Equilibrium points for Optimal Investment with Vintage Capital

TL;DR

This paper investigates equilibrium points in infinite-dimensional optimal investment problems with vintage capital, providing conditions for their existence and explicit computations in economic models.

Contribution

It introduces a framework for identifying and computing equilibrium points in infinite-dimensional boundary control problems applied to vintage capital investment.

Findings

Sufficient conditions for equilibrium existence are established.

Explicit equilibrium solutions are computed for specific economic models.

The approach demonstrates effective use of infinite-dimensional optimal control techniques.

Abstract

The paper concerns the study of equilibrium points, namely the stationary solutions to the closed loop equation, of an infinite dimensional and infinite horizon boundary control problem for linear partial differential equations. Sufficient conditions for existence of equilibrium points in the general case are given and later applied to the economic problem of optimal investment with vintage capital. Explicit computation of equilibria for the economic problem in some relevant examples is also provided. Indeed the challenging issue here is showing that a theoretical machinery, such as optimal control in infinite dimension, may be effectively used to compute solutions explicitly and easily, and that the same computation may be straightforwardly repeated in examples yielding the same abstract structure. No stability result is instead provided: the work here contained has to be considered as a first step in the direction of studying the behavior of optimal controls and trajectories in the long run.