Cobb Douglas Production Function, Returns To Scale

Illustration 40

Given the production function O = f (a1, a2) = a1^2 * a2^3 where a1, a2 are two inputs and O is production. Does this depict invariable, declining or enhancing returns to scale?

Solution

O = a1^2 * a2^3

Incrementing both inputs a1 and a2 by x, we obtain the following:

O’        =          (xa1) ^2 * (xa2) ^3

=          x^2 * x^3 . a1^2 * a2^3

O’        =          x^5 * a1^2 a2^3

=          x^5 . O

Therefore, increment in inputs by ‘a’ causes productivity to enhance by x^5. Thus, enhancing returns to scale triumph in this crate. The sum of exponents of two inputs a1 and a2 is 2 + 3 = 5.

As described above, in the crate of Cobb-Douglas production function, if the sum of exponents is greater than one, increasing returns to scale happens. It is to be noted in addition that O = a1^2.a2^3 is function of production f Cobb-Douglas diversity.

Illustration 41

Consider production function O = 10A^0.5.K2^0.3. Does it represent enhancing decreasing or invariable returns to scale? A is labour and B is capital.

Solution

Incrementing both sides that is A and B by x, we get the following,

O’        =          10(xA)^0.5 . 2(xB)^0.3

=          x^0.5 . x^0.3 * 10A^0.5 . 2B^0.3

O’        =          x^0.8 . O

Therefore, by incrementing labour A and capital B by x, productivity enhances by x^0.8, that is less than x. Therefore, in the ease declining returns to scale happens.

Illustration 42

Does each of the following production function exhibit declining, invariable or incrementing returns to scale?

(1) O = 0.4BA             (2) O = 4B + 6A

Solution

(1) O = 0.4BA

Incrementing B and A by x, we procure:

O’        =          0.4xB . xA

=          x.x * 0.4BA

=          x^2 * 0.4BA

O’        =          x^2*O

Incrementing B and A by x, productivity increments by x^2, that is enhancing returns to scale function in this case,

(2) O = 4B + 6A

Incrementing B and A by x, we obtain:

O’        =          4xB + 6xA

=          x(4B + 6A)

O’        =          x.O

That is incrementing B and A by x in this production function consequents in enhancement in productivity also by x. That is invariable returns to scale happens in this crate.

Illustration 43

In the event a product is manufactured with two inputs, labour and capital and the production function is given by

O         =          5 * √A.B

Where O is productivity and A and B are amounts of labour and capital. What type of returns to scale does it display?

Solution

O         =          5A^1/2 . B^1/2

Therefore, this is a Cobb-Douglas production function. To depict the nature of returns to scale, let us increment A and B by λ.

O’        =          5 (λA)^1/2 * (λB)^1/2

=          λ^1/2 * λ ^ ½ * A ^ ½ B ^ ½

=          λ.O

Incrementing A and B by λ results in incrementing in productivity O by λ. Hence this depicts invariable returns to scale.

Illustration 44

Econometric analysis of the cotton industry in a developing country depicts that Cobb-Douglas production function can be functional and the advocate of labour is 0.42 and exponent of capital is 0.07. If both labour and capital are incremented by one percent by what amount of output of jute will enhance?

Solution

As represented above, an exponent of raw material in Cobb-Douglas production function can be interpreted as the percentage increment in productivity consequents from one per cent in increase in raw material.

Therefore the exponent of 0.42 of labour entails that one percent increase in labour leads to 0.42percent extension in productivity of cotton. Likewise, the exponent of 0.07 of capital entails one percent extension in capital tends to 0.07 per cent enhancement in productivity.

Therefore, one percent increment in both labour and capital will consequent in 0.42 + 0.07 = 0.49 percent extension in productivity.

Illustration 45

Presume you ascertain that a sugarcane mill utilises capital and labour is subject to the production function O = A^0.75B^0.25. Will it be correct to say that in the sugarcane mill, productivity per employee will be a function of capital per worker?

Solution

It is to be noted that capital per worker entails, capital-labour ration (B/A). The provided production function O = A^0.75B^0.25 is a Cobb-Douglas production function where aggregate of exponents parities to one (0.75 + 0.25).

Let us assume as follows:

Let m represent 0.75 then 0.25 will be 1-a. Now, restructuring the production function as below:

O = A^m.B^1-m

Now, average product of Labour A would be           =         O         =          A^m.B^1-m
A                              A

Now dividing ‘A’ both numerator and denominator, we get,

=          APA or  O        =          A^m-m.B^1-m
A                          A^1-m

As A^m-m = A^0 = 1, we procure,

=          APA     =          B^1-m =          (B) ^1-m
A^1-m             (A)

As m is an invariable, it follows that average product of labour or productivity per worker is based on capital labour proportion, that is capital per worker.

Illustration 46

Presume we are interested in ascertaining the returns to scale of a Consumer company whose production function is as follows:

O = 20MN – 5M^2 – 2N^2

Where O is productivity or output, M and N are inputs.

Solution

Multiplying each input in the above production function by an invariable a, we have the following:

O’        =          20(aM).(aN) – 5(aM)^2 – 2(aN)^2

=          20a^2.MN – 5a^2.M^2 – a^2.N^2

=          a^2 (20MN – 5M^2 – 2N^2)

O’        =          a^2.O

This entails that output or productivity of a consumer company depicts enhancing returns to scale.

Illustration 47

The production function of an industry is as follows:

O = 0.4S + 0.2T

Where O = output, S and T are inputs. You are required to ascertain the returns to scale.

Solution

To ascertain returns to scale we are required to increment or multiply each input by an invariable aspect u.

Therefore,

O         =          0.4uS + 0.2uT

u can be factored out and thus we arrive at the following:

O’        =          u(0.4S + 0.2T)

O’        =          u.O

Therefore, enhancing each raw material input by an invariable ‘u’ tends to enhance in productivity by the same invariable u. Therefore, it represents that production function is homogenous of first magnitude and returns to scale are invariable.

Illustration 48

Ascertain the returns to scale of the following production function.

O         =          mA + nB         where ‘A’ and ‘B’ are labour and Capital respectively.

Solution

To ascertain returns to scale we increment each input by an invariable factor c.

Therefore,

O’        =          m.cA + m.cB

=          c (mA + mB)

=          c.O

This evidences that returns to scale are invariable.

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