Economics 335 Problem Set 6 Spring 1998 February 17, 1999 1. Consider a monopolist with the following cost and demand functions: q ö D(p) ö 120 p C(q) ö 900 ø 0.5q 2 a. What is the marginal cost function? b. What is the average cost function? c. What is the inverse demand function (What is the price as a function of output)? d. What is revenue as a function of output? e. What is marginal revenue as a function of output?
f. Find the competitive level of output q c (where marginal cost equals the price). g. What is the price p c at this level of output? h. What is the firm s cost at this level of output? i. What is the firm's profit at this level of output? j. Now find the level of output q m, that this firm will choose as a monopolist (where marginal cost equals marginal revenue). k. What is the price p m at this level of output?
l. What is marginal cost at this level of output? m. What is marginal revenue at this level of output? n. What is the firm's profit at this level of output? o. What is the firm's gain from monopoly (the difference between the answer to n and the answer to i)? p. Calculate the elasticity of demand 0 D at the monopolistic output level q m. Note that 0 D ö 0q 0p p q. q. Verify that 0 D > 1. Intuitively, why is this the case? r. Using your answer to (m), verify that MR ö p 1 ø 1. 0 D
2. Now look at the following figure which represents the situation described above: $ 120 MC A B p m C D E p c F G H MR D q m q c 120 Q a. The consumer surplus from the competitive output y c is the area A+B+C+D+E. What is this area? b. When the monopolistic level of output y m is chosen, the consumer surplus is only A+B. What is this area? c. What is the loss to consumers from monopolistic pricing? (This is the same as the area of C+D+E.)
d. Now consider the profits of the firm. This is harder to calculate exactly from the graph because of the fixed cost. The firm gains the area C+D due to the higher price, and it loses the area H due to lower quantity. What is the area C+D? What is the area H? (Note that marginal revenue is equal to marginal cost when marginal cost is equal to 40). What is the firm's gain from monopoly? Verify that this is the same as your answer to 1o above. e. The dead-weight loss from monopoly is the loss to consumers minus the gain to the firm. What is this dead-weight loss in this problem? f. One can also calculate this deadweight loss directly from the graph as the area of E+H. Calculate the area of this triangle and verify that it is the same as your answer to 2e.
3. Consider the following production function for a firm using two inputs x 1 and x 2. y ö 20x 1 ø 15x 2 x 2 1 x 2 2 Assume that the price of output is $100.00 and the prices of inputs are w 1 = $600 and w 2 = $100. a. What is an algebraic expression for the firm s profit in terms of x 1 and x 2? Œ = b. Use calculus to determine the profit maximizing levels of x 1 and x 2, y, and profit. 4. Now assume that the firm is monopsonist in the purchase of input 1. Assume that the price of input 1 is an increasing function of the amount bought. Suppose that w 1 ö 200 ø 50x 1 a. What is an algebraic expression for the firm s profit in terms of x 1 and x 2? Œ = b. Use calculus to determine the profit maximizing levels of x 1, x 2, y, and profit.
5. We now compare the monopsonistic solution to the efficient one. In the efficient outcome, the firm takes the price as given and sets MP 1 =w 1. That is, the firm takes the price w 1 to be given, and ignores the effect that it behavior has on this input price. a. What is an algebraic expression for the firm s profit in terms of x 1, x 2, and w 1? Œ = b. Use calculus to determine the profit maximizing levels of x 1, x 2 as a function of w 1. c. Now use the equation of x 1 from b and the input supply equation w 1 ö 200 ø 50x 1 to find the equilibrium levels of x 1 and x 2. Verify that demand for x 1 is higher than in the monopsonistic solution, but demand for x 2 is unchanged. d. What are the profit maximizing levels of y and profit? Verify that profit is lower than in the monopsonistic outcome.
6. Consider an amusement park operating as a monopoly. The figure below shows the demand curve of a typical consumer at the park. There are no fixed costs. The marginal cost associated with each ride is constant. It is comprised of two parts, each also a constant. There is the cost per ride of labor and equipment, k, and there is the cost per ride of printing and collecting tickets, c. A management consultant has suggested two alternative pricing policies for the park. Policy A: Make entry to the park free but charge a fixed fee per ride Policy B: Charge a fixed admission fee, T, and a fee per ride of p. (multi-part pricing) a. For pricing policy A, show on the graph the per ride price, p that will maximize profits. Illustrate the levels of consumer and producer surplus on the graph as well. Price The Amusement Park Problem 5 V Demand k + c e c f g 0 Q 1 Q 2 Q 3 Quantity
b. For pricing policy B, show on a similar graph the admission fee, T and price per ride p, that will maximize profits. Illustrate the levels of consumer and producer surplus. Price The Amusement Park Problem 5 V Demand k + c e c f g 0 Q 1 Q 2 Q 3 Quantity c. Compare the two policies. Which is better for the monopolist? Which is better for consumers? Which leads to the most efficient outcome?