Is Cobb-Douglas Strictly Quasi-Concave?

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.

How do you know if a function is quasi-concave?

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 does strictly quasi-concave mean?

Strictly Quasi-concave. It is described as the real-valued function which is defined on a convex subset or on an interval. A function is said to be strictly quasi-concave if every point on any line segment except the endpoints. The inverse image of any set or subset of the functional form is depicted as a convex set.

What type of function is Cobb-Douglas?

The Cobb-Douglas (CD) production function is an economic production function with two or more variables (inputs) that describes the output of a firm. Typical inputs include labor (L) and capital (K). It is similarly used to describe utility maximization through the following function [U(x)].

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. It is strictly quasi-convex iff −f is strictly quasi-concave.

Is a linear function strictly quasi concave?

▶ Linear functions are concave but not strictly concave. maximizers: if x and y are local maxima, then so are all of the points on the line segment connecting x to y. We say a function f is quasi concave over an interval X ⊂ R if ∀x,y ∈ X and δ ∈ (0,1), we have f (δx + (1 − δ)y) ≥ min{f (x),f (y)}.

Are linear functions 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 is the difference between concave and strictly concave?

Functions of a single variable
If f is twice-differentiable, then f is concave if and only if f ′′ is non-positive (or, informally, if the “acceleration” is non-positive). If its second derivative is negative then it is strictly concave, but the converse is not true, as shown by f(x) = −x⁴.

Is quasi-concave the same as convex?

f concave implies the upper contour set is convex. f quasi-concave is equivalent to the upper contour set being convex. If f : Rn → R is concave, then f is quasiconcave. If f : Rn → R is convex, then f is quasiconvex.

Can a convex function be quasi-concave?

All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity.

Are Cobb-Douglas functions 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 special about Cobb-Douglas?

A Cobb-Douglas production function models the relationship between production output and production inputs (factors). It is used to calculate ratios of inputs to one another for efficient production and to estimate technological change in production methods.

What are the properties of Cobb-Douglas?

Major Properties of the Cobb-Douglas Production Function

• Q=A.K^αL^β
• The C-D Production Function Can be Used to Measure the Returns to Scale.
• The Factor Intensity (A Relative Importance of Factor in Production Process)
• Average Physical Productivity of Inputs.

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.

Is the sum of two quasi concave functions necessarily quasi concave?

Let f1,f2 be two smooth quasiconcave functions defined on a convex subset of Rd. It is known that f1+f2 is not necessarily quasiconcave.

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!

How do you determine if a function is strictly convex or concave?

To find out if it is concave or convex, look at the second derivative. If the result is positive, it is convex. If it is negative, then it is concave. To find the second derivative, we repeat the process using as our expression.

Is entropy strictly concave?

This is verified just as we verified that the Shannon entropy is strictly concave. In particular, we can use this to conclude that the relative entropy is always non-negative: D(p q) ⩾ 0 for all p, q ∈ ∆Ω, with equality if and only if p q.

Is a linear function strictly convex?

Linear functions are convex, but not strictly convex. Lemma 1.2.

Are quasi-linear functions convex?

A characteristic feature of quasi-linear preferences is that they are not strictly convex.

What is quasi-linear production function?

A production function is called quasi-sum if there are continuous strict monotone functions F , h 1 , … , h n with such that f ( x ) = F ( h 1 ( x 1 ) + ⋯ + h n ( x n ) ) (cf. Aczél and Maksa (1996) [1]). A quasi-sum production function is called quasi-linear if at most one of F , h 1 , … , h n is a nonlinear function.