The Optimal Control Approach in Problems of the Income Distribution into Consumed and Invested Parts

Abstract

In this paper, we investigate dynamic problems of the income distribution into consumed and invested parts using optimal control methods. Here it is assumed that the basic economic identity holds true, i.e., the income is the sum of the consumption and the investment. The income is understood either in the pure form or the gross domestic product, or the national income. The consumption is also understood in different ways: in its pure form or the aggregate consumption. We start with the analysis of the Harrod–Domar macromodel with the capital intensity of the income growth depending on the continuous time. In particular, it is shown that the balance equation for accumulated household savings also satisfies the basic economic identity and the capital intensity of the income growth can depend on time. Since households are the best savers and they demonstrate the best survival, we modify the considered macromodels replacing the given production functions with the problem of maximizing the integral discounted utility of the consumption. In this case, the utility function satisfies the Arrow–Pratt condition of the relative risk aversion.