INTRODUCTION
MORAL HAZARD ;
When one party to a transaction may undertake certain actions that ;
a- affect the other party’s valuation of the transaction but that ;
b- the second party can not monitor or enforce perfectly .
A classical example is fire insurance , where the insure may or may not
exhibit sufficient care while storing flammable materials. The solution to a
problem of moral hazard is the use of incentives.
This is structuring the transaction so that the party who undertakes the
action will , in his own best interests, take actions the second party
would prefer . Ex; partial insurance
Another examples are ;leasing a car for a specific period and return it
after that ,or hiring a person to do a hard work at the tasks that are set
for him
In each of these cases it is possible to monitor , but perfect monitoring
and enforcement may be impossible .
But the transaction might be restructured in such a way that the party
taking the action has relatively greater incentives to act in away the
second party prefers .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
1
INTRODUCTION
From the incentive point of view the transaction should be
restructured in such a way that the party who is taking the hidden action
bears fully the consequence of his action . For example ;
1-The insurance company may only insure only up to 90 percent . Or
the company may refuse to insure the company .
2-The party who lease the car could share a part of the amount of sale if
the car is sold after leasing . Or the company may sell the car instead of
leasing it.
3-The worker’s wage may be tied to some observable measures of
hardworking . For example piece rate paying system .
But there are always inefficiencies in these kind of restructuring
contracts . For example ;
1- Selling the car may bear more tax than leasing it .
2- A company with few individual share holders may be less willing to
bear the risk of having a fire .
3- The piece rate paying system may not be feasible . Quality as well as
quantity may be important . Piece rate system may subject the worker
to risks that he may be less well equipped .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
2
EFFICIENT INCENTIVES ; EXAMPLE
principle ; the first party who hires a second party ;
agent the second party which is performing some specific task for
the principle .
reservation level of utility ; the utility which the agent could get
from the next best opportunity . All other things being equal the
agent is not willing to work hard . Whether the agent works hard or
not determines the value to the principle of having this agents work .
Reservation wage ; a wage high enough so that combined with not
working hard the agent’s net utility exceeds the reservation level of
utility . If the agent does work hard , then the principle will get
enough out of the transaction to make it worthwhile for both sides .
U(w,a) = W1/2 – a = the agent’s overall utility from work.
W = the amount which the agent receive .
a = 5 , if the agent works hard .
a = 0 , if the agent does not work hard .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
3
EFFICIENT INCENTIVES ; EXAMPLE
Reservation level of utility = U* = 9
If the agent works hard , the accomplished task worth $270 to the
principle and if does not work hard it worth $70 to the principle .
1- If the agent does not work hard , ( U ≥ 9 , a=0 ) , then ;
W1/2 ≥ 9 , or W ≥ $81 , so cost to the principle ≥ 81 , worth of work
to principle = 70 , there will be no deal .
2- if the agent works hard , (U ≥ 9 , a = 5) , then ;
W1/2 - 5 ≥ 9 , → W ≥ $196 , cost is $ 196 , and worth of work is
equal to $ 270 . The deal is worth to do for the principal .
The principle should try to make the agent to work hard . HOW ?
The principle could pay an amount greater than 196 and less than
270 to induce him to work hard . But how could he monitor his hard
working ?
Another possibility is to offer a contract that calls for the agent’s
pay depend on how much effort he puts in .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
4
EFFICIENT INCENTIVES ; EXAMPLE
For example the principle could sign a contract to pay the agent
any amount greater than $196 and less than 270 if he works
hard and any amount less than $81 if he does not work hard .
But , is this contract enforceable? For many reasons which
could deal with high transaction cost and high cost of
monitoring , this kind of contract might not be feasible , either for
principle or for agent .
Even if the principle cannot tie the worker’s wage directly to his
level of effort , the principle might be able to find some indirect
measure of effort to which wages can be tied . For example
suppose that :
Agent is a salesman , who will be representing the
principal to a particular client .
There are two cases ; the agent ( the salesman) can or
can not work hard;
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
5
EFFICIENT INCENTIVES ; EXAMPLE
1- if the agents work hard at making the sale , then ;
a $400 sale result with 0.6 probability
a $100 sale result with 0.3 probability
a $0
sale result with 0.1 probability.
2 - if the agents does not work hard then ;
a $400 sale result with 0.1 probability
a $100 sale result with 0.3 probability
a $0 sale result with 0.6 probability.
The probability
distribution is the
same for principle
and agent
The size of the sale is observable and the agent’s wages can be
made contingent upon this variable . The principle is always risk
neutral . The worth of the agent’s work ( $270 when working hard
and $70 when does not working hard ) is the expected gross profit
. Unless the principle can observe the effort level of the agent
directly , the data receive do not tell conclusively what level of
effort was put in because data could be the outcome of case 1 or
2 . ( $400 sale could be the result of working hard or working not
hard)
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
6
EFFICIENT INCENTIVES ; EXAMPLE
Case 1 – A risk neutral agent
Utility function = U(w,a)= w – a
Reservation level of utility =U*= $81
High effort level → a = 25
and low effort level → a=0
If high work effort could be guaranteed , then principal should pay an
amount to agent which leave him a utility greater than the reservation
level
→
a=25 , w ≥ (U*+ a) → w ≥ 106 then ;
Principal profit = 270 – 106 = 164 at most .
If the hard work can not be guaranteed and the principal has to pay
the agent a wage which leaves him a utility at least greater than the
reservation level
→
a=0 , w ≥ (U*+ a)=81 then ;
Principal’s profit = 70 – 81 = - 11 , is there any other solution ?
The difficulty and cost of monitoring the high level of effort is present
The principal could offer the agent the following contract ;
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
7
EFFICIENT INCENTIVES ; EXAMPLE
1 - If the agent makes no sale he should pay $164 to principle
2 - If the agent makes small sale (worth $100) he should pay $ 64
(164 – 100)
to principle .
3 - If the agent makes large sale (worth $400) the principle pay
agent $236 = (400 – 164)
$164 worth of profit (,maximum profit when agent works hard)
should be guaranteed for the principal any way.
The agent could do one of the followings ;
A- turn down the contract and get reservation level equal to $81.
B- take the contract and put in low effort; his net expected utility is;
(0.1)(236) + (0.3)(-64) + (0.6)(-164) -0 = - 94
C- take the contract and work hard ; his net expected utility is ;
(0.6)(236) + (0.3)(-64) + (0.1)(-164) -25 = 81
The agent is indifferent between option (a) and (c) , because the
reservation level of utility is $81 . If principal could make option (c)
a little more attractive option (c) will be chosen.
What the principal has to do is to get the agent to internalize
the effort of his decision .
D.M. KREPS CH 16
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INCENTIVES
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EFFICIENT INCENTIVES ; EXAMPLE
Case 2 – A risk averse agent
Agent’s utility = U = W1/2 – a . Reservation level = U*=9
Working hard → a=5 , not working hard → a= 0
If we could write a contract based upon a effort level ,
If the agent works hard , the accomplished task worth $270 to the principle
and if does not work hard it worth $70 to the principle the best one is ;
A- the agent gets $196 or a little bit more , if he works hard, since
if the agent works hard , (U ≥ 9 , a = 5) , then ;
W1/2 - 5 ≥ 9 , → W ≥ $196 , cost is $ 196 , and worth of is equal to $
270 for principle then principle’s profit will be $270 – $196 = $74
B- the agents gets almost zerto if he does not work hard .there will be
no deal The profit is zero .
So the expected profit in this contract ( taking both A and B into
account ) is equal to $74 . The cost of monitoring is still present
But the principal can make a contract (pay agent’s wage) based upon the
size of the sale .
When the agent was risk neutral it was possible to make the principal
as well off as if principal could write a contract contingent on the actual
effort level . But in this case it is not possible.
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
9
EFFICIENT INCENTIVES ; EXAMPLE
in this case when the principal is risk neutral and the agent is risk averse.
the most efficient arrangement is the one in which the agent’s wage is
certain . Because ; when one party to a transaction is risk averse and
the other is risk neutral , then it is efficient for the risk neutral party to
bear all the risks .The intuitive reasoning is as follows ;
If the principle pays the agent a random wage (different wages with different
probabilities ), then the agent evaluates the wage according to his expected
utility . Being risk averse ,if the wage is at all risky the agent values it at less
than its expected value ( because of the concave shape of the utility
function ). But being risk neutral , the principal values the cost of the wages
paid at their expected value (utility function is a straight line ).
If we imagine that the agent’s wage had expected value equal to W* ;
The principal see this as an outflow from his pocket equivalent to W*
The agent see this as an inflow to his pocket of something less than
W*, as long as there is any risk at all .
On the other hand if the wage was riskless the agent has no incentive to
work hard , and in this case the principal does not want to enter the
transaction . What should be done ?
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
10
EFFICIENT INCENTIVES ; EXAMPLE
To answer this suppose that the agent is paid as follows ;
A- the agent is paid Xo2 if the no sale is obtained (prob. = 0.1)
B- the agent is paid X12 if the small sale is made (prob. = 0.3)
C- the agent is paid X22 if the large sale is made (prob. = 0.6)
Offered this contract the agent has three choices ;
1- refuse the contract and get reservation level equal to U* = 9 .
2- take the contract and put it in high effort and get an expected
utility equal to ;
E(U) =ΣΠi(Ui) = ΣΠi (wi1/2 – 5 )= 0.1(xo -5) + 0.3(x1 -5) + 0.6(x2 -5) =
0.6x2 + 0.3x1 + 0.1x0 -5
3- take the contract and put it in low effort ; then ;
E(U)= 0.6x0 + 0.3x1 + 0.1x2
The best possible contract from the principle point of view is the
one in which the agent take the contract and put in a high level
of effort and optimum wage level for the principal is the
one in which the expected wage was minimum subject to
providing a utility level for the agent more than the reservation
level U* = 9 ;
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
11
EFFICIENT INCENTIVES ; EXAMPLE
EXP (W) = 0.1xo2 + 0.3x12 + 0.6x22 (for high effort)
0.1xo + 0.3x1 + 0.6x2 - 5 ≥ 9
0.1xo + 0.3x1 + 0.6x2 - 5 ≥ 0.6xo + 0.3x1 + 0.1x2
the solution for xo , x1 , x2 should be such that the expected utility
obtained be greater than reservation level (first constraint ), and
expected utility for high level of effort (hard working ) be greater
than the low level of effort (second constraint ) . The solution is as
follows ;
x02 =$29.46 , if no sale is made .
x12 = 196 , if $100 sale is made .
x22 = 238.04 , if $400 sale is made .
The expected wage bill is
0.1(29.46) + 0.3(196) + 0.6(238.04) = 204.56
Principal expected profit = 270 – 204.56 = 65.44
Min
Sub to
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
12
x
2
EFFICIENT INCENTIVES ; EXAMPLE
comparing this with the contract that the principle gives
the agent a wage of $196 and relies on trust (or the
compulsion of monitoring scheme) to ensure that the
agent puts in a high level of effort reveals that ,
To give the agent right incentive we had to have
him bear some of the risk by rewarding him in case
of the outcome that is more likely if he puts in
greater effort . This cost the principal an expected
amount of $8.56 . (204.56 – 196 ) .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
13
Finitely many action and outcomes
Suppose an agent who chooses an action α from the set of
A = {α1 , α2 , …. αn ).
The action is not observed by principal ; instead the principal sees
an(imperfect) signal s (sale or profit for example) from
S = {s1 , s2 , s3 ….. SM }
Πnm = the probability that signal sm is produced , if the agent
M
chooses action a n . Where
for each n .
1
 
m1
nm
Assumption 1 . The probability Πnm >0 for all n and m . Every
outcome is possible.
agent utility = U(w , a ) = u(w) – d(a ) . Where w is wage and a is
the action taken by the agent . Agent is a risk averse agent .
B(a) = gross benefit to the principle of hiring the agent if action a
is chosen by the agent.
Assumption 2 ; The function U is strictly increasing , continuously
differentiable , and concave .
Principle net benefit = B(a) – exp(wage paid).
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
14
Finitely many action and outcomes
Solving the basic problem ;
Step 1 ; for each an € A , what is the minimum expected wage that must
be paid to induce the agent to take the job and choose the action a n
.
If the signal chosen by agent is s m , and if w(sm ) is the wage paid to
the agent , we could define xm as following for m=1,,,,M.
xm = u{w( sm )} = value of utility as a function of wage paid when
signal s m is choosen .
xm = decision variable which is the level of wage utility ( utility of
wage ) paid to agent as a function of signal s
And if v be the inverse of u , then w(sm ) = v(xm ).= wage paid as a
function of utility obtained when signal sm is chosen.
.Expected wage which the principle must pay in terms of utility if the
agent takes action an :

D.M. KREPS CH 16
M
m1
 nmv( x m)  exp(w).
MORAL HAZARD AND
INCENTIVES
15
Finitely many action and outcomes
The agent should receive his reservation level of utility . Which is Type
equation here.;
(1) : 𝑀
∶ 𝑝𝑎𝑟𝑡𝑖𝑐𝑖𝑝𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡
𝑚=1 𝜋𝑛𝑚 𝑥𝑚 − 𝑑 𝑎𝑛 ≥ 𝑈0
We should note that the first part is the expected utility gain by the
agent when he receives a wage equal to w(sm ) m=1,2,3….M , and
the second part is the disutility lost by the agent when he takes
action an .
Choosing an should be better than choosing any other action a n*
− 𝑑 𝑎𝑛 ≥ 𝑀
𝑚=1 𝜋𝑛𝑚 − 𝑑 𝑎𝑛∗
relative incentive constraint for any action other than an
step I : for each an
M
Minimize expected wage:
Subject to (1) and (2)
(2)
𝑀
𝑚=1 𝜋𝑛𝑚

m 1
16
nm
v( xm )
Finitely many action and outcomes
C(an )= value of the expected wage at the optimal solution = the
minimum expected cost of inducing the agent to select action an .
For each action (ai , i=1,..n) taken by the agent we could find
C(an ) and the principle will have an amount of benefit equal to
B(an ) . Step 2 ; we should search for the action (ai ) which
maximize net benefit of the principle ,B(a) - C(an ). So, first we find
the minimum cost way to induce action a for each a and then we
choose the optimal a by comparing the benefits and costs
BASIC RESULTS AND ANALYSIS
The agent is risk averse and the principal is risk neutral . So the
cheapest way of guaranteeing the agent a given level of utility is
with a fixed payment. So the cheapest way (C0 (an )) to convince the
agent to accept the action an is to offer him :
0
C
(an )  Cn (an )
It is evident that
C 0 (an )  v(u0  d (an ))
for all n . Since this way is the cheapest way .
17
Finitely many action and outcomes
We should then maximize B ( an )  C ( an ) for all n . The effort level that
solves this is called the first best level of effort .
0
M
B(an )    nmSm ; the expected benefit for principle when action an is chosen
m1
with signals (profits) S
, m=1,2,3... M
m
Preposition 1
If the agent is risk averse , then the unique efficient risk-sharing
arrangement is for the principal to bear all the risk ; the agent gets a sure
wage . But if the agent is given a wage that is independent of the signal
( C0 (an ) ), he will choose the least onerous ( hardest) action . Because
as it is mentioned the principal will choose to give the agent cheapest
wage which is equal to C0 (an ) the reservation level.
Preposition 2
For a risk neutral agent , maxa B(a) – C0(a) = maxa [B(a) – C(a)]. In fact , if
a* achieves the maximum in , maxa B(a) – C0(a), one scheme that
imements this action is for the principal to pay the agent sm -B(a*)+ C(a*) if
the gross profit s are sm , so the principle receives [sm - {sm - B(a*) +
C(a*) }= B(a*) - C(a*) ] for sure and the agent bears all the risk ( since for
each a there are m signals and it is not clear that which one will be 18
observed).
Finitely many action and outcomes
It is as the principle “sold the venture “ to the agent , who is now
propitiator and is working for himself , and who now chooses the
optimal action in his own sole and best interest .
Now suppose that the actions are levels of effort that the agent might
select ordered in terms of increasing utility according to the utility
function U(w , a ) = u(w) – d(a) or ; U(w , a ) = u(w) – a
d(a1 ) < d(a2 ) < ….. < d(aN ) or a1 < a2 < ….. < aN
we also assume that higher level of efforts result in higher gross
expected profit for the principle . To write this assumption we should
have ; s1 < s2 < …. < sM . To assume that higher level of effort
leads to higher level of profit we have to assume that
m
is increasing in n . But some thing should
 S

m1
nm m
be assumed that is Increase in effort results in higher probability of
higher levels of profit .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
19
Finitely many action and outcomes
Assumption 3 ;. For all pairs n and n’ such that n’ > n and for all
m= 1,2,….M , we should have ;
𝑀
𝑀
𝑛𝑛𝑖
𝑖=𝑚
≤
𝜋𝑛, 𝑖
𝑖=𝑚
with for each n and n’ a strict inequality for at least one m . It
means that increased effort result s in a first order stochastic
increase in the levels of gross profit. This means that ; for each n
and n’ we should have at least inequality for at least one m
it is necessary to ask ; Is the wage paid , or equivalently the wageutility level xm , non-decreasing with increase in the level of profit ?.
The answer is no in general . Take an example where it fails ;
three possible levels of profits S1 = $1 , S2 = $2 , S3 =$10000
Two possible effort levels
a1 = 1 , a2 = 2
For each a any of the S could happen ;
20
‘
MORAL HAZARD AND INCENTIVES
Finitely many action and outcomes
Choosing effort lelvels
a1 : S1 with PS1  0.5, and S2 with PS2  0.3 and S3 with PS3  0.2
a2 : S1 “
“= 0.4 “ S2 “
0.1
S3 “
0.5
By assumption 3 higher effort level leads to higher probabilities better
outcomes .
Exp( gross profit for a1 ) = 2001.5
Exp ( gross profit for a2 ) = 5000.6 ,so he chooses a2 by Ass. 3.
The first order condition for the expected wage utility minimization
(page 16) will be as follows ;
V ( xm )     n' 1 n* (1 
'
N
v’ (x1 ) = λ + η(1- (0.5/0.4)) = λ – 0.25 η
v’ (x2 ) = λ + η(1- (0.3/0.1)) = λ – 2η
v’ (x3 ) = λ + η(1- (0.2/0.5)) = λ + 0.6 η
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
 n'm
 nm
)
21
Finitely many action and outcomes
Where x1 is the wage utility corresponding to s1 = $1
“
x2
“
s2 = $2
‘
x3
“
s3 = $10000
λ is the multiplier on the participation constraint
η is the multiplier on the relative incentive constraint .
if η>0 → v’ (x1 ) > v’ (x2), → x1 > x2 → w(x1)>w(x2) which is not . In
order for this not to happen the ratio of probabilities in brackets
should change in right way as stated in Ass. 4 ;
Assumption 4 : the monotonic-likelihood ratio property . For any two
'
effort level an and an' such that n n , and for any two gross
profit level Sm , Sm'  m m' , the relative likelihood of the better
outcome under the higher effort level to the lower is at least as large
as this likelihood ratio for the lower outcome. Or ;
 n'm'

'
D.M. KREPS CH 16nm

 n' m
 nm
MORAL HAZARD AND
INCENTIVES
22
Finitely many action and outcomes
Is assumption 4 enough to get an affirmative answer to the question ?
We will see that it is not . To see why , we should take into account
the first order condition ;
N
V ( xm )  V ( xm' )  n' [
'
'
n' 1
 n ' m'
 nm'

 n' m
 nm
]
As it is seen as long as n’ < n , and the term in the bracket is negative
by assumption 4 ,. Since m’ > m and V’ (xm) < V’ (xm,), n' is
positive and assumption 4 holds , the left and right hand side of the
above relation are both non positive .But if for only some n’ >n we
have  n'  0 while m’ > m , then the above relation may not hold.
What is needed in addition to assumption 4 is an assumption that the
only binding relative incentive constraint for the optimal effort level an
are constraints corresponding to levels of effort lower than an .
Assumption 5 : if an , an' , an" are effort levels such that
a '   a  (1   ) a " 0    1 for each m=1,2,3….M then :
n
n
n
23
Finitely many action and outcomes
M

i m
n 'i
M
M
i m
I m
    ni  (1   )   n"i
This is called the concavity of distribution function condition . Let ;
M
 m (an )  i m  ni = probability of seeing a gross profit level at least
as large as sm if effort level an is taken .
The assumption says that increase in the effort level ( measured by
disutility) have decreasing marginal impact ( concave shape) on the
probabilities of better outcome (an” is the lowest level of effort)
1
If   , then we could see that , an  a '  a '  a "  0.
n
n
n
2
an" to the
Increase in disutility in moving from the lowest level of effort
intermediate level an ' is the same as increase in moving from an'
to
an . And according to the assumption it should be;
 m (an' )   m (an" )   m (an )   m (an' ) when signal m is seen
and  
1
2
24
Finitely many action and outcomes
Preposition 3 : if assumptions 1 ,..5 all hold and U is strictly concave ,
then the optimal wage incentive scheme for the principal has wages
that are non decreasing functions of the level of gross profit.
Proof is not needed .
THE CASE OF TWO OUTCOME
An intuitive case might be made that the principal would always choose
to implement a level of effort less than or equal to the first-best level
, since risk-sharing between the agent and the principal entails
shielding the risk averse agent from some of the consequences of
his effort in terms of profits, while he bears fully its utility.
Suppose gross profits can take only two forms ; s1 and s2 > s1 .
Agent’s effort results in either failure or success of some venture .
Checking for the assumptions ;
Ass 1 ; both success and failure outcomes are possible.
Ass 2 ; utility function should be strictly increasing and continuously
differentiable and concave .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
25
Finitely many action and outcomes
Ass 3 ; higher levels of efforts leads to increased chances of success.
Ass 4 ; is not needed, when Ass.3 holds .
Ass 5 ; The probability of success should be a concave function of the
effort level . In this special case with Ass. 3 Ass. 5 is not needed.
With Ass. 3 alone we can formulate the problem as follows ;
The agent has a base wage denoted by b1 to which added a bonus of
b2 ≥ 0 in the event of success .
b
If we write
 2
(s  s )
2 1
b  b2  b  (s  s )
The agent wage with success outcome is
1
1
2 1
 should be positive if principal wants to implement any efforts
level above minimum ( s1).
Preposition 4 ; if U is strictly concave , then  1 at the optimal wageincentive schedule . Otherwise the principle prefers the outcome to
be failure which can not be a scheme. Proof is not needed .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
26
Finitely many action and outcomes
An example which shows how the first best solution may not work .
Two outcome and two possible effort level ;
a1 = 0 with PF = 0.9 and PS = 0.1 and slightly different effort level ,
a2 = 0.1 with PF = 0.85 and PS = 0.15
U( w ,a ) = ln ( w) – a , with u0 = 0 , and
Success worth $10 and Failure worth $0 to the principal .
1- To implement the effort level a1 , the principle expected profit ;
Exp(profit) = (0.9)($0) + ( 0.1)($10) = $1 , and wage cost is ;
ln(w)=0 , w = e0 =1 and net profit = 1-1 = 0
2- To implement the effort level a2 = 0.1 , the principle should pay a
wage equal to ln(w)= 0.1 , w= e0.1 = $1.105 , regardless of action
and his exp(profit) = 0.85($0) + (0.15)($10) = $1.5 , when a2 =0.1 ,
and his gross expected profit = 1.5 – 1.105 = $0.395
hence the first best effort level is a2 .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
27
Finitely many action and outcomes
Working out the optimal wage incentive program ( effort level a2 ) and
solving the first order equations we will get w1 = 0.8187 if outcome
is failure and w2 = 6.04965 if outcome is success . Now
exp(wage )= (0.8187)(0.85) + (6.04965)(0.15)=$1.6034
effort level a2 is chosen so expected profit = $1.5
net expected profit = $1.5 - $1.6034 = - $0.1034 .
So the principle has to choose the other option ( effort level a1 )
which is not optimal , that is paying the fixed wage of $1 . So as it is
seen the principle choose an effort level less than the first-best one .
CONTINOUS ACTION : First order approach
the agent can choose any effort level drawn from some interval {a0 ,a1 ]
and that the range of possible gross profit levels is R or some
appropriate subinterval .
as an example we use the model of two possible outcome ; success
and failure and action choice that bears the interpretation of effort .
D.M. KREPS CH 16
MORAL HAZARD AND
INCENTIVES
28
CONTINOUS ACTION
Effort is chosen from an interval [ a0 , a1 ] where effort is measured in terms
of disutility .
U(w,a) = u(w) – a . If effort level a is chosen , then the probability of a success
outcome is π(a) , where π(.) is a strictly increasing function .
If we want to discover the minimal cost of implementing a* , we should solve
Min (1- π(a*)) v(xf ) + π(a*)v(xs )
sub (1- π(a*)) xf + π(a*) xs - a* ≥ u0 ,
(1- π(a*)) xf + π(a*) xs - a* ≥ (1- π(a)) xf + π(a) xs - a for all a € [ a0 , a1 ] .
The last constraint contains of infinite number of constraints . In order to make
it operational the last constraint could be written as follows ;
U(a) = (1- π(a)) xf + π(a) xs - a , should be maximized at a = a* , or
π’(a*) [xs - xf ] =1 . Then substitute this for the last constraint ;
Min (1- π(a*)) v(xf ) + π(a*)v(xs )
sub (1- π(a*)) xf + π(a*) xs - a* ≥ u0
π’(a*) [xs - xf ] =1,
two equations in two unknowns could be solved if we assume that both
MORAL HAZARD AND
29
constraints are binding
INCENTIVES