Adverse Selection. Chapter 3

Chapter 3 Adverse Selection Adverse selection, sometimes known as The Winner s Curse or Buyer s Remorse, is based on the observation that it can be bad news when an o er is accepted. Suppose that a buyer makes an o er to buy a good from a seller who knows more about its value than the buyer does. This is called a situation of asymmetric information. The buyer must beware because a seller with a high-quality good will probably not sell it at a given price, while a seller with a low-quality good will be happy to unload it at the same price. There are two related approaches to equilibrium analysis in the context of adverse selection. A general equilibrium analysis requires assumptions about the relative market power for buyers and sellers; these assumptions specify the division of gains from trade, if any, and thus identify equilibrium price(s). It is still necessary to adjust the re ne the description of general equilibrium to incorporate the buyer s uncertainty about the quality of products she wishes to buy. 1 As a result, some of the properties of general equilibrium (e.g. the First Welfare Theorem) do not hold in the context of adverse selection, and in particular, equilibrium outcomes may not be Pareto optimal for buyer and seller. A game theoretic analysis requires speci c information about the nature of interactions between buyers and sellers so that it is possible to translate that information into an extensiveform game and solve for a Nash equilibrium. The game theoretic analysis does not directly 1 Note that with a random selection of goods sold at any price (so that there is no correlation between the availability of a good at a given price and the quality that good), uncertainty could be incorporated into a general equilibrium analysis that allows for risk aversion in individual preferences. 92

specify the relative market power of buyers and sellers. Instead, the nature of market power emerges endogenously on the basis of the order of moves within the extensive form game. These two distinct approaches to equilibrium analysis tend to coincide in their results. However, both require quite speci c assumptions in order to identify an equilibrium. It is also possible to consider a third approach, simply identifying all market prices and allocations of goods that yield gains from trade to both buyer and seller. This third approach identi es all outcomes that could plausibly result in equilibrium for some de nition of market power or description of the trading game played between buyer and seller. We utilize all three approaches during the course of this chapter, emphasizing the similarities and di erences between the game theory and general equilibrium predictions. 3.1 Motivating Examples We consider three classic examples of adverse selection where a buyer values a good more than its current owner (the seller). We use the most general approach - identifying the range of possible equilibrium prices without specifying market power (as necessary for a general equilibrium approach) or the order of moves (as necessary for a game theory approach) in the rst two examples, but revert to a more speci c approach for the third example of insurance to ease algebraic computation. 3.1.1 The Market for Lemons Suppose that the buyer is a consumer without a car and the seller is a used car owner. This example was originally suggested by George Akerlof in a paper that is so widely read that this example is sometimes called Akerlof s Lemons Market. With probability, the used car is high quality (a Peach ) while with probability 1 the used car is low quality (a Lemon ). A Peach is worth $3,000 to a buyer and $2,500 to a seller. A Lemon is worth $2,000 to a buyer and $1,000 to a seller. 2 If the car is sold at price p, then the buyer s payo is v b p, where v b is the true value to the buyer, and the seller s payo is p v s, where v s is the true value to the seller. 2 These values match the ones given in Kreps (1990). Akerlof (1970) did not analyze the two-type case in any detail, considering instead an example that more closely resembles the Takeover Game. 93

We begin with a general analysis to identify possible equilibrium prices without specifying the exact game that the buyer and seller play to arrive at a particular equilibrium price. Since the buyer cannot distinguish between a Peach and a Lemon, the market price must be the same for peaches and lemons. 3 A seller who owns a Peach will only sell it if the market price is at least $2,500, while a seller who owns a Lemon will sell it if the market price is at least $1,000. 4 With only two types of cars, there is a discontinuity in the expected value of cars sold as a function of the price, as shown in Figure 3-1. The average value of a car for a buyer conditional on a sale at a price of $2,501 (a combination of Lemons and Peaches) is much higher than the average value of a car for a buyer conditional on a sale at a price of $2,499 (just Lemons). At a price of $2,500 or more, owners of Lemons and owners of Peaches will sell their cars, so the average value to buyers is 3000 + (1 ) 2000 = 2000 + 1000: At a price between $1,000 and $2,499, owners of Lemons will sell their cars, but owners of peaches will not, so the average value to buyers is simply the value of a lemon, $2,000. Finally, at a price below $1,000, no owners will sell their cars (but if there existed some car even worse than a lemon, its owner might sell it at this price, so it is appropriate to graph a value below $2,000 for these market prices). A market price is a possible equilibrium price with positive probability of trade if 1) each type of seller trades only if the price is greater than the value of the car to the owner and at least one type of seller accepts the o er; 5 2) the price yields an expected pro t to the buyer given the knowlege of which types of sellers will transact at that price. The requirement that the buyer anticipates the decisions of sellers at the time that she formulates her o er is commonly known in general equilibrium analysis as a Rational Expectations condition. Rational expectations is already implicit in the best response requirement for Nash equilibrium in a game theoretic formulation. Thus Rational Expectations Equilibrium (REE) and Nash Equilibrium should be expected to produce similar predictions for super cially di erent approaches to analyzing the bilateral interaction between buyer and seller. 3 In the remainder of this example, we describe the trading outcome as though there are many buyers and many sellers. The same calculations apply for the case of one buyer and one seller. 4 For simplicity, we assume that the seller with a peach will accept an o er of exactly $2,500, while a seller with a lemon will accept an o er of exactly $1,000. 5 With a continuous distribution of types for the seller, we would rephrase this requirement to say that the seller accepts the price with positive probability. 94

Value to Buyer E(v v < x) $(2000 + 1000q) $2000 Bid (x) $1000 $2500 Figure 3-1: Expected Value of Cars Sold We now consider each price range in turn to identify the set of possible equilibrium prices these are prices that could result in a Rational Expectations Equilibrium or a Nash Equilibrium if we provided enough information to specify a particular REE or Nash outcome. A market price below $1,000 cannot produce an equilibrium with positive probability of trade because the price is not su ciently high for either type of seller to trade. A market price between $1,000 and $2,499 will induce sales of Lemons but will be rejected by owners of Peaches: the pro t to buyers at price p in this range is then $2,000 - p. So, if the price is between $1,000 and $2,000, it yields an expected pro t to the buyer, who recognizes that any car that is available at this price must be a lemon but is still willing to pay up to $2,000 for it. But if the price is between $2,001 and $2,499, the same logic implies that the buyer will lose money on any purchase. So market prices between $1,000 and $2,000 could produce an equilibrium where only lemons are sold, but market prices between $2,001 to $2,499 do not produce an equilibrium. 6 A market price of $2,500 or more will induce sales of both types of cars. Then buyer s expected value of a car conditional on a sale depends on the parameter : E(v b ) = (3; 000) + 6 The given de nition of an equilibrium requires that the buyer be willing to transact at the equilibrium price. So, while there is no trade at a price less than $1,000 and also that there would be no trade at a price between $2,001 and $2,499, prices less than $1,000 are equilibrium prices because buyers would be willing to transact at those prices. 95

(1 )(2; 000) = 1; 000+ 2; 000:Subtracting the sale price, the buyer makes an expected pro t at price p if 1,000 + 2; 000 > p. At the minimum price in this range, p = $2,500, the buyer makes an expected pro t if > 1/2. So if < 1/2, then the buyer loses money for each price of $2,500 or more. If > 1=2, then there is a range of possible equilibrium prices from p = $2,500 to p = $(1,000 + 2,000). But if < 1/2, then no price of $2,500 produces an equilibrium because buyers lose money on average at these prices. The conclusion is that if < 1/2, then the only possible equilibrium outcomes are 1) only lemons are traded and the equilibrium price is between $1,000 and $2,000; 2) no cars are traded and the equilibrium price is between $0 and $999. Yet, both types of cars are worth more to the buyer and the seller, so the outcome is economically ine cient if < 1/2. If 1/2, there is a third possibility: 3) both peaches and lemons are traded and the equilibrium price is between $2,500 and $3,000. In this case, trade is assured and the outcome is e cient, but peach owners gain less than lemon owners from trade since the transaction price is the same for both types of cars. This is known as cross-subsidization. The price of a peach is arti cially depressed and the price of a lemon is arti cially in ated because buyers cannot distinguish one from the other. Price Range Seller s Response Buyer s Pro t p < $1; 000 No Trade Zero Pro t $1; 000 p $2; 000 Trade Lemons Positive $2; 000 < p $2; 500 Trade Lemons Negative $2; 500 p $3; 000 Trade Lemons, Trade Peaches Positive if 1=2 p > $3; 000 Trade Lemons, Trade Peaches Negative Table 1: Prices and Results for the Lemons Market Example Table 1 summarizes the results for the Lemons Market example with these values. Prices between $1,000 and $2,000 are always possible equilibrium prices where only lemons are sold. Prices between $2,500 and $3,000 are possible equilibrium prices where both types of cars are traded, but only if the proportion of peaches is su ciently high. The condition 1=2 highlights the tradeo produced by adverse selection. Whether the car is a lemon or a peach, the buyer values it more than the seller, so there are possible gains from trade from 96

transacting. However, these gains are reduced from the buyer s perspective because of the buyer s informational disadvantage. There can be an equilibrium where both types are traded so long as the buyer s gains from trading both cars are large enough to overcome the costs of adverse selection for the buyer which occurs when the proportion of peaches is su ciently large. 7 The Takeover Game Suppose that the buyer is a business executive who knows that he can improve the value of an existing company by 50%. The seller knows the exact value of the company, but the buyer does not. The buyer s best assessment is that the current value, v, is equally likely to be any number between 0 and 100. That is, v is uniformly distributed on (0; 100). We say that v is the seller s type, which is private information to the seller. If the seller of type v accepts an o er of price p to sell the company, then the buyer receives a payo of 1:5v p, while the seller receives a payo of v. If the seller rejects the o er, the buyer receives a payo of 0 and the seller retains the company at its current value for a payo of v. Both parties are assumed to be risk neutral. On average, the company is worth E(v) = 50 to the seller and E(1:5v) = 75 to the buyer. Since the value of the company always increases after a sale, it is natural to imagine a solution where the buyer pays 62.5 to the seller and each side gets an average pro t of 12.5 because of the gain in e ciency - transferring the company to the more e ective management team. But adverse selection precludes this option, for the seller will not accept any o er unless p > v, so sellers of types 62.5 or more would not sell at a price of 62.5. The only market price that ensures a transaction is p = 100, but this price cannot be sustained in an equilibrium. At price p = 100, the buyer loses an average of 25 because he is paying 100 for a company with expected value 75 after accounting for the 50% gain in value due to the sale. In fact, no positive market price can be sustained as an equilibrium. At any positive price 7 Note that the cost of adverse selection is a function of downside risk (i.e. the probability of a Lemon) rather than of uncertainty. For example, a distribution with 80% probability of a Lemon and 20% probability of a Peach has the same variance for the seller s value as does a distribution with 80% probability of a Peach and 20% probability of a Lemon. So these distributions of types embody similar levels of uncertainty, but only the more favorable distribution can produce an equilibrium where Peaches are traded. 97

p, the seller accepts the o er only if his type is less than or equal to p: 8 Conditional on the seller s willingness to sell at a price of p, the buyer updates his assessment to conclude that v is uniformly distributed between 0 and p. This is the bad news element of adverse selection - for any o er, the buyer only receives the lowest quality goods. In expected value, the buyer s anticipates the seller s decision rule; at a market price of p, the buyer s rational expectation is that the expected value of v for a company that is available at a price of p is p=2 (the expected value for a uniform distribution ranging from 0 to p). 9 Similarly, the average pro t to the buyer at market price p is E((p)) = [Probability(p is accepted) [E(1:5vjp is accepted) p] = ( p ) [E(1:5vjx > p) 100 p] = ( p 100 ) (3p 4 p) = p 2 400 : That is, the buyer loses money on average for each p > 0 and loses even more money the higher the o er. Although the company is known to be worth more to the buyer than to the seller, there is no equilibrium with a positive probability of trade - regardless of whether the interaction is modeled as a game with a Nash equilibrium or as a general equilibrium trading market with a Rational Expectations Equilibrium. Adverse selection provides such a large obstacle that it precludes trade in this example despite the obvious bene ts from improving the management of the company. bene ts were even larger, then trade could occur. But if the For example, if the new management would triple the value of the company, then a price of 100 would be accepted by each type of seller and would provide the buyer with an expected pro t of 50, since the expected value of the company would be 150 with new management. from trade and cost from informational asymmetries. This highlights, once again, the tradeo between gains If the gains from trade are su ciently 8 We assume that values and o ers can take any real value from 0 to 100. Then the seller s action when v = p does not a ect the overall average payo s because v = p occurs with probability 0. 9 Similarly, the buyer s assessment of v jumps to (1 + x)=2 if the seller rejects the o er, but this conditional expected value has no e ect on the payo s in the game. 98

large, then an e cient outcome (trade with probability 1) would still result despite the buyer s losses due to adverse selection if v is close to zero. Experimental Evidence: The term "Takeover Game" indicates that this situation has been commonly modeled as a sequential move game with the buyer making a take it or leave it o er to the seller. The only Nash (or subgame perfect) equilibrium is for the buyer to make no o er at all, and so the company is never sold. Samuelson and Bazerman (1985) tested the equilibrium prediction of this model with MBA students making bids for the company. More than 70% of the students made bids of 50 or more; fewer than 10% of the students bid 0. 10 A Simple Insurance Market Suppose that a risk averse consumer has wealth W and faces a risk of losing it all with probability t. The consumer has utility function u(w) = p w. Since she is risk averse, the consumer would like to insure against a loss with a (risk neutral) insurance company. The di culty is that the consumer knows t, but the insurance company does not. Suppose that the insurance company assesses a uniform distribution on t across all values from 0 to 1. Once again, there are obvious gains from trade for these two parties, but the gains are imperiled by the presence of asymmetric information. Suppose that all insurance is full insurance and that the market price is p. If the consumer accepts the policy, she receives a nal wealth of W p whether or not she has a loss. (If there is a loss of wealth W, the consumer pays p to the company as a premium and is reimbursed for the entire loss.) If the consumer does not accept the policy, she receives a nal wealth of W if there is no loss and a nal wealth of 0 if there is a loss. Given a price of p, the consumer will buy insurance if 10 These experimental results provide a strong critique of the Rational Expectations requirement; it is not clear that participants understand that they should reduce their assessment of v when a bid for the company is accepted. See Thaler, "The Winner s Curse," chapter 5, for broad discussion of this phenomenon and Eyster and Rabin, "Cursed Equilibrium" for a recent technical paper that summarizes relevant experimental results and provides an alternative model to Rational Expectations / Nash equilibrium. 99

p W p (1 t) p W + t p 0 r W W p 1 t t 1 r W W p That is, the consumer chooses full insurance at price p if the risk is high enough. This phenomenon poses a problem for the insurance company, because the company can rationally anticipate that the consumer will only buy insurance if she is a relatively bad risk. Note that adverse selection occurs in this example because the seller (the insurance company) lacks information about the buyer (the consumer), in contrast to the Lemons and Takeover examples. But the company can adjust the price to manage the tradeo between sales and adverse selection. At a price p = W, the consumer never buys insurance (the condition for purchase simpli es to t 1), while at a price p = 0, the consumer always buys insurance (the condition for purchase simpli es to t 0). 11 So, at a market price of p for full insurance, the consumer will accept if t is between 1 p (W p)=w and 1. The market price necessary to guarantee that all consumers will insure is p = 0, but at this price, the insurance company loses money for certain. So there cannot be an equilibrium where every type of consumer would buy insurance. For any price p > 0; not all types of consumers buy insurance. With a uniform distribution on 0 to 1 for t, the consumer accepts the o er p with probability p (W t)=w. Further, the expected value of t conditional on purchasing insurance is therefore: E(t j bought insurance, price p) = 1 r 1 W 2 W p Given a sale, the company expects to pay an average value of W E(t j bought insurance) to the consumer in exchange for the consumer s insurance premium of p. For the company to 11 In practice, the insurance company can also o er partial insurance of losses or can add stipulations, such as a deductible, to the policy to discourage some types of consumers from buying insurance. 100

at least break even on insurance, then the price must satisfy: p W 1 r 1 W 2 W! p This is a complicated equation (it can be expanded into a fourth-order polynomial in p) with the solution that p 3W 4, meaning that the possible equilibrium prices all involve relatively high prices. Insurance is limited to consumers with t 1 2 if p = 3W 4, or to even fewer consumers if p > 3W 4.12 3.1.2 Summary of Examples These examples illustrate the principal tradeo involved with Adverse Selection. Two parties can conduct a trade that would obviously be bene cial to both of them - if only they can agree on a price. But due to asymmetric information, the choice of the price limits the bene ts to one side, which may even lose on the transaction in some instances with a poor choice of price. The nature of the resulting equilibrium depends on the relative importance of the gains from trade in comparison to the severity of asymmetric information. The general model in the next section highlights this tradeo. 3.2 A General Model We now present a general model that encompasses each of the examples above. A seller has a good that is valued by buyers. The seller s value is V, drawn from a known distribution f(v) with corresponding cumulative distribution F (v) = P (V < v). We use V to denote the random variable for the seller s value and v to denote speci c values of V. We say that v is the seller s type and we assume that the seller is male and that the buyer is female. If the seller s value is v, then any buyer s value for the same good is b = h(v), where h is a (weakly) increasing function. The seller knows the true value v, while the buyer just knows 12 Of course, consumers with t < 1=2 are technically able to purchase insurance, but the price they would have to pay is too high to make it worthwhile for them. In an e cient market, these individuals would be able to buy full insurance at a fair price for their particular value of p. 101

its distribution f(v), with a minimum value of v L and a maximum value of v H. If a buyer and seller trade at price p then the buyer receives a payo of h(v) p and the seller receives a payo of p. If the o er is rejected, the buyer receives a payo of 0 and the seller receives a payo of v. 13 Combining the outcomes for buyer and seller, the net surplus of the transaction is h(v) v. From the perspective of social e ciency, the good should be sold if h(v) > v: In the examples discussed above, h(v) > v for all possible values of s, but this is not a necessary characteristic of all adverse selection models, and we do not assume it here. This formulation incorporates each of the examples described above, with the caveat that the insurance case requires negative values for v and h(v). To simplify exposition, we will assume that there are many buyers competing to make a purchase from a single seller, thereby giving full market power to the seller. We consider two di erent de nitions of equilibrium. The assumption of complete market power for the seller reduces the set of possible equilibrium prices in the motivating examples (using either equilibrium concept) to those where the buyer receives zero expected pro t. 14 In a Rational Expectations Equilibrium, the transaction price equals the expected value to buyers conditional on a trade; the seller enjoys all of the gains from trade in equilibrium as pro ts. Thus, any price that satis es the zero-pro t condition for buyers produces a Rational Expectations Equilibrium. The game theoretic approach incorporates the market power of the seller by assuming that the buyers move rst and making simultaneous price o ers to the seller. Formally, there are I buyers, with buyer i o ering price p i. The seller identi es the higher o er p, and rejects all lower o ers. In (a Nash) equilibrium, the seller accepts the highest o er p if p v. 15 If more than one buyer makes that highest o er, then the seller randomizes among them with equal probability (assuming p v). This is a sequential move game and so the appropriate 13 It would be equilvalent to assume that the seller s payo is p v for a sale at price p and 0 for no trade. In either case, the seller pro ts only by selling at a price with p > v. 14 For example, if > 1=2 in the Lemons example, then the equilibrium price when all goods are traded and the seller has full market power is p = 1; 000 + 2000: At this price, all buyers, including the one who is able to buy a car at this price, make zero expected pro ts. 15 The same result would occur if there are many buyers and many sellers, but the number of buyers is su ciently large to give complete market power to the sellers. 102

solution concept is either Bayes Nash equilibrium or subgame perfect equilibrium. 16 While buyers have incomplete information about the seller s type, they do not have to respond to an action by the seller so we do not use Perfect Bayesian Equilibrium as the solution concept. Proposition 1 There may be multiple Rational Expectations Equilibria. In almost every case, there is a unique subgame perfect equilibrium with price equal to the highest Rational Expectations Equilibrium price. Subgame perfect equilibrium analysis of this game is straightforward because it is a weakly dominant strategy for the seller to accept a price p if p > v, so the seller follows this strategy in every subgame perfect equilibrium; however, it is possible that the seller deviates from this strategy o the equilibrium path in a Bayes-Nash equilibrium. 17 In either case, if the higher price o ered by a buyer is p, then the probability that the o er is accepted is equal to F (p ), the probability that the seller has a value that is lower than p. Given price p, then the expected value to the buyer conditional on a sale is E(h(v) j v p ). Combining these values, the expected pro t to the buyer who makes the highest price o er, p, is (p ) = P (seller accepts o er p ) E( b j seller accepts bid) (p ) = F (p ) [E(h(v)jV p ) p ]. Thus, the buyer who o ers the highest price, p, makes an expected pro t if (p ) is positive, meaning that the conditional expected value E(h(v)jV p ), is greater than the bid price p. There cannot be a Subgame Perfect equilibrium with expected pro ts to the buyers, for otherwise some buyer would do better by deviating from the proposed equilibrium strategy to an o er of p + ". So the price in any SPE is a zero-pro t price for buyers. 16 We use the terms Bayes-Nash equilibrium and Nash equilibrium interchangeably in this chapter. Note that in the case of adverse selection, the seller s type in uences the payo to the buyer even given the seller s action - a contrast to the discussion of Bayes-Nash equilibrium in the previous chapter. 17 We assume that the seller accepts the o er if v = p, which makes the seller indi erent between accepting and rejecting the o er. As in the earlier example, this tiebreaking rule has no e ect on expected pro ts if f is continuous. assumes that the seller accepts the o er in the case of a tie with v = p. As in the earlier example, the tiebreaking rule has no e ect on expected pro ts if f is continuous. 103

In fact, in almost every instance, there is a unique SPE price equal to the highest zero-pro t price for buyers. If there are multiple zero-pro t prices, p 1 and p 2 with p 2 > p 1 then both then p 1 and p 2 are REE prices, but there is almost always a price between p 1 and p 2 that yields an expected pro t to buyers. In this case, there cannot be a subgame perfect equilibrium with equilibrium price p 1 because buyers would have a pro table deviation to a price between p 1 and p 2. The only exception is if somehow E(h(v)j v p 1 ) = p 1, E(h(v)j v p 2 ) = p 2, but E(h(v)j v < p) < p for all other prices with p > p 1. In this exceptional case there could be multiple subgame perfect equilibria with di erent zero pro t prices. The seller s actions o the equilibrium path are less restricted in a Bayes-Nash equilibrium. For this reason, it is possible to construct a Bayes-Nash equilibrium for any zero-pro t price for the buyers. In fact, it is also possible to use strategies of the form described in the text to produce a Bayes-Nash equilibrium with any price that gives the buyer an expected pro t. For example, we can sustain a Bayes-Nash equilibrium at a price p 0 that is pro table for buyers by assuming that the seller responds appropriately to price p 0, selling if v p 0, but will not accept any other o er. 3.2.1 Systematic Analysis of the Lemons Model We can produce much of the intuition for our general results in the context of the Lemons model, which does not require complicated notation or computation of conditional expectations. In the Lemons model, there are just two types of sellers, those that own peaches and those that own lemons. The distribution F consists of just one parameter in this case, the probability that any single car is a peach, or P (Peach) =. A Peach is worth b H to a buyer and r H to a seller, while a Lemon is worth b L to a buyer and r L to a seller with b H > b L and r H > r L. We denote the value to the seller by r to indicate the fact this is the seller s "reservation value" for selling a car. We also assume b H > r H otherwise, peaches will never be sold and negotiations between buyer and seller are just about the price for a lemon. In equilibrium with b H > b L and r H > r L, there can be only two possible prices: 1) equilibrium price = the value of a lemon, b L if only lemons are sold; 2) equilibrium price = the buyer s unconditional expected value of a car, E(b), if both lemons and peaches are sold. Thus, there are two possible two zero-pro t prices (for buyers) with a positive probability of sale: p = b L and p = E(v). In addition, 104

there is always the possibility of an equilibrium with no sale technically all prices p < r L are zero-pro t prices because neither type of seller would accept a price in this range. We focus on the possibility of a no-trade equilibrium with price just below the seller s reservation value for a lemon, p = r L. We distinguish between two types of zero-pro t REE s in our analysis. An REE is unstable if a small deviation in the price o ered by one buyer would increase the payo s for both that buyer and the seller. 18 An REE is stable if no small deviation in price by one buyer would increase the payo s for both that buyer and the seller. If a single buyer deviates to a price o er slightly below a given REE level, no seller would consider this o er. For this reason, we only need to check whether a buyer could pro t by o ering a price slightly higher than a given REE level to determine if each REE price is stable or unstable. No trade with p = r L is always an REE, but it is only stable if b L r L. Otherwise, a small deviation to a price slightly more than r L would lead to sales of lemons that produce gains from trade to both buyer and seller. Similarly, trade of lemons with p = b L produces an REE if b L r L and r H > b L. Under both of these conditions, a seller who owns a lemon will sell it at price b L, but a seller who owns a peach will not sell at that price. Then a small increase in price will not produce a pro t for the buyer, since this would only cause the buyer to overpay for a lemon, and is not a pro table deviation for a buyer from an REE with p = b L. Trade of both types of cars with p = E(b) is an REE if E(b) r H. This is always a stable equilibrium because a small increase in price would simply guarantee a sale at a price greater than the expected value to buyers. Note that if r H b L, which precludes an REE with sale of lemons only, then E(b) > b L > r H and so there must be an REE with sale of both types of cars. Summarizing these results: 1) if b L < r L and E(b) < r H, then there is a unique REE with p = r L and no trade, and this equilibrium is stable; 2) if b L < r L and E(b) > r H, then there are two stable REE s, one with with p = r L and no trade and one with p = E(b) where both types of cars are traded; 3) if b L > r L and E(b) < r H, there is an unstable REE with p = r L 18 By de nition, an unstable Nash equilibrium is based on actions o the equilibrium path that are not best responses for small changes in price. Otherwise, at least one buyer would o er a slightly di erent price to increase payo from the equilibrium result. 105

and a stable REE with p = r L where only lemons are sold; 4) if b L > r L and E(b) > r H, there is an unstable REE with p = r L and two stable Nash equilibria, one with p = r L where only lemons are sold and one with p = E(b) where both types of cars are traded. Subgame perfect equilibrium requires buyers to anticipate the pro t-maximizing responses by the seller to all possible deviations in price. In cases 1) and 3) above where there is a unique stable REE, then this is also the unique SPE, because this is the highest possible zero-pro t price. In 2) and 4) above, where there are multiple stable REE s, then there is a unique SPE at the highest REE price. When there are multiple equilibria, the lower price generally fails because a deviation to a slightly higher price would be pro table for one of the buyers. 19 As mentioned above, any REE price can be sustained in a Nash equilibrium, The comparison between E(b) and r H summarizes the tradeo s inherent in the adverse selection model. Here, E(b) = b H + (1 )b L, so the condition for both types of cars to be traded is b H + (1 )b L r H. By rearranging this condition, we can highlight the importance of each of several parameters. Isolating, the condition for trade of all cars is (r H b L ) =(b H b L ). So for any xed values r H ; b H, and b L satsifying r H > b L and b H > b L, there is a cuto value between 0 and 1 so that the condition for full trade can be written as >. 20 Translated into words, if the number of peaches is su ciently large relative to the number of lemons, then there is a subgame perfect equilibrium where all cars are sold. Similarly, xing other factors and isolating either b H or b L, the condition for trade of all cars can be written as b H [r H (1 )b L ]= or as b L (r H b H )=(1 ). So there are cuto values b H and b L such that b H > b H ( xing and b L) or b L b L ( xing and b H) is su cient for all cars to be sold. These conditions indicate that if the gains from trade are su ciently large, then adverse selection does not prevent the trade of all cars. In the numerical example at the start of the chapter, b L > r L and b H > r H, so when a fulltrade equilibrium exists, it is e cient. However, if b L < r L, it is possible to have a full-trade equilibrium that is ine cient. In this case, it is e cient for peaches but not for lemons to be traded. Regardless of and the speci c values to buyer and seller, no Nash equilibrium in the 19 The one exception to this statement occurs when one of the equilibrium conditions holds with equality. For example, if b L > r L and E(b) = r H, then there are two REE outcomes, p = r L and p = E(b), which can both be sustained in SPE s. Given the strict equality, E(b) = r H, a deviation from price p = r L to a price just below E(b) will be unpro table because it will not induce peach owners to sell. 20 As noted above, if r H < b L, then the condition for full trade holds for any. 106

adverse selection game is e cient if b L < r L and b H > r H. Systematic Analysis of the Continuous Case The intuition from the two-type lemons market case carries over to the continuous case with an in nite number of types. The main point is that adverse selection involves tradeo s between 1) gains from trade and 2) cross-subsidization. If the gains from trade are large enough, then it is possible to sustain an equilibrium with a positive probability (possibly a certainty) of trade despite the problems due to adverse selection. We now proceed with equilibrium analysis for the continuous model. We focus on Rational Expectations equilibrium analysis in the discussion below because this is su cient to identify the subgame perfect equilibrium price as well. As we described in our general analysis for the Lemons market example, any REE is also a Nash equilibrium where all buyers o er the equilibrium price and the seller accepts that o er. 21 However, only the highest REE price can be sustained in a subgame perfect equilibrium except for the unusual case where there are multiple zero-pro t REE prices, but no prices between them where the buyer pro ts on average from trade. Boundary Equilibrium There are two possible boundary outcomes, one with (Rational Expectations) equilibrium price equal to the lowest possible value for the seller, v L, and another where the (Rational Expectations) equilibrium price is at least as large as the highest possible value for the seller, v H. In the rst case, p i = v L, the good is sold with probability zero 22 If h(v L ) > v L, then it would be pro table for one buyer to deviate to an o er slightly higher than v L, but still less than h(v L ). Thus, the condition for a stable REE with no one buying the good is simply h(v L ) < v L. In the second case, the good is sold with probability one. The average value to the buyers is then the unconditional expected value E(b) = E[h(v)]. If E[h(v)] v H, then it is unpro table 21 It is also possible for some buyers to o er prices less than the equilibrium price, p, in a Nash Equilibrium. But there must be at least two buyers who o er the REE price OR the seller must refuse all o ers below the REE price, for otherwise there would be a pro table deviation to a price below p. 22 Technically, the good would be sold if the seller s value is exactly v L, but this occurs with probability zero. 107

for a buyer to buy the good with probability 1 (at a price p = v H ). Thus, the condition for a stable REE in which the good is always sold is E[h(v)] v H, and the corresponding equilibrium price is p = E[h(v)]. The intuition implicit in this REE is that while all types of sellers would be willing to trade at p = v H, competition among buyers will drive the price to the higher zero-pro t price, p = E[h(v)]. Interior Equilibrium In an interior (Rational Expectations) equilibrium, the price falls in the range between v L and v H, meaning that some types of sellers will trade and others will not. Then there is a probability between 0 and 1 of a sale at price p, and the zero pro t condition for a REE is E[h(s)jS p ] = p : 23 This condition identi es a stable REE if a small deviation to a higher price is unpro table and an unstable Nash equilibrium if a small deviation to a higher price is pro table. If E[h(v)jV p] lies above p just to the left of equilibrium price p and lies below p just to the right of equilibrium price p, we call the intersection a "crossing from above", referring to the fact that the expected value curve lies above the price just before the intersection of these two curves. A "crossing from above" corresponds to a stable equilibrium, whereas a "crossing from below" corresponds to an unstable equilibrium, where the buyer could make a pro t at a price slightly higher than p. 3.2.2 Examples of Rational Expectations Equilibria with Adverse Selection The set of Rational Expectations outcomes can take any of several di erent forms, as shown in the following gures. In each of these gures, the position of the function E[h(v)jV p ] relative to p (the 45 line) determines whether an o er of p would be pro table to the buyers if that is the highest 23 Note that E[h(v)jV p ] p is the expected pro t-per-sale to the buyer, not the overall expected pro t for the buyer. The overall expected pro t to the buyer who makes the highest o er p takes into account the probablity that the o er is accepted by the seller: (p ) = P (V b ) [E(h(v)jV p ) p ]. 108

$ h(v) E[h(v) V< b*] 45 v L v H b* Figure 3-2: Adverse Selection Case #1 o er. If E[h(v)jV p ] lies above p, an o er of p would be pro table, while if E[h(v)jV p ] lies below p, an o er of p would be unpro table. Figures 3-2 through 3-4 depict three common cases when h(v L ) < v L. In Case 1, E[h(v)jV p ] < p for each value p in the range from v L to v H. the buyer, so there is a unique stable REE with p = v L : Every possible price is unpro table for In Case 2, E[h(v)jV p ] = p at two values, v 1 and v 2. Prices between v 1 and v 2 are pro table, while o ers below v 1 and above v 2 are unpro table for the buyer. There is a boundary REE with p = v L and two interior equilibria, one at p = v 1 and another at p = v 2. The equilibria at p = v L and p = v 2 are stable, while the equilibrium at p = v 1 is unstable. In Case 3, E[h(v)jV p ] = p at v 1. Prices below v 1 are unpro table for the buyer while prices between v 1 and v H are pro table for the buyer. There is a boundary REE with p = v L and an interior equilibrium with p = v 1. In addition, there is a REE with p = E[h(v)], with certainty (i.e. probability 1) of trade at the expected value to the buyer. The two boundary equilibria are stable, while the interior equilibrium is unstable. Figures 3-5 through 3-7 depict three common cases when h(v L ) > v L. In Case 4, E[h(v)jV p ] is greater than p for each value b p in the range from v L to v H. Every possible price in this range is pro table for the buyer, so there is a unique stable REE with p = E[h(v)] 109

$ h(v) E[h(v) V< b*] 45 v L v 1 * v 2 * v H b* Figure 3-3: Adverse Selection Case #2 $ h(v) E[h(v) V< b*] 45 v L v 1 * v H b* Figure 3-4: Adverse Selection Case #3 110

$ h(v) E[h(v) V< b*] 45 v L v H b* Figure 3-5: Adverse Selection Case #4 and probability 1 of a sale. (Technically there is also an unstable boundary equilibrium with p = v L.) In Case 5, E[h(v)jV p ] = p at a single value v1. Prices below v 1 are pro table for the buyer, while prices above v1 are unpro table for the buyer. There is a unique stable REE with p = v1. In Case 6, E[h(v)jV p ] = p at two values, v1 and v 2. Prices below v 1 and prices between v2 and v H are pro table for the buyer, prices between v1 and v 2 are unpro table for the buyer. In this case, there are two interior equilibria at p = v1 and p = v2 and a boundary equilibrium at p = E[h(v)]. The rst interior equilibrium and the boundary equilibrium are stable, while the second interior equilibrium is unstable. 3.2.3 Comparison to E cient Benchmark: The First-Best Outcome With perfect information, competition among the buyers would drive the price to h(v) when the seller has a value of v. The seller would accept an o er of h(v) if h(v) v and reject it otherwise. This would be socially e cient because the good would always be allocated to the person with the highest value for it. This socially e cient equilibrium is sometimes called the First-Best equilibrium, to distinguish it from ine cient equilibrium - the Second-Best - that 111

$ h(v) E[h(v) V< b*] 45 v L v 1 * v H b* Figure 3-6: Adverse Selection Case #5 $ h(v) E[h(v) V< b*] 45 v L v 1 * v 2 * v H b* Figure 3-7: Adverse Selection Case #6 112

results from the information asymmetry. In general, an REE with incomplete information is socially ine cient, as shown in Figures 3-2 through 3-7. The socially e cient outcome calls for a sale whenever h(v) v, but each REE consists of a cuto v (which is sometimes v L or v H ), with sales occurring whenever v v. In Case 1, h(v) > v for some values, but the good is never sold. Similarly, in Cases 2 and 3, there is an ine cient equilibrium with b = v L. In addition, each equilibrium with a positive probability of sales in Cases 2 and 3 involves sales of goods with values close to v L, yet h(v L ) < v L so those sales are ine cient. The most common cases of e cient transactions are the simplest ones. For example, in Case 4, h(v) > v for each v. The unique stable REE calls for a certain sale at price p = E[h(v)] and that is e cient because the buyer is known to value the good more than the seller at any price. Similarly, if h(v) < v for each v, any REE will have a price p < v L and that equilibrium will be e cient because there is no possibility of a sale. 3.2.4 Determinants of Adverse Selection Figures 3-2 through 3-7 subsume the calculation of E[h(v)jV p ]. This placement of this conditional expectation relative to v determines the e ect of adverse selection in a Nash equilibrium. In turn, this conditional expectation depends on the magnitude of h(v) and the nature of the distribution function f(v). For any o er p that the seller accepts, the buyer gets a pro t of h(v) p. Thus, the buyer prefers: 1) functions h that give large di erences between h(v) and v; 2) distribution functions that place maximum weight on higher values of v since h(v) is increasing in v. In the Takeover Game example, h(v) > v for each v, but the di erence h(v) v was not large enough to overcome the degree of adverse selection inherent in the uniform distribution f(v). To see the e ect of the interplay between h(v) and f(v), consider a version of the Takeover Game where v L = 0 and v H = 1, h(v) = av, and f(v) = 1 c+2cv, where a and c are constants with a 1 and 1 c 1. Here, we have two parameters. The constant a determines the degree of surplus available from trade. Higher values of a increase the value to the buyer, relative to the seller, thereby reducing adverse selection. The value of c adjusts the slope of the distribution of v. The 113

F(v) F(v) 2 2 c = 1 1 v c = 1 1 v Figure 3-8: Distribution of Payo s distribution is always linear in c, but it can have a positive or negative slope. For c < 0, higher values of v are progressively less likely than lower values of v. For c > 0, higher values of v are more likely than lower values. As c increases, the conditional expectation E[h(v)jV p ] increases for each p, once again reducing adverse selection. The distribution is still not completely exible: The cases with lowest and highest values are given by c = 1 and c = 1, depicted in Figure 3-8. In the Takeover Game discussed at the outset of this chapter, h(v) = 1:5v, meaning a = 1:5, and f(v) = 1, meaning c = 0. To see the general relationship between the buyer s valuation function and the distribution 114

function, consider the expected pro t to a buyer at price p: b (p) = (E[h(v)jV p ] p ) P (v p ) R p! 0 h(v) f(v)dv = R b 0 f(v)dv p P (v p ) R p! 0 av[1 c + 2cv]dv = R p 0 [1 c + 2cv]dv p P (v p ) 0" 1 = @ 2 a(1 # 1 c)v2 + 2 p 3 acv3 (1 c)v + cv 2 p A P (v b ) 0 1 2 = a(1! c)p + 2 3 ac(p ) 2 (1 c) + cp p P (v p ) = 1 3a(1 c)p + 4ac(p ) 2 6p (1 c) 6c(p ) 2 6 (1 c) + cp P (v p ) The denominator is always positive for p v H = 1, given that 1 c 1. Furthermore, P (v b ) must be between 0 and 1 since it is a probability. Overall, then, (p) 0 if: 3a(1 c)p + 4ac(p ) 2 6p (1 c) 6c(p ) 2 0 p (3a 6)(1 c) + (4ac 6c)(p ) 2 0 Full E ciency: When a > 1, the e cient outcome is for the buyer to purchase the good in all cases - i.e. p = 1. This is pro table if: (3a 6)(1 c) + (4ac 6c) 0 3a 6 3ac + 6c + 4ac 6c 0 3a 6 + ac 0 115

So, when the following condition holds a 6 3 + c or c 6 3a a competition will drive the price to E[h(v)] and the good will be sold with probability 1 in a stable REE. That is, adverse selection disappears in the equilibrium if a and c are jointly large enough to reduce the problem. Note that a larger value of c (which reduces adverse selection) reduces the cuto needed for the value of a to achieve full e ciency and vice versa. In the case of a uniform distribution, c = 0, and then a 2 is the condition for e ciency. Similarly, if a = 1:5, then c 1 is the condition for full e ciency. So the combination of a = 1:5 and c = 0 corresponding to the Takeover Game described at the outset of the chapter - does not yield the e cient outcome. 3.2.5 Commentary on Subgame Perfect Equilibrium The discussion of Cases 1 through 6 of the general model identi ed Nash equilibrium outcomes without solving for strategies o the equilibrium path. In a subgame perfect equilibrium, the seller s follows her dominant strategy both on and o the equilibrium path, accepting any o er b v. Working backwards, a zero-pro t price cannot yield a subgame perfect equilibrium if a higher price would yields pro ts to the buyers. This rules out all but the highest zero pro t price, which is the unique subgame perfect equilibrium (SPE). Note that a subgame perfect equilibrium must exist in this case. Once the seller s strategy is xed to accept o ers b v, it is possible to calculate net pro ts (b ) to buyers for each price (the highest o er) b. If (v H ) > 0, then E[h(v)] > v H and the subgame perfect equilibrium calls for a boundary solution with b = E[h(v)]. If (v H ) < 0, then the subgame perfect equilibrium simply calls for the highest price b < v H that gives zero expected pro ts to the buyers. If there is no such price, then b = v L, and no trades occur. If there are multiple Nash equilibria with ascending prices b 1, b 2, b 3,..., b n, then each of the equilibria with prices below b n requires noncredible beliefs by the buyers. For example, b 1 is a Nash equilibrium price based on a threat by the seller(s) not to accept any o er if v > b 1. But subgame perfection rules out buyers believing this empty threat, so that there is a pro table 116

deviation (generally to just below b n) for buyers. 24 3.2.6 Pareto E ciency If there are multiple Nash equilibrium prices, the highest equilibrium price Pareto dominates all lower prices. The buyers receive zero expected pro ts at any equilibrium price, so they do not have a preference among these outcomes. The sellers on the other hand, are better o when o ered higher prices, so they (at least the ones who accept the o er) strictly prefer the highest equilibrium price. That is, subgame perfection selects the most preferred equilibrium based on the Pareto criterion. 3.3 Labor Market Adverse Selection An alternate formulation reverses the relationship between seller s and buyer s valuation. In this model, the value to the buyer is, while the value to the seller is r(). Of course, the results in this formulation are identical to the above results, though recast from h(v) and v to and r(). MWG presents adverse selection in the context of workers and rms who do not know the qualities of each worker. In e ect, the rms are the buyers and the workers are the sellers, where each worker is hoping to sell his own labor to the rm. Formally, each worker has an ability (or type), where worker i knows i. The rms cannot observe the individual i s, but they know what the overall distribution of s in the workforce looks like. Suppose that ranges from L to H according to continuous distribution f() and that productivity of worker i is exactly i. Firms maximize total pro ts, which can be written for each rm j as (j) = (# of workers at rm j) [E j () E j (w)] where E j () is the average productivity of the workers at rm j and E j (w) is the average wage of the workers at rm j. Firms compete in Bertrand fashion for workers. The rm that o ers 24 In an extreme case, it is possible that b n gives a local maximum in expected pro ts to the buyer so that expected pro ts are negative for all prices greater than b n 1, except for b n. In this unusual instance, both b n and b n 1 can be sustained as subgame perfect equilibrium prices. 117

the highest wage gets all the workers who are willing to accept that wage. If multiple rms o er the same wage, then they divide the supply of workers at that wage. With many rms competing to hire each worker, each rm o ers the same wage, w, and each rm receives zero pro ts. Each worker i accepts the o er w if it is above a reservation wage r i, which may vary with the worker s ability i. We write the reservation wage as r(), meaning that the reservation wage is the same function of ability for each worker. Intuitively, we imagine that the reservation wage is the known wage in another industry (possibly selfemployment) where wages are at least partially contingent on productivity. By assumption, there is some correlation between ability in one job and ability in another job. Adverse selection arises if there is positive correlation between ability in the two industries, meaning that workers who are above average in one job also tend to be above average in the other job. Note that the correlation can be imperfect so that r() is closer to average productivity than is. Then for a given wage o er, workers with the highest abilities also have the highest outside option, meaning that the workers who accept the o er will be the ones with the lowest abilities. This assumption rules out some pairs of jobs (e.g. economist and painter) where you might conclude those who are talented at one job is unlikely to be talented at the other. To examine the e ciency implications, we again look to the full information case as a benchmark. If the ability of each worker is known, then each rm would bid up to i to hire worker i. Worker i would accept the o er if i > r( i ), exactly the condition for social e ciency. All the workers take the job where they are most e cient - this assumes that the reservation wage re ects the true value of the worker in the other industry. This condition i > r( i ) is the benchmark for e ciency. In the asymmetric information equilibrium, the expected ability of the workers who accept wage w is the conditional expectation E[ j w > r()]. If r is strictly increasing and thus invertible, the conditional expectation can be rewritten as E[ j < (w )], where (w ) is de ned to be equal to r 1 (). The condition for accepting the wage o er, < (w ), highlights the element of adverse selection in the process - the rm gets the lowest ability workers at this wage. The pro t per worker for each rm o ering wage w is then 118

Z(w ) = E[ j < (w )] w. Note that expected productivity, E[ j < (w )], is increasing in w. An increase in wage attracts workers of slightly higher productivity than anyone who is currently working, thus increasing average productivity. We stated above without proof that an equilibrium must give zero pro ts for each rm. The reasoning is identical to that from the Bertrand case. If the rms are making pro ts at w, then one rm can gain by deviating to a slightly higher wage. If the rms are losing money at w, then each should drop out of the market by o ering a lower wage. 3.3.1 Conditions for Interior Equilibrium For an interior solution equilibrium, with some workers accepting the wage w and others rejecting it, the zero-pro t condition holds with equality: E[ j < (w )] = w. One approach to solve for an interior equilibrium is to graph wage on the horizontal axis and expected productivity (the conditional expectation) on the vertical axis. This is a very direct approach, but it obscures the choice of workers and the cuto value. Another approach is to observe that by de nition, r[ (w )] = w, so that we can rewrite the zero-pro t condition as E[ j < (w )] = r[ (w )]. Then we can graph the worker s type on the horizontal axis and expected productivity on the vertical axis. 25 This presentation has the approach of presenting expected productivity as a conditional expectation on the y-axis as a function of the value on the x-axis. In this context, any intersection between the functions E( j < ) and r( ) corresponds to an REE. 26 Here, is simply the value on the x-axis. 25 This reverses the axes from a similar presentation in MWG. 26 In some cases, we suppress the dependence of on w for notational convenience. 119

3.3.2 Conditions for a Boundary Equilibrium There may also be a boundary equilibrium where either no workers accept the wage o er w or all accept it. No One Working - For none of the workers to accept the o er w, it must be that w < r( L ). Since the lowest possible wage is L, this condition corresponds to L < r( L ). Whenever this condition holds, there is an REE with wage w = L and no one working. In fact, there is a continuum of equivalent equilibria with wages ranging from L to r( L ) and no one working. (There is also a nonsensical way to create an equilibrium when L > r( L ) if we allow wage o ers below L. For example, it could be that ranges from 10 to 20 and r(10) = 5. Then there is an unstable REE with wage w = 4 and no one working.) Everyone Working - For all of the workers to accept the o er w, it must be that w r( H ). The conditional expected productivity at such a high wage is equal to the unconditional expected productivity E(), since all workers accept this o er. This wage is unpro table for rms unless E( j < ) w, or E() w r( H ). Combining these conditions, it is possible for the rms to hire all workers pro tably if E() r( H ). Then competition between the rms would drive the wage up to w = E(). Thus, when E() r( H ), there is a stable REE with w = E() and all workers working, and this is the unique SPE. 3.3.3 Examples of Adverse Selection Equilibria The simplest example of an adverse selection equilibrium occurs with constant reservation value: r() = r, where r is a known constant. This case does not t all of our assumptions because r() is not strictly increasing and is not invertible. But this is a particularly simple case because everyone works if w > r and nobody works if w < r. 27 If E() > r, then the unique 27 We ignore the possibility that some workers may accept an o er of w = r while others reject it. It is possible to get an interior equilibrium with r() = r and wage w = r if we assume that the only workers who accept the wage r are those with the lowest possible abilities. In e ect, that assumption changes the example to be as if r() is increasing. 120

Wage w* r (θ) 45 θ * θ Figure 3-9: Equilibrium with No One Working stable REE has w = E() and everyone working. If E() < r, then there are many equivalent equilibria with w < r and no one working. Graphically, it is easy to depict the two di erent equilibrium possibilities. In Figure 3-9, no one works. As shown, workers with > have productivities higher than their reservation wages r(), while workers with < has productivities less than their reservation wages, but this equilibrium outcome is ine cient because workers with > should be working in this industry and they are not doing so. In Figure 3-10, everyone works and the outcome is ine cient for exactly the opposite reason: Workers with < are working in this industry when they would be more productive in the other industry. The fact that all workers get the same wage (in a competitive market, w = E(), from the zero-pro t condition), is described as cross-subsidization. The higher productivity workers are being paid less than they are producing, and the lower productivity workers are being paid more than they are producing - that is, the higher productivity workers are subsidizing the wages of the other workers. 121

Wage w* r (θ) 45 θ * θ Figure 3-10: Equilibrium with Everyone Working 3.4 Appendix: Calculating Expectations in a Continuous Model The following is an example of the kind of calculation that allows us to solve for the conditional expectation function, E[ j < (w )] and the competitive equilibrium. First, we must be given the reservation wage as a function of ability, r(), and a probability distribution function for ; these are the two exogenous parameters of the problem. reservation wage r() = 4 + 1 4. f(): Let the Let the distribution of be de ned by the following function f() = 3 2 if 0 1 Then we can integrate to obtain the cumulative density function, F (): F () = Z 0 f(x)dx = Z 0 3x 2 dx = x 3 j 0 = 3 And now we can derive the conditional expectation function. 122

E( j < ) = E( j < ) = E( j < ) = R 0 x f(x)dx R 0 f(x)dx R 0 3x 3 dx F ( ) 3 4 ( ) 4 ( ) 3 = 3 4 The equilibrium depends on the intersection of this conditional expectation with the reservation wage, i.e. where r() = E( j < ). function: It will be useful to convert r() into its inverse r() = 4 + 1 4 (r) = 4r 1 r 1 (w) = 4w 1 So, for a given wage w, workers with ability from = (0; 4w 1) will work (unless 4w 1 > 1, in which case everyone will work). Combining the two equations we have solved, we get: E[ j < r 1 (w)] = 3 (4w 1) 4 And at last, we solve for the equilibrium: r() = E( j < ) w = 3 (4w 1) 4 w = 3 8 To check this result, we can plug in for w = 3 8 the conditional expectation (the zero-pro t condition): and make sure that the wage in fact equals 123

(w) = 4w 1 ( 3 8 ) = 1 2 E( j < ) = 3 4 E( j < 1 2 ) = 3 8 124