Quasi-concave (function)

Definition

Interpretation
u is quasi-concave means:

(1) the upper contour set is strictly convex.
Or if is convex for all

(2) for , :

strictly quasi-concave

f is strictly quasi-concave if strict inequality holds for λ in (0, 1) and

If f is concave, then it is quasi-concave.

A quasi-concave function CANNOT be "U" shape.

See also

• quasi-convex

Properties of Quasi-Concave functions

• Any concave (convex) function is quasi-concave (quasi-convex) but not vice versa
• If f is a linear function, then it is both quasi-convex and quasi-concave
• If f is quasi-concave, then -f is quasi-convex and vice versa
• Any monotone non-decreasing transformation of a concave function results in a quasi-concave function
• Quasi-concave and quasi-convex functions are not necessarily continuous in the interior of their domains.
• Quasi-concave (quasi-convex) functions can have local maximum (minimum) that are not global maximum (minimum)
• First-order conditions are not sufficient to identify even local optima under quasi-concavity.

Quasi-concave function and maximum

Suppose is a strictly quasi-concave function on the convex set . Then, any local maximum of f on is also a global maximum of f on . Moreover, the set of maximizes of f on is either empty or a singleton.

Let f and be (continuously differentiable) quasi-concave functions mapping the open and convex set into . Define:

Suppose there exists and such that the Kuhn-Tucker first-order conditions are met:

• KT-1:
• KT-2: and for

Note: is the ith constraint
D means the derivative with respect to

Then maximizes f over provided at least one of the following conditions holds:

• QC-1:
• QC-2: f is concave.

Note

Example