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.