THE ACT OF GOD CLAUSE

Surajeet Chakravarty and David KelseyDepartment of EconomicsUniversity of Exeter.

University of Exeter

NOVEMBER 2008

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 1 / 29

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 2 / 29

INTRODUCTION

In a 5 year period between 1989 and 1994, 5 natural disasters causeda damage of 75 billion dollars in USA.

In US, wind and �re related risk insurance amount to 30 trilliondollars in both residential and commercial exposures while forearthquake related risks the number is only 3.4 trillion dollars.

A substantial number of residents in California do not buy earthquakeinsurance and signi�cant number of residents in Florida do not buyhurricane insurance

Agents do not seem to share risks as well as they could.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 3 / 29

INTRODUCTION

In a 5 year period between 1989 and 1994, 5 natural disasters causeda damage of 75 billion dollars in USA.

In US, wind and �re related risk insurance amount to 30 trilliondollars in both residential and commercial exposures while forearthquake related risks the number is only 3.4 trillion dollars.

A substantial number of residents in California do not buy earthquakeinsurance and signi�cant number of residents in Florida do not buyhurricane insurance

Agents do not seem to share risks as well as they could.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 3 / 29

INTRODUCTION

In a 5 year period between 1989 and 1994, 5 natural disasters causeda damage of 75 billion dollars in USA.

In US, wind and �re related risk insurance amount to 30 trilliondollars in both residential and commercial exposures while forearthquake related risks the number is only 3.4 trillion dollars.

A substantial number of residents in California do not buy earthquakeinsurance and signi�cant number of residents in Florida do not buyhurricane insurance

Agents do not seem to share risks as well as they could.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 3 / 29

INTRODUCTION

In a 5 year period between 1989 and 1994, 5 natural disasters causeda damage of 75 billion dollars in USA.

In US, wind and �re related risk insurance amount to 30 trilliondollars in both residential and commercial exposures while forearthquake related risks the number is only 3.4 trillion dollars.

A substantial number of residents in California do not buy earthquakeinsurance and signi�cant number of residents in Florida do not buyhurricane insurance

Agents do not seem to share risks as well as they could.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 3 / 29

NON-PERFORMANCE CLAUSES

Insurance contracts commonly contain an �Act of God�clause.

This says the insurance company does not have to pay if the loss iscaused by an unforeseen event.

Incentive contracts often allow for non-performance if an unforeseenevent occurs.

e.g. Builders are allowed to abandon a contract if costs are raisedsigni�cantly for some unforeseen reason.

The contract between Southwest Water and its regulator requires the�rm to prevent pollution from entering rivers but allows fornon-performance when �exceptional�events happen.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 4 / 29

NON-PERFORMANCE CLAUSES

Insurance contracts commonly contain an �Act of God�clause.

This says the insurance company does not have to pay if the loss iscaused by an unforeseen event.

Incentive contracts often allow for non-performance if an unforeseenevent occurs.

e.g. Builders are allowed to abandon a contract if costs are raisedsigni�cantly for some unforeseen reason.

The contract between Southwest Water and its regulator requires the�rm to prevent pollution from entering rivers but allows fornon-performance when �exceptional�events happen.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 4 / 29

NON-PERFORMANCE CLAUSES

Insurance contracts commonly contain an �Act of God�clause.

This says the insurance company does not have to pay if the loss iscaused by an unforeseen event.

Incentive contracts often allow for non-performance if an unforeseenevent occurs.

e.g. Builders are allowed to abandon a contract if costs are raisedsigni�cantly for some unforeseen reason.

The contract between Southwest Water and its regulator requires the�rm to prevent pollution from entering rivers but allows fornon-performance when �exceptional�events happen.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 4 / 29

NON-PERFORMANCE CLAUSES

Insurance contracts commonly contain an �Act of God�clause.

This says the insurance company does not have to pay if the loss iscaused by an unforeseen event.

Incentive contracts often allow for non-performance if an unforeseenevent occurs.

e.g. Builders are allowed to abandon a contract if costs are raisedsigni�cantly for some unforeseen reason.

The contract between Southwest Water and its regulator requires the�rm to prevent pollution from entering rivers but allows fornon-performance when �exceptional�events happen.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 4 / 29

NON-PERFORMANCE CLAUSES

Insurance contracts commonly contain an �Act of God�clause.

This says the insurance company does not have to pay if the loss iscaused by an unforeseen event.

Incentive contracts often allow for non-performance if an unforeseenevent occurs.

e.g. Builders are allowed to abandon a contract if costs are raisedsigni�cantly for some unforeseen reason.

The contract between Southwest Water and its regulator requires the�rm to prevent pollution from entering rivers but allows fornon-performance when �exceptional�events happen.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 4 / 29

CONTRACT LAW

Even when a contract does not contain an explicit act of God clause,common law will often not enforce the contract if an unforeseen eventoccurs.

In contract law there are number of legal doctrines that excuseperformance if extreme events occur.

Impossibility: In modern contract law �Suez�cases form the basis ofthis doctrine.

One contracting party bears all extreme risk in the contractualrelationship.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 5 / 29

CONTRACT LAW

Even when a contract does not contain an explicit act of God clause,common law will often not enforce the contract if an unforeseen eventoccurs.

In contract law there are number of legal doctrines that excuseperformance if extreme events occur.

Impossibility: In modern contract law �Suez�cases form the basis ofthis doctrine.

One contracting party bears all extreme risk in the contractualrelationship.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 5 / 29

CONTRACT LAW

Even when a contract does not contain an explicit act of God clause,common law will often not enforce the contract if an unforeseen eventoccurs.

In contract law there are number of legal doctrines that excuseperformance if extreme events occur.

Impossibility: In modern contract law �Suez�cases form the basis ofthis doctrine.

One contracting party bears all extreme risk in the contractualrelationship.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 5 / 29

CONTRACT LAW

Even when a contract does not contain an explicit act of God clause,common law will often not enforce the contract if an unforeseen eventoccurs.

In contract law there are number of legal doctrines that excuseperformance if extreme events occur.

Impossibility: In modern contract law �Suez�cases form the basis ofthis doctrine.

One contracting party bears all extreme risk in the contractualrelationship.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 5 / 29

POSSIBLE EXPLANATIONS

Extreme event insurance is di¤erent.

Capital market imperfections.

Moral hazard and asymmetric information.

Behavioural explanations:

Fairness,Dread,Ambiguity aversion.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 6 / 29

POSSIBLE EXPLANATIONS

Extreme event insurance is di¤erent.

Capital market imperfections.

Moral hazard and asymmetric information.

Behavioural explanations:

Fairness,Dread,Ambiguity aversion.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 6 / 29

POSSIBLE EXPLANATIONS

Extreme event insurance is di¤erent.

Capital market imperfections.

Moral hazard and asymmetric information.

Behavioural explanations:

Fairness,Dread,Ambiguity aversion.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 6 / 29

POSSIBLE EXPLANATIONS

Extreme event insurance is di¤erent.

Capital market imperfections.

Moral hazard and asymmetric information.

Behavioural explanations:

Fairness,Dread,Ambiguity aversion.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 6 / 29

POSSIBLE EXPLANATIONS

Extreme event insurance is di¤erent.

Capital market imperfections.

Moral hazard and asymmetric information.

Behavioural explanations:

Fairness,

Dread,Ambiguity aversion.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 6 / 29

POSSIBLE EXPLANATIONS

Extreme event insurance is di¤erent.

Capital market imperfections.

Moral hazard and asymmetric information.

Behavioural explanations:

Fairness,Dread,

Ambiguity aversion.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 6 / 29

POSSIBLE EXPLANATIONS

Extreme event insurance is di¤erent.

Capital market imperfections.

Moral hazard and asymmetric information.

Behavioural explanations:

Fairness,Dread,Ambiguity aversion.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 6 / 29

CATASTROPHE BONDS

ISSUED BY INSURANCE COMPANIES

FORGIVE INTEREST AND PRINCIPAL IN THE EVENT OF ASPECIFIED CATASTROPHE.

NOT APPARENTLY CORRELATED WITH MARKET RISK.

CAPM IMPLIES THE YIELD ON CATASTROPHE BONDS TO BEAPPROXIMATELY EQUAL TO THE EXPECTED LOSS.

IN FACT, THE YIELD ON CATASTROPHE BONDS IS SEVENTIMES THE EXPECTED LOSS.

THUS WE CAN ALSO SEE AMBIGUITY-AVERSION IN MARKETEVIDENCE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 7 / 29

CATASTROPHE BONDS

ISSUED BY INSURANCE COMPANIES

FORGIVE INTEREST AND PRINCIPAL IN THE EVENT OF ASPECIFIED CATASTROPHE.

NOT APPARENTLY CORRELATED WITH MARKET RISK.

CAPM IMPLIES THE YIELD ON CATASTROPHE BONDS TO BEAPPROXIMATELY EQUAL TO THE EXPECTED LOSS.

IN FACT, THE YIELD ON CATASTROPHE BONDS IS SEVENTIMES THE EXPECTED LOSS.

THUS WE CAN ALSO SEE AMBIGUITY-AVERSION IN MARKETEVIDENCE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 7 / 29

CATASTROPHE BONDS

ISSUED BY INSURANCE COMPANIES

FORGIVE INTEREST AND PRINCIPAL IN THE EVENT OF ASPECIFIED CATASTROPHE.

NOT APPARENTLY CORRELATED WITH MARKET RISK.

CAPM IMPLIES THE YIELD ON CATASTROPHE BONDS TO BEAPPROXIMATELY EQUAL TO THE EXPECTED LOSS.

IN FACT, THE YIELD ON CATASTROPHE BONDS IS SEVENTIMES THE EXPECTED LOSS.

THUS WE CAN ALSO SEE AMBIGUITY-AVERSION IN MARKETEVIDENCE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 7 / 29

CATASTROPHE BONDS

ISSUED BY INSURANCE COMPANIES

FORGIVE INTEREST AND PRINCIPAL IN THE EVENT OF ASPECIFIED CATASTROPHE.

NOT APPARENTLY CORRELATED WITH MARKET RISK.

CAPM IMPLIES THE YIELD ON CATASTROPHE BONDS TO BEAPPROXIMATELY EQUAL TO THE EXPECTED LOSS.

IN FACT, THE YIELD ON CATASTROPHE BONDS IS SEVENTIMES THE EXPECTED LOSS.

THUS WE CAN ALSO SEE AMBIGUITY-AVERSION IN MARKETEVIDENCE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 7 / 29

CATASTROPHE BONDS

ISSUED BY INSURANCE COMPANIES

FORGIVE INTEREST AND PRINCIPAL IN THE EVENT OF ASPECIFIED CATASTROPHE.

NOT APPARENTLY CORRELATED WITH MARKET RISK.

CAPM IMPLIES THE YIELD ON CATASTROPHE BONDS TO BEAPPROXIMATELY EQUAL TO THE EXPECTED LOSS.

IN FACT, THE YIELD ON CATASTROPHE BONDS IS SEVENTIMES THE EXPECTED LOSS.

THUS WE CAN ALSO SEE AMBIGUITY-AVERSION IN MARKETEVIDENCE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 7 / 29

CATASTROPHE BONDS

ISSUED BY INSURANCE COMPANIES

FORGIVE INTEREST AND PRINCIPAL IN THE EVENT OF ASPECIFIED CATASTROPHE.

NOT APPARENTLY CORRELATED WITH MARKET RISK.

CAPM IMPLIES THE YIELD ON CATASTROPHE BONDS TO BEAPPROXIMATELY EQUAL TO THE EXPECTED LOSS.

IN FACT, THE YIELD ON CATASTROPHE BONDS IS SEVENTIMES THE EXPECTED LOSS.

THUS WE CAN ALSO SEE AMBIGUITY-AVERSION IN MARKETEVIDENCE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 7 / 29

POLICY QUESTIONS

Can private insurance provide cover against extreme risk?

Some economists argue that insurance markets for extreme risk havebeen destroyed by excessive regulation.

We suggest that extreme risks are ambiguous. Ambiguity-aversionprovides a barrier to risk-sharing.

It is plausible that there is ambiguity concerning 500 year losses 250year losses or even 100 year losses.

In most cases there is insu¢ cient data to assign frequentistprobabilities

Even for 100 year losses their impact on a technologically advancedsociety is unknown.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 8 / 29

POLICY QUESTIONS

Can private insurance provide cover against extreme risk?

Some economists argue that insurance markets for extreme risk havebeen destroyed by excessive regulation.

We suggest that extreme risks are ambiguous. Ambiguity-aversionprovides a barrier to risk-sharing.

It is plausible that there is ambiguity concerning 500 year losses 250year losses or even 100 year losses.

In most cases there is insu¢ cient data to assign frequentistprobabilities

Even for 100 year losses their impact on a technologically advancedsociety is unknown.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 8 / 29

POLICY QUESTIONS

Can private insurance provide cover against extreme risk?

Some economists argue that insurance markets for extreme risk havebeen destroyed by excessive regulation.

We suggest that extreme risks are ambiguous. Ambiguity-aversionprovides a barrier to risk-sharing.

It is plausible that there is ambiguity concerning 500 year losses 250year losses or even 100 year losses.

In most cases there is insu¢ cient data to assign frequentistprobabilities

Even for 100 year losses their impact on a technologically advancedsociety is unknown.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 8 / 29

POLICY QUESTIONS

Can private insurance provide cover against extreme risk?

Some economists argue that insurance markets for extreme risk havebeen destroyed by excessive regulation.

We suggest that extreme risks are ambiguous. Ambiguity-aversionprovides a barrier to risk-sharing.

It is plausible that there is ambiguity concerning 500 year losses 250year losses or even 100 year losses.

In most cases there is insu¢ cient data to assign frequentistprobabilities

Even for 100 year losses their impact on a technologically advancedsociety is unknown.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 8 / 29

POLICY QUESTIONS

Can private insurance provide cover against extreme risk?

Some economists argue that insurance markets for extreme risk havebeen destroyed by excessive regulation.

We suggest that extreme risks are ambiguous. Ambiguity-aversionprovides a barrier to risk-sharing.

It is plausible that there is ambiguity concerning 500 year losses 250year losses or even 100 year losses.

In most cases there is insu¢ cient data to assign frequentistprobabilities

Even for 100 year losses their impact on a technologically advancedsociety is unknown.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 8 / 29

POLICY QUESTIONS

Can private insurance provide cover against extreme risk?

Some economists argue that insurance markets for extreme risk havebeen destroyed by excessive regulation.

We suggest that extreme risks are ambiguous. Ambiguity-aversionprovides a barrier to risk-sharing.

It is plausible that there is ambiguity concerning 500 year losses 250year losses or even 100 year losses.

In most cases there is insu¢ cient data to assign frequentistprobabilities

Even for 100 year losses their impact on a technologically advancedsociety is unknown.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 8 / 29

LITERATURE REVIEW IRisk sharing and Ambiguity

Chateauneuf, Dana and Tallon (2000).Dana (2004), Carlier and Dana (2003).

General equilibrium model with complete markets.All perceive the same events to be ambiguous.Any equilibrium with ambiguity is also an equilibrium with SEUpreferences.In their model ambiguity does not cause failures of risk-sharing

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 9 / 29

LITERATURE REVIEW IRisk sharing and Ambiguity

Chateauneuf, Dana and Tallon (2000).Dana (2004), Carlier and Dana (2003).

General equilibrium model with complete markets.

All perceive the same events to be ambiguous.Any equilibrium with ambiguity is also an equilibrium with SEUpreferences.In their model ambiguity does not cause failures of risk-sharing

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 9 / 29

LITERATURE REVIEW IRisk sharing and Ambiguity

Chateauneuf, Dana and Tallon (2000).Dana (2004), Carlier and Dana (2003).

General equilibrium model with complete markets.All perceive the same events to be ambiguous.

Any equilibrium with ambiguity is also an equilibrium with SEUpreferences.In their model ambiguity does not cause failures of risk-sharing

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 9 / 29

LITERATURE REVIEW IRisk sharing and Ambiguity

Chateauneuf, Dana and Tallon (2000).Dana (2004), Carlier and Dana (2003).

General equilibrium model with complete markets.All perceive the same events to be ambiguous.Any equilibrium with ambiguity is also an equilibrium with SEUpreferences.

In their model ambiguity does not cause failures of risk-sharing

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 9 / 29

LITERATURE REVIEW IRisk sharing and Ambiguity

Chateauneuf, Dana and Tallon (2000).Dana (2004), Carlier and Dana (2003).

General equilibrium model with complete markets.All perceive the same events to be ambiguous.Any equilibrium with ambiguity is also an equilibrium with SEUpreferences.In their model ambiguity does not cause failures of risk-sharing

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 9 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.Ambiguity is modelled by incompleteness of preferences.Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.Ambiguity is modelled by incompleteness of preferences.Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.Ambiguity is modelled by incompleteness of preferences.Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.

Ambiguity is modelled by incompleteness of preferences.Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.Ambiguity is modelled by incompleteness of preferences.

Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.Ambiguity is modelled by incompleteness of preferences.Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.Ambiguity is modelled by incompleteness of preferences.Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.Ambiguity is modelled by incompleteness of preferences.Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.Ambiguity is modelled by incompleteness of preferences.Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

LITERATURE REVIEW II

Mukherji and Tallon (2003).

Show that ambiguous risks cannot be diversi�ed.

Rigotti and Shannon (2001)

Use the Bewely model of ambiguity.Ambiguity is modelled by incompleteness of preferences.Incentives

Mukherji (1997), (2003).

Cost plus contracts may be desirable with ambiguity.

Chakravarty and Macleod (2005).

Incentive contracts in the building industry.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 10 / 29

THE ELLSBERG PARADOX

AN URN CONTAINS 90 BALLS, 30 OF WHICH ARE RED R. THEREMAINDER ARE EITHER BLUE B OR YELLOW Y . THEPROPORTION OF BLUE AND YELLOW BALLS IS NOT SPECIFIED.

30 60R B Y

Choice 1 a 100 0 0p

b 0 100 0Choice 2 c 100 0 100

d 0 100 100p

MOST SUBJECTS PREFER a TO b AND d TO c .

THIS IS NOT COMPATIBLE WITH MAXIMISING EXPECTEDUTILITY.

NOR IS IT COMPATIBLE WITH ANY OTHER THEORY WHICHREPRESENTS BELIEFS AS PROBABILITIES.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 11 / 29

THE ELLSBERG PARADOX

AN URN CONTAINS 90 BALLS, 30 OF WHICH ARE RED R. THEREMAINDER ARE EITHER BLUE B OR YELLOW Y . THEPROPORTION OF BLUE AND YELLOW BALLS IS NOT SPECIFIED.

30 60R B Y

Choice 1 a 100 0 0p

b 0 100 0Choice 2 c 100 0 100

d 0 100 100p

MOST SUBJECTS PREFER a TO b AND d TO c .

THIS IS NOT COMPATIBLE WITH MAXIMISING EXPECTEDUTILITY.

NOR IS IT COMPATIBLE WITH ANY OTHER THEORY WHICHREPRESENTS BELIEFS AS PROBABILITIES.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 11 / 29

THE ELLSBERG PARADOX

AN URN CONTAINS 90 BALLS, 30 OF WHICH ARE RED R. THEREMAINDER ARE EITHER BLUE B OR YELLOW Y . THEPROPORTION OF BLUE AND YELLOW BALLS IS NOT SPECIFIED.

30 60R B Y

Choice 1 a 100 0 0p

b 0 100 0Choice 2 c 100 0 100

d 0 100 100p

MOST SUBJECTS PREFER a TO b AND d TO c .

THIS IS NOT COMPATIBLE WITH MAXIMISING EXPECTEDUTILITY.

NOR IS IT COMPATIBLE WITH ANY OTHER THEORY WHICHREPRESENTS BELIEFS AS PROBABILITIES.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 11 / 29

THE ELLSBERG PARADOX

AN URN CONTAINS 90 BALLS, 30 OF WHICH ARE RED R. THEREMAINDER ARE EITHER BLUE B OR YELLOW Y . THEPROPORTION OF BLUE AND YELLOW BALLS IS NOT SPECIFIED.

30 60R B Y

Choice 1 a 100 0 0p

b 0 100 0Choice 2 c 100 0 100

d 0 100 100p

MOST SUBJECTS PREFER a TO b AND d TO c .

THIS IS NOT COMPATIBLE WITH MAXIMISING EXPECTEDUTILITY.

NOR IS IT COMPATIBLE WITH ANY OTHER THEORY WHICHREPRESENTS BELIEFS AS PROBABILITIES.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 11 / 29

CONCLUSIONS FROM THE ELLSBERG PARADOX

BELIEFS ARE NOT REPRESENTED BY PROBABILITIES.

PEOPLE RESPOND TO UNCERTAINTY BY BEHAVINGCAUTIOUSLY.

NEURO-ECONOMICS

SUBJECTS ARE ASKED TO MAKE ECONOMIC DECISIONSWHILE INSIDE BRAIN SCANNING MACHINES.

IT IS FOUND THAT AMBIGUOUS DECISIONS ARE PROCESSEDBY DIFFERENT PARTS OF THE BRAIN TO RISKY DECISIONS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 12 / 29

CONCLUSIONS FROM THE ELLSBERG PARADOX

BELIEFS ARE NOT REPRESENTED BY PROBABILITIES.

PEOPLE RESPOND TO UNCERTAINTY BY BEHAVINGCAUTIOUSLY.

NEURO-ECONOMICS

SUBJECTS ARE ASKED TO MAKE ECONOMIC DECISIONSWHILE INSIDE BRAIN SCANNING MACHINES.

IT IS FOUND THAT AMBIGUOUS DECISIONS ARE PROCESSEDBY DIFFERENT PARTS OF THE BRAIN TO RISKY DECISIONS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 12 / 29

CONCLUSIONS FROM THE ELLSBERG PARADOX

BELIEFS ARE NOT REPRESENTED BY PROBABILITIES.

PEOPLE RESPOND TO UNCERTAINTY BY BEHAVINGCAUTIOUSLY.

NEURO-ECONOMICS

SUBJECTS ARE ASKED TO MAKE ECONOMIC DECISIONSWHILE INSIDE BRAIN SCANNING MACHINES.

IT IS FOUND THAT AMBIGUOUS DECISIONS ARE PROCESSEDBY DIFFERENT PARTS OF THE BRAIN TO RISKY DECISIONS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 12 / 29

CONCLUSIONS FROM THE ELLSBERG PARADOX

BELIEFS ARE NOT REPRESENTED BY PROBABILITIES.

PEOPLE RESPOND TO UNCERTAINTY BY BEHAVINGCAUTIOUSLY.

NEURO-ECONOMICS

SUBJECTS ARE ASKED TO MAKE ECONOMIC DECISIONSWHILE INSIDE BRAIN SCANNING MACHINES.

IT IS FOUND THAT AMBIGUOUS DECISIONS ARE PROCESSEDBY DIFFERENT PARTS OF THE BRAIN TO RISKY DECISIONS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 12 / 29

CONCLUSIONS FROM THE ELLSBERG PARADOX

BELIEFS ARE NOT REPRESENTED BY PROBABILITIES.

PEOPLE RESPOND TO UNCERTAINTY BY BEHAVINGCAUTIOUSLY.

NEURO-ECONOMICS

SUBJECTS ARE ASKED TO MAKE ECONOMIC DECISIONSWHILE INSIDE BRAIN SCANNING MACHINES.

IT IS FOUND THAT AMBIGUOUS DECISIONS ARE PROCESSEDBY DIFFERENT PARTS OF THE BRAIN TO RISKY DECISIONS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 12 / 29

CONCLUSIONS FROM THE ELLSBERG PARADOX

BELIEFS ARE NOT REPRESENTED BY PROBABILITIES.

PEOPLE RESPOND TO UNCERTAINTY BY BEHAVINGCAUTIOUSLY.

NEURO-ECONOMICS

SUBJECTS ARE ASKED TO MAKE ECONOMIC DECISIONSWHILE INSIDE BRAIN SCANNING MACHINES.

IT IS FOUND THAT AMBIGUOUS DECISIONS ARE PROCESSEDBY DIFFERENT PARTS OF THE BRAIN TO RISKY DECISIONS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 12 / 29

CONCLUSIONS FROM THE ELLSBERG PARADOX

BELIEFS ARE NOT REPRESENTED BY PROBABILITIES.

PEOPLE RESPOND TO UNCERTAINTY BY BEHAVINGCAUTIOUSLY.

NEURO-ECONOMICS

SUBJECTS ARE ASKED TO MAKE ECONOMIC DECISIONSWHILE INSIDE BRAIN SCANNING MACHINES.

IT IS FOUND THAT AMBIGUOUS DECISIONS ARE PROCESSEDBY DIFFERENT PARTS OF THE BRAIN TO RISKY DECISIONS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 12 / 29

KNIGHTIAN UNCERTAINTY

PEOPLE DO NOT KNOW THE PROBABILITIES NOR DO THEYBEHAVE AS IF THEY DO.

WE USE A MODEL OF KNIGHTIAN UNCERTAINTY DUE TOSCHMEIDLER (ECONOMETRICA 1989), WHICH REPRESENTSBELIEFS AS CAPACITIES.

PREFERENCES ARE REPRESENTED BY MAXIMISING THEEXPECTED VALUE OF UTILITY WITH RESPECT TO ACAPACITY.

THE EXPECTATION IS EXPRESSED AS A CHOQUET INTEGRAL.

ACT a IS PREFERRED TO ACT b IF :

minp2C

Epu(a) > minp2C

Epu(b),

WHERE C DENOTES THE PROBABILITY DISTRIBUTIONSBELIEVED TO BE POSSIBLE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 13 / 29

KNIGHTIAN UNCERTAINTY

PEOPLE DO NOT KNOW THE PROBABILITIES NOR DO THEYBEHAVE AS IF THEY DO.

WE USE A MODEL OF KNIGHTIAN UNCERTAINTY DUE TOSCHMEIDLER (ECONOMETRICA 1989), WHICH REPRESENTSBELIEFS AS CAPACITIES.

PREFERENCES ARE REPRESENTED BY MAXIMISING THEEXPECTED VALUE OF UTILITY WITH RESPECT TO ACAPACITY.

THE EXPECTATION IS EXPRESSED AS A CHOQUET INTEGRAL.

ACT a IS PREFERRED TO ACT b IF :

minp2C

Epu(a) > minp2C

Epu(b),

WHERE C DENOTES THE PROBABILITY DISTRIBUTIONSBELIEVED TO BE POSSIBLE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 13 / 29

KNIGHTIAN UNCERTAINTY

PEOPLE DO NOT KNOW THE PROBABILITIES NOR DO THEYBEHAVE AS IF THEY DO.

WE USE A MODEL OF KNIGHTIAN UNCERTAINTY DUE TOSCHMEIDLER (ECONOMETRICA 1989), WHICH REPRESENTSBELIEFS AS CAPACITIES.

PREFERENCES ARE REPRESENTED BY MAXIMISING THEEXPECTED VALUE OF UTILITY WITH RESPECT TO ACAPACITY.

THE EXPECTATION IS EXPRESSED AS A CHOQUET INTEGRAL.

ACT a IS PREFERRED TO ACT b IF :

minp2C

Epu(a) > minp2C

Epu(b),

WHERE C DENOTES THE PROBABILITY DISTRIBUTIONSBELIEVED TO BE POSSIBLE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 13 / 29

KNIGHTIAN UNCERTAINTY

PEOPLE DO NOT KNOW THE PROBABILITIES NOR DO THEYBEHAVE AS IF THEY DO.

WE USE A MODEL OF KNIGHTIAN UNCERTAINTY DUE TOSCHMEIDLER (ECONOMETRICA 1989), WHICH REPRESENTSBELIEFS AS CAPACITIES.

PREFERENCES ARE REPRESENTED BY MAXIMISING THEEXPECTED VALUE OF UTILITY WITH RESPECT TO ACAPACITY.

THE EXPECTATION IS EXPRESSED AS A CHOQUET INTEGRAL.

ACT a IS PREFERRED TO ACT b IF :

minp2C

Epu(a) > minp2C

Epu(b),

WHERE C DENOTES THE PROBABILITY DISTRIBUTIONSBELIEVED TO BE POSSIBLE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 13 / 29

KNIGHTIAN UNCERTAINTY

PEOPLE DO NOT KNOW THE PROBABILITIES NOR DO THEYBEHAVE AS IF THEY DO.

WE USE A MODEL OF KNIGHTIAN UNCERTAINTY DUE TOSCHMEIDLER (ECONOMETRICA 1989), WHICH REPRESENTSBELIEFS AS CAPACITIES.

PREFERENCES ARE REPRESENTED BY MAXIMISING THEEXPECTED VALUE OF UTILITY WITH RESPECT TO ACAPACITY.

THE EXPECTATION IS EXPRESSED AS A CHOQUET INTEGRAL.

ACT a IS PREFERRED TO ACT b IF :

minp2C

Epu(a) > minp2C

Epu(b),

WHERE C DENOTES THE PROBABILITY DISTRIBUTIONSBELIEVED TO BE POSSIBLE.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 13 / 29

minp2C

Epu(a) > minp2C

Epu(b),

Intuition WHEN THE DECISION-MAKER DOES NOT KNOWTHE TRUE PROBABILITIES (S)HE CONSIDERS A NUMBER OFPROBABILITY DISTRIBUTIONS TO BE POSSIBLE ANDBEHAVES CAUTIOUSLY.

ACTS ARE EVALUATED BY THE LEAST FAVOUR-ABLEPROBABILITY DISTRIBUTION.

ADVANTAGES

EXPECTED UTILITY IS A SPECIAL CASE.RELATIVELY EASY TO USE IN ECONOMIC APPLICATIONS.

CORPORATE PLANNING MANAGERS ARE ENCOURAGED TOPLAN FOR A NUMBER OF ALTERNATIVE SCENARIOS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 14 / 29

minp2C

Epu(a) > minp2C

Epu(b),

Intuition WHEN THE DECISION-MAKER DOES NOT KNOWTHE TRUE PROBABILITIES (S)HE CONSIDERS A NUMBER OFPROBABILITY DISTRIBUTIONS TO BE POSSIBLE ANDBEHAVES CAUTIOUSLY.

ACTS ARE EVALUATED BY THE LEAST FAVOUR-ABLEPROBABILITY DISTRIBUTION.

ADVANTAGES

EXPECTED UTILITY IS A SPECIAL CASE.RELATIVELY EASY TO USE IN ECONOMIC APPLICATIONS.

CORPORATE PLANNING MANAGERS ARE ENCOURAGED TOPLAN FOR A NUMBER OF ALTERNATIVE SCENARIOS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 14 / 29

minp2C

Epu(a) > minp2C

Epu(b),

Intuition WHEN THE DECISION-MAKER DOES NOT KNOWTHE TRUE PROBABILITIES (S)HE CONSIDERS A NUMBER OFPROBABILITY DISTRIBUTIONS TO BE POSSIBLE ANDBEHAVES CAUTIOUSLY.

ACTS ARE EVALUATED BY THE LEAST FAVOUR-ABLEPROBABILITY DISTRIBUTION.

ADVANTAGES

EXPECTED UTILITY IS A SPECIAL CASE.RELATIVELY EASY TO USE IN ECONOMIC APPLICATIONS.

CORPORATE PLANNING MANAGERS ARE ENCOURAGED TOPLAN FOR A NUMBER OF ALTERNATIVE SCENARIOS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 14 / 29

minp2C

Epu(a) > minp2C

Epu(b),

Intuition WHEN THE DECISION-MAKER DOES NOT KNOWTHE TRUE PROBABILITIES (S)HE CONSIDERS A NUMBER OFPROBABILITY DISTRIBUTIONS TO BE POSSIBLE ANDBEHAVES CAUTIOUSLY.

ACTS ARE EVALUATED BY THE LEAST FAVOUR-ABLEPROBABILITY DISTRIBUTION.

ADVANTAGES

EXPECTED UTILITY IS A SPECIAL CASE.

RELATIVELY EASY TO USE IN ECONOMIC APPLICATIONS.

CORPORATE PLANNING MANAGERS ARE ENCOURAGED TOPLAN FOR A NUMBER OF ALTERNATIVE SCENARIOS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 14 / 29

minp2C

Epu(a) > minp2C

Epu(b),

Intuition WHEN THE DECISION-MAKER DOES NOT KNOWTHE TRUE PROBABILITIES (S)HE CONSIDERS A NUMBER OFPROBABILITY DISTRIBUTIONS TO BE POSSIBLE ANDBEHAVES CAUTIOUSLY.

ACTS ARE EVALUATED BY THE LEAST FAVOUR-ABLEPROBABILITY DISTRIBUTION.

ADVANTAGES

EXPECTED UTILITY IS A SPECIAL CASE.RELATIVELY EASY TO USE IN ECONOMIC APPLICATIONS.

CORPORATE PLANNING MANAGERS ARE ENCOURAGED TOPLAN FOR A NUMBER OF ALTERNATIVE SCENARIOS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 14 / 29

minp2C

Epu(a) > minp2C

Epu(b),

Intuition WHEN THE DECISION-MAKER DOES NOT KNOWTHE TRUE PROBABILITIES (S)HE CONSIDERS A NUMBER OFPROBABILITY DISTRIBUTIONS TO BE POSSIBLE ANDBEHAVES CAUTIOUSLY.

ACTS ARE EVALUATED BY THE LEAST FAVOUR-ABLEPROBABILITY DISTRIBUTION.

ADVANTAGES

EXPECTED UTILITY IS A SPECIAL CASE.RELATIVELY EASY TO USE IN ECONOMIC APPLICATIONS.

CORPORATE PLANNING MANAGERS ARE ENCOURAGED TOPLAN FOR A NUMBER OF ALTERNATIVE SCENARIOS.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 14 / 29

EXAMPLE

Let S1 = fs 01, s 001 g ,S2 = fs 02, s 002 g and S = S1 � S2.Assume that there are two individuals, 1 and 2, who have endowments

s 01, s02 s 01, s

002 s 001 , s

02 s 001 , s

002

ω1 2 2 0 0ω2 2 0 2 0

.

Without ambiguity the two individuals would choose to share risk.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 15 / 29

Assume

Each individual views the shock to his/her own endowment asunambiguous.

The shock to the other person�s endowment is ambiguous.

Both individuals have utility function u (x) = ln (1+ x) .

Then competitive equilibrium allocations and prices are:

s 01, s02 s 01, s

002 s 001 , s

02 s 001 , s

002

xA 2 1+ 2γ 1� 2γ 0xB 2 1� 2γ 1+ 2γ 0

p 1 32(1�γ)

32(1�γ)

3(1+γ)(1�γ)

.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 16 / 29

Assume

Each individual views the shock to his/her own endowment asunambiguous.

The shock to the other person�s endowment is ambiguous.

Both individuals have utility function u (x) = ln (1+ x) .

Then competitive equilibrium allocations and prices are:

s 01, s02 s 01, s

002 s 001 , s

02 s 001 , s

002

xA 2 1+ 2γ 1� 2γ 0xB 2 1� 2γ 1+ 2γ 0

p 1 32(1�γ)

32(1�γ)

3(1+γ)(1�γ)

.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 16 / 29

Assume

Each individual views the shock to his/her own endowment asunambiguous.

The shock to the other person�s endowment is ambiguous.

Both individuals have utility function u (x) = ln (1+ x) .

Then competitive equilibrium allocations and prices are:

s 01, s02 s 01, s

002 s 001 , s

02 s 001 , s

002

xA 2 1+ 2γ 1� 2γ 0xB 2 1� 2γ 1+ 2γ 0

p 1 32(1�γ)

32(1�γ)

3(1+γ)(1�γ)

.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 16 / 29

Assume

Each individual views the shock to his/her own endowment asunambiguous.

The shock to the other person�s endowment is ambiguous.

Both individuals have utility function u (x) = ln (1+ x) .

Then competitive equilibrium allocations and prices are:

s 01, s02 s 01, s

002 s 001 , s

02 s 001 , s

002

xA 2 1+ 2γ 1� 2γ 0xB 2 1� 2γ 1+ 2γ 0

p 1 32(1�γ)

32(1�γ)

3(1+γ)(1�γ)

.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 16 / 29

The equilibrium has the following properties:

If there is no ambiguity, γ = 0, risks are shared equally.

For 0 6 γ 6 12 trade is reduced by ambiguity.

For high levels of ambiguity, γ > 12 , there is no risk-sharing in

equilibrium.

Each individual has higher consumption in the state which iscomplementary with his/her endowment. This reduces the amount ofambiguity which (s)he faces.

As ambiguity increases, the price of the consumption good increasesin all states in which it is relatively scarce.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 17 / 29

The equilibrium has the following properties:

If there is no ambiguity, γ = 0, risks are shared equally.

For 0 6 γ 6 12 trade is reduced by ambiguity.

For high levels of ambiguity, γ > 12 , there is no risk-sharing in

equilibrium.

Each individual has higher consumption in the state which iscomplementary with his/her endowment. This reduces the amount ofambiguity which (s)he faces.

As ambiguity increases, the price of the consumption good increasesin all states in which it is relatively scarce.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 17 / 29

The equilibrium has the following properties:

If there is no ambiguity, γ = 0, risks are shared equally.

For 0 6 γ 6 12 trade is reduced by ambiguity.

For high levels of ambiguity, γ > 12 , there is no risk-sharing in

equilibrium.

Each individual has higher consumption in the state which iscomplementary with his/her endowment. This reduces the amount ofambiguity which (s)he faces.

As ambiguity increases, the price of the consumption good increasesin all states in which it is relatively scarce.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 17 / 29

The equilibrium has the following properties:

If there is no ambiguity, γ = 0, risks are shared equally.

For 0 6 γ 6 12 trade is reduced by ambiguity.

For high levels of ambiguity, γ > 12 , there is no risk-sharing in

equilibrium.

Each individual has higher consumption in the state which iscomplementary with his/her endowment. This reduces the amount ofambiguity which (s)he faces.

As ambiguity increases, the price of the consumption good increasesin all states in which it is relatively scarce.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 17 / 29

The equilibrium has the following properties:

If there is no ambiguity, γ = 0, risks are shared equally.

For 0 6 γ 6 12 trade is reduced by ambiguity.

For high levels of ambiguity, γ > 12 , there is no risk-sharing in

equilibrium.

Each individual has higher consumption in the state which iscomplementary with his/her endowment. This reduces the amount ofambiguity which (s)he faces.

As ambiguity increases, the price of the consumption good increasesin all states in which it is relatively scarce.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 17 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.

There are n individuals 1 6 i 6 n and one physical commodity.Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.There are markets in all state contingent commodities.The endowment of individual i , ωi (si ) is independent of s�i .Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.The risks di¤erent individuals face are independent.Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.There are n individuals 1 6 i 6 n and one physical commodity.

Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.There are markets in all state contingent commodities.The endowment of individual i , ωi (si ) is independent of s�i .Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.The risks di¤erent individuals face are independent.Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.There are n individuals 1 6 i 6 n and one physical commodity.Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.

The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.There are markets in all state contingent commodities.The endowment of individual i , ωi (si ) is independent of s�i .Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.The risks di¤erent individuals face are independent.Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.There are n individuals 1 6 i 6 n and one physical commodity.Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.

There are markets in all state contingent commodities.The endowment of individual i , ωi (si ) is independent of s�i .Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.The risks di¤erent individuals face are independent.Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.There are n individuals 1 6 i 6 n and one physical commodity.Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.There are markets in all state contingent commodities.

The endowment of individual i , ωi (si ) is independent of s�i .Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.The risks di¤erent individuals face are independent.Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.There are n individuals 1 6 i 6 n and one physical commodity.Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.There are markets in all state contingent commodities.The endowment of individual i , ωi (si ) is independent of s�i .

Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.The risks di¤erent individuals face are independent.Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.There are n individuals 1 6 i 6 n and one physical commodity.Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.There are markets in all state contingent commodities.The endowment of individual i , ωi (si ) is independent of s�i .Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.

The risks di¤erent individuals face are independent.Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.There are n individuals 1 6 i 6 n and one physical commodity.Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.There are markets in all state contingent commodities.The endowment of individual i , ωi (si ) is independent of s�i .Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.The risks di¤erent individuals face are independent.

Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.There are n individuals 1 6 i 6 n and one physical commodity.Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.There are markets in all state contingent commodities.The endowment of individual i , ωi (si ) is independent of s�i .Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.The risks di¤erent individuals face are independent.Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .

Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

THE ECONOMY

We consider an exchange economy with uncertainty.There are n individuals 1 6 i 6 n and one physical commodity.Each individual�s endowment is subject to shocks, which (s)heperceives as being non-ambiguous but which appear ambiguous toothers.The state space S , is a Cartesian product, S = S1 � S2 � ...� Sn.There are markets in all state contingent commodities.The endowment of individual i , ωi (si ) is independent of s�i .Si is a set of factors which a¤ect i�s endowment but not theendowment of any other individual.The risks di¤erent individuals face are independent.Individual i has beliefs represented by a convex capacity νi on S withan additive marginal πi on Si . This implies thatZ

f (si , s�i ) dνi (s) = ∑si2Si

πisi

Zf (si , s�i ) dµi (s�i ) ,

where µi is a convex capacity on S�i .Let π be the additive probability distribution on S , which is theindependent product of the marginals πi for 1 6 i 6 I .

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 18 / 29

LINEAR UTILITY

Assume that utility is linear, which removes any reason forrisk-sharing other than a response to ambiguity.

We make the following assumption which says that there is limitedagreement between the beliefs of di¤erent individuals.

Assumption

π�i 2 int C�µi�, where C

�µi�denotes the core of the capacity µi .

If individuals are risk neutral and ambiguity-averse then, in any ParetoOptimum, each individual consumes his/her endowment plus ariskless lump sum tax or subsidy.

Hence ambiguous risks are not shared.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 19 / 29

LINEAR UTILITY

Assume that utility is linear, which removes any reason forrisk-sharing other than a response to ambiguity.

We make the following assumption which says that there is limitedagreement between the beliefs of di¤erent individuals.

Assumption

π�i 2 int C�µi�, where C

�µi�denotes the core of the capacity µi .

If individuals are risk neutral and ambiguity-averse then, in any ParetoOptimum, each individual consumes his/her endowment plus ariskless lump sum tax or subsidy.

Hence ambiguous risks are not shared.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 19 / 29

LINEAR UTILITY

Assume that utility is linear, which removes any reason forrisk-sharing other than a response to ambiguity.

We make the following assumption which says that there is limitedagreement between the beliefs of di¤erent individuals.

Assumption

π�i 2 int C�µi�, where C

�µi�denotes the core of the capacity µi .

If individuals are risk neutral and ambiguity-averse then, in any ParetoOptimum, each individual consumes his/her endowment plus ariskless lump sum tax or subsidy.

Hence ambiguous risks are not shared.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 19 / 29

LINEAR UTILITY

Assume that utility is linear, which removes any reason forrisk-sharing other than a response to ambiguity.

We make the following assumption which says that there is limitedagreement between the beliefs of di¤erent individuals.

Assumption

π�i 2 int C�µi�, where C

�µi�denotes the core of the capacity µi .

If individuals are risk neutral and ambiguity-averse then, in any ParetoOptimum, each individual consumes his/her endowment plus ariskless lump sum tax or subsidy.

Hence ambiguous risks are not shared.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 19 / 29

LINEAR UTILITY

Assume that utility is linear, which removes any reason forrisk-sharing other than a response to ambiguity.

We make the following assumption which says that there is limitedagreement between the beliefs of di¤erent individuals.

Assumption

π�i 2 int C�µi�, where C

�µi�denotes the core of the capacity µi .

If individuals are risk neutral and ambiguity-averse then, in any ParetoOptimum, each individual consumes his/her endowment plus ariskless lump sum tax or subsidy.

Hence ambiguous risks are not shared.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 19 / 29

LINEAR UTILITY

Assume that utility is linear, which removes any reason forrisk-sharing other than a response to ambiguity.

We make the following assumption which says that there is limitedagreement between the beliefs of di¤erent individuals.

Assumption

π�i 2 int C�µi�, where C

�µi�denotes the core of the capacity µi .

If individuals are risk neutral and ambiguity-averse then, in any ParetoOptimum, each individual consumes his/her endowment plus ariskless lump sum tax or subsidy.

Hence ambiguous risks are not shared.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 19 / 29

Proposition

If ui is linear for 1 6 i 6 n, then an allocation x = hx1, ..., xni is Paretooptimal if and only if it has the form xi (s) = ωi (si ) + ηi , where∑ni=1 ηi = 0.

The proposition holds even if individuals only display small degrees ofambiguity-aversion.

In many contracts some of the parties are large �rms or insurancecompanies which one would expect to be risk-neutral in the usualsense of having linear utility.

This result would be applicable to the usual situation in theprincipal-agent model where the principal is assumed to have linearutility.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 20 / 29

Proposition

If ui is linear for 1 6 i 6 n, then an allocation x = hx1, ..., xni is Paretooptimal if and only if it has the form xi (s) = ωi (si ) + ηi , where∑ni=1 ηi = 0.

The proposition holds even if individuals only display small degrees ofambiguity-aversion.

In many contracts some of the parties are large �rms or insurancecompanies which one would expect to be risk-neutral in the usualsense of having linear utility.

This result would be applicable to the usual situation in theprincipal-agent model where the principal is assumed to have linearutility.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 20 / 29

Proposition

If ui is linear for 1 6 i 6 n, then an allocation x = hx1, ..., xni is Paretooptimal if and only if it has the form xi (s) = ωi (si ) + ηi , where∑ni=1 ηi = 0.

The proposition holds even if individuals only display small degrees ofambiguity-aversion.

In many contracts some of the parties are large �rms or insurancecompanies which one would expect to be risk-neutral in the usualsense of having linear utility.

This result would be applicable to the usual situation in theprincipal-agent model where the principal is assumed to have linearutility.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 20 / 29

Proposition

If ui is linear for 1 6 i 6 n, then an allocation x = hx1, ..., xni is Paretooptimal if and only if it has the form xi (s) = ωi (si ) + ηi , where∑ni=1 ηi = 0.

The proposition holds even if individuals only display small degrees ofambiguity-aversion.

In many contracts some of the parties are large �rms or insurancecompanies which one would expect to be risk-neutral in the usualsense of having linear utility.

This result would be applicable to the usual situation in theprincipal-agent model where the principal is assumed to have linearutility.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 20 / 29

CONCAVE UTILITY

Now assume that ui is concave for 1 6 i 6 n. Apart from ambiguity,our assumptions are standard in risk sharing models.

With concave utility there is a trade-o¤.

Ambiguity-aversion creates a barrier to sharing ambiguous risks.Diminishing marginal utility of wealth makes it desirable to smoothconsumption across states.

If there is su¢ cient ambiguity-aversion, at any Pareto optimum, eachindividual consumes his/her endowment plus a riskless lump sum taxor subsidy.It follows that competitive equilibrium is unique and involves no trade.The result would also apply to e¢ cient Coasian bargaining.

Theorem

If ui is concave for 1 6 i 6 n, there exists γ̄ such that if 1 > γi > γ̄, for1 6 i 6 n, then an allocation x = hx1, ..., xni is Pareto optimal if and onlyif it has the form xi (s) = ωi (si ) + ηi , where ∑n

i=1 ηi = 0.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 21 / 29

CONCAVE UTILITY

Now assume that ui is concave for 1 6 i 6 n. Apart from ambiguity,our assumptions are standard in risk sharing models.With concave utility there is a trade-o¤.

Ambiguity-aversion creates a barrier to sharing ambiguous risks.Diminishing marginal utility of wealth makes it desirable to smoothconsumption across states.

If there is su¢ cient ambiguity-aversion, at any Pareto optimum, eachindividual consumes his/her endowment plus a riskless lump sum taxor subsidy.It follows that competitive equilibrium is unique and involves no trade.The result would also apply to e¢ cient Coasian bargaining.

Theorem

If ui is concave for 1 6 i 6 n, there exists γ̄ such that if 1 > γi > γ̄, for1 6 i 6 n, then an allocation x = hx1, ..., xni is Pareto optimal if and onlyif it has the form xi (s) = ωi (si ) + ηi , where ∑n

i=1 ηi = 0.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 21 / 29

CONCAVE UTILITY

Now assume that ui is concave for 1 6 i 6 n. Apart from ambiguity,our assumptions are standard in risk sharing models.With concave utility there is a trade-o¤.

Ambiguity-aversion creates a barrier to sharing ambiguous risks.

Diminishing marginal utility of wealth makes it desirable to smoothconsumption across states.

If there is su¢ cient ambiguity-aversion, at any Pareto optimum, eachindividual consumes his/her endowment plus a riskless lump sum taxor subsidy.It follows that competitive equilibrium is unique and involves no trade.The result would also apply to e¢ cient Coasian bargaining.

Theorem

If ui is concave for 1 6 i 6 n, there exists γ̄ such that if 1 > γi > γ̄, for1 6 i 6 n, then an allocation x = hx1, ..., xni is Pareto optimal if and onlyif it has the form xi (s) = ωi (si ) + ηi , where ∑n

i=1 ηi = 0.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 21 / 29

CONCAVE UTILITY

Now assume that ui is concave for 1 6 i 6 n. Apart from ambiguity,our assumptions are standard in risk sharing models.With concave utility there is a trade-o¤.

Ambiguity-aversion creates a barrier to sharing ambiguous risks.Diminishing marginal utility of wealth makes it desirable to smoothconsumption across states.

If there is su¢ cient ambiguity-aversion, at any Pareto optimum, eachindividual consumes his/her endowment plus a riskless lump sum taxor subsidy.It follows that competitive equilibrium is unique and involves no trade.The result would also apply to e¢ cient Coasian bargaining.

Theorem

If ui is concave for 1 6 i 6 n, there exists γ̄ such that if 1 > γi > γ̄, for1 6 i 6 n, then an allocation x = hx1, ..., xni is Pareto optimal if and onlyif it has the form xi (s) = ωi (si ) + ηi , where ∑n

i=1 ηi = 0.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 21 / 29

CONCAVE UTILITY

Now assume that ui is concave for 1 6 i 6 n. Apart from ambiguity,our assumptions are standard in risk sharing models.With concave utility there is a trade-o¤.

Ambiguity-aversion creates a barrier to sharing ambiguous risks.Diminishing marginal utility of wealth makes it desirable to smoothconsumption across states.

If there is su¢ cient ambiguity-aversion, at any Pareto optimum, eachindividual consumes his/her endowment plus a riskless lump sum taxor subsidy.

It follows that competitive equilibrium is unique and involves no trade.The result would also apply to e¢ cient Coasian bargaining.

Theorem

If ui is concave for 1 6 i 6 n, there exists γ̄ such that if 1 > γi > γ̄, for1 6 i 6 n, then an allocation x = hx1, ..., xni is Pareto optimal if and onlyif it has the form xi (s) = ωi (si ) + ηi , where ∑n

i=1 ηi = 0.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 21 / 29

CONCAVE UTILITY

Now assume that ui is concave for 1 6 i 6 n. Apart from ambiguity,our assumptions are standard in risk sharing models.With concave utility there is a trade-o¤.

Ambiguity-aversion creates a barrier to sharing ambiguous risks.Diminishing marginal utility of wealth makes it desirable to smoothconsumption across states.

If there is su¢ cient ambiguity-aversion, at any Pareto optimum, eachindividual consumes his/her endowment plus a riskless lump sum taxor subsidy.It follows that competitive equilibrium is unique and involves no trade.

The result would also apply to e¢ cient Coasian bargaining.

Theorem

If ui is concave for 1 6 i 6 n, there exists γ̄ such that if 1 > γi > γ̄, for1 6 i 6 n, then an allocation x = hx1, ..., xni is Pareto optimal if and onlyif it has the form xi (s) = ωi (si ) + ηi , where ∑n

i=1 ηi = 0.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 21 / 29

CONCAVE UTILITY

Now assume that ui is concave for 1 6 i 6 n. Apart from ambiguity,our assumptions are standard in risk sharing models.With concave utility there is a trade-o¤.

Ambiguity-aversion creates a barrier to sharing ambiguous risks.Diminishing marginal utility of wealth makes it desirable to smoothconsumption across states.

If there is su¢ cient ambiguity-aversion, at any Pareto optimum, eachindividual consumes his/her endowment plus a riskless lump sum taxor subsidy.It follows that competitive equilibrium is unique and involves no trade.The result would also apply to e¢ cient Coasian bargaining.

Theorem

If ui is concave for 1 6 i 6 n, there exists γ̄ such that if 1 > γi > γ̄, for1 6 i 6 n, then an allocation x = hx1, ..., xni is Pareto optimal if and onlyif it has the form xi (s) = ωi (si ) + ηi , where ∑n

i=1 ηi = 0.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 21 / 29

CONCAVE UTILITY

Now assume that ui is concave for 1 6 i 6 n. Apart from ambiguity,our assumptions are standard in risk sharing models.With concave utility there is a trade-o¤.

Ambiguity-aversion creates a barrier to sharing ambiguous risks.Diminishing marginal utility of wealth makes it desirable to smoothconsumption across states.

If there is su¢ cient ambiguity-aversion, at any Pareto optimum, eachindividual consumes his/her endowment plus a riskless lump sum taxor subsidy.It follows that competitive equilibrium is unique and involves no trade.The result would also apply to e¢ cient Coasian bargaining.

Theorem

If ui is concave for 1 6 i 6 n, there exists γ̄ such that if 1 > γi > γ̄, for1 6 i 6 n, then an allocation x = hx1, ..., xni is Pareto optimal if and onlyif it has the form xi (s) = ωi (si ) + ηi , where ∑n

i=1 ηi = 0.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 21 / 29

CONCAVE UTILITY

Now assume that ui is concave for 1 6 i 6 n. Apart from ambiguity,our assumptions are standard in risk sharing models.With concave utility there is a trade-o¤.

Ambiguity-aversion creates a barrier to sharing ambiguous risks.Diminishing marginal utility of wealth makes it desirable to smoothconsumption across states.

If there is su¢ cient ambiguity-aversion, at any Pareto optimum, eachindividual consumes his/her endowment plus a riskless lump sum taxor subsidy.It follows that competitive equilibrium is unique and involves no trade.The result would also apply to e¢ cient Coasian bargaining.

Theorem

If ui is concave for 1 6 i 6 n, there exists γ̄ such that if 1 > γi > γ̄, for1 6 i 6 n, then an allocation x = hx1, ..., xni is Pareto optimal if and onlyif it has the form xi (s) = ωi (si ) + ηi , where ∑n

i=1 ηi = 0.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 21 / 29

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I2

Figure:

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 22 / 29

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I 01

Figure: E¤ect of Ambiguity

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 23 / 29

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I1

I2

Figure: Mutual Advantage

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 24 / 29

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I2

I1

Figure: No trade

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 25 / 29

INCENTIVE CONTRACTSWork in Progress

We consider a principal agent framework.

The agent makes a widget for the principal.

Both the value of the widget to the principal and the cost of makingthe widget are uncertain.

The principal views the cost of making the widget as ambiguous.

The agent views the value of the widget to the principal asambiguous.

The agent can exert e¤ort which will increase the chance that costwill be low and value will be high.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 26 / 29

INCENTIVE CONTRACTSWork in Progress

We consider a principal agent framework.

The agent makes a widget for the principal.

Both the value of the widget to the principal and the cost of makingthe widget are uncertain.

The principal views the cost of making the widget as ambiguous.

The agent views the value of the widget to the principal asambiguous.

The agent can exert e¤ort which will increase the chance that costwill be low and value will be high.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 26 / 29

INCENTIVE CONTRACTSWork in Progress

We consider a principal agent framework.

The agent makes a widget for the principal.

Both the value of the widget to the principal and the cost of makingthe widget are uncertain.

The principal views the cost of making the widget as ambiguous.

The agent views the value of the widget to the principal asambiguous.

The agent can exert e¤ort which will increase the chance that costwill be low and value will be high.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 26 / 29

INCENTIVE CONTRACTSWork in Progress

We consider a principal agent framework.

The agent makes a widget for the principal.

Both the value of the widget to the principal and the cost of makingthe widget are uncertain.

The principal views the cost of making the widget as ambiguous.

The agent views the value of the widget to the principal asambiguous.

The agent can exert e¤ort which will increase the chance that costwill be low and value will be high.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 26 / 29

INCENTIVE CONTRACTSWork in Progress

We consider a principal agent framework.

The agent makes a widget for the principal.

Both the value of the widget to the principal and the cost of makingthe widget are uncertain.

The principal views the cost of making the widget as ambiguous.

The agent views the value of the widget to the principal asambiguous.

The agent can exert e¤ort which will increase the chance that costwill be low and value will be high.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 26 / 29

INCENTIVE CONTRACTSWork in Progress

We consider a principal agent framework.

The agent makes a widget for the principal.

Both the value of the widget to the principal and the cost of makingthe widget are uncertain.

The principal views the cost of making the widget as ambiguous.

The agent views the value of the widget to the principal asambiguous.

The agent can exert e¤ort which will increase the chance that costwill be low and value will be high.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 26 / 29

When both parties display maximum ambiguity-aversion we get a �atcontract, in which the principal provides no incentives.

PropositionIf Agents are ambiguity averse, such that, γA = 1, and the principal isambiguity averse, such that, γP = 1, then the principal o¤ers a �atcontract.

The next result retains the assumption that the agent displays maximalambiguity-aversion but allows the ambiguity-aversion of the principal tovary.

Proposition

If γA = 1, and γP 2 [0,γ�P ), then the optimal contract is a cost pluscontract and if γA = 1, and γP 2 [γ�P , 1], then the optimal contract is a�at contract.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 27 / 29

When both parties display maximum ambiguity-aversion we get a �atcontract, in which the principal provides no incentives.

PropositionIf Agents are ambiguity averse, such that, γA = 1, and the principal isambiguity averse, such that, γP = 1, then the principal o¤ers a �atcontract.

The next result retains the assumption that the agent displays maximalambiguity-aversion but allows the ambiguity-aversion of the principal tovary.

Proposition

If γA = 1, and γP 2 [0,γ�P ), then the optimal contract is a cost pluscontract and if γA = 1, and γP 2 [γ�P , 1], then the optimal contract is a�at contract.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 27 / 29

RISK NEUTRALITY

Assume that both principal and agent are risk neutral.

Proposition

The optimal contract is P21 =πchc1+(1�πch)c2+eh

πch,P22 = P11 = P12 = 0,

where there is trade only is high value and low cost is realized.

This is e¤ectively a contract with a non-performance clause.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 28 / 29

CONCLUSION

Explains the lack of risk sharing for extreme events.

The model provides an insight into why excusing performance underextreme events makes sense.

Public policy implication.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 29 / 29

CONCLUSION

Explains the lack of risk sharing for extreme events.

The model provides an insight into why excusing performance underextreme events makes sense.

Public policy implication.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 29 / 29

CONCLUSION

Explains the lack of risk sharing for extreme events.

The model provides an insight into why excusing performance underextreme events makes sense.

Public policy implication.

David Kelsey (University of Exeter) THE ACT OF GOD CLAUSE NOVEMBER 2008 29 / 29