Maximizing expected exponential utility of consumption with a constraint on expected time in poverty

We compute the optimal investment and consumption strategies for an individual who wishes to maximize her expected discounted exponential utility of lifetime consumption, while imposing a constraint on the expected time her wealth spends below a poverty threshold b. First, we compute the optimal strategies for the corresponding (unconstrained) problem with a running penalty for time that wealth spends below b. This penalty acts as a Lagrange multiplier for our original constrained problem, so we recover the optimal strategies for our original problem from the recast problem. We show that (1) if the current wealth is greater than b, then the optimal investment strategy becomes more conservative as the poverty constraint becomes sharper; and (2) if the current wealth is less than b, then the optimal investment strategy is either independent of the poverty constraint or becomes more aggressive as the poverty constraint becomes sharper, depending on the value b. We also show that the optimal rate of consumption (weakly) decreases as the poverty constraint becomes sharper.