The Graduation of Income Distributions

A FUNCTION representing the distribution of income in a society can serve a number of purposes. It may be used to smooth out irregularities in the observed income distribution caused by the misreporting of income. In this role it is similar to the graduation formulae used in demographic work to correct an age distribution distorted by mistatements of age. An income function may also form the basis of a model explaining how an income distribution is generated. Interest here lies in the success with which the model generates a distribution close to that of the observed values and in the meaning that can be ascribed to the parameters in the model. In addition, an income function can assist in the analysis of income distributions by highlighting the more important characteristics of such distributions and providing measures, which can be compared spatially or temporally, of those characteristics. Other uses can no doubt be suggested. For a function to serve these purposes adequately, it is desirable that it should approximate observed distributions of income closely when particular values, usually estimated from the observed data, are given to the parameters. This criterion is the least satisfied by the formulae that have been suggested to date except, perhaps, over limited segments of the income range. The Pareto curve fits income distributions at the extremities of the income range but provides a poor fit over the whole income range. The log normal (or Gibrat) distribution fits reasonably well over a large part of the income range but diverges markedly at the extremities. A function suggested