Question

In: Economics

For the following two functions prove whether each of the production functions has increasing, decreasing, or...

For the following two functions prove whether each of the production functions has increasing, decreasing, or constant returns to scale. Then find whether the MPL is increasing, decreasing or constant with L.

A. ? = ?^1/3?^1/3

B. ? = 3?^3/2

Expert Solution

Ans. To check whether the function has increasing, decreasing or constant returns to scale, we will increase the each input by proportion t and if after this, output increases by t then there is constant returns to scale, more than t then there is increasing returns to scale and less than t then there is decreasing returns to scale.

And MPL is increasing if it is positive, decreasing if it is negative and constant if MPL is equal to constant.

a) q = L^1/3 * K^1/3

For returns to scale,

q' = (tL)^1/3 * (tK)^1/3 = t^2/3 *K^1/3 * L^1/3 = t^2/3 *q < tq

So, decreasing returns to scale.

Marginal product of labour, MPL = dq/dL = 1/3*K^1/3 * L^(-2/3) > 0

Hence, MPL is increasing

b) q = 3L^3/2

For returns to scale,

q' = 3*(tL)^3/2 = t^3/2 * 3*L^3/2 = t^3/2 q > tq

So, increasing returns to scale.

Marginal product of labour, MPL = dq/dL = 9/2 * L^1/2 > 0

Hence, MPL is increasing.

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Rahul Sunny answered 2 years ago

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