use of the logarithm of income, considered as a moral wealth. Looking at the results from another viewpoint, one may say that the approach is based upon a new kind of diachronic factor analysis, the use of which will be justified in detail. (2) I wish to point out the wider role which I believe that these new stochastic processes will eventually play in linear economics, for example in certain problems of aggregation. The reader will also find that the results are translatable with little effort into terms of theories of variation of various economic quantities similar to income. It may even turn out that this approach will be more reasonable for some other such quantities or that the empirical fit will be better in other cases. As a result, the tools to be introduced may be as important as the immediate results which are hopefully to be achieved. In particular, the problem of the distribution and variation of city sizes is very similar to the problem raised by income. A much less obvious generalization