What is strictly quasi-concave?
Strict quasiconcavity That is, a function is strictly quasiconcave if every point, except the endpoints, on any line segment joining points on two level curves yields a higher value for the function than does any point on the level curve corresponding to the lower value of the function.
Does concavity imply strict quasi concavity?
As can be expected from the definition, strong quasi-concavity implies strict quasi-concavity. Theorem 5 Let f（x）be a function of class C2 defined on an open convex set S. If f（x）is strongly quasi-concave, then f（x）is strictly quasi-concave as well. (Proof) Suppose f（x）is not strictly quasi-concave.
What is strictly quasi-concave utility function?
Definition: A function f is strictly quasi-concave if for any two points x and y, x = y, in the domain of f, whenever f(x) ≤ f(y), then f assigns a value strictly higher than f(x) to every point on the line segment joining x and y except the points x and y themselves.
How do you find quasi concavity?
In summary, f is quasiconcave if and only if either a > 0 and c ≥ b2/3a, or a < 0 and c ≤ b2/3a, or a = 0 and b ≤ 0. Use the bordered Hessian condition to determine whether the function f(x,y) = ye−x is quasiconcave for the region in which x ≥ 0 and y ≥ 0.
Is Cobb Douglas strictly concave?
If our f(x, y) = cxayb exhibits constant or decreasing return to scale (CRS or DRS), that is a + b ≤ 1, then clearly a ≤ 0, b ≤ 0, and we have thus the Cobb-Douglas function is concave if and M1 ≤ 0, M1 ≤ 0, M2 ≥ 0, thus f is concave.
How do you determine if a function is convex or concave Hessian?
We can determine the concavity/convexity of a function by determining whether the Hessian is negative or positive semidefinite, as follows. if H(x) is positive definite for all x ∈ S then f is strictly convex.
How do you prove a utility function is quasi concave?
Thus if preferences are convex, the utility function is quasi-concave. To prove the converse, suppose that xº y and x¹zy. Then U(xº) ≥ U(y) and U(x¹) ≥ U(y). If U is quasi-concave, it follows that U(x^) ≥ Min {U(xº), U(x¹)} ≥ U(y).
Why is Cobb-Douglas function is concave?
How do you prove strict convexity?
(1) The function is strictly convex if the inequality is always strict, i.e. if x = y implies that θf ( x) + (1 − θ)f ( y) > f (θ x + (1 − θ) y). (2) A concave function is a function f such that −f is convex.
How do you determine if a function is strictly concave?
- f is concave if and only if H(x) is negative semidefinite for all x ∈ S.
- if H(x) is negative definite for all x ∈ S then f is strictly concave.
- f is convex if and only if H(x) is positive semidefinite for all x ∈ S.
- if H(x) is positive definite for all x ∈ S then f is strictly convex.
Is a linear function strictly quasi concave?
* A function that is both concave and convex, is linear (well, affine: it could have a constant term). Therefore, we call a function quasilinear if it is both quasiconcave and quasiconvex. Example: any strictly monotone transformation of a linear aTx.
What does quasi linear mean in economics?
Definition in terms of preferences In other words: a preference relation is quasilinear if there is one commodity, called the numeraire, which shifts the indifference curves outward as consumption of it increases, without changing their slope.
Is Cobb-Douglas concave or convex?
What is a in Cobb-Douglas function?
K = capital input (a measure of all machinery, equipment, and buildings; the value of capital input divided by the price of capital) A = total factor productivity. α and β are the output elasticities of capital and labor, respectively. These values are constants determined by available technology.
What is the difference between convexity and strict convexity?
Geometrically, convexity means that the line segment between two points on the graph of f lies on or above the graph itself. See Figure 2 for a visual. Strict convexity means that the line segment lies strictly above the graph of f, except at the segment endpoints.
What is the difference between a quasiconcave and concave function?
If you graph a mathematical function and the graph looks more or less like a badly made bowl with a few bumps in it but still has a depression in the center and two ends that tilt upward, that is a quasiconcave function. It turns out that a concave function is just a specific instance of a quasiconcave function—one without the bumps.
What is an example of quasiconcavity?
This function is quasiconcave (its upper level sets are the sets of points above rectangular hyperbolae), but is not concave (for example, f (0, 0) = 0, f (1, 1) = 1, and f (2, 2) = 4, so that f ( (1/2) (0, 0) + (1/2) (2, 2)) = f (1, 1) = 1 < 2 = (1/2) f (0, 0) + (1/2) f (2, 2)). Why are economists interested in quasiconcavity?
Is the converse of strictly quasiconcave (quasiconvex) true?
It should be clear from this that any strictly quasiconcave ( strictly quasiconvex) function is quasiconcave (quasiconvex), but the converse is not true. For a better grasp, let us examine the illustrations in Fig. 12.3, all drawn for the one-variable case.
Is quasiconcave a topological property?
Quasiconcave as a Topological Property. Quasiconcave is a topological property that includes concavity. If you graph a mathematical function and the graph looks more or less like a badly made bowl with a few bumps in it but still has a depression in the center and two ends that tilt upward, that is a quasiconcave function.