Question

Prove that the Production Function cannot be transformed into a Brief Function of Solo’s model without having a constant return to scale. Y = KαLβ

Answer 1

The given production function is

Y = (K)^a(L)^b

Now, in Solow's brief production function is the output per worker as function of capital per worker only i.e.

y = f(k) [Where, y = Y/L and k = K/L]

Now, from the given production function we can write,

Y/L = (K)^a(L)^b-1

or, Y/L = (K/L)^a.(L)^a+b-1

or, y = (k)^a(L)^a+b-1

Now, if the function is Solow's brief production function, then y must be a function of k only i.e.

y = k^{​​​​​​a}

Hence,

L^{​​​​​​a+b-1} = 1

or, a+b-1 = 0

or, a + b = 1

Hence, a + b = 1 implies that the given production function is Constant Returns to Scale.

if, a + b > 1 or a + b < 1, the term [L^{​​​​​​a+b-1}] will not be equal to 1.

Hence, the Production Function cannot be transformed into a Brief Function of Solow’s model without having a constant return to scale.

Hope the explanation is clear to you my friend.