Multivariate risk attitude: a comparison of alternative approaches in sustainability policies

Risk aversion has an unambiguous meaning in the univariate context: But, what does it mean to be risk averse in the multivariate case? Concave Risk Aversion (CRA) and Multivariate Risk Aversion (MRA) are relevant extensions of the risk aversion concept used in the univariate case to the multivariate case, corresponding to concave and ultramodular utility classes, respectively. Although CRA and MRA can coexist, they are dramatically different in some ways, leading to opposite preferences under some circumstances, as in the face of irreversible risks. We introduce the notions of purely concave and purely multivariate risk aversion, related to disjoint utility classes. We apply the purely risk aversion notions to the field of sustainability, where catastrophic and irreversible outcomes can be faced, in order to highlight and compare the consequences of the two approaches on sustainability policies. In this respect, we provide three main results. First, the kind of risk aversion determines the pursued goal. Second, the principle of rejecting any fair bet is not always preserved. Third, sustainability policies induced by different risk aversions, if repeated, produce final states in which mean-variance criterion holds.