John Riley
D.5
6 April 2017
MORAL HAZARD
1.
Efficient contracts
1
2.
Principal Agent problem (two outcomes)
4
3.
The general case
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Appendix 1: Comparison of Finite Distributions
Note: Draft D.5 corrects a number of errors in section 3.
19 pages
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John Riley
Econ 201C
1. Efficient contracts
The model
The owner of a firm (the “principal”) contracts with a manager (the “agent”) to operate
the firm. The value of the firm’s output is a random variable y( x) that depends on the agent’s
action x  X  {x1 ,..., xn } where x1  ...  xn . Higher values of x are more costly to the agent. A
more costly action (higher x ) shifts the distribution of the random variable to the right and so
raises the expected value of the firm’s output. We assume that there are S possible
realizations y  ( y1 ,..., yS ) where y1  y2  ...  yS . The probability of each of these outcomes
is the probability vector  ( x)  ( 1 ( x),...,  S ( x)) .
The principal observes the firm’s value y s and thus can infer the “state” s . However
the principal does not observe the action of the manager. Thus the wage contract
w  ( w1 ,..., wS ) , offered to the agent is a function only of the state. Let the state s payment
to the agent be ws . The revenue retained by the principal is therefore rs  ys  ws .
The Von Neumann-Morgenstern (VNM) utility functions are vP ( rs ) and vA ( ws )  K ( x) .
Note that K ( x) is the utility cost to the agent of taking action x . The set of possible actions is
the agent
Any contract w offered to the agent induces some action by the agent, x(w) and hence
expected payoffs U P ( w) and U A ( w) to the principal and agent. The feasible set of such
payoffs is depicted below. An efficient contract then lies on the boundary of this set.
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Fig. 1.1: Feasible payoffs
John Riley
Econ 201C
Consider the two outcome (2 state) case
Figure 1.2 below depicts a level set for the principal, if the agent takes action x .
U P (r )   1 ( x)vP (r1 )   2 ( x)vP (r2 )  u P
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Fig. 1.2: Level set of the principal
Note first that the principal’s marginal rate of substitution is
MRSP (r ) 
1 ( x) vP (r1 )
 2 ( x) vP (r2 )
Thus along the 45 line in Figure 1-2, the MRS is equal to the probability ratio (or odds).
Note second, that the intersection of the level set and the 45 line is the point
(r1 , r2 )  (r , r ) where vP (r )  u P
For the agent taking action x , expected utility is
U A ( w, x)   1 ( x)v A ( w1 )   2 ( x)v A ( w2 )  K ( x)
Therefore the level set U A ( w, x)  u A can be written as follows:
 1 ( x)vA ( w1 )   2 ( x)v A ( w2 )  u A  K ( x) .
Define w( x) to satisfy
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John Riley
Econ 201C
v A ( w( x))  u A  K ( x )
Arguing exactly as for the principal, the level set of the agent is depicted below.
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Fig. 1.3: Level set of the agent
Pareto Efficient contract when the action is observable
For the two state case we can depict the possible outcomes and level sets in an
Edgeworth-Box diagram. Total revenue in state 1 exceeds total revenue in state 2 ( y1  y 2 ).
Thus the 45 line of the agent is to the left of the 45 line of the principal.
lines
Fig. 1.4: PE allocations
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Econ 201C
For any point above the 45 line of the agent, w1  w2 and r1  r2 . Therefore
MRS A (w) 
 1 ( x)
 MRSP (r ) .
 2 ( x)
For any point below the 45 line of the principal, w1  w2 and r1  r2 . Therefore
MRS A (w) 
 1 ( x)
 MRSP (r ) .
 2 ( x)
Thus no allocation in either of the shaded regions is a PE allocation. Hence the set of PE
allocations are all in the unshaded region between the two 45 lines.
Pareto inefficient contract with observable action
Starting from the PE contract w* , consider the contracts ŵ and ŵˆ on the level set of the agent
South-East of w* . Note that the further the contract is from w* , the lower is the expected
utility of the principal, i.e. the greater the extent of the inefficiency
Fig. 1.5: increasing inefficiency
We shall appeal to this observation in section 2.
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John Riley
Econ 201C
Exercise: Risk neutral agent
(a) Consider the S state case. If the agent is risk neutral show that the PE contract for a given
action x is a fixed rent contract r  (r1 ,..., rS )  (r ,..., r )
(b) Characterize the PE contract over the set of actions X .
(c) If the principal leaves the agent to choose the action what will he choose?
2. Principal- agent problem1 when the action is unobservable
We make one additional assumption
Assumption 1: Monotone likelihood ratio property holds2
 t ( x)  t ( x)
for all s  t and all x  x .

 s ( x)  s ( x)
That is, the relative probability of higher firm revenue (lower state) is greater if the agent takes
a more costly action.
For the two state case this reduces to the assumption that  1 ( x2 )   1 ( x1 ) .
2A. The risk neutral case
As a preliminary, consider the simplest case in which both principal and agent are risk
neutral. There are two states and two actions.
U P (r )   1 ( x)r1   2 ( x)r2 , U A ( w, x))   1 ( x) w1   2 ( x) w2  K ( x)
The sum of the payoffs is therefore
U A ( w, x))  U P (r )   1 ( x) y1   2 ( x) y2  K ( x)
1
In 2016 Bengt Holmstrom was awarded the Novel Prize for his ground-breaking work on incentives. We will
consider a version of his model.
2
See the Appendix for a technical discussion of this assumption
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John Riley
Econ 201C
Ignoring the informational constraint, the PE action maximizes expected total revenue less the
cost of the action. In Figure 2.1 below, the low cost action x1 is efficient.
Fig. 2.1: Action
is efficient
Consider the point (u A , uP ) on the Pareto frontier. Suppose that the principal offers a fixed wage
offer,
( w1 , w2 )  ( w, w)  (u A  K ( x1 ), u A  K ( x1 ))
The agent’s payoff if he chooses action x is
U A ( w1 , w2 , x)  w  K ( x) .
He therefore chooses the least costly action (i.e. the PE action) and his payoff is w  K ( x1 )  u A .
Figure 2.2 depicts the case in which the more costly action is efficient.
Fig. 2.2: Action
is efficient
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John Riley
Econ 201C
The fixed wage contract
( w1 , w2 )  ( w, w)  (u A  K ( x2 ), u A  K ( x2 ))
is no longer efficient, since without an incentive to choose a more costly action, the agent
chooses the cost minimizing action x1 .
There are, however, many contracts that do achieve efficiency. The payoff to taking
action x1 is
U A ( w, x1 )   1 ( x1 ) w1   2 ( x1 ) w2  K ( x1 )
  1 ( x1 ) w1  (1   1 ( x1 )) w2  K ( x1 )   1 ( x1 )( w1  w2 )  w2  K ( x1 )
By the same argument,
U ( w, x2 )   1 ( x2 )( w1  w2 )  w2  K ( x2 )
Therefore the agent is indifferent between the two actions if and only if
U A ( w, x2 )  U A ( w, x1 )  ( 1 ( x2 )   1 ( x1 ))( w1  w2 )  ( K ( x2 )  K ( x1 ))  0 .
By the same argument the agent strictly prefers action x 2 if
U A ( w, x2 )  U A ( w, x1 )  ( 1 ( x2 )   1 ( x1 ))( w1  w2 )  ( K ( x2 )  K ( x1 ))  0
Thus the agent is only incentivized to take action x 2 if the wage difference across states
satisfies the following inequality:
w1  w2 
K ( x2 )  K ( x1 )
1 ( x1 )  1 ( x2 )
The contracts satisfying this inequality are those in the dotted region in Figure 2.3
below. Note that the level set for action x1 intersects the 45 line at ( w( x1 ), w( x1 )) . Then
U A ( w( x1 ), w( x1 ))   1 ( x1 )v A ( w( x1 ))   2 ( x1 )v A ( w( x1 ))  K ( x1 )
 vA ( w( x1 )  K ( x1 )  u A
Also, the level set for action x 2 intersects the 45 line at ( w( x2 ), w( x2 )) . Then
U A ( w( x2 ), w( x2 ))  v A ( w( x2 )  K ( x2 )  u A .
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John Riley
Econ 201C
Since K ( x2 )  K ( x1 ) , it follows that w( x2 )  w( x1 ) .
line
action is
Fig. 2.3: Level sets of the agent
The agent is indifferent between the two actions at the contract B . By our earlier argument, at
this contract,
w1  w2 
K ( x2 )  K ( x1 )
1 ( x1 )  1 ( x2 )
If the wage difference is larger, then the agent prefers the more costly action, x 2 .
In the language of incentive theory, the more costly action is incentive compatible in the
dotted region (including its boundary) but is not incentive compatible elsewhere in the positive
quadrant.
Remark: If the agent is completely moral he will tell the principal which action he selects. In
economics we rarely rely on such a moral perfection and assume that the agent will pursue his
self-interest.
We are seeking a contract that maximizes the expected payoff to the principal given
that it also satisfies the constraint that the agent’s expected payoff must be at least u A .
U A ( w, x2 )  u A
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John Riley
Econ 201C
Both constraints are satisfied in the grid-lined region in Figure 2.4.
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action is
Fig. 2.4: Both constraints are satisfied
In Figure 2.5a we superimpose Fig. 2.4 on an Edgeworth Box. Looking at the points where the
two level sets cross the 45 line of the principal, it follows that the low cost action is efficient.
Hence the fixed wage contract is both incentive compatible and efficient.
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Fig. 2.5a: Efficient action is
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John Riley
Econ 201C
By the same argument, the high cost action is efficient in Fig. 2.5b. To induce the agent to take
action x 2 , the principal incentivizes the agent by paying him sufficiently more when the
outcome is more favorable.
line
Fig. 2.5b: Efficient action is
All the wage contracts on the level set U A ( w, x2 )  u A are PE allocations. But only those
in the shaded region satisfy the incentive compatibility constraint. Thus the incentive
compatible PE contracts are those on the level set U A ( w, x2 )  u A to the right of B .
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John Riley
Econ 201C
2B. Risk-averse agent
Figure 2.3 is reproduced below.
line
action is
Fig. 2.6: Wage contracts for which the agent chooses
The only significant difference is that the superlevel sets of the agent are now strictly
convex. The agent is incentivized to take action x 2 only for contracts in the dotted region.
Inducing the more costly action
We seek the best contract for the principal that induces the agent to take action x 2 , and also
ensures that the agent has a utility of at least u A . The contract must satisfy the incentive
compatibility constraint
U A ( w, x2 )  U A ( w, x1 )  0
and be in the super level set, i.e.
U A ( w, x2 )  u A
The principal then chooses the contract that maximizes his utility subject to these two
constraints. These are both satisfied in the grid-lined region in Figure 2.7.
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John Riley
Econ 201C
line
action is
Fig. 2.7: Both constraints satisfied
A contract like ŵ in the interior of the grid-lined region is not the best for the principal because
it is possible to lower the wage paid to the agent in both states and still be in the region. Then
the best contract for the principal is on the boundary, i.e. in the level set
U A ( w, x2 )  u A
In section 1 we argued that the degree of inefficiency increases around a level set
moving away from the PE contract. Since the PE contract on the level set is the contract N ,
the principal’s incentive compatible expected payoff is maximized at B*
Alternatively the principal can choose to induce action x1 . This is achieved with the
fixed wage contract ( w( x1 ), w( x1 )) , i.e. the point N in Figure 2.7.
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John Riley
Econ 201C
The two contracts are shown below in an Edgeworth Box diagram.
Fig. 2.8: Best action is
Class Exercises:
1. Explain why the best action is x 2 in Figure 2.8.
2. As the agent’s aversion to risk approaches zero, explain why the efficient contract
approaches the contract B .
3. What is the best action if the agent is highly risk averse?
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John Riley
Econ 201C
3. The general case with S states3
We consider the case of a risk neutral principal and seek to characterize best incentive
compatible outcomes. We break the problem down into two steps:
Step 1: For each action fix the expected utility of the agent and solve for the contract that is
best for the principal among all incentive compatible contracts.
Step 2: Select the most favorable contract for the principal.
We will focus on Step 1.
Step 1: Constraints
Lower bound for the agent’s payoff4
U A ( w, x)  u A
Downward Incentive constraints
For action x  X to be incentive compatible,
U A ( w, x)  U A ( w, xˆ ) for all x̂  x .
Upward Incentive constraints
For action x to be incentive compatible,
U A ( w, x)  U A ( w, xˆ ) for all x̂  x
Relaxed problem:
Ignore upward constraints. Intuitively, these are unlikely to be binding.
3
See EM section 7.4
4
Alternatively we can interpret this as the agent’s participation constraint
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John Riley
Econ 201C
Simply to minimize notation we assume that only one downward constraint is binding. Let this
action be x̂
Designer’s problem
Max{U P (r )  U P ( y  w) | U A ( w, x)  U A ( w, xˆ), U A ( w, xˆ)  u A} .
w
The Lagrangian of this maximization problem is
L( w,  )  U P ( y  w)   (U A ( w, x)  U A ( w, xˆ ))   (U A ( w, x)  u A )
 U P ( y  w)  (   )U A ( w, x)  U A ( w, xˆ )  u A
S
S
s 1
s 1
   s ( x)( ys  ws )  (   )[  s ( x)vA (ws )  K ( x)]  constants
L
( x,  )   s ( x)  (   ) s ( x)vA ( ws )   s ( xˆ)vA ( ws )  0 .
ws
Dividing by  s ( x ) we obtain the following equation.
 ( xˆ) 
1  (   )u A (ws )   s
v (w )  0 .
 s ( x) A s
Rearranging this equation
1
vA ( ws )
 (   )  
 s ( xˆ )
 s ( x)
(3.1)
Appealing to the MLRP,
for all s  t
 t ( xˆ )  t ( x)
since x̂  x .

 s ( xˆ )  s ( x)
Thus
for all s  t ,
 s ( xˆ)  t ( xˆ)
.

 s ( x)  t ( x)
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John Riley
Econ 201C
Therefore the right hand side of (3.1) decreases as s increases. Thus the left hand side
decreases with s and so vA ( ws ) increases with s . Since v ( ws ) is concave (the agent is risk
averse) it follows that {ws }sS is strictly decreasing.
Proposition: Risk sharing (Holmstrom)
If the binding constraints are downward constraints and { ys }sS is strictly decreasing, then
{ws }sS is also strictly decreasing. Thus the payment to the agent is strictly increasing in the
value of the firm’s output.
Exercise: Multiple downward constraints
The proof above is for a single binding constraint. Show that the result still holds if multiple
downward constraints are binding.
Remark: Sufficient conditions for the upward constraints to not be binding were derived by
Jewett “Justifying the First-Order Approach to Principal Agent Problems”, Econometrica, (Sept.
1988) pp. 1177-1190. For finite type examples my experience with a spread-sheet suggests that
for a wide range of plausible assumptions, binding upward constraints are hard to find.
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John Riley
Econ 201C
Appendix 1: Comparison of Finite Distributions
First order stochastic dominance property
More weight in the left tail
q  qˆ  Fs (q )  Fs (qˆ ), s  S
Conditional stochastic dominance property
More weight in the left tail for every right-truncated distribution
q  qˆ 
Fs (q) Fs (qˆ )

, t  1, s  S
Ft (q) Ft (qˆ )
Lemma: Ratio Rule
For a, b  0, ai  bi
a1 a2
a a a a a a
  1 2 1 2 2 1 .
b1 b2
b1 b2  b1 b2 b2  b1
Proof: Exercise.
HINT: Define ri 
ai
.
bi
Proposition: The strict CSD property holds if and only if
q  qˆ 
f s (q) f s (qˆ )

, s  1 .
Fs (q) Fs (qˆ )
Proof:
q  qˆ 
Fs (q) Fs 1 (q) Fs (q)  f s 1 (q)
.


Fs (qˆ ) Fs 1 (qˆ ) Fs (qˆ )  f s 1 (qˆ )
Appealing to the Ratio Rule
Fs (q) Fs 1 (q) Fs (q)  f s 1 (q) Fs (q)  f s 1 (q)  Fs (q) f s 1 (q)
.




Fs (qˆ ) Fs 1 (qˆ ) Fs (qˆ )  f s 1 (qˆ ) Fs (qˆ)  f s 1 (qˆ)  Fs (qˆ) f s 1 (qˆ)
QED
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John Riley
Econ 201C
Monotone likelihood property
f1 (q) f 2 (q) f3 (q)


...
f1 (qˆ ) f 2 (qˆ ) f3 (qˆ )
Proposition: MLRP  CSDP
Proof: Appealing to the Ratio Rule
F1 (q) f1 (q) f1 (q)  f 2 (q) F2 (q) f 2 (q)




F1 (qˆ ) f1 (qˆ ) f1 (qˆ )  f 2 (qˆ) F2 (qˆ) f 2 (qˆ)
Also
F2 (q) f 2 (q) f3 (q)
F (q) F2 (q)  f3 (q) F3 (q) f3 (q)


 2



F2 (qˆ ) f 2 (qˆ ) f3 (qˆ )
F2 (qˆ ) F2 (qˆ )  f3 (qˆ ) F3 (qˆ ) f3 (qˆ)
We have proved that
F1 (q) F2 (q) F3 (q)
.


F1 (qˆ ) F2 (qˆ ) F3 (qˆ )
The proof for higher states is very similar.
QED
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