Difference between “Cobb-Douglas” and “Leontief” Production Functions

Production functions can be of various types. Here we describe two types of production function:

(a) Cobb-Douglas production function and

(b) Leontieff production function.

1. Cobb-Douglas Production Function:

The most widely used production function is Cobb-Douglas production function. It is expressed as:

Q = A L^α K^β; Q, A, L, K, α, β > 0

Where

Q = Quantity of output

A = A constant which is technology specific and determines productivity

L = Quantity of labour

­K = Quantity of capital

α = Elasticity of output with respect to labour

β= Elasticity of output with respect to capital

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There are some important characteristics of Cobb-Douglas production function. First, sum of exponents of labour and capital (i.e., α + β) determines returns to scale. The relation α + β =1 implies constant returns to scale, α + β > 1 indicates increasing returns to scale and α + β < 1 implies decreasing returns to scale.

Secondly, Cobb-Douglas production function is also applicable where more than two factors of production are used. Thirdly, marginal productivity of labour and capital derived from Cobb-Douglas production function are dependent on quantities of labour and capital employed in the production process. This characteristic is very realistic.

2. Leontieff Production Function:

The production function which represents employment of fixed proportion of inputs and does not allow any substitution among inputs is known as Leontieff production function. The isoquants which represent Leontieff production function are ‘L’ shaped. Leontieff production function is expressed as:

Q = Min. (I₁ / a, I₂ / b)

Where

Q = Quantity of output

I₁, I₂ = quantities of inputs employed

a, b = requirements of I₁ and I₂ respectively to produce one unit of output

Let us take a numerical example to illustrate the meaning of the symbolical representation of Leontieff production function.

Suppose, 20 units of input I and 50 units of input II are employed to manufacture a particular product. If 10 units of input I and 5 units of input II are required to produce one unit of output, then using Leontieff production function, quantity of output can be determined as

Q = Min. (20 / 10, 50 / 5)

= Min. (2, 10)

= 2 units.

Total quantity of output is 2 units. So, total quantity of input I and 10 units of input II will be utilised.