  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. We show
that Pareto optima exist and are comonotone, and 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 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
Pareto-optimal risk-sharing for some special types of coherent risk measures.
