Table of Contents
Is a quasi concave function convex?
A function with the property that for every value of a the set of points (x, y) such that f(x, y) ≥ a—the set of points inside every contour on a topographic map—is convex is said to be quasiconcave.
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.
When a function is called quasi concave?
Advertisements. Let f:S→R where S⊂Rn is a non-empty convex set. The function f is said to be quasiconvex if for each x1,x2∈S, we have f(λx1+(1−λ)x2)≤max{f(x1),f(x2)},λ∈(0,1) For example, f(x)=x3.
Is e x quasi convex?
If f : Rn → R is convex, then f is quasiconvex. ex is quasiconcave but not concave. In fact it is also convex and quasiconvex.
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.
Can a function be both quasi concave and quasi convex?
Definition and properties A quasilinear function is both quasiconvex and quasiconcave. The graph of a function that is both concave and quasiconcave on the nonnegative real numbers.
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.
What are quasi concave preferences?
Right: Preferences are strictly quasiconcave. Quasiconcavity implies that an individual’s indifference curves are convex; see Figure 3 for two examples.
Are quasi linear preferences convex?
A characteristic feature of quasi-linear preferences is that they are not strictly convex. Under such preferences, the existence and uniqueness of an interior optimal allocation is not, in general, guaranteed.
Is a linear function quasi concave?
What are quasi linear goods?
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.
What is quasi-linear equation?
What are Quasi-linear Partial Differential Equations? A partial differential equation is called a quasi-linear if all the terms with highest order derivatives of dependent variables appear linearly; that is, the coefficients of such terms are functions of merely lower-order derivatives of the dependent variables.
Is quasi-linear monotonic?
Thus, utility function (c) is also a quasi-linear function, because it is just a monotonic function of (b). This is worth keeping in mind because utility function (c) is concave in w, so it represents a risk-averse agent. Thus, the marginal rate of tradeoff between w and e depends on both e and w.
What are Cobb Douglas preferences?
Cobb-Douglas preferences are the standard example of indifference curves that look well-behaved, and in fact the formula describing them is about the simplest algebraic expression that generates well-behaved preferences.
Is Cobb-Douglas convex?
Recall that we work in the first orthant x > 0, y> 0, and a, b, c > 0. 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.
Are Cobb-Douglas preferences convex?
Note how different values of the parameters c and d lead to different shapes of the indifference curves. Cobb-Douglas indifference curves look just like the nice convex mono-tonic indifference curves that we referred to as “well-behaved indifference curves” in Chapter 3.