Chapter 3

The Moral Hazard Problem

Stefan P. SchleicherUniversity of Graz

Economics of InformationIncentives and Contracts

1. Introduction (1)

Asymmetric information creates incentive to use information advantage

Moral hazard problemAgent has information advantage since his effort is not observable

1. Introduction (2)

P designsthe contract

A accepts(or rejects)

A suppliesnon-verifiable effort

N determinesstate of the world

Outcomeand

pay-offs

Agent’s action is not verifiable oragent receives private information after the relationship has been initiated Informational asymmetry after the contract has been signed Examples

Labor market, research projectAutomobile insurance

1. Introduction (3)

))((1 MINMIN evUuw

Under a fixed wage contract the agent will always choose the smallest possible effort.

Therefore the principal will choose a wage which exactly compensates the agent for this effort.

2. The Moral Hazard Problem (1)

Asymmetric information with respect to effort

Effort is not verifiable

Solution concept

Subgame perfect equilibrium

2. The Moral Hazard Problem (2)

Stage 3: Agent chooses his effort

Incentive restriction or

Incentive compatibility constraint

n

i iie evxwuepMaxe1ˆ )ˆ())(()ˆ(arg

2. The Moral Hazard Problem (3)

Stage 2:Agent decides about acceptance of thecontract proposed by the principal

Participation constraint or

Individual rationality condition

n

i ii Uevxwuep1

)())(()(

2. The Moral Hazard Problem (4)

Stage 1:Principal designs the contract,anticipating the agent’s behaviour

n

i iiixwe xwxBepMaxnii 1])}({,[ ))(()(

,...,2,1

n

i iie evxwuepMaxe1ˆ )ˆ())(()ˆ(arg

[P3.1]

n

i ii Uevxwuepts1

)())(()(..

3. Agent chooses between two effort levels (1)

Risk-averse agent can choose between only two effort levels,

H high

L low

LH eee ,

)()( LH evev

Efforts and disutilities

3. Agent chooses between two effort levels (2)

nxxx ...21

Ordered set of results

Probability of outcome i under high or low effort

)( Li

Li epp )( H

iHi epp

pH first order stochastically dominates pL

(bad results are more likely under lower effort)

1...,,111

nkpp

k

i

Li

k

i

Hi

n

i

Li

n

i

Hi pp

111

3. Agent chooses between two effort levels (3)

(1) Principal demands low effort eL

Wage that guarantees reservation utility sufficient

)()()()( HLLL evwuevwu

)((1 LL evUuw

No true moral hazard problem exists.

Symmetric information contract continues to be optimal.

3. Agent chooses between two effort levels (4)

(2) Principal demands high effort eH

Wage needs to depend on outcome

Expected utility gain for agent must be greaterthan the implied increase in disutility.

n

i

Li

Ln

i

Hi

H evxwupevxwupii 11

)())(()())((

n

i

HLi

LH evevxwuppii1

)()())(()(

3. Agent chooses between two effort levels (5)

Principal solves

n

i

HLi

LH evevxwuppii1

)()())(()(

n

i iiH

xw xwxepMaxinii 1)}({ ))()((

,...,2,1

n

i

Hi

H Uevxwuptsi1

)())((..

3. Agent chooses between two effort levels (6)

Lagrangean

n

i

HLi

LH

n

i

Hi

H

n

i iiH

i

evevxwupp

Uevxwup

xwxpxwL

ii

i

i

1

1

1

)()())(()(

)())((

))((),,)((

3. Agent chooses between two effort levels (7)

First-order conditions

nixwupp

xwupp

iLi

Hi

iHi

Hi

...,,1,))((')(

))(('

0,0))(('1

n

ii

Hi

xwu

p

Hi

Li

i p

p

xwu1

))(('

1

3. Agent chooses between two effort levels (8)

Shadow price of participation restriction

0

Hi

Li

i p

p

xwu1

))(('

1

0))(('1

n

ii

Hi

xwu

p

Shadow price of incentive restriction

Wage will be larger the smaller the likelihood ratioand this will induce larger effort

3. Agent chooses between two effort levels (9)

Hi

Li

i p

p

xwu1

))(('

1

Wage will be larger the smaller the likelihood ratioand this will induce larger effort

Hi

Li

i

p

puxw

1

1)'()( 1

4. Solution using the first-order approach (1)

replaced by first-order condition

n

i iie evxwuepMaxe1ˆ )ˆ())(()ˆ(arg

n

i ii evxwuep1

0)('))(()('

Attempt to handle effort as a continuous variable

4. Solution using the first-order approach (2)

Modified optimization problem of the principal

n

i iiixwe xwxepMaxnii 1])}({,[ ))(()(

,...,2,1

[P3.2]

n

i ii Uevxwuepts1

)())(()(..

n

i ii evxwuep1

0)('))(()('

4. Solution using the first-order approach (3)

First-order conditions w.r.t. w(xi)

nixwuep

xwuepep

ii

iii

...,,10))((')('

))((')()(

i

i

i p

p

xwu

'

))(('

1

4. Solution using the first-order approach (4)

Wages increiase with increasing likelihood quotient(with high probability a good effort was exerted)

i

i

i p

p

xwu

'

))(('

1

5. A simple case with continuous effort

Probability function of the agent’s effort

Li

Hii pepeep )1()(

Condition of linearity of the distribution function

Can be solved by the first-order approach

6. Moral Hazard with hidden information

After contract is agreed upon the agent obtains informationon the environment that will determine which effort level isthe most adequate.

This information is not observable or not verifiable by the principal.

P designsthe contract

A accepts(or rejects)

N determines state of the world

which is onlyobserved by A

A supplieseffort

Outcomeand

pay-offs

7. Some comments on simpleMoral Hazard models (1)

7.1 The value of information

Loss of informtion about the agent’s effort impliesa cost to the principal-agent relationship.

Principal is interested in any signals that revealinformation on the agent’s effort.

Therefore, the contract can contain many contingencies, e.g. state of nature.

The sufficient statistic result.

7. Some comments on simpleMoral Hazard models (2)

The sufficient statistic result.

A contract should exploit all available informationin order to filter out risk optimally.

Principal may want to pay for control activities.

7. Some comments on simpleMoral Hazard models (3)

7.2 Mechanisms based on severe punishments

For results that are a perfect signal that the demanded effort was not excerted.

The threat should be sufficient.

Should lower the costs of controls.

7. Some comments on simpleMoral Hazard models (4)

7.3 The stratigic effects of contracts

Relevant, if contract effects third parties.

Then the contract is not only an incentive devicebut also an instrument to influence the behaviorof third parties.

7. Some comments on simpleMoral Hazard models (5)

7.4 What happens when it is the agent who offersthe contract

If the agent designs the contract, he must take intoaccount the fact that the pricipal will only accept believable contracts.

The only difference – the pricipal instead of the agent – is put at her reservation utility level.

B.1 Incentives for managers (1)

Objective of the shareholdes is to maximize profits,and this depends on the effort of the managers.

Shareholders hire managers.

The shareholders cannot control the effort ofthe managers.

B.1 Incentives for managers (2)

Manager’s utility

)()(),( evwuewU 0)('',0)(' wuwu 0)('',0)(' evev

Shareholder’s profits

wxcxpwxB ),(

Density distribution of sales

),( exf

B.1 Incentives for managers (3)

(1) Symmetric information

X

Xxw

Uevdxexfxwuts

dxexfwxcxpMax

)(),())((..

),()()(

Shareholder problem

B.1 Incentives for managers (4)

0),())(('),( exfxwuexf First-order conditionj w.r.t. w(x)

w(x) is constant

)((1 evUuw

Xe evUudxexfxcxpMax ))((),()( 1

B.1 Incentives for managers (5)

Xe evUudxexfxcxpMax ))((),()( 1

Rewritten shareholder problem

First-order conditionj w.r.t. e

X e evUuu

evdxexfxcxp

))(('

)('),(')(

1

B.1 Incentives for managers (6)

(2) Asymmetric information

X e

X

Xxw

evdxexfxwu

Uevdxexfxwuts

dxexfwxcxpMax

0)('),('))((

)(),())((..

),()()(

Shareholder problem

B.1 Incentives for managers (7)

),(

),('

))(('

1

exf

exf

xwue

i

First-order condition w.r.t. w(x)

0),('))(('

),())(('),(

exfxwu

exfxwuexf

e

Managers’ wages w(x) increase with sales xand the likelihood quotient (indicator for agent’sefforts).