Sellers goods have unobservable quality to buyers. Borrowers ability to repay is unobservable to banks who lend

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1 Information Sellers goods have unobservable quality to buyers Borrowers ability to repay is unobservable to banks who lend Insurees health unobservable to insurance companies The lemons model There are sellers whose goods are lemons, and their quality is q = 0, or peaches, and their quality is q = v The proportion of lemons in the market is p If a seller keeps their good, they get a payoff sq; if they sell, they get the price, t If a buyer makes a purchase, the buyer gets q t, where q is the good s quality and t is the price; if they don t buy, they get a payoff of zero The lemons models We want to know what happens when you include private or incomplete information: how does it impact market efficiency? The key idea is that being willing to participate in a trade reveals information about you The lemons model While the buyer s payoff is q t, he doesn t observe q at the time of trade: it s random, like a coin flip with a weighted coin but he knows that the seller is willing to trade at the prevailing market price. But how should we model the buyer s payoff? For bet based on a coin flip, the outcomes are H and T, the probabilities are p(h) = 1/2 and p(t ) = 1/2, and the utilities to the agent are u(h) and u(t ). The expected utility is 1 2 u(h) u(t ) For the buyer, the utility of a peach is v t, the utility of a lemon is 0 t, so the expected utility of a purchase is (1 p)(v t) + p(0 t) = (1 p)v t. 1

2 (Expected value in general) To model randomness, we imagine that (1) events occur (lemon or peach, rain or sunshine, etc), those events have (2) probabilities of occuring which (3) determine utilities for the agents The events are e 1, e 2,..., e N, the probability of each event e k is p(e k ), and the utility to the agent is u(e k ) Then expected value is given by (Expected value in general) E[U] = N p(e k )u(e k ) k=1 Imagine rolling a six-sided die, and getting a dollar for each pip The events are 1, 2, 3,..., 6, the probability of each event 1/6, and the utility to the agent is u(1), u(2),..., u(6) Then expected value is given by E[U] = 1 6 u(1) u(2) u(3) u(4) u(5) u(6) The market for lemons We ll set the market price equal to the social benefit of the marginal trade, just like in a price-taking equilibrium (it also sets the payoff of all buyers to zero) The price then depends on who participates, which in turn depends on the price itself: { t = E[q t p0 + (1 p)v, Peaches and lemons traded ] = 0, Only lemons traded Trade in the lemons market There s two possibilities: 1. The market functions: peaches and lemons trade, t sv 0 and t 0 2. The market collapses or unravels : only lemons trade, t < sv and t 0 2

3 Trade in the lemons market If the market is functioning, the expected quality of the goods is p0 + (1 p)v = (1 p)v. Lemons are obviously willing to trade, but peaches are only willing to trade if (1 p)v sv, or 1 s p. If the market collapses, the expected quality of the goods is 0, so only lemons trade, and p > 1 s. Trade in the lemons market 3

4 t v t=p 0+(1 p)v 1 s 1 p Inefficiency Notice that the outcome is inefficient: since 0 < s < 1, all sellers wish they could sell 4

5 The problem is that making a purchase carries risk, which wouldn t be there if sellers could observe quality The presence of bad goods drives down the market price, destroying the gains from trade for peaches You can think of it as an informational externality that justifies intervention or creates an opportunity for social entrepreneurs Adverse selection When the presence of bad trades reduces the efficiency of a market, we say it suffers from adverse selection The great recession: bad mortgages were mixed with good ones to create financial instruments. Once it was learned that the bad mortgages were really bad, and risks were not indepedent across the mortgages, the market collapsed. ACA health exchanges: pools of insurees were much less healthy than insurers initially believed. After experiencing losses, many insurers have left the exchanges, and many have single insurers left. Public schools: wealthier families leave public schools for private/charter/magnet schools, and the kids remaining have fewer resources, leading more families to leave... Institutions and Market Design This leads to two responses by the government, firms, or entrepreneurs: Signaling: take a costly action to convince others that you are the good type (the side with private information acts) Screening: offer an agent with private information a set of choices to incentivize them into self-selecting and revealing their types (the side without private information acts) Costly Verification Suppose there are three quality types: low, medium and high. Low quality goods are worth 0 and happen with probability l, medium quality goods are worth m and happen with probability p m, and high quality goods happen with probability p h and are worth h, h > m > 0. If a seller retains his good, he gets a payoff of sq, 0 < s < 1, so there are gains from trade. If he sells, he gets the price. If a buyer makes a purchase, she gets the true quality of the good minus the price. 5

6 What are the prices if only low quality goods are sold? Low and medium? Low and high? When do only low quality goods trade? Low and medium? All types? If the high-types pull their goods off the market, can it collapse? Costly Verification Suppose any seller could pay c to reveal their type and receive the true value of their good in the market When are high types willing to pay for verification? What happens in the market for low and medium quality goods if high types leave? If a social entrepreneur invented a technology to reveal type and started selling it, what problems might arise? Unintended Consequences and Verification Statistical discrimination: Ban-the-box legislation Pre-existing conditions: information destroys opportunities for insurance Who monitors the monitors? Moody s, LIBOR, etc. Costly Signaling What if, instead, objective verification wasn t possible. Can peaches still separate themselves out? Suppose there are low-skilled, who occur with probability p, and highskilled workers, who occur with probability 1 p. College costs c for low-skilled types, but only sc with 0 < s < 1 for high-skilled types: they find it easier to get through tough classes Firms decide what to pay workers based on their degree, not their true type: college graduates get w ug and high school grads get w hs High-skilled workers produce profits π H for the firm and low-skilled workers produce profits π L (if type were observable and labor markets were perfectly competitive, these would be the wages). 6

7 Costly Signaling What must be true for high types to prefer to go to college? w ug sc w hs What must be true for low types to prefer to go to high school? Putting these together yields w ug w hs s w hs w ug c c w ug w hs, so there is always a pair of wages (w ug, w hs ) that separate the high- from low-skill workers, even if college does nothing for productivity. This is a separating equilibrium. If the above condition fails because c is too low, it is a pooling equilibrium where all types adopt the same behavior. Costly Signaling How inefficient is this? The cost c must be weakly greater than the discounted exected gain in wages from going to college rather than highschool (why has tuition gone up so much over the 20th century?) What if you are a credit-constrained high-type and can t afford to pay c? If c is sufficiently high and workers are paid their marginal product, w ug = π H and w hs = π L. If c is too low (c < w ug w hs ), however, both types might go to college, and the signal gives no information. Firms then pay expected marginal product, w = pπ H + (1 p)π L and the value to high types of a college degree falls. Notice that everyone is still going to college (this is a rat race ). What are the incentives for the universities? Other examples? Screening When the side without private information acts to mitigate the consequences of adverse selection, it is called screening We ll consider the simple case of a monopolist who faces consumers with different values for a good, and is trying to design the prices for a product line 7

8 Price Discrimination There is a seller who offers two types of goods: A high-quality one with price t h, and a low-quality one with price t l (but nothing is stopping t h from equalling t l ). For example, a car dealership might have both BMWs and Camrys (Camries?) on the lot. There are two types of buyers in the population: High types, who occur with probability p, and low types, who occur with probability 1 p. The high-type s valuation of the high-quality good is v + h, and of the low quality good is v + l, with h > l. The low-type s valuation of the high-quality good is v, and of the low quality good is v. If an agent buys the good, his payoff is his valuation minus the price, and if he does not buy the good, he gets a payoff of zero. (For example, a high-type buyer getting the high quality good gets a payoff v + h t h ) How should the seller decide on prices to maximize profits? Implementation Our goal is to find prices that implement a particular pattern of behavior by the buyers. In particular, we might want to 1. Sell low quality goods to all the types. 2. Sell high quality goods to all the types. 3. Sell high quality goods to the high type, and low quality goods to the low type. 4. (Why won t Sell low quality goods to the high type, and high quality goods to the low type work?) So we say things like, Prices t h = 10 and t l = 3 implements the high type buying the high quality good, and the low type buying the low quality good. Individual Rationality First issue: Buyers can always get a payoff of zero by withdrawing from the market. This means that if the high type buyer is supposed to buy the high quality good, IR h : v + h t h 0 and if the high type buyer is supposed to buy the low quality good, IR h : v + l t l 0 Likewise if the low type buyer is supposed to buy the high quality good, IR l : v t h 0 8

9 and if the low type buyer is supposed to buy the low quality good, IR l : v t l 0 So every type will have its own IR constraint, saying that, The seller can t ask for more money from the buyer for the good than what it is worth to buyer. Perfect Information Suppose buyers types are observable by the seller. What prices should be picked? The high type is willing to pay v + h for the high quality good, v + l for the low quality good, and 0 for nothing. Setting t h = v + h and offering the high quality good maximizes the seller s profits from the high types, and the high types don t refuse to participate. The low type is willing to pay v for the high quality good, v for the low quality good, and 0 for nothing. Setting t h = v and offering the high quality good maximizes the seller s profits from the low types, and the low types don t refuse to participate. So when a rich businessperson in a suit and expensive shoes comes to the dealership, they are shown BMWs and quoted a high price. When a less wealthy person comes in, the dealer shows them the BMW as well, but quotes a lower price. Note that the outcome is efficient (since total surplus is p(v+h)+(1 p)(v), which is greater than any other allocation of goods), and the seller gets all the gains from trade (perfect price discrimination). Imperfect Information But what if the buyers privately know their own types? Does offering the high quality good to the high types and the low types still work? The high type wants to imitate the low type, and pay v instead of v + h, since v + h v }{{} Low type price v + h (v + h) }{{} = 0 High type price Incentive Compatibility If types are no longer observable, the seller must offer a single price for the high quality good, and a single price for the low quality good. If we want to implement the high type buying the high quality good, we need them to prefer buyer the high quality good at the high quality price to the low quality good at the low quality price, IC h : v + h t h v + l t l 9

10 and we need the low quality type to prefer the low quality good at the low quality price to the high quality good at the high quality price, IC l : v t l v t h These are called incentive compatibility constraints. IR and IC constraints The individual rationality (IR) and incentive compatibility (IC) constraints together are actually an extraordinarily powerful idea. In general, The IR constraints say, Every type is better off participating the way the seller wants than dropping out altogether and taking a payoff of zero The IC constraints say, Every type is better off participating the way the seller wants than imitating another type If all the IR and IC constraints are satisfied, then the outcome the seller wants will actually be implemented (right?). The Seller s Problem Suppose we want implement the high types buying the high quality good at price t h and the low types buying the low quality goods at price t l in the most profitable way possible for the seller. Then we need to solve: max t h,t l pt h + (1 p)t l subject to IR h : v + h t h 0 IR l : v t l 0 IC h : v + h t h v + l t l IC l : v t l v t h But what do we do with all these constraints? There are only two prices and four constraints. Some of the constraints must be equalities binding constraints and some must be strict inequalities slack constraints. IC h + IR l IR h We are going to show that IR h is slack, or irrelevant, since some of the other constraints already imply it. v + h t h v + l t l by IC h > v t l since l > 0 0 by IR l This means v + h t h > 0, so IR h is slack, and we can drop it. 10

11 The Seller s Problem (2) We re left with: subject to max t h,t l pt h + (1 p)t l IR l : v t l 0 IC h : v + h t h v + l t l IC l : v t l v t h IC l is slack Suppose IC l and IR l are both binding, so that and v t l = 0 t l = v v t h = v t l t h = t l Then IC h is v + h t h = h > 0, meaning that the high type strictly prefers buying the high quality good to the low quality good. The problem here is, we can now raise the price on the high types and make a higher profit without the low types dropping out, so this can t be profit-maximizing. Basically, IC l is an irrelevant constraint that forces the seller to hold back from really squeezing money from the high types. The IC/IR Picture 11

12 Two Types Three Types High type Low type : Low type doesn t drop out :High type doesn t want to imitate the low type High type Medium type Low type Low type doesn t drop out High type doesn t want to imitate the medium type Medium type doesn t want to imitate the low type The Seller s Problem (3) We re left with: subject to max t h,t l pt h + (1 p)t l IR l : v t l = 0 IC h : v + h t h = v + l t l Then we must have t l = v and t h = v + h l. This gives the seller profits of p(v + h l) + (1 p)(v). Things to notice The high type gets a payoff of v + h t h = l, and the low type gets a payoff of v t l = 0. In general, the higher types get higher payoffs, since they have the option of imitating the low types. The high type gets the efficient, high quality good while the low type gets the inefficient, low quality good. In general, the higher your type is, the closer you are to getting the efficient quantity. 12

13 Offering anything to the low type is not always profit maximizing. We could offer the high quality good at a price t h = v + h and get profits of p(v + h), which might be better than p(v + h l) + (1 p)v. Private information gives the buyers more power to bargain against the seller, but it leads to more inefficiency as the seller combats this by reducing the quality of the goods offered to some types. Quality Distortion The fact that the high type gets the efficient quality and the low type does not is very important: It is not because of the few thousand francs which would have to be spent to put a roof over the third-class carriages or to upholster the third-class seats that some company or other has open carriages with wooden benches... What the company is trying to do is prevent the passengers who can pay the second-class fare from traveling third-class; it hits the poor, not because it wants to hurt them, but to frighten the rich... And it is again for the same reason that the companies, having proved almost cruel to third-class passengers and mean to second-class ones, become lavish in dealing with first-class passengers. Having refused the poor what is necessary, they give the rich what is superfluous. Quality distortion Quantity discounts: The unit price of the large popcorn at the movie theater is much smaller than the medium and small sizes. Large jugs of olive oil are much cheaper per ounce than smaller jugs. Entry-level products are often deficient: Entry level guitars are almost unnecessarily poorly made (and often really ugly), just to push people up to the middle-of-the-range guitars. Why do top universities/corporations offer so many amenities, while lower level ones do not? What would these results imply about health care and insurance provision? Adverse selection When agents hold information about themselves that is unobservable to others, it leads to inefficiencies and potential market unraveling This leads to two potential remedies: Signaling: the side with private information acts to reveal its type, by verification or costly signaling 13

14 Screening: the side without private information sets prices or provides incentives for types to reveal their information honestly (e.g., business versus economy tickets, quantity-discounting, etc.) 14