Vol.2, No.9, 1110-1119 (2010) Health
doi:10.4236/health.2010.29164
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/HEALTH/
Effect of user fee on patient’s welfare and efficiency in a
two tier health care market
Eugenia Amporfu
Kwame Nkrumah University of Science and Technology, Kumasi, Ghana; eamporfu@gmail.com
Received 26 March 2010; revised 27 May 2010; accepted 29 May 2010.
ABSTRACT
This is a theoretical paper examining the effect
of user fee on patients’ welfare and social wel-
fare under three forms of provider reimburse-
ments: full cost, prospective payment and cost
sharing. The paper extends Rickman and
McGuire (1999) by introducing user fee to the
public sector and maintaining the assumption
that providers can work in both the private and
public health sectors. Contrary to previous
studies, this study shows that efficiency is
possible under the full cost reimbursement. The
paper also shows the conditions under which
efficiency is possible under each reimburse-
ment scheme. Patient’s welfare can improve
with the introduction of user fee when services
in the public and private sector are comple-
mentary.
Keywords: User Fee; Two Tier Healthcare; Mixed
Financing; Prospective Payment; Cost Sharing
1. INTRODUCTION
Many countries have the two tier health care system: the
coexistence of public and private health care sectors.
And over the years, public health care reform in many
countries has involved a switch from fully financed sys-
tem to mixed ones [1,2]. Such a switch is observed in
both industrialized and developing countries. The mixed
financing system involves making the patient bear at
least part of the cost of care provided in the public sector.
The intention is to partly relieve the government of the
burden of funding public health care and at the same
time reduce excessive use of care that might exist under
the fully funded system. Various payment schemes are
also used in the public health sector to pay health care
providers. The purpose of this paper is to find the impact
of patient direct payment at the point of purchase, in the
public health sector, on the efficiency of three payment
schemes: full cost reimbursement, prospective payment
and cost sharing in a two tier health care sector.
Example of mixed financing systems in industrialised
countries consists of a combination of compulsory social
security system covering a package of essential services
and private insurance policy to cover the rest. The pa-
tient then has to pay premium and co-payment [3]. Some
of the industrialised countries that have adopted this
include Australia, Italy, and the United States (in the
Medicare plan) [3]. In developing countries the mixed
financing system in general involves introduction of user
fee. Many Sub-Saharan countries such as Cote d’Ivoire,
Kenya, and Nigeria have adopted this system.
There is considerable theoretical literature on the ef-
fect of private insurance (with co-payment) on quality of
care and the efficient provision of services (e.g., [4-8].
While these analyses centred more on the effect of in-
surance on quality and efficiency of services than on the
reimbursement schemes, others examined the role of
different reimbursement schemes in the efficient provi-
sion of services [9-17]. The question then is does the
effect of mixed financing system on efficiency depend
on the type of reimbursement scheme to providers.
To answer the question, the current paper extended a
model on a two tier health care system in [16] that ex-
amined the effect of the private health care sector on the
efficiency of provider reimbursement schemes. The ex-
tension involves the introduction of a user fee to the
public sector. Reference [16] in turn, was an extension of
[18], which examined the effect of reimbursement
scheme on the supply of services in the public sector. In
their study, the public health care was a fully funded
system (e.g., the National Health Services in the Britain)
and so the patient did not pay for the services received.
Reference [16] extended [18] by including a private
sector while maintaining the full funding system in the
public sector. The specific interest in the paper then is to
examine how their results of the two previous papers
would be affected when a user fee is included in the
public sector.
E. Amporfu / HEALTH 2 (2010) 1110-1119
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1111
Earlier studies that examined health production in
private and public sector with user fee in the public sec-
tor include [18-20]. Reference [18] presented an empiri-
cal model in which effort was not observable and pa-
tients had to choose medical contracts for health care
provision from government and mission hospitals in the
Cameroun. Reference [20] also presented an empirical
study, which examined how the introduction of user fee
to the public sector affects quality and accessibility of
services. In a theoretical model in which public sector
services are covered partly by compulsory social insur-
ance, partly by private insurance and partly by
out-of-pocket, [21] examined the conditions for optimal
rates of social insurance and private coinsurance. Fol-
lowing [16] the current paper focuses on three forms of
provider-reimbursement schemes: full-cost reimburse-
ment, prospective payment and cost sharing.
2. METHOD
This is a model of mixed health care market, public and
private, that allows physicians to work in both sectors.
Let q represent publicly provided health care received by
a representative patient and s represent privately pro-
vided health care. The patient receives benefit, B(q,s),
from treatment and pays for it according to the marginal
benefit it provides: Bs(.)s, for treatment in the private
sector and (1-
)Bq(.)q for treatment in the public sector,
where 0 <
< 1 is the fraction of the fee paid by the
government. These fees do not have to equal the full cost
of treatment. The patient could have partial insurance
and bear only part of the cost of treatment. What is im-
portant is that the patient bears some cost for treatment
in both sectors. B(.) has positive marginal products, with
unique maxima in q and s, Bqq < 0, Bss < 0. Treatments, q
and s, can be complements, Bsq(.) > 0, substitutes, Bsq(.)
< 0 or unrelated Bsq(.) = 0. Treatments are complements
if, for example, the physician uses the public sector to
request the patient to do some tests in the private sector
to help with diagnoses in the public sector. When the
physician uses, for example, the private sector to treat an
illness that can be treated in the public sector then q and
s are substitutes. The patient’s net benefit is:
(,)(,)(,)(1)(,)
sq
NqsBqsB qssBqsq
 (1)
With the exception of the last term on the RHS, which
represents the fee in the public sector, (1) is identical to
the patient’s net benefit in [16]. The marginal net bene-
fits are:
(,)(,)(1)(,)(,) 0
qq qqsq
NqsB qsqBqssBqs
 
1
(2)
(,)(,) (1)(,)0
sss qs
NqssB qsBqsq
 
(3)
Physicians are regulated and are often expected to
follow a code of ethics with the purpose of taking the
patient’s interest into account when choosing treatment.
Following [22] it is assumed that the physician cares
about the ethics of treatment2. Thus, the physician cares
about the well-being of his patient, N(q,s), as well as the
profit of his public hospital,
h and private sector profit,
p:
() (1) (,)()
h
q
RqB qsqcq

  (4)
where R(q) is the revenue that the public hospital re-
ceives from the government. Again with the exception of
(1 –
)Bq(q,s)q (4) is identical to the public hospital
profit in [16]. The private profit, however, is the same as
that in [16]:
(,) ()
p
s
Bqss cs

(5)
where c’(i) > 0, c’’(i) > 0 (i = q, s),3 and
h and
p have
unique maximum in q and s respectively, the marginal
profits are:
(1)(,) or < 0
h
ssq
Bqsq

 (6)
'() (1)(,)
(,)'() or > 0
h
qq
qq
RqB qs
qBqscq



(7)
where R’(q) > 0.
(,) 0
p
qsq
Bqss
(8)
(,)(,)'() 0
p
ss ss
BqssB qscs
 (9)
The physician’s utility function is U(
h,
p, N), with
UN > 0, 0
h
U
, 0
p
U
, UNN < 0, 0
hh
U

, and
0
PP
U

. The physician chooses q and s to maximize
his utility. The first order conditions are:
0
hp
hp
qqNq
UU UN


 (10)
0
hp
hp
ssNs
UU UN



(11)
1Note that Nq can be negative when services are complements and sis
very large. This case is not examined.
2I do not use the assumption in [16] that patients do not search among
alternative physicians but because physicians care about the welfare o
f
p
atients they do not charge a monopolist price. In the current model, the
p
hysician’s care for the patient represents his care for ethics of treat-
ment .This prevents the physician from charging monopolist price. As
shown in (11) the physician would charge a monopolist price if UN =
0. This is consistent with [23], where the physician is constrained by
p
atient information. In [23] even though, patients cannot evaluate the
marginal benefit from treatment from a given physician they can evalu-
ate the absolute utility upon treatment. Patients can observe the utility
of other patients after treatment. If patients of one physician end up
with a lower utility on average than others then the physician looses
p
atients.
3In order to avoid the implied requirement that the marginal cost in the
two sectors be equal, I do not assume constant marginal costs for the
p
rovision of treatment as in [16].
E. Amporfu / HEALTH 2 (2010) 1110-1119
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1112
h
p
h
s
Ns
p
s
UUN
U
 (12)
By substituting (12) into (10) and rearranging pro-
duces:
h
p
hp
qqsq
h
qs
pp
N
s
ss
N
MRS NN







(13)
where MRS N
h = UN /U
h > 0. Eq.13 defines a locus {q,s}
that maximize the physician’s utility. With the exception
of the second term on the RHS, (13) is identical to what
[16] obtained. This second term appears here because the
inclusion of a fee in the public sector makes
s
h, which is
zero in [16], positive, negative or zero, in the current
model, depending on whether q and s are complements,
substitutes, or unrelated respectively. The terms in bra-
ckets are the magnitudes of the slopes of the private
sector iso-profit (
q
p/
s
p) and the patient’s indifference
curve (Nq/Ns).
The welfare function, W(q, s) = B(q, s) – c(q) – c(s), is
used to find efficiency under the various reimbursement
schemes. It is assumed that the welfare function is con-
cave and has unique maxima in q and s. Efficiency re-
quires that the following first order conditions for the
maximization of the welfare function are satisfied:
(,) 0
h
q
Bqs c (14)
(,) 0
p
s
Bqsc (15)
4
The three physician reimbursement rules in the public
sector and their effect on equilibrium q and s provided
by the physician are now examined. The reimbursement
rules are full-cost reimbursement, prospective payment,
and cost sharing. The patients in [16] did not have to pay
fees in the public sector and so the full-cost reimburse-
ment involved the government providing enough reve-
nue to cover the cost of production. In the current model,
however, the full-cost reimbursement, involves the pub-
lic hospital receiving R(q) from the government to cover
part of the cost not covered by the user fee. Prospective
payment involves the government giving a fixed amount
of revenue, G, to the hospital regardless of the total cost
of production and of the total fees collected. The
cost-sharing rule is a combination of prospective pay-
ment and cost reimbursement. The government gives
fixed revenue, G, and then pays for a fraction of the cost
of production. Under each rule, (13) is used to examine
how the q and s chosen by the physician affect optimal-
ity from the points of view of the patient and society.
The patient’s indifference curves are downward slop-
ing for both complements and substitutes5. This is be-
cause the slope of the indifference curve is –Nq/Ns and
with Nq > 0, Ns > 0 regardless of the relationship be-
tween q and s. Figures 1(a) and 1(b) shows the iso-profits
for the physician’s private profit. The slope of the
iso-profit (-
q
p/
s
p) depends on whether treatments are
substitutes or complements. As shown in (8) and (9),
q
p
is positive when treatments are complements and nega-
tive when they are substitutes;
s
p is positive when s is
very small and becomes negative as it is increases. Thus,
when treatments are substitutes, the iso-profit slopes up-
wards when s is small and downwards when s increases
with profit increasing as q falls. The opposite occurs
when treatments are complements. When treatments are
complements the iso-profit slopes downwards when s is
small and upwards as s increases with profit increasing
in q. These are shown in Figure 1.
3. RESULTS AND DISCUSSION
3.1. Full Cost Reimbursement
Under this rule, the government provides the revenue
required to cover part of the cost of production that the
user fee could not cover. Hence
h = 0 and so
q
h = 0; i.e.,
the physician chooses q to maximize public hospital
profit. Equation (13), then, becomes:
0h
p
hp
qqsq
spp
N
s
ss
N
MRS NN







(16)
Rearranging (16) produces:
h
h
hp
Ns s
qq
p
N
ss
s
MRS NN
M
RS NN
(17)
Notice that when
s
h = 0, the two slopes in brackets in
(16) are equal, and the results are identical to those in
[16], i.e., the physician chooses q and s to equate the
slopes of the iso-profit and the patient’s indifference
curves. Such result is possible in the current model if q
and s are unrelated, i.e., if Bqs(q, s) = 0 implying that
s
h
= 0. It is of interest to examine the optimality of equilib-
rium q and s from the patient’s and society’s perspective
when services are complements,
s
h > 0, and when they
are substitutes,
s
h < 0. Eq.17 shows that in equilibrium
the difference between the slopes of the iso-
profit and indifference curves depends on the sign of
s
h.
When treatments are substitutes,
s
h < 0, the iso-profit
curve is steeper than the patient’s indifference curve. The
resulting q lowers the patient’s welfare compared to [15]
as well as [16]. Reference [15] showed that in the ab-
4where ch and cp represent the marginal cost of q and s respectively.
5Because the marginal net benefits in this model differ from those in
Rickman and McGuire the patient’s indifference curves in this model
are also different. The indifference curves in Rickman and McGuire
slope downward when q and s are substitutes and upward when they are
complements (U-shaped indifference curves). In Rickman and McGuire
N
s >0, whether q and s are substitutes or complements; Nq> 0 when
they are substitutes and
N
q
< 0 when they are complements.
E. Amporfu / HEALTH 2 (2010) 1110-1119
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1113
(a)
(b)
(c)
Figure 1. (a) Private iso-profit curves for substitutes; (b)
private iso-profit curves for complements; (c) patient’s
indifference curves for substitutes and complements.
sence of a private sector and a user fee, the optimal q
chosen by the physician coincides with what the patient
would choose under full information, i.e., at the point
where net marginal benefit is zero. However, by includ-
ing the private sector, [16] showed that the physician
reallocates treatments between the private and public
sector such that the patient’s marginal net benefit in
equilibrium is positive; hence, from the patient’s per-
spective, the q chosen by the physician is sub-optimal. In
the current model, the inclusion of a user fee further in-
creases the patient’s marginal net benefits. The patient,
then, gets less q and lower welfare than in [16]. Figure
2(a) shows the equilibrium treatments from each sector
when treatments are substitutes.
For complements Ns > 0, Nq > 0 and
s
h > 0 and so in
equilibrium the patient’s indifference curve is steeper
than the iso-profit. The patient, then, receives less supply
of both services than he would have chosen himself.
This is contrary to [16] where the patient received over-
supply of q and undersupply of s. It is thus not clear if
the patient is worse off or better off in this model than in
[16] when services are complements. The equilibrium is
shown in Figure 2(b).
The difference between the sign of
s
h in [16] and the
current paper makes an important point. With
s
h = 0 in
[16], the quantity of s chosen by the physician does not
affect the profit of the public hospital. The introduction
of user fee in the current paper, however, makes the pub-
lic hospital profit dependent on the quantity of services
supplied in the private sector with public profit decreas-
ing in s when services are substitutes and increasing s
when services are complements. When services are sub-
stitutes, the two sectors compete for services and so
every unit of private treatment supplied represents a loss
of fee to the public sector. With complementary services,
however, the two sectors become partners and so an in-
crease in supply of private treatment is accompanied by
an increase in supply of public treatment and thus in-
creases the amount of fees received by the public hospi-
tal. Since the physician is agent to the public as well as
the private hospitals, a change in the physician’s behav-
iour that depends on the relationship between the ser-
vices represents the conflict of interest that exist in the
agency relationship. Such conflict of interest can ad-
versely affect the public hospital if the physician puts
more weight on private profit than public profit when
services are substitutes. When services are complements
the two hospitals becomes partners and so any conflict
of interest is eliminated. Thus, the standard notion that
physician has incentive to create artificial shortages in
the public sector in order to increase services in the pri-
vate sector is relevant when services are substitutes.
The efficiency of the equilibrium is now considered
by using (16):
E. Amporfu / HEALTH 2 (2010) 1110-1119
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1114
(a)
(b)
Figure 2. (a) Full cost equilibrium for substitutes; (b) full
cost equilibrium for complements.
0h
p
hp
qqsq
Ns
pp
s
ss
N
MRS NN







(16)
By substituting (9) into (16) and rearranging (16) be-
comes:
h
h
h
Ns s
p
qss
Nq
MRS N
vsB
MRS N

(18)
where v = Bs – cp. Assuming efficiency in the private
sector, v = 0, it is of interest to find out if efficiency in
the public sector is also achievable. This is done by set-
ting v = 0 and substituting (2), (3), (6) and (8) into (18)
and solving for Bq:
(1 )(1)
(1 )1h
h
sq N
qq sq
q
ss N
qB MRS
qB sB
BsB MRS





(19)
The sign of Bq depends on whether services are com-
plements or substitutes as well as on the size of MRS N
h.
For substitutes, Bq < 0 < ch when MRSN
h 1 and for
complements, Bq < 0 < ch is possible when MRSN
h < 1.
However, Bq > 0
ch is possible when services are sub-
stitutes and MRSN
h < 1 or when services are comple-
ments and MRSN
h > 1. For complements when MRSN
h
= 1, Bq < 0 < ch when q > s and demand for q is inelastic6.
Similarly, Bq > 0
ch when q < s and demand for q is
elastic. MRSN
h = 1, means that the physician puts equal
weight on the public hospital’s surplus and patient’s util-
ity, when choosing treatment. Reference [15], call this
behaviour perfect agency. Recall that MRSN
h = UN /U
h,
and so MRSN
h >1 when the physician puts more weight
on the patients net benefit than public hospital surplus.
Similarly, MRSN
h < 1 when the physician puts more
weight on the public hospital surplus than the patient’s
net benefit. As described in [15], such imperfect agency
is likely, because the hospital often has a stronger bar-
gaining power on the physician than the patient. Hence
the analysis will focus on the case where MRSN
h 1.
Perfect agent is used to refer to the case in which MRSN
h
= 1 while imperfect agent refers to the case in which
MRSN
h < 1.
For substitutes, (19) shows that when the supply of s
is efficient the physician oversupplies q if he is a perfect
agent. In the same way the physician can oversupply q
when for complementary services he is an imperfect
agent or is a perfect agent, q > s, and demand for q is
inelastic. For both substitutes and complements effi-
ciency in both sectors can be ruled out under these cir-
cumstances. However, efficiency is possible when pro-
viding services that are substitutes and the physician is
an imperfect agent. When services are complements
perfect agency combined with elastic demand for q and a
less supply of q than s is required for efficiency.
These results are important in several respects. First,
contrary to [16], where efficiency in both sectors is not
possible under the full-cost reimbursement rule, effi-
ciency in both sectors is possible in the current model
under the full cost. This gives credibility to the argument
that cost control policy on the demand-side in the form
of user fee, coinsurance, and deductibles is essential to
reducing the excessive use of care that exist under the
full cost reimbursement scheme with full insurance in
the public sector. However, even though efficiency is
6Note that price elasticity of demand for q is: qq
qqq
qq q
B
B
B
qB q

E. Amporfu / HEALTH 2 (2010) 1110-1119
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1115
possible it is achieved at the expense of the patient’s
welfare. This is not surprising because the user fee forces
the patient rather than the provider to internalize any
externality that existed in the market. As already ex-
plained, the user fee reduces the equilibrium supply of
services and this makes the patient worse off. Secondly,
imperfect agency is required for efficiency. Unlike [16]
as well as [15] where the agency role is not able to
achieve efficiency under full cost reimbursement, im-
perfect agency is crucial for efficiency in the current
model. In [16], efficiency in the private sector is only
achievable at the expense of oversupply of services in
the public sector. The possibility of efficiency in the
current model then implies that when services are sub-
stitutes, the introduction of user fee in the public sector
constraints the physician from oversupplying services to
the extent of supplying the amount that maximizes social
surplus as long as the physician is an imperfect agent.
The user fee then transfers any cost caused by the im-
perfection of the agency unto the patient. Thirdly, the
elasticity of demand for public service is important for
efficiency when services are complements. For comple-
mentary services, the user fee maximizes social surplus
by reducing oversupply as long as the physician is a
perfect agent when demand for q is elastic and the equi-
librium supply of q is less than s. When services are
complements and there are no close substitutes the re-
sulting inelastic demand for q makes even perfect agen-
cy unable to achieve efficiency in the public sector. This
weakens the ability of the user fee to achieve the effi-
ciency and so weakens the argument for cost control on
the demand side when services are complements.
The results are also contrary to what [24] and [25]
found in comparing full cost reimbursement with pro-
spective payment. They show that cost-reimbursement
like fee-for-service is characterized by oversupply be-
cause it does not provide the provider incentive to
economize on the quantity of services. Like [15] they did
not have user fee in their models.
3.2. Prospective Payment
Under this rule, the government gives a fixed amount of
revenue, G, to the public hospital regardless of cost of
the production and revenue collected from user fee. Thus,
(4) becomes:
(1)( , )( )
h
q
GBqsqcq

 (20)
The resulting marginal profit with respect to q is neg-
ative:
(1)( , )(1)(, )'( )0
h
qq qq
BqsqB qscq
 
  (21)
Eq.13, then, becomes:
(1)( , )(1)(, )'( )
h
qqq
php
qqsq
Ns
pp
s
ss
BqsqB qscq
N
MRS NN



 





(22)
Rearrange to get:
'()(1)(,)(1)(,)
h
h
h
qq q
Ns
hp
Nss
qq
p
Ns s
s
cqqBqsB qs
MRS N
MRS NN
MRS NN

 

(23)
With the exception of the first term on the left hand
side (LHS), (23) is identical to (13). This (positive) term
determines the difference between the equilibrium q and
s under the full-cost payment scheme and the prospec-
tive payment scheme. Again, the relationship between
the slopes of the indifference curve and the iso-profit
depends on whether services are substitutes or comple-
ments. For substitutes, the coefficient of the iso-profit’s
slope is less than one. Thus, (23) shows that, under the
prospective payment scheme, the equilibrium q and s
occurs at a point where the iso-profit is steeper than the
indifference curve but not as steep as under the full-cost
scheme. This is shown in Figure 3(a) as (ii) which has
less q and more s than the full cost equilibrium7. This
result is equivalent to [16]. When services are comple-
ments, the coefficient of the iso-profit’s slope in (23) is
greater than one implying that in equilibrium the iso-
profit is flatter than the indifference curve but not as flat
as under the full cost. As shown in Figure 3(b) as ii, the
equilibrium q and s are both less than those of the full
cost and for a given level of private profit, the patient
ends up on a lower indifference curve than the full cost
equilibrium. This again is consistent with the results of
[16].
To determine the efficiency of this equilibrium (2), (3),
(6), (8), and (9) are substituted into (23) set Bs = cp (i.e.,
efficiency in the private sector), and rearranged to pro-
duce:
2
(1)(1 )
h
sqqq ss
Nq
ss
BBB
kMRS BqB


(24)
where k = Bq – ch = 0 is required for efficiency in the
public sector. Eq.24 shows that k = 0 when MRSN
h = 1.
However, k > 0 when MRSN
h < 1 and k < 0 when
MRSN
h > 18. Efficiency in the public sector depends
7Note Note that a fall in q causes an increase in private profit when
services are substitutes. By using the positive sloped portion of the
iso-profit it is possible to obtain similar results if the private iso-profit
curve is allowed to shift to the left and indifference curve remains un-
changed.
8The underlying assumption is that
2
0
sqssqq
ss
BBB
B
, i.e., the q and s
maximize patient benefit if cost were zero, making the term in the
s
q
uare bracket
p
ositive.
E. Amporfu / HEALTH 2 (2010) 1110-1119
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/HEALTH/
1116
(a)
(b)
Figure 3. (a) Prospective payment equilibrium for substitutes
relative to that of full cost; (b) Prospective payment equilib-
rium for complements relative to that of full cost.
on the agency behaviour of the physician and not on the
relationship between the services. Perfect agency is re-
quired for efficiency in the public sector while imperfect
agency produces too little q in equilibrium. Given that
the physician is an imperfect agent, (24) shows that term
in the square bracket increases when the price elasticity
of demand for s increases9. As the demand for services
in the private sector becomes more elastic the imperfect
agent supplies an optimal amount of private services at
the expense of too little supply of public services re-
gardless of the relationship between the services.
These results are also important. First, compared with
the cost reimbursement case, the disincentive that ac-
companies the provision of q under the prospective
payment causes the imperfect agent to cut down the
oversupply of q that exist under the cost reimbursement
when services are complements as well as the optimal
supply of q when services are substitutes such that too
little q is produced whether services are complements or
substitutes. Second, the results here are contrary to [16]
where efficiency in both sectors is possible under the
prospective payment when services are substitutes re-
gardless of the agency type of the physician. Third, the
results are similar to [15] in that perfect agency is re-
quired for efficiency. Reference [15] argues that the need
for perfect agency weakens the argument in favour of
prospective payment scheme. The influence of hospitals
on physicians’ behaviour is manifested in the change in
services provided by physicians in accordance with
changes in the payment scheme to hospitals. Thus, phy-
sicians are more likely to be imperfect agents than per-
fect agents implying that efficiency cannot be actualized
under the prospective payment. The results also provide
interesting comparison with [10], [16] and [12]. Refer-
ence [10] presents a model in which all firms (hospitals)
produce homogeneous product but differ by the cost of
production. He concludes that prospective payment, be-
cause it makes payment independent of the hospital’s
cost, is optimal. Reference [16] considers a model with
heterogeneous patients, in terms of costliness, that can
be treated with varying efforts with demand responding
to the variation. Managerial effort is required for the
enhancement of quality and reduction of cost. His results
show that prospective payment can elicit the efficient
effort if the provider has to treat all patients. Selden’s
results, however, showed that prospective payment is not
optimal even under full insurance.
The question then is what size of
can lead to effi-
ciency in the public sector, under the prospective pay-
ment, given efficiency in the private sector. This is found
from (24) by setting k = 0 and solving for
:
2
*
2
()
()
sqss qq
s
qssqqqss
qBBB
qBBBBB

(25)
Thus, there is an optimal
at which efficiency in both
sectors is possible under the prospective payment re-
gardless of the type of agency role played by the physi-
cian as well as the relationship between services. Obvi-
ously,
* < 1. Note, however, that
* increases as the
elasticity of demand for private service falls regardless
of the relationship between the private and public ser-
9Note that
s
s
sss
s
ss
B
B
B
s
Bs
, substituting this into (24) and differ-
entiating with respect to εs yields
2
(1 )0
sq
s
B
sq B
.
E. Amporfu / HEALTH 2 (2010) 1110-1119
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/HEALTH/
1117
vices10. This is interesting because (24) shows that given
that the physician is imperfect, a fall in the elasticity of
demand for private services causes the physician to in-
crease services in the public sector given an optimal
supply of services in the private sector. Thus, (25) im-
plies that as the services in the private and/or the public
sector become necessities (and so less elastic) efficiency
demands that the government reduces the user fee for
public services. This is consistent with [16] that when
= 1 efficiency is possible under prospective payment
regardless of the type of agency.
3.3. Cost Sharing
Under this rule, the government makes a fixed payment,
G, and covers a fraction,
, of the cost of production:
() ()Rq Gcq
 (26)
The public hospital profit becomes:
(1)(,)(1) ()
h
q
GBqsqcq
 
 (27)
where 0 <
< 1. The government can increase, G, and
reduce
such that the total payment remains constant.
Note that
= 0 implies prospective payment while G = 0
and
= 1 represents full cost reimbursement. The mar-
ginal profit is:
(1)(,)(1)(,) (1)'()0
h
qq qq
BqsqB qscq
 
 
(28)
Eq.13 becomes:
(1)'()(1)(,)(1)(, )
h
h
h
qq q
Ns
hp
Nss
qq
p
Ns s
s
cqqBqsB qs
MRS N
MRS NN
MRS NN
 
 

(29)
With the exception of -
c’(q), (29) is identical to (23).
Eq.29 shows that for substitutes the iso-profit is steeper
than the patient’s indifference curve but not as steep as
under the prospective payment. When services are com-
plements, however, the iso-profit is flatter than the pa-
tient’s indifference curves but by lower degree than un-
der the prospective payment. These are shown as iii in
Figures 4(a) and 4(b):
Thus, patients are worse off under cost sharing than
under full cost but are not as worse off as under prospec-
tive payment.
Following [16], it is important to consider the size of
(a)
(b)
Figure 4. (a) Cost sharing equilibrium for substitutes, relative
to those of full cost and prospective payment; (b) Cost sharing
equilibrium for complements relative to those of full cost and
prospective payment.
at which optimality can be achieved in the public sector
given that the private sector is optimal. This is obtained
by substituting (2), (3), (6), (8) and (9) into (29) and
setting k = 0:
2
*(1 )()
(1 )'( )
h
sqss qq
N
ss
qBBB
MRS cqB



 



(30)
Clearly
* is zero when the physician is a perfect
agent in which case prospective payment is optimal.
10The elasticity of demand for s is:
s = ;
ss
ss
ss s
B
B
B
s
Bs

2
*
22
0
[() ]
qsq
sssqssqqqss
qB B
BqBBBBB

.
E. Amporfu / HEALTH 2 (2010) 1110-1119
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/HEALTH/
1118
However,
* cannot be one11, i.e., cost sharing can be
used to ensure optimal supply of services. It is optimal
for the government to use a combination of fixed fee and
incentive payment to reward the hospital because this
forces providers to internalize the externalities that leads
to undersupply under prospective payment (Selden 1990;
[15] 1986) and oversupply under full cost reimburse-
ment. Equation (30) shows that the size of * does not
depend on the relationship between services i.e., it is
optimal for the government to pass on the same fraction
of cost for substitutes and complements. In [16] effi-
ciency in both sectors requires that
* = (1- MRSN
h).
This is identical to what [15] recommend for efficiency
in the public sector when there is no private sector. Thus,
in Rickman & McGuire, the optimal fraction of cost that
is passed on to providers does not depend on the rela-
tionship between the services.
The term in the square bracket in (30) represents the
effect of the user fee in the public sector on the size of *.
Note that when
= 1 (30) becomes
* = (1 - MRSN
h),
which is the same as in [16] as well as in [15]. With the
term in the square bracket less than one,
* in the current
model is less than that in [16]. Thus, when there is a user
fee in the public sector, efficiency requires that less rev-
enue be retained to induce optimal provision of public
services for both substitutes and complements than when
there is no user fee. Note that the second term in the
square bracket increases in the elasticity of services in
both sectors12. The
* here is subject to the same setback
as in [15] as well as [16] in that information on an un-
observable variable, the physician’s utility, is required.
However, the effect of elasticity on
* in the current
model reduces the dependence on information on the
physician’s utility. The presence of the positive term in
the bracket which depends on elasticity reduces the
range within which
* falls. Thus, under the user fee
system in the public sector in a two-tier system, cost
sharing can ensure efficiency in both the public and pri-
vate sector regardless of the relationship between ser-
vices in the two sectors. This is also intuitive. Compared
to the full cost reimbursement system (where G = 0 and
= 1), the fall in marginal revenue resulting from letting
* < 1 gives the imperfect agent the incentive to reduce
the oversupply of complementary services while main-
taining supply of substitutes at the efficient level. For
comparison with prospective payment, (where G > 0 and
= 0) the increase in marginal revenue from setting
* <
1 induces the imperfect agent to increase services.
4. CONCLUSIONS
Many recent health care reforms introduce user fees to
the public sector. This paper examined patient’s welfare
and efficiency under different provider reimbursement
schemes in the public sector in a mixed health care sys-
tem where the patient bears cost for treatment in both the
public and private sectors. The paper extended previous
studies by introducing user fee in the public sector. The
provider reimbursement schemes examined are full-cost
reimbursement, prospective payment and cost sharing.
The results show that efficiency is possible under the
full cost reimbursement scheme if the physician trades
off public hospital surplus for patient net benefit when
services are substitutes and trading off patient’s net ben-
efit for public hospital surplus or being a perfect agent
when services are complements. This is contrary to the
results in previous studies where there is oversupply of
services in equilibrium in the public sector under the full
cost reimbursement. Under the prospective payment
efficiency is only possible when the fraction of cost not
covered by the user fee is at its optimal level. Similarly,
under the cost-sharing scheme, efficiency occurs only if
the fraction of cost that the government passes on to
providers is at its optimal level. This is similar to the
results in previous studies; however, the optimal fraction
in this paper is less than that in previous studies. Of the
three reimbursements schemes, the patient is worst off
under the prospective payment and has the highest utility
under the full-cost reimbursement. In general the intro-
duction of user fee in the public sector makes the patient
worse off when services are substitutes. However, it is
not clear whether the user fee makes patients worse off
or better off when services are complements.
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