Abstract
This chapter considers the Monge–Kantorovich problem in the one-dimensional case, when both the worker and the job are characterized by a scalar attribute. The important assumption of positive assortative matching, or supermodularity of the matching surplus, is introduced and discussed. As a consequence, the primal problem has an explicit solution (an optimal assignment) which is related to the probabilistic notion of a quantile transform, and the dual problem also has an explicit solution (a set of equilibrium prices), which are obtained from the solution to the primal problem. As a consequence, the Monge–Kantorovich problem is explicitly solved in dimension one under the assumption of positive assortative matching.
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