Assume that a firm M (the manufacturer) sells an input (a lawn mower) to firm R (the retailer). Now R sells the lawn mower to the public, incurring a constant cost of $5 per lawn mower for its services. Fixed-proportions production holds for R. Let X therefore represent the number of lawn mowers. If both M and R are monopolists and P_L is the lawn mower price charged to the public with demand X=100-P_L

a) Find the derived demand for lawn mowers facing M. Hint: Find the marginal revenue equal marginal cost condition for R, where R's marginal cost is the sum of the $5 and the price P_X that it must pay M per lawn mower. Solving for X gives the derived demand.

b) If M's total cost function is 10+5X+X² , find the equilibrium prices and quantities P_L,P_X and X (Hint: MC = 5 + 2X)

c) Assume now that M and R form a single vertically integrated firm, M-R. Find the equilibrium values of P_L and X and the profit of M-R.

d) Compare the unintegrated case in part b with the integrated case in part c. Is it true that both the firms and the public would prefer the case in part c? Explain.