Q: Find the Returns to Scale of the following production functions: 
1) f(L,K)=L1/2K1/3 

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)= 11/211/3=1 and f(2,2)=21/221/3=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.