The Distribution of an Indivisible Resource: Optimal Control and Prices

We study a dynamical system of economics with discrete time, whose states at every moment are characterized by integer nonnegative points in a two-dimensional vector space. There are two types of goods and two different production facilities, in each of which the state of the system can change by some random integer vector with various collections of probabilities. By a control we understand choosing at every moment one of the available collections of probabilities. The aim of control is to minimize the probability of leaving the positive quadrant. We study the existence of prices for the goods which agree with the optimal control.