Consumer buying dynamics for a single commodity using flow and stock utilities and a principle of minimum total utility imbalance

Abstract

A consumer unit is modeled as possessing a flow utility and a stock utility with respect to money and the quantity of a single commodity. A principle of minimum total utility imbalance is proposed to model the intertemporal dynamics of buying. The consequences of this principle are analyzed when unused money is deposited at interest, purchased stocks may deteriorate and a variable consumption rate is permitted. Second-order differential equations are derived. These equations are generally non-linear and considerably more difficult than equations of planar classical mechanics. From the solutions of these equations the actual spending and buying behavior can be calculated easily. To illustrate the potential of this approach, a quadratic non-interactive approximation for the flow utility and quadratic interactive approximation for the stock utility are chosen. The differential equations turn out to be linear, low-order coupled and driven by consumption. They become linear with constant coefficients if the persistence coefficients (in the flow utility) and the preference coefficients (in the stock utility) are also constant. Standard methods permit initial or boundary value problems based on the minimum imbalance principle to be solved explicitly, based on four exponential modes, two oscillatory modes or one oscillatory and two exponential modes. Other utility formulas permit approximations by quadratic splines. Further applications are made to the problem of managing the buying to increase wealth and decrease unused material stocks, given interest rate, deterioration rate, commodity price (assumed constant), and consumption patterns. Comments on other microeconomic uses and on the relation of this theory to other buying dynamics theories are made.