Adverse selection, signaling & screening

Transcription

1 , signaling & screening Applications of game theory 2 Department of Economics, University of Oslo ECON5200 Fall 2009 Seller Buyer Situation 1: Symmetric info One market 1 2 prob high quality 1 2 prob high quality 1 2 prob low quality 1 2 prob low quality Situation 2: Complete info Two markets high quality high quality low quality low quality Situation 3: Asymmetric info??? 1 high quality 2 prob high quality 1 low quality prob low quality 2

2 Asymmetric information and adverse Examples and outline Examples: One worker One car driver One used car seller One business start-up Many firms Many insurance firms Many potential used car buyers Many investors Outline: under asymmetric information for a competitive Example: One worker Many firms (1) Firms produce output with CRTS technology, maximize profit, and are risk neutral. Worker s productivity: θ [θ, θ] R, where0 θ < θ <. Non-degenerate distribution function: F (θ). Opportunity cost: r(θ). Set of types who accept employment at wage rate w: Θ(w) ={θ r(θ) w} Definition In the competitive labor market model with unobservable worker productivity levels, a competitive is a wage rate w and a set Θ(w ) such that w = E[θ θ Θ(w )].

3 for a competitive Example: One worker Many firms (2) Opportunity cost r(θ) independent of θ. w = E(θ) θ, w θ θ θ 45 degree r(θ) θ θ, w for a competitive Example: One worker Many firms (3) Opportunity cost r(θ) increasing in θ. θ, w 45 degree θ r(θ) E(θ) E[θ θ Θ(w)] w θ θ, w r(θ) θ w r( θ) θ

4 Game-theoretic (1) Structure: (1)Twofirmsannouncewageoffers (2) Nature draws the productivity of the worker (3)Workerdecideswhethertoworkandforwhom Equilibrium concept: Subgame-perfect. Why? Proposition Let r( ) be strictly increasing with r(θ) θ for all θ [θ, θ] and F ( ) have an assoc. density f ( ) with f (θ) > 0 for all θ [θ, θ]. Let W denote the set of competitive wages and let w =max{w w W }.Ifw > r(θ) and ɛ s.t. E[θ r(θ) w ] > w for all w (w ɛ, w ), then there is a unique pure strategy SPE, where firms offer w and the worker accepts if θ Θ(w )={θ r(θ ) w }. Game-theoretic (2) Proof. Part 1: In an SPE, both firms earn zero profit. Suppose M workers are hired at w with total profit Π = M(E[θ r(θ) w]) > 0. Hence, M > 0and w r(θ). At least one firm earns no more than Π/2. By offering w + α for small α>0, this firm earns Π β, whereβ>0 is small; this is a profitable dev. Any SPE outcome with employment is a comp.. Part 2: In an SPE, both firms offer w. Suppose highest wage rate offered w w, implying that w < w.why? But then, profitable dev. to w (w ɛ, w ). If one offers w < w, then profitable dev. by other to w (w ɛ, w ). Part 3: Both firms offering w = w is an SPE. Offering w > w yields negative profit; w < w zero profit.

5 Constrained Pareto-efficiency Can a central authority achieve a Pareto-improvement relative to a market, without having the ability to observe worker types? Proposition Let r( ) be strictly increasing with r(θ) θ for all θ [θ, θ] and F ( ) have an assoc. density f ( ) with f (θ) > 0 for all θ [θ, θ]. Let W denote the set of competitive wages and let w =max{w w W }. The competitive where the firms offer w is constrained Pareto-efficient. A competitive where the firms offer w w is not constrained Pareto-efficient. If a costless and accurate test exists, then every worker with θ>θwill submit to it in. What if test is costly? Education is a costly, inaccurate and otherwise useless test. Two types: 0 <θ L <θ H λ = Prob(θ = θ H ) (0, 1) c(e,θ) c(0,θ)=0 c e (e,θ) > 0 c θ (e,θ) < 0 c ee (e,θ) > 0 c eθ (e,θ) < 0 u(w, e θ) =w c(e,θ) r(θ H )=r(θ L )=0 Without signaling, all workers are employed at w = E[θ], which is Pareto-efficient. Highlights inefficiencies created by signaling.

6 Equilibrium concept Single-crossing property Structure: (1) Nature draws worker type (2) Worker chooses education level (3)Twofirmsannouncewageoffers (4)Workerdecideswhethertoworkandforwhom Game tree is illustrated in figure 13.C.1 What concept to use? Why is subgame-perfect inappropriate? Sequential ensures that firms have identical out-of- beliefs of worker type. They will offer the same wage in response to any education level: w(e) =μ(e)θ H +(1 μ(e))θ L where μ(e) is firms prob. that worker type is θ H. Single-crossing property holds since c eθ (e,θ) < 0 In a separating, the two types of workers choose different education levels: e (θ L ) e (θ H ). Observations: 1 In any sequential separating, w (e (θ H )) = θ H and w (e (θ L )) = θ L. 2 In any sequential separating, e (θ L )=0. 3 In any sequential separating, e (θ H ) ẽ (see figures 13.C.5 7). effects: The separating with (e (θ L ), e (θ H )) = (0, ẽ) Pareto-dominates any other separating. The separating with (e (θ L ), e (θ H )) = (0, ẽ) may be Pareto-dominated by the no-signaling outcome.

7 In a pooling, the two types of workers choose the same education level: e (θ L )=e (θ H )=e. Observations: 1 In any sequential pooling, w (e )=E[θ]. 2 In any sequential pooling, 0 e e (see figures 13.C.9 10). 3 There exists a pooling that implements the no-signaling outcome (with e =0). effects: are weakly Pareto-dominated by the no-signaling outcome. Multiple and refinement Second-best market intervention The Cho-Kreps (1987) criterion picks out the separating with (e (θ L ), e (θ H )) = (0, ẽ). Without having the ability to observe worker types, a central authority may be able achieve a Pareto-improvement relative to the sep. with (e (θ L ), e (θ H )) = (0, ẽ) even in the case where this is not Pareto-dominated by the no-signaling outcome. This is achieved by cross-subsidization.

8 Can the uninformed parties (the firms) screen the informed individuals (the worker)? Two types: 0 <θ L <θ H λ = Prob(θ = θ H ) (0, 1) Task level, t, is a costly and useless. c(t,θ) c(0,θ)=0 c t (t,θ) > 0 c θ (t,θ) < 0 c tt (t,θ) > 0 c tθ (t,θ) < 0 u(w, t θ) =w c(t,θ) r(θ H )=r(θ L )=0 Without screening, all workers are employed at w = E[θ], which is Pareto-efficient. Highlights inefficiencies created by screening. Equilibrium concept Structure: (1) Two firms simultaneously announce menus of contracts; a contract is a pair (w, t); amenuconsistsofafinite#ofcontracts. (2) Nature draws the productivity of the worker (3) The worker choose whether to accept a contract, and if yes, which. (How are ties resolved?) What concept to use? Why is subgame-perfect appropriate? With observable types: In any SPE of the screening game with observable types, aworkeroftypeθ i accepts (wi, t i )=(θ i, 0), andfirmsearnzeroprofits.

9 Analysis with unobservable types (1) Consider now on the screening game with unobservable types. 1 In any SPE, whether pooling or separating, both firms must earn zero profit. 2 No pooling exist. 3 If (w L, t L )and(w H, t H ) are the contracts signed by the lowand high-ability workers in a separating, then both contracts yield zero profits; i.e., w L = θ L and w H = θ H. 4 In any sep. equil., the low-ability workers accept (θ L, 0); i.e., they receive the same contract as under observable types. 5 In any sep. equil., the high-ability workers accept (θ H, ˆt H ), where ˆt H satisfies θ H c(ˆt H,θ L )=θ L c(0,θ L ). Analysis with unobservable types (2) properties of screening Proposition In any SPE of the screening game, low-ability workers accept (θ L, 0) and high-ability workers accept (θ H, ˆt H ), where ˆt H satisfies θ H c(ˆt H,θ L )=θ L c(0,θ L ). An (in pure strategies) does not exist if λ = Prob(θ = θ H ) is sufficiently large. Observations on welfare properties: 1 In an SPE, low-ability workers are worse-off compared to the when screening is not feasible. 2 In an SPE, high-ability workers are as well-off compared to the when screening is not feasible. 3 If an SPE exists, a Pareto-improving pair of contracts cannot be constructed; even if it involves cross-subsidization.