Review of Population and Social Policy, No. 8, 1999, 11–25.

An Empirical Study of the Physician-Induced
Demand Hypothesis—The Cost Function Approach to Medical
Expenditure of the Elderly in Japan—
Nobuyuki IZUMIDA*, Hiroo URUSHI** and Satoshi NAKANISHI***

Abstract
Increased medical expenditures for the elderly have become a social
problem in Japan. One of the most powerful arguments to explain this increase is
the physician-induced demand hypothesis. In this article, we adopt theexpenditure function approach to examine the physician-induced demand
hypothesis. In this approach, the improvement of health due to medical treatment
is adjusted in the model and the nonvalid use of medical services can bemeasured. Furthermore, we allow for the possibility of substitution between
hospital services and outpatient care. A change in the physician-population ratio
may lead to a change in the ratio of inpatient service to outpatient care, and thismay result in the medical expenditure. Therefore, to determine the degree of thereflection, we have allowed for the possibility of the substitution. We conclude
that when the number of physicians per capita increases, the use of inpatient
services and of outpatient services increases with statistical significance,
respectively. The substitution between hospital services and outpatient services
also exists. Moreover, the demand for inpatient or outpatient services issignificantly influenced by the self-payment price.

1. Introduction
According to the physician-induced demand hypothesis, a physician can induce apatient to undergo more intensive medical treatment based on the fact that thephysician has more medical information than the patient. This asymmetric
information raises a serious problem. In the ordinary market, as the number ofsuppliers increases, the market becomes more competitive and the price falls.
But when physician-induced demand is prevalent in the medical care market, an
increase in the number of physicians per capita raises the demand for medical
services as a result of the physician’s discretionary power.
This is the English version of a paper presented at the annual conference of the Japan Economics
Association in 1998. Here, we revise the estimation model and method originally published in
Kikan Shakai Hosho Kenkyu (Quarterly of Social Security Research) 33(4), Spring 1998.
* Research Fellow, National Institute of Population and Social Security Research.
** Hiroo Urushi died on 4 November 1998. The authors deeply regret his passing.
*** Associate Professor, Department of Economics, Nihon Fukushi University.

[ 11 ]

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Nobuyuki IZUMIDA, Hiroo URUSHI and Satoshi NAKANISHI

In evaluating public policy for the training of physicians, for example, it isimportant to determine whether the physician-induced demand hypothesis holds
or not. If this hypothesis is valid, the policy that promotes competition among
physicians leads to unnecessary medical services. Although the physician-induced
demand hypothesis has important political implications, a comprehensive
empirical study of this theory has not been conducted in Japan. Therefore, in thispaper we examine the physician-induced demand hypothesis positively.
Feldstein (1970) found that the number of physicians per capita correlated
positively with the payments they received, and he pointed out that this positive
correlation contradicted the results shown by the competitive market. Because
the number of physicians per capita can be thought of as the degree ofcompetition within the physician market in a particular region, we would assume
that in a more competitive region, the supply of physicians will be large. And
when competition among physicians exists, it follows that payments tophysicians for services rendered must fall relative to other regions. But Feldsteinfound the opposite correlation.
It is Fuchs (1978) who provides a theoretical interpretation of thephenomenon described by Feldstein. According to Fuchs, the physicians whoface a decline in income due to an increase in the physician-population ratio
raise the demand for their services through their influence on patients. As aresult, the drop in income due to the increased physician-population ratio iscompensated. Based on this argument, Fuchs found that the number ofoperations rose 0.3% as the surgeon-population rate increased by 1%.
There is, however, criticism of the physician-induced demand hypothesis.1
Most important, the physician-initiated demand cannot be distinguished from thepatient-initiated demand. The physician-induced demand hypothesis claims thatan increase in the physician-population ratio leads to an increase in the1

Criticism is classified into three types. First, Auster and Oaxaca (1981) point out that Fuchs’s
research does not satisfy the identification condition in terms of econometrics. They insist that,to meet this condition, much variation in factor price is needed in estimating the physician
supply function. The second type of criticism involves the estimation method. Dranove andWehner (1986) examine the hypothesis that physician-induced demand exists as to the numberof births by employing the instrumental variable method, also used by Fuchs. As a result, an
irrational result that physician-induced birth is obtained. They say that the source of this problem
is either ignoring the effect of border crossing or choosing the wrong instrumental variable.
Third, Reinhardt (1985) insists that under the theoretical premise on which the physicianinduced demand hypothesis is based, the process of medical treatment is not fully captured. He
further contends that doing positive research under such a theory does not clarify the physicianinduced demand. In contrast to this argument, Dranove (1992) suggests the theoretical modelthat includes the physician’s process of medical treatment and the patient’s process of receiving
medical treatment. Although his theoretical model leads to a proposition that can be proved, it isdifficult to suggest counterevidence due to the lack of data.

An Empirical Study of the Physician-Induced Demand Hypothesis—The Cost Function Approach to Medical Expenditure of the Elderly in Japan—

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physician-initiated demand for treatment. Even if the physician has nodiscretionary power, the real price that a patient pays falls because of a decline
in the cost of access to the medical service; as a result, the patient-initiateddemand may increase. Rossiter and Wilensky (1983, 1984, 1987), Wilensky andRossiter (1981, 1983), and Escarse (1992) also call attention to this problem. If thepossibility of the patient-initiated demand is allowed for, then the effect of thephysician-induced demand is overestimated.
The number of first contacts with physicians, or the rate of receiving medical
treatments, can be thought of as the proxy of patient-initiated demands that are notsubject to physician influence. On the other hand, the number of second contacts
with physicians, or treatment intensity, may have the effect of the physician-induced
demand. Therefore, to estimate physician-induced demand, we must divide thedemand for medical services into the number of times of first and second contact,
or into the rate of receiving treatment and the intensity of treatment.
According to Rossiter-Wilensky and Escarse, the physician-induced demanddoes not exist, or it is extremely small even if it does exist. But the studies of
Rossiter-Wilensky and Escarse have a common pitfall. Based on their arguments,
consumers are more likely to see their doctor when access cost falls due to a risein the physician-population ratio. Generally speaking, the medical conditions oflatecomers to hospitals are less severe than those of earlier arrivals. In this case,
if medical expenditures per patient do not fall, it means that a hidden physicianinduced demand exists.2
There are some studies of physician-induced demand in Japan.3 In early
research on this issue, Nishimura (1987) employed the medical care cost per claim
as the explained variable in his model. He observed that the physician-population
ratio had a positive significant effect on medical care cost per number. But thisresearch cannot avoid the criticism that patient-initiated demand has not been
2

3

All the above research was done in North America, where the cost of medical care isdetermined by the market. On the other hand, the research in countries where price ispolitically determined is interesting in view of its application to Japan. Birch (1988), e.g.,presents a positive analysis using English data, and Grytten, Dorthe and Laake (1990) do sousing Norwegian data. Because England and Norway have adopted the fee-for-service system,
we assume the existence of physician-induced demand in these countries. Birch and Grytten, etal., observe physician-induced demand, whereas Grytten, Carlsen and Sorensen (1995), do not.
In addition to the research mentioned here, Yamada (1994) empirically examined whether
physician-induced demand existed in dental services for the elderly by using the equilibrium
analysis model and disequilibrium model. He found that physician-induced demand was
supported in the equilibrium analysis model but not in the disequilibrium model. Ikegami, etal. (1997), examined the effect of dentist intensity on the amount of treatment for cavities bylinking the claim data of nursery school children with data on a group examination in one
prefecture. In this study, although the coefficient of dentist population ratio was positive, it was
not significant; hence the physician-induced demand hypothesis was not supported.

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Nobuyuki IZUMIDA, Hiroo URUSHI and Satoshi NAKANISHI

distinguished from physician-induced demand.
Suzuki (1997), whose subject is the outpatient care of elderly people can be said
to perform the Rossiter-Wilensky-Escarse type research. According to her findings,
although expenditures for outpatient services increase due to an increase in thephysician-population ratio, their effect is not dominant. Moreover, a coefficient ofphysician intensity to the expenditure per outpatient is not statistically significant,and the physician-induced demand hypothesis cannot be supported.
To overcome faults in the original research of Fuchs as well as the RossiterWilensky-Escarse type of study, we have adopted the expenditure function
approach to examine the physician-induced demand hypothesis. The expenditure
function can measure the level of medical service needed to produce some levelof health. Therefore, improvement of health due to a rise in the number of
medical treatments is adjusted in the model, and the nonvalid input of medical
services can be measured. The health level being adjusted, if it is confirmed thatan increase in the number of physicians raises the demand for medical services,
this implies that physician-induced demand is actually prevalent.
Furthermore, we allow for the possibility of substitution between the inpatient
and the outpatient, which Suzuki (1997) ignores. A change in the physicianpopulation ratio may lead to a change in the ratio between inpatient service andoutpatient care, and this may result in the medical expenditure. Therefore, toanalyze the degree of the reflection, we allow for the possibility of the substitution.
According to the estimation results in this paper, substitution between thehospital service and the outpatient service exists. We find, in addition, that thedemand for inpatient or outpatient services is significantly influenced by the selfpayment price. Finally, when the number of physicians per capita increases by
1%, the hospital demand increases by 0.8% and the outpatient demand increases
by 0.4%, while holding the health level constant.
In section 2 the model is constructed. In sections 3 and 4, after examining thedata, we perform our statistical estimation. Finally, in section 6 we examine theresult of the estimation and state the implications for the policy of planning for
local medical care.

2. A Model
Is it possible to achieve the same health level by selecting a different
combination of medical services? If for some technical reason, an alternative caremix cannot be offered, then only one care mix can be used for healthimprovement. On the other hand, if an alternative mix is available, a patient canselect an outpatient-service-intensive combination of medical services when theAn Empirical Study of the Physician-Induced Demand Hypothesis—The Cost Function Approach to Medical Expenditure of the Elderly in Japan—

15

outpatient service is relatively cheaper than the inpatient one. Similarly, when
there is only one treatment for a particular disease, it is impossible to reduce themedical expenditure by changing the care mix unless we allow for a lower
quality of medical treatment. On the other hand, if the treatment methodology is
flexible, then it is possible to reduce costs by changing the combination oftreatments/services.
It is important for medical policymakers to know whether treatments andservices are flexible or not. The degree of flexibility is measured by the elasticityof alternative care mix substitution. When substitution is impossible, the elasticity
becomes zero. The greater the elasticity, the more flexible will be the alternatives
for care.
In a case where alternative inpatient and the outpatient services can besubstituted, we consider the care mix that will minimize the expenditure forthese services but provide an identical level of health. A locus of the combinationof inpatient and outpatient services that will produce some health level is called
the iso-quant. In Figure 1, the iso-quant is represented by I0. This implies that, byincreasing the inpatient service, the outpatient service can be saved, andtherefore that these services are substitutable.
A locus of the combinations of the quantities of medical services bought forsome fixed amount of money is called the iso-expenditure line. If p1 denotes the
Outpatient service

E1

0

X1
0

X0

E0

I1
I0
1

X0

1

X1

Inpatient service

Figure 1 Medical care input and level of health

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Nobuyuki IZUMIDA, Hiroo URUSHI and Satoshi NAKANISHI

self-payment price of inpatient service, and p0 denotes the price of outpatient
service, then the combination of inpatient services x1 and outpatient services x0–
that can be purchased for some expenditure m is–
m = p 1x 1+ p 0x 0
Rearranging (1), we obtain the iso-expenditure line:–m – p 1x 1
0
x =
p0

(1)

(2)

The iso-expenditure line shows the combinations of inpatient and outpatient
services that can be purchased for the same amount of money. Of course, themore northeast the iso-expenditure line locates, the higher the expenditure is.
When a patient chooses E 0 where the iso-quant line I 0 touches the isoexpenditure line B0, the expenditure is minimized. Then, both the outpatient
service x00 and the inpatient service x01 are used respectively. If a combination
other than E0 is chosen on the iso-quant line, the expenditure will increase. On
the other hand, the health level corresponding to I0 cannot be produced for lessmoney. Therefore, the expenditure is minimized on E0.
Now, we suppose that the self-payment price of the inpatient service rises andthat the slope of the iso-expenditure line becomes steep. Then, new point E'owhere the iso-expenditure line B'0 comes in contact with the iso-quant I0 ischosen. As a result, less inpatient services and more outpatient services aredemanded than before. The higher a patient is willing to raise his/her healthlevel, the more medical services will be needed. Therefore the iso-quantcorresponding to that level is located more northeast than I0, and more medical
services and a higher medical expenditure will be needed.
Suppose that the number of physicians increases and competition among
physicians becomes severe. Then, can a physician induce a patient to receive
medical service that is unnecessary? This is the question suggested by thephysician-induced demand theory. Now let us consider the physician-induced
demand hypothesis within the framework used here. A physician’s influence canbe thought of as moving the subjective iso-quant to the northeast which thepatient faces. Suppose that the actual technology that produces some health levelis located at I0. If a physician makes a patient believe that more medical serviceis needed and a patient misunderstands that it is located at I1, then the patient
will select point E 1 and request an unnecessary medical service, which isunrelated to the diagnosis. In this case, the inpatient service x 11 and theoutpatient service x10 are demanded. These medical services do not improve the1100
health. Therefore, the inpatient service, x 1 –x 0 and the outpatient service, x 1 –x 0

An Empirical Study of the Physician-Induced Demand Hypothesis—The Cost Function Approach to Medical Expenditure of the Elderly in Japan—

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represent excess consumption or pure waste.
Patients’ medical expenditures are determined both by the self-payment priceof the medical service and by the desirable level of health. Of course, a rise in
the desired health level or a rise in the self-payment price will increase themedical expenditure. If the physician-induced demand theory is correct, then thenumber of physicians per capita also will have a positive effect on medical
expenditures. This relationship between medical expenditures and self-payment
costs or between medical expenditures and health level or between medical
expenditures and the number of physicians per capita is expressed by thefollowing expenditure function:4m = m h, p, z

(3)where m = the medical expenditure that a patient pays himself, p = the vector of
the self-payment price, h = the realized health level, and z = the number ofphysicians per capita.
To estimate the expenditure function, we must specify the form of thefunction. Here, we apply a generalized Leontief production function that is usedwidely as well as the Trans-Log production function:5m = h A0 = 1
2

2
i=1

2
j=1

A ij

pi

pj +

2
i=1

Bi p i z .

(4)

Using the Shepard’s lemma,6 we obtain a demand function,∂m∂p i=x i =h A i +A ij

Pj+B i z .

(5)

PiWhen estimation is done using time-series data, because both the explained
variable and the explaining variable have a trend, we often see a large value ofcoefficient of determination even if the formalization has no meaning from an
economic viewpoint. Therefore, it is needed to eliminate the factor of trend.
Moreover, the economic model gives a causal relation in the long-term steadystate. But the data that can be used in empirical analysis are nonsteady state datathat have a trend. It is difficult to obtain the robust parameter even if theordinary regression analysis is done using these data. To avoid these problems,
4

5

6Grossman (1972) was the first investigator to consider the input of medical care service as a
derived demand for the production of health.
For the generalized Leontief function, see Diewert (1971). McAvinchey and Yannopoulos (1993)
estimate the elasticity of substitution between private medical care and public medical care.
For the Shephard’s lemma, see Shephard (1970).

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Nobuyuki IZUMIDA, Hiroo URUSHI and Satoshi NAKANISHI

econometrics researchers developed the Error-Correction Model.7 Converting thedemand function of the medical service (5) into ECM,∆xih=a i ∆Pj
Pi+β i ∆z–γ i

x i,–1h –1–A i –A ij

P j,–1
P i,–1–B i z –1(6)

where ∆ represents the difference between the previous period and the current
period, α is the parameter that denotes the short-run effect of a price on the
demand for medical service, and A is the parameter that denotes the long-run
effect. Similarly, β and B denote the effect of the change of the number ofphysicians on the medical service demand; γ denotes the adjustment speed,which means the degree of the effect of the number of physicians or the price on
the demand for medical service.
We estimated the demand function of the inpatient service and the outpatient
service simultaneously, to obtain the parameters. In the estimation, eachexplanatory variable was divided by its mean value.

3. Data
Our analysis is based on regional data, according to prefecture, collected from1984 to 1994. The sources of these data are summarized in Table 1.

Table 1 Sources of data
Variable

SourceNumber of physicians

Survey of Physicians, Dentists, and Pharmacists,
Ministry of Health and Welfare

Medical expenditure

Annual Report on Health Care for the Elderly,
Ministry of Health and Welfare

Individuals eligible for health insurance
Amount of self-payment
Days of care
Population

Population Estimation, Management andCoordination Agency

Death rate

Vital Statistics, Ministry of Health and Welfare

7

A brief commentary on the Error-Correction Model is provided in the Appendix.

An Empirical Study of the Physician-Induced Demand Hypothesis—The Cost Function Approach to Medical Expenditure of the Elderly in Japan—

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For medical care expenditures, we used the total amount of self-payment thatwas reported in the Annual Report on Health Care for the Elderly, published by
the Health and Welfare Bureau for the Elderly in the Ministry of Health andWelfare. As the cost of medical care, we used the amount of self-payment of
inpatients and outpatients per day. As the indicator of the level of health, weused the alive rate above age 70. The alive rate was obtained from the number ofdeaths above age 70 reported in Vital Statistics of the Ministry of Health andWelfare and Population Estimation of Management and Coordination Agency.
To calculate the physician population rate, we used “the number of physicians
engaged in a medical institution,” according to prefecture, reported in the Surveyof Physicians, Dentists, and Pharmacists by the Ministry of Health and Welfare.
Because the latter survey was conducted every other year, there are years forwhich data were not obtained. For those years, we calculated values by thelinear complement method. Descriptive statistics for these variables are
summarized in Table 2.

Table 2 Descriptive statistics for variables
Mean

S.D.

Min.

Max.

Physician-population ratio

1.591

0.334

0.285

2.418

Alive rate

0.944

0.005

0.886

0.957

Demand for hospital services (days)

22.45813.9342.413

263.657

Demand for outpatient service (days)

41.075

8.103

5.313

64.225

Self-payment price of hospital services (¥1,000)

8.336

4.450

1.716

17.153Self-payment cost of outpatient services (¥1,000)

0.726

0.448

0.299

9.946

4. Estimation Results
The estimation results of parameters are shown in Table 3. The short-run
parameter related to prices is not necessarily good. The parameter related to thenumber of physicians is positive in the case of both inpatient services andoutpatient services, and the former is significant. The long-run parameters havethe same sign as the theory predicted and are statistically significant. The
adjustment speed is negative, as the theory predicted. Hence we can conclude
that these results are fairly good estimates.
Next, from the estimated parameters we calculated the elasticity of
substitution, the price elasticity of demand, and indicators related to physicians

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Nobuyuki IZUMIDA, Hiroo URUSHI and Satoshi NAKANISHI

Table 3 Estimation results
Parameter

Coefficientt-value

α1–5.73565–3.68647

α20.10849

0.15857

β1

3.13111

1.95623β20.651071.44497

A1–5.66492–1.53561

A12

9.29415

3.33756

A2

15.13080

3.52132B1

16.99160

10.21110B2

17.20820

3.92278

γ1–0.37220–2.78960

γ2–0.08325–4.49970

Short-run ParameterLong-run Parameter

Adjustment Speedand the demand for medical care. The Hicks-Allen partial elasticity evaluated atmean can be obtained from the following equation.
AES ij =

mm ij

(7)

m im jThe estimation results of the elasticity of substitution are shown in Table 4. The
elasticities of the medical service inputs are negative, as the theory predicted, andthey are statistically significant. The elasticity of inpatient/outpatient substitution issignificantly positive. Therefore, we can conclude that these services aresubstitutable. The rise in the self-payment cost of a particular medical service
leads to a decrease in the demand of that service and an increase in the demand
for another service.
Table 4 Elasticity of substitution
Estimates

t-value

AES11–0.68035–3.01837

AES22–0.16691–3.43587

AES12

0.33698

3.22418

The elasticity of demand is shown in Table 5. The short-run price elasticity of
inpatient service demand—0.14—is positive. The short-run elasticity of outpatient
service—0.001—is negative, but it is insignificant. On the other hand, the longrun price elasticity of medical care demand is negative, as the theory predicted,

An Empirical Study of the Physician-Induced Demand Hypothesis—The Cost Function Approach to Medical Expenditure of the Elderly in Japan—

21

and significant. A 1% rise of the self-payment price results in a 0.2% decrease in
inpatient service demand and a 0.1% decrease in inpatient service and outpatient
service demand, respectively.
Table 5 Elasticity of demand
estimates

t-value

Inpatient (short-run)

0.13907

3.81879

Outpatient (short-run)–0.00130–0.15846

Inpatient (long-run)–0.22536–3.15502

Outpatient (long-run)–0.11620–3.36558

Next, we measure the elasticities of medical care demands with respect to thenumber of physicians. The effect of the number of physicians on medical careservice is shown in Table 6. According to our estimates, in the short-run, a 1%
increase in the number of physicians will lead to a 0.2% increase in the demand for
inpatient services and a 0.02% increase in the demand for outpatient services. But,for outpatient services, the standard error of estimates is large and insignificant.
Table 6 The effect of the number of physicianson medical care service
Estimates

t-value

Inpatient (short-run)

0.15184

1.96539

Outpatient (short-run)

0.015641.44647

Inpatient (long-run)

0.82400

9.55939

Outpatient (long-run)

0.41333

4.04451

On the other hand, in the long run, a 1% increase in the number of physicians
will lead to a 0.8% increase in inpatient service demand and a 0.4% increase in
outpatient service demand, respectively. We can think that the short-run effect on
the demand for medical care services is small when the number of physicians
changes and that the effect becomes gradually strong. According to ourestimation, the increase in medical care services derived from an increase in thenumber of physicians is unrelated to the improvement of health and can be
regarded as a pure waste of resources.

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Nobuyuki IZUMIDA, Hiroo URUSHI and Satoshi NAKANISHI

5. Conclusion
Allowing for inpatient/outpatient substitution, we estimate the generalized
Leontief expenditure function to be as follows: a 1% increase in the number ofphysicians per capita leads to a 0.8% increase in inpatient demand and a 0.4%
increase in outpatient demand. The point of our analysis is that we have used theexpenditure function for an estimation model. Because we estimate the effect ofthe physician-population ratio on the demand for medical care services and on
expenditures adjusting for the level of health, the measured increase in the
demand for care indicates an increase in the demand induced by physicians. As aresult, we can determine the extent of the physician-induced demand. As far as
we know, this is the only study of physician-induced demand in which theexpenditure function approach is used.
Now, let us examine the implications of the estimation results for actual
government policy. To narrow the regional gap in medical care services byprefecture, the regional medical project has been enforced since 1988. But it
regulates only the upper limit of the number of beds in each prefecture; it does
not regulate the number of physicians. Entry regulations that limit the number ofphysicians have been regarded as one of the most realistic prescriptions forlimiting medical care expenditures. On the other hand, it has been pointed outthat a significant weakness of such regulation is that it has the effect of protecting
the benefits of existing physicians and lowering the incentive for competition
among physicians to provide high quality medical care.
Based on this research, increased demand for medical services does notnecessarily lead to improved health. Although our findings justify the policy that
constrains increases in the number of physicians, we must note the following
points. First, we have analyzed only health care for the elderly. Because in that age
group the rate of chronic disease is relatively high compared to the rate for youth,
there is much room for a physician’s discretion. This may produce a greater
demand for medical care and thus substantially increase the pool of physicians.
Therefore, to evaluate the policy restraining the number of physicians
comprehensively, positive analysis of medical care services for youth is needed.
Finally, the argument of the physician-induced demand hypothesis includes
asymmetrical information about physician and patient. If we return to the starting
point of this argument, constraining the number of physicians may be only asymptomatic response to the problem. It is a policy that excludes asymmetric
information and can become an essential solution. For our purposes, a policythat reduces the asymmetry of information such as medical education for theusers of medical care services or disclosure of medical care information to patientsis important.

An Empirical Study of the Physician-Induced Demand Hypothesis—The Cost Function Approach to Medical Expenditure of the Elderly in Japan—

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Appendix: The Error-Correction Model
Suppose that Y and X satisfy the following equation in long-run equilibrium:
Y=α 0 +α 1 X+u(A-1)

But it takes time to approach equilibrium and the adjustment process has a timelag. The short-run relationship, for example, is expressed by the following
distribution lag model:
Y=β 0 +β 1 X+β 2 X –1 +λY –1 +ε(A-2)

If the economy reached the steady state, from (A-2):
β0

Y=

+

β 1 +β 2

1–λ

X+

1–λ

ε

(A-3)

1–λ

Because (A-3) must equal (A-1), we obtainβ0
1–λ=α 0 ,

β 1 +β 21–λ=α 1 ,

ε

=u.

1–λ

or
β 0 =α 0 1–λ , β 2 =α 1 1–λ –β 1 .
Substituting this relation into (A-2),
Y=α 0 1–λ +β 1 X+ α 1 1–λ –β 1 X –1 +λY –1 +ε.

(A-4)

Subtracting Y from both sides of (A-4) and rearrange, we have
∆Y=β 1 ∆X– 1–λ Y –1 –α 0 –α 1 X –1 +ε.

(A-5)

The second term of (A-5) expresses the following error collection. When Y islarge and the error is positive in the previous term, Y becomes smaller than Y in
the current term and adjustment to equilibrium is acted. In contrast case,adjustment to opposite direction takes place. Like this, Y is changed by ∆ X andthe error collection in ECM.
It is possible to think of ECM as a partial adjustment model that has been often
used in positive analysis:

24

Nobuyuki IZUMIDA, Hiroo URUSHI and Satoshi NAKANISHI

Y t –Y t –1 = γ Y *t –Y t –1 , 0< γ <1

(A-6)

Y *t =α 0 +α 1 X t
where Y* denotes an optimal level of Y at long-run equilibrium and γ denotes theparameter that expresses the adjustment speed. It is clear that
α 1 X t =α 1 ∆X t +α 1 X t –1 .

(A-7)

From (A-6) and (A-7)∆Y t =β 1 ∆X t –γ Y t –1 –α 0 –α 1 X t –1

(A-8)

This is the same as (A-5).

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