EconS First-Degree Price Discrimination

Transcription

1 EconS First-Degree Price Discrimination Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 41

2 Introduction We are moving on to our second unit of the semester; an examination of pricing techniques available to monopolists. We ll begin with a discussion of price discrimination, which allows a monopolist to charge di erent prices to di erent groups based on their identifying characteristics. Today, we ll look at rst-degree price discrimination, also known as personalized pricing, or perfect price discrimination. Eric Dunaway (WSU) EconS 425 Industrial Organization 2 / 41

3 Price Discrimination Up until now, we ve been using uniform pricing. Uniform meaning that we can only charge one price to everyone in the market. Price discrimination allows us to charge multiple prices to di erent groups of people, based on observable characteristics. Examples can range from students, to the elderly, to women, etc. Eric Dunaway (WSU) EconS 425 Industrial Organization 3 / 41

4 Price Discrimination We classify price discrimination into three types, based on how much information is available to the monopolist. First-degree price discrimination occurs when the monopolist has perfect information about the market. Second-degree price discrimination occurs when the monopolist is aware that people are members of certain groups, but does not know which groups people belong to. Third-degree price discrimination occurs when the monopolist can identify which people are within each group. Eric Dunaway (WSU) EconS 425 Industrial Organization 4 / 41

5 Price Discrimination Why should a monopolist price discriminate? It increases pro ts! Price discrimination converts signi cant portions of consumer surplus into producer surplus. In some cases, it also eliminates some or all of the deadweight loss created by uniform pricing. Eric Dunaway (WSU) EconS 425 Industrial Organization 5 / 41

6 Price Discrimination A quick example: A theater in town is holding a showing of a new movie this weekend. Students will see the movie if the price is 10 dollars or less. Senior Citizens will see the movie if the price is 5 dollars or less. Let s say that there are 10 students and 5 seniors. Depending on what price the theater charges, the movie theater s pro ts look like Price Student π Senior π Total π $ $ In this case, the rm would do better by charging a higher price and losing the senior market. Could they do better? Eric Dunaway (WSU) EconS 425 Industrial Organization 6 / 41

7 Price Discrimination What if the theater charged $10 to students and $5 to seniors? Its pro t would be Price Student π Senior π Total π $ $ $10/$ this way, the rm could increase its pro ts by charging less to the seniors and getting them to come see the movie, too. The lower willingness to pay for the senior citizens is equivalent to having a lower price elasticity of demand (more elastic). Their atter demand curve requires a lower price from the monopolist. Eric Dunaway (WSU) EconS 425 Industrial Organization 7 / 41

8 Price Discrimination Real world example: Disneyland. They charge $199 to out of state residents for a 3 day pass. But they only charge $154 to California residents. Why? People who are travelling from out of state are likely on vacation and are planning on going to Disneyland regardless. They are less sensitive to the price than the local residents who have already been several times. Eric Dunaway (WSU) EconS 425 Industrial Organization 8 / 41

9 Price Discrimination There are three requirements a rm must meet in order to implement price discrimination. The rm must have market power. The rm must be able to di erentiate consumers (and they must have di erent valuations of the good). The rm must be able to prevent arbitrage (resale). Eric Dunaway (WSU) EconS 425 Industrial Organization 9 / 41

10 Price Discrimination The rst two points may seem a bit obvious, but let s talk about the third. If a rm cannot prevent arbitrage, consumers who can obtain a lower price via their own valuation can resell the good to consumers with a higher valuation. Potentially, they could compete with the rm, o ering a price lower than the monopolist s price, and cutting into their pro ts. Most arbitrage prevention is done by identi cation veri cation or quantity controls. Eric Dunaway (WSU) EconS 425 Industrial Organization 10 / 41

11 First-Degree Price Discrimination In rst-degree price discrimination, a rm is able to observe every consumers willingness to pay (reservation price) for their product. Naturally, the rm wants to charge every consumer exactly their willingness to pay. Intuitively, the rm will sell to anyone it can make a pro t from, selling to those consumers whose valuations are at least the marginal cost of the rm. We ll prove this in a little bit. Eric Dunaway (WSU) EconS 425 Industrial Organization 11 / 41

12 First-Degree Price Discrimination Before we get into the details of rst-degree price discrimination, we need to take a step back and talk about the demand function. Remember that the demand function, in essence, is an ordering of consumers based on their willingness to pay for the good. At the top of the demand curve, we have the consumer who is willing to pay the most for the good, and the valuation decreases from there. Most importantly, we have assumed that consumers are distributed uniformly along the demand curve. Eric Dunaway (WSU) EconS 425 Industrial Organization 12 / 41

13 First-Degree Price Discrimination p q D q Eric Dunaway (WSU) EconS 425 Industrial Organization 13 / 41

14 First-Degree Price Discrimination Number of Consumers q D Eric Dunaway (WSU) EconS 425 Industrial Organization 14 / 41

15 First-Degree Price Discrimination Number of Consumers Uniform Distribution q D Eric Dunaway (WSU) EconS 425 Industrial Organization 15 / 41

16 First-Degree Price Discrimination Number of Consumers f (x ) q D Eric Dunaway (WSU) EconS 425 Industrial Organization 16 / 41

17 First-Degree Price Discrimination Another facet of rst-degree price discrimination is that we could have consumer willingness to pay arbitrarily distributed along the demand curve. We can even have consumers with the same valuations. This can pose some challenges for calculating pro ts, as we ll see. Let v(x) represent the willingness to pay for consumer x. We say that the consumers are distributed along some function f (x). Let v(0) = v, i.e., the rst consumer along our spectrum has the highest valuation of v. As we move along our valuation (demand) curve, v(x) decreases, i.e., dv (x ) dx < 0. Eric Dunaway (WSU) EconS 425 Industrial Organization 17 / 41

18 First-Degree Price Discrimination To start, we will assume that every consumer will either buy one unit of the good, or none at all. This is known as unit demand. The rm will sell to consumers whose valuations are high enough. At some point, consumer s will have a valuation where they are indi erent between buying the good or not. We are interested in nding consumer s. Everyone with valuations equal to or above s will purchase the good, while everyone with valuations lower than s will not. Eric Dunaway (WSU) EconS 425 Industrial Organization 18 / 41

19 First-Degree Price Discrimination We can obtain the total number of units sold in this market, q, by integrating the density function up until our indi erent consumer s, q = Z s 0 f (x)dx This is equivalent to saying that we just add up everyone who buys the good. In the uniform case, f (x) = 1, and we nd that q = Z s 0 1dx = xj s 0 = s 0 = s Eric Dunaway (WSU) EconS 425 Industrial Organization 19 / 41

20 First-Degree Price Discrimination Number of Consumers f (x ) s q D Eric Dunaway (WSU) EconS 425 Industrial Organization 20 / 41

21 First-Degree Price Discrimination With this, we can set up an extremely general version of the rm s pro t maximization problem. With non-uniform pricing and a non-uniform distribution of consumers, we have Z s max s Z s 0 v(x) f (x) dx {z} {z} p q c 0 f (x)dx And from here, we can make some simpli cations. If we impose uniform pricing, v(x) = v(s) for all x, and we can pull it out of the integral, max s Z s v(s) f (x)dx 0 Z s c 0 f (x)dx Eric Dunaway (WSU) EconS 425 Industrial Organization 21 / 41

22 First-Degree Price Discrimination If we wanted to have non-uniform pricing, but a uniform distribution of our consumers, we would have q = s, Z q Z q Z q max v(x)dx c dx = max v(x)dx c (q) q q 0 0 Or if we wanted to have both uniform pricing and a uniform distribution of our consumers, again, we would have q = s, Z q Z q max v(q) dx c dx = max v(q)q c(q) q 0 0 q which is essentially the pro t function we have seen since the beginning. 0 Eric Dunaway (WSU) EconS 425 Industrial Organization 22 / 41

23 First-Degree Price Discrimination Let s go back to our original, general de nition. Z s Z s max v(x)f (x)dx c f (x)dx s 0 We can di erentiate this function with respect to s to obtain v(s)f (s) c 0 (q)f (s) = 0 v(s) = c 0 (q) Here is that result that I spoke of before. The indi erent consumer s will have a valuation that is equal to the rm s marginal cost at the equilibrium quantity. 0 Eric Dunaway (WSU) EconS 425 Industrial Organization 23 / 41

24 First-Degree Price Discrimination E ciency wise, under rst-degree price discrimination, we have a situation where every individual whose valuation is at least as high as the marginal cost of the rm purchases the good. That sounds pretty e cient to me. In fact, under rst-degree price discrimination, there is no deadweight loss. There isn t any consumer surplus either. Since every consumer pays their valuation, they receive no surplus. All surplus in this case is producer surplus. Eric Dunaway (WSU) EconS 425 Industrial Organization 24 / 41

25 First-Degree Price Discrimination p q S CS PS DWL MR q D q Eric Dunaway (WSU) EconS 425 Industrial Organization 25 / 41

26 First-Degree Price Discrimination p MC PS v(x ) x Eric Dunaway (WSU) EconS 425 Industrial Organization 26 / 41

27 First-Degree Price Discrimination To obtain the welfare level, we simply integrate the di erence between the price paid (the consumer s valuation) and the marginal cost of each consumer up until consumer s, PS = Z s 0 v(x) c 0 (x) f (x)dx or we could simply use a triangle formula when it is convenient. Eric Dunaway (WSU) EconS 425 Industrial Organization 27 / 41

28 First-Degree Price Discrimination Consider a banana market where a single rm acts as a monopolist. There are four consumers, Stuart, Kevin Bob, and Dave. Stuart values a banana at 5, Kevin and Dave value a banana at 3, and Bob values a banana at 1. It costs 2 to make a banana. If the banana monopolist knew all the reservation prices and could prevent the consumers from reselling bananas, what price would it charge to each of them? Eric Dunaway (WSU) EconS 425 Industrial Organization 28 / 41

29 First-Degree Price Discrimination The banana monopolist would charge Stuart 5, Kevin and Dave 3, and wouldn t sell a banana to Bob. Poor Bob. In this case, we don t have an indi erent consumer s since our consumers are discrete. If we did have one, their valuation would have to be 2, the marginal cost of the banana monopolist. Furthermore, the banana monopolist captures all of the surplus in this market, PS = (5 2) + (3 2) + (3 2) = 5 Eric Dunaway (WSU) EconS 425 Industrial Organization 29 / 41

30 First-Degree Price Discrimination First-degree price discrimination is very costly to implement. Typically, the cost to acquire all of the information necessary for each consumers valuation would o set anything gained by non-uniform pricing. Furthermore, consumers will have strong incentives to hide their true valuations from the rm. Only a few rms, like personalized accounting and some legal practices, are able to implement this sort of thing. Be careful not to confuse rst-degree price discrimination with bargaining. Eric Dunaway (WSU) EconS 425 Industrial Organization 30 / 41

31 Two-Part Pricing Thus far, we have limited our discussion of rst-degree price discrimination to consumers with unit demand, i.e., they only buy at most one unit of the good. The method above becomes more challenging when we have consumers who would want to purchase multiple units of the same good. Willingness to pay for the second unit is typically less than the rst, etc. We can allow for rst-degree price discrimination while consumers are able to purchase multiple units of the good by employing two-part pricing. Eric Dunaway (WSU) EconS 425 Industrial Organization 31 / 41

32 Two-Part Pricing Two-part pricing breaks down the price a consumer pays into a xed and variable part. T (q) = A + pq where consumers must pay the xed cost A for the right to purchase the good, and then a (usually small) price p for each unit purchased. Two-part pricing is fairly common in the real world. Taxi services, phone plans. Costco, bars. Eric Dunaway (WSU) EconS 425 Industrial Organization 32 / 41

33 Two-Part Pricing T (q) = A + pq The most challenging aspect of two-part pricing is designing the xed cost, A, such that the consumer is willing to pay it. We require the xed cost to be incentive compatible, i.e., the better choice for the consumer. We can gure out the value of A by looking at the surplus received by an individual consumer x, S x (q) = v(x, q) p(x)q Eric Dunaway (WSU) EconS 425 Industrial Organization 33 / 41

34 Two-Part Pricing S x (q) = v(x, q) p(x)q If consumer x doesn t pay the xed cost, their surplus is 0 (no change). Thus, in order to get them to buy the product, their surplus must be positive. Thus, A S x (q). For simplicity. We are going to assume that if the consumer is indi erent between buying and not buying, they choose to buy. This lets us set A = S x (q). Intuitively, the rm can charge consumer x their entire surplus for the right to purchase the good. Eric Dunaway (WSU) EconS 425 Industrial Organization 34 / 41

35 Two-Part Pricing Looking at just consumer x, the monopolist then needs to gure out how much to charge per unit, p. Setting up the pro t maximization problem, max A + p(x)q c(q) q and we can substitute A = S x (q) = v(x, q) to obtain pq into this expression max q max q v(x, q) p(x)q + p(x)q c(q) v(x, q) c(q) Di erentiating with respect to q, v(x, q) q c 0 (q) = 0 Eric Dunaway (WSU) EconS 425 Industrial Organization 35 / 41

36 Two-Part Pricing v(x, q) q c 0 (q) = 0 v (x,q) Let s talk about that rst term, q. This is how consumer x s valuation of their quantity purchased changes as the quantity increases. That sounds a lot like an inverse demand function for consumer x. In fact, that s exactly what it is. v (x,q) We can replace q with p(x), the price for consumer x, and obtain p(x) c 0 (q) = 0 p(x) = c 0 (q) Eric Dunaway (WSU) EconS 425 Industrial Organization 36 / 41

37 Two-Part Pricing Thus, a rm simply wants to charge each individual the marginal cost per unit that they purchase. Again, this leads to an e cient solution (no deadweight loss). To make up for charging such a low unit price, the rm sets the xed cost to exactly the consumer s surplus, thus claiming all of the surplus for themself. It s basically the same outcome as we saw in rst-degree price discrimination. Eric Dunaway (WSU) EconS 425 Industrial Organization 37 / 41

38 Two-Part Pricing This is very much how Costco prices its products. Yearly membership fee, then marginal cost for everything else. The only di erence between this method and our model is that the xed price is uniform; Costco doesn t di erentiate between its consumers. This could lead to some small distortions in the market, but such is the case in reality. Eric Dunaway (WSU) EconS 425 Industrial Organization 38 / 41

39 Summary First-degree price discrimination, while hard to implement, allows a rm to charge an individual price to each consumer based on their willingness to pay. It s also e cient! Eric Dunaway (WSU) EconS 425 Industrial Organization 39 / 41

40 Next Time Block Pricing and Third Degree Price Discrimination Organizing consumers into groups is much easier than looking at everyone individually. Reading: 5.3 Eric Dunaway (WSU) EconS 425 Industrial Organization 40 / 41

41 Homework 2-3 Consider a market with a single rm, that faces the following inverse demand function, p = 100 5q and a constant marginal costs of MC = If the rm decided to implement a two-part pricing strategy, what would the monopolist charge for the entry xed cost and the price per unit? Eric Dunaway (WSU) EconS 425 Industrial Organization 41 / 41