The Society for Economic Studies
The University of Kitakyushu
Working Paper Series
No.2022-4(accepted in March 23, 2023)

An Alternative Endogenous Ineﬃciency Structure in
Stochastic Frontier Analysis under Soft Budget
Atsushi Fujii ∗March 23, 2023

Abstract
This note presents a hospital cost ineﬃciency index model as a consequence of somesubjective equilibrium. We model the hospital subsidy depends on its scale, not thedeﬁcit. With a simpliﬁed model speciﬁcation in which the role of subsidy is emphasized,
the subsidy per deﬁcit forms the distribution of cost ineﬃciency which corresponds tothe ineﬃciency term in stochastic frontier analysis. We check the distribution with actual
data controlling the location of the hospitals (a key source of ineﬃciency in many previousanalyses).

∗

Faculty of Economics and Business Administration, University of Kitakyushu, 8028577 Japan.
E-mail: afujii@kitakyu-u.ac.jp
Acknowledgment: This work was supported by JSPS KAKENHI Grant Number JP18K01548.

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Introduction

This note is an attempt to reconsider the distributional assumption of ineﬃciency instochastic frontier analysis (SFA). In SFA, half-normal distribution is often used to capture
ineﬃciency, while some works consider positive mode models (for example, [Greene, 1990];[Fujii, 2001] for hospital ineﬃciency analysis). The distributional assumption on the ineﬃciency is argued because it summarizes the mechanism through which the ineﬃciency
is generated and observed.
To do a preliminary analysis to build an appropriate ineﬃciency distribution assumption in municipal hospitals case, [Fujii, 2016] introduced [Sano, 2007]’s hospital behavior
model together with the soft-budget problem ([Kornai, 1986]) discussed in [Yamauchi, 2009].
The model sets the government subsidy to a hospital as a function of its deﬁcit and foundpositive mode distributions are more suitable as an ineﬃciency distribution.
On the other hand, the approach seems somewhat like short-run settings. There isalso a long-run problem: The geometric distribution of public hospitals in Japan. Sometimes municipal hospitals are regarded as operating on larger scales than their optimum([Fuj and Ohta, 1999], [Otani and Fukuda, 2019]). In such a situation, the mechanism
through which subsidy directly stimulates scale should be considered. The subsidy isnecessary to maintain the healthcare provision system, while it becomes a major ﬁnancial
concern in Japan ([Kawaguchi et al., 2014]). So quantitative method to understand therelation in SFA is required, even if the eﬀect of the subsidy is not just “ineﬃciency.”
Also, there have been advances in the non-parametric estimation of SFA (say, [Kneip et al., 2015],[Forchini and Theler, 2023]). These analytical techniques enable us to relax our assumption on the ineﬃciency distribution and to encompass speciﬁc structures in a speciﬁcindustry (like soft-budget in public entities).
To prepare such an analysis, we introduce in this note an alternative model of thesoft-budget problem that emphasizes the relationship between scale and subsidy. The
model is very simple, so does not require statistical estimation.

2Methodology

As an example of a municipal hospital behavior model, let us consider a modiﬁed versionof [Fujii, 2016]. [Fujii, 2016] utilizes a simpliﬁed model of [Sano, 2007], which emphasizes
the eﬀect of subsidy. Our model is written in a general form:Maximize

U (B, ΠS )(1)

ΠS = Π(B) + S(B).

(2)

with respect to B subject to

The hospital is assumed to maximize its utility function (1) deﬁned over the space ofscale variable (number of beds, B) and the net income ΠS (gross proﬁt Π(B) plus expected subsidy S(B)). To encompass the idea of a soft-budget problem ([Kornai, 1986],2[Yamauchi, 2009]), each hospital is assumed to know the level of S in advance. B stands
for the quantitative aspects of healthcare provision by the municipal hospital, whereasΠS appears as an argument of utility because each hospital is a ﬁnancially independententity.
The diﬀerence between our model and [Fujii, 2016] is that we assume S depends on B
whereas S depends only on the deﬁcit in [Fujii, 2016]. Our model relatively emphasizes
the role of public hospitals in a regional area, while [Fujii, 2016] focuses mainly on theﬁnancial soft-budget problem.
The optimality condition of the hospital’s subjective equilibrium isdΠ∂UdS∂U= 0.
(3)++∂B ∂ΠS dB dB
This equation implies that the production scale is expanded if a subsidy is enhanced.
In order to understand quantitatively, [Sano, 2007] assumes a speciﬁc functional form.
We adopt the following speciﬁcation:
Π = pB − rB, S(B) = sB, U = α log B + ΠS ,(4)where pB is the expected revenue; rB is the cost that depends on B; α, s, p and r are
non-negative constants.
We assume the observed values of B and the cost C are determined in the equilibrium
and is characterized as:
B ∗ (s) =ααr, C ∗ (s) = rB ∗ =.
r−p−s
r−p−s

(5)

The ineﬃciency index I ∈ (0, 1) is deﬁned as:
I=sC ∗ (s) − C ∗ (0)=.
∗C (s)
r−p(6)

If the cost is observed as C ∗ (s) = C ∗ (0)eu , where u is the non-negative ineﬃciency termas modeled in SFA, then the value of u can be computed as:
u = − log(1 − I).

(7)

We check the distribution of u using data from Annual Statistics of Public Enterprises(Chihou Kouei-Kigyou Nenkan in Japanese) Vol 65, edited by Local Public Financial
Bureau, Ministry of Internal Aﬀairs and Communications, the Government of Japan. This
source reports data for the 2017 ﬁscal year. We limit our sample to operating hospitals
that receive subsidies, that are located in an unproﬁtable area, and that have a net deﬁcit,
resulting in a sample of 738 hospitals. We also excluded those hospitals with I ≥ 1, since
our functional speciﬁcation cannot approximate their behavior. The ﬁnal sample size is570. The descriptive statistics are shown in Table 2.

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Location
Number of Beds B
Subsidy per bed S/B (in thousand yen)Ineﬃciency term u
Ineﬃciency index I

Mean
234.7142884.441.286996.6367521

Std. Dev.
Min.
176.620252350.166 32.49495.914121 .0286657.2135114 .0282587

Max.
880
20448.8
7.027087.9991125

Table 1: Descriptive Statistics

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Result and Comments

Table 2 summarizes the data used in our analysis. The mean of I is as high as 0.64, which
may be a result of limiting our sample to the unproﬁtable areas. Although the soft-budget
mechanism is the only source of ineﬃciency considered in our model, the result implies
the signiﬁcance of the problem. For empirical applications, the shape of the distribution
is extremely helpful. Figure 3 illustrates the distribution of u computed by (7). Clearly,
we observe a positive mode for the distribution. This result suggests we better consider
positive mode ineﬃciency for SFA of municipal hospitals in Japan. Parametric as well
as non-parametric approaches, such as [Forchini and Theler, 2023] or [Kneip et al., 2015]
seem applicable in our case.
There are some drawbacks to the approach shown in this note, however. We observed
about 23% loss in sample size (1 − 570/738) by imposing I < 1 restriction on the sample. Looking at (6), the linearity assumption of the subsidy-bed relation (6) may havecaused the loss. Also, the absence of ﬁxed cost and scale economy structure should betreated for empirical application. Nonetheless, the simplicity of our model well depicts
the distributional assumption of ineﬃciency under soft-budget circumstances.

References[Forchini and Theler, 2023] Forchini, G. and Theler, R. (2023). Semi-parametric modelling of ineﬃciencies in stochastic frontier analysis. Journal of Productivity Analysis,59(2):135–152.
[Fuj and Ohta, 1999] Fuj, A. and Ohta, M. (1999). Stochastic cost frontier and costineﬃciency of japanese hospitals: a panel data analysis. Applied Economics Letters,
6(8):527–532.
[Fujii, 2001] Fujii, A. (2001). Determinants and probability distribution of ineﬃciency inthe stochastic cost frontier of japanese hospitals. Applied Economics Letters, 8(12):807–812.
[Fujii, 2016] Fujii, A. (2016). What ineﬃciency distribution is ‘generated’ when behavior
principle is given?: A short note on endogenous economic ineﬃciency modeling ofmunicipal hospitals in japan. The Society for Economic Studies Working Paper Series
No.2015-9, The University of Kitakyushu.

4Figure 1: Distribution of u[Greene, 1990] Greene, W. H. (1990). A gamma-distributed stochastic frontier model.
Journal of Econometrics, 46(1):141–163.
[Kawaguchi et al., 2014] Kawaguchi, H., Tone, K., and Tsutsui, M. (2014). Estimationof the eﬃciency of japanese hospitals using a dynamic and network data envelopment
analysis model. Health Care Management Science, 17(2):101–112.
[Kneip et al., 2015] Kneip, A., Simar, L., and Van Keilegom, I. (2015). Frontier estimation in the presence of measurement error with unknown variance. Journal of Econometrics, 184(2):379–393.
[Kornai, 1986] Kornai, J. (1986). The soft budget constraint. Kyklos, 39(1):3–30.
[Otani and Fukuda, 2019] Otani, Y. and Fukuda, H. (2019). The eﬀects of public hospital
restructuring on management performance. Journal of the Japan Society for Healthcare
Administration, 56(1):17–27.
[Sano, 2007] Sano, H. (2007). Analysis of consistency in japanese municipal hospitalsubsidies using an economic model. Hospital Administration, Journal of the JapaneseSociety on Hospital Administration, 44(3):5–16.
[Yamauchi, 2009] Yamauchi, Y. (2009). The institutional positioning and source-ofrevenue problems of municipal hospitals. International Public Policy Studies, 13(2):51–63.

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