4. Adverse Selection - Economic Theory · 4. Adverse Selection Klaus M. Schmidt LMU Munich Contract Theory, Summer 2010 Klaus M. Schmidt (LMU Munich) 4. Adverse Selection Contract

4. Adverse Selection

Klaus M. Schmidt

LMU Munich

Contract Theory, Summer 2010

Klaus M. Schmidt (LMU Munich) 4. Adverse Selection Contract Theory, Summer 2010 1 / 51

Basic Readings

Basic Readings

Textbooks:

Bolton and Dewatripont (2005), Chapters 2 and 9

Fudenberg and Tirole (1991, Chapter 7)

Laffont and Martimort (2002), Chapter 2

Schmidt (1995), Chapter 4

Papers:

Baron and Myerson (1982)

Laffont and Tirole (1988)


Introduction

Introduction

Definition of adverse selection?

Adverse selection in markets and in bilateral contracts.

Examples for adverse selection?

Why is it important that the uninformed party makes a take-it-or-leave-itoffer?

Is this assumption a restriction or without loss of generality?


The Two-Types Case

The Tow-Types Case

Price discrimination: A monopolist does not know the type of his customer(s):

c constant marginal cost of production

q quantity of production

θ type of the customer, θ ∈ {θ, θ}

p probability that the monopolist assigns to the event that the customeris of type θ.

t total payment of the customer to the monopolist.

The customer knows his own type, the monopolist knows only the ex anteprobabilities (p,1 − p) for each type.


The Two-Types Case

We will assume here that there is only one customer. Alternatively, we couldassume that there is a continuum of customers with mass 1, the fraction p ofwhich is of type θ. The formal analysis is the same.

Payoff functions:U(q, t , θ) = θu(q)− t

Π(q, t) = t − c · q

Both parties are risk neutral. If the consumer does not consume anything, hegets a reservation utility of 0.


The First Best Allocation

If there was no informational asymmetry, the principal would offer a contractthat solves the following problem:

maxq,t

t − cq

subject to

(PC) θu(q)− t ≥ 0 .

In the optimal solution (PC) must hold with equality. (Why?) Thus, the firstbest allocation is characterized by

t = θu(q)

andθu′(q) = c .


The First Best Allocation

Remarks:

1. The monopolist extracts all the rents from his customer

2. He chooses a production level such that marginal utility (or rather theMRS between this good and the expenditures for all other goods) equalsmarginal cost.

3. t and q are both functions of θ. This can also be written as an non-linearpricing scheme t(q) (second degree price discrimination).


The Revelation Principle

What kind of contract should the principal offer if he does not know the agent’stype?

There are infinitely many types of contracts (“mechanisms”) that could beoffered:

a non-linear pricing scheme t(q)

a revelation mechanism {t(θ̂),q(θ̂)}

a lottery: the customer buys lottery tickets at certain prices thatdetermine his probability of “winning” a certain amount q.

a multi-stage contract: the agent first has to pay an entrance fee, thensome multi-stage game between the principal and the agent is played,the outcome of which determines the allocation

etc.



Thus, the problem is that the set of contracts over which the principal wants tomaximize his utility is not well defined, and if we wanted to define it, if wouldbe an incredibly complex object.However, the Revelation Principle tells us, that any allocation that can beimplemented at all, can also be implemented by using a direct revelationmechanism. Therefore, without loss of generality, we can restrict attention todirect revelation mechanisms which is a much simpler set of objects.The Revelation Principle is extremely important. To fully understand it, weneed a few definitions:

Definition 1An allocation , (q(θ), t(θ)), is a function that assigns to each possible type ofthe agent a consumption quantity q and a payment t .



Definition 2A mechanism is a game form, i.e. a set of strategies for the agent and afunction (q(s), t(s)) that assigns to each possible strategy of the agent anoutcome (q, t).

What is the difference between a game form and a game?

Definition 3

A revelation mechanism (or direct mechanism) (q(θ̂), t(θ̂)) is a mechanism,in which the agent is asked to announce his type, i.e., the agent’s strategyspace S is the set of possible types Θ. Furthermore, a revelation mechanismhas to be incentive compatible , i.e., for each possible type of the agent ithas to be optimal to announce his type truthfully (θ̂ = θ).



Definition 4A mechanism {S, (q(s), t(s))} implements an allocation (q(θ), t(θ)) if andonly if for every possible type θ ∈ Θ there exists an optimal strategy s∗(θ),such that

(q(s∗(θ)), t(s∗(θ))) = (q(θ), t(θ))



Proposition 4.1 (Revelation Principle)The allocation function (q(θ), t(θ)) can be implemented by some (arbitrarilycomplicated) mechanism if and only if it can also be implemented by a (direct)revelation mechanism.

Proof: Let (q(θ), t(θ)) be an allocation that can be implemented by amechanism {S, (q(s), t(s))}. Then there exists for each type θ ∈ Θ an optimalstrategy s∗(θ), such that

(q(s∗(θ)), t(s∗(θ))) = (q(θ), t(θ))

Let us now construct another mechanism{

Θ, (q(θ̂), t(θ̂))}

as follows:

If the agent claims to be type θ̂, then the allocation(q(s∗(θ̂)), t(s∗(θ̂))) will be implemented.

Note that if each type tells the truth (θ̂ = θ), then this direct mechanismimplements exactly the same allocation as the indirect mechanism we started



with. It remains to be shown that truthtelling is indeed optimal for each type ofagent:Note that s∗(θ) is an optimal strategy in the original mechanism. Therefore wehave:

θu(q(s∗(θ)))− t(s∗(θ)) ≥ θu(q(s)))− t(s) ∀ s ∈ S

In particular:

θu(q(s∗(θ)))− t(s∗(θ)) ≥ θu(q(s∗(θ̃)))− t(s∗(θ̃)) ∀ θ̃ ∈ Θ

By the definition of our direct mechanism it follows that:

θu(q(θ))− t(θ) ≥ θu(q(θ̃))− t(θ̃) ∀ θ̃ ∈ Θ

But this means that it is indeed optimal for each type θ ∈ Θ to announce θ

truthfully. Q.E.D.



Remarks:

1. We have proved the revelation principle in the specific context of the pricediscrimination problem. However, it is straightforward to generalize thisresult to any other problem with adverse selection.

2. There are different types of implementation, corresponding to differentnotions of equilibrium, e.g.:

dominant strategy implementation (equilibrium in dominated strategies)full implementation (unique Nash equilibrium)truthful implementation (truthtelling is just one of potentially many Nashequilibria)Bayesian Nash implementationsubgame perfect implementation (truthtelling is a unique, subgame perfectequilibrium)virtual implementationetc.

Most of these concepts coincide if there is only one agent. Why?However, with multiple agents these concepts are quite different. SeeMoore (1992) for a brilliant survey of this literature.



3. The revelation principle requires that the principal can fully commit to theterms of the contract. If this is not the case, an indirect mechanism, whichallows for some commitment, may strictly outperform any direct revelationmechanism, which allows for no commitment.


How to Solve the Two-Types Case

How to Solve the Tow-Types Case?Let q = q(θ), q = q(θ), etc. The Second Best Problem can be written as thefollowing optimization problem:

maxq,q,t,t

p(t − cq) + (1 − p)(t − cq)

subject to:

(IC1) θu(q)− t ≥ θu(q)− t ,

(IC2) θu(q)− t ≥ θu(q)− t ,

(PC1) θu(q)− t ≥ 0 ,

(PC2) θu(q)− t ≥ 0 ,



We will solve this problem in a sequence of steps:

Step 1: (PC2) is redundant and can be ignored:

θu(q)− t ≥ θu(q)− t (1)

≥ θu(q)− t (2)

≥ 0 (3)

(1) is (IC2), (2) is implied by θ > θ and (3) holds by (PC1).

Step 2: (PC1) must be binding in the optimal solution. If this was not the case,then it would be possible to increase t and t by ε > 0 without violating anyconstraint. This would further increase the principal’s payoff, a contradiction.



Step 3: (IC2) must be binding in the optimal solution. If this was not the case,the principal could increase t without violating any other constraint, acontradiction. To see this, illustrate the indifference curves of the differenttypes graphically in the q, t space:

θu(q)− t = const ⇒ t = θu(q)− const

The indifference curves are concave (u′′(·) < 0), and for each q I(θ) issteeper than I(θ). This implies that the indifference curves of the differenttypes can cross only once. This is the famous “single crossing property” .Without this property, adverse selection problems are very hard to solve.Note that I(θ) must pass through the origin of the diagram, because we knowalready that (PC1) must be binding at the optimum.



Let point A represent the contract that is supposed to be chosen by type θ.Consider the indifference curve I(θ) through this point.

The contract for type θ (point B) cannot be to the left/above of I(θ),otherwise type θ would choose contract A.

It cannot be to right/below I(θ) either, otherwise type θ would choosecontract B.

Hence, q ≤ q. This monotonicity condition is a direct implication ofincentive compatibility.

Suppose, B is strictly below I(θ). Then we could increase t a little bit withoutviolating the incentive compatibility or participation constraint for type θ. Thiswould improve the principal’s payoff, a contradiction.


How to Solve the Two-Types CaseStep 4: If q > q, then (IC1) is not binding. This is obvious from the graphicalillustration in this example. In other examples it may be more difficult to show.In this case it may be better to ignore (IC1), to solve the relaxed problem, andto check afterwards whether the solution to the relaxed problem satisfies(IC1).

Step 5:(PC1) ⇒ t = θu(q)

(IC2) ⇒ t = θu(q)−[θu(q)− θu(q)

]

︸︷︷︸

information rent

Hence, our optimization problem reduces to:

maxq,q

p[θu(q)− cq

]+ (1 − p)

[θu(q)− θu(q) + θu(q)− cq

]

FOCs:∂

∂q= u′(q)

[θ − (1 − p)θ

]− pc =

{

≤ 0 if q = 0= 0 if q > 0



∂

∂q= u′(q)(1 − p)θ − (1 − p)c =

{

≤ 0 if q = 0= 0 if q > 0

The second order conditions are globally satisfied. To guarantee an interiorsolution we assume that θ > (1 − p)θ (otherwise q = 0 is optimal), andlimq→0 u′(q) = ∞ (otherwise it would not be guaranteed that u′(q) issufficiently large to guarantee an interior solution.In an interior solution, the FOCs hold with equality:

θu′(q) =c

1 − (1−p)p

(θ−θ)θ

θu′(q) = c

It is now straightforward to check algebraically that (IC1) holds.


Interpretation

Interpretation of the Optimal Solution

1) q = qFB(θ), i.e., “no distortion at the top”. This property holds in alladverse selection models in which the single crossing property issatisfied.

2) q < qFB(θ), i.e., the “bad” type is distorted and consumes too little.

3) The intuition for these results is as follows: Note first that qFB(θ) < qFB(θ).Suppose now that q 6= qFB(θ). The we can move along the indifferencecurve of type θ to qFB(θ) without violating any constraint. This leaves bothtypes of agent indifferent and increases the principal’s payoff.


Interpretation

Why is q < qFB(θ)?The principal could implement the first best and offer a contract A’ to the badcustomer which induces him to consume efficiently.However, in this case he would have to offer the contract B’ to the goodcustomer in order to prevent him from choosing A’ as well. This means thatthe good type has to get a higher information rent .Suppose the principal reduces q starting from qFB(θ) a little bit along theindifference curve I(θ). This does not affect the agent and results in asecond order welfare loss. Hence, the principals loss is also of second order.On the other hand he can now increase t which yields an additional firstorder profit.Hence, it is always optimal for the principal to distort the quantity of the badtype a little bit in order to reduce the rent that has to be left to the good type.


Interpretation

4) The degree of the distortion depends on the relative likelihood of facingthe bad customer. If his probability is close to 1, then the distortion will beclose to 0. In this case the expected information rent is very smallanyway. On the other hand, the smaller p, the larger is the optimaldistortion. There exists an p, such that q = 0 if p < p. If q = 0, theprincipal does not have to pay any information rent to the agent:

t = θu(q)−[θu(q)− θu(q)

]

︸︷︷︸

=0

.

5) In a standard adverse selection problem, there is a trade-off betweenachieving allocative efficiency and minimizing the agent’s rent .


A Continuum of Types

A Continuous Type Model

V = v(x , θ)− t principal’s payoff function

U = u(x , θ) + t agent’s payoff function

x ∈ [0, x ] agent’s action, observable and verifiable

t ∈ < transfer from P to A (reverse notation!)

θ ∈ [0,1] agent’s type (private information)

F (θ) probability distribution over θ

f (θ) density of θ, f (θ) > 0 ∀ θ ∈ [0,1].

Remarks:

1. Both payoff functions are linear in t , i.e., both players are risk neutral.2. Specification is quite general. Examples:

procurement problems (e.g. Laffont and Tirole, 1993)price regulation of a monopolist with unknown cost (e.g. Baron andMyerson, 1982)monopolistic price discrimination (e.g. Maskin and Riley, 1984)


A Continuum of Typesoptimal taxation (e.g. Mirrlees, 1972)

Assumption 4.1

A1: ∂u(x,θ)∂θ

> 0.

A2: “Single Crossing Property”: ∂2u(x,θ)∂x∂θ > 0.

A3: u(x , θ) and v(x , θ) are both concave in x, i.e., ∂2u(x,θ)∂x2 < 0, ∂2v(x,θ)

∂x2 < 0.

A4: ∂2v(x,θ)∂x∂θ ≥ 0, ∂3u(x,θ)

∂x2∂θ≥ 0, ∂3u(x,θ)

∂x∂θ2 ≤ 0.

A5: “Monotone Hazard Rate”: ∂∂θ

(f (θ)

1−F (θ)

)

≥ 0.

A6: For any θ ∈ [0,1] ∃ x < x, such thatx ∈ arg maxx {v(x , θ) + u(x , θ)}.


A Continuum of Types

Remarks:

1. A1 requires that the types can be ordered in the unit interval such thatu(·) is monotonic in θ. It does not matter whether it is monotonicallyincreasing or decreasing. This assumption restricts us toone-dimensional type spaces.

2. A2 is very important. It requires that utility and marginal utility go in thesame direction as θ increases. Plausible in many cases, but not always.

3. A3 is necessary to make sure that the optimization problem is concave.

4. A4 is ugly because it is an assumption on third derivatives. Purelytechnical assumption with no natural economic interpretation. However,these assumptions are required to guarantee that the second orderconditions are always satisfied.

5. A5 is very controversial. We will see later what happens if thisassumptions does not hold.

6. A6 guarantees that the first best problem has a solution.


First Best

The First Best:

If the principal can observe θ he will offer a contract which is a solution to thefollowing problem:

maxx(θ),t(θ)

{v(x(θ), θ)− t(θ)}

subject to u(x(θ), θ) + t(θ) ≥ 0. The first best action is fully characterized by:

∂v(xFB(θ), θ)

∂x+

∂u(xFB(θ), θ)

∂x= 0 ,

and tFB(θ) = −u(xFB(θ), θ).


Second Best

The Second Best:

By the revelation principle, we can restrict attention to direct revelationmechanisms. Thus, the principal has to solve the following problem:

maxx(θ),t(θ)

∫ 1

0[v(x(θ), θ)− t(θ)] dF (θ)

subject to:

u(x(θ), θ) + t(θ) ≥ u(x(θ̂), θ) + t(θ̂) ∀θ, θ̂ ∈ [0,1] (IC)

u(x(θ), θ) + t(θ) ≥ 0 ∀θ ∈ [0,1] (PC)

We proceed in two steps: First, we characterize the set of all allocations{x(θ), t(θ)} that are implementable. Then we ask, which of these feasibleallocations maximizes the utility of the principal.


Second Best

Definition 5An allocation {x(θ), t(θ)} is implementable if and only if it satisfies (IC) for allθ, θ̂ ∈ [0,1].

Proposition 4.2An allocation {x(θ), t(θ)} is implementable if and only if

dx(θ)dθ

≥ 0 ∀θ ∈ [0,1] (4)

∂u(x(θ), θ)∂x

·dx(θ)

dθ+

dt(θ)dθ

= 0 ∀θ ∈ [0,1] (5)

Proof: “⇒” First we show that (IC) implies conditions (4) and (5). FOCs of theagent’s maximization problem:


Second Best

∂u(x(θ̂), θ)∂x

·dx(θ̂)

d θ̂+

dt(θ̂)

d θ̂= 0 (6)

∂2u(x(θ̂), θ)∂x2

(

dx(θ̂)

d θ̂

)2

+∂u(x(θ̂), θ)

∂x·

d2x(θ̂)

d θ̂2+

d2t(θ̂)

d θ̂2≤ 0 (7)

If truthtelling is an optimal stragety, then (6) and (7) must be satisfied at θ̂ = θ.If we substitute θ for θ̂ in (6), then we get (5). Note that (5) must hold for allvalues of θ ∈ [0,1]. Therefore, by differentiating (5) with respect to θ, we get:

∂2u(x(θ), θ)∂x2

(dx(θ)

dθ

)2

+∂2u(x(θ), θ)

∂x∂θdx(θ)

dθ

+∂u(x(θ), θ)

∂xd2x(θ)

dθ2 +d2tdθ2 = 0 (8)

Using (8) and substituting θ for θ̂ in (7), we can write (7) as follows:


Second Best

∂2u(x(θ), θ)∂x∂θ

dx(θ)dθ

≥ 0 (9)

The single crossing property (A2) implies that the first term is strictly positive.Hence, (9) implies (4).

“⇐” Now we have to show that any allocation {x(θ), t(θ)} that satisfies (4)and (5) is implementable, i.e., (IC) is satisfied. Let U(θ̂, θ) = u(x(θ̂), θ) + t(θ̂).Suppose there exists an θ̂, such that U(θ̂, θ) > U(θ, θ). Then we have

U(θ̂, θ)− U(θ, θ) =

∫ θ̂

θ

∂U(τ, θ)

∂τdτ (10)

=

∫ θ̂

θ

[∂u(x(τ), θ)

∂x·

dx(τ)dτ

+dt(τ)dτ

]

dτ (11)

> 0 . (12)


Second Best

Suppose θ̂ > θ. By (4) we know that dx(τ)dτ ≥ 0, and (A2) requires that

∂u(x(τ),τ)∂x ≥ ∂u(x(τ),θ)

∂x (note that τ ≥ θ). Hence, (5) implies

U(θ̂, θ)− U(θ, θ) ≤

∫ θ̂

θ

[∂u(x(τ), τ)

∂xdx(τ)

dτ+

dt(τ)dτ

]

dτ = 0 , (13)

a contradiction. Analogously, suppose that θ̂ < θ. Then we have by (A2)∂u(x(τ),τ)

∂x ≤ ∂u(x(τ),θ)∂x (because now τ ≤ θ), and we get

U(θ̂, θ)− U(θ, θ) = −

∫ θ

θ̂

[∂u(x(τ), θ)

∂xdx(τ)

dτ+

dt(τ)dτ

]

dτ

≤ −

∫ θ

θ̂

[∂u(x(τ), τ)

∂xdx(τ)

dτ+

dt(τ)dτ

]

dτ

= 0 , (14)

again a contradiction. Hence, it is optimal to announce θ̂ = θ, and {x(θ), t(θ)}is incentive compatible. Q.E.D.


Second Best

We have shown that (IC) is equivalent to (4) und (5). Therefore, we canreformulate the principal’s problem as follows:

maxx(θ),t(θ)

∫ 1

0[v(x(θ), θ)− t(θ)] dF (θ)

subject to:dx(θ)

dθ≥ 0 (4)

∂u(x(θ), θ)∂x

dx(θ)dθ

+dt(θ)dθ

= 0 (5)

u(x(θ), θ) + t(θ) ≥ 0 (PC)


Second Best

Let U(θ) ≡ U(θ, θ) = u(x(θ), θ) + t(θ). Using (5) we get:

dU(θ)

dθ=

∂u(x(θ), θ)∂x

dx(θ)dθ

+∂u(x(θ), θ)

∂θ+

dt(θ)dθ

=∂u(x(θ), θ)

∂θ(15)

Integrating both sides of this equation yields:

∫ θ

0

dU(τ)

dτdτ =

∫ θ

0

∂u(x(τ), τ)∂τ

dτ (16)

or:

U(θ) = U(0) +∫ θ

0

∂u(x(τ), τ)∂τ

dτ (17)

By Assumption 1 ∂u(x(τ),τ)∂τ

≥ 0, i.e., the agent’s utility is monotonicallyincreasing with his type. Thus, if (PC) holds for the worst type θ = 0, then itmust also hold for all other types as well. Because the principal wants to


Second Best

minimize the payment to the agent, (PC) must be binding for θ = 0. (17) and(PC) binding for type θ = 0 are equivalent to

U(θ) =

∫ θ

0

∂u(x(τ), τ)∂τ

dτ (18)

U(θ) is the information rent that the principal has to pay to the agent of typeθ in addition to his reservation utility, in order to induce him not to mimic anyother type.Because U(θ) = u(x(θ), θ) + t(θ) we get:

t(θ) = −u(x(θ), θ) +∫ θ

0

∂u(x(τ), τ)∂τ

dτ (19)

Hence, we can rewrite the principal’s problem:

maxx(θ)

∫ 1

0

[

v(x(θ), θ) + u(x(θ), θ)−∫ θ

0

∂u(x(τ), τ)∂τ

dτ

]

dF (θ) (20)


Second Best

subject to:dx(θ)

dθ≥ 0 (4)

The principal maximizes expected total surplus minus the expectedinformation rent that has to be paid to the agent. (Compare to moral hazard,where the principal maximizes total surplus minus risk premium to the agent.)Consider the relaxed problem (without (4)) first. If the solution to the relaxedproblem satisfies (4), we are done. Partial Integration of the last term in (20)yields:1


Second Best

∫ 1

0

u(x)︷︸︸︷

∫ θ

0

∂u(x(τ), τ)∂τ

dτ

v ′(x)︷︸︸︷

f (θ) dθ

=

[

u(x)︷︸︸︷

∫ θ

0

∂u(x(τ), τ)∂τ

dτ ·

v(x)︷︸︸︷

F (θ)

]1

0

−

∫ 1

0

u′(x)︷︸︸︷

∂u(x(θ), θ)∂θ

v(x)︷︸︸︷

F (θ) dθ

1Digression: Partial Integration. Let u, v : [a, b] → R be two continuously differentiablefunctions. Then: ∫ b

au(x)v ′(x)dx = u(x)v(x)|ba −

∫ b

av(x)u′(x)dx .

Proof: Applying the product rule to the function U := uv we getU′(x) = u′(x)v(x) + u(x)v ′(x)

Integrating up both sides from a to b, we get:∫ b

au′(x)v(x)dx +

∫ b

au(x)v ′(x)dx = U(x)|ba = u(x)v(x)|ba .


Second Best

=

∫ 1

0

∂u(x(θ), θ)∂θ

dθ −∫ 1

0

∂u(x(θ), θ)dθ

F (θ)dθ

=

∫ 1

0

∂u(x(θ), θ)∂θ

·[1 − F (θ)]

f (θ)f (θ)dθ . (21)

Hence, the principal maximizes

∫ 1

0

[

v(x(θ), θ) + u(x(θ), θ)−(

1 − F (θ)

f (θ)

)∂u(x(θ), θ)

∂θ

]

f (θ)dθ . (22)

Let us ignore the possibility of a corner solution for a minute. Pointwisedifferentiation of (22) with respect to x yields:

∂v(x , θ)∂x

+∂u(x , θ)

∂x−

(1 − F (θ)

f (θ)

)∂2u(x , θ)∂x∂θ

= 0 (23)


Second BestTo show that this condition is not only necessary but also sufficient, we haveto check whether the principal’s payoff function is globally concave in x :

∂2(22)∂x2 =

∂2v∂x2︸︷︷︸

<0

+∂2u∂x2︸︷︷︸

<0

−

(1 − F (θ)

f (θ)

)

︸︷︷︸

≥0

∂3u(x , θ)∂x2∂θ︸︷︷︸

≥0︸︷︷︸

<0

< 0 . (24)

Note that we made use of ∂3u(·)∂x2∂θ

≥ 0 (Assumption A4) here.Hence, (23) characterizes the solution of the relaxed problem.


Second Best

Now we have to check whether (4) is indeed satisfied. The implicit functiontheorem implies:

dx∗(θ)

dθ= −

∂(23)∂θ

∂(23)∂x

(25)

The denominator must be negative because of the second order condition(24). Thus, dx∗(θ)

dθ ≥ 0 iff

∂2v(·)∂x∂θ︸︷︷︸

>0

+∂2u(·)∂x∂θ︸︷︷︸

>0

−∂

∂θ

(1 − F (θ)

f (θ)

)∂2u(·)∂x∂θ︸︷︷︸

>0

−

(1 − F (θ)

f (θ)

)

︸︷︷︸

≥0

∂3u(·)∂x∂θ2︸︷︷︸

≤0

≥ 0 . (26)

Note that we used here that ∂3u(·)∂x∂θ2 ≤ 0 (A4).


Second Best

Obviously, ∂∂θ

(1−F (θ)

f (θ)

)

≤ 0 is a sufficient condition for this inequality to hold.

Note that this condition is equivalent to the “Monotone Hazard Rate”Assumption A5. Many standard distributions satisfy A5 (e.g. the uniform,normal, exponential, logistic, chi-square, or Laplace distribution). However,there is no economic reason why this assumption should be satisfied in ourcontext. If this assumption does not hold, some intervals of types have to bepooled and will be offered the same contract.

Note, that if A5 holds, then dxdθ > 0.

Given Assumptions A1-A5 equation (23) fully characterizes x∗(θ). If wesubstitute x∗(θ) in equation (19), we get:

t∗(θ) = −u(x∗(θ), θ) +

∫ θ

0

∂u(x∗(τ), τ)

∂τdτ (27)


Interpretation of the Second Best Solution


The optimal contract is characterized by (23) and (27):

∂v(x∗(θ), θ)

∂x+

∂u(x∗(θ), θ)

∂x−

(1 − F (θ)

f (θ)

)∂2u(x∗(θ), θ)

∂x∂θ= 0 (23)

t∗(θ) = −u(x∗(θ), θ) +

∫ θ

0

∂u(x∗(τ), τ)

∂τdτ (27)

Remarks:

1. x∗(1) = xFB(1), i.e., “no distortion at the top” (because 1 − F (1) = 0).

2. x∗(θ) < xFB(θ) fof all θ < 1.

3. The agent’s information rent U(θ) =∫ θ

0∂u(x∗(τ),τ)

∂τdτ is increasing in θ.

4. Trade-off between maximizing total surplus and minimizing the agent’sinformation rent.



5. What happens if the Monotone Hazard Rate property does not hold? Inthis case we have to consider (4) explicitly. This is technically quitecomplicated (see e.g. Guesnerie and Laffont (1984) or Baron undMyerson (1982)).Intuition: If A5 does not hold, then x∗(θ) is not increasing everywhere.However, the solution must be monotonic. Therefore, intervals of typeshave to be pooled (“bunching”), i.e., all types in one interval get the samecontract and choose the same action. Note that “bunching” can neveroccur in a neighborhood of 1, since

∂

∂θ

(f (θ)

1 − F (θ)

)

=f ′(θ) [1 − F (θ)] + f (θ)2

[1 − F (θ)]2≥ 0 (28)

if and only if f ′(θ)[1 − F (θ)] + f (θ)2 ≥ 0, which must be true if θ is closeenough to 1.

6. It can be shown that it is never optimal to use a lottery in the contract aslong as A4 holds. If A4 is violated, counter examples are possible.



7. We assumed that the reservation utility of the agent is independent of histype. This is not always the case. If the reservation utility is increasingwith θ, the technical problem arises that (PC) need not be binding for theworst type anymore (see Lewis und Sappington (1989)). Illustrategraphically.

8. The standard adverse selection model has the same problem as thestandard moral hazard problem: The optimal contract crucially dependson the ex ante probability assessment of the principal which is notobservable. Hence, it is difficult to obtain any testable predictions.

9. The model makes a clear prediction about the direction of the distortion:There will be underproduction for all types except for the best type. Inprinciple this is emprically testable, and there are some indications thatthis is correct.

10. An important problem of a direct mechanims is the possibility ofrenegotiation. After the agent has revealed his type truthfully, both partiescan be made better off by renegotiating the initial contract to the efficientquantity. However, if the agent anticipates renegotiation, his incentives toreveal his type truthfully are distorted (see Bester and Strausz (2001)).



11. Usually, we do not observe direct revelation mechanisms in reality.However, if A5 holds and if x∗(θ) is strictly increasing, we can replace therevelation mechanism by an indirect mechanism of the form

T (x) =

{

t(θ) if ∃θ, such that x = x(θ)0 otherwise

This type of contract has the advantage that it offers sometimes someprotection against renegotiation.


Repeated Adverse Selection

Repeated Adverse Selection ProblemsSuppose that the same adverse selection game is being played several timesbetween the principal and the agent. The optimal contract depends on thecommitment possibilities of the principal:

full commitment

no commitment

limited commitment (no commitment not to renegotiate)


Full Commitment

Full Commitment

Proposition 4.3If the principal can fully commit to a long-term contract, then he will offer acontract that is the T -fold repetition of the optimal one-period contract.

Remarks:

1. The proof if simple. If another contract would do better in the T -periodmodel then scaling down this contract by factor 1

T would also do better inthe one-period problem, a contraction.

2. Repetition does not improve efficiency in an adverse selection context.

3. With full commitment the repeated problem is essentially a staticproblem. Everything is determined in period 1 already.

4. Note that if the principal can fully commit, the revelation principle applies.This is no longer the case if the principal cannot fully commit.


No Commitment

No Commitment

In many context the “no commitment” assumption is more plausible than the“full commitment” assumption:

Present government cannot bind future governments

Labor contracts have to be short-term contracts (no slavery)

“Incomplete” contracts

With no commitment, the ratchet effect arises: Because the principal cannotcommit not to exploit the information that he learns in the first period toexpropriate the agent’s information rent in the second period, the agent will bevery reluctant to reveal his type truthfully.


No Commitment

Laffont and Tirole (1988) consider a two-period model with a continuum oftypes and show that there is no separating equilibrium in which all typesreveal themselves truthfully in period 1. Incentive constraints may bind in bothdirections. Problem: “Take-the-money-and-run strategies”.

Efficiency goes down as compared to full commitment. Why?

Examples for the ratchet effect?

Hart and Tirole (1988, 2 types) and Schmidt (1993, N types) considerT -period models. Main results:

Complete pooling except for the last finite number of periods.

Principal’s bargaining power is considerably reduced.


Limited Commitment

Limited Commitment

The most realistic case seems to be that the principal and the agent can writea long-term contract, but that they cannot commit not to renegotiate thiscontract.

Laffont and Tirole (1990) consider the case of a two-period model with twotypes. They show:

1. “Take-the-money-and-run strategies” are not a problem. Therefore,incentive constraints bind only in one direction.

2. Three different types of equilibria, which are difficult to characterize.

3. With a continuum of types full separation is possible but never optimal forthe principal.


Documents

4. Adverse Selection - Economic Theory · 4. Adverse Selection Klaus M. Schmidt LMU Munich Contract Theory, Summer 2010 Klaus M. Schmidt (LMU Munich) 4. Adverse Selection Contract