Concepts and Properties of Substitute Goods ∗ Paul Milgrom

We distinguish two notions of substitutes for discrete inputs of a firm. Class substitutes are defined assuming that units of a given input have the same price while unitary substitutes treat each unit as a distinct input with its own price. Unitary substitutes is necessary and sufficient for such results as the robust existence of equilibrium, the robust inclusion of the Vickrey outcome in the core, and the law of aggregate demand, while the class substitutes condition is necessary and sufficient for robust monotonicity of certain auction/tâtonnement processes. We analyze the concept of pseudo-equilibrium which extends, and in some sense approximates, the concept of equilibrium when no equilibrium exists. We characterize unitary substitutes as class substitutes plus two other properties. We extend the analysis to divisible inputs, with a particular focus on robustness of the concepts and their relation to the generalized law of aggregate demand. ∗We are grateful to John Quah, Kevin Roberts and all participants of the Nuffield College Theory Workshop for useful comments. All remaining errors are our own. Milgrom’s research has been supported by the National Science Foundation. †Department of Economics, Stanford University. Email: [email protected]. ‡Nuffield College, Oxford University. Email: [email protected].