List of production functions explained

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):

Y=AF(K,L)

Y=F(K,AL)

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=\left[\partial(slope)
\partial(L/K)
L/K
slope

\right]-1

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

slope=-\partialF(K,L)/\partialK
\partialF(K,L)/\partialL

.

Returns to scale

Returns to scale can be

Some widely used forms

Y=A[\alphaK\gamma+(1-\alpha)L\gamma]

1
\gamma

, with

\gamma\isin[-infty,1]

which includes the special cases of:

Y=A[\alphaK+(1-\alpha)L]

when

\gamma=1

Y=AK\alphaL1-\alpha

when

\gamma\to0

Y=Min[K,L]

when

\gamma\to-infty

\gamma=0

ln(Y)=ln(A)+aLln(L)+aKln(K)+bLLln2(L)+bLKln(L)ln(K)+bKKln2(K)

z

) of each output
n(x
Y=A\prod
i-z
\alphai
i)

Some Exotic Production Functions

Y=AKa[L+baK](1-a)v

a1K+a2L
Y=Ae

K1-bLb

Y=AK\alphaL1-\alpha-mL

y=m-A

n
\prod
i=1
xi
a
i

Y=min\{Y*,\beta1+\beta2L,\beta2+\beta4K\}

where

Y*

is the maximal yield (considers capacity limits).

C(p,y)=\sumibii\left(

byi
y

pi+\sumj:jbij\sqrt{pipj}

by
y

\right)

.

where

c

denotes the cost per unit output, the unit cost,

bij=bji

, and

\sumibij=1

. This cost function reduces to the well-known Generalized Leontief function of Diewert[6] when

byi=0

for all inputs.

By applying the Shephard's lemma, we derive the demand function for input

i

,

xi

:

xi={\partialC\over\partialpi}=bii

byi
y

+

m
style\sum
ij

bij\sqrt{pi/p

by
j}y
Here,

ai

denotes the amount of input

i

per unit of output.

Notes and References

  1. Chambers, R. G. (1988). Applied Production Analysis: A Dual Approach. New York, NY: Cambridge University Press.
  2. https://assets.cambridge.org/97811070/36161/frontmatter/9781107036161_frontmatter.pdf Sickles, R., & Zelenyuk, V. (2019). Measurement of Productivity and Efficiency: Theory and Practice. Cambridge: Cambridge University Press. doi:10.1017/9781139565981
  3. DECISION ASPECTS OF THE SPILLMAN PRODUCTION FUNCTION Janusz Jaworski 1977 https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1744-7976.1977.tb02884.x
  4. Nakamura, Shinichiro. "A nonhomothetic generalized Leontief cost function based on pooled data." The Review of Economics and Statistics (1990): 649-656.
  5. Fuss, Melvyn, and Daniel McFadden, eds. Production economics: A dual approach to theory and applications: Applications of the theory of production. Elsevier, 2014.
  6. Diewert, W. Erwin. "An application of the Shephard duality theorem: A generalized Leontief production function." Journal of political economy 79.3 (1971): 481-507.