Why Prices Need Algorithms

Understanding when equilibria are guaranteed to exist is a central theme in economic theory, seemingly unrelated to computation. This paper shows that the existence of pricing equilibria is inextricably connected to the computational complexity of related optimization problems: demand oracles, revenue-maximization, and welfare-maximization. This relationship implies, under suitable complexity assumptions, a host of impossibility results. We also suggest a complexity-theoretic explanation for the lack of useful extensions of the Walrasian equilibrium concept: such extensions seem to require the invention of novel polynomial-time algorithms for welfare-maximization.