Question

Davy Metal Company produces brass fittings. Davy's engineers estimate the production function represented below as relevant for their long-run capital-labor decisions.

? = 500? 0.6? 0.8 where Q = annual output measured in pounds, L = labor measured in person-hours, K = capital measured in machine hours. The marginal products of labor and capital are:

??? = 300? −0.4? 0.8 ; ??? = 400? 0.6? −0.2

Davy's employees are relatively highly skilled and earn $15 per hour. The firm estimates a rental charge of $50 per hour on capital. Davy forecasts annual costs of $500,000 per year, measured in real dollars.

a. Determine the firm's optimal capital-labor ratio, given the information above.

b. How much capital and labor should the firm employ, given the $500,000 budget? Calculate the firm's output.

c. Davy is currently negotiating with a newly organized union. The firm's personnel manager indicates that the wage may rise to $22.50 under the proposed union contract. Analyze the effect of the higher union wage on the optimal capital-labor ratio and the firm's employment of capital and labor. What will happen to the firm's output?

d. Graph the optimal bundles before and after wage change in the same diagram. K on y axis and L on x axis.

Answer 1

Part A

Here Given,

Now the MRTS is,

Then equate this with w and r

Part B

Here given

from the optimal ratio,

or Hour

Substitute the value of L to find out K,

or

now,

Part C

Based on MRTS,

equating the MRTS,

Substitute it with C,

or

Substitute it with C and find K,

or

Here K is constant, now output is,

After the wage change output falls from 157,568,202.50 to 123,541,771.8

Part D

The optimal output bundle are given below,