Consider a perfectly competitive industry with many identical firms, each producing according to the production function:

Q = √KL

Labor and capital are supplied to the industry according to the supply curves L = W and K = 4R.

a. Suppose that the industry produces Q units of output, using K units of capital and L of labor. Write a formula for L in terms of Q, W, and R and for K in terms of Q, W, and R.

b. Write two equations expressing the conditions of equilibrium in the two factor markets. Use these equations to get a numerical value for W/R.

c. Show that the industry's long-run total cost curve is given by:

Q = P

d. Suppose that the demand curve for the industry's output is given by:

Q = 1/5,000 − P

What are the price and quantity of output? How much labor is hired, and at what wage? How much capital is rented, and at what rental rate?

e. Under the conditions of part (d), calculate the producers' surplus in the output market. How much producers' surplus is earned by labor and how much by capital? How much profit is earned by firms? Is your answer consistent with your answer to Numerical Exercise (f)?