Adverse Selection, Signaling, Screening Introduction : One of the implicit assumption of the...

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening

Introduction Introduction : :

One of the implicit assumption of the fundamental welfare theorems is that the characteristics of all commodities are observable to all market participants. Without this condition , distinct markets cannot exist for goods having different characteristics , and so the complete markets assumption cannot hold . In reality , however , this kind of information is often asymmetrically held by market participants

1- used car markets

2- firms hiring the worker and does not know the innate ability of the workers ,

3- automobile insurance

13 MAS_COLLEL CH1ADVERSE SELECTION SIGNALING

SCREENING

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening 1 -How do we characterize market equilibrium in the presence of

asymmetric information .

2- what are the properties of this equilibrium.

3- Are there possibilities for welfare-improving market equilibrium.

I.In the presence of asymmetric information , market equilibrium often fails to be pareto-optimal .

II.Constrained pareto-optimal allocation are all allocations that cannot be pareto improved upon by a central authorities who, like market participants , cannot observe individual’s privately held information .

III.Signaling ; informed individuals may find ways to signal information about their unobservable knowledge through observable actions. Seller of a used car could offer to allow a prospective buyer to take the car to a mechanic .

IV.Screening ; the possibility that uninformed parties may develop mechanism to distinguish , or screen , informed individuals who have different information . An insurance company may offer two policies ; one with no deductible at a high premium and another with a high deductible at a much lower premium . 13 MAS_COLLEL CH2

ADVERSE SELECTION SIGNALING SCREENING

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Informational Asymmetric and Adverse Selection ; Informational Asymmetric and Adverse Selection ;

Assumptions ;

I.Many identical potential firms that can hire workers .

II.Constant return to scale technology . Labor is the only input .

III.Firms are risk neutral , Expected Profit maximize and price taker

IV.Price of the firm’s output equals to one .

V. θi is the expected level of productivity for each labor i.

VI.[θL , θH ] RR denotes the set of possible worker productivity levels ls where 0 ≤ θL ≤ θH ≤

VII.F(θ) = proportion of workers with productivity of θ or less .

VIII. there are at least two types of workers . Total number of workers is N

IX. workers seek to maximize the amount of income they earn .

X. r(θ) is the opportunity cost to a worker of type θ of accepting employment .


SCREENING


1-a- Competitive equilibrium when worker’s productivity level is 1-a- Competitive equilibrium when worker’s productivity level is observable.observable.

I. There is a distinct equilibrium wage w* (θ) for each type of worker θ.

II . Given the competitive and constant return to scale nature of the firm in the equilibrium we will have w* (θ) = VMP = (1) (θ) = θ

As would be expected from the first fundamental welfare theorem , this competitive equilibrium is Pareto-Optimal . Since the perfectly competitive equilibrium is prevailed and in this relation wage rate is equal to the value of marginal product . Workers are willing to be employed if the opportunity cost of getting employed for them is less than the wage they will be paid. So when productivity level is observed , each worker will be paid based her VMP and the result is pareto-optimal.


SCREENING

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening 1-b- Competitive equilibrium when worker’s productivity level is not 1-b- Competitive equilibrium when worker’s productivity level is not observable observable

I.When worker’s type in terms of their productivity is not observable , the wage rate should be independent of a worker’s type and we will have a single wage rate for all the workers.

II.The set of worker types who are willing to accept employment at wage rate w are those ; (w) = { θ:r(θ)≤ w}

III.If a firm believes that the average productivity of workers who accept employment is μ , its demand for a labor is given by z(w) :

z(w) = 0 , if μ < w

z(w) = [0 , ] , if μ= w

z(w) = ∞ , if μ > wIV.If worker type in the set are accepting employment offers in a competitive equilibrium . Then μ= E[θ ; θ ]

V.This implies that demand for labor will be equal to supply of labor if and only if μ= w = E [θ ; θ ]


SCREENING

**

*

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening definition 13.B.1:definition 13.B.1:

In the competitive labor market model with unobservable worker productivity levels , a competitive equilibrium is a wage rate w* and a set

worker’s types who accept employment such that ;

condition 13.B.6

The above condition involves rational expectations on the part of the firms. That is , firms correctly anticipate the average productivity of those workers who accept employment in the equilibrium .

Asymmetric Information and Pareto Inefficiency : Asymmetric Information and Pareto Inefficiency :

typically a competitive equilibrium defined in the 1-a section will fail to be pareto-optimal in Asymmetric Information case .

Suppose that r(θ)= r for all θ . And F(r) = (0,1). That is ; pareto optimal allocation of labor has workers with θ ≥ r accepting employment at a firm , and those with θ < r not doing so .


SCREENING

** *

* *

{ : ( ) }

[ ]

r w

w E


If we consider a competitive equilibrium when r(θ) = r for all θ . The equilibrium wage rate (w* ) should be equal to the expected productivity level which is E[θ]. So if E[θ] ≥ r , then all the workers accept employment at a firm , and if E[θ] < r , then none will accept . Which type of equilibrium arises depends on the relative fraction of good and bad workers.

a- If there is high fraction of low-productivity workers , because firms cannot distinguish good from bad one , they will be unwilling to hire any workers at a wage rate ( at least r )that is sufficient to have them accept employment . Too few workers are employed compared to pareto- optimal allocation.

b- on the other hand if there are very few low productivity workers , then the average productivity of workers will be above r , and so the firms will be willing to hire workers at a wage rate that they willing to accept . Too many workers are employed compared to pareto-optimal allocation.

Cause of this failure of competitive allocation to be pareto-optimal is the inability of firms to distinguish among workers of different productivities which is the reason why market cannot allocate workers efficiently between firms and the best alternative situation .


SCREENING


Adverse selection and market unraveling Adverse selection and market unraveling r(θ) rises with θ . Adverse selection is said to occur when an informed individual trading decision depends on her unobservable characteristics in a manner that adversely affects the uninformed agent in the market When a less capable workers are willing to accept a firm’s employment offer at any given wage . Adverse selection can have a striking effect on market equilibrium . To see the power of adverse selection, suppose that r(θ)≤ θ for all θ, and pareto-optimal allocation is prevailing. And r(.) is a strictly increasing function . Which means that workers who are more productive at firm are also more productive at home . 13 MAS_COLLEL CH8



It is this assumption that generates adverse selection . Because the payoff of home production is greater for more capable workers , only less capable workers accept employment at any given wage rate . The expected value of worker productivity depends on the wage rate . As wage increase , more productive workers become willing to accept employment at a firm , and average productivity of those workers accepting employment will increase .F(.) has an associated density function f(.) with f(θ) >0 for all θ . This assures that the average productivity of those workers willing to accept employment , E[θ ; r(θ)≤w], varies continuously with wage rate . Equilibrium wage rate = w* which is equal to ; w* = E[θ; r(θ)≤ w* ]

89ADVERSE SELECTION SIGNALING

SCREENING

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening We can see the equilibrium in the following figure :


SCREENING

W

45o

E[θ; r(θ)≤ w ] = expected value of θ who would choose to work for a firm when the prevailing wage is w . Has a minimum value of θ when w= r(θL ) and maximum when w ≥ r(θH ) .

θHθL

r (θH) r (θL ) w*

w*

θH

θL

E[ θ ]

Competitive equilibrium at which w* = E[θ; r(θ)≤ w* ]All of these workers have average productivity equal to w* .

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening We can see from the figure that the market equilibrium need not

be efficient .

To get the best workers to accept employment at a firm , we need the wage to be at least r(θH ). But according to the figure , because of the inability of the firms to observe the productivity of the workers , they could have an expected output of E[θ] < r(θH ) from each worker . This makes the firm loose . The presence of the enough low productivity workers therefore forces the wage down below r(θH ) and best workers are driven out of the market . Therefore the average productivity of the workers will fall . Therefore firms are willing to pay further lower wages . This will follow with the next best workers driven out of the market . How far this process can go ? This process can continue until only type θL (worst type) accept employment with w* = θL . But in this case the firm may not be willing to employ any labor . 13 MAS_COLLEL CH11


Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening The competitive equilibrium which is shown in the page 10 need not to be unique . The expected average productivity curve may intersect the 45 degree

line in more than one point .


SCREENING

W1* W2

* W3* r(θL )

θL

θH

θH

W’

E[θ; r(θ)≤ w ]

Note that the equilibrium in this figure can be pareto ranked . Firms can earn zero profits in any equilibrium , and workers are better off if the wage rate is higher . Thus the equilibrium with the higher wage pareto dominates all the others . The low wage , pareto –dominated equilibrium may arise because of a coordination failure. The wage is too low because firms expect that the productivity of workers accepting employment is poor and at the same time , only the bad workers accept employment precisely because the wage is low .

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening A game theoretical approach Consider the equilibrium wage rate w2

* on page 12 . In this equilibrium the firm with a small increase in the wage rate to w’> w2

* would raise its profit because it would attract workers with average productivity E[θ ; r(θ)≤w’]>w’ ( average productivity is above 45 degree line ) . This could be repeated for any equilibrium point like w1

* .

Hence it seems unlikely that a model in which firms could change their offered wages would ever lead to this equilibrium (w1

* or w2* )

outline .

To be more formal about this idea , we could consider the more formal game theoretical model ; in stage 1 two firma simultaneously announce their wage offers. Then in stage 2 workers decide whether to work for a firm , and , if so , which one .

13 MAS_COLLEL CHADVERSE SELECTION SIGNALING SCREENING13

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Constrained pareto-optimal and market intervention Constrained pareto-optimal and market intervention

presence of asymmetric information often results in market equilibrium that fails to be pareto optimal . Central authority who knows all the agent private information and can engage in lump-sum transfers among agents in the economy , can achieve a pareto improvement over these outcomes .

In practice , however , a central authority may be no more able to observe agents’ private information than market participant . So the authorities will face additional constraints in achieving the pareto optimality.

For pareto improving market intervention to be possible in this case , a more sophisticated test must therefore be passed .

An allocation that can not be pareto improved by an authority who is An allocation that can not be pareto improved by an authority who is unable to observe agent’s private information is known as a unable to observe agent’s private information is known as a

constrained ( or second-best ) pareto optimum constrained ( or second-best ) pareto optimum .We might as well simply think of intervention schemes in which authorities run the firm

herself and tries to achieve the pareto improvement for the workers . Since they can not distinguish among different types of workers , any differences in lump-sum transfer to or from a worker can depend upon whether the worker is employed .

13 MAS_COLLEL CHAVERSE SELECTION SIGNALING SCREENING14

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening The authorities offering a wage of we to those accepting employment , and

unemployment benefit of wu to those who do not accept the employment .

Can the competitive equilibria of our adverse selection model be pareto-improved upon in this way .

Consider first the competitive equilibria that are pareto dominated by some other competitive equilibria . The equilibrium with wage rate w1

* shown in the figure in page 12 . A central authority who is unable to observe workers types can always implement the best competitive equilibrium . The authorities need to set we = w* , the highest competitive wage , and wu =0 . All workers whose their opportunity cost is less than w* will be employed in the firm and the autorities could balance the budget .

Thus , the outcome in such an equilibrium is not a constrained one , since the planner is essentially able to step in and solve the coordination failure that is keeping the market at the low wage equilibrium .


Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening what about the highest-wage competitive equilibrium; as it could be

proved any such equilibrium is a constrained pareto optimum in this model .

If all workers are employed in the highest wage competitive equilibrium then the outcome is fully pareto-optimumal. So , suppose some are not employed . Note that for any wage we and unemployment benefit wu offered by the central authorities the set of worker types accepting employment has the form [θL , θH ] for some

[ it is {θ ; wu + r(θ) ≤ we ]

Suppose , then, that the authorities attempts to implement an outcome in which worker types θ ≤ accept employment . To do so, she must choose ;

wu + r( ) = we

To balance the budget we should have ;

we F( ) + wu (1- F( )) =

13 MAS_COLLEL CHADVERSE SELECTION SIGNALING

SCREENING16

( )H

L

f d

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening from these two functions we could find :

Or equivalently :

(1)

(2)


ˆ ˆ ˆ( ) ( ) ( ) ( )H

Luw f d r F

ˆ ˆ ˆ( ) ( ) ( )(1 ( ))

H

Lew f d r F

ˆ ˆ ˆ ˆ( ) ( )( [ ] ( ))uw F E r ˆ ˆ ˆ ˆ ˆ( ) ( )( [ ] ( )) ( )

( )ˆ, [ ]

ˆ( )

H

L

ew F E r r

f dwhere E

F

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening If θ* denotes the highest worker type who accept employment in the highest wage competitive equilibrium . We know that ;

r(θ* ) = E[θ; θ≤ θ* )] . Substituting this into (1) and (2) we will find that ;

wu (θ*) =0 , and we (θ*) = r(θ* ) . Thus when the authorities set we will get exactly the same result as in the highest-wage competitive equilibrium .

It could be examined and see that a pareto improvement is not possible by setting . Indeed in the adverse selection model the authority in unable to create a Pareto Improvement as long as the highest wage competitive equilibrium is the market outcome .

More generally , whether pareto-improvement market intervention is possible in situation of asymmetric information depends on the specific of the market under study and on which equilibrium result .


*

*

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening SignalingSignaling

one might expect mechanism to develop in the market place to help firms distinguish among workers . This is plausible , since both firms and high-ability workers have incentives to do this task. The mechanism that will be examined called signaling in which the high-ability workers may have actions they can take to distinguish themselves from low abilities.

Consider the following model of two types of workers with productivities of θL and θH . And λ = prob (θ=θH ) . Before entering the market a worker can get some education and the amount of education is observable . Education does nothing for a worker’s productivity .it is assumed that both cost and marginal cost of education are assumed to be lower for high-ability workers. U(w, e ; θ) denote the utility of type θ with his wage equal to w and the education level equal to e . If c(θ ,e ) is the cost of education of type θ worker ,then ; U(w, e ; θ) = w - c(θ ,e ) . r(θ) is the opportunity cost of the worker of type θ


Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening we shall see that this otherwise seemingly useless education may

serve as a signal of unobservable worker productivity. Equilibrium emerge in which high-productivity workers choose to get more education than low productivity workers and firms correctly take differences in education levels as a signal of ability . It should be kept in mind that worker’s welfare may be reduced if they are compelled to engage in a high level of signaling activity .

For simplicity suppose that r(θ) =0 for both type of workers . The unique equilibrium that arise under this assumption and in the absence of ability to signal is that all workers will be employed at a w = E[θ] and it is pareto efficient . It will be shown that how signaling may create inefficiencies . It will be shown that how alternative assumption about the r(.) function may generate efficiencies under signaling .


Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening initially , a random move of nature determine whether a worker is of high

or low ability . Then , conditional on her type , the worker choose how much education to obtain . After obtaining her chosen educational level , the worker enters the market . Conditional on the observed educational level of the worker , two firms simultaneously make wage offers to her . Finally , the worker decide whether to work for a firm , and , if so, which one .

In contrast with the model in which r(.) is rising , in this model here we explicitly model only a single worker of unknown type . The equilibrium that will be followed is that of a weak perfect Bayesian equilibrium . The firm believes that for each possible choice of education e , there exist a number μ(e) є [ 0 , 1] . Such that ; first ; firm’s 1 belief that the worker is of type θH after seeing her choice e is μ(e) . And second ; after the worker has chosen e , firm 2’s belief that the worker is of type θH and that firm 1 has chosen wage offer w is precisely μ(e) σ1

*

(w ; e) , where σ1* (w ; e) is firm’s 1’s equilibrium probability of

choosing wage offer w after observing education level e .


Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening A set of strategies and a belief function μ(e) є [ 0 , 1] giving the firms common probability assessment that the worker is of high ability after observing education level e is a PBE if;

1 -the worker’s strategies is optimal given the firms strategies.

2- the belief function μ(e) is derived from the workers strategy using Bayes’ rule where possible .

3- the firms’ wage offer following each choice “ e “ constitute a Nash equilibrium of the simultaneous –move wage offer game in which the probability that the worker is of high ability is μ(e) .

Suppose that after seeing some educational level e , the firms attach a probability of μ(e) that the worker is of type θH . In this case the expected productivity of the worker is μ(e) θH + ( 1- μ(e)) θL .

How about the worker’s equilibrium strategy ? Her choice of equilibrium depends on her type . The following figure shows an indifference curve for each type of the two workers with the single crossing property . It arises here because the worker’s marginal rate of substitution between wage and education is decreasing in θ . That is

(dw/de)u=cte = ce (e , θ) is decreasing since ceθ (e , θ) < 0 .


SCREENING22



w

e

ˆ ˆ{( , ) : ( , ) ( , )L Lw e u w e u w e

ˆ ˆ{( , ) : ( , ) ( , )H Hw e u w e u w e

e

w Note that since in any PBE w(e) = μ(e)θH + (1- μ(e)) θL for the equilibrium belief function μ(e) , the equilibrium wage offer resulting from any choice of e must lie in the interval [ θL , θH )] . A possible wage offer curve can be shown in the following figure . w

e

θH

θL

W(e)

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening separating eqilibrium separating eqilibrium

Lemma 1 ; in any separating equilibrium , w* (e* (θH )) = θH and w* (e* (θL )) = θL , that is , each worker type receives a wage equal to her productivity level .

This implies that based upon seeing education level e*(θL ) or e* (θH ) , firm must assign probability one to the worker being type θL or θH . The resulting wage are then exactly θL or θH .

Lemma 2 ; in any separating equilibrium , e* (θL) =0 ; that is , a low-ability worker chooses to get no education .

This implies that if low ability worker can choose any education level e’ and know that he will be paid θL , why not choosing the zero education level . So the low ability indifference curve should cross the vertical axis as it is shown in the following figure .

using the following figure we could construct a separating equilibrium as follows ;


Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening --


w*(e*(θH)) = θH

w*(e*(θL)) = θL

Type θL Type θH

e*(θL )= 0 e*(θH)) = e’

w* (e)

w

e

The firms’ equilibrium beliefs following education choice e are μ*(e) = (w*(e) - θL)/(θH - θL) , we should note thatμ*(e) є [0,1] for all e ≥0 , since w*(e) є [θL , θH ] . To verify that this is indeed a PBE , note that we are completely free to let firms have any belief when e is neither 0 or e’ .

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening what about the worker’s strategy ? Given the wage function w* (e) , the

worker is maximizing her utility by choosing e = 0 when she is type θL and by choosing e=e’ when she is type θH . Because we have so much freedom to choose the firms’ belief off the equilibrium path , many wage schedules can arise that support these education choices. The following figure shown another equilibrium . Firm’s believe that the worker is certain to be of high quality if e>e’ , and is certain to be of low quality if e<e’ . The resulting wage schedule has w* (e) = θH if e>e’ and w* (e) = θL if e<e’ .

the fundamental reason that education can serve as a signal here is that the marginal cost of education depends on a worker’s type in a sense that the marginal cost of education is higher for a low-ability worker .






w*(e*(θH)) = θH

w*(e*(θL)) = θL

Type θL Type θH

e*(θL )= 0 e*(θH)) = e’

w* (e)

w

e

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Screening

in this section , we consider an alternative market response to the problem of unobservable worker productivity in which the uninformed parties take steps to try to distinguish or , screen, the various types of individuals on the other side of the market. This possibility first studied by Rothschild and Stigltiz and Wilson in the context of insurance market . Assumptions;

1-two types of workers ; θL , θH with θH > θL > 0

2-fraction of workers who are of type θH = λ є ( 0 , 1) .

3-opportunity cost for both type of workers (best alternative) = 0

4-jobs may differ in task level required of the worker . For example jobs could differ in the number of hours per week that the worker is required to work. Or the task may represent the speed at which the production line is run in a factory. Higher task level adds nothing to the productivity of the worker ,rather , the only effect is that to lower the utility . The output of a type θ is therefore θ regardless of worker’s task (task acts like education in the signaling section. Workers signal their quality by their education

level and employers screen the workers by different task levels)


Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Utility of type θ worker = U(w , t ; θ ) = w – c(t , θ ) where w is the wage

rate , t is the task level and c(t , θ ) is disutility of task level t for worker of type θ . Like the role of education in signaling model we assume that ; C(0,θ) =0 , Ctt (t , θ ) >0 , Ct ( t , θ ) >0 , Cθ (t , θ)<0 , Ctθ (t , θ)<0

Here we study the Pure Sub-game Perfect Nash Equilibrium of the following type ;

stage 1 ; two firms simultaneously announce set of offered contracts ( w , t )

stage 2 ; given the offers made y the firms , workers of each type choose whether to accept a contract , and , if so, which one . If the worker is indifferent between two contracts , she always chooses the one with lower task level. If a worker’s most preferred contract is offered by both firms, she accept each firm’s offer with probability 1/2 . Different types of the workers may then end up choosing different contracts.


SCREENING29

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Case 1 ; Worker’s type is observable. . The firm can offer a contract ( wL , tL )

solely to type θL and ( wH , tH ) solely to type θH .

Preposition 1 ; In any SPNE model of the screening game with observable workers types, a type θi worker accepts contract (wi

* , ti* ) = ( θi , 0 )

and the firm earn zero expected profit .

proof ; zero expected profit → wi* = θi .

if wi* > θi the firm makes a loss and will not offer any contract .

if wi* < θi , then there will be an aggregate profit Π>0 earned by the two firms on

workers of type θi . Supposedly each will not earn more than Π/2 . Competition will wiped out this excessive profit. In a sense that each of the firms will try to absorb all the workers by increasing the wage by a very small amount . Increasing the wage in this way will equalize wi = θi .

Now suppose that (wi* , ti

* ) = (θi , t’ ) for some t’ >0 . As shown in the following figure either firm could deviate and earn strictly positive profit by offering a contract in the shaded area of the figure such as ( w’ , t’ ). The competition will wipe out this profit. The only contract at which there are no profitable deviation is (wi

* , ti* ) = ( θi , 0 )


Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening observable productivity;


Type θi indifference curve

W

t

(θi , t’ )

)w’ , t( ’

(wi* , ti

* )

θi

This is a higher indifference curve for type θi . So workers of type θi will be attracted to new contract ( w’ , t’ ). This new contract will be profitable for the firms because w < θi .

There can not be any indifference curve higher than this one.

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Case 2 ; worker’s type are not observable.

Complete information outcome indentified in the previous section can not be concluded in this cased . Firms can not offer two types of contracts to both types of workers and let them to choose the , because in this case every low ability worker prefer the high ability contract (θH,0 ) to low ability (θL,0 ) contract and firm will end up loosing money

Lemma 1 ; in any equilibrium, whether pooling or separating , both firms must earn zero profit.

Proof ; suppose that the two firms aggregate profits are Π>o . One firm must be making no more than Π/2 . Any firm could deviate and offer a very small amount over the wage levels to both types of the workers , ( L and H ) , this will attract all the workers . In this way the competition will wipe out any profit which is remained in the market . So in any equilibrium no firm can have a deviation that allows it to earn strictly positive profit.

Lemma 2 ; No pooling equilibrium exist .

Suppose that there is a pooling equilibrium contract (wp , tp) as shown in the following figure .




Type θH

Type θL

(wp , tp)

θH

θL

E(θ)

)w’ , t( ’

t

w

Suppose that firm j offers this contract

Firm K could offer this contract in the shaded region and attract all the θH workers and non of the θL workers. And this firm makes profit since w’ < θH . So (wp , tp) is not an equilibrium .

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Lemma 3 ; if ( wL , tL) and (wH , tH ) are the contracts signed by

the low and high ability workers in a separating equilibrium, then both contracts yield zero profit. That is , wL =θL and wH = θH . In other words all the contracts accepted in the separating equilibrium must yield zero profit.

Proof ; suppose that wL < θL . Either firm could earn strictly positive profit by offering the contract (w’ , t L ) where wL < w’ < θL . All the low ability workers accept this contract , and the deviating firm makes a profit from any worker . Since by lemma 1 there should not be such a deviation in the equilibrium , we should have wL = θL .

Now suppose that wH < θH . As it is shown in the figure type θL contract must lie somewhere in the blue area . To see this we note that since type θH worker choose contract (wH , tH ) , then contract ( wL , tL) must lie on or below the type θH indifference curve through (wH , tH ).




w

t

θH

θL

)wL , tL(

)wH , tH (

Type θL type θH

)w’ , t( ’

Suppose that firm j propose the ( wL , tL)Contract to low ability workers and (wH , tH )To high ability workers. Then firm k could propose a contract in the green region like ( w’ , t’ ) and attract all the high ability workers and none of the low ability workers . Firm k still is making profit and the previous separating equilibrium is no ore valid . So wH = θH

By the first lemma we should have wH = θH and also we

should have wL = θL

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Lemma 4 : in any separating equilibrium , the low ability workers

accept contract (θL , 0 ) ; that is they receive the same contract as when no informational imperfections are present in the market.


w

t

θH

θL

Type θHType θL

)wH , tH (

)w’ , t( ’

)θL, t’L (

Suppose that this point is the accepted contract .

Then a firm can make strictly positive profit by offering only a contract lying in the shaded area such as ( w’ , t’ ) . All low ability workers accept this contract , and the contract yields the firm strictly positive profits from any worker who accept it . So the only point which this move is not possible is when tL = 0

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Lemma 5 ; In any separating equilibrium , the high ability workers

accept contract (θH , t”H ) which satisfies θH - c (t”H , θL ) = θL - c( 0 , θL ) .


θH

θL

(wH , tH )

( wL , tL)

w

tt”H

)w’ , t( ’

Type θHType θL

As is seen ( wL , tL) = (θL , 0 ) and that wH = θH .if type θL are willing to accept the (θL, 0) contract tH must be as large as t”H . Suppose that high ability contract is such that tH

> t”H . Then either firm can earn positive profit by also offering a contract lying in the shaded region of the figure such as ( w’ , t’ ) . This contract attracts all the high ability workers and does not change the choice of low ability workers. Thus in any separating equilibrium the high-ability contact must be (θH , t”H ).

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Proposition 2 . In any sub-game Perfect Nash Equilibrium of the

screening game , low ability workers accept contract (θL , 0 ) and high ability workers accept contract (θH , t”H ) , where

θH - c (t”H , θL ) = θL - c( 0 , θL ) .


θL

θH

Type θLType θH

(wH , tH )

( wL , tL)

w

t

E(θ)

Suppose that both firms are offering two contracts as shown in the figure . No firm can earn strictly positive profit by deviating in a manner that attracts either only high or low ability workers . But not a contract which attract all the workers. A contract will attract all the workers if it lies in the shaded area . But this region is not profitable for the firm since it lies above the breakeven line (E(θ)).t”H

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening However if some of the shaded area lies below the breakeven line then

there will be a profitable deviation and no equilibrium would exislt .




θL

θH

Type θLType θH

(wH , tH )

( wL , tL)

w

t

E(θ)

( w’ , t’ )

This contract will be profitable and could demolish the old equilibrium .

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening Even when no single pooling contract breaks the separating equilibrium, it is

possible that a profitable deviation involving a pair of contracts may do so. For example a firm can attract both type of workers by offering the contract s (w”L , t”L ) and ( w”H , t”H ) as shown in the following figure .



13 MAS_COLLEL CHADVERSE SELECTION SIGNALING SCREENING40 13 MAS_COLLEL CHADVERSE SELECTION SIGNALING SCREENING40

θL

θH

Type θL Type θH

(wH , tH )

( wL , tL)

w

t

(w”L , t”L )

( w”H , t”H )

As it is seen both type of workers prefer the new contracts over the old ones . Because it raises their utility . These new contracts will be profitable for the firm if the breakeven line lies in such away to bring profit for the firm . (above the wage level for high ability worker)

Adverse Selection , Signaling , ScreeningAdverse Selection , Signaling , Screening welfare properties of the screening equilibrium

in the screening model high ability worker ends up signing contracts that make them engage in completely unproductive and disutility-producing tasks merely to distinguish themselves from their less able counterparts .

As in the signaling model , the low ability worker are always worse off when screening is possible than when it is not . One difference from the signaling model is that in cases where an equilibrium exist, screening must make the high-ability workers better off precisely in those cases where it would not that a move to a pooling contract breaks the separating equilibrium . ( shown in page 40 ) indeed it is a constrained pareto-optimal outcome .


Adverse Selection, Signaling, Screening Introduction : One of the implicit assumption of the...

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