Advanced Microeconomics

Advanced Microeconomics

ECON5200 - Fall 2014

Advanced Microeconomics

Adverse Selection

Adverse SelectionGeneral Framework

I q ∈ Q and θ ∈ Θ ≡[θ, θ]distributed according to F (·);

I Agent’s preference U (q, t; θ) = V (q, θ)− t;

I Principal’s preference t − C (q, θ);

I P’s problem under full info:

max(t ,q)

t − C (q, θ)

s.t.:V (q, θ)− t ≥ 0

I FB allocation is Cq(qFB (θ) , θ

)= Vq

(qFB (θ) , θ

)and

tFB (θ) = V(qFB (θ) , θ

).

Advanced Microeconomics

Adverse Selection

Adverse SelectionGeneral Framework: Implementability

I Let r (θ) ≡ V (q (θ) , θ)− t (θ) the agent’s rent;I Then r (θ) ≥ 0 and r (θ) = maxθ̃ V

(q(θ̃), θ)− t

(θ̃);

I Spence-Mirrlees condition (i.e. single-crossing property)Vqθ (q, θ) ≥ 0 for each q, θ;

TheoremIf single-crossing property holds, then (q (·) , r (·)) is incentivecompatible iff qθ (θ) ≥ 0

r (θ) = r (θ) +∫ θ

θVθ (q (s) , s) ds

Proof.(See notes!).

Advanced Microeconomics

Adverse Selection

Adverse SelectionGeneral Framework: Optimality

P’s problem under incomplete info:

max(t(·),q(·))

∫ θ̄

θ[t (θ)− C (q (θ) , θ)] f (θ) dθ

s.t.:

V (q (θ) , θ)− t (θ) ≥ 0, ∀θ

V (q (θ) , θ)− t (θ) ≥ V(q(θ̃), θ)− t

(θ̃), ∀θ, θ̃

Advanced Microeconomics

Adverse Selection

Adverse SelectionGeneral Framework: Optimality

By using the Theorem, the P’s problem is:

max(t(·),q(·))

∫ θ̄

θ[V (q (θ) , θ)− C (q (θ) , θ)− r (θ)] f (θ) dθ

s.t.:

r (θ) ≥ 0, ∀θ

r (θ) = r (θ) +∫ θ

θVθ (q (s) , s) ds, ∀θ, θ̃

qθ (θ) > 0

Additional assumption Vθ (q (θ) , θ) ≥ 0. (See notes!).

Advanced Microeconomics

Markets and Information

Markets and Information

I Fundamental welfare theorems rely on perfect observability ofall commodities to all market participants;

I Often in a transaction one party knows something that otherparties don’t know;

I We study three types of equilibria:

i. Adverse selection: An informed individual’s trading decisionsadversely affects uninformed market participants;

ii. Signaling : Informed individuals signal information about theirunobservable knowledge through observable signal;

iii. Screening : Uninformed parties develop mechanisms to screeninformed individuals with different information.

Advanced Microeconomics

Markets and Information

Adverse Selection (Akerlof, 1970)

I Consider a labor market with many identical potential firms;

I Firms want to hire workers to produce final good by adoptinga constant return to scale technology with labor as the soleinput;

I Firms are risk-neutral, price-taker and profit maximizer;

I The price of the firm’s output is equal to one;

I Workers differ in productivity, θ ∈[θ, θ]distributed according

to F (θ);

I r (θ) is the reservation wage of workers, i.e. the gain theymight obtain by working at home.

Advanced Microeconomics

Markets and Information

Adverse SelectionPerfect Observability

I Due to the assumptions of perfect competition and CRS thefirms would set a wage equal to:

w ∗ (θ) = θ

I Only the workers with {θ|r (θ) ≤ θ} would accept the offer;

I The equilibrium is Pareto optimal.

Advanced Microeconomics

Markets and Information

Adverse SelectionImperfect Observability

DefinitionIn a competitive labor market with unobservable productivitylevels, a competitive equilibrium is the pair {w ∗,Θ∗} such that:

Θ∗ = {θ|r (θ) < w ∗}w ∗ = E (θ|θ ∈ Θ∗)

I The equilibrium is characterized by a fixed point;

I Typically this equilibrium will not be Pareto optimal.

Advanced Microeconomics

Markets and Information

Adverse Selection

I Suppose r (θ) = r and F (r) ∈ (0, 1);

I The Pareto optimal allocation is: if θ ≥ r accept employmentand if θ < r not accept;

I If w > r then Θ∗ =[θ, θ], if w < r then Θ∗ = ∅;

I In both cases w = E (θ): Firms are unable to distinguishamong workers’productivity. Thus, the equilibrium outcomeis ineffi cient:

i. If the share of good workers is large enough, then E (θ) > rand too many workers are hired;

ii. If the share of bad workers is large enough, then E (θ) < r andtoo few workers are hired.

Advanced Microeconomics

Markets and Information

Adverse Selection

I Adverse selection occurs when an informed individual’sdecision depends on her unobservable characteristics andadversely affects the uninformed agents;

I In the labor market context, adverse selection arises when onlyrelatively less capable workers accept a firm’s employmentoffer at any given wage;

I If r(θ) is no longer constant adverse selection arises(specifically if it is increasing!).

Advanced Microeconomics

Markets and Information

Adverse SelectionSuppose r(θ) ≤ θ for all θ and r ′(θ):

Advanced Microeconomics

Markets and Information

Adverse SelectionPossibility of multiple equilibria:

Advanced Microeconomics

Markets and Information

I Both the firms and the high-ability workers have incentives totransmit information;

I Market responses to the problem of adverse selection:

i. Signaling : Informed individuals (workers) choose their level ofeducation to signal information about their ability touninformed parties (the firms);

ii. Screening : Uninformed parties (firms) take steps to screen thevarious types of individuals on the other side of the market(workers).

Advanced Microeconomics

Markets and Information

Signaling (Spence 1973, 1974)

I Two types of workers: high ability, θH , and low ability, θL;

I The probability of high type workers ( exogenous share in thepopulation) is λ ∈ (0, 1);

I Perfect competitive markets (zero profits), whose profit’sfirms is π = θ − w .

Advanced Microeconomics

Markets and Information

SignalingFull Information

I Bertrand competition outcome;

I (wL,wH ) such that wH = θH and wL = θL;

I If abilities are not observable, then w = E (θ|θ ∈ Θ∗) whereΘ∗ is equal to the set of types accepting the contract.

Advanced Microeconomics

Markets and Information

SignalingImperfect Observability

I Suppose that before entering the job market a worker can getsome level of education, e, and the firms observe it;

I Assume that:

i. Education does not increase workers’productivity;

ii. Education is more costly for the low ability worker than for thehigh ability worker, also at the margin (single crossingproperty);

I Workers’payoff u = w − c (e, θ) with c (0, θ) = 0,ce (e, θ) > 0, cθ (e, θ) < 0 and ceθ (e, θ) < 0.

Advanced Microeconomics

Markets and Information

SignalingWelfare

Welfare effect is generally ambiguous:

I Signaling can lead to a more effi cient allocation of workers’labor and to a Pareto improvement;

I Signaling is a costly activity and workers’welfare may bereduced.

Advanced Microeconomics

Markets and Information

SignalingTiming

I Nature determines worker’s type;

I Worker chooses an education level contingent on his type;

I Conditional on the education level, firms make wage offerssimultaneously

I Worker decides which offer to accept, if any.

Advanced Microeconomics

Markets and Information

SignalingPerfect Bayesian Equilibrium

I We use the concept of Perfect Bayesian Equilibrium;

I The worker’s strategy is optimal given the firm’s strategy;

I µ (e) are the up-dated firms’beliefs that the worker ishigh-type;

I Each firm’s wage offer, following the choice e, is optimal giventhe belief µ (e), the worker’s strategy and the other firms’strategy.

Advanced Microeconomics

Markets and Information

SignalingPerfect Bayesian Equilibrium

The firms’(pure strategy) Nash equilibrium wage offers equal theworker’s expected productivity:

w (e) = µ (e) θH + (1− µ (e)) θL

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Markets and Information

SignalingSeparating Equilibrium

I Let e∗(θ) the educational equilibrium choice of type θ andw ∗(e) the equilibrium wage of a worker who displays aneducational level of e;

I In any separating equilibrium e∗(θH ) 6= e∗(θL) andw ∗ (e∗(θH )) = θH and w ∗ (e∗(θL)) = θL;

I e∗(θL) = 0, which implies that the low ability worker receivesutility equal to θL at the separating equilibrium.

Advanced Microeconomics

Markets and Information

Signaling

Example (15 - Education and Signaling - See notes!)

I θH = 2, θL = 1, c (e, θ) = eθ ;

I Find the separating PBNE.

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Markets and Information

SignalingPooling Equilibrium

I The two types of workers choose the same level of educatione∗(θL) = e∗(θH ) = e∗;

I This implies that w ∗ (e∗) = λθH + (1− λ) θL;

I Any education level between 0 and e ′ can be sustained as apooling equilibrium. e ′ corresponds to the level of educationsuch that low type receives zero utility with w = E [θ], e ′:w − c(e ′(θL), θL) = 0.

Advanced Microeconomics

Markets and Information

Screening

I Workers’outside option is zero r(θi ) = 0;

I Jobs may differ in the “task level” t > 0 required. (Ex.different number of ours per week);

I The output of a type-θi worker is θi regardless of the worker’stask level. Higher task levels only affect workers’utility. Theyare costly for the workers;

I Workers’payoff u = w − c (t, θ) with c (0, θ) = 0,ct (t, θ) > 0, cθ (t, θ) < 0 and ctθ (t, θ) < 0.

Advanced Microeconomics

Markets and Information

ScreeningTiming

I Stage 1 : Firms simultaneously announce a menu of contracts(w , t). Each firm may announce any finite numbers ofcontracts;

I Stage 2: Workers decide whether they want to sign a contractand which one to sign:

- If indifferent between signing and not signing a contract, theworker will sign;

- If indifferent between two types of contract, the worker willchoose the contract with the lower task level.

Advanced Microeconomics

Markets and Information

ScreeningFull Observability

If abilities are observable equilibrium entails firms offering adifferent contract to each type:

I (w ∗H , t∗H ) = (θH , 0) for high ability workers;

I (w ∗L , t∗L ) = (θL, 0) for low ability workers;

I Workers accept contracts and firms earn zero profits.

Advanced Microeconomics

Markets and Information

ScreeningBreak-Even Line

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Markets and Information

ScreeningPooling Equilibria

No pooling equilibria exists:

Advanced Microeconomics

Markets and Information

ScreeningSeparating Equilibria

I If (wL, tL) and (wH , tH ) are the contracts signed respectivelyby the low- and the high-ability workers in a separatingequilibrium, then both contracts yield zero profits: wL = θLand wH = θH ;

I This implies that separating equilibria does not allow forcross-subsidies. Separating equilibria are on the break-evenlines;

I In any separating equilibrium, the low-ability workers acceptcontract (wL, tL) = (θL, 0). They receive the same contractas under full information.

Advanced Microeconomics

Markets and Information

ScreeningSeparating Equilibria

I In any separating equilibrium the high ability workers acceptcontract (wH , tH ) = (θH , t̃H ) where t̃H is such that:θH − c(t̃H , θL) = θL − c(0, θL) = θL;

I This means that low type is indifferent between contract(θH , t̃H ) and contract (0, θL);

I If tH > t̃H , firms can offer contracts which attract high abilityworkers and make positive profits.

Advanced Microeconomics

Markets and Information

ScreeningSeparating Equilibria

I A separating equilibrium exists if the pooled break-even line issuffi ciently far from θH ;

I This means that λ must be suffi ciently low;

I As in the signaling model, asymmetric information leads toPareto ineffi cient outcomes.