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)[1] and Sickles and Zelenyuk (2019, Chapter 6).[2]
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):
Solow-neutral technology, or "capital augmenting":
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
Stone-Geary, a variation of the Cobb-Douglas production function that considers existence of a threshold factor requirement (represented by ) of each output
Some Exotic Production Functions
Variable Elasticity of Substitution Production Function (VES)
Spillman Production Function (This function is referenced in Agricultural Economics Research)
von Liebig Production Function
where is the maximal yield (considers capacity limits).
The
Generalized Ozaki (GO) Cost Function[4] (because of the duality between cost and production functions, a specific technology can be represented equally well by either the cost or production function[5]).
.
where denotes the cost per unit output, the unit cost, , and . This cost function reduces to the well-known Generalized Leontief function of Diewert[6] when for all inputs.
By applying the
Shephard's lemma, we derive the demand function for input , :
Here, denotes the amount of input per unit of output.
References
^Chambers, R. G. (1988). Applied Production Analysis: A Dual Approach. New York, NY: Cambridge University Press.
^Nakamura, Shinichiro. "A nonhomothetic generalized Leontief cost function based on pooled data." The Review of Economics and Statistics (1990): 649-656.
^Fuss, Melvyn, and Daniel McFadden, eds. Production economics: A dual approach to theory and applications: Applications of the theory of production. Elsevier, 2014.
^Diewert, W. Erwin. "An application of the Shephard duality theorem: A generalized Leontief production function." Journal of political economy 79.3 (1971): 481-507.
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)[1] and Sickles and Zelenyuk (2019, Chapter 6).[2]
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):
Solow-neutral technology, or "capital augmenting":
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
Stone-Geary, a variation of the Cobb-Douglas production function that considers existence of a threshold factor requirement (represented by ) of each output
Some Exotic Production Functions
Variable Elasticity of Substitution Production Function (VES)
Spillman Production Function (This function is referenced in Agricultural Economics Research)
von Liebig Production Function
where is the maximal yield (considers capacity limits).
The
Generalized Ozaki (GO) Cost Function[4] (because of the duality between cost and production functions, a specific technology can be represented equally well by either the cost or production function[5]).
.
where denotes the cost per unit output, the unit cost, , and . This cost function reduces to the well-known Generalized Leontief function of Diewert[6] when for all inputs.
By applying the
Shephard's lemma, we derive the demand function for input , :
Here, denotes the amount of input per unit of output.
References
^Chambers, R. G. (1988). Applied Production Analysis: A Dual Approach. New York, NY: Cambridge University Press.
^Nakamura, Shinichiro. "A nonhomothetic generalized Leontief cost function based on pooled data." The Review of Economics and Statistics (1990): 649-656.
^Fuss, Melvyn, and Daniel McFadden, eds. Production economics: A dual approach to theory and applications: Applications of the theory of production. Elsevier, 2014.
^Diewert, W. Erwin. "An application of the Shephard duality theorem: A generalized Leontief production function." Journal of political economy 79.3 (1971): 481-507.