This paper studies the relation between concavity, stochastic or state-dependent utility functions, and risk aversion. Using the common definition of risk aversion, but modified for state-dependent preferences, we show that concavity does not imply risk aversion. Instead, it implies a weaker version of risk aversion, defined herein, and called risk aversion for independent gambles. Furthermore, to characterise the economic meaning of concavity, we define two new risk aversion notions, called uniform risk aversion and uniform risk aversion for independent gambles, respectively. We show that concavity is equivalent to uniform risk aversion for independent gambles, and that concavity plus some additional conditions are equivalent to uniform risk aversion.

本文研究了凹度，随机或依赖于状态的效用函数与风险规避之间的关系。使用风险规避的通用定义，但针对状态依赖的偏好进行了修改，我们表明凹度并不意味着风险规避。取而代之的是，这意味着本文中定义的风险规避的一种较弱的版本，并且被称为独立赌博的风险规避。此外，为了表征凹面的经济含义，我们定义了两个新的风险规避概念，分别称为独立赌博的统一风险规避和统一风险规避。我们证明，凹性等同于独立赌博的统一风险规避，并且凹度加上一些附加条件等同于统一风险规避。