A Financial Model for Sustaining the Medical Home

Ping HuangRegenstrief Center for Healthcare Engineering, Purdue University

West Lafayette, IN 47907, USATel.: +1-765-494-9049 Fax: +1-765-494-3023 Email: huang74@purdue.edu

Mark LawleyWeldon School of Biomedical Engineering, Purdue University

West Lafayette, IN 47907, USA

Ken MusselmanRegenstrief Center for Healthcare Engineering, Purdue University

West Lafayette, IN 47907

Fei PanKrannert School of Management, Purdue University

West Lafayette, IN 47907, USA

Steve WitzRegenstrief Center for Healthcare Engineering, Purdue University

West Lafayette, IN 47907

March 1, 2010

Title: A Financial Model for Sustaining the Medical Home

Abstract

Because primary care is perceived to be under-valued, many see the need for new modes of

delivery to improve both its effectiveness and appeal. The Patient-Centered Medical Home

is an emerging strategy that augments fee-for-service consultation with coordinated care ser-

vices provided by a physician led team. Proponents believe this will improve patient access

and outcomes, while reducing avoidable usage. Financing will come through fee-for-service,

capitation, and incentive. Our goal is to determine the conditions under which the medical

home is financially sustainable. We create a model to help identify the appropriate levels

of capitation, incentive, and resource investment for medical home services. Because these

decisions are distributed between payers and providers, our objective is to characterize an

equilibrium representing the set of strategies that neither party has reason to modify.

Keywords: Primary Care, Healthcare Reimbursement, Medical Home, Payment Structure

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1 Introduction

Evidence suggests that primary care is the key to effective healthcare delivery (Starfield et

al. 2005). Access to primary care is consistently associated with improved health outcomes

and lower costs (Franks and Fiscella 1998). For example, in both England and the U.S.,

each additional primary care physician per 10,000 persons is associated with a decrease

in mortality rate of up to 10% (Gulliford 2002; Macinko et al. 2007). Also, preventable

hospitalizations are about twice as likely in areas with a shortage of primary care physicians

(Parchman and Culler 1994; 1999). Unfortunately, primary care is in crisis (American College

of Physicians 2006; Bodenheimer 2006; Rosenblatt et al. 2006). Only 30% of the nation’s

doctors practice primary care (as opposed to half in 1960), and there is currently a 30%

shortage of primary care physicians (Association of American Medical Colleges 2009; Halsey

2009). In 2008, fewer than 10% of graduating medical students chose to enter primary care,

with the rest pursuing more lucrative specialties (American Academy of Family Physicians

2009). This is occurring at a time when the population is aging and chronic disease is

reaching epidemic proportions (Health Resources and Services Administration 2008; Institute

of Medicine 2008; Phillips and Starfield 2003).

For these reasons, there is a growing movement to restructure primary care through

the Patient-Centered Medical Home (PCMH) (Berenson et al. 2008). First championed by

the American Academy of Pediatrics, the PCMH is broadly defined as primary care that

is “accessible, continuous, comprehensive, family-centered, coordinated, compassionate, and

culturally effective” (Patient Centered Primary Care Collaborative 2008). In 2007, four

primary care specialty societies, representing more than 300,000 internists, family physicians,

pediatricians, and osteopaths, agreed on the joint principles of the PCMH (Abrams 2008).

These included “personal physician; whole-person orientation; safe and high-quality care

(e.g., evidence-based medicine, appropriate use of health information technology); enhanced

access to care; and payment that recognizes the added value provided to patients who have

a patient centered medical home” (American College of Physicians 2008).

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In the ideal PCMH, each patient would have a primary care physician who heads a

team of professionals, including nurses, pharmacists, and nutritionists, to provide round-

the-clock access to care. The primary care physician would help patients get specialty

care when needed and would keep track of treatments and inform specialists of patient

progress. Using electronic medical records, doctors and patients would have easy access to

medical information, and patients with chronic ailments would routinely receive help from

the medical home team in managing their conditions (Barr and Ginsburg 2006; Grumbach

and Bodenheimer 2002).

Currently, private insurers are sponsoring more than twenty PCMH demonstration ini-

tiatives around the country. Medicaid has pilot programs in thirty-one states, and Medicare

is launching eight demonstration programs (Patient Centered Primary Care Collaborative

2008). These pilots are intended to verify the anticipated benefits to the whole healthcare

system, including patients, providers and payers (Colorado Clinical Guidelines Collabora-

tive 2009; Sepulveda et al. 2008). Early evidence indicates that clinical care coordination,

on-going communication with patients, concentration on evidence based measures, and early

intervention on emerging problems can lead to better care at a lower cost (Patient Centered

Primary Care Collaborative 2008). For example, IBM employees using PCMH have health-

care premiums 6% lower for family coverage and 15% lower for single coverage, and they pay

26% to 60% less in overall health costs than employees at other large companies (American

Academy of Family Physicians 2008). Also, the Geisinger Medical Home Initiative found a

nearly 20% reduction in hospital admissions and approximately 7% savings across the board

in medical costs (Paulus et al. 2008).

While most healthcare stakeholders believe that the PCMH will bolster primary care and

its associated benefits, reimbursement questions are in a state of flux (Abrams et al. 2009).

The suggested approach is to provide the primary care clinic with three sources of revenue,

(1) fee-for-service (FFS) reimbursements for patient visits and other coded activities, (2)

per member per month (PMPM) capitation payments to cover non-billable PCMH services,

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and (3) incentive payments for improved quality and reduction in avoidable usage (Patient

Centered Primary Care Collaborative 2008). Although there has been some activity in

estimating PMPM, with estimates ranging from $3 to $150 (American Medical Association

2010; Deloitte 2008; Kuraitis 2009; Zuckerman et al. 2009), we are aware of no research

that addresses how these three revenue streams should be structured and integrated into a

sustainable package that is financially attractive for both payers and providers.

Thus, the purpose of our research is to investigate how the PCMH should be financed

and reimbursed. Specifically, we are interested in determining the reimbursement conditions

under which the PCMH is financially sustainable. By “financial sustainability” we mean

that the financial benefits accruing to both payers and providers are sufficient to keep the

medical home continuing in practice. These benefits can only come from the improvements

in patient outcomes that lead to more effective use of the healthcare system. This change in

how patients use the system alters the financial flows that payers and providers experience.

Payers still collect premiums from patients and their employers, and patients still use the

healthcare system according to their needs. But, for the PCMH to be sustainable, proactive

medical home practices must bring reductions in higher cost healthcare utilization (avoidable

inpatient admissions, emergency department visits, surgeries, redundant tests, etc.) so that

a sufficient portion of the reimbursements currently going to non-primary care providers

finds its way into primary care.

Our goal is to create a model that helps answer a variety of questions, such as:

• What are the appropriate levels of PMPM payments and performance incentives?

• How much should the primary care provider invest in medical home services?

• How much improvement in patient outcomes is necessary to sustain the medical home?

These and other fundamental questions can only be analyzed by modeling the complex

interactions that occur among payers, providers, and patients. Game theory (Fudenberg and

Tirole 1991) provides one approach for doing this. It assumes a set of interacting players

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with different goals who select strategies to advance their own objectives. The purpose of

the analysis is to characterize the “equilibrium” point, that is, the set of player strategies

that yields a solution that no player benefits from modifying. In the PCMH, this includes

the levels of (1) capitation and incentive given by the payer and (2) resources dedicated by

the provider to PCMH services, two key elements that must be balanced if the PCMH is to

be sustainable.

The remainder of the paper is structured as follows. Section 2 develops a model of

healthcare utilization reduction to capture the improved patient outcome and then constructs

the above game by defining financial objective functions for the payer and the provider

that represent their respective revenues and costs. Section 3 develops the “best response

function” for both payer and provider and then characterizes an equilibrium point, which

is in closed form and is unique for this model. For illustrative purposes, Section 4 applies

this model to a “representative” clinic constructed using data from the Medical Expenditure

Panel Survey provided by the Agency for Healthcare Research and Quality (AHRQ 2007)

and from the Physician Production Survey provided by the Medical Group Management

Association (MGMA 2008). This application is used to illustrate the workings of the model

and to develop managerial insights useful for decision makers. Finally, Section 5 provides a

conclusion. Derivations, proofs, and other technical details are given in the appendix.

2 A Financial Model for the Medical Home

Figure 1 provides a conceptual framework for the patient and financial flow that need to be

captured. We consider the key stakeholders to be the primary care practice (referred to as

the “provider”) and the insurer (referred to as the “payer”) who reimburses the practice’s

services, since they are the two parties most affected by the medical home reimbursement

structure. While we recognize that, in reality, a provider interacts with many payers, we

assume here the provider negotiates with one payer for a reimbursement and incentive pack-

5

age to cover that payer’s patients. Conversely, the payer would normally interact with

many providers, but for the purposes of this model we assume a single provider. Additional

stakeholders to be considered include specialists, hospitals, pharmacies, laboratories, and so

forth that provide services for medical home patients. These must be considered since those

services generate costs that the payer must cover for medical home patients.PatientsEmployers / Purchasers Insurers / PayersShare of Premium Premium PaymentCo-pay, DeductiblePremium Primary CareSpecialist Rx Lab Hospital HomeCare

Figure 1: Patient Centered Medical Home Conceptual Framework

In the current system, the provider is paid through Fee-for-Service (FFS) for face-to-face

patient consultations (Centers for Medicare and Medicaid Services 2009). Reimbursement

depends on the Relative Value Units (RVUs) coded for services provided during the patient

visit. Under this setting, the provider is not paid for time spent on activities such as coor-

dinating patient care arrangements, tracking patient adherence to best practices, ensuring

patient therapeutic compliance, or on-going patient communications, since such activities

are team-based and not captured in RVU systems (Zuckerman et al. 2009). Further, the

current system offers no financial incentive for improved patient outcomes.

For the purposes of this model, we assume a hybrid reimbursement package that includes

(1) FFS for coded patient visits to the clinic, (2) a PMPM capitation fee to cover medical

home services beyond those included in FFS, and (3) performance incentive for better patient

outcomes. Table 1 provides notation required for the models developed in the following

subsections.

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Table 1: Selected Notation

COP Provider operating cost per monthCIT Additional monthly IT cost required by medical home modeld Insurance premium per patient per monthk1 RN/LPN compensation per hourk2 Physician compensation per hourN Number of patients in the practice panelr Revenue per Total RVUR Number of Total RVUs produced per month from FFS before medical home modelt Number of hours worked by a physician per montht(non−FFS) Number of hours spent by physician on non-FFS medical home services per monthw Portion of savings from reduced healthcare utilization used as incentive, 0 < w < 1x Portion of the total medical home fee used for medical home services, 0 < x < 1y Per-member, per-month (PMPM) medical home feez Percent of physician time spent on medical home service, z = t(non−FFS)/tα Provider cost of medical home service per hour, α = k1 + ρk2

β Payer’s portion of FFS reimbursement, β = 1− CoPay, 0 < β < 1γ Percent of the practice panel covered by medical home fee, 0 < γ < 1η RN/LPN cost per Total RVUξ Payer’s expenditure on healthcare utilization before medical home modelξ(MHz) Payer’s expenditure on healthcare utilization after medical home model at level z∆ξ(z) Functional model of healthcare utilization reduction, (ξ(MHz) − ξ)/ξρ Physician time over RN/LPN time on medical home services, 0 < ρ < 1τ Revenue after provider cost per Total RVU

2.1 Modeling Reduction in Healthcare Utilization

As discussed in Section 1, there is growing evidence that better primary care leads to im-

proved patient outcomes, fewer emergency department visits, and inpatient hospitalizations.

We assume that the payer uses a portion of the savings from reduced healthcare utiliza-

tion as performance incentive. This portion is specified by an incentive rate, w, where

w ∈ (0, 1). To compute the incentive, we define a healthcare utilization reduction func-

tion, ∆ξ(z), that approximates the mapping from the proportion of physician time used

for non-FFS medical home (MH) services, z = t(non−FFS)/t, to the reduction in healthcare

utilization, (ξ(MHz) − ξ)/ξ.

In mathematical analysis, the Weierstrass approximation theorem states that every func-

tion defined on a closed interval [a, b] can be uniformly approximated as closely as desired

by a polynomial function (Browder 1995). To model the reduction in healthcare utilization,

we posit the following polynomial: ∆ξ(z) = a3z3 + a2z

2 + a1z + a0, where, as noted above,

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z = t(non−FFS)/t is the proportion of physician time used for non-FFS MH services, and

the ai coefficients are parameters to be estimated from data collected by the medical home

stakeholders. Polynomials of either higher or lower degree can also be used, depending on

the precision desired.

This polynomial function (illustrated in Figure 2) captures the assumptions that (1) if

the provider spends zero time on MH services, then the resulting reduction in other higher

cost healthcare services will be zero. Furthermore, if the provider spends all his/her time on

MH services, then the usage of other healthcare services might actually be higher than before

since clinic access for face-to-face visits billed as FFS will be severely restricted; (2) as the

amount of provider time on MH services increases away from zero, the utilization of avoidable

healthcare services decreases; (3) as the amount of provider time on MH services continues to

increase, there is some critical point at which a maximum reduction in avoidable healthcare

utilization is realized; (4) beyond that critical point, additional increases in provider time

spent on MH services yield no further benefit and might actually increase the occurrence of

avoidable healthcare utilization due to more limited access to FFS visits.

Figure 2: Functional Model of Reduction in Healthcare Utilization

Now let us discuss ∆ξ(z) = a3z3 + a2z

2 + a1z + a0, z ∈ [0, 1]. We have the following

conditions so that ∆ξ satisfies the four assumptions listed above. First, for assumption (1),

we must have ∆ξ(0) = 0 and ∆ξ(1) > 0. For assumption (2), we have ∆ξ′(0) < 0. For

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assumption (3), there exists a point, say z∗, at which ∆ξ reaches its minimum. By the

nature of ∆ξ, z∗ is the unique value such that ∆ξ′(z∗) = 0. Last, for assumption (4), the

cubic model implies a point, say z, at which ∆ξ shifts from concave up to concave down,

that is, ∆ξ′′(z) = 0.

To summarize the above, we provide the following:

Definition 1. (A functional model of healthcare utilization reduction) Let the

healthcare utilization reduction, (ξ(MHz) − ξ)/ξ, be approximated by ∆ξ(z) = a3z3 + a2z

2 +

a1z +a0, where z ∈ [0, 1] is the proportion of physician time used for non-FFS medical home

services. The function ∆ξ(z) satisfies the following conditions:

(1) ∆ξ(0) = 0 (i.e., a0 = 0) and ∆ξ(1) > 0 (i.e., a1 + a2 + a3 > 0);

(2) ∆ξ′(0) < 0 (i.e., a1 < 0);

(3) ∆ξ′(z) has a unique root z∗ ∈ (0, 1) such that ∆ξ′(z∗) = 0, and ∆ξ(z∗) ≥ −1;

(4) ∆ξ′′(z) has a root z ∈ (0, 1) such that ∆ξ′′(z) = 0 (i.e., a2a3 < 0 and |a2| < 3|a3|).

Given the definition, we now formulate the “best-proportion” of physician time used for

non-FFS medical home services. By “best-proportion” we mean the derived MH services

time that maps to the maximum reduction of healthcare utilization.

Proposition 1. Let ∆ξ(z) satisfy the conditions of Definition 1. Then ∆ξ(z) reaches its

minimum (the maximum reduction is realized) at

z∗ =−a2 +

√a2

2 − 3a1a3

3a3

,

and condition (3) in Definition 1 is equivalent to the following:

(3’) a22 − 3a1a3 ≥ 0, a1 + 2a2 + 3a3 > 0, and 27a2

3 + a2(2a22 − 9a1a3)− 2(a2

2 − 3a1a3)32 ≥ 0

Remark 1. If the estimated values do not satisfy conditions (1) - (4), then it indicates that

the medical home implementation does not satisfy the four basic assumptions given above.

In this case, we believe that the implementation is inherently unsustainable.

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Earlier discussions suggest that spending too much time on MH services might actually

increase the occurrence of avoidable healthcare utilization due to limited access to FFS visits.

The following proposition formulate the upper bound of physician time used for MH services.

Proposition 2. Let ∆ξ(z) satisfy the conditions of Definition 1. There exists a point z0 > 0

such that ∆ξ(z0) = 0, that is, the reduction becomes zero at

z0 =−a2 +

√a2

2 − 4a1a3

2a3

.

Remark 2. This suggests that the proportion of physician time used for MH services, z =

t(non−FFS)/t , should not exceed z0, i.e., 0 < z ≤ z0.

Although early evidence from pilot programs indicates that PCMH activities lead to

favorable reductions in avoidable healthcare utilization, there is little evidence on how in-

vestments in MH services truly map to improvements in patient outcomes and cost reduction

(Barr 2008). Thus, our posited model represents a “best-guess” (and a “first-guess” since

there is no precedent in the literature). If this guess does not provide a reasonable approx-

imation of the true relationship, then MH services are not likely to lead to cost savings

through reduction in avoidable healthcare utilization, and the PCMH is unlikely to be sus-

tainable. We will therefore use this model as a working hypothesis, noting that the on-going

medical home pilot studies may employ this model to evaluate the demonstration projects.

2.2 Revenues and Costs for the Provider

We assume that the provider gets monthly payments from the payer which include FFS

reimbursement for patients seen in face-to-face visits, a PMPM capitation fee, $y, for each

patient enrolled for MH services, and a performance incentive with a rate w ∈ (0, 1). Here,

w is the portion of the payer’s savings in payments for avoidable healthcare utilization used

as incentive. The payer intends to fund both PMPM and incentives from an anticipated

improvement in patient outcomes, which results in less healthcare usage.

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Let N be the number of patients in the practice panel covered by the payer, and let γ be

the percentage of these patients covered by PMPM. (We note that PMPM payments might

be provided only for selected classes of patients with high avoidable usage such as those

with chronic conditions and co-morbidities.) Then the provider receives a PMPM payment

totaling $yNγ from the payer. In a medical home, the physician will typically supervise a

care management team that includes nurses and staff to track patient history and adher-

ence, review and discuss cases, maintain electronic health records (EHR), communicate with

patients, etc. Let α be the hourly provider cost for these services, including both physi-

cian and non-physician staff time. Then the total number of hours dedicated to providing

these services will be xyNγ/α, where x is the proportion of PMPM revenue allocated to

support the physician-led team. Our review of the literature indicates that for every hour

the non-physician staff team spends on these MH services, the corresponding physician will

have to spend ρ hours, where ρ ∈ (0, 1). For example, for every hour on MH services, if

the physician spends 15 minutes with the team, then ρ = 0.25. Therefore, the resources to

provide MH services include xyNγ/α non-physician staff hours and t(non−FFS) = ρxyNγ/α

physician hours.

In current practice, physician productivity is measured by Total RVUs (or Work RVUs)

generated by FFS visits (Centers for Medicare and Medicaid Services 2009). Assume that

a physician works t hours and produces R Total RVUs per month, and let τ represent

the revenue after provider cost per RVU. Then τR is the revenue from FFS under current

practice. With the PCMH, a physician spends t(non−FFS) hours on MH services (non-FFS

activities), and thus, the revenue from FFS is τR · (1 − t(non−FFS)/t) = τR · (1 − z) under

the PCMH.

We will now develop objective functions for the provider and payer to set up the game

model. In these objectives, we will assume that z ∈ (0, z0] (see Remark 2 in section 2.1).

The provider’s objective consists of the cost of providing FFS patient consultations, the

cost of providing non-FFS MH services, the cost of maintaining MH information technology

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hardware and software along with depreciation or amortized acquisition, and other operat-

ing costs which exclude physician/staff compensation. Further, when the physician spends

t(non−FFS) hours on MH services, those hours will not be used for FFS visits, and thus non-

physician staff time on FFS will decrease as well. Let η denote non-physician staff cost per

Total RVU. Then the provider cost can be offset by ηR · (t(non−FFS)/t) = ηRz, which comes

from the reduction in FFS visits.

Thus, the provider’s objective function (for a given time unit, possibly one month) is

given as:

f(x, y) = τR · (1− z) + yNγ + wξ · |∆ξ(z)|+ ηRz − xyNγ − COP − CIT (1)

where: τR · (1− z) is revenue from FFS visits;

yNγ is revenue from PMPM payments for MH services;

wξ · |∆ξ(z)| is revenue from incentive payments;

xyNγ is the provider cost of providing MH services;

ηRz is the reduction in non-physician staff cost for FFS visits;

COP is operating cost excluding physician/staff compensation;

CIT is cost of additional information technology required by MH model.

2.3 Revenues and Costs for the Payer

The payer’s revenue is from insurance premiums, totaling $dN . (For simplicity, we assume

every patient has the same premium. This is not necessary and only requires expanding

the formalism.) The payer’s cost consists of FFS reimbursements for face-to-face visits

performed by the provider, PMPM payments covering the provider’s MH services, incentive

payments to the provider, and payments to other healthcare facilities for services such as

inpatient admissions, outpatient procedures, emergency department visits, surgeries, lab

tests, pharmaceuticals, and so forth.

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Under the PCMH, the payer’s monthly payment to the provider for FFS visits is βrR ·(1 − z), where β = 1 − CoPay, r = reimbursement per Total RVU, and R = number of

Total RVUs generated by the physician per month before medical home implementation.

The PMPM fee is $y for each covered patient, making the total payment $yNγ, where γ

is the percentage of patients covered by PMPM. The performance incentive is funded from

an anticipated reduction in avoidable healthcare utilization. Instead of paying $ξ for the

total healthcare system usage, the payer’s payment is reduced to $ξ · (1 + ∆ξ(z)), where

∆ξ(z) < 0 due to improved health outcomes in the patient population. The payer uses a

specified portion, w, of the savings to reward the provider. The performance incentive to

the provider is $wξ · |∆ξ(z)|.Thus, the payer’s objective function (for a given time unit, possibly one month) is given

as:

g(x, y) = dN − βrR · (1− z)− yNγ − wξ · |∆ξ(z)| − ξ · (1 + ∆ξ(z)) (2)

where: dN is revenue from insurance premiums from the provider’s panel;

βrR · (1− z) is FFS payment to the provider;

yNγ is PMPM payments to the provider for MH services;

wξ · |∆ξ(z)| is incentive payments to the provider;

ξ · (1 + ∆ξ(z)) is payments to non-primary care providers.

We have now developed payer and provider objective functions that capture the costs

and revenues for each party. The following section focuses on analyzing the relationships

between these and finding the combination of payer reimbursement and provider investment

that is beneficial to both parties.

3 Analysis of the Financial Flow Model

We note that the objective functions of the payer and the provider interact, in that the

payer’s costs are included in the provider’s revenues. We assume that for the PCMH to be

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sustainable, neither the provider nor the payer should be financially worse off than before

the PCMH implementation. Thus, from the payer perspective, PMPM fees and incentives

must be covered by reductions in non-primary care healthcare utilization. On the other

hand, the provider sacrifices some FFS revenue to engage in MH services, and the resulting

revenue loss must be covered by the PMPM fee and incentive payments. Our goal is to

find the payer’s reimbursement structure and the provider’s medical home investment level

that satisfies these conditions. In fact, we want to identify the payer reimbursement /

provider investment combination that neither party has a financial incentive to change. This

combination can be thought of as an “equilibrium point”, which provides a strong indication

that the medical home concept is financially sustainable.

To accomplish this, we apply game theory. We design a simultaneous game in which two

players determine the best action strategies at the same time to maximize their respective

profits. The action of the payer is to determine the amount of the PMPM fee ($y) to pay

the provider, and the action of the provider is to determine how much (the portion, x) of

the total monthly PMPM fee to allocate to support MH services. To include performance

incentives, we analyze this game for various levels of incentive rate, w.

The provider’s objective function can be developed as follows:

max f(x, y) = τR · (1− z) + yNγ + wξ · |∆ξ(z)|+ ηRz − xyNγ − COP − CIT

= b0 + b1xy + b2(xy)2 + b3(xy)3 + yNγ

where,

b0 = τR− COP − CIT

b1 = −a1ξwk + R · (η − τ)k −Nγ, where k = ρNγ/αt

b2 = −a2ξwk2

b3 = −a3ξwk3

where a1, a2, and a3 are parameters of ∆ξ(z).

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The payer’s objective can be developed in a similar fashion as follows:

max g(x, y) = dN − βrR · (1− z)− yNγ − wξ · |∆ξ(z)| − ξ(1 + ∆ξ(z))

= c0 + c1xy + c2(xy)2 + c3(xy)3 − yNγ

where,

c0 = dN − βrR− ξ

c1 = −a1ξ(1− w)k + βrRk, where k = ρNγ/αt

c2 = −a2ξ(1− w)k2

c3 = −a3ξ(1− w)k3

where a1, a2, and a3 are parameters of ∆ξ(z).

With these objective functions, we are able to derive tight bounds of the incentive rate, w,

for the provider and the payer, respectively. The solution details are given in the appendix.

Lemma 1. To make the PCMH sustainable for the provider, it is necessary that

w >(τ − η)R + αt/ρ

−a1ξ,

where w is the incentive rate set by the payer.

Lemma 2. To make the PCMH sustainable for the payer, it is necessary that

w < 1− βrR− αt/ρx

a1ξ,

where x is the portion of the total PMPM fee that the provider allocates for MH services,

and w is the incentive rate set by the payer.

To solve for the equilibrium as discussed earlier, we develop best response functions for

each player. In game theory, the best response is the strategy which produces the most

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favorable outcome for a player, taking other players’ strategies as given (Fudenberg and

Tirole 1991; Gibbons 1992). The concept of a best response is central to John Nash’s best-

known contribution, the Nash equilibrium, the point at which each player in a game has

selected the best response to the other players’ strategies (Nash 1950).

The best response function for the provider, x∗(y), specifies the optimal portion x∗ of the

total PMPM fee to use for financing MH services when the payer’s PMPM is at amount y.

Proposition 3. Let y be the amount of PMPM given by the payer, then the provider’s best

response function is

x∗(y) =−a2 +

√a2

2 − 3a1a3 − 3a3[αtρ

+ (τ − η)R]/wξ

3a3ky, if w >

(τ − η)R + αt/ρ

−a1ξ.

The best response function for the payer, y∗(x), determines the optimal PMPM payment

y∗ when the provider uses x portion of the total PMPM fee to support MH services.

Proposition 4. Let x be the portion of the total PMPM fee that the provider uses for MH

services, then the payer’s best response function is

y∗(x) =−a2 +

√a2

2 − 3a1a3 − 3a3(αtρx

+ βrR)/(1− w)ξ

3a3kx, if w < 1− βrR− αt/ρx

a1ξ.

The above two best response functions can be solved simultaneously to find the equi-

librium point, (xN , yN), the pair of strategies that neither the provider nor the payer has

incentive to change given the other’s decision. The following theorem derives a unique Nash

equilibrium point in the closed form.

Theorem 1. There exists a unique Nash equilibrium point, (xN , yN), for the provider and

the payer, if the incentive rate w is in the range of

((τ−η)R+αt/ρ

−a1ξ, 1

). The pair of strategies

is given by

xN =αt/ρ

βrR + [αt/ρ + (τ − η)R]( 1w− 1)

,

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and

yN =

βrR + [αt/ρ + (τ − η)R]( 1

w− 1)

− a2 +

√a2

2 − 3a1a3 − 3a3[αtρ

+ (τ − η)R]/wξ

3a3Nγ.

We will use numerical examples to demonstrate these results in next section.

Before we wrap up the theoretical discussions, here is one more property worth to men-

tioning.

Proposition 5. The equilibrium point, (xN , yN), varies with the incentive rate, w. It holds

that xN increases with w and yN decreases with w after some threshold.

4 Illustrative Example

In this section, we create a “representative” clinic by using data from the Medical Expen-

diture Panel Survey provided by the Agency for Healthcare Research and Quality (AHRQ

2007), and from the Physician Production Survey provided by the Medical Group Manage-

ment Association (MGMA 2008). For all data, we use median values, as recommended by

AHRQ and MGMA. We assume that the medical team consists of one physician and several

RN/LPNs. Table 2 lists the model parameters obtained from these data sources. The results

presented in this section are for illustration only. They do not represent recommendations

or guidelines since each individual practice will have its own unique conditions.

Figure 3 illustrates the best responses for the provider and the payer, respectively. It

also provides the Nash equilibrium point, i.e., the optimal strategy pair, given a maximum

reduction in avoidable utilization of 10% and an incentive of 50%. Note that at the equi-

librium point, the PMPM is $10.15 (Total PMPM = 2200× 0.75× $10.15 = $16, 747), and

the amount invested in MH services is 38%. This yields 0.38× 16, 747 = $6, 364 per month

to cover the physician-led team for MH services. This is approximately 0.95 non-physician

FTEs and 0.24 physician FTEs per month based on the data given above. The remainder

17

Table 2: Model Parameter Values

Parameter Description ValueCOP Provider operating cost per month $13,981.00CIT Additional IT cost due to MH per month $1,500.00d Insurance premium per patient per month $402.00k1 RN/LPN compensation per hour $15.00k2 Physician compensation per hour $107.00N Number of patients in the practice panel 2200r Revenue per Total RUV $50.34R Number of Total RV Us per month from FFS prior to MH 694.58t Number of hours worked by physician per month 160α Provider cost of MH services per hour, α = k1 + ρk2 $42.00β Payers portion of FFS reimbursement, β = 1− CoPay 0.80γ Percent of the practice panel covered by MH fee 0.75η RN/LPN cost per Total RV U $5.53ξ Payers expenditure on healthcare utilization prior to MH $576,503.00ρ Physician time over RN/LPN time on MH services 0.25τ Revenue after provider cost per Total RUV $28.97

of the PMPM revenue can be used to cover additional IT cost required by the PCMH and

to enhance the primary care provider’s compensation.

Figure 3: Best Response Functions of Provider and Payer (assuming ∆ξ(z∗) = −10%, w =50%)

From Table 3, the total savings at this equilibrium point is $53,730, which is 9.32% (Table

3, row *) of ξ = $576, 503 (Table 2). Note that using the data from Table 2, the monthly net

income prior to the PCMH implementation is calculated to be $6,141 and $279,924 for the

provider and payer, respectively. The improvements in net income at the equilibrium point

18

for the provider and payer are $32,029 and $16,613 (calculated from data in Table 3, row *).

The Nash equilibrium concept guarantees that (38%, $10.15) is the pair of strategies that

neither the provider nor the payer has incentive to change given the other’s decision.

Table 3: Incentive Rate Maximizing g(xN , yN), denoted w (assuming ∆ξ(z∗) = −10%)

w xN yN f(xN , yN ) g(xN , yN ) ∆ξ11.9% 7.71% $3.27 $10,059 $280,082 -1.01%12.8% 8.32% $6.43 $15,343 $280,614 -2.06%13.8% 9.05% $9.20 $20,082 $281,548 -3.09%15.1% 9.91% $11.45 $24,059 $282,845 -4.06%16.7% 11.03% $13.30 $27,532 $284,619 -5.02%19.0% 12.66% $14.70 $30,568 $287,096 -6.04%22.2% 14.98% $15.27 $32,731 $290,061 -7.00%27.9% 19.24% $14.72 $34,288 $293,818 -8.01%32.7% 22.98% $13.74 $34,964 $295,710 -8.51%40.7% 29.55% $12.00 $36,138 $297,008 -9.00%

w 42.8% 31.37% $11.56 $36,525 $297,060 -9.09%48.3% 36.23% $10.50 $37,696 $296,767 -9.27%

* 50.2% 38.18% $10.15 $38,170 $296,537 -9.32%57.2% 44.58% $9.03 $40,075 $295,302 -9.46%61.8% 49.18% $8.37 $41,518 $294,182 -9.54%70.1% 58.01% $7.33 $44,422 $291,703 -9.63%77.4% 66.29% $6.56 $47,210 $289,174 -9.69%80.1% 69.47% $6.31 $48,279 $288,180 -9.71%85.6% 76.30% $5.82 $50,563 $286,030 -9.74%

Figure 4 plots the columns of Table 3. Note that the provider’s decision, xN , increases

with incentive rate, w. Conversely, after some lower threshold on incentive, the payer’s

decision, yN , will decrease with increasing w.

Recall that the above calculations assume a maximum reduction of 10% and an incentive

of 50%. As shown in Table 3, different incentive rates produce different equilibrium strategies.

The payer might ask which incentive will bring the highest net income to the payer’s own

objective function. As indicated in Table 3, this occurs at an incentive of 42.8% for this

example. We will denote this incentive as w.

Table 4 lists the so defined w along with the corresponding equilibrium strategies and the

objective function values for various utilization reduction scenarios, ranging from as low as

5% to as high as 30%, for our representative clinic. Ideally, the maximum reduction, ∆ξ(z∗),

can be estimated from collections of claims data generated by patients receiving MH services.

19

Figure 4: Nash Equilibrium Points Vary with Incentive Rate (assuming ∆ξ(z∗) = −10%)

As noted earlier, evaluation studies are under way to determine utilization reduction under

the PCMH model.

Table 4: Model Performance for Various Utilization Reduction Scenarios

∆ξ(z∗) w xN yN f(xN , yN )− f g(xN , yN )− g ξ(1 + ∆ξ(kxNyN ))-5.0% 57.3% 44.7% $6.50 $14,978 $4,299 $553,021-7.5% 48.4% 36.3% $9.26 $23,289 $10,147 $538,432

-10.0% 42.8% 31.4% $11.56 $30,384 $17,135 $524,103-12.5% 38.8% 28.0% $13.63 $36,853 $25,020 $509,574-15.0% 35.9% 25.6% $15.44 $42,772 $33,487 $495,053-17.5% 33.5% 23.7% $17.17 $48,230 $42,431 $480,552-20.0% 31.6% 22.2% $18.73 $53,329 $51,717 $466,086-22.5% 29.9% 20.8% $20.30 $58,088 $61,312 $451,671-25.0% 28.5% 19.8% $21.65 $62,604 $71,074 $437,337-27.5% 27.5% 18.9% $22.88 $66,979 $81,130 $422,854-30.0% 26.2% 18.0% $24.28 $71,086 $91,385 $408,455

Figure 5 plots the data given in Table 4. It is interesting to note that the maximum

utilization reduction, ∆ξ(z∗), seems to be positively correlated with PMPM and inversely

correlated with incentive.

20

Figure 5: Nash Equilibrium Points Vary with Incentive Rate (assuming ∆ξ(z∗) = −10%)

5 Conclusion

This paper develops a model for reimbursing the medical home. It assumes reimbursements

to the provider come from fee-for-service payments, capitation payments for medical home

services, and incentives for improved patient outcomes as reflected by reductions in avoidable

healthcare system utilization. The paper develops an intuitively appealing functional form

for modeling utilization reduction due to medical home services and derives several properties

of this functional form. The paper then develops a payer-provider game, which models the

interaction between objective functions and decision strategies of the two players. This

game yields a variety of results, the most significant being a unique Nash equilibrium point

which specifies a PMPM capitation payment from the payer to the provider along with the

percentage of the PMPM payment that the provider should use to support medical home

services. We then apply these results to a representative clinic, constructed using data from

AHRQ and MGMA, which allows us to make some observations on the practical behavior

of the model. Overall, we believe the paper provides a theoretical foundation for developing

new reimbursement structures that benefit both payer and provider, a necessary condition

21

for medical home sustainability.

Acknowledgements

We would like to thank Dr. Michael Barr and other professionals from American College of

Physicians for introducing us to this problem and providing feedback on our work.

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Appendix

Solution details of Proposition 1:

Let ∆ξ′(z) = 3a3z2+2a2z+a1 = 0, then z =

−a2±√

a22−3a1a3

3a3, where a2

2−3a1a3 ≥ 0. According

to Definition 1, a1 < 0, and a2a3 < 0. For a2a3 < 0, there are two cases:

case (i), a2 < 0, a3 > 0, so√

a22 − 3a1a3 > |a2|. Thus, z =

−a2−√

a22−3a1a3

3a3cannot be a root.

case (ii), a2 > 0, a3 < 0, so −a2 −√

a22 − 3a1a3 < −a2 +

√a2

2 − 3a1a3 < 0.

Thus, 0 <−a2+

√a22−3a1a3

3a3<

−a2−√

a22−3a1a3

3a3.

Since we assume that ∆ξ′(z) has a unique root z∗ ∈ [0, 1], therefore, in both cases, we have

z∗ =−a2+

√a22−3a1a3

3a3. Also because z∗ is the unique root of ∆ξ′(z) in [0, 1], and ∆ξ′(z) > 0

when z ∈ (z∗, 1]. Thus, we have ∆ξ′(1) > 0 , i.e., a1 + 2a2 + 3a3 > 0.

Now let us find the value of ∆ξ(z∗).

∆ξ(z∗) = a3(z∗)3 + a2(z

∗)2 + a1z∗ = 0, where z∗ =

−a2 +√

a22 − 3a1a3

3a3

, (3)

26

and

∆ξ′(z) = 3a3z∗2 + 2a2z

∗ + a1 = 0. (4)

Let Eq. (3) x 3 - Eq. (4) x z*, we get 3∆ξ(z∗) = a2z∗2 + 2a1z

∗ and thus, ∆ξ(z∗) =

z∗(a2z∗+2a1)3

=a2(2a2

2−9a1a3)−2(a22−3a1a3)

32

27a23

.

Therefore, ∆ξ(z∗) ≥ −1 is equivalent to the following

a2(2a22 − 9a1a3)− 2(a2

2 − 3a1a3)32

27a23

≥ −1,

or,

27a23 + a2(2a

22 − 9a1a3)− 2(a2

2 − 3a1a3)32 ≥ 0.

Solution details of Proposition 2: (Similar to Proposition 1.)

Solution details of Lemma 1:

The objective function of the provider is as follows,

max f(x, y) = b0 + b1xy + b2(xy)2 + b3(xy)3 + yNγ (5)

where b0 = τR−COP −CIT , b1 = −a1ξwk +R(η− τ)k−Nγ, b2 = −a2ξwk2, b3 = −a3ξwk3,

and k = ρNγ/αt. Hence, we have

∂f(x, y)

∂x= b1y + 2b2y(xy) + 3b3y(xy)2.

27

To make the medical home sustainable for the provider, it is necessary that ∂f(x,y)∂x

|x=0 > 0.

This is equivalent to b1 > 0, i.e.,

b1 = −a1ξwk + (η − τ)Rk −Nγ > 0

⇔ −a1 − [αt

ρ+ (τ − η)R]/wξ > 0

⇔ w >αt/ρ + (τ − η)R

−a1ξ.

Solution details of Lemma 2: (Similar to Lemma 1.)

Solution details of Proposition 3:

From solution details of Lemma 1, we have ∂f(x,y)∂x

= b1y+2b2y(xy)+3b3y(xy)2. As ∂f(x,y)∂x

= 0,

and y 6= 0, we obtain

b1 + 2b2(xy) + 3b3(xy)2 = 0, (6)

xy =−b2 ±

√b22 − 3b1b3

3b3

.

Substitute b1 = −a1ξwk + R(η− τ)k−Nγ, b2 = −a2ξwk2, b3 = −a3ξwk3, and k = ρNγ/αt

back to the pervious equation, we have

xyk =−a2 ±

√a2

2 − 3a1a3 − 3a3[αtρ

+ (τ − η)R]/wξ

3a3

or,

xyk =−a2 ±

√a2

2 + 3a3Ω

3a3

, (7)

28

where Ω = −a1 − [αtρ

+ (τ − η)R]/wξ = b1ξwk

. It follows from Lemma 1 that b1 > 0, thus, we

conclude that Ω > 0. The left-hand-side of Eq. (8) is xyk = xy(ρNγ/αt) = t(non−FFS)/t = z,

and z ∈ (0, 1). Now, let us take a look at the right-hand-side of Eq. (8). We understand

that any value does not belong to (0, 1) has to be excluded.

Consider case (i) which has a2 < 0, a3 > 0, we have

√a2

2 + 3a3Ω > |a2| ⇒ −a2 −√

a22 + 3a3Ω

3a3

< 0.

Notice that the above value is less than 0, thus, Eq. (8) becomes xyk =−a2+

√a22+3a3Ω

3a3, where

Ω = −a1 − [αtρ

+ (τ − η)R]/wξ.

In case (ii) which has a2 > 0, a3 < 0, we have 3a3Ω > 3a3(−a1) since Ω < −a1, this is

true because −a1 = Ω+ [αtρ

+(τ − η)R]/wξ. From Proposition 1, we have 3a3 +2a2 + a1 > 0

which leads to 3a3(−a1) > 2a2a1 + a21. From these two conditions, 3a3Ω > 3a3(−a1) and

3a3(−a1) > 2a2a1 + a21, we can obtain the following relationship:

3a3Ω > 2a2a1 + a21 ⇒ a2

2 + 3a3Ω > a22 + 2a2a1 + a2

1 = (a1 + a2)2. (8)

From Definition 1, we have a1 + a2 + a3 > 0; and in this case (ii), a3 < 0. Thus, a1 + a2 > 0.

Therefore, from Eq. (9), we have

a22 + 3a3Ω > (a1 + a2)

2 ⇒√

a22 + 3a3Ω > |a1 + a2| = a1 + a2

⇒ a2 +√

a22 + 3a3Ω

−3a3

>a2 + a1 + a2

−3a3

=a1 + 2a2

−3a3

. (9)

Now going back to Proposition 1’s 3a3 + 2a2 + a1 > 0, we derive the following

3a3 + 2a2 + a1 > 0 ⇒ 2a2 + a1

−3a3

> 1. (10)

29

Combine (10) and (11), we get

xyk =a2 +

√a2

2 + 3a3Ω

−3a3

>2a2 + a1

−3a3

> 1.

Recall that the value of the right-hand-side of Eq. (8) can be only between 0 and 1, thus in

both cases, we have xyk =−a2+

√a22+3a3Ω

3a3, where Ω = −a1 − [αt

ρ+ (τ − η)R]/wξ, or

xyk =−a2 +

√a2

2 − 3a1a3 − 3a3[αtρ

+ (τ − η)R]/wξ

3a3

.

Thus, the best response function of the provider is

x∗(y) =−a2 +

√a2

2 − 3a1a3 − 3a3[αtρ

+ (τ − η)R]/wξ

3a3ky, if w >

(τ − η)R + αt/ρ

−a1ξ.

Solution details of Proposition 4: (Similar to Proposition 3)

Solution details of Theorem 1:

From Eq. (12), we have

x =Nγ

c1 + 2c2(xy) + 3c3(xy)2.

Substitute c1 = −a1ξ(1 − w)k + βrRk, c2 = −a2ξ(1 − w)k2, c3 = −a3ξ(1 − w)k3 and

k = ρNγ/αt back to the above equation, we have

x =Nγ

−a1ξ(1− w)k + βrRk − 2a2ξ(1− w)k2(xy)− 3a3ξ(1− w)k3(xy)2. (11)

30

From Eq. (7), we have

−b1 = 2b2(xy) + 3b3(xy)2.

Substitute b1 = −a1ξwk + R(η− τ)k−Nγ, b2 = −a2ξwk2, b3 = −a3ξwk3, and k = ρNγ/αt

back to the above equation, we have

a1ξwk + R(η − τ)k + Nγ = −2a2ξwk2(xy)− 3a3ξwk3(xy)2. (12)

Multiple Eq. (16) by (1− w)/w, and substitute it into Eq. (15), we obtain

xN =Nγ

βrRk + [Nγ + R(τ − η)k](

1−ww

) =αt/ρ

βrR + [αt/ρ + (τ − η)R](

1w− 1

) . (13)

Combine Eq. (17) and Eq. (14), we get

yN =

βrR + [αt/ρ + (τ − η)R]

(1w− 1

) − a2 +

√a2

2 − 3a1a3 − 3a3

[αtρ

+ (τ − η)R]/wξ

3a3Nγ.

Recall that from Lemma 1, we require that

w >(τ − η)R + αt/ρ

−a1ξ,

also from Lemma 2, it is necessary that

w < 1− αt/ρxN − βrR

−a1ξ.

Substitute (17) in above, we get

w < 1−αtρ∗ βrR+[αt/ρ+(τ−η)R]( 1

w−1)

αt/ρ− βrR

−a1ξ= 1− [αt/ρ + (τ − η)R]

(1w− 1

)

−a1ξ.

31

This leads to

[αt/ρ + (τ − η)R](

1w− 1

)

−a1ξ< 1− w, and w 6= 1.

Therefore, we have

(τ − η)R + αt/ρ

−a1ξ< w < 1.

Solution details of Proposition 5:

Since

xN =αt/ρ

βrR + [αt/ρ + (τ − η)R](

1w− 1

) ,

and

dxN

dw=

αt[αt + (τ − η)Rρ]βrRρ + [αt + (τ − η)Rρ]

(1w− 1

) 2

w2

> 0,

we obtain that xN is an increasing function of w. We apply similar proof to yN , and yN is

a decreasing function of w after some threshold.

32