What are quasi concave preferences?
What are quasi concave preferences?
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.
Why quasi concavity is important?
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.
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.
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).
How do you show quasi concavity?
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.
Is quasi concavity ordinal?
The next theorem states that any monotonic transformation of a quasiconcave function is quasiconcave. This means that quasiconcavity is in fact an ordinal property!
What is 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.
What is a quasi concave production function?
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.
Are quasi linear functions convex?
* 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.
Is Cobb Douglas 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.
What is quasi-concave utility function?