How do you show 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. Show that the function f defined by f(x) = g(U(x)) is quasiconcave. Suppose that f(x) ≥ f(y).
What is quasi-concave utility function?
In microeconomics, quasiconcave utility functions imply that consumers have convex preferences. Quasiconvex functions are important also in game theory, industrial organization, and general equilibrium theory, particularly for applications of Sion’s minimax theorem.
Is linear function strictly quasiconcave?
In view of Theorem II, a linear function must also be both quasiconcave and quasiconvex, though not strictly so. In the case of concave and convex functions, there is a useful theorem to the effect that the sum of concave (convex) functions is also concave (convex).
Is x1x2 convex?
It is not quasiconcave or concave. 1/x2 on R × R++. [ 1 −x1/x2 ] ≽ 0. Therefore, f is convex and quasiconvex.
How do you know if a utility function is quasiconcave?
Definition: A function is quasiconcave if all of its upper contour sets are convex. Definition: A function is quasiconvex if all of its lower contour sets are convex. So in most of the economics you do, the assumption you will see is that utility functions are quasi-concave.
What are quasi-concave preferences?
A utility function is quasi–concave if and only if the preferences represented by that utility function are convex. A utility function is strictly quasi–concave if and only if the preferences represented by that utility function are strictly convex.
Is a linear function quasi concave?
~ Theorem ill (linear function) If f(x) is a linear function, then it is quasi concave as well as quasiconvex. Theorem I follows from the fact that multiplying an inequality by -1 reverses the sense of inequality. Let f(x) be q~iconcave, with f(v) ::: f(u).
Is the MAX function quasiconcave?
Convexity of follows directly from the following. Theorem : Pointwise maximum of convex functions is convex, and therefore also quasiconvex.
How do you prove ex is convex?
Convex: see the following figure. The function ex is differentiable, and its second derivative is ex > 0, so that it is (strictly) convex. Hence by a result in the text the set of points above its graph, {(x, y): y ≥ ex} is convex.
Is XA convex function?
The absolute value function f(x)=|x| is convex (as reflected in the triangle inequality), even though it does not have a derivative at the point x=0. Now we know that f′(x)=1, for x>0 and f′(x)=−1, for x<0. Considering all values of x≠0, we can still conclude that f″(x)=0 for all x≠0.
Is Cobb Douglas a quasiconcave?
Thus, we can write any such Cobb-Douglas function as a monotonic transformation of a concave (also Cobb-Douglas) function, which proves that the function is quasiconcave.