Derived demand for capital and labor cost minimization

Question

Derived demand for capital and labor: Cost minimization

General Motors produces cars by using labor and capital. The production function for cars can be expressed as:

Q = 10K^0.4L^0.6

General Motors can rent its equipment and hire workers at competitive rates. Equipment needed for this operation can be rented at $30 per hour, and labor can be hired at $8 per worker hour. General Motors has allocated $100,000 for the initial run of car production.

Where Q represents number of cars produced, K denotes capital input (units per hour), and L denotes labor input (units of worker time per hour). The marginal products of labor and capital are as follows:

MPL = (0.6)(10)K^0.4L^-0.4

MPK = (0.4)(10)K^-0.6L^0.6

a. Using cost minimization determine the appropriate input mix (L, K, values in the long run) to get the greatest output for an expense of $100,000 for a production run of cars. Also, compute the level of output.

b. Suppose that the government imposes a $1 tax per hour worked. All else equal, explain how this would affect General Motors demand for labor and capital, and the level of cars produced.