Carroll and Kimball (1996) have shown that, in the class of utility functions that are strictly increasing, strictly concave, and have nonnegative third derivatives, hyperbolic absolute risk aversion (HARA) is sufficient for the concavity of consumption functions in general consumption-saving problems. This paper shows that HARA is necessary, implying the concavity of consumption is not a robust prediction outside the HARA class.

中文翻译：

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消费函数凹的双曲线绝对风险厌恶的必要性

Carroll 和 Kimball (1996) 表明，在效用函数是严格递增的、严格凹的和具有非负三阶导数的一类中，双曲线绝对风险厌恶 (HARA) 足以满足一般消费储蓄中消费函数的凹性。问题。本文表明 HARA 是必要的，这意味着消费的凹度不是 HARA 类之外的稳健预测。