Overview
Although it is unlikely to get an exact analytical form for the production function of any given firm, it is useful to have a set of examples that can be fit to the firm's data, or for instructional purposes.
Cobb Douglas Production Function
A common function used to model production is the Cobb Douglas Production function
{% Y = AL^{\beta}K^{\alpha} %}
- {% Y %} - total production
- {% L %} - labor input
- {% K %} - capital input
- {% A %} - total factor productivity
- {% \alpha %},{% \beta %} elasticities
The cobb douglas production function is typically used for explanatory purposes in micro economic theory, but the function could be fit to a firms data, if the company feels that it approximates their production curves well.
The cobb douglas function can be recast as follows:
{% log Y = log A + \beta \times L + \alpha \times K %}
from which the coefficients can be estimated using
standard
regression techniques.
The challenge to fitting Cobb Douglas is to define the variables K and L in a way that is measurable.
Constant Returns to Scale
Constant returns to scale occurs when you proportionally scale all inputs, and the output is then scaled proportionally.
In mathematical terms, this occurs when the exponents in the Cobb Douglas production function sum to one.
{% Y = AL^{\alpha}K^{1 - \alpha} %}