Optimal Consumption and Investment with Welfare Constraints

In this study, we investigate an optimal consumption and investment problem of an economic agent who faces a welfare constraint; the agent does not accept her expected utility (continuation value) falls below a certain fixed level regardless of the time and state. This optimization problem involves an infinite number of constraints. Using a duality approach, we transform infinitely many constraints into a single constraint and define the dual problem, which becomes a two-dimensional singular control problem. The dual problem provides its associated Hamilton-Jacobi-Bellman (HJB) equation with a gradient constraint. Under a general class of utility functions, we obtain an explicit solution to the HJB equation and provide optimal strategies by establishing a duality theorem. As an example, we consider hyperbolic absolute risk aversion (HARA) utility, which may incorporate a government subsidy or a basic support, and provide the solution and its implications