2. Static Moral Hazard
Klaus M. Schmidt
LMU Munich
Contract Theory, Summer 2010
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
1 / 64
Basic Readings
Basic Readings
Textbooks:
Bolton and Dewatripont (2005), Chapter 4
Laffont-Martimort (2002), Chapter 4
Schmidt (1995), Chapter 2
Papers:
Grossman and Hart (1983)
Holmström (1982)
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The Standard Hidden Action Model
The Standard Hidden Action Model
A principal (P) and an agent (A) can jointly generate some surplus. P controls
a technology, F (x; a), that has to be run by A.
x
a
F (x; a)
=
=
=
outcome (publicly observable)
action (private information of the agent)
cumulative distribution function over x given a
This is called the “parameterized distribution function approach”. In some
early papers it is assumed that x = x(a, θ) and that θ is distributed with cdf
G(θ). However, this “state space formulation” is less general and turns out to
be more difficult to deal with.
Feasible Contracts: The principal (and the courts) do not observe a but only
the noisy signal x. Therefore, the only contracts that are feasible are contracts
contingent on x, denoted by w(x).
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The Standard Hidden Action Model
Time Structure:
principal makes a “take-it-or-leave-it" offer
agent accepts or rejects
if agent rejects, game ends and both parties get their outside option
utilities.
if agent accepts, he chooses action a
outcome x is realized
payoffs according to contract
The assumption that the principal makes a “take-it-or-leave-it offer” is without
loss of generality in a moral hazard model. Why?
However, in models of adverse selection this assumption is important! Why?
Klaus M. Schmidt (LMU Munich)
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Contract Theory, Summer 2010
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The Standard Hidden Action Model
Assumption 2.1
The utility functions of P and A are given by
P(x, a, w(x))
=
x − w(x)
U(w(x), a)
=
V (w(x)) − G(a)
where
(1) V (·) is a continuous, strictly increasing and concave function defined on
(w, ∞), w ∈ {−∞, <}.
(2) limw→w V (·) = −∞
(3) ∀a ∈ A ∃w ∈ (w, ∞), such that V (w) − G(a) ≥ U, where U is the agent’s
reservation utility.
(4) G(·) is a positive function defined on A.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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The Standard Hidden Action Model
Remarks:
1. The agent is assumed to be risk averse while the principal is risk-neutral.
Risk-neutrality of the principal is mainly assumed for simplicity, but it also
has a natural economic interpretation if P is the owner and A the
manager of a firm. Which one?
The agent’s utility function is assumed to be additively separable in effort
costs and utility from income. What does this mean?
2. What is the meaning of assumption (2)? Is this assumption realistic?
3. What is the meaning of assumption (3)?
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2. Static Moral Hazard
Contract Theory, Summer 2010
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The Standard Hidden Action Model
Assumption 2.2
x ∈ {x1 , x2 , . . . , xn },
a ∈ A, A is a finite set,
Pn
f : A → S = {f ∈ <n | f > 0, i=1 fi = 1}, where f (a) = (f1 (a), . . . , fn (a))
is a probability distribution over the xi , given a.
Remarks: This discrete formulation looks a bit clumsy, but a continuous
formulation has several problems:
1. If x is continuous, it may happen that an optimal solution to the moral
hazard problem does not exist (see the Mirrlees example below).
2. If A is infinite, an optimal solution to the MH problem does exist, but there
is no general technique how to solve it. See however the “First Order
Approach” below.
3. If some outcomes have zero probability, the solution to the MH problem
may become trivial. See Proposition 2.2 below.
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2. Static Moral Hazard
Contract Theory, Summer 2010
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The First Best Allocation
The First Best Allocation
Which allocation (a, w1 , . . . , wn ) would maximize the utility of the principal
subject to the constraint that the agent receives at least his reservation utility?
Note: Here we do not care how to implement the first best action. It would be
easy to implement the first best if a was observable and verifiable and could
be contracted upon. Even though this is not the case we are interested in, the
first best serves as a useful benchmark.
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2. Static Moral Hazard
Contract Theory, Summer 2010
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The First Best Allocation
The first best problem:
max
a,w1 ,...,wn
n
X
fi (a) [xi − wi ]
i=1
subject to
n
X
(PC)
fi (a)V (wi ) − G(a) ≥ U
i=1
Question: Why do we maximize only the principals utility? What would be the
result if we maximized the agent’s utility, or a weighted sum of the two utility
functions?
The Lagrangian for this problem is:
L =
n
X
i=1
Klaus M. Schmidt (LMU Munich)
"
fi (a) [xi − wi ] − λ U −
n
X
i=1
2. Static Moral Hazard
fi (a)V (wi ) + G(a)
#
Contract Theory, Summer 2010
9 / 64
The First Best Allocation
FOCs:
∂L
= −fi (a) + λfi (a)V 0 (wi ) = 0
∂wi
→
V 0 (wi ) =
1
λ
for all i ∈ {1, . . . , n}.
What do these FOCs imply for the first best wage contract?
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
10 / 64
The First Best Allocation
Let us now characterize the first best action: Define
h ≡ V −1 (w) ,
i.e. h(V ) tells us how much the agent has to be paid in order to achieve the
utility level V . In the optimal solution (PC) must be binding. Why?
V (w) − G(a) = U
Define
C FB (a) = h(U + G(a))
Thus, the first best problem reduces to
max
a
n
X
fi (a)xi − C FB (a)
i=1
Given that A is finite, a solution aFB to this problem exists and can be found.
Why? How?
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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The Second Best Problem
The Second Best Problem
We are now looking for the optimal contract to be offered by P to A given that
a is A’s private information.
max
a,w1 ,...,wn
n
X
fi (a) [xi − wi ]
i=1
subject to
(IC)
(PC)
a ∈ arg max
â∈A
n
X
n
X
fi (â)V (wi ) − G(â)
i=1
fi (a)V (wi ) − G(a) ≥ U
i=1
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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The Second Best Problem
Remarks:
1. The principal chooses a in the sense that he chooses which a to induce
the agent to take. However, in order to do so he has to satisfy the
incentive compatibility constraint (IC).
2. The principal cannot force the agent to participate but has to offer the
agent at least an expected utility of the agent’s reservation utility, U. This
is the participation constraint (PC).
Two reference cases:
1. Suppose the principal can observe a and verify it to the courts. In this
case there exists a simple contract that implements the first best
allocation and gives all the surplus to P. How does this contract look like?
2. Suppose the principal cannot observe a but the agent is risk neutral. In
this case there exists another simple contract which implements the first
best. Which one?
Thus, a problem arises only if a cannot be contracted upon and if the agent is
risk averse. In this case there is a fundamental tradeoff between insurance
and incentives.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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The Second Best Problem
How to solve the principal’s problem? Note that these are two
maximization problems that are intertwined: P maximizes her utility subject to
the constraint that A maximizes his utility. In order to solve this we proceed in
two steps:
Step 1: For every a ∈ A, find the cheapest incentive scheme {wi } that
implements a.
min
w1 ,...,wn
subject to
(IC)
−
n
X
fi (a)V (wi ) + G(a) +
i=1
(PC)
n
X
fi (a)wi
i=1
n
X
fi (ã)V (wi ) − G(ã) ≤ 0 ∀ã ∈ A
i=1
−
n
X
fi (a)V (wi ) + G(a) + U ≤ 0
i=1
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
14 / 64
The Second Best Problem
This is a Kuhn-Tucker problem. Note that the (IC) are not convex. In order to
make sure that the Kuhn-Tucker conditions fully characterize the optimal
solution, we have to use a trick:
Let vi = V (wi ), so h(vi ) = wi . Note that h0 > 0 und h00 > 0, because h ≡ V −1
and V (·) is an increasing and concave function. Thus, we can write the
Kuhn-Tucker problem as follows:
max −
v1 ,...,vn
n
X
fi (a)h(vi )
i=1
subject to
(IC)
−
n
X
fi (a)vi + G(a) +
i=1
Klaus M. Schmidt (LMU Munich)
n
X
fi (ã)vi − G(ã) ≤ 0 ∀ã ∈ A
i=1
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Contract Theory, Summer 2010
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The Second Best Problem
(PC)
−
n
X
fi (a)vi + G(a) + U ≤ 0 ,
i=1
This is a concave maximization problem with linear constraints. Let {v ∗ (a)}
denote a solution to this problem (if it exists). Define:
(P
n
∗
∗
i=1 fi (a)h (vi (a)) if v (a)exists
C(a) =
∞
otherwise
Remarks:
1. For a concave maximization problem with linear constraints the
Kuhn-Tucker conditions are necessary and sufficient for the optimal
solution. Without the assumption that U(w, a) = K (a)V (w) − G(a) this
“trick” would not have been possible.
2. The {xi } do not play any role for the optimal incentive scheme which
implements a given a.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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The Second Best Problem
3. It is possible that there does not exist a vector {v1 , . . . , vn } that
implements a. Example: Consider two actions, a and â, such that
f (a) = f (â) and G(a) > G(â). In this case it is impossible to implement a.
Why?
Step 2:
Pn
Let B(a) = i=1 fi (a)xi denote the expected profit of P if A chooses a. Thus,
the principal’s problem now reduces to find:
a∗ ∈ arg max B(a) − C(a)
a∈A
There always exists at least one a for which C(a) < ∞.
Why?
Hence, given that A is finite, this problem has a solution.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
17 / 64
Properties of the Optimal Contract
Properties of the Optimal Contract
Unfortunately, the optimal contract has very few general properties:
Proposition 2.1
The optimal contract must satisfy:
n
X
fi (a)V (wi ) − G(a) = U .
i=1
How can you prove this?
Which assumption drives this result?
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
18 / 64
Properties of the Optimal Contract
Proposition 2.2
The first best allocation can be implemented if one of the following conditions
holds:
1. V (·) is linear
2. There exists a first best action aFB , such that ∀i:
fi (aFB ) > 0
→
fi (a) = 0 ∀a ∈ A, a 6= aFB
3. There exists aFB ∈ A and i ∈ {1, . . . , n} , such that
- fi (aFB ) = 0, and
- fi (a) > 0 ∀a ∈ A, a 6= aFB .
4. There exists an aFB ∈ A which minimizes G(a).
What is the intuition for each of these results?
Klaus M. Schmidt (LMU Munich)
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Contract Theory, Summer 2010
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Properties of the Optimal Contract
These are very weak properties. More interesting questions would be:
1. Suppose that x1 < x2 < . . . < xn . Under which conditions is the optimal
incentive contract strictly increasing in x?
2. Under what conditions is the optimal incentive contract a linear function of
x?
Unfortunately, in the current framework there does not exist an economically
interesting answer to this question. To see what drives the shape of the
optimal incentive contract consider the two-action case:
a ∈ {aL , aH },
G(aH ) > G(aL )
In this case there is only one incentive constraint. If the principal wants to
implement aH , the Lagrangian of the principals cost minimization problem is:
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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Properties of the Optimal Contract
L =
−
n
X
fi (aH )h(vi ) − λ[U −
i=1
−µ[
n
X
fi (aH )vi + G(aH )]
i=1
n
X
(fi (aL ) − fi (aH ))vi + G(aH ) − G(aL )]
i=1
Note that we do not have to worry about corner solutions. Why not?
The FOCs of this problem are:
−fi (aH ) · h0 (vi ) + λfi (aH ) + µ [fi (aH ) − fi (aL )] = 0
for all i ∈ {1, . . . , n} and µ > 0.
These conditions can be written as:
h0 (vi ) = λ + µ − µ
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
fi (aL )
.
fi (aH )
Contract Theory, Summer 2010
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Properties of the Optimal Contract
These conditions imply that wi increases if and only if fi (aH )/fi (aL ) goes up:
fi (aH )
↑ ⇔
fi (aL )
⇔
Klaus M. Schmidt (LMU Munich)
fi (aL )
↓
fi (aH )
RHS ↑
⇔
⇔
LHS ↑
vi ↑ (because h is convex)
⇔
wi ↑ (because V (w) is increasing)
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Contract Theory, Summer 2010
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Properties of the Optimal Contract
Remarks:
1. fi (aH )/fi (aL ) is the so called “likelihood ratio”. Thus, w(x) is increasing in
x if and only if the likelihood ratio is increasing in i (i.e. in x). This is
called the Monotone Likelihood Ratio Property (MLRP). This property
is satisfied by many standard distribution functions (e.g. uniform, normal,
etc.) but not by all.
2. The optimal incentive scheme looks like the solution to a statistical
inference problem: The wage goes up if the signal about the agent’s effort
suggests that it is more likely that the agent has chosen the right action.
However, this is not at all a statistical inference problem. Why not?
3. There is no hope to get a monotonic relation between x and w in general,
because the principal’s cost minimization problem is independent of the
xi . Only the probabilities of the xi enter this problem, but not the absolute
value of the xi .
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2. Static Moral Hazard
Contract Theory, Summer 2010
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Properties of the Optimal Contract
4. MLRP implies First Order Stochastic Dominance (FOSD), but not vice
versa. F1 (x) dominates F2 (x) if and only if F1 (x) ≤ F2 (x) ∀x. Example
that FOSD 6⇒ MLRP:
f (aL )
f (aH )
f (aH )
f (aL )
=
=
=
(0.1, 0.9, 0)
(0.05, 0.05, 0.9)
1 5
( ,
, ∞)
2 90
F (aH ) dominates F (aL ), but MLRP is violated.
5. We can enforce that 0 ≤ w 0 (x) ≤ 1 if we assume
free disposal: the agent can always destroy profits unnoticed.
profit boosting: the agent can always increase profits out of his own pocket.
Both assumptions make a lot of economic sense, but they may
complicate the maximization problem considerably.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
24 / 64
The Mirrlees Example
The Mirrlees Example
The following example illustrates the problem that can arise if x is continuous.
Suppose:
U(w, a) = −e−w − a
x = a + ˜, a ∈ {aL , aH }
a ∈ {aH , aL }
˜ ∼ N(0, σ 2 )
This example is very natural. It assumes CARA and an additively separable
utility function, and a normally distributed noise term. Nevertheless, Mirrlees
(1974) has shown that there does not exist a solution to the principal’s
problem.
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2. Static Moral Hazard
Contract Theory, Summer 2010
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The Mirrlees Example
The idea is as follows:
There exists a sequence of step contracts w m (x) that implement aH at cost
C m (aH ), such that C m (aH ) > C FB (aH ) for all m, but
lim Cm (aH ) = C FB (aH )
m→∞
i.e. the first best can be approximated arbitrarily closely but cannot be
reached.
Illustrate graphically!
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Contract Theory, Summer 2010
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The Mirrlees Example
Intuition:
Pay a fixed wage to the agent for almost all outcomes, but punish him
severely if x falls below a certain threshold.
The normal distribution has the property, that
f (x, aH )
=0,
x→−∞ f (x, aL )
lim
i.e., if x becomes very small, than the likelihood that the agent has
chosen the right action gets arbitrarily close to 0.
This makes a harsh punishment very efficient, because the probability
that this punishment has to be carried out almost vanishes if the agent
chooses the right action.
You are supposed to show this in one of the exercises.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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The Mirrlees Example
The Mirrlees example considers a very natural case. Therefore it raises two
deep concerns about the standard moral hazard model:
1. Technical problem: There need not exist a solution to the principal’s
maximization problem if x is continuous.
2. Economic problem: If we can get arbitrarily close to the first best solution,
the moral hazard does not seem to be very important.
What would you conclude from this?
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2. Static Moral Hazard
Contract Theory, Summer 2010
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The Sufficient Statistics Result
The Sufficient Statistics Result
Even though optimal incentive schemes have very few general properties, the
model has very strong implications about which observable variables should
be contracted upon. Consider again the two action case and suppose that the
principal can observe not only the profit level x, but also some additional
signal y. Suppose that (x, y) is distributed with joint density g(x, y; a).
Furthermore, suppose that the principal wants to implement aH . Under what
conditions should the optimal contract be conditional on y?
The optimal incentive scheme is characterized by
w(x, y) :
1
g(x, y; aL )
=
λ
+
µ
−
µ
V 0 (w(x, y))
g(x, y; aH )
Thus, w(x, y) does not vary with y if and only if
g(x, y; aL )
= α(x)
g(x, y; aH )
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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The Sufficient Statistics Result
Suppose this is the case. Then we can write
g(x, y; a) = A(x, y) · B(x; a)
by defining: A(x, y) = g(x, y; aH ), B(x; aH ) = 1, and B(x; aL ) = α(x).
If it is possible to separate g(·) this way, then x is called a sufficient statistic
for (x, y) with respect to a, i.e., x contains all relevant information about a, and
y cannot be used to make additional inferences about a.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
30 / 64
The Sufficient Statistics Result
More intuitively: If g(x, y; a) = A(x, y) · B(x; a) then we can split up the
lottery over (x, y) in two sublotteries:
a lottery over x which depends on a
a lottery over (x, y) which does not depend on a
A(x,y )
B(x;a)
a
z}|{
=⇒
x
z}|{
=⇒ (x, y)
Note that the second lottery does not contain any information about a but is
rather white noise.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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The Sufficient Statistics Result
The sufficient statistics result (Holmström, 1982) says that if x is some
random vector of observable signals, and if T (x) is a sufficient statistic for x,
then the optimal contract can be written as a function of T (x) only.
Conversely, this result says that if there is an additional signal that does
contain additional statistical information about the agent’s effort, then the
optimal contract should be conditional upon it.
This implies that the optimal contract is far more complicated than the actual
contracts that we observe in reality.
Can you give some examples to illustrate this?
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
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The First-Order Approach
The First-Order Approach
A different approach for solving the principal agent problem goes back to
Holmström (1979). This approach is more elegant but, unfortunately,
considerably less general:
a∈A⊂<
x ∈<
f (x, a) is the density of x given action a
Thus, the agent’s action and the observed outcome are continuous variables.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
33 / 64
The First-Order Approach
The principal’s problem is:
max
w(·),a
Z
[x − w(x)] f (x, a)dx
subject to:
(IC)
a ∈ arg max
(PC)
Z
Z
V (w(x))f (x, a)dx − G(a)
V (w(x))f (x, a)dx − G(a) ≥ U
The First Order Approach replaces IC by the FOC of the agent’s maximization
problem:
(IC 0 )
Klaus M. Schmidt (LMU Munich)
Z
V (w(x))fa (x, a)dx − G0 (a) = 0
2. Static Moral Hazard
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The First-Order Approach
Solving this problem we get as the FOC for the optimal incentive contract:
−f (x, a) + µV 0 (w(x))fa (x, a) + λV 0 (w(x))f (x, a) = 0
⇔
fa (x, a)
f (x, a)
1 = V 0 (w(x)) · µ
+λ
f (x, a)
f (x, a)
⇔
1
fa (x, a)
=
λ
+
µ
V 0 (w(x))
f (x, a)
Remark: This is the differential version of the condition we had in the discrete
problem with a very similar interpretation.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
35 / 64
The First-Order Approach
Problems of the First Order Approach:
1. The Mirrlees example applies here as well. If f (x, a) is normal, we can
approach the first best arbitrarily closely.
2. The agent’s FOC is only necessary for a global optimum of the agent’s
problem. To be also sufficient, the agent’s objective function would have
to be concave in a. But this depends on w(x), which is determined
endogenously!
3. Thus, if we use the First Order Approach, you first have to solve for the
optimal w(x). Then you have to check whether the agent’s objective
function, given this w(x) is indeed globally concave. This is ugly, because
we do not know in advance whether the First Order Approach can be
used.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
36 / 64
The First-Order Approach
4. There are some (quite strong) conditions on f (x, a) that guarantee the
validity of the First Order Approach.
Linear Distribution Function Condition (LDFC):
f (x, a) = afH (x) + (1 − a)fL (x) ,
a ∈ [0, 1] .
Convexity of the Distribution Function Condition (CDFC):
F (x, λa + (1 − λ)a0 )) ≤ λF (x, a) + (1 − λ)F (x, a0 ),
∀a, a0 , x, 0 ≤ λ ≤ 1.
requires that F (x, a) is convex in a.
See Rogerson (1985) and Jewitt (1988) for details.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
37 / 64
Limited Liability Constraints
Limited Liability Constraints
So far we assumed that there is no limit to the punishments that can be
imposed on the agent. It would be much more realistic to assume that these
punishments are bounded, e.g. because the agent is wealth constrained and
protected by limited liability.
Remarks:
1. Limited liability means that the additional constraint wi ≥ w has to be
satisfied, with V (w) > −∞.
2. With this additional constraint the participation constraint may become
slack: it may be optimal to pay a rent to the agent.
3. Thus, the principal has to deal with three problems simultaneously:
To give the right incentives to the agent
To achieve efficient risk-sharing
To minimize the rent that has to be left to the agent.
This problem is almost intractable.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
38 / 64
Limited Liability Constraints
4. However, if we assume that the agent is risk-neutral, a nice
characterization of the optimal incentive scheme is possible.
Innes (1990): Assume that
the agent is risk neutral
the agent is wealth constrained and protected by limited liability, i.e.
w ≥0
agent is
free disposal and profit boosting, i.e., 0 ≤ w 0 (x) ≤ 1.
if a1 < a2 , then F (x, a2 ) dominates F (x, a1 ) is the sense of FOSD.
Then the optimal contract looks like a debt contract:
(
0
if x < x
w(x) =
x − x if x ≥ x
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
39 / 64
Limited Liability Constraints
Intuition:
FOSD means that a higher effort level shifts probability mass to higher
outcomes.
Thus, if the agent’s reward is shifted to higher outcomes as well, he has a
stronger incentive to spend effort.
The idea is to shift his rewards to higher outcomes as much as possible.
Because of 0 ≤ w 0 (x) ≤ 1 the contract that shifts the agent’s rewards to
higher outcomes as much as possible is a debt contract.
Without this constraint it would pay positive wages only for the highest
possible outcome and 0 otherwise.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
40 / 64
Limited Liability Constraints
Proof sketch:
Consider any non-debt contract w(x) which induces the agent to spend
some effort a.
Now consider any debt contract w̃(x) which gives the same expected
wage to the agent if the agent chooses the same effort level a.
Note that the non-debt contract and the debt contract can intersect only
once, because w 0 (x) ≤ 1 and w̃ 0 (x) = 1.
Note further that the non-debt contract provides weaker incentives for the
agent. A marginal increase in effort shifts probability mass to the right.
The debt contract gives full marginal incentives to the agent for the high
outcomes, while the non-debt contract gives less than full marginal
incentives.
Thus, for the same expected wage the debt contract induces the agent to
spend more effort than any non-debt contract.
Hence, a non-debt contract cannot be optimal.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
41 / 64
Limited Liability Constraints
Empirical Implications:
1. Model explains why simple debt contracts may be optimal.
2. Model gives a partial justification for share options to be given to
managers. However, a typical incentive contract with share options gives
only some fraction of the marginal return in the very good states to the
manager.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
42 / 64
Team Production
Team Production
Holmström (1982) considers a model where the principal deals with several
agents:
m > 1 agents, j = 1, . . . , m.
actions aj are private information of the agents
principal observes only x̃ = x̃(a1 , . . . , am )
The team production problem arises even if output is deterministic and/or if
the agents are all risk neutral.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
43 / 64
Team Production
Deterministic Output
x = x(a1 , . . . , am ), with x(0, . . . , 0) = 0.
Uj = wj − Gj (aj )
wj (x) (“sharing rule”)
Note that with deterministic output we can assume w.o.l.g. that the agent is
risk neutral.
The first best action, aFB maximizes
x(a1 , . . . , am ) −
m
X
Gj (aj )
j=1
and is characterized by
∂x
= Gj0 (aj ) .
∂aj
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
44 / 64
Team Production
To see the problem let us start with the case where there is no
and
Pprincipal
m
let us assume that the sharing rule has to be balanced, i.e., j=1 w(x) = x.
Proposition 2.3
There does not exist a balanced sharing rule which implements aFB .
Proof: We prove this result only for the case of differentiable sharing rules.
The full proof can be found in Holmström (1982).
Agent j maximizes
wj (x(aj , a−j )) − Gj (aj ) .
The FOCs for individually optimal effort choices in a Nash equilibrium require:
wj0 (x) ·
Klaus M. Schmidt (LMU Munich)
∂x
= Gj0 (ai )
∂aj
2. Static Moral Hazard
Contract Theory, Summer 2010
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Team Production
The First Best can only be implemented if
wj0 (x) = 1
for all agents j. But this contradicts the requirement of a balanced budget
because:
X
X
wi (x) = x ⇒
wi0 (x) = 1
i
i
Q.E.D.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
46 / 64
Team Production
What happens if we do not require the sharing rules to be balanced, but only
that they never yield a deficit?
Proposition 2.4
There are sharing rules {wi (x)} such that aFB is a Nash equilibrium of the
induced game between the agents and such that there never is a deficit (on
and off the equilibrium path).
Proof: Consider
wi (x) =
where bi > Gi (aiFB ) > 0 and
P
(
bi
0
if x ≥ x(aFB )
if x < x(aFB )
bi = x(aFB ).
There always
exist bi satisfying these conditions because
P
x(aFB ) − Gi (aiFB ) > 0 (otherwise aFB would not be efficient). Given this
sharing rule aFB is a Nash equilibrium.
Q.E.D.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
47 / 64
Team Production
Remarks:
1. An external principal could play an important role here by acting as a
“budget breaker”. He could get the surplus off the equilibrium path.
Possible explanation for why capital hires labor (and not vice versa), i.e.
why residual profits go to capital owners rather than workers.
2. Unfortunately, however, this is not the only N.E.. Thus, this sharing rule
does not “fully implement” aFB .
3. Furtermore, this sharing rule is not “collusion-proof”. Who could collude
in order to exploit the other participants?
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
48 / 64
Team Production
Stochastic Outcomes and Risk Neutral Agents
With stochastic outcomes and risk neutral agents another sharing rule has to
be used to implement the first best:
wi (x) = x − Ki
P
where E(x) − Ki > Gi (aiFB ) and
Ki = E((n − 1)x).
Remarks:
1. This is a simple version of a “Groves mechanism”. At the margin, each
agent gets the entire social surplus. Therefore, each agent wants to
maximize social surplus.
2. The principal “sells the store” to each agent. In order for the principal to
break even in expectation, it has to be the case that
PN
Ex + i=1 Ki = n · Ex.
3. However, there is a strong incentive for the agents to collude against the
principal. What would they do?
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
49 / 64
Team Production
Conclusion: With team production there arise interesting new moral hazard
problems (collusion, renegotiation, full implementation, etc.) even if risk
sharing is not an issue. See also Mookherjee (1984), Ma, Moore and Turnbull
(1988), Holmström and Milgrom (1990), Itoh (1992) and the references given
in Bolton and Dewatripont (2006).
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
50 / 64
Relative Performance Evaluation
Relative Performance Evaluation
Holmström (1982) points to an additional problem if there are several agents.
Consider the following model with
a risk neutral principal
n risk averse agents, Ui (w, a) = Vi (wi ) − Gi (ai ), V 0 > 0, V 00 < 0
a vector of signals x = (x1 , . . . , xn )
F (x, a) cdf of x given a = (a1 , . . . , an )
f (x, a) density of x given a.
Question: Should the wage of agent i depend on the performance of agent j?
The general answer should be clear from Holmström’s “Sufficient Statistics
Result”.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
51 / 64
Relative Performance Evaluation
Some examples:
1. Suppose that xi = xi (ai , i ), and i and j are stochastically independent
for all i, j, i 6= j. Then xi is a sufficient statistic for x with respect to ai and
the optimal incentive scheme for agent i is of the form wi = wi (xi ).
2. Suppose that xi = ai + η + i , i = 1, . . . , n, where η is a common shock
and the i are ideosyncratic shocks. Suppose further that η, 1 , . . . , n
are all independently normally distributed with mean 0 and variances
ση2 , σ21 , . . . , σ2n . Let τi = σ12 be the “precision” of i and define
i
x=
n
X
i=1
αi xi ,
τi
αi = ,
τ
τ=
n
X
τi .
i=1
Then an optimal contract has the form wi (xi , x).
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
52 / 64
Relative Performance Evaluation
Remarks:
1. The intuition for the result is simple. We are looking for a good estimator
of η. We have n independently and normally distributed signals about η,
so the weighted average of the signals (where the weights correspond to
the precisions) should be used.
2. Note: The result does not say that an optimal incentive scheme exists!
Whenever you work with normal distributions, the Mirrlees problem may
arise! However, we can avoid this problem by assuming limited liability. In
this case it may become quite difficult to compute the optimal incentive
scheme, but the above result would still hold.
3. A similar result holds if xi (ai , θi ) = ai · (η + i ).
4. If σ2i → 0 ∀i, we can approach the first best. Why?
5. If n → ∞, it also possible to filter out η completely (law of large numbers).
In this case the optimal incentive scheme becomes wi (x) = wi (xi − x).
Note that his has much more structure than wi (x) = wi (xi , x).
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
53 / 64
Relative Performance Evaluation
6. A special form of relative performance evaluation are rank order
tournaments (Lazear and Rosen). However, tournaments are rarely
optimal, because they ignore a lot of additional information that would be
available (namely the absolute performance levels). However,
tournaments may be optimal if outcomes are observable but not
verifiable. In this case the principal has a problem to commit to pay the
agents according to the agreement. How can this problem be solved by a
tournament.
7. In many situations relative performance evaluation is a good idea, but
there are also several important problems with it:
multiple equilibria
collusion between agents
incentives to create a bad reference group
sabotage
positive production externalities cannot be exploited
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
54 / 64
Renegotiation
Renegotiation
Suppose that the parties agreed to a contract that turns out to be inefficient ex
post. In this case there is a strong incentive to renegotiate the initial contract
in order to achieve ex post efficiency. However, from an ex ante perspective
an ex post inefficient contract may be optimal. In this case the anticipation of
renegotiation may distort the incentives of the involved parties.
Why is there an incentive to renegotiate in a moral hazard context?
What is the ex post inefficiency?
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
55 / 64
Renegotiation
Fudenberg and Tirole (1990):
Example:
a ∈ {aH , aL }
q ∈ {S, F }
f (S, aL ) =
f (S, aH ) =
1
4
3
4
3
4
= 41
, f (F , aL ) =
, f (F , aH )
In order to implement aH , it has to be the case that wS > wF . Suppose there
exists an incentive scheme which induces the agent to choose aH with
probability 1. After the agent has chosen his action, both parties have an
incentive to renegotiate the initial contract to a fixed wage contract, w, such
that
Klaus M. Schmidt (LMU Munich)
w
<
V (w)
>
3
1
wS + wF
4
4
3
1
V (wS ) + V (wF )
4
4
2. Static Moral Hazard
Contract Theory, Summer 2010
56 / 64
Renegotiation
Such an w must exist if the agent is risk averse and the principal is risk
neutral.
Time structure with renegotiation:
contract w(x)
agent chooses a
principal makes renegotiation offer
outcome x realized
payoffs
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
57 / 64
Renegotiation
If the agent anticipates that he will eventually get a fixed wage, he has no
incentive to choose aH . This proves:
Proposition 2.5
If the parties are free to renegotiate, then there does not exist a contract that
induces the agent to choose aH with probability 1.
Remarks:
1. There does exist an equilibrium in mixed strategies:
The agent randomizes between aH and aL .
The principal does not know which action has been taken by the agent.
Renegotiation with asymmetric information (adverse selection problem).
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
58 / 64
Renegotiation
2. Fudenberg and Tirole (1990) characterize the optimal contract with
renegotiation and show that it is strictly worse than the optimal contract if
renegotiation is impossible.
3. The result by Fudenberg and Tirole is troubling. There is a strong
incentive for managers who are paid in stock options to either renegotiate
the terms of these options when they retire or to hedge the risk involved
with these options. This is in fact what we sometimes observe.
4. In this model renegotiation is bad for both parties. The reason is that the
optimal contract has to satisfy an additional constraint: it has to be
renegotiation proof.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
59 / 64
Renegotiation
Definition 1
A contract is renegotiation proof if there is no state of the world in which the
involved parties have an incentive to renegotiate the contract.
The “renegotiation principle” says that if the parties are free to renegotiate,
then we can restrict attention without loss of generality to renegotiation proof
contracts.
The reason is that if the contract is renegotiated in some state of the world,
then the parties could have included the result of the renegotiation process
(that they can perfectly anticipate) into the original contract already.
However, the renegotiation principle requires that the parties do not get
access to additional information when it comes to the renegotiation stage.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
60 / 64
Renegotiation
Hermalin and Katz (1991) consider a model where renegotiation is good
because the parties can contract on new information. They assume
that the principal and the agent both observe a,
but the courts only observe x.
In this case the following proposition holds:
Proposition 2.6
Suppose the principal can observe a and then make a take-it-or-leave-it
renegotiation offer to the agent. Then it is possible to implement the first best.
Klaus M. Schmidt (LMU Munich)
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Contract Theory, Summer 2010
61 / 64
Renegotiation
Proof: Consider a contract w(x) that implements aFB without renegotiation.
For any action this contract offers a lottery to the agent. After the agent has
taken the action, the principal will offer to replace the lottery by a fixed
payment w(a) such that
V (w(a)) = E (V (w(x)) | a) .
Hence, the agent’s utility after each action is unaffected. Thus, the agent is
still induced to choose aFB , but he does not bear any risk. This is the first
best.
Q.E.D.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
62 / 64
Renegotiation
Remarks:
1. The idea is that the principal first “sells the store” to the agent and then,
after he observed the agent’s effort choice, he buys back the store in
order to better insure the agent.
2. The first best can also be implemented when the agent can make a
take-it-or-leave-it offer in the renegotiation game.
3. Hermalin and Katz describe an extreme case, where the principal is
equally good informed as the agent, while the courts observe only x. The
more realistic case would be were the principal observes a more precise
signal about the agent’s action than the courts do. This case is quite
difficult but Hermalin and Katz offer some nice characterizations for
several special cases.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
63 / 64
Renegotiation
Conclusion:
1. Fudenberg and Tirole show that renegatiation is bad, if it makes it more
difficult to the parties to commit to the terms of the contract.
2. Hermalin and Katz show that renegotiation is beneficial, if the parties can
exploit additional non-verifiable information at the renegotiation stage.
Klaus M. Schmidt (LMU Munich)
2. Static Moral Hazard
Contract Theory, Summer 2010
64 / 64