Understanding Preferences: “Demand Types”, and the Existence of Equilibrium with Indivisibilities

We propose new techniques for understanding agents’ valuations. Our classification into “demand types”, incorporates existing definitions (such as substitutes, complements, “strong substitutes”, etc.), and permits additional distinctions. We obtain an easy-to-check necessary and sufficient condition for the existence of a competitive equilibrium for indivisible goods. Our condition generalises many existing results, and provides new insights: contrary to much popular belief, there are more classes of purelycomplements preferences than classes of purely-substitutes preferences for which competitive equilibrium always exists. Our techniques are also powerful when equilibrium cannot be guaranteed from the “demand type”. For a specific set of individual valuations, we often can check for equilibrium existence by simply counting the number of intersection points of the geometric objects we study! Our methods also have applications to matching, and to the Product-Mix Auction, introduced by the Bank of England in response to the financial crisis.