The dynamic programming approach for a family of optimal investment models with vintage capital is here developed. The problem falls into the class of infinite horizon optimal control problems of PDE’s with age structure that have been studied in various papers (Barucci and Gozzi, 1998, Barucci and Gozzi, 2001, Feichtinger et al., 2003 and Feichtinger et al., 2006) either in cases when explicit solutions can be found or using Maximum Principle techniques.
The problem is rephrased into an infinite dimensional setting, it is proven that the value function is the unique regular solution of the associated stationary Hamilton–Jacobi–Bellman equation, and existence and uniqueness of optimal feedback controls is derived. It is then shown that the optimal path is the solution to the closed loop equation. Similar results were proven in the case of finite horizon by Faggian, 2005b and Faggian, 2008a. The case of infinite horizon is more challenging as a mathematical problem, and indeed more interesting from the point of view of optimal investment models with vintage capital, where what mainly matters is the behavior of optimal trajectories and controls in the long run.
Finally it is explained how the results can be applied to improve the analysis of the optimal paths previously performed by Barucci and Gozzi and by Feichtinger et al.
JEL classification
C61;
C62;
E22
AMS classification
49J20;
49J27;
35B37
The aim of the paper is to develop the dynamic programming (DP) approach for a family of optimal investment models with vintage capital, having infinite time horizon, with particular attention to the behavior of optimal paths in the long run.
We briefly recall that optimal investment models with vintage capital1 have been studied in various papers in recent years, and modeled in various ways. That of optimal control of linear age structured equations is one of the possible approaches undertaken in literature. Such framework has been introduced in Barucci and Gozzi, 1998 and Barucci and Gozzi, 2001 and then studied in various works, among which we highlight (Faggian, 2005b, Faggian, 2008a, Faggian and Gozzi, 2004, Feichtinger et al., 2003, Feichtinger et al., 2004 and Feichtinger et al., 2006). The optimal investment problem with vintage capital is there treated in two main cases:
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In Barucci and Gozzi, 2001, Barucci and Gozzi, 1998, Faggian and Gozzi, 2004 and Feichtinger et al., 2004 the production function is linear and the representative investor is price taker (corresponding to an objective function which is linear in the capital stock). The value function is then linear and the optimal investment strategies, together with the corresponding capital stock trajectories, can be explicitly calculated. Consequently, a deep qualitative analysis of the problem can be performed, including that of the long run behavior of the capital stock.
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In Feichtinger et al. (2006) the production function is linear and a large representative investor is considered (which leads to an objective function which is nonlinear in the capital stock). The value function is then nonlinear and the optimal investment strategies cannot be explicitly calculated. In Feichtinger et al. (2006) Feichtinger et al. make use of a particular version of Maximum Principle (first introduced in Feichtinger et al. (2003)) to analyze the optimal investment strategies, highlighting an anticipation effect. The paper does not analyze the long run behavior of the capital stock.
This second case is undertaken here too, as it is more interesting from the economic point of view, and yields challenging mathematical issues due to the lack of explicit solutions. Since the only paper with similar assumptions is Feichtinger et al. (2006), we clarify the contribution of the present work with respect to Feichtinger et al. (2006):
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We aim to study the long run behavior of the optimal capital stock, while in Feichtinger et al. (2006) the authors analyze the optimal investment dynamics (with no particular attention to properties in the long run);
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we use a DP approach, as the Maximum Principle appears less efficient in the study of optimal trajectories; this drives to the study of a family of Hamilton–Jacobi–Bellman (HJB from now on) equations that do not fit into the existing theory.
More precisely we develop the theory of a general mathematical model that may applies to a variety of examples and that encloses investment with vintage capital as a subcase. Such general mathematical problem is stated in Section 3.1. The main results are contained in Theorem 5.2 and Theorem 5.3 where it is shown that the Value Function is the unique solution of a suitable HJB equation, and in Theorem 5.4 where existence and uniqueness of optimal paths in closed loop form is proven.
Although the present work concerns mainly the theoretical matters, we would like to make clear that it adds both to mathematics and economics: on the one hand, Theorem 5.2 and Theorem 5.3 extend the existing theory of regular solutions of HJB equations in Hilbert spaces to a new set of problems; on the other hand, Theorem 5.4 allows to investigate the properties of the optimal state-control pairs (especially the long run behavior) in the vintage capital problem and in those other applications that can be framed into the same setting.
The paper is organized as follows. In Section 2 we introduce the model of investment with vintage capital, in Section 3 we rewrite the model as an abstract mathematical problem summarizing the main mathematical difficulties. We also review the existing literature on HJB equations in Hilbert spaces.
Then we come to the technical part. In Section 4 we introduce the notation, recall the definition of strong solution and the results on existence and uniqueness of strong solutions in the finite horizon case, as they appear in Faggian (2005b). In Section 5 we study the general mathematical problem and we state the main results. Proofs are postponed in Appendix A. In Section 6 we apply the results to optimal investment with vintage capital, and in Section 7 we drive the conclusions
In this paper we have considered an optimal investment model with vintage capital where the revenue function R is nonlinear. This is motivated, e.g. by the study of the case of large representative investors. We have embedded the problem in a class of optimal control problems in infinite dimension that has not been treated so far in the literature since it contains various nontrivial technical difficulties to overcome. Using the dynamic programming approach we have proven that the value function is the unique solution of the associated HJB equation and, consequently, the existence of optimal feedback controls. We have proved that such results apply to our vintage capital problem and observed that this provides a solid basis to study the long run behavior of the optimal capital trajectory. This will require additional nontrivial work, due to the infinite dimensionality of the problem and will be done in a subsequent paper.
