Linear Pricing Mechanisms for Markets without Convexity

We introduce two linear pricing mechanisms for quasilinear economies in which market-clearing prices may not exist. Electricity markets, fisheries markets, and many others include producers with start-up costs, ramping costs, or other fixed costs that fail the convexity assumptions traditionally used to prove that clearing prices exist. Each mechanism relaxes a condition of Walrasian equilibrium. While the Walrasian mechanism determines payments among buyers and sellers using a single price vector, our markup mechanism allows one more parameter -- a multiplier -- that marks up the prices paid by buyers above those paid to sellers. These markups allow the mechanism to avoid budget deficits even when non-convexities lead to failures of market-clearing. And while the Walrasian mechanism assigns each producer its preferred production plan, our rationing mechanism carefully rations some buyers. Both mechanisms always produce feasible allocations, avoid budget deficits, and are computationally tractable. The proportion of efficient surplus lost in the markup mechanism is O(1/N), where N is the number of buyers and sellers. When agents on the buyer side have convex preferences and strongly monotone demand, the rationing mechanism suffers a smaller welfare loss, namely O(1/N2-ε) for all ε>0. Importantly, both mechanisms have good large-market incentive properties similar to those of the Walrasian mechanism. Key to our construction of these mechanisms and of some independent interest is our new Bound-Form First Welfare Theorem for quasilinear economies, which gives an upper bound on the deadweight loss of any feasible allocation ω in terms of any positive price vector p. It asserts that the welfare loss is bounded above by B+R, where B is the budget deficit from ω at prices p, which is non-zero when supply strictly exceeds demand, and R is the sum of the rationing losses suffered by each individual agent n when its allocated bundle ωn is different from its preferred bundle at price vector p. The Bound Form First Welfare Theorem takes its name from its implication that the welfare loss is zero when (p,ω) is a competitive equilibrium. The full paper is available at https://mitchwatt.github.io/files/PricingMechanismsNonConvex.pdf.