Efficiency in Pure‐Exchange Economies With Risk‐Averse Monetary Utilities

We study Pareto efficiency in a pure‐exchange economy where agents' preferences are represented by risk‐averse monetary utilities. These coincide with law‐invariant monetary utilities, and they can be shown to correspond to the class of monotone, (quasi‐)concave, Schur concave, and translation‐invariant utility functionals. This covers a large class of utility functionals, including a variety of law‐invariant robust utilities. Given that Pareto optima exist and are comonotone, we provide a crisp characterization thereof in the case of law‐invariant positively homogeneous monetary utilities. This characterization provides an easily implementable algorithm that fully determines the shape of Pareto‐optimal (PO) allocations. In the special case of law‐invariant comonotone‐additive monetary utility functionals (concave Yaari‐dual utilities), we provide a closed‐form characterization of Pareto optima. As an application, we examine risk‐sharing markets where all agents evaluate risk through law‐invariant coherent risk measures, a widely popular class of risk measures. In a numerical illustration, we characterize PO risk‐sharing for some special types of coherent risk measures.