An Optimal Matching Problem

Suppose we are given three goods, X,Y , and Z. They are not homogeneous, but come in different qualities, x ∈ X , y ∈ Y and z ∈ Z. Goods X and Y are used for the sole purpose of producing good Z, which we are interested in. To obtain one piece of good Z, one has to assemble one piece of good X and one piece of good Y . More precisely, one can obtain a piece of quality z by assembling one piece of quality x and one piece of quality y, yielding a benefit of u (x, z) + w (y, z). Given the distributions μ and ν of goods X and Y , one wishes to minimize the total benefit of production. This translates into the following optimization problem: find maps s : X → Z and t : Y → Z such that s (μ) = t (ν) (this is the matching condition) and the integral