03_PrincAgent

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Principal-Agent Models

Laffont, J.-J. and D. Martimort [2002]

The Theory of Incentives, Princeton Univ. Press

Salani B. [1997],The Economics of Contracts, A Primer, MIT, Cambridge MA:

ch. 2 (2.1, 22.), pp. 11-26


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What is P-A-theory about?

Why is health insurance so expensive?

Why does nobody pay me for what I am really

worth?

How much should I bid for this oil field? Should insurance companies be allowed to check

my genes before offering me a contract?

Why are there banks (and other financialintermediaries)?


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Why principal-agent?

One party (the principal) contracts anotherparty (the agent) to perform some action orto take some decision.

The agent has an information advantage

he knows something the principal does not

know,

will know something the principal does not

know,

he can take secret actions.

The principal knows that the agent has this

advantage (making it a potential handicap!).


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Examples

owner manager

insurance company insured

creditor debtor

firm salesmen

voters government

investor portfolio manager


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Two basic stories

Story1

A discovers

his type

P offers a

contract

A accepts or

refuses

The contract

is executed

time

t=0 t=1 t=2 t=3

Hidden Information Adverse Selection

Story2

P offers a

contract

A accepts

or refuses

A exerts an

effort or not

The outcome is

realized and the

contract is

executed

t=0 t=1 t=2 t=3

Hidden Action Moral Hazard

time

Source: Laffont und Martimort (2002)


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You consider buying a second hand vehicle.

You know that there are different qualities

the cars' owners know their car's quality

you cannot distinguish cars

Owners do not sell for less than the true value

You do not bid more than the expected value

What will happen?

Adverse selection


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V = 7500 E[V | V p]

50% good: Value = $ 1000050% bad: Value = $ 5000

45E[V]

5000

10000

5000

Price

offered10000

Quantity=50%

Case 1: two qualities

Partial market failure

but no efficiency loss


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V = 7500 E[V | V p]

Uniform distribution of values between $ 5'000-10'000

45E[V]

5000

10000

5000 10000

Case 2: Continuum of qualities

Price

offered


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V = 7500 E[V | V p]

Uniform distribution of values between $ 5'000-10'000

45E[V]

5000

10000

5000 10000

Quantity = 0

Case 2: Continuum of qualities

Total market failure

but no efficiency loss

Price

offered


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E[aVs | Vs p]

45

5000

10000

5000 10000

Case 3: Gains from tradeValue for seller uniform distribution between $ 5'000-10'000

a = valuation buyer / valuation seller = vb/ vs =1.2

7500

E[Vb]

Price

offered


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E[aVs | Vs p]

45

5000

10000

5000 10000



Quantity=50%

7500

1.2 6'250

= 7500

E[Vb]

Price

offered


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E[aVs | Vs p]

45

5000

10000

5000 10000



Quantity=50%Partial market failureEfficiency loss best cars not optimally used (worst cars subsidized)

7500

1.2 6'250

= 7500

E[Vb]

Price

offered


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E[aVs | Vs p]

45

5000

10000

5000 10000



Quantity=50%

Higher a smaller range of qualities unsold higher efficiency loss for best cars

Partial market failureEfficiency loss best cars not optimally used (worst cars subsidized)

7500

1.2 6'250

= 7500

E[Vb]

Price

offered


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Informational asymmetries create inefficiencies.

Quantities traded are smaller than under first best;market may fail completely.

Less informed party knows that they know less

and anticipate opportunistic behavior bybetter informed party.

The information advantage

is a handicap for the owners of the best qualities.

is an advantage for the owners of the worst qualities

(they can hide behind the intermediate qualities).

The presence of bad cars creates a

negative externality to the owners of good cars.

Preliminary conclusion


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an attempt to overcome informational asymmetries

one party proposes contract (mostly principal)

the counterparty accepts or rejects

contract creates incentives:

good types accept, bad types reject (hidden info)exert effort (hidden action)

mechanics: offering party maximizes utility

subject to constraints:participation

incentives

wealth, feasibility

Optimal contracts


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For several months not a drop of rain has fallen in Bilbao. Thedesperate mayor is contacted by a sourcerer who says he is able tomake rain.

If the sourcerer is a bluffer the chance of rain for the next week stays at2/100. If he really is a sourcerer, the chance of rain goes up to 20/100.

The pretending sourcerer has a utility function of u(w) = w0.5.

If he is a sourcerer, he accepts a contract if he gets at least u=10.If he is a bluffer, he accepts if he gets u>1.

The mayor does not want to be caught with a bluffer.

a) What contract should the mayor offer?

b) What is the cost of the information asymmetry to the city of Bilbao?

Source:I. Macho-Stadler und J.D. Perez-Castrillo, An Introduction to the Economics of

Information, Oxford University Press, 1977, p. 161

Rain in Bilbao


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With what type of agent do you want to work?

sourcerer: yes

bluffer: no

Find all contracts that

the sourcerer accepts

the bluffer rejects

Choose the cheapest of these contracts and offer it on

a take-it-or-leave-it basis

Compare cost of contract to cost of the cheapestcontract under symmetric information:

Difference is cost of information asymmetry.

The solution in a nutshell


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Hidden Information: Game Tree

good type

reject

bad type

c

EU(c) EU(rej)

rejectc

EU(c) EU(rej)

offer contract c


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How to get the good type only?

good type

reject

Participation Constraints

bad type

c

EU(c) EU(rej)

rejectc

EU(c) EU(rej)

offer contract c


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pro memoria: assumptions

Agent's utility: u = w0.5

Agent accepts contract:

- sourcerer: if uS 10 = uS- bluffer: if uB > 1 = uB

p(R|S) = 0.20

p(R|B) = 0.02


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contract space Contracts can define payments that are contingent on

observable (and verifiable) outcomes. Type of agent (S, B): not observable (not even ex post)

Outcome (R, 0): observable

Contract can specify payments for R and 0.

This is the contract space.

The principal should use her contract space.

The general form of the contract thus is c = {wR, w0}

The optimal contract c* = {w*R, w*0}

maximizes the principal's utility

subject to constraints (S accepts, B rejects)


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Participation constraints

Sourcerer: 0.20(wR0.5) + 0.80(w

0

0.5) 10= uS

Bluffer: 0.02(wR0.5) + 0.98(w

0

0.5) 1 = uB

both constraints bind, because if not

- for sourcerer: we could have him cheaper- for bluffer: sourcerer bears unnecessary risk

(we could make outcomes more similar,

thus paying sourcerer less in expected

terms)


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Graphical solution

0

45w0

uSuB

uS

uB

wR2500


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Numerical solution

Optimal contract when types are unobservable:w*

R= 2500, w*

0= 0

Optimal contract when types are observable

(first best):wf = wR

= w0

= 100 (only offered to sourcerer)

The cost of the information asymmetry?Cost = Ew* Ewf = 0.2(2500) -100 = 400is here paid by the principal.


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The example is a bit too simple

negative payments are excluded by assumption

(utility is a function of square root of w).

The bluffer is useless by definition

=> the principal does not need to look at

menus with a contract for each type

It pays to hire the sourcerer by definition

Only one contract satisfies both constraints

=> this contract is automatically the optimal contract=> the principal's objective function is passive


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The general case

Four cases: hire S, B, both or neither?

What is the optimal contract for each case?(cheapest contract that attracts exactly the targeted type(s))

Which of the four optimal contracts maximizes the

principal's utility?=> the overall optimal contract


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Our example slightly modified

Utility: u = w0.5

Participation:- sourcerer if: uS 20 = uS- bluffer if: uB > 11 = uB


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Graphical solution

0

45

uS

uB

OF

w0

wR3600

100

N i l l i


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Numerical solution

Optimal contract when types are unobservable:w*

R= 3600, w*

0= 100

Interpretation:

performance wage100 = fixed wage; 3500 = bonus for rain

3600 = fixed wage; 3500 = penalty if no rain

l


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Guarantee Health Insurance: Franchise

Mortgage Credit

Baby 81 King Salomo's Judgement

Examples

T b i i


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Two basic stories

Story1

A discovers

his type

P offers a

contract

A accepts or

refuses

The contract

is executed

timet=0 t=1 t=2 t=3

Hidden Information Adverse Selection

Story2

P offers a

contract

A accepts

or refuses

A exerts an

effort or not

The outcome is

realized and the

contract is

executed

t=0 t=1 t=2 t=3

Hidden Action Moral Hazard

time

Source: Laffont und Martimort (2002)

idd i G


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effort

accept

no effort

reject

EU(eff) EU(no)

EU(acc+eff) EU(reject)EU(acc+no) < EU(reject)

Hidden Action: Game Treeoffer contract c

ParticipationConstraints

Incentive

Constraint

O l difi d


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Our example modified once more

The mayor only faces one person.He is a sourcerer, but only increases theprobability of rain if he exerts a special effort.

Utility: u = w0.5 e

Participation if: u 11 = u

Cost of effort: e = 9

Hidd A ti R lt


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Hidden Action: Result

The optimal contract is only "second best":Agent must bear some risk as an incentive.

If he is more risk averse than the principal,

this means a cost (often paid by the principal).

Optimal effort level under hidden effort is

normally smaller than under observable effort

E i I ti C t t


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You ask a broker to sell some real estate for you.

The outcome is either good ($ 50'000) or bad ($ 25'000)

The broker can influence the outcome by exerting effort:

effort e1(low) e

2(medium) e

3(high)

cost of e (utility units) 5 20 40

probability ($ 50'000) 25 50 75probability ($ 25'000) 75 50 25

You cannot observe the broker's effort, but you observe the outcome

(the price at which you can sell).

You are risk neutral (i.e. you maximize expected profit)

The broker maximizes U = w1/2- e, where w is financial income and e

effort cost. He only becomes active if he gets U=120

What contract do you offer the broker?

What are the effects of unobservability of effort?

Exercise: Incentive Contract

Hi t


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What are the best contracts for each possible effort level when

effort is not observable?(Draw the constraints in a graph with w1/2-axes!!!)

Which among these best contracts yields the highest profit

overall?

What effort level would P want to get if effort was observable?What contract would he offer?

What difference does unobservability make on:

the optimal effort level?

A' welfare?

P's welfare?

Hints:

A t t d i i t lli t t


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A big law firm is interested to invite you to give a

lecture on Contract Design.You are supposed to make

a proposal as to a speaker's fee, but you are quite

uncertain about what they are ready to pay.

You value your best alternative (leisure or lecturingsomewhere else) to $ 10'000.

What do you propose if you

maximize your expected feealso try show that you know some contract theory

assume that their valuation is V~U[0,50'000]

A contract design intelligence test

Hi t


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An important idea in the economics of contracts

(or: mechanisms) is the Revelation Principle The Revelation Principle says (roughly):

whatever outcome you can achieve, you can achieve by

giving the counterparty an incentive to tell the truth.you do not loose anything by making the counterparty

tell the truth.

Example:

thesecond price auction (also Vickreyauction):

the highest bidder gets the good

for the amount of the second highest bid

Hints:

Wh t i P A th b t?


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What is P-A-theory about?

Why is health insurance so expensive? Why does nobody pay me for what I am really

worth?

How much should I bid for this oil field? Should insurance companies be allowed to check

my genes before offering me a contract?

Why are there banks (and other financialintermediaries)?

03_PrincAgent

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