The Society for Economic Studies
The University of Kitakyushu
Working Paper Series No.2012-10(accepted in March 22, 2013)

1

Cost eﬃciency of Japanese municipal hospitals:
Three-component input distance error model
Atsushi Fujii ∗March 22, 2013

Abstract
The aim of this paper is to analyze the allocative and technical ineﬃciency of themunicipal hospitals in Japan by estimating the input distance function system. To
capture the unobserved heterogeniety across the hospitals in relatively short panel
data, this paper applies three-component error model. The result shows that technical ineﬃciency in unproﬁtable area is 6.5%, which implies elimination of technicalineﬃciency is quite small relative to the hospital’s budget and physicians’ shortage inrural area in Japan. Analysis of allocative ineeﬁciency reveals the overuse of capital
equipments and the underuse of nursing labor power. These results are consistent
with the non-cost-minimizing behavior model of hospitals.

Keywords:

Hospital cost, input distance function, allocative ineﬃciency, technical ineﬃciency.

JEL Classiﬁcations: I11, C10∗

This work partly was supported by The Ministry of Education, Culture, Sportts, Science and Technol-

ogy Grant-in-Aid for Young Scientists (B) 08103103. All the possible remaining errors are mine. Faculty of
Economics and Business Administration, The University of Kitakyushu. Kitagata 4-2-1 Kokuraminamiku,
Kitakyushu City, Fukuoka 8028577 JAPAN. E-mail:afujii@kitakyu-u.ac.jp2

1

Introduction

In this paper we analyze the technical and allocative ineﬃciency of public hospitals
in Japan by means of estimating the input distance function. There has been a lot ofliterature on the analysis of the hospital cost ineﬃciency, such as (Ros01), (BE04),
and (HPR02). Most of these recent researches employ either the stochastic cost
frontier (SCF) approach, or the data envelopment analysis (DEA) approach applied
to the input distance function. One of the characteristic features of the former is thatit produces a direct estimate of hospital cost ineﬃciency index which is a mixtureof allocation and technical ineﬃciencies. On the other hand, the latter produces
technical ineﬃciency index estimate, without assuming any functional form on thehospital service production technology.
These approaches have been applied to the data of Japanese hospitals by many
researchers. Among others, (Nak04) reports the technical eﬃcency of Japanese public
hospitals is 90% in 2002 based on data envelopment analysis. On the stochastic cost
frontier approach, (Fuj01) reports the cost ineﬃciency of Japanese public hospitals
is at least 21.1% in 1995.
As noted earlier, the cost ineﬃciency is considered to be a mixture of the technicalineﬃciency (radial shrinkage of inputs without reducing the output), and the allocative ineﬃcency (corresponds to the diﬀerence between the marginal rate of technicalsubstitution and the observed relative input price). (YBV95) measures the allocative
ineﬃcency by estimating the non-minimum cost function. This approach, however,considers no explicit technical ineﬃciency in the hospital service production process.
On the separate estimation of technical and allocative ineﬃciency, (RAFBL04)
recently suggest to estimate modiﬁed version of the system of the input distancefunction and the cost share functions. (RAFBL04) derive this model by assuming
bureaucratic behavior principle for public hospitals, and enable us to measure allocative ineﬃciency as explained later. Besides this preferable aspect of the model,
it also has a practical usefulness in obtaining the two ineﬃciency estimates: The
technical ineﬃciency appears only in the input distance function while the allocative
ineﬃciency appears only in the share equations, so that we can assume a fairly simple
distributional structure for the two sorts of ineﬃciency. These characteristics make3

the model estimation much easier. For the Japanese data, (Fuj06) applies this method
to ﬁnd there was almost no technical ineﬃciency but some allocative ineﬃciency.
As a matter of econometric speciﬁcation, we improved (Fuj06) in the way that we
utilize the panel structure of the data by means of the method proposed by (FO99).
(FO99) adds an extra disturcance term to the composite error of the (cost) function
to capture the random eﬀect.
For the source of ineﬃciency, it seems that we can classify most of those previous
studies listed above in three categories: Over-equipped in capital inputs, failure inallocation of labor inputs, and physician-induced demand. Among those, we focus on
the ﬁrst two problems, both of which are uniformly illustrated within the frameworkof input distance function system.
This paper is organized as follows. Section 2 presents the econometric model of
estimating the technical and the allocative ineﬃciency of the public hospitals in Japan
based on (RAFBL04). Section 3 explains the data used in this study, and Section 4reports estimation results of the input distance function and derived measure of thetwo ineﬃciency separately. In the ﬁnal section we summarize our ﬁndings.

2

Model

We ﬁrst specify the translog input distance function as a modiﬁed version of (Nak03):ln 1 = ln D(Xht , Yht , Zht )∑1 ∑∑αk ln Xhtk +βkl ln Xhtk ln Xhtl
= α0 +
2
k

k

l

1
2
+ αY ln Yht + βY Y ln Yht
2∑+
βkY ln Xhtk ln Yht
k

+

∑∑
k

βkj ln Xhtk ln Zhtj +

j

(1)
∑

βY j ln Yht ln Zhtjj+ εht .
The main diﬀerences between our speciﬁcation and that of (Nak03) are that we focus
on the medical labor inputs and that we include observable service quality variables

4

in the input distance function. The symbols are deﬁned as follows: The kth labor
input and the output of hospital h are denoted as Xhtk and Yht respectively. We consider that the hospital service is diﬀerent across hospitals, so Zhj s captures observed
characteristics of the hospital services. Since the input distance function in homogeneous of degree one in inputs, we put the following restriction on the parameter:∑∑k αk = 1 andl βkl = 0 for all k. We also put the symmetric restriction: βkl = βlk
for all k and l.
The error term εht is decomposed into three parts following by (FO99):
εht = vht + uh + wh .

(2)

The ﬁrst term in this error decomposition equation, vht , is the usual random disturbance term, and is assumed to have N[0, (σv )2 ]. The second term, uh , stands for thetechnical ineﬃciency of the hospital, so that uh distributes over the range of uh ≥ 0.
The third term wh captures unobserved heterogeniety in hospital production. Given(Xht , Yht , Zht ), the value of the distance function is equal to exp(uh ), so we employ
T IE = E[exp(uh )] as our technical ineﬃciency index.
We assume uh has an exponential distribution of which the density is f (u) =exp(−u/σu )/σu , where σu = E(u). By doing so, the probability density of εh is given
by
) (
)−1
)
(
(
T
∏
√ ε̄
1
σ0
σ0
εt − ε̄f (ε) = √
T
−√
ϕ √
,
Φ
exp(−ε̄/σu )ϕσ0σv
T σu σvT −1
T σu
T σu
t=1
(3)
√
∑
T
2 , ε̄ =where σ0 =σv2 + T σw
t=1 εt /T , and ϕ and Φ are the standard normal
density and ditribution functions, respectively. With this assumption, the technicalineﬃciency index deﬁned above is given explicitly by T IE = 1/(1 − σu ).
For the cost share equations, we set up the model following (RAFBL04). The
T = (X T , . . . , X T ) is obtained by radially shrinktechnically eﬃcient input mix Xht
htK
ht1
T =X
ing the observed input mix. Therefore it satisﬁes Xhtk
htk /dht for every k, where

dht is the value of the input distance function (dht ≥ 1). It is clear that the observed
∑
∑
T
T /
cost share Shtk satisﬁes Shtk = Whtk Xhtk / l Whtl Xhtl ≡ Whtk Xhtk
l Whtl Xhtl ,where Whtk is the observed price of the kth input.

5When there is no allocative ineﬃciency (that is, there is no pricing error in deT is equal to the cost minimizing input mix, X ∗ =termining the input mix), Xht
ht∗ , . . . , X ∗ ), so that
(Xht1
htK
T
Whtk Xhtk∂ ln D(Xht , Yht , Zht )∑=+ ε0htk ,
T
∂
ln
X
W
X
htk
htl htll(4)where the random disturbance term ε0htk has zero mean. If, on the other hand,
there exists persistent allocative ineﬃciency, the disturbance term has non-zero mean:E(ε0htk ) = µk ̸= 0.
Thus, in order to capture the persistent divergence of Shtk (or the left hand sideof (4)) from ∂ ln D(Xht , Yht , Zht )/∂ ln Xhtk , we employ the following equation for theobserved cost share:
Shtk = µk + αk +∑βkl ln Xhtl + βkY ln Yht +∑βkj ln Zhtj + εhtk ,

(5)

j

l

for every k, where the new disturbance term εhtk now has zero mean. The positive(negative) value of µk means the hospital uses too much (too few) Xk relative to the
cost-minimizing input Xk∗ , therefore implies the existence of allocative ineﬃciency.
We assume εhtk is composed of two disturbances:εhtk = vhtk + whk ,(6)2 ) and w
2
where vhtk ∼ N(0, σvk
hk ∼ N(0, σwk ), and are independent each other. With

this distributional assumption, we are able to estimate all the parameters by the
maximum likelihood estimation of the system (1) and (5).

3

Data

We collected our data from The Yearbook of Public Firms, Edition for Hospital (Chihou Kouei Kigyou Nenkan Byouinhen, in Japanese), Vol. 52 through 54 , edited by
the Research Association of Local Public Firm Management (Chihou Kouei Kigyou
Keiei Kenkyu Kai, in Japanese), published by the Institute of Local Finance (Chihou Zaimu Kyoukai, in Japanese). It reports ﬁnancial data of the public hospitals inJapan for 2004-2006 ﬁscal years. Throughout this period 792 public hospitals existed.

6

Among those, there are speciﬁc-purpose hospital such as sanitariums and mental hospitals. Following (Nak03) we excluded those hospitals and limit our sample to general
hospitals in order to control the quality of hospital services.
The input factors are, the number of physicians (XDOC ), the number of registerednurses (XRN S ), the number of practical nurses (XP N S ), and the ﬁxed asset per bed(XAST ). The output is measured by the number of beds.
As the observed quality of hospital services, we use the number of clinical examinations per 100 patients (ZEXM ), the number of X-ray examinations per patients(ZRAD ). These two variables are aﬀected by the types of diseases the patients suﬀer,
so they are used to control the hospital’s average severeness of diseases.
The cost share of factor input (Shtk ) are obtained by dividing the cost on the input
by sum of the labor costs on those three medical labor inputs and the cost associatedto holding asset. The asset holding cost is calculated as the sum of depreciation cost,bond interest payment, bond issueing cost, and deplition const.
The data sources report each hospital was located either in the proﬁtable area
(mainly urban area) or in the unproﬁtable area (mainly rural area). It is natural toassume the ineﬃciency is aﬀected by the hospital location, hence we estimate ourmodel for each of the location groups. The ﬁnal sample sizes are 123 for unproﬁtablearea group, and 526 for the proﬁtable area group.
Table 1 shows the descriptive statistics for each location group. The numberof beds diﬀers greatly between the two area groups: Hospitals in proﬁtable areas
have ﬁve times beds than those in unproﬁtable areas when compared at the meanof each sample. The numbers of clinical and X-ray examinations, both of which canbe interpreted as indices of patients’ average severeness of diseases, are about 1.75
and 2.24 times larger in the proﬁtable areas than the corresponding values in the
unproﬁtable areas. In spite of the large diﬀerences in scale and content of medical
services between the two area groups, the value of medical equipments per bed diﬀers
only 1.7%. This can be interpreted as one of the characteristic results of national
health policy in Japan so far placing importance on equity in regional hospital service
provision.
In estimating our equations, all the right-hand side variables are divided by their7

Deﬁnition

Unroﬁtable Area

Proﬁtable Area

Mean

Mean

Std. Dev.

Std. Dev.

Number of physicians (XDOC )

3.664

1.449

32.16526.946

Number of registered nurses

15.5998.755

165.478130.194

nurses

8.295

3.833

13.062

10.870

Fixed asset (thousand yen per

18454.444

31180.679

18772.247

12988.740

Number of beds (Y )61.75320.689305.337184.971

Number of clinical examinations

172.88196.706302.404169.150

12.5959.098

28.33182.055(XRN S )Numberofpractical(XP N S )bed) (XAST )per 100 patients per day (ZEXM )Number of X-ray examinations
per 100 patients per day (ZRAD )Sample size

123

526

Table 1: Deﬁnition and Descriptive Statistics of Variables
sample mean before taken logarithms.

4

Empirical Results

We estimated equations (1) and (5) by the maximum likelihood estimation method.
The results are presented in Tables 2. We present the results for the conventional
panel frontier model in which there is no systematic diﬀerences in unobserved hospitalquality (hospital-speciﬁc eﬀects wh and whk s) as well, for comparison, by putting
σw = σw,DOC = σw,RN S = σw,AST = 0. All of the estimates are correctly signed.

8Location
Hospital speciﬁc eﬀect
Parameter
α0

Unproﬁtable Area
Yes
No
Estimate
Estimate
0.480
0.590∗∗( 0.399 )( 0.235 )αDOC
0.198∗∗∗
0.203∗∗∗( 0.039 )( 0.028 )αRN S
0.629∗∗∗
0.622∗∗∗( 0.040 )( 0.026 )
αAST
0.003
0.038
( 0.052 )( 0.027 )αY−1.048∗∗−0.378( 0.410 )( 0.252 )αEXM−0.0480.042( 0.123 )( 0.095 )αRAD−0.007−0.160( 0.208 )( 0.226 )βDOC,DOC
0.114∗∗∗
0.113∗∗∗( 0.006 )( 0.005 )βDOC,RN S−0.070∗∗∗−0.078∗∗∗( 0.004 )( 0.003 )βDOC,AST−0.021∗∗∗−0.021∗∗∗( 0.005 )( 0.003 )βDOC,Y−0.027−0.026∗∗∗( 0.018 )( 0.007 )βDOC,EXM−0.010−0.007∗∗( 0.006 )( 0.003 )βDOC,RAD−0.002−0.009∗( 0.008 )( 0.005 )βRN S,RN S
0.148∗∗∗
0.176∗∗∗( 0.006 )( 0.004 )βRN S,AST−0.016∗∗∗−0.038∗∗∗( 0.004 )( 0.002 )βRN S,Y−0.004−0.043∗∗∗( 0.020 )( 0.009 )βRN S,EXM−0.003−0.005( 0.006 )( 0.005 )βRN S,RAD
0.007−0.007( 0.007 )( 0.004 )βAST,AST
0.055∗∗∗
0.089∗∗∗( 0.006 )( 0.003 )βAST,Y
0.067∗
0.103∗∗∗( 0.036 )( 0.017 )βAST,EXM
0.013
0.012
( 0.009 )( 0.008 )βAST,RAD
0.002
0.025∗∗( 0.016 )( 0.010 )βY,Y−0.2490.262( 0.275 )( 0.178 )βY,EXM−0.0310.003
9( 0.078 )( 0.063)
βY,RAD
0.000−0.064( 0.133 )( 0.144 )(continues to the next page.)

Proﬁtable Area
Yes
No
Estimate
Estimate−0.065∗∗∗−0.220∗∗∗( 0.017 )( 0.006 )0.036∗∗∗
0.036∗∗∗( 0.009 )( 0.009 )0.795∗∗∗
0.754∗∗∗( 0.011 )( 0.010 )0.032∗∗∗
0.039∗∗∗( 0.007 )( 0.005 )−1.052∗∗∗−0.967∗∗∗( 0.018 )( 0.009 )−0.032∗∗∗−0.049∗∗∗( 0.005 )( 0.004 )−0.007−0.011∗∗( 0.005 )( 0.005 )0.123∗∗∗
0.125∗∗∗( 0.002 )( 0.002 )−0.091∗∗∗−0.095∗∗∗( 0.002 )( 0.001 )−0.022∗∗∗−0.027∗∗∗( 0.001 )( 0.001 )−0.035∗∗∗−0.034∗∗∗( 0.002 )( 0.001 )0.001
0.002∗( 0.001 )( 0.001 )0.000
0.005∗∗∗( 0.001 )( 0.001 )0.160∗∗∗
0.198∗∗∗( 0.003 )( 0.002 )−0.050∗∗∗−0.079∗∗∗( 0.001 )( 0.001 )−0.031∗∗∗−0.067∗∗∗( 0.004 )( 0.002 )−0.0030.000( 0.002 )( 0.001 )−0.004∗∗∗−0.013∗∗∗( 0.001 )( 0.001 )0.082∗∗∗
0.116∗∗∗( 0.002 )( 0.001 )0.086∗∗∗
0.128∗∗∗( 0.003 )( 0.002 )0.003−0.003∗∗( 0.002 )( 0.001 )0.006∗∗∗
0.013∗∗∗( 0.001 )( 0.001 )0.012
0.067∗∗∗( 0.022 )( 0.016 )−0.011−0.005( 0.013 )( 0.012 )−0.010∗0.005( 0.005 )( 0.005 )Location
Hospital speciﬁc eﬀect
Parameterσv
σv,DOC
σv,RN S
σv,AST
σw
σw,DOC
σw,RN S
σw,AST
µDOC
µRN S
µAST
T IE
Log likelihood
Number of Observation

Unproﬁtable Area
Yes
No
Estimate
Estimate∗∗∗
0.075
0.083∗∗∗( 0.003 )( 0.003 )0.022∗∗∗
0.042∗∗∗( 0.001 )( 0.001 )0.021∗∗∗
0.043∗∗∗( 0.001 )( 0.001 )0.038∗∗∗
0.072∗∗∗( 0.002 )( 0.002 )0.237∗∗∗( 0.074 )0.037∗∗∗( 0.003 )0.043∗∗∗( 0.005 )0.069∗∗∗( 0.007 )0.074∗∗∗
0.050∗∗( 0.028 )( 0.023 )−0.169∗∗∗−0.190∗∗∗( 0.030 )( 0.024 )0.268∗∗∗
0.258∗∗∗( 0.026 )( 0.011 )1.065∗∗∗
1.747∗∗∗( 0.374 )( 0.260 )2236.490
1903.080
123

123

Proﬁtable Area
Yes
No
Estimate
Estimate∗∗∗
0.064
0.076∗∗∗( 0.001 )( 0.001 )0.013∗∗∗
0.032∗∗∗( 0.000 )( 0.000 )0.018∗∗∗
0.049∗∗∗( 0.000 )( 0.001 )0.028∗∗∗
0.064∗∗∗( 0.000 )( 0.001 )0.143∗∗∗( 0.007 )0.031∗∗∗( 0.001 )0.051∗∗∗( 0.002 )0.061∗∗∗( 0.003 )0.194∗∗∗
0.197∗∗∗( 0.009 )( 0.009 )−0.342∗∗∗−0.309∗∗∗( 0.013 )( 0.010 )0.213∗∗∗
0.212∗∗∗( 0.010 )( 0.005 )1.189∗∗∗
1.433∗∗∗( 0.024 )( 0.050 )11012.000
8764.730
526

526

Note: ***, **, * indicate p < 0.01, p < 0.05, and p < 0.10, respectively. Standard errors inparentheses.
Table 2: Maximum Likelihood Estimation of Input Distance Function and Cost Share
Equations10

The comparison of models with and without hospital-speciﬁc eﬀects reveals thatthe computed technical ineﬃciency reduces greatly if we consider the presence of
hospital-speciﬁc eﬀect. In unproﬁtable area, the computed amount of cost associatedto the technical ineﬃciency becomes eleven times larger if we do not take into accountthe hospital-speciﬁc eﬀect. This is consistent with the large standard deviations of
hospital-speciﬁc errors (w, wk s): They are more than twice as large as those of v andvk s. Therefore the error decomposition model (2) and 5 well capture the unobserved
service hetergeniety across the hospitals in our relatively short panels.
The estimated technical ineﬃciency index, T IE, of unproﬁtable area group islower than that of proﬁtable area group: The amount of cost that can be saved byeliminating technical ineﬃciency (which possibly causes reduction in quality) is only
6.5% in the former group and 19% in the latter group. In this sense, the input scalein unproﬁtable area is closer to the eﬃcient level than in proﬁtable area.

1

The

diﬀerence between the intercept estimates for two areas might be partly explained by
the diﬀerence in the operation scales.
On the allocative ineﬃciency, in both areas, the parameters µDOC and µAST aresigniﬁcantly positive while µRN S is signiﬁcantly negative. Since µRN S , µP N S < 0 <µDOC < µAST holds in both areas, it should be interpreted that the hospitals spend
too much on ﬁxed asset and too little on nursing labors in terms of cost minimization.
As (RAL04) point out, it is natural to assume public hospitals to have non-cost
minimizing behavior principle, such as preference not only on the cost or proﬁt but
also on the combination of input mix and output. Our estimate of the allocative
ineﬃciency parameters are thus consistent with non-cost minimizing behavior rather
than the simple cost-minimizing behavior. It also suggests the public hospitals may
have stronger preference on the real asset (medical equipments or buildings) than on
nursing in Japan.

1This result for unproﬁtable area is similar to (Nak03)’s DEA result, which reports the average technicaleﬃciency (inverse of technical ineﬃciency) is 0.889.

11

5

Concluding Remarks

In this paper we considered input distance function of the public hospitals in Japan
as well as the cost share functions derived from it, inspired by (RAFBL04). We
estimated those as a system equations and obtained technical and allocative ineﬃency
measure. The main results are (i) the computed technical ineﬃciency gets muchlarger if we do not control the unobserved heterogeniety across the public hospitals
even in the short panel. (FO99)’s three-component error model which was developed
for cost frontier model plays good role in the analysis of technical ineﬃciency using
input distance function system; (ii) the estimated technical ineﬃciency is 6.5% in the
unproﬁtable area sample. The input scale may not be reduced without damaging
quality given the alleged lack of physicians in rural areas in Japan; (iii) the patternof allocative ineﬃciency shows underuse of nursing labors and overuse of assets like
medical equipments and buildings.
The future research may include developing hospital utility model that coincides
with the pattern of allocative ineﬃciency presented in this paper. The balanced budget behavior and subsidy ﬁnanced investment on ﬁxed asset would be an importantview.
Finally, relaxing the assumption on the technical ineﬃciency distribution is yet
important. Though the operation scales are quite diﬀerent between the unproﬁtableand proﬁtable area group, the big diﬀerence between the intercept estimates of thetwo samples implies alternative assumptions like truncated normal may ﬁt well thedata, hereby bring more reasonable diﬀerence between the estimates of intercepts andtechnical ineﬃciencies.

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