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SIZE DISTRIBUTIONS IN ECONOMICS 99 5 tribution. There must be a stabilizing influence to offset the tendency of the variance to increase; indeed, a distinguishing feature of the various theories presently to be reviewed lies in the kind of stabilizer they introduce to offset the diffusion. Two interesting cases may be noted here. One possibility is to modify the law of proportionate effect and assume that the chances of growth de cline as size increases. This approach has been taken by Kalecki (1945), who assumes a negative correlation between the size and the jump and obtains a Gibrat law with constant variance. An other possibility is to combine the diffusion process of the random walk with a steady inflow of new, small units (firms, cities, incomes). Some units may continue indefinitely to increase in size, but their weight will be offset by that of a continuous stream of many new, small entrants, so that both the mean and the variance of the distribution will remain constant. This approach, which leads to the Pareto law, has been taken by Simon (1955). Review of various models. Descriptions of var ious models will illustrate the methods employed. Models differ with regard to the distribution law explained, the field of application (towns, incomes, etc.), and the type of stochastic process used. Champernoivne’s model. Champernowne (1953 ) presents a model that explains the Pareto law for the size distribution of incomes. The stochastic process employed is the so-called Markov chain [see Markov chains]. The model is based on a matrix of probabilities of transition from one income class to another in a certain interval of time, say a year. The rows are the income classes of one year, the columns the income classes of the next year. The income classes are chosen in such a way that they are equal on the logarithmic scale (for example, incomes from 1 to 10, from 10 to 100, etc.). The probability of a jump from one income class to the next income class in the course of a year is assumed to be independent of the income from which the jump is made (the law of proportionate effect). The number of income earners is constant. The number of income earners in income class s is then determined as follows. The number of in comes in class s at time £ + 1 is f(s, t+l) = Sf(s- W„t)p(ti), u--n where s, it, and t take on integer values, p(u) is the probability of a jump over u intervals (i.e., the transition probability), and the size of the jump is constrained to the range +1, —n. In the steady state equilibrium reached after a sufficiently long time has passed, the action of the transition matrix leaves the distribution unchanged. We then have f(s) = Y,f(s-u)p(u), s >°. u = -n as This difference equation is solved by putting f(s) = z\ The characteristic equation g(z) = Z z 1 U p(u) -z = o w=-n has two positive real roots, one of which is unity. To assure that the other root will be between 0 and 1, Champernowne introduces the following stability condition: g'(i) = - S up(u) > o. M = -n The relevant solution is f(s) = b*, 0 < b < 1, which gives the number of incomes in income class s. If the lower bound of this class is the log of the income Y a , then the probability of an income ex ceeding Y s is given by log P(Y f ) = s log b. Since s is determined by log Y, = sh + log Y mjn) where h is the class interval and Y min is the lower boundary of the lowest income class, it follows that log P(Y*) = y-arlogY,, where the parameters y and a are determined by b, h, and Y min . This is the Pareto law with Pareto coefficient a. Champernowne’s stability condition implies that the mathematical expectation of a change in in come is negative. This counteracts the diffusion. How can the stability condition be justified on economic grounds? It may be connected with the fact that in this model every income earner who drops out is replaced by a new income earner. Since, in practice, young people have on the aver age lower and more uniform incomes than old people, the replacement of old income earners by young ones usually means a drop in income. Thus, Champernowne’s stability condition, as far as its economic basis is concerned, is very similar to the entry of new, small units that act as a stabilizer in Simon’s model. Rutherford's model. Rutherford’s model (1955) leads, in his opinion, to the Gibrat law for the size distribution of incomes. Newly entering income earners, assumed to be log-normally distributed at the start, are subject to a random walk and thus