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...

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...

This paper analyzes collective decision making when individual preferences evolve through learning.  Votes are affected by their anticipated effect on future preferences.  The analysis is conducted in a two-arm bandit model with a safe alternative and a risky alternative whose payoff...

In many economic applications involving comparisons of multivariate distributions, supermodularity of an objective function is a natural property for capturing a preference for greater interdependence.  One multivariate distribution dominates another according to the supermodular stochastic...

The single crossing property plays a crucial role in monotone comparative statics (Milgrom and Shannon (1994)), yet in some important applications the property cannot be directly assumed or easily derived.  Difficulties often arise because the property cannot be aggregated: the sum of two...

Whether they are financial, economic, or psychological, discount rates affect most economic decisions: investment and savings, hirings and firings, defaults and refinancing, financial and economic reforms, learning and experimentation, and any other decision with long-term consequences, such as...