Adverse Selection, Signaling, and Screening in Markets

Transcription

1 BGPE Intensive Course: Contracts and Asymmetric Information Adverse Selection, Signaling, and Screening in Markets Anke Kessler Anke Kessler p. 1/27

2 Stylized Facts: Market Failure used cars, even if they are like new, sell far below their dealership price laid-off workers experience longer spells of unemployment than workers for different reasons without a job (e.g. military) private health care for the elderly is essentially unavailable young drivers pay very expensive insurance premiums what do these empirical regularities have in common? these markets are characterized by informational asymmetries and suffer from adverse selection Anke Kessler p. 2/27

3 Productivity Uncertainty in the Labor Market 2 identical firms F, 1 representative worker W worker s productivity is θ [θ, θ], private information of W, c.d.f F(θ) is continuous and has full support utilities u W = wq + r(θ)(1 q), where q = 1 if W works and q = 0 otherwise w IR = wage paid by firm r(θ) = W s reservation utility, r > 0 u F = (θ w)q timing: t = 0: W observes θ t = 1: F offer fixed-wage employment contracts t = 2: W accept/rejects payoffs Anke Kessler p. 3/27

4 Equilibrium Analysis Equilibrium under Full Information both worker and firms know θ firms can offer productivity dependent wages w(θ) competition between firms gives w (θ) = θ in equilibrium W accepts employment if w (θ) = θ r(θ) equilibrium is efficient Equilibrium under Asymmetric Information only W knows θ, F s only know F(θ) assume r(θ) θ and r(θ) < θ, θ > θ efficient employment has q(θ) = 1 θ Anke Kessler p. 4/27

5 Equilibrium under Asymmetric Information firms do not know θ offer (same) fixed wage contract w in equilibrium utility of worker of type θ who accepts wage offer of w is u W = w comparing this with outside utility r(θ) gives supply Θ(w) = {θ r(θ) w} note: average productivity E{θ θ Θ(w)} depends on price expected profit of firm in the market is E{θ θ Θ(w)} w competition gives zero profits; necessary equilibrium condition is w = E{θ θ Θ(w )} = E{θ r(θ) w } Anke Kessler p. 5/27

6 Market Equilibrium graphic illustration: 45 E(θ) E{θ r(θ) w } θ r(θ) w r( θ) w Anke Kessler p. 6/27

7 Properties of the Equilibrium equilibrium exists equilibrium is generically unique: if there are multiple solutions to equation w = E{θ θ Θ(w )} = E{θ r(θ) w }, equilibrium is highest w satisfying condition welfare properties: if r(θ) E(θ), equilibrium is efficient if r(θ) > E(θ), equilibrium is inefficient if r(θ) = θ, can have complete market breakdown intuition: F cannot break even at wage w = E(θ) < r(θ) w falls even more high-productivity workers drop out of market wage must drop further (alternatively : market participation of individual worker introduces externality) Anke Kessler p. 7/27

8 Government Intervention Can free-market outcome be improved upon? assume: social planner/government does not know workers productivity θ either (is subject to same informational constraints as firms) can only devise policies based on whether people choose to work or not (from IC s) let w e (resp. w u ) be transfer to worker if q = 1 (resp. q = 0) w u + r(ˆθ) = w e (IC) Claim. There is no (balanced) policy {w e, w u } that yields to a Pareto improvement over the market outcome where workers with θ θ work and workers θ > θ do not. Anke Kessler p. 8/27

9 Proof. Government Intervention equilibrium is Pareto efficient if r(θ) E(θ) if r(θ) > E(θ), let ˆθ be cut-off type worker given {w e, w u } budget balance requires w e F(ˆθ) + w u (1 F(ˆθ)) = substituting for w u + r(ˆθ) = w e gives Z ˆθ θ θf(θ)dθ = F(ˆθ)E(θ θ ˆθ) w u (ˆθ) = F(ˆθ)[E(θ θ ˆθ) r(ˆθ)] w e (ˆθ) = F(ˆθ)[E(θ θ ˆθ) r(ˆθ)] + r(ˆθ) note: ˆθ = θ gives market outcome w u = 0 and w e = r(θ ) ˆθ < θ decreases social surplus suppose θ < ˆθ < θ. Since w > w E(θ r(θ) w) < w and w = r(θ ) with r > 0, we get E(θ r(θ) r(ˆθ)) = E(θ θ ˆθ) < r(ˆθ) and, hence, w u < 0 unemployed are worse off ˆθ = θ implies w e = E(θ) highest type workers are worse off since r( θ) > E(θ). Anke Kessler p. 9/27

10 Other Markets with Adverse Selection formal model translates into goods markets and insurance markets: labor consumption good (cars) insurance workers sellers insuree firms buyers insurer θ buyer s value -(exp. payments to insuree) r(θ) seller s value -(inverse CE of risk) w price -(insurance premium) other markets where adverse selection relevant credit market (θ = default risk of debtor) dating and marriage market (θ = attractiveness of partner) stock market and corporate equity market (IPO s) (θ = firm value) Anke Kessler p. 10/27

11 Conclusion adverse selection can lead to total market failure if trade occurs, it will be less than efficient in markets with adverse selection (asymmetric information) prices are correlated with quality prices serve dual role of info transmission and market clearing insitutional/market responses against market failure caused by adverse selection signaling and screening devices, e.g. warrantees reputation (brand names and chains) experts, inspections, standards, licensing mandatory insurance (health, automobile) liability laws Anke Kessler p. 11/27

12 Using a Signal asymmetric information causes market failure participants have incentives to develop ways to reduce informational asymmetries signaling: informed market participants move first to convey info screening: uninformed market participants move first to elicit info Signaling some market participants may be worse off as a result of their privately held information ( sellers in lemons market, consumers in insurance market) would want to reveal this information to others problem: information revealed must be credible use of signals examples: warranties, lineups, peacock tail but: for the signal to work (be credible), it must be costly to fake Anke Kessler p. 12/27

13 Education as a Signal in Labor Markets (Spence) 2 identical firms F, 1 representative worker W worker W with ability (=produtivity) θ {θ L, θ H }, θ L < θ H, private information of W with p = Prob{θ = θ H } set r(θ H ) = r(θ L ) = 0 for simplicity worker can invest in education e IR + 0 marginal cost of education of θ i -type is c i with 0 < c H < c L education does not improve productivity θ utilities: u F = θ i w and u W = w c i e i = H, L timing: t = 1: W learns θ i, chooses e t = 2: F s observe e, form beliefs µ(e) = Prop{θ = θ H e} t = 3: F s offers (same) wage w Anke Kessler p. 13/27

14 General Structure of Signaling (and Cheap Talk) Game dynamic game of incomplete information with sender S and receiver R timing t = 0 nature draws type θ i Θ = {θ 1, θ 2,...,θ n } for sender; with p i = p(θ i ) = Prob{θ = θ i } > 0, θ i Θ t = 1 sender observes own type θ i, chooses message m j M = {m 1, m 2,...,m N } t = 2 receiver observes message m j (but not θ i ) and chooses action x X t = 3 payoffs u S (x, m j, θ i ) and u R (x, m j, θ i ) realized signaling game: u S depends on m j cheap talk game: u S (and u R ) is independent of m j Anke Kessler p. 14/27

15 Perfect Bayesian Nash Equilibrium Definition A Perfect Bayesian Equilibrium (PBE) is a pair of strategies m (θ i ) and x (m j ) and a belief µ (θ i m j ) such that a) m (θ i ) maximizes S s utility given R s strategy: m (θ i ) = arg max mj M u S(x, m j, θ i ) θ i Θ b) x (m j ) maximizes R s utility given beliefs about S s type: x (m j ) = arg max x X µ(θ i m j )u R (x, m j, θ i ), θ i c) R forms consistent beliefs that are calculated by Bayes Rule whenever possible: θ i Θ µ(θ i m j ) = 1, m j µ(θ i m j ) = where Θ(m j ) {θ i Θ m (θ i ) = m j }. p(θ i ) θ i Θ(m j ) p(θ i) Anke Kessler p. 15/27

16 Benchmark Cases Equilibrium under Full Information both worker and firms observe ability θ i, i = H, L firms offer wage w i = θ i independent of e worker chooses e H = e L = 0 efficient Imperfect but Symmetric Information neither worker nor firms observe θ i, i = H, L wage w can no longer depend on θ firms offer w(e = 1) = w(e = 0) = pθ H + (1 p)θ L independent of e worker chooses e H = e L = 0 efficient Anke Kessler p. 16/27

17 Imperfect and Asymmetric Information only worker not firms observe θ i, i = H, L wage w can no longer depend on θ Separating Equilibrium two types of workers choose different education levels e H e L firms have beliefs µ(e H ) = θ H and µ(e L ) = θ L firms offer wages w(e H ) = θ H and w(e L ) = θ L e H e L optimal for W requires (from (IC) s) e L = 0 and e H > 0 where c H e H < θ H θ L c L e H ( ) off equilibrium beliefs?... let µ(e) = 0 for e < e H, µ(e) = 1 for e e H Anke Kessler p. 17/27

18 Education as a Signal graphic illustration of a separating equilibrium: w {(w, e) u(w, e; θ L ) = u(w H, e H ; θ L )} θ H {(w, e) u(w, e; θ H ) = u(w H, e H ; θ H )} θ L 0 note: W s preferences satisfy single crossing property: e H dw de ū in θ or u e (w, e; θ L ) u e (w, e, ; θ H ) = c L c H > 0 (SC) e Anke Kessler p. 18/27

19 Pooling Equilibrium Other Types of Equilibria both types of workers choose same education level e H = e L = ẽ firms have beliefs firms offer wage µ(ẽ) = pθ H + (1 p)θ L w(ẽ) = θ H + (1 p)θ L off equilibrium beliefs?... let µ(e) = 0 for e < ẽ, µ(e) = p for e ẽ note: both ẽ = 0 and ẽ > 0 may be possible Semi-Separating Equilibrium some education levels are chosen by both types, and some by only one type no type chooses more than two levels of e with positive probability three types of semi-separating equilibria, depending on who makes two different choices with positive probability Anke Kessler p. 19/27

20 Properties of Signaling in Markets equilibrium with e H = e L = 0 always exists wasteful signaling in separating equilibria where e H > 0 wasteful education also possible in some pooling equilibria where e H = e L > 0 multiple (types) of equilibria cannot be unambiguously ranked by the Pareto criterion out-of-equilibrium beliefs µ(e) are critical but not always plausible equilibrium refinements Intuitive Criterion (Cho-Kreps): if an out-of-equilibrium deviation is dominated for some type θ i, but not for all types, then out-of-equiloibrium beliefs should put zero probability of that type action dominated = whatever the receiver s action to the deviation, type θ i is made worse off by the deviation selects unique least-cost separating equilibrium this equilibrium may be Pareto dominated, however (intuition = informational externality) Anke Kessler p. 20/27

21 Conclusion signaling can least to wasteful resource allocation and the market outcome may thus be inefficient in markets with signaling privately informed individuals use signal to reveal their information signal only works (is credible) if sending the same signal is too costly for other individuals other markets where wasteful signaling is relevant consumer products (signal = warranty, advertisements, price) corporate equity and start-ups (signal = equity/own money invested) legal disputes (signal = pre-trial settlement demands) bargaining (signal = rejection of offer/delay) live entertainment and restaurants (signal = lineups) marriage and dating (signal = fancy car) poker (signal = stakes) Anke Kessler p. 21/27

22 Competitive Screening Screening = contractual arrangements originating from uniformed side of market to elicit information from informed market participants 2 identical firms F, 1 representative worker W worker W with ability (=produtivity) θ {θ L, θ H }, θ L < θ H, private information of W with p = Prob{θ = θ H }, r(θ i ) = 0 firm can set task difficulty level t IR + 0 utilities: u F = θ i w and u W = w c i t i = H, L task difficulty does not influence productivity and costs workers effort c i with 0 < c H < c L timing: t = 1: W learns θ i t = 2: F s offer menu of contracts {(w, t)} t = 3: W picks firm/contract Anke Kessler p. 22/27

23 Equilibrium Analysis note: W s preferences satisfy single crossing property: dw dt ū in θ or u t (w, t; θ L ) u t (w, t, ; θ H ) = c L c H > 0 (SC) Equilibrium under Full Information both worker and firms observe ability θ i, i = H, L firms offer wage/task contracts (w i, t i ) = (θ i, 0) worker accepts employment SPE is efficient Imperfect but Symmetric Information neither worker nor firms observe θ i, i = H, L wage w can no longer depend on θ firms offer wage/task contract (w, t) = (pθ H + (1 p)θ L, 0) worker accepts efficient Anke Kessler p. 23/27

24 Imperfect and Asymmetric Information only worker not firms observe θ i, i = H, L wage w can no longer depend on θ find equilibrium in steps let (w H, t H ) and (w L, t L ) be the lowest-wage contracts accepted by workers of type i in equilibrium Step 1: in any SPE, firms earn zero profits Step 2: there is no SPE in which the high-type worker accepts a wage w H < θ H with positive probability all low types must earn w L = θ L and the equilibrium must be separating in a separating SPE, must have t L = 0 and θ H c L t H = θ L, i.e., the only candidate for equilibrium is the least-cost separating one Anke Kessler p. 24/27

25 Screening with Task Assignments graphic illustration of the separating equilibrium: w type θ L θ H E[θ] (w H, t H ) type θ H θ L (w L, t L ) 0 note: no deviation profitable; in particular, any contract attracting both types of workers lies above break even line t Anke Kessler p. 25/27

26 Screening with Task Assignments graphic illustration of the separating equilibrium: w type θ L θ H E[θ] ( w, t) (w H, t H ) type θ H θ L (w L, t L ) 0 note: deviation profitable; a contract ( w, t) attracting both types of workers lies below break even line if least-cost separating equilibrium is Pareto dominated by pooling, then no equilibrium exists t Anke Kessler p. 25/27

27 Properties of Market Screening equilibrium (in pure strategies) may not exist if equilibrium exist, is unique and identical to least-cost separating equilibrium equilibrium separation is necessary; otherwise, rival firms can do cream skimming and attract only high-types away form the firm wasteful screening with t H > 0 occurs in equilibrium outcome is Pareto inefficient other screening devices in labor markets probation periods seniority wages performance based compensation Anke Kessler p. 26/27

28 Conclusion sceening can least to wasteful resource allocation and the market outcome may thus be inefficient in markets with screening uninformed individuals use screening devices to make informed individuals reveal their information by choice of (preferred) contract from menu screening device only works (separates types) if accepting same contract is undesirable for other individuals other markets where wasteful screening is relevant consumer products (screening device = warranty, price) credit (screening device = collateral, incomes) marriage and dating (screening device = household chores) poker (screening device = stakes) Anke Kessler p. 27/27