Heterogeneous hospital response to per diem prospective payment

system �

Galina Besstremyannaya Dmitry Shapiro

March 20, 2013

Abstract

The paper provides theoretical analysis for hospitals�heterogeneity in the response to the

change from the the fee-for-service (FFS) system to a per diem prospective payment system

with a length-of-stay dependent step-down rate (SDR): hospitals with shorter (longer) average

length of stay under FFS have longer (shorter) average length of stay under SDR. We also

show that for FFS hospitals with longer average length of stay the planned readmission rate is

to increase under the SDR. Using a recent administrative database for 684 Japanese hospitals

in 2007-2011, we conduct estimations with dynamic panel data and �nd an empirical support

for the predictions of our theoretical model. The results suggest that step-down rate contributes

to di¤erential hospital�s response to a per diem prospective payment system and might lead to

insu¢ cient cost containment.

�Besstremyannaya: gbesstre@ce�r.ru, Center for Economic and Financial Research at New Economic School,

Nakhimovsky prospekt 47, Moscow 117418 Russia; Shapiro: dashapir@uncc.edu, Belk College of Business, University

of North Carolina, 9201 University City Boulevard, Charlotte NC 28223-0001, USA.

1

1 Introduction

Health care is an example of an industry in which providers have a strong in�uence on consumers�

choice of medical services (Christianson and Conrad 2011; Mayes 2007). Combined with the volume-

based fee-for-service (FFS) reimbursement, the power of health care providers leads to supplier-

induced demand, overuse of resources, and overspending. Although providers may take initiative

in exerting cost-reducing e¤orts and raising e¢ ciency of medical treatment (Borghans et al 2012),

cost-reducing e¤orts are not immediately veri�able. Therefore, the task of devising a reimbursement

mechanism that encourages e¢ ciency falls on the governments, who act as social planners concerned

over welfare issues (Chalkley and Malcomson 2000; Holmstrom and Milgrom 1991).

A particularly signi�cant example of such reimbursement mechanism is the one based on the

diagnosis-related groups (DRGs) which were developed in the U.S. in the 1960s. DRGs is a classi-

�cation of diseases into medically justi�ed groups with a stable distribution of resources required

to treat patients in each group (Thompson et al. 1979). Providers receive a �xed reimbursement

amount for each episode of medical care to a patient with a given DRG. This system is called

prospective payment system (PPS) and it promotes cost e¢ ciency since hospitals start bearing the

�nancial burden of excessive medical treatment.

Countries which are not yet ready to introduce the genuine version of PPS (e.g., owing to

high variation of medical practices, historical di¤erences in hospital reimbursement or lack of stan-

dardized data on patient cases) might favor a per diem PPS as the system that contains certain

incentives for cost containment. Under the per-diem PPS hospitals have incentives to limit the

daily resource use; however, their incentives with regards to the length of hospital stay and total

cost are not directly a¤ected. Such per diem PPS may be regarded as a cost-sharing system, which

allows for an appropriate balance between cost-e¢ ciency and quality (La¤ont and Tirole 1993).

Among the developed countries, Germany and Japan employ the per diem system. Furthermore,

the Japanese version on the inpatient PPS contains explicit incentives to shorten the average length

of stay (ALOS). For each group of diagnoses �which are called diagnosis-procedure combinations,

DPCs �the amount of the inclusive per diem payment is a step-wise decreasing function of the pa-

tient�s length of stay. While Germany exploited per diem PPS in 1996�2003 as a transitory system

to the prospective reimbursement, Japan keeps preserving the per diem character of its PPS.

Originally introduced in 2003, the Japanese PPS immediately resulted in the decline of the

ALOS at the hospital and at the national levels (MHLW 2005). Since ALOS is often treated as

a proxy for hospital e¢ ciency (Lopes et al., 2004; Rapoport et al., 2003; Heggestad, 2002), one

could argue that a fall in the ALOS was, in fact, associated with increased e¢ ciency (Kuwabara et

al. 2011). Yet, both technical and cost e¢ ciency of Japanese hospitals demonstrate only a minor

improvement owing to the reform (Besstremyannaya 2012) and the impact on hospitals costs is

ambiguous (Nishioka 2010; Yasunaga et al. 2006; Yasunaga et al. 2005a). Notably, the e¤ect of

the PPS introduction on ALOS was not uniform and for some hospitals the ALOS has increased

(Nawata and Kawabuchi, 2012).

The Japanese PPS also resulted in quality deterioration re�ected in the rise of early readmission

2

rate (by Japanese de�nition - readmissions within 42 days after discharge, Hamada et al. 2012; Ya-

sunaga et al. 2005a), as well as in the growing prevalence of "remission" report and the decreasing

prevalence of "healing" report1 to the discharged patients (Besstremyannaya, 2010). The major

reason for the rise in early readmission rate is the increase in planned readmissions (Besstremyan-

naya 2010; Okamura et al. 2005)2 which was, in turn, caused by the LOS-dependendent stepdown

per diem PPS tari¤ (Kondo and Kawabuchi 2012).

Motivated by the Japanese PPS reform and prior evidence regarding its e¤ect, in this paper we

provide a theoretical and empirical analysis of how the reform a¤ected hospitals��nancial incentives

and its impact on the ALOS and planned readmission rates. In the theoretical part, we develop a

model to compare the outcomes under the fee-for-service reimbursement scheme, FFS, and a per

diem PPS with a LOS-dependend step-down rate, SDR. The former corresponds to the pre-reform

system and the latter corresponds to the post-reform system. To separate the e¤ect of the per

diem system per se from the e¤ect of LOS-dependent per diem rates we also study an intermediate

reimbursement system with the �at per diem rate, PD.

In our model we assume that heterogeneity among hospitals results in a variation in the hospital-

level ALOS, which is consistent with the data. We show that a change from the FFS to the per

diem system has a di¤erential e¤ect, depending on hospital�s ALOS under the FFS system. Specif-

ically, we demonstrate that hospitals with shorter ALOS have incentives to prolong the ALOS,

and hospitals with longer ALOS prefer to decrease their ALOS under PD. Adding LOS-dependent

reimbursement rates such that initial stay is reimbursed under higher tari¤, as in SDR, has unam-

biguously perverse incentives on hospitals. Higher initial tari¤ increases hospitals�marginal bene�t

from longer stay while does not a¤ect marginal cost. E¤ectively, all hospitals, except for those with

the longest ALOS, �nd it pro�table to treat patients longer.

In order to model the e¤ect on planned readmission rate, we allow hospitals to choose whether

to treat a patient with one or two admissions. We speci�cally focus on �nancial incentives to use

planned readmission, as we assume that medical reasons for readmissions remain una¤ected by the

1The following categories of outcomes are speci�ed in the MHLW (2009) �Explanatory materials for the 2008

survey on the E¤ect of DPC introduction�: 1) Healing (Chiyu): There is no need in outpatient treatment after

discharge. 2) Improvement (Keikai): Improvement was achieved in the course of treatment. In principle, there is

a need for continuous outpatient care after discharge. 3) Remission (Kankai): Radical treatment (e.g., as in case

of blood diseases) was applied during hospital stay, and there is temporary improvement; yet, there is a chance of

disease reoccurrence. 4) No change (Fuhen): No improvement was reached in the course of the relevant treatment in

hospital. 5) Worsening (Dzouaku): Worsening was noticed in the course of the relevant treatment in hospital.2According to MHLW (2005) readmissions are classi�ed into planned, anticipated, and unplanned. The reasons for

anticipated readmissions are: 1)Anticipated worsening of medical condition; 2)Anticipated worsening of comorbidity;

3)Patient was temporarily discharged to raise his/her QOL; 4)Discharged due to patient�s desire during previous

hospital stay; 5)Other.

The reasons for planned readmissions are: 1)operation after preliminary tests; 2) planned operation or procedures;

3) chemical or radioactive treatment; 4)planned examinations/tests; 5) examination/operation was stopped during

previous treatment, and patient was discharged; 6)patient was sent home to recover before the operation

The reasons for unplanned readmissions are: 1)Non-anticipated worsening of medical condition; 2) Non-anticipated

worsening of co-morbidity; 3)Emergence of other acute medical condition; 4)Other.

3

reform. We show that from the �nancial point of view, hospitals with longer ALOS under the

FFS have stronger incentives to treat patients using planned readmission. Since each admission is

reimbursed separately, planned readmission enables hospitals to bene�t twice from higher initial

rates under the SDR system. The implication of this result is two-fold. First, we should expect

an increase in the planned readmission rate for hospitals with the longest ALOS. Second, hospitals

with the longest ALOS can use planned readmissions to decrease the reported ALOS, even though

the full treatment takes longer.

In the second part of the paper we empirically test the predictions of our model. We use a

recently released administrative database for 684 Japanese hospitals in 2007-2011, with most hos-

pitals gradually joining the PPS reform in 2009-2011. The data come from the Japanese Ministry of

Health, Labor, and Welfare�s (MHLW) database on PPS hospitals. We supplement it with the data

on hospitals� characteristics taken from the Handbook of Hospitals, and �nancial characteristics

used from the Ministry of Internal A¤airs�s database. The empirical analysis is conducted for each

Major Diagnostic Category (MDC). The Japanese MDCs aggregate groups of certain diagnoses

(e.g., circulatory system diseases) and are constructed on the basis of the International Classi�-

cation of Diseases (ICD) with minor modi�cations. In addition, we estimate the model for the

pooled data without the separation by MDCs. We �nd strong evidence supporting heterogeneity

in hospitals�response. In each of 15 MDCs, as well as in the pooled data, we observe that hospitals

with the shortest ALOS (in the �rst quartile, when ordered by ALOS) signi�cantly increase their

ALOS after the introduction of the SDR, whereas hospitals with the longest ALOS (those in the

fourth quartile) signi�cantly decrease it. Regarding the planned readmission rate, only for two

MDCs out of �fteen3 we observe a drop in planned readmission rates in hospitals with the longest

ALOS, as compared to ten MDCs for which the readmission rate signi�cantly goes up as predicted

by the model.

A certain amount of empirical literature pays attention to a di¤erential response of hospital�s

ALOS to the change from FFS to PPS (Sood et al. 2008; Ellis and McGuire 1996; Gold et al.

1993; Coulam and Gaumer 1991) and per diem PPS (Grabowski et al. 2011). Yet, a moral hazard

explanation of larger supply of LOS to patients with longer LOS (Ellis and McGuire 1996) and

Yasunaga et al.�s (2006; 2005b) statistical comparison of per diem pro�ts for DPCs with high and

low material costs are, to the best of our knowledge, the only attempts to theoretically exploit the

potential sources for heterogeneity in the dynamics of ALOS.

The �ndings of our analysis may be relevant not only for the country-level, but also for the

medical specialty level generalizations. Indeed, along with the experience of Germany and Japan,

prospective per diem PPS is currently employed in Medicare�s psychiatric hospitals, skilled nursing

facilities and hospices, as well as in Medicaid�s psychiatric inpatient facilities.

The remainder of the paper is structured as follows. Section 2 provides a description of the

major features of Japanese inpatient prospective payment system. Section 3 sets up a theoretical

model for a pro�t-maximizing hospital as a supplier of health care and quality. Section 4 describes

3MDC3 "Ear, nose, and throat" and MDC12 "Female reproductive system".

4

the data, and Section 5 provides speci�cations for the empirical analysis. Section 6 presents the

results of the estimations, and the follow-up discussion is given in Section 7.

2 Japanese inpatient prospective payment system

The issue of cost containment became on the agenda of Japanese health care policy makers in 1970s,

when the rate of health care expenditure growth started to exceed the rate of growth in GDP (Fujii

and Reich 1988). The main factors causing soaring costs of the Japanese health care system are

aging population, decrease of the labor force, and the physician-induced demand combined with

the development of medical technologies. In fact, the Japanese social health insurance system

has always been highly subsidized. In 2012, for example, central government �nanced 25.3% of

health care expenditure (MHLW 2012c), which contributed to 10.2 % of the government�s budget

(Ministry of Finance, 2012). By early 2000s the e¤ects of increased coinsurance rates and lowered

fees in the uni�ed fee schedule as the measures to decrease health care costs have been exhausted

(Ikegami 2009). Consequently, the Ministry of Health, Labor, and Welfare decided to introduce an

inpatient prospective payment system for acute care hospitals in order to create incentives for cost

containment.

The �rst attempt to employ an inpatient PPS in Japan was implemented in 1990, when inclusive

per diem rates were introduced in 50% of geriatric hospitals (MHLW 2012a; Ikegami 2005; Okamura

et al. 2005). Then, inpatient PPS was piloted in 10 acute care national hospitals in 1998. Finally,

in 2003 the PPS was introduced in 82 speci�c function hospitals, which provide high-technology

health care (80 public and private university hospitals as well as two national centers: for cancer and

cardiovascular diseases). The subsequent years saw an increasing number of hospitals, voluntarily

joining the PPS. As of July 2010, 18 percent of acute care (general) hospitals, which account for

50 percent of hospital beds in Japan, are �nanced according to PPS.

The Japanese inpatient PPS is essentially a mixed system. The two-part tari¤ is the sum of

DPC and fee-for-service components. The DPC component is constructed as a per diem step-

down rate, related to hospital�s length of stay. For each DPC, the amount of the daily inclusive

payment is �at over each of the three consecutive periods: period 1 represents the 25-percentile

of ALOS calculated for all hospitals submitting the data to MHLW,4 period 2 contains the rest of

the ALOS, and period 3 encloses two standard deviations from the ALOS. After period 3 expires

hospitals are reimbursed according to the FFS system. To create incentives for shorter length of

stay, per diem DPC payment in the �rst period is established 15% larger than the standard per

diem reimbursement (Figure 1).

The �rst version of DPCs consisted of 2552 groups of diagnoses. Most of the groups (1860) had

su¢ cient cases and were rather homogeneous (Ikegami 2005). For these groups, which corresponded

to about 90% of admission cases, the rates were set. The numbers of diagnoses and DPCs are

gradually increasing since 2003, and as of 2012 there are 2927 groups of diagnoses and 2241 DPCs.

4The initial rates were set on the basis of 267,000 claim data on patients discharged from 82 targeted hospitals in

July-October 2002.

5

Figure 1: Step-down per diem payment scheme for a given DPC. Source: MHLW (2011).

Along with the diagnosis, each DPC incorporates three essential issues: algorithm, procedure,

and co-morbidity. Diagnoses are coded according to ICD-10 and the Japanese Procedure Code

(commonly used under FFS reimbursement) is employed for coding procedures (Matsuda et al.

2008, MHLW 2004).

The DPC component covers basic hospital fee, hospital expenditures on examinations, diag-

nostic images, pharmaceuticals, injections, and procedures costing less than 10,000 yen. The fee-

for-service component reimburses the cost of medical teaching, surgical procedures, anaesthesia,

endoscopies, radioactive treatment, pharmaceuticals and materials used in operating theatres, as

well as procedures worth more than 10,000 yen (MHLW 2012a; Yasunaga et al. 2005a).

The introduction of inpatient PPS is a voluntary reform for each Japanese hospital. We con-

ducted a thorough investigation of potential administrative tools and found that there was no

pressure. The records of the Ministry of Health, Labor, and Welfare, and anecdotal evidence (e.g.,

Okuyama 2008) demonstrate that participation in PPS is voluntary: the decision is made by the

hospital itself. There are several eligibility criteria: a hospital has to meet the threshold value of

MHLW nurse sta¢ ng ratio of 2 inpatients per nurse; has to follow the methodology for accounting

inpatient expenditure; and has to collect standardized data on prescribed drugs. In particular, the

methodology for accounting inpatient expenditure implies employment of special administrative

sta¤, detailed book keeping, ICD-10 coding, and data processing (Sato 2007).

The Japanese PPS resulted in the decrease of the ALOS in participating hospitals (MHLW

2012). Case studies demonstrate that the Japanese hospitals use the classic measures of reduc-

ing ALOS through raising e¢ ciency of medical treatment (Borghans et al. 2012), which include

shortening the diagnostic and tests procedures (Suwabe, 2004). However, a combination of a ret-

rospective and a prospective fee might not have shorted the ALOS in certain cases (Yasunaga et

al. 2006).

6

3 Theoretical model

The section develops a theoretical framework to analyze hospitals� incentives in response to the

introduction of a per diem PPS. We consider three reimbursement systems: the fee-for-service

(FFS), which corresponds to the system used before the reform; the per diem prospective system

(PD); and the per diem prospective system with stepdown rate (SDR), which corresponds to the

post-reform reimbursement, as explained in the previous section. The PD system is an intermediary

between the FFS and the SDR, and enables us to isolate the e¤ects of the switch to a per diem

system from the e¤ects of di¤erent per diem rates.

In this section we restrict our attention to the treatment of a patient with a given diagnosis

(DPC). We assume that there is a variety of medical procedures and input combinations that could

be used to treat a given condition, and it is up to a hospital or a physician to choose a particular

input combination. Since the major goal of this paper is to understand the e¤ect of reforms on the

ALOS, it is natural to classify inputs and procedures based on their impact on the LOS for a given

patient. Inputs that decrease the LOS are labeled D, and inputs that increase the LOS are labeled

I. Owing to the medical constraints on the minimal value of the LOS for a given diagnosis, we set

an upper limit for the value of D by assuming that D 2 [0; �D]. As for increasing inputs, we assumethat I 2 [0;1). Note that I are not necessarily wasteful in terms of patient�s health. For example,such inputs could include appropriate precautionary treatments and follow-up tests.

Given levels of I and D, we de�ne a function L(I;D) that determines the length of stay. As

hospitals deal with many cases of a given diagnosis, we can think of L(I;D) as the average length of

stay for the diagnosis in a hospital. Hospital�s cost is given by a function g(L), where g is strictly

increasing and convex.5 We assume that di¤erent hospitals have di¤erent , so that the cost (and

marginal cost) is higher for hospitals with higher . The heterogeneity parameter may re�ect

the di¤erence in equipment costs, human capital, or opportunity costs due to personnel availability

or bed occupancy rates. Alternatively, heterogeneity could be introduced through the production

function L(I;D; ) as long as heterogeneity parameter, , ranks hospitals in terms of their costs

and marginal costs (the required technical conditions are L0 > 0 and L00I > 0).

The model is based on the following theoretical approaches in the literature on hospital eco-

nomics, regulation and provider incentives. First, we model a hospital as a pro�t-maximizing

supplier of health care and quality (Hodgkin and McGuire 1994; Ellis and McGuire 1996; Ma 1998;

Grabowski et al. 2011). Second, we focus on the intensity of treatment (McClellan 1996; McClellan

1997; Grabowski et al. 2011), which is particularly relevant for a per diem PPS with the two-part

tari¤, where procedures are given a special emphasis (MHLW 2004; Busse and Schwartz 1997).

Finally, the existence of the heterogeneity parameter in the cost function is analogous to La¤ont�s

and Tirole�s (1993) technological parameter, re�ecting hospital�s e¢ ciency.

5The assumption that g is a strictly increasing function of L is not innocuous one because one can imagine that

a faster treatment can be considerably costlier as it might require modern and more expensive equipment. Thus,

the situation where g declines at �rst and becomes an increasing function later is conceivable. Note, however, that

neither the FFS nor the PD systems will lead to a choise of L at the interval where g declines.

7

3.1 Length of stay

3.1.1 Fee-for-service system

We model the fee-for-service system, which reimburses a given predetermined price for each unit

of input. Denote the price of decreasing inputs as pD and the price of increasing inputs as pI . The

maximization problem is

max pDD + pII � g(L);

and so it follows that D� = �D. Intuitively, higher D raises revenues, pDD, and decreases the costs

because of lower LOS. The optimal level of I is determined from the FOC and satis�es:

pI = g0(L)L0I : (1)

An immediate property of I� to be used later is that it is a decreasing function of . The

implicit function theorem applied to (1) produces:

@I�

@ =

g0(L) � L0I� [g00(L)L0I + g0(L)L0II ]

< 0:

Here, the denominator is the second derivative of the objective function in (1) and, by the second-

order condition, is negative at the optimum. The numerator is positive since g(�) is convex andL0I > 0 by assumption. The intuition is straightforward: for hospitals that incur higher cost for a

given L it is optimal to use lower amount of inputs that increase L.

3.1.2 Per diem prospective payment system

The maximization problem under the per diem PPS is

max �dL(I;D)� g(L(I;D)); (2)

where �d is a per diem rate received by the provider. In Japan, the value of �d is determined according

to the average per diem reimbursement under the pre-reform fee-for-service system. Speci�cally,

for a given hospital denote the optimal LOS under the fee-for-service system LFFS = L(I�; �D).

Then the average per diem reimbursement is

d =pD �D + pII

�

LFFS;

where I� depends on . Taking the average over all hospitals we get the expression for �d:

�d = E

�pD �D + pII

�

LFFS

�: (3)

The FOC for the maximization problem (2) is

�d� g0(LPD) = 0; (4)

and, in particular, it implies that higher values of �d, ceteris paribus, lead to longer LOS.

8

To compare LFFS and LPD recall that from (1)

g0(LFFS) =pI

(LFFS)0I;

and, therefore,

LFFS 7 LPD if and only ifpI

(LFFS)0I7 �d:

The term �d on the right-hand side does not depend on . As for the fraction term on the left-hand

side, its derivative with respect to is�pI

(LFFS)0I

�0

= �pI(LFFS)00 � (@I�=@ )

[L0FFS ]2

: (5)

Since @I�=@ < 0 the monotonicity of (5) as a function of is determined by the convexity of

L(�; �D). When L(�; �D) is concave it is a decreasing function of and when L(�; �D) is convex it isan increasing function.

Thus three options are possible. IfpI

(LFFS)0Iis greater (less) than �d for every , then for all

hospitals the LOS will decrease (increase) after the reform. Clearly, if the per diem rate is too low,

all hospitals �nd it pro�table to discharge patients earlier; on the other hand, if the per diem rate

is too high, all hospitals prefer to keep patients for as long as possible. The most interesting case

arises for intermediate values of �d, when there exists 0 such thatpI

(LFFS)0I

����� = 0

= �d. Then the

hospital�s response depends on and convexity of L as summarized in the Table below:

< 0 > 0

L is concave LFFS > LPD LFFS < LPD

L is convex LFFS < LPD LFFS > LPD

The intuition is as follows. Consider, for example, the case when L(�; �D) is a convex function.Look at a hospital with high so that L and, by convexity, L0I are low. Under the PD system, the

marginal cost should always equal to �d which is the marginal revenue. Under the FFS the marginal

cost is g0(L)L0I . If a hospital decides to choose the LPD under the FFS then its marginal cost is�dL0I and because of convexity of L it is low. Therefore, under the FFS for hospitals with higher it

is optimal to choose L greater than LPD because the marginal cost, �dL0I , is lower and the marginal

bene�t, pI is the same.

Proposition 1 When the reimbursement rule changes from the fee-for-service to the per-diem

PPS it will have the following impact on the LOS. If per diem rate is too low (high) all hospitals

will decrease (increase) the length of treatments. For intermediate values of the per diem rate the

response will be heterogeneous. When the LOS is a concave function of increasing inputs then

hospitals with high pre-reform LOS (and low ) will decrease the LOS as the results of the reform;

hospitals with low pre-reform LOS will increase the LOS.

9

Figure 2: The LOS under the PD (solid line) and FFS (dashed line) systems.

In the empirical section we establish that hospitals�response is heterogeneous and is consistent

with L being a concave function of I: the average length of stay increases for hospitals with lower

pre-reform ALOS and increases for hospitals with higher ALOS.

We illustrate Proposition 1 with the following stylized example. Let � U [1; 3]; g(L) = L4 andL(I;D) =

pI �D + 1, so that L is a concave function of I. Assume that pI = 2 and pD = 1 and

that �D = 1. Then the cost function under the FFS system is

2I +D � � (I �D + 1)2:

The optimal level of D-inputs equals to �D which is 1. The optimal level of I-inputs is given from

the FOC, I =1

, and LOSFFS =

1p . The average daily payment to a hospital with a given is

pDD + pII

L(I;D)=

1 + 2 � 1

1p

;

and taking the average over all hospitals we get that �d � 3:39.Under the PD system the maximization problem is

�dL� L4:

From the �rst order condition we get that LPD =3

s�d

4 . Figure 2 shows the lengths of stay under

the FFS and the PD systems. As proved in Proposition 1, hospitals with longer LOS under the

FFS decrease the LOS, whereas the e¤ect is opposite for hospitals with shorter LOS.

10

3.1.3 Per diem prospective payment system with a step-down rate

The previous section analyzed the impact of the switch from the FFS to PD reimbursement rules

on the length of stay. In this section we add an additional feature to the PD reimbursement to

capture the speci�cs of the health care reform in Japan, which employs the step-down per diem

rate (SDR).

Let �L denote the the average LOS under the FFS system. We assume that there are two per

diem rates under SDR: a higher per diem rate, q �d, during the initial ��L days, where q > 1 and

� � 1; and a regular per diem rate, �d, afterwards.6

The hospital�s pro�t function under the SDR is:

�(L) =

(q �dL� g(L) if L � ��L(q �d) � ��L+ �d(L� ��L)� g(L) if L > ��L

(6)

From (6), �(L) is a continuous function of L, however, it has a kink at point L = ��L. Therefore,

the optimum is either reached at the point where �0(L) = 0 or at ��L. Let L�1( ) denote the

unconstrained maximum of the �rst part of (6) and and L�2( ) denote the unconstrained maximum

of the second part of (6). Formally, L�1( ) satis�es q �d = g0(L) and L�2( ) satis�es �d = g0(L).

Note that since (q �d)��L does not depend on L, L�2( ) is equal to LPD( ) from the previous section.

alphaL L2*

Low γ (γ<γ2)

Pro

fit

LalphaL

Intermediate γ (γ2<γ<γ

1)

Pro

fit

LL1* alphaL

High γ (γ>γ1)

Pro

fit

L

Figure 3: Graphical representation of hospital�s pro�t function for low, intermediate and high values

of .

It follows from the convexity of g that L�1( ) > L�2( ), and that both are decreasing functions of

. Let 2 be such that L�2( 2) = ��L and 1 be such that L�1( 1) = �

�L. Since L�i ( ) are decreasing

functions we have that 2 < 1. Depending on the value of three cases are possible (see Figure

3):

i) when < 2 then LPD( ) = L�2( ) > ��L. This is because if < 2 then L�1( ) > L

�2( ) >

L�2( 2) = ��L. The optimum of the second part is reached at point L�2 such that L

�2 > �

�L. It is the

6 In Section 7 we discuss the predictions of our theoretical model when it is expanded into a model with three per

diem rates which is an exact analogue of Japanese per diem PPS with thresholds a, c, and d, as is shown on Figure

1.

11

global optimum because �(L) is an increasing function for L < ��L. When is low, introducing

higher premium for shorter stay does not a¤ect hospital�s behavior compared to the PD system.

Intuitively, with low the cost associated with LOS is small so that extra bene�ts from shorter

stay are not su¢ cient to a¤ect hospital�s behavior.

ii) when 2 < < 1 then the optimum is reached at ��L. For this range of �s the �rst function

in (6) is increasing and the second function is decreasing on their respective domains. Compared to

the PD system, the LOS goes up, since L�2( ) < ��L. For intermediate values of a higher per diem

rate makes hospitals willing to keep patients longer than they would under PD, however, only up

until the moment when the higher per-diem expires.

iii) when > 1 then the maximum is reached at point L�1( ) < ��L. For high values of ,

hospitals will try to discharge the patients before less favorable per-diem rate is being paid. The

di¤erence with previous case is that is too high and is not worthwhile to keep patients until ��L is

reached. Importantly, as compared to the PD case the LOS still goes up. The main reason being

that the marginal bene�t for longer stay is higher, due to premium q, but the marginal cost is the

same as under the PD.

The analysis above shows that the e¤ect of introducing step-down rate, where initial per diem

rate has a premium, has actually perverse incentives on hospitals as in almost all cases the LOS

increases instead of going down. The Table below summarizes the e¤ect of the change from PD to

SDR reimbursement systems.

< 2 2 < < 1 > 1

E¤ect on LOS LPD = LSDR LPD < LSDR LPD < LSDR

It is well-documented that in Japan the ALOS is the highest among developed countries, which

is why one of the reform�s goals was to provide incentives for quicker discharges. An important

policy insight from our analysis is that having a higher per diem rate, whether for the entire stay

or for some initial period, has an unambiguously opposite e¤ect. Longer stays are more pro�table

for all hospitals.

The combined e¤ect of the change from FFS to SDR reimbursement systems depends on the

sizes of FFS ! PD and PD ! SDR e¤ects, as well as on relation between 0; 1 and 2. To

highlight the incentives that the FFS ! SDR change generates for hospitals we look at the extreme

cases of hospitals with very low and very high values of �s. For such hospitals the combined e¤ect

is summarized in the table below.

low high

L is concave LFFS > LSDR LFFS < LSDR

L is convex LFFS < LSDR ambiguous

For example, consider a hospital with low , where low means < minf 2; 0g; and assume Lis concave. The change to the PD system will decrease the LOS and introducing the premium with

the SDR system will not a¤ect it. Thus, the total change in LOS is negative and LFFS > LSDR.

12

On the other hand when is high so that > maxf 1; 0g then both a change to the PD system

and the premium on the per diem rate as prescribed by SDR will lead to an increase in LOS. The

total e¤ect is, therefore, positive.

3.2 Quality

Although there is still a large inconsistency in economic research about association between read-

mission and inpatient care (Ashton and Wray, 1996), a number of studies demonstrate that early

readmissions may serve as an indicator of quality for hospital performance (Halfon et al., 2006;

Lopes et al., 2004; Weissman et al., 1999; Ashton et al., 1997). In our model we focus on the

planned readmission rate, assuming that there are strong personal relations and high degree of

trust between doctor and patient (Muramatsu and Liang, 1996). Therefore, the patient would

tolerate being discharged sick at the decision of the hospital. Moreover, the patient would seek the

continuation of the inpatient care at the same hospital.

Planned readmission rate is in direct relation to the ALOS. Hospitals can use planned read-

mission to shorten the average length of stay since each readmission, even if the same patient is

readmitted with the same diagnosis, is recorded and reimbursed as a separate instance. Needless to

say, most common reasons for planned readmission are of medical nature. Nonetheless, we think it

is important to understand hospital�s �nancial incentives regarding planned readmission and how

the FFS and the SDR reimbursement systems a¤ect these incentives.

The possibility of readmission changes hospital�s optimization problem as follows. In addition

to determining the optimal length of stay the hospital needs to decide whether to treat a patient

using one admission or two admissions where the second one would be the planned readmission.

For simplicity, we assume that the decision regarding the a number of admissions is made in the

beginning of the treatment.

If a hospital chooses to treat a patient with one admission its cost is g(L). If a hospital chooses

to treat a patient with planned readmission and L1 is the LOS during the �rst admission and L2is the LOS during the planned readmission, then hospital�s cost is g(L1 +L2) +F . Here F � 0 isthe �xed cost due to the planned readmission. We assume that F is a random variable, distributed

with cdf �(�). The reason for the assumption is two-fold. First, with a deterministic F , a plannedreadmission is a 0=1 decision, which is di¤erent from what we observe in the data. Second, random

F captures the idea that the cost of readmission can vary depending on the circumstances such as

patient condition or hospital occupancy rate. Both L1 and L2 are functions of input combinations

used by hospitals that is L1 = L(I1; D1) and L2 = L(I2; D2). Finally, in this section we restrict our

attention to the case of L being a concave function of I which is consistent with the evidence from

our data.

3.2.1 Fee-for-service system

First, we look at hospitals��nancial incentives to use planned readmission under the FFS system.

Having the possibility of a readmission does not change the fact that it is strictly optimal for the

13

hospital to use as much D-inputs as possible. This is because under the FFS systems D-inputs

reduce hospital�s cost associated with the duration(s) of stay, and, accordingly, increase hospital�s

revenue. In what follows it will be convenient also to introduce inverse function I(L;D) which is a

convex function of L. Hospital�s pro�t then is

maxL1;L2

pII(L1; �D) + pII(L2; �D)� g(L1 + L2)� F

if a planned readmission is used. Without the readmission the pro�t is

maxL1

pD �D + pII(L; �D)� g(L):

The next proposition shows that under the FFS there are no �nancial incentives to use planned

readmission. It immediately follows from the convexity of I(L; �D).

Proposition 2 If F � 0 then planned readmission is suboptimal.

Proof. Assume not. Let L1 > 0 and L2 > 0 be the optimal LOS under the �rst and second

admissions. Without loss of generality we can assume that L1 � L2. For a small " > 0 then

pII(L1 + "; �D) + pII(L2 � ")� g(L1 + "+ L2 � ")� F > pII(L1) + pII(L2)� g(L1 + L2)� F ,which is a contradiction to L1 and L2 being optimal. The inequality follows from the convexity of

I(�). Thus the two strict optima are (L�; 0) and (0; L�), and therefore to avoid cost F it is optimalto use one readmission.

3.2.2 Per diem prospective payment system with a step-down rate

Next we study hospital�s �nancial incentives to have planned readmissions under the SDR system.

Recall that under the step-down rate the reimbursement is as follows. There is a basic per diem

rate �d which is augmented by factor q > 1 during initial ��L days. Without the planned readmission

the pro�t is given by (6) and with the planned readmission it is

� F +

8><>:q �d(L1 + L2)� g(L1 + L2) if L1; L2 � ��L2(q �d) � ��L+ �d(L1 + L2 � 2��L)� g(L1 + L2) if L1; L2 � ��Lq �dLj + (q �d) � ��L+ �d(Li � ��L)� g(Li + Lj) if Li > ��L > Lj

(7)

Pro�t (7) is calculated under the assumption that the two admissions are treated and reimbursed

independently of each other. That is, the initial phases of both stays, up to ��L, are compensated

under higher rate q �d and stays longer than that are compensated with per diem rate �d. The �rst

line in (7) corresponds to the pro�t when the length of both admission is short so that the hospital

is reimbursed under the premium per-diem rate q �d. The second line corresponds to the case when

both admissions are longer than ��L and end up receiving daily payment �d. The last line is hospital�s

pro�t when one admission is long7 and another is short.

Let �1 denote the optimal pro�t without the readmission and �2 is the optimal pro�t with

the readmission without the �xed cost F . Planned readmission is more pro�table if and only if7 In what follows �long� is a label for the LOS greater than ��L and �short� for the LOS shorter than ��L.

14

�2��1 > F , that is when gain in pro�t is higher than the cost of the second admission. On averagethen, for a given hospital the likelihood of readmission is �(�2 � �1). Note that the likelihood ofreadmission is a readmission rate, which is observable in the MHLW�s administrative database.

The next statement is the main result of this section. It consists of two parts. The �rst part

shows that �2 � �1 is a decreasing function of , which means that hospitals with low have

stronger incentives to use planned readmission than with high . The immediate and testable

corollary of this result is that hospitals with higher LOS, are more likely to start using planned

readmission. The second part, concerns the length of stay. Recall from the previous section that the

SDR reimbursement encourages longer stays. This is because the marginal bene�t for extra day

is increased by factor q during initial ��L days. However, as we show with the planned readmission

hospitals can split treatment between two stays, thereby reducing the LOS per admission.

Proposition 3 Let L� be the optimal LOS without readmission and L�1 and L�2 be two LOS with

planned readmission. Then

i) �2(L�1; L�2)� �1(L�) is a decreasing function of .

ii) (L�1 + L�2)=2 � L� � L�1 + L�2. The former inequality is strict for hospitals with low . The

latter inequality is strict for hospitals with intermediate values of .

The proof of the proposition is somewhat technical and is given in the Appendix. The intuition,

however, is straightforward. For hospitals with high , the LOS has to be so short that with or

without the planned readmission the per diem rate is q �d and thus there is no gain from using the

planned readmission. For hospitals with low , on the other hand, the gain is substantial as a long

LOS can be split in two, thus doubling the number of days for which hospitals is compensated

under the higher rate q �d.

In the proof of Proposition 3 we consider �ve di¤erent cases, depending on the value of .

While the exact conditions that determine each range are speci�ed in the Appendix; the Table

below summarizes the e¤ect on the LOS for each -range. Cases with lowers numbers correspond

to higher values of . The second column is the di¤erence between the average length of stay

with and without planned readmissions. Note that since each admission is considered and reported

separately, the average length of stay will be the average of L�1 and L�2. The third column shows

the di¤erence in actual number of days that a patient would have to stay in the hospital.

(L�1 + L�2)=2 vs. L

� L�1 + L�2 vs. L

� Planned Readmission

Case 1 (highest ) = = No

Case 2 < > Yes

Case 3 = > Yes

Case 30 < > Yes

Case 4 < > Yes

Case 5 (lowest ) < = Yes

As one can see from the Table above, apart from a few exceptions, the �nancial incentives

related to the planned readmission have undesirable e¤ects. With exception of hospitals with the

15

highest (and, therefore, shortest LOS), all hospitals prefer to use planned readmission. If planned

readmission is used as a response to �nancial incentives, then for all but most extreme cases the

total number of days the patient spends in a hospital goes up even though the recorded LOS, i.e.

LOS per admission, declines. All these negative e¤ects arise owing to the premium per diem rate

during the initial stay, as hospitals have strong �nancial incentive to double the number of days for

which they receive the premium rate.

3.3 Pro�t

We conclude the theoretical part of the paper with a simple comparison of hospital�s pro�tability

under di¤erent reimbursement systems. First of all, for every hospitals are better o¤ under the

SDR system than under the PD system. This is because the initial period under the SDR is

reimbursed with a premium rate q > 1 and therefore �SDR(L) > �PD(L) for every L. A possibility

of using the planned readmission increases hospitals�pro�t even further. This is because the planned

readmission would be used by a hospital only if it is more pro�table. Thus the only non-trivial

comparison is between FFS and PD systems. Recall that under the PD system the per diem rate

is determined based on the average daily payments under the FFS system, that is

�d = E

�pDD + pII

L(I;D)

�:

Let d(L) be the average daily payment that a given hospital receives under the FFS:

d(L) =pDD + pII

L(I;D):

Thus we can rewrite the FFS maximization problem as

maxLd(L)L� g(L);

and the PD maximization problem is as before

maxL

�dL� g(L):

Owing to the similarity of the two expressions above, we obtain the following result:

Proposition 4 Let L�PD denote the LOS in a given hospital under PD and L�FFS denote the LOS

in the same hospital under FFS.

i) if d(L�PD) > �d then �FFS > �PD, i.e. � #;ii) if d(L�FFS) < �d then �FFS < �PD, i.e. � ";

Proof. Consider the �rst case:

�PD = �dL�PD � g(L�PD) < d(L�PD)L�PD � g(L�PD) � �FFS :

The �rst equality comes from the fact that L�PD is optimal LOS under PD. The next inequality

comes from the fact that d(L�PD) > �d and the last inequality comes from the fact that the highest

16

pro�t under FFS has to be greater or equal than the pro�t the hospital can achieve using L�PD.

For the second case

�FFS = d(L�FFS)L

�FFS � g(L�FFS) < �dL�FFS � g(L�FFS) � �FFS ;

which is similar to the �rst case.

As one could expect, the value of the average daily payment under the FFS relative to �d

determines, whether the pro�t is higher under the PD or FFS. In particular, if the average daily

payment under the FFS is less than �d, a hospital bene�ts from the switch to the PD system.

Finally, we examine whether it is hospitals with low or high that are likely to bene�t from

the PD. It follows from the Proposition above that the answer to this question depends on the

function d(�) and, in particular its monotonicity as a function of . For instance, if d(�) is adecreasing function of , then it is hospitals with high that are likely to su¤er from the reform.

Taking the derivative of d(L) with respect to and looking at its sign, we obtain that

sgn�pD �D + pII

L( �D; I)

�0

= sgn�pD �D + pII

L( �D; I)

�0I

@I�

@ = �sgn

�pI(L( �D; I))� (pD �D + pII)L0I

�:

Re-arranging the terms we get

pI

�L( �D; I)� L( �D; 0)� I � L0I

�+ pIL( �D; 0)� pD �DL0I : (8)

The expression in brackets is positive for concave L and, therefore the sign of (8) depends on

relative values of the �rst two positive terms and the third negative term. For example, when

L( �D; I) = I� then the sign of (8) is negative at I close to zero (corresponds to high values) and

positive for large I. This means that d(L) is non-monotone and takes higher value at extremes, i.e.

when �s are low or high, thereby implying that it is the hospitals with longest and the shortest

ALOS that would su¤er from the reform. More generally, in the case of concave L( �D; �) term L0Ideclines with I and therefore (8) is likely to be positive for high values of I (low values of ), so

that hospitals with the longest ALOS are likely to be hurt by the switch to the PD system.

4 Data

4.1 Sample

We employ a recently released administrative database from Japan�s Ministry of Health, Labor,

and Welfare (Aug 16, 2012) on annual aggregated information for hospital�s patients, discharged

in July-December of each corresponding year 2002-2011.8 The data are voluntarily sent to MHLW

by hospitals, which plan to join the PPS reform. Hospitals may join the PPS reform after the trial

8According to the MHLW decision, the data on hospital discharges is annually collected for the period of July to

December .

17

period (normally after one or two years), may postpone the decision and continue submitting the

data to the MHLW, or may choose to never join the reform and stop sending their data.

The annual �les allow us to retrieve the full (i.e. two year) pre-PPS information only for

hospitals, which joined the PPS in the year 2009.9 Merging MHLW�s annual �les by hospital name

(checking for the change of name due to restructuring, mergers, and closures), we construct an

unbalanced panel of 684 such hospitals, which submit the data to MHLW since 2007. 566 hospitals

introduced PPS in 2009, 33 hospitals � in 2010, and 14 hospitals � in 2011. The rest remained

in the FFS system. Note that 14 FFS hospitals left the database in 2010 and 2 hospitals � in

2011.10

Hospital characteristics (the binary variables for rural, emergency, university hospitals, the

number of hospital departments and the presence of MRI and CT scanners) come from the 2011

online version of the Handbook of Japanese Hospitals. Using the data from Japan Council for

Quality Health Care (2012) we construct a binary variable, which equals unity if the hospital is

given accreditation by the beginning of the corresponding �nancial year.11 The MHLW (2012b) data

are employed to create a binary variable, with unity value for hospitals, which received the status of

designated hospital by the beginning of the �nancial year.12 We use �nancial data on hospital�s costs

from the Ministry of Internal A¤airs (The Yearbook of Local Government Enterprises, Hospitals,

Vol.47-56, 1999-2009). Since ownership is shown to be a signi�cant determinant of LOS (Kuwabara

et al. 2006) and e¢ ciency (Motohashi 2009), we construct a binary variable for public hospitals.13

As the MHLW�s database does not provide the hospital ALOS and quality by each DPC, we

use the aggregation at the level of MDCs. The Japanese MDCs are constructed on the basis of

International Classi�cation of Diseases (ICD), with occasional aggregation or disaggregation of

certain diagnoses as explained in Table 2.14

It should be noted that there were 16 MDCs in Japan before 2008. The 16th MDC, which

9Although the one-year pre-PPS data is available for 82 speci�c function hospitals, as well as for 358 hospitals

that joined the PPS in 2008, we do not include them in the analysis. Indeed, the former produces speci�c type of

health care services. As for the latter group, the database does not report hospital names in the �rst year, when they

started to submit their information to the MHLW.10The distributions of ALOS for FFS hospitals that left the database and remained in the database are similar.11Since 1997, Japanese hospitals are given a third-party accreditation if they ful�ll seven standards: 1) mission,

policy, organisation and planning; 2) community needs; 3) medical care and medical care support systems; 4) nursing

care; 5) patient satisfaction and safety; 6) administration; 7) speci�c standard for rehabilitation and psychiatric

hospitals (Hirose et al. 2003).12Prefecture grants the status of a designated hospital to a local public hospital if it satis�es the following require-

ments: 1) has over 200 beds; 2) the share of patients referred from other facilities is over 60%; 3) shares its beds and

expensive equipment (e.g. MRI, CT scanner) with other hospitals; 4) educates local health care o¢ cials; 5) has an

emergency status. Designated local public hospitals receive a support of 10,000 yen per each admission.13Public hospitals in our sample include national (kokuritsu), prefectural (kenritsu, douritsu, furitsu),

city (shimin, shiritsu), town (chouritsu), village (sonritsu), municipal (kouritsu) hospitals, as well as hos-

pitals within the system of National Health Insurance (kokuho) and the system for health care of workers

(roudoushakenfukushikikou).14The Japanese MDC6 encompasses MDC6 and MDC7 in ICD, MDC11 incorporates MDC11 and MDC12 in ICD,

MDC12 combines MDC13 and MDC14 in ICD, MDC13 includes MDC16 and MDC17 in ICD. At the same time,

MDC9 in ICD is disaggregated into the Japanese MDC8 and MDC9.

18

Variable Definition Obs Mean St Dev Min MaxPPS =1 if joined PPS by correspondning

financial year 3388 0.52 0.50 0 1beds total number of beds 3388 294 169 30 1196departments total number of departments 3388 15 6 1 33urban =1 if urban hospital 3388 0.89 0.31 0 1public =1 if public hospital 3388 0.28 0.45 0 1designated =1 if granted the status of designated

local public hospital by correspondningfinancial year 3388 0.08 0.27 0 1

accredited =1 if given independent accreditation byJapan Council for Quality Healthcare 3388 0.62 0.49 0 1

emergency =1 if emergency hospital 3388 0.84 0.37 0 1university =1 if university hospital 3388 0.02 0.13 0 1mri_ct =1 if has MRI or CT scanner 3388 0.93 0.25 0 1

Table 1: Descriptive statistics for the unbalanced panel in 2007-2011.

encompassed all unclassi�ed diseases, was subdivided into three categories in 2008: �Trauma,

burns, poison� (new MDC 16); �Mental diseases and disorders� (new MDC 17), and �Others�

(new MDC 18). Therefore, to analyze the MDC-level data in 2007-2011 we use only 15 MDCs.

The variables of our interest are average length of stay and the number of planned readmissions

(i.e., planned readmissions within 42 days after discharge). While the values of ALOS are available

at the MDC-level, the database reports the prevalence of planned readmissions only at the hospital

level. The MDC-level data are available only for three major reasons of planned readmissions: �Op-

eration after preliminary tests�, �Planned operation or treatment�, and �Chemical and radioactive

treatment�, which account for 72-82 percent of all planned readmissions. We impute the total

number of planned readmissions for each MDC assuming that the share of these three reasons for

planned readmissions is constant across all MDCs and equals to the hospital-level share.

19

Table2:ALOSandreadmissionrateforeachMDCinhospitalswhichimplementedPPSin2009.

NOTE:ThenumbersofMDCsaregivenasof2008.Consequently,in2007thevaluesforMDC16(other)aregivenintherow,

correspondingtonewMDC18.

20

5 Empirical speci�cation

Our theoretical model gives predictions about length of stay for a patient with a given DPC. Given

data availability, the empirical analysis deals with the hospital-level data, and therefore, the testable

hypotheses are formulated in terms of the average length of stay.

Using the longitudinal data on Japanese local public hospitals, we estimate a panel data �xed

e¤ect model with logarithm of ALOS as a function of several hospital inputs (taken in logs):

numbers of doctors, nurses, hospital beds, amount of medical materials (measured in yen), and

examinations per patient. We discover that the sum of coe¢ cients for inputs that increase ALOS

is less than unity and conclude that ALOS is a concave function of increasing inputs. Given the

concavity of the ALOS, our theoretical model yields the following testable hypotheses.

5.1 Hypotheses

H1: The change from a fee-for-service (FFS) to a per diem PPS with a step-down tari¤ (SDR),

increases the ALOS in more e¢ cient hospitals and reduces ALOS in less e¢ cient hospitals.

H2: The change from a fee-for-service (FFS) to a per diem PPS with a step-down tari¤ (SDR),

increases the prevalence of planned readmissions in hospitals with longer ALOS.

5.2 Dynamic panel data model

In our analysis we estimate two models, both based on the following speci�cation:

yit � � = �1(yi;t�1 � �) + �2(yi;t�1 � �) � PPSit + �Xit + �i + "it: (9)

Here, PPSit is the reform dummy which equals unity if hospital introduced PPS in year t, Xit arehospital control variables, �i are �xed e¤ects, "it are i.i.d. with zero mean. The dependent variable,

yit, is ALOS of hospital i in period t for the �rst speci�cation, and planned readmission rate of

hospital i in period t for the second speci�cation. We assume that there is �attraction point�� so

that the e¤ect of the PPS reform depends on whether yit is greater than � or not. In other words,

the e¤ect of the PPS reform for hospitals with the pre-reform value of yit greater (smaller) than �

monotonically approaches the e¤ect for hospitals with yit equal to � �from above�(�from below�).

For convenience we estimate an equivalent speci�cation

yit = �0 + �1yi;t�1 + �2yi;t�1 � PPSit + �3PPSit + �Xit + �i + "it; (10)

where �0 = �(1 � �1) and � = ��3=�2: The identi�cation condition for the existence of theattraction point (i.e., the fact that yit follows an AR(1) process) is 0 < �1 < 1: Given the

identi�cation condition holds, the signi�cance of � implies the presence of an attraction point. In

particular, in case of the speci�cation with ALOS, the estimated value of � may be contrasted to

21

the actual values of the thresholds of a piece-wise tari¤. If an additional condition 0 < �1+�2 < 1

holds, then the attraction point does not change with time (i.e. in the pre-PPS and post-PPS

period).

Since yi;t�1 is a factor of the cross-term (yi;t�1 � PPSit), we treat the cross-term as a prede-

termined variable. Given voluntary participation in the PPS reform, we assume that a hospital

decides about introducing PPS, taking into consideration the value of hospital�s ALOS in the pre-

reform year. Consequently, PPSit must be regarded as a predetermined variable, too. The

time-varying hospital controls in Xit are accreditation dummy and designated hospital dummy.

Equation (10) is estimated using Arellano�Bover (1995)/Blundell�Bond (1998) estimator,15 with

robust variance-covariance matrix (Windmeijer 2005). Lagged levels and lagged di¤erences of yit,

PPSit and (yi;t�1 � PPSit) are taken as instruments for the di¤erenced equation. Arellano-Bond(1991) test does not reject the hypothesis about the absence of serial correlation at order two in

the �rst di¤erenced errors.16 When included in (10), annual dummies proved to be insigni�cant for

most MDCs. Therefore, we do not use time dummies as regressors in our empirical analysis.

We estimate model (10) for each MDC with y being equal to ALOS or planned readmission

rate. Recall that the theoretical parameter of heterogeneity is inversely related to ALOS (i.e.,

hospitals with the shortest ALOS proxy hospitals with the highest ). Therefore, to test for

heterogeneity of hospitals�response we divide all hospitals into four quartiles based on their ALOS

in the pre-reform year (2008).17 We study the changes in the �tted values of the dependent

variable in the k post-reform years and the pre-reform year. More precisely, for each k = 1:::3 let

Dy;i;k =Pks=1 yi;t+s � yi;t; where y is the �tted value of the corresponding dependent variable, y,

i.e. ALOS or planned readmission rate, estimated in (10) and t = 2008. If H1 holds, DALOS;i;k is

positive for the lower quartiles of ALOS and negative for the upper quartiles of ALOS. If H2 holds,

Dplanned readmission rate ;i;k is positive for the highest quartile of ALOS.

To check robustness of the analysis with the panel data in assessing H1 and H2, we measure

DALOS;i;k and Dplanned readmission rate ;i;k by estimating cross-section analogues of equations (10) for

each t = 2009; 2010; 2011. The cross-section speci�cation enables incorporating time-invariant

hospital characteristics in Xit, which are di¤erenced out in the estimations with dynamic panel

data.

6 Results

6.1 Average length of stay

The results of our estimations reveal that the identi�cation condition for dynamic panel data

analysis holds for the speci�cation when y is the average length of stay: �1 belongs to the interval

15Which is more e¢ cient than Arellano�Bond (1991) estimator.16Except for MDC6 in case of the regression with planned readmission rate.17 In case of the hospital-level analysis (the analysis for the average of all MDCs) and for MDC 4 we estimate the

dynamic panel with the �rst di¤erences of ALOS. Therefore, the �tted value of the dependent varibale in 2008 is

unde�ned. Consequently, we construct the quartiles based on the actual values of ALOS in 2008.

22

HospitalsMDC 1 648     0.548*** (0.103) ­0.422*** (0.099)    0.126*** (0.050)    22.230*** (0.588)MDC 2 403     0.877*** (0.042) ­0.260*** (0.041)    0.617*** (0.034)    5.063*** (0.349)MDC 3 598     0.758*** (0.057) ­0.255*** (0.061)    0.503*** (0.054)    6.666*** (0.560)MDC 4 643    ­0.061 (­0.134) ­0.377*** (­0.137)   ­0.438*** (0.025)   ­2.630 (1.866)MDC 5 630     0.421*** (0.064) ­0.332*** (0.062)    0.089** (0.039)   14.530*** (0.520)MDC 6 643     0.892*** (0.060) ­0.513*** (0.059)    0.379*** (0.034)   13.950*** (0.283)MDC 7 630     0.595*** (0.081) ­0.512*** (0.079)    0.083** (0.049)   19.630*** (0.548)MDC 8 496     0.520*** (0.123) ­0.602*** (0.125)   ­0.082 (0.048)   12.490*** (0.377)MDC 9 350     0.335*** (0.075) ­0.528*** (0.080)   ­0.192 (0.051)   10.650*** (0.591)MDC 10 645     0.446*** (0.092) ­0.506*** (0.097)   ­0.060 (0.056)   16.170*** (0.469)MDC 11 631     0.453*** (0.090) ­0.315*** (0.084)    0.138*** (0.039)   12.960*** (0.914)MDC 12 330     0.488*** (0.086) ­0.286*** (0.088)    0.202*** (0.061)   11.720*** (0.560)MDC 13 508     0.645*** (0.074) ­0.488*** (0.063)    0.156*** (0.039)   24.290*** (0.906)MDC 14 272     0.783*** (0.070) ­0.148*** (0.050)    0.635*** (0.058)   14.140*** (1.981)MDC 15 582     0.139 (0.098) ­0.367*** (0.096)   ­0.228 (0.042)     7.497*** (0.328)all MDCs 684     0.030 (­0.091) ­0.391*** (0.097)   ­0.361*** (0.030)    ­2.936*** (­1.146)

α1 α2 α1+α2 μ

Table 3: Dynamic panel data estimations with ALOS as the dependent variable.

Note: In case of all MDCs and MDC4 the table reports the results of the estimations for the �rst

di¤erences of yi;t and yi;t�1, since the process in levels proved to be non-stationary.

(0; 1) for fourteen MDCs out of �fteen. Furthermore, it is signi�cant at 0:01 level for thirteen MDCs,

which means that the attraction point, �, exists for each of those MDCs. The sum �1+�2 belongs

to the interval (0; 1) and is statistically signi�cant for eleven MDCs out of �fteen.18 Arguably, for

the remaining four MDCs the "attraction point" varies over time. Estimated values for �1; �2 as

well as of the attraction point � are given in Table 3.

Table 4 reports estimated changes in ALOS for hospitals that introduced PPS in 2009. As

mentioned earlier, hospitals are sorted based on their ALOS in 2008 so that hospitals in quartile

1 are those with the shortest ALOS. For each MDC and for the average of all MDCs the values

of DALOS;i;k are positive in the lower quartiles of ALOS and negative in the upper quartiles. This

is consistent with H1 regarding a di¤erential response of hospitals to the introduction of the PPS

reform. The results indicate that the values of DALOS;i;k are higher in the lower quartiles of ALOS

and lower in the upper quartiles for the average of all MDCs and for each MDC, which is consistent

with H1 regarding heterogeneity in the change of hospital�s ALOS. As for the robustness check, we

conducted cross-section calculations and discovered that the values of DALOS;i;k are positive in the

lower quartiles of ALOS for 10 out of 15 MDCs; and are negative in the highest quartile for each

MDC as well as for the average of all MDCs.19 Therefore, the results of the cross-section analysis

correspond to the predictions of H1.

It should be noted that our result with the aggregated data (MDC-level data) corresponds to

the �nding about larger reduction in ALOS in hospitals with larger pre-reform ALOS with the data

for patients with the same diagnosis (Nawata and Kawabuchi 2012).

18Statistical insigni�cance is found for MDCs 8, 9, 10 and 15.19Statistical signi�cance is found for MDCs 1, 3, 4, 5, 7, 8, 9, 13, 14 and 15.

23

MD

C1   

3.43

0***

   (0

.125

)    

1.29

7***

   (0

.074

)  ­0

.426

***

   (0

.080

)  ­

3.24

3***

   (0

.169

)   3

.484

***  

 (0.1

24) 

  1.2

43**

*   

(0.0

71) 

 ­0.4

62**

*   

(0.0

75)

  ­3.

298*

**   

(0.1

63) 

  3.5

29**

*   

(0.1

26) 

  1.2

15**

*   

(0.0

70) 

 ­0.4

37**

*   

(0.0

74)

  ­3.

305*

**   

(0.1

61)

MD

C2   

0.45

4***

   (0

.055

)   ­

0.12

9*   

(0.0

82) 

 ­0.6

62**

*   

(0.0

72)

  ­1.

770*

**   

(0.1

11)  

 0.4

15**

*   (0

.057

)  ­0

.291

***

   (0

.083

)  ­0

.918

***

   (0

.079

)  ­

2.20

0***

   (0

.121

)   0

.373

***

   (0

.058

)  ­0

.397

***

   (0

.079

)  ­1

.070

***

   (0

.080

)  ­

2.36

6***

   (0

.121

)

MD

C3   

0.79

2***

   (0

.075

)    

0.24

8***

   (0

.085

)  ­0

.635

***

   (0

.091

)  ­

1.82

0***

   (0

.167

)   0

.747

***  

 (0.0

67) 

  0.1

17*

   (0

.071

)  ­0

.677

***

   (0

.073

)  ­

1.98

1***

   (0

.176

)   0

.709

***

   (0

.063

)   0

.040

   (0

.061

)  ­0

.802

***

   (0

.070

)  ­

2.08

3***

   (0

.180

)

MD

C4   

0.45

5***

   (0

.099

)   ­

0.10

3   

(0.1

24) 

 ­0.0

34   

(0.1

33)

  ­1.

240*

**   

(0.1

72)  

 0.3

51**

*   (0

.105

)   ­

0.13

2   

(0.1

09) 

 ­0.2

09**

*   

(0.1

29)

  ­1.

420*

*   

(0.1

59) 

  0.5

99**

*   

(0.1

33) 

 ­0.0

17   

(0.1

47) 

 ­0.3

20**

*   

(0.1

61)

  ­1.

786*

**   

(0.2

05)

MD

C5   

1.34

7***

   (0

.074

)    

0.39

1***

   (0

.051

)  ­0

.835

***

   (0

.058

)  ­

3.21

5***

   (0

.160

)   1

.292

***  

 (0.0

76) 

  0.3

26**

*   

(0.0

55) 

 ­0.9

15**

*   

(0.0

62)

  ­3.

336*

**   

(0.1

60) 

  1.2

53**

*   

(0.0

77) 

  0.2

96**

*   

(0.0

58) 

 ­0.9

60**

*   

(0.0

66)

  ­3.

374*

**   

(0.1

62)

MD

C6   

1.23

8***

   (0

.113

)   ­

0.41

6***

   (0

.070

)  ­1

.509

***

   (0

.065

)  ­

3.50

4***

   (0

.139

)   1

.073

***  

 (0.1

15) 

 ­0.6

36**

*   

(0.0

61) 

 ­1.7

38**

*   

(0.0

61)

  ­3.

826*

**   

(0.1

37) 

  1.0

41**

*   

(0.1

17) 

 ­0.7

33**

*   

(0.0

59) 

 ­1.8

43**

*   

(0.0

60)

  ­4.

008*

**   

(0.1

39)

MD

C7   

2.49

1***

   (0

.145

)    

0.11

7   

(0.1

03) 

 ­1.6

65**

*   

(0.0

98)

  ­4.

980*

**   

(0.1

87)  

 2.5

14**

*   (0

.144

)   0

.030

   (0

.105

)  ­1

.671

***

   (0

.094

)  ­

5.08

2***

   (0

.184

)   2

.508

***

   (0

.129

)   0

.033

   (0

.106

)  ­1

.703

***

   (0

.093

)  ­

5.11

8***

   (0

.186

)

MD

C8   

1.81

3***

   (0

.081

)    

0.71

9***

   (0

.072

)  ­0

.221

**   

(0.0

99)

  ­1.

906*

**   

(0.1

87)  

 1.8

30**

*   (0

.085

)   0

.817

***

   (0

.080

)  ­0

.155

*   

(0.0

98)

  ­1.

804*

**   

(0.1

85) 

  1.8

25**

*   

(0.0

92) 

  0.8

72**

*   

(0.0

90) 

 ­0.1

77**

   (0

.099

)  ­

1.78

6***

   (0

.186

)

MD

C9   

0.76

2***

   (0

.221

)   ­

0.03

2   

(0.1

85) 

 ­1.3

56**

*   

(0.1

42)

  ­4.

534*

**   

(0.3

04)  

 0.7

64**

*   (0

.209

)   0

.063

   (0

.181

)  ­1

.214

***

   (0

.129

)  ­

4.24

6***

   (0

.292

)   0

.850

***

   (0

.204

)   0

.137

   (0

.174

)  ­1

.172

***

   (0

.130

)  ­

4.20

1***

   (0

.287

)

MD

C10

   1.

416*

**   

(0.0

93) 

  ­0.

159*

*   

(0.0

71) 

 ­1.2

59**

*   

(0.0

82)

  ­3.

740*

**   

(0.1

48)  

 1.3

68**

*   (0

.091

)  ­0

.167

***

   (0

.070

)  ­1

.250

***

   (0

.079

)  ­

3.73

9***

   (0

.150

)   1

.341

***

   (0

.092

)  ­0

.196

***

   (0

.070

)  ­1

.252

***

   (0

.078

)  ­

3.74

4***

   (0

.150

)

MD

C11

   0.

790*

**   

(0.0

63) 

  ­0.

212*

**   

(0.0

75) 

 ­1.1

95**

*   

(0.0

87)

  ­3.

122*

**   

(0.1

82)  

 0.7

62**

*   (0

.063

)  ­0

.268

***

   (0

.074

)  ­1

.252

***

   (0

.082

)  ­

3.20

0***

   (0

.169

)   0

.766

***

   (0

.063

)  ­0

.286

***

   (0

.072

)  ­1

.308

***

   (0

.080

)  ­

3.25

4***

   (0

.165

)

MD

C12

   0.

670*

**   

(0.0

60) 

   0.

033

   (0

.038

)  ­0

.419

***

   (0

.065

)  ­

1.89

6***

   (0

.153

)   0

.666

***  

 (0.0

56) 

 ­0.0

01   

(0.0

35) 

 ­0.5

05**

*   

(0.0

56)

  ­1.

985*

**   

(0.1

51) 

  0.6

36**

*   

(0.0

54) 

 ­0.0

18   

(0.0

34) 

 ­0.5

43**

*   

(0.0

55)

  ­2.

050*

**   

(0.1

50)

MD

C13

   4.

680*

**   

(0.4

07) 

  1.9

22**

*   

(0.3

58) 

 ­0.7

51**

*   

(0.2

90)

  ­6.

315*

**   

(0.4

05)  

 4.9

64**

*   (0

.410

)   2

.092

***

   (0

.340

)  ­0

.731

***

   (0

.297

)  ­

6.55

3***

   (0

.404

)   5

.168

***

   (0

.405

)   2

.125

***

   (0

.342

)  ­0

.662

**   

(0.2

92)

  ­6.

563*

**   

(0.4

04)

MD

C14

   1.

254*

**   

(0.1

42) 

  0.6

08**

*   

(0.1

61) 

 ­0.2

48*

   (0

.190

)  ­

1.61

0***

   (0

.284

)   1

.252

***  

 (0.1

36) 

  0.5

94**

*   

(0.1

59) 

 ­0.2

40   

(0.1

88)

  ­1.

545*

**   

(0.2

91) 

  1.2

72**

*   

(0.1

23) 

  0.6

35**

*   

(0.1

70) 

 ­0.2

33   

(0.1

92)

  ­1.

687*

**   

(0.2

94)

MD

C15

   0.

307*

**   

(0.0

50) 

  0.0

66*

   (0

.044

)  ­0

.237

***

   (0

.045

)  ­

0.62

8***

   (0

.086

)   0

.331

***  

 (0.0

43) 

  0.0

62**

   (0

.036

)  ­0

.205

***

   (0

.040

)  ­

0.64

2***

   (0

.078

)   0

.343

***

   (0

.040

)   0

.089

***

   (0

.032

)  ­0

.167

***

   (0

.038

)  ­

0.59

2***

   (0

.076

all M

DCs

   ­0

.196

*** 

  (0.

065)

  ­0.

323*

**   

(0.0

67) 

 ­0.2

86**

*   

(0.0

71) 

  ­0.

573*

**   

(­0.

079)

 ­0.3

25**

*   (

0.06

6)  ­

0.61

3***

   (0

.068

)  ­0

.541

***

   (0

.077

)­0

.858

***

   (0

.082

) ­0

.347

***

   (0

.077

)  ­0

.756

***

   (0

.078

)  ­0

.716

***

   (0

.084

)   ­

1.07

9***

   (­

0.09

9)

D1

quar

tile 

1qu

artil

e 2

quar

tile 

3qu

artil

e 4

quar

tile 

1qu

artil

e 4

D2

D3

quar

tile 

2qu

artil

e 3

quar

tile 

4qu

artil

e 1

quar

tile 

2qu

artil

e 3

Table4:ChangesinALOSforhospitalsthatintroducedPPSin2009.HospitalsaresortedbasedontheirALOSin2008.Quartile1

correspondstotheshortestALOSandquartile4tothelongest.

24

6.2 Planned readmission rate

The estimations of (10) with planned readmission rate as the dependent variable (Table 5) indicate

that the identi�cation condition for dynamic panel data analysis holds for the average of all MDCs

as well as for ten MDCs out of �fteen.20 The attraction point, �, exists for the average of all MDCs

and for eight MDCs out of �fteen. Finally, the sum �1 + �2 belongs to the interval (0; 1) and is

statistically signi�cant for the the average of all MDCs and eight MDCs out of �fteen.

Table 5: Dynamic panel data estimations with planned readmission rate as the dependent variable

Dplanned readmission rate ;i;k is positive in the highest quartile of ALOS in case of the average of all

MDCs as well as for thirteen out of �fteen MDCs (exceptions are MDCs 3 and 12). We observe a

signi�cantly positive increase for the average of all MDCs in 2009-2011; for nine MDCs in 2009, ten

MDCs in 2010, and eleven MDCs in 2011.This may be interpreted as the proof of H2. The results

of cross-section estimations similarly indicate that H2 holds for the average of all MDCs and for

most MDCs.

Note that the MDC-level estimations with the planned readmission rate are based on the as-

sumption that the share of the three reasons for planned readmission is the same for all MDCs.

The assumption is justi�ed exclusively by the desire to conduct reasonable approximation in the

absence of available data. In particular, the assumption is likely to be questionable for MDC 12

"Female reproductive system, abnormal pregnancy", which would not have many planned readmis-

sions, and they should not be for chemical and radioactive treatment. Overall, since we can not

quantitatively assess the assumption, the MDC-level results with planned readmission rate may be

treated only as tentative �ndings.

20Exceptions are MDC4, MDC8, MDC10, MDC14, and MDC15.

25

Table5:ChangesintheplannedreadmissionrateforhospitalsthatintroducedPPSin2009.

26

7 Discussion

A number of theoretical approaches forecasts potential heterogeneity in hospital�s length of stay

owing to the introduction of a per diem PPS. Yasunaga et al. (2006) argue that although the

Japanese inpatient PPS decreases the mean ALOS for all participating hospitals, the reduction

in ALOS occurs only in large hospitals, which deal primarily with surgical patients.21 A hint

at explaining heterogeneous response of hospital�s ALOS to per diem PPS may be found in the

hypothesis about longer LOS as a result of per diem rates set above marginal costs (Lave 2003;

Frank and Lave 1989; Frank and Lave 1986). As for heterogeneity in hospital�s quality, Kondo and

Kawabuchi (2012) suggest that key issues are the step-down tari¤ and the composition of diagnoses

in a hospital. In particular, patients which require long treatment (e.g. rehabilitation after surgery

owing to hip fractures) are opt to premature discharges owing to the incentives within the step-

down per-diem inclusive payment. Consequently, the quality of care for such DPCs is likely to

deteriorate.

In this paper we present a theoretical model, which explicitly incorporates the essential features

of the per diem PPS with LOS-dependent step-down tari¤, as implemented Japan. The empirical

analysis con�rms the predictions of the model about heterogeneous response of hospital�s ALOS

and quality to the per diem PPS. Our results indicate that the average length of stay increases

for more e¢ cient hospitals and declines for less e¢ cient hospitals, owing to change from a fee-for-

service reimbursement to a per diem PPS. As for the planned readmission rate, it becomes more

prevalent in the least e¢ cient hospitals.

It should be noted that establishing diminishing rates with due incentives is a subtle task for

a per diem PPS (Monrad Aas 1995). Using a simpli�ed case of a per diem PPS with two steps

and a degressive rate, we demonstrate that it contains incentives for an increase in ALOS in more

e¢ cient hospitals. Indeed, the marginal bene�t of treating a patient in the �rst period (with the

highest per diem rate) is larger than the marginal bene�t in the potential case with a �at per diem

rate in both periods. Therefore, the Japanese inpatient PPS makes hospitals raise their ALOS up

till the end of the �rst period. Our �nding is similar to Okamura et al.�s (2005) conclusion about

a disincentive for a sharp decline in ALOS within the Japanese per diem tari¤. In this regard, the

average outcomes of Japanese inpatient PPS may be contrasted to those of the German PPS in

late 1990s, where per diem rate is not degressive (Busse and Schwartz 1997). As for the planned

readmission rate, the economic theory suggests that it increases when a readmitted patient has a

higher revenue-to-cost margin compared to a potential patient who might have been admitted to

sustain the same bed occupancy rate (Hockenberry et al. 2013). Our model reveals that the per

diem PPS with a step-down rate serves as a perfect example to such case.

In 2012 in an attempt to �ne-tune the step-down per diem rates, Japan introduced a modi�cation

of the reimbursement schedule: regardless of hospital�s position in the empirical distribution of

ALOS, no more than 50% of days for each hospital stay may be reimbursed according to the

21 In this reagrd, Kuwabara et al. (2006) demonstrate that surgical procedures explain a large variation in resourse

use.

27

highest rates. Based on our model, we predict that this change has no e¤ect on less e¢ cient

hospitals. However, the incentives of more e¢ cient hospitals to keep patients longer are weakened.

Therefore, the attempt to loosen the stimuli within the step-down per diem rate become bene�cial

for a social planner.

As we mentioned earlier, the actual reimbursement system in Japan has three per diem rates.

The highest rate is paid during the �rst quartile of the ALOS, where the average for a given DPC is

taken over all hospitals. The second per diem rate is paid until the length of stay reaches the ALOS,

after which hospitals are reimbursed with the lowest per diem rate. The extension of our model

from a two-step to a three-step per diem rate does not a¤ect our results. According to the model,

the LOS increases for most hospitals, owing to the premium coe¢ cient q, which is greater than 1.

This coe¢ cient raises the marginal bene�t of a longer stay while the marginal cost is una¤ected.

Whether a hospital uses planned readmission or not, it has more incentives to take advantage of the

higher initial per diem rate and, therefore, two premia q1 > q2 > 1 lead to a qualitatively similar

results.

Finally, we should note the limitations of our analysis. First, we employ the MDC-level data

and therefore, implicitly assume that the composition of DPCs within each MDC is the same in

all analyzed hospitals. Second, our estimations with the MDC level planned readmission rate are

based on an approximation and therefore, may be used only for tentative conclusions. Third,

unavailability of the long time series data for the pre-reform years does not enable us to introduce

time-trends in the estimations with dynamic panel data. Lastly, the fact that the Japanese DPC

database contains the data only for those FFS hospitals which plan to employ PPS in the nearest

future introduces a selection bias in the estimations. Therefore, our analysis of the e¤ect for the

quartiles of the reformed hospitals is based on an assumption that the bias is the same in each

quartile.

8 Conclusion

The paper presents a theoretical model to compare hospital�s incentives under three reimbursement

policies: a standard fee-for-service system (FFS); a per diem PPS system with the per diem rate

equal to the average daily payments under the FFS system; a per diem PPS with a step-down

tari¤ (SDR), where the per diem rate during the initial period of stay is higher than for the rest

of patient�s stay. Our main focus is two-fold. First, we compare the hospital�s e¢ ciency, proxied

by the average length of stay. Second, we study hospital�s �nancial incentives to use a planned

readmission as a response to di¤erent reimbursement rules and, in particular, as a way to reduce

the hospital�s ALOS.

Our main results are as follows. We show that the introduction of any form of the per diem PPS

(either with a �at or with a length-of stay dependent tari¤), leads to a heterogeneity in hospital�s

response. Under some technical assumptions speci�ed in the paper, hospitals with the shortest

ALOS under the FFS lengthen patients�treatment under the PPS, and hospitals with the longest

ALOS under the FFS reduce the length of treatment under the PPS. Contrary to the expectations,

28

higher per diem rate for initial periods, e.g. for the �rst 25% of ALOS, does not generate incentives

to shorten the length of stay. Instead, hospitals prefer to treat patients longer in order to fully

bene�t from the higher per diem rate. Finally, given the emphasis on the shorter ALOS under the

SDR system, hospitals have incentives to use planned readmission to shorten the reported length of

stay. This can be done, since each admission, whether repeated or not, is reimbursed on a separate

basis. Most importantly, we show that it is hospitals with the longer ALOS under the FFS system

have the strongest incentives to treat patients with planned readmissions.

Using a recently released administrative database for 684 Japanese hospitals in 2007-2011, we

conduct estimations with dynamic panel data and �nd an empirical support for the predictions of

the theoretical model. The majority of the 684 hospitals from our data joined the PPS reform in

2009-2011, which involved a change from the FFS to the SDR reimbursement systems. The results

of the empirical analysis con�rm theoretical predictions. We show that ALOS increases in more

e¢ cient hospitals (with shorter pre-reform ALOS), and decreases in less e¢ cient hospitals (with

longer pre-reform ALOS). Furthermore, planned readmission rate rises, particularly for hospitals

with a longer pre-reform ALOS.

9 Appendix

Proof of proposition 3: Let L� be the optimal LOS without readmission and L�1 and L�2 two

LOS with planned readmission. Then

i) �2(L�1; L�2)� �1(L�) is a decreasing function of ;

ii) (L�1 + L�2)=2 � L� � L�1 + L�2.

The former inequality is strict for hospitals with low . The latter inequality is strict for

hospitals with intermediate values of .

Proof. First, it follows from (7) that it is never optimal to have one admission short and

another admission long, that is Li > ��L > Lj . Given that daily payment on admission j are higher

than on i it is strictly optimal to decrease Li and increase Lj by the same amount. Thus, we only

need to consider cases on the cases where either both admissions are long or both are short. In the

former case, if the optimum is interior then it is reached when q �d = g0(L1 + L2). In the latter

case, if the optimum is interior it is reached when �d = g0(L1 + L2).

Next, we will consider several cases for di¤erent values of . We start with hospitals with the

highest .

1. is such that g0(2��L) > g0(��L) > q �d. For these parameter values the pro�t with read-

mission has the global maximum at point where g0(L�1 + L�2) = q �d, and the pro�t with

readmission has the global maximum at point g0(L�) = q �d. This means that even without

readmission the hospital can ensure the daily payment at a premium rate, q �d, which e¤ec-

tively means that unless F = 0, hospitals with such high have no �nancial incentives to use

planned readmission.

29

2. is such that g0(2��L) > q �d > g0(��L) > �d. The �rst inequality means that if a hospital

to use planned readmission it is optimal to use two planned readmissions. In particular,

L�1 + L�2 < 2�

�L. The last two inequalities mean that without readmission the optimal LOS

is ��L and L�1 + L�2 > �

�L since otherwise the optimal solution without readmission would be

less than ��L. This shows that the LOS drops when the readmission is used, but the total

number of days for a given patient goes up.

Next we look at the pro�t di¤erence.

Lemma 1 When g0(2��L) > q �d > g0(��L) > �d the pro�t di¤erence with and without read-

mission is positive and is a strictly decreasing function of .

Proof. The pro�t di¤erence is

�� =

�q �d(L�1 + L

�2)� g(L�1 + L�2)

���q �d��L� g(��L)

�> 0:

By the envelope theorem the derivative of the �rst term with respect to is �g(L�1 + L�2).Since ��L is a constant and does not depend on the derivative of the second term is �g(��L).For the highest in our range, which is when g0(��L) = q �d it is the case that L�1 + L

�2 = �

�L

and so �� = 0. For lower values of we have L�1 + L�2 > �

�L and so @��=@ is

�g(L�1 + L�2) + g(��L) < 0

Thus, �� is a strictly decreasing function of which is equal to 0 at the upper limit of the

interval and therefore is positive on the remaining part of the interval.

For lower values of two cases are possible. When we move away from Case 2 by decreasing

either it g0(��L) becomes equal to �d �rst, or g0(2��L) becomes equal to q �d �rst. We consider

each of these two cases separately.

3. is such that q �d > g0(2��L) > g0(��L) > �d. Then the optimal solution with the readmission

is to have L�1 = L�2 = �

�L. The optimal solution without the readmission is as before, that is

L� = ��L. In this case the LOS is the same and equal to ��L, but L�1 + L�2 > L

�.

The pro�t di¤erence is

�� =

�q �d(2��L)� g(2��L)

���q �d��L� g(��L)

�and it is positive. The reason is that by continuity the value of �� at the upper bound (that

is for such that q �d = g0(2��L)) the value of the pro�t di¤erence should be the same as its

value at the lower bound in the previous case, which is positive. Given that �� is decreasing

with it will remain positive as decreases.

3� Alternatively we consider the case when is such that g0(2��L) > q �d > �d > g0(��L). In

this case without readmission hospitals would go for long admission and with readmission the

30

hospital would go for two short admissions. By the same logic as in case 2 we conclude that

2��L > L�1 +L�2 > �

�L. Furthermore, L�1 +L�2 > L

� because g0(L�) = �d < q �d = g0(L�1 +L�2).

Thus, the LOS will decrease and the total number of days per case goes up. As for pro�t

di¤erence,

�� =

�q �d(L�1 + L

�2)� g(L�1 + L�2)

���q �d��L+ �d(L� � ��L)� g(L�)

�Its derivative with respect to is �g(L�1 + L�2) + g(L�) < 0.

4. is such that q �d > g0(2��L) > �d > g0(��L). The solution with the readmission is L�1 = L�2 =

��L and without is such that g0(L�) = �d, where ��L < L� < 2��L. Thus, the LOS declines

and the total stay goes up. Pro�t di¤erence is

�� =

�q �d(2��L)� g(2��L)

���q �d��L+ �d(L� � ��L)� g(L�)

�;

and its derivative is �g(2��L) + g(L�) < 0 and thus as before we can conclude that �� > 0and is a decreasing function of .

5. is such that �d > g0(2��L) > g0(��L). In this case g0(L�1 + L�2) = g0(L�) = �d and so

L�1 + L�2 = L

�. The pro�t di¤erence is�q �d(2��L)+ �d(L�1+L

�2� 2��L)� g(L�1+L�2)

���q �d��L+ �d(L����L)� g(L�)

�= (q� 1) �d��L

It does not depend on and is positive. While the total LOS does not change, the average

LOS does. This is because it is worth to receive the premium payment q �d twice rather than

once and so hospitals have incentives to use planned readmission.

This concludes the proof of the Proposition.

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