Econ 452
LM on Moral Hazard
Moral Hazard (ch 4 in LM)
In PA problem: A's effort level, after contract
accepted, not costlessly observable - hence not
contractible ex ante.
If outcome of delegation of activity also subject to
random fluctuations, A's effort level not recoverable
ex post.
NB: as with adverse selection, only a problem
if/because P's and A's objectives do not coincide;
incentive issue is to align A's goals more closely with
those of P.
Risk aversion of A important: if A is risk-neutral,
asymmetric info adds no additional cost to production
- only issue is who bears risk, and incentives "right" if
A does. This is costly if A is RA, so revelation of
asymmetric info uses up resources.
Basic Model:
1. agent
A chooses effort level
e ∈ {0,1}
disutility of effort: ψ (e) : ψ (0) = ψ 0 = 0;
ψ (1) = ψ 1 = ψ
Payment for effort: transfer t from P
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LM on Moral Hazard
A's utility U (t , e) = u (t ) − ψ (e)
as usual, u'>0>u"; normalize u(0)=0
2. production technology:
random; output can be high ( q ) or low ( q )
difference is ∆q
probability of high depends on effort level
Prob(q= q |e=0)=π 0 ; Prob(q= q |e=1)=π 1 > π 0
difference between probabilities = ∆π
[NB: production increases in effort - in sense of first
order stochastic dominance: Prob q less than q* is
decreasing in e, for any q*.
implication - if P's utility is increasing in output,
will prefer more to less effort]
3. Principal's objective?
S (q ), with S (q ) = S > S = S ( q )
Assume: P is risk neutral
Incentive feasible contracts?
Form of contract: in general, a function t(q); with only
two possible output levels, contract is a pair of
transfers, one for each (observable) output level.
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LM on Moral Hazard
Given A's effort, P's EU is
V1 = π 1 ( S − t ) + (1 − π 1 )( S − t )
if e=1
or
V0 = π 0 ( S − t ) + (1 − π 0 )( S − t ) < V1
if e=0
P's problem?
i) How to induce high effort (e=1)?
ii) Is it worth it to induce high effort?
Constraints?
i)
Participation:
π 1u ( t ) + (1 − π 1 )u ( t ) − ψ ≥ 0
(given normalization of A's reservation utility)
ii)
moral hazard incentive constraint:
π 1u ( t ) + (1 − π 1 )u ( t ) − ψ ≥ π 0u ( t ) + (1 − π 0 )u ( t )
(recall lower level of effort =0, with no utility cost)
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Econ 452
LM on Moral Hazard
Benchmark: full information:
If effort observable, only constraint is participation - P
pays sum dependent on effort exerted
P's problem: choose ( t
, t ) to solve
max π 1 ( S − t ) + (1 − π 1 )( S − t )
+ µ[π 1u ( t ) + (1 − π 1 )u ( t ) − ψ ]
FOC's?
Solution yields
which implies
µ=
1
1
=
u '( t *) u '( t *)
t * = t * = t * is always a solution.
Interpretation?
with no info problems, and P risk neutral,
providing A with full insurance against intrinsic
uncertainty always optimal.
if A is risk neutral, this is fine
if A is risk averse, this solution is unique.
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Econ 452
LM on Moral Hazard
When is inducing effort (e=1) better than not (e=0)?
if e=1, A receives res'n utility =0, so
gain to P from e=1 vs e=0?
t* = h(ψ )
∆π∆S − h(ψ )
Ans: if expected gain ≥ cost of effort
[Pictures for this?}
Second best results: with moral hazard
Now P needs incentive constraint as well as
participation constraint.
1. A is risk neutral:
assume u(t)=t and h(ψ)=ψ
(up to an affine transformation)
Then constraints facing P become
i) Partcipation: π 1t
+ (1 − π 1 ) t − ψ ≥ 0
ii) Incentive compatibility:
π 1t + (1 − π 1 ) t − ψ ≥ π 0 t + (1 − π 0 ) t
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Econ 452
LM on Moral Hazard
P has all bargaining power - both constraints satisfied
as equalities - can solve 2 eq'ns in 2 unknowns
t*=
(1 − π 0 )
ψ
∆π
and
t*= −
π0
ψ
∆π
Note: 1) A's payment depends on outcome - A is
bearing risk;
2) if outcome low (so q = q ) , A's payment <0
This will yield first-best effort level, with A receiving
EU just sufficient to compensate for effort (res'n u=0).
So: 1st best outcome obtainable under asym info wrt
effort. If high effort optimal under full info, can be
implemented with asym. info when agent is risk
neutral.
Interpretation of
t * < 0?
franchise: A makes up front payment to P, st A's
expected utility =0; then A is residual claimant ex
post.
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2. Agent is risk averse:
- no longer feasible to transfer all risk to agent at 1st
best payment
a) Limited liability: risk neutral for t>0; floor on t<0
b) Risk aversion:
a) Limited liability constraints: (4.3)
- most obvious is bankruptcy - law permits
declaration of bankruptcy
Form of constraints:
as before: incentive compatibility + participation
plus t ≥ −l , t ≥ −l , for some l > 0
When is this an issue? What are the consequences?
π0
ψ , then transfer to agent if low
- if 0 ≤ l ≤
∆π
st
outcome which induces 1 best effort under asym.
info not feasible - lowest this transfer can be is −l .
- with risk neutrality, the key element is the
difference between A's pay-offs with high and low
output; if limited liability is binding, then payment in
good state is forced up to maintain the gap:
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t * = −l ,
ψ
t * = −l +
∆π
Then A's EU is
EU
LL
A
ψ
= π 1 ( −l +
) + (1 − π 1 )(−l ) − ψ
∆π
π0
ψ
>0, since 0 ≤ l ≤
∆π
Thus: with limited liability as a binding constraint on
P's transfers to A, 1st best effort level only attainable
(under asym info on effort) if P transfers some surplus
to A - a limited liability rent .
Hence: P's surplus under full info must be larger than
before, to make it worth while inducing 1st best effort.
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