## LeoMessi 3 years ago What type of returns to scale does the following production function exhibit?

1. LeoMessi

\[q = 10K^{2/3} + L^{1/2}\] K = # machines, L = # labor units.

2. dpflan

OK, there are (i.) decreasing returns, (ii.) increasing returns, and (iii.) constant returns. Your function is one of K and L so \[q=f(K,L)\]. if we increase K and L by 2, is the quantity produced greater than, less than, or equal to 2 * the production function?

3. dpflan

\[2*F(K,L) versus F(2*K, 2*L)\]

4. LeoMessi

\[F(2*K, 2*L) = 10*(2K)^{2/3} + (2*L)^{1/2} = 10 * 2^{2/3} * K + 2^{1/2} * L\] and \[2*F(K,L) = 2 *(10 * K^{2/3} + L^{1/2}\]

5. dpflan

right, to easily compare the 2 factor out a 2 in \[F(2*K, 2*L)\] \[(2) * (10 * 2^{-1/3} * K + 2^{-1/2} * L)\] given the exponents, that is obviously less than \[2*(10*2^{2/3}*K + 2^{1/2}*L)\]

6. LeoMessi

yep, so decreasing returns to scale

7. LeoMessi

right on, so it seems safe to assume that when the exponents are less than 1, decreasing returns are most likely, and when the exponents are 1 then constant, and if greater than 1 than increasing

8. LeoMessi

cool stuff, thanks man

9. dpflan

yeah, I think you get the idea, try out some more problems