Constant elasticity of substitution

As a CES function ( short for English- constant elasticity of substitution - "constant elasticity of substitution " ) is called in economics a class of functions, which are characterized in that they have at any point in its domain of definition the same elasticity of substitution. This property is in a variety of economic applications - be it in the micro - or macro-economic area - advantageous. For certain parameter constellations go from the general CES function also special classes of functions produced that are also widely used.

In scientific practice, see CES functions, among others, as demand functions ( CES demand function ), utility functions ( CES utility function ) and the production function ( CES production function) use.

Definition

As the CES function is generally described as a function

With; and for all as well.

This is ( yet to be explained reason) the degree of homogeneity of the function and the elasticity of substitution. Almost always puts you and usually also.

If you use the function as a production function, they are referred to on a regular basis with y ( instead of z ) to express that it indicates the quantity of a good produced. The are then available for the amount of input factor i used, and there are precisely n input factors. It is often used as example, the two-factor CES production function ( sometimes also with the default), where K is the capital and L is labor input; is in a introduced by Robert Solow in the theory of growth Version field.

When used as a utility function (usually u) denotes the amount of the consumed goods i

Properties

It can be shown that the CES function in the defined sense is just homogeneous of degree. Furthermore, it is quasiconvex, for quasikonkav. For, while it is also concave and even strictly concave.

Special cases

It can be shown that the CES function overrides for a function in the Cobb -Douglas type () and for in a Leontief () function.

Specific parameter constellations allow for further clarifications. For example, with the CES type with elasticity of substitution. For converges and z is reduced to linear homogeneous Cobb - Douglas function. For the following turn and it result in the limit the Leontief function.