A characterization of two-agent Pareto representable orderings

Partial orders defined on a nonempty set $X$ admitting a two-agent Pareto representation are characterized. The characterization is based upon the fulfillment of two axioms. The first one entails the existence, for any point $x\in X$, of a very particular decomposition of the points which are incomparable to $x$. The second one encodes a separability condition. Our approach is then applied to show that if the cardinality of $X$ is, at most, 5, then a two-agent Pareto representation always exists whereas this need not be the case otherwise. The connection with the concept of the dimension of a poset is also discussed. Certain examples are also presented that illustrate the scope of our tools.