Adverse
selection
Geir B.
Asheim
Introduction
Adverse selection, signaling & screening
Competitive
equilibrium
Applications of game theory 2
Signaling
Screening
Geir B. Asheim
Department of Economics, University of Oslo
ECON5200
Fall 2009
Introduction
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
Signaling
Screening
Seller
Buyer
Situation 1: Symmetric info
1
2 prob high quality
1
2 prob low quality
One market
1
2 prob high quality
1
2 prob low quality
Situation 2: Complete info
high quality
————
low quality
Two markets
high quality
————
low quality
Situation 3: Asymmetric info ? ? ?
1
high quality
2 prob high quality
————
1
low quality
2 prob low quality
Asymmetric information and adverse selection
Examples and outline
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
Signaling
Screening
Examples:
One worker
Many firms
One car driver
Many insurance firms
One used car seller
Many potential used car buyers
One business start-up
Many investors
Outline:
Competitive equilibrium under asymmetric information
Signaling
Screening
Conditions for a competitive equilibrium
Example: One worker — Many firms (1)
Adverse
selection
Geir B.
Asheim
Firms produce output with CRTS technology,
maximize profit, and are risk neutral.
Introduction
Competitive
equilibrium
Conditions
Gametheoretic
analysis
Welfare
analysis
Signaling
Worker’s productivity: θ ∈ [θ, θ̄] ⊂ R, where 0 ≤ θ < θ̄ < ∞.
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 }
Screening
Definition
In the competitive labor market model with unobservable worker
productivity levels, a competitive equilibrium is a wage rate w ∗
and a set Θ(w ∗ ) = ∅ such that w ∗ = E [θ | θ ∈ Θ(w ∗ )].
Conditions for a competitive equilibrium
Example: One worker — Many firms (2)
Adverse
selection
Geir B.
Asheim
Introduction
Opportunity cost r (θ) independent of θ.
θ, w
θ̄
45 degree
Competitive
equilibrium
Conditions
Gametheoretic
analysis
Welfare
analysis
w ∗ = E (θ)
r (θ)
Signaling
Screening
θ
θ
θ̄
θ, w
Conditions for a competitive equilibrium
Example: One worker — Many firms (3)
Adverse
selection
Geir B.
Asheim
Introduction
Opportunity cost r (θ) increasing in θ.
θ, w
θ̄
45 degree
Competitive
equilibrium
Conditions
Gametheoretic
analysis
Welfare
analysis
r (θ)
E (θ)
Signaling
Screening
w∗
E [θ | θ ∈ Θ(w )]
θ
r (θ) θ
w∗
r (θ̄) θ̄
θ, w
Game-theoretic analysis (1)
Adverse
selection
Geir B.
Asheim
Structure: (1) Two firms announce wage offers
(2) Nature draws the productivity of the worker
(3) Worker decides whether to work and for whom
Introduction
Competitive
equilibrium
Conditions
Gametheoretic
analysis
Welfare
analysis
Signaling
Screening
Equilibrium concept: Subgame-perfect equilibrium. Why?
Proposition
Let r (·) be strictly increasing with r (θ) ≤ θ for all θ ∈ [θ, θ̄] and
F (·) have an assoc. density f (·) with f (θ) > 0 for all θ ∈ [θ, θ̄].
Let W ∗ denote the set of competitive equilibrium wages
and let w ∗ = max{w | w ∈ W ∗ }. If w ∗ > r (θ)
and ∃ s.t. E [θ | r (θ) ≤ w ] > w for all w ∈ (w ∗ − , 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 analysis (2)
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
Conditions
Gametheoretic
analysis
Welfare
analysis
Signaling
Screening
Proof.
Part 1: In an SPE, both firms earn zero profit. Suppose M
workers are hired at w̄ with total profit Π = M(E [θ | r (θ) ≤ w̄ ])
> 0. Hence, M > 0 and w̄ ≥ r (θ). At least one firm earns no
more than Π/2. By offering w̄ + α for small α > 0, this firm
earns ≥ Π − β, where β > 0 is small; this is a profitable dev.
⇒ Any SPE outcome with employment is a comp. equilibrium.
Part 2: In an SPE, both firms offer w ∗ . Suppose highest
wage rate offered w̄ = w ∗ , implying that w̄ < w ∗ . Why?
But then, profitable dev. to w ∈ (w ∗ − , w ∗ ). If one offers
w < w ∗ , then profitable dev. by other to w ∈ (w ∗ − , w ∗ ).
Part 3: Both firms offering w = w ∗ is an SPE. Offering
w > w ∗ yields negative profit; w < w ∗ zero profit.
Constrained Pareto-efficiency
Adverse
selection
Geir B.
Asheim
Introduction
Can a central authority achieve a Pareto-improvement
relative to a market equilibrium,
without having the ability to observe worker types?
Competitive
equilibrium
Conditions
Gametheoretic
analysis
Welfare
analysis
Signaling
Screening
Proposition
Let r (·) be strictly increasing with r (θ) ≤ θ for all θ ∈ [θ, θ̄] and
F (·) have an assoc. density f (·) with f (θ) > 0 for all θ ∈ [θ, θ̄].
Let W ∗ denote the set of competitive equilibrium wages
and let w ∗ = max{w | w ∈ W ∗ }. The competitive equilibrium
where the firms offer w ∗ is constrained Pareto-efficient.
A competitive equilibrium where the firms offer w = w ∗ is not
constrained Pareto-efficient.
Signaling
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
Signaling
Gametheoretic
analysis
Separating
equilibria
Pooling
equilibria
Screening
If a costless and accurate test exists, then every worker with
θ > θ will submit to it in equilibrium. What if test is costly?
“Education” is a costly, inaccurate and otherwise useless test.
Two types: 0 < θL < θH
c(e, θ)
λ = Prob(θ = θH ) ∈ (0, 1)
c(0, θ) = 0
ce (e, θ) > 0 cθ (e, θ) < 0
cee (e, θ) > 0 ceθ (e, θ) < 0
u(w , e | θ) = w − c(e, θ)
r (θH ) = r (θL ) = 0
Without signaling, all workers are employed at w ∗ = E [θ], which
is Pareto-efficient. Highlights inefficiencies created by signaling.
Equilibrium concept
Single-crossing property
Introduction
Structure: (1) Nature draws worker type
(2) Worker chooses education level
(3) Two firms announce wage offers
(4) Worker decides whether to work and for whom
Competitive
equilibrium
Game tree is illustrated in figure 13.C.1
Adverse
selection
Geir B.
Asheim
Signaling
Gametheoretic
analysis
Separating
equilibria
Pooling
equilibria
Screening
What equilibrium concept to use?
Why is subgame-perfect equilibrium inappropriate?
Sequential equilibrium ensures that firms have identical
out-of-equilibrium beliefs of worker type. They will
offer the same wage in response to any education level:
w (e) = μ(e)θH + (1 − μ(e))θL
where μ(e) is firms’ prob. that worker type is θH .
Single-crossing property holds since ceθ (e, θ) < 0
Separating equilibria
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
In a separating equilibrium, the two types of workers choose
different education levels: e ∗ (θL ) = e ∗ (θH ).
Observations:
1
Signaling
Gametheoretic
analysis
Separating
equilibria
Pooling
equilibria
2
3
In any sequential separating equilibrium, w ∗ (e ∗ (θH )) = θH
and w ∗ (e ∗ (θL )) = θL .
In any sequential separating equilibrium, e ∗ (θL ) = 0.
In any sequential separating equilibrium, e ∗ (θH ) ≥ ẽ
(see figures 13.C.5–7).
Screening
Welfare effects:
The separating equilibrium with (e ∗ (θL ), e ∗ (θH )) = (0, ẽ)
Pareto-dominates any other separating equilibrium.
The separating equilibrium with (e ∗ (θL ), e ∗ (θH )) = (0, ẽ)
may be Pareto-dominated by the no-signaling outcome.
Pooling equilibria
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
In a pooling equilibrium, the two types of workers choose the
same education level: e ∗ (θL ) = e ∗ (θH ) = e ∗ .
Observations:
1
Signaling
Gametheoretic
analysis
Separating
equilibria
Pooling
equilibria
Screening
2
3
In any sequential pooling equilibrium, w ∗ (e ∗ ) = E [θ].
In any sequential pooling equilibrium, 0 ≤ e ∗ ≤ e (see figures 13.C.9–10).
There exists a pooling equilibrium that implements the
no-signaling outcome (with e ∗ = 0).
Welfare effects:
Pooling equilibria are weakly Pareto-dominated by the
no-signaling outcome.
Multiple equilibria and equilibrium refinement
Second-best market intervention
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
The Cho-Kreps (1987) criterion picks out the separating
equilibrium with (e ∗ (θL ), e ∗ (θH )) = (0, ẽ).
Signaling
Gametheoretic
analysis
Separating
equilibria
Pooling
equilibria
Screening
Without having the ability to observe worker types,
a central authority may be able achieve a Pareto-improvement
relative to the sep. equilibrium with (e ∗ (θL ), e ∗ (θH )) = (0, ẽ)
even in the case where this equilibrium is not
Pareto-dominated by the no-signaling outcome.
This is achieved by cross-subsidization.
Screening
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
Signaling
Can the uninformed parties (the firms) screen the informed
individuals (the worker)?
Two types: 0 < θL < θH
λ = Prob(θ = θH ) ∈ (0, 1)
“Task level”, t, is a costly and useless.
Screening
Gametheoretic
analysis
c(t, θ)
c(0, θ) = 0
ct (t, θ) > 0 cθ (t, θ) < 0
ctt (t, θ) > 0 ctθ (t, θ) < 0
u(w , t | θ) = w − c(t, θ)
r (θH ) = r (θL ) = 0
Without screening, all workers are employed at w ∗ = E [θ], which
is Pareto-efficient. Highlights inefficiencies created by screening.
Equilibrium concept
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
Signaling
Structure: (1) Two firms simultaneously announce menus
of contracts; a contract is a pair (w , t);
a menu consists of a finite # of contracts.
(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?)
Screening
Gametheoretic
analysis
What equilibrium concept to use?
Why is subgame-perfect equilibrium appropriate?
With observable types:
In any SPE of the screening game with observable types,
a worker of type θi accepts (wi∗ , ti∗ ) = (θi , 0),
and firms earn zero profits.
Analysis with unobservable types (1)
Adverse
selection
Geir B.
Asheim
Introduction
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 equilibria exist.
Competitive
equilibrium
Signaling
Screening
Gametheoretic
analysis
3
4
5
If (wL , tL ) and (wH , tH ) are the contracts signed by the lowand high-ability workers in a separating equilibrium, then
both contracts yield zero profits; i.e., wL = θL and wH = θH .
In any sep. equil., the low-ability workers accept (θL , 0); i.e.,
they receive the same contract as under observable types.
In any sep. equil., the high-ability workers accept (θH , t̂H ),
where t̂H satisfies θH − c(t̂H , θL ) = θL − c(0, θL ).
Analysis with unobservable types (2)
Welfare properties of screening equilibria
Adverse
selection
Geir B.
Asheim
Introduction
Competitive
equilibrium
Signaling
Screening
Gametheoretic
analysis
Proposition
In any SPE of the screening game, low-ability workers
accept (θL , 0) and high-ability workers accept (θH , t̂H ),
where t̂H satisfies θH − c(t̂H , θL ) = θL − c(0, θL ).
An equilibrium (in pure strategies) does not exist if
λ = Prob(θ = θH ) is sufficiently large.
Observations on welfare properties:
1
In an SPE, low-ability workers are worse-off
compared to the equilibrium when screening is not feasible.
2
In an SPE, high-ability workers are as well-off
compared to the equilibrium 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.