# Adverse Selection. Chapter 3

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10 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 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

18 \$ 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

19 \$ 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

20 \$ 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 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

21 \$ 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

23 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

24 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 ) = a(1! c)p 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

25 So, when the following condition holds a 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 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

26 deviation (generally to just below b n) for buyers 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

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