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,
Q=K^(1/3)[2zK/w]^(2/3)
solve for K (when K is optimized)
K*= [(2z/w)^(-2/3)]Q
If you do the same steps for L you get
L*= [(2z/w)^(1/3)]Q
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.
To find the long run cost function, we need to determine the optimal combination of inputs (capital and labor) that minimizes the cost of producing a given level of output in the long run.
a. Long Run Cost Function:
In the long run, the firm can adjust both capital (K) and labor (L) to minimize costs. The cost function is given by the equation:
C = wL + rK
where C represents the cost, w is the wage rate, L is the amount of labor, r is the rental rate on capital, and K is the amount of capital.
To find the optimal combination of K and L, we need to minimize this cost function, subject to the production function Q = K^(1/3) * L^(2/3).
b. Short Run Cost Function:
In the short run, the firm can only adjust the amount of labor while the capital is fixed at 10 units. The production function remains the same: Q = K^(1/3) * L^(2/3). However, the firm can only vary L to produce the desired level of output.
To find the short run cost function, we need to substitute K = 10 into the production function and solve for L.
c. Profit-Maximizing Level of Output:
To determine the profit-maximizing level of output, we need to take into account the price of the good (P), the wage rate (w), and the rental rate on capital (r).
The profit-maximizing level of output occurs when the marginal cost of production equals the market price of the good. The marginal cost (MC) is the derivative of the cost function with respect to output (Q).
MC = (∂C/∂Q)
To maximize profit, we set MC equal to the market price (P) and solve for Q. Substituting the production function into MC, we have:
MC = (∂C/∂Q) = (w/2L) * Q
Setting MC equal to P gives us:
(w/2L) * Q = P
Solving for Q, we can express the profit-maximizing level of output as a function of the price of the good (P), the wage rate (w), rental rate on capital (r), the amount of capital (K), and the quantity of labor (L).