The structure and existence of solutions of the problem of consumption with satiation in continuous time.

Abstract

With the help of the method of Lagrange multipliers and KKT theory, we investigate the structure and existence of optimal solutions of the continuous-time model of consumption with satiation. We show that the differential equations have no solutions in the C1 class but that solutions exist in a wider space of functions, namely, the space of functions of bounded variation with non-negative Borel measures as controls. We prove our theorems with no additional assumptions about the structure of the control Borel measures. We prove the conjecture made in the earlier literature, that there are only three types of solutions: I-shaped solutions, with a gulp of consumption at the end of the interval and no consumption at the beginning or in the interior; U-shaped solutions, with consumption in the entire interior of the interval and gulps at the beginning and the end; and intermediate (J-shaped) solutions, with an initial interval of abstinence followed by a terminal interval of distributed consumption at rates and a gulp at the end. We also establish the criteria that permit determination of the solution type using the problem's parameters. When the solution structure is known, we reduce the problem of the existence of a solution to algebraic equations and discuss the solvability of these equations. We construct explicit solutions for logarithmic utility and CRRA utility.