Supermodularity

Supermodularity is a concept in economics and mathematics that describes a specific type of convexity for functions of multiple variables. A function is supermodular if it exhibits increasing differences. Mathematically, for a function f defined on a subset of R^n, it is supermodular if for any x, y, and any two coordinates i and j, the following inequality holds: f(x + e_i + e_j) - f(x + e_i) >= f(x + e_j) - f(x), where e_i and e_j are standard basis vectors. This means that the marginal benefit of increasing one variable is non-decreasing as the other variable increases. In simpler terms, if you have two factors that contribute to an outcome, the more you have of one factor, the more beneficial it becomes to have more of the other factor.

This property has significant implications in various fields. In economics, it is crucial for analyzing situations involving complementarities. For example, in production, if labor and capital are complements, their joint productivity might exhibit supermodularity. This means that as you add more capital, the productivity gain from adding an extra unit of labor increases. Supermodular functions are also important in game theory, particularly in the study of games with strategic complementarities, where a player's best response function is non-decreasing in the other players' actions. The existence of supermodularity often simplifies optimization problems and guarantees certain properties of equilibria.