What is the condition of quasi concavity?
The function f of many variables defined on a convex set S is quasiconcave if every upper level set of f is convex. (That is, Pa = {x ∈ S: f(x) ≥ a} is convex for every value of a.) Similarly, every convex function is quasiconvex.
How do you determine if a function is quasi concave?
Reminder: A function f is quasiconcave if and only if for every x and y and every λ with 0 ≤ λ ≤ 1, if f(x) ≥ f(y) then f((1 − λ)x + λy) ≥ f(y). Suppose that the function U is quasiconcave and the function g is increasing.
What is the second order condition for convexity?
According to 2nd-order conditions: for twice differentiable function f, it is convex if and only if ∇2f(x)≥0,∀x∈domf.
How do you prove quasi concavity?
The function f is strictly quasi-concave iff for any x, x ∈ C, if x = x and f(x) ≥ f(x) then for any θ ∈ (0,1), setting xθ = θx + (1 − θ)x, f(xθ) > f(x). The function f is quasi-convex iff −f is quasi-concave.
What is concave utility function?
In expected utility theory for choice under uncertainty, cardinal utility functions of risk averse decision makers are concave. In microeconomic theory, production functions are usually assumed to be concave over some or all of their domains, resulting in diminishing returns to input factors.
How do you determine if a function is convex or concave?
For a twice-differentiable function f, if the second derivative, f ”(x), is positive (or, if the acceleration is positive), then the graph is convex (or concave upward); if the second derivative is negative, then the graph is concave (or concave downward).
What is convex and concave?
Concave means “hollowed out or rounded inward” and is easily remembered because these surfaces “cave” in. The opposite is convex meaning “curved or rounded outward.” Both words have been around for centuries but are often mixed up. Advice in mirror may be closer than it appears.
Is Lnx a Quasiconcave?
ln(x) is (strictly) concave. A function f can be convex in some interval and concave in some other interval.
Why is utility function concave?
(ii) The concavity of the utility function implies that the person is risk averse: a sure amount would always be preferred over a risky bet having the same expected value; moreover, for risky bets the person would prefer a bet which is a mean-preserving contraction of an alternative bet (that is, if some of the …