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.
To find the long-run and short-run cost functions, we need to understand the relationship between inputs and outputs in the production function Q = K^(1/3) * L^(2/3).
a. Long-run cost function:
In the long run, both capital (K) and labor (L) are variable inputs. To minimize cost, the firm chooses the combination of inputs that produces a desired level of output (Q) at the lowest cost. The cost function represents the minimum cost of producing a given level of output.
To find the long-run cost function, we need to solve for the minimum cost combination of inputs (K and L) that satisfies the production function Q=K^(1/3) * L^(2/3). We can use the concept of isoquants to find the optimal input combination. Isoquants are curves that represent different levels of output with varying combinations of inputs.
Using the production function, we can rewrite it in terms of the ratio of inputs as Q/K^(1/3) = L^(2/3). By taking the natural logarithm of both sides, we get ln(Q/K^(1/3)) = ln(L^(2/3)).
Taking the derivative of both sides with respect to K and L, we get:
(1/3)(dQ/dK)/(Q/K^(1/3)) = (2/3)(dL/dL)/(L^(2/3))
(dQ/dK)/Q = (2K)/(3L)
(dQ/dL)/Q = (4L)/(3K)
The marginal rate of technical substitution (MRTS) is the ratio of the marginal product of labor to the marginal product of capital. It indicates how much labor (L) needs to be substituted to maintain a constant level of output (Q) when there is a change in capital (K).
Setting the MRTS equal to the ratio of input prices (wages/capital rental rate), we can write:
(w/r) = (4L)/(3K)
Solving this equation for L, we get:
L = (3/4) * (w/r) * K
Now, substitute this value of L back into the production function to find Q as a function of K:
Q = K^(1/3) * [(3/4) * (w/r) * K]^(2/3)
Simplifying further, we get:
Q = (3/4)^(2/3) * (w/r)^(2/3) * K
Therefore, the long-run cost function is:
C(Q, w, r) = C(K, w, r) = [(3/4)^(2/3) * (w/r)^(2/3)] * K
b. Short-run cost function if capital is stuck at 10 units:
In the short run, one input (in this case, capital) is fixed, and only the other input (labor) is variable. Given that the capital (K) is stuck at 10 units, the short-run cost function depends solely on the cost of labor (w).
Using the production function Q = K^(1/3) * L^(2/3) and substituting K = 10, we can isolate L as a function of Q and K:
Q = (10)^(1/3) * L^(2/3)
L = [(Q / (10)^(1/3))]^(3/2)
Substituting this value of L back into the production function, we get the short-run cost function:
C(Q, w) = C(L, w) = [(Q / (10)^(1/3))]^(3/2) * w
c. Profit-maximizing level of output:
To find the profit-maximizing level of output, we need more information such as the price of the good, wages, rental rate on capital, and the amount of capital. Without additional data, it isn't possible to provide a specific function to determine the profit-maximizing output level in this case.