GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 325

Patient Dumping, Outlier Payment, and Optimal
Healthcare Payment Policy under Asymmetric
Information

Tsuyoshi Takahara

December 2013; revised October 2014

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Patient Dumping, Outlier Payments, and Optimal
Healthcare Payment Policy under Asymmetric
Information∗
Tsuyoshi Takahara†‡
Graduate School of Economics, Osaka University
October 22, 2014

Abstract
We analyze a rationale for official authorization of patient dumping in the prospective payment policy framework. We show that when the insurer designs the healthcare
payment policy to let hospitals dump high-cost patients, there is a trade-off between thedisutility of dumped patients (changes in hospitals’ rent extraction due to low-severity
patients) and the shift in the level of cost reduction efforts for high-severity patients.
We also clarify the welfare-improving conditions by allowing hospitals to dump highseverity patients. Finally, we show that if the efficiency of the cost reduction efforts
varies extensively and the healthcare payment cost is substantial, or if there are many
private hospitals, the patient dumping policy can improve social welfare in a wider environment.

JEL Classification Code: I13, I18, L51
Keywords: Patient dumping; healthcare payment policy; adverse selection∗

During the course of this study, I talked to many people whose knowledge and ideas contributed significantly to the analysis in this paper. In particular, I thank Junichiro Ishida, Shingo Ishiguro, and Noriaki
Matsushima for their comments and the helpful discussions I had with them. I also thank Akifumi Ishihara,
Hiroshi Kitamura, Akira Miyaoka, Keizo Mizuno, Ryo Ogawa, Wataru Tamura, and participants at the Applied Microeconomic Theory Workshop, the Japan Association for Applied Economics 2012 spring meeting,
the Japanese Economic Association 2012 spring meeting, the Contract Theory Workshop Summer Camp 2012
in Shinshu, the Applied Economic Theory Workshop for Alumni, the Workshop in Macroeconomics for Young
Economists, the 13th International Meeting of the Association for Public Economic Theory, and the Lunchtime
Workshop at Kyoto University for their constructive comments. I also acknowledge the financial support fromthe Research Fellowship for Young Scientists of the Japan Society for the Promotion of Science (JSPS). This
work is supported by a Grant-in-Aid for JSPS Fellows 12J02394.
†
Graduate School of Economics, Osaka University. Address: 1-7 Machikaneyama, Toyonaka, Osaka 5600043, Japan. E-mail: mge012tt@student.econ.osaka-u.ac.jp‡
JSPS Research Fellow

1Introduction

Government agencies in many countries would like to introduce social security systems that
decrease healthcare payments while providing high-quality medical services. Two types ofhealthcare reimbursement policies achieve this end: the retrospective payment system andthe prospective payment system.1 The retrospective payment system is a cost-based system
in which insurers pay the entire treatment cost to hospitals. Under the prospective payment
system, insurers pay a fixed amount defined by a government agency for each diagnosis per
admission. The latter system incentivizes hospitals to reduce treatment costs and may yieldsocially optimal cost reductions by hospitals.2 Accordingly, some countries have introduced
prospective payments to reduce the cost of social security. 3
Under the prospective payment system, however, hospitals incur risk when treating extraordinarily expensive patients (also called outlier patients). Hospitals that admit outlier
patients incur losses even if they make socially optimal efforts to reduce treatment costs.
Subsequently, an incentive naturally arises for hospitals to refuse treatment to avoid this financial risk, a phenomenon called the dumping problem in the literature.4 The social costof patient dumping is obvious. For one thing, it triggers potentially fatal treatment delays.
Further, as pointed out by Newhouse (1983), the dumping problem stimulates patient convergence on particular hospitals, particularly public hospitals, leading to crowding and longer
treatment delays.
Given these social costs associated with the possibility of patient dumping, a clear alternative is to insure hospitals for some fraction of the extra costs incurred to treat each outlier
patient.5 We call this the outlier payment policy for expositional clarity. This policy is precisely what the United States adopted in the 1990s to alleviate the dumping problem. Under
this policy, the insurer pays an additional amount equaling some part of the cost exceeding1

For instance, see Newhouse (1996) for this classification.
Stephen and Berger (2003) noted that a patient’s pathway (preclinical medical plan) shortens her hospital
stay and reduces total treatment cost.
3
For instance, the United States adopted the prospective payment system in 1983. Some countries, notably
Japan, still adopt the retrospective payment system.
4This problem was discussed in Ma (1994).
5
If insurers pay all of the treatment cost, hospitals take no risk; such a healthcare payment system is equivalent to the retrospective payment system.
2

1the fixed payment when hospitals admit outlier patients. This additional payment can reduce
hospitals’ financial risk, thereby contributing to reduced numbers of dumped patients.
The overall welfare effect of the outlier payment policy is not necessarily clear, however,as gains might arise from allowing hospitals to dump patients at their discretion. We argue
that adopting the outlier payment policy is not always justified, even though hospitals are lesslikely to dump outlier patients when they are insured against such patients. To substantiate
this argument, we consider a canonical model of adverse selection in which there are two
hospitals, called private and public for expositional clarity. The sole difference between thetwo is that the insurer may induce the private hospital to dump its patients whereas it cannot
allow the public hospital to do so, perhaps because of legal restrictions. We assume thatpatients randomly visit one of the hospitals that privately observes the treatment cost for eachpatient.6 In this setting, the insurer devises a contract for healthcare payment that is based on
the hospital’s level of effort. Given the contract, the hospital decides which patients are to be
dumped and decides its level of cost reduction efforts accordingly.
To observe the potential welfare consequences of patient dumping, suppose that the insurer chooses not to adopt the outlier payment policy and instead allows the private hospital to dump high-severity patients selectively. Under this patient dumping policy, it is too
expensive for the private hospital to treat high-severity patients; consequently, the bulk ofthem is eventually transferred to the public hospital. Although patient dumping is in itself
welfare-reducing, it also endogenously changes the distribution of patients across hospitals
and initiates a sorting effect that substantially alleviates information asymmetry regarding patient types. This sorting effect is potentially welfare-improving because it is instrumental in
reducing information rent and consequently in realizing more efficient levels of cost reduction for high-severity patients in equilibrium. We show that gains from the sorting effect canoutweigh the social cost of patient dumping under some conditions, suggesting that there aresituations in which some degree of patient dumping should be tolerated for the betterment ofsociety.
This study yields several findings. First, if the difference in cost reduction efficiency be6

This assumption of asymmetric information is common in the literature. See Mougeot and Naegelen (2008);
Sappington and Lewis (1999); Beitia (2003); Siciliani (2006); Chalkley and Khalil (2005); Marchand et al.
(2003); Glazer and McGuire (2000); Chalkley and Malcomson (1998).

2tween high-severity and low-severity patients is large, the patient dumping policy is optimal
in a wider environment even when the ratio of high-severity patients is high.7 Intuitively,
under this circumstance, the information rent is large, thus making patient dumping an advantageous payment policy. Second, if the healthcare payment cost (administrative cost for ahealthcare payment system) is large, the outlier payment policy is preferred over the patient
dumping policy, 8 because an increase in the healthcare payment cost reduces the information rent, and therefore favors the outlier payment policy over the patient dumping policy.
Third, if the number of patient dumping hospitals (e.g., private hospitals) is large, the patient
dumping policy is more welfare-improving. If government agencies adopt the patient dumping policy, they need not pay information rent to hospitals that dump high-severity patientswhereas government agencies pay information rent to all hospitals under the outlier payment
policy. As such, we insist that insurers should consider the circularity of private hospitals
when choosing between the outlier payment policy and the patient dumping policy.9
The main contribution of this study is to show that patient dumping can be an optimal
healthcare payment policy. Many studies have analyzed patient dumping (Newhouse (1983),
Dranove (1987), Eze and Wolfe (1993), Newhouse (1996), and Meltzer et al. (2002)). Newhouse (1983) showed that patient dumping may occur under the prospective payment system
and under competition between hospitals. Dranove (1987) noted that the patient dumping
policy can be efficient due to specialization among hospitals and concentration of patients.
Eze and Wolfe (1993) also showed the optimality of the patient dumping policy using theexample of the United States Veterans Affairs hospital inpatient services. Results of these
two studies parallel ours. However, efficiencies from patient dumping are “gains from specialization” in both studies whereas in the present study, efficiencies from patient dumping
are “gains from information acquisition.”
7

The difference in cost reduction efficiency is, for instance, actuated by preventive medicine. The effect ofpreventive medicine on treatment cost was studied in Cohen et al. (2008) with data from Tufts Medical Center.
8
Woolhandler and Himmelstein (1991) and Woolhandler et al. (2003) compared the administrative cost for
healthcare programs per capita between the United States and Canada. Skinner et al. (2005) investigated thedeterminants of inefficiency in the Medicare program in the United States.
9
The Emergency Medical Treatment and Active Labor Act inhibits patient dumping in any region of theUnited States, and the Centers for Medicare & Medicaid Services, a division of the United States Department
of Health and Human Services, reimburses the treatment cost under the Medicare program. (This payment canbe considered an outlier payment.)3

We assume that insurers offer severity-dependent contracts to hospitals as a healthcare
payment policy. Allen and Gertler (1991) discusses the optimality of this selective payment
policy.10 We, too, assume hospitals treat patients selectively. This assumption is consistent
with the theoretical conclusion of Ellis (1998).11
The outlier payment policy can be interpreted as a severity-dependent contract. Keeler
et al. (1998) showed that it acts as insurance for hospitals.12 Many prior works investigate
the optimal scheme under outlier payment (Ellis and McGuire (1990), Ma (1994), Mougeot
and Naegelen (2008), Jack (2005), and Jack (2006)). Ellis and McGuire (1990) analyzed
a consumer-welfare-maximizing outlier payment scheme. Other works study the optimal
ratio for the outlier payment. Ma (1994) investigated optimal outlier payments under theassumption of two-dimensional efforts (cost reduction and treatment quality) by hospitals.
He revealed that insurers should reimburse all of the treatment cost. Mougeot and Naegelen(2008) studied the optimal outlier payment under asymmetric information between insurers
and hospitals and concluded that insurers should reimburse all treatment costs, even underasymmetric information.

2

Model

2.1

Environment

We consider a healthcare payment system in which a public insurer offers contracts to a
private and a public hospital for treatment of patients with a specific diagnosis. Throughout
the analysis, we denote each hospital by j, where j = pr indicates the private hospital andj = pu, the public hospital. As stated, the only difference between them is that the insurer
may induce the private hospital to dump patients selectively, whereas it cannot allow the
public hospital to do so. The reasoning underlying this assumption is that if a public hospital
dumps a specific type of patient, he is unlikely to receive any medical attention; in fact, public
10

Many works examine the optimality of outlier payments: Glazer and McGuire (2002); Ellis and McGuire(1986); Selden (1990); Ellis (1998); Newhouse (1996); Meltzer et al. (2002); Glazer and McGuire (2000);
Barros (2003); Keeler et al. (1998); Jack (2006); Chalkley and Malcomson (1998); Eggleston (2005); Glazer
and McGuire (2000).
11
See also Frank et al. (2000); Ellis and McGuire (1996,?); Siciliani (2006); Eggleston (2000).
12
See also Marchand et al. (2003).

4hospitals in the United States cannot dump any patient. Apart from this distinction, the two
hospitals are assumed to have identical technological acumen.
To simplify, we assume that decision making by patients is given exogenously: λ ∈ (0, 1)
people select the private hospital, and (1 − λ) people select the public hospital.13 After apatient chooses a hospital, the chosen hospital privately observes patient severity i ∈ {H, L},
where i = H denotes a high-severity and i = L, a low-severity patient. The insurer knows thatany given patient is of the high-severity type with probability φ ∈ (0, 1), which is common
knowledge.

2.2

Hospitals

Each hospital can reduce its treatment cost by exerting effort.14 The marginal productivityof any cost-reducing effort depends on patient severity, and we assume that cost reduction ishigher for low-severity than high-severity patients for the same level of cost reduction efforts.
Cost reduction efforts potentially have adverse consequences for hospitals, such as extended
duty hours. For a hospital with patient severity i, total cost C(i, e) can be written as
1
C(i, e) = c − θi e + e2 ,2i ∈ {L, H}.

(1)

Here, the cost of treatment, c, is the same for any patient type and is assumed to be sufficiently
large. We also assume θL > θH > 0, which means that it is easier to reduce the treatment costfor low-severity patients. The last term represents the hospital’s disutility from cost reduction
efforts. Letting w denote the payment collected from the insurer, the payoff function of a
hospital with a type-i patient can be written as151π̃(i, w, e) = w − c + θi e − e2 ,213i ∈ {L, H}.

(2)

We assume that each patient does not know his or her severity and chooses a hospital based solely on
exogenous factors such as proximity. Of course, we can obtain qualitatively similar results as long as exogenous
factors have some effect.
14
For example, the degree of preventive care by doctors can be interpreted as such a variable.
15
All propositions hold so long as the hospital’s payoff function has the Spence–Mirrlees single crossing
property.

5

2.3

Insurer

The insurer offers each hospital a take-it-or-leave-it contract to each hospital that specifies a
healthcare payment and a level of cost reduction effort for each patient severity as reportedby the hospital. The contract specifies the payment wîj and the level of cost reduction effortseîj for a given report î. Define ωîj ≡ (eîj , wîj ) and ω j ≡ (ωHj , ωLj ).
The insurer seeks to maximize social welfare, assumed to consist of (i) patients’ utility,(ii) the social cost of treatment, and (iii) payment by the insurer. To achieve this, the insurer
devises a contract contingent on the hospital’s report î about patient severity. Given the pairof menu contracts (ω pr , ω pu ), assuming truth-telling, the insurer’s payoff is given byW(ω pr , ω pu ) = λ{φ{[1 − C(H, eHpr ) − ηwHpr ](1 − d)+ [γ − C(H, eHpu ) − ηwHpu ]d} + (1 − φ)[1 − C(L, eLpr ) − ηwLpr ]}+ (1 − λ){φ[1 − C(H, eHpu ) − ηwHpu ] + (1 − φ)[1 − C(L, eLpu ) − ηwLpu ]},(3)

where d is an indicator function that takes d = 1 when the private hospital dumps highseverity patients and d = 0 otherwise. We normalize patients’ utility when they receive
immediate medical treatment to 1. In contrast, the utility of dumped patients is given byγ ∈ (−∞, 1), which captures the ill-effects of patient dumping, such as delayed attention andadditional treatment cost.16 Finally, η ∈ [1, ∞) represents a healthcare payment cost.

2.4

Timing

The timing of the game is summarized as follows:
1. the insurer offers contracts to hospitals;
2. a fraction λ of patients select the private hospital, and the remaining fraction 1 − λ ofpatients select the public hospital, with no patient being aware of his/her severity;
3. the hospitals observe the severity of the patients;
4. they decide whether or not to dump the patients, and if yes, which patients to dump;
16Additional treatment cost includes the social cost indicated by Newhouse (1983).

6

5. they set the level of cost reduction efforts;
6. they report patients’ severity to and charge healthcare payments from the insurer, andthe contract is implemented.

3

Optimal Healthcare Payment under Symmetric Information

This section characterizes the first-best healthcare payment system as a benchmark. Withsymmetric information, we suppose that the insurer can observe patient severity and thereby
can impose its preferred cost reduction efforts on the hospital without information rent. It iseasily seen that the insurer prefers no patient dumping, and the first-best contract must satisfy
the following participation constraint for each i = L, H and each j = pr, pu:
1 2
wij − c + θi eij − eij ≥ 0.
2

(PCij )For each i = L, H and each j = pr, pu, the insurer’s problem is defined as follows:max λ{φ[1 − C(H, eHpr ) − ηwHpr ] + (1 − φ)[1 − C(L, eLpr ) − ηwLpr ]}jωi+ (1 − λ){φ[1 − C(H, eHpu ) − ηwHpu ] + (1 − φ)[1 − C(L, eLpu ) − ηwLpu ]},(4)

subject to (PCij ).
All constraints obviously are binding at the optimal solution. Further, there is no reason totreat patients differently as the hospitals are symmetric. This implies that by substituting theparticipation constraint, the optimization problem for each hospital j can be rewritten as
1 2
1 2
max φ[1 − (1 + η)(c − θH eHj + eHj )] + (1 − φ)[1 − (1 + η)(c − θL eLj + eLj )].
jj22eH ,eL(5)

Solving this optimization problem, we now obtain the first-best allocation. As we assume nodisparity in technology, the solution is symmetric between hospitals.
Proposition 1. In the absence of asymmetric information between the insurer and hospitals, the optimal cost reduction efforts and the optimal cost reduction efforts (the first-best7contract) are as follows:
eHpr

FB

pr FB

eL

FB

= θH ,(6)

puFB= θL ,(7)= eHpu= eL1= c − θ2H ,
2
FB
FB
1
wLpr = wLpu = c − θ2L .
2
wHpr

FB

= wHpu

FB

(8)
(9)

4

Optimal Healthcare Payment under Asymmetric Information

4.1

Optimal outlier payment policy

In this subsection, we obtain the optimal healthcare payment under outlier payments, or simply the optimal outlier payment policy, under asymmetric information. Formally, any outlier
payment policy requires the insurer to devise a contract that satisfies all participation constraints. Furthermore, since the insurer cannot observe patient severity, the optimal contract
must satisfy the following incentive compatibility constraint for each i = L, H and eachj = pr, pu:

1 2
1 2
wij + θi eij − eij ≥ wĩj + θi eĩj − eĩj , i , ĩ.
22(ICij )

Because the insurer designs the payment system to bar patient dumping, d = 0 in (3), and theinsurer’s problem can be written as
max λ{φ[1 − C(H, eHpr ) − ηwHpr ] + (1 − φ)[1 − C(L, eLpr ) − ηwLpr ]}jωi+(1 − λ){φ[1 − C(H, eHpu ) − ηwHpu ] + (1 − φ)[1 − C(L, eLpu ) − ηwLpu ]},(10)

subject to (PCij ) and (ICij ),for each i = L, H and each j = pr, pu. The following lemma, which is well known in theliterature,17 is helpful in solving this optimization problem.
Lemma 1. At the optimal solution, (PC Hpr ), (PC Hpu ), (IC Lpr ), and (IC Lpu ) are binding.
17

For example, see Salanié (2005).

8

This lemma implies that the following equations must be satisfied:
1 2
wHj = c − θH eHj + eHj ,
2
1 j2
j
j
wL = c − θL eL + eL + eHj ∆θ,
2

(11)
(12)

for j = pr, pu, where ∆θ ≡ θL − θH > 0. Note, also, that the problem faced by one hospital isagain independent from and identical to that faced by the other hospital because there is notechnology gap in treatment. The optimization problem then can be rewritten as
1 pr2
pr
e )
max
λ{φ[1
−
(1
+
η)(c
−
θ
e
+
H
H
pr pr pu pu
2 H
eH ,eL ,eH ,eH
1 2+ (1 − φ)[1 − (1 + η)(c − θL eLpr + eLpr ) − ηeHpr ∆θ]}2
1 2
pu
+ (1 − λ){φ[1 − (1 + η)(c − θH eH + eHpu )
2
1 pu2
pu
+ (1 − φ)[1 − (1 + η)(c − θL eL + eL ) − ηeHpu ∆θ]}.
2(13)

Using the above, we obtain the optimal cost reduction effort under the outlier payment policy:
eHpr,O∗ = eHpu,O∗ = θH −

η
P∆θ,
1+η

eLpr,O∗ = eLpu,O∗ = θL ,

(14)
(15)

η
where P ≡ 1−φ. Here, we assume θH − 1+η
P∆θ > 0 to assure the existence of an interiorφsolution.
Next, we obtain the optimal healthcare payment using (11), (12), (14), and (15). It is
straightforward to show that1
1 η
wHpr,O∗ = wHpu,O∗ = c − θ2H + (P∆θ)2 ,
2
2 1+η1
ηwLpr,O∗ = wLpu,O∗ = c − θ2L + (θH −
P∆θ).
2
1+η

(16)
(17)

Comparing (6) and (14), we observe that the level of cost reduction efforts for high-severity
patients under the outlier payment policy is distorted downward. Since the level of cost reduction efforts under the outlier payment policy is smaller than the first-best level, the totaltreatment cost and the optimal healthcare payment for high-severity patients are larger (the9third term in (16)). In contrast, the optimal cost reduction efforts for low-severity patients
under the outlier payment policy is not distorted, and the insurer needs to set a higher healthcare payment (the third term in (17)). This, too, is an effect of information asymmetry. We
summarize the optimal contract under the outlier payment policy as follows.
Proposition 2. When the insurer constructs the healthcare payment system so as not to dump
any patient, the optimal healthcare payment compared to the first-best case is such that
eHpr,O∗ = eHpu,O∗ < eHpr

FB

= eHpu ,FB

eLpr,O∗ = eLpu,O∗ = eLpr

FB

= eLpu ,FBwHpr,O∗ = wHpu,O∗ > wHpr

FB

= wHpu ,FB

wLpr,O∗ = wLpu,O∗ > wLpr

FB

= wLpu .

FB∗

We denote the optimized social welfare in the case of the outlier payment policy as W O .

4.2

Optimal Patient Dumping Policy

We now examine the optimal payment policy when the private hospital is induced to dump
high-severity patients. In this case, the insurer sets the healthcare payment for the private
hospital with a participation constraint with respect to low-severity patients only. The insurerthen obviously offers the first-best contract for low-severity patients. The profit of the private
hospital when it admits high-severity patients is given by
1 FB
FB
FB
π(H, eFB
L ) = wL − c + θH eL − eL = θL (θH − θL ) < 0.
2

(18)

Hence, the private hospital would refuse to treat high-severity patients, which can be observedby the insurer. The insurer’s objective function then can be written as
max λ{φ[γ − C(H, eHpu ) − ηwHpu ] + (1 − φ)[1 − C(L, eLpr ) − ηwLpr ]}jωi+(1 − λ){φ[1 − C(H, eHpu ) − ηwHpu ] + (1 − φ)[1 − C(L, eLpu ) − ηwLpu ]},subject to (PC Lpr ), (ICipu ), and (PCipu ),

10

(19)

for each i = L, H and each j = pr, pu. Note that the public hospital still is barred from patient
dumping, and thus, the participation constraint for high-severity patients must be satisfied forthe public hospital.
To solve this problem, we reapply Lemma 1 and obtain
1 2
wLpr = c − θL eHpr + eLpr ,2
1 2
wHpu = c − θH eHpu + eHpu ,2
2
1
wLpu = c − θL eLpu + eLpu + eHpu ∆θ.
2

(20)
(21)
(22)

Given these, the problem can be rewritten as
1 pr2
pr
max
λ(1
−
φ)[1
−
(1
+
η)(c
−
θ
e
+
e )]
L
L
pr pu pu
2 L
eL ,eH ,eL1
1 2+ (1 − λ)(1 − φ)[1 − (1 + η)(c − θL eLpu + eLpu ) − ηeHpu ∆θ]
2
2
1 pr2
pr
+ φ[λγ + (1 − λ) − (1 + η)(c − θH eH + eH )].
2(23)This problem yields the optimal cost reduction efforts under a patient dumping policy:
eLpr,D∗ = θL ,

(24)

η
P∆θ,
eHpu,D∗ = θH − (1 − λ)1+ηeLpu,D∗ = θL .

(25)
(26)

Further, the optimal healthcare payment can be obtained by substituting (24), (25), and (26)
into (20), (21), and (22):1
wLpr,D∗ = c − θ2L ,21ηη
wHpu,D∗ = c − [θH − (1 − λ)P∆θ][θH + (1 − λ)
P∆θ],
2
1+η
1+η
1
η
P∆θ]∆θ.
wLpu,D∗ = c − θ2H + [θH − (1 − λ)
2
1+η

(27)
(28)
(29)

Unlike previous cases, the optimal contract in this case is asymmetric between hospitals eventhough we assume no asymmetry in technology. The key is that the insurer need not provide
information rent to the private hospital but still pays it to the public hospital. Comparing (17)11and (29), we observe that information rent for the public hospital is higher under the patient
dumping policy than under the outlier payment policy. In contrast, comparing (14) and (25),
we also show that distortion in the level of cost reduction efforts for high-severity patients issmaller under the patient dumping policy than under the outlier payment policy. The optimal
healthcare payment system in the case of the patient dumping policy is summarized by thefollowing proposition.
Proposition 3. The optimal healthcare payment policy under the patient dumping policy issuch that
eLpr,D∗ = eLpu,D∗ =eLpr,O∗ = eLpu,O∗ = eLpr

FB

FB

= eLpu ,FB

eHpu,D∗ >eHpu,O∗ > eHpu ,FBwLpr,O∗ >wLpr,D∗ = wLpr ,FB

wHpu,O∗ >wHpu,D∗ > wHpu ,FB

wLpu,D∗ >wLpu,O∗ > wLpu .
∗We denote the optimized social welfare in the case of the patient dumping policy as W D .

5

Welfare Analysis

5.1

Welfare Comparison

We thus far have characterized optimal contracts under two distinct regimes: outlier payment
and patient dumping. Given the indicated results, we now are ready to compare social welfare
between the two policies to illustrate whether a degree of patient dumping should be tolerated
and, if so, under what conditions. To this end, we first compute the welfare difference between

12the two policies (hereafter, welfare difference) as follows:1η
W D∗ − W O∗ = φλ(1 + η)(P∆θ)2 (2 − λ)
2
1+η
|
{z
}
Heavy-severity patients−φλ(1 − γ)
| {z }

Patient dumping costη+λ(1 − φ)η(θH −P∆θ∆θ)
1+η
|{z}
Low-severity patients in the private hospitalη−(1 − λ)(1 − φ)ηλP∆θ2 .
1+η
|
{z
}

(30)

Low-severity patients in the public hospitalObviously, the patient dumping policy is preferred over the outlier payment policy when thisdifference is strictly positive. The first term gives the welfare difference associated with thetreatment cost and the payment cost when i = H (which for expositional simplicity we callthe welfare difference for high-severity patients). As mentioned, the optimal level of cost
reduction efforts for high-severity patients is higher under the patient dumping policy, which
always contributes to welfare improvement. The second term gives the welfare difference
associated with patients’ utility when i = H and j = pr (the welfare difference for highseverity patients in the private hospital). It is always negative because the utility of dumped
patients is discounted to γ. The third term gives the welfare difference when i = L andj = pr (the welfare difference for low-severity patients in the private hospital). It is always
positive because the insurer need not pay information rent to the private hospital under thepatient dumping policy. Finally, the last term reflects the welfare difference when i = L
and j = pu (the welfare difference for low-severity patients in the public hospital). It is
negative because the insurer must provide a larger information rent in this contingency under
the patient dumping policy.
The patient dumping policy clearly is less likely to be optimal when its cost is relatively
large (γ is small). We can subsequently conjecture that there is a threshold level γ̄ such thatthe outlier payment policy is optimal if and only if γ̄ > γ. By rearranging (30), the threshold13

is computed asγ̄ = 1 −1 η2 2 2P ∆θ (2 − λ) −
21+η
|
{z
}
High-severity patients= 1 − ηP∆θ{θH −ηηP(θH −P∆θ)∆θ
(1 + η)|{z}

Low-severity patients in the private hospital+η2(1 − λ)P2
∆θ2
1+η
|
{z
}

Low-severity patients in the public hospitalη1P∆θ(1 − λ)}.
1+η
2

(31)

η
η
P∆θ > 0 and λ ∈ (0, 1) by assumption, θH − 1+η
P∆θ(1 − 21 λ) > 0. We also
Since θH − 1+ηassume η and ∆θ are positive, and by definition P is positive. This implies that γ̄ < 1. We
then obtain the following result, which is not surprising by itself but still clarifies that thepatient dumping policy can be optimal under some conditions.18
Proposition 4. There is a threshold patient dumping cost γ̄ that satisfies γ̄ ∈ (−∞, 1).

5.2

The Optimal Cost Reduction Efforts and the Information Rent Analysis

We now assess the impact of changes in external conditions on the optimal healthcare payment system via the optimal level of cost reduction efforts and the change in information
rent. We particularly focus on changes in the ratios of low-/high-severity patients and public/private hospital patients.
5.2.1

Higher Ratio of High-Severity Patients

We begin with the effect of the ratio of low-/high-severity patients, as captured by P, andexamine how a change in P affects the threshold γ̄. A change in P generally has three effects
on differences in social welfare. The first effect is the number effect, that is the effect on thenumber of patients for which the insurer pays information rent to the hospital. If the numberof high-severity patients rises, the number of patients for whom the insurer pays information
rent to the hospital declines. The second effect is the distortion effect, which is shown in(14) and (25). It can be seen that the extent of distortion in the level of cost reduction efforts
diminishes as the number of high-severity patients rises. The third effect is the information
We so far assume that the private/public patient ratio λ is unaffected by the insurer’s policy choice. In the
Appendix, we show that the following proposition is satisfied even when some patients who select the private
hospital under the outlier payment policy select the public hospital under the patient dumping policy.
18

14

rent effect, which is shown in (17) and (29). The magnitude of the information rent shrinks
with an increase (decrease) in the number of high-severity (low-severity) patients.
To evaluate the impact on welfare of a change in P more precisely, it is instructive todecompose welfare differences into three elements as above: the welfare difference for (i)high-severity patients, (ii) low-severity patients in the private hospital, and (iii) low-severity
patients in the public hospital. Using algebraic techniques, we obtain
−

∂γ̄
η2
=
P∆θ2 (2 − λ)∂P 1 + η|{z}High-severity patientsη+η[θH − 2∆θP]∆θ
1+η
|{z}

Low-severity patients in the private hospitalη2−2(1 − λ)P
∆θ2
1+η
|
{z
}

.

(32)

Low-severity patients in the public hospital

1. For high-severity patients, only the distortion effect influences welfare. As seen in (14)and (25), the distortion effect is larger under the outlier payment policy. As such, if P
decreases (i.e., there are many high-severity and few low-severity patients), the gap inwelfare under the two cases shrinks, and γ̄ increases (shown by the first term in (32)).
2. For low-severity patients in the private hospital, the number effect and information rent
effect are influential. If there are many high-severity patients, the number of patientsfor whom the insurer pays information rent to the private hospital is small. This effectmoves γ̄ upward, since it increases social welfare under the outlier payment policy;
however, this effect does not affect social welfare under the patient dumping policy.
In contrast, the information rent effect moves γ̄ downward since it is milder under theoutlier payment policy. Hence, if the number effect is weaker than the information rent
effect, a decrease in P moves γ̄ downward. When the information rent per patient isηP∆θ is small), the number effect is weaker (shown by the second
smaller (i.e., θH − 1+η

term in (32)).
3. For low-severity patients in the public hospital, only the information rent effect is applicable. As seen in (17) and (29), this effect is weaker under the outlier payment policy,
which moves γ̄ downward.

15

The overall welfare impact of the patient dumping policy is determined by these tradeoffs.
In particular, one crucial factor yields the difference in efficiency of cost reduction efforts
between the two patient types. We summarize this observation as follows.
∂γ̄
∂γ̄
Proposition 5. There exists θ̄H such that if θH > θ̄H , ∂P
≥ 0, and if θH < θ̄H , ∂P
≤ 0.

This result asserts an important policy implication. From Proposition 5, there exists a
possibility of welfare improvement by abolishing the outlier payment policy even when thenumber of high-severity patients is large. If the variance in efficiency of cost reduction efforts
is large (i.e., θH is small against ∆θ), the patient dumping policy is preferred over the outlier
payment policy for more diseases (i.e., γ̄ moves downward).
According to Cohen et al. (2008), screening all 65-year-olds for diabetes (low-cost patients) is more cost-reducing than screening 65-year-olds with hypertension for diabetes(high-cost patients). This situation implies that in the case of diabetes, θH is smaller than∆θ. In such a case, the patient dumping policy is preferred over the outlier payment policy,
even when the number of diabetic patients with hypertension is large.
[Figure 1 about here.]∂γ̄Figure 1 depicts the region in which ∂P< 0 holds. The region tends to shrink as healthcare

payment cost η increases. Intuitively, if the healthcare payment cost is large, the optimal
information rent is smaller, and the number effect weakens.
Proposition 6. γ̄ rises as the healthcare payment cost η increases when P is high.
Woolhandler and Himmelstein (1991) and Woolhandler et al. (2003) empirically investigated per capita administrative cost for US and Canadian healthcare payment programs andfound that it is higher in the United States than in Canada. As such, our model implies thatthe patient dumping policy is more advantageous in Canada.
5.2.2

Higher Number of Patients in the Private Hospital

We now examine how a change in the proportion of patients who visit the private hospital, as
captured by λ, affects the threshold. To this end, it is straightforward to obtain∂γ̄
1 η2 2 2=−P ∆θ ≤ 0.
∂λ
21+η
16

(33)

Note that a change in λ does not affect social welfare under the outlier payment policy since
the optimal contract is symmetric between the two hospitals and thus independent of λ. This
is not the case under the patient dumping policy, however, as the optimal contract is asymmetric. If the number of patients who select the private hospital increases (i.e., λ increases),
the distortion in the cost reduction efforts decreases under the patient dumping policy (25).
This moves γ̄ upward, thereby favoring the patient dumping policy over the outlier payment
policy. However, as shown in (28), the optimal information rent increases as λ increases,
which yields a countervailing effect and moves λ downward. We can show that this latter effect is generally stronger than the former; hence, an increase in λ always reduces γ̄ as shownin (33).
Proposition 7. γ̄ decreases as λ increases.
It is intuitively clear that patients’ initial choice of hospitals, given exogenously in ourmodel, could influence the optimal healthcare payment scheme significantly. According to
the proposition, under the assumption that patient dumping is allowed only for private hospitals, social welfare can be improved by abolishing the outlier payment policy against many
diseases in areas where more private hospitals operate. In practice, this implies that a regulatory agency should admit selected regional variations in the healthcare payment scheme as
the number of private institutions is expected to be high in urban areas and low in rural areas.

6

Conclusion

The main result of this paper is that there are cases in which insurers should not reimburse
additional payments to hospitals that admit expensive patients even though doing so maytrigger socially expensive patient dumping. A payment scheme that insures against outlier
patients exacerbates the extent of information asymmetry between insurers and hospitals andconsequently results in less-efficient effort for cost reduction. When this cost is sufficiently
significant, insurers should instead allow hospitals to dump expensive patients to specific hospitals as a second-best alternative to the outlier payment policy. We show that such a payment
scheme, which tolerates a degree of patient dumping, can ease information asymmetry andimprove efficiency under some conditions.
17One important limitation of our model is the assumption that patients are allocated randomly to hospitals. Although we do not expect that relaxing this assumption alters our maincontention in any qualitative way, the analysis would be more complete, though certainly
more complicated, if we explicitly modeled patients’ choice of hospitals. In the future, it
might be of interest to explore three-way interactions among insurers, hospitals, and patients.

Appendix
In Section 5, we assume that the private/public patient ratio λ is not affected by policy change
since we assume that the patients cannot recognize the severity of their condition. However,if the insurer introduces the patient dumping policy, and the patient knows this, some fractionof patients who select the private hospital under the outlier payment policy may select the
public hospital ab initio so as not to be dumped and not to suffer the dumping cost even when
they do not know the severity of their condition.
Then, in this appendix, we assume that a fraction λ of patients select the private hospital
under the outlier payment policy, and a fraction αλ of patients select the private hospital
under the patient dumping policy where α ∈ (0, 1) (patient movement). Finally, we show that
Proposition 4 is satisfied under this assumption.
If the ratio of patients who select the public hospital is increased under the patient dumping policy, three effects emerge.19 The first is the self-selection effect. Under the patient
dumping policy, the expected patient dumping cost is λγ. However, if some fraction ofpatients who select the private hospital under the outlier payment policy select the public
hospital first, the expected patient dumping policy is reduced to αλγ. Then, the relative superiority of the patient dumping policy becomes substantial. The second is the optimal contracteffect. As seen in (25) and (29), the optimal contract for the public hospital depends on λ.
As previously demonstrated, under the optimal contract, the cost reduction efforts for highseverity patients and the information rent for low-severity patients are larger under the patient
dumping policy than under the outlier payment policy. If λ changes to αλ under the patient
dumping policy, these effects weaken. Accordingly, the relative superiority of the patient
19

Note that the ratio of patients λ has no effect on social welfare under the outlier payment policy.

18

dumping policy is reduced. The third is the number effect. Under the patient dumping policy, the optimal information rent for low-severity patients in the private hospital is reduced tozero. Then, if λ changes to αλ, this information rent-saving effect weakens due to the change
in the number of patients, and the superiority of the patient dumping policy diminishes.
Next, we aggregate these effects. We term the optimized social welfare under the patient
dumping policy when a fraction of αλ select the private hospital as W D . We compute the∗∗welfare difference between the two policies as follows:
1∗∗∗
W D − W O = αλφ(1 + η)A2
2|
{z}
High-severity patients−φαλ(1 − γ)
| {z }

Patient dumping cost+αλ(1 − φ)η(θH − A)∆θ|{z}

Low-severity patients in the private hospital−(1 − φ)η∆θ{−(1 − αλ)(θH − A) + (1 − αλ)[θH − (1 − αλ)A]} .
|
{z
}

(34)

Low-severity patients in the public hospital

We denote the threshold γ as γ̃α between the two policies when patient movement occurs. By
rearranging (34), we obtain1γ̃α = 1 − (1 + η)A2 − ηP{θH − A(αλ − 2)}∆θ.
2(35)

The second and third terms are negative by assumption. And we obtain that there exists a
threshold patient dumping cost γ̃α that satisfies γ̃α ∈ (−∞, 1) even when patient movement
occurs.

References
Allen, Robin and Paul Gertler, “Regulation and the provision of quality to heterogenous
consumers: The case of prospective pricing of medical services,” Journal of Regulatory
Economics, 1991, 3 (4), 361–375.
Barros, Pedro Pita, “Cream-skimming, incentives for efficiency and payment system,” Journal of Health Economics, 2003, 22 (3), 419 – 443.
19

Beitia, Arantza, “Hospital quality choice and market structure in a regulated duopoly,” Journal of Health Economics, 2003, 22 (6), 1011–1036.
Chalkley, Martin and Fahad Khalil, “Third party purchasing of health services: Patient
choice and agency,” Journal of Health Economics, 2005, 24 (6), 1132 – 1153.
and James M. Malcomson, “Contracting for health services when patient demand does
not reflect quality,” Journal of Health Economics, 1998, 17 (1), 1 – 19.
Cohen, Joshua T., Peter J. Neumann, and Milton C. Weinstein, “Does Preventive Care
Save Money? Health Economics and the Presidential Candidates,” New England Journalof Medicine, 2008, 358 (7), 661–663.
Dranove, David, “Rate-Setting by Diagnosis Related Groups and Hospital Specialization,”
The RAND Journal of Economics, 1987, 18 (3), 417–427.
Eggleston, Karen, “Risk Selection and Optimal Health Insurance-Provider Payment Systems,” The Journal of Risk and Insurance, 2000, 67 (2), 173–196.
, “Multitasking and mixed systems for provider payment,” Journal of Health Economics,
2005, 24 (1), 211 – 223.
Ellis, Randall P., “Creaming, skimping and dumping: provider competition on the intensive
and extensive margins,” Journal of Health Economics, 1998, 17 (5), 537 – 555.
and Thomas G. McGuire, “Provider behavior under prospective reimbursement: Cost
sharing and supply,” Journal of Health Economics, 1986, 5 (2), 129 – 151.
and, “Optimal payment systems for health services,” Journal of Health Economics,

1990, 9 (4), 375 – 396.
and, “Hospital response to prospective payment: Moral hazard, selection, and practice-style effects,” Journal of Health Economics, 1996, 15 (3), 257 – 277.

20Eze, Pius and Barbara Wolfe, “Is Dumping Socially Inefficient? An Analysis of the Effectof Medicare’s Prospective Payment System on the Utilization of Veterans Affairs Inpatient
Services,” Journal of Public Economics, 1993, 52 (3), 329–344.
Frank, Richard G, Jacob Glazer, and Thomas G McGuire, “Measuring adverse selection
in managed health care,” Journal of Health Economics, 2000, 19 (6), 829 – 854.
Glazer, Jacob and Thomas G. McGuire, “Optimal Risk Adjustment in Markets with Adverse Selection: An Application to Managed Care,” The American Economic Review, 2000,
90 (4), 1055–1071.
and, “Setting health plan premiums to ensure efficient quality in health care: minimum

variance optimal risk adjustment,” Journal of Public Economics, 2002, 84 (2), 153 – 173.
Jack, William, “Purchasing health care services from providers with unknown altruism,”
Journal of Health Economics, 2005, 24 (1), 73 – 93.
, “Optimal risk adjustment with adverse selection and spatial competition,” Journal of
Health Economics, 2006, 25 (5), 908 – 926.
Keeler, Emmett B, Grace Carter, and Joseph P Newhouse, “A model of the impact of
reimbursement schemes on health plan choice,” Journal of Health Economics, 1998, 17(3), 297 – 320.
Ma, Albert C., “Health Care Payment Systems: Cost and Quality Incentives,” Journal of
Economics & Management Strategy, 1994, 3 (1), 93–112.
Marchand, Maurice, Motohiro Sato, and Erik Schokkaert, “Prior Health Expenditures
and Risk Sharing with Insurers Competing on Quality,” The RAND Journal of Economics,
2003, 34 (4), 647–669.
Meltzer, David, Jeanette Chung, and Anirban Basu, “Does Competition under Medicare Prospective Payment Selectively Reduce Expenditures on High-Cost Patients?,” The
RAND Journal of Economics, 2002, 33 (3), 447–468.

21

Mougeot, Michel and Florence Naegelen, “Supply-side risk adjustment and outlier payment
policy,” Journal of Health Economics, 2008, 27 (5), 1196–1200.
Newhouse, Joseph P., “Two prospective difficulties with prospective payment of hospitals,
or, it’s better to be a resident than a patient with a complex problem,” Journal of Health
Economics, 1983, 2 (3), 269 – 274.
, “Reimbursing Health Plans and Health Providers: Efficiency in Production Versus Selection,” Journal of Economic Literature, 1996, 34 (3), 1236–1263.
Salanié, Bernard, A economics of contracts: primer, Cambridge, Massachusetts: The MIT
Press, 2005.
Sappington, David E. M. and Tracy R. Lewis, “Using Subjective Risk Adjusting to Prevent Patient Dumping in the Health Care Industry,” Journal of Economics & Management
Strategy, 1999, 8 (3), 351–382.
Selden, Thomas M., “A model of capitation,” Journal of Health Economics, 1990, 9 (4), 397– 409.
Siciliani, Luigi, “Selection of treatment under prospective payment systems in the hospital
sector,” Journal of Health Economics, 2006, 25 (3), 479–499.
Skinner, Jonathan S., Elliott S. Fisher, and John Wennberg, “The Efficiency of Medicare,”in David A. Wise, ed., Analyses in the Economics of Aging, NBER Book Series - The
Economics of Aging, University of Chicago Press, August 2005, chapter 6, pp. 129–160.
Stephen, Antonia E. and David L. Berger, “Shortened length of stay and hospital cost
reduction with implementation of an accelerated clinical care pathway after elective colon
resection,” Surgery, 2003, 133 (3), 277 – 282.
Woolhandler, Steffie and David U. Himmelstein, “The Deteriorating Administrative Efficiency of the U.S. Health Care System,” New England Journal of Medicine, 1991, 324(18), 1253 – 1258.

22

, Terry Campbell, and David U. Himmelstein, “Costs of Health Care Administration in
the United States and Canada,” New England Journal of Medicine, 2003, 349 (8), 768 –
775.

23

θH

θ̄H =

η
P ∆θ(2 − λ)
1+η

∂γ̄
≥0
∂P

∂γ̄
≤0
∂P

0

∆θ

Figure 1: Effect of P on γ̄24