The factors determining the distribution of personal incomes are both numerous and interrelated. However it is common, even in complicated problems, to discover that the basic relationship between the fundamental factors are simple. In the belief that an understanding of the theory of income distribution at its simplest level is a pre-requisite for progress in elucidating its detailed aspects, a model has been developed which contains the necessary basic features and within which the choices facing society are described in a precise manner. Our starting point is to consider the benefit or utility which a worker derives from his income and leisure. This leisure is determined by the time he spends in producing an outpus and his income is also determined by his output. Thus a worker's utility is a function of his output and we assume that he produces that output which maximizes his utility. His utility, output, income and leisure are then determined. The next concept we need is that of an individual worker's ability to produce. This will determine the time taken and leisure lost in producing a given output. The utility, output, income and leisure are therefore all functions of the ability to produce and its distribution in society will be a fundamental function in the model. For application of the theory to real systems it would often be convenient to define the ability to produce as the earning rate (per unit time). The point in the above description where society makes the crucial choice is in the income which it allocates to a given output. This choice essentially determines all quantities, including the magnitude and the distribution of incomes. This then is the central point of interest, and the value of the model which we shall use is that it will enable us to study the factors which might be considered in making the choice of the income function (of output) and to derive the consequences of a given choice. As an example of the possibilities, we can determine that income function which would maximize the sum of individual utilities. We would then have a system in which workers maximized their individual utilities with respect to their output and society maximized the total utility with respect to the income function. Another way of selecting an income function would be to require that the function be such that any changes would receive the support and incur the opposition of an equal number of workers. One might expect such a solution to be an approximation to that which would be chosen in a democracy. In order to illustrate the use of the model we shall consider a pair of simple functions to describe utility and leisure, and investigate algebraically the consequences of varying the income function. This again is done in a simple way by choosing a particular functional form with two parameters, variation of which varies incentives. One of these measures the lowest income in the society and the other determines the income function at higher incomes, that is the progressiveness of the taxation system. The use of the model for more