Editing Income distribution (section)

== Development of income distribution as a stochastic process ==
It is difficult to create a realistic and not complicated theoretical model, because the forces determining the distribution of income (DoI) are varied and complex and they continuously interact and fluctuate.

In a model by Champernowne,<ref>{{cite journal |last1=Champernowne |first1=D. G. |title=A Model of Income Distribution |journal=The Economic Journal |date=1953 |volume=63 |issue=250 |pages=318–351 |doi=10.2307/2227127 |jstor=2227127 }}</ref> the author assumes that the income scale is divided into an enumerable infinity of income ranges, which have uniform proportionate distribution. The development through time of the DoI between ranges is regarded to be a [[stochastic process]]. The income of any person in one year may depend on the income in the previous year and on a chance of progress. Assuming that to every "dying" income receiver, there is an heir to his or her income in the following year, and vice versa. This implies that the number of incomes is constant through time.

Under these assumptions any historical development of the DoI can be described by the following vectors and matrices.

* <math>X_r(0)</math> ... number of the income receivers in range r = 1, 2, ... in the initial year <math>Y_0</math>
* <math>p'_{rs}(t)</math> ... matrix, that contains proportions of the occupants of r-th range in the year <math>Y_t</math> shifted to the s-th range in the following year <math>Y_{t+1}</math>

The vector of the DoI can be expressed as

<math>X_s(t+1)=\sum_{r=0}^\infty X_r(t) p'_{rs}(t).</math>

The elements of proportion matrices can be estimated from historical data.