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994 SIZE DISTRIBUTIONS IN ECONOMICS SIZE DISTRIBUTIONS IN ECONOMICS The size distributions of certain economic and socioeconomic variables—incomes, wealth, firms, plants, cities, etc.—display remarkably regular pat terns. These patterns, or distribution laws, are usu ally skew, the most important being the Pareto law (see Allais 1968) and the log-normal, or Gibrat, law (below). Some disagreement about the pat terns actually observed still exists. The empirical distributions often approximate the Gibrat law in the middle ranges of the variables and the Pareto law in the upper ranges. The study of size distribu tions is concerned with explaining why the ob served patterns exist and persist. The answer may be found in the conception of the distribution laws as the steady state equilibria of stochastic processes that describe the underlying economic or demo graphic forces. A steady state equilibrium is a macroscopic condition that results from the bal ance of a great number of random microscopic movements proceeding in opposite directions. Thus, in a stationary population a constant age struc ture is maintained by the annual occurrence of approximately constant numbers of births and deaths—the random events par excellence of hu man life. The steady state explanation is evidently inspired by the example of statistical mechanics in which the macroscopic conditions are heat and pressure and the microscopic random movements are per formed by the molecules. Characteristically, the steady state is independent of initial conditions, i.e., the initial size distribution. In economic appli cations this is important because it means that the pattern determined by certain structural constants tends to be re-established after a disturbance is imposed on the process. This will only be the case, however, if the process leading to the steady state is really ergodic, that is, if the influence of initial conditions on the state of the system becomes negligible after a certain time; and it will be rele vant in practice only if this time interval is suffi ciently short. The idea that the stable pattern of a distribution might be explained by the interplay of a multitude of small random events was first demonstrated in the case of the normal distribution. The central limit theorem shows that the addition of a great number of small independent random variables yields a variable that is normally distributed, when properly centered and scaled. A stochastic process that leads to a normal distribution is the random walk on a straight line with, for example, a 50 per cent probability each of a step in one direction and a step in the opposite direction. It is only natural that attempts to explain other distribution patterns should have started from this idea. The first exten sion was to allow the random walk to proceed on a logarithmic scale. The resulting distribution is log-normal on the natural scale and is known as the log-normal or Gibrat distribution. The basic as sumption, in economic terms, is that the chance of a certain proportionate growth or shrinkage is independent of the size already reached—the law of proportionate effect. This law was proposed by J. C. Kapteyn, by Francis Galton, and, later, by Gibrat (1931). O Let size (of towns, firms, incomes) at time t be denoted by Y(t), and let e(t) represent a random variable with a certain distribution. We have Y(t) = (1 + e(t))Y(t — 1) = Y(0)(l+e(l))---(l+e(t)), where Y(0) is size at time 0, the initial period. For small time intervals the logarithm of size can be represented as the sum of independent random variables and an initial size which will become negligible as t grows: log Y(t) = log Y(0) +6(1)+ 6(2) • • •+€(*). If the random variables e are identically distributed with mean m and variance cr 2 , the distribution of log Y(t) will be approximately normal with mean mt and variance a 2 t. This random walk corresponds to the process of diffusion in physics which is illustrated by the so- called Brownian movement of particles of dust put into a drop of liquid. Since it implies an ever growing variance, the idea of Gibrat is not itself enough to provide an explanation for a stable dis