Can Cobb-Douglas Function Be Concave?

Now, let us apply a monotonically increasing transformation to G – the exponential function: exp{G(x,y)} = Axayb = F(x,y). 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 Cobb-Douglas production function concave or convex?

concave function
Since the closure of a convex set is convex, the hypograph of f must be a convex set. Hence, f is a concave function.

What are the limitations of Cobb-Douglas production function?

Since, the Cobb-Douglas (CD) function has been (and is still) abundantly used by economists because it has the advantage of algebraic tractability and of providing a fairly good approximation of the production process. Its main limitation is to impose an arbitrary level for substitution possibilities between inputs.

How do you know if a production function is concave?

This means that when we move to the right along the graph of the production function, the slope of the curve decreases. A function with this property is said to be concave.

What type of function is Cobb-Douglas?

In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly physical capital and labor) and the amount of output that can be produced by

Is PPC concave or convex?

concave
Production possibility curve (PPC) is concave to the origin because marginal opportunity cost (Loss of output of YGain of output of X) of shifting resources from commodity Y to commodity X tends to rise.

Why is PPC concave not convex?

Since resources are use specific, therefore every time when one more unit of a commodity is produced more units of the other commodity are sacrificed that results in increasing marginal opportunity cost which leads to the concave shape of the production possibility curve.

Can a PPC be concave?

PPC is concave to the origin because of increasing Marginal opportunity cost. This is because inorder to increase the production of one good by 1 unit more and more units of the other good have to be sacriced since the resources are limited and are not equally efficient in the production of both the goods.

How do you tell if a function is concave or convex?

A convex function has an increasing first derivative, making it appear to bend upwards. Contrarily, a concave function has a decreasing first derivative making it bend downwards.

Which functions are concave?

A function that has an increasing first derivative bends upwards and is known as a convex function. On the other hand, a function, that has a decreasing first derivative is known as a concave function and bends downwards. We also describe a concave function as a negative of a convex function.

What does it mean when the PPC is concave?

When the PPC is a straight line, opportunity costs are the same no matter how far you move along the curve. When the PPC is concave (bowed out), opportunity costs increase as you move along the curve. When the PPC is convex (bowed in), opportunity costs are decreasing.

Can production possibility curve convex?

It is well-known that if returns to scale differ in different output ranges of the same commodity, the production possibility curve may change its shape from concave to convex to the origin.

Why does PPC have concave shape?

Production Possibility Curve (PPC) is concave to the origin because of the increasing opportunity cost. As we move down along the PPC, to produce each additional unit of one good, more and more units of other good need to be sacrificed. That is, as we move down along the PPC, the opportunity cost increases.

What function has no concavity?

If the graph of a function is linear on some interval in its domain, its second derivative will be zero, and it is said to have no concavity on that interval.

What functions are both concave and convex?

A linear function will be both convex and concave since it satisfies both inequalities (A. 1) and (A. 2).

Is a concave function 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).

What are the limitations of production?

Some of the major limitations of large scale production are as follows: 1. Extent of the Market 2. Nature of Industry 3. Nature of Demand 4.

What is production function uses and limitations?

It is a mathematical function that relates the maximum amount of output that can be obtained from a given number of inputs – generally capital and labor. The production function, therefore, describes a boundary or frontier representing the limit of output obtainable from each feasible combination of inputs.

What are the properties of Cobb-Douglas production function?

The powers of labor and capital (that are β and α) in the C-D production function measure output elasticities of labor (L) and capital (K) respectively. The output elasticity of a factor shows the percentage change in output due to a given percentage change in the number of factor inputs.

What are its useful properties of Cobb-Douglas production function?

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.