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.

中文翻译：

---

最大化预期的消费指数效用，并限制贫困中的预期时间

我们为希望最大化其终生消费的预期折现指数效用的个人计算最佳投资和消费策略，同时对其财富花费低于贫困阈值b的预期时间施加了限制。首先，我们针对财富花费低于b的时间，对相应的（不受约束的）问题进行最优策略的计算，并对其进行持续性的惩罚。这个惩罚充当我们原始约束问题的拉格朗日乘数，因此我们从重铸问题中恢复了针对我们原始问题的最优策略。我们证明（1）如果当前财富大于b，那么随着贫困约束的加剧，最优投资策略将变得更加保守；（2）如果当前财富小于b，则最佳投资策略要么独立于贫困约束，要么随着贫困约束变得更加尖锐而更加激进，具体取决于价值b。我们还表明，随着贫困约束的加剧，最佳消费率（弱）下降。