The Society for Economic Studies
The University of Kitakyushu
Working Paper Series No.2015-9(accepted in March 30, 2016)

What ineﬃciency distribution is ‘generated’ when behavior principle is given?: A shortnote on endogenous economic ineﬃciency modeling of municipal hospitals in Japan

Atsushi Fujii

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What ineﬃciency distribution is ‘generated’ whenbehavior principle is given?: A short note on
endogenous economic ineﬃciency modeling ofmunicipal hospitals in Japan
Atsushi Fujii**

Faculty of Economics and Business Administration, University ofKitakyushu, Kitagata 4-2-1 Kokuraminami Kitakyushu 8021577 JAPAN.
March 30, 2016

Abstract
This note presents an attempt to check the shape of ineﬃciency distribution instochastic frontier estimation, by regarding ineﬃciency as a consequence of somesubjective equiibrium of economic agent. As an example, application to the cost
data of municipal hospitals in Japan is considered. The influence of subsidy has been
focused in this field. With simplified functional forms in which the role of subsidy isemphasized, we find the degree of deﬃcit shrinkage by subsidy forms the distributionof cost ineﬃciency. We also check the distribution with actual data. The result
seems compatible with the monotonic truncated ineﬃciency distribution. However,
the result also implies the lower bound of the ineﬃciency should be reconsidered, as
it is not zero.

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Introduction

This short note attempts to provide an example of alternative measurement of inefficiency for the case of municipal hospitals in Japan. We asume hospital behavioral
assumption, instead of ineﬃciency distribution in stochastic frontier analyses (SFA).
The standard procedure to analyze industry-mean ineﬃciency in SFA is to assume one of plausible ineﬃciency distributions such as half-normal or exponential(Aigner, Lovell and Schmidt (1977), Wim Meeusen and van Den Broeck (1977)).
In their framework, the assumed ineﬃciency distribution summarizes all the latent
information on ineﬃciency, including the mechanism through which the ineﬃciency
is generated.
The biggest reason (except for goodness of fit to the data) for most SFA users toemploy these distribution is that they are one-sided: A clearly consistent property
with economic theory.
Then, do we have to leave everything beyond the one-sidedness to the goodnessof fit? One of the possible theoretical approaches to this problem is to specify themechanism that generates ineﬃciency. As summarized in Alvarez and Arias (2014)asHowever, equating ineﬃciency with poor management does not imply that the role of management in production is being correctly modelled.(Section 2.1, page 4)
understanding on behavioral principle of decision making unit (DMU) is essential.
The widepread manner, which relates the predicted individual ineﬃciency to theexogenous covariates, helps to select relevant ineﬃency factors. But the selection of
the covariates often lacks theoretical background.
Building a model that generates ineﬃciency (that is, ineﬃciency is described as
an endogenously determined equilibrium) for each industry is itself cumbersome. But
once the mechanism is specified, individual ineﬃciency is determined as a functionof exogenous variables. This function provides information not provided in usual
two-step SFA. The former approach helps evaluation not only of the eﬀect of various
covariates, but also of behavior principle.
Ineﬃciency is observed because DMUs need to do something more than profit
or cost optimization, not because we append ineﬃciency term to frontier. This may
be the case especially in public utility or non-profit industries. Healthcare provisionsystem in many coutries are typical examples. Moreover, in such industries, government financial aid is sometimes considered in the form of, say, subsidy, allowance,or lendng.
Previous studies on government-owned hospitals includes Rodrı́guez-Álvarez andLovell (2004) which presents the expense preference behavior. A government hospital is described as a section of a government organization, so it maximizes its
objective (utility) function under a given financial constraint. For municipal hospitals in Japan, Sano (2007) incorporates the role of subsidy to the hospital utility

3maximization model. 1 In these studies, hospitals are assumed to have preference
on the quality of their service and financial status. 2In this note we try to numerically and visually understand the cost ineﬃciencydistribution which emerges as a consequence of hospital utility maximization framework given in Sano (2007).

2Methodology and Example

As an example of municipal hospital behavior model, let us consider Sano (2007)
which emphasizes the eﬀect of subsidy. The model is written in general form as:Maximizeu = U (x, π s ),(1)

π s = (1 − s) (p(x) − w · x) .

(2)

with respect to x subject toThe symbols are: x is a vector of input factors of production per bed. The perpatient revenue p(x) is assumed to depend on x. y is the number of patients, andw is the input price vector. It is assumed that hospitals expect they receive subsidy
proprtional to its (negative) profit. The proportion is exogenous to each hospital,and is denoted by s: Each hospital is assumed to know the level of s (Kornai (1979),Kornai (1986), Yamauchi (2009)).
The optimality condition of the hospital’s subjective equilibrium is∂U
∂U+· (1 − s)∂x∂π(∂p−w∂x)
= 0.

(3)

Notice ∂p/∂x = w holds when ∂U/∂x = 0 (or, equivalently, when ∂U/∂π → ∞).
Large x implies aﬄuent healthcare environment. As a consequence of the regulatedindustry, When ∂p/∂x < w as a consequence of the regulated industry, positive sworks to oﬀset the diﬀerence so that it equates to the hospital’s money value of theaﬄuent environment (∂U/∂x)/(∂U/∂π).
For (3), we define three symbols: xob is the solution for given s and U . This
corresponds to the observed x (with disturbance); xns is the hypothetical value ofthe solution when s = 0 and given U . This expresses what if there is no subsidysystem; xnq is the hypothetical value of the solution when ∂U/∂x ≡ 0. This is theeﬃcient x when structure aspect of the healthcare quality is ignored.
Corresponding cost predictions and ineﬃciency measures are also defined:3
1

Though not in an explicit optimization form, Oshima and Ishida (2011) measures the imapct ofsubsidy on the cost structure in Japanese municipal hospitals.
2Rodrı́guez-Álvarez and Lovell (2004) and Sano (2007) rather focus on the structure (economic) aspectof the input resource allocation in hospitals, than overall quality considered in medical sciences (say,Donabedian (1966).
3
These measures can be computed for every hospital in the sample, to form a distribution. In cost
frontier models, the ineﬃciency term η is appended to the frontier function G in the form of log C = G+η,4

C ob = w · xob y, C ns = w · xns y, C nq = w · xnq y,
I nq =(4)C ob − C nq ns C ob − C ns
C ns − C nq,
I=,
I=.
0C nq
C nsC nq

(5)

When quality is not accounted for, I nq is computed. I0 is attributable to subsidy
system: It is not avoidable as long as hospitals have preference on quality. The
simple relation between these ineﬃciency measures:
1 + I ns =1 + I nq,
1 + I0(6)

implies that the usual measure I nq should be discounted by the ‘inevitable ineﬃciency’ factor I0 before interpretation.
In order to understand quantitavely, Sano (2007) assumes specific functionalform. A modified version is:4
U (xh , πhs ) =∑αi log(xhi − θi ) + πhs , p(xh ) = β0 +i∑

βi xhi .

(7)

iThe subscript h denotes hospital, and i denotes input. Then (3) yields(Observed Cost per Bed) = A + BI nq =where A =∑i wi θi , B =∑1,1−sγsγ, I ns =·, I0 = γ,1−s1−s 1+γ

(8)
(9)

i αi /(wi − βi ), and γ = B/A.

We check the distribution of I nq using data from Annual Statistics of Public Enterprises (Chihou Kouei-Kigyou Nenkan in Japanese) Vol 61, edited by Local Public
Financial Bureau, Ministry of Internal Aﬀairs and Communications, the Government of Japan. This source reports data for 2013 fiscal year. We limit our sample toprefectural hospitals that report negative gross profit. We excluded those hospitals
that receives more subsidy than its deﬃcits (s ∈ (0, 1)). Final sample size is 37.
The mean and standard deviation of 1/(1 − s) are 0.58 and 0.08, respectively, withminimum of 0.50 and maximum of 0.85.
The distribution of 1/(1 − s), which is proportional to I nq , is shown in Figure
1. Since I ns and I nq satisfy the relation I nq = (1 + γ)I ns + γ in our example, thedistribution of I ns is expected to have smaller mean and variance than that of I nq
has.
The distribution seems nearly monotonic.5 Those familiar simiple one-sided distributions such as half-normal and exponential work well as long as they have positive
truncation point.
and is assumed to have some one-sided non-negative distribution. The ineﬃciency indices defined in (5)are related to cost frontier ineﬃciency term η as η = log(1 + I).
4We introduced the minimum input parameter θi , and this prevents our model from having equilibrium
inputs being zero. Marginal utility of π s is also changed to constant in this note. We treat number ofbeds as production scale variable.
5Nevertheless, we better mention that there is another small mode around 1/(1 − s) = 0.7.

5Figure 1: Distribution of 1/(1 − s).

3Remarks

Since in our methodology the only exogenous variable is s, the positive truncation
is clearly determined by lower bound of s. The range is limited by our selection ofsample to fit the model.
Actual distribution of s has wider range, and this should be the future research
topic.

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