Adverse Selection,Signaling, Screening

Adverse Selection, Signaling, Screening


Timo Grote

December 1, 2013


1 Introduction

2 Adverse Selection

3 Signaling

4 Screening

5 Conclusion


Introduction

Motivation

Where do we have perfect information in our daily life?

How important are information assymetries in buisnessand economics?

Typical Examples:

Second-hand markets (used car market)Labor marketinsurance marketcredit market in developing countries


Introduction

Motivation

The economics of information were developed in the 70swith:

”The market of Lemons”(1970) by George AkerlofMichael Spence´s ”Job Market Signaling”(1972)Joseph Stiglitz essay: ”The Theory of Screening,Education and the Distribution of Income” (1975)

These Contributions were awarded with the nobel prize in2001


Adverse Selection

The Model

Adverse Selection: exists whens information betweeninteracting units are assymetrically distributed and determinewelfare inefficient market outcomesAssumption of an enhanced Akerlof model (1970):

Many identical (potential) firms hiring workersIdentical constant returns to scale technology, with laboras the only input=output: θMaximization of profits under risk neutrality andpricetaking behaviourWorkers differ in productivity level θ ∈ (θ, θ) ⊂ R with0 ≤ θ < θ <∞Distribution function F (θ) associated with the continousdensity function f (θ) = F 8(θ) > 0Workers can earn r(θ) when they work at homeTotal number of workers: N


Adverse Selection

The Model

Worker´s perspectiveare maximizing their profits facing a trade off betweenworking in a firm or at homeas productivity is publically observable workers work infirms for w ∗(θ) = θnumber of workers employed is {θ : r(θ) ≤ θ}The sum of aggregate surplus:∫ θ

θ

N[I (θ)θ + (1− I (θ))r(θ)]dF (θ) where I (θ) is either 0

or 1Sum of agregate surplus is maximized if:

I(θ)=

{1 if θ ≥ r(θ)0 otherwise

if the given condition holds an equilibrium is attained


Adverse Selection

The Model

If worker productivity is not observable:w(θ) = w irrespective of type θ

now firms are only hiring workers under imperfectinformation

Definition 13.B.1:In the competitive labor market model with unobservableworker productivity levels, a competitive equilibrium is a wagerate w ∗ and a set Θ∗ of worker types who accept employmentsuch that

Θ∗ = θ : r(θ) ≤ w ∗ Condition (13.B .4)

w ∗ = E [θ|θ ∈ Θ∗] Condition (13.B .4)


Adverse Selection

Pareto Inefficiency

Assume that r(θ) = r and F (r)ε(0, 1)

In an equilibrium Θ∗ is either [θ, θ] for w≥r or 0 for w< r

As the allocation of workers is not distinctive the ocurringsituation is pareto inefficient


Adverse Selection

Competitive equilibrium with adverse selection

Additional Assumption: r(θ) varies with θ and r 8 > 0

The result is a competitive equilibrium with adverseselection(Figure 13.B.1):


Adverse Selection

Complete market failure

One extreme case of adverse selection is that just lowability workers are accepting w

The result is complete market failure(Figure 13.B.2):


Adverse Selection

Multiple competitive equilibria

Another extreme case are multiple competitive equilibriaas the firm offers wages with a function E [θ|r(θ) ≤ w ]The slope of the function varies with the density ofworkers accepting the wage (Figure 13.B.3):


Adverse Selection

Constrained Pareto Optima and Market

Intervention

The Constrained(Second-best) Pareto optimum is marketequilibrium that can not be improved by an external agent

Proposition 13.B.2: In the adverse selection labor marketmodel (where r(.) is strictly increasing with r(θ) ≤ θ for allθ ∈ [θ, θ] and F (.) has an associated density f(.) withf (θ) > 0 for all θ ∈ [θ, θ]), the highest-wage competitiveequilibrium is a constrained Pareto optimum


Adverse Selection


Intervention

Intuition:

If government could observe θ, it could implement a fullyPareto optimal equilibrium

If all workers accept employment in the highest wageequilibrium it is fully (and constrained)Pareto optimal

To move all workers to become employed in the highestwage equilibrium can implement w e = w ∗ = r(θ∗) andwu = 0

Aditionally the unemployed can be taxed to force them towork wu < 0 (which is not Pareto improving)


Adverse Selection


Intervention

As the Pareto Improvement is not possible, the aggregatesurplus can be increased by

scheduling that every worker accepts employment

firms pay wage w = E [θ]

which leads to the social welfare function:∫ θ

θ

N[I (θ)θ + (1− I (θ))r(θ)]dF (θ)

Whether an allocation is a constrained optimum dependon the point at which the welfare evaluation is conducted


Signaling

Model

Signaling: is the idea that the better informed partycredibly conveys the other one of its quality underasymmetrical informationModel assumptions

Same assumptions as beforeBut only two types of workers:θH , θL with θH > θL > 0and λ = Prob(θ = θH ∈ (0, 1))Workers receive education level e for cost of c(e,θ) withc(0, θ) = 0, ce(e, θ) > 0, cee(e, θ) > 0, cθ(e, θ) <0 for alle > 0 and ce,θ < 0No human capital effectUtility:

u(w , e|θ) = w − c(e, θ)

Home production r(θ) = 0 for all θ


Signaling

Worker allocation process with signaling

Allocation of workers works as follows

1 Nature determines type of productivity2 Worker choosing e dependent on his type3 Firm makes offer conditional on e4 Worker chooses rationally the best offer

Result: Perfect Bayesian equilibrium


Signaling

Result

A set of strategies and a belief function µ(e) ∈ [0, 1]giving the propability assesment that the worker has ahigh ability given e is a PBE

The worker´s strategy is optimal given the firm´sstrategyThe belief functionµ(e) is derived from the worker´sstrategy using Bayes ruleThe firms wage offers determined by e constitute a Nashequilibrium of the simulatenous-move wage offer game inwhich the propability of a high ability worker is µ(e)


Signaling

The indifference curves cross once which is called thesingle crossing property

Firms offer a wage w(e) = µ(e)θH + (1− µ(e))θL


Signaling

Seperating Equilibrium

Seperating equilibrium: Worker types choose differenteducation levels

Lemma 13.C.1:In any seperating perfect Bayesian equilibrium the optimalwages are:

w ∗(e∗(θH)) = θHw ∗(e∗(θL)) = θL


Signaling

Lemma 13.C.2: In any seperating perfect Bayesian equilibriume∗(θL) = 0

Since education is shows productivity and is costlyL typ choose e = 0H type choose e > 0


Signaling

Welfare implications

Firms always make zero profits

Workers with productivity θL are worse off

Wokers with productivity θH are either better or worse off

High ability workers are worse off if

In a seperating equilibrium (0,E [θ]) is no longer availableIf they choose e=0, they expect to receive w = θLthen forbidding signaling activity is Pareto improving

If the fraction of productive workers λ increase thepropability that θL workers are worse off increases


Signaling

In a pooling equilibrium all workers choose the same levelof education e∗(θH) = e∗(θL) = e∗

The optimal equilibrium wage isw ∗(e∗) = λθH + (1− λ)θL = E [θ]

Pooling equlibria with e > 0 are PBE but the Paretodominant education level is e=0


Signaling

The intuitive criterion

Different types of equilibria can be sustained as PBE

Problem of the refinement, the ”Intuitive criterion” byCho and Kreps (1987):

The outcome of an equilibrium can only be Paretooptimal if low ability workers have no incentive to signalwronglyIf the ”Intuitive criterion” applies only the bestseperating equilibrium survives


Signaling

Even the best seperating equilibrium can be paretodominated by a state without signaling activity

As high productive workers are better off, crosssubsidization in favor of low ability workers can paretoimprove the situation


Screening

Screening: The uninformed side of the market takesobservations to reduce information assymmetryModel:

Same framework as in Screening chapter

No education signaling, but firms create jobs withdifferent task levelsTask levels do not affect output, but decrease workerutilityUtility function of θ:

u(w , t|θ) = w − c(t, θ)

with c(0, θ) = 0, ct > 0, ctt > 0, cθ < 0 and ctθ < 0


Screening

The Situation is viewed as a two-stage game:

1 Two firms simultaneously offer a set of contracts(w,t)

2 The workers decide on whether and which contract toaccept

Proposition 13.D.1: In any SPNE of the screening game withobservable worker types a type θi worker accepts contract(w ∗

i , t∗i ) = (θi , 0) and firms earn zero profits

Implications:

if w ∗ > θi firms make losses

if w ∗ < θi firms gain profits

if (w ∗i , t

∗i ) = (θi , t

p) with t p > 0 the firm could deviateand underpay to gain positive profits


Screening

Other Case: θ is unobservable for firms

Two types of equilibria:

pooling equilibriaseperating equilibria

Lemma 13.D.2 No pooling equilibria exit

In any pooling equilibrium workers of type θH would beunderpaid

Lemma 13.D.3 If (wL, tL) and (wH , tH) are the contractssigned by the low- and high-ability workers in a seperatingequilibrium, then both contracts yield zero profits; that is,wL=θL and wH=θH


Screening

Lemma 13.D.3 If (wL, tL) and (wH , tH) are the contractssigned by the low- and high-ability workers in a seperatingequilibrium, then both contracts yield zero profits; that is,wL=θL and wH=θH


Screening

Proposition 13.D.2 In any subgame perfect Nash equilibriumof the screening game, low-ability workers accept contract(θL,0) and high ability workers accept (θH , t

pH) where t pH

satisfies θH − c(t pH , θL) = θL − c(0, θL)


Screening

Welfare considerations (only equilibrium cases):

By and large the same as for signalingL types are worse offH types are better off


Conclusion

Adverse Selection:

market failure tendenciesθH workers often worse offstate intervention unrealistic

Signaling:

θH workers are better offintuitive criterion as a good instrument

Screening

θH workers are better offbut equilibrium in model can be undermined


Conclusion

References

Akerlof,G.(1970). The market for lemons: Qualityuncertainty and the market mechanism.Quaterly Journalof Economics 89:488-500Macho-Stadler, Ines, and J. David Perez-Castrillo(2001).Introduction to the Economics of Information:Incentives and Contracts. Oxford: Oxford UniversityPress.Mas-Colell, Andreu, Michael D. Whinston, and Jerry R.Green (1995). Microeconomic Theory. New York, NY:Oxford University Press.Rothschild, M. and J. E. Stiglitz. (1976). Equilibrium incompetitive insurance markets: An essayy in theeconomics of imperfect information.Quarterly Journal ofEconomics 80:629-649Spence, A. M. (1974). Market Signaling.Cambridge,Mass: Harvard University Press.

Adverse Selection,Signaling, Screening

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