posted by janet on .
Suppose a firm produces output using the technology Q=K1/3 L2/3 Find
a. The long run cost function
b. The short run cost function if capital is stuck at 10 units.
c. The profit maximizing level of output as a function of the price of the good, wages, rental rate on capital, the amount of capital, and some other numbers.
Since this is at least the second post of this question, I think I better answer it.
How is your calculas. Mine is a bit rusty. But here goes. (I hope there are no typos below).
Let w be the price of labor (L), z be the price of capital (K). (Let y be the lagrangian multiplier. Let 6 be the sign for partial derivitive)
TC = wL + zK
So, for any level Q, we want to:
min(wL+zK) subject to Q=K^(1/3)L^(2/3)
Set up the lagrange minimization equation:
LA = wL + zK + y(Q - f(Q,L))
first orders are:
6LA/6L = w - y(6f/6L) = 0
6LA/6K = z - y(6f/6K) = 0
6LA/6y = Q - f(Q,L) = 0
6f/6L is the marginal product of labor.
6f/6K is the marginal product of capital
Using the first two first-order equations, we get y = w/MPl = z/MPk where
MPl = (2/3)K^(1/3)L^(-1/3)
MPk = (1/3)K^(-2/3)L^(2/3)
So, MPl/MPk = w/z = 2K/L
rearrange terms to get L=2zK/w
Now then plug this L into the original production function,
solve for K (when K is optimized)
If you do the same steps for L you get
now plug these into a total cost functions when L and K are optimized.
TC = wL* + zK*
TC = w[(2z/w)^(1/3)]Q + z[(2z/w)^(-2/3)]Q
you could collapse terms to get a single Q. But essentially, you are done. TA DA.
From here, with K fixed at 10, optimization should be a breeze.