Price quasi-equilibrium with transfers ($y{\succ }_{a}f(a)⟹p\cdot y\ge p\cdot f(a)$)

A price quasi-equilibrium with transfers is an exactly feasible consumption allocation $f$ and a nonzero price vector $p$ for which, almost everywhere, every strictly preferred bundle has cost at least $p\cdot f(a)$. The transfer-adjusted wealth is $p\cdot f(a)$; transfers $p\cdot (f(a)-\omega (a))$ sum to zero. Equal-cost strict improvements may remain, which distinguishes a quasi-equilibrium from a utility-maximizing equilibrium. Nonnegative prices need extra assumptions if goods may be undesirable.

For an atomless measure of consumers with measurable, complete, transitive and continuous preference relations, every exactly feasible Pareto efficient consumption allocation admits nonzero quasi-equilibrium supporting prices. Countably many rational strict-improvement directions remove all exceptional null sets simultaneously. If their convex hull contained zero, sufficiently small disjoint cohorts could make those improvements in exactly balancing proportions, violating Pareto efficiency. A supporting hyperplane theorem then supplies the prices. Individual convexity of preferences is unnecessary.