On endogenous cost of altruism

ＬＢｾｆｕｎｄａￇａｯ＠

. ' " GETULIO VARGAS

EPOE

Esco\a óe P68-Graduaçao em Economia

"On Endogenous Cost of Altruism"

j

Prof. Marcos Lisboa

(EPGE)

I,OCAIJ

Fundação Getulio Vargas

Praia de Botafogo, 190 - 1

<r

andar - Auditório

IJATA

18/11/99 (58 feira)

1l0ltÁIUO

16:00h

ＭＭＭＭＭＭＭｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｾｉ＠

On Endogenous Cost of Altruism

Marcos

B.

Lisboa

*

Humberto Moreira

t

November 18, 1999

Abstract

The paper extends the cost of altruism model, analyzed in Lis-boa (1999). There are three types of agents: households, providers of a service and insurance companies. Households have uncertainty about future leveIs of income. Providers, if hired by a household, have to choose a non-observable leveI of effort, perform a diagnoses and privately learn a signal. For each signal there is a procedure that maximizes the likelihood of the household obtaining the good state of nature. Finally, insurance companies offer contracts to both providers and households. The paper provides suflicient conditions for the exis-tence of equilibrium and shows the optimal contract induces providers to care about their income and also about the likelihood households will obtain the good state of nature, which in Lisboa (1999) was stated as altruism assumption. Equilibrium is inefficient in comparison with the standard moral hazard outcome whenever high leveIs of effort is chosen precisely due to the need to incentive providers to choose the least expensive treatment for some signals. We show, however that an equilibrium is always constrained optimal.

"Escola de Pós-Graduação em Economia da Fundação Getúlio Vargas.

..

,.

ｾＭＭ ｾｾ＠ - - -

-1

Introduction

The standard health economics model usuaHy assumes the existence of

asym-metric information between insurance companies and patients.1 _Patients

may know more about their health then insurance companies do and may like to intensively use health services in case they have access to a fuH in-surance contract. These assumptions generate some of the stylized facts ob-served in health insurance contracts such as the charge of co-payment and the supply of partial insurance contracts, even though households typically are risk-averse, while insurance companies may be risk-neutral.

In the last twenty years, however, the typical health insurance contract has been subject to several transformations not explained by the standard models. A central aspect of these transformations, usually labelled as the managed care revolution, was the introduction of risk-sharing contracts be-tween doctors and insurance companies.2 _{Since doctors are typically}

risk-averse, this introduction suggests the need to provide incentives for doctors to take into account the cost of treatment while accessing the patient health and choosing the appropriate treatment. Doctors, however, are typically absent of the standard health insurance model.

The basic feature of contracts between doctors and insurance compa-nies in the managed care motivated the altruism model presented in Lisboa (1999), which provides an alternative model for insurance market. There are three types of individuaIs: patients, doctors and insurance companies. Patients have uncertainty about future leveIs of income. Doctors, if hired by a patient, perform a diagnoses, which provides a signal about the patient's health that is only observed by the doctor. For each signal, there is a treat-ment that maximizes the.likelihood of the patients obtaining the good state of nature in the final period. The paper assumes that doctors care about their income and also about the likelihood that the patient will obtain the

good state of nature (altruism assumption). There are risk neutral insurance

companies that offer contracts to both the doctors and the patients. In that

paper, the altruism assumption leads to a loss of efficiency in the market outcome. However, given this asymmetry of information, this outcome is constrained optimal, or second best. The optimal contract in that frame-work presents the following feature: the doctor payment is inversely related to the cost of proposed treatment: the more expensive the treatment, the less the doctor receives.

lFor a survey on health economics literature, see Zweifel and Breyer (1997).

2 Andrade and Lisboa (1999) summarizes the contracts under the managed care regime.

.'

This paper extends the model presented in Lisboa (1999). Instead of the altruism assumption, we assume that, before providing the diagnoses, do c-tors have to choose an effort leveI. High effort leveIs increase the probability of the good state of nature for every signal and treatment chosen. Insurance companies, on the other hand, offer contracts to doctors with payments con-tingent both on the treatment chosen and the state of nature realized in the final período The paper provides sufficient conditions for the existence of equilibrium. In order to induce doctors to make effort, the optimal contract must specify higher payments in the good state of nature and thus it induces them to care about the patient's outcome. This last equilibrium property result is equivalent to the altruism assumption used in Lisboa (1999).

There-fore, this model endogenizes that assumption by introducing a moral hazard

problem in the relation between doctors and insurance companies. In

partic-ular, all basic equilibrium properties of the altruism model also hold in this modeI. The equilibrium outcome is characterized by a loss of welfare for the households, in comparison with the first best outcome, due to the need to incentive providers to choose the least expensive treatment for some signals. However, the equilibrium outcome is still constrained optimaI.

2 The basic model

Consider a partial equilibrium model with a single commodity, three periods

- ex-ante, interim, ex-post - and three types of individuaIs: patients,

doc-tors and insurance companies. A patient faces uncertainty about her initial

endowment in the ex-post period: there are two individual states of nature and her endowment in the second period is strictly larger than her endow-ment in the first period. We refer to the patient endowendow-ment in state s by

Ws , s E {B, G}. Her preferences over consumption bundles is represented

by a state independent utility function u : ?R+ ｾ＠ ?R, strictly increasing and concave in the differentiable sense:3

(Hl) u E C2 _{and satisjies: i) du> O ; ii) d}2_u

_<

_{O. Moreover,}

O<WB <Wa

Under these assumptions of random endowments and strict risk-aversion, the patient is willing to buy an insurance contract, provided that the price of the contract is not much higher than the actuarially fair price. The stan-dard insurance literature assumes the existence of a finite collection of risk

3From now on, the symbol d represents the differential of the respective function.

..

neutral insurance companies, who offer insurance contracts simultaneously and independently. Under these assumptions, the outcome of the mo deI with at least two companies is to offer actuarially fair contracts, which are accepted by the patients.

The model proposed here departs from the standard literature by as-suming the existence of a third type of individual, whom we refer to as

doctor. A doctor examines a patient and chooses an action, or treatment,

in the interim period that affects the patient's probability of the good state of nature. This action is suppose to be perfectly observable, however its effectiveness depends on a signal privately observed by the doctor.

To make the argument precise, suppose that in the interim period the doctor observes a private signal, s E [O, 1

J.

The probability of a signal s is described by the cumulative probability function F (s) E Cl, dF( s)

>

O for every S E (0,1). Without any loss of generality, we assume that F is the

uniform distribution. There are 2 types of actions available for the doctor

Action i costs ai to be implemented in addition to the doctors' payments, O =

aO

<

a

l

. Action

ao

should be interpreted as the doctor choosing no

action. Given this signal, the probability of the good state of nature for a

patient depend upon the action chosen and it is given by 1r (a, s) for every a. We assume:

(H2) For each a E A the function 1r (a,.) E C2. Furthermore, for every

SE(O,l)

and

0< 1r(a\O)<1r(aO,O)<l

0< 1r(aO,l)<1r(a

l

_,l)<l

Therefore, the higher the signal the lower is the probability of the good state and there is a signal

s*

E (0,1) such that both actions generate the same probability of the good state of nature

1r

(a\s*)

=

1r

(aO,s*)

.'

Moreover, for every signal s

<

s* action aO generates the higher probability

of the good state of nature in the second period, while for every signal s

>

s* the reverse happens. We refer to the action that maximizes the probability of the good state of nature as the best action.

Before observing the signal, doctors have to choose an effort level, e E

{O, I},

with associated costs given by

0= Co

<

CI = C

Let

7r0 (a, s) = 7r (a, s) E (0,1)

be the probability of the good state given a, s and e = 0, and

7r1 (a, s)

=

TI (a, s) E (0,1)

be the probability of the good state given s, a and e

=

1. We assume that

(H2) holds for each 7ri and the following:

(H3) (i) 7r(aj, s)

<

TI (aj, s) ,j

=

0,1, Vs E [0,1]; and (ii) the function

is non-increasing in s.

dIT (aI, s)

dIT (ao, s)

The rate of the probability of the good state given e

=

°

and e

=

1 is less than 1 and the ratio of the marginal improvement in the probability of the good state given treatment decreases with the s (the technology of the signal has decreasing return of scale).

Observe that (H2) and (ii) of (H3) is equivalent to assume that the function T: [TI

_ü

1(1),TI

_ü

1(0)] -- [0,1] given by

is an increasing convex function that gives the trade off between the prob-ability of the good state given treatment and no treatment. The superior (inferior) extreme of the graphics of T corresponds to the signal

°

(1).

,"

Let v : R+ ---; R E C2 be the doctor's utility function strictly concave in the differentiable sense:4

(H4J

dv

>

0,

d

2

_v::;

_0,

_v

_:;::::

°

_{and lim}

_v(x)

_{= 00.}

x-+oo

A doctor has a reservation utility denoted by ti.

3

The optimal contract

3,1 Incentives and contracts

We start the analysis of the model investigating the principal's offers to the doctor which specify how much he receives in the interim period. As we will see later, in the optimal contract may be optimal to make the doctors' payments contingent upon the action chosen. Let

ri

be the payment the principal makes to the doctor if he chooses a treatment j E {O, I} and the nature chooses w E {B, G}. As usual in the literature of moral hazard problems, we will write the contracts in terms of the doctor's utility, i.e.,

vi

:=

v(rj).

We define also the power of the contract: tiVj =

vf

-

v!

and

tivw := vY' - V

o

that give the incentives for effort and choice of treatment.

Definition 1 We say that a contract

{vi}

generate a partition

(Sb,

Si)

if the treatment aj ís chosen in S3 given the effort i, i. e.,

aj E arg max

v!

+

7["i (aj, s) ti Vj if and only if s E

Si.

Thus

Si

is a Borel set,

Sb

n

SI

=1=

0

and

Sb

U

SI

=

[0,1]. If

s

is a signal where the doctor change the treatment, then

Moreover, if

s

represent the change from treatment (j = 1) to no treat-ment (j

=

O), then

The doctor's expected utility on the contract

{ri}

given the partition

(So,St) and the effort i is:

4We could assume that limv(x) = -00 without changing the results that follow.

x-o

6

or

where ai :=

JSi

_o ds is the ex-ante probabiIity that the doctor chooses no treatment and effort i and

is the ex-ante probability of the good state given the treatment aj and effort

1,.

There are I

>

1 principaIs, or insurance companies, that provide insur-ance for the patients and intermediate their reIation with the doctors. We suppose that these principaIs are risk-neutral. In the ex-ante period they

hire doctors and offer contracts for the doctors and patients. Like in Roth-schiId and StigIitz (1976), we assume that the companies are free to entry or exit the market. If an insurance company offers a contract

r

=

{rj

}

to the doctor and d

=

{dB, - da} to the patient and both accepts the contract, then its expected profit will be

i

Cid _{a -} (1 _- Ci)d _{B -} {[ _a i _{- 1fo} -il _ro

B+

_1fo -i _ro

a

+

[1-

ai -

7fil

(rp

+

a)

+

7fi(rf

+

a)}

where Ci =

?ro

+

?rI

is the probability of the good state given the contract and effort i. If the patient accepts this offer her utility is then given by

Suppose the remaining companies offered contracts (r, d). The patient would

."

li"

and

since the patient may always reject all offers.

Observe that given a contract (dB , da) between the patient and the

in-surance company, we can redefine a new contract (dB , da) such that

Both contracts give the same expected profit

and the patient weakly prefer the tilde contract. Therefore, we can assume that contract between the insurance company and the patient smooths the consumption across the states of nature (the wealth of patient is constant). On the other hand, the company should guarantee the participation of the doctor and incite the optimal effort of the doctor. A doctor who decides to make an effort i will accept this contract if the participation constraint is satisfied:

and it induce the effort i if it satisfies the incentive constraint:

ￜｩｖｾ＠

+

(1 - üi)vP

+

Ｗｲ｢ｾｖｏ＠

+

7ri ｾｖｬ＠ - Ci ｾ＠

Ｍｩｖｾ＠

+

(1 - ü-i)vP

+

7r_Üi ｾｶｯ＠

+

7r

₁

i ｾｖｬ＠ - C-i (lC_{i )}

where -i =1= i is the alternative effort. The insurance company may also

decide not to hire a doctor, in which case it can simply offer r

=

O and the

probability of the good state of nature is given by 1[" (null contract). In this

case, the household just buy a full insurance contract that transfer income

from the bad to the good state of nature and does not go to see a doctor. Let R be the set of contracts that satisfy the incentive and participation constraints and the null contract.

Therefore, besides the participation constraint of the doctor, there will be also the incentive constraints that must be satisfied. In the next subsections we deal with this problem. What is important here is that the decision of effort is taken after he signed the contract and before the doctor see the signal in the interim period. As we see below, this willlead to the endogenization of the altruism assumption used in Lisboa (1999).

3.2 When no effort is the eificient outcome

First we consider the situation where no effort of the doctor maximizes the social welfare. In this case the optimal contract between doctors and insurance company has no incentive. More precisely:

Lemma 2 lf i = O is the second best optimal effort, then the optimal con-tmct is constant: there is r such that rj = r for all w and j.

Proof. By the concavity of v,

Define:

and

."

."

and

We showed that every contract is dominated by a constant one, since the (1Co) is satisfied. _

By lemma 6, the doctor is indifferent when he is going to change the treatment. In this case we assume that, in equilibrium, the doctor chooses what is the best for the patient.

Define the inverse of the doctor's utility function: h = V-I. It is easy

to see that the cost of implementing the change of treatment at

s

with no effort is

By assumption H2, the best way to implement this change is using the

partition: 80

=

[O,

s]

and 81

=

[s,

1] .

Proposition 3 lf there is a second best equilibrium with no effort and the optimal change of treatment

8°

is interior, then

Reciprocally, if

8°

E [0,1] satisfies the equation above and no effort is second best optimal, then

8°

is the optimal change of treatment.

Proof. It is enough to show that such

8°

satisfies the first order condition of the following program:

max el

+

(e2 - ed CO(s) - cO(s)

sE[O,l]

what is straightforward. Observe that assumption (H2) implies that the program above is a concave one. _

.0

iO

3.3 The cost of implementing the change of treatment with effort

Define the inverse of the doctor's utility function: h

=

v-lo The cost of implementing the change of treatment in

s

with effort is the value function of the following program:

sot

min alh

ＨｶｾＩ＠

+

(1-aI)

[h

(vP)

+

a]

+

TIo

[h

Ｈｶｾ＠

+

ｾｶｯＩ＠

-

h

ＨｶｾＩ｝＠

{vj}

+

TIl

[h

(vP

+

ｾｖｉＩ＠

-

h

(vf)]

(IR)

(CT)

where the last constraint is the first order condition of the change of treat-ment (CT)o

Lemma 4 For each

s,

there exist a solution for the program aboveo M ore-over, the I R constraint is binding at the optimal contract.

Proof. Since the doctor's utility is bounded from below (see (H4)), it is sufficient to show that the set of feasible contracts is not emptyo Observe that we can easily construct a contract that satisfies (CT) o We can also find ｟ｾｶｂ＠ sufficiently large such that

(IC

_{I) is trueo By} (H4) we can choose

vp

to satisfy (I R) o Finally, we claim that the (I R) constraint is binding at the optimal contracto If it is not the case, we can reduce

v{f

and

vf

by t,

keeping ｾｖｪ＠ constant and without breaking the (ICI ) and (CT) constraints.

This reduces the cost of implementing the change of signal at

s,

what is a contradiction. The proof is then complete. _

.0

IToôvo

+

ITIÔVI - V - c

+

vf - aI ÔvB -II (ao,

s)

Ôvo

+

II (aI,

s)

ÔVI

+

ÔvB

This implies that ÔVi is affine function of (vff, vf) o

c (lCI)

- O (lR) O (CT)

Lemma 5 Let J..l be the multiplier associated with the (lC_{I )} constrainto

Then, at any solution

s

of the above problem (ICI ) is binding and J..l

> 00

The proof is provided in the appendixo

Therefore, our problem is to find the solution of the first order conditions associated with the above problem, which consists of a non linear system with five equations and variableso

Recall we have defined the function T : [IIOI (1), IIOI (O)] -+ [0,1] satisfies

which gives the trade off between the probability of the good state of nature given treatment and no treatmento Also,

Remark 6 Observe that the restriction (CT) in the above problem

-ÔvB - II (ao,

s)

Ôvo

+

II (aI,

s)

ÔVI ;? O

is satisfied if and only if

where

Since the graphics of T is an increasing convex curve, the hyperplane H crosses the graphics of T twice at mosto

Remark 7 By the Maximum Theorem cl (s) is a continuous function of

s

o

Lemma 8 In the cost of implementing program, the optimal contract in-duces just one change of treatment: no treatment below

s

and treatment above

s

o

..

The lemma is proved in the appendix.

Under the assumption that follows, we can show that the optimal con-tract between the insurance company and the doctor pays more in the good state of nature, i.e., f).Vj 2': O, j

=

0,1.

(H5)

11

8, 8 E [0,1] is such that

TI (ao,

8)

=

7r(ao,

i)

then

For instance, it is easy to check that assumption (H5) holds if there exist p E (0,1) such that TI (aj,s)

=

7r(aj,ps), for all s E [0,1] and j = 0,1.

Proposition 9 For each 8 E [0,1], the optimal contract that implements 8 is such that f).Vj 2': O, j

=

O, 1.

Proof. From the lemma above, f).vo 2':

°

is the only possibility at the optimal contract that implements a signal 8. Now, suppose that f).vl

< O.

Consider the following path:

t>

°

--+ (vp - t, f). vl

+

ｉｉＨ｡ｾＬｳＩＩ＠

Along this path the following properties hold:

(1) (CT) is satisfied by the definition of the path.

(2) By assumption (H2),

1 - 8 - (TI (aI, 8))-1 11 TI(al, s)ds

>

°

Thus (I R) is satisfied for t sufficiently small.

(3) By assumption (H5),

{I

{I

1 - 8 - (TI (aI, 8))-1 Js TI(al, s)ds

>

1 -

i -

(TI (aI, 8))-1

_ｊｾ＠

7r(al, s)ds

- - -*

where 8 is the change of signal given no effort (observe that 8 2': 8 , where

-*

..

,.

Taking the derivative of the objective function of cost of implementing program with respect to

t

(along the path) and evaluating at

t

=

°

we get:

because 0<

dh(vf)

<

dh (vf),

when ÂV1 <

o.

This is a contradiction with

the optimality of the contract. _

3.4 Leading examples:

Let us consider two important cases:

Case 1: Doctors are risk neutral

Assume that doctor utility function v(x) = x. It is easy to see that the

cost of implementing the change of treatment at 8 with effort is

By assumption H2, the best way to implement this change is using the partition: So

=

[0,8] and SI

=

[8,1] .

Case 2: There exist p E (0,1) such that II (aj,s) = 7r(aj,ps) , for all s E [0,1].

This means that there exists an uniform diagnose lag between effort and no effort, i.e., for each signal and treatment, the probability of the good state given the doctor's is equivalent to the one given no effort but for a signal p times less (and, therefore, a more favorable signal). It is immediate

from (CT) that oP = pa.1 and by a simple variable change in the integral we have 7rj

=

pllj, for j

=

0,1. From the (leI) and (IR),

v -

c -

vf

=

cj

(1 -

p)

Therefore, our problem is to find the solution of the following equations:

.'

B -

-Ｍｾｶ＠

+

ｉｉｯｾｶｯ＠

+

ｉｉｬｾｖｬ＠ =

c/

(1 - p)

｟ｾｖｂ＠ ＭｉｉＨｳＬ｡ｯＩｾｶｯＫｉｉＨｳＬ｡ｬＩｾｖｬ＠ = O

B - P

v = v---c

1 1-P

dh

ＨｶｾＩ＠

+

ｾ＠

{

(1 -

ＨｬＭｾ｜Ｉｾ＠

) dh (vff) - dh

ＨｶｾＩ＠

+

jL

ｰＨＱｅｾｩＩｾ＠

} - jL

(/--$)

=

dh (vf)

+

ｬｾｾｩ＠

{

(1-

ｑｾｾＩ＠

dh(vf) - dh (vf)

+

ｪｌｰｾｬｑｾ＠

}

After determine the solution, we can compute the cost function c1(s)

and solve the following problem

max el

+

(e2 -

ed

C1(s) - c1(s) sE[O,l)

to determine the second best contract with effort.

4

Existence of equilibrium and welfare

Definition 10 An equilibrium is a collection of strategies {(ri, di)J where

for each i the contract (ri, di) solves the problem

maxLi _{(r, d)}

subject to U (r, d) ｾ＠ U (rk, dk) , k =1= i

U(r,d) ｾ＠ U(O,O)

r E R

.'

..

at equilibrium doctors receive a fixed payment independently of the treat-ment chosen, and which provides them the reservation utility. Insurance companies, due to competition, make zero profits and the optimal signal is chosen to maximize the households welfare. We refer to this outcome as household-efficient.

In our model, on the other hand, in some cases insurance companies have to induce doctors to choose a high effort leveI by offering more rewards in the case the good state of nature happens to occur. These incentives induce doctors to choose the most expensive treatment when it maximizes the probability of the good state of nature. However, this behavior may not be optimal from the household perspective since she also takes into account its effect on the expected cost of treatment and, therefore, the expected cost of insurance. Therefore, the optimaI contract may also have to provide incentives for doctors to choose the Ieast expensive treatment in some cases by reducing their payment in case they choose the more expensive, and more effective from the health perspective, treatment.

We will show later, however, that the market outcome is always at least second best, in the sense provided below. Let RO denote the set of incentive compatible contracts which can be contingent upon both the effort leveI and signal observed by the doctor, and let RS be the set of incentive compatible contracts which can be contingent only upon the signal observed. The set

RO correspond to the mo deI with perfect, symmetric, information, and RS correspond to the standard moral hazard mode!.

Definition 11 We say that an equilibrium {(ri,

di)J

is first best if there is no other contract (r, d) satisfying

r E RO, ｌＨｾＬ＠ d)

2:

O and U (r, d)

>

max U (ri, di)

z

We say that an equilibrium {(ri,di)i} is second best if there is no other contract (r, d) satisfying

r E RS

, L(r, d)

2:

O and U (r, d)

> max

U (ri, di)

z

Finally, we say that an equilibrium {(ri,

ddJ

is third best, or constrained optimal, if there is no other contract (r, d) satisfying

r E R, L(r, d)

2:

O and U (r, d)

>

max U (ri, di)

z

Proposition 12 Every equilibrium is constrained optimal. lf doctors choose

effort then the outcome is not second best.

.'

Proof. This proposition folIows immediately from the continuity of the insurance companies problem and the assumption that I

>

1. Suppose there is an equilibrium which is not second best. Without loss of gener-ality, suppose the household is buying a contract from insurance company 1 while insurance company 2 is offering an alternative contract which pro-vides the same or less indirect utility for the household. Since the equilib-rium is not second best, there is an alternative contract, {dE, -de}, that

strictly increases the household welfare and satisfies the incentive restric-tions. By charging {dE

+

é, -

(de

+

é)} ,

where

é >

O is smalI enough, this contract still strictly increases the household welfare and provides strictly positive profits. Since company 2 was making zero profits in trading with this household, it strictly prefer to offer this contract then the one it was offering originally. Therefore, the proposed set of strategies was not an equilibrium, which is the desired resulto That any equilibrium where doc-tors choose effort is not first best follows from the standard moral hazard mo de!. Moreover, as we saw in the last section, in this case we must have flvj

#

O for some j. The result then folIows from the strict risk aversion of the doctors. _

Proposition 13 There is an equilibrium.

Proof. The set R is trivially bounded and closed. Moreover, it always contains, at least, the nulI contract. Therefore, there is at least one second best contract. It is simple to verify that this contract must provides zero profits. Consider the folIowing strategy profile for the insurance compa-nies. AlI companies offer the this second best contract, and households buy only the contract from company 1. The existence of a profitable deviation for any company would violate the very definition of second best contract. Therefore, the proposed profile is an equilibrium. _

5

The dynamic model

To be written.

6

Appendix

6.1 Proof of lemma 5

.'

TI

ô/1vo

TI

Ô/1V1 o-ô B

+

1 -Ô B

V_o V_o

( _ ô/1vo (_) Ô/1V1

-TI

ao,

s)

--B-

+

TI

aI,

s

--B-ôVo ôVo

=

1

Analogous expression can be obtained taking the derivative with respect to vf (the only difference is that the constant of the linear system will be

-(I-aI) and -1, respectively. Define the determinant of coefficient matrix:

By Cramer rule,

-

-/1

:= TIoTI

(aI,

s)

+

TI1TI

(ao,

s)

>

O

= -

(TIl

+

aI

TI

(aI,

s) )

//1,

(TIl -

(1-

aI)

TI

(aI,

s))

//1

(TIo - a

1_TI

_(ao,

s))

_//1,

= -

(TIo -

(1 -

a

1_)TI

_(ao,

s))

_//1

Thus, the first order condition of the program above iS:5

aI dh

ＨｶｾＩ＠

+

TIo {

｛ｾｾｾｑ＠

+

1]

dh( v{f) - dh

ＨｶｾＩ＠

}

+

TIl

dh(

vf)

ｾｾｾｑ＠

-fL

[(aI

-

aO)

+

(TIo

ＭＷｲｯＩￔ￴ｾｾｑ＠

+

(TIl

ＭＷｲｬＩￔ￴ｾ＿｝＠

= O

(O)

(1 -

aI)

dh (vf)

+

TIl {

｛ￔ￴ｾｲ＠

+

1]

dh(

vf) -

dh (vf) }

+

TIodh

ＨｶｾＩ＠

ￔ￴ｾｦｑ＠

[

1 o - - ôt:;. - - ôt:;. ] -fL -(a - a )

+

(TIo - 1l"0)

ôvfº

+

(TIl - 1l"1)

ôvfl

=

O

5Let s' be the signal where the change of treatment occurs given no effort. By the Envelope Theorem, the devivative of the lagrangian through s' is nul!.

18

.'

Adding up the two equations, we have that

Multiplying by (1 - aI) and aI equations(O) and (1), respectively, and subtracting one from the other, we have

dh

ＨｶｾＩ＠

+

ｾ＠

{

(1 -

ＨＱＭｾＱＩ｢Ｌ＠

) dh

(vr?)

-

dh

ＨｶｾＩ＠

+

/-l (I-:\)b, } - /-l

a?tl-=-:oi)

ｾ＠

dh

(vr)

+

ｬｾｾｩ＠

{

(1 -

｡ｾＧＩＮ＠

)

dh(vf)

-

dh (vf)

+

i'

J'). }

(3)

If /-l = 0, then from (j) we conclude that

vf

=

v!'

Le., !::l.Vj = 0, what

is a contradiction with the (Iel ) constraint. Thus /-l> O. The proof is then

complete.

6.2 Proof of lemma 8

By assumption (H3), Tis an increasing convex function. Assume that Tis

strictly convex.6 _{First, observe that (H3) implies that there are two changes}

of treatment at mosto Then, define a function cp : [0,1] - t [0,1] such that

cp(s) is the other signal'where the treatment changes (besides s), when it exists, or 1, otherwise. This function has the following properties:

1) cp is continuous.

By the Maximum Theorem (see Berge (1959, pp. 115-117)), the (unique) optimal solution of the cost of implementing program is a continuous func-tion of S. Since cp(s) is the other parameter where the hyperplane defined in

the remark above intercepts the graphics of T, and both vary continuously

in s, we get the continuity of cp.

6 We can easyly extend the proof for the case where T is just a convex function. The

only modification is that 'P wilJ be an upper hemi-continuous correspondence.

.'

•.

..

1. (2) <.p is non-increasing and <.p2(s) = s, for all s

E [O,lJ

such that <.p(s)

<

Since the solution of the cost of implementing program is unique, the so-lution in s and in <.p(s) should be the same when <.p(s)

<

1. Thus, <.p(<.p(s))

=

S. We claim that <.p is an injective function on { s E

[O,

1

J;

<.p(s)

< I}: given

Si E [0,1] with <.p(Si)

<

1, i

=

1,2, such that <.p(SI)

=

<.p(S2), we have that SI

=

<.p2(sI}

=

<.p2(S2)

=

S2. The continuity of <.p implies that <.p is non-increasing on {s E

[O,

1]; <.p(s)

<

I}.

(3) <.p

==

1 (what concludes the first part of the proof!)

Otherwise, there exist So E (0,1) fixed point of <.p. In this case, sois the parameter where the hyperplane defined in the remark above (for s = so)

and the graphics of T are tangent. This means that there is no change of treatment at so, i.e., this is the solution of cost of implementing program with no treatment always or treatment always. However, this is a contradic-tion with the uniqueness and continuity of the solucontradic-tion, because there exist a neighborhood of S

=

°

or S

=

1 such that for all S in this neighborhood, the solution at

s

is far from the solution at so(here we are using the strict convexity of <.p).

By the continuity, for s dose to

°

(or 1) we have that the optimal contract consists in no treatment for signal below s and in treatment for signal above S. By the first part of the lemma, this property must hold for all s (otherwise, by the continuity, there exist some signal s such that

<.p(s)

<

1).

7

References

Andrade, M. and M. B. Lisboa (1999): "Seguro privado de saúde: lições do caso americano"; EPGE-FGV.

Berge, C. (1959): Topological Spaces; english translation 1963 from the french second edition (1962), republished by Dover, 1997.

Lisboa, M. B. (1999): "On the cost of altruism: an alternative model for insurance markets"; EPGE-FGV.

Rothschild, M. and J. Stiglitz (1976): "Equilibrium in competitive insurance markets"; Quarterly Journal of Economics, 90: 629-650.

Zweifel, P. and F. Breyer (1997): Health Economics, Oxford

Uni-versity Press.

FUNDAÇÃO GETULIO VARGAS

BIBLIOTECA

ESTE VOLUME DEVE SER DEVOLVIDO À BIBLIOTECA NA ÚLTIMA DATA MARCADA

N.Cham. P/EPGE SPE L 7690

Autor: Lisboa, Marcos de Barros. Título: On endogenous cost of altruism.

1 I11111111111111111111111111111111111111

FGV - BMHS

303128 X3S12

N" Pat.:ABFS7

ｾ＠... ｟ｾ＠ ... _ ... _._ ...

-

... _ ..

-

...

-

" ..

000303128