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Unformatted text preview: Economics 101, UCLA Jernej Copic Lecture 5. Some Pricing Examples and Pareto Efficiency. Example 1: Markets and intermediaries - a case study In this example, we will take things to a new level. Instead of simply trying to analyze a given game (after we have formulated it as an abstraction of a story), we will take an economic interaction, and try to examine it carefully, using the concepts we have learned so far. Members of No1's team want to exchange flower pots. Alan and Bob want to buy exactly one flower pot each, and Cariatide and Grunf are each able to produce exactly one flower pot. Alan, being a romantic soul, really wants the flower pot, so his valuation of the flower pot is 100 in dollar equivalents. That is, if Alan got the flower pot for free he would be as happy with that as if he was given 100 bucks. On the other hand, if he were to pay $x for the flower pot, his utility out of that transaction would be 100 - x. Bob doesn't care nearly as much for the flower pot, and his valuation is 51 in dollar equivalents (hence if he pays a price $x, his utility out of that transaction is 51 - x. On the production side, Cariatide is really good at producing flower pots and he really enjoys doing it, so his cost in dollar equivalents is literally 0 (in principle, he needs to pay 50 cents for the materials but the enjoyment from working compensates this; the opportunity cost of his time is also pretty much 0 as he would otherwise be lying around in idle boredom). Cariatide's utility from selling the flower pot for a price of $x would therefore be x - 0 = x. Grunf, on the other hand, is much less apt at producing flower pots, and he also has better things to do in life, so that his production cost in dollar equivalents is $49, and the utility he would get from selling the pot at a price $x would be x - 49. If any of these guys doesn't get an agreeable price offer for either selling or buying, then he simply gets a utility of 0 (a producer in that case doesn't produce the pot at all, and a buyer doesn't get to enjoy it but doesn't have to pay for it either). No1 understands that his team does not function well without supervision, so he appoints Oliver to oversee the flower pot exchange as an intermediary who can set prices. The only requirement that No1 puts to Oliver is that he may not discriminate as that may unsettle the team spirit. Finally, since all of them know each other very well, they know exactly what utilities they get from different transactions. A way to think about this is that by setting prices, Oliver will set up a game between the traders. By non-discrimination, he must do that anonymously (so that he cannot offer different prices based on a person's name), and a way to do that is to simply write prices on a board (as is done for example at NYSE), and whoever ends up agreeing to these will then participate in the exchange, provided that he can find a counter-party for the trade. 1 In other words, for every set of prices that Oliver, the intermediary (or the market maker ), posts on the board he sets up a different game between the traders. The traders then simply shout "Buy" or "Sell" in a simultaneous-move game. One possibility for Oliver would be to simply write one price on the board. Notice that it would make no sense for Oliver to post a price at which say 2 buyers agreed to buy but only one producer agrees to sell, for instance, a price of $40 would do that then Oliver would only face issues in having to solve the dispute between the buyers. He can avoid this by posting a price that matches supply and demand, that is a price that clears the market. For example, the price $50 would do just fine for that purpose.1 But Oliver is a man of more ingenious designs. He realizes that the non-discriminatory requirement does not preclude him from posting two prices on his board: one for the buyers, and one for the sellers: he then buys the pots from all the sellers who are willing to sell at the ask price, pask , and sells the pots to the buyers who are willing to buy at the bid price, pbid . As before, he makes sure to be able to deliver on his promises, so that these prices again have to clear the market. Conveniently, of course, Oliver pockets the difference (or provides a subsidy if he must, since he is a good-hearted soul - note that would only happen if Oliver were buying the flower pots from the producers for more money than what he were selling them for to the buyers). He could for instance post a price to buy the pots at pask = $49.01, and sell them at pbid = $50.99. At these prices, both sellers would be willing to sell the pots, and both buyers would be willing to buy them, and Oliver would be making a profit of $1.98 on each flower pot, so that his total gain from these transactions would be 2 $1.98 = $3.96. Note also that this would be a Pareto optimal allocation if we take the four traders as well as Oliver into account but would not be Pareto-optimal if we consider just the four traders. But Oliver can do better than that. Oliver's game is as follows. He posts the ask price pask = $0.01, and a bid price of pbid = $99.99, and then waits at his desk with a sign "a not-for profit organization" for all the traders who are able and willing. Each trader thus has two possible actions: "Yes" and "No"; for a seller a "Yes" means that he will bring a flower pot and receive the cash money amount of $0.01 in exchange, for a buyer a "Yes" means that he is willing to bring the cash money payment of $99.99 and will subsequently receive a flower pot, and a "No" means that nothing happens to the particular trader (so he simply obtains a utility of 0). A Nash equilibrium of this game set up by Oliver is that Alan and Cariatide say "Yes" while Bob and Grunf say "No".2 Notice also that these prices satisfy the market clearing condition and
Show that if the objective was to maximize the sum of traders utilities, then $50 would be a price that does it. Does any other price do that? 2 There are also other equilibria. For instance, when they all shout "No", regardless of the prices that Oliver posts, no trader would have any incentive to change his behavior - since he would not be able to find a counter-party to trade with, he would not trade anyways so he may just as well shout "No". In fact, this may be a real problem, for instance when a new exchange is established, and nobody shows up (that being
1 2 that Oliver thus obtains a benefit of $99.98. 3 Example 2: Monopoly price. In this example, we still have Alan who wants to buy flower pots, but now he wants a hundred flower pots to decorate his own apartment. His valuation for the first pot he buys is $100, he values the second pot at $99, the third at $98, and so on he is only willing to get the 101st pot if it comes for free, and he doesn't want the 102nd flower pot at all. Alan being a simple and honest man, then simply announces this on the board in search of possible flower-pot vendors. First, suppose that Grunf was sent to a study-abroad trip in order to better learn the technology of flower pot making 4 Cariatide is then the only producer and suppose that he is only allowed to charge the same price for every flower pot that he sells (so he cannot offer quantity discounts). In other words, he is a monopolist producer, his production cost is $0, and he is facing Alan's demand for the flower pots. To make things simple, let's assume that Cariatide will only consider the prices in whole dollars in order not to fiddle around with pennies. Cariatide then pauses and thinks about the price he should charge and notices the following: he could sell 100 pots at the price of $1 and since his production cost is $0, he would thus make a profit of $1 on each unit, which makes a total of 100 $1 = $100; but if he sells pots at $2 he will sell only 99 but he will make a bigger profit. In fact, the his biggest profit is when he sells 50 flower pots and charges a price of p = $50, in which case
equivalent to shouting "No" in our little game) because everyone expects nobody to show up - hence why show up... In the 1980s when CBOE was creating new markets for derivatives, they at some point started giving away TV sets so that the traders would show up. 3 Questions: 1. In the game between the four traders, is this equilibrium in dominant strategies? 2. Is the final allocation of utilities that the traders obtain Pareto optimal among the traders? 3. Suppose we included Oliver in the consideration of Pareto optimality. That is, is the final allocation Pareto optimal for Alan, Bob, Cariatide, Grunf, and Oliver? 4. Which of the following would restore Pareto optimality among traders: (A) If No1 allowed Oliver to present traders with personalized prices (B) if No1 required Oliver to write only one price (C) if No1 allowed Oliver to do whatever he pleased (D) if No1 allowed traders to negotiate prices by themselves without Oliver's control of the exchange (E) if No1 appointed Cariatide to be the intermediary instead of Oliver. 5. Suppose now that instead of posting prices on the board, Oliver decided to only write one price on the board and then charge the same fee to every trader who wanted to participate on the exchange (as a compensation for his effort and time). What price would Oliver post and what would be the fee that he charges? Would the resulting allocation be Pareto optimal? 6. What if the only requirement No1 imposed on Oliver were that every trader obtains the same utility. Would the resulting allocation maximize the total sum of traders' utilities? Would it be Pareto-optimal among the traders and Oliver? What requirement should No1 in your opinion impose on Oliver in order for the exchange to be fair (assume that Oliver's operating cost of the exchange is 0)? Is there a simple way in which No1 could verify that this requirement had been met? 4 Financed by Oliver who was forced by the disgruntled No1 to pay $2 for the cost of Grunf's study and airfare, and pay $92 to No1 so that No1 would in the future again consider him for his splendid services as a market maker. 3 he makes a profit of $2500. One way to see that is to simply consider all the possibilities, but then we would have to compute the profit of Cariatide at 100 different prices. A simpler way is to think of Alan's demand as a continuous function - so that it's the linear function y(p) = 100 - p, where y represents the number of units Alan would buy if the price he faces is p. Clearly, for every integer amount of flower pots this is the same function as the one described in the previous paragraph. Now if Cariatide puts a price tag of p on his flower pots he will sell y(p) units at that price and thus make a profit of (p) = y(p)(p - cost) = (100 - p)p. His profit will be largest where this function has a maximum. Since this is a quadratic with a negative coefficient in front of the quadratic term, it has an inverted U-shape, so that we can compute its maximum by taking the first derivative and obtaining the optimal price by setting this derivative equal to 0. That is, (p) = 100 - 2p, and 100 - 2p = 0, so that p = 50. With this price level, Alan will get a surplus of 1250 in terms of his utility in dollar equivalents - he gets a surplus of 50 on the first pot, 49 on the second one, and so on, which sums up to 1250. This is precisely the area under the triangle between his demand and the price of 50 charged by Cariatide.5 Example 3: Bertrand competition. Grunf made it back safely from his study abroad. Due to his newly acquired knowledge and enthusiasm his cost of producing the flower pots is now also 0 in dollar equivalents. Now Cariatide and Grunf each post on the board a price at which he is willing to offer the flower pots with the understanding that they will deliver as many flower pots as necessary to satisfy their customers at those price. Alan can then decide from which one of them he wants to buy his flower pots, after he considers the prices he faces. Since Grunf and Cariatide produce exactly the same sort of flower pots (in terms of aesthetics, quality, ecological concerns and any other possible practical or impractical issues), Alan would clearly be silly to purchase his flower pots from the producer with the higher price offer. But Grunf and Cariatide both know that. So Grunf thinks to himself:"If I post a price of $49, i will undercut Cariatide's price and make a handy profit myself. But hey, then Cariatide will know that and undercut me by posting a price at $48, oh but then he will again undercut me... Aha, so I guess I should post a price at $0!" In other words, both producers understand that unless they price at their production cost they would be undercut by the other producer, so that they both
Questions: 1. Is this allocation Pareto efficient? Why? 2. What would happen if Cariatide was allowed to charge different prices for each flower pot (this is called price discrimination) - what prices would he charge? Would the resulting allocation be Pareto efficient? 3. What are all the Pareto-efficient allocations for this exchange between Alan and Cariatide?
5 4 post a price at $0.6 This example is called Bertrand competition (after a French economist who went by the last name of Bertrand, amazingly enough), and it is in fact a very general example. It shows that whenever there are at least two firms who have similar production costs and compete by setting prices, the resulting price is at their production costs. Of course, under such circumstances, Alan buys his 100 flower pots (or 101, since the price is 0 so that he is indifferent between buying the 101-st pot), and he makes all the surplus he could possibly make, which amounts to 5000 dollar equivalents. Clearly this is then a Pareto-optimal allocation, since Alan could never be made better off.7 Questions: 1. Model this situation as a sequential game in which first Cariatide and Grunf simultaneously decide what prices to set, and then Alan decides from what producer to buy his flower pots. 2. Assume that if Alan faces the same price from both of them, he buys half of his flower pots from Cariatide and the other half from Grunf. Assume also that Cariatide and Grunf only set prices in whole dollars and show that this game has 2 prices that arise in subgame-perfect Nash equilibria in pure strategies: p = $1 and p = $0. Note that if the prices were allowed to be set in pennies these equilibrium prices would be $0.01 and 0 which is why we say that under Bertrand competition the producers price at the production cost. 7 Question (a tough one): What would happen if Grunf and Cariatide had different production costs? For example, assume that Cariatide due to his drinking problem loses the touch and got lazy and is hence only able to produce the flower pots at a cost of 33 dollar equivalents. Can you figure out what the possible equilibrium prices would then be? If you feel up to it, try to be very formal and model this again as sequential game and precisely describe each player's strategy. (hint: Alan does not need to buy half of his costs from one producer and half from the other when he is indifferent - he can decide anyway he wants) 6 5 ...
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This note was uploaded on 02/06/2012 for the course ECON 101 taught by Professor Buddin during the Spring '08 term at UCLA.
- Spring '08