# List of production functions

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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.

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):

• 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": ${\displaystyle \ Y=F(AK,L)}$

## 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:

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

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

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.

## Some widely used forms

${\displaystyle Y=A[\alpha K^{\gamma }+(1-\alpha )L^{\gamma }]^{\frac {1}{\gamma }}}$, 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 ${\displaystyle \ \gamma =1}$
${\displaystyle \ Y=AK^{\alpha }L^{1-\alpha }}$ when ${\displaystyle \gamma \to 0}$
${\displaystyle \ Y={\text{Min}}[K,L]}$ when ${\displaystyle \gamma \to -\infty }$
• Translog, a linear approximation of CES via a Taylor polynomial about ${\displaystyle \gamma =0}$
${\displaystyle \ln(Y)=\ln(A)+a_{L}\ln(L)+a_{K}\ln(K)+b_{LL}\ln ^{2}(L)+b_{LK}\ln(L)\ln(K)+b_{KK}\ln ^{2}(K)}$