In simple words, production function refers to the functional relationship between the quantity of a good produced (output) and factors of production (inputs).

“The production function is purely a technical relation which connects factor inputs and output.” Prof. Koutsoyiannis

Defined production function as “the relation between a firm’s physical production (output) and the material factors of production (inputs).” Prof. Watson

In this way, production function reflects how much output we can expect if we have so much of labour and so much of capital as well as of labour etc. In other words, we can say that production function is an indicator of the physical relationship between the inputs and output of a firm.

The reason behind physical relationship is that money prices do not appear in it. However, here one thing that becomes most important to quote is that like demand function a production function is for a definite period.

It shows the flow of inputs resulting into a flow of output during some time. The production function of a firm depends on the state of technology. With every development in technology the production function of the firm undergoes a change.

The new production function brought about by developing technology displays same inputs and more output or the same output with lesser inputs. Sometimes a new production function of the firm may be adverse as it takes more inputs to produce the same output.

Mathematically, such a basic relationship between inputs and outputs may be expressed as:

Q = f( L, C, N )

Where Q = Quantity of output

L = Labour

C = Capital

N = Land.

Hence, the level of output (Q), depends on the quantities of different inputs (L, C, N) available to the firm. In the simplest case, where there are only two inputs, labour (L) and capital (C) and one output (Q), the production function becomes.

Q =f (L, C)

**Features of Production Function**

Following are the main features of production function:

**Substitutability**

The factors of production or inputs are substitutes of one another which make it possible to vary the total output by changing the quantity of one or a few inputs, while the quantities of all other inputs are held constant. It is the substitutability of the factors of production that gives rise to the laws of variable proportions.

**Complementarity**

The factors of production are also complementary to one another, that is, the two or more inputs are to be used together as nothing will be produced if the quantity of either of the inputs used in the production process is zero.

The principles of returns to scale is another manifestation of complementarity of inputs as it reveals that the quantity of all inputs are to be increased simultaneously in order to attain a higher scale of total output.

**Specificity**

It reveals that the inputs are specific to the production of a particular product. Machines and equipment’s, specialized workers and raw materials are a few examples of the specificity of factors of production. The specificity may not be complete as factors may be used for production of other commodities too. This reveals that in the production process none of the factors can be ignored and in some cases ignorance to even slightest extent is not possible if the factors are perfectly specific.

Production involves time; hence, the way the inputs are combined is determined to a large extent by the time period under consideration. The greater the time period, the greater the freedom the producer has to vary the quantities of various inputs used in the production process.

In the production function, variation in total output by varying the quantities of all inputs is possible only in the long run whereas the variation in total output by varying the quantity of single input may be possible even in the short run.

**TYPES OF PRODUCTION FUNCTIONS**

Production function is the mathematical representation of relationship between physical inputs and physical outputs of an organization.

There are different types of production functions that can be classified according to the degree of substitution of one input by the other.

**Cobb-Douglas Production Function**

Cobb-Douglas production function refers to the production function in which one input can be substituted by other but to a limited extent. For example, capital and labor can be used as a substitute of each other, but to a limited extent only.

**Cobb-Douglas production function can be expressed as follows:**

Q = AK^{a}L^{b}

Where, A = positive constant

a and b = positive fractions

b = 1 – a

**Therefore, Cobb- Douglas production function can also be expressed as follows:**

Q = ak^{a}L^{1-a}

**The characteristics of Cobb- Douglas production function are as follows:**

(i) Makes it possible to change the algebraic form in log linear form, represented as follows:

log Q = log A + a log K + b log L

This production function has been estimated with the help of linear regression analysis.

(ii) Makes it possible to change the algebraic form in log linear form, represented as follows:

log Q = log A + a log K + b log L

This production function has been estimated with the help of linear regression analysis.

(iii) Acts as a homogeneous production function, whose degree can be calculated by the value obtained after adding values of a and b. If the resultant value of a + b is 1, it implies that the degree of homogeneity is 1 and indicates the constant returns to scale.

(iv) Makes use of parameters a and b, which signifies the elasticity’ coefficients of output for inputs, labor and capital, respectively. Output elasticity coefficient refers to the change produced in output due to change in capital while keeping labor at constant.

(v) Represents that there would be no production at zero cost.

**Leontief Production Function**

Leontief production function uses fixed proportion of inputs having no substitutability between them. It is regarded as the limiting case for constant elasticity of substitution.

**The production function can be expressed as follows:**

q= min (z_{1}/a, Z_{2}/b)

Where, q = quantity of output produced

Z_{1} = utilized quantity of input 1

Z_{2} = utilized quantity of input 2

a and b = constants

For example, tyres and steering wheels are used for producing cars. In such case, the production function can be as follows:

Q = min (z_{1}/a, Z_{2}/b)

Q = min (number of tyres used, number of steering used).

**CES Production Function**

CES stands for constant elasticity substitution. CES production function shows a constant change produced in the output due to change in input of production.

**It can be represented as follows:**

Q = A [aK^{β} + (1-a) L-^{β}]-^{1/β}

Or,

Q = A [aL-^{β} + (1-a) K-^{β}]-^{1/β}

CES has the homogeneity degree of 1 that implies that output would be increased with the increase in inputs. For example, labor and capital has increased by constant factor m.

**In such a case, production function can be represented as follows:**

Q’ = A [a (mK)-^{β} + (1-a) (mL)-^{β}]-^{1/β}

Q’ = A [m-^{β} {aK-^{β} + (1-a) L-^{β}}]-^{1/β}

Q’ = (m-^{β})-^{1/β} .A [aK-^{β} + (1-a) L-^{β})-^{1/β}

Because, Q = A [aK-^{β} + (1-a) L-^{β}]-^{1/β}

Therefore, Q’ = mQ

This implies that CES production function is homogeneous with degree one.

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