ADDITIONAL MONOPOLY TOPICS: PRICING STRATEGIES AND PUBLIC POLICY

Self-Test Problems

1. Suppose adults have the demand curve Pa = 30 − (1/5)qa for tickets to a local community theater. Children have a demand curve of Pc = 10 − (1/10)qc. Furthermore, MC = AC = 2.

a. Solve for the price charged and number of tickets sold in each separate market. Draw a graph.b. Solve for total consumer surplus, total producer surplus, and monopoly profit. Label these areas on your graph.c. Now, assume the theater wants to charge a single, uniform price in both markets. Solve for the combined demand curve. (Remember, it will have a kink.)d. Calculate the uniform price charged and the total number of tickets sold. Draw a graph of the combined demand, and show the uniform price charged and total amount sold.e. Solve for consumer surplus, producer surplus, and monopoly profit with a uniform price. Label these areas on the graph you created in response to (d), above.f. Compare profits in (b) to profits in (e). Which pricing method does the producer prefer?g. Prove that, in this case, third-degree price discrimination decreases total welfare.

Answer:

a. See the graphs above. Set MR = MC in each market. In the adult market, MR = 30 – (2/5)qa, using the twice-as-steep rule. Set MRa = MC; 30 – (2/5)qa = 2; (2/5)qa = 28; qa = 70; Pa = 30 – (1/5)(70) = 16. In the children's market, MR = 10 – (1/5)qc. Set MRc = MC; 10 – (1/5)qc = 2; (1/5)qc = 8; qc = 40; Pc = 10 – (1/10)(40) = 6.

b. In the adult market, consumer surplus = (1/2)(30 – 16) × 70 = 490, labeled area A in graph (a), and producer surplus = (16 – 2) × 70 = 980, labeled area B in graph (a). In the children's market, consumer surplus = (1/2)(10 – 6) × 40 = 80, labeled area C in graph (b) and producer surplus = (6 – 2) × 40 = 160, labeled area D in graph (b).

Total consumer surplus is 570 and total producer surplus/profit is 1,140. (Because MC = AC, profits and producer surplus are equal.) Total surplus is 570 + 1,140 = 1,710.

c. See the graph below. We need to horizontally sum the two demand curves, Pa = 30 – (1/5)qa; (1/5)qa = 30 – Pa; qa = 150 – 5Pa; Pc = 10 – (1/10)qc; (1/10)qc = 10 – Pc; qc = 100 – 10Pc; Q = qa + qc = 150 – 5P + 100 – 10P; Q = 250 – 15P for P ≤ 10. The kink will occur at P = 10. Note that, at P = 10, qa = 150 – 5(10) = 100 and Q = 250 – 15(10) = 100. For prices less than 10, the market demand curve is Q = 250 – 15P or:

P = 16.67 – (1/15)Q or P = 16.67 – 0.067Q MR = 16.67 – (2/15)Q or MR = 16.67 – 0.133Q

d. Now, set MR = MC and get 16.67 – (2/15)Q = 2; 14.67 = (2/15)Q; Q = 110. (Note that, depending on how you round 16 2/3, you might get a small decimal place after 110. But, we know from our answer to (a), above, that Q = 110, so we'll round to 110).

P = 16.67 – (1/15)(110) = 9.34. (See the graph above.) Note that, at Q = 100, MR in the adult market equals –10. MR from the market demand curve is 3.34. There's a big jump in the MR curve at Q = 100.

e. Producer surplus = profit = (9.34 – 2) × 110 = 807.4 (area YZVW on the graph above). Consumer surplus = the area of trapezoid XSTY plus the area of triangle SZT. Using the formula (1/2)(h1 + h2) × b, where h1 and h2 represent the two different heights of the trapezoid, we get consumer surplus = (1/2)(20.66 + .66) × 100 + (1/2)(0.66) × 10 = 1,066 + 3.3 = 1,069.3. Total surplus is 807.4 + 1,069.3 = 1,876.7.

f. The monopolist prefers separating the two markets, as its profits are equal to 1,140 under that method and only 807.4 without price discrimination.

g. Total welfare is 1,710 under price discrimination and 1,876.7 under single-pricing policy. In this case, total welfare is reduced by 166.7 with price discrimination.

2. Consider a golf course where each golfer has an identical annual demand function P = 200 − 4Q, where P is price per round and Q is rounds per year. Furthermore, MC = AC = 8. Assume, for simplicity, FC = 0.

a. What is the profit-maximizing single price and output the course should charge each golfer? Calculate profit per golfer.b. Show that the monopolist can make more profit using a two-part tariff, with P = MC. Solve for the fixed fee and profit per golfer.c. Show that even a $1 increase in price (per round) would cause profit per golfer to fall from that in (b) above.

Answer:

a. Using the twice-as-steep rule, MR = 200 – 8Q. Set MR = MC; 200 – 8Q = 8; 192 = 8Q; Q = 24; P = 200 – 4(24) = 104. Profit = (104 – 8) × 24 = 2,304, area B in graph (a), above.

b. See graph (a). If the monopolist charges P = MC, then 200 – 4Q = 8; 192 = 4Q; Q = 48; P = 200 – 4(48) = 8 = MC. The tariff would be equal to consumer surplus = (1/2)(200 – 8) × 48 = 4,608 (areas A + B + C). Thus, profits would be 4,608, which is higher than 2,304 (area B), in the previous answer.

c. In graph (b), if price equals 9, then 9 = 200 – 4Q; 191 = 4Q; Q = 47.75. In this case, the optimal fixed fee would be (1/2)(200 – 9) × 47.75 = 4,560.125 (area E). In addition, the course earns a profit of 47.75 (9 – 8) = 47.75 (area F) from a price of 9. So total profits equal the sum of areas E and F, or 4,607.875, which is still less than 4,608, with P = 8. The difference is the small triangle, G, in graph (b).

3. Consider a monopolist facing two consumers who have two different demand curves for its product, P1 = 200 − 4Q1 and P2 = 122 − 6Q2. Assume FC = 0 and MC = AC = 8.

a. The monopolist decides to use a two-part tariff that permits both consumers to stay in the market. Assume it prices at marginal cost. Solve for each consumer's output and fixed fee under this scenario. What is the monopolist's profit?b. Show that the monopolist is better off eliminating the low-volume consumer from the market. Solve for the output, the fixed fee, and the monopolist's profit if the low-volume consumer is eliminated.

Answer:

a. If the monopolist charges P = MC = 8, then 8 = 200 – 4Q1; 4Q1 = 192; Q1 = 48 and 8 = 122 – 6Q2; 6Q2 = 114; Q2 = 19. The fixed fee can be only what the consumers of the lower demand will pay; thus, it's (1/2)(114) × 19 = 1,083. The monopolist's profit is 2 × 1,083 = 2,166.

b. If the monopolist eliminates the low-volume consumer, it can charge a fee the high-volume consumer will pay. Consumer surplus for the higher demand is (1/2)(192)(48) = 4,608. The low-volume consumer would never pay this fee; thus, the low-volume consumer is eliminated. Profits to the monopolist will be 4,608. The monopolist earns more profit by eliminating the low-volume user than by charging a low fixed fee to keep the consumer in the market.

Questions and Problems for Review

4. Consider a monopolist with the demand curve P = 200 − 2Q. The monopolist's marginal cost (MC) is equal to 2, which also equals average cost (AC).

a. Solve for the non-discriminating monopolist's price and quantity.b. Now, assume the monopolist practices first-degree price discrimination. What is the total number of goods sold under this pricing policy? What is the lowest price charged for the good?c. Figure 1 in the text assumes a discrete number of units of the good to show consumer surplus and producer surplus. (Thus, consumer surplus is a collection of rectangles, rather than one big triangle.) Instead, assuming a continuous demand curve [similar to the analysis in Figure 5(b) in the chapter Further Topics in Consumer Theory], calculate total welfare under the two pricing schemes calculated in (a) and (b), above.d. Is consumer surplus higher using the pricing method in which total welfare is higher or lower? Explain.

Answer:

a. The non-discriminating monopolist's price and quantity are P = 101 and Q = 49.5, respectively.

b. Q = 99 is the total number of goods sold under first-degree price discrimination. The lowest price charged is at MC, so P = 2.

c. For the non-discriminating monopolist, consumer surplus is 2,450.25 and producer surplus is 4,900.50, so total welfare is 7,350.75. Under first-degree price discrimination, the producer extracts all consumer surplus for itself. Consumer surplus is 0 and producer surplus is 9,801, so total welfare is 9,801.

d. Although total welfare is higher under first-degree price discrimination, consumer surplus isn’t (because it equals zero). Consumer surplus is higher using the pricing method (non-discriminating) in which total welfare is lower.

5. For Broadway show tickets, adults have the demand curve Pa = 200 − qa and children have the demand curve Pc 100 − (1/2)qc. MC = AC = 20. Assume that Broadway theaters want to charge a single uniform price in both markets .

a. Solve for the combined demand curve. (Remember, it will have a kink.)b. Calculate the uniform price charged and the total number of tickets sold. Draw a graph of the combined demand, and show the uniform price charged and total number of tickets sold.c. Solve for consumer surplus, producer surplus, and monopoly profit. Label these areas on your graph.d. Does price discrimination increase or decrease total welfare?

Answer:

a. See the graph above. The demand curve is P = 133.33 – (1/3)Q for prices less than 100. The kink occurs at the point (100,100).

b. See the graph above. The uniform price charged and the total number of tickets sold are P = 76.67 and Q = 170, respectively.

c. Consumer surplus is labeled as area A, and producer surplus is labeled as area B. Consumer surplus = (1/2)(100)(100) + (1/2)(23.33)(100 + 170) = 5,000 + 3,149.55 = 8,149.55, and producer surplus = profit = 9,633.9. Total welfare is 17,783.45.

d. With price discrimination, consumer surplus = 4,050 + 1,600 = 5,650 and producer surplus = profit = 8,100 + 3,200 = 11,300. Total surplus = 16,950. Total surplus without discrimination, as calculated in (c), is higher, at 17,783.45 . Thus, price discrimination, in this case, reduces total welfare.

6. Comment on the statement, "Third-degree price discrimination always decreases welfare."

Answer: Third-degree price discrimination generally reduces welfare. However, if one group doesn't consume a good without discrimination, then price discrimination increases total output and results in a net welfare gain.

7. For each of the following scenarios, identify which type of price discrimination is described (first-degree, second-degree, or third-degree).

a. Although illegal, a local car dealership researches the income, number of children, and debt load of each consumer before negotiating on price. They then offer a price for the car based on each consumer's characteristics.b. A local retail store has a "buy one, get the second one at half price" deal on all remaining seasonal clothing items.c. A phone company charges different rates based on the time of day the phone call is placed (lower prices evenings and weekends than weekdays).

Answer:

a. First-degree price discrimination.

b. Second-degree price discrimination.

c. Third-degree price discrimination.

8. A local department store offers 10-percent discounts on Tuesdays for all senior citizens, for any purchase they make. As an economist, you're perplexed by this policy, because you don't believe their method of price discrimination is effective (increases the store's profit). You recommend they stop the price discrimination policy. Explain why.

Answer: To successfully discriminate, the price discrimination must prevent the transfer of the good from one group to another. A senior citizen could simply purchase a good for a non-senior citizen.

9. Consider a monopolist facing the following two demand curves for its product:

For simplicity's sake, assume fixed cost (FC) = 0 and MC = AC = 4.

a. The monopolist decides to price using a two-part tariff. Assume price equals marginal cost. Solve for output in each market (round q2 to two decimal places), the fixed fee, and monopoly profit under this scenario.b. Now, consider an increase in the per-unit price, from $4 to $5. Calculate the total output sold in each market, under this scenario, and the fixed fee charged. What is monopoly profit?c. Does pricing at MC = 4 and maximizing the fixed fee maximize profit? Explain, using your numbers from (a) and (b) above.

Answer:

a. At P = MC = 4, Q1 = 196 and Q2 = 175.79. The fixed fee is $14,678.465, and profit is $29,356.93.

b. At P = 5, Q1 = 195 and Q2 = 174.74. The fixed fee is 14,503.42. Profit = 2(Fixed Fee) + (p – mc)(q1 + q2) = 2(14,503.42) + (5 – 4)(195 + 174.74) = $29,376.58.

c. The firm is better off raising the price by $1. Profits are higher under this scenario, by $19.65. (Our analysis is similar to Figure 7 in the text. In this case, area E is larger than area C.)

10. Consider a monopolist facing the following two consumer demand curves for its product:

Assume FC = 0 and MC = AC = 4.

a. If the monopolist uses a two-part tariff that permits both types of consumers to remain in the market, solve for output, fixed fee, and monopoly profit.b. Now, show that the monopolist is better off eliminating the low-volume consumer. Solve for output, fixed fee, and monopoly profit if the low-volume consumer is eliminated.

Answer:

a. Q1 = 196, Q2 = 58, the fixed fee is $3,364, and monopoly profit is $6,728.

b. Q1 = 196, the fixed fee is $19,208, and monopoly profit is $19,208. Thus, monopoly profit is higher when the low-volume consumer is eliminated.

11. Nathaniel and Jonathan both love baseball. The minor league team in their town is almost guaranteed to make it to the playoffs. The following table represents the maximum prices Nathaniel and Jonathan will pay for regular season tickets and playoff tickets.

a. Compare total revenue for the baseball team under a separate pricing policy and under a pure bundled price.b. Which pricing scheme should the team pick? Explain how your answer is a result of the correlation of the demand curves. How are Nathaniel and Jonathan's demand curves correlated in this example? Explain.

Answer:

a. Total revenues under a separate pricing policy and a pure bundled policy are the same, $480.

b. Both pricing schemes generate the same revenue. Nathaniel and Jonathan’s demand curves are positively correlated. Jonathan is willing to pay a higher price than Nathaniel for both the regular season and the playoffs.

12. Consider a local fast food restaurant. The table below shows the maximum price Alex and Anna will pay for two products—chicken nuggets and fries.

Assume the marginal cost of chicken nuggets is $1.00 and the marginal cost of fries is $0.50.

a. Are Alex and Anna's demands negatively or positively correlated? Explain.b. Compare all four possible single-item pricing policies to find the profit-maximizing single-item pricing policy.c. Solve for profit if the restaurant engages in pure bundling.d. Assume the restaurant engages in mixed bundling and charges $3.00 for a bundle, $2.52 for chicken nuggets, and $1.50 for fries. Show that the profit obtained under mixed bundling will be higher than that under pure bundling or single-item pricing.

Answer:

a. Alex and Anna's demands are negatively correlated, because Anna will pay more for chicken nuggets than Alex and Alex will pay more for fries than Anna.

b. Profits are highest by charging $2.55 for chicken nuggets and $1.50 for fries. Profit is $2.55, in this case.

c. Pure bundling profits are $3.00.

d. Under mixed bundling. Anna will buy only chicken nuggets, and Alex will buy a bundle. Profit is $3.02, higher than in the cases of pure bundling or single-item pricing.

13. Consider a local fast food restaurant. The table below shows the maximum price Alex and Anna will pay for two products—chicken nuggets and fries.

Assume the marginal cost of chicken nuggets is $1.00 and the marginal cost of fries is $0.50.

a. Compare all four possible single-item pricing policies to find the profit-maximizing policy.b. Solve for profit if the restaurant engages in pure bundling.c. Assume the restaurant engages in mixed bundling and charges $3.50 for a bundle, $2.45

for chicken nuggets, and $2.00 for fries. Show that the profit obtained under mixed bundling isn't higher than that under pure bundling.d. Explain why mixed bundling doesn't yield higher profit, in this case.

Answer:

a. For nuggets at $1.50 and fries at $2, profit is $2.50. For nuggets at $1.50 and fries at $1, profit is $2. For nuggets at $2.50 and fries at $2, profit is $3. For nuggets at $2.50 and fries at $1, profit is $2.50. Profit is the highest ($3) when charging $2.50 for chicken nuggets and $2 for fries.

b. Pure-bundling profits are $4.

c. The mixed-bundling profit is $3.45, which is lower than the pure bundling profit.

d. Because both consumers have reservation prices above marginal cost, for both nuggets and fries, mixed bundling doesn't increase profit.

14. Explain what we mean by "X-inefficiencies." In what way does direct price regulation help to cause X-inefficiencies? What other problems are there with price regulation?

Answer: Recall, from the chapter Monopoly and Monopolistic Competition, that X-inefficiencies refer to a situation in which less than the maximum output is produced from a given set of inputs, because managers have objectives other than profit maximization. X-inefficiencies often exist in regulated firms, as regulated firms often assume the regulatory commission will guarantee the firm a profit and, therefore, the firm's managers have less incentive to minimize costs. In fact, under a rate-of-return method of regulation, managers may have an incentive to incur higher costs than necessary, so that the firm's profits and rate of return will decrease, and the firm can then request a rate hike. Rate-of-return price regulation might also encourage firms to use too much capital, relative to other inputs, to raise the regulated firm's rate base.