List of production functions

This is a list of production functions that have been used in the economics literature. Production functions are a key part of modelling national output and national income. For a much more extensive discussion of various types of production functions and their properties, their relationships and origin, see Chambers (1988) and Sickles and Zelenyuk (2019, Chapter 6).
The production functions listed below, and their properties are shown for the case of two factors of production, capital (K), and labor (L), mostly for heuristic purposes. These functions and their properties are easily generalizable to include additional factors of production (like land, natural resources, entrepreneurship, etc.)
Technology.
There are three common ways to incorporate technology (or the efficiency with which factors of production are used) into a production function (here "A" is a scale factor, "F" is a production function, and "Y" is the amount of physical output produced):
Elasticity of substitution.
The elasticity of substitution between factors of production is a measure of how easily one factor can be substituted for another. With two factors of production, say, "K" and "L", it is a measure of the curvature of a production isoquant. The mathematical definition is:
where "slope" denotes the slope of the isoquant, given by
Returns to scale.
Returns to scale can be