1. The following cubic equation is a long-run production function for a firm:

Q = 0.002K³L³+ 6K²L²

Suppose the firm employs 10 units of capital.

a. What are the equations for the total product, average product, and marginal product of labor curves?

b. At what level of labor usage does the marginal product of labor begin to diminish?

c. Calculate the marginal product and average product of labor when 10 units of labor are being employed.

Now suppose the firm doubles capital usage to 20 units.

d. What are the equations for the total product, average product, and marginal product of labor curves?

e. What happened to the marginal and average product of labor curves when capital us-age increased from 10 to 20 units? Calculate the marginal and average products of labor for 10 units of labor now that capital usage is 20 units. Compare your answer to part c. Did the increase in capital usage affect marginal and average product as you expected?

2.The production function is Q = AL^aK^b, wherewhere a > 0 and b > 0.

a. The marginal product of labor is MP_L= _______.

b. The marginal product of capital is MP_K= _______.

c. The marginal rate of technical substitution is MRTS = ______.

d.Show that the isoquants for this production function are convex. [Hint: Show that MRTS diminishes as L increases. (Why?)]

e. Derive the equation for the long-run expansion path.

3. For the production function in exercise 1, let the price of labor be w and the price of capital be r.

a. The efficient usage function for labor is L* = _______.

b. The efficient usage function for capital is K* = ________ .

c. Find the long-run cost functions: LTC, LAC, and LMC.

d. Show that both LAC and LMC increase at any Q when either w or r increases.