Pindyck and Rubinfeld, Chapter 13 Sections 13.1, 13.2, 13.3 and 13.6 continued

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1 Pindyck and Rubinfeld, Chapter 13 Sections 13.1, 13.2, 13.3 and 13.6 continued In deciding whether a threat is credible or not, reputation can play a role. For example, in the product choice game, if Far Out develops the reputation of being a little crazy, then Race Car Motors might believe a threat that Far Out makes to produce only big engines no matter what Race Car does. Thus, when a game is repeated several times, players sometimes find it advantageous to behave irrationally the first few times to develop such a reputation. Another kind of game: How to buy a dollar bill. This game was invented by Martin Shubik: A dollar bill is auctioned in the following way. The highest bidder gets the dollar in exchange for their bid. The second-highest bidder also has to give up the amount they bid, without getting anything in return. Experiments done in the classroom show that students bid more than a dollar for the dollar bill. This is because one player often ends up bidding 90 cents, and another player bids a dollar. The player who has bid 90 cents either will lose 90 cents if he loses the auction, or lose 10 cents if he bids $1.10 and wins the auction. The students ended up losing money because they did not think through the likely responses of their opponents. We have been looking at noncooperative games until now. Cooperative games differ from noncooperative games An example of a cooperative game could be a seller and buyer bargaining over a rug using Nash bargaining. Suppose the cost of producing the rug was $100 and the buyer values the rug at $200. The Nash bargaining solution finds x, the price to be paid for the rug, to maximize (200 x)(x 100). So we maximize the product of the two players surpluses. The solution is x = 150. Nash bargaining over a trade with two players results in the even division of the surplus generated by the trade. This game is called a constant sum game because the sum of the two players payoffs is constant whatever the distribution of payoffs. Another example of a cooperative game is where two firms negotiate a joint investment in research. Assume that neither firm has the ability to succeed alone. If the firms can sign a binding contract on how to divide the profits from the new technology, both parties can be made better off. This is a nonconstant sum game, because the sum of the payoffs depends on how the contract is formulated (one firm might work harder in the joint venture if it is promised more of the profits). Chapter 13, page 519, exercise 8. Answer. a. Each player will choose output to maximize profit given what the other player chooses. So you choose Q 1 to maximize (30 Q 1 Q 2 )Q 1 given Q 2 (because marginal cost is zero, profits equal revenue. The solution 1

2 to this is Q 1 = (1/2)(30 Q 2 ). By symmetry, your competitor s best output is Q 2 = (1/2)(30 Q 1 ). Now we must solve these two equations for Q 1 and Q 2. Plugging in (1/2)(30 Q 1 ) for Q 2 in the first equation, we get Q 1 = (1/2)(30 (1/2)(30 Q 1 )) = (1/2)( (1/2)Q 1 ). So 2Q 1 = 15 + (1/2)Q 1, 1.5Q 1 = 15, and Q 1 = 10. By symmetry, Q 2 = 10 also. Each firm s profits are ( )(10) = 100. b. Your competitor again maximizes profits, (30 Q 1 Q 2 )Q 2 given your output choice, so your competitor s output is (1/2)(30 Q 1 ). But now you can maximize your profits given that the competitor is doing this. So you maximize (30 Q 1 (1/2)(30 Q 1 ))Q 1. This can be simplified to (15 (1/2)Q 1 )Q 1 whose derivative is (15 (1/2)Q 1 ) (1/2)Q 1 = 15 Q 1 = 0, so Q 1 = 15. Then Q 2 = (1/2)(30 15) = 7.5. Your profits are (7.5)(15) = Your competitor s profits are (7.5) 2 = So it is an advantage to announce first, and you would pay anything less than dollars for the ability to go first. Chapter 11, Sections 1,2 How can firms with market power best use it to maximize their profits? Such firms can often increase their profits if they can charge different prices to different customers. To do this, the firms need to know about demands of different groups. Price discrimination is the practice of charging different prices to different customers, sometimes for the same good and sometimes for slight variations of the good. Another method is the two-part tariff. This requires customers to pay for the right to buy a good later, for some price. For example, customers pay a monthly subscription cost for long-distance telephone service and pay per minute of time. Bundling involves selling different products together as a package. For example a computer that comes with software packages. All these pricing strategies are ways of taking consumer surplus (and deadweight loss) and giving it to the producer. Consider the situation of a firm with market power. If it has to charge the same price to everyone, it chooses output Q where marginal cost equals marginal revenue and sets price P at the height of the demand curve at Q. There are consumers who would be willing to pay more than P for the good. Those consumers are represented by the portion of the demand curve to the left of Q. There are also consumers who would be willing to buy the good at a price higher than the firm s marginal cost, but are not willing to buy the good at the price P. If the firm could charge different prices to these different groups of consumers, it would make more profit than by charging just one price to everyone. For example it could charge P 1 to everyone willing to pay more than P 1, P to everyone willing to pay more than P, and P 2 to everyone willing to pay more 2

3 than P 2. The problem is to identify the different customers and to separate them so that the high-valuation ones aren t able to pay the low price. There are three types of price discrimination: first, second and third-degree price discrimination. If a firm could, it would charge each customer the maximum amount they are willing to pay for that unit. It would charge different amounts for different units bought by the same person if the person s reservation price (maximum amount they are willing to pay) decreases as they have more of the good. The practice of charging each customer their reservation price is called perfect first-degree price discrimination. Let us compare the firm s profit when it practices perfect first-degree price discrimination to its profit when it charges a single price to everyone. Consider first the case where a firm charges a single price. The firm s variable profit equals profit plus fixed costs. It is represented by the area between the marginal cost and marginal revenue curves where marginal cost is less than or equal to marginal revenue. Consumer surplus is everything below the demand curve and above the price, shown by the bold triangle in the graph. 3

4 Pmax P* Pc MR D=AR Q* Q** Now consider the case where the firm can perfectly price discriminate. Now the marginal revenue on each additional unit of the good is the price charged for that unit, as the firm does not have to reduce price for everyone when it reduces price for one customer. So the new marginal revenue curve is the demand curve. Variable profit is now the whole area above marginal cost and below the new marginal revenue. This area is bigger than the variable profit when the firm charged a single price. Consumer surplus is now zero, as each customer is charged his or her reservation price. In reality, it is impractical to charge everyone a different price, and it is difficult to find out the reservation price of each customer. Therefore perfect first-degree price discrimination is almost never possible. But firms can often imperfectly price discriminate. Doctors may charge less to lower-income patients whose ability to pay is lower and who have worse insurance. Accountants, lawyers and architects also may price discriminate, as they usually can get to know their client s ability and willingness to pay. Car salespeople might give a deal to someone who seems not in a hurry to buy from that company, but charge the full price to someone who is in a hurry to buy. Colleges offer scholarships or subsidized loans based on ability and willingness 4

5 to pay (parents are required to disclose financial information when applying for aid). With imperfect price discrimination, some consumers may be better off than if only one price was charged. For example, some of the people who were not willing to pay the single price are now charged a lower price and buy the good. Price Imperfect first-degree Price discrimination MR D=AR Second degree price discrimination is charging different prices for different quantities of the same good or service. For example, electricity companies charge lower prices for high amounts of electricity and higher prices for low amounts. The same is true for water and heating fuel. They do this because people tend to have lower reservation prices for higher amounts of electricity, but high reservation prices for a minimal amount. This is true for other goods as well, but for electricity and water the sellers can design a pricing system that gives buyers an incentive to buy only from that seller. This way the seller can keep track of what each individual buyer has bought (and thus figure out their entire demand curve) and charge a price for each unit that is close to the demand for that unit. discounts are another example of second-degree price discrimination. For many goods, like cereal or film, the price of a big box is lower per unit than the price of a small box. 5

6 Price P1 P0 D P2 P3 AC MR Q1 Q0 Q2 Q3 In the graph, there are three blocks, for each of which a different price is charged. Consumers and producers are made better off from this second-degree price discrimination relative to a situation where only one price is charged. Third-degree price discrimination divides customers into groups with different demand curves for each group. Then the same good is sold to these different groups at different prices. For example, a liquor company produces vodka, which it sells at $16 per bottle, advertised to be the best-tasting vodka. It also sells the same vodka for $8 a bottle under a different name. The company is dividing people into groups, one which cares about a name brand and is willing to pay a high amount for that, and one that is not willing to pay very much. Similar seats on an airplane can be sold for very different prices. Also, student and senior discounts. The ability to third-degree price discriminate depends on being able to divide consumers into groups based on some characteristic. For many goods, students and senior citizens are willing to pay less than the rest of the population. Also, it can be determined who is a student/senior citizen by checking a student ID or driver s license. 6

7 For airplane tickets, businesses are usually willing to pay more than private people, but have more restrictive travel times, and may need to travel at short notice. So to separate business from private travelers, airplane companies require advance purchase of the ticket or a Saturday night stay. How does the firm decide how much to charge each group? The total output should be divided between the groups so that marginal revenues for each group are equal. If the marginal revenue from group 1 exceeded the marginal revenue from group 2, the firm could make more profit by shifting some of the output from group 2 to group 1. It would do this by lowering the price for group 1 and raising it for group 2. Total output should be such that marginal revenue from each group equals the marginal cost of production. If for example marginal revenue from the two groups equalled each other but exceeded the marginal cost of production, the firm could get more profits by increasing total output, while keeping marginal revenues from both groups the same. It would decrease price to both groups, letting marginal revenues from the two groups decrease and come closer to marginal cost. These two results can also be shown algebraically. Let P 1 (Q 1 ) be the price charged to the first group (as a function of output), P 2 (Q 2 ) the price charged to the second group, and C(Q 1 +Q 2 ) the total cost of producing output Q 1 +Q 2 = Q T. Total profit is π = P 1 (Q 1 )Q 1 + P 2 (Q 2 )Q 2 C(Q 1 + Q 2 ). Taking the first derivatives with respect to Q 1 and Q 2 and setting them equal to zero, we get (d/dq 1 )(P 1 (Q 1 )Q 1 ) C (Q 1 + Q 2 ) = 0 = MR 1 (Q 1 ) (Q 1 + Q 2 ), (d/dq 2 )(P 2 (Q 2 )Q 2 ) C (Q 1 + Q 2 ) = 0 = MR 2 (Q 2 ) (Q 1 + Q 2 ). So MR 1 = MR 2 = gives the profit-maximizing output levels (for the two groups). It may be easier for managers to calculate relative prices that should be charged to each group of consumers and to relate these to elasticities of demand. From rearranging the expression for marginal revenue (see 10.1), we have that MR = P (1 + 1/E d ), where E d is the elasticity of demand. So when there are two groups with possibly different elasticities of demand, and the firm sells to both groups, we have MR 1 = P 1 (1 + 1/E 1 ) and MR 2 = P 2 (1 + 1/E 2 ). Therefore P 1 /P 2 = (1 + 1/E 2 )/(1 + 1/E 1 ) if the firm sells to both groups. This gives an expression for the relative price in terms of the elasticities. In some cases, though, it is not worth it for the firm to sell to all groups of consumers if some of them have small demand for the good. 7

8 Price D2 MR1 D1 MR2 In the graph, there is no quantity where the marginal revenue of sale to group 1 exceeds the marginal revenue of sale to group 2. On the other hand, with the graph below it is worth it to sell to both groups, because as you get to quantities close to the = MR output, the marginal revenue gotten from selling to group 1 exceeds the marginal revenue gotten from selling to group 2. 8

9 Price D2 MR1 D1 MR2 9