# List of production functions

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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.)

A production function can also be seen as the dynamics of national output/national income. This list is to collect production functions & the dynamics of national output/income that have been used in literature & textbooks.

## Technology

There are three common ways to incorporate technology (or the efficiency with which factors of production are used) into a production function:

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

## 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:

$\ \Epsilon=[\frac {\partial(slope)} {\partial(L/K)} \frac {L/K} {slope}]^{-1}$

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

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

## 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 Famous forms

$Y = A[\alpha K^\gamma + (1-\alpha) L^\gamma]^{\frac{1}{\gamma}}$, with $\gamma \isin [-\infty,1]$ which includes the special cases of:

• Linear production (or perfect substitutes)
$\ Y=A[\alpha K+ (1-\alpha) L]$ when $\ \gamma=1$
$\ Y=AK^\alpha L^{1-\alpha}$ when $\gamma \to 0$
$\ Y=Min[K,L]$ when $\gamma \to -\infty$
• Translog (generalization of the above)

$\ ln(Y)=ln(A)+aL*ln(L)+aK*ln(K)+bLL*ln(L)*ln(L)+$

$\ +bKK*ln(K)*ln(K)+bLK*ln(L)*ln(K)$

• The PAR production technology

The PAR production functions have such flexible limiting properties which are lacking in the CES technology. The Cobb-Douglas production function (a=0) as well as the Leontief production function (a→∞) are special cases of the PAR technology. The general form of the PAR production function is as follows:

Y = A*[c*(D^\−a −L^\−a)/((D/L)^\-a*c-1)]^\−(1/a) (a ≠ 0)(c ≠ 0)(D ≠ L)

  = A**[c*(D^\−a −L^\−a)/(-a*ln(D/L))]^\−(1/a)             (a ≠ 0)(c = 0)(D ≠ L)

  = A* D^\((1-c)/2)* L^\((1+c)/2)                  (a = 0)(-1≤c≤1)(D ≠ L)

  = A*D                            (D = L)


If a=-1, then

Y = A*c*(D-L)/[( D/L)^\c-1] (a = -1)(c ≠ 0)(D ≠ L)

= A*(D-L)/[ln(D/L)] (a = -1)(c = 0)(D ≠ L)

In the above equation, Y is the physical output of production, D is input 1, L is input 2, “c” is the distribution limit parameter and “a” is the substitution parameter. Features of PAR technology are thoroughly examined by Tenhunen (1990).

Lauri Tenhunen, The CES and Par Production Techniques, Income Distribution, and the Neoclassical Theory of Production. Acta Universitatis Tamperensis ser A vol 290. Tampere 1990. http://acta.uta.fi/pdf/978-951-44-8581-7.pdf

## Factors

Some authors also propose "indirect" production function. Production function can be assumed to have more factors in play, and with some factors refined. For instance, $Y=A\,F(Kf, Kw, Lw, Lm, business\, confidence, \, consumer \, confidence)$ while $Kf$ is the fixed capital, $Kw$ is the working capital, $Lw$ is the working labor, $Lm$ is the management labor.