Quiz 4

Q: Find MC if f(L,K)=LK^{1/2}, w=1, and v=4.

A: For a C-D production function we use RTS=w/v. In this problem
RTS is 2K/L and w/v is 1/4. So the optimal condition is 8K=L. Since
q=LK^{1/2}, we know that q=(8K)K^{1/2} or q=8K^{3/2}.
Dividing both sides by 8 gives q/8=K^{3/2}. Raising both sides to
the 2/3 power gives (q^{2/3})/4=K.

Costs are wL+vK=1*(8K)+4K= 12K=12*(q^{2/3})/4 = 3(q^{2/3}).
MC =dTC/dq= 2q^{-1/3}.

*Additional Note: Notice that this function has increasing returns to
scale and ATC=TC/q=3q ^{-1/3}, which is below MC. So ATC is falling
as we would expect. You can also see that ATC is falling by taking the
derivative of ATC. That is dATC/dq = -q^{-4/3}, which is less than
0 when q > 0. *