The Impact of Reimbursement Policy on Patient Welfare

The Impact of Reimbursement Policy on Patient Welfare,

Readmission Rate and Waiting Time in a Public Healthcare System:

Fee-for-Service vs. Bundled Payment

Pengfei Guo

Faculty of Business, the Hong Kong Polytechnic University, Hong Kong, pengfei.guo@polyu.edu.hk

Christopher S. Tang

Anderson School of Management, University of California, Los Angeles, California 90095,

[email protected]du

Yulan Wang

Faculty of Business, the Hong Kong Polytechnic University, Hong Kong, yulan.wang@polyu.edu.hk

Ming Zhao

Faculty of Business, the Hong Kong Polytechnic University, Hong Kong; and

School of Economics and Management, Southwest Jiaotong University, Chengdu, China,

lighting.zhao@connect.polyu.hk

Abstract

This paper examines the impact of two reimbursement schemes on patient welfare,

readmission rate, and waiting time in a three tiered public healthcare system compris-

ing (a) a public funder who decides on the reimbursement rate to maximize patient

welfare, (b) a public healthcare provider (HCP) who decides on the service rate (which

aﬀects readmission rate and operating cost), and (c) a pool of (waiting time sensitive)

patients who decide whether or not to seek elective treatments. We focus our analysis

on (1) a Fee-For-Service (FFS) scheme under which the HCP receives payment each

time a patient is admitted (or readmitted); and (2) a Bundled Payment (BP) scheme

under which the HCP receives a lump sum payment for the entire episode of care for

each patient (regardless of the number of readmissions). By considering an M/M/1

queueing model with endogenous arrivals and readmissions, we analyze a three-stage

Stackelberg game to determine the patient’s initial admission rate, the HCP’s service

rate (which aﬀects the readmission rate), and the funder’s reimbursement rate. This

analysis enables us to compare the equilibrium outcomes (patient welfare, readmission

rate and waiting time) associated with the FFS and BP schemes. We ﬁnd that, when

the patient pool is large, the BP scheme dominates in terms of higher patient welfare

and lower readmission rate, but the FFS scheme dominates in terms of waiting time.

However, when the patient pool is small, the BP scheme dominates the FFS scheme in

all three performance measures.

Keywords: Healthcare operations, Fee-For-Service, Bundled Payment, Queueing.

1 Introduction

Public healthcare systems are facing many challenges: operating cost escalates, service qual-

ity deteriorates, and waiting time lengthens. For example, the cost of public healthcare

insurance for the average Canadian family increased by 48.5% from 2005 to 2015, which is

1.6 times of the national salary increase over the same period (Palacios et al. 2015). At the

same time, Canadian patients often wait for 18.2 weeks for elective treatments (Barua and

Fathers 2014). In the UK, waiting times for elective surgery are considered by the public as

the second most important failing of the public healthcare system: the average waiting time

is 95 days for knee replacements, 68.8 days for cataract surgeries, and 80.7 days for hernia

repairs (Hurst and Sicilliani 2003).

In Hong Kong, the waiting time for cataract surgery is

longer than eight months.

Because excessive long waiting time causes patient dissatisfac-

tion, some public healthcare systems (such as the UK) include waiting time (especially for

elective surgeries) as a key performance measure (Dimakou 2013), and others (such as South

Australia) are committed to reduce waiting time.

Many healthcare professionals believe that an eﬀective reimbursement scheme can entice

healthcare providers (HCPs) to reduce waiting time, contain cost and improve service quality.

Currently, the predominant scheme is called Fee-For-Service (FFS) under which a HCP

receives payment each time a patient is admitted (or re-admitted). The FFS scheme creates

incentives for HCPs to urge their doctors to rush through their appointments so as to treat

more patients per day (Rabin 2014), even though it is known to be an eﬀective scheme for

reducing waiting time (Blomqvist and Busby 2013). Without resolving patients’ problems

completely, higher readmissions will ensue (Kociol et al. 2012); and the FFS scheme creates

major concerns including: (1) excessive treatments (Davis 2007); (2) high readmissions

(Fenter and Lewis 2008); and (3) low service quality at high cost (Calsyn and Lee 2012).

To improve service quality and contain cost, the Centers for Medicare and Medicaid

Services (CMS) in the United States is shifting gradually from the FFS scheme to the Bundled

Payment (BP) scheme under which the HCP receives a lump sum payment for the entire

episode of care (within a speciﬁed time window), regardless of the number of times a patient

is readmitted (Tsai et al. 2015). A recent survey study claimed that, relative to FFS, the

BP reimbursement scheme can reduce the cost per episode of care by 3% (Japsen 2015).

See “NHS patients waiting longer for routine operations under coalition” at

http://www.theguardian.com/society/2014/jul/04/nhs-patients-waiting-longer-for-routine-operations-

under-coalition.

See “Waiting Time for Cataract Surgery”, released on the Hong Kong government website at

http://www.ha.org.hk/visitor/ha visitor text index.asp?Parent ID=214172&Content ID=214184.

See “Elective surgery services”, posted on the South Australian Government website at

http://www.sahealth.sa.gov.au/wps/wcm/connect/Public+Content/SA+Health+Internet/Health+services

/Elective+surgery+services/.

At the same time, Ontario (Canada) is examining the eﬀectiveness of the BP scheme since

2011

; and the Australian government is considering the BP scheme in 2015

While the BP scheme has been adopted by some public healthcare systems, many public

systems continue to operate under the FFS scheme because the underlying implications are

not well understood. Therefore, it is important to gain a deeper understanding about the

implications of these two schemes on certain performance measures including patient wel-

fare and service quality (readmissions and waiting time). In this paper, we compare these

performance measures associated with the FFS and BP schemes for providing outpatient

elective care services in a public healthcare system that consists of a funder, a HCP and a

population of patients. To facilitate our comparative analysis, we use a three-stage Stack-

elberg game to capture the dynamic interactions among all three parties. Speciﬁcally, in

our model, the funder acts as the ﬁrst leader who determines the reimbursement rate to

maximize the patient welfare. Given the reimbursement rate, the HCP acts as the second

leader who decides on the service rate to maximize its proﬁt, where a higher service rate

yields a higher readmission rate. Finally, given the HCP’s service rate, each patient decides

whether or not to seek elective care from the HCP by taking other patients’ admissions into

consideration. Hence, the patient’s admission rate is endogenously determined according to

a Nash equilibrium.

Embedded in our three-stage Stackelberg game is an M/M/1 queueing model with en-

dogenous patient arrivals and readmissions. This queueing model enables us to determine

the service rate (decided by the HCP) and the corresponding patient arrival rate, readmis-

sion rate, waiting time and patient welfare for any given reimbursement rate (speciﬁed by

the funder) under both FFS and BP schemes. By comparing the equilibrium outcomes (the

patient welfare, readmission rate and waiting time) associated with these two schemes, we

ﬁnd that the dominance of one scheme over the other depends heavily on the size of the

patient population as follows:

1. When the patient population is suﬃciently large, the BP scheme dominates the FFS

scheme in terms of higher patient welfare and lower readmission rate. However, the

See “Ontario Funds Bundled Care Teams to Improve Patient Experience” at

https://news.ontario.ca/mohltc/en/2015/09/ontario-funds-bundled-care-teams-to-improve-patient-

experience.html.

See the report of the 2015 Primary Health Care Advisory Group, released on the Australian government

website at http://www.health.gov.au/internet/main/publishing.nsf/Content/primary-phcag-report.

For elective care service in a public system, each patient can seek help from the HCP or elsewhere.

For example, starting in 2013 and partly in order to address the issue of long waiting time, the European

Union (EU) has decided to grant European citizens the freedom to choose the member-state from which

they receive care while being entitled to reimbursement from their home insurance systems (Andritsos and

Tang 2014).

FFS scheme outperforms the BP scheme in terms of both shorter waiting time per visit

and shorter total waiting time in the system.

2. When the patient population is suﬃciently small, the BP scheme dominates the FFS

scheme in terms of higher patient welfare, lower readmission rate and shorter waiting

times.

3. When the patient population is medium, we identify exact conditions under which the

BP scheme and the FFS scheme yield identical performance.

This paper makes two contributions to the healthcare operations literature. First, our

paper represents a new attempt to examine the implications of two reimbursement schemes

by incorporating issues of endogenous patient elected admissions and random readmissions

arising from a public healthcare system that provides elective care. Second, our analysis

provides insights regarding the conditions under which one scheme outperforms the other in

terms of patient welfare, readmission rate and waiting time.

This paper is organized as follows. §2 reviews the relevant literature. In §3, we present

our queuing model and establish some preliminary results. In §4, we analyze our three-stage

Stackelberg game by determining the equilibrium outcomes associated with the FFS and

BP schemes when the patient population is large, while in §5, we compare the equilibrium

outcomes under two schemes when the patient population is small. Concluding remarks are

provided in §6. All proofs are relegated to the online Appendix A.

2 Literature Review

This paper is related to the healthcare operations management literature that examines

the performance of diﬀerent payment schemes. Speciﬁcally, there is a stream of literature

that examines various performance-based payment schemes. So and Tang (2000) examine

the impact of an outcome-oriented drug reimbursement policy on the patient’s health. By

using a dynamic principal-agent game theoretic model, Fuloria and Zenios (2001) ﬁnd that

a patient outcome-based reimbursement scheme is eﬀective for improving service quality.

Lee and Zenios (2012) empirically show that an evidence-based payment system with risk

adjustment can induce the HCP to improve its service quality. Other research papers in this

stream include Jiang et al. (2012), Ata et al. (2013) and Bavafa et al. (2013).

As public funders in diﬀerent countries are contemplating whether they shall change the

payment scheme from FFS to BP, researchers are developing diﬀerent models to compare

the performance measures associated with these two schemes. The ﬁrst paper in this area is

by Adida et al. (2014). They consider a healthcare system in which a risk-averse HCP can

select the type of patients to admit and decide the treatment intensity for each admitted

patient. By analysing a two-stage model, they examine the impact of the FFS and BP

schemes on patient selection and treatment intensity. They ﬁnd that, due to risk aversion of

the HCP, the HCP has the incentive to provide excessive treatments under FFS and to incur

suboptimal patient selection under BP. To alleviate the shortcomings of FFS and BP, they

propose two alternative payment systems that may induce system optimal decisions. Next,

in a diﬀerent setting, Andritsos and Tang (2015) consider a situation in which the patient

care can be co-managed by the HCP and the patient so that the readmission depends on the

eﬀort exerted by both the HCP and the patient. They show that the BP scheme outperforms

the FFS scheme in terms of patient welfare because the BP scheme can induce the HCP and

the patient to exert more readmission-reduction eﬀorts. Finally, Gupta and Mehrotra (2015)

examine the BP scheme for Care Improvement (BPCI) initiative initiated by the CMS. The

BPCI invites HCPs to propose bundles of services along with target payments per episode,

quality targets, etc. By considering the proposal selection process adopted by the BPCI,

they derive an optimal strategy for the CMS to consider.

Although we also focus on the comparison of performance measures associated with the

FFS and BP schemes, our paper complements the above work in the following manner. First,

unlike the setting examined in Adida et al. (2014) in which the HCP selects which type of

patients to admit, we consider a situation in which patients are sensitive to waiting time and

they can elect not to seek elective care from the public HCP so that the patient’s arrival rate

is endogenously determined by the patients (not the HCP). Second, unlike those two-stage

models developed by Adida et al. (2014) and Andritsos and Tang (2015) in which a patient

can only be readmitted at most once, we use a queueing model with random readmissions

over time to determine the patient’s total waiting time in the system. Third, while Gupta

and Mehrotra (2015) examine the auction-like mechanism adopted by the CMS, we are

interested in comparing the patient welfare, readmission rate, and waiting times associated

with FFS and BP.

Besides the healthcare operations management literature, our paper is related to the

queueing literature that examines the issue of speed-quality trade-oﬀ (i.e., the service quality

depends on the service rate) so that the service rate is endogenously determined. First,

when the service quality depends on the service rate, Hopp et al. (2007) ﬁnd that capacity

expansion can make waiting time longer. Second, when the service quality is decreasing in

the service rate and when the arrival rate is endogenously determined by the customers,

Anand et al. (2011) investigate the optimal pricing strategy for the service provider. While

Anand et al. (2011) ﬁnd that a lower service rate will increase both the waiting time and the

service quality, it is interesting to note that this ﬁnding continues to hold in our model when

the patient population is large. However, due to the fact that there are two inter-related

customer arrival streams (initial admissions and readmissions) in our model, we obtain a

diﬀerent result when the patient population is so small that all patients will seek (initial)

admissions. In this case, a (slightly) lower service rate will not aﬀect the initial admission

rate; however, it will reduce the waiting time due to a lower readmission rate (i.e., higher

service quality).

Along the same vein, Kostami and Rajagopalan (2014) analyze the quality-speed trade-oﬀ

in a dynamic setting. Tong and Rajagopalan (2014) compare the ﬁxed fee and time-based

fee schemes and identify conditions under which one scheme dominates the other. Li et

al. (2016) consider the quality-speed trade-oﬀ with bounded rational customers. While the

above papers examine the issue of speed-quality trade-oﬀ in a general context, there are

papers that deal with this issue in industry-speciﬁc contexts including diagnostic services

(Pa¸c and Veeraraghavan 2010, Wang et al. 2010, Alizamir et al. 2013), service quality

variability (Xu et al. 2015), call center (de Vericourt and Zhou 2005, Hasija et al. 2009) and

health care staﬃng (Yom-Tov and Mandelbaum 2014). While de Vericourt and Zhou (2005),

Chan et al. (2014) and Yom-Tov and Mandelbaum (2014) consider returning customers, we

consider the case where the arrival process is endogenously determined by the patients while

the readmission rate is endogenously determined by the HCP (via its selection of service

rate). Also, our focus is on the comparison of various performance measures associated with

the FFS and BP schemes, and our results enable us to specify the conditions under which

one scheme dominates the other.

3 Model Preliminaries

Consider a public healthcare system consisting of a funder who sets the reimbursement

rate subject to a limited budget, a HCP who determines its service rate, and a pool of

homogeneous patients who decide whether or not to seek elective treatments from the public

HCP

. The HCP provides a single outpatient elective treatment (e.g., hernia repairs). We

model the HCP operation as an M/M/1 queue with random readmissions (via Bernoulli

trials). Speciﬁcally, we consider the case when potential patients arrive at the HCP according

to a Poisson process with a rate of Λ. However, due to balking patients, the initial arrival

rate to the HCP is exogenously given by λ in the base model. (However, we shall examine

the case when the initial arrival rate to the HCP is endogenously determined by the patients

The assumption about homogeneous patients is reasonable given that, in many countries such as France,

Germany and the United States, patients are classiﬁed into diﬀerent diagnosis-related groups according to

their respective symptoms, and the patients in the same group demand similar resources and services (e.g.,

Street et al. 2011).

in §4.1.)

In our queueing model, the HCP serves patients on a ﬁrst-come-ﬁrst-serve (FCFS) basis,

where the service takes an exponentially distributed service time at rate µ.

Akin to Anand

et al. (2011), we shall assume that the HCP’s manpower capacity (i.e., the number of

doctors) is ﬁxed.

However, the HCP can change its service rate µ by adjusting the service

time per patient. To capture the speed-quality trade-oﬀ, we shall consider the case in which

the patient readmission is more likely to occur when the HCP increases its service rate.

After discharge, the patient is either cured or readmitted with probability δ(µ). For

tractability, we make the following assumptions about the readmission rate δ(µ):

Assumption 1: The readmission rate δ(µ) is increasing in the service rate µ, where δ(µ) ∈

[0, 1], δ(0) = 0 and δ(∞) = 1.

Assumption 2: The cure rate (1 − δ(µ)) is logconcave in µ; i.e., log(1 − δ(µ)) is concave so

that g(µ) = δ

(µ)/(1 − δ(µ)) is increasing in µ.

Assumption 1 captures an empirical fact that the readmission rate is increasing in the

service rate µ (Kociol et al. 2012). By noting that the elasticity of the cure rate (1 − δ(µ))

equals µg(µ), assumption 2 guarantees that the cure rate is more sensitive to the change in

the service rate when the service rate is larger. Observe that the logistic function δ(µ) =

1/(1 + e

−aµ+b

) with parameters a > 0 and b > 0 satisfy both assumptions, where the

logistic function is a standard approach to measure the relationship between the readmission

rate and other variables in the healthcare management literature (e.g., Fethke et al. 1986,

Morrow-Howell and Proctor 1993).

Based on above assumptions, we can model the healthcare system as an M/M/1 queue

with random readmissions (via Bernoulli trials) as depicted in Figure 1. Note that the

service rate for new patients and that for readmitted patients are assumed to be the same.

This assumption is reasonable for outpatient elective care service (such as hernia repair

operations) where the appointment block for each patient is normally ﬁxed, regardless of

whether the patient is new or readmitted.

Both the Poisson arrival process and exponential service time have been well-tested in the healthcare

operations management literature. For instance, Kim et al. (1999) empirically verify that the arrival process

to a hospital intensive care unit follows a Poisson process, and the service time follows an exponential

distribution.

In practice, the capacity change due to increasing the number of doctors is costly and time consuming.

For example, the supply of primary care physicians in the United States, measured by the number per

100,000 population, remains stable from 2002 to 2012 (Hing and Hsiao 2014).

Kociol et al. (2012) ﬁnd empirical evidence that the readmission rate is increasing in the service rate µ.

For example, the appointments with primary care doctors are normally scheduled at 15-minute

intervals in the United States (see https://www.washingtonpost.com/opinions/when-medical-care-

is-delivered-in-15-minute-doses-theres-not-much-time-for-caring/2015/11/13/85ddba3a-818f-11e5-a7ca-

joining

cured patients

1 − δ(µ)

uncured patients

δ(µ)

balking

potential initial arrival

rate

effective arrival/admission

rate

initial arrival/admission

rate

Figure 1: A Schematic of the Model

3.1 Cure Service Rate

Observe from Figure 1 that the probability that a patient is cured after a visit is equal to

1 − δ(µ), where µ is the HCP’s service rate. Therefore, µ(1 − δ(µ)) is the eﬀective service

rate that the HCP cures its patients. This observation motivates us to introduce a term that

we refer to as the cure service rate o(µ), where

o(µ) = µ · (1 − δ(µ)). (1)

As we shall see in our subsequent analysis, the cure service rate o(µ) allows us to interpret

our results intuitively.

Let µ

be the service rate that maximizes the cure service rate o(µ); i.e., µ

= argmax{o(µ) :

µ ≥ 0}. Using the ﬁrst order condition along with assumption 2, we get the following result.

Lemma 1. The cure service rate o(µ) is quasi-concave in µ. The optimal µ

is attained

when the elasticity of the cure rate equals 1; i.e., when

· g(µ

) = 1. (2)

Also, the cure service rate o(µ) is concave in µ for µ ≤ µ

Lemma 1 shows that the cure service rate o(µ) has a unique mode µ

that has the “elasticity”

of cure rate 1 − δ(µ) (i.e., µ · g(µ) equals one). This result can be explained by using the

following intuition. When µ · g(µ) < 1, i.e., when the cure rate 1 − δ(µ) is inelastic, 1 − δ(µ)

changes slowly so that an increase in the service rate µ will cause a net increase in the cure

service rate o(µ). By using the same logic, an increase in the service rate µ will cause a net

decrease in the cure service rate o(µ) when µ · g(µ) > 1. Consequently, the optimal point is

attained at the service rate that has µ · g(µ) = 1.

6ab6ec20f839 story.html). In the United Kingdom, the physicians allocate almost the same amount of time

for the initial visits (i.e., slightly less than 11 minutes) and the follow-up ones (i.e., slightly less than 10

minutes) (Konrad et al. 2010).

3.2 Total Waiting Time

By considering the queueing network as depicted in Figure 1, we now determine the total

waiting time that a patient spends in the system before being cured. Here, the total waiting

time includes the waiting time of the initial admission and the waiting time of all potential

subsequent readmissions during a medical episode. Let λ and λ

denote the patients’ initial

arrival rate (i.e., the arrival rate of newly admitted patients)

and the patients’ eﬀective

arrival rate (that includes initial admissions and all subsequent readmissions), respectively.

In steady state, the departure rate of the system is equal to the eﬀective arrival rate λ

which, in turn, equals the sum of the initial arrival rate λ and the arrival rate associated

with the readmissions (which is equal to δ(µ) · λ

). Therefore,

= λ + δ(µ) · λ

⇒ λ

1 − δ(µ)

. (3)

Given µ, let N represent the number of visits that a patient endures before being cured. It

can be shown that the expected number of visits that a patient endures before being cured,

denoted by n(µ) can be expressed as (see Ross 2007, Example 2.18)

n(µ) = E[N] =

1 − δ(µ)

. (4)

From (3) and (4), we have λ

= n(µ) · λ, which implies that the eﬀective arrival rate equals

the initial arrival rate λ times the expected number of visits per medical episode n(µ).

By considering an M/M/1 queue with instantaneous Bernoulli feedback (Ross (2007)),

it is well known that the average number of customers in the system is equal to L =

µ−λ

o(µ)−λ

. Let W and T denote the expected waiting time per visit and the expected total

waiting time per medical episode; respectively. By using the Little’s law, we have L =

· W = λ · T . By combining these two observations, we get:

W (λ, µ) =

µ − λ

1 − δ(µ)

o(µ) − λ

, (5)

T (λ, µ) =

o(µ) − λ

. (6)

Note that we can interpret T given in (6) as the expected waiting time associated with the

classic M/M/1 queue with a corresponding arrival rate λ and service rate o(µ).

To be consistent with the terminology used in the healthcare industry, we shall refer the eﬀective arrival

rate of newly admitted patients as the “initial admission rate” throughout this paper.

3.3 Patient Utility

For any given service rate µ, a waiting time sensitive patient who seeks admission from the

HCP derives her utility U(λ, µ), where

U(λ, µ) = R − [n(µ) · t + θ · T (λ, µ)] = R −

1 − δ(µ)

−

o(µ) − λ

, (7)

in which n(µ) is the number of admissions a patient expects to experience per medical episode

given in (4), T (λ, µ) is the expected total waiting time per medical episode given in (6), R is

the patient’s reward for being cured after the entire episode, t is the patient’s non-pecuniary

disutility associated with each admission, and θ is the imputed cost associated with waiting.

Here, we assume that the waiting time will not cause adverse eﬀects (i.e., worsening patients’

symptoms). This assumption is reasonable for elective surgeries and it is supported by the

empirical evidence established by Hurst and Siciliani (2003).

Knowing the readmission rate and waiting time,

each patient will seek admission if and

only if the utility associated with the admission U(λ, µ) ≥ 0.

By using the fact that the

utility U(λ, µ) given in (7) is strictly decreasing in λ, we can determine the initial admission

rate

λ(µ) when admissions are endogenously decided by the patients. First, consider the case

when the potential initial admission rate Λ (i.e., the potential arrival rate of newly admitted

patients) is suﬃciently large so that U(Λ, µ) < 0 (because U(λ, µ) is strictly decreasing in

λ). In this case, the initial admission rate

λ in equilibrium satisﬁes U(

λ, µ) = 0, where

λ < Λ, and the balking rate equals (Λ −

λ) (Hassin and Haviv 2003). We shall refer to

this case as the partial coverage scenario. Next, consider the case when the potential initial

admission rate Λ is suﬃciently small so that U(Λ, µ) ≥ 0. In this case, all potential patients

will seek admissions so that

λ = Λ. We shall refer to this case as the full coverage scenario.

To avoid repetition and to ease our exposition, we shall present our analysis for the cases

where potential patients are either partially covered or fully covered under both schemes

in the main text, and provide similar analysis for the case where patients are fully covered

under one scheme but are partially covered under the other scheme in the online Appendix

So far, we have established the relationships among the initial admission rate λ, the

readmission rate δ(µ), the patient utility U(., .), the patient’s waiting time per visit W(., .),

In many countries, the readmission rate and waiting time are common knowledge. For example, the Aus-

tralian government releases the Australian hospital statistics report at http://www.aihw.gov.au/publication-

detail/?id=60129553174, in which the information regarding the waiting time and the readmission rate for

the elective surgery can be found on page 35 and page 52, respectively.

When the imputed disutility associated with each admission or the waiting cost is large enough such

that R−θ ·T (0, µ)−t ·n(µ) < 0 for all µ > 0, the patients’ utility is always negative and therefore, no patient

will seek admission. To avoid this trivial case, hereafter we assume that max

µ>0

{R − θ · T (0, µ) − t · n(µ)} > 0

so that some patients will seek admission in equilibrium.

and the patient’s total waiting time T (., .). Next, we are going to use these relationships to

analyze the three-stage Stackelberg game that involves the funder, the HCP and the patients.

In this game, the funder (the government or a private insurer) ﬁrst selects the payment

scheme (FFS or BP) and the reimbursement rate. Anticipating the funder’s reimbursement

rate, the HCP determines its service rate µ. Finally, given the service rate µ, the patients

decide whether to seek elective treatments from the HCP or not (i.e., patients may balk).

4 Reimbursement Schemes under Partial Coverage: FFS

and BP

In this section, we consider the case in which potential patients are partially covered. This

case is commonly observed in many overcrowding public healthcare systems with long waiting

time. We shall use backward induction to analyze the three-stage Stackelberg game for each

payment scheme. First, each patient will decide whether or not to seek admission based on

her expected utility. Anticipating the patients’ initial admission rate and eﬀective admission

rate in equilibrium

λ(µ) and

(µ), we shall derive the HCP’s service rate decisions and the

funder’s reimbursement decisions under the FFS and BP schemes, respectively. Speciﬁcally,

for each scheme s, s = f, b (where f and b represent the FFS scheme and the BP scheme,

respectively), we ﬁrst determine the HCP’s optimal service rate ˜µ

. Then, by anticipating the

HCP’s service rate ˜µ

and the corresponding admission rates

λ(˜µ

) and

(˜µ

), we determine

the funder’s optimal reimbursement rate ˜r

. Table 1 describes the decision sequences under

the FFS and BP schemes.

Table 1: Reimbursement Schemes and Sequence of Decisions

Fee-For-Service (FFS) Bundled Payment (BP)

1. The funder determines the optimal reimbursement 1. The funder determines the optimal reimbursement

rate r

for each visit. rate r

for each episode.

2. The HCP determines the optimal service rate µ

. 2. The HCP determines the optimal service rate µ

3. The patients decide to join or balk and the initial 3. The patients decide to join or balk and the initial

admission rate in equilibrium is equal to

λ(µ

). admission rate in equilibrium is equal to

λ(µ

Under the FFS scheme, the HCP receives payment for diﬀerent types of treatments it provides during

each visit. However, as our model focuses on a single type of treatment, we can treat the FFS payment as

the payment per visit.

4.1 Patients’ Joining Decision under Partial Coverage

Under the partial coverage scenario, it is well known that the patient utility in equilibrium

equals zero (Hassin and Haviv 2003). By considering the utility function given in (7) and

solving U(

λ, µ) = 0, we can obtain the initial admission rate

λ in equilibrium under the

partial coverage case as

λ(µ) = o(µ) −

θ(1 − δ(µ))

R(1 − δ(µ)) − t

. (8)

Note that

λ(µ) is decreasing in t, the disutility associated with each admission, which is

intuitive.

By accounting for the number of visits over the entire episode n(µ) given in (4), we can

use (3) and (8) to obtain the eﬀective admission rate

in equilibrium as follows:

(µ) = n(µ) ·

λ(µ) = µ −

R(1 − δ(µ)) − t

. (9)

It is worth noting from (5) that, to ensure the stability of the system (i.e.,

(µ) ≤ µ), the

optimal service rate selected by the HCP should satisfy R(1 − δ(µ)) > t.

In the next section, we shall utilize the initial admission rate

λ(µ) and the eﬀective

admission rate

(µ) in equilibrium to derive the HCP’s optimal service rate under diﬀerent

payment schemes. In preparation, let us diﬀerentiate

λ(µ) and

(µ) given in (8) and (9)

with respect to µ, getting:

Corollary 1. Under the partial coverage scenario (i.e., U(Λ, µ) < 0), in equilibrium both

the initial admission rate

λ(µ) and the eﬀective admission rate

(µ) are unimodal in µ.

Moreover, the mode of

λ(µ) is smaller than that of

(µ).

Corollary 1 is induced by the two opposite eﬀects caused by the service rate µ. On one

hand, a higher service rate enables the HCP to treat more patients per unit time, which may

reduce the waiting time and encourage more patients to seek admissions (Rabin 2014). On

the other hand, a higher service rate will cause a higher readmission rate, which discourages

patients from seeking admissions.

Since the cure rate (1 − δ(µ)) is log-concave, it is less

sensitive to the change in µ when µ is small than when it is large. Therefore, when µ is

small (large, respectively), the ﬁrst (second, respectively) eﬀect dominates such that

λ(µ)

and

(µ) are increasing (decreasing, respectively) in µ. Hence,

λ(µ) and

(µ) are unimodal

in µ as depicted in Figure 2.

The readmission rate can aﬀect a patient’s joining-or-balking decision. For example, Varkevisser et al.

(2012) show that patients prefer hospitals with low readmission rates and a 1% reduction in the readmission

rate is associated with a 12% increase in hospital demand.

0 4

(µ)

λ(µ)

Figure 2: Initial and eﬀective admission rates



δ(µ) =

1+e

−µ+2

, θ = 0.5, t = 1, R = 8



Corollary 1 (along with Figure 2) has two implications. First, when µ is large, having

the physicians to work faster can discourage patients to seek admissions (i.e., both

λ(µ) and

(µ) will decrease) since the readmission rate is too high. Second, when µ is moderate, as

the mode of

λ(µ) is smaller than that of

(µ), having the physicians work faster can reduce

the initial admission rate

λ(µ) but it can increase the eﬀective admission rate

(µ) (due to

the signiﬁcant increase in the readmission rate δ(µ)). Therefore, when choosing the service

rate µ under diﬀerent payment schemes, the HCP should take into account the impact of

µ on the initial admission rate and the eﬀective admission rate in equilibrium. We shall

consider this issue in the next section.

4.2 The HCP’s Service Rate Decision under Partial Coverage

Given any reimbursement rate r

, s ∈ {f, b}, and anticipating the equilibrium initial admis-

sion rate

λ(µ) and the equilibrium eﬀective admission rate

(µ) as given in (8) and (9), the

HCP needs to determine its service rate µ

to maximize its expected proﬁt that is composed

of two components: (a) the total amount of reimbursement received from the funder; and

(b) the variable cost associated with each patient. First, recall that the HCP is paid r

for

each admission under the FFS scheme and r

for each episode under the BP scheme. Hence,

the HCP receives r

(µ) under the FFS scheme and r

λ(µ) under the BP scheme from the

funder. Second, recall that the capacity (i.e., the number of doctors) under our setting is

ﬁxed. Hence, the variable cost is mainly attributed to the length of patients’ outpatient

visits (i.e., personnel, nurse, consumable items, etc.) so that the variable cost per patient

visit is c · (1/µ), where 1/µ is the mean length of outpatient visit and c is the correspond-

ing unit time variable cost incurred by the HCP for treating the patient. Combining these

observations, we can formulate the HCP’s problem under the two schemes as

(F F S) max

(µ) = r

(µ) − c ·

(µ) =



−



(µ); (10)

(BP ) max

(µ) = r

λ(µ) − c ·

(µ) =



−

o(µ)



λ(µ). (11)

By considering the ﬁrst order conditions, we get the following results.

Proposition 1. For any given reimbursement rate r

, s ∈ {f, b}, the HCP’s expected proﬁt

(µ) is unimodal in the service rate µ. In equilibrium,

1. the optimal service rate ˜µ

) under the FFS scheme is the unique solution that solves

d log

(µ)

dµ

cµ − r

, (12)

and ˜µ

) must be larger than the mode of

(µ).

2. the optimal service rate ˜µ

(˜r

) under the BP scheme is the unique solution that solves

d log

λ(µ)

dµ

c · o

(µ)

o(µ) · (c − r

· o(µ))

, (13)

and ˜µ

) must be larger than the mode of

λ(µ) and smaller than µ

, the service rate

that maximizes o(µ), i.e., ˜µ

) < µ

. Furthermore,

λ(r

) =

λ(˜µ

)) >

λ(µ

First, observe that the unimodality of Π

(µ) for s = f, b follows immediately from the

unimodality of

(µ) and

λ(µ) as stated in Corollary 1. Second, by substituting the optimal

˜µ

) (given in (12) and (13) respectively) into δ(µ),

λ(µ) given in (8),

(µ) given in

(9), and Π

(µ) given in (10) and (11), we can express these quantities as functions of the

reimbursement rate r

so that δ(r

) = δ(˜µ

)),

λ(r

) =

λ(˜µ

)), and Π

) = Π

(˜µ

)).

(For ease of exposition, we shall suppress the arguments when convenient.) We shall use

these quantities to determine the funder’s reimbursement decision r

, s = f, b, under the two

schemes. In preparation, let us examine the properties of these quantities with respect to

the reimbursement rate r

Corollary 2. The reimbursement rate r

, s ∈ {f, b}, has the following impact on the fol-

lowing quantities:

1. The HCP’s optimal service rate ˜µ

) and the corresponding readmission rate δ(r

)

are decreasing in r

2. The initial admission rate

λ(r

), the waiting time per visit W (r

), the total waiting

time T (r

), and the HCP’s proﬁt Π

) are increasing in r

The ﬁrst statement of Corollary 2 reveals that, under both the FFS and BP schemes,

when the funder oﬀers a higher reimbursement rate r

, the HCP will set a lower service rate

so that physicians spend more time on treating each patient and therefore, less patients are

readmitted to the system. To explain this result intuitively, with a higher reimbursement

rate r

, the HCP has a stronger desire to attract more patients to seek admissions (i.e., to

increase the eﬀective admission rate

(µ) under the FFS scheme and to increase the initial

admission rate

λ(µ) under the BP scheme). Having this desire in mind, it is easy to observe

from (12) and (13) that, in order to increase

(µ) under FFS and

λ(µ) under BP, the HCP

has to lower its service rate µ. This explains the ﬁrst statement.

Next, the second statement informs us that a higher reimbursement rate will encourage

more patients to seek admission (i.e., the initial admission rate

λ(r

) is increasing in r

thereby increasing the congestion level of the healthcare system (i.e., both the waiting time

per visit W (r

) and the total waiting time T (r

) are increasing in r

). This result can be

explained as follows. Recall from above that, with a higher reimbursement rate r

, the HCP

will reduce its service rate so as to increase the eﬀective admission rate

(µ) under FFS

and increase the initial admission rate

λ(µ) under BP. Also, with a lower service rate and a

higher eﬀective admission rate

(µ) under FFS, it is easy to check from (3) that the initial

admission rate

λ(µ) under FFS is also higher. Therefore, when the funder oﬀers a higher

reimbursement rate r

, the HCP will lower its service rate so that the initial admission

rate and the eﬀective admission rate become higher under both schemes. Consequently,

the waiting time per visit and the total waiting time will increase. However, the HCP will

earn a higher proﬁt due to higher admission rate and higher reimbursement rate under both

schemes.

In summary, Corollary 2 highlights the trade-oﬀ between readmission rate and waiting

time that the funder needs to strike a balance when considering the reimbursement rate,

which we analyze next.

4.3 The Funder’s Reimbursement Decision under Partial Cover-

age

Anticipating the HCP’s service rate ˜µ

), s ∈ {f, b} given in (12) and (13) respectively,

we now turn our attention to the funder’s decision regarding the reimbursement rate r

Essentially, the funder selects r

to maximize the patient welfare S(r

). When the patient

initial admission rate is endogenous, the patient welfare generally consists of two components:

(a) the utility of those patients who seek admissions; and (b) the penalty cost associated

with those patients who balk (without seeking admissions). As stated in the introduction, it

is common that patients who demand for public care abandon and seek help elsewhere due

to the long waiting time. Therefore, accessibility is central to the performance of the public

healthcare systems (Levesque et al. 2013, and Adritsos and Tang, 2014). To incorporate

accessibility into the funder’s objective, we impose a penalty cost β for each balking patient.

Under the partial coverage case, we know that

λ(r

) is the initial admission rate, and

(Λ −

λ(r

)) is the balking rate. By noting that each admitted patient obtains a utility

λ(µ), µ) and each balking patient incurs a disutility β, the patient welfare S(r

) =

λ(r

) ·

λ(r

), ˜µ

))−β(Λ−

λ(r

)). By taking the funder’s budget B and the HCP’s participation

constraint into consideration, we can formulate the funder’s problem as follows:

max

S(r

) =

λ(r

)U(

λ(r

), ˜µ

)) − β(Λ −

λ(r

)) = −β(Λ −

λ(r

)), (14)

s.t. Budget constraint :

(

(F F S) r

) ≤ B,

(BP ) r

λ(r

) ≤ B,

(15)

P articipation constraint : Π

) ≥ 0.

Recall from §4.1 that U(

λ(r

), ˜µ

)) = 0 under the partial coverage case, it is easy to

check from (14) that the funder’s objective is equivalent to maximizing the initial admission

rate

λ(r

This is consistent with the practice that, in many public healthcare systems

that serve a large patient population, the government’s overarching objective is to improve

accessibility by maximizing

λ(r

) (Aboolian et al. 2016). Combining this observation and

the fact that

λ(r

) is increasing in r

(Corollary 2), we obtain the following result.

Proposition 2. The patient welfare S(r

) given in (14) is increasing in r

. The funder’s

budget constraint (15) is binding so that the funder’s optimal reimbursement rate ˜r

satisﬁes:

˜r

(˜r

) = B under the FFS scheme, and ˜r

(˜r

) = B under the BP scheme.

Proposition 2 implies that when serving a large patient population (e.g., the UK), the public

healthcare system can only provide partial coverage, and the funder always exhausts its

budget under both FFS and BP schemes (Donnelly and Swinford 2013).

We can further

establish the following comparative statics with respect to the budget B.

Corollary 3. Under both the FFS and BP schemes, the HCP’s optimal service rate ˜µ

(˜r

)

and the corresponding readmission rate δ(˜r

) are decreasing in B. However, the optimal

Under the full coverage case, each patient can obtain a positive utility (i.e., U(Λ, µ) > 0). Therefore,

the funder’s objective is no longer equivalent to maximizing the initial admission rate.

See “NHS is about to run out of cash top oﬃcial warns” at http://www.telegraph.co.uk/news/health/

news/10162848/NHS-is-about-to-run-out-of-cash-top-oﬃcial-warns.html.

reimbursement rate ˜r

, the initial admission rate

λ(˜r

), the waiting time per visit W (˜r

) and

the total waiting time T (˜r

) are increasing in B.

Corollary 3 reveals that, under the partial coverage case, the funder can aﬀord to oﬀer a

higher reimbursement rate with a higher budget. By considering this key result, we can apply

Corollary 2 to interpret all other results as stated in Corollary 3. To avoid repetition, we

omit the details. Hence, under both the FFS and BP schemes, increasing the funder’s budget

B can improve the patient access and reduce the readmission rate, but it can increase the

waiting time. This implication will play a role when we compare the performance between

the FFS scheme and the BP scheme.

4.4 Performance Comparison under Partial Coverage: FFS v.s.

So far, we have derived the equilibrium outcomes of diﬀerent performance metrics (the patient

welfare, initial admission rate, service rate, readmission rate and waiting times) associated

with both payment schemes under the partial coverage scenario as presented in Propositions

1 and 2. We now compare these performance outcomes between the FFS scheme and the

BP scheme. Speciﬁcally, our comparison yields the following results.

Proposition 3. Consider the case in which potential patients are partially covered under

both FFS and BP schemes. Then, relative to the FFS scheme,

1. both the patient welfare and the initial admission rate are higher under the BP scheme

(i.e., S(˜r

) > S(˜r

) and

λ(˜r

) >

λ(˜r

));

2. both the service rate and the readmission rate are lower under the BP scheme (i.e.,

˜µ

(˜r

) < ˜µ

(˜r

) and δ(˜r

) < δ(˜r

)); and

3. both the waiting time per visit and the total waiting time are higher under the BP

scheme (i.e., W (˜r

) > W (˜r

) and T (˜r

) > T (˜r

)).

Proposition 3 implies that when potential patients are partially covered under both FFS

and BP schemes, the BP scheme dominates the FFS scheme in terms of the patient welfare

and service quality, but the FFS scheme outperforms the BP scheme in terms of the waiting

time. These results can be explained as follows. Recall that the BP scheme pays the HCP a

ﬁxed amount for each admitted patient no matter how many times the patient is readmitted

to the system. Hence, the HCP under the BP scheme has incentives to reduce the service

rate so as to reduce the readmission rate. However, reducing the readmission rate attracts

higher initial admission rate under the BP scheme. Furthermore, from (7), if the potential

patients are partially covered, then the initial admission rate solves U(

λ, µ) = 0. Due to

a lower readmission rate, admitted patients under the BP scheme can tolerate a longer

waiting time in equilibrium such that W (˜r

) > W (˜r

) and T (˜r

) > T (˜r

). As maximizing

the patient welfare is equivalent to maximizing the initial admission rate under the partial

coverage scenario, the patient welfare under the BP scheme is also larger.

Proposition 3 is consistent with the ﬁndings of previous studies showing that the FFS

scheme is eﬀective for reducing the waiting time but not for improving service quality. For

example, Blomqvist and Busby (2013) show that the FFS scheme is eﬀective for reducing

the waiting time in Canada. Mot (2002) ﬁnds that, in the Netherlands, the abolition of the

FFS scheme has caused the waiting time to increase for the elective surgery.

In summary, by considering the partial coverage case that occurs when the patient popu-

lation is large, we ﬁnd that there is no dominant scheme. The BP scheme is more eﬀective for

improving the patient welfare and reducing the readmission rate; however, the FFS scheme

is more eﬀective for reducing the waiting time. Next, we examine the full coverage case

that occurs when the patient population is small. As we shall see, the results as stated in

Propositions 2 and 3 no longer hold.

5 Reimbursement Schemes under Full Coverage: FFS

and BP

We now consider the case when the potential patients are fully covered under both FFS and

BP schemes. This case is suitable for the public HCP in the rural area, which normally

has a low patient volume (DiChiara 2015).

Recall that under the full coverage scenario,

U(Λ, µ) ≥ 0 (see §4.1). This condition can be further simpliﬁed as

λ(µ) ≥ Λ after some

algebra, where

λ(µ) is given in (8). In other words, when the condition under the partial

coverage scenario is infeasible (i.e.,

λ(µ) ≥ Λ), the healthcare system will achieve the full

coverage scenario.

Under the full coverage scenario, the initial admission rate is given by Λ and the eﬀective

admission rate is Λ/(1 − δ(µ)). By noting that the average variable cost associated with

each patient’s visit is c · (1/µ), we can formulate the HCP’s problems under the full coverage

See “Rural Hospitals Address Medicare Reimbursement Cut Concerns” at

http://revcycleintelligence.com/news/rural-hospitals-address-medicare-reimbursement-cut-concerns.

scenario as follows:

(F F S) max

(µ) =



−



1 − δ(µ)

, (16)

s.t.

λ(µ) ≥ Λ,

(BP ) max

(µ) =



−

o(µ)



Λ, (17)

s.t.

λ(µ) ≥ Λ,

where the constraint

λ(µ) ≥ Λ guarantees that the healthcare system oﬀers full coverage to

all potential patients.

Proposition 4. Consider the case in which all potential patients are fully covered under

both FFS and BP schemes. Then, for any given reimbursement rate r

, s ∈ {f, b},

1. the HCP’s optimal service rate under the FFS scheme satisﬁes ˜µ

= max{µ, subject to,

λ(µ) ≥

Λ}, where

λ(µ) is given as in (8).

2. the HCP’s optimal service rate under the BP scheme satisﬁes

˜µ

(

, if

λ(µ

) ≥ Λ,

max{µ :

λ(µ) ≥ Λ} if

λ(µ

) < Λ.

Under both schemes, the optimal service rate ˜µ

, s = f, b, is independent of the funder’s

reimbursement rate r

The intuition behind Proposition 4 is as follows. First, observe from (16) that, the HCP’s

variable cost c/µ decreases in µ while its eﬀective arrival rate Λ/(1 − δ(µ)) increases in µ.

Hence, the HCP’s proﬁt Π

(µ) under the FFS scheme is always increasing in µ. These

observations imply that the HCP will choose the largest service rate in equilibrium that

ensures that the potential patients are fully covered (i.e., max{µ :

λ(µ) ≥ Λ}). Our results

reveal that, when the patient population is small, the FFS scheme creates an incentive for

the HCP to increase its service rate so as to generate as much revenue as possible.

Next, under the BP scheme, observe from (17) that the HCP’s objective is equivalent

to minimizing the variable cost c/o(µ). According to Lemma 1, the cure service rate o(µ)

is unimodal in µ and reaches its maximum at µ

. Therefore, Π

(µ) given in (17) is also

unimodal in µ and its corresponding mode is also µ

. When µ

is feasible under the full

coverage scenario (i.e.,

λ(µ

) ≥ Λ), it is natural that the HCP will choose µ

in equilibrium.

Whereas, when µ

is infeasible under the full coverage scenario(i.e.,

λ(µ

) < Λ), the HCP

will choose the largest service rate that ensures that the potential patients are fully covered

(i.e., max{µ :

λ(µ) ≥ Λ}).

Finally, unlike the partial coverage scenario, Proposition 4 implies that, under the full

coverage scenario, the HCP’s optimal service rate is independent of the funder’s reimburse-

ment rate. Because the funder cannot regulate the HCP’s service rate decision under both

schemes, the funder will select the smallest feasible reimbursement rate. Because all of the

performance metrics that we are going to compare such as the patient welfare, the initial

admission rate and the total waiting time only depend on the service rate and the initial ad-

mission rate, for ease of exposition, we thus omit the analysis of the funder’s reimbursement

rate decisions under the full coverage case. (See the online Appendix B for details.)

To guarantee that the healthcare system in equilibrium achieves the full coverage, the

equilibrium outcome under the partial coverage must be infeasible; that is, Λ ≤

λ(˜r

s ∈ {f, b}, where

λ(˜r

) represents the initial admission rate under the partial coverage case.

According to Proposition 3, this condition is equivalent to Λ ≤

λ(˜r

). By comparing diﬀerent

performance metrics associated with the FFS and BP schemes under the full coverage (i.e.,

Λ ≤

λ(˜r

)), we get the following results.

Corollary 4. Suppose that potential patients are fully covered under both FFS and BP

schemes (i.e., when Λ ≤

λ(˜r

)). Then,

1. If the potential patient population Λ is medium so that

λ(µ

) ≤ Λ ≤

λ(˜r

)

, then

the optimal service rates under the FFS and BP schemes are equal (i.e., ˜µ

= ˜µ

max{µ :

λ(µ) ≥ Λ}). Consequently, the patient welfare, the readmission rate, the

initial admission rate, the waiting time per visit and the total waiting time are the

same under both FFS and BP schemes.

2. If the potential patient population Λ is small so that Λ < min{

λ(µ

λ(˜r

)}, then the

BP scheme dominates the FFS scheme in terms of the patient welfare, service quality

and the total waiting time (i.e., S(˜µ

) < S(˜µ

), δ(˜µ

) > δ(˜µ

) and T (˜µ

) > T (˜µ

)).

The ﬁrst statement of Corollary 4 reveals that the FFS and BP schemes are equally

eﬃcient if the patient population is medium (i.e.,

λ(µ

) ≤ Λ ≤

λ(˜r

)). However, when the

patient population is very small (i.e., Λ < min{

λ(µ

λ(˜r

)}), the BP scheme dominates the

FFS scheme in terms of the patient welfare, service quality and the congestion level. These

results can be explained as follows. When

λ(µ

) ≤ Λ ≤

λ(˜r

), Proposition 4 reveals that

the HCP will select the largest service rate that makes the potential patients fully covered

under both FFS and BP schemes so that ˜µ

= ˜µ

. Therefore, the FFS and BP schemes are

equally eﬃcient in terms of all the performance metrics.

Note that the existence of

λ(µ

) ≤ Λ ≤

λ(˜r

) implicitly requires that

λ(µ

) ≤

λ(˜r

). We have shown in

Appendix A that there exist thresholds

t and

B (where the expression of

t and

B can be found in (27) in

Appendix A) such that

λ(µ

) ≤

λ(˜r

) if and only if t ≥

t and B ≥

Next, when Λ < min{

λ(µ

λ(˜r

)}, µ

is feasible under the full coverage and thus,

˜µ

= µ

. Because the optimal service rate under the FFS scheme ˜µ

is the largest one that

makes the potential patients fully covered, ˜µ

> ˜µ

= µ

. As the readmission rate δ(µ) is

increasing in µ, δ(˜µ

) > δ(˜µ

). From (6) we can easily know that the total waiting time T is

decreasing in o(µ). As o(µ) is maximized at µ

, T (˜µ

) < T (˜µ

). Thus, compared with the

FFS scheme, the service quality is better and the total waiting cost is smaller under the BP

scheme. Therefore, the patient welfare under the BP scheme is also larger than that under

the FFS scheme.

In summary, we can conclude from Corollary 4 that under the full coverage scenario, when

the patient population is medium, both schemes yield the same performance. However, when

the patient population is very small, the BP scheme dominates the FFS schemes in terms of

the patient welfare, service quality and the total waiting time.

Remark. When

λ(˜r

) < Λ ≤

λ(˜r

) so that the potential patients are fully covered under

the BP scheme but are partially covered under the FFS scheme, the comparison results are

similar to Proposition 3 and Corollary 4.

Speciﬁcally, when

λ(µ

) ≤

λ(˜r

), the results given

in Proposition 3 still hold: the BP scheme dominates the FFS scheme in terms of the patient

welfare and service quality but the FFS scheme outperforms the BP scheme in terms of the

congestion level. However, when

λ(µ

) >

λ(˜r

), there exists a threshold

Λ ∈ [

λ(˜r

λ(µ

)]

such that when Λ is relatively large (i.e., Λ >

Λ), the results given in Proposition 3 hold;

and when Λ is relatively small (i.e., Λ ≤

Λ), the results given in the second statement of

Corollary 4 hold (i.e., the BP scheme dominates the FFS scheme in terms of the patient

welfare, service quality and the total waiting time).

6 Conclusion Remarks

In this paper, we have presented a queueing model with endogenous arrival rate selected by

the patient and endogenous readmission rate controlled by the HCP (via the selected ser-

vice rate). By analysing a three-stage Stackelberg game with an embedded queueing model,

we compare the performance associated with the FFS and BP reimbursement schemes. By

considering the trade-oﬀ between service rate and service quality (in terms of the readmis-

sion rate), we obtain the following managerial insights. First, when potential patients are

partially covered, we ﬁnd that a higher service rate may reduce both the initial admission

rate and the eﬀective admission rate under both schemes. Second, we show that under the

partial coverage, a higher reimbursement rate can improve the service quality in terms of the

See Proposition B7 in the online Appendix B for details.

readmission rate but it can increase the waiting time under both FFS and BP reimbursement

schemes.

More importantly, by investigating the funder’s reimbursement decisions and comparing

the equilibrium outcomes associated with the two schemes, we ﬁnd that the BP reimburse-

ment scheme may not always dominate the FFS reimbursement scheme. Speciﬁcally, when

the potential patients are partially covered, the BP scheme dominates the FFS scheme in

terms of the patient welfare and service quality (i.e., the readmission rate); however, the

FFS scheme outperforms the BP scheme in terms of the waiting time per visit and the total

waiting time.

When the potential patients are fully covered, the BP scheme weakly dominates the

FFS scheme in terms of the patient welfare, service quality and the total waiting time. In

particular, when the patient population is medium, the FFS and BP schemes are equally

eﬃcient in terms of all performance metrics including the readmission rate, the waiting time

per visit, the total waiting time and the patient welfare. Overall, the implications of our

ﬁndings are as follows. First, when the size of the patient population is large, shifting from

FFS to BP can improve the patient welfare and reduce readmissions, but it can increase the

waiting time. Second, when potential patients are fully covered and the size of the patient

population is moderate, the two schemes yield the same outcomes. In such case, it seems

unnecessary to move from FFS to BP. However, when the size of the patient population is

very small, the BP scheme dominates the FFS scheme.

Our analysis represents an initial attempt to examine the performance of the healthcare

reimbursement scheme by capturing the strategic interactions among the patients, the HCP,

and the funder and by taking into account the relationship between service quality (in terms

of the readmission rate) and service speed. However, our model has several limitations that

we shall leave them as future research for further investigation. First, we have assumed

that patients have perfect information about the HCP’s service quality and it is of interest

to examine a situation when there is information asymmetry between patients and HCPs.

Second, our model is more suitable for elective non-urgent outpatient care. However, future

research is needed to examine the implications of diﬀerent payment schemes on urgent care

especially when the patient’s health condition may deteriorate rapidly over time so that

longer waiting time may adversely aﬀect patient’s health quality outcomes.

Another important area is to consider the competition among diﬀerent HCPs. In the

presence of the market competition, the non-cured patients may seek admissions from other

HCPs. (For example, Andritsos and Tang (2014) examine the impact of competition between

the private and public HCPs on the patient welfare in Europe.) Therefore, the HCP may

not generate more demand by reducing readmission rate. In view of this, it is of interest

to investigate the impact of competition on the HCPs’ choices of service rate and other

performance metrics such as the waiting time, the patient welfare and the admission rate.

Finally, another possible extension is to add the mortality rate into our analysis. An implicit

assumption in our model is that patients can deﬁnitely be cured in the long run. This

assumption is realistic for the elective surgeries. However, for serious illness such as diabetes

mellitus, the increase in the mortality rate is an inevitable consequence of the low service

quality (i.e., high readmission rate and/or long waiting time).

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Online Appendices

“The Impact of Reimbursement Policy on Patient Welfare,

Readmission Rate and Waiting Time in a Public Healthcare System:

Fee-for-Service vs. Bundled Payment”

Appendix A: Proofs of Lemmas and Propositions

Proof of Lemma 1. Taking the ﬁrst order condition (FOC) of o(µ) over µ yields

do(µ)

dµ

= 1 − δ(µ) − µδ

(µ) = 0,

which can be rewritten as

(1 − δ(µ))(1 − µg(µ)) = 0.

As g(µ) is increasing in µ, do(µ)/dµ crosses zero only once from above. Therefore, o(µ) is

quasi-concave in µ and the optimal service rate µ

solves µ

g(µ

) = 1.

Next, we prove that when µ ≤ µ

, o(µ) is concave in µ. As g(µ) is increasing in µ,

dg(µ)

dµ

(µ)(1 − δ(µ)) + (δ

(µ))

(1 − δ(µ))

> 0, (18)

which yields δ

(µ)(1 − δ(µ)) + (δ

(µ))

> 0. Then we can show that

o(µ)

dµ

= −2δ

(µ) − µδ

(µ)

= −2δ

(µ) +

µ(δ

(µ))

1 − δ(µ)

−

µδ

(µ)(1 − δ(µ)) + µ(δ

(µ))

1 − δ(µ)

= −δ

(µ) −

(µ)

1 − δ(µ)

do(µ)

dµ

−

µδ

(µ)(1 − δ(µ)) + µ(δ

(µ))

1 − δ(µ)

As o(µ) is increasing in µ when µ ≤ µ

, the above equation is negative for all µ ≤ µ

Therefore, o(µ) is concave in µ for µ ≤ µ

Proof of Corollary 1. Denote µ

= max{

λ(µ)} and µ

= max{

(µ)}. By noting that

o(µ) = µ(1 − δ(µ)) and o

(µ) = 1 − δ(µ) − µδ

(µ), we have o(µ) − µo

(µ) = µ

(µ). Hence,

taking derivative of

λ(µ) with respect to µ yields

λ(µ)

dµ

= o

(µ) −

θo

(µ)

Ro(µ) − µt

θo(µ)(Ro

(µ) − t)

[Ro(µ) − µt]

= o

(µ) −

θδ

(µ)t

[R(1 − δ(µ)) − t]

. (19)

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

From Lemma 1, we can show that d

λ(µ)/dµ < 0 for µ ≥ µ

. Thus µ

< µ

. Using Lemma

1 again, o

(µ

) < 0. Recall from (18) that δ

(µ)(1 − δ(µ)) + (δ

(µ))

> 0. Therefore,

λ(µ)

dµ



µ=µ

= o

(µ

) −

θtδ

(µ

)

[R(1 − δ(µ

)) − t]

−

2Rθ(δ

(µ

))

[R(1 − δ(µ

)) − t]

= o

(µ

) +

θt(δ

(µ

))

(1 − δ(µ

))[R(1 − δ(µ

)) − t]

−

2Rθ(δ

(µ

))

[R(1 − δ(µ

)) − t]

−

θt[δ

(µ

)(1 − δ(µ

)) + (δ

(µ

))

]

(1 − δ(µ

))[R(1 − δ(µ

)) − t]

= o

(µ

) −

θt[δ

(µ

)(1 − δ(µ

)) + (δ

(µ

))

]

(1 − δ(µ

))[R(1 − δ(µ

)) − t]

−

θt(δ

(µ

))

(R(1 − δ(µ

)) + t)

(1 − δ(µ

))[R(1 − δ(µ

)) − t]

(20)

< 0,

which shows that

λ(µ) is quasi-concave in µ. Taking derivative of

(µ) with respect to µ

and using (18),

(µ)

dµ

= 1 −

θRδ

(µ)

[R(1 − δ(µ)) − t]

, (21)

(µ)

dµ

= −

θRδ

(µ)

[R(1 − δ(µ)) − t]

−

2θR

(δ

(µ))

[R(1 − δ(µ)) − t]

−Rθ

(1 − δ(µ))



(δ

(µ))

(R(1 − δ(µ)) + t)

[R(1 − δ(µ)) − t]

(µ)(1 − δ(µ)) + (δ

(µ))

[R(1 − δ(µ)) − t]



< 0. (22)

Thus

(µ) is concave in µ. We next show that the mode of

λ(µ) is smaller than the mode

(µ), that is, µ

< µ

. By noting that o

(µ) = 1 − δ(µ) − µδ

(µ), (21) can be rewritten as

(µ)

dµ

= 1 −

µ[R(1 − δ(µ)) − t]

θ(Ro

(µ) − t)

µ[R(1 − δ(µ)) − t]

(µ)

θ(Ro

(µ) − t)

µ[R(1 − δ(µ)) − t]

Obviously, the maximum eﬀective admission rate should be positive; that is,

(µ

) > 0.

Because µ

should satisfy the FOC of

(µ, ), namely, d

(µ

)/dµ = 0, we have Ro

(µ

)−t <

0. Using (21) and substituting d

(µ

)/dµ = 0 into (19), we have

dλ(µ)

dµ



µ=µ

(µ

) − t

< 0, (23)

which implies that µ

> µ

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

Proof of Proposition 1. With a slight abuse of notion, we interchangeably use ˜µ

) and

˜µ

. From (10), the FOC of Π

(µ) can be written as

dΠ

(µ)

dµ

(µ)



−



(µ)

dµ

= 0, (24)

which can be rewritten as (12). Furthermore, the optimal service rate ˜µ

) should satisfy

(µ)

dµ

µ=˜µ

< 0. By recalling from Corollary 1 that

(µ) is concave in µ, ˜µ

) is larger

than the mode of

(µ). Furthermore,

(µ)

dµ



µ=˜µ

˜µ

(µ)

dµ



µ=˜µ

−

(˜µ

)

˜µ



−

˜µ



(µ)

dµ



µ=˜µ

< 0,

which shows that Π

(µ) is quasi-concave in µ and therefore, ˜µ

maximizes Π

(µ).

From (11), taking the derivative of Π

(µ) with respect to µ yields

dΠ

(µ)

dµ

(µ)

λ(µ)

(o(µ))



−

o(µ)



λ(µ)

dµ

. (25)

From (19) and Lemma 1, we can know that o

(µ) < 0 and d

λ(µ)/dµ < 0 for µ > µ

. To

ensure that Π

(µ) > 0, r

> c/o(µ) is required. Hence, if ˜µ

≥ µ

, dΠ

(µ)/dµ < 0. Therefore,

the optimal service rate selected by the HCP satisﬁes ˜µ

< µ

. According to Lemma 1, then

we have o

(˜µ

) > 0 and o

(˜µ

) < 0. Then from (20) and (25), we can get

λ(˜µ

)

dµ

< 0;

λ(˜µ

)

dµ

< 0.

λ(µ) is unimodal in µ, ˜µ

is larger than the mode of

λ(µ). From (19), we have

λ(µ

)

dµ

< 0 as

(µ

) = 0. Because ˜µ

< µ

and

λ(µ) is unimodal in µ, we have

λ(µ

) <

λ(˜µ

). Furthermore,

(µ)

dµ



µ=˜µ

c(o

(˜µ

)o(˜µ

) − 2(o

(˜µ

))

)

λ(˜µ

)

(o(˜µ

))

2co

(˜µ

)

(o(˜µ

))

λ(˜µ

)

dµ



−

o(˜µ

)



λ(˜µ

)

dµ

< 0.

Therefore, Π

(µ) is unimodal in µ and the optimal service rate ˜µ

satisﬁes dΠ

(˜µ

)/dµ = 0,

which can be rewritten as (13).

Proof of Corollary 2. With a slight abuse of notion, we interchangeably use ˜µ

) and

˜µ

. Diﬀerentiating (24) with respect to r

, we have

∂

(µ)

∂µ∂r

(µ)

dµ

. Recall from the proof

of Proposition 1 that

(µ)

dµ

µ=˜µ

< 0. According to the implicit function theory,

d˜µ

= −

∂

(µ)

∂µ∂r

∂

(µ)

∂µ



µ=˜µ

< 0.

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

Therefore, ˜µ

is decreasing in r

. As δ(µ) is increasing in µ, δ(r

) is also decreasing in r

Next, substituting ˜µ

into (9) and taking the derivative of

) over r

yields

)

(˜µ

)

dµ

d˜µ

> 0.

By using (3),

λ(r

)

= −δ

(˜µ

)

d˜µ

(˜µ

) + (1 − δ(˜µ

))

)

> 0.

Diﬀerentiating (25) with respect to r

, we have

∂

(µ)

∂µ∂r



µ=˜µ

λ(˜µ

)

dµ

. Recall from the proof

of Proposition 1 that

λ(˜µ

)

dµ

< 0 and

(˜µ

)

dµ

< 0. Using the implicit function theory again,

we have

d˜µ

= −

∂

(µ)

∂µ∂r

∂

(µ)

∂µ



µ=˜µ

< 0.

Therefore, ˜µ

is decreasing in r

. As δ(µ) is increasing in µ, δ(r

) is also decreasing in r

Next, substituting ˜µ

into (8) and taking the derivative of

λ(r

) over r

yields

λ(r

)

d˜µ

λ(˜µ

)

dµ

> 0.

Finally, plugging

(˜µ

) into (5) and (6), we get

W (r

) =

R(1 − δ(˜µ

))

−

; T (r

) =

−

θ(1 − δ(˜µ

))

Diﬀerentiating W (r

) and T (r

) with respect to r

yields

dW (r

)

= −

Rδ

(˜µ

)

d˜µ

> 0;

dT (r

)

= −

tδ

(˜µ

)

θ(1 − δ(˜µ

))

d˜µ

> 0.

Proof of Proposition 2. According to Corollary 2,

λ(r

) and r

) are increasing in

, and

λ(r

) and r

λ(r

) are increasing in r

. From (14), the funder’s objective is equivalent

to maximizing the initial admission rate

λ(r

). Therefore, the budget constraint (15) will

be binding and the funder’s optimal reimbursement rate satisﬁes ˜r

(˜r

) = B under FFS

scheme and ˜r

λ(˜r

) = B under BP scheme.

Proof of Corollary 3. According to Proposition 2, ˜r

(˜r

) = B and ˜r

λ(˜r

) = B. Recall

from the proof of Corollary (2) that

) is increasing in r

, and

λ(r

) is increasing in r

Therefore, as B increases, ˜r

should increase. Using Corollary (2) again, we can easily show

that the total waiting time T (˜r

), the waiting time per visit W (˜r

), and the initial admission

rate

λ(˜r

) are increasing in B, while ˜µ

(˜r

) and δ(˜r

) are decreasing in B.

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

Proof of Proposition 3. We show

λ(˜r

) <

λ(˜r

) by contradiction. Suppose this is not

true so that

λ(˜r

) ≥

λ(˜r

). Recall from Corollary 2 that

λ(r

) is increasing in r

. As ˜µ

)

is decreasing in r

and

λ(r

)

λ(˜µ

))

dµ

d˜µ

)

we can obtain

λ(˜µ

))

dµ

< 0.

From the proof of Proposition 1 we have

λ(˜µ

(˜r

))

dµ

< 0. Because

λ(µ) is unimodal in µ and

λ(˜r

) =

λ(˜µ

(˜r

)) ≥

λ(˜µ

(˜r

)) =

λ(˜r

), this implies that ˜µ

(˜r

) ≤ ˜µ

(˜r

). Furthermore,

according to Propositions 2, under the partial coverage, the budget constraints associated

with both FFS and BP schemes are binding, that is, ˜r

(˜r

) = ˜r

λ(˜r

) = B. Recall that

(˜r

) =

λ(˜r

)/(1 − δ(˜µ

(˜r

))). Then we have

˜r

λ(˜r

)

(˜r

)

λ(˜r

)

λ(˜r

)

(1 − δ(˜µ

(˜r

)) ≤ 1 − δ(˜µ

(˜r

))

since

λ(˜r

) ≥

λ(˜r

). Taking a close look at (10) and (11), when r

= ˜r

, we can utilize (3) to

rewrite the proﬁt function of the HCP under the BP scheme as

(µ) =



˜r

−

˜r

1 − δ(µ)



λ(µ) + Π

(µ).

Note that ˜µ

(˜r

) maximizes Π

(µ) and

λ(˜µ

(˜r

))

dµ

< 0. As

˜r

≤ 1 − δ(˜µ

(˜r

)),

dΠ

(µ)

dµ



µ=˜µ

(˜r

)



˜r

−

˜r

1 − δ(˜µ

(˜r

))



λ(µ)

dµ



µ=˜µ

(˜r

)

−

˜r

(˜µ

(˜r

))

λ(˜r

)

(1 − δ(˜µ

(˜r

)))

< 0.

As Π

(µ) is unimodal in µ (Proposition 1), this implies that ˜µ

(˜r

) > ˜µ

(˜r

), which leads

to a contradiction. Thus,

λ(˜r

) <

λ(˜r

) and ˜µ

(˜r

) > ˜µ

(˜r

). As δ(µ) is increasing in µ,

δ(˜r

) > δ(˜r

). From (14), we have

S(˜r

) = −β(Λ −

λ(˜r

)) < −β(Λ −

λ(˜r

)) = S(˜r

W (µ) =

R(1 − δ(µ))

−

; T (µ) =

−

θ(1 − δ(µ))

, (26)

it can be easily shown that both W (µ) and T (µ) are decreasing in µ. Since ˜µ

(˜r

) > ˜µ

(˜r

W (˜r

) < W (˜r

) and T (˜r

) < T (˜r

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

Proof of Proposition 4. Diﬀerentiating Π

(µ) given in (16) with respect to µ yields

dΠ

(µ)

dµ

1 − δ(µ)



−



Λδ

(µ)

(1 − δ(µ))

> 0.

Thus, Π

(µ) is increasing in µ. The HCP will select the largest service rate that satisﬁes the

full coverage requirement (i.e., max{µ|

λ(µ) ≥ Λ}).

Diﬀerentiating Π

(µ) given in (17) with respect to µ yields

dΠ

(µ)

dµ

(µ)

Λ,

which equals zero at µ = µ

. According to Lemma 1, o

(µ

) < 0. Thus,

(µ)

dµ



µ=µ

(µ

)

(µ

)

Λ < 0.

That is, Π

(µ) is unimodal in µ. Therefore, when µ

satisﬁes the full coverage requirement

(i.e., λ(µ

) ≥ Λ), the HCP’s optimal service rate under the full coverage equals µ

. Otherwise,

λ(µ

) < Λ, the full coverage requirement

λ(µ) ≥ Λ should be binding. Since

λ(µ) is

unimodal in µ, there exist two solutions that satisfy

λ(µ) = Λ. We denote these two solutions

as µ and ¯µ, respectively. Without loss of generality, we assume that µ < ¯µ. Taking the

derivative of the second term of

λ(µ) in (8) with respect to µ, we have

dµ



θ(1 − δ(µ))

R(1 − δ(µ)) − t



tθδ

(µ)

(R(1 − δ(µ)) − t)

> 0.

Therefore,

θ(1 − δ(µ))

R(1 − δ(µ)) − t

θ(1 − δ(¯µ))

R(1 − δ(¯µ)) − t

Because

λ(µ) =

λ(¯µ) = Λ, o(µ) < o(¯µ). Since Π

(µ) is increasing in o(µ), Π

(µ) < Π

(¯µ).

Thus, when

λ(µ

) ≤ Λ, the HCP’s optimal service rate under the full coverage equals

¯µ = max{µ|

λ(µ) ≥ Λ}.

Proof of Corollary 4. According to Proposition 4, when

λ(µ

) ≤

λ(˜r

) and

λ(µ

) ≤ Λ ≤

λ(˜r

), ˜µ

= ˜µ

= max{µ|

λ(µ) ≥ Λ}. While when Λ < min{

λ(µ

λ(˜r

)}, ˜µ

= µ

and

˜µ

= max{µ|

λ(µ) ≥ Λ}. Since max{µ|

λ(µ) ≥ Λ} is the larger solution of

λ(µ) = Λ and

Λ <

λ(µ

), ˜µ

= µ

< ˜µ

. And from (6) we have that the total waiting time T is decreasing

in o(µ). As o(µ) is maximized at µ

, T (Λ, ˜µ

) < T (Λ, ˜µ

). Also, since δ(µ) is increasing in

µ, δ(˜µ

) > δ(˜µ

In addition, we can show that n(˜µ

) < n(˜µ

) as n(µ) is increasing in µ. As n(˜µ

) < n(˜µ

)

and T (Λ, ˜µ

) < T (Λ, ˜µ

), we can show the following relationship about the patient’s utility:

U(Λ, ˜µ

) = R − [n(˜µ

) · t + θ · T (Λ, ˜µ

)] < R − [n(˜µ

) · t + θ · T (Λ, ˜µ

)] = U(Λ, ˜µ

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

In consequence, S(˜µ

) = Λ · U(Λ, ˜µ

) < Λ · U(Λ, ˜µ

) = S(˜µ

Below we will show that there exist thresholds

t and

B such that under the partial

coverage scenario,

λ(˜r

) ≥

λ(µ

) if and only if (iﬀ) t ≥

t and B ≥

B. To this end, we ﬁrst

show that when

(µ)

dµ



µ=µ

≥ 0,

λ(˜r

) <

λ(µ

). We have shown in the proof of Proposition

1 that

(µ)

dµ



µ=˜µ

(˜r

)

< 0. Because

(µ) is unimodal in µ,this implies that µ

< ˜µ

(˜r

Also, from the proof of Proposition 3 we have

λ(˜µ

(˜r

))

dµ

< 0. In addition, from (19), we get

λ(µ

)

dµ

< 0 since o

(µ

) = 0. As

λ(µ) is unimodal in µ and µ

< ˜µ

(˜r

), we have

λ(˜r

) <

λ(µ

Thus,

λ(˜r

) ≥

λ(µ

) requires that

(µ)

dµ



µ=µ

< 0.

Next, we show that

(µ)

dµ



µ=µ

< 0 iﬀ t > R · (1 − δ(µ

)) −

θRδ

(µ

). Substituting

t = R · (1 − δ(µ

)) −

θRδ

(µ

) into (21), we have

(µ)

dµ



µ=µ

= 0. From (21),

∂

(µ)

∂µ∂t

= −

2θRδ

(µ)

[R(1 − δ(µ)) − t]

< 0.

Thus,

(µ)

dµ



µ=µ

< 0 iﬀ t > R · (1 − δ(µ

)) −

θRδ

(µ

Finally, we show that when

(µ)

dµ



µ=µ

< 0, there exists a

B such that

λ(˜r

) ≥

λ(µ

) iﬀ

B ≥

B. To this end, we ﬁrst show that there exists a r

such that ˜µ

) = µ

. First, we

can show that when r

= c/µ

, the HCP’s corresponding optimal service rate should satisfy

˜µ

)|r

= c/µ

> µ

as the HCP’s marginal proﬁt (i.e., r

− c/µ) shall be positive. Next,

the FOC (24) implies that when r

goes to inﬁnity,

(˜µ

))

dµ

→ 0. As now

(µ)

dµ



µ=µ

< 0,

lim

→+∞

˜µ

) < µ

. Since ˜µ

) is decreasing in r

(Corollary 2), there exists a r

such

that ˜µ

) = µ

. Recall from Proposition 2 that in equilibrium, the budget constraint under

the partial coverage is binding, that is, ˜r

(˜r

) = B. Because

λ(r

) is increasing in r

and

˜µ

) = µ

, when B ≥ r

(µ

), ˜r

≥ r

. As ˜µ

) is decreasing in r

, ˜µ

(˜r

) ≤ ˜µ

) =

. We have shown in the proof of Proposition 3 that

λ(˜µ

(˜r

))

dµ

< 0. And from (19), we get

λ(µ

)

dµ

< 0 as o

(µ

) = 0. As

λ(µ) is unimodal in µ and ˜µ

(˜r

) ≤ µ

λ(˜r

) ≥

λ(µ

). Similarly,

when B < r

(µ

), we can show that ˜r

< r

, ˜µ

(˜r

) > ˜µ

) = µ

and

λ(˜r

) <

λ(µ

Let

t = R · (1 − δ(µ

)) −

θRδ

(µ

) and

B = r

(µ

), where ˜µ

) = µ

. (27)

Then

λ(˜r

) ≥

λ(µ

) iﬀ t ≥

t and B ≥

Appendix B: The Analysis of the Scenario When the

Potential Arrival Rate is Small

In this appendix, we study the scenario under which the potential patient size is relatively

small such that it is possible that no patients will leave without being treated. Again we

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

apply the backward induction to derive the patients’ queueing-joining decisions, the HCP’s

service rate decisions and the funder’s reimbursement decisions under both the FFS and BP

schemes.

B.1: Endogenous Patient Initial Admissions

Recall that if the potential initial admission rate is relatively large such that U(Λ, µ) < 0,

the potential patients are partially covered and the health care system reaches the partial

coverage scenario. While if the potential initial admission rate is relatively small such that

U(Λ, µ) ≥ 0, all the patients choose to seek admissions from the HCP and the health care

system reaches the full coverage scenario. We have derived the initial admission rate

λ(µ) and

the eﬀective admission rate

(µ) under the partial coverage case in the main context of this

paper. In this appendix, we still use

λ(µ) and

(µ) to represent the initial admission rate

and the eﬀective admission rate under the partial coverage scenario, respectively. However,

to avoid confusion, hereafter we let

λ(µ) and

(µ) represent the initial admission rate and

the eﬀective admission rate in equilibrium, respectively. Taking into account that the full

coverage case may occur in the equilibrium, based on (7), (8) and (9), we then have

λ(µ) = min{Λ,

λ(µ)} and

(µ) = min{n(µ)Λ,

(µ)}. (28)

Next, we will use (28) to derive the HCP’s service rate decision and the funder’s reimburse-

ment decision. To facilitate our analysis, we shall analyze both the partial coverage scenario

and the full coverage scenario separately. Since we have analyzed the partial coverage sce-

nario in the main context, here we will focus on analyzing the full coverage scenario. To be

consistent, we still use “

· ” to indicate the variables associated with the partial coverage

scenario and use “

· ” to indicate the equilibrium outcomes.

B.2 The FFS Scheme

We ﬁrst consider the FFS scheme. In this section, we ﬁrst determine the HCP’s optimal

service rate ˆµ

. Then, by anticipating the HCP’s service rate decision and the patient’s

admission behavior, we determine the funder’s reimbursement decision ˆr

B.2.1 The HCP’s Service Rate Decision

From (28), we know that the potential patients are fully covered iﬀ

λ(µ) ≥ Λ. When the

patients are fully covered,

λ(µ) = Λ and

(µ) = Λ/(1 − δ(µ)). By substituting

λ(µ) = Λ

and

(µ) = Λ/(1 − δ(µ)) into (10) and taking the full coverage requirement

λ(µ) ≥ Λ into

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

account, we can write the the HCP’s optimization problem under the full coverage case as

max

(µ) =



−



1 − δ(µ)

, (29)

s.t.

λ(µ) ≥ Λ. (30)

A close look of (29) reveals that as the service rate µ increases, the variable cost c/µ decreases

while the demand Λ/(1 − δ(µ)) increases. Therefore, the HCP’s proﬁt Π

(µ) is increasing in

µ. Because

λ(µ) is unimodal in µ (according to Corollary 1), there exist at most two service

rates such that the constraint (30) is binding (i.e.,

λ(µ) = Λ). And

λ(µ) ≥ Λ iﬀ µ is located

between them. As the HCP’s proﬁt Π

(µ) increases in µ, the optimal service rate under

the full coverage shall be the larger root of

λ(µ) = Λ. In other words, the HCP chooses the

largest service rate that satisﬁes the full coverage requirement.

Proposition B1. Suppose the potential patients are fully covered such that

λ(µ) = Λ. Under

the FFS scheme, for a given reimbursement rate r

, the HCP’s proﬁt Π

(µ) given in (29) is

increasing in µ. The optimal service rate equals ¯µ, where ¯µ is the larger root of

λ(µ) = Λ

and is independent of r

B.2.1.1 Equilibrium Service Rate

So far we have derived the HCP’s optimal service rate and the corresponding expected proﬁt

under both partial and full coverage scenarios. By comparing the HCP’s expected proﬁt

under these two scenarios, we can derive the HCP’s optimal service rate under the FFS

scheme as follows.

Proposition B2. Under the FFS scheme, for a given reimbursement rate r

, the HCP’s

optimal service rate satisﬁes

ˆµ

) =

(

˜µ

), if

λ(r

) < Λ,

¯µ, if

λ(r

) ≥ Λ.

(31)

Proposition B2 reveals that if the potential patient size is relatively large (i.e.,

λ(r

) < Λ),

then some patients will balk and the system reaches the partial coverage. While if the

potential patient size is relatively small (i.e.,

λ(r

) ≥ Λ), then the system ends up being

fully covered.

B.2.2 The Funder’s Reimbursement Decision

Anticipating the HCP’s service rate ˆµ

) given in (31), we now turn our attention to

determining the funder’s reimbursement decision. Similarly, we ﬁrst discuss the partial

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

coverage scenario and the full coverage scenario separately. Then, by comparing the patient

welfare associated with these two scenarios, we can ﬁgure out under which conditions which

scenario appears as the equilibrium outcome. Since we have analyzed the partial coverage

scenario in the main context, we now focus on the full coverage scenario. Plugging

λ(µ) = Λ

and ¯µ into (14), the funder’s problem under the full coverage scenario is

max

S(r

) = Λ · U(Λ) = Λ · (R − θ · T (Λ, ¯µ) − t · n(¯µ)), (32)

s.t. r

1 − δ(¯µ)

≤ B, (33)

Recall from Proposition B1 that the HCP’s optimal service rate ¯µ under the full coverage

is independent of r

. Therefore, the funder fails to regulate the HCP’s service rate decision

when the potential patients are fully covered. However, the threshold

λ(r

) (in (31)) which

determines when the full coverage scenario occurs does depend on r

. Thus, the funder can

indirectly control the full coverage scenario by regulating the threshold

λ(r

B.2.3 Equilibrium Outcome

So far we have derived the optimal equilibrium outcomes associated with the partial coverage

scenario and the full coverage scenario, respectively. Below we further investigate under

which conditions which scenario appears as the equilibrium outcome.

Proposition B3. The equilibrium outcome associated with the FFS scheme can be charac-

terized as follows.

1. When

λ(˜r

) < Λ, the potential patients under the FFS scheme in equilibrium are

partially covered. Therefore, in equilibrium, the funder’s reimbursement rate ˆr

satisﬁes

ˆr

= ˜r

and the HCP’s service rate ˆµ

(ˆr

) satisﬁes ˆµ

(ˆr

) = ˜µ

(˜r

2. When

λ(˜r

) ≥ Λ, the potential patients under the FFS scheme in equilibrium are fully

covered. Thus, in equilibrium, the funder’s reimbursement rate ˆr

satisﬁes ˜µ

(ˆr

) = ¯µ

and the HCP’s service rate ˆµ

(ˆr

) = ¯µ.

Proposition B3 reveals that when the potential initial arrival rate Λ is large enough such

that

λ(˜r

) < Λ, some patients in equilibrium will leave without being treated and the health

care system will end up with the partial coverage (i.e., ˆr

= ˜r

and ˆµ

(ˆr

) = ˜µ

(˜r

)). While

if the potential initial arrival rate Λ is relatively small such that

λ(˜r

) ≥ Λ, all the patients

can receive the treatment and the health care system will end up with the full coverage.

Recall from Proposition B1 that the HCP’s optimal service rate under the full coverage ¯µ

satisﬁes

λ(¯µ) = Λ and is independent of r

. The patient welfare under the full coverage

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

remains the same for any feasible reimbursement rate r

. However, the funder will not be

more generous than it has to be. Therefore, it will choose the smallest feasible reimbursement

rate under the full coverage. Since

λ(r

) is increasing in r

(according to Corollary 3), the

funder will select the reimbursement rate that just satisﬁes the full coverage requirement

λ(˜µ

(ˆr

)) =

λ(¯µ) = Λ.

B.3 The BP Scheme

In this section, we study the BP scheme. Similar to §B.2, to solve the equilibrium outcome

associated with the BP scheme, we ﬁrst determine the HCP’s optimal service rate ˆµ

. Then

by anticipating the HCP’s service rate decision and the patient’s admission behavior, we

determine the funder’s reimbursement decision ˆr

. Also, we will consider the partial coverage

scenario and the full coverage scenario separately. Since the partial coverage scenario has

been analyzed in the main context, we will focus on the full coverage scenario in the following.

B.3.1 The HCP’s Service Rate Decision

From (28), we know that to ensure the full coverage of potential patients, the HCP’s service

rate should satisfy

λ(µ) ≥ Λ. Plugging

λ(µ) = Λ and

(µ) = Λ/(1 − δ(µ)) into (11) and

taking the full coverage requirement

λ(µ) ≥ Λ into consideration, the HCP’s problem under

the full coverage becomes

max

(µ) = max



−

o(µ)



Λ, (34)

s.t.

λ(µ) ≥ Λ. (35)

In (34), the HCP’s objective under the full coverage is equivalent to minimizing the variable

cost c/o(µ). According to Lemma 1, the cure service rate o(µ) is unimodal in µ and reaches

its maximum at µ = µ

. Therefore, Π

(µ) given in (34) is also unimodal in µ and reaches

its maximum at µ = µ

. When the size of potential patients is small (i.e.,

λ(µ

) ≥ Λ), the

service rate µ

that maximizes Π

(µ) given in (34) is feasible under the full coverage. Thus,

the HCP will set µ = µ

in equilibrium. Whereas, when the potential initial admission

rate is large (i.e.,

λ(µ

) < Λ), µ

is infeasible and therefore, the full coverage requirement

λ(µ) ≥ Λ will be binding. Since

λ(µ) is unimodal in µ, there exist two roots of

λ(µ) = Λ.

As one can easily show that the second term of

λ(µ) in (8) (i.e., θ(1 − δ(µ))/R(1 − δ(µ)) − t)

is increasing in µ, o(µ) associated with the larger root of

λ(µ) = Λ shall be larger. Recall

that under the full coverage scenario, the HCP’s objective associated with the BP scheme

is equivalent to minimizing the variable cost c/o(µ). The HCP then shall choose the larger

root of

λ(µ) = Λ. Thus, we obtain the following results.

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

Proposition B4. Suppose the potential patients are fully covered such that

λ(µ) = Λ. Under

the BP scheme, for a given reimbursement rate r

, the HCP’s proﬁt Π

(µ) given in (34) is

unimodal in µ. Furthermore,

the HCP

s optimal service rate =

(

¯µ, if

λ(µ

) < Λ,

, if

λ(µ

) ≥ Λ,

(36)

where ¯µ is the larger root of

λ(µ) = Λ and is independent of r

B.3.1.2 Equilibrium Service Rate

In this section, we will derive the HCP’s optimal service rate under the BP scheme by

comparing the HCP’s expected proﬁt associated with the partial and full coverage scenarios,

respectively. When the potential initial admission rate is very large (i.e.,

λ(r

) < Λ), the

optimal service rate ˜µ

) under the partial coverage is feasible and it is the equilibrium

service rate. When the potential initial admission rate is relatively small such that the

optimal service rate under the partial coverage is infeasible (i.e.,

λ(r

) ≥ Λ), the system ends

up with the full coverage. In this case, by noting from Proposition 1 that

λ(µ

) <

λ(r

then if the potential initial admission rate is small enough (i.e.,

λ(µ

) > Λ), the HCP will

select the service rate µ

to minimize its variable cost. But if the potential initial admission

rate is in a moderate range (i.e,

λ(µ

) ≤ Λ ≤

λ(r

)), the full coverage requirement will be

binding (i.e.,

λ(˜µ

)) =

λ(¯µ) = Λ). We then have the following results.

Proposition B5. Under the BP scheme, for a given reimbursement rate r

, the HCP’s

optimal service rate satisﬁes

ˆµ

) =











˜µ

), if

λ(r

) < Λ,

¯µ, if

λ(µ

) < Λ ≤

λ(r

, if

λ(µ

) ≥ Λ.

(37)

B.3.2 The Funder’s Reimbursement Decision

We now turn our attention to determining the funder’s reimbursement decision under the

BP scheme. Anticipating the HCP’s service rate ˜µ

) given in (37), the funder selects

the reimbursement rate r

to maximize the patient welfare S(r

) in equilibrium. Plugging

λ(µ) = Λ and the HCP’s optimal service rate given in (36) into (14), the funder’s problem

under the full coverage becomes

max

S(r

) = Λ · U(Λ) =

(

Λ · (R − θ · T (Λ, ¯µ) − t · n(¯µ)), if

λ(µ

) < Λ,

Λ · (R − θ · T (Λ, µ

) − t · n(µ

)), if

λ(µ

) ≥ Λ,

(38)

s.t.

(

1−δ(¯µ)

≤ B, if

λ(µ

) < Λ,

1−δ(µ

)

≤ B, if

λ(µ

) ≥ Λ.

(39)

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

Because both ¯µ and µ

are independent of r

, the funder fails to regulate the HCP’s service

rate decisions when the potential initial patients are fully covered. However, the threshold

λ(r

) in (37) which determines when the full coverage scenario occurs does depend on r

Therefore, the funder can control the occurrence of the full coverage scenario by regulating

the threshold

λ(r

B.3.2.1 Equilibrium Outcome

Based on the optimal decisions of the patients, the HCP, and the funder associated with the

partial and full coverage scenarios, we now investigate under which conditions which scenario

appears as the equilibrium outcome.

Proposition B6. The equilibrium outcome associated with the BP scheme can be charac-

terized as follows.

1. When

λ(˜r

) < Λ, the potential patients under the BP scheme in equilibrium are partially

covered . Therefore, in equilibrium, the funder’s reimbursement rate ˆr

satisﬁes ˆr

= ˜r

and the HCP’s service rate ˆµ

(ˆr

) satisﬁes ˆµ

(ˆr

) = ˜µ

(˜r

2. When

λ(µ

) < Λ ≤

λ(r

), the potential patients under the BP scheme in equilibrium

are fully covered. In equilibrium, the funder’s reimbursement rate ˆr

satisﬁes ˜µ

(ˆr

) = ¯µ

and the HCP’s service rate ˆµ

(ˆr

) = ¯µ.

3. When Λ ≤

λ(µ

), the potential patients under the BP scheme in equilibrium are also

fully covered. In equilibrium, the funder’s reimbursement rate ˆr

satisﬁes ˜µ

(ˆr

) = µ

and the HCP’s service rate ˆµ

(ˆr

) = µ

Proposition B6 shows that when the results under the partial coverage is feasible (i.e.,

λ(˜r

) < Λ), the health care system will end up with the partial coverage (i.e., ˆr

= ˜r

and ˆµ

(ˆr

) = ˜µ

(˜r

)). Otherwise, all the potential initial patients in equilibrium will seek

admissions from the HCP. Recall that the patient welfare under the full coverage (given in

(38)) is independent of r

. Note that the funder will not be more generous than it has to be.

Thus, under the full coverage it will choose the smallest feasible reimbursement rate. Since

λ(¯µ) = Λ and

λ(r

) is increasing in r

, ˆr

that satisﬁes ˜µ

(ˆr

) = ¯µ is the smallest feasible

reimbursement rate under the full coverage.

B.4 Performance Comparison: FFS vs BP

So far, we have obtained the equilibrium outcomes of diﬀerent performance metrics associated

with the FFS and BP schemes, respectively. Because both the FFS and BP schemes may

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

achieve either the partial or the full coverage scenario, to compare the equilibrium outcomes

under these two schemes, we shall ﬁrst ﬁgure out the conditions under which which scenario

may occur. The corresponding conditions are characterized in the following lemma.

Lemma B1. 1. When Λ >

λ(˜r

), the potential patients are partially covered under both

FFS and BP schemes. Also, the initial admission rate associated with the BP scheme

is higher than that of the FFS scheme; i.e.,

λ(˜r

) >

λ(˜r

2. When

λ(˜r

) < Λ ≤

λ(˜r

), the potential patients are fully covered under the BP scheme

but are partially covered under the FFS scheme.

3. When Λ ≤

λ(˜r

), the potential patients are fully covered under both FFS and BP

schemes so that the endogenous initial arrival rate equals Λ.

Lemma B1 implies that to compare the performance of the two schemes, we should

consider three cases: 1) when Λ >

λ(˜r

), the potential patients are partially covered under

both FFS and BP schemes; 2) when Λ <

λ(˜r

), the potential patients are fully covered

under both FFS and BP schemes; 3) when

λ(˜r

) < Λ ≤

λ(˜r

), the potential patients are

fully covered under the BP scheme but are partially covered under the FFS scheme. We

have examined the ﬁrst two cases in the main body of this paper. In the following we will

only examine the third case

λ(˜r

) < Λ ≤

λ(˜r

Proposition B7. When

λ(˜r

) < Λ ≤

λ(˜r

) such that the potential patients are fully covered

under the BP scheme but are partially covered under the FFS scheme, relative to the FFS

scheme,

1. when max{

λ(˜r

λ(µ

)} ≤ Λ ≤

λ(˜r

), the BP scheme dominates the FFS scheme in

terms of the patient welfare (i.e., S(ˆr

) > S(ˆr

)) and service quality (i.e, the readmis-

sion rate δ(ˆr

) < δ(ˆr

)), while the FFS scheme outperforms the BP scheme in terms

of the waiting time per visit and the total waiting time (i.e., W (ˆr

) > W (ˆr

) and

T (ˆr

) > T (ˆr

)).

2. if

λ(˜r

) <

λ(µ

), there exists a

Λ ∈ [

λ(˜r

λ(µ

)] such that

(a) if Λ ∈ [

λ(˜r

Λ], the BP scheme dominates the FFS scheme in terms of the

patient welfare, service quality and the total waiting time (i.e., S(ˆr

) < S(ˆr

δ(ˆr

) > δ(ˆr

) and T (ˆr

) > T (ˆr

)).

(b) if Λ ∈ [

Λ,

λ(µ

)], the BP scheme dominates the FFS scheme in terms of the

patient welfare and service quality (i.e., S(ˆr

) < S(ˆr

), δ(ˆr

) > δ(ˆr

)), while the

FFS scheme outperforms the BP scheme in terms of the total waiting time (i.e.,

T (ˆr

) < T (ˆr

)).

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

FFS

Partial Coverage ˆµ

= ˜µ

Partial Coverage

ˆµ

= ˜µ

Full Coverage

λ(˜r

)

Full Coverage ˆµ

= ¯µ

λ(µ

)

λ(˜r

)

II: ˆµ

= ¯µ

I: ˆµ

= µ

(a) t <

t or/and B <

B; ¯µ = max{µ :

λ(µ) = Λ}

FFS

Partial Coverage ˆµ

= ˜µ

I: ˆµ

= µ

Partial Coverage ˆµ

= ˜µ

II: ˆµ

= ¯µ

λ(˜r

)

λ(˜r

)

Full Coverage ˆµ

= ¯µ

Full Coverage

λ(µ

)

(b) t ≥

t and B ≥

B; ¯µ = max{µ :

λ(µ) = Λ}

Figure 3: The Comparison of the Equilibrium Outcomes: FFS vs BP

As illustrated in Figure 3, when Λ ≥ max{

λ(˜r

λ(µ

)}, ˆµ

(ˆr

) = ¯µ and ˆµ

(ˆr

) = ˜µ

(˜r

In this case, the ﬁrst statement of Proposition B7 implies that the results for the case under

which the potential patients are partially covered uner both the FFS and BP schemes still

hold. That is, the BP scheme dominates the FFS scheme in terms of the service quality and

the patient welfare, while the FFS scheme is better in terms of the congestion level. However,

when

λ(˜r

) <

λ(µ

) and Λ ∈ [

λ(˜r

λ(µ

)], ˆµ

(ˆr

) = µ

(see Figure 3(a)). Then the second

statement of Proposition B7 implies that if the potential arrival rate Λ is relatively large

(i.e., Λ ∈ [

Λ,

λ(µ

)]), the results for the case under which the potential patients are partially

covered uner both the FFS and BP schemes hold. While if the potential arrival rate Λ

is relatively small (i.e., Λ ∈ [

λ(˜r

Λ]), the results for the case under which the potential

patients are fully covered uner both the FFS and BP schemes hold.

Proof of Propositions in Appendix B

Proof of Proposition B3. When

λ(˜r

) < Λ, the results given in Proposition 2 are feasible.

Therefore, ˆr

= ˜r

and ˆµ

(ˆr

) = ˜µ

(˜r

). When

λ(˜r

) ≥ Λ, the results under the partial

coverage scenatio are infeasible. The health care system will end up with the full coverage in

equilibrium. Recall that the patient welfare under the full coverage is independent of r

; see

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

(32). Therefore, the patient welfare remains the same for any given r

under the full coverage.

However, the funder will not be more generous than it has to be. Therefore, it will choose

the smallest feabile reimbursement rate. We next show that ˆr

, where ˜µ

(ˆr

) = ¯µ, is the

smallest one in the feasible set of the funder’s problem under the full coverage. According to

Proposition B1, the HCP’s optimal service rate ¯µ under the full coverage satisﬁes

λ(¯µ) = Λ.

Since

λ(ˆr

) =

λ(¯µ) = Λ, and

λ(r

) is increasing in r

, ˆr

is the smallest one that satisﬁes

the full coverage requirement. Furthermore, because

λ(ˆr

) = Λ ≤

λ(˜r

), ˜r

≥ ˆr

. Thus,

ˆr

λ(ˆr

) ≤ ˜r

λ(˜r

) = B, which implies that ˆr

also satisﬁes the budget constraint (33).

Therefore, ˆr

is the feasible solution under the full coverage scenario. As ˆr

is the smallest

one that satisﬁes the full coverage requirement, it is also the smallest one in the feasible set

of the funder’s problem under the full coverage.

Proof of Proposition B6. When

λ(˜r

) < Λ, the results given in Proposition 2 are feasi-

ble. Therefore, ˆr

= ˜r

and ˆµ

(ˆr

) = ˜µ

(˜r

). When

λ(˜r

) ≥ Λ, the results under the partial

coverage are infeasible. The health care system will end up with the full coverage in equi-

librium. As the patient welfare under the full coverage is independent of r

(see (38)), the

patient welfare remains the same for any given feasible r

under the full coverage. However,

the funder will not be more generous than it has to be. Therefore, under the full coverage it

will choose the smallest feasible reimbursement rate. We next show that when Λ ≤

λ(˜r

), ˆr

where ˜µ

(ˆr

) = ¯µ, is the smallest feasible reimbursement rate. Since

λ(ˆr

) =

λ(¯µ) = Λ and

λ(r

) is increasing in r

, ˆr

is the smallest one that satisﬁes the full coverage requirement.

Furthermore, as

λ(ˆr

) = Λ ≤

λ(˜r

), ˆr

≤ ˜r

. Thus, ˆr

λ(ˆr

) ≤ ˜r

λ(˜r

) = B, which implies that

ˆr

also satisﬁes the budget constraint (39). As ˆr

is the smallest one that satisﬁes the full

coverage requirement, it is also the smallest one in the feasible set of the funder’s problem

under the full coverage.

Proof of Proposition B7. When

λ(˜r

) < Λ ≤

λ(˜r

), the potential patients are partially

covered under the FFS scheme and are fully covered under the BP scheme. We ﬁrst consider

the case when

λ(˜r

) ≥ Λ >

λ(˜r

) ≥

λ(µ

). From Propositions B3 and B6, we know that

in equilibrium, ˆµ

(ˆr

) = ˜µ

(˜r

), ˆµ

(ˆr

) = ¯µ, and

λ(ˆr

) =

λ(˜r

) < Λ =

λ(ˆr

). Because

λ(µ) is unimodal in µ, ¯µ is the larger root of

λ(µ) = Λ,

λ(˜µ

(˜r

))

dµ

< 0 (shown in the

proof of Proposition 3) and

λ(˜r

) ≤ Λ, ¯µ < ˜µ

(˜r

). Hence, δ(ˆr

) < δ(ˆr

). Since both

W (µ) and T (µ) are decreasing in µ (see (26)), W (ˆr

) = W(˜r

) < W(¯µ) = W(ˆr

) and

T (ˆr

) = T (˜r

) < T (¯µ) = T (ˆr

). As n(µ) increases in µ, n(ˆr

) < n(ˆr

). When patients are

fully covered, U(Λ, ˆµ

(ˆµ

)) > 0. Based on (14) and (38), we have S(ˆr

) = −β(Λ − ˜µ

(˜r

)) <

0 < Λ · U(Λ, ˆµ

(ˆµ

)) = S(ˆr

We next consider the scenario

λ(˜r

) <

λ(µ

). First, when

λ(˜r

) ≥ Λ >

λ(µ

), from

Propositions B3 and B6, ˆµ

(ˆr

) = ˜µ

(˜r

), ˆµ

(ˆr

) = ¯µ, and

λ(ˆr

) =

λ(˜r

) < Λ =

λ(ˆr

Comparison of Reimbursement Schemes: FFS vs BP Online Appendices

Similar to the above analysis, we can show that δ(ˆr

) < δ(ˆr

), W(ˆr

) < W (ˆr

), T(ˆr

) < T (ˆr

)

and S(ˆr

) < S(ˆr

When Λ ∈ [

λ(˜r

λ(µ

)], utilizing Propositions B3 and B6 again, we have ˆµ

(ˆr

) =

˜µ

(˜r

), ˆµ

(ˆr

) = µ

λ(ˆr

) =

λ(˜r

) < Λ =

λ(ˆr

). Because ¯µ is the larger solution of

λ(µ) = Λ,

λ(µ) is unimodal in µ and Λ ≤

λ(µ

), µ

< ¯µ. We have shown above that

¯µ < ˜µ

(˜r

). Thus, ˆµ

(ˆr

) = µ

< ˜µ

(˜r

) = ˆµ

(ˆr

) and δ(ˆr

) < δ(ˆr

). Based on (14) and

(38), we have S(ˆr

) = −β(Λ − ˜µ

(˜r

)) < 0 < Λ · U(Λ, ˆµ

(ˆµ

)) = S(ˆr

Finally, we show that there exists a

Λ such that T (ˆr

) < T (ˆr

) iﬀ Λ <

Λ. First, because

potential patients are fully covered under the BP scheme but are partially covered under the

FFS scheme, from (6) and (26), we get

T (ˆr

) = T (Λ, ˆµ

(ˆr

)) = T (Λ, µ

) =

o(µ

) − Λ

T (ˆr

) = T (

λ(˜r

), ˜µ

(˜r

)) =

−

θ(1 − δ(˜r

))

where T (ˆr

) is increasing in Λ while T (ˆr

) is independent of Λ. Therefore, if there exist Λ

and Λ

such that T (ˆr

) = T (Λ

, ˆµ

(ˆr

)) > T (ˆr

) and T (ˆr

) = T (Λ

, ˆµ

(ˆr

)) < T (ˆr

), then

there must exist a

Λ such that T(ˆr

) < T (ˆr

) iﬀ Λ <

Λ. Denote Λ

λ(µ

) and Λ

λ(˜r

Then, when Λ = Λ

, we can show that

T (ˆr

) = T (

λ(µ

), µ

) =

−

θ(1 − δ(µ

))

−

θ(1 − δ(˜r

))

= T (ˆr

where the inequality is due to that δ(µ) is increasing in µ and µ

< ˜µ

(˜r

). When Λ = Λ

based on (6) and (8), we can show that

T (ˆr

) =

o(˜r

) −

λ(˜r

)

o(µ

) −

λ(˜r

)

= T (

λ(˜r

), ˆµ

(ˆr

)) = T (ˆr

where the inequality is due to that o(µ) achieves its maximum at µ

. We thus complete the

proof.