Proportional growth with or without diffusion, and other explanations of scaling

Abstract

However useful and “creative” scaling may be, it is not accepted as an irreducible scientific principle. Several isolated instances of scaling are both unquestioned and easy to reduce to more fundamental principles, as will be seen in Section 1. There also exists a broad class of would-be universal explanations, many of them variants of proportional growth of U, with or without diffusion of log U. This chapter shows why, countering widely accepted opinion, I view those explanations as unconvincing and unacceptable.

The models to be surveyed and criticized in this expository text were scattered in esoteric and repetitive references. Those I quote are the ear?liest I know. Many were rephrased in terms of the distribution of the sizes of firms. They are easily translated into terms of other scaling random variables that are positive. The two-tailed scaling variables that represent change of speculative prices (M 1963b{E14}) pose a different problem, since the logarithm of a negative change has no meaning.