Solving constrained consumption–investment problems by decomposition algorithms

Consumption–investment problems with maximizing utility agents are usually considered from a theoretical viewpoint, aiming at closed-form solutions for the optimal policy. However, such an approach requires that the model be relatively simple: even the inclusion of nonnegativity constraints can prevent the derivation of explicit solutions. In such cases, it is necessary to solve the problem numerically, but standard dynamic programming algorithms can only solve small problems due to the curse of dimensionality. In this paper, we adapt the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve dynamic constrained consumption–investment problems with stochastic labor income numerically. Unlike classical dynamic programming approaches, SDDP allows us to analyze problems with multiple assets, and an internal sampling procedure allows the problems to have a very large, or even infinite, number of scenarios. We start with a simpler problem for which a closed-form solution is known and compare it to the optimal policy obtained by SDDP. We then illustrate the flexibility of our approach by solving a defined contribution pension fund problem with multiple assets, for which no closed-form solution is available.