Competition in the Physicians’Market: A Search
Theoretical Approach
Février (2009)
Ko¢ A. KPELITSE*
Abstract:
In this paper, I examine whether and to what extent a patient’s search for a competing physician may
encourage e¢ ciency in a cost containment prospective-payment setting. More speci…cally, I examine a twoperiod model where: i) the patient’s illness severity is measured with error by both patients and physicians,
ii) patients, after health care consumption, evaluate their physicians’ diagnosis precision, and may decide
(following a new health shock) to stay with him or to search for an outside physician and potentially switch.
I …nd that the patient’s switching decision depends on his level of ignorance about the illness severity, and
on his social network’s evaluation of the alternative physician. Depending on the extent of these elements,
patients make type I and II errors. I also …nd, under some assumptions that the treatment provided depends
on the physician’s information about the patient’s type. Hence, at the equilibrium, the fear of loosing a
patient in a competitive market does not systematically induce the physician to treat appropriately the
latter.
* Phd candidate, Institut d’économie appliquée, HEC Montréal, 3000, Chemin de la Côte-Sainte-Catherine,
Montréal, H3T 2A7; Tél: 514-340-6451; ko¢ -ahoto.kpelitse@hec.ca
1
1
Introduction
The health care market is characterized by some market imperfections such as information
asymmetry between patients and providers. These market imperfections, the presence of
health insurance and di¤erent physician payment mechanisms can lead to ine¢ cient consumption of health care. Health insurance, which reduces the out-of-pocket costs paid by
consumers, leads to the well-known ex post moral hazard problem. That is, people have
incentives to demand more services than what they would otherwise if they were fully responsible for the costs of their medical treatment.1 The traditional fee-for-service payment
mechanism (a mechanism in which providers are paid for each service they provide) coupled
with the presence of information asymmetry can also lead to over consumption of medical
services and consequently, to excessive health-care costs.2 In this context, it has been shown
that only contracts or payments based on health states are e¢ cient (Arrow 1963). However, because such types of contracts are infeasible because health states are unveri…able
(or, at least are costly to identify), several solutions have been proposed to overcome these
phenomena.
Among these solutions are demand-side and supply-side mechanisms. On the supply side,
prospective payment mechanisms such as capitation (payment system in which physicians
receive a …xed payment for each patient they enlist into their practice) may be an e¤ective way
to deal with excessive provision of health-care services and rising costs. They are, however,
often associated (at least theoretically) with the under provision of health care.3 However, the
theoretical models of capitation generally ignore the e¤ect of competition between providers
1
Although there are two types of moral hazard problem, in this paper we are only concerned with the ex
post moral hazard.
2
Due to his informational advantage, the physician can in‡uence the patient’s demand by inducing ine¢ cient treatment. This is termed in the literature as “physician-induced-demand”.
3
In a capitation system, physicians receive a …xed fee for each patient they enrol. Since providing more
care to the patient is costly and thus decreases the net bene…t per patient, physicians have incentives to
reduce care.
2
which can encourage the e¢ cient provision of health care. More speci…cally, the fact that
patients can leave their current physician if they are unsatis…ed with the level of care they
receive, may reduce physicians’willingness to underprovide medical services to their patients
in the presence of prospective payment mechanism. In light of this critique, the aim of this
paper is to examine whether and to what extent a patient’s search for competing physician
may encourage e¢ ciency in a cost containment prospective-payment setting.
Although demand-side mechanisms have been put forth to decrease the ex post moral
hazard problem, their inconvenience is that patient generally bear a greater risk of illness and
monetary loss (Ellis and McGuire, 1993).4 On the supply side, capitation, a pure prospective
payment system in which the physician’s revenue is independent of the services he provides,
is often used. Because physicians are often the key decision makers in most of health care
resources’utilization, it has been argued that supply-side cost control mechanisms are likely to
be more e¢ cient than those on the demand side. First, as noted by Ellis and McGuire (1993),
providers have more information about risks and bene…ts of health services utilization than
patients and are better able to make appropriate tradeo¤s. Moreover, supply-side cost sharing
can reduce utilization without shifting costs to patients. However, as noted previously, the
principal inconvenient of the capitation system is the well- known problem of under provision
of care, both in quantity and quality.
5
In fact, since the level of treatment provided is not
related to the payment, physicians may have strong incentives to reduce services’utilization.
A few papers in the literature suggest di¤erent mechanisms to achieve e¢ cient delivery
of health care under a prepaid system. These include monitoring by third parties (Léger,
2000), liability for medical malpractice (Danzon, 2000) and competition between physicians
for patients (Rochaix, 1989; Ma and McGuire, 1997; Allard, Léger and Rochaix, henceforth
4
Demand-side rules are in the form of deductibles, co-payments, limits on coverage and try to make patient
responsible of part of his services utilization.
5
Another well-known problem of capitation system is the selection problem; that is physician can select
patients based on their illness severity and avoid costly cases.
3
ALR, 2008). It is important to note that, in the presence of altruism or ethical concerns,
physicians may also provide care appropriately to the patient without any regulation (Ma
and McGuire, 1997; Jack, 2006; ALR, 2008).
In this paper, I also examine the role of competition between providers but in a patient’s
search framework. More speci…cally, I present a model which examines the patient-physician
relationship. In the model, I assume that physicians who are di¤erentiated by their precision in treatment’s recommendation, estimate patient’s illness severity and choose a level
of treatment. This model is particularly well suited to the relationship between patients
with chronical illnesses and their specialists. Following health care consumption and posttreatment health state, patients decide either to continue to seek care from their current
physician or switch to an alternative one. Unlike ALR (2008), the present work focuses on
a search framework which includes the elements of search theory. In fact, allowing patients
who choose to leave, to search for the outside physician may help to explain some stylized
facts which cannot be described in the traditional models without aditional assumptions.
Initially, in the …rst version of the model, I assume that physicians’diagnosis skills are
exogeneous and …xed over time and that they always provide the appropriate treatment
given their diagnosis. I …nd that the patient’s switching decision depends on his level of
ignorance about the illness severity, and on his social network’s evaluation of the alternative
physician. Hence, depending on the extent of theses two elements, some patients stay with
their physicians even if physicians’diagnosis skills are relatively low (and consequently receive
a lower quality of care) and others may leave physicians with higher diagnosis skills to poorer
ones. I also …nd that even a perfectly informed patient may make type I and II errors.
The model provides thus predictions which are consistent with the evidence in the medical
literature (Rossister et al., 1989; Schlessinger et al., 1999) where patients continue to seek care
from a physician they believe is incompetent. Although ALR (2008) also found some unstable
4
patient-physician relationships, this result cannot be possible without their assumption of
the presence of uncertainty between treatment and health outcome. I also …nd that the
heterogeneity in the ignorance about the illness severity leads to heterogeneity in the search
intensity. In the second version of the model, I allow physicians to act strategically. In
this setting, in contrast with the previous works, I …nd that the fear of loosing a patient
in a competitive market does not systematically induce the physician to treat appropriately
the latter. Hence, one may observe that even if physicians do not treat appropriately their
patients, they can however keep them.
This paper is also related to some work in the literature on competition (in non-price
dimension) in the physicians’ market (Dranove 1988, Rochaix 1989, ALR 2008). These
authors analyze the e¤ect of competition between physicians in a setting where physicians can
use their informational advantage to in‡uence the provision of health care. The general results
are that including competition can help to reach the second best and that the negative e¤ect
of capitation or prospective payment is attenuated. This may explain why the capitation
system is still widely used even though it has theoretically predicted to have negative e¤ects
on health quality.6
In Dranove (1988), physicians compete on the level of "inducement of the demand".
If a physician is too aggressive in his treatment’s recommendations, patients can leave his
practice and since a physician’s income depends on the number of patients who agree to
receive the treatment, patient’s willingness to switch can provide a disincentive for inducement behaviour. Dranove however remains silent on how patients choose their alternative
providers. In Rochaix (1989), a patient’s threat to seek a second opinion in competitive
market serves as a monitoring technique for physicians, and this threat induces physicians to
choose a level of treatment which is close to what their patients would choose if they were
6
In an international comparison of doctors’ remuneration in public health systems in OECD countries
(except Mexico and the United States), Esmail and walker (2005) observe that twelve countries …nance at
least partly their general practitioners care under a capitation system.
5
well informed. Rochaix deals with the patient’s outside options but she supposes a given
exogenous level of utility in the case where the patient doesn’t agree with his current physician treatment’s recommendations. So the patient’s outside option in Rochaix (1989) is not
modelled explicitly. Although the two previous models generate interesting results about the
e¤ect of competition in physicians market on treatment recommendations, the fundamental assumption in these models, namely the patient’s decision to agree with the treatment
is based on the gap between his expectation about the illness severity and his physician’s
message, is somewhat questionable. Even if medical services are classi…ed as credence goods,
patients can (partially) evaluate their physicians’quality only after health-care services are
consumed.7 Consequently, this limitation is taken into account in the model proposed here
by basing patient’s switching decision on the di¤erence between the expected and the current
treatment.
ALR (2008) study the repeated interaction between patients and physicians in a theoretical
model. In their model, physicians are di¤erentiated by an unobserved individual-speci…c
ethical constraint which speci…es the minimal amount of e¤ort he is willing to provide to
the patient. They show that under some assumptions the equilibrium is characterized by
physicians supplying optimal e¤ort although this input (e¤ort) is not contractible. Although
their formulation of patient’s alternative choice is novel compared to the previous models,
it may not be appropriate to describe stylized facts. In fact, in the real world the e¤ect
of “social network” is important.8 Gourash (1978) in a literature review on people’s helpseeking behaviour concludes that, indeed, at least 75% of individuals are in‡uenced by family
members, friends or co-workers in their needs of medical goods.
The remainder of this paper is organized as follows. Section 2 presents the model. In
7
Darby and Karni (1973) de…ne credence good to be any good for which quality can neither be assess in
normal use nor after its acquisition.
8
Social network is described in the literature as individuals or groups with whom a particular individual
is in contact.
6
section 3, I derive the main results of the model when physicians’ type are assumed to be
exogenous and …xed. In section 4 I look for the role of competition between physicians. The
last section contains some concluding remarks.
2
The model
In this section, I present a model of the relationship between patients, physicians and an
insurance company. In the model, the insurer collects premiums and co-payments, and pays
physicians on a capitated basis. I assume that the insurer signs a contract with physicians and
patients before the patient’s illness is revealed. In the …rst period, if an individual becomes
ill, he seeks care from the physician. I …rst assume that physicians di¤er in their diagnosis
precision (as in Jelovac 2001; Allard, Jelovac and Léger, 2007) and this precision is also
assumed to be …xed for each physician. I relax this assumption later by allowing physician to
exert e¤ort in order to improve his diagnosis precision. The quantity of treatment provided by
the physician is assumed to be observable by both the patient and insurer. After health-care
services are consumed, the patient observes his post-treatment health.
In the second period, conditionning on receiving a new health shock which is drawn from
the same illness severity distribution as the …rst, the patient must decide whether to continue
to seek care from the same physician or to search for a new one.9 It is assumed that if a
patient does not receive a new health shock, there is no need for care, and consequently no
search.10 Patient’s switching decision serves as the basis of competition in physicians’market.
More speci…cally, competition (between physicians) is introduced in the model by adopting
9
Even if the new health shock is assumed to be drawn from the same illness distribution, the level of
severity may di¤er. That is, I do not impose any restriction on the relationship between the …rst and the
second health shock.
10
This may not be true in a multi-period setting because a risk averse patient may search as a form of
precaution.
7
the same formulation as in ALR (2008): patient and physician have repeated interactions
and in every period the patient can move from one provider to another. However, as noted
above, when a patient decides to switch he will "search" for a new physician instead of being
"randomly matched" to a new one as supposed in the previous work. Also, once a physician is
met, his switching decision depends on the social network’s e¤ect (modeled explicitly below).
I examine, in the …rst version of the model, the simple case where the illness severity
is measured with error by both patients and physicians and where physicians’ diagnosis
precision is exogenous and …xed. In this context, patients, after evaluating their physicians’
types (or equivalently, their physicians’diagnosis precision), decide to stay or to search and
potentially switch. This basic formulation will be extended in the second version of the
model to incorporate competition in the physicians’market. More speci…cally, in the second
version, I suppose that physicians can improve their diagnosis precision by exerting costly
e¤ort.
I describe in the following sections the timing of the model and the agents’preferences.
2.1
Timing
The timing of the game is as follows:
Period 1
Step 1
The insurance company o¤ers contracts to the physicians and patients. These contracts
specify the capitation payment P for the provider and the premium
dividual. The premium
for the insured in-
is assumed to be actuarially fair. I suppose, for simplicity, that
patients have full insurance and thus there is no co-payment.
8
Step 2
With probability , the patient receives a health shock and requires treatment, and with
(1
), he is healthy. If a patient receives a health shock, is drawn from a known distribution
function F ( ) with support [
H
L; H ]
where
L
corresponds to the lowest level of severity and
the highest level. I assume that there is an appropriate (exogenous) level of care t ( )
which is associated with every , and that this is public information.11 I also assume that
the patient, given his symptoms, estimates his illness severity with error as
thus expects to receive t ( p ).
over time. In other words,
health condition: if
under-estimator. Thus
the mean of
p
=
+
and
is a parameter which denotes the patient’s type and is …xed
captures the level of ignorance of each patient about his true
> 0 then the patient is an over-estimator while
< 0 refers to an
can be thought of as an individual-speci…c e¤ect. I assume that
is equal to zero in the population of the patients, and that patients’types are
distributed according to a distribution function T ( ).12
Step 3
The physician, given his medical knowledge and clinical information, observes the patient’s
illness severity imprecisely as
d
. After observing
the appropriate treatment for his evaluation of
d
, the physician is assumed to provide
i.e. t ( d ). For simplicity, I assume that
physician’s types are exogenous (an assumption which will be relaxed in the second version
of the model). It is important to note that, in this version of the model physicians are
assumed to always choose the appropriate treatment conditional on their estimate, i.e., I
assume that there is no gaming on the physician’s side. More speci…cally, I assume that
t ( d ) = t ( ) where
is a …xed and positive parameter which captures the physicians’
11
The major implication of this assumption is that patients and physicians have the same health treatment
function. This is reasonable because the function t is exogeneous and represents the clinically appropriate
treatment.
12
The assumption that E( ) = 0 seems reasonable in our context. Even if some patients, in reality are
hypochondriacs, their over-estimation may be compensated by those who under-estimate their illness severity.
9
unobserved heterogeneity. I assume that 0 <
, where
designs the upper bound
and may be greater than one. The physician types are distributed according to a known
distribution function G( ); and g( ) denotes the density function.
Step 4
Given that the patient can observe t ( d ), he can compare his expected treatment t ( p ) to
his current treatment t ( d ) and form an estimate of his physician’s type.13
Period 2
Step 5
If the patient receives a new health shock of the same illness, he may decide either to stay
with the same physician or to search for another one.14 If the patient is healthy, then the
game ends. Following the new health shock, if the patient decides to search for another
provider, he chooses how intensively to …nd a physician and after that he collects (from his
social network) the information about that particular physician. If the signal received about
that physician’s type is acceptable then the patient switches, otherwise he stays with his
current physician.
Figure 1: the timing
Insurance
Contract
Sick and
Stay
Sick
1
3
4
Health State
Treatment
2
5
Healthy
Sick and
Search
Healthy
13
I assume that the treatment-outcome relationship is certain
Search for another physician is costly and since the second period’s health care consumption is conditionally on receiving a new health shock, patients do not automatically search after the …rst period’s treatment.
14
10
2.2
Patients
As discussed above, patients have only one kind of disease but are heterogeneous relative to
the level of severity of their health shock. Each patient is characterized by a
parameter
which is considered as the patient’s type. Patients’ utility is assumed to depend on two
elements: health state h and income available for consumption goods x; and is increasing
and concave on both elements. The patient’s health state depends negatively on the illness
severity and positively on the quantity of treatment received. After health care consumption,
the patient evaluates his physician diagnostic precision. If the patient is not satis…ed with the
health care received, and following a new health shock, he can search for another physician
who can give him a level of utility at least greater than what he has obtained with the current
physician.
Formally, the patient’s per-period expected utility is:
p
U = (1
where x = y
2.3
)U (x; h(0; 0)) +
Z
U (x; h( ; t ( ))dF ( )
(1)
and y is the individual’s income.
Physician
As noted above, physicians are heterogeneous in terms of their diagnostic precision, i.e.,
their capacity to diagnose accurately the disease and to propose an appropriate treatment
which will …t patient’s expectations. In other words, physicians are di¤erent relative to their
precision in treatment recommendation which is assumed …xed in the …rst version of the
model. I refer to this as the physician’s type . In the model proposed here, each physician
is assumed to provide the appropriate treatment conditional on his diagnosis. As de…ned
11
previously, the treatment chosen by a speci…c physician is a function of the appropriate
treatment for the true level of illness
=
t ( d)
:
t ( )
Since
can be greater than 1, I allow overprovision of health care in the model.
The type parameter
by a physician. Since
performance,
and the physician’s type ; i.e. t ( d ) = t ( ) or
is interpreted as the proportion of the appropriate treatment provided
can be interpreted also as a measure of the degree of physician’s
= 1 corresponds to the case of the provider with perfect diagnosis skills and
who provide the appropriate treatment. A
< 1 ( > 1) corresponds to a physician who
under-treats (over-treats) the patient. The heterogeneity in physicians’market can be seen
as: for a same level of severity, patients may receive di¤erent treatments since the choice of
the treatment depends on the physician’s type.
2.4
Insurance provider
The insurance market is assumed to be perfectly competitive and the insurer, who is assumed
to be a pro…t maximizing …rm, collects premiums and pays physicians on capitated basis. I
assume that the insurer does not restraint patients in their choice of health-care providers.
Since the insurer can observe neither physician type nor patient’s illness severity, writing
contracts based on physician type or illness severity is infeasible (the …rst best is unachievable)
but patients may be able to encourage physicians to provide them with adequate care by
credibly threatening to switch to another health-care provider.
3
Patient’s search behaviour
In this section, I model explicitly the patient’s switching and search behaviour in an imperfect
information (about the illness severity) setting and where the physicians’types are assumed
exogenous and …xed over time.
12
As noted before, I assume that the patient cannot perfectly observe his illness severity
he estimates it as
p
i.e.
: Hence, after health-care consumption, the patient cannot attribute all
the di¤erence between his expected post-treatment health and his current post-treatment
health to the physician (and he is aware of this). In other words, some of the di¤erence
may be attributed to the patient’s incorrect estimation of his illness severity. In this setting,
the patient compares the expected treatment t ( p ) to his current treatment t ( d ) to decide
whether or not to search.
I have assumed that
p
=
+
where the patient’s type
is drawn from a known
distribution T ( ). Hence, the patient’s expected and current treatment can be rewritten as
t ( + ) and t ( ); respectively, and the corresponding expected post-treatment health and
current post-treatment health as h( + ; t ( + )) and h( ; t ( )); respectively. As assumed
above, patients evaluate their physicians’ types by comparing their expected treatment to
their current treatment, i.e., they compare t ( + ) to t ( ). From this comparison the
patient can estimate his current physician’s type. Let denotes by
of his physician’s type, then
may arise. (i) if
EV
= 0, then
=
EV
t ( )
.
t ( + )
=
t ( )
t ( + )
> 1 and hence
EV
i.e., when the patient estimates perfectly the illness
EV
< 0, then t ( ) > t ( + ) or
> ; i.e., when the patient under-estimates the illness
severity, he will over-evaluate his physician’s type.15 (iii) if
as above, we have
the patient’s evaluation
Depending on the values of , three possible cases
severity, he will perfectly infer his physician’s type. (ii) if
equivalently
EV
> 0, using the same rationale
< , i.e., when the patient over-estimates the illness severity, he will
under-evaluate his physician’s type.
For a given
and
d
; the patient’s ex post-treatment’s utility level is:
U p = U x; h( ; t ( d )) = U [x; h( ; t ( ))].
After the care, the patient decides whether to switch or to continue to seek care from his
15
I assume that t is increasing in :
13
current physician if he has another health shock. Furthermore, because I also assume that
physicians’ types are exogenous and the patients’ type is also …xed over time, the patient
knows with certainty that if he receives another health shock, his current physician will treat
him with the same proportion of the appropriate treatment. Thus, the patient knows that if
he stays with the current physician, he will get a level of utility equal to U (x; h( ; t ( ))).
By deciding to switch, the patient searches …rst for an alternative physician and after that
he collects information about that speci…c physician by consulting his social network. The
information collection process and the matching technology are given in more detail below.
3.1
The patient’s search strategy
I start by describing the patient’s search strategy. Much of my description of patient’s search
behaviour is borrowed from the abundant literature on assignment, search and matching
models.16 Although search theory is extensively developed, I restrict my attention to a few
papers which are very close to the present work.
Using standard search framework ( Mortensen(1986), Burdett & Coles(1999) and Wolinsky A.(1987)), I present a single-sided search model where the patients search for a better
physician. The matching technology is described as follows: a patient contacts a physician
according to a Poisson process with a parameter : So the probability that an individual
i will meet a physician during short interval
is
i
. This probability can be raised by
the patient, but raising the meeting probability is costly. Mortensen (1982) termed
i
as
person speci…c search intensity. I follow Mortensen (1982) by assuming that the probability
of a new match formation is determined by the search intensity chosen by the patient. An
interpretation of search intensity in this model is how intensively the patient searches for a
16
Search theory is used to explain the rate of unemployment, wage dispersion among homogeneous workers
(in labor market); price dispersion of homogeneous goods ( in product market); the matching process between
men and women ( in marriage market);...
14
new provider.
The information collection process is de…ned as follows: after …nding an potential alternative physician, a patient contacts a subset of the sample of people in his social network and
ask for detail about the type of that speci…c physician. The information obtained represents
a signal of the type of the latter, and is denoted by ~.17 It is assumed that all patients in this
speci…c subset seek care from that particular physician or have already used the services of
this physician. This assumption, however, does not mean that people in the social network
are homogeneous. In fact, patients with di¤erent kinds of illness may receive services from
a same physician and, independently of their illness, they can give their overall appreciation
about that particular physician. Also, even if it is where the case that people in the social
network have the same illness, because of their di¤erent levels of ignorance about their illness
severity, they can not be considered as homogeneous. I assume that ~ = + " where " is i.i.d
with mean zero. More speci…cally, the signal obtained by the patient depends on the type of
the people contacted in the social network. Hence if " > 0 then the patient has collected the
information from the sample of patients who on average over-evaluate their physician’s type
while if " < 0 the information is collected from patients who on average under-evaluate their
physician’s type. For the purpose of this paper, I allow patients to recall a past observation,
that is if the search process is unfruitful, the patient can return to get care from his previous
physician.
The equilibrium analysis is done in the next section (3.2) by assuming that physicians are
not capacity constrained. More speci…cally, I assume that physicians can always provide care
to a new patient, i.e, once a patient meets a physician’s whose perceived type is relatively
high, the latter will provide care to him.
17
A this stage, it is important to note that once a physician is contacted, the information obtained from
the social network is assumed to be costless. In other words, I assume that the information collection process
is exogenously given. One may however endogenous it by assuming that the signal obtained depends on the
patient’s search intensity through his social network.
15
3.2
Equilibrium analysis
Ex ante, the patient has no information about his physician’s type but ex post, he can form
beliefs about his physician’s type by comparing t ( + ) to t ( ): Henceforth, to simplify
notation, I rewrite the patient’s post-treatment health to be a function of his illness severity
and the type of the physician he patronizes i.e. h = h( ; ):
Let V p denote the expected present value from search,
c
the true type of the current physician
(and thus U p = U (x; h( ; c )) the patient’s current level of utility), then the patient would
stay with his current physician if and only if: U (x; h( ; c ))
V p.
In order to maximize his expected utility from receiving care from the alternative physician,
the patient chooses for every level of evaluation
physician type
;
, a search intensity
and a critical
which maximize the following expression:
V p = max
(
EV
8
>
>
>
>
>
>
>
<
)>
>
>
>
>
>
>
:
c( ) +
1
1+r
2
6
6
6
6
6
6
6
4
0
~>
E V j~ >
B Pr
@
Pr ~
Up
p
+
) U p + o( )
(1
1 39
>
+ C >
>
>
A 7
>
7>
=
7>
7
7
7>
>
7>
>
5>
>
>
;
(2)
where r denotes the discount rate, c( ) the cost of search incurred by the patient during
the time interval
with c0 > 0; c00 > 0; c(0) = c0 (0) = 0; and o( ) is the probability that
the patient meets more than one physician in the interval
where o( ) tends to 0 when
approaches 0.
Equation (2) states that during the short time interval
to …nd a physician. With probability
; the patient pays search cost c( )
he meets a physician. Once a physician is met,
the patient collects from his social network the information about that physician’s type and
16
thus obtains a signal about his diagnosis ability ~: The social network’s evaluation of that
physician can be either acceptable or not. If the evaluation is acceptable (i.e. ~ >
) then
the patient will switch and his utility will be V p , otherwise he will stay and will get U p (i.e.
if ~
).18 Finally, with probability (1
) he does not meet any alternative physician
and therefore stays with his current physician and consequently gets U p :
Equation (2) can be rewritten as (for the Proof, see the Appendix A):
V p = max
;
(
)
(
"
U p + o( )
1
c( ) +
+
1+r
1+r
Z
(V p
The …rst-order conditions (F.O.C.) of maximization with respect to
Z h
1
c( )+
1+r
0
@
hR
@
U p ) dG(~)
and
i
U p ) dG(~)
0
(2 )
are respectively:
i
U p (x; h( ; c )) dG(~) = 0
V p x; h( ; ~)
(V p
#)
(3)
(4)
=0
which can be reduced respectively as19 :
18
If physicians are capacity constrained, then even if a patient, in his search, …nds a physician with a
relatively high diagnosis precision, there is still an uncertainty that this physician will provide care to him.I
have examined the model with this new feature by assuming that there is an exogenous probability that if
the patient …nds a physician with an acceptable diagnosis skills (i.e. ~ > ), he will receive care from the
latter. But the main intutions and results of the model remain almost unchanged.
1
is very small then lim 1+r
= 1.
R
b
0
Equation (4 ) follows from Leibniz’s rule. In fact, its states that if J(a; b) = a F (t; x)dt where t is the
variable of integration and x is a variable which is neither the variable of integration nor a limit of integration,
19
Equation (30 ) follows from the fact that if the time interval
then
@J
@a
=
F (t; x)
=
F (a; x) i.e. the derivative of a de…nite integral with respect to its lower limit of
t=a
integration his the negative of
evaluated
at that point.
i thehintegrand
i
R
R
nh
@
(V p U p )dG(~)
@
(V p U p )g(~)d~
=
=
V p (x; h( ; ~))
Hence,
@
@
=
Since
follows.
@V p
@
f[Vhp (x; h( ;
= 0 is equivalent to
@
R
))
p
U
(x; h( ;
i
(V p U p )dG(~)
@
c
))] g( )g :
U p (x; h( ;
c
i
o
)) g(~)
~=
= 0, and using the fact that g( ) > 0, the equation (4’)
17
0
c( )=
Z h
V p x; h( ; ~)
V p [x; h( ;
)]
i
U p (x; h( ; c )) dG(~)
0
(3 )
U p [x; h( ; c )] = 0
(40 )
0
Equations (3 ) and (40 ) describe implicitly the optimal critical physician’s type and the search
0
intensity choice. The term on the right-hand side of (3 ) is the expected gain attributable to
…nding an acceptable alternative (Mortensen (1986)) (i.e. the expected gain when a patient
…nds a physician whose perceived type is at least equal to the type of his current physician).
0
Thus equation (3 ) shows that the optimal search intensity is such that the marginal cost of
search is equal to the expected gain generated by the optimal search strategy. Equation (40 )
states that at the equilibrium, the critical physician type is such that the expected utility
from search is equal to his present level of utility. Furthermore, since the patient perceives his
current physician’s true type
c
as
EV
, then equation (4) states also that at the optimum, the
critical physician type is equal to the patient’s perception of the type of his current physician.
In order to …nd a more tractable relationship between
and
0
; I integrate (3 ) by parts,
which yields:
0
c ( ) = V (x; h( ; ))
p
c
U (x; h( ; )) G( )
[V (x; h( ;
))
p
U ] G( )
Z
G(~)V 0 d~
Using (4) and the fact that G( ) = 1, we have:
0
c ( ) = V (x; h( ; ))
p
c
U (x; h( ; ))
Z
G(~)V 0 d~
(5)
The term in the square brackets on the right-hand side of (5) is the di¤erence between
18
the levels of utility procured by the physician whose diagnosis precision is perceived as the
c
highest ( = ) and the current one ( =
). The last term is the marginal gain from search
when the patient changes physicians weighted by the corresponding probabilities. In other
words, equation (5) states that: at equilibrium, the optimal search intensity and the critical
physician type are such that the marginal cost of search is the di¤erence between the gain
from searching for a physician with perceived perfect diagnosis precision and the expected
marginal gain attributable to …nding a physician whose perceived type is greater than
:
0
By totally di¤erentiating both sides of equation (3 ) and using the earlier application of
Leibniz’s rule I get the following result:
c00 ( )d
=
c00 ( ) dd EV =
f[V p (x; h( ;
f[V p (x; h( ;
U p (x; h( ; c ))] g( )g d
))
))
U p (x; h( ; c ))] g( )g.20 Every patient who switches be-
lieves that his critical physician’s type
U p (x; h( ; c ))
0. Since c00
=)
is such that
c
or equivalently V p (x; h( ;
0 and g( ) > 0 then it follows that:
d
d EV
))
0; which means
that the search intensity chosen by the patient is decreasing in his perception of the current
physician’s type. In other words, if the patient has a physician whose diagnosis precision is
perceived as relatively high, then he has little incentive to search actively. Thus, depending
on their perceptions of their physicians’diagnosis skills, patients with the same "true" health
condition and which have the same physician search di¤erently i.e., the heterogeneity in the
ignorance about the illness severity leads to an heterogeneity in the search intensity.
From equation (4) I show that: by following the optimal search strategies, a patient will leave
his physician if and only if the expected value from search and the current utility are equal.
This also means that the critical physician type which will induce the patient to switch has
the following characteristics: leaves if ~ >
and stays if ~
20
. Remember also that in this
Because, at the equilibrium the patient sets his critical physician’s type
of his current physician’s type then d = d EV :
19
equal to his evaluation
EV
section, physicians are not permitted to react to their patients’ behaviour by changing for
example the level of care they provide to them (because physician types are assumed to be
…xed), and as a result patients do not also change their search strategies (in particular their
critical physician type). This is simply because they know that their physicians’types will
remain the same in the next period. Hence, if a patient in his …rst search process does not
…nd a better alternative, he will return to the market in the next period by using the same
search behaviour.
From my assumption about the patient’s evaluation of his current physician’s type, it follows
that
EV
is di¤erent from
c
unless
= 0: In other words, patients can perfectly infer their
physician’s type if and only if they perfectly estimate their illness severity (i.e.
p
= ), I call
this the perfect information case. Depending on the patients’type, three possible cases may
arise. In what follows, each case is analyzed seperately with much detail. It is important to
note that in the following cases, I compare: (i) for patients who do not switch, the quality
of care they would receive if they left with that received and, (ii) for those who switch, the
quality of care received with that which they would receive if they stay with their physician.
More speci…cally, I investigate whether the search process can or not help patients taking
appropriate decisions.
Case 1: Patients with type
If
p
=0
= , then after health care consumption, the patient can attribute the entire di¤erence
between his expected treatment and his current treatment to the physician and thus can
perfectly infer his current physician type. In this case, patients’switching decision depends
only on the type of the information collected about the outside physician.
Proposition 1:
When the patient is perfectly informed about his illness severity, he always receives better care
if: (i) he collects information from people who are on average over-evaluators and he does not
20
switch, (ii) the information is collected from under-evaluators and the patient switches; otherwise the care received depends on the extent of the patient’s social network’s mis-evaluation.
Proof:
We have assumed that ~ =
+ " so if the patient collects information from people who
are on average over-evaluators of their physician’s type, then ~ is higher than the true type
of the alternative physician. Since patients base their switching decision on ~, then for
patients who do not switch ~ is lower than
since
is lower than ~ thus
EV
(which is equal to
is always lower than
c
c
because
= 0). Hence,
(see Figure 2-a) and consequently
V p (x; h( ; )) < U p (x; h( ; c ).21 Thus patients stay with a relatively high type; and since
physicians are assumed to provide the appropriate treatment given their diagnosis, thus the
patient will receive a better care from the current physician than what he will receive if he
switches. For those who switch,
people who under-evaluate (i.e. ~
c
is lower than ~; and if the information is collected from
) then
c
is always greater than
whatever the value
of " (see Figure 3-b). However, if " > 0 and the patient switches, then his decision is
appropriate if and only " is not relatively high (see Scenario 1 in Figure 2-b). Likewise, if
" < 0 but the patient stays with his current physician, then his decision is appropriate if and
only " is relatively low (see Scenario 1 in Figure 3-a). QED
Proposition 1 states that a patient who is perfectly aware about his illness severity may
nevertheless take an inappropriate decision, i.e., leave a physician whose type is relative high
for one whose type is low, or stay with a physician whose type is relatively low. This result
come from the fact that the patient’s social network evaluation of the alternative physician is
imperfect. If it is were the case that the evaluation was perfect, then the search process will
help the patient to take an appropriate decision. Hence, one possibility is to make information
on physicians’s performance available to patients. This can help patients in their decisions.
21
All …gures are in Appendix B.
21
I move now to the more interesting case where patients do not perfectly observe their true
health condition. In this setting, the switching decision will depend not only on the type of
the information givers but also on the type of the patient. This means that the switching
and ~.
decision will depend on both
EV
Case 2: Patients with type
<0
The patient under-estimates the illness severity and consequently, he will over-evaluate his
physician’s type. In other words,
EV
>
c
and consequently the optimal critical physician
type will also be over-evaluated, that is patients set their reservation type greater than the
true type of their physician. Hence, the patient’s switching decision depends on the extent of
his over-evaluation of the physician’s type and also on the type of the information collected
about the outside physician.
Proposition 2:
When the patient under-estimates his illness severity, for a given level of his evaluation (of
his current physician), he always receives better care if and only if he collects information
from people who are on average under-evaluators and he switches; otherwise the care received
depends on the di¤erence between patient’s evaluation and his social network’s perception of
the alternative physician, or on the extent of the patient’s social network’s mis-evaluation or
on both.
Proof:
If the patient collects information from people who are on average under-evaluators of their
physician’s type, then ~ is lower than the true type
of the alternative physician. We
know that a patient switches if and only if ~ is greater than
than ~, then for a patient who switches,
EV
. Hence, since
is always greater than
c
is higher
whatever the value of
" (see Figure 5-b) and consequently V p (x; h( ; )) > U p (x; h( ; c ). Patients who receive
22
information from average under-evaluators but do not switch make an appropriate decision if
and only if ~ is far less than
EV
and " is very small (see Scenario 1 in Figure 5-a). However,
if " > 0 and the patient does not switch, then his decision is appropriate if and only if ~
is far less than
EV
, or ~ is close to
EV
and " is relatively high (see Scenarios 1 and 2 in
Figure 4-a). If however " > 0 but the patient leaves his current physician, then his decision
is appropriate if and only if " is not too high (see Scenarios 1 and 2 in Figure 4-b). QED
Proposition 2 states that search may be a good tool to …nd a better physician only if the
illness severity is under-estimated and the signal about the physician’s type is obtained from
under-evaluators (on average), and the patient switches. The main problem here is that
the patient faces two types of uncertainty: the ignorance about his health condition and the
mis-evaluation of the social network. Thus, even if one try to make perfect the social network
evaluation, there may still a problem. So patients must be more informed about their health
conditions.
Case 3: Patients with type
>0
The patient over-estimates the illness severity and consequently he under-evaluates his physician’s type i.e.
EV
<
c
. The optimal critical physician type will also be under-evaluated,
that is patients set their reservation type lower than their physician’s true type. As in the
case of under-estimators, I integrate the social network’s information.
Proposition 3:
When the patient over-estimates his illness severity, for a given level of his evaluation (of his
current physician), he always receives better care if and only if he collects information from
people who are on average over-evaluators and he does not switch; otherwise the care received
depends on the di¤erence between the patient’s evaluation and his social network’s perception
of the alternative physician, or on the extent of the patient’s social network’s mis-evaluation
or on both.
23
Proof:
If the patient collects information from people who are on average over-evaluators of their
physician’s type, then ~ is higher than the true type
of the alternative physician. We know
that a patient switches if and only if ~ is greater than
EV
switch, since ~ is greater than (because " > 0) and
is always lower than
c
EV
. Hence, for a patient who does not
is lower than
c
(because
> 0) then
(see Figure 6-a) and consequently V p (x; h( ; )) < U p (x; h( ; c ).
For a patient who receive information from average over-evaluators and switches, he makes an
appropriate decision if and only if ~ is far greater than
EV
and " is very small (see Scenario
5 in Figure 6-b). However, if " < 0 and the patient does not switch, then his decision is
appropriate if and only if " is not too high (see Scenarios 1 and 2 in Figure 7-a). If however,
the patient leaves his current physician, then his decision is appropriate if and only if ~ is
not too close to
EV
and " is not very small (see Scenarios 2 and 3 in Figure 7-b). QED
As in the case of patients with type
< 0, information about health condition and availability
of physician’s performance are two requirements which can help patients to take appropriate
decisions.
.
24
4
Competition between physicians
In this section, I focus on physicians’ response given patients’ search behaviour. In the
previous sections, I assumed that physicians’ types were exogeneous and …xed and that
physicians always provided the appropriate treatment given their estimate of the patient’s
illness severity. Although this enabled me to derive patients’optimal search strategies, a more
realistic model should consider physicians’behaviour as endogenous. I do this by allowing
physicians to choose an unobservable input which increases the quality of care provided (as
in Ma and McGuire 1997). By doing so, I allow gaming on the physicians’side.
I consider a two-period model in which physicians can choose an unobservable level of e¤ort
and the latter determines the quality of care they provide to their patients. At the beginning
of the …rst period, and after the …rst meeting between the physician and the patient, the
physician chooses how much e¤ort he is willing to invest in the treatment of the patient. I
assume that the level of e¤ort is chosen only at the …rst period and that the latter determines
quality of care in both periods. In other words, if the patient stays with the physician in
the second period, there is no need for e¤ort in that period. This e¤ort (which is denoted
by e) can be thought of as any costly activity that a¤ects the patient’s valuation of the care
received (as in Ma and McGuire 1997). I assume that there is an appropriate (exogenous)
level of e¤ort ea ( ) which is associated with every , and that this is public information. ea ( )
can simply be considered as the illness-speci…c e¤ort which produces a level of treatment that
is socially desirable. The physician bears a cost C(e) when he chooses a level of e¤ort e where
C is assumed to be an increasing and convex function. The level of e¤ort is assumed to be
between emin ; emax , where emin and emax denote respectively the minimum and the maximum
e¤ort level a physician can choose.22 I assume for simplicity that for a given illness severity
, C is the only per-patient cost borned by the physician. I assume that the physician has a
22
One may set the minimum level of e at 0 and interpret it as if the physician refused to treat the patient.
But this would be unrealistic in our context.
25
utility function separable in income and e¤ort, hence the physician’s …rst period net utility
per-patient is P
C(e), where P is the capitation fee as de…ned previously.
The physician is assumed to be a pure income maximizer and thus I rule out the possibility
for elements such as altruism or ethical consideration or heterogeneity in the underlying
talent of physicians. The physician’s utility is increasing in the net payment received and
decreasing in e¤ort. To ensure physician participation, the optimal payment design must
give a minimum non-negative pro…t to the physician.
In this model, it is assumed that the e¤ort is unobservable to an insurer and thus noncontractible. Patients do not observe the level of e¤ort chosen by their physician but they
observe a signal of physician’s e¤ort. I assume that the quality of care is increasing with the
level of e¤ort. Physicians are assumed to maximize the total (discounted) expected utility.
Finally I assume that the physician’s future revenue depends on the likelihood that the
patient stays with him. This likelihood (which is detailed below) depends on the patient’s
outside option and on the type of people contacted in the social network. As before, patients,
after their …rst-period health care consumption, evaluate their physician’s type and decide
whether to stay with him or to seek care from a competing physician in the second period.
If the patient decides to search, he follows the optimal search strategy derived in section 3.
4.1
The physician’s behaviour
I showed from section 3 that a patient will wish to switch if his evaluation
physician’s type is lower than a critical physician type
EV
of his current
: After searching, the patient will
e¤ectively switch if in his search process, he receives a signal ~ of an alternative physician
which is greater than
EV
,
. Let’s denote by eEV , e , and e~ the level of e¤ort associated with
, and ~, respectively. More speci…cally, eEV is the patient’s evaluation of the level of
e¤ort chosen by his current physician, e is the critical level of e¤ort (i.e., the minimal e¤ort
26
the patient would like to choose if he were himself the physician), and e~ is the perceived e¤ort
level of a speci…c alternative physician.
The physician chooses a level of e¤ort e to maximize the total expected utility W . Thus, the
physician maximizes:
W =P
(6)
C(e) + P !(e)
which can be rewritten as:
W = P (1 + !(e))
0
(6 )
C(e)
where !(e) is the probability that the patient will stay with the physician in the second
period, and is assumed to be increasing in e. From (6), one can see that a higher e¤ort
reduces physicians’ current pro…t but increases the probability that the patient stays with
the latter. Without loss of generality, we assume that the discount factor is equal to one. It
follows from (60 ) that the expected utility for a physician who loses his patient in the second
period is simply W0 = P
!(e) =
C(e) ( i.e. !(e) = 0). I now de…ne !(e) in detail.
Pr(eEV > e ) + (1
)+
Pr(~
e
eEV )
(7)
Equation (7) states that with probability ; the patient becomes ill in the second period and:
(i) will not wish to search for information about a competing physician if his evaluation of the
current physician’s e¤ort is greater than his critical physician’s e¤ort (i.e., if eEV > e ); (ii) if
eEV
e , the patient searches and with probability (1
) he does not …nd any alternative
physician and therefore stays with his current physician; (iii) …nally, with probability
he
…nds a physician but his social network’s average perception of that physician is lower than
the patient’s current physician’s type and consequently will not leave his physician.
27
Denote W0min = P
C(emin ) as the physician’s …rst period per-patient utility evaluated at
emin , where emin is the minimum e¤ort level a physician can choose. In order to rule out
trivial solutions, I make the two following assumptions: (i) the capitation fee is set such that
for a given , if the physician exerts the appropriate e¤ort ea , his …rst period per-patient
utility is non-negative; (ii) the physician’s …rst period per-patient utility evaluated at emin is
lower than the total expected utility evaluated at ea . The assumption (i) is interpreted as the
physician’s participation constraint while (ii) ensures that the physician will not always set
the e¤ort level at its minimum, and loose the patient but will derive the maximum utility.
Given these assumptions, the physician will try to keep the patient into the second period
if his total expected utility when he maintains the patient into the second period is greater
than that derived from loosing him.
4.2
Equilibrium analysis
I solve the model by assuming that, before the treatment occurs, the physician observes
perfectly the patients’ beliefs about their health condition. In other words, I assume that
the physician observes their patients’ types. Although this assumption may be somewhat
questionable, I believe that in many circumstances, the physician, by interacting with their
patients can at least know if a patient is either an under-estimator or rather an over-estimator.
I relax this assumption later on.
4.2.1
The case where patients’type is observable
As I assume that the physician observes perfectly the patient’s type, he would like to maintain
the patient into the second period if, conditionning on the patient’s expectation about the
value of e, W0
0, and W > W0min . Since under these conditions, the physician’s total
28
expected utility increases with the continuation of the relationship, his optimal choice is to
choose for every type, the e¤ort level which induces a greater level of satisfaction for the
patient and consequently does not induce the latter to switch. This strategy however will be
optimal if and only if e¤ort is costless. The fact that the latter is costly for the physician
imposes limits on what he can or may be willing to do even if I assume that providing more
than the appropriate e¤ort is not harmful for patients.
Given that the appropriate e¤ort is ea , a patient who under-estimates (over-estimates) his
illness severity, expects to receive a lower (greater) e¤ort than ea . Since the e¤ort cost
is increasing on the level of e¤ort chosen, the physician has no incentive to choose more
e¤ort than what the patient expects to receive from him. Moreover, choosing more than the
appropriate e¤ort may make physicians’s …rst-period per-patient pro…t negative.
Remember that I assumed that the capitation fee is set such as even if the physician exerts
the appropriate e¤ort (i.e. e = ea ), he must still earn a positive or at least zero pro…t. The
physician’s problem is to choose for a given level of illness severity, the level of e¤ort which
maximizes (60 ) knowing that the probability that the patient will stay with him into the
second period is given by (7).
Proposition 4:
When the physician observes perfectly the patient’s type, for: (i) an under-estimator, he
chooses a level of e¤ort lower than both the critical and the appropriate levels of e¤ort but
may keep the patient into the second period; (ii) an over-estimator, the physician may choose
more than the appropriate e¤ort but less than the critical one.
Proof:
Since for an under-estimator the critical e¤ort is lower than the appropriate e¤ort (i.e.,
e < ea ), then P
C(e ) < P
C(ea ) and thus W0 is always non-negative. The physician
29
may provide to the patient who under-estimates his illness severity the appropriate e¤ort
given the patient’s estimate and keep him into the second period. But, since the physician
knows that patients in their search process do not switch automatically (due to the fact that
they do not always …nd an outside physician, or that the signal received about the outside
physician’s type is lower than that of their current one), he may choose a lower e¤ort than the
patient’s critical e¤ort and may keep him into the second period even if the latter’s outside
options are unknown to the physician. Thus, for a given illness severity, the physician’s will
choose a level of e¤ort lower than e . And since e < ea , then the equilibrium e¤ort level will
be less than both e and ea , and must be between emin ; e .
For an over-estimator, the equilibrium will be completely di¤erent. In fact, the physician
knows that even if he provides to the patient the appropriate care, the patient will judge this
to be insu¢ cient and consequently will search and may switch. In this case, the physician’s
strategy will depend on the extent of the over-estimation. More speci…cally, even if choosing
the patient’s desired level of e¤ort e , P
C(e ) is still positive, then the physician can choose
the patient’s desired level of e¤ort if P
C(emin ) < P (1 + !(e ))
C(e ), otherwise, he will
choose emin .23 For the same reasons as in the case of an under-estimator, even if P
C(e )
is still positive, the fact that the switching probability when the patient searches is less than
one give some power to the physician and consequently he may choose less than what the
patient expects. That is, the physician chooses a level of e¤ort lower than e . Since e is
greater than ea , hence, depending on the magnitude of , the physician’s e¤ort e may be
lower than ea or between (ea ; e ). But if the patient’s estimation is high enough such that
P (1 + !(e ))
C(e )
0, then the physician will always provide the minimum e¤ort and
will loose the patient. QED
In other words, the physician will not treat appropriately the patient who under-estimates
23
P C(emin ) < P (1 + !(e )) C(e ) means that the gain from choosing the minimum e¤ort and looses
the patient is lower than the gain from choosing the patient’s desired level of e¤ort.
30
his illness severity but the latter may stay with him.
Thus, patients who over-estimate their illness severity will receive either more or less than
the appropriate level of care and stay or just the minimum and consequently will search and
may switch.
4.2.2
The case where patients’type is unobservable
In this section, the model is analysed by assuming that the physician does not know the type
of the patient he faces.24 So, choosing the same strategy as in the case of the observable type
may not be optimal. Since the physician is not able to observe his patient’s type, I assume
that he may nevertheless form a belief about the latter. More speci…cally, the physician
knows that E( ) = 0 i.e., the physician knows that, in expectation, the patient estimates
perfectly his illness severity. I examine how this uncertainty about the patient’s type a¤ects
the above results.
Proposition 5:
When the patient’s type is unobservable by the physician, the physician chooses less than the
appropriate e¤ort.
Proof:
Since the physician believes that the patient observes perfectly his illness severity, he believes
also that if he exerts the appropriate e¤ort, he can keep him into the second period. But the
physician knows that even if he does not chooses the appropriate e¤ort, the patient may not
switch due to the fact that the probability for a patient to …nd a physician whose perceived
type is greater than hers is less than one. So the equilibrium e¤ort will be less than the
appropriate e¤ort. QED
24
In the present model, there is no learning.
31
Thus, if the patient is an under-estimator, he will receive more than what he expects but
lower than the appropriate and will stay with the physician. The over-estimator will receive
also lower than the appropriate and he will judge it insu¢ cient and consequently will search
and may leave. Hence, uncertainty about the patient’s type forces the physician to over-treat
some types of patients. Moreover, exerting the appropriate e¤ort does not prevent the risk
of loosing a patient.
When the patient’s type is unobservable, the physician will exerts lower than the appropriate
e¤ort but can keep some patients.
5
Conclusion
The question that motivated this paper is whether a patient’s search for a competing physician may encourage e¢ ciency in a health care sector even in the presence of a prospective
payment mechanism. To this end, I examine a two-period model where: i) the patient’s illness severity is measured with error by both patients and physicians, ii) patients, after health
care consumption, evaluate their physicians’diagnosis precision, and may decide (following
a new health shock) to stay or to search and switch. My main contribution resides in the
formulation of patients’outside options.
In the …rst version of the model I assume that physicians’diagnosis skills are exogeneous
and …xed over time and that physicians always provide the appropriate treatment given their
diagnosis. I …nd that due to their ignorance about the true illness severity, some patients
stay with their physicians even if their diagnosis precision are relatively low while other leave
physicians with better diagnosis skills to poorer ones. That is patients make type I and II
errors. Considering patients who switch, I …nd that the treatment they receive in the second
period depends in most of the cases on either the extent of their ignorance or on that of
32
their social network’s mis-evaluation, or on both. More speci…cally, a patient’s search for an
outside physician may induce (in some circumstances) the patient to patronize a physician
whose type is lower than that of his …rst period’s physician and as a result, the patient will
receive a relatively lower quality of care.
In the second version of our paper, I relax the restrictions imposed on physicians’behaviour
by allowing them to choose an unobservable level of e¤ort or investment which determines
the quality of care they provide to their patients. I show, under some assumptions that
the treatment provided depends on the physician’s information about the patient’s type.
Hence, at the equilibrium, the fear of loosing a patient in a competitive market does not
systematically induce the physician to treat appropriately the latter. More speci…cally, if
the patient’s type is observable by the physician, a patient who under-estimates his illness
severity will be under-treated by the physician while a patient who over-estimates his illness
severity may be over-treated. If however the patient’s type is unobservable, the physician
will under-treat the patient.
33
References
1. Allard Marie, Jelovac Izabela and Léger Pierre Thomas (2007) “A dynamic model of
health”, unpublished paper.
2. Allard Marie, Léger Pierre Thomas and Rochaix Lise (2008) ‘Provider competition in
a dynamic setting’, Journal of Economics and Management Strategy, Forthcoming.
3. Arrow Kenneth (1963) ‘Uncertainty and welfare economics of medical care ‘, American
Economic Review 53(5), 941-973.
4. Burdett K. and Coles M. (1999): “Long-term partnership formation: marriage and
employment,”Economic Journal 109(406), F306-34.
5. Danzon Patricia (2000) ‘Liability for medical malpractice’in Handbook of Health Economics.
6. Dranove David (1988) ‘Demand inducement and physician-patient relationship, Economic Inquiry 26(2), 281-298.
7. Esmail Nadeem and Walker Michael (2005) ‘How good is canadian health care? 2005
report’, Critical Issues Bulletin (2005),The fraser Institute.
8. Ellis P. Randall and McGuire Thomas (1993) ‘Supply-side and demand-side cost sharing
in health care’, Journal of Economic Perspectives 7(4), 135-151.
9. Gourash Nancy (1978) ‘Help-seeking: a review of the literature’, American Journal of
Community Psychology 6(5), 413-423.
10. Jack William (2005) ‘Purchasing health care services from providers with unknown
altruism’, Journal of Health Economics 24, 73-93.
34
11. Jelovac Isabela (2001) ‘Physicians’payment contracts, treatment decisions and diagnosis accuracy’, Health Economics 10, 9-25.
12. Léger Pierre Thomas (2000) ‘Quality control mechanisms under capitation payment for
medical services’, Canadian Journal of Economics 33, 564-88.
13. Ma Ching-to Albert and McGuire Thomas (1997) ‘Optimal health insurance and provider
payment’, American Economic Review 87, 685-704.
14. Mortensen Dale (1982) ’ The matching process as a noncooperative game’, in J.J.
McCall ed, Economics of Information and Uncertainty, 233-258.
15. Mortensen D. T. (1986) ‘Job search’, in Handbook of Labor Economics.
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17. Rochaix Lise (1989) ‘Information asymmetry and search in the market for physicians
services‘, Journal of Health Economics 8, 53-84.
18. Rossiter Louis, Langwell Kathryn, Wan Thomas and Rivnyak Margaret (1989) ‘Patients
satisfaction among elderly enrollees and disenrollees in medicare health maintenance organization: Results from the national medicare competition evaluation’, JAMA 262(1),
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19. Schlesinger Mark, Druss Benjamin and Tracey Thomas (1999) ‘No exit? The e¤ect of
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20. Wolinsky Asher (1987) ’Matching, search and bargaining’, Journal of Economic Theory
42(2), 311-333.
35
Appendix A
Proof of equation (2)’
V p = max
;
(
= max
(
8
>
>
>
>
<
>
>
:
= max
(
;
1
c( ) +
1+r
)>
>
>
>
:
2
R
6
6
6
6
4
)
(
U p + o(
c( ) +
1+r
+
2
6
6
6
6
4
1+r
U
39
>
>
7>
>
7=
7
7>
5>
>
>
;
p
U p + o( )
R
V dG(~) + G( )
p
U
p
+
Up
"Z
V p dG(~) + (G( )
1)U p
39
>
>
7>
>
7=
7
7>
5>
>
>
;
#)
R
dG(~) =) 1 G( ) =
dG(~), and (G( ) 1)U p = U p (1
R
G( )) hence(G( ) 1)U p = U p
dG(~): Finally it should be noted that U p (the patient’s
We know that G( ) =
R
+
V dG(~) + G( )
p
Up
1
U p + o( )
+
c( ) +
1+r
1+r
)>
>
;
8
>
>
>
>
<
0
current level of utility) is …xed and is independent of the alternative physician’s type, so
R
R
Up
dG(~) =
U p dG(~): By using this fact, it follows that:
V p = max
(
;
)
(
1
U p + o( )
+
c( ) +
1+r
1+r
36
"
Z
(V p
#)
U p ) dG(~)
Q.E.D.
Appendix B: Figures
Figure 2–a:
= 0 and " > 0, patients who do not switch.
Figure 2–b:
= 0 and " > 0, patients who switch.
Figure 3–a:
= 0 and " < 0, patients who do not switch.
37
Figure 3–b:
= 0 and " < 0, patients who switch.
Figure 4–a:
< 0 and " > 0, patients who do not switch.
Figure 4–b:
< 0 and " > 0, patients who switch.
38
Figure 5–a:
< 0 and " < 0, patients who do not switch.
Figure 5–b:
< 0 and " < 0, patients who switch
Figure 6–a:
> 0 and " > 0, patients who do not switch.
39
Figure 6–b:
> 0 and " > 0, patients who switch.
Figure 7–a:
> 0 and " < 0, patients who do not switch.
40
Figure 7–b:
> 0 and " < 0, patients who switch.
41