INTRODUCTION Greshmans Law bad money derives out good

INTRODUCTION Greshman’s Law ; bad money derives out good money. The holder of a coin is able to shave a bit of gold from it in a way that is undetectable without careful measurement ; the gold so obtained can then be used to produce new coins. The holder of an unshaved coin will therefore withhold the coin from trade ; only the shaved coins will circulate. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 1

AKERLOF’S MODEL OF LEMONS Bad used cars derives out good cars out of market. suppose that there are two types of used care ; peaches and lemons 1 - a peach worth $3000 to a buyer & $2500 to a seller. The supply of peach cars is fixed and the demand of possible buyers are infinite , so that the equilibrium price in the peach market is $3000. 2 - a lemon worth $2000 to a buyer and $1000 to a seller. There are twice as many lemons as peaches. We can summarize the worth of different kinds in the following table , Sellers Buyers lemons 1000 2000 peaches 2500 3000 Different kinds of transaction could be recognized ; KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 2

AKERLOF’S MODEL OF LEMONS 1 - if buyers and sellers both had the ability to look at the car and recognize the type of the car , peaches will be sold for $3000 and lemons will be sold for $2000. 2 - if neither buyer nor seller knew whether a particular car was a peach or lemon and assuming risk neutrality we would have no problem. a- a seller thinking that she has a peaches with probability of 1/3 and a lemon with the probability of 2/3. She also expect that she might have a car with the expected value equal to $1500 = (1/3)(2500) + (2/3) (1000) b- a buyer assuming the same probability expect that she might buy a car with expected value equal to 2333. 3 = (1/3)(3000) + (2/3)(2000) Assuming inelastic supply and elastic demand the market clears at a price equal to $2333. 3 But in reality the market is not like neither of these. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 3

AKERLOF’S MODEL OF LEMONS In the market of used cars , the seller have lived with car for a while and knows whether the car is peach or lemon. But the buyer does not have inside information about the car , expect for the quick inspection when he wants to buy it. If we assume that the buyer’s can not tell at all , the peaches market breaks down. Suppose that cars are offered for sale at any price above $1000. All the lemons will appear in the market. Peaches would appear in the market at prices above $2500 since lemons peaches Sellers 1000 2500 Buyers 2000 3000 At a price between $1000 and $2500 the rational buyer suppose that peaches does not appear in the market So the cars worth only $2000 to the buyers. At any price above $2500 , the expected value (price) of the car to the buyer is equal to 2333. 3 = (1/3)(3000) + (2/3)(2000) <2500. There will not be any offer for peaches. Only lemons will be sold for $2000. (demand is perfectly elastic and supply is perfectly inelastic. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 4

AKERLOF’S MODEL OF LEMONS Suppose that there were two peaches for every lemon. So probabilities are as follows ; lemons ( P = 1/3) peaches (P = 2/3 ) Sellers $ 1000 $2500 Buyers $2000 $3000 Exp (buyer) = (1/3)(2000) + (2/3)($3000) = 2666. 67 > 2500 This is enough for the owners of peaches to sell and we get the market clearing price at 2666. 67 with all the cars for sale. The buyer buys a car at 2666. 67 and he does not know whether it is lemon or peach. Owners of peaches are not pleased about the lemons ; without them , peach owners would be getting an extra of 333. 33=3000 -2666. 67 for their peaches. But at least peaches can be sold. If a particular good or service comes in many different qualities , and in a transaction one side but not the other knows the quality in advance , the other side must worry that it will get an adverse selection out of entire population. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 5

AKERLOF’S MODEL OF LEMONS The classical example of this is in life /health insurance. If premiums are set at fair rates for the population as a whole , insurance may be a bad deal for healthy people , who then will refuse to buy. The problem noted above becomes worse the greater the number of qualities of cars and the smaller the valuation gap , the difference between what a car is worth to a buyer and a seller , assuming they have the same information. Note the following example ; Imagine for example , that the quality spectrum of used cars runs from real peaches , worth $2900 to sellers and $3000 to buyers , down to real lemons , worth 1900 to sellers and 2000 to buyers. Between the two extremes are cars of every quality level , always worth $100 more to buyers than sellers. The distribution of quality levels between these two levels are uniform. Suppose that there are 10001 cars one of which worth $1900 to its owner and $2000 to buyers, a second worth $1900. 10 to its owner and $2000. 10 to buyers , and so on. Assuming inelastic supply and elastic demand at every level of quality , what will be the equilibrium. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 6

AKERLOF’S MODEL OF LEMONS Total number of cars = 10001 types Each one represents a different ( quality ) kind Quality of cars Worth to seller Worth to buyer Number offered Real Peaches 2900 3000 10001 Tenth quality level 1901 better 2001 11 …………. . ………. . Second quality level better 1900. 2 2000. 2 3 First quality level better 1900. 1 2000. 1 2 Real Lemon 1900 2000 1 KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 7

AKERLOF’S MODEL OF LEMONS Supply curve would be as follows; Min supply price for lemons to be offered = 1900 → only one car will be offered If p = 1900. 1 → 2 cars will be offered. If p = 1900. 2 → 3 cars will be offered. If p = 1901 → 11 cars will be offered. If p = 1902 → 21 cars will be offered. If p = 2900 → all 10001 cars will be offered for sale. supply price 2900 2100 1900 demand quantity 2001 KREPS CH 17 10001 ADVERSE SELECTION & MARKET SIGNALING 8

AKERLOF’S MODEL OF LEMONS Demand curve would be perfectly elastic as follows; Buyers are willing to pay $2000 for real lemons and $3000 for real peaches. Demand price will be between $2000 & $3000. If a car be offered at a price equal to P where 2000≤ P ≤ 3000 buyers assume that only sellers who value their own cars at P or less are willing to sell. Hence a car being sold at P has a quality level that makes it worth on average between $2000 and $(P+100) to buyers , with each value in this range equally likely. The average car being sold is worth A= $(2000+P+100)/2 to the buyer. (Since minimum price = 2000 and actual one is ( p +100 ) ) If P= 2100 , then A=P , { A=(2000+2100+100)/2 = 2100 } If P>2100 , then A < P , there will not be any demand. If P< 2100 , then A > P , there will be infinite demand. So demand is horizontal at p=2100 (figure slide 7) Equilibrium price will be at p=2100 , with q=2001 supply of cars. As it is seen , only 2001 out of 10001 of the total stock of car will be offered for sale in the market. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 9

AKERLOF’S MODEL OF LEMONS Now suppose that , between the two extremes are cars of every quality level , always worth $50 more to buyers than sellers. in this case the quality spectrum of used cars runs from real peaches , worth $2950 to sellers and $3000 to buyers , down to real lemons , worth 1900 to sellers and 1950 to buyers. The equilibrium price will be P = 2000 with 1001 cars offered for sale in the market, [ A = $(1950+P+50)/2 ] Now suppose that some item like durable goods or some services comes in N different qualities, q 1, q 2, …. q. N. The supply for each n depends upon the price level (P) , and is given by an upward sloping supply curve Sn(p). As it is explained demand depends on price (P) and average quality in the market ( A) which is equal to ; average value = q n = quality level of type n S n ( p) = number of quality level of type n Let D( p , qavg (p) ) be the demand function. It is decreasing in p and increasing in qavg(p). d D ( p , qavg (p) ) /dp = ∂D/∂P + ( ∂D/ ∂q )( ∂qavg (p)/ ∂p ) (most likely positive ∂qavg (p)/ ∂p > 0 ) , ∂D/ ∂q >0 , ∂D/∂P < 0 → it might be d D ( p , qavg (p) ) /dp > 0 if the second term is big enough , KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 10

AKERLOF’S MODEL OF LEMONS It is possible that for some price levels lower than p* the slope of demand curve becomes positive [ d D( p , qavg (p)/dp > 0 ] and does not intersect the supply curve as it is shown in the following figure. in these cases there will not be an equilibrium price and market breaks down as it is in some health insurance market. price demand supply P* quantity KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 11

SIGNALING QUALITY In reality all manners of markets subject to adverse selection do function. Why ? Often because the side to the transaction that has the superior information will do something that indicates the quality of the good being sold 1 - in life insurance the high quality buyer distinguish himself from the low quality buyer and take actions that is not worthwhile for low quality buyer. 2 - in used car market The high quality seller may take actions which distinguish him from the low quality seller and is not worthy for them. with used cars ; a- seller may offer partial warranty or b- may get the car checked by an independent party. In life insurance ; a- medical checkups are sometimes required. Or ; b- golden age policy may be used. No one will be turn down but only the benefits will greatly reduced for the beginning years of the insurance plan. If the buyer of the insurance knows that he is dying soon , or if he wants to use the benefits very soon , he will not gain the whole benefit. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 12

SIGNALING QUALITY Quality seller and his actions are not worthwhile for low quality workers. One can think of adverse selection as an special case of moral hazard and market signals as a special case of incentive schemes. The incentives could be direct or indirect ; 1 - direct incentives ; the seller of the low quality car may be tossed into jail if he misrepresent the quality of car sold. 2 - indirect incentives ; the seller of the used cars could be asked to give a six month warranty if he says that the car is peach. Or the buyer may offer $X if the seller offers him a warranty and $Y ( X>Y) , if he does not offer him a warranty. The two classic model of adverse selection in the literature are 1 - SPENCE’S model of job market signaling and 2 - ROTCHILD & STIGLITIZ model of insurance market. In the present analysis the job market model (Spence’s setting )will be used to illustrate both cases. Imagine a population of workers with two qualities with equal numbers ; First – high quality with high innate ability , denoting t=2 Second – low quality with low innate ability , denoting with t=1 KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 13

SIGNALING QUALITY There are firms that will hire the workers and operates in competitive market and making zero pure profit. Before going to workers seek education that enhances their productivity. But education can not move workers from low to high quality level , because it is intrinsic. Each worker chooses a level of education from a set say from e=0 to e=16 (the numbers of schooling). A worker of type t ( t=1 , 2) and with education level of e ( e=1 , … 16 ) worth te to a firm. note that for every level of education higher quality worker worth twice as much as lower quality ones. Firms when they hire a worker are unable to tell whether the worker is able or not. But they do get to see the worker’s CV and learn how many years they have gone to school and make wages contingent on the number of years they have spent in school. Workers dislike education , and less able dislike education more. Workers also want higher wages to be sure. Workers want to maximize the following utility. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 14

SIGNALING QUALITY Max U=Ut ( w , e ) ; where “w” is wage rate , “e” is the education level and “t” is the type of the worker. Utility is strictly increasing in “w” , and strictly decreasing in “e”. Wage paid Worker’s indifference curve is the locus of the combination of wage and education level which brings a constant utility level. It is upward Shape of the worker’s indifference curve ; sloping and strictly convex. Take any point like P (or any other point). Notice that at least one indifference Indifference curve of w low ability workers curve for low ability worker and one indifference curve for high ability worker would pass from this point The indifference curve of high ability worker shows a higher level Direction of of utility than low ability one at any increasing point like P. To compensate a preference Indifference curve worker for a given increase in of high ability education requires a greater workers P increase in wages for a low ability worker than for a high ability worker. So , the indifference curve e of the low ability worker is always more steeply sloped than high Education level KREPS CH 17 ADVERSE SELECTION &ability. 15 MARKET SIGNALING

SIGNALING QUALITY This assumption known as a single-crossing property For example ; Ut ( w , e ) = f(w) – Kt g(e) , where ; f(w) is increasing and concave function , g(e) is increasing and strictly convex function , Kt is positive constant with K 2 >K 1. What would happen if the ability of workers were observable (full information model ). ) The high ability workers with education level e would be paid 2 e , while the low ability worker with the same education level would be paid e. As showed in the figure in page 17 the high and low ability workers will pick different education levels which maximizes their utility levels given these wages. In the equilibrium education level of high ability workers (e 2 ) is greater than low ability workers (e 1). So the firm could realize the type of the workers from their education level But we have to deal with situations in which ability level is not directly observable. We will consider two cases ; 1 - First ; job market signaling (Spence model ) 2 - Second ; worker self selection from a list of menu ( Rotchild and Stigltiz model). KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 16

SIGNALING QUALITY, ability of workers are observable Indifference curve of high ability workers W=2 e Indifference curve of low ability workers Wage paid Direction of increasing preference W=e e 1 e 2 Education level KREPS CH 17 The low ability worker knows that he should move on the W=e line and the high ability worker knows that he should move on the W=2 e line, because the ability is observable by the firm. In other words each worker knows that his ablity is observable and he can not hide it. So , what he does is just a utility maximization concerning his constraint ( w=e or w=2 e ). e ADVERSE SELECTION & MARKET SIGNALING 17

SPENCE’S MODEL ; JOB MARKET SIGNALING How many years Workers choose to go to school ? . They do so anticipating some wage function w(e) that gives wages for every level of education. Formally an equilibrium in the sense of Spence consist of ; a – an anticipated wage function w(e) that workers anticipate will be paid to any one obtaining this education level e and , b- probability distribution πt(e) on the set of education levels for each type of worker , t= 1 , 2 , such that the following two conditions are met ; b 1 - For each type t worker and education level e → πt(e)> 0 only if ut [w(e) , e] achieves the maximum value of ut [w(e) , e] over all e Condition b 1 says that based upon their anticipation , the workers choose that wage level which maximizes their utility level b 2 - for each education level e such that; π1(e) + π2(e) > 0 ; w(e) = e [0. 5π1(e) ] / [0. 5π1(e) + 0. 5π2(e)] + 2 e [0. 5π2 (e) ] / [0. 5π1(e) + 0. 5π2(e)] πt(e) gives the proportion of type t workers who select education level e in equilibrium. So πt(e) is the probability of type t worker with education level of e. 0. 5 = probability of type t in the population (opulation of workers with two qualities are assumed to be the same) Condition b 2 express that in the equilibrium which is attained under perfect competition , the wage rate , w(e) is equal to the conditional expected value of a worker presenting education level e. Since ability of workers are not observable , the eqilibrium wage should be represented in terms of expected wage (both from workers and employers point of view). For the simplicity we assume that all the workers of a given type choose the same level of education : KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 18

SIGNALING QUALITY Two types of equilibrium - non observable quality (ability) ; 1 -First type ; separating (screening ) equilibrium ; All workers of type t=1 choose an education level e 1. All workers of type t=2 choose an education level e 2. Firms seeing the education level e 1 , knowing the worker is of type 1 , pay a wage equal to e 1. And if they see the education level e 2 , knowing the worker is of type 2 , pay a wage equal to 2 e 2. 2 -Second type ; pooling equilibrium ; workers all pool at a single education level , e* , and the equilibrium wage will be equal to ; firms by seeing educational level e* thinks that the probability of worker from type 1 or 2 is 1/2. w(e) = ( e* ) ( 1/2) + 2( e* ) ( 1/2) = 1. 5 e* {π1(e) = π2(e) } 1 -First type - Separating equilibrium (non observable quality); In order to have an separating equilibrium each type of workers will choose different level of education ; workers of type 1 would rather choose the wage education pair ( w=e 1 , e = e 1 ) than ( w = 2 e 2 , e = e 2 ) and vice versa. Such an equilibrium could be seen in the following figure. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 19

Wage level SIGNALING QUALITY both kinds of labors will maximize their utility levels with respect to the expected wage constraint w(e) and solutions are E and F. F Indifference curve of high ability workers W= e =2 W E e Indifference curve of low ability workers e 1 W(e) curve must lie every where at or below the indifference curves of the workers at the points that the workers select because E and F are the maximization points; this is the self selection condition. e 1 and e 2 are education level which are selected by low and high ability worker in the equilibrium and observed by the firm as a signal e 2 Education level point E on w(e) is selected by low ability worker , so indifference curve of low ability worker and W=e line should pass through point E and point F on w(e) is selected by high ability worker so indifference curve of high ability worker and W=2 e pass SELECTION through point F KREPS CH 17 line should ADVERSE & MARKET SIGNALING 20

What restriction should be placed on the points that are not selected No specific restriction is needed except that ; 1 - W(e) should be nonnegative for every level of e 2 - we require that e ≤ w(e) ≤ 2 e for all values of e. 3 - it is also required that w(e)/e is non decreasing. (convex w(e) function) first condition is self explanatory. Condition 2 says that if firms are unsure about the worker’s ability level , firms can not be certain about the level of education ( e or 2 e ). But according to the competition among the firm they never pay more than 2 e and not less than e. As for condition 3 , workers assume that firms will asses a worker who chooses education level e will be of low ability with probability a(e) Assuming that firms all are risk neutral the wage which will be paid in this competitive situation will be ; w(e) = a(e) e + [ 1 - a(e) ] 2 e or w(e)/e = a(e) + [ 1 - a(e) ]2 = 2 – a(e). if w(e)/e is non decreasing , the a(e) should be non increasing This means that with increase in the level of education , the probability of the worker to have low ability is decreasing. This means that tan(α) is increasing and w(e) wil be convex towards the horizental axis. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 21

SIGNALING QUALITY We could rephrase the condition 2 in the following manner ; a(e) is the probability of low ability worker choosing education level e. So firms assessments given by a(e) are confirmed at education levels that workers do select in the equilibrium ; a(e) = [ 0. 5 π1(e) ] / [ 0. 5 π1(e) + 0. 5 π2 (e) ] πt(e) gives the proportion of type t workers who select education level e in equilibrium. population of workers with two qualities are in equal numbers ; . Note that in the figure in slide no 20 the w(e) function does not lie between e and 2 e. So it does not satisfy condition (b). In the following figure we introduce an equilibrium which satisfies condition b. In this equilibrium the education level which is chosen by the worker is e^1. That is when w=e , the education level is equal to e^1. In other words the indifference curve of low ability worker is tangent to line w=e when e= e^1. In contrast to figure in slide no. 20 the low ability worker get higher utility in the equilibrium. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 22

Wage level SIGNALING QUALITY Indifference curve of high ability workers W(e) curve e W= Indifference curve of low ability workers which is higher than before. e =2 W e^ 1 KREPS CH 17 e 1 e 2 ADVERSE SELECTION & MARKET SIGNALING Education level 23

Preposition 1 ; in any separating equilibrium for which the function w(e) satisfies condition (b) , low ability workers choose precisely e^1 and get the corresponding wages. In any equilibrium (separating or not) for which the function w(e) satisfies (b) , low ability workers get at least the utility that they get from ( w=e^1 , e=e 1 ). (this is the utility level when he introduce himself as low quality so in pooling equilibrium the utility level should not be lower than this) Pooling equilibrium ; in the pooling equilibrium we assume that all the workers choose a single level of education say e*. Numbers of high and low ability workers are the same and each have education e*. then a(e*) = 0. 5 since ; a(e) = [ 0. 5 π1(e) ] / [ 0. 5 π1(e)+0. 5 π2 (e) ] , if π1(e)= π2 (e) {pooling eqilibrium condition}, then a(e) =1/2 w(e) = a(e) e + [ 1 - a(e) ] 2 e → w(e*) = 1. 5 e* workers assume that firms will asses a worker who chooses education level e will be of low ability with probability a(e) πt(e) gives the proportion of type t workers who select education level e in equilibrium An equilibrium of this sort is given by the following figure in slide no. 25. note that the function w(e) has a kink at the pooling point. This is necessary. Without the kink , we could not have w(e) underneath both types of indifference curves and touching those curves at pooling point. As we can see point E could be anywhere on the w=1. 5 e* line and there could be many equilibrium. 24

Wage level SIGNALING QUALITY W=1. 5 e Indifference curve of high ability workers W(e) curve e =2 W E e W= Indifference curve of low ability workers which is higher than before. e* KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING Education level 25

Rothschild and Stiglitz model ; Workers self selection from a model ; The previous assumptions are holding ; that is ; 1 - two types of workers ( low and high ability ) in equal numbers. 2 - πi(e) = probability of choosing education level e by worker of type i. Proportion of i workers who choose education level e. In the Spence model workers choose education levels in anticipation of offers from the firm , w(e) , and we assume that those anticipation are correct , at least for the education levels actually selected. Let’s turned this around and suppose that the firms moves first. Suppose that the firms offer to workers a number of contracts of the form ( w, e ) before the workers go off to school , content in the knowledge of what wage they will get once the school is done ( assuming they complete the number of years for which they have contracted). (We could also think of insurance market instead with insurance contracts being offered to people before hand). The context with which the model is dealing. Now consider an equilibrium in This context. An equilibrium in the context of Rothchild and Stiglitz consist of ; KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 26

SIGNALING QUALITY a- a menu of contracts , a set of pairs [ ( w 1, e 1) , ( w 2 , e 2), … , (wk, ek) ] For some finite integer. And b- a selection rule by which workers are assigned to contracts or , for each type of workers ( t ) a probability distribution πt over [1, 2, 3, …k], that satisfy three conditions as follows ; (1) – Each type of worker are assigned to the contract that is best for That worker in the menu. In symbols , πt(j) > 0 only if ut(wj , ej ) achieves the maximum of ut (wj’ , ej’ ) for j’ = 1, 2, …. k and t = 1, 2. (2) – Each contract in the menu to which workers are assigned at least breaks even on average Otherwise , firms offering that contract would withdraw from the contract. In other words wage offered by firm is not greater than the average anticipated wage level. In symbols for each j = 1, 2, 3, …k if π 1(j) + π2(j) > 0 , then ; Wj ≤ [ 0. 5 π1(j) ]/[0. 5 π1(j) + 0. 5 π2(j) ] ej + [ 0. 5 π2(j) ]/[0. 5 π1(j) + 0. 5 π2(j) ] 2 ej (3) – No contract can be created that if offered in addition to those in the menu would make strictly positive profits for the firm offering it , assuming that workers choose among contracts in a manner consistent with rule (1). KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 27

SIGNALING QUALITY Preposition 2 – In an equilibrium , any contract that is taken up by workers must earn precisely zero expected profit per worker (competition leads the profit towards zero). Suppose that (w’ , e’ ) is offered and it is taken by some workers , and earns an expected profit of ε for the firm per worker who take it. Then because of competition among the firms there could be some firms Which offers an amount equal to ( w’ + ε/2 , e’). This new contract will attract the workers who previously were attracted by (w’ , e’ ) and will still return a profit of ε/2 , which is contradicting the condition 3. Now assume that the new contract may attract others besides those who previously took (w’ , e’ ) and those others may render the profitability of this contract. If the new workers are low ability workers these low ability workers may be able to render the profitability of new contract and the contradiction with condition 3 could be wipe out. But if the new workers are high-ability workers , then it could be shown that ; Proposition 3 ; there could not exist a pooling equilibrium ; KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 28

SIGNALING QUALITY Wage level W=1. 5 e Indifference curve of high ability workers W’ E 1. 5 e* e W= e =2 W Indifference curve of low ability workers. KREPS CH 17 e* e’ ADVERSE SELECTION & MARKET SIGNALING equilibrium point E (contract E) is chosen among different points (different contracts ) because it has the highest utililty among four different dotted points Education level 29

SIGNALING QUALITY Suppose that the firm are offering a menu of contracts that causes all the workers to choose (1. 5 e* , e* ) as shown in the figure. In the figure in page 29 the filled-in dots are the menu of contracts that we suppose Are offered. As it is seen the (1. 5 e* , e* ) contract is the best , because the utility levels are higher. From this position , any one of the firms can offer the contract (w’ , e’ ) that is shown as a filled triangle. This is a contract that has a slightly higher wages and education levels than (1. 5 e* , e* ) , where the increased wages more than compensate a high-ability worker but don’t compensate a low-ability worker relative to the pooling contract. It will only attract the high ability workers. All the low ability workers prefer the (1. 5 e* , e* ) contract. But if all high ability workers flock to ( w’ , e’ ) contract and non of low ability workers do so , then each worker who chooses ( w’ , e’ ) worth 2 e’ to the firm that makes this offer. The firm makes a profit and according to proposition 2 we can not have a pooling equilibrium. From this point we can reach to the following proposition ; Proposition 4. It is impossible , in eqilibrium , that any contract (w , e) is taken by positive fractions of high- and low ability workers both. So , only possible equilibrium is separating equilibrium. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 30

SIGNALING QUALITY Proposition 5 ; There is only a single candidate for a separating equilibrium. In this candidate equilibrium. Low-ability workers choose the contract ( w=e^1 , e=e^1 ) , where e^1 is defined as before ( page 23) and High-ability worker get w=2 f ^2 , e= f ^2 , (page 33) First let us concentrate on page 33 and on the first part of the proposition. If low ability workers are separated and choosing any other contract other than ( w=e^1 , e=e^1 ) , a firm could add to the menu of contracts a contract like ( w=e^1 – δ , e=e^1 ) for δ>0 small enough so that the low ability workers prefers this to the contract they are choosing now (other than ( w=e^1 , e=e^1 ) ) in the supposed equilibrium. But then for any value of δ>0 , this contract must be profitable. For explaining the second part of the proposition we should consider the f ^2 which solve the maximization of u 2(2 e , e ) subject to e≥ f ^1. That is f ^2 is the education level that high-ability workers would choose if they could have any point along the ray w=2 e for e ≥ f ^1. We want education level to be greater than low ability worker when w=2 e, because we expect high ability workers get higher education than low ability workers and separate themselves from them. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 31

SIGNALINAG QUALITY Let’s now concentrate on the high-ability workers and show that high quality workers get (w=2 f ^2 , e= f ^2 ). Why ? If high quality workers are separated by any other contract other than (w=2 f ^2 , e= f ^2 ) , some firm could offer a contract like (w=2 f ^2 - δ , e= f ^2 ) that for sufficiently small δ would be more attractive to high ability workers than the contract they are taking at the supposed equilibrium. Since f ^2 ≥ f ^1 , this contract is less appealing than the contract ( w = e^1 , e = e^1 ) for low ability workers (lower utility level for low ability workers ). Hence this contract will precisely attract the high ability workers and thus for any δ >0 , this contract is precisely profitable. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 32

Wage level SIGNALING QUALITY W = 1. 5 e Indifference curve of high ability workers. w’ e = W E e =2 W e^1 KREPS CH 17 Indifference curve of low ability workers. e’ f ^1 = f ^2 ADVERSE SELECTION & MARKET SIGNALING Education level 33

SIGNALINAG QUALITY Considering the figure in page 33 we have there depicted separating equilibrium in a case where f ^2 = f ^1. It could be shown that no firm would try to break this equilibrium point by offering a contract in a position such as (w’ , e’ ) that is shown with w’ a bit less than 2 e’ and e’ a bit less than f ^1 (point E lies above the w=1. 5 e line ). This contract when added to the menu would certainly attract the high ability workers because it offers them a higher utility level than the one they are enjoying with ( w=2 f ^2 e= f ^2 ) contract. Every high ability worker would be profitable at this contract because wage is less than 2 e’. But this will also attract the low ability workers because the utility level of low ability workers are also higher than the one which they are enjoying now. Now in this pooling equilibrium the firm should pay a wage less than 1. 5 e’ to earn a profit which is not the case in the figure. Now consider the figure in page 35 in which point E located below the w=1. 5 e line. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 34

SIGNALING QUALITY Wage level W=1. 5 e Indifference curve of high ability workers. e = W w’ e =2 W E e^1 e’ KREPS CH 17 Indifference curve of low ability workers. f ^1 = f ^2 ADVERSE SELECTION & MARKET SIGNALING Education level 35

SIGNALINAG QUALITY In page 35 the indifference curve of the high ability workers through the point ( w=2 f^2 , e= f^2 ) dips below the line w= 1. 5 e. In this case a firm could offer a contract such as (w’ , e’ )as shown below the line w=1. 5 e , but still above the high-ability worker’s indifference curve. This would break the equilibrium , because even though it attracts all the workers , high ability and low ability , it is still profitable. So there can not be an equilibrium at all. Any sort of pooling is inconsistent with the equilibrium , (as proved by proposition 3). And the only possible separating equilibrium can also be broken. Proposition 6 ; In the formation of Rothschild and Stiglitz , there is at most one equilibrium. In the candidate equilibrium , low ability workers choose ( w=e^1 , e=e^1 ) and high ability worker choose the education level f ^2. Such that ( w=2 f ^2 e= f ^2 ) is their most preferred point along the ray w=2 e for e≥ f ^1. If the indifference curve for high ability workers through the point ( w=2 f ^2 e= f ^2 ) dips below the pooling line w=1. 5 e, then there is no equilibrium at all. if this indifference curve stays above ( or just touch) the pooling line , then this single candidate equilibrium is an only equilibrium. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 36

SIGNALINAG QUALITY The stories of Riley and Wilson In many markets where signaling takes place , such as insurance , it seems natural to suppose that uninformed parties put a menu of offers on the table from which informed parties must choose. So conclusion of Rothchild and Stiglitz analysis , namely that there may be no equilibrium at all , is rather troublesome. A number of authors have suggested that the problem lies in the notion of equilibrium proposed by Rothchild and Stiglitz. Two of the writers are Riely and Wilson. Riley advocates the notion of reactive equilibrium , in which to destroy a proposed equilibrium , it must be possible to add a contract to the menu that will be strictly profitable and that will not become strictly unprofitable if other firms allowed to add still more contracts to the menu. In this sort of scheme , our argument for why there are no pooling equilibrium stands up ; we broke a pooling equilibrium by adding a contract that attracted high-ability workers only. Given that the pooling contract remained to attract low-ability workers. There is no way to make this pool-breaking contract strictly unprofitable as long as the old pooling contract is remains , because as long as the old KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 37

SIGNALINAG QUALITY Pooling contract is present , only the high ability workers would ever consider taking the new contract. But you are at much more risk when adding a contract that attempts to break a separating equilibrium by pooling high and low ability workers. For in this case , still another contract can be added that siphons from your contract the high ability workers , leaving you with only low ability workers. Hence , Riley concludes that reactive equilibrium always exist , and they correspond to the single candidate separating equilibrium. Wilson , on the other hand , proposed a notion called anticipatory equilibrium. In this case , to break a proposed equilibrium it must be possible to add a contract that is strictly profitable and that does not become strictly unprofitable when unprofitable contracts from the original menu are withdrawn. Now if you try to break a pooling equilibrium by skimming the high ability workers , you are at risk ; the pooling contract that you destroy becomes unprofitable because after your addition it attracts only low ability workers. But if it is withdrawn , all those low ability workers may flock to your contract. On the other hand if you break a separating equilibrium with a pooling contract , you are not at risk ; because you have created a contract that attracts all the workers , You do not care if other, now unused contracts ADVERSE SELECTION & 38 are withdrawn MARKET SIGNALING

SIGNALINAG QUALITY So, Wilson concludes , anticipatory equilibrium always exist ; sometimes there is more than one ; and pooling is possible as an equilibrium outcome. How do we sort between Rothschild and Stiglitz , Riley , and Wilson ? Think in Terms of an insurance market that is regulated by some regulatory authorities. If you think that firms can not withdraw contracts because the regulatory committee forbid this , then Riley’s equilibrium equilibriu concept seems the more reasonable. If you think that the regulatory authorities permit the firms to withdraw unprofitable contracts and are very tough on adding contracts that potentially affect the profitability of others , then Wilson’s notion seems more reasonable. As for Rothcilds and Stiglitz , think of regulators who call for firms to register simultaneously and independently the contracts they wish to offer , with no room to add or subtract subsequently. KREPS CH 17 ADVERSE SELECTION & MARKET SIGNALING 39