Question

A US glass manufacture’s production function in the long run (based on Hsieh, 1995) is

expressed as
= (K, ) = 100.5K0.5
Its marginal product functions are P = 0.5
, and PK = 0.5
K. Suppose the wage () is $1 per
hour and the rental cost of capital () is $4.
a. What is the cost minimizing ratio of labor and capital for this firm?
b. If the firm chooses to produce = 100, how much labor and capital will it employ?
c. Draw an accurate figure showing how the glass firm minimizes its costs of production?
d. What is the equation of the long run expansion path? Illustrate this path in the graph above.
e. Derive the long run cost curve as a function of output ()

Answer 1

The cost-minimizing ratio of labor to capital for the US glass manufacturer's production function is 4, meaning that for every unit of capital, four units of labor should be used. To produce an output level of Q = 100, the firm should employ 20 units each of labor and capital. The long-run expansion path is L = 4K, and from this, the long run cost curve can be determined.

Step-by-step explanation:

The question revolves around the concept of production functions in long run models and applying this knowledge to a hypothetical US glass manufacturer. Given the production function Q = 100.5K0.5L0.5 and the marginal product functions PL = 0.5 Q/L and PK = 0.5 Q/K, we aim to find the cost-minimizing combination of labor (L) and capital (K).

For cost minimization, the ratio of the marginal product of labor to wage should equal the ratio of the marginal product of capital to the rental cost of capital. Therefore, we have (PL/w) = (PK/r), which simplifies to (0.5 Q/L)/1 =(0.5 Q/K)/4. After simplifying, it tells us that the cost-minimizing combination (L/K) should be 4. So, for every unit of capital, the firm should use four units of labor.

To find the amount of labor and capital employed to produce Q = 100, we substitute Q into the production function and use the L/K ratio found earlier. After solving, we find that both labor and capital should amount to 20 units each to produce 100 units of output efficiently.

Unfortunately, without the ability to draw figures directly, we cannot satisfy the request for a graphical representation of cost minimization. However, the long-run expansion path shows the combination of inputs that minimizes cost for any level of output, which is L = 4K in this example. Consequently, the long run cost curve, C(Q), can be derived by substituting the equation L = 4K into the original production function and considering the costs of L and K.