Quasikonkavität

Quasikonkavität is a property of functions that is related to concavity but is a weaker condition. A function is quasiconcave if its upper level sets are convex. An upper level set of a function f at a value c is the set of all points x such that f(x) is greater than or equal to c. In simpler terms, if you take any two points within an upper level set, the line segment connecting them must also lie entirely within that set.

This property is important in economics, particularly in the analysis of consumer preferences and utility functions. A utility function is quasiconcave if it represents preferences that are convex. This means that a consumer is indifferent between any two bundles of goods if they provide the same utility, and they prefer any weighted average of these two bundles. This reflects a sense of diminishing marginal utility or diversification.

Mathematically, a function f: X -> R, where X is a convex set, is quasiconcave if for any x1, x2 in X and any lambda in [0, 1], the following inequality holds: f(lambda * x1 + (1 - lambda) * x2) >= min(f(x1), f(x2)). This means that the value of the function at a convex combination of two points is greater than or equal to the minimum of the function's values at those two points.

While all concave functions are quasiconcave, the converse is not always true. Quasiconcave functions do not necessarily have the property that the sum of their weighted values is greater than or equal to the value at the weighted sum, which is characteristic of concavity.