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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
Advanced Microeconomics
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.
Advanced Microeconomics
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
Advanced Microeconomics
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.