The Optimal Allocation of Risk
James Mirrlees
Chinese University of Hong Kong
At Academia Sinica, Taipei
8 October 2010
Risky contracts
Insurance
Outcome conditional on accident
Pensions
Outcome could be conditional on capital market
performance; or on GDP
“Securities”
conditional on companies' performance or default
Loans
conditional on own experience – might default

Optimality
Arrow and Debreu are supposed to have shown
that an optimum can be implemented by a full set
of free markets in securities, with income
redistribution. Does not even need to be a full set.

Second-best considerations imply the need for
taxes on goods and services and securities, but
do not affect the main Arrow/Debreu argument.


But the claim is wrong.
Moral Hazard

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
Familiar objection.
People's actions are not always observable, or
deducible from outcomes, and contracts cannot
be made conditional on them.
Intuitively, one guesses contracts should be
conditional on outcomes, as imperfect proxies
for actions.
But that is not enough.
Optimal contracts with moral hazard.





Let insurers sell contracts to people with moral
hazard.
The contracts should pay conditional on all
observables that are influenced by, or influence
the hidden action.
The contracts exclude clients from other
contracts conditioned on these observables.
Insurers maximize expected profit, and
compete.
An optimum can be achieved with these
contracts.
Feasibility?





CDOs. These are traded contracts, i.e. there is
no exclusion.
Wrong incentives for the client, in this case a
mortgage lender, for example.
Health insurance.
Coinsurance proposed. But thought that people
can still insure the coinsurance payment. Then
coinsurance has no effect, and incentives are
wrong.
Conclusion? State monopoly?
Multiple relevant observables

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Car insurance.
Miles driven is observable, for some contracts –
e.g. buying petrol. Clearly associated with
accident risk.
It is too costly for insurers to include miles
driven in contracts.
That can be corrected, imperfectly, with higher
petrol taxation.
In many cases, complexity rules out intervention
of this kind. Case for regulation.
Another flaw in Arrow/Debreu: ignorance.

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Arrow/Debreu uses individual preference, e.g.
expected utility.
– in terms of whatever probabilities individuals
happen to believe.
Surely welfare must be judged by outcomes, if
not entirely, then mainly.
Two people, identical preferences, opposite
beliefs – sun for sure, rain for sure. They will
trade so that one starves if rain, the other if sun.
Plainly not an optimum.
Back to fundamentals

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
Karl Borch considered allocation where
everyone has the same beliefs. Aggregate
output varies across states of nature, and has
to be allocated to individuals. He showed that
each individual's allocation is an increasing
function of total output.
It can be achieved with Arrow/Debreu
securities. There would be no short trading.
If probabilities differ, utilities are the same,
equal sharing of output is optimal: i.e. no
consumer choice.
The general problem


Two major difficulties. To define optimality, one
must have true probabilities. One feels these
are not always known (though since they should
correctly describe uncertainty, they should allow
for that ignorance). Anyway, fully informed
people will disagree.
Secondly, it is technically challenging.
Varied risk preferences.
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
If second-best taxation creates optimal incentives for labour supply, incomecontingent variability of r may be quite small, relative to belief-variance.
11
Varying risk preferences

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
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In reality people have different false beliefs, and
also different risk preferences, and they find
themselves with different prospects – hence
hedging and insurance seems desirable.
One can ask when it is desirable to restrict
choice, e.g. by suppressing such gambling as
(some kinds of) CDSs.
Answer (roughly): when the variance of riskpreferences is less than the variance of
probabilities.
Really, it should depend on the individual
gambler's prospects, hard to observe.
12
Policy?


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The conclusion is that in certain circumstances,
probably many, though difficult to identify,
government should ensure that certain kinds of
trades should not take place.
The technical theorem so far gives little help in
identifying asset classes that shoul be severely
regulated.
The constraint might take the form of saying
that only certain kinds of contracts should be
traded. Prohibiting short trading is an example.
And it may be a case for prescribing the form of
pension contracts.
13