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.
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To find the long run cost function, we need to minimize the cost of producing a given level of output when both capital (K) and labor (L) can vary. In this case, the production technology is given by Q = K^(1/3) * L^(2/3).
a. Long run cost function:
In the long run, the firm can choose any combination of capital and labor that minimizes the cost of producing a given level of output. Assuming the firm faces constant input prices (wages and rental rate), the long run cost function (C) can be derived by solving the cost minimization problem.
The cost minimization problem can be expressed as:
Minimize C = wL + rK
subject to: Q = K^(1/3) * L^(2/3)
Here, w represents the wage rate and r represents the rental rate on capital.
To solve this problem, we can use the Lagrangian method:
L = wL + rK - λ(Q - K^(1/3) * L^(2/3))
Taking partial derivatives with respect to L, K, and λ and setting them equal to zero, we can solve for L and K in terms of Q.
b. Short run cost function:
In the short run, capital (K) is stuck at a fixed level, while only labor (L) is variable. The short run cost function is then derived by finding the cost of producing a given level of output, given the fixed capital level.
Using the production function Q = K^(1/3) * L^(2/3) and the fixed capital level K = 10 units, we can substitute K value into the production function.
Q = 10^(1/3) * L^(2/3)
Q^(3/2) = 100 * L
L = (Q^(3/2))/100
Substitute this value of L into the cost function C = wL + rK, where K = 10. This will give you the short run cost function.
c. Profit maximizing level of output:
To determine the profit-maximizing level of output, we need to consider the marginal cost (MC) and marginal revenue (MR). The profit-maximizing output level occurs where MR equals MC.
The marginal cost can be derived from the cost function by taking the derivative of the cost function with respect to output (Q). The marginal revenue, on the other hand, depends on the price (P) of the good.
If we assume that MR equals P, we can set MR equal to MC and solve for the output level (Q) that maximizes profit.
Profit-maximizing output level: Set MR = MC
P = MC
Now, you can solve the equation P = MC to find the profit-maximizing level of output as a function of the price of the good (P), wages (w), rental rate on capital (r), the amount of capital (K), and other relevant variables or numbers specific to the problem.