Discussion Papers In Economics
And Business

The Dynamic Welfare Cost of Stagnation: An Alternative
Measure to the Lucas–Obstfeld Model

Tatsuyoshi Miyakoshi, Masakatsu Okubo and Junji Shimada

Discussion Paper 07-26

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Dynamic Welfare Cost of Stagnation: An Alternative
Measure to the Lucas–Obstfeld Model

Tatsuyoshi Miyakoshi, Masakatsu Okubo and Junji Shimada

Discussion Paper 07-26

June 2007
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Dynamic Welfare Cost of Stagnation: An Alternative
Measure to the Lucas–Obstfeld Model*
byTatsuyoshi Miyakoshi a , Masakatsu Okubob and Junji Shimada cabOsaka School of International Public Policy, Osaka University

Graduate School of Systems and Information Engineering,University of TsukubacSchool of Management, Aoyama Gakuin University

Abstract
This paper proposes an alternative measure to the Lucas–Obstfeld model to analyze thewelfare costs of stagnation, and provides a practical illustration of both theLucas–Obstfeld model and the alternative model. Compared with the Lucas–Obstfeld
model, the alternative model can evaluate: (i) whether the policy was implemented in atimely fashion, (ii) whether the policy cost was expensive compared with the cost of
stagnation, and (iii) whether the policy implemented was effective or whether an
additional policy is required.

Keywords: Lucas model; dynamic welfare cost; time-varying parameters
JEL Classification Number: C32; C50; E32; E20

Correspondence: Tatsuyoshi Miyakoshi, Osaka School of International Public Policy
Osaka University,1-31, Machikaneyama-machi, Toyonaka,Osaka, 560-0043, Japan.
tel:+81-6-6850-5638; fax:+81-6-6850-5656. E-mail: miyakoshi@osipp.osaka-u.ac.jp* Earlier versions of this paper were presented at Osaka University and Hitotsubashi University
in 2003, City University of Hong Kong and Tokyo University in 2004, 2004 Western Regional
Science Association Conference (in Hawaii), 2004 Western Economic Association Conference(in Vancouver), 2004 East Asian Economic Association Conference (in Hong Kong). We thankKenneth S. Chan, Makoto Saito, Yum K. Kwan, Yong Wang, Eiji Ogawa, Yoshiro Tsutsui, Yuzo
Honda, Shinsuke Ikeda, Soyoung Kim, and Joshua Aizenman for useful suggestions. The firstauthor’s research was supported by Grant-in-Aid 16530204 from the Ministry of Education,
Culture, Sport, Science, and Technology of Japan.

1. Introduction
Diagnoses of the seriousness of stagnation and the appropriate recovery process areimportant because they enable a government to evaluate its policy effects andimplement additional policy if required. Several measures for diagnoses exist. A
standard measure is the reduction in GDP (or its growth). Other measures are proposedby Chow and Kwan (1996) and Kwan and Chow (1996), who measured the effects ofpolitical movements in China (the Great Leap Forward Movement of 1958–1962 andthe Cultural Revolution of 1966–1969) on output, consumption, and investment. They
compared the ‘hypothetical’ time paths of these variables (i.e., the time paths in theabsence of the political movements) with the actual time paths and computed the ratio
between both time paths. This provided a measure of stagnation relating to the politicalmovements. A third measure for diagnosis was developed by Lucas (1987) and Obstfeld(1994).
Lucas and Obstfeld measured compensations that would leave consumers
indifferent to a decline in economic growth and an increase in economic instability. We
refer to both types of compensation (for the decline in economic growth and theincrease in instability) as the ‘welfare cost of stagnation’ and the model as the‘Lucas–Obstfeld model’. The welfare costs based on utility will equal the costs thatpeople are willing to pay to prevent the outbreak of stagnation. Then, we can evaluate
whether the cost of the policy to prevent stagnation, which a government would financefrom tax revenues, is higher than the welfare cost of stagnation. The practical use ofmeasuring this cost had not been explored previously. However, Lucas and Obstfeld did
not define the cost of stagnation. Rather, their concern was with the latter compensation(the cost of the economic instability) and they did not provide any analysis of the cost of1

stagnation: they only derived the formulation for the cost of stagnation. This also
applies to work by Dolmars (1998), Krusell and Smith (1999), Storestetten et al. (2001),
Beaudry and Pages (2001), and Pallage and Robe (2003).
We consider that an alternative to the Lucas–Obstfeld model is needed tomeasure the cost of stagnation. Lucas and Obstfeld assumed that: (i) a stagnation
economy is characterized by constant growth over time and instability, (ii) an agent
knows the moments of the distribution for the economy (i.e., they assume rational
expectations), and (iii) an agent measures the cost from the beginning of stagnation to
the future as the ‘whole cost’ of stagnation compared with a hypothetical economy (i.e.,
an economy without stagnation). Alternatively, we argue that it is natural to assume that:(i) the stagnation economy has more complex processes of time-varying growth andinstability, (ii) an agent knows the moments of the distribution for today’s economy only
and he/she expects (or believes) these moments will continue forever (i.e., myopic
rational expectations, meaning that an agent bases tomorrow’s moments on tomorrow’s
coming information, and the same for future moments), and (iii) an agent measures thecost of stagnation from period t to the future as a ‘ cost at period t’ compared with ahypothetical economy. Both cost measures are based on different setups. The alternativemodel can reflect decreasing costs period by period in the recovery process ofstagnation. Other alternatives combining the above three aspects (i)–(iii) seem
implausible and are difficult to derive by applying the previous contributions of Lucas(1987), Obstfeld (1994), and Kwan and Chow (1996).
The purpose of this paper is to propose an alternative measure to theLucas–Obstfeld model for analyzing the welfare cost of stagnation, and to provide apractical illustration of the alternative model compared with the Lucas–Obstfeld model2

by using Japanese data from 1975 to 2002.
By using this alternative model, we can evaluate: (i) whether the policy was
implemented in a timely fashion, (ii) whether the policy cost was expensive comparedwith the cost of stagnation, and (iii) whether the policy was effective or whether
additional policy is required. These specific exercises cannot be carried out under the
existing frameworks, including the Lucas–Obstfeld model. However, how should wecompare the Lucas–Obstfeld model and the alternative model? The ‘whole cost’ ofstagnation by the Lucas–Obstfeld model is evaluated as being 25,390 yen per month(the cost ratio is 26% of the monthly consumption of 99,373 yen in January 1990, thestarting period of stagnation). On the other hand, the alternative model evaluates thecost as being 35,798 yen per month (35%) in January 1992, and determines that the costreaches a peak of 59,400 yen per month (51%) in December 2001, and falls to 55,610
yen per month (48%) in the most recent period, August 2002. The cost does not show
any clear sign of a decrease.
This paper is organized as follows. In Section 2, we sketch the Lucas–Obstfeld
model, adding the analyses for the properties of the welfare costs. In Section 3, we
provide an alternative model. In Section 4, we show the practical use of both models.
Section 5 concludes the paper.

2. The Lucas–Obstfeld Model
2.1. Framework3We sketch the Lucas–Obstfeld model.1 Following Lucas (1987), the representative
agent lives infinitely and maximizes an expected utility function V by choosing realconsumption Ct at time t. The agent has preferences specified by:⎤⎡∞
1
V = E ⎢∑ β t
Ct1−γ ⎥ ,1−γ⎦
⎣ t =0(1)

where β∈(0,1) is a constant discount factor and γ>0 is the constant coefficient ofrelative risk aversion. Here, we consider a pure exchange economy with no production,
no storable goods, and no borrowing. Then, the optimal consumption Ct for an agent issubject to exogenous income It in each period and hence is equal to income: Ct = It for
all t.
Lucas and Obstfeld assumed a class of exogenous income. Hence, the optimal
consumption streams Ct with trend and cycle components, are given by:

Ct = λ (1+ µ ) e
t

1
− σ2
2

zt ,

,
1
i. e., ln Ct = ln λ - σ 2 + t ⋅ ln (1+ µ ) + ln zt : ln zt ~ N (0,σ 2 )2(2)where µ is the growth rate of consumption and ln zt ~ N(0, σ2). In addition, Lucas and

1

The Lucas–Obstfeld model was not constructed for practical use. Thus, it must be revisedslightly, and so the cost that we derive is a little different from the original cost derived by Lucas
and Obstfeld.
.

4

Obstfeld assume that an agent has rational expectations, which implies that an agent
knows those moments of the consumption distribution, and then maximizes anunconditional expectation of utility (1): the subscript of time t is not attached on V in (1).
Owing to the property of the log-normal distribution, E(zt•exp(–σ2/2)) = 1, the mean
consumption is:E(Ct ) = λ (1+ µ )t ,(3)

where the mean consumption at t = 0 is λ.
Thus, Lucas and Obstfeld assumed that the stagnation process of exogenous
income (consumption) can be expressed by constant moments over time of the
distribution of consumption, λ, µ, and σ2, and that an agent has rational expectations.
Under the above setup, we can calculate the indirect utility given the
consumption process described by (2) and denote it by V(λ, µ, σ2|γ,β). This is derived asfollows:

V (λ, µ,σ 2 | γ , β ) =1γ⎧⎫exp⎨(1− γ )(ln λ − σ 2 )⎬ if φ < 1,
(1− γ )(1− φ )
2
⎩
⎭
(4)

φ = exp{ln β + (1− γ ) ln(1+ µ)}.

Details of the derivation are given in the Appendix.
We consider two economies. One is called the stagnation economy, in which

5consumption growth µ s and its variance σs2 are calculated based on the data in thestagnation period. We denote the resulting indirect utility as V( λ S , µ S , σ S2 | γ , β ) . The
other economy is called the hypothetical economy (i.e., the economy without stagnation),
which is based on expected consumption under the assumption that the growth rate andthe variance in the prestagnation period are maintained during the stagnation period.
The resulting indirect utility is V( λ H , µ H , σ H2 | γ , β ) . The intuition behind thiscomparison is shown in Figure 1. Owing to (3), the λH is mean consumption at the
beginning of the stagnation period for the hypothetical economy. Thus, we compare
both economies from the beginning of the stagnation period (denoted by t = 0 in Figure1). Although γ and β may differ between the prestagnation and stagnation periods, we
assume that they remain constant over time at ( γ , β ) .

[INSERT Figure 1]

Using the indirect utilities under these economies, we define the ‘whole cost’ ofstagnation as follows.

Definition 1. The cost of stagnation is given by λ*, which satisfies the following
equation:

V (λS + λ*, µS , σ S2 | γ , β ) = V (λH , µ H ,σ H2 | γ , β ) ,(5)where the subscripts S and H denote the stagnation economy and the hypothetical6

economy, respectively.
The key concept relating to the whole cost of stagnation is the following. The
consumption parameters are different between the stagnation and the hypothetical
economies. Consumer preferences, given by ( γ , β ) , transform the difference in
consumption parameters into a difference in utility levels. The whole cost of stagnationis measured by the compensation required to leave consumers indifferent between thetwo economies. This compensation is uniform across all periods. The whole cost
implies the cost from the beginning of the stagnation to the future.
The calculation of λ* is given by:

λ* = exp{Ψ} − λS ,
Ψ=⎫1 ⎧ 1 - φS
) + (1− γ )(ln λH − γσ H2 /2)⎬ + γσ S2 /2 , (6)⎨ln(
1 − γ ⎩ 1- φH⎭

where:

φS = exp{ln β + (1− γ ) ln(1+ µS )} and φH = exp{ln β + (1− γ ) ln(1+ µ H )}.

The derivation is given in the Appendix.2

2.2. Properties of Welfare Costs
We derive the partial derivatives of the cost λ∗in terms of the time preference2

Obstfeld (1994, pp. 1474–75) completely derived the cost of stagnation in the framework ofLucas, and the formulation of the costs is the mostly same as in (6). However, his concern was
only with the cost of the economic instability. Obstfeld evaluated whether the persistency of
consumption shocks affects Lucas’s finding that the cost of economic instability may be slight.
Therefore, in his application, only the cost of the economic instability was examined, and noanalysis of the cost of the stagnation was provided (see Section 4 of Obstfeld, 1994).

7

β and the relative risk aversion γ by using equation (6) under γ > 1 :φH − φS∂λ∗,= exp{Ψ}∂β(1 − γ ) β (1 − φ H )(1 − φ S )

(7)

⎫
⎧ 1
∂λ∗
1 2
2
{ln
ln
}
= exp{Ψ}⎨
−
+
−
µ
µ
(
σ
σ
)
⎬,
S
H
S
H
2
∂γ
2
(
1
)
−
γ
⎩
⎭

(8)

and

where φ H < 1 and φ S < 1 , as shown in equation (4).3 When the growth rate of
consumption falls in a stagnation period (i.e., µH>µS, which means that φ H < φ S ), theincrease in the time preference (i.e., the agent evaluates the future more) increases thewelfare cost, which is uniform across all periods, to compensate for the reduced growth
to the distant future: ∂λ∗ / ∂β >0. When the reduced growth rate of consumption in astagnation period is less than that in a prestagnation period plus the difference between
both variances (which are estimated in this paper as mostly equal), a decrease in relativerisk aversion increases the cost: ∂λ∗ / ∂γ <0. In addition, the signs of the derivatives interms of µH, µS σH, σS, λH, and λ S are obvious: ∂λ*/∂µH>0, ∂λ*/∂µS<0, ∂λ*/∂σH <0,∂λ*/∂σS >0, ∂λ*/∂λH >0, and ∂λ*/∂λS <0. The derivation is given in the Appendix.

3. An Alternative Model: The Dynamic Welfare Cost

Alternatively, it is natural to assume that the stagnation economy for consumption can

3

So far, Lucas (1987), Obstfeld (1994), and others have only roughly investigated therelationship between the cost and the preference parameters by calibration. Nevertheless, our8be expressed by the following time-varying intercept and slope:4

lnCt = at + bt ⋅ t + ln zt :ln zt ~ N (0,σ 2 )1at ≡ ln λt − σ 2 ; bt ≡ ln(1+ µt )2
⎡a t ⎤ ⎡α0 ⎤ ⎡α1 0 ⎤ ⎡a t −1 ⎤ ⎡ηa,t ⎤⎢ ⎥= ⎢ ⎥+ ⎢⎥ ⎢b ⎥ + ⎢η ⎥ :bβ
0β⎢⎣ t ⎥⎦ ⎣ 0 ⎦ ⎣ 1 ⎦ ⎢⎣ t−1 ⎥⎦ ⎢⎣ b, t ⎥⎦(9)⎛⎛ 0⎞ ⎡qa2 0 ⎤ ⎞⎡ηa,t ⎤⎢ ⎥~ N⎜⎜⎜⎜ ⎟⎟, ⎢ 2 ⎥ ⎟⎟ (10)⎢⎣ηb,t ⎥⎦⎝⎝ 0⎠ ⎢⎣0 qb ⎥⎦ ⎠t=0,1,2,….
Then, we assume that an agent has myopic rational expectations, which implies
that an agent knows only the moments of the distribution of today’s consumption
parameters, σ2, at (λt) and bt (µt). Moreover, an agent expects (or believes) these
moments of today will continue forever for future consumption. Thus, based on today’s
information, an agent obtains the conditional moments for today and for future
consumption (i.e., the stagnation economy). The indirect utility function of (4)conditioned on today’s information includes the subscript of time t.

Vt (λt , µt ,σ 2 | γ , β ) =1γ ⎫⎧exp⎨(1− γ )(lnλt − σ 2 )⎬ if φt < 1,(1−γ )(1−φt ) ⎩
2 ⎭
(11)

φt = exp{lnβ + (1−γ ) ln(1+ µt )}.

theoretical analysis requires that stringent results be obtained.
4
We assume the variance of shocks is constant over time, considering Lucas’s finding that thecost of business cycles may be slight. In addition, we assume that the preference parameters areconstant over time, following the Lucas–Obstfeld model.

9At period t, an agent gets these parameters and computes indirect utility fromperiod t to the future, based on the coming information at period t. Here, we compare
this indirect utility at period t with the indirect utility for the hypothetical economy
mentioned in the Lucas–Obstfeld model. The cost of stagnation at period t is given by
λt*, which satisfies the following equation:

Vt (λS ,t + λ∗t , µS , t , σ S2 | γ , β ) = Vt (λH ,t , µ H ,σ H2 | γ , β ) ,(12)where the subscripts S and H denote the stagnation and the hypothetical economies,
respectively. The parameters for the stagnation economy are changed period by period,
whereas the parameters for the hypothetical economy are the same and constant overeach period, as in the Lucas–Obstfeld model. The parameter λH,t is the exception
because it is an intercept of consumption and it changes at period t.
Then, an agent recomputes the ‘cost at period t’ of stagnation in each period,
based on the updated information.

λ∗t = exp{Ψt } − λS , t ,Ψt =⎫1 ⎧ 1 - φS , t) + (1− γ )(ln λH , t − γσ H2 /2)⎬ + γσ S2 /2 ,⎨ln(
1 − γ ⎩ 1- φH⎭(13)

where:

φS , t = exp{ln β + (1− γ ) ln(1+ µS , t )} and φH = exp{ln β + (1− γ ) ln(1+ µ H )}.

Keeping the variance constant in the stagnation period but letting the consumption

10intercept and growth fluctuate, the agent recomputes the cost at period t, comparing
with the hypothetical economy. This cost of stagnation is different period by period.
Figure 2 is a schematic diagram that describes this idea and summarizes ouralternative model. An agent knows only the present period (T1)’s intercept λS,T1 andslope µS,T1 in (11) from information at period T1, and he or she expects (or believes)these parameters λS,T1 and µS,T1 will continue forever. On this basis, the agent constructs
the stagnation economy from period t to the future. Thus, the stagnation economies
expected from period T1 to the future and from period T2 to the future will be different,
and as a result, the costs between T1 and T2 are different. The ‘cost at period t’
calculated in this way may be referred to as the dynamic cost of stagnation, in contrast
to the ‘whole cost’ computed from the beginning of the stagnation to the future by theLucas–Obstfeld model. In general, the cost at period t will be higher at the start andlower at the end of the stagnation period.

[INSERT Figure 2]

By using a ‘cost at period t’, we can evaluate: (i) whether the policy was
implemented immediately in the period when the highest welfare cost arose, (ii)whether the cost of the implemented policies was expensive compared with the welfare
costs of stagnation, and (iii) whether the cost of stagnation decreased gradually—that is,whether the policies were effective—or whether additional policies are required, which
will be ascertained by determining the current welfare costs. These specific exercises
cannot be undertaken under the existing frameworks, including under the
Lucas–Obstfeld model.

11

Although an alternative model can compute the cost period by period, how can it
derive the whole cost of stagnation that occurred in an era? We may consider the ‘cost at
the starting period t’ as the ‘whole cost’ of stagnation. However, as an agent has myopic
rational expectations about the stagnation economy expressed by time-varying
parameters, he or she cannot predict the future economy. Thus, the ‘cost at the starting
period’ cannot correctly capture the ‘whole cost’ from the starting period to the future.
On the other hand, it is a very bold and implausible assumption that an agent has
rational but not myopic rational expectations of the stagnation economy expressed bytime-varying parameters. Moreover, it is technically difficult to derive the ‘ whole cost’and the ‘cost at period t’ by applying the Lucas–Obstfeld model to an economy
expressed by constant parameters. Thus, the two cost measures are derived based on thetwo different conceptions of the stagnation economy. However, we propose acompromise. If we value a long-run viewpoint, it is natural to assume a stagnation
economy with constant growth over time and instability. In contrast, if we value a
short-run viewpoint, the assumption of a stagnation economy with time-varying
parameters is more natural. In this sense, when we value the long-run viewpoint andevaluate the ‘whole cost’ of stagnation that has occurred in an era, the Lucas–Obstfeld
measure may be appropriate.

4. A Practical Illustration of the Lucas–Obstfeld Model and the Alternative Model
4.1. Data, Unit Root, and Structural Change

The data used in this paper are monthly data from January 1975 to August 2002 (i.e.,
1975:M1 to 2002:M8), which provides 332 observations. To estimate the parameters (λ,µ, and σ2) for consumption in the model, we use total consumption expenditure for12workers’ households from the Monthly Report on Family Income and Expenditures
Survey (FIES). As the FIES reports nonseasonally adjusted data, we apply the census

X-11 method to obtain the seasonally adjusted series. The per capita series isconstructed by dividing consumption expenditure by the number of family members in
the household. These data are converted to real values by using the consumer priceindex (for general prices in 2000) from the Monthly Report of the Consumer Price Index.
All data are taken from the NIKKEI NEEDS CD-ROM.
We partition the whole sample (1975:M1 to 2002:M8) into two subsamples. The
first subsample is from 1975:M1 to 1989:M12 and the second is from 1990:M1 to
2002:M8. The first subsample is the prestagnation period. Our objective is to estimate
the cost of stagnation during the second period (the stagnation period) by comparing itwith the hypothetical economy. This partition of periods seems appropriate, and isconsistent with previous research, including Hayashi and Prescott (2002).5 Moreover,
we implement the tests for a unit root of consumption and for the structural change byusing equation (14) of Perron (1989, p.1380):k∆ logCt = a1 + a2 DUt + a3 D(TB)t + b1t + b2 DTt + ρ logCt −1 + ∑ξi ∆ logCt −i + ε t ,

(14)

i =1

5

We suggest that stagnation began in 1990:M1 because the NIKKEI 225 peaked at a stock
price of 38,926 yen in 1989:M12. Since that date, it has fallen gradually, as has the price of land.
These events are said to define the collapse of the so-called Japanese bubble economy. In linewith previous research, we maintain the importance of these events in defining the starting pointof stagnation. Figure 2 plots consumption in logs, which shows the behavior of the data in themodel. It suggests a trend break in the log of consumption (which reduces consumption growth)around 1990:M1. These figures suggest an obvious structural change.

13where DUt=1 if t>TB and 0 otherwise, D(TB)t=1 if t= TB+1 and 0 otherwise, DTt=t if
t>TB and 0 otherwise. This model allows a sudden change in the level followed by ashift in the slope of the trend function at time TB (1< TB<T). TB=1989:M12. Both thenull model of the unit root (ρ=0,b1=b2=0) and the alternative of trend stationarity (ρ<0)
are nested in (14). At the same time, this nested model allow us to test the structuralchange at t=TB+1. In fact, the structural change did occur because a2 and b2 seem to besignificant, but a3 is not significant and hence there is no jumping of data series. Table 1
rejects the null hypothesis of a unit root for consumption and the null hypothesis of no
structural change at TB+1=1990:M1.The visual inspection is given in Figure 3.

[INSERT Figure 3 and Table 1]

4.2. Preference Parameters

In order to compute the welfare cost of stagnation, the preference parameters are set
exogenously, following convention. For the coefficient of relative risk aversion, we use
values of γ ∈ {0.5,1.5, 2.5, 3.5, 5.0} . Previous studies suggest that, as long as theconstant relative risk aversion preferences are used, the value of γ falls into this range(see, e.g., Kitamura and Fujiki (1997), Hamori (1998), and Nakano and Saito (1998)).
We choose γ = 2.5 as a base value and adopt a strategy that evaluates how the welfare
cost is affected by the other choice of γ . For the discount factor, we take values ofβ ∈ {0.994, 0.995, 0.996, 0.997, 0.998} and choose β = 0.996 as a base value. Similarly,
we check the robustness of the welfare cost to the other values of β .

14

4.3 Estimated Cost for a Lucas-Obstfeld Model
Normality of Stochastic Term and Estimated Parameters

The parameters for consumption in the prestagnation period (1975:M1–1989:M12) andthe stagnation period (1990:M1–2002:M8) are estimated by applying an ML
methodology to the whole sample from 1975:M1 to 2002:M8:

1
lnCt = ln(λB (1− DUt ) + λS DUt ) - (σ B2 (1− DUt ) +σ S2 DUt )2+ t ⋅ ln(1+ µB (1− DUt ) + µS DUt ) + lnzt :lnzt ~ N(0,σ (1− DUt ) +σ DUt )
2
B

,

(15)

2
S

where DUt=1 if t>TB (that is, TB=1989:M12) and 0 otherwise. As shown in Table 2A,
each test statistic for mean=0, skewness=0, excess kurtosis=0, and nonserial correlationof the squared error term with lags of up to 6 are not statistically significant at the 1%
level, except for excess kurtosis in the stagnation period, supporting the normality of ln
zt. In Panel (A) of Table 2B, the parameters in both the prestagnation and the stagnation
periods are estimated and denoted as the estimated parameters for the hypothetical
economy and the stagnation economy. The exception is an estimate for λ in ahypothetical economy, which is replaced with one for a stagnation economy for the sakeof simplicity (the difference is negligible because a3=0 in Table 1, which means nojumping of the data series). Thus, an agent perceives that the estimated parameters inboth periods are the consumption parameters for both economies. All parameter
estimates for consumption are statistically significant at the 1% level. The estimatedmonthly consumption growth rate falls from 0.1059% in the prestagnation period to15

–0.002018%. The variance (which represents instability) of the error term in the log of
consumption increases from 2.339E(–4) to 3.927E(–4). The difference in consumption
growth and variance between both economies is statistically significant at the 1% level,
as is shown in Panel (B) of Table 2B.

[INSERT Tables 2A and 2B]

Estimated Cost

What is the estimated cost of stagnation according to this model? As Table 3 shows, byusing these parameters in equation (5), we obtain a utility level of –3.813E(–06) for the
hypothetical economy and one of –5.362E(–06) for the stagnation economy. Thisimplies that stagnation reduces utility. Our cost measure λ* enables us to convert the
reduction in the utility level into a level of compensation in Japanese yen, which is thesame for all periods. The cost of stagnation is 25,390 yen. The cost ratio (cost/λH) is theratio of the cost to the consumption of the hypothetical economy at the starting period. It
is 26%, which shows the relative amounts and hence an actual impact.
To check the robustness of the results, we calibrate the preference parameters in(1). We use γ = 0.5,1.5, 2.5, 3.5, and 5.0 and β = 0.994, 0.995, 0.996, 0.997, and 0.998
around the base values in Table 4. Note here that γ>0 implies risk aversion. The ratios ofcosts range from 17% to 49% as β increases at γ =2.5, and from 20% to 33% as γ
decreases at β =0.996, as the theoretical analysis suggests in (7) and (8). The ratios ofcosts reach a maximum at 87% and a minimum at 15%. These numbers are not small
and the cost ratio seems to be relatively sensitive to the time preference.
Thus, the proposed ‘whole cost’ λ*, which is particularly relevant to consumers,
16

suggests that the cost of stagnation is large for Japanese consumers. As this represents
what people are willing to pay to prevent the outbreak of stagnation, it indicates thatpeople were prepared to pay a great deal to avoid the stagnation.

[INSERT Tables 3 and 4]

4.4. Estimated Cost for an Alternative Model
Estimated Parameters

The estimation results for α0, α1, β0, and β1 and other parameters in model (9) are
obtained by the Kalman filter algorithm for the time varying parameter model (see the
Appendix). The estimation results are shown in Table 5. Moreover, the estimated at andbt over time, the actual values of log Ct, and the estimated at+ t⋅bt values over time areshown in Figure 4.6 By using the estimated parameters in Table 5, the estimatedcoefficients over time are at =0.609+0.947 at-1 and b t=0+0.496 bt-1 , where someestimated parameters are not significant at the 5% or 10% levels. However, thedynamics of coefficients converge to lim t→∞a t =0.609/(1-0.947) =11.518 and lim t→∞b t=0. In addition, the dynamic process of those coefficients are shown in Figure 4, andthus the time-varying coefficient estimates without the growth component (b t =0) can
capture the actual values of consumption, as seen in the third column of Figure 4.7 Ondate T1 in Figure 2, an agent can know the moments of distribution of the consumption

6

In general, the normality test of a stochastic term is difficult. This will be a task for the future.
Miyakoshi, Okubo, and Shimada (2006) have analyzed the welfare cost of the 1997 Asiancrisis for the Asian countries by using this model. The time-varying parameter estimates cannot
capture the recovery process for each country without the growth component because the Asian
economy has grown strongly after the crisis.

7

17

process on period T1, that is, aT1, bT1, and σ2 in (9). The agent expects that theseparameters will continue forever, and computes the welfare cost on this basis. However,on period T1+1, the agent recomputes a different welfare cost, based on aT1+1, bT1+1, andσ on period T1+1. As seen in Figure 5, the estimated growth parameter µST and theinitial level of consumption λST are less than those values for the hypothetical economy
for nearly all periods, which results in a higher cost of stagnation. On the other hand, thefinding that the instability σ2 (=0.013 *0.013 =0.00017 in Table 5) of the stagnation
economy is smaller than that of the hypothetical economy σ2 (=2.339E(–4) in Table 2B)
results in a lower cost.

[INSERT Table 5, Figure 4, and Figure 5]

Estimated Costs

We can compute the welfare cost at each period by applying equation (13). As seen in
Figure 6, the dynamic cost ratio (i.e., cost at each period/ λH, where λH =99,373 yen isfor 1990:M1 in (9)) shows the four peaks, which are 42% (44,260 yen per month) in
February 1995, 42% (45,268 yen) in May 1997, 47% (51,992 yen) in February 1999,
and 51% (59,399 yen) in December 2001. In addition, the cost remains high at 48% in
the last period of August 2002. The cost of stagnation is large (more than 30% or 29,812
yen per month) for all periods and shows no sign of decreasing.8[INSERT Figure 6]

8

However, Miyakoshi, Okubo and Shimada (2006) have found that the welfare cost of the 199718

Characteristics of the Alternative Model

We evaluate whether the policy was implemented immediately around February 1995,
when the first peak of welfare cost arose. Table 6 shows the rapid policy response soonafter, which involved establishing a special bank to rescue failed banks. At the sametime, the Ministry of Finance announced a decrease in the budget by 2.9%. Althoughthese two policies seem to conflict, they were implemented immediately, and therecovery process seemed to proceed slowly until the second peak in May 1997.
However, the policy did not prevent the subsequent accidental failures of two big
financial companies (Hokkaido Takushyoku Bank and Yamaichi Security Company) in
November 1997. The ineffectiveness of this policy induced the third peak in February
1999. The timing of policy implementation in February and March 1999 was rapid forthe third peak, but ineffective, which induced the fourth peak of 51% in December 2001.
Rather than the timing of policy implementation being the problem, the policies
themselves were not effective.
We evaluate whether the cost of the policies implemented was cheap incomparison with the size of the welfare costs. That is, if the public money injection (thepolicy costs) of 7 trillion yen on March 19, 1999 could rescue the economy immediately,
would people agree to pay this policy cost? The welfare costs (51,992 yen in February1999) implies that people pay 51,992 yen per month forever (many months). On theother hand, if 7 trillion yen in policy costs can sustain the hypothetical economy, howmany months should people pay the cost of 51,992 yen per month to collect 7 trillion

Asian crisis decreased gradually despite repeated upturns and downturns of costs.

19

yen? That is, (policy costs/number of workers in February 1999)/the welfare cost equals
the number of months. Then, (7 trillion yen/53,250,000 workers)/51,992 yen/month =2.5 months.9 Less than three months is a very short period, meaning people will agree
to pay the cheaper policy cost than the welfare cost. If the effectiveness of the policy is
confirmed, the government can publicly announce the cheap cost of the policy. However,actually it is not confirmed.
We evaluate whether the cost decreased gradually, that is, whether the policies
were effective, or whether additional policies are now required by checking the current
welfare costs. The cost of stagnation at period t increased rapidly to 30% in June 1990,
and did not show any sign of decreasing, suggesting that the policies were ineffective.
Finally, the welfare cost remained high at 48% in the last period of August 2002. Thissuggests that an additional policy is required.

[INSERT Table 7]

5. Concluding Remarks

This paper has proposed an alternative measure to the Lucas–Obstfeld model to analyze
the welfare cost of stagnation. In addition, it has provided a practical illustration of thealternative model in comparison with the Lucas–Obstfeld model, by using Japanese datafrom 1975 to 2002. The cost of the stagnation is worth measuring, but the practical useof measuring this cost had not been explored previously.
Using the Lucas–Obstfeld model, which is based on a long-run viewpoint and

9

The data for 53,250,000 workers (employees) were compiled from the Labor Force Survey of20

assumes a stagnation process with constant growth over time and instability, we foundthat the ratio of the ‘whole cost’ is high at 26% of the consumption at the starting periodof stagnation (99,373 yen). We proposed an alternative model, which is based on ashort-run viewpoint and assumes a stagnation process with time-varying consumption.
Based on this model, the cost of the policies implemented immediately after thestagnation cost hit the third peak was 7 trillion yen in March 1999, which is not
expensive compared with the corresponding welfare cost in February 1999. However, in
the absence of effective policies, the cost did not decrease, but remained at 55,610 yen(48%) in August 2002, suggesting that an additional policy is required. Thus, two costmeasures, the ‘whole cost’ and the ‘cost at period t’ of stagnation, proposed by bothmodels, seem to be complementary in evaluating the cost of stagnation.

the Statistic Bureau of Japan.

21

Appendix
Derivation of equation (4)Taking the logarithm of consumption in (2) and defining the first three terms as αt, we
obtain the following:
1 2
σ
2
.

ln C t = α t + ln z t , α t = ln λ + t ⋅ ln (1 + µ ) -(A-1)

Clearly, E(lnCt) = αt and Var(lnCt) = σ2. In addition, because of the property oflog-normality, E(Ct) = exp{αt +σ2 /2}. In the utility function (1), we take logarithms andobtain the following:ln E [ β c
t

1− γ
t

] = ln E [ β η
t

1− γ
t

z

1− γ
t

] , η t ≡ λ (1 + µ ) e
t

1
− σ2
2

.

(A-2)

Using the above property of log-normality, we can rewrite the above equation as:
ln E [ β t η 1t−γ z t1−γ ] = E [ln ( β t η 1t−γ z t1−γ )] +Var [ln ( β t η 1t−γ z t1−γ )]/2

.

(A-3)

Extensive rearrangement yields the following:E[ln ( β tη 1t−γ z t1−γ )] = t ln β + (1 − γ )( ln λ + t ln (1 + µ ) − σ 2 γ /2 ),Var[ ln ( β tη 1t−γ z t1−γ )] = (1 - γ ) 2 σ 2 .

(A-4)

Hence,

ln E [β t η1t−γ z t1−η ] = [exp(ln β + (1− γ ) ln µ)] t exp{(1− γ )( lnλ − σ 2γ /2 )}

.

(A-5)

Thus, inserting these relations into (1) yields the following indirect utility function:∞1 ∞1
t 1−γ 1−γ
2
[
]
β
η
=
exp
{(1
−
γ
)(
log
λ
σ
γ
/2
)}
φ t,
E
z
∑
∑
t
t
(1 − γ ) t =0
(1 − γ )t =0
φ = exp { ln β + (1 − γ ) ln(1 + µ )}.
(A-6)

V=

■

22

Derivation of equation (6)Substituting (4) into (5) yields:

γ
γ
1
1
⎧
⎫
⎧
⎫
⋅ exp⎨(1 − γ )(ln(λS + λ*) − σ S2 )⎬ =⋅ exp⎨(1 − γ )(ln λS − σ H2 )⎬ . (A-7)
1 − φS22⎩⎭ 1 − φH⎩⎭
Taking logarithms of (A-7) yields:ln ( λ S + λ *) =⎫ γ
1 ⎧ 1-φSγ) + (1 - γ )(ln λ S − σ H2 ) ⎬ + σ S2 .
⎨ ln ((1 − γ ) ⎩ 1 - φ H
2
⎭ 2

(A-8)

We replace the right-hand side of (A-8) with Ψ to obtain equation (6).

Derivation of equations (7) and (8)The Ψ in equation (6) is rewritten as follows:Ψ=1{ln(1- φS ) − ln(1-φH )}+ (1− γ ) ln λH + γ (σ S2 − σ H2 ) .
1− γ
2

(A-9)

We take derivatives of equation (6). First, ∂φ H / ∂β = φ H / β , ∂φ S / ∂β = φ S / β . Second,using these relations, we obtain:(φH - φS )∂Ψ=.
∂β (1− γ )β (1- φS )(1- φH )(A-10)

Finally, inserting this relation into ∂λ∗ / ∂β = exp(Ψ) ⋅ ∂Ψ / ∂β , we obtain equation (7).
The φH and φS in (6) are approximated by a Taylor expansion around β = 1 andµ=0. They then become:φ H = β + (1− γ )µ H,

φ S = β + (1− γ )µ S .

(A-11)

Then, we obtain the relation: (1 − φ H )µ S − (1 − φ S )µ H = (1 − β )(µ S − µ H ) . We obtain the23

following partial derivatives: ∂φ H / ∂γ = −µ H , ∂φ S / ∂γ = −µ S . Finally, we take thederivative of λ*. That is,∂λ∗∂Ψ. Here:= exp{Ψ} ⋅
∂γ
∂γ

⎫⎪
⎧− ∂φS / ∂γ
∂φ / ∂γ1∂Ψ
=⋅(1−γ ) + H(1−γ ) + ln(1−φS ) − ln(1−φH )⎬ + (σ S2 −σ H2 ) / 22 ⎨∂γ
1− φ H(1−γ ) ⎩ 1−φS⎪⎭⎫
1=
⋅ {(1− β )(µS − µ H )(1−γ ) + ln(1−φS ) − ln(1−φH )⎬ + (σ S2 −σ H2 ) / 2
2(1−γ )⎭. (A-12)When we evaluate this derivative at β = 1 (its common parameter between bothparadigms), its derivative leads to equation (8).
However, the signs of derivatives in terms of the µH, µS σH, σS, λH, and λ S are
directly derived without approximation around β = 1 and µ=0.

■

The Kalman filter algorithm for the time-varying parameter model

We briefly describe the Kalman filter algorithm for the time-varying parameter inour model’s equations (9) and (10). We derive the smoothing estimates of thetime-varying parameter.
The time-varying parameter model for (9) and (10) can be regarded as the state
space model.
The observation equation:yt = Ht x t + ε t : ε t ~ N (0,σ 2 ) ,(A-13)

where yt ≡ ln Ct, εt ≡ ln zt, at ≡ ln λt-σ2/2, bt ≡ ln(1+µt), Ht = (1, t), and xt = (at, bt)'.

The state equation:24

x t = α + β x t -1 + η t , η t ~ NID(0 , Ση ) ,⎛αwhere α = (α0, β0)', β = ⎜⎜ 1
⎝0

0⎞
⎛ qa2
⎟ , ηt = (ηa,t, ηb,t)', 0 = (0, 0)', and Ση = ⎜⎜
β1 ⎟⎠
⎝0

(A-14)

0⎞⎟.
qb2 ⎟⎠We set the initial conditions a1 and b1 to be the estimates of regression model:ln Ct = a + b⋅ t + ln zt :ln zt ~ N (0,σ2 ) ,(A-15)

obtained from the first four samples.
The unknown parameters are θ = (α0, α1, qa, β0, β1, qb, σ, a0, and b0). Applying astandard Kalman filter algorithm, we can obtain the estimates of parameters and statevariables. See Hamilton (1994) and Harvey (1989) for a detailed explanation of the
Kalman filter algorithm.
The smoothing densities of at and bt given the full sample YT = (yT, ..., y1) are:at YT ~ N(at|T , Pt|Ta ) ,bt YT ~ N(bt|T , Pt|Tb ) ,(A-16)

where a t T ≡ E[a t YT ] , b t T ≡ E[b t YT ] , PtaT ≡ Var [a t YT ], and PtbT ≡ Var [b t YT ] .
So, the smoothing estimates of λt and µt are obtained by the expectations of log-normal
density:

1
1 ⎞⎛E(λt YT ) = exp⎜ at + σˆ 2 + Pt|Ta ⎟ ,2
2 ⎠⎝(A-17)1 ⎞⎛E(µt YT ) = exp⎜bt + Pt|Tb ⎟ −1.
2 ⎠⎝(A-18)

25

References

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value of unemployment insurance”, European Economic Review 45:
1545–1572.
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Hamori, S. (1998), “Defying the conventional wisdom: US consumers are found to be
more risk averse than those of Japan”, Economic Modeling 15:217–235.
Harvey, A.C. (1989) Foresting, Structural Time Series Models, and the Kalman Filter,
Cambridge University Press, Cambridge UK.
Hayashi, F. and E.C. Prescott (2002), “The 1990s in Japan: a lost decade”,
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Kitamura, Y. and Fujiki, H. (1997), “Sapurai-saido jyoho o riyoushita shyohi ni
motozuku Shihon-Shisan Kakaku model No Suikei (in Japanese)”, “Estimationsof C-CAPM Based on Supply-Side Information(in English) ”Kinyu Kenkyu
16(4):137–154.
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Kwan, Y. K. and Chow, G. (1996), “Estimating economic effects of political movementin China”, Journal of Comparative Economics 23:192–208.
Lucas, R.E. (1987), Models of Business Cycles, Blackwell, Oxford.

26

Miyakoshi, T., Okubo, M. and Shimada, J. (2006), “The Dynamic Welfare Costs of the1997 Asian Crisis", mimeo, Tohoku University, Sendai, Japan.
Nakano, K. and Saito, M. (1998), “Asset pricing in Japan”, Journal of the Japanese andInternational Economies 12:151–166.
Obstfeld, M. (1994), “Evaluating risky consumption paths: The role of intertemporal
substitutability”, European Economic Review 38:1471–1486.
Pallage, S. and Robe, M.A. (2003), “On the welfare cost of economic fluctuations indeveloping countries”, International Economic Review 44: 677–698.
Perron, P.(1989),”The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis”,
Econometrica 57:1361–1401.
Storestetten, K., Telmer, C.I., and Yaron, A. (2001), “The welfare cost of business cycles
revisited: Finite lives and cyclical variation in idiosyncratic risk”,
European Economic Review 45:1311–1339.

27

Table 1. Perron’s t test for Unit Root Hypothesis and Structural Change in the Trend
Break at 1989:M12

a1

a2

a3

b1

b2

ρ

ξ1

ξ2

4.124

0.090

0.001

3.920E(–4)–4.102E(–4)–0.366–0.348–0.155(5.85) (5.33)(0.08)(5.64)(–5.43)(–5.84)(–5.30)(–2.81)–x

Note: E(–x) denotes 10 . The numbers in parentheses denote t statistics. The order of the six lags isdetermined by SBIC and AIC, and the second lag is chosen unanimously by both criteria. The 1%
significance point of the t statistic for ρ=0 is –4.90 from Table VI.B of Perron (1989, p. 1377), inview of the fact that 180/332≈0.5.
Table 2A. Normality of the Stochastic Term

Period

NOBS

STDEV

SKEW

PS-Period 180–0.000

0.015

0.230

0.800

2.898

S-Period

0.000

0.020

0.471

1.653*

6.006

152

EXKURT

Q2(6)

MEAN

Note: PS-Period and S-Period denote the prestagnation period and the stagnation period, respectively.
NOBS, MEAN, STDEV, SKEW, and EXKURT denote the number of observations, the mean,
standard deviation, skewness, and excess kurtosis, respectively. The symbol ** denotes statisticallysignificance at the 1% level for each test statistics for mean=0, skewness=0, and excess kurtosis=0.
The Ljung-Box Q-Statistics, Q2(6), are distributed as χ2 (6) under the null hypothesis of nonserial
correlation with up to six lags. The 1% critical value is 16.8.

Table 2B. Estimated Parameters for Consumption(A) Consumption Parameters
λ
µ
σ

2

H-ECO

S-ECO

99373 yen

99373 yen(619.43)(619.43)1.059E(–3)–2.018E(–5)(101.91)(–3.33)2.339E(–4)

3.927E(–4)(10.95)(11.50)(B) Test Hypotheses
Hypothesis

χ2(1)H1: µH=µS,8049.48

H2: σ2H=σ2S,

15.54

Note: H-ECO and S-ECO denote the hypothetical economy and the stagnation economy,
respectively. The λ is consumption in yen in 1990:M1 for both economies. All coefficients weresignificant at the 1% level. The critical value of the χ2(1) distribution is 6.64 at the 1% level.

Table 3. Welfare Costs (γ*=2.5, β*=0.996)
Parameters

Indirect Utility
H-ECO

γ=γ*,β=β*–3.813E(–06)

Cost
S-ECO

Level (yen)–5.362E(–06)

28

25,390

Ratio=λ*/λH(%)

26

Table 4. Welfare Costs (ratio) based on Various Preference Parameters (%)
β

γ

0.994

0.995

0.996

0.997

0.998

0.5

21

25

33

48

87

1.5

19

23

29

39

61

2.5

17

21

26

34

49

3.5

16

19

23

30

42

5.0

15

17

20

25

34

Note: The welfare cost (ratio) is λ*/λH (%).

Table 5. Estimated Parameters in (10)Estimates
t values

σ

α0

α1

qa

β0

β1

qb

0.013

0.609

0.947

0.003

0.000

0.496

0.000

(9.04)(0.91)(16.19)(3.16)(-0.40)(1.15)(2.71)

Table 6. Main Economic Policies around the Peaks of Welfare Costs
Date
12/1/1994
12/9/1994
12/20/1994

Events and the Contents of Policy
The credit unions began to fail.
The government decided to establish a special bank to rescue failed banks.
The Ministry of Finance announced a decrease in the budget by 2.9% after a 40
year period without any reductions in the budget.

5/1/1997
5/30/1997
6/27/1997

Security companies began to fail.
The Acts related to stock options and mortgage securitization were issued.
The Fidelity and the Hyunde security companies (foreign companies) obtained
security dealers’ licenses from the Ministry of Finance.

2/1/1999
2/26/1999
3/12/1999

The big banks began to fail.
The think tank of the prime minister announced the rescue packages.
The government injected public money of 7 trillion yen to rescue the big banks.

12/1/2001
12/18/2001

Many private companies began to fail.
Of the companies having special status (Tokushu-hojin, in Japanese), only 17 wereabolished and 45 privatized.
2/28/2002
The Bank of Japan increased the amount of its open market operations(purchasing), from 0.8 trillion yen per month to 1 trillion yen per month.
Note: We checked policies implemented within two months of the peaks in costs. The date of thecost peaks and the main events following within two months are denoted in bold.
Sources: Nikkei Kinyu Nenpo 2005 (Summer), and Nippon Kezai Senbun Shya (in Japanese).

29

Figure 1. Hypothetical and Stagnation Economies using the Lucas–Obstfeld Model
logCt

H-ECO

logλH

log λS

PSP

S-ECO

0

SP

t

Note: PSP (H-ECO) and SP (S-ECO) denote the prestagnation period (the hypothetical economy)
and the stagnation period (the stagnation economy), respectively. Parts of the intercept, –(1/2)σ2H
and –1/2)σ2S, are neglected because they are negligible.

Figure 2. Hypothetical and Stagnation Economies using the Alternative Model
logCt
H-ECO
logλ Ht0●logλ St0●
S-ECO

•T1

PSP

0

•T2

SP

t

Note: See the note for Figure 1.

30

Figure 3. Per Capita Total Consumption in Logarithm (Aggregate)
Workers' Households
11.7
11.6
11.5
11.4
11.3
11.2
11.1
11.0
01/75

01/79

01/83

01/87

01/91

01/95

01/99

Figure 4. Estimated at and bt over Time: Estimation Values vs Actual Values
bt

at
11.55

0.001011.50

0.000511.45

0.0000

11.40
01/90

-0.0005
01/90

01/94

01/98

01/02

01/94

01/98

01/02

log(Ct) vs a+tb
11.60

log(Ct)
a＋tb

11.55
11.50
11.45
11.40
01/90

01/94

01/98

31

01/02

Figure 5. Estimated µt and ln(λt) over time: Hypothetical vs Stagnation

11.70
11.65
11.60
11.55
11.50
11.45
11.40
01/90

log(λt)µt0.00150.00100.00050.0000

01/94

01/98

01/02

-0.0005
01/90

01/94

01/98

01/02

Note: The dotted and solid lines are for the hypothetical and stagnation economies,
respectively.

Figure 6. The Ratio of Cost at Each Period
0.6
0.5
0.4
0.3
0.2
0.1
0
01/90

01/94

01/98

32

01/02