Cobb Douglas Production Function

1 year ago

Atiqur Rahman

9 minutes

In Economics, the Cobb-Douglas production function is widely used to represent the relationship between inputs and output in an economy. The two most important neoclassical production functions are the Constant Elasticity of Substitution (CES) and the Cobb Douglas.

The Cobb-Douglas production function was created by Charles Cobb (Mathematician) and Paul Douglas (Economist) in 1927. But its functional form was proposed by Knut Wicksell (Economist) in the 19^th Century.

The Cobb-Douglas production function lies between linear and fixed proportion production functions with the elasticity of substitution equal to one. It is very popular among economists because of its flexibility and ease of use.

Mathematical Form:

The mathematical form of the Cobb-Douglas production function for a single output with two factors can be written as:

𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾^𝛼𝐿^1−𝛼

Where,

Y: Output

K: Capital input

L: Labour input

A: Level of technology or total factor productivity (A>0)

α: Constant between 0 and 1( 0< α <1)

Constant Returns to Scale:

The return to scale is a long-run concept when all the factors of production are variable. In the long run, output can be increased by increasing all the factors of production. An increase in scale means that all factors are increased in the same proportion, and output will increase, but the increase may be at an increasing rate or at a constant rate or at a decreasing rate.

Three Situations of Return to the Scale:

(i). Increasing Returns to Scale:

Increasing return to scale occurs when output increases in a greater proportion than the increase in inputs. If all factors are increased by 20%, then output increases by, say, 30%. So by doubling the factors, output increases by more than double.

(ii). Constant Returns to Scale:

Constant Return to Scale occurs when output increases in the same proportion as the increase in input. If all factors are increased by 20%, then output also increases by 20%. So doubling of all factors causes a doubling of output, then returns to scale are constant. The constant return to scale is also called a linearly homogenous production function.

(iii). Decreasing Returns to Scale:

Decreasing return to scale occurs when output increases in a lesser proportion than the increase in inputs. If all the factors are increased by 20%, then output increases by less than 20%.

The Cobb-Douglas production function exhibits constant returns to scale. Constant returns to scale occur when output increases in the same proportion as the increase in input. Under constant returns to scale, the sum of two exponents for capital and labour is one, i.e. α + (1-α) =1.

The Constant Return to Scale occurs when output increases in the same proportion as the increase in input.

It is shown in Fig 15.1. The labour and capital are shown on X-axis and Y-axis. Y1, Y2 and Y3 are the isoquant curves showing different levels of output. Under constant return to scale, the distance between successive isoquants remains the same as we expand output from 100 to 200 to 300 units. On a straight line OR starting from the origin, the distance OA, AB and BC are all equal.

If the sum of the two exponents for capital and labour is greater than one, then the function exhibits increasing returns to scale. And if the sum of the two exponents for capital and labour is less than one, then the function exhibits decreasing returns to scale.

Isoquant is Convex to the Origin:

Under the Cobb-Douglas production function, isoquants are convex to the origin. Fig 15.2 represents an isoquant map. An isoquant map refers to the family of isoquant curves where the higher the isoquant, the higher the level of production.

In the figure, labour is measured on X-axis and capital on Y-axis. Y1, Y2 and Y3 are the isoquant curves showing various possible combinations of inputs physically capable of producing a given level of output. If you can operate production activities independently, then weighted averages of production plans will also be feasible. Hence the isoquants will have a convex shape.

The isoquant Y1 represents 100 units of output, whereas isoquant Y2 represents 200 units of output, and the level of output is higher on isoquant Y2 than Y1. Isoquant Y3 shows 300 units of output which is higher than the level of output as shown by isoquant Y1 and Y2, and so on. So, the higher the isoquant, the higher the level of output.

The isoquants under Cobb Douglas production function are convex to the origin. This occurs because of the diminishing marginal rate of technical substitution.

Here the isoquant is:

Downward sloping.
Convex to the origin: Diminishing MRTS.
Higher the isoquant, the higher the level of output.

Marginal Products and Average Products:

(i). Under Cobb Douglas production function, the average product and marginal products of factor depend upon the ratio in which the factors are combined to produce output.

𝑀𝑃_𝐾 = ∂Y/∂K = 𝐴𝛼𝐾^𝛼−1 𝐿^1−𝛼 = 𝐴𝛼 (𝐾/𝐿)^𝛼−1

𝐴𝑃_𝐾 = Y/𝐾 = 𝐴𝐾^𝛼𝐿^1−𝛼/𝐾 = 𝐴 (𝐾/𝐿)^1−𝛼

The average product of capital depends on the ratio of capital and labour (K/L) and does not depend upon the absolute quantities of the factors used. The same is true for labour.

(ii). The marginal product is proportional to the output per unit of its factor.

𝑀𝑃_𝐾 = ∂Y/∂K = 𝐴𝛼𝐾^𝛼−1 𝐿^1−𝛼 = 𝐴𝛼 (𝐾/𝐿)^𝛼−1 = 𝛼 (Y/𝐾)

𝑀𝑃_L = ∂Y/∂L = 𝐴 (1-𝛼) 𝐾^𝛼 𝐿^−𝛼 = 𝐴 (1-𝛼) (𝐾/𝐿)^𝛼 = (1-𝛼) (Y/L)

Linear in Logarithm:

The Cobb-Douglas production function is linear in logarithm.

𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾^𝛼𝐿^1−𝛼

Now applying log on both sides, we get

ln 𝑌 = ln 𝐴 + 𝛼 ln 𝐾 + (1−𝛼) 𝑙𝑛 𝐿

Where,

α is the partial elasticity of output w.r.t capital. It measures the percentage change in output for, say, one percentage change in the capital input holding the labour input constant.
(1-α) is the partial elasticity of output w.r.t labour. It measures the percentage change in output for, say, one percentage change in the labour input holding the capital input constant.

The Cobb-Douglas production function is linear in parameter. It can be estimated using the least squares method.

Properties of Cobb Douglas Production Function

(i). Constant Returns to Scale:

The Cobb-Douglas production function exhibits constant returns to scale. If the inputs capital and labour are increased by a positive constant, λ, then output also increases by the same proportion, i.e. 𝑓(λK, λL, A) = λ f(K, L, A) for all λ>0.

Y = 𝑓 (K, L, A) = AK^α L^1-α

𝑓 (λK, λL, A) = A (𝑓(λK)^α (λL)^1-α

= Aλ^α K^α λ^1-α L^1-α

= λA K^α L^1-α

= λY

● If the function exhibits decreasing returns to scale, then 𝑓(𝜆𝐾, 𝜆𝐿, 𝐴) < 𝜆𝑌 𝑓𝑜𝑟 𝑎𝑛𝑦 𝜆 > 1.

● If the function exhibits increasing returns to scale, then 𝑓(𝜆𝐾, 𝜆𝐿, 𝐴) > 𝜆𝑌 𝑓𝑜𝑟 𝑎𝑛𝑦 𝜆 > 1.

(ii). Positive and Diminishing Returns to Inputs:

The Cobb Douglas production function is increasing in labour and capital, i.e. positive marginal products.

(a). ∂Y/∂K > 0 𝑎𝑛𝑑 ∂Y/∂L > 0

𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾^𝛼𝐿^1−𝛼

𝑀𝑃_𝐾 = ∂Y/∂K = 𝐴𝛼𝐾^𝛼−1𝐿^1−𝛼

𝑀𝑃_𝐿 = ∂Y/∂L = 𝐴(1−𝛼)𝐾^𝛼𝐿^−𝛼

Assuming A, L and K are all positive and 0 < 𝛼 < 1, the marginal products are positive.

(b). Diminishing Marginal Products with respect to each Input: ∂²Y/∂K² < 0 𝑎𝑛𝑑 ∂²Y/∂L² < 0

∂²Y/∂K² = 𝐴𝛼 (𝛼−1) 𝐾^𝛼−2 𝐿^1−𝛼 < 0 if α<1

Here, any small increase in capital will lead to a decrease in the marginal product of capital. Any small increase in capital cause output to rise but at a diminishing rate. The same is true for labour.

(iii). Inada Conditions

(iv). The Cobb Douglas Production Function has Elasticity of Substitution Equal to Unity

(v). Constant Income Shares of Output:

The exponent of capital (labour), α (1- α), represents the contribution of capital (labour) to output. This is the same as the portion of output distributed to capital (labour), i.e. capital (labour) income share.

𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾^𝛼𝐿^1−𝛼

The real wage of labour (w) is calculated by partially differentiating Y w.r.t. L, which is nothing but the marginal product of labour (𝑀𝑃_𝐿).

𝑤 = 𝑀𝑃_𝐿 = ∂Y/∂L = 𝐴 (1−𝛼) 𝐾^𝛼𝐿^−𝛼 = 𝐴 (1−𝛼) (𝐾/𝐿)^𝛼 = (1−𝛼) (Y/𝐿)

Total Wage Bill = 𝑤. 𝐿 = 𝑀𝑃_𝐿. 𝐿 = 𝐴 (1−𝛼) 𝐾^𝛼𝐿^1−𝛼

The labour share in real National Product is = 𝑇𝑜𝑡𝑎𝑙 𝑊𝑎𝑔𝑒 𝐵𝑖𝑙𝑙/𝑅𝑒𝑎𝑙 𝑁𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑃𝑟𝑜𝑑𝑢𝑐𝑡

= 𝑇𝑜𝑡𝑎𝑙 𝑊𝑎𝑔𝑒 𝐵𝑖𝑙𝑙/𝑅𝑒𝑎𝑙 𝑁𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 = 𝑤𝐿/𝑌 = (𝐴 (1−𝛼) 𝐾^𝛼 𝐿^1−𝛼)/𝐴𝐾^𝛼𝐿^1−𝛼 = 1−𝛼

In the same way, the capital income share remains constant at 𝛼.