Optimal stopping time, consumption, labour, and portfolio decision for a pension scheme

In this work we solve in a closed form the problem of an agent who wants to optimise the inter-temporal recursive utility of both his consumption and leisure by choosing: (1) the optimal inter-temporal consumption, (2) the optimal inter-temporal labour supply, (3) the optimal share of wealth to invest in a risky asset, and (4) the optimal retirement age. The wage of the agent is assumed to be stochastic and correlated with the risky asset on the financial market. The problem is split into two sub-problems: the optimal consumption, labour, and portfolio problem is solved first, and then the optimal stopping time is approached. We compute the solution through both the so-called martingale approach and the solution of the Hamilton–Jacobi–Bellman partial differential equation. In the numerical simulations we compare two cases, with and without the opportunity, for the agent, to work after retirement, at a lower wage rate.