Quasiconcave

Definition: A function f(x) mapping from the reals to the reals is quasiconcave if it is nondecreasing for all values of x below some x₀ and nonincreasing for all values of x above x₀. x₀ can be infinity or negative infinity: that is, a function that is everywhere nonincreasing or nondecreasing is quasiconcave.

Quasiconcave functions have the property that for any two points in the domain, say x₁ and x₂, the value of f(x) on all points between them satisfies:

f(x) >= min{f(x₁), f(x₂)}.

Equivalently, f() is quasiconcave iff -f() is quasiconvex.

Equivalently, f() is quasiconcave iff for any constant real k, the set of values x in the domain of f() for which f(x) >= k is a convex set.

The most common use in economics is to say that a utility function is quasiconcave, meaning that in the relevant range it is nondecreasing.

A function that is concave over some domain is also quasiconcave over that domain. (Proven in Chiang, p 390).

A strictly quasiconcave utility function is equivalent to a strictly convex set of preferences, according to Brad Heim and Bruce Meyer (2001) p. 17.

(Econterms)

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