Production Function - Specifying The Production Function

Specifying The Production Function

A production function can be expressed in a functional form as the right side of

where:

quantity of output

quantities of factor inputs (such as capital, labour, land or raw materials).

If Q is not a matrix (i.e. a scalar, a vector, or even a diagonal matrix), then this form does not encompass joint production, which is a production process that has multiple co-products. On the other hand, if f maps from Rn to Rk then it is a joint production function expressing the determination of k different types of output based on the joint usage of the specified quantities of the n inputs.

One formulation, unlikely to be relevant in practice, is as a linear function:

where and are parameters that are determined empirically.

Another is as a Cobb-Douglas production function:

The Leontief production function applies to situations in which inputs must be used in fixed proportions; starting from those proportions, if usage of one input is increased without another being increased, output will not change. This production function is given by

Other forms include the constant elasticity of substitution production function (CES), which is a generalized form of the Cobb-Douglas function, and the quadratic production function. The best form of the equation to use and the values of the parameters vary from company to company and industry to industry. In a short run production function at least one of the 's (inputs) is fixed. In the long run all factor inputs are variable at the discretion of management.