Convexity in economics

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Convexity is a geometric property with a variety of applications in economics.^[1] Informally, an economic phenomenon is convex when "intermediates (or combinations) are better than extremes". For example, an economic agent with convex preferences prefers combinations of goods over having a lot of any one sort of good; this represents a kind of diminishing marginal utility of having more of the same good.

Convexity is a key simplifying assumption in many economic models, as it leads to market behavior that is easy to understand and which has desirable properties. For example, the Arrow–Debreu model of general economic equilibrium posits that if preferences are convex and there is perfect competition, then aggregate supplies will equal aggregate demands for every commodity in the economy.

In contrast, non-convexity is associated with market failures, where supply and demand differ or where market equilibria can be inefficient.

The branch of mathematics which supplies the tools for convex functions and their properties is called convex analysis; non-convex phenomena are studied under nonsmooth analysis.