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·'

N9 ·77

.-.EXISTENCE

..

OP セN@ SOLUTION TO THE

PRINCIPAL • S PROBLEM

*

Carlos Ivan Simonsen Leal

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Cbapter ill: Existence ofa

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1. Introduçtion

page·52

Chapter W: Existcnçe

ora

Solution to the Prinçipal's ProbIem.

The principal's problem, due to Ross (1973), is one ofthe focal results in the theory

of assymetric information. This modeI studies the relantionships between a principal and bis

agent The principal faces lhe following problem. He is to determine a sharing rule for a

monetary random payoff, the distribution of which depends on lhe agent's actions. Thc

agent's utility depends on the amount of money he receives and on the action he tak:es. Both

the utility functions of the principal and the agent are cornrnon knowledge. The action taken

by the agent is unknown by the principal. 111e principal faces the problem of maximizing his

expected utility subject to: a) the agent takes bis action to maximize bis own utility given the

sharing rule; b) the agent is to be セウオイ・、@ of a minimum levei of satisfaction, lhe çertainty

CQJ1ivalenl leveI. The purpose of this papel' is to give a general existence theorem for this

problem.

In thc literature, the usual approach for discussing the existcncc of a solution (Ros:):

1973; Shavell 1979) has bcen to ignore the certainty equivalent constraint AIso, it is

standard to assume thal for each sharing rule there exists one optimal action for the agent In

this case, provided some tcchnical conditions are satisfied, one uses the implicit function

theorem on the agent's first- order conditions to obtain bis action as a function of the sharing

role. Substitution of this function into the principal's utility function transforms the

constrained maximization problem we

h8d

before into an unconstrained one. Unfortunately,

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that if one cannot guarantee the uniquencss of the agent's maximwn. then the fmt- order

conditions of the agent are not even a necessary condition for an optimum.

More recently. Grossman and Hart (1983) have talcen a new approach to the

existence of a solution to the principars problem. Rather than appea1ing to differential

ca1culus techniques. they use lhe lower- semicontinuity properties of the solution of the dual

of a linear program to establish the existence of a solution to the principal's problem. Our

procedure follows their lead, but does not use linear programming. Grossman and Hart,

because they were interested in doing comparative statics, had to make strong assumptions

on the nature of the utility functions of the principal and agenl We only worry about the

existence of a solution and lhus we are able to consider a larger class of utility functions. The

only assumptions required are of a technical nature, like differentiability of the agent's utility

function and compactness of lhe space of actions. Unlike

Orossman

and Hart

wc

do not

require any concavity assumptions, the utility functions of the principal and agent can be

quite general.

In Section 2, we present the basic skeleton of our model} where the only source of

uncertainty considcred concems the agent's actions. This assumption makes the problem

mathematically more treatable. In Section 3, we develop the mathematical troIs needed for

studying the existence of solution. Because we are interested in lhe situation where the

number of possibJe outcomes is infinite, we give special attention to lhe consttuction of the

space of sharing roles with an adequate topology. This topology will make the space of

sharing roles a compact metric spacc. Moreover, when the number of outcomes is fmite the

space of sharing roles is contained in a fmite dimensional real vector space and this topoJogy

coincides with the euclidean topology. No matter what is the number of possiblc outcomes,

we use Berge's Theorem of the Maximum to solve lhe セ「ャ・ュ@ of feasibility for the

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page- 54

of existence of a solution is lhen obtained in a purely topological way.

In Section 4, we allow for uncertainty on different states of nature andIor uncertainty on the individual's type. In lhe fll'St case, the random payoff of the game will also depend on the state of nature, which both principal and agent are uncertain about. In the second one, the

total payoff of the garoe depends on some characteristic, type, of the agcot. The type is

known by the agent, but unknown by the principal. The main result of thia scctioo ia to show that obtaining lhe solution of the principal's problem in these two cases ia formally equivalent

to what is done in scction 3.

The last section covers two distinct topics: fU'St, imposing constraints on the sharing rules, that is, limiting the values they can take at each outcomc; and second, requiring that the sharing rule be a continuous function. Pinally, we have an appendix where we discusa some of the difficulties in computing a solution of lhe problem. .

2.

MathematicaJ

Model

The sp&ee of outcomcs, that is the quantity of money available, is a compact interval

[a,pl,

where

a

<

p.

We consider the set M([a,P]) of regular probability measures over

[a,Pl. Thia set is a topological space with the weak*· topology. In this topology two

measures セ@ and ro are c10se if and only Ü for any continuous function f:

[a,pl

セ@ 9t the values of the integraIs

I

ヲ、セ@

and

I

fdro are close. lt is possible to prove that M([

a,pn

ia a compact and metric space (see the fmt chapter of this thesis).

The agent's sction space ia a compact metric ウセ@ A. We are given a continuous

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action

lhe

laalt takea.

n

ia known by both the aaent and lhe principal. The aaent has a

continuOUl Von Neumann- Morgenatem utiUty function U: Ax[a,P]

-+

91 that ia alao continuously differentiab1e in lhe second variab1e. The dependence on

a

reflects the possible diautiUty of effort.

1be priIlcipal has I continuoua Von Neumann- Morgenstem utility function V: {。LセQ@

-+

9l.

ma

problem is to finei a function h:[a,P]

-+

[0,1] the sharina rule, that solves

a) a maximizes

J

U(a,(l-h(a»·s) dcon(,) (8)

b)

a

Batiafiea

J

U(a,(l-h(s» .. ) d':'TI(I) (I)

セ@

Ã.},

whcre

Ã. is tbe ccrtainty equivalent

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page - 56

1.-

1 2 3

i

1

11(1)

-

A

-

O

2 ; i 2

TI(2)

-

3

I

O

-

1

4 4

I

n(3) O 1

-

1

2 2

and that the certailit)' equívalent levei is Â.= 3/2. The utility function of the principal is V(I)

=-1.

In this example, the solution ofthe principal's problem is a triple (bl' h2' h3) e

[0,1]3, where hi is lhe sharing role for s = i. The agent's payoff is a function of (hI , h2' h3)

and lhe action a, we have

Suppose for a moment L1at action a

=

1 is optimal for the agent at lhe solution of the

principal's probJem, then the principal's problem is to choose hI' h2 as to

Maximize

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(1/2)(1-hl)2 + 2(1-h2)2:2: (213)(1-h2)2 + (3i2)(l-h)2

(1I2)(l-hl)2 + 2(l-h2)2:2: 3/2,

where the eonstraint inequalities say that the agent preffers a :::: 1 to a

=

2 and to a

=

3

respectively and that the certainty equivalent is satisfied. TIle problem is depicted on the

figures below. For h3 between 1-11-./3 and 1 the only active constraint is the third, that is, Ú1e

certainty equivalent We have the situation of Figure la. For h) between 2/3 and 1-11-./3 the

second constraint is active and we have something like Figure lb. Finally, for h3 smaller

than 213 the three constraints are active, Figure le. The principal is maxirnizing his utility in a

non- eompact non- convex set lf a

=

1, then a solution to the prineipal's maximization

problem is h1

=

1, h2 = 1-0.5...J3 and h3 ::: 1, while his utility is EV ::: 0.5 + 1 -0.S-./3.

(0,1-

J. )

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Maximintion cf the PI1nc1pal's UUl1ty vben a- 1

ceI"lta1ntv equivalent

.. - 1 i:n41ffll1lnt セ@ a - 2

2/3

s

h3 Si· 11./3

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page - 58

The figure below depicts the same problem when we suppose that the action taken i;;

a

=

3. The maximization of the principal's utility leads him to the point (h 1,h2,h3>

=

(1,1,0),

where he forces the agent to pick a :;:: 3 and where EV

=

1. The reader should check that this

is actually lhe solution to the principal's problern by checking a

=

2.

a-3 .... _, ... ..

(0,1-./5/9)

The following consideration gives another example of the principal- agent problem,

it shows that it actually generalizes the 1Vª&e ince.ntiY.c problem, which is a part of the

problem of designing optimaJ wage contracts (see Hart, 1983).

Example 2: Suppose that A c 9t and that each action a is a measure of the effort the agent

puts in his work. Suppose

n

is such that ッセiH。IHウI@ is the measure that gives weight 1 to the

point f(a), where f is a continuous function f: A -+ [a,Pl. If we ignore the certainty

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[0,1] that solves

Maximize V (h(f(a»ef(a)

subject to a maximizes V(a,(l-h(f(a»)ef(a».

セzZ@ Evcn Jfi easy problern like the one above may not have a solution. Suppose A "

(0,1]. If we particulariz.e 1'(a)

=

a, V(I):: 1 - 111 and V(a,1) = 12/8, then the problem does no:

have a solution. Indeed, the optimal sharing role satisfies dU(a,(l-h(a»ea)/da

=

O. This

ímplies that hl(a)= (2ar1(1-h(a», which integrated yields h::: l-C-.Ja, where C セ@ O is a

constant chosen by the principal. If C > O the agent picks a ::: 1, but then the principal will

always have an incentive to reduce C. 1f C ::::. oセ@ lhen lhe agcnt is indifferent between any

action a and the principal cannot ensure the maximil,ation of because 111 ::; 1/a -t - when a

-+0.

Remark 3: From Example 2 it is easy to see the necessity of some of our hypotheses. If A lS

not compact lhen lhe maximization problem of the agent can very well not have a solution.

On the other hand. the assumption that the space of outcome8 is a compact interval is

unecessary 。セ@ long as the map

n

is continuous. Indeed, we are only interested in the set

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NNMMMMMMMMMMMMMMMMMMセセMMMMMMセMMM

-page - 60

3. Mathematics

In order to discuss the existence of a soIution to lhe principal- agent problem we WI!J

gíve a topology for the space H of sharing rules. Fintt, we shall assume that H consists of

the measurable functions ィZ{HャNセ}@ -+ [0,11. Then, we define a function d:Hx.H -+

9t

by

putting d(h,h'):= supdlh-h'ldCO: co E

mH{。Lセ}I@

}, we claim that this function is a distance

function. Indeed, it is trivial that d(h,h'): d(h',h) and that d obeys the triangle inequality,

also by considering lhe probability measure that gives weight 1 to a point s we see that if

d(h,h')=O, then Ih(s)-h'(s}i セ@ d(h,h')=O, since s is arbitrary we have h=h'. The topology of

H is generated wi!..h thí5 distance function, H is a metric space.

In our discussion, we shall also need Berge's Theorem of the Maximum (see

Hildenbrand, 1974, page 26). We recaI1 that a com;s,pondence is a map m: X -+ 2 Y that to

each point ofX associates a subset (possibly void) of Y. When X and Y are compact metric spaces we say that m is up,per- semiçontinuous (u.s.c.) if for any convergiog sequence

(Xn'Yn) e graph(m):=: { (x,y): ye mexI } the lintit poiot (x'" ,y*) also belongs to graph(m).

Theorem 4: (Theorem of the MaxilnUffi) Let q>: Xxi -+ 9\ be a continuous function. Then the

function m: X -+ 9\ defined by m(x):= max{ cp(x,y) : y E Y } is continuous and the

correspondencc \P: X -+ 2 Y defined by \f'(x):= { y : q>(x,y)= m(x) } is

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In the next two propositions we study the properties of two 」セウーッョ、・ョ」・ウ@ that

detennine the existence of a solution.

Proposition 5: The correspondence '1': HIR Nセ@ A sue.h dlat \I'(h) consists of the sets of a's

that maximize !U(a,(l-h(S».S)dIDn<a)(S) ís U.s.c. and nonvoid always.

fnlQf; The map r: AxH -+

9t

defined by r(a,h):= IU(a,(l-h(S».S)dWn<a>(S) is continuous.

iョ、セ@

The frrst term on the right hand side of this inequation is bounded by Kp d(h,h'), where K:=

max{DU2(a,x): a E A and x E [a,b] }. The second term is bounded by

max{\U(a',x)-U(a,x)1 : x E

[a,pl }

This is enough to prove that r is continuous.

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NMMMMMM⦅MMMMMMMMMMMMMMMMMMセMMセM -

---page - 62

FinaUy, the upper-semicontinuity of'l' is implied by Berge's Maxirnum Theorem.

QED

Pfol>osition 6: The correspondence r: H -+ A defined by r(h) := {a : a satisfies

juH。セHャMィHウᄏ。ウI@

dOln<a)(S)

セ@

 } is U.s.C ..

fmgf: The continuity of r irnplies that the cOlTCspondence nh)

= {

a : r(h) セ@ Â} has closed

graph, since A is compact and metric this is equivalent to saying that this correspondence is

u.S.c .. QED

Notice that r(h) may very well be empty. This means that for the sharing role h the

certainty equivalent is not satisfied. In many ーイッ「ャ・ュウセ@ wc will have that r(h)

=

0 for alI h

and the principal- agent problem is never solvable.

PropositioIl1: The principal- agent problem has a solution provided that lhe set C of the h's

such that r(h) fi "P(h):I: 0 is compact and nonvoid

fmQf: First, notice that r(h) fi 'I'(h) is nonvoid if r(h) is nonvoid. This is trivia1. Second,

let I: C -+ 9t be the correspondence defined by l(h)= dV(h(s)as)d"TI<I)(S): a e nh)

n

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upper-page - 63

semicontinuous. Indeed, let [hnl -+ h and Yn

=

Max 1([hnD E l([hnJ), then take Ynl セ@

lirn sup Yn' because of the upper- semicontinuity of I one has that lim sUPYnE I(lim[hn'D

=

I(h) and therefore lim sUPYn セ@ Max I(h). An upper- semicontinuous function defined

over a compact set always reaches its rnaximum and therefore the principal- agent problem

has a solution. QFD

Remark 8: If the certainty equivalent is always satisfied, that is if r(h) セ@ (2) for alI h, then C=

li Grossman and Hart adopt an assumption that implies this one (assumption A2 of their

paper).

The following proposition says that usua1ly the solution of the principal- agent

problem is enforceable, that is, that the agent will only pick one action at the solution.

PrQposition 9: For adense set in the space of utility functions of the agent the solution of the

principal- agent problem coincides with lhe solution of the following probIem: choose h as to

Maximize

f

V(h(s)·s) dWo(a)(s):

subject to a) a maximizes

J

U(a,(1-h(s»es) dWr!(8) (s) (*)

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page - 64

fmQf: The reader should convince hirnself that a solution of the principal- agent problem is a

solution to (*) Ü r(h*) n 'P(h*) is a singleton. Let h* be a solution of the principal- agent

problem, it is possible to take U- close to U 8ueh that r(h*) n 'I'(h*) becomes a singleton

while h* remains a solution of the principal agent problem when U is replaeed by U ....

lndeed. take a* E r(h*) n 'P(h*) sueh that IV(h(s)eS)dCOo(a*)(S)

=

Max I(h*) and replace U(a,(1-h(s»es) by U-(a.(1-h(s»es):= (1-(l/K)d(a"',a»U(a,(l-h(s»es), where K > O ean

be taken as bjg as we want. Because A is eompact we have that max{d(a"',a): a E A} < c>O.

For K big enough this implies lhat J(1-(lIK)d(a*,a»U(a.(1-h*(s»eS)dcon(a*)(S)

セ@

(l-(l/K)d(a* ,a»e JU(a,(l-h(s»eS)dCon(a.)(S). And it is clear that (l-(lIK)d(a* ,a»e

QED

4. ExternaI Uncertainty

The moral hazard problem and the adverse selection problem are nol jncluded in lhe

fonnulation of the last section. In this scction we correct this. we aIlow for externaI

uncertainty, that is for uneertaínty that is not controlled by the players. Two forms of externai

uncertainty are to be considered. The fIrst says that the aggregated profIt of the garoe depends not only on the agent's actions. but also on the state of nature l. One has a continuous profit

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compact metric space. This case can be subdivided into three subcases:

1) 80th principal and agent セッキ@ which probability distribution セ@ the variable z

obeys. The principal's problem is to find a sharing rule h: [a,P]xZ セ@ [0,1] such that h

solves

Maximize Maxdf V(h(s)·L(Stz

»

dcJ>n{I)(!-.}dJ..l(Z):

a) a maximizes

ff

U(a,(l-h(s»· L(s,z) )

、wイiH。IHウI、セHコI@

b) a satisfies

fi

U(a,{l-h(s)r L(Stz) )

、wョH。IHウI、セHコI@

セ@

Â.},

2) The principal knows m adopted by nature and the 'guess' of m used by the agent.

This problem is equivalent to 1.

3) Tbe principal knows the true セ@ and is capable of influencing the guess of the

agent.

Example 13: Subcase 1 inc1udcs the moral hanni examples. For example, let us suppose that

the principal is an insurmce company and that the agent ia an individual who would like to

buy insurance. There are two states of nature: accident and no accident; and lhe agent has two

possible actions: to take care and not to take care. If care is taken the probability of an

accident is 1/10, otherwise it is 112. The utility function ofthe agent is U(c,m)= (l-e-m)/c,

where c= 1 if no care is taken and c= 1.005 if cate is エ。ォ・ョセ@ If no accident occurs m= 100 and

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page· 66

neutral and that competition on lhe insurance 1tW'ket makes profits zero. In this case, the arnount i of insurance offered will be the solution to the equation oNsHQセᄋャPPI@ + oNsHャM・セゥI@

=

O.9«(l-e- H)()}l1.05) + 0.1 «(l-e-i)/LOS). It is easy to check that i < 100.

The discussion of the ex.istencc of a 801ution to the principal's problem as formulare j

in the subcases 1 and 2 above canbe handlcd in the &ame way we did on the last section.

Indeed, using Fubini's theorem (Halmos, page 148), we can substitute lhe double integraIs

above for inlegrals in

[a,PlxZ.

this permits U8 to obtain correspondencea analogous to the

correspondcnccs

r

and 'P of propositions 8 and 9. Existencc of a solution can then be ensurcd unde! conditions similar to those of proposition la. The on1y step that ODe has to

prove is

Lemma 10: H

fi:

A セ@ M([a,pn is continuous, thcn

n*:

A セ@

M([a,PlxZ)

dcfineei by

n*(a):= "TI<a)XJ.l is continuolts (for

a

definition of the product measure coxlJ. sce Halmos, page 143).

P.mslf:

Both A and M([ a,

111

xZ) are metric spaces, therefore to prove the continuity of

n

it is

enough to consider sequcnccs. We have to provc that ir In セ@ a, then n*(In) セ@ n*(a) and,

from lhe definition of the weak*· topology, this implica that we have to prove that for any

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jヲHsNzI、oャQH。IxセャN@

Usíng Fubini's theorem we have

f

f(s,z)d( Wo<In)XJL)= ! (!f(s,Z)dWn(ID»)dJJ.

-4

jjヲHセエzI、ッョH。ᄏI、セャ@

=

jヲHsLzI、HッョHiIxセI@

This proves the lemma. QF.D

In lhe subease 3 above, the principal has a certain controI over the agent's guesses, one may suppose that the principal will use this power to achieve two objectives. First. to

make lhe agent think that the ceruunty equivalent constraint is satisfied. He may for example

impose that lhe set Z*(l1):= { z e Z:

J

U(a,(1-h(s»·L(s,z» dIDn(a)(s)

セ@

11 }, where 11 > O

and

M't

< 1, be guessed with probability セGHzNHQャᄏ@ > ÃIrl. Double intcgration then gives that

IJU(a,(l-h(s». L(s,z) )

、wョH。IHウI、セGHコI@

セ@ セ@

One can dlink of ウゥエオ。エゥッョセ@ where the movements of nature constrain the 8.Ctions that the agent ean tak:e. In this situation, we ean still ensure the existente of a solutiOll to the

principal's problem if the age.nt is forced to talte his actions such that エッョHNIクセ@ Hes on a

eompact subset ofM([a,Plxz}. A sufficient condition for this is that the set ofprobability

distribution that the agent ean take consists of alI the probability distributions over a compact

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page-68

Lemma11: Let K be a compact subset of [a,1l1x.Z, then the set of measures with support in

K is compacto

fIslQ!; Let EM(K) be the set of extreme points ofM(K), that is the set of measures that give

weight 1 to points of K. By the Krein-MilJman theorem (Rudin, 1973) M(K) is the convex

hulI of EM(K). If K is compact, then EM(K) is compact and therefore M(K) is compact

QED

The second fonn of ullcertainty considered in this SCCtiOll concerns the adverse

selection problem. This happens when the principal is not sure of the agent's type. TIle

principal only knows a probability distribution J.I. over the possible types of the agent The

principal- agent problem is then written as the problem of fmding a sharing rule h that solves

Maximize

MaxdJ

Y(h(s)-L(s,z» dmn<i)(s)dJ,t(z)

a) a maximizes

JJ

U(a,(1-h(s»- L(s,ZO) ) dmn(a)(s)

b) a satisfies

JJ

U(a,(1-h(s»- L(s,ZO) ) dmnCi)(s)

セ@

"'},

where

zo

is the actual type of the agent The reader should check that this new formulation of

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5. Introducin& Consttaints

We have not yet imposed any type of constraint on lhe sharing rules. In lhe wage

incentive problem, for example, it is not uncommon lhat due to some previous hlstory the

employer be limited on his choice or which sharing role hecan adopl This is the case of a

contract imposed by an union or when monitoring the agent's actions is costly. In the

insurance examples one should recall that there are laws telling you the minimum coverage

you can get anel limiting the liability of the insurance companies.

The way we choose to model these constraints is to say lhal there is a partiti<m of the

set of outcomes into a finite number of disjoint measurable subsets S 1 ,S2, .... ,Sn and then

add the following extra condition to the principal'! problem

lt is clear that undcr (c) the a solution or lhe principal problem may not existo A

necessary condition for existence is. of COU1'Se, lhat ri S 1 and Ri セ@ O. In this case, we

redefine our space H: we say that H is the set of measurabJe functions h: {。Lセ}@ セ@ [0,11 lhat

obey the constraints in (c). One may then go through the same machinery or sections 2 and 3

to discuss the existence of a solution.

Another type of constraint occurs when one imposes lhat the sharing rule has to be

continuous. This changes our formulation completely. We define the space H by

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page - 70

be lhat yielded by lhe sup distancc. however. under this topology H is oot sequentially

compacto Indeed, by the Ascoli theorem (see Munkrees, page 277) a set is sequentiaJly

compact in lhis topology if it is a ctosed. bounded and equicontinuous set of fW1ctions.

To find an H we can work with we shall assume lhat there is a constant K > O sue h

that a11 the admissible sharing rules obey Ih(s')-h(s)1 S Kd(s',s) for any s,s'. This of COUI"Se

ensures equicontinuity. Boundness is uiviaJly ensured by O セ@ h(s) S J. We may then use lhe

converse of Ascolrs theorem, lhe At7.ela's theorem (Munkres, page 279), and state that H ili

scquentially compact. b・」Nセ。オウ・@ H is a metric space H, it wilJ be a compact space ir it is

sequentially compacto

Final1y, we define a funcuon r. AxH -+ 9t like that of proposition 8. The proof that r is continuous is even easier here than it was there. We are then able to establish a proposition

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ARm'0dix:

Remark

on thc CoJnputatiOQ of

ih,

Sbari0l Rule.

To calculate lhe solution to lhe principal- agent problem is in general much more

difficult than proving that it exists. For instancc, in Example 1 we had to maximize the

expected utility of lhe agent subjected to constraiots lhat were non-convex. In this appendix

we are concerned with what happens when the number of possible actions of the agent is

infmite. We are goiog to show that we cao approximate the solution ofthe principal· agent

problem.

We recaI1 that we have assumed that the space A of 8gent's actions was 8 compact.

metric space. therefore A countable anel dense set {8 l' 82' ... }. Let Ak be the set dcfmcd by

Ak: = {aI' a2, ... , &)c} and consider the solutions of the principalM agenl problem when the

agent is restrictcd to pick bis actions from Ar By Proposition 6, for each k, we can obtain a

solution [h:t] of such a problem, we have

Pxgposition A: If h is ao accumulation point for the sequence [hk]' then h is a 801ution of of

lhe principal- agent problem when the agent can play any action in A .

.fn'lof:

Suppose that h is not 8 501ution of (*), then there exists h* and E > O such tbat

Max I(h*) -E > Max I(h). Let a* belong to r(h*) ,.., 'I'(h*) be 5uch lhat IV(h*(8).S)droma.)

(23)

page - 72

because IV(h*(s>-S>dmn<a) iB a continuous function of a and lhe

Illc

are adense set Therefore

we have that

ivHィNHウIセ^、」ッョH。エI@

> Max l([htn, a contradiction and the proposition is

proved.

(24)

BibJiQKrapb)'

Berge, C., Espaces J'o»olo&iQ.Ues, Dunod, Paris, 1966.

Bil1ingsley, P., Conyer&ence QfEmbabi1ity Measures. JoOO Wiley & Sons, 1968.

Grossman, S. J. and O. D. Hart: "An analysis of the Principal-Agent Problem", Sconometrica.lL. No. 1, 1983.

Halmos, P.,

Measure

ThcOQ. Springer-Verlag, New York, 1974.

Hart, O., "Optimal Labour Contracts under Asynunetric lnfonnation: An lntroduction",

Revia

of EcouQmic Studics. 1983.

Mirlees, J. A., "The Thcory ofMoral Hazard and Unobserved Behaviuour- Part f', Nuffield College, Oxford, mimeo, 1975.

Munkres, J., Topolo&y. A Fiat Course. Prentiee Hall, 1975.

Ross, S., "The Economic Theory of Agency: The Principal's Problem", American Economjc Reyiew.

a

1973, 134-139.

Rudin. W., FunctiQDal Analysjs. MacGraw-Hill. New York, 1973.

(25)

ENSAIOS ECONOMICOS DA EPGE

1. ANALISE COMPARATIVA DAS ALTERNATIVAS DE POLTTICA COMERCIAL DE UM PAIS EM PRO-CESSO DE INDUSTRIALIZAÇAo - Edmar Bacha - 1970 (ESGOTADO)

2. ANALISE ECONOMtTRlCA DO MERCADO INTERNACIONAL DO CAFt E DA POlrTICA BRASILEi-RA DE PREÇOS - [drrar 8acha - 1970 (ESGOTADO)

30

A ESTRUTURA ECONOMICA セrasileira@ - Mario HenrIque Simonsen - 1971 (ESGOTADO)

40

O PAPEL DO INVESTIMENTO EM EDUCAÇAO E TECNOLOGIA NO PROCESSO DE DESENVOlVIMEN TO ECONOMICO - Carlos Geraldo Langonl - 1972 (ESGOTADO)

Se A EVOLUçAo DO ENSINO DE ECONOMIA NO BRASil - Luiz de Freitas Bueno - 1972

6. POLrTICA ANTI-INFLACIONARIA - A CONTRISUICAo BRASILEIRA - Mar/(J Henrlquü

Si-monsen - 1973 (ESGOTADO)

-7. ANALISE DE SrRIES DE TEMPO E MODELO DE formaᅦセo@ DE EXPECTATIVAS - José Luí l Carvalho - 1973 (ESGOTADO)

8

0 OISTRIBUIÇAO DA RENDA E DESENVOLVIMENTO ECONOMICO DO 8RASIL: UMA reafirmaᅦセo@

Carlos Geraldo Langoni - 1973 (ESGOTADO)

9, UMA NOTA SOBRE A populaᅦセo@ OTlMA DO BRASIL - Edy Luiz Kogut - 1973

10, ASPECTOS DO PROBLEMA DA ABSORÇAO DE MAO-DE-OBRA: SUGESrüES PARA PESQUISAS José Luiz Carvalho - 1974 (ESGOTADO)

11. A FORÇA 00 TRABALHO セo@ BRASIL - Mario Henrique Sirnonsen - 1974 (ESGOTADO)

12. O SISTEMA BRASILEIRO DE INCENTIVOS FISCAIS - Mario Henrique Simonsen - 1974 (ESGOTADO)

13. MOEDA - Antonio Maria da Silveira - 1974 (ESGOTADO)

(26)

160

ANALISE DE CUSTOS E BENEFTclOS SOCIAIS I - Edy LuIz Kogut .. 1974 (ESGOTADO)

17. DISTRIBUIÇ"-O DE RENDA: RESUMO ·DA EVIDtNCfA - Carlos Geraldo Langonl - 1974

(ESGOTA.DO)

18. O MODELO ECONOMETRICO DE ST, LOUIS APLICADO NO BRASIL: RESULTADOS PRELIMINA

RES - AntonIo Carlos Lemgruber - 1975

19. OS MODELOS

clセssicos@

E NEOCLAsSICOS DE DALE W. JORGENSON - El 1seu R. de

An-drade Alves - 1975

20. DIVID: UM. PROGRAMA FLEXrVEL PARA CONSTRUÇ"O DO QUADRO DE EVOlUÇ"O DO ESTUDO

DE UMA DrVIDA - Clõvls de Faro - 1974

210 ESCOLHA. ENTRE OS REGIMES DA TABELA PRICE E DO SISTEMA DE AMORTIZAÇOES CONSTAN

TES: PONTO-DE-VISTA DO

mutuセrio@

- Clovis de Faro -

1975

-22. ESCOLAR.lDADE, EXPERIENCIA NO TMB·ALHO E SALARIOS NO BRASIL .. José Julio

Sen-na - 1975

230

PESQUISA QUANTITATIVA NA ECONOMIA - Luiz de Freitas Bueno - 1978

2". UMA ANA.L I SE EM CROSS-SECTI ON DOS GASTOS FAMI LI ARES EM CONEXAo COM NUTRI ç"O,

SAODE, FECUNDIDADE E CAPACIDADE DE GERAR RENDA - José LuIz Carvalho - 1978

25'. DETERMINAC"O DA TAXA DE JUROS IMPLTCITA EM ESQUEMAS GENI:RICOS DE

FINANCIA-MENTO: COMPARAÇ"O ENTRE OS ALGORrTIMOS DE WllD E DE

nevtonMセphson@

- Clovis

de Faro· - 1978

26. A URBAHIZAÇ"O E O CrRCULO VICIOSO DA POBREZA: O CASO DA CRIANÇA URBANA NO

BRASIL - José Luiz

Carvalho

e

Urrei

de Magalhães -

1979

27. MICROECONOMIA - Parte I - FUNDAMENTOS DA TEORIA DOS F'REÇOS • MarIo Henrique

Slmonsen - 1979

(27)

- - -MMMMMセMMMMMMMMMMMMM

29. CONTRADIÇAo APARENTE - Octivlo gッオカセ。@ de Bulh5es - 1979

30. M!CROECONOMIA - Parte 2 - FUNDAMENTOS DA TEORIA DOS PREÇOS - Mario Henrique

Símonsen - 1980 (ESGOTADO)

31. A CORREÇÃO MONETÃRIA NA JURISPRUDtNCIA BRASILEIRA- Arnold Wald - 1980

32. MICROECONOMIA - Parte A - TEORIA DA DETERMINAÇAo DA RENDA E DO NfvEL DE PRt ÇOS -'José Julio Senna - 2 Volumes - 1980

33. ANALISE DE CUSTOS E BE'NEFfCIOS SOCIAIS III - Edy Luiz Kogut - 1980

34.

MEDIDAS DE CONCENTRAÇAo - Fernando de Holanda Barbosa - 1981

35.

crセoito@ ruセlZ@ PROBLEMAS ECONOMICOS E SUGESTOES DE MUDANÇAS - Antonio

Sala-zar Pessoa Brandão e Uriel de Magalhães - 1982

3(,.

determinaᅦセo@ NUMrRICA DA TAXA INTERNA DE RETORNO: CONFRONTO ENTRE ALGORflt MOS DE BOUlDING E DE WILO - Clovis de Faro - 1983

-37.

MODELO DE EQUAÇOES SIMULTANEAS - Fernando de Holanda Barbosa - 1983

38.

A EflCltNCIA MARGINAL DO CAPITAL COMO CRITtRIO DE AVALIAÇAo EeOHOMICA DE PRO JETOS DE INVESTIMENTO - Clovis de Faro - 1983 (ESGOTADO)

39.

salaセio@ REAL E inflaᅦセo@ (TEORIA E ILUSTRAÇAO EMPTRICA) - Raul José Ekerman

- 1984

400 TAXAS DE JUROS EFETIVAMENTE PAGAS POR TOMADORES DE emprセstimos@ JUNTO A BAN

COS COMERCIAIS - Clovis de Faro - 1984

-41. REGULAMENTAÇAO E DECISOES DE CAPITAL EM BANCOS comeセciaisZ@ REVISAo DA LITI

RATURA E UM ENFOQUE PARA O BRASIL - Urlel de Magalhaes - 1984

42. INDEXAÇAO E AMBltNCIA GERAL DE NEGOCIOS - Antonio Maria da SilveIra - 1984

(28)

45. SUBSrDIOS eREolTrelos A exportaᅦセo@ - gイ・セVイャッ@ F.L. Stukart - 1984

46. PROCESSO DE desinflaᅦセo@ - Antonio C. Porto Gonçalves - 1984

47. INDEXAÇAO E pNealャmentaᅦセoinflacionaria@ - Fernando de Holanda Barbosa - 1984

48.

SALARIOS MlDiOS E SAlARIOS INDIVIDUAIS NO SETOR INDUSTRIAL: UM ESTUDO DE DI

ferenciaᅦセo@ $;\LJl.RIAL ENTRE FIRMAS E ENTRE INDIVTnuos - Raul José Ekerman

e

Uriel de m。ァセゥィセ・ウ@ - 1984

49. THE OEVELOPING-COUNTR'f DEBT PROBLEM - Mario Henrique Simonsen - 1984

50. JOGOS DE infoセセaᅦaッ@ INCOMPLETA: UMA INTRODUçAO - Sérgio Ribeiro da Costa

Werlang - 1984

51, A TEORIA monetセria@ MODERNA E O EQUILrBRIO GERAL WALRASIANO COM UM NOMERO

INFINITO DE BENS - A. Araujo - 1984

52.

A INDETERMINAÇÃO DE MORGENSTERN - Antonio Maria da SilveIra - 1984

53.

O PROBLEMA DE CREDIBILIDADE EM POLTTICA ECONOMICA Rubens Penha Cysne

-1984

54. UMA anセlise@ ESTATTsTICA DAS CAUSAS DA EMISsAo DO CHEQUE SEM FUNDOS:

FORMU-LAÇÃO DE UM PROJETO PILOTO - Fernando de Holanda Barbosa, Clovis de Faro e Alorsio Pessoa de Araujo - 1984

55. pOLTnCA HACROECONOt11CA NO BRASIL: 1964-66 - Rubens Penha Cysne - 1985

56, EVOLUCAo DOS PLANOS BAs I COS DE F I NANC I AMENTO PARA AQU I S I ç7S,O DE CASA prセーイ[、@ i.,

DO BANCO NACIONAL DE HABITAÇAO: 1964 - 19840 - Clovis de Faro - 1385

57. MOEDA INDEXADA - Rubens p, Cysne - 1985

(29)

59. O ENFOQUE MONETARIO DO BALANÇO DE PAGAMENTOS: UM RETROSPECTO - Valdir Ramalho

de Melo - 1985

60. MOEDA E PREÇOS RELATIVOS: EVIOtNCIA EMPfRICA - Antonio Salazar P. Brandão _

1985

61.

interpretaᅦセo@

ECONOMICA,

inflaᅦセo@

E INDEXAÇÃO - Antonio Maria da Silveira

セ@

1985

VRセ@ セacroeconomia@ - CAPITULO I - O SISTEMA MONETARIO - Mario Henrique Simonsen

e Rubens Penha Cysne - 1985

63. MACROECONOMIA - CAPfTULO 11 - O BALANÇO DE PAGAMENTOS _

Simonsen e Rubens Penha Cysne - 1985 Mar i o Henr j que

64.

MACROECONOMIA - CAPTTULO III - AS CONTAS NACIONAIS - Mario Henrique Slmonsen

e Rubens Penha Cysne - 1985

65. A DEMANDA POR DIVIDENDOS: UMA JUSTIFICATIVA TEORICA - Tommy Chin-Chiu Tan e

Sergio Ribeiro da Costa Werlang - 1985

66. BREVE RETROSPECTO DA ECONOMIA BRASILEIRA EN1RE 1979 e QYXセ@ - Rubens Penha

Cysne - 1985

67. CONTRATOS SALARIAIS JUSTAPOSTOS E POLrTICA ANTI-INFLACIONARIA - Mario Henrique

Si monsen - 1985

68. INFLAÇAO E POlfTleAS DE RENDAS - Fernando de Holanda Barbosa e Clovis de

Faro - 1985

69 BRAZIL INTERNATIONAL TRADE AND ECONOMIC GROWTH - Mario Henrique Simonsen - 1986

70. CAPITALIZAÇÃO CONTINUA: APLICAÇÕES - Clovis de Faro 1986

71. A RATIONAL EXPECTATIONS PARADOX - Mario Henrique Simonsen - 1986

72. Ã BUSINESS CYCLE STUDY FOR THE U.S. FORM 1889 TO 1982 - Carlos Ivan Simonsen

(30)

7'" COMMON KNOWLEOGE ANO GAME THEORY - Sergi o Ri bei ro da Cos ta Wer1 ang - 1986 75. HYPERSTABllITY OF N.ASH EQUILlBRIA - Carlos Ivan Slmonsen leal - 1986

76.

THE BROWN-VON NEUMANN DIFFERENTIAl EQUATION FOR BIMATRIX GAMES Carlos Ivan Slmonsell Leal - 1986

77.

EXISTENCE OF A SOlUTION TO THE PRINCIPAL'S PROBlEM - Carlos Ivan Slmonsen Leal - 1986

000048216

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