# Optimal Investment with Vintage Capital:Equilibrium Distributions

## TL;DR

This paper develops a method to analyze equilibrium distributions in optimal investment models with vintage capital, using PDEs and infinite-dimensional control theory, providing explicit formulas and conditions for existence and uniqueness.

## Contribution

It introduces a general approach to compute equilibrium distributions in infinite-dimensional PDE control problems, specifically applied to vintage capital investment models.

## Key findings

- Existence and uniqueness of long-run equilibrium distributions are established.
- Explicit formulas for optimal controls and trajectories are derived.
- The method applies broadly to linear PDE control problems with convex criteria.

## Abstract

The paper concerns the study of equilibrium points, or steady states, of economic systems arising in modeling optimal investment with \textit{vintage capital}, namely, systems where all key variables (capitals, investments, prices) are indexed not only by time $\tau$ but also by age $s$. Capital accumulation is hence described as a partial differential equation (briefly, PDE), and equilibrium points are in fact equilibrium distributions in the variable $s$ of ages. Investments in frontier as well as non-frontier vintages are possible. Firstly a general method is developed to compute and study equilibrium points of a wide range of infinite dimensional, infinite horizon boundary control problems for linear PDEs with convex criterion, possibly applying to a wide variety of economic problems. Sufficient and necessary conditions for existence of equilibrium points are derived in this general context. In particular, for optimal investment with vintage capital, existence and uniqueness of a long run equilibrium distribution is proved for general concave revenues and convex investment costs, and analytic formulas are obtained for optimal controls and trajectories in the long run, definitely showing how effective the theoretical machinery of optimal control in infinite dimension is in computing explicitly equilibrium distributions, and suggesting that the same method can be applied in examples yielding the same abstract structure. To this extent, the results of this work constitutes a first crucial step towards a thorough understanding of the behavior of optimal controls and trajectories in the long run.