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2 min read**Tutorial Topic:** 3 Types of Production Functions: Cobb-Douglas, Leontief and constant elasticity substitution (CES) production function.

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There are different types of production functions that can be classified according to the degree of substitution of one input by the other. Figure 7.14 illustrates different types of production functions:

## Cobb-Douglas production function

The Cobb-Douglas production function, given by American economists, Charles W. Cobb and Paul. H Douglas, studies the relation between the input and the output.

The cobb-douglas production function is that type of production function wherein an input can be substituted by other to a limited extent.

For example, capital and labour can be used as a substitute of each other, however to a limited extent only. Cobb-Douglas production function can be expressed as follows:

Q = AKa Lb

Where, A = positive constant a and b = positive fractions b = 1â€“a

### Cobb- Douglas production function Properties

The main attributes of Cobb- Douglas production function are discussed as follows:

- It enables the conversion of the algebraic form into 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.
- It 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.
- It makes use of parameters a and b, which signifies the elasticity coefficients of output for inputs, labour and capital, respectively. Output elasticity coefficient is the change in output that occurs due to adjustment in capital while keeping labour at constant.
- It depicts the non-existence of production at zero cost.

## Leontief Production Function

Leontief production function, evolved by W. Wassily Leontif, uses fixed proportion of inputs having no substitutability between them.

It implies that if the input-output ratio is independent of the scale of production, there is existence of Leontief production function. It assumes strict complementarity of factors of production. Leontief production function is also called as fixed proportion production function.

This production function can be expressed as follows:

q= min (z1/a, z2/b)

where, q = quantity of output produced z1 = utilised quantity of input 1 z2 = utilised quantity of input 2 a and b = constants Minimum implies that the total output depends upon the smaller of the two ratios.

## CES Production Function

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

It is expressed as: Q = A [aKâ€“Î² + (1â€“a) Lâ€“Î²]â€“1/Î² Or, Q = A [aLâ€“Î² + (1â€“a) Kâ€“Î²]â€“1/Î² CES has the homogeneity degree of 1 that implies that output would be increased with the increase in inputs. For example, labour and capital has increased by constant factor m.

### Properties of CES Production Function

The properties of CES function are as follows:

- The value of elasticity of substitution depends upon the value of a.
- The marginal products are positive and slope downwards.

### Merits of CES Production Function

The merits of CES production are as follows:

- Covers a number of parameters, such as efficiency and substitutability
- Easy to estimate
- Free from unrealistic assumptions, such as fixed technology, etc.

### Demerits of CES function

The demerits of CES production system are as follows:

- Fails to fit for manufacturing industries
- Cannot be generalised in case of n factors of production
- Fails to give correct economic implications