Substitute Valuations, Auctions, and Equilibrium with Discrete Goods

For economies in which goods are available in several (discrete) units, this paper identifies two notions of substitutes. The weaker notion guarantees monotonicity of tatonnement processses and convergence of clock auctions to a pseudo-equilibrium, but only the stronger notion, which treats each unit traded as a distinct good with its own price, guarantees that every pseudo-equilibrium is a Walrasian equilibrium, the Vickrey outcome is in the core, and the law of aggregate demand is satisfied. The paper provides several characterizations and properties of weak and strong substitutes.