3. dynamic moral hazard - economic · pdf fileresidual claimant on the margin. ... (which...

3. Dynamic Moral Hazard

Klaus M. Schmidt

LMU Munich

Contract Theory, Summer 2010

Klaus M. Schmidt (LMU Munich) 3. Dynamic Moral Hazard Contract Theory, Summer 2010 1 / 64

Basic Readings

Basic Readings

Textbooks:

Bolton and Dewatripont (2005), Chapter 10

Schmidt (1995), Chapter 3

Papers:

Holmström and Milgrom (1987, 1991)

Hellwig and Schmidt (2002)


Main Questions

Main Questions

The literature on dynamic moral hazard problems has focused on threequestions:

1. Is it possible to achieve a more efficient allocation if principal and agentinteract more than once?

2. What are the advantages of writing a long-term contract rather than asequence of short-term contracts?

3. Do optimal contracts in a repeated relationship have more structure thanoptimal contracts in a one-shot relationship?


An Infinitely Repeated Principal-Agent Problem


Suppose the agent is maximizing either

T∑

t=1

V (w ti )− G(at)

T, T → ∞

or

(1 − δ)∞∑

t=1

δt [V (w ti )− G(at)

], δ → 1 .

The principal maximizes either his average profit or the discounted NPV of allfuture profits, but he is assumed to be risk neutral.



Proposition 3.1If T → ∞ (δ → 1) then the first best can be approximated arbitrarily closely.

This result is due to Radner (1981) and Rubinstein (1981), and the intuition isfairly simple:

The parties write a long-term contract that gives a fixed wage to theagent unless the distribution of outcomes generated by his actionsdeviates “too much” from the distribution that would have beengenerated had the agent always taken the first best action. The Lawof Large Numbers implies that any systematic shirking of the agentcan be detected with a probability arbitrarily close to one. In thiscase the agent is punished very harshly.



This result is also familiar from the literature on the Folk Theorem in repeatedgames with public monitoring.

However, the agent has to be very patient for this result to hold.

In the chapter on “Relational Contracts” we consider the question what theparties can implement for a discount factor that is bounded away from 1.


Long-term vs. Short-term Contracts

Short-term vs. Long-term Contracts

The Radner-Rubinstein result seems to suggest that long-term contracts aremuch more powerful than short-term contracts. However, Fudenberg,Holmström and Milgrom (1990) demonstrated that this is not the case. Thefirst best can also be implemented with a sequence of short-term contracts,where the agent gets w t = x t − K in each period, i.e. the agent is maderesidual claimant on the margin.In order to get optimal insurance, the agent could follow the followingself-insurance strategy:

Starve and save until you have accumulated a wealth level of S.Thereafter consume the first best level of consumption and smoothincome shocks by using your savings.

Illustrate graphically.


Long-term vs. Short-term Contracts

Remarks:

If the discount factor is close enough to 1, the first finite number ofperiods in which the agent starves do not count very much, and we canapproach the first best arbitrarily closely.

The problem here is again that the case where δ → 1 is not veryinteresting. Real life is short and agents are impatient.


The Two-Period Problem


To address the question of long-term vs. short-term contracts with impatientagents, look at a two-period moral hazard problem. Does a long-term contractstrictly outperform a sequence of short-term contracts.

Lambert (1983) and Rogerson (1985) argued, that a long-term contract(which makes the wage in the second period conditional on the output of thefirst and the second period) is strictly better than two short-term contracts,each of which is conditional only on this periods output.However, their results rely on the assumption that the agent cannot save orget a credit.

The following result is due to Malcomson and Spinnewyn (1988) andFudenberg, Holmström and Milgrom (1990):



Proposition 3.2Suppose that the actions taken by the agent in perid t affect the output ofperiod t only, and suppose that the principal and the agent have equal andunlimited access to the capital market, then a sequence of short-termcontracts is as efficient as a long-term contract.

Remarks:

1. With unequal access to the capital market, a long-term contract could beused to smooze consumption but this has nothing to do with themoral-hazard problem.

2. This result says that with moral hazard there is no benefit to long-termcommitments. This is an important difference to adverse selectionproblems, where commitment is very important.


Robustness and Linear Contracts

Robustness and Linear Contracts

Incentive schemes should be robust and give proper incentives, even if thereal world is slightly different from the model that we consider. In particular, itcould be the case that the action space of the agent is richer than assumed inthe model.

Examples:

1. Free disposal and profit boosting

2. Intertemporal arbitrage

3. Interregional arbitrage

4. Continuous effort adjustment of the agent in the Mirrlees example.


The Holmström-Milgrom Model


Holmström and Milgrom (1987) consider a repeated moral hazard problemwhich is somewhat different from the ones we have seen so far.

HM want to capture the idea that the agent does not choose his action atone single point in time, but that he rather adjusts his effort level more orless continuously over some time period.

Thus, they keep the total time interval fixed, but subdivide it into more andmore periods, so that the agent can choose his action more and morefrequently.

In the limit, the agent controls the drift rate of a multidimensionalBrownian motion.



This model is very important:

It shows that a linear incentive scheme is optimal in a natural class ofproblems.

It offers a framework, in which it is very easy to compute the optimalincentive scheme.

The model can be used as a building block in more complicated appliedmoral hazard problems.

The exposition is based on Schmidt (1996) and Hellwig and Schmidt (2002).



Holmström and Milgrom impose the following assumptions:

1. The agent consumes only at the end of the total time interval. Thus, thereis no scope for consumption smoothing or self-insurance, and it is notpossible to implement the first best as the number of periods goes toinfinity.

2. The agent (and the principal) has constant absolute risk aversion. Thus,the agent’s risk preferences are unaffected by the history of pastperformance.

3. The technology is also stationary and not affected by the history of pastperformance.


The One-Period Problem


The agent chooses a probability vector p = (p0, . . . ,pN) ∈ P, which generatesa probability distribution over possible profit levels π ∈ {π0, . . . , πN}. P is the Ndimensional simplex. The agent’s cost function is c(p) which is convex in p.

Proposition 3.3Suppose the principal wants to implement an action p in the interior of P, suchthat the agent’s expected utility corresponds to the certainty equivalent w. If pis implementable, then there exists a unique incentive scheme thatimplements p:

si = w + c(p)− 1r

ln

1 − rci + rN∑

j=0

pjcj

for i = 0, . . . ,N.



Proof: For a given incentive scheme s = {s0, s1, . . . , sN}, the agent’smaximization problem is:

maxp0,...,pN

−N∑

i=0

pie−r(si−c(p))

subject to∑N

i=0 pi = 1. The Lagrange function of this problem is:

L = −N∑

i=0

pie−r(si−c(p)) − λ[

N∑

i=0

pi − 1]

The first order conditions require:

dLdpj

= −e−r(sj−c(p)) −N∑

i=0

pi rcj(p)e−r(si−c(p)) − λ = 0 .



Multiply this with pj and take the sum over all j :

−N∑

j=0

pje−r(sj−c(p))

︸︷︷︸

=−UA(w)

−rN∑

j=0

pjcj(p)N∑

i=0

pie−r(si−c(p))

︸︷︷︸

=−UA(w)

= λ .

Note that the agent will be hold down to his reservation utility, so it must bethe case that:

−N∑

i=0

pie−r(si−c(p)) = UA(w) = −e−rw

Hence, we can write:

UA(w) + rUA(w)N∑

j=0

pjcj(p) = λ .



Substituting this expression in the first order condition, we get:

rcj(p)UA(w)− e−r(sj−c(p)) − rUA(w)

N∑

j=0

pjcj(p)− UA(w) = 0

which is equivalent to:

e−r(sj−c(p)) = e−rw

1 − rcj(p) + rN∑

j=0

pjcj(p)

Taking the logarithm on both sides yields:

−rsj + rc(p) = −rw + ln

1 − rcj(p) + rN∑

j=0

pjcj(p)



or, equivalently:

sj = w + c(p)− 1r

ln

1 − rcj(p) + rN∑

j=0

pjcj(p)

Q.E.D.

Remarks:

1. This result is remarkable: If an action can be implemented, then there isa unique incentive scheme implementing it!

In the models we considered so far there existed many incentiveschemes implementing a certain action. This is why we had to look forthe cost minimizing such incentive scheme which is not necessary here.

2. Note that the agent does not choose an action, but the probabilitydistribution over outcomes directly.



3. In a model where the agent chooses a one-dimensional effort level thatgives rise to a probability distribution p, the set of feasible probabilitydistributions is a one-dimensional curve in P.

4. If the agent chooses p directly out of P, his action space is much richer.He can now move freely in the N-dimensional simplex. Thus, theincentive compatibility constraints become more restrictive: the agent hasto be deterred from any deviation in any direction, i.e., not just in onedimension but in N dimensions. This is the reason, why there exists atmost one incentive scheme implementing p. Illustrate graphically.


The Multi-Period Problem


We now consider the case where the above one-period problem is repeatedm times.

Suppose that in the one period problem it is optimal for the principal toimplement p∗ by using s(p∗).

Let Aτi ∈ {0,1} be a random variable that is equal to 1 if πτ = πi and 0

otherwise.



Proposition 3.4The optimal incentive scheme for the m-period problem is the m-fold repetitionof the optimal incentive scheme of the one-period problem. Hence, theprincipal wants to implement p∗ in every period and offers the incentivescheme

s =n∑

i=0

(

si(p∗)

(m∑

τ=1

Aτi

))

.

Note that this incentive scheme is a linear function of the number of timeseach outcome materialized.

Proof sketch: Why is this rather trivial?


Discrete Approximation of a Continuous-Time Model


We don’t want to have a repeated moral hazard model, we rather want thatthe agent is able to adjust his action very frequently over time. Thus, we haveto increase the number of periods while holding the total time period constant.In the limit, the agent should be able to adjust his effort continuously.

How to set up such a model is not trivial. Let me illustrate the problems for themost simple case:

Suppose that there are m = 1∆ periods, each of length ∆ ∈ {1, 1

2 ,13 , . . .},

indexed by τ ∈ {1, . . . , 1∆}.

In each period τ , there is a Bernouilli random variable

X τ =

{

+1 ·∆X “success”−1 ·∆X “failure”



In each period the agent chooses the probabilitypτ = Prob(X τ = +1 ·∆X ) at cost c(pτ ). We know already that theincentive scheme of the agent will be linear in the number of successesand that he will take a constant action pτ = p over time.

Note that

E(X τ ) = (2p − 1)∆X

Var(X τ ) = 4p(1 − p)∆X 2

The random variable X (τ) = X 1 + X 2 + · · ·+ X τ follows a binomialdistribution with

E(X (τ)) = τ · (2p − 1)∆X

Var(X (τ)) = τ · 4p(1 − p)∆X 2



Now we want to take the limit as the number of periods goes to ∞ and ∆ → 0.

Let t ∈ [0,1] be a measure of “real time”. Ignoring integer problems, if thelength of a time period is ∆ < 1, then there are t

∆ periods between date 0and date t .

At the same time we have to reduce profits in each periods and adjust theeffort cost of the agent. Otherwise, profits and costs could go to infinity.

To make the one-period problem and the m-period problem comparable,we want that in the limit E(X ( t

∆ )) and Var(X ( t∆ )) are linear in t .

To see the problems involved consider the following two (unsuccessful)attempts:



Attempt 1: ∆X = ∆t → 0

lim∆t=∆X→0

Var(

X(

t∆t

))

= lim∆t=∆X→0

(∆X )2 · t

∆t· 4p(1 − p)

= lim∆t→0

∆t · t · 4p(1 − p)

= 0

Thus, we get a degenerate stochastic process with a variance of 0 in the limit.This process is deterministic and it is trivial to implement the first best!



Attempt 2: ∆X =√∆t → 0. Now the variance does not converge to 0, but:

lim∆X=

√∆t→0

E(

X(

t∆t

))

= lim∆X=

√∆t→0

∆X · t∆t

· (2p − 1)

= lim∆t→0

√∆t · t∆t

· (2p − 1)

= lim∆t→0

t√∆t

· (2p − 1) =

{+∞ if p > 1

2−∞ if p < 1

2

Thus, the expected value goes either to plus or minus infinity.



How to do it properly?

In order to avoid that E(X ( t∆t )) → ∞, we have to redefine the action of the

agent as follows:

Let ∆X =√∆t → 0 and p = 1

2 + 12µ

√∆t . Note that if ∆t → 0, p → 1

2 , but pgoes to 1

2 more slowly than ∆t goes to 0. In the following we take µ to be theaction of the agent.

lim∆X=

√∆t→0

E(

X(

t∆t

))

= lim∆X=

√∆t→0

∆X · t∆t

· (2p − 1)

= lim∆t→0

√∆t · t∆t

· (1 + µ√∆t − 1)

= lim∆t→0

t∆t

·∆t · µ= µ · t



lim∆X=

√∆t→0

Var(

X(

t∆t

))

= lim∆X=

√∆t→0

(∆X )2 · t∆t

· 4p(1 − p)

= lim∆t→0

∆t · t∆t

· 4(

12+

12µ√∆t)(

12− 1

2µ√∆t)

= lim∆t→0

t · 4[

14− 1

4µ2∆t

]

= t

Note that

E(X ( t∆t )) and Var(X ( t

∆t )) are linear in t ,

Var(X ( t∆t )) is independent of µ.

In each Intervall [t , t ′] we have that X (t ′)− X (t)− (t ′ − t)µ is a sum oft′−t∆t i.i.d. random variables with finite variance and mean 0. Thus, as∆t → 0, this is random variable is normally distributed with mean 0 and avariace that is proportional to (t ′ − t).



This is a driftless Brownian motion!

Definition 3.1A stochastic process [X (t), t ≥ 0] is called a (driftless) Brownian motion if itsatisfies the following properties:

X (0) = 0

The increment X (t + s)− X (t) is normally distributed with mean 0 andvariance σ2s.

For any two disjunct intervals [t1, t2] , [t3, t4] with t1 < t2 < t3 < t4 it mustbe the case that the increments X (t4)− X (t3) and X (t2)− X (t1) arestochastically independent.

It can be shown that a Brownian motion is continuous everywhere butnowhere differentiable. Thus, it is impossible to predict the movements of aBrownian motion.


Approximation of the Brownian Motion Model


The discussion of the previous section motivates what we have to do in ourmultidimensional model:

There are m = 1∆ periods, each of length ∆ ∈ {1, 1

2 ,13 , . . .}, indexed by

τ ∈ {1, . . . , 1∆}

Fix a “standard” p ∈ P, p � 0. This “standard” corresponds to p = 12 in

the previous section. It is going to determine the variance of theBrownian motion.

Given this standard we can normalize profits such that

N∑

i=0

piπi = 0

.



In each period profit levels are given by:

π∆i = πi∆

12 ∀ i ∈ {0, . . . ,N} .

Note thatN∑

i=0

piπ∆i = ∆

12

N∑

i=0

piπi = 0

In each period the agent chooses µ∆ = (µ∆1 , . . . , µ

∆N ) which determines

an associated action vector p∆(µ∆):

p∆i (µ∆) = pi + µ∆

i∆

12

ki∀ i ∈ {1, . . . ,N} ,

p∆0 (µ∆) = 1 −

N∑

i=1

p∆i = p0 −

N∑

i=1

µ∆i∆

12

ki

where ki = πi − π0.



The agent’s cost of taking action p∆ are given by

c∆(p∆) ≡ ∆ · c(

p0 +p∆

0 − p0

∆12

, . . . , pN +p∆

N − pN

∆12

)

= ∆c

(

p0 −N∑

i=1

µ∆i

ki, p1 +

µ∆1

k1, . . . , pN +

µ∆N

kN

)

≡ ∆c(µ∆)



What is the point of this specification?

1. Suppose the agent chooses a constant vector p∆ (or, equivalently, aconstant µ∆) in each period. Then the expected profit over all periods isgiven by:

1∆

N∑

i=0

p∆i π∆

i =1∆

N∑

i=0

(p∆i − pi)(πi − π0)∆

12

=

N∑

i=0

(πi − π0)p∆

i − pi

∆12

=

N∑

i=1

µ∆i .

Note that the expected profit is independent of the length of each period.In particular, if the agent chooses a constant µ∆ = µ in each period(independent of period length), then total profit is the same for all ∆.



2. Similarly, if the agent chooses the same µ independent of period length,then his total cost are always

1∆∆c(µ) = c(µ) .

3. Thus, the one-period and the m-period problem are in fact comparable.

We know by Proposition 3.3 that there exists at most one incentive schemethat implements p∆(µ∆). Assuming that w = 0 and substituting µ∆ and c(µ∆)we have:

s∆i = ∆c(µ∆)− 1

rln

1 − r ciki∆12 + r

N∑

j=0

p∆j cjkj∆

12

,

where ci =ci−c0

kiis the partial derivative of c with respect to µ∆

i , and c0 = 0.



Consider now the multi-period model with T = 1∆ periods of length ∆. We

know that the only incentive scheme that implements any given sequence ofactions µ∆

t is the sum of the incentive schemes that implement these actionsin the one-period problem.

If we take the limit of the multi-period model als T → ∞ and ∆ → 0 we getafter several simplifications (Taylor expansions) and reformulations our firstmain result:



Theorem 3.1Consider a sequence of discrete models with period length ∆,∆ = 1, 1

2 ,13 , . . .. Suppose that, as ∆ → 0, the time path of actions µ∆(t)

converges uniformly to some continuous function µ(t), t ∈ [0,1]. As ∆ → 0,

(a) the stochastic process of cumulative deviations from the meanX∆(t) = (X∆

1 (t), . . . ,X∆N (t)) converges in distribution to X (·) which is a

driftless N-dimensional Brownian motion with covariance matrix

Σ =

k21 p1(1 − p1) −k1k2p1p2 · · · −k1kN p1pN

−k2k1p2p1 k22 p2(1 − p2) · · · −k2kN p2pN

......

. . ....

−kNk1pN p1 −kNk2pN p2 · · · k2N pN(1 − pN)

and starting point X (0) = 0;

(b) the total cost to the agent converges to∫ 1

0 c(µ(t))dt;



Theorem 3.1 (cont.)

(c) the incentive payments that serve to implement µ∆(t) converge indistribution to

s =

∫ 1

0c(µ(t))dt +

∫ 1

0c′(µ(t))dX +

r2

∫ 1

0c′(µ(t))˚[c′(µ(t))]T dt

where c′(·) = (c1(·), . . . , cN(·)).

Remarks:

1. The proof is not that difficult, but the notation and some of thecomputations are a bit messy. See H&S (2002).

2. The optimal incentive scheme is equivalent to the optimal incentivescheme that Holmström and Milgrom derive for the Brownian model.



3. The difference to H&M is that H&M start with the Brownian model, whileH&S show, how to derive this result as the limit of a sequence of discretemodels. Thus, there is no discontinuity between the discrete and thecontinuous model. Note that this need not be the case. There are otherways how to approximate the Brownian model that are discontinuous(Mirrlees).

4. Theorem 1 considers any converging sequence of µ∆(t). However, weknow already that in each discrete model the principal wants toimplement the a constant µ∆ in each period. Thus, the sequence ofoptimal action paths converges to a constant µ∗. For µ(t) = µ∗ we have:

s = w + c(µ) + c′(µ)X +r2

c′(µ)˚[c′(µ)]T

where X denotes the vector of cumulative deviations from the mean attime t = 1.



5. Note that this incentive scheme is linear in X and has an interestinginterpretation:

The first two terms compensate the agent for his outside option and hisdisutility of effort.The third term is the incentive payment. It is linear in the final outcome X .The last term is the risk premium, that has to be paid to the agent. Note thatthis is a constant which is independent of the action that the principal wantsto implement.

6. The “standard” p determines the variance-covariance matrix Σ.


Linearity in Aggregates


So far, the optimal incentive scheme is linear in the “accounts”, that count theinstances of the different possible profit levels. There may be many suchaccounts in which case the optimal contract would still look much morecomplicated than most contracts in the real world.

It would be much nicer to have a result where the optimal contract is linear inaggregates, such as total profits. This would obtain if there are only twopossible output levels in each period.

With more than two profit levels, H&M offer a slightly different approach to getlinearity in aggregates. Consider the Brownian model and assume that theprincipal does not observe each individual account over time, but only anaggregate (e.g. the sum) of all accounts over time. H&M show that thisimplies that the optimal contract must be linear in the aggregate.



Remarks:

1. This is the result that we are really after.

2. Unfortunately, however, this result cannot be derived as the limit of asequence of discrete models. The reason is, that for any ∆ > 0 theprincipal can compute the time paths of the accounts for all possible profitlevels from the observation of the time path of the aggregate profit level.Just compute πt − πt−1 and you get the profit level in period t .



Hellwig and Schmidt (2002) consider a somewhat different model, in whichthe linearity in aggregates result can be approximated by a sequence ofdiscrete models. They assume that

the agent can destroy output unnoticed

the principal observes total profits only at date 1, but he does not observethe time path of total profits.

They show that if ∆ → 0, then the optimal contract of the Brownian model isε-optimal in the discrete model with period length ∆ if ∆ is sufficient small.

Note: This result nicely corresponds to the intuition provided by Holmströmand Milgrom’s introduction. The linear scheme is optimal because the agenthas a much richer action space than can be controlled by the principal.


The Holmström-Milgrom Model in Action


Consider first the one-dimensional case where the principal observes totalprofits x(t), t ∈ [0,1], over time. We know by Holmströöm and Milgrom thatthe optimal incentive scheme must be linear in x =

∫ 10 x(t)dt :

s(x) = αx + β .

This scheme implements the same action at each point in time, i.e., µ(t) = µ

for all t ∈ [0,1]. Thus, total profits at date 1 are normally distributed with meanµ and some exogenously given variance σ2. Finally, we assume that

c(µ) =K2µ2 .

The First Best Contract offers full insurance to the agent and solves thefollowing problem:

maxµ

{E(x)− c(µ)} ⇔ maxµ

{

µ− K2µ2}


The Holmström-Milgrom Model in ActionFOC:

1 − Kµ = 0 ⇒ µFB =1K

Let the certainty equivalent of the agent’s reservation utility be w = 0. Thenthe agent’s participation constraint requires:

0 = s − K2µ2

FB ⇒ s =1

2K.

Consider now the Second-Best Problem:Which action is the agent going to choose if he is offered a linear incentivescheme s(x) = αx + β? Consider first the certainty equivalent of his impliedutility if he chooses action µ:

EU =

∫ ∞

−∞−e−r [αx+β− K

2 µ2] · 1√

2πσe− 1

2 (x−µ

σ)

2

dx = −e−rCE

1√2πσ

∫ ∞

−∞−e−rαx− 1

2 (x−µ

σ)

2

dx = −e−rCE+rβ−r K2 µ

2



Digression:

−rαx − 12

(x − µ

σ

)2

= −12

x2 − 2µx + µ2 + 2rαxσ2

σ2

= −12

x2 − 2x(µ− rασ2) + (µ− rασ2)2 + 2µrασ2 − r2α2σ4

σ2

= −12

[x − (µ− rασ2)

σ

]2

− rα(2µ− rασ2)

2

If we plug this in above we get:

−e− rα(2µ−rασ2)

2 ·∫ ∞

−∞

1√2πσ

e− 1

2

[

x−(µ−rασ2)

σ

]2

dx︸︷︷︸

=1

= −e−r[CE−β+ K2 µ

2]



The integral is equal to one because we are simply integrating over the fullrange of the density function of a normal distribution with mean (µ− rασ2)and variance σ2. Thus:

− rα(2µ− rασ2)

2= −r

[

CE − β +K2µ2]

αµ− r2α2σ2 = CE − β +

K2µ2

Hence:

CE = αµ+ β︸︷︷︸

expected wage

− K2µ2

︸︷︷︸

effort cost

− r2α2σ2

︸︷︷︸

risk premium



Maximization of the agent’s expected utility is equivalent to maximizing thecertainty equivalent of this utility. Note that this expression is concave in µ forany (α, β). Therefore, the optimal action of the agent is fully characterized bythe FOC:

µ =α

K

If we want to induce the agent to choose the action µ, we have to choose α

such thatα = µK

If the CE of the agent’s reservation utility is 0, the participation constraintrequires:

0 = µ · Kµ+ β − K2µ2 − r

2µ2K 2σ2

Therefore, β has to be chosen such that:

β = −K2µ2 +

r2µ2K 2σ2


The Holmström-Milgrom Model in ActionThe principal maximizes E(x − s(x)) = µ− s(µ). Hence, we can write hisproblem as:

maxµ

µ− µK︸︷︷︸

α

µ︸︷︷︸

µ

+K2µ2 − r

2µ2K 2σ2

︸︷︷︸

−β

FOC requires:1 − Kµ− rK 2σ2µ = 0

which implies:



µ∗ =1

K (1 + rKσ2)

α∗ =1

1 + rKσ2

π∗ = µ∗ − s(µ∗) =1

2K (1 + rKσ2)

This is a very nice and intuitive result:

1) If r = 0 or σ2 = 0, then α∗ = 1, µ∗ = 1K , π∗ = 1

2K , i.e., we get the first best.

2) The incentive intensity as expressed by α∗ increases if K decreases, σ2

decreases, or r decreases.

3) The agent works harder (µ∗ increases), if K decreases, σ2 decreases, orr decreases.

4) The agent works to little µ∗ < µFB.


Multiple Tasks

Multiple Tasks

In this section we will discuss some interesting features that arise if weconsider multiple-tasks principal-agent problems. Holmström and Milgrom(1991) offer a general analysis of these problems, but we will restrict attentionto a few examples:


Example 1: Contracts for Teachers


Two tasks:

t1 Teach basic skills: reading, writing, basic math

t2 Promote the students creativity and social skills

The principal can observe the students’ test scores at the end of the yearwhich give an imperfect signal about the teacher’s effort in teaching basicskills, but there is no test for creativity, i.e.:

x1 = t1 + ε1, σ21 < ∞,

x2 = t2 + ε2, σ22 = ∞, σ12 = 0.


Example 1: Contracts for TeachersWhat is the optimal incentive scheme?

maxt1,t2

B(t1, t2)− C(t1, t2)−12

rα21σ

21

subject to:

(*) C1(·) = α1 = α

(**) C2(·) = α2 = 0

Note that α2 = 0 must be optimal. Otherwise the principal would impose aninfinitely high risk on the agent. Nevertheless we assume that there is aninterior solution for the principal’s problem, i.e., ∂C(t)

∂t2< 0 for some t2.

Interpretation?Let t1 = t1(α) and t2 = t2(α). Then we can maximize with respect to α ratherthan with respect to (t1, t2). FOC:

B1∂t1∂α

+ B2∂t2∂α

− C1∂t1∂α

− C2∂t2∂α

− rασ21 = 0



Totally differentiating (*) and (**), we get:

C11∂t1∂α

+ C12∂t2∂α

= 1

C21∂t1∂α

+ C22∂t2∂α

= 0 ⇒ ∂t1∂α

= −C22

C21

∂t2∂α

⇒ C11

(

−C22

C21

∂t2∂α

)

+ C12∂t2∂α

=∂t2∂α

(

C12 −C11C22

C21

)

= 1 .

This implies:

∂t1∂α

= − C22

C212 − C11C22

∂t2∂α

=1

C12 − C11C22C21

.


Example 1: Contracts for TeachersSubstituting this in the FOC we get:

(B1 − C1︸︷︷︸

=α

)

(C22

C11C22 − C212

)

+ (B2 − C2︸︷︷︸

=0

)

(C21

C212 − C11C22

)

− rασ21 = 0

Solving for α, we get:

α =B1 − B2

C21C22

1 + rσ21

(

C11 − C212

C22

)



Interpretation:

Suppose that C12 = 0. The we have α = B11+rC11σ

21. This is exactly the

same expression that we got in the single task case.

Suppose that, starting at C12 = 0, C12 decreases. Then the numeratorincreases while the denominator decreases, so α goes up. C12 < 0means that the two tasks are complements: the more basic skills theteacher teaches, the easier it is to promote their creativity. In this casethe teacher should get stronger incentives to promote basic skills than inthe single task case.

Suppose that C12 has the same absolute value as before but is positive.In this case the numerator becomes smaller, while the denominator isunchanged, so α decreases. C12 > 0 means that the two tasks aresubstitutes: the more effort the teacher spends on teaching basic skills,the more costly it becomes for him to teach creativity (e.g. because thereis less time left). In this case the teacher should get weaker incentives ascompared to the case where the tasks are complements.



Remarks:

1. If two tasks are complements, then giving stronger incentives for one taskimproves the agent’s performance on the other task, while if the tasks aresubstitutes the reverse holds.

2. There are two ways how to give stronger incentives for task i :increase αi

increase αj , if task j is a complement of task i , or decrease αj , if task j is asubstitute for task i .

3. Even if a task is not measurable, it is possible to affect the agent’s efforton this task by changing the incentives for other tasks that arecomplements or substitutes.



Other examples of multi-task principal agent problems with a similar structureinclude:

Construction companyt1 Complete house in timet2 Do it properly

Workert1 produce outputt2 take care of machine

Sales agentt1 sell contractst2 improve customer satisfaction


Example 2: When Are No Incentives Optimal?


Consider the following specification of the teacher’s problem:

C(t) = C(t1 + t2), C′′(·) > 0.

There exists a t > 0, such that C′(t) ≤ 0 for all t ≤ t and C′(t) = 0.

Furthermore, we have:

t1 cannot be measured (σ21 = ∞),

x = t2 + ε, ε ∼ N(0, σ2),

s(x) = αx + β,

B(t1, t2) is strictly increasing in t1 and t2,

B(0, t2) = 0 ∀t2, i.e., the non-measurable task is very important.



Proposition 3.5α = 0 is optimal, even if the agent is risk neutral.

Proof: The agent maximizes:

maxt1,t2

αt2 + αε+ β − C(t1 + t2)

If α = 0 the agent will choose t1 + t2 = t and he is indifferent how to allocatehis time between tasks. Hence, (t1, t2) ∈ arg max B(t1, t2) is optimal.If α > 0, the agent will choose t1 = 0 and t2 > t . Total surplus is:

B(0, t2)︸︷︷︸

=0

−C(t2)︸︷︷︸

≥C(t)

− 12α2σ2

︸︷︷︸

>0

< −C(t) < B(t1, t2)︸︷︷︸

>0

−C(t)

which cannot be optimal.



If α < 0, the agent will choose t1 = t and t2 = 0. But, by definition, we haveB(t1, t2) > B(t ,0). The agent’s cost are the same as by α = 0 but he mustbear some additional risk. Hence, this cannot be optimal either. Q.E.D.

Note, that in this example it is not possible to write a contract on B(·). Hence,even if the agent is risk neutral, it is not possible to implement the first best,because we cannot make the agent residual claimant on social surplus.


Example 3: Incentives in Firms and Markets

Example 3: Incentives between Firms vs. Incentives withinFirms

t1: produce output,

t2: maintain machine,

B(t1, t2),

C(t1, t2) = C(t1 + t2).

An independent contractor typically owns the machine he is working with,while an employee typically does not own “his” machine. Why?Suppose that the owner of the machine benefits from the increase in valuethat is due to proper maintenance, which is not possible for an employee (e.g.because the value of the machine is not verifiable, no second hand market).


Example 3: Incentives in Firms and Markets

Proposition 3.6Suppose that maintenance is sufficiently important. In this case the optimalincentive scheme pays a fixed wage to the employee (α = 0). The optimalcontract for an independent contractor has α > 0. Both types of contracts maybe optimal (depending on parameters). If α = 0 is optimal given (r , σ2

1 , σ22),

then this type of contract is also optimal for higher values of these parameters.

Proof: See Holmström and Milgrom (1991).The intuition is simple: If the agent does not own his machine, then thesmallest incentives to produce more output will result in zero maintenance.Thus, if maintenance is sufficiently important, it is better to give no incentivesat all. However, if the agent owns his machine, he will maintain it even if hegets very strong incentives to produce output. The disadvantage of having anindependent contractor is that it is impossible to insure him againstfluctuations of the value of his machine.


Empirical Evidence

Empirical Evidence

1) Anderson and Schmittlein (1988): independent sales agents get muchstronger incentives to sell than employed sales agents. The difference isthat the independent ones typically own the “client list” and have anincentive to invest in customer relations, while the employed sales agentsdon’t. Anderson and Schmittlein show that a given sales agent is morelikely to be an employee the more difficult it is to measure hisperformance. Furthermore, the commission rate for independent salesagents is much higher.

2) Krueger (1991) shows that 30% of all McDonalds Restaurants are ownedby McDonalds while 70% are franchises. A franchised restaurant ownergets 90% of marginal revenues, while an employed restaurant managergets no financial incentives.


3. dynamic moral hazard - economic · pdf fileresidual claimant on the margin. ... (which...

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