What is the meaning of quasi-concave?
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 show a function is quasi-concave?
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.
What is a 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.
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.
How do you know if a function is quasi convex?
Understanding Quasiconcave and Quasiconvex Functions – YouTube
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.
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!
Are quasi linear functions 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 Cobb Douglas function 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.
Is a linear function 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.
Is Cobb Douglas function continuous?
The type 2 and the type 1 Cobb-Douglas functions are equal, i.e. It can be noted that , and therefore , is always positive, independent of k, in lines 3, 4, and 5. In line 6, since the function is continuous and therefore bounded, it is Riemann-integrable.
How do you know if a function is concave or convex PDF?
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.
What are the limitations of Cobb-Douglas production function?
Limitations of Cobb Douglas Production Function
It is not possible to combine the different factors due to the scarcity of factors and due to their indivisibility. 2) No single producer raises output for the sake of getting constant returns. The producer aims not at constant returns but at achieving increasing returns.
Why Cobb-Douglas production function is used?
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.
How do you find out if a function is 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.
Can a function be both concave and convex?
(A. 1) or (A. 2) for any X1 = X2. A linear function will be both convex and concave since it satisfies both inequalities (A.
Is Cobb-Douglas function is homogeneous?
The Cobb-Douglas is homogeneous of degree = (+ ).
Is Cobb-Douglas a linear function?
The Cobb-Douglas production function is based on the empirical study of the American manufacturing industry made by Paul H. Douglas and C.W. Cobb. It is a linear homogeneous production function of degree one which takes into account two inputs, labour and capital, for the entire output of the .
Can a function be convex and concave at the same time?
Notice that a function can be both convex and concave at the same time, a straight line is both convex and concave. A non-convex function need not be a concave function. For example, the function f(x)=x(x−1)(x+1) defined on [−1,1].
When a function is concave?
What is concavity? Concavity relates to the rate of change of a function’s derivative. A function f is concave up (or upwards) where the derivative f′ is increasing. This is equivalent to the derivative of f′ , which is f′′f, start superscript, prime, prime, end superscript, being positive.
Are concave functions continuous?
This alternative proof that a concave function is continuous on the relative interior of its domain first shows that it is bounded on small open sets, then from boundedness and concavity, derives continuity. Theorem 1. If f : C → R is concave, C ⊂ Rl convex with non-empty interior, then f is continuous on int(C).
Is Square Root concave?
The sqrt function is concave and increasing, which means it can only take a concave argument.
Why do we use Cobb-Douglas?
How do you check if a function is convex or concave?
How do you check if a function is strictly concave?
Determining the concavity of a function – YouTube