Econ 1 - Problem Set 4 With Solutions

Problem Set 4 Hard copies of your answers are due at the beginning of your section, either on Thursday, October 20, or Friday, October 21. For example, if your section starts at 10:00am on Friday, you should submit your answers to your TA in your section classroom at 10:00am on Friday, October 21. Late problems earn zero points. Note: you can work on these problems or your own, or in a small group with other current Econ 1 students. If you choose to work in a group, each student needs to hand in a separate, individual copy to his/her TA. 1. Federico can produce lemonade for $1 per gallon (his marginal cost and average cost both equal $1 per gallon). Federico has a local monopoly on the production of lemonade, and he faces the demand schedule in the following table: Price Quantity demanded $7.00 1 $6.00 2 $5.00 3 $4.00 4 $3.00 5 $2.00 6 $1.00 7 $0.50 8 a. If Federico cannot discriminate prices, how much should Federico produce and what price should he charge to maximize his profit? b. How big is his profit? c. How much consumer surplus do his customers get? d. What is the monetary value of the deadweight loss from Federico’s monopoly? e. Answer parts a through d assuming that Federico can perfectly pricediscriminate. 2. Consider a firm that uses the following rule to decide how much output to produce: If the profit margin (price minus short-run average total cost) is positive, the firm will produce more output. Use the firm’s short-run curves to evaluate this approach. Draw the firm’s short-run supply curve and compare it to the short-run supply curve of a profit-maximizing firm.1

1

From Economics, Principles and Tools by Arthur O’Sullivan and Steven Sheffrin. Fourth edition. Pearson/Prentice Hall. 2006.

3. The National Park Service grants a single firm the right to sell food and other goods in Yosemite National Park. Discuss the trade-offs associated with this policy.2 4. Consider a regulated natural monopoly with an initial price (equal to average total cost) of $3 per unit. Suppose the demand for the monopolist’s product decreases. What will happen to the price? How does this differ from the effects of a decrease in demand for a product produced in a perfectly competitive market?3 5. Consider a market for bottled water served by only two firms, Aquapura and Mountain Spring. Each firm can draw water free of charge from a mineral spring located on its own land. Customers supply their own bottles, and the firms have zero marginal costs. Rather than compete with one another, the two firms decide to collude by selling water at the price a profit-maximizing pure monopolist would charge. Under their agreement, each firm would produce and sell half the quantity of water demanded by the market at the monopoly price. Since the agreement is not legally enforceable, each firm actually has the option of charging less than the agreed price. If one firm sells water for less than the other firm, it will capture the entire quantity demanded by the market at the lower price. (a) Draw a typical demand and marginal revenue curve for a single-price monopolist, and show the equilibrium price and quantity if Aquapura and Mountain Spring abide by the agreement. (b) What would happen if the firms do not abide by the agreement? Describe the thought processes of the managers of the firms, and show (verbally and graphically) the equilibrium of this game. (c) Do you expect the firms to collude? Why or why not? Explain carefully! 6. Two firms, Faster and Quicker, are the only two producers of sports cars on an island that has no contact with the outside world. The firms collude and agree to share the market equally. If neither firms cheats on the agreement, each firm makes $3 million economic profit. If either firm cheats, the cheater can increase its economic profit to $4.5 million, while the firm that abides by the agreement incurs an economic loss of $1 million. If both firms cheat, they earn zero economic profits. Neither firm has any way of policing the actions of the other. a. What is the payoff matrix of the game that is played just once? b. What is the equilibrium if the game is played only once? Explain. c. What do you think will happen if the game can be played many times? Why? d. What do you think will happen if a third firm came in the market? Would it be harder or easier to achieve cooperation among the three firms? Why? 2 3

Ibid. Ibid.

Extra practice problems (completely optional; no points awarded) A. TotsPoses, Inc., a profit-maximizing business, is the only photography business in town that specializes in portraits of small children. Diane, who owns and runs TotsPoses, expects to encounter an average of eight customers per day, each with a reservation price shown in the following table. Customer A B C D E F G H

Willingness to pay ($/photo) 50 46 42 38 34 30 26 22

(a) If the marginal cost of a photo portrait is $12 and there are no fixed costs, how much should Diane charge if he must charge a single price to all customers? At this price, how many portraits will Diane produce each day? What will be her economic profit? (b) How much consumer surplus is generated each day at this price? (c) What is the socially efficient number of portraits? (d) Diane is very experienced in the business and knows the willingness to pay of each of her customers. If she can charge any price she likes to any of her consumers, how many portraits will she produce each day, and what will her economic profit be? (e) In this case, how much consumer surplus is generated each day? (f) Suppose Diane can charge two prices. She knows that customers with a willingness to pay above $30 never bother with coupons, whereas those with a reservation price of $30 or less always use them. At what level should Diane set the list price of a portrait? At what level should she set the coupon discount? How many photo portraits will she sell at each price? (g) In this case, what is Diane’s economic profit, and how much consumer surplus is generated each day? B. Suppose that the town theater is a local monopoly whose demand curve for adult tickets on Saturday night is P = 12 – 2Q (MR = 12 – 4Q), where P is the price of a ticket in dollars and Q is the number of tickets sold in hundreds. The demand for children’s tickets on Sunday afternoon is P = 8 – 3Q (MR = 8 – 6Q), and for adult tickets on Sunday afternoon, P = 10 – 4Q (MR = 10 – 8Q). On both Saturday night and Sunday afternoon, the marginal cost of an additional patron (child or adult) is $2. What price should the cinema charge in each of the three markets if its goal is to maximize profits?

PROBLEM SET 4 SOLUTIONS Question 1 Using the table provided in the question, we can construct additional columns for Total Revenue (TR), Marginal Revenue (MR), and Marginal Cost (MC): Price ($) 7 6 5 4 3 2 1 0.5 (a)

(b)

(c)

(d)

(e)

Demand 1 2 3 4 5 6 7 8

TR ($) 7 12 15 16 15 12 7 4

MR ($) 7 5 3 1 -1 -3 -5 -3

MC ($) 1 1 1 1 1 1 1 1

As a monopolist, Federico maximizes profits by producing at the quantity such that his MR equal to his Marginal Cost (MC). At any other quantity, he could do better by either lowering or raising the quantity produced. As MC=$1, Federico should produce four units. (Note that I assumed Federico produces the fourth unit even though profits would be the same if he produced 3 units.) Hence Q = 4 and P = $4. Profits are equal to Total Revenue (TR) - Total Cost (TC). Since Federico is producing 4 units, TR = $16. As marginal cost is constant and equal to average cost, there are no fixed costs. Therefore TC = $4. Hence Profit = $12. Consumer surplus (CS) is calculated by summing up the differences between the Marginal Benefit (MB) and price of each unit. The MB of the first unit is $7, $6 for the second unit, etc. Hence CS = (7-4) + (6-4) + (5-4) + (4-4) = $6. Deadweight loss (DWL) is equal to the sum of any unrealized surpluses. Though Federico will only produce 4 units, the MB of the fifth, sixth, and seventh units is greater than or equal to their cost (which is $1). Hence there are three unrealized trades that have non-negative surpluses. DWL is therefore equal to (3-1) + (2-1) + (1-1) = $3. Note that quantities in this problem are measured in discrete, not continuous, units. Therefore, any calculation of surplus should be made using discrete numbers, as above. In other examples, when quantity demanded and supplied are measured in continuous units (e.g. demand and supply are represented by smooth curves), surplus should be calculated as the area between the relevant curves. (a) If Federico can perfectly price-discriminate, he will charge each consumer their exact willingness-to-pay, as long as willingness-to-pay is greater than his marginal cost. Therefore, he will charge the first consumer $7, the second consumer $6, etc. Federico will only supply the first seven customers and will not supply the eight consumer since the latter’s willingness-to-pay ($0.50) is less than the marginal cost of producing that unit ($1). (Note that I am assuming consumers only purchase one unit of lemonade.) (b) Profit will be equal to (7+6+5+4+3+2+1) – 7 = $21.

(c) CS is zero since all purchasing customers are charged their willingness to pay. (d) There is no DWL since all consumers with a willingness-to-pay greater than marginal cost are sold lemonade. Question 2 This strategy is not profit-maximizing. To see this, consider the cost curves below.

The profit-maximizing quantity is Q* (where P* = MC). However, this firm will produce at Q’ which results in losses equal to the shaded area. Each unit after Q* has a (marginal) cost greater than its (marginal) revenue which equals P*. So every unit produced after Q* results in a loss. At Q’, the firm is making zero economic profit while a profitmaximizing firm is making a positive short-run economic profit.

A profit-maximizing firm’s short-run supply curve is the MC curve above SRAVC (shaded). Note that it will not supply in the short-run if price is below the minimum of the SRAVC curve. The supply curve for this firm is equal to the SRATC right of its intersection with MC (shaded). It will not supply in the short-run if price is below the minimum of the SRATC curve. Question 3 The possible losses from the National Park Service granting a single firm the right to sell food and other goods in Yosemite National Park are: - As a monopolist, the single firm will set a price such that at the quantity demanded, its marginal revenue will equal its marginal cost. Price may be higher and quantity lower relative to the case of a perfectly competitive market. This means the license may result in a deadweight loss to society. - As the equilibrium price is higher with a monopoly, consumer surplus will be lower than it would be under competition. If the NPS values consumers more than firms, the monopoly outcome may not be desirable. - The firm may have little incentive to improve the quality of its products and service since it faces no competition. The possible benefits are: - With only one firm, there may be efficiency gains. For example, it may be that the firm needs to employ fewer employees to service all of the visitors to Yosemite relative to the combined number of employees given competition in the market. Hence average total costs may be lower with only one firm. It is therefore possible that the firm posts lower prices and sells more units yielding a surplus to society. - To preserve the natural beauty of the National Parks, it may be desirable to limit the number of food facilities.

Question 4 Consider the diagram below. Note that MC is constant in the diagram but it need not be (though it does need to lie below the ATC curve since we are considering a natural monopoly). Price is initially $3 and quantity is Q1. P

D2

D1

P2 ATC

P1=$3

MC

Q2

Q

Q1

If demand decreases and the regulator continues to price at ATC, the quantity supplied will fall from Q1 to Q2 and price will rise from P1=$3 to P2. Note that this is different to the case of a perfectly competitive market where a fall in demand leads to a fall in quantity but a fall in price. Question 5 Dollars

Demand MR

p*

Q*

MC

Quantity

Single price Monopolist

(a)

Marginal revenue has three properties that must be captured by the graph: - marginal revenue declines as the quantity of output rises

- the marginal revenue curve lies below the demand curve - marginal revenue can be negative. The marginal cost in this case is equal to zero as shown on the chart. The quantity sold is the one of the intersection between marginal cost and marginal revenue. Then we find the corresponding price using the demand curve. (b)

If the firms do not abide by the agreement, then the firm which sets the lower price will sell to the whole market and the other one will sell nothing. Thus, the managers of either company always have an incentive to decrease their price just below the level set by the other company so that they can sell to the whole market. True, the firm’s operating profit (i.e. profit before we take into account fixed costs, interest payments and taxes) from its half of the market will fall due to the decrease in price, but this loss is outweighed by the increase in operating profit realized by now serving the whole market (instead of half of it). Operating profit Dollars Operating profit when sharing the market Operating profit when one company sets its price right under the one of its competitor

P other Demand

MC

Q/2

Quantity

Q

Price war

The firms will repeat this process in their minds, and the price will decrease until reaching the marginal cost of the companies. In this case since MC=0, this means that the price will decrease until reaching p=0 as shown below. Dollars

Demand MR

p*

Q*

MC

When companies cheat

Quantity

The equilibrium price and quantity will be the same as those in a competitive market. (c) Whether collusion is sustainable depends on the number of times this game is played. One shot game

In this case, companies have an incentive to cheat as explained previously because the incentive to serve the whole market is very high. Repeated game

If the game is played a large number of times, the firms can build strategies that could lead to a collusive agreement by making the long-run payoffs of following the strategy larger than the short-run payoff of deviating from the agreed strategy. A great example is the “tit for tat” strategy. The idea is for one player to regularly match in the next game the actions of the other player in the current game. Thus there will be an incentive for the other player to abide by the agreement, thus maximizing the profits of both companies. Another possible strategy is the “Grim Trigger Strategy”. In this case, a company cooperates until the other one defects but once it defects, it defects forever (explaining the name of this strategy). Thus there is a strong incentive for both companies to respect the agreement.

Question 6 a)

The payoff matrix is as follows: Faster \ Quicker Collude Not Collude

b)

c)

Collude 3,3 4.5,-1

Not Collude -1,4.5 0,0

Denote the strategies played by the firms in this order: (Strategy played by Faster, Strategy played by Quicker). Then, the Nash Equilibrium is clearly (Not Collude, Not Collude). To check why, note that no firm has an incentive to unilaterally deviate from playing this strategy, given what the other firm is doing. For example, if Faster is considering playing “Collude”, given that Quicker plays “Not Collude”, she gets a payoff of –1, whereas playing “Collude”, she gets a payoff of zero. So she doesn’t have an incentive to move from that strategy. The same argument applies to Quicker. In any other strategy profile, this doesn’t hold. For example, to check that (Collude, Collude) is not a Nash Equilibrium in the one-period game, it suffices to see that, given Quicker is playing “Collude”, Faster can unilaterally deviate and play “Not Collude”, and get a payoff of 4.5, violating the condition for this profile to be a Nash Equilibrium. The answer to this question really depends on how “many” those multiple times are going to be. For example, if the game were to be played only twice, we can

treat the second time that the game is played as a one-shot game in itself, for which we already know the Nash Equilibrium: (Not Collude, Not Collude). Given that the firms know this will be played the second time, they don’t have an incentive to cooperate the first time they interact, since they know they will be cheated in the second period. That is to say, we have to take care of the oneperiod game again only, and we know what the Nash Equilibrium for that is: (Not Collude, Not Collude). Actually, the same logic applies to any number of repetitions in this game. Assume the game is played N times, then on the N-th period, firms know this is the last time the game is played, and hence both will play a Nash Equilibrium. So this N-th period is irrelevant to see what will happen in the (N-1)-th period. Here, period N-1 is the last period, and the same logic applies, until we reach the first period. Therefore, cooperation will not be sustainable. However, if the firms don’t know for how long the game will be played, some kind of collusion could develop. If a firm cheats for a period, then the other one could punish her by cheating in the next period. If both firms know they will be in the market forever, then they might realize it is better for them to cooperate. By cooperating they’ll get a payoff of three forever, whereas by cheating they get zero. So collusion could be sustained. d) If we have three firms instead of two, and the Nash Equilibrium is still (Not Cooperate, Not Cooperate, Not Cooperate) and is unique, then the same logic as in the previous item will apply, so this will not help to attain cooperation. Now consider the case when cooperation could be sustained as argued in the previous item. Then with a third firm, it would be hard to identify which one deviated from the colluding agreement. So a cheated firm would not know who to punish or for how long. The chances of deviating from cooperation are bigger, so collusion will be less likely to arise. (For the curious: If, however, with a third firm the game is such that we have multiple Nash Equilibria, some cooperation could be sustained, even if the game is played only a finite period of times).