List of production functions

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):

• Hicks-neutral technology, or "factor augmenting": ${\displaystyle \ Y=AF(K,L)}$
• Harrod-neutral technology, or "labor augmenting": ${\displaystyle \ Y=F(K,AL)}$
• Solow-neutral technology, or "capital augmenting": ${\displaystyle \ Y=F(AK,L)}$

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:

${\displaystyle \ \epsilon =\left[{\frac {\partial (slope)}{\partial (L/K)}}{\frac {L/K}{slope}}\right]^{-1}}$

where "slope" denotes the slope of the isoquant, given by

${\displaystyle \ slope=-{\frac {\partial F(K,L)/\partial K}{\partial F(K,L)/\partial L}}.}$

Returns to scale 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.

• Constant elasticity of substitution (CES) function:

${\displaystyle Y=A[\alpha K^{\gamma }+(1-\alpha )L^{\gamma }]^{\frac {1}{\gamma }}}$, with ${\displaystyle \gamma \in [-\infty ,1]}$

which includes the special cases of:

• Linear production (or perfect substitutes)

${\displaystyle \ Y=A[\alpha K+(1-\alpha )L]}$ when ${\displaystyle \ \gamma =1}$

• Cobb–Douglas production function (or imperfect complements)

${\displaystyle \ Y=AK^{\alpha }L^{1-\alpha }}$ when ${\displaystyle \gamma \to 0}$

• Leontief production function (or perfect complements)

${\displaystyle \ Y={\text{Min}}[K,L]}$ when ${\displaystyle \gamma \to -\infty }$

• Translog, a linear approximation of CES via a Taylor polynomial about ${\displaystyle \gamma =0}$

${\displaystyle \ln(Y)=\ln(A)+a_{L}\ln(L)+a_{K}\ln(K)+b_{LL}\ln ^{2}(L)+b_{LK}\ln(L)\ln(K)+b_{KK}\ln ^{2}(K)}$

• Stone-Geary,a variation of the Cobb-Douglas production function that considers existence of a threshold factor requirement (represented by ${\displaystyle z}$) of each output

${\displaystyle Y=A\prod _{i=1}^{n}(x_{i}-z_{i})^{\alpha _{i}}}$

• Zellener-Revankar Production Function (ZRPF)

${\displaystyle \ln Y+\alpha Y=\beta _{0}+\beta _{1}\ln K+\beta _{2}\ln L}$