Optimal Contracts under Adverse Selection

When principals compete for agents

How is this different from the previous model?

• In the previous model, we studied a case where one principal wanted to hire one agent

• Now, we will study the case where there are many principals that are competing to attract agents

• As a result, each principal will have to offer the agent greater than his reservation utility so that her offer will be accepted above the offers of the other principals

How is this different from the previous model?

• In the previous model:– There was no risk– Effort was a choice variable

• In this one:– There will be risk involved– Effort will not be a choice variable. It will be

unique

• In the previous model, we used effort to separate the types of agents, in this one, we will use risk as a separation device

Description of the model that we will use:

• Production process can result in:– Success (S), or– Failure (F)

– Gross revenues for the P if S: xS

– Gross revenues for the P if F: xF

– ws= payments to the agent if S

– wf= payments to the agent if F

Description of the model that we will use:

• Two types of agents:– G= more productive– B= less productive– pG=prob. of success for type G– pB=prob. of success for type B– pG>pB !!!!!!– U(w)= concave utility function, identical for both types– We assume that effort is unique, so the P cannot

separate the agents by demanding different amounts of effort to each type

Pictures:

• Let’s draw the isoprofit for the G-types (the combinations of (ws

G,wFG) that gives to the principal

the same expected profits of E(Π) )• Failure in vertical axis and Success in the horizontal

( ) ( ) (1 )( ).

1[ (1 ) ( ) ]

1

Decreasing line, with slope: 1

If ( ) 0, and then:

(1 ) average rev.

If ( ) 0, and =0 then:

G G G GS S F F

G G G G GF S F SG

G

G

G GS F

G G G GS F S F

G GF S S

E p x w p x w

w p x p x E p wp

p

p

E w w

w w p x p x

E w w x

1 G

FG

px

p

Pictures:

• For the B-types, it will be same with the obvious changes:

( ) ( ) (1 )( ).

1[ (1 ) ( ) ]

1

Decreasing line, with slope: 1

If ( ) 0, and then:

(1 ) average rev.

1If ( ) 0, and =0 then:

B B B BS S F F

B B B B BF S F SB

B

B

B BS F

B B B BS F S F

B BF S S

E p x w p x w

w p x p x E p wp

p

p

E w w

w w p x p x

E w w x

B

FB

px

p

Pictures:

Slope for the B-type: 1

Slope for the G-type: 1

1 1

Isoprofit for the G-type is steeper than

for the B-type

B

B

G

G

G B

G B

p

p

p

p

p p

p p

Let’s draw them !!!!

Picture those with zero profits, and make use you understand them… the same point can yield to profits or losses depending on who chooses it…

Pictures:

( ) ( ) (1 ) ( )

In order to compute the MRS:

( )

'( ) (1 ) '( )

0

'( )*

(

If

1 )

t

'

h

( )

:

(1

en

)

G G G G GS F

G

G

G GS F

G G G G GS S F F

GG GSF

G G GS F

G GFG GS

E U p U w p U w

dE U

p U w dw p U w dw

U wdw pMRS

dw p U w

dw pMRS

dw p

w w

Consumer’s indifference curves:

Pictures:

If then

Notice, in the risk free line, the slope of the

G's

So:

'

type indifference curves coincide with

the slope of the i

( )*

(1 ) '( )

:

(

soprofit f

1 )

or

G

GG GSF

G G GS F

G GFG

GS

GS

F

U wdw pMRS

dw p U w

dw pMRS

d p

w w

w

the G's type

Consumer’s indifference curves:

Pictures:

If then

Notice, in the risk free line, the slope of the

B's type indifference curves coincide with

the slope of the isoprofit for the B's

( )

( ) (1 ) ( )

:

(

y e

1

t

)

p

B B B B BS F

B BFB BS

B BS F

E U p U w p U w

dw

w

pMRS

w

dw p

Consumer’s indifference curves:

We would do the same for the B-type

Pictures:

( ) ( ) (1 ) ( )

:

'( )

(1 ) ( )

So:

(1 )

If then

Given a con

(1

tract, G's type indifference curves are steeper than

B's type ! !

)

! !

B B B B BS F

BB BSF

B B BS F

G

G

B BS F

B

B

E U p U w p U w

U wdw pMRS

dw p U w

p

p p

w w

p

Consumer’s indifference curves:

We would do the same for the B-type

Picture

• Failure in vertical, Success in horizontal axis• Isoprofit are lines (constant slope)• G’s type isoprofits are steeper than B’s type• Given a contract, G’s type indifference curves

are steeper than B’s type • In the risk free line, each type indifference curve

has the same slope than its respective isoprofit (tangency)

• Let’s draw the whole picture with zero expected profits– Notice the relative situation of the zero isoprofits

The objective…

• In previous lectures, our objective was to find the optimal contract that maximizes the Principal’s profits

• However, we are now studying a market situation where Principals compete for agents

• So, we must find out the market equilibrium !!!!

What is an equilibrium?

• A equilibrium is a menu of contracts:– {(wS

G, wFG),( wS

B wFB)}

– Such that no other menu of contracts would be preferred by all or some of the agents,

– and gives greater expected profits to the principal that offers it

• The competition among Principals will drive the principal’s expected profits to zero in equilibrium

Classification of Equilibriums

• An equilibrium must be:– {(wS

G, wFG),( wS

B wFB)}

• We call it pooling if:– Both types choose the same contract

– (wSG, wF

G)=( wSB wS

B)

• We call it separating if:– Each types chooses a different contract

Equilibrium under Symmetric Information

• Principal can distinguish each agent’s type and offer him a different contract depending on the type

• As the P can separate, we can study the problem for each type separately

• Show graphically that the solution is full insurance– The eq. must be in the zero isoprofit line– If the contract with full insurance is offered, not any

other contract in the zero isoprofit will attract any consumer

– Fig. 4.6

Can the equilibrium under Symmetric Information prevail under AS?

• Show in the graph (Fig. 4-6) that:– Only the contract intended for the G-type will

attract customers– Principals will have losses with B types

contracts– This cannot be an equilibrium– Notice that in this case, it is the B types the

one that has valuable private information to sell !!!!

How is the Eq. under AS?

• Before doing this, we need to study how is the isoprofit line of a contract that is chosen by both types

• Probability of good type=q( ) [ ( ) (1 )( )]

(1 )[ ( ) (1 )( )].

( ) ( ) (1 )( ),

where is the average success probability:

* (1 )*

This isoprofit line is also a line,

with slope

G GS S F F

B BS S F F

I IS S F F

I

I G B

E q p x w p x w

q p x w p x w

E p x w p x w

p

p q p q p

1

which is somewhere between the isoprofit

of the G type and the B type

I

I

p

p

How is the Eq. under AS?

• Can an equilibrium be pooling?• Fig 4.7• Draw the 3 isoprofits• Choose a point (pooling contract) in zero profits in

the pooling isoprofit line• Draw the indifference curves. Remember G type is

steeper• Realize that there is an area of contracts that is

chosen only by G-types and it is below the zero isoprofit for G-type

• Any firm offering this contract will get stricitly positive profits

• The potential pooling eq. is broken !!!!!!• Pooling equilibrium cannot exist !!!!!!!!!!!!!!!!

How is the Eq. under AS?

• What menu of contracts will be the best candidate to be the equilibrium?

• Fig 4.8• Show first that the contract for the B type

must be efficient• We also know that must give zero profits• So, the eq. contract that is intended for the B

type is the same as in Symmetric Information

How is the Eq. under AS?

• Finding the eq contract for the G type is easy– It must give zero profits– Do not be better for the B-type than the contract intended

for the B-type– Show the graph…

• Notice that this is just a candidate, as there might exist a profitable deviation that breaks the equilibrium

• This profitable deviation exists if the percentage of B types is small

• Intuition: in this candidate G types are treated very badly because of the presence of B types.

• Intuitively, this cannot constitute an equilibrium if B are a low percentage…

How is the Eq. under AS?

• What is the equilibrium candidate?– Zero profits to each type– Full insurance for B type– Incomplete insurance for G type– For the G type, the contract of the G-type zero

isoprofit that gives to B the same utility that the contract that is intended for him

– Equations in page 124

• Notice, that the equilibrium will not exist if the proportion of G types is very large !!!

• If the proportion of G types is very low, then the candidate is certainly an equilibrium

How is the Eq. under AS?

• Notice the contract for the G type will not be efficient, it gets distorted !!!

• Show in the graph that is not Pareto Efficient• Analogy with the case of 1 principal and 1

agent:– The type that has valuable information is the one

that gets the efficient contract– There is non distortion at the top !!!– In AS models, the top agents are those for whom

no one else wants to pass themselves off (and not necessarily the most efficient ones !!!)

How is the Eq. under AS?

• Notice, the contract for the G type will not be efficient, it gets distorted !!!

• In particular, the contract for the G type is not of full insurance:– Utility depends on outcomes though there is no

moral hazard

• This shows that having utility depending on outcomes is not a strict consequence of moral hazard, but it also can occur due to adverse selection

An application to competition among insurance companies

• We can use the same framework to understand the consequences of competition among insurance companies in the presence of adverse selection

An application to competition among insurance companies

• Main ingredients of the model:

– Many insurance companies. Risk Neutral

– Consumers are risk averse

– Two types: • High probability of accident. Bad type• Low probability of accident. Good type

1

2

1 2

2 1

1 2

21 2

1

Two types of consumers: high and low risk

Probability of accident=

Wealth if no accident=

Wealth if accident=

(1 ) ( ) ( )

'( )1

'( )

1In the certainty line: ;

H L

W

W

U U W U W

dW U W

dW U W

dWW W

dW

1 1

The indifference curves of the Low risk are steeper !!!!

H L

H L

0

0 1

0 2

Initial wealth=

Loss= , with probability

Insurance premium =

What the insurance company pays in case of accident=

Wealth if no accident=

Wealth if accident=

Zero isoprofit conditions

w

l

P

R

w P W

w P l R W

0 1 2 0

0 1 2 1

2 1 0

imply that:

, substituting:

( )

( )

Zero isoprofit:

(1 )

P R

w W W w P l

w W W W l

W W w l

2 1 0

1 0 2 0

2

1

Zero isoprofit:

(1 )

Let's represent the case with no insurance:

If ;

dW (1 )Slope:

dW

We will have two types: and

So, two different isoprofits, with different slopes !!!!H L

W W w l

W w W w l

but the case of no insurance is the same, no matter the type!

certainty line

W1

W2

W 0

W0*- l

E=Point of no insurance

E

certainty line

W1

W2

W 0

W 0 - l E

F

Gl

l

)1(

slope

H

Hslope

)(1

The low-risk person willmaximize utility at pointF, while the high-riskperson will choose G

certainty line

W1

W2

W 0

W 0 - l E

F

Gl

l

)1(

slope

H

Hslope

)(1

The low-risk person willmaximize utility at pointF, while the high-riskperson will choose G

Draw the indifference curves to show the equilibrium under symmetric information.

Notice the tangency between the indifference curve and the isoprofit in the certainity line

Adverse Selection

• If insurers have imperfect information about which individuals fall into low- and high-risk categories, this solution is unstable– point F provides more wealth in both states– high-risk individuals will want to buy

insurance that is intended for low-risk individuals

– insurers will lose money on each policy sold

Adverse Selection

certainty line

W1

W2

W 0

W0 - l E

F

G

One possible solution would be for the insurer to offer premiums based on the average probability of loss

HSince EH does notaccurately reflect the trueprobabilities of each buyer,they may not fully insureand may choose a pointsuch as M

M

Point M (which is a pooling candidate) is not an equilibrium because further tradingopportunities exist for low-risk individuals

UH

UL

Adverse Selection

certainty line

W1

W2

W0

W 0 - l E

F

G

HM

An insurance policysuch as N would beunattractive to high-risk individuals, butattractive to low-riskindividuals and profitable for insurers

N

Adverse Selection

• If a market has asymmetric information, the equilibria must be separated in some way– high-risk individuals must have an incentive to

purchase one type of insurance, while low-risk purchase another

Adverse Selection

certainty line

W1

W2

W 0

W0 - l E

F

G

Suppose that insurers offer policy G. High-risk individuals will opt for full insurance.

UH

Insurers cannot offer any policy that lies above UH becausethey cannot prevent high-risk individuals from taking advantage of it

Adverse Selection

certainty line

W1

W2

W *

W * - l E

F

GUH

The policies G and Jrepresent a separating equilibrium

The best policy that low-risk individuals can obtain is one such as J

J

Adverse Selection

certainty line

W1

W2

W *

W * - l E

F

GUH

The policies G and J represent a separating equilibrium. Notice that the Low risk only gets an

INCOMPLETE insurance. So, we can have results that depend on outcomes even if there is no moral

hazard!!!!

J

Parallelisms…

• Workers model• SI:

– High constant wage for G type (productive)

– Low constant wage for B type (unproductive)

• If offered under AI:– Type B will pass

himself off as G type– Type B has

“something to sell”

• Insurance companies:• SI:-Full ins. with low premium

for G type (low p. ac.)

-Full ins. with high premium for B type (high p. of ac.)

• If offered under AI:– Type B will pass

himself off as G type– Type B has

“something to sell”

Parallelisms…

• Workers model • Type B has “something to

sell”

• AS:– B: fixed wage: full ins.

Same contract as under SI

– G: no full insurance. Distorted contract. Worse off due to AS

• Insurance companies:• Type B has “something to

sell”

• AS:- B (high prob. acc): full

ins. Same contract as under SI

– G (low prob. acc): no full insurance. Distorted contract. Worse off due to AS !!!

We can see how it is the type with low probability of accident the one that ends up having incomplete insurance !! It is the one worse off due to AS !!!!

Insurance contracts• Menu of contracts: one with full insurance,

another one with incomplete insurance.• This is what we observe in reality with most

types of insurance contracts (car, health…)– Insurance contracts usually have an “excess”.

But the “excess” can be eliminated by paying an additional premium

• Insurance excess (from this link) Applies to an insurance claim and is simply the first part of any claim that must be covered by yourself. This can range from £50 to £1000 or higher. Increasing your excess can significantly reduce your premium. On the other hand a waiver can sometimes be paid to eliminate any excess at all.