Question

A) What are the returns to scale for the following production function: q=5^(1/2)K^(1/2)+19^(1/2)L^(1/2)

B) What type of returns to scale does 3K^(1/3)L^(2/3)+3 have?

Answer 1

a) q= 5^(1/2)K^(1/2)+19^(1/2)L^(1/2)

Returns to scale is the proportion by which the output increases , when the inputs are increased.

Assuming that the inputs are increasing by x , the new inputs would be xL and xK.

the new production function would be

q= F(K,L)= 5^(1/2)(xK)^(1/2)+19^(1/2)(Lx)^(1/2) = x^1/2(5^(1/2)K^(1/2)+19^(1/2)L^(1/2) )= x^1/2F(K,L)

so when the inputs are increased by x, the output increases by x^1/2 ie the increase in output is less than the increase in inputs .

Since F(xK,xL)>xF(K,L)

This is decreasing returns to scale.

b) q= 3K^(1/3)L^(2/3)+3= F(K,L)

Assuming that the inputs are increasing by x , the new inputs would be xL and xK.

the new production function would be

q= 3(Kx)^(1/3)(Lx)^(2/3)+3

q= x(3K^(1/3)L^(2/3)+3/x)

Since only the term with inputs is increasing and the rest of the term ie "3" is constant , it means that when the inputs are increased, only a part of production increases by that proportion and not the whole function.

Since F(xK,xL)>xF(K,L)

Hence it is decreasing returns to scale.

(You can comment for doubts )