On social preferences and the intensity of risk aversion

We study the relative risk aversion of an individual with particular social preferences: his wellbeing is influenced by his relative wealth, and by how concerned he is about having low relative wealth. Holding constant the individual's absolute wealth, we obtain two results. First, if the individual's level of concern about low relative wealth does not change, the individual becomes more risk averse when he rises in the wealth hierarchy. Second, if the individual's level of concern about low relative wealth intensifies when he rises in the wealth hierarchy and if, in precise sense, this intensification is strong enough, then the individual becomes less risk averse: the individual's desire to advance further in the wealth hierarchy is more important to him than possibly missing out on a better rank.