This is a list of
production function
In economics, a production function gives the technological relation between quantities of physical inputs and quantities of output of goods. The production function is one of the key concepts of mainstream neoclassical theories, used to define ...
s that have been used in the
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
literature. Production functions are a key part of modelling
national output and
national income
A variety of measures of national income and output are used in economics to estimate total economic activity in a country or region, including gross domestic product (GDP), gross national product (GNP), net national income (NNI), and adjusted nat ...
. 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
Capital may refer to:
Common uses
* Capital city, a municipality of primary status
** List of national capital cities
* Capital letter, an upper-case letter Economics and social sciences
* Capital (economics), the durable produced goods used fo ...
(K), and
labor
Labour or labor may refer to:
* Childbirth, the delivery of a baby
* Labour (human activity), or work
** Manual labour, physical work
** Wage labour, a socioeconomic relationship between a worker and an employer
** Organized labour and the la ...
(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
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
, ''F'' is a production function, and ''Y'' is the amount of physical output produced):
*
Hicks-neutral technology, or "factor augmenting":
* Harrod-neutral technology, or "labor augmenting":
* Solow-neutral technology, or "capital augmenting":
Elasticity of substitution
The elasticity of substitution between
factors of production
In economics, factors of production, resources, or inputs are what is used in the production process to produce output—that is, goods and services. The utilized amounts of the various inputs determine the quantity of output according to the rel ...
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
In economics, returns to scale describe what happens to long-run returns as the scale of production increases, when all input levels including physical capital usage are variable (able to be set by the firm). The concept of returns to scale arises ...
can be
* Increasing returns to scale: doubling all input usages more than doubles output.
* Decreasing returns to scale: doubling all input usages less than doubles output.
* Constant returns to scale: doubling all input usages exactly doubles output.
Some widely used forms
*
Constant elasticity of substitution
Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions. Several economists have featured in the topic and have contributed in the final finding of the constant. They include Tom M ...
(CES) function:
::
, with