Cobb–Douglas production function

As a Cobb - Douglas function is called in economics a class of functions that is often used for the formulation of utility and production functions. In this case, the application field extends on both micro - and macroeconomic applications.

Historical Background

The Cobb -Douglas function is based on intelligence that is collected Johann Heinrich von Thünen in the first half of the 19th century in agriculture. With its per capita capital gains function p = HQN with h as a level parameters, p and q as income or capital per worker and n he has developed the first indirectly formulated Cobb -Douglas production function as the elasticity of substitution of capital.

In the history of the Cobb - Douglas function often two early works are cited: Coordination of the Laws of Distribution (1894 ) by Philip Wicksteed and Lectures (1901 ) by Knut Wicksell (or Ekonomisk Tidskrift, 1900). In spite of these publications can be shown that Wicksell its functional relationship implicitly in 1895 and 1900 explicitly used.

Thus succeeded to the Swedish economist Knut Wicksell (1851-1926) to formulate the relationships between input and output in an existing elasticity of substitution as a production function in the form known today.

The U.S. economist Paul Howard Douglas began in 1927 with a first formulation of the Cobb -Douglas production function. Douglas looked at the example of the manufacturing industry in the United States 1899-1922 empirically the question of what is the connection between the local production (Y ) and the capital stock (K ) and the number of workers employed (L ) exists. Looking for a functional description of this relationship - a so-called production function - he consulted with a colleague, the mathematician Charles Wiggins Cobb, who previously used to it by Wicksell and Wicksteed function

Suggested.

1928 calculated by the method of least squares for k - the so-called production elasticity of labor - a value of 0.75; the result could later be replicated approximate also the National Bureau of Economic Research with a value of 0.741.

Definition

As a Cobb - Douglas function is generally described as a function, given by

With; and for all.

Regularly in the literature is to dispense with the level parameter (or assumed equal ), since it is obsolete with appropriate scaling of the other factors.

A commonly encountered in particular in the two-goods case, the Cobb -Douglas utility function limitation provides that the exponents add up to just one, so that consequently. Ensures this assumption can be shown, that the function has constant returns to scale. Its justification is that ordinal utility functions can be transformed any positive monotonic by assumption; there can now be but just to show that for any a transformation can be found, according to which the sum of the exponents is actually one.

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 where there are n input factors. It is often used as example, the two-factor Cobb -Douglas production function ( sometimes simplified to with ), where K is the capital and L is labor input.

When used as a utility function (usually u) the amount of the consumed good i called for the two-goods case, one usually uses the form; because of the monotonous transformability of ordinal utility functions of the factor would be obsolete anyway. The exponents found here one to preserve the property of constant returns to scale (which is as described above also justified by the transformability of the utility function ).

Properties

The central feature of the functions of the Cobb -Douglas type is that the exponent of a direct interpretation are available, it is still at just about the elasticity of z with respect. Consider, for example, the Cobb -Douglas production function, so it is in the so -called production elasticity of input factor i - they are approximate to to increase the percentage of the output y as the amount of factor i employed increased by one percent will.

Demand functions, which are obtained from a Cobb -Douglas utility function, have the property that households always spend a constant share of their income for the goods. This application represents an example of why the issue raised in the remaining section property often makes dealing with the function; then that can be directly interpreted as the exponent of the searched constant proportion.

The returns to scale are constant ( the example is, for the case of a production function that the output doubled when doubling the inputs), for decreasing ( a doubling of inputs leads to less than a doubling of output ) and increasing ( doubling the inputs leads to more than a doubling of output ).

The scale elasticity - ie the approximate change in production in percent as a result of a one percent increase in all inputs - is the sum of the exponents.

With a Cobb -Douglas production function, the elasticity of substitution is - that is the ratio of the relative change in the ratio of factor inputs to the relative change in the marginal rate of substitution - always one.

In general, moreover, that the Cobb - Douglas function in the sense defined homogeneous of degree is. She is also quasikonkav for all; concave if and even strictly concave if.

Example

The figure shows a linear homogeneous Cobb -Douglas production function is shown as " production mountains". The area of ​​the rock is made up of straight lines emanating from the origin ( 0,0,0). If you hold one factor of production is constant and increasing the other factor of production, then also increases the output, but always to a lesser extent, the partial marginal productivity of a factor decreases with increasing amounts of this factor from. The partial marginal productivity is the slope of the production mountains, if you move on it perpendicular to the axis of the static factor of production.

Moves the economy along a " contour ", then the use of a factor of production by that of the other is substituted .. It is the law of diminishing marginal rate of technical substitution.