Different Types Of Production Functions

1. Cobb-Douglas Production Function

This production function was proposed by C. W. Cobb and P. H. Dougles. This famous statistical production function is known as Cobb-Douglas production function. Originally the function is applied on the empirical study of the American manufacturing industry from 1899 to 1922. Cobb – Douglas production function takes the following mathematical form.

The above production function shows that 1% change in labor input, capital remaining the same, is associated with a 0.75 percent change in output. Similarly, 1% change in capital, labor remaining the same, is associated with a 0.25 percent change in output.

Assumptions:

1. The function assumes that output is the function of two factors viz. capital and labour.
2. It is a linear homogenous production function of the first degree
3. There are constant returns to scale (b+c=1)
4. All inputs are homogenous
5. There is perfect competition
6. There is no change in technology
7. Both L&K should be positive for Q to exist. If either of these is zero, Q will be zero. This implies that both labor and capital will be combined to get output.

2. Leontief Production Function

Leontief production function, also known as Fixed Proportion 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:

Where, q = quantity of output produced, Z1 = utilized quantity of input 1, Z2 = utilized quantity of input 2, a and b = constants

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

Suppose that the inputs “tires” and “steering wheels” are used in the production of a car (for simplicity of the example, to the exclusion of anything else). Then in the above formula q refers to the number of cars produced i.e., one in our example, z1 refers to the number of tires used, and z2 refers to the number of steering wheels used. Assuming that each car is produced with 4 tires and 1 steering wheel, the Leontief production function is

In the above given figure, OR shows the fixed Tyres-Wheels ratio, if a firm wants to produce 1 car, then 4 tyres and 1 wheel must be used. Similarly, for the production of 3 cars and 4 cars, 12 tyres and 3 wheel and 16 tyres and 4 wheel must be employed respectively.

3. CES Production Function

Definition: The Constant Elasticity of Substitution Production Function or CES implies, that any change in the input factors, results in the constant change in the output. In CES, the elasticity of substitution is constant and may not necessarily be equal to one or unity.

In constant elasticity of substitution production function, all the input factors are taken into the consideration such as raw material, technology, labor, capital, etc. The marginal product of one factor increases with the increase in the value of the other factors of production. Also, the marginal product of labor and capital will be positive in case of constant returns to scale.

The constant elasticity of substitution production function can be expressed algebraically as:

Where, Q = output, K = Capital and L = Labor

A = efficiency parameter that shows the organizational aspects of production and the state of technology.

The Constant elasticity of substitution production function shows, that any change in the technology or organizational aspects, the production function changes with a shift in the efficiency parameter.

α = distribution parameter or capital intensity factor coefficient concerned with relative factors in the total output.

β = substitution parameter, that determines the elasticity of substitution

The homogeneity of the production function can be determined by the value of the substitution parameter (β), if it is equal to one, then it is said to be linearly homogeneous i.e. the proportionate change in the input factors results in the increase in the output in the same proportion.

For example:

Consider the following production function.

It means 1 unit K produces 5 units and 2 units of L produce 10 units.

If we assume that, initially K = 1, L = 2, the total output may be obtained by substituting 1 for K and 2 for L in the below equation. Thus,

When inputs are doubled (i.e., K = 2 and L = 4). Then,

Thus, the given production functions, when inputs are doubled, the output is also doubled.