Transcript Slide 1

Optimal Contracts under
Moral Hazard
What does it mean Moral Hazard?
•We will use much more often the notion of Moral Hazard
as hidden action rather than ex-post hidden information
•Moral Hazard means that the action (effort) that the A
supplies after the signature of the contract is not verifiable
•This means that the optimal contract cannot be contingent
on the effort that the A will exert
•Consequently, the optimal contracts will NOT have the
form that they used to have when there is SI::
•If e=eopt then principal (P) pays w(xi) to agent (A) if not,
then A will pay a lot of money to P
What does the solution to the SI case does
not work when there is Moral Hazard?
•Say that a Dummy Risk Neutral Principal offers to a Risk
Averse A the same contract under moral hazard that he
would have offered him if Information is Symmetric
If e=eo then principal (P) pays to agent (A) the
fixed wage of: w0  u 1 (U  v(e0 ))
if not, then A will pay a lot of money to P
-Threat is not credible because e is no verifiable
-…plus Wage does not change with outcome: no incentives.
-RESULT: Agent will exert the lowest possible effort instead
of eo
Anticipation to the solution to the optimal
contract in case of Moral Hazard
•Clearly, if the P wants that the A will exert a given level of
effort, she will have to give some incentives
•The remuneration schedule will have to change
according to outcomes
•This implies that the A will have to bear some risk
(because the outcome does not only depend on effort
but also on luck)
•So, the A will have to bear some risk even if the A is
risk averse and the P is risk neutral
•In case of Moral Hazard, there will not be an efficient
allocation of risk
How to compute the optimal contract under MH
•For each effort level ei, compute the optimal wi(xi)
•Compute P’s expected utility E[B(xi- wi(xi)] for each effort
level taking into account the corresponding optimal wi(xi)
•Choose the effort and corresponding optimal wi(xi) that
gives the largest expected utility for the P
•This will be eopt and its corresponding wi(xi)
•So, we break the problem into two:
•First, compute the optimal wi(xi) for each possible
effort
•Second, compute the optimal effort (the one that max
P’s utility)
Moral Hazard with two possible effort levels
Moral Hazard with two possible levels of effort
•For simplification, let’s study the situation with only two
possible levels of effort: High (eH) and Low (eL)
•There are N possible outcomes of the relationship. They
follow that:
•x1<x2<x3<….<xN
•That is, x1 is the worst and xN the best
•We label piH the probability of outcome xi when effort is H
•We label piL the probability of outcome xi when effort is L
H
p1H , p2H , p3H ,..., pN
L
p1L , p2L , p3L ,..., pN
Moral Hazard with two possible levels of
effort
•For the time being, let’s work in the case in which P is risk
neutral and the A is risk averse
B( x  w)  x  w
U ( w)  v(e )
Now, we should work out the optimal remuneration
schedule w(xi) for each level of effort:
-Optimal w(xi) for L effort ( this is easy to do)
-Optimal w(xi) for H effort ( more difficult)
Optimal w(xi) for low effort
•In the case of low effort, we do not need to provide any
incentives to the A.
•We only need to ensure that the A want to participate (the
participation constraint verifies)
•Hence, the w(xi) that is optimal under SI is also optimal
under MH, that is, a fixed wage equals to:
1
w  u (U  v(e ))
L
L
Why is it better this fixed contract that one than a risky one that pays
more when the bad outcome is realized?
Optimal w(xi) for High effort
•This is much more difficult
•We have to solve a new maximization problem…
Optimal w(xi) for High effort
•We must solve the following program:
n
Max  p
i 1
H
i
( xi  w( xi ))
n
st : p H i [U ( w( xi ))  v(e H )]  U
i 1
n
p
i 1
n
H
[U ( w( xi ))  v(e )]   p i [U ( w( xi ))  v(e )]
H
i
L
i 1
The first constraint is the Participation Constraint
The second one is called the Incentive compatibility
constraint (IIC)
L
About the IIC
•The Incentive Compatibility Constraint tell us that
n
n
i 1
i 1
H
H
L
L
p
[
U
(
w
(
x
))

v
(
e
)]

p
[
U
(
w
(
x
))

v
(
e
)]
 i
 i
i
i
The remuneration scheme w(xi) must be such that the
expected utility of exerting high effort will be higher or equal
to the expected utility of exerting low effort
In this way, the P will be sure that the A will be exerting High
Effort, because, given w(xi), it is in the Agent’s own interest
to exert high effort
About the IIC
•The IIC can be simplified:
n
H
H
p
[
U
(
w
(
x
))

v
(
e
)] 
 i
i
i 1
n
L
L
p
[
U
(
w
(
x
))

v
(
e
)]
 i
i
i 1
So:
n
n
i 1
i 1
H
H
H
p
U
(
w
(
x
))

p
v
(
e
)] 
 i
 i
i
n
n
i 1
i 1
L
L
L
p
U
(
w
(
x
))

p
v
(
e
 i
 i )
i
n
so, wehave : [ p  p ]u(w( xi ))  v(e )  v(e )
i 1
H
i
L
i
H
L
Optimal w(xi) for High effort
•Rewriting the program with the simplified constraints:
n
Max  p
i 1
H
i
( xi  w( xi ))
n
st : p H iU ( w( xi ))  v(e H )  U
i 1
 p
n
i 1
H
i
p
L
i
U (w( x ))  v(e
i
H
)  v (e )
The first constraint is the Participation Constraint
The second one is called the Incentive compatibility
constraint (IIC)
L
Optimal w(xi) for High effort
•The Lagrangean would be:
n
L   p H i ( xi  w( xi )) 
i 1
 n

H
H
    p iU ( w( xi ))  v(e )  U  
 i 1

 n
H
L
H
L 
     p i  p i  U ( w( xi ))  v (e )  v (e ) 
 i 1

Taking the derivative with respect to w(xi), we obtain
the first order condition (foc) in page 43 of the book.
After manipulating this foc, we obtain equation (3.5)
that follows in the next slide…
Optimal w(xi) for High effort
Equation (3.5) is:
pH i
= pHi + pHi -pLi 
U '(w( xi ))
By summing equation (3.5) from i=1 to i=n, we get:
H
p i
 i 1 U '( w( x )) =
i
n
so   0
The PC is binding !!
This means that in the optimum, the constraint will hold with
equality(=) instead of (>=)
Optimal w(xi) for High effort
•Notice that (eq 3.5) comes directly from the first order
condition, so (eq. 3.5) characterizes the optimal
remuneration scheme
•Eq. (3.5) can easily be re-arranged as:
 pLi 
1
= + 1- H 
U '( w( xi ))
 p i
We know that λ>0. What is the sign of μ?
-It cannot be negative, because Lagrange Multipliers
cannot be negative in the optimum
-Could μ=0?
Optimal w(xi) for High effort
If μ was 0, we would have:
1
=
U '( w( xi ))
We know that this implies that:
w( x1 )  w( x2 )  w( x3 )  ...  w( xn ) (Full Ins.)
because of the strict concavity (RA) of U
Intuitively, we know that it cannot be optimal that the Agent
is fully insured in this case (see the example of the dummy
principal at the beginning of the lecture)
So, it cannot be that μ was 0 is zero in the optimum.
Optimal w(xi) for High effort
Mathematically: If μ was 0, we would have:
We know that this implies that:
w( x1 )  w( x2 )  w( x3 )  ...  w( xn )  w # (Full Ins.)
substituting in the left hand sinde of the IC:
H
L
H
L
#
p

p
U
(
w
(
x
))

p

p
U
(
w
)0
 i i 
 i i 
i
n
n
i 1
i 1
So, the PC will be:
0  v (e H )  v (e L )
what cannot be true !!!
Optimal w(xi) for High effort
In summary, if μ was 0 the IC will not be verified !!!
We also know that μ cannot be negative in the optimum
Necessarily, it must be that μ> 0
This means that the ICC is binding !!!
So, in the optimum the constraint will hold with (=), and we
can get rid off (>=)
Optimal w(xi) for High effort
Now that we know that both constraints (PC, and ICC)
are binding, we can use them to find the optimal W(Xi):
n
H
H
i
i
i 1
p
 p
U ( w( x ))  v(e )  U
n
i 1
H
i
p
L
i
U (w( x ))  v(e
i
H
)  v (e )
Notice that these equations might be enough if we only
have w(x1) and w(x2).
If we have more unknowns, we will also need to use the
first order conditions (3.5) or (3.7)
L
Optimal w(xi) for High effort
•The condition that characterizes optimal w(xi) when P is
RN and A is RA is (3.5) and equivalently (eq 3.7):
 pLi 
1
= + 1- H 
U '( w( xi ))
 p i
-This ratio of probabilities is called the likelihood ratio
-So, it is clear that the optimal wage will depend on the
outcome of the relationship because different xi will
normally imply different values of likelihood ratio and
consequently different values of w(xi) ( the wage do
change with xi) !!!!
Optimal w(xi) for High effort
•The condition that characterizes optimal w(xi) when P is
RN and A is RA is (eq 3.7):
 pLi 
1
= + 1- H 
U '( w( xi ))
 p i
We can compare this with the result that we obtained
under SI (when P is RN and A is RA):
1

= =constant
U '( w( xi )) a
So the term in brackets above show up because of
Moral Hazard. It was absent when info was symmetric
What does the likelihood
L

p
ratio: i 
 H
p i 
mean?
The likelihood ratio indicates the precision with which the
result xi signals that the effort level was eH
Small likelihood ratio:
-piH is large relative to pLi
-It is very likely that the effort used was eH when
the result xi is observed
Example: X  10, p H  0.2, p L  0.8
1
1
1
X 2  100, p2H  0.8, p2L  0.2
p1L
p2L
1
 4; H 
H
p1
p2
4
Clearly, X2 is more informative than X1 about eH was exerted, so it has a smaller
likelihood ratio
What is the relation between optimal w(xi)
and the likelihood ratio when effort is high?
 pLi 
1
= + 1- H 
U '( w( xi ))
 p i
λ >0, we saw it in the previous slides.
μ>0, we saw it in the previous slides
Notice: small likelihood ratio (signal of eH) implies high w(xi)
An issue of information
Assume a RN P that has two shops. A big shop and a small
shop. In each shop, the sales can be large or small.
For each given of effort, the probability of large sales is the
same in each shop
The disutility of effort is also the same
However, the big shop sells much more than the small shop
For the same level of effort, will the optimal remuneration
scheme be the same in the large and small shop?
A question of trade-offs…
•P is RN and A is RA. This force will tend to minimize risk to
the Agent
•Effort is no verifiable: This force will tend to make payments
to the agent vary according to actual xi (introducing risk), as
long as actual xi gives us information about the effort
exerted
•The optimal remuneration schedule trades off these two
forces
•Notice that it would not make sense to make the contract
contingent on a random variable that:
•The agent cannot influence
•It is not important for the value of the relationship
When will w(xi) be increasing with xi?
 pLi 
1
= + 1- H 
U '( w( xi ))
 p i
If the likelihood ratio is decreasing in i, that is, if higher xi are
more informative about eH than lower levels of effort. This is
called the monotonous likelihood quotient property. Notice
that this property does not necessarily have to hold:
ph
pl
pl/ph
x1
0.2
0.4
2
x2
0.1
0.4
4
x3
0.7
0.2
2/7
Is the solution Pareto Efficient?
 pLi 
1
= + 1- H  .
U '( w( xi ))
 p i
So, we have that:
U '( w( x1 )) U '( w( x1 ))
...

U '( w( x2 )) U '( w( x3 ))
The tangency condition does not verify.
The solution will not be Pareto efficient in general...
Graphical analysis: P is RN and A is RA.
Two outcomes: x1 and x2
Max
p H ( x2  w2 )  (1  p H )( x1  w1 )
st :
(1) p H u ( w2 )  (1  p H )u ( w1 )  v(e H )  U
(2) p H u ( w2 )  (1  p H )u ( w1 )  v(e H ) 
p L u ( w2 )  (1  p L )u ( w1 )  v(e L ).
(1) is PC and (2) is the IC.
After writting the IC with equality,
we can write the IC as:
v (e H )  v ( e L )
u ( w2 )  u ( w1 ) 
pH  pL
This give us the combinations of w 2 , w1, eH , eL
where the constraint is binding
Graphical analysis: P is RN and A is RA.
Two outcomes: x1 and x2
v(e H )  v(e L )
The IC is: u ( w2 )  u ( w1 ) 
pH  pL
To satisfy it, if w1 increase, w2 also increases
Draw f(w2,w1) in Fig 3.3, page 59
Now, we need to know how to draw the indifference curves
dw1
u ' ( w2 ) p H
H
slope for e :
 '
dw2
u ( w1 ) 1  p H
'
L
dw
u
(
w
)
p
1
slope for e L :
 ' 2
dw2
u ( w1 ) 1  p L
if w1  w2 , Indifference curve for e H is steeper
Draw the indifference curves as in Fig 3.3, page 59
Graphical analysis: P is RN and A is RA.
Two outcomes: x1 and x2
The expected profit lines for the principal are:
for e H : p H ( x2  w2 )  (1  p H )( x1  w1 )
for e L : p L ( x2  w2 )  (1  p L )( x1  w1 )
Their slopes are:
slope for e H
d ( x1  w1 )
pH
:

d ( x2  w2 )
1 pH
d ( x1  w1 )
pL
slope for e :

d ( x2  w2 )
1 pL
L
The expected profit line for e H is steeper
Graphical analysis: P is RN and A is RA.
Two outcomes: x1 and x2
x1-w1
x1-w1=x2-w2
For eH
For eL
x2-w2
Picture of the expected profit lines
We can invert the axis, and make Fig 3.4
Graphical analysis: P is RN and A is RA.
Two outcomes: x1 and x2
Draw Figures 3.5 and 3.6
-First draw contracts L and H (but call them A & B)
-Draw the Indifference curve for low effort through them (in
order to measure the utility), and high effort through B
-Then say that A will be the optimal contract under SI for eL.
-Then say that B is the optimal contract under SI for eH
-Explain why A and B are not incentive compatible under MH
-Draw the optimal contract under MH: (H’)
-Show it is not Pareto Efficient
Optimal Contract with two levels of effort
After we have computed the optimal remuneration scheme
for High and Low effort,
The principal will assess if she prefers High or Low effort
levels
The optimal contract will be the one that implements her
preferred level of effort
Optimal Contract with Moral Hazard
•So far, we have studied the case where P is RN and A is
RA. If the P wants to implement High Effort, the SI solution
(fixed wage) is not incentive compatible, hence a new
optimal contract that takes into account the ICC must be
computed
•Notice that if P is RA and A is RN, then the optimal solution
in case of SI (the P will get a fixed rent, and the A will get the
outcome minus the rent) is incentive compatible (the A will
exert high effort). Consequently:
•Moral Hazard does not create problems when the P is RA
and the A is RN. The SI solution can be implemented
Moral Hazard with continuous effort
Just to remind you…
Given the differentiable function F(x). If the point x0
is its maximum, then it must be the case that the
first derivative of F(x) evaluated at x0 is equal to
zero.
That is F’(x0)=0
However, other points that are not a maximum, can
also satisfy the condition that the first derivative
evaluated at them is zero (do a graph…)
Problem with continuous effort.
Optimal contract to implement e0
n
Max  pi (e )( xi  w( xi ))
0
i 1
n
st : pi (e0 )U ( w( xi ))  v(e 0 )  U
i 1
n
e  arg Max{e}{ pi (e)U ( w( xi ))  v(e)}
0
i 1
The (IIC) is the last one. It tell us that e0 should
maximize the agent’s expected utility given w(xi), so
that it is in the Agent’s own interest to carry out e0
Problem with continuous effort.
Optimal contract to implement e0
The problem is very difficult to solve as it is because it is a
maximization problem within another maximization problem.
To simplify it:
-If e0 maximizes the agent’s expected utility, it must
be the case that the derivative of the agent’s expected utility
with respect to effort, evaluated at e0 is zero, that is:
n
 p ' (e )u(w( x ))  v '(e )  0
0
i 1
i
0
i
Is this restriction equivalent to the ICC of the previous slide?
No always… draw a concave and a non-concave function…
In a non-concave function, the effort levels that satisfy this
second restriction are more than the ones that satisfy the ICC
Problem with continuous effort.
Optimal contract to implement e0
Substituting the real ICC by the simplified constraint is
called the First Order Approach.
When this approach is correct, economists says that the
conditions for the first order approach verifies
If the expected utility function is concave, then the First
Order Approach is valid
Problem with continuous effort.
Optimal contract to implement e0
Using First Order Approach:
n
Max  pi (e )( xi  w( xi ))
0
i 1
n
st : pi (e0 )U ( w( xi ))  v(e 0 )  U
i 1
n

i 1
p 'i (e0 )u ( w( xi ))  v '(e 0 )  0
Problem with continuous effort.
Optimal contract to implement e0
Using First Order Approach:
We form the Langragean:
n
L   pi (e0 )( xi  w( xi ))
i 1
 n

0
0
    pi (e )U ( w( xi ))  v(e )  U  
 i 1

 n
0
0 
    p 'i (e )u ( w( xi ))  v '(e ) 
 i 1

The first order condition:
 pi (e0 )   pi (e0 )u ' ( w( xi ))   p 'i (e 0 )u ' ( w( xi ))  0
Problem with continuous effort.
Optimal contract to implement e0
Using First Order Approach:
From the first order condition:
 pi (e0 )   pi (e0 )u ' ( w( xi ))   p 'i (e 0 )u ' ( w( xi ))  0
We obtain that:
p 'i (e 0 )
1

'
u ( w( xi ))
pi (e 0 )
Notice that w(xi) will depend on the result (sales)
because the ratio of the right hand side depends on
the results. So, the agent is not fully insured
Problem with continuous effort.
Optimal contract to implement e0
Using First Order Approach:
The previous analysis has given us the optimal
remuneration scheme for a given level of effort (e0)
Now, we would have to study the optimal level of
effort but we will not do that because it is too
complicated from a mathematical point of view.
Other issues in optimal
contracts under Moral Hazard
•Limited liability
•Value of information
•Contracts based on severe punishments
•What happens when it is the agent who
offers the contract?
Limited liability
• Contracts where a P is RN and A is RA under SI followed
the following scheme:
– If the agent exerts effort e0, he will get the fixed wage w0 if he
exerts another effort, he will have to pay to the principal a large
sum of money
• This contract incorporates a threat to penalize the agent.
This threat ensures that the agent does not find
attractive to exert a level of effort that is not desired by
the principal.
• Sometimes, the penalization is not legal or is not
credible:
– An employee cannot pay to the firm. The firm has always to obey
the minimum wage
– A bank cannot make the shareholders of a company to pay the
company debts if the company goes bankrupt
Limited liability
• If the penalization is not legal or it is not credible,
the agent can exert a low level of effort even if:
– Information is symmetric (no MH)
– P is requesting a high level of effort
• So, the P will have to use the Incentive
Compatibility constraint even if information is
symmetric
• So, when there is limited liability, the optimal
contract might give incentives to the agent even
if the P is RN and information is symmetric
The value of information under MH
• So far, we have studied that the contract will be
contingent only on the result of the relationship
(sales). This has been done for simplicity.
• Clearly, the principal is interested in using in the
contracts signals that reveal new information on
the agent’s effort
• These signals could be:
– Other’s agents results
– Control activities
– State of Nature (let’s see an example with this)
Example with state of nature..
Sales H Sales L
Effort
H
0.6
0.4
Effort
L
0.15
0.85
In this case, it might be very costly to provide incentives
so that the agent exerts high effort. This is because even
if the agent exerts high effort, the probability of low sales
is quite high.
This might be because the probability of raining is too
high…
Example cont…
If it rains
If it does not rain
Sales H Sales L Sales H
Sales L
Effort H 0.3
0.7
0.9
0.1
Effort L
0.8
0.1
0.9
0.2
In this case, if it does not rain, the sales are quite good
predictors of the effort, so it will not be very risky for the agent
to exert high effort when it is not raining
The optimal contract will depend on the sales level and
whether it is raining or not
Conditioning on the state of nature is useful because it allows
better estimations of the agent’s effort thus reducing the risk
inherent in the relationship
The value of information under MH
– On one side, a contract should exploit all
available information in order to reduce the
risk inherent in the relationship
– On the other side, one must also consider the
cost of obtaining the information
• Knowing whether it rained or not is free
• However, monitoring activities are not free
• Conditioning the contract in other’s agent results is
not free (they could collude)
Mechanisms based on severe
punishments
• Assume that the P wants that the A exerts
high effort
• Sometimes, very bad results are only
possible if effort exerted is low
• In this case, a optimal contract could
include very bad punishment in case the
result obtained is very bad
• In this case, the P will ensure that the A
does not exert low effort
What happens when it is the agent
who offers the contract?
• In some situations, it is the person that is
going to carry out the job the one that
offers the contract (ie. State agents when
they are hired to sell a house)
• The Problem would be:
– MAX Agent Expected Utility
st (1) Principal expected utility >= reservation utility
(2) Incentive compatibility constraint for the Agent
What happens when it is the agent
who offers the contract?
(2) Needs to be taken into account because the P will only
accept those contracts that are credible, that is, those
contracts in which it is credible that the agent is going to
exert the level of effort that he claims is going to exert
The solution to this problem will have the same features
than the one that we have studied (P will offer the
contract to the agent) in terms of incentives and risk
sharing, but what changes is who obtains the reservation
utility…
Multitask
• So far, we have analysed the case where
the A works in one task
• However, it could be that the A will need to
carry out two tasks (or more, but let’s
consider just two…)
• How will the optimal contract be in those
circumstances?
Multitask
• We can consider that the task are
substitute or complements
• Complements: having exerted an effort for
task 1, the effort for task 2 is reduced
• Substitutes: when exerting more effort on
one increases the cost of the other
Multitask
• If tasks are Complements, the principal is
interested in motivating task 1, since in this way
she simultaneously motivates the agent to work
on task 2
• If the tasks are Substitutes, then giving
incentives for one task can be achieved in two
ways:
– Through the payments associated with each task
– By reducing the opportunity cost through reductions in
the incentives of the other tasks that the agent must
do
Multitask
• Multitasking can explain why incentive
schemes might not be used even if there
is MH…let’s see why…
• Consider two substitute tasks, task 1
provide results that can be measured, but
task 2 does not
• Hence, the principal could only give
explicit incentives for Task 1 but not for
Task 2
Multitask
• For instance:
– Task 1: carry out hip surgeries
– Task 2: treat patients well, study about new illnesses,
carry out medical research
• The principal must think what is best:
– Provide strong incentives for Task 1 knowing that the
A will abandon Task 2 at all
– Do not provide incentives for Task 1, knowing that the
Agent A will exert low effort in Task 1 but he will not
abandon Task 2 so much
• The optimal solution might be not to give
incentives at all, even if there is MH
Multitask
• Other examples:
– Bureaucratic systems: filling forms correctly, filling
forms correctly… cannot be measured, so it might be
better not to provide incentives for cases attended
– Finishing dates for home construction: if we give
incentives for the builder to finish the work by some
date… it might happen at the expense of quality
which is difficult to measure
– These are examples where incentives might no be
optimal even if there is MH because there is
multitasking and the result of one Task cannot be
measured
Multitask
• Multitask is also relevant for the following:
– The A can work in the task that gives
profits to the principal
– And in a private task that gives profits to
himself
– The A has to exert an effort for each task
– Example: doctor that works for the NHS
and works in his private practice
• Will the P allow the A to carry out his
private task?
Multitask
• Will the P allow the A to carry out his private
task?
• If she does, The P will have to pay less to the A if
she allows him to carry out his private task
• The final decision depends on a trade off…
• The P will not allow the A to carry out his private
task if it is difficult to motivate the A to exert effort
in the activity that he must carry out for the P,
probably due to measurement problems