Q: Find the Returns to Scale of the following production
functions:

1) f(L,K)=L^{1/2}K^{1/3}

2)f(L,K)=min(L,K)

A: To determine the returns to scale we need to pick two inputs bundles, one of which has exactly twice as much of each good as the other bundle. The easiest ones are typically (1,1) and (2,2).

1) f(1,1)= 1^{1/2}1^{1/3}=1 and f(2,2)=2^{1/2}2^{1/3}=1.78
< 2. Since doubling the inputs less than doubles the output this
function exhibits decreasing returns to scale. *There is a general rule
for Cobb-Douglas production functions which relates the returns to scale to the
sum of the exponents. if the exponents are a and b then a+b>1 yields
increasing returns, a+b=1 yields constant returns, and a+b<1 yields decreasing
returns.*

2) f(1,1)=min(1,1)=1 and f(2,2)=min(2,2)=2. Since
doubling the inputs doubles the output this function exhibits constant returns
to scale. *There is also a general rule for fixed proportion production
functions. f(L,K)=min(AL,BK) will always exhibit constant returns to
scale. *