Compromise Solution in an Economy Model with Replacement of Equipment

In this paper, we formalize the problem on the optimal strategy for replacing equipment in a branch of economy in the presence of price expectations and expectations of Harrod-neutral technological progress. The problem is to determine a strategy for the purchase and write-off of equipment at each moment of time minimizing the integral cost taking into account the discount. In the particular case where the equipment is not subject to aging, the problem is formalized as an optimal control problem with an additional condition of increasing the function. An analytical solution of this problem is obtained by the time discretization method and subsequent solving by the dynamic programming method. In the general case, dynamic programming is not applicable, and time discretization provides only an approximate numerical solution. Further, we assume that various branches have different discount functions describing the dependence of utility on cost dynamics over time. As an optimality principle, we use a compromise solution. In this case, time discretization also allows one to obtain an approximate numerical solution. © 2022, Springer Science+Business Media, LLC, part of Springer Nature.