On the cost of sympathy : a alternative model of insurance markets

' " #,"FUNDAÇÃO ' " GETULIO VARGAS EPGE

Escola de Pós-Graduação em Economia

"On the Cost of Sylllpathy:

An

Alternative

Model for Insurance Markets"

Prof. Marcos B. Lisboa

(EPGEIFGV)

LOCAL

Fundação Getulio Vargas

Praia de Botafogo, 190 - 10° andar - Auditório

DATA

24/06/99 (53 feira) HORÁRIO

On the Cost of Sympathy:

An

Alternative Model for Insurance

Markets*

Marcos B. Lisboa

t

June 24, 1999

Abstract

The paper provides an alternative model for insurance market with three types of agents: households, providers of a service and insurance compa-nies. Households have uncertainty about future leveIs of income. Providers, if hired by a household, 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. The paper assumes that providers care about their income and also about the likelihood households will obtain the good state of nature (sympathy assumption). This

assump-tion is satisfied if, for example, they care about their reputaassump-tion or if there are possible litigation costs in case they do not use the appropriate proce-dure. Finally, insurance companies offer contracts to both providers and households. The paper provides sufficient conditions for the existence of equilibrium and shows that the sympathy assumption 1eads to a 10ss of welfare for the households due to the need to incentive providers to choose the least expensive treatment.

*To my father, Rodolpho Paulo Rocco.

1.

Introduction

The theoreticalliterature on health economics usually relies on the adverse sele c-tion model to explain the apparent inefficiency observed in the provision of health insurance. In this class of model, there are several types of agents, or patients, and risk-neutral insurance companies. Patients face individual uncertainty, are strictly risk-averse and have a private information about their own risk type, which is not known by the insurance companies.

There would be obvious gains from trade to be exploited case the information about patients' risk types were common knowledge: Insurance companies would be willing to provide insurance at actuarially fair prices, which would strictly improve each patient's welfare. However, due to private nature of information, insurance companies cannot distinguish between patients of different risk types. One can classify the equilibria in this class of model as separating, where different

types of agents accept different contracts, and pooling, when at least two distinct

types of agents accept the same insurance contract.

Rothschild and Stiglitz (1976) showed that in this adverse selection mo deI an equilibrium may not existo Moreover, equilibria, when they exist, are necessarily separating. The low highest risk type essentially receives the same contract he would receive in the absence of asymmetry of information. The lower risk types, on the other hand, receives worse contracts, from their welfare perspective, than they would receive in the absence of asymmetry of information, in order to prevent the higher risk types to misrepresent themselves.

The adverse selection model, however, is not capable of addressing many ques-tions which are in the core of recent debates on health economics. First, the basic featur:e of model is the fact that patients know more about their health than insurance companies do. However, due to pre-existing contingencies insurance companies are able to avoid this risk. Second, if some agent has a better informa-tion in the health sector is the doctor who examines a patient. However, doctors are surprisingly absent from the adverse selection mo deIs used in this ｡ｮ｡ｬｹｳｾｳＮ＠

This absence is even more surprising since is precisely the change in the contracts between doctors are insurance companies that lies in the core of the recent changes in the health sector, particularly the development of HMOs:1

A central aspect of these changes is a nsk-sharing contract between the insurance company and the primary care physician of the patient's expected costs of treatment. The existing

1 Andrade e Lisboa (1999) review the recent evolution of the american health insurance

adverse selection mo deIs are unable to address the motivations for this change or to study their welfare implications.

The paper provides an alternative model for insurance market, in particular health insurance. 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, which is observed only by the doctor. For each signal, there is a treatment 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 (sympathy assumption). This assumption is satisfied if, for example,

doctors care about their reputation or, in case they do not use the appropriate treatment, there is a probabi1ity of litigation (defensive medicine). There are risk neutra1 insurance companies that offer contracts to both the doctors and the patients. We show that the sympathy assumption leads to a 10ss of efficiency in the market outcome. However, given this asymmetry of information, this outcome is constrained optimal, or second best. The optimal contract in this framework 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.

The basic structure of the model seems to apply to any provision of any spe-cia1ized service in which the proposed service depends upon a diagnosis performed by the provider himse1f. Households may benefit from buying an insurance against the necessity of having to buy such a service. The provider of the insurance, on the other hand, has to design a contract with the provider taking into account that he has a privi1eged information about the service that may be needed. Moreover, due to gains due to reputation, the provider of the service may have incentives, in the presence of a contract that pays for service performed, to a1ways choose the service that minimizes the probability that a new problem may happen again, independently of the cost of such service. Despite these possib1e applications of

2. The basic model

2.1. Uncertainty, patients and doctors

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

-ex-ante, interim, ex-post - and three types of individuaIs: patients, doctors and insumnce 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 endowment in the first period. We refer to the patient endowment in state s by eS

• Her preferences over consumption

bundles is represented by a state independent utility function u : lR+ -+ ｾＬ＠ strictly

increasing and concave in the differentiable sense:

(Hl) u E C2 and satisfies: i) du

>

O ; ii) d2u

<

O. Moreover,

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 con-tract is not much higher than the actuarially fair price. The standard insurance literature assumes the existence of a finite collection of risk neutrals insurance companies, who offer insurance contracts simultaneously and independently. Un-der these assumptions, the outcome of the model with at least two companies is that they offer actuarially fair contracts, which are accepted by the patients.

The model proposed here departs from the standard literature by assuming 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 [0,1]. The probability of a signal s is described by the cumulative probability function F (8) E CI, dF(8)

>

O for every s E (0,1).

There are 2 types of actions available for the doctor

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

this signal, the probability of the good state of nature for a patient depend upon the action chosen and it is given by 7r (a, s) for every a. We assume:

(H2) For each a the function 7r (a,·) E C2, for every s E (0,1)

d7r (aI, s)

>

d7r (aO, s)

and

0< 7r(aI,O) <7r(aO,O) <1

0< 7r (aO, 1) <7r(a1,1) <1

Therefore, there is a signal s* E (0,1) such that both actions generate the same

probability of the good state of nature

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.

We assume that doctors care about their income and about choosing the best action for the patients, given the signal they observe. This second assumption, which we refer to as sympathy assumption, is the distinctive assumption of our

mo dei and it implies the following resulto Suppose the doctor income does not de-pend upon the action chosen. Then he always choose the best action. N otice that the term best here has no economic meaning: the best treatment can actually be very costly and patients may prefer an alternative treatment, which is less effective but cheaper. This result is trivially consistent with the stylized facts from health

insurance markets, in particular the overuse, from the economic perspective, of exams and treatments the system if patients are provided a full insurance system. The sympathy assumption can be derived in a dynamic game where the his-" torical rate of patients achieving the good state of nature is used by the doctor as a signal either to differentiate himself from different types of doctors or to show his commitment in choosing the appropriate action for a certain class of signals. Alternatively, one can generate the sympathy assumption by allowing the possi-bility of doctors being sued in case they do not use the best treatment (defensive medicine) In order to keep the model simple, we focus on the static game where

More formally, if the doctor has income r in the ex-post period, observes signal

s and chooses action a, his utility is given by

v ClT" (a, s) -

ｭ｡ｾｸ＠

7r (a' , s) , r )

where v : [-1, O] x ｾＫ＠ ｾ＠ ｾ＠ is strictIy increasing and strictIy concave in the dif-ferentiabIe sense:

(H3) v E C2 and satisfies: i) dv

»

O ; ii) d2v is negative definite; and iii)

Iim v (7r (a, s) - max 7r (a' , s)

,Y)

= - 00

y-,O a'

A doctor has a reservation utiIity denoted by 'Ü and there is r such that

v (O, r)

=

'Ü. Let _ｾ＠ 7r (a, s) : = 7r (a, s) - maxa, 7r ( ai, s) .

There are J

>

1 principaIs, or insurance companies, that provide insurance 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. The doctor's contract specify how much the principal will pay for the doctor 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.

The patient's contract specify how much she pays for the insurance company in each individual state of nature and the type of contract the insurance has established with the doctor. Indeed, in choosing the contract that maximizes her expected utility, the patient takes into account that the doctor's choice of action does depend on his contract with the insurance company.

Suppose a patient has accepted a particular principal offer in the first period. In the second period the patient meets the doctor who is the only one who observes the private signal s. Given this signal and the contract the doctor has with the

principal he chooses an action. Both the principal and the patients correctly anticipates how the doctor behaves for each possible signal for each given contract established with the principaI.2

The timing of the model is as follows. The principaIs simultaneously offer contracts to the patients and doctors. Household and doctor choose to accept the

contracts that maximizes their respective expected utility. The interim period come and the doctor observes the private signal and choose the action. Household uncertainty is resolved and contracts are enforced.

2.2. The optimal contract

We start the analysis of the model investigating the insurance companies' offers to the doctor. Let ri be the payment the principal makes to the doctor if he chooses action ai. In what follows we restrict the analysis to the case where ri :::; rO.3

Suppose the doctor has accepted the offer and has observed signal s. He, then,

solves the problem

This problem has at least one solution. Moreover, from the monotonicity of the probability and utility function, there is a criticaI signal s(r)

>

O such that for all s

<

s(r) the doctor chooses ao and for all s

>

s(r) the doctor chooses ai' At s = s(r) the doctor is indifferent between the two actions, provided that s(r)

<

l.

Therefore, the doctor problem reduces to choosing the criticaI signaI s(r). Let

More formally, for each given contract r the doctor chooses the criticaI signal s(r)

to solve the problem

ｭｾｸ＠ V (rO, ri,

s)

s

The set R of contracts for the doctor that satisfy participation constraint can be described as follows

where fj is the doctor reservation utility.

3Suppose a principal offer rO _{< r1• In this case, the doctor, if have accepted the principal's}

Since rO

2

rI, the doctor chooses s(r)

2

s*

>

O. The optimal solution of the doctor problem must satisfy the following first order equations

v (671' (8, aO) ,rO)

ｾｾ＠

(8) - v (O, rI)

ｾｾ＠

(8) - /-lI = O

min {/-lI' 1 - 8} = O

To simplify the notation, and prevent the doctor from choosing s(r) = 1, we

impose the following simplifying assumption:

(H3) lim v (71' (1, aO) ,r)

<

fi.

r-+oo

Therefore, the solution of the doctor problem for any contract in the set Ris

completely characterized by the following equation

since dF(8)

>

O for every 8 by assumption. Moreover, an easy application of the implicit function theorem shows that this equation has a uni que solution

8(r) E

e

l . Let

s I

G(8(r)):=

J

1I'(s,aO)dF(s)+

J

1I'(s,al)dF(s)

°

s

be the probability of the good state of nature given the contract offered which results in the choice of 8.

Suppose insurance company j offers a contract rj for the doctor, rj E R, and that the patient pays

d;

for the insurance company in the patients good state of nature and ｴｨｾｬｴ＠ the insurance company pays

dj

to the patient in the bad state of nature. The company j's payoff is then given by

rr

(rj, dj) := G (8 (rj)) d;-(1 - G (8 (rj))) d}-F (8 (rj)) rJ-(1 - F (8 (rj))) (r} +al) If the patient accepts this offer her utility is then given by

U (rj, dj ) := (1 - G (rj)) U

(e

1

+

d})

+

G(rj)u (e2 - d;)

Suppose the remaining companies had offered contracts (r j', dj') j'#)' The patient

and

U(rj,dj ) 2:: U(O,O)

since the patient may always reject all offers. Let (ro, do) := (O, O).

Definition 2.1. An equilibrium is a collection of strategies {(rj, dj)j} where for each j the contract (rj, dj) solves the problem

m&,'{ IIj (r, d)

subject to U (r, d) 2:: . max U (rj', di') J=O, ... ,J

r E R

Consider an alternative model which the same primitives, except that doctors do not care for the patients. In this case one can easily verify that at equilibrium doctors receive a fi.xed payment independently of the treatment 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, insurance companies have to induce the doctors to choose a least expensive treatment by increasing their payment in case they choose the cheaper, and less effective, treatment. Therefore, any equilibrium with critical signal

s

<

1 cannot be household-efficient in this modelo

This loss of efficiency is due, precisely, to the fact that doctors care about choosing the best treatment, where best is defined by likelihood of the patient achieving the good state of nature. However, the patient, and the insurance company, also has to incorporate the cost of the treatment, which is not done by the doctors. Thus, the inefficiency, in our model, results from the asymmetry of information by the doctors and the fact that they have incentives to provide the best treatments. We will show, however, that the market outcome is still second best, in the sense provided below.

Definition 2.2. We say that an equilibrium {(rj, dj)j} is second best if there is no other contract (r, d) satisfying

Notice that in our model, as in the standard moral hazard model, the criticaI signal chosen by the doctor can be inferred from the contract offered. This suggest an alternative approach to analyzing the doctor problem, which is summarized in the next lemma.

Lemma 2.3. There is an open and non-empty set S C [8*,1] such that for each

8 E S there is a minimum expected cost, C(8), that the insurance company has

to incur in order to induce the doctor to choose s as the criticai signal. Moreover,

C E Cl for every 8 E S and

C(8) -+ 00 as 8 -+ 1

In the appendix we prove the lemma and provide a characterization of the

function C ( 8 ) .

In order to characterize the optimal contract is convenient to define the

pa-tient's utility leveI associated with refusing the insurance companies contracts which provides doctors service for the patients and just buying an insurance con-tracts that smooth her consumption over the individual states of nature

U(l) := u

(e

lG(l) +

e

2 (1 - G(l)))

We define

C(l) := O

Lemma 2.4. The contract (r*, d*) is the outcome of an equilibrium, with

asso-ciated critical signal 8*, if and only if (d* , s*) solves the following problem

ｾ｡ｸｵ＠ (eI

+

_{dI) (1 -} G(s))

+

_{u (e}2 _{- d}2 ) G(s) s,x

d2G(s) _{- dI (1 -} G(s)) _- C(s) _{:::: O}

el

₊

_dI

_>

_{- O e2 - d}2

>

_O

,

-By the previous lemma, at any equilibrium the patient must have perfectly smooth consumption over the individual states of nature and the budget constraint must be biding

el

+

_dI

and thus the patient's income in any state of nature is given

But then, it follows that the patient's optimal contract must be associated with the criticaI signal that maximizes the above income.

Lemma 2.5. The critical signal

s*

associated with the outcome of an equilibrium (d* , r*) must solve the following programing problem

max el (1 - G(s))

+

e2G(s) - C(s)

sE[O,I]

Proof. Suppose the claim is not true, and therefore there are

s

and Si, and

corresponding (dV , d21 ) and (dI, d2), satisfying

(e2 - el) G(8')

+

C(8')

<

u

(e

l

+

dI) G(8')

+

u (e2 - d2) (1-G(8'))

>

where d2 (1 - G(s)) - dIG(s)

(e2 - el) G(s)

+

C(s)

u

(e

l

+

dI) G(s)

+

u (e2 - d2) (1 - G(s)) C(s)

As before, we can restrict the anaIysis to contracts that leads to state independent income for the patient

(

1 d2(1- G(s)) _ C(s))

e

+

G(s) G(s)

(e

2 _ d2)

(G(s)el

₊

_{d2 (1- G(s)) - C(s)) G(s)}

_(e

2 - d2)

d2 _ (e2 _- eI) G(s)

+

C(s)

and thus

(e2 - el) G(8')

+

C(8')

>

(e2 - eI) G(s)

+

C(s) which is a contradiction. O

Proposition 2.6. Every equilibrium is second best.

This proposition follows immediateIy from Iemma 2.1 and the definition of second best allocation.

Proof. Let

5:= {s E [8*,1)/ V(r,s) = V}

As we show in the appendix, 5 is non-empty and bounded. Moreover, the func-tions G(s) and O(s) are continuous for s E 5. If there is no solution to the above

problem it means that there is a sequence sn ----lo S

ti:

5, {Sn} C 5, such that

(e2 - el) G (sn)

+

O (sn)

>

(e2 - el) G (Sn+l)

+

O (Sn+l)

for every n. By definition of 5, this implies ro (sn) ----lo 00, otherwise, taking a

subsequence if necessary, Sn ----lo S E [0,1], r (sn) ----lo r and thus by continuity

V(sn,r(sn)) ----lo V(s,r)

=v

which implies sE 5, a contradiction. Therefore, ro (sn) ----lo 00, and since rI (r(Sn), Sn) ｾ＠

O for every n, we obtain

which is an absurdo Therefore, the above minimization problem has a solution. O

Proposition 2.8. At an optimal contract r associated with a critical signal S

<

1

we must have ro

>

rI and at any such contract, O(s) is locally a decreasing function of S. Moreover, there is a threshold level of expected income below which households do not hire a doctor.

Proof: From lemma 2.5, at the optimal contract (ro, rI) associated with a

criticaI signal S

< 1 we must have

where

Therefore

(d2 +dl ) dG(s) dO(s)

dG(s)

. =? (e2 - el

) dG(s) = dO(s)

(71" (s,aO) -71" (s,al )) dF(s) : b.71" (s) dF(s)

dG (s)

<

O=? dO(s)

<

O

Moreover, from Lemma 2.5 household will only decide to hire doctors if maxel (1- G(s))

+

e2G(s) - O(s) ｾ＠ el (1-G(l))

+

e2G(1) - 0(1)

>

O

sES

3. Appendix

Fix a signal 8 E (8*, 1) and an doctor's payment in case he chooses action aI, rI'

Consider the equation

V (71" (8, ao) - 71" (8) ,x) - v (O, rI) =

°

It is simple to verify that this equation has a uni que solution for 8 dose to 8*,

denoted by ro (rI, 8). Moreover, since drv ＨｾＷＱＢＬ＠ r)

>

0, by the implicit function theorem ro (.) E C1. Let

S 1

V (rI, 8) :=

J

v

ＨｾＷＱＢ＠

(8, ao) ,ro (rI, 8)) dF (8)

+

J

v (O, rI) dF(8)

o ;s

Consider the set

S := {8 E [8*,1)/ V (r, 8) = v}

Since V (7', 8*) =

v,

where v (7') =

v,

it is simple to verify that S is non-empty.

Moreover, by construction for each 8 E S there is a unique rI (8) satisfying

V (r1(8), 8) =

v

where rI (8) E C1 since

and

dv(O,q)

dq

°

dv(f:r.1!"(s),ro)

>

dTO

Therefore, for every 8 E S there is a neighborhood of 8,

Vs,

such that V;s

n

[8*,1] C S. Moreover, for each 8 E S there is a unique corresponding contract

that induces the doctor to choose the signal 8 and that provides an utility leveI identical to the reservation utility,

r(8) := (ro(r1(8), 8), rI (8))

The expected cost for the insurance company of implementing this contract is given by

4. References

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

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Título: On the cost of sympathy : an alternative model for,

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