Production Function

Production involves the transformation of inputs into outputs. Production involves the transformation of inputs into physical output. The output is thus a function of factors which are also called inputs. The term ‘production function’ refers to the relationship between the inputs and outputs produced by them. The functional relationship between physical inputs and physical output of a firm is known as production function. Algebraically, production function can be expressed as,

Q = f ( Ld, L, K, M…. )
Where, 
Q stands for the quantity of output, Ld, L, K and M stand for the land, labor, capital and management respectively.

The above equation shows that the quantity (Q) of output produced depends upon the quantities of the factors used. Simply, production function expresses the relationship between the quantity of output and the quantities of the various inputs used for the production. More precisely, the production function states the maximum quantity of output that can be produced with
by given quantities of various inputs.

In economic theory, there are two types of production functions on time basis. The production function when the quantities of some inputs such as capital and labor are kept constant and the quantity of one input such as labor is varied. This kind of production function [Q = f (K, L)] is known as short-run production function. The study of short-run production function is the subject-matter of the law of diminishing returns which is also called the law of variable proportion.

Secondly, we study production function (input – output relation) by varying all inputs, and this is called long-run production function and can be expressed as Q = f (Ld, L, K, M ). This form is the subject-matter of the law of returns to scale. Generally, the terms constant and increasing returns are used with reference to constant and increasing returns to scale.

Besides these, there are other production function, such as

1) Linear Production Function

A linear production function would take the form:

Total production Y = a + bx

From this function for the managerial production will be,

Y/X.a/x+b

The equation for the marginal product will be,

∆Y/∆X = b

2) Power Function

A power function expresses output Y, as a function of input in the form:

y = AXα

It contains certain properties:
a) The exponents are the elasticity of production. Thus, in the above function, the exponent α represents the elasticity of production.
b) The equation is linear in the logarithms, i.e., it can be expressed as, 
log Y = logA + αlogX
when the power function is expressed in logarithmic form as above, the coefficient α represents the elasticity of production.

c) If one input is increased while all others are held constant, marginal product will be decline.

3) Quadratic Production Function

The production function may be quadratic, taking the following form:

Y = a + bx – cx2

Where, the dependent variable Y shows total output and the independent variable X represents input. The small letters are parameters; their probable values are determined by a statistical analysis of the data.

Properties of quadratic function:
a) The minus sing in the last term denotes diminishing marginal returns.
b) The equation allows for decreasing marginal product but not for both increasing and decreasing marginal products.
c) The elasticity of production is not constant at all points along the curve as in a power function, but declines with input magnitude.
d) The equation never allows for an increasing marginal product.

4) Cubic Production Function

The cubic production function takes the following form:

Y = a + bx + cx2 – dx3

Some important special properties of a cubic production function are:
a) It allows for both increasing and decreasing marginal productivity.
b) The elasticity of production varies at each point the curve.
c) Marginal productivity decreases at an increasing rate in the later stages.

5) Power Production Function (Cobb-Douglas Function)

Power functions have been employed in a large number of empirical production studies, particularly since Charles W. Cobb and Paul H. Douglass’s pioneering work in the late 1920s. The impact of this work was so great that power production functions are now frequently referred to as Cobb-Douglas production functions.

Cobb-Douglas production function can be expressed as,

Q = AKα Lβ
Where, Q = total output
L = index of employment of labor in manufacturing
K = index of fixed capital in manufacturing

The exponents α and β are the elasticities of production i.e., α and β measure the percentage response of output to percentage changes in labor and capital respectively.

Properties of Power Function
a) Power functions allow the marginal productivity of a given input to depend upon the levels of all inputs employed a condition that often holds in actual production systems.
b) They are linear in logarithms and thus can be easily analyzed using linear regression analysis.
log Q = log A + α log K + β log L
The least squares technique can be used to estimate the coefficients of equation and thereby the parameters of equation.

c) Power functions facilitate returns to scale estimation. Returns to scale are easily calculated by summing the exponents of the power function. If the sum of the exponent is less than one, (α + β < 1) diminishing returns are included. A sum greater than one (α + β > 1) indicates increasing returns. Finally, if the sum of the exponent is exactly one (α + β = 1), returns to scale are constant.

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