Discussion Paper No.

641

RATIONAL ADDICTION WITH OPTIMALINVENTORIES: THEORY AND EVIDENCE
FROM CIGARETTE PURCHASESIN JAPAN

Junmin Wan
The Seventh ISER-Moriguchi Prize (2005)
Awarded Paper

August 2005

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Rational Addiction with Optimal Inventories: Theory andEvidence from Cigarette Purchases in Japan∗
Junmin Wan†‡
The Institute of Social and Economic Research
Osaka University
August, 2005
(1st: Dec. 2003)∗ This paper received the Osaka University Institute of Social and Economic Research Moriguchi Prize on February 9, 2005. (WEB address) http://www.iser.osakau.ac.jp/moriguchi/7/7syo.html† I wish to thank Kazuo Ogawa for providing guidance and advice while I was writing
this paper. I also wish to thank Koichi Futagami, Charles Yuji Horioka, Kaoru Hosono,
Daiji Kawaguchi, Kazuo Mino, Tsutomu Miyagawa, Fumio Ohtake, Yoshiyasu Ono, Xiangyu
Qu, Masaya Sakuragawa, and Wako Watanabe, as well as seminar participants at Osaka
University and Gakushuin University, for their valuable advice and comments. This paper
was presented at the 2004 Spring Meeting of the Japanese Economic Association, which took
place at Meiji Gakuin University on June 12, 2004; moreover, I wish to thank the discussant,Noriko Hashimoto of Kansai University, and the seminar participants for their useful advice.
My thanks go also to several anonymous referees for their helpful comments. Finally, I give
special thanks to Shinsuke Ikeda of Osaka University and Keunkwan Ryu of Seoul National
University for their extremely helpful suggestions. Any remaining errors are my own.
‡ Correspondence: Junmin Wan, The Institute of Social and Economic Research, Osaka
University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. Tel: +81(6)6879-8568; Fax:
+81(6)6878-2766; E-mail: wan@iser.osaka-u.ac.jp

Abstract
A model of rational addiction (RA) with optimal inventories is developed and empirically tested using data on purchases in Japan. If a
consumer has information regarding a future price increase, then she may
hoard addictive goods; in this case, the optimal inventory period increases
with the price hike but decreases with the inventory cost. Owing to thecreation of such inventories by consumers, the absolute value of the priceelasticity of demand is smaller in the case of a price increase than in thatof a price decrease, and this difference is especially salient in the short-run.
The evidence provided by daily cigarette purchases is consistent with thisasymmetric price effect. Monthly cigarette purchase data do not support
the RA hypothesis when inventory is ignored, as inventory becomes anomitted variable that correlates with price; however, this hypothesis doesfind support if inventory is identified in the demand equation.

JEL classification: C12; D11; D12; H31
Keywords: rational addiction, tax increase, hoarding, optimal stopping, asymmetric price effect, omitted variable1

Introduction

The rational addiction (RA) model developed by Becker and Murphy (1988)
has been seen as a radical and powerful way of explaining the consumption of
addictive goods, such as cigarettes (see Chalouphka, 1991 and Becker et al.,1994). In empirical studies, econometricians often lack consumption data, butthey do tend to have information on purchases. However, if purchase data areused for consumption data without considering rational inventory behavior, as
is the case in the existing literature, the resulting estimators may be seriously
biased due to omitted inventory effects. For example, Gruber and Köszegi (2000,
2001) replicated the analysis of Becker et al. (1994) to report that the effect on
cigarette “consumption” of the following year’s tax rate was positive, thereby
casting doubt on the validity of the RA hypothesis. By examining monthly
cigarette sales in California, Keeler et al. (1993) reported that the coefficient on
lagged “consumption” was negative and seemingly inconsistent with RA. These
results may be due to the omitted variable problem. To test the validity of theRA hypothesis effectively, it is necessary to incorporate the omitted inventory
effect into the structural model.
By incorporating optimal inventory creation into the framework of Becker etal. (1994), the objectives of this paper are to develop a testable RA model, basedon aggregate purchase data, and to empirically test the model using Japanesedaily and monthly cigarette purchase data. In doing so, to examine whether
incorporating inventory actually resolves the omitted variable problem, the empirical results with and without the assumption of optimal inventory selection
are compared.
In setting up the optimal inventory model of RA, I first characterize optimal behavior as follows. First, I obtain the closed form solution to the optimal
inventory period, thereby showing that (i) if the consumer has certain informa-1

tion on a future price increase, she has an incentive to hoard storable addictive
goods, and she decides how many periods’ worth of goods she should store bysolving an optimal stopping problem; and (ii) the optimal inventory period increases with the price hike, and decreases with the inventory cost. Secondly, I
find that the effects on consumption of a price decrease and price increase areasymmetric. The absolute value of the price elasticity of demand is smaller inthe case of a price increase than in that of a price decrease, and this differenceis especially salient in the short-run.
I examine the empirical validity of the RA model in two ways. First, I
empirically examine the hypothesis of an asymmetric price effect using a daily
cigarette purchase dataset unique to Japan. The results suggest that consumption does not decrease significantly during the inventory period but does once
the inventory has been exhausted. This evidence is consistent with the existenceof asymmetric price effects.
Second, I examine whether incorporating inventory actually resolves theomitted variable problem by comparing the empirical results with and without the distinction between consumption and purchase. I test the model using monthly cigarette purchases in Japan, where the central government completely controls the price of cigarettes and cigarette prices are exogenous to
consumers. Because a new law must be passed before cigarette taxes can bechanged, Japanese cigarette consumers have perfect foresight concerning future
prices and hoard large amounts just before a tax increase. If the hoarding associated with a price or tax change is not included in the estimation equation
but is included in the error term, consistent estimator will not be obtained.
I indeed show that the monthly cigarette purchase data do not support theRA hypothesis when inventory is ignored, but do support it when inventory is
considered.

2Related literature
Several theoretical and empirical issues surrounding RA have yet to be resolved.
The first concerns whether it is reasonable to assume that addicts are forwardlooking. Elster and Skog (1999) pointed out that the addict paradoxically engages in voluntary self-destruction. In fact, it is very difficult to insist that the
addict is either fully rational (forward-looking) or fully irrational. For example, an addict always uses future price information in deciding how to hoard
addictive goods in preparation for her future consumption. This behavior isforward-looking, even though it may be self-destructive to some extent.
Second, it is difficult to test empirically the hypothesis of RA, because many
estimation issues arise when actual data are used. Gruber and Köszegi (2000,
2001) pointed to a number of problems with Becker et al.’s (1994) formulation,noting that ‘very few price increases are announced far in advance (as much as
one year)’ and that ‘forward-looking behavior does not imply time inconsistentpreference’; they also noted the presence of a ‘sizable hoarding effect’ and a‘wrong signed coefficient of next year’s tax rate.’ Keeler et al. (1993) also
noted the hoarding effect and a negative coefficient on lagged consumption.
Furthermore, Auld and Paul (2004) pointed out that aggregate data tend toyield spurious evidence in favor of the RA hypothesis.
Third, even in the absence of RA, a problem still arises in the estimationof normal demand and supply equations, primarily for two reasons. First, theprice becomes endogenous because demand and supply are determined simultaneously. Therefore, it is very difficult to estimate the demand or supply equation
using aggregate data. A natural experiment constitutes a good approach forsolving this problem. For example, Angrist et al. (2000) used stormy weather
as an instrument for price in estimating fish demand. Second, the price may correlate with some unobserved factors in the error term; for example, consumer’s

3

hoarding under the condition of a price increase. If a retailer or consumer obtains the weather report and hoards fish in anticipation of a price increase caused
by bad weather, the possibility exists that unobserved inventory may correlatewith stormy weather.1
Recently, several papers have analyzed consumer inventory creation. Feenstra and Shapiro (2001) have pointed out that the Consumer Price Index (CPI)
cannot be calculated precisely if inventory creation is not considered, and they
have tested this conclusion using data on canned tuna. In addition, Hendel
and Nevo (2001) have analyzed supermarket sales and consumer inventories,
and tested their model using data on soft drinks. Nevertheless, some important
points have not been analyzed in detail: 1) the correlation between the inventory in the error term and the price change, 2) the timing of hoarding, 3) a
suitable proxy for inventory, 4) addictive goods, and 5) perishable goods. This
paper makes an attempt to address these points.
The rest of the paper is organized as follows. Section 2 presents the theoretical framework. Evidence from daily and monthly purchases are presented in
Section 3. My conclusions are presented in Section 4.

2

Theoretical framework

2.1

Rational addiction with optimal inventories

A representative consumer is assumed to consume two types of goods: services,which cannot be stored, and addictive goods, such as cigarettes, which can be
stored for a limited time.2 The consumer has to choose her optimal consumption, purchases, and inventory in every period to maximize utility. There are
1 In Angrist et al. (2000), moving averages of the three days’ wind speeds and wave heights
preceding the trading day were measured as the dummy instrument, stormy.
2 Here, I do not analyze the issues of self-control and time-inconsistent preferences presented
by Winston (1980) and Gruber and Köszegi (2001). In addition, I do not analyze the influenceof chemical dependency, which was discussed by Cameron (2000).

4so many choices that the consumer’s problem is very complex. To simplify the
problem, I make several assumptions without loss of generality. First, the shopping cost is assumed to be zero; this is reasonable because cigarette vending
machines are ubiquitous in Japan. Under this assumption, the consumer has
no incentive to store cigarettes if she knows that the price will decrease; in thiscase, purchases equal consumption. By contrast, she stores cigarettes if she
knows that the price will increase; in this case, purchases exceed consumption.
Second, the per-unit cost of storage per unit of time is assumed to be z. If one
unit of cigarettes is stored for period t, the present value of the inventory costiszβ1 + zβ2 + ... + zβt = ztXβ w = z(β − β t+1 )/(1 − β),w=1

where β is the time discount factor. Therefore, the consumer will hoard cigarettes
just before a price increase. Third, the price increase is assumed to occur attime 1, while the consumer can purchase cigarettes (consumption from time 1
to time τ ) at time 0. The consumer will choose the rates of consumption andpurchases, and the inventory period τ .3 I assume that there are no new tax
increases or decreases during the period after a tax or price change; in otherwords, P0 < P1 = P2 = ...P t... = P∞.
I also assume that the price is known with perfect foresight. This is the casein Japan, because the central government regulates cigarette prices, and theDiet must enact a new law before any tax increase can be implemented. Sevencigarette tax changes have been enacted during the postwar period. According
to the Asahi Shimbun, the dates on which tobacco tax increase proposals passedin the Diet were April12, 1968; October 25, 1975; February 7, 1980; February
3 This problem is somewhat similar to the timing of retirement; see Kingston (2000) andSamwick (1998) for details.

5

1, 1983; March 8, 1986; October 6, 1998; and March 4, 2003. The respective
dates on which these proposals went into effect were May 1, 1968; December 18,
1975; April 22, 1980; May 1, 1983; May 1, 1986; December 1, 1998; and July 1,
2003. The consumer can stockpile cigarettes after the announcement of a tax
increase; therefore, the consumer’s budget function can be expressed as

P0

τ
X

Ct +

t=1

τ
X
t=1

∞
∞
t

 X
X
X
β t Yt = A0
β t Pt Ct +
βw +
Ct z
w=1

t=1

t=τ+1

β = 1/(1 + r),

where Ct is the quantity of cigarettes consumed in period t. Yt is the consumption of the composite commodity in period t. The composite commodity, Y ,is taken as the numeraire, so the price of cigarettes in period t is denoted by
Pt . The rate of interest is assumed to equal the rate of time preference; thus,
the time discount factor β=1/(1+r). τ is the inventory period. The budgetfunction can also be written in a more intuitive form as∞X

β t (Yt + Pt Ct ) = A0 +

t=1

t=1

In equation (1),

t=τ
X

Pt=τ


β t Ct (Pt − P0 β −t ) − z(β 1−t − β)/(1 − β)].

(1)

t
−t
t=1 β Ct (Pt − P0 β ) is the present value of the total gain

caused by inventory, while

Pt=τ

t=1 zCt (β − βt+1)/(1 − β) is the present value ofthe total inventory cost. Thus, the second part of the right side of equation (1)is the present value of the net gain from inventory.
According to Becker et al. (1994), the consumer is assumed to be infinitely
long-lived and to maximize her lifetime utility and income, which are discounted
at the rate r. This utility has two components: the utility from addictive goods,
such as cigarettes, and that from services. The consumer’s problem can be6expressed as

max(Ct ,Yt ,τ)∞X

β t−1 U (Ct , Ct−1, Yt , et ).

t=1

s.t. equation (1)Here, et reflects the impact of unmeasured life-cycle variables on utility. The
initial condition for the consumer in period 1, C 0 , measures the level of cigarette
consumption in the period before the one under consideration.
The Lagrangian function can be written as

L{Ct ,Yt ,τ} =

∞
X
t=1

∞
nX
β t U (Ct , Ct−1, Yt , et ) − λβ t (Yt + Pt Ct ) − A0

−

t=1

t=τ
X
t=1


o
β t Ct (Pt − P0 β −t ) − z(β 1−t − β)/(1 − β) .

(2)

The associated first-order conditions are4

Uy (Ct , Ct−1, Yt , et ) = λ,(3)

U1 (Ct , Ct−1, Yt , et ) + βU2 (Ct+1 , Ct, Yt+1 , et+1 )= λ[P0 β −t + z(β 1−t − β)/(1 − β)], for t ∈ [1, τ ],(4)

= λPt , for t ∈ (τ, ∞],(5)Pτ − P0 β −τ = z(β 1−τ − β)/(1 − β).

(6)4 This is an optimal stopping problem, and the value-matching and smooth-pasting conditions are implicitly imposed here. The proof in the continuous time framework is presented
in Appendix A.

7

From equation (6),5 I calculate the optimal inventory period:hτ ∗ = ln 1 +i−1/lnβ
Pt − P0.
P0 + zβ/(1 − β)(7)Cigarettes are perishable and have a best-before date, or time limit. This limit
is assumed to be T l . Therefore, τ = τ ∗ for T l ≥ τ ∗ , and τ = T l for τ ∗ > T l . In
consequence, the following proposition obtains.

Proposition 1: If a price or tax increase occurs at time 1 and the priceremains constant thereafter, the optimal inventory period increases with theprice but decreases with the inventory cost.6 The optimal inventory period is
also bounded by the time limit T l , τ = τ ∗ for T l ≥ τ ∗ , and τ = T l for τ ∗ > T l .
Proof: See the continuous time framework in Appendix A.

Consider a utility function that is quadratic in Yt , Ct , and et . A linear
difference equation can be derived by solving the first-order condition for Yt
and substituting it into the first-order condition for Ct :Ct = θ0 + θCt−1 + βθCt+1 + θ1(t) Pt + θ2 et + θ3 et+1

(8)5 Here τ is represented in discrete time, and I approximately differentiate the Lagrangianfunction with respect to τ . However, it is possible to obtain τ in a continuous-time frameworkusing the same model, if the optimal inventory period in the discrete-time case converges to
h
i1/r
−P0
in the continuous-time framework.
one. That is, τ ∗ = ln 1 + PPt+z/r
0

6 The effects of the time discount factor and the interest rate on the optimal inventory

period are not uniquely determined.

8where,θ0=−λ(uy1 + βuy2 ),θ=−(u12 uyy − u1y u2y ),(u11 uyy − u21y ) + β(u22 uyy − u22y )

θ1(t)

=

uyy λ[P0 β −t /Pt + z(β 1−t − β)/((1 − β)Pt )], for t ∈ [1, τ ],(u11 uyy − u21y ) + β(u22 uyy − u22y )

=

uyy λ
= θ1 ,
2(u11 uyy − u1y ) + β(u22 uyy − u22y )θ2=−(uyy u1e − u1y uey ),(u11 uyy − u21y ) + β(u22 uyy − u22y )θ3=−β(uyy u2e − u2y uey ).
(u11 uyy − u21y ) + β(u22 uyy − u22y )for t ∈ (τ, ∞],Proposition 2: If a price or tax increase occurs at time 1 and the priceremains constant thereafter, the shadow price of the addictive goods during the
optimal inventory period increases with time and is lower than the practicalprice, as a result of consumer inventory creation.
Proof: This result is obvious from θ1(t) in equation (8).

The consumption good is addictive if θ > 0, and the degree of addiction
increases with θ. The roots of difference equation (8) are

φ1 =1 + (1 − 4θ2 β)1/2
1 − (1 − 4θ2 β)1/2, φ2 =
,
2θ
2θ

(9)

and the stability conditions are
4θ2 β < 1, φ1 < 1, φ2 > 1.

(10)

Given these roots, the short-run price effect (of a price decrease, no inventory)
9

is
θ1
dCt
,
=
∗
dP
θ(1 − φ1 )φ2(11)

which is defined as the impact of a temporary change in the current and all
future prices on current consumption,7 with past consumption held constant.
The long-run price effect (of a price decrease, no inventory) is
θ1
dC∞
=
,
dP
θ(1 − φ1 )(φ2 − 1)(12)which is defined as the effect of a permanent change in prices in all periods.8

Proposition 3: If the consumer can hoard addictive goods in anticipationof a price increase, but cannot short-sell them prior to a price decrease, theeffects of a price decrease and a price increase on consumption following theprice changes are asymmetric. The absolute value of the price elasticity ofdemand is smaller for a price increase than for a price decrease, as a result ofconsumer inventory creation; this difference is most salient in the short-run.
Proof: This result also is obvious from θ1(t) in equation (8), under the condition that the consumer cannot consume goods at a lower price in the future.

The essential reason for this asymmetry is market incompleteness. For example, if no futures market exists, then a consumer cannot short-sell cigarettes,
even when she knows that the price will fall in the next period.

7 When the price increases, the short-run price effect differs and will be smaller than in thecase of a price decrease, because of inventory creation.
8 When the price increases, the long-run price effect differs and will be smaller than in thecase of a price decrease, partly as a result of inventory creation. If τ is sufficiently small, thelong-run price effect of a price increase is similar to that of a price decrease.

10

2.2

Issues in empirical analysis

The consumption set (C1 , ..., CT ) is decided optimally, but in the context of empirical analysis, C is very difficult to observe. Economists often lack consumption data but have aggregate data on purchases. Nevertheless, purchases do notequal consumption, especially in the short-run. The intertemporal substitution of purchases may offset the intertemporal complementarity of consumption
during estimation, if consumption is not distinguished from purchases.

2.3

Solutions to the issues raised in the empirical model

The consumption equation is derived in the theoretical model. However, theaggregate quantity consumed is not observable, and only purchase data Qt areavailable. The relationship between purchases Qt and consumption Ct is given
by

Qt

=

It − It−1 + Ct = ∆It + Ct ,(13)

where It denotes inventory at time t. Equation (13) may be rewritten as Ct =Qt − ∆It . Similarly, Ct−1 = Qt−1 − ∆It−1 , Ct+1 = Qt+1 − ∆It+1 . Then thepurchase Qt equation may be used, and purchases substituted for consumption
Ct in equation (8). Therefore, Qt can be represented in the following form:

Qt=θ0 + θQt−1 + βθQt+1 + θ1(t) Pt + θ2 et + θ3 et+1+(∆It − θ∆It−1 − βθ∆It+1 ).

(14)

Note that (∆It − θ∆It−1 − βθ∆It+1 ) is the effect of inventory, and that itcorrelates with the price. Since inventory also correlates with either the tax
change or the lead and lag of the price, the error term also correlates with theprice change (or tax rate) when (∆It −θ∆It−1 −βθ∆It+1 ) is not included on the11right side of the structural model. Such a typical endogeneity problem is causedby the omission from the model of a variable correlated with the explanatory
variable. Moreover, this type of endogeneity bias caused the coefficient of next
year’s tax rate to be positive in the analysis of Gruber and Köszegi (2000), andthe coefficient on lagged consumption to be negative in the study of Keeler etal. (1993).
The optimal inventory and ∆It in every period are derived in the theoreticalframework. Therefore, I use Tl dummies to indicate ∆It , one dummy for thehoarding effect, and Tl − 1 dummies for storage effects subsequent to hoarding.
According to the theoretical model, the extent of hoarding is a function of tax
increases, and the tax increase rate is a good proxy for hoarding.9 Therefore,
each of the seven tax increases is used to examine the effects of hoarding andstorage.

Store2t ,Store3t ...... StoreTtl

T ime,

P ricet , Hoardingt , Store1t ,

t-2

P1

0

0

0

0

.......

0

t-1

P1

0

0

0

0

.......

0t

P1(P2 / P1 )

0

0

0

.......

0

t+1

P2

0

(P2 / P1 )00.......

0

t+2

P2

0

0

(P2 / P1 )0.......

0

t+3

P2

0

0

0

t+4

P2

0

0

0

0

t+Tl

P2

0

0

0

0....... (P2 / P1 )t+Tl + 1

P2

0

0

0

0

(P2 / P1 ) ......
.......

.......

000

According to ‘The History of the Japanese Tobacco Monopoly’, the shelf life
9 When the inventory cost z is sufficiently small and the rate of tax increase is not large,
the optimal inventory period τ ∗ ≈ (Pt − P0 )/(rP0)=rate of tax increase/r.

12

of cigarettes is about five months, and distribution requires about two months;
therefore, the consumer’s maximum storage period is about three months. Thus,
the purchase equation can be written asQt= θ0 + θQt−1 + βθQt+1 + θ1(t) Pt + θ2 et + θ3 et+1+(∆Hoarding t + ∆Store1t + ∆Store2t + ∆Store3t )−θ(∆Hoarding t−1 + ∆Store1t−1 + ∆Store2t−1 + ∆Store3t−1 )−βθ(∆Hoarding t+1 + ∆Store1t+1 + ∆Store2t+1 + ∆Store3t+1 ).(15)

Since the inventory has the characteristics presented in Figure 1, it can be
written as∆Hoarding t=−a∆Store1t+1 ,=−b∆Store2t+2 ,=−c∆Store3t+3 , for 1 < a, b, c, .

Therefore, the variables on the right side can be written as∆Hoarding t−1=−a∆Store1t ,∆Store1t−1=(b/a)∆Store2t ,∆Store2t−1=(c/b)∆Store3t∆Store1t+1=(−1/a)∆Hoarding t ,∆Store2t+1=(a/b)∆Store1t ,∆Store3t+1=(b/c)∆Store2t , .

13

Then Qt can be written as

Qt=θ0 + θQt−1 + βθQt+1 + θ1(t) Pt + θ2 et + θ3 et+1+(−θ)∆Hoarding t+1+(1 + βθ/a)∆Hoardingt+(1 + θa(1 − β/b))∆Store1t+(1 − θb/a − βθb/c)∆Store2t+(1 − θc/b)∆Store3t+(−θ)∆Store3t−1 .

(16)

Before a price increase, purchases exceed consumption; thus, hoarding has apositive effect on purchases. Therefore, ∆Hoarding t+1 > 0 and ∆Hoarding t >0. Moreover, because 0 < θ < 1 and 0 < β < 1, (−θ)∆Hoarding t+1 < 0
and (1 + βθ/a)∆Hoardingt > 0. After the price increase, because the inventory must have a negative effect on purchases, ∆Store1t < 0, ∆Store2t < 0,
∆Store3t < 0, and ∆Store3t−1 < 0. Therefore, (1 + θa(1 − β/b))∆Store1t < 0,(−θ)∆Store3t−1 > 0, and the signs of (1 − θb/a − βθ)∆Store2t and (1 −θc/b)∆Store3t are undetermined.10 The estimation of purchase equation (16),
Qt is discussed in the following section, as data on these variables and suitableproxies for the inventories are available.

10 θ
1(t) is a function of time. When the timing of θ1(t) is considered, equation (16) can be
re-written by using the constant coefficient θ1 and adding another positive term θ1 [−(Pt −P0 β −t ) + z(β 1−t − β)/(1 − β)].

14

3Evidence

3.1

Cigarette price and tax in Japan

The price of cigarettes is totally controlled by the Japanese government, and special laws are passed to enact cigarette tax changes. Seven cigarette tax increases
have been passed by the Japanese government, not because of fluctuations in the
demand for cigarettes, but because of a large public deficit.11 Thus, the priceof cigarettes is an exogenous variable for the Japanese consumer. Furthermore,
to implement every tax change, a new law must be created. Hence, cigarette
tax increases can be considered a natural experiment that can be analyzed inorder to examine consumer responses to price changes.

3.2

Evidence from daily purchases

3.2.1

Daily purchases before and after a tax increase

I now discuss a recent tax change: a new cigarette tax increase law was passed
on March 4, 2003, and came into effect on July 1 of the same year. Daily purchase data for cigarettes in Japan,12 for the period from April 1 to September
30 are illustrated in Figure 2. The horizontal axis indicates the purchase date,
and the vertical axis shows daily purchases. Figure 2 suggests that a big increase in purchases began on June 23, about one week before the price increase,which peaked on June 30, one day before the price increase. This behavior
corresponds to the hoarding effect, due to the tax increase, and is consistent
with the predictions of the theoretical model of the previous section. After the
price increase on July 1, purchases declined. This is the storage effect due to
11 See “The History of the Japanese Tobacco Monopoly” (in Japanese, Niho Tabako Senbaishi).
12 Thedaily data were downloaded from The Japan Statistics Bureau by
the author in 2003.
(http://www.stat.go.jp/data/kakei/200309/zuhyou/a615.xls;
http://www.stat.go.jp/data/kakei/200309/zuhyou/a616.xls) The data may no longer beavailable via download, but they are available from the author upon request.

15

hoarding and is also consistent with the predictions of the theoretical model.
Table 1 compares purchases before and after the tax hike with average purchases. The average daily purchase per family was 37.43 yen between April 1
and September 30. However, purchases on June 30 were 8.89 times greater than
the average daily purchase, and purchases decreased markedly following the tax
hike.
3.2.2

Testing the asymmetric price effect using daily data

According to the predictions of the theoretical model, consumption should notdecrease sharply in the short-run, even when the price increases in practicebecause of consumer inventory creation. This hypothesis was tested using dataon one-year daily purchases that occurred before and after the tax change. In
particular, I used Japanese household daily data for the period from April 1,
2003, to March 31, 2004. These data were downloaded from The Japan Statistics
Bureau. The results are provided in Table 2.
As mentioned in the previous section, the shelf life of cigarettes is about
three months for a consumer; thus, a rational consumer’s consumption should
not decrease significantly during the inventory period (about three months) following a tax increase. I used data on purchases per three months, both before
and after the tax increase, to test whether consumption decreased during thistime. From the results of the difference tests provided in Table 2, it is clearthat purchases made between April 1 and June 20, 2003, were equal to thosemade between June 21 and September 31, 2003, while purchases made between
January 1 and March 31, 2004, decreased significantly relative to those madebetween October 1 and December 31, 2003. These results imply that consumption during the inventory period did not decrease; however, consumption did
decrease after the hoard of cigarettes had been exhausted.13 This is evidence of
13 Possible seasonal effects were omitted.

If seasonal effects were taken into account, con-16

an asymmetric price effect.

3.3

Evidence from monthly purchases

3.3.1

Data set for an econometric model

The following data consist of monthly series spanning the period from January
1954 to September 2003. 14(Cigarette purchases by Japanese worker households) P urchaset isthe monthly total of cigarette purchases, in packs, per capita. The data weretaken from the ‘Annual Report of Family Income and Expenditure Survey’ andwere seasonally adjusted using X-12 ARIMA.
(Price) P ricet is the real average retail cigarette price per pack in montht. It equals the Tobacco Price Index divided by the CPI. These data weretaken from the ‘Annual Report on the Consumer Price Index’ and the ‘Monthly
Report on the Retail Price Survey.’ Prices were seasonally adjusted using X-12
ARIMA and are measured in 100’s of yen, as of 1995, per pack.
(Disposable income) Yt is the real monthly worker household disposable
income per capita. These data were taken from the ‘Annual Report of Family
Income and Expenditure Survey.’ This measure equals the total disposable
income per family, divided by the total population per family and the CPI.
Moreover, these data were seasonally adjusted using X-12 ARIMA, and aremeasured in 1000’s of yen, as of 1995, per capita.
(First difference of disposable income) ∆Yt is the first difference ofmonthly disposable income and is measured in 1000’s of yen, as of 1995, percapita.
The summary statistics of these variables are presented in Table 4.
sumption would presumably be found to have decreased more sharply, because January is the
beginning of the New Year for the Japanese. The New Year vacation is the longest holiday inJapan, and related consumption therefore always increases in the first quarter.
14 The details are given in Appendix B.

17

3.3.2Monthly purchase frequency and purchases before and after a
tax increase

Table 3 indicates that the frequency of purchases in June 2003, the month justbefore the tax increase, increased markedly, while it decreased markedly in July.
This observation is consistent with the predictions of the theoretical model.
From Table 3 and Figure 3, it is clear that per family cigarette expenditure
increased markedly in June, while it decreased markedly in July 2003. Thisobservation also is consistent with optimal inventory theory.
3.3.3

Hoard size versus the tax hike

The amount of hoarding before every tax increase was calculated. According to
the prediction of the theoretical model, the amount of hoarding approximately
equals the purchase difference between the two months just before the tax increase, if consumption in those two months did not change drastically. The
results of these calculations are presented in Table 5. In particular, the amountof hoarding increased with the rate of the tax increase.

tax elasticity of hoarding==(average hoarding)/(average purchase)average rate of tax increase
1.126/1.475
× 100%
0.183= 418.067%.

In addition, the estimated tax elasticity of hoarding is astonishingly large. Thisimplies that the consumer should hoard more than four times as much as theaverage monthly purchase, if the tax rate increases by 100 percent.

18

3.3.4

Unit root tests

If any variables are non-stationary, problems can arise from statistical inferencesusing ordinary least-squares (OLS) or two-stage least-squares (2SLS) methods.
Therefore, I tested whether each variable was stationary, using the ADF andPhillips-Perron tests. The test results are reported in Table 6.
Unit root hypotheses for P urchaset with time were rejected at the 1% significance level. In addition, unit root hypotheses for P ricet were rejected at the
5% significance level. Since unit root hypotheses for Yt could not be rejected
at any conventional level of significance, Yt is clearly not stationary. Since unit
root hypotheses for ∆Yt with time were rejected at the 1% significance level,∆Yt may be considered stationary over time.
3.3.5

Estimation techniques

OLS and 2SLS were used to obtain the parameter estimates. The OLS estimates
may not be consistent because of the endogeneity of past and future consumption(or purchases) and serial correlation among the residuals. Therefore, to ensure
consistent estimates, I also used 2SLS.
The 2SLS estimates are consistent and were derived using instrumental variables.15 As noted above, the price of cigarettes is an exogenous variable for theJapanese consumer. Furthermore, price strongly correlates with cigarette consumption and, therefore, is a good instrument for cigarette consumption (orpurchases). Lagged prices and taxes were used to instrument for past cigarette
consumption (or purchases), while the leads of price and taxes were used as
instruments for future cigarette consumption (or purchases).
The Wu test was used to determine whether the OLS estimates were con15 To some extent, the estimation procedure used is subject to the critiques of Auld andGrootendorst (2004). But Auld and Grootendorst (2004) also mentioned that instrumentalvariable estimates of the lag and lead of consumption are consistent if prices are exogenous;see pages 1124-25.

19

sistent. The over-identification test (OID) was used to test the validity of theoveridentifying restrictions.
3.3.6

Estimation without distinction between purchases and consumption

Equation (8) was estimated, and the resulting estimates are reported in Table
7. Since P urchaset , P ricet , and ∆Yt are stationary with time, the time trend
was included in the estimation equation. In the OLS column, the estimates
are inconsistent because of the endogenous explanatory variables. In the 2SLS
column, the coefficient on P ricet is negative and significant. The coefficientson P urchaset−1 and P urchaset+1 likewise are negative and significant. In addition, the sign of the estimated coefficients on P urchaset−1 and P urchaset+1
does not satisfy the addiction condition. These results appear to indicate adurability effect. Therefore, the problem of deriving consistent estimates in thiscase is similar to that faced by Gruber and Köszegi (2000) and Keeler et al.
(1993).
According to the OID test, the set of instruments is invalid in this case.
This means that the instruments correlate with the error term, which causes
the problem of sign inversion. Therefore, the influence of inventory is serious.
If it is ignored, consistent estimates cannot be obtained.
3.3.7

Estimation with distinction between purchases and consumption

Equation (16) was estimated, and the results are reported in Table 8. Since someof the explanatory variables are stationary with time, the time trend was also
included in the estimation equation. The set of instruments used in the 2SLS
is valid, according to the OID test. The hypothesis that the OLS estimates are
consistent could be rejected at the 5% level by means of the Wu test. Therefore,
20the 2SLS estimates are consistent.
In the 2SLS column, the coefficient on P ricet is negative and significant.
The coefficients on P urchaset−1 and P urchaset+1 are also negative and significant. Moreover, the sign of the estimated coefficients on P urchaset−1 andP urchaset+1 now satisfies the addiction condition. The estimated values alsosatisfy the stability conditions. However, the coefficients on Store2t , Store3t ,and Store3t−1 are not significant.16 The coefficients on Hoardingt+1 , Hoardingt ,and Store1t are significantly negative, positive, and negative, respectively.17
These estimation results, which were derived from data on Japanese monthly
cigarette purchases, are consistent with the predictions of the RA model with
optimal inventories. I subsequently used the estimated coefficients and sample
means provided in Table 4 to estimate the short- and long-run price elasticity
given in rows ‘short−run ǫ′ and ‘long −run ǫ′ , respectively. The table indicatesthat the long-run price elasticity is about 2.681 times greater than the short-run
value.18

4

Conclusion

This paper has obtained the following findings. First, I have built a RA modelwith optimal inventories that provides a theoretical explanation about why the
16 The model incorporates six dummies that indicate tax increases; thus, ordinarily there
should have been 42 dummies for the seven times that a tax increase took place. According
to theoretical prediction, the optimal inventory period (or quantity) is approximately a linear
increasing function of the rate of tax increase. In consequence, to minimize the total numberof coefficients estimated, I assumed that inventory was an increasing linear function of therate of tax increase; thus, it was reasonable that I used only six dummies (corresponding torates of tax increase) for the seven times that tax increases occurred.
17 −θ∆Hoarding
t+1 and −θ∆Store3t−1 have the common coefficient θ, but it was expected that the absolute value of −θ∆Hoardingt+1 should be much larger than that of−θ∆Store3t−1 , because the dummy for ∆Hoardingt+1 captures the effect of hoarding, while
the dummy for ∆Store3t−1 captures the effect of store changes after hoarding. A comparisonof the signs of the two estimated coefficients is also meaningful in this case.
18 The absolute value of the elasticity is in this case higher than that estimated by Wan(2004), perhaps because the data set differs, or because the effects of health information havenot been considered.

21

estimated coefficient on lagged “consumption” may be negative and the estimated coefficient on the next year’s tax rate may be positive. Such results indicate that the consumer uses information on future prices to decide how manystorable addictive goods to hoard in preparation for her future consumption.
The optimal inventory period increases with the price hike, but decreases with
the inventory cost and is also bounded by the shelf life of the goods. Because
purchases are not equal to consumption in every period, the intertemporal substitution of purchases generally offsets, and may even exceed, the intertemporalcomplementarity of consumption that derives from the hypothesis of RA.
Second, the effects of a price decrease and a price increase are asymmetric,
and market demand is kinked, owing to inventory creation. In other words, theabsolute value of the price elasticity of demand is smaller in the case of a priceincrease than in that of a price decrease, and this difference is most salient in
the short-run. Evidence from a unique data set on daily cigarette purchases inJapan is consistent with asymmetric price effects.
Thirdly, I have empirically examined the RA model using monthly cigarettepurchases in Japan, both with and without the distinction between consumption
and purchase, since peculiar policies on cigarette price and tax changes provide
a suitable context for testing consumer responses to price changes, in terms ofpurchases and consumption. The estimated coefficients are sign-reversed when
consumption is not distinguished from purchases, because inventories become
an omitted variable in the error term and correlate with prices and taxes. I
have obtained evidence that is consistent with the predictions of a model of RA
with optimal inventories; this evidence was obtained through the substitutionof purchases for consumption and the inclusion of inventory dummies in thepurchase equation.
When the government uses new taxes as a means of regulating the consump-

22tion of harmful, addictive goods, such as cigarettes or heroin, tax effects may
be smaller than those anticipated because of consumer inventory creation, especially in the short-run. This important point should be considered in the designof regulations. In addition, the model presented in this paper has other possible applications. For example, inventory effects matter not only in the case ofcigarette tax increases, but also in those of changes in the taxes levied on otherstorable goods. Even in the absence of consumption data, structural consumption equations can be consistently estimated on purchase data if inventories are
identified.
In future research, it would be challenging and interesting, following Gruber and Köszegi (2000), to incorporate time inconsistent preferences into therational addiction model with optimal inventories; this could be accomplishedby considering inventory as a self-control device. Another promising topic deserving future investigation involves consumer responses to other factors, suchas health information. 19 In addition, the consumption and inventory of oil may
conveniently be analyzed in this framework, as its substitutes are expensive andit may be viewed as an addictive good. These extensions are left for future
work.

Appendix A: Continuous time framework
Consider the model in a continuous-time framework, and then solve the consumer’s problem for the optimal inventory period τ ∗ and the consumption
set (Ct∗ , Yt∗ ). This is a standard optimal stopping problem; thus, the traditional solution approach used here involves the conversion of time to two periods, the post-inventory period (τ , ∞] and the pre-inventory period (0, τ ].
P0 < P0+ = Pt for t ∈ (0, ∞].
19 These responses have been partially analyzed by Wan (2004).

23

Post-inventory problem
Suppose that the consumer makes a consumption plan for the post-inventory
period (τ , ∞] at time 0. Then, the consumer’s problem is given byττ

Γ(A , S , τ ) = max(Yt ,Ct )Z ∞e−rt U (Yt , Ct , St )dt.

(A.1)

τ

s.t. Ȧt = rAt − Pt Ct − Yt ,(A.2)

Ṡt = Ct − σSt ,(A.3)

where the S means the consumption capital of addictive goods and the σ means
the instantaneous depreciation rate. This setting is the same as Becker andMurphy (1988). The Lagrangian function is

L{Yt ,Ct } =Z ∞he−rt U (Yt , Ct, St ) − λt (Ȧt − rAt + Pt Ct + Yt )τi− µt (S˙t − Ct + σSt ) dt.

(A.4)

The first-order conditions are

Uy (Yt , Ct , St ) =λt ert ,Uc (Yt , Ct , St ) =Pt λt ert − ert µt ,

λt

=

µt

=

λ0 e−rt ,Z ∞e−(r+σ)w+σt Us (Yt , Ct , St )dw,t

where λ and µ are the Lagrange multipliers for t∈ (τ, ∞].
If the utility function U is assumed to be quadratic in C, Y and S, the optimal
solution (steady states) of C, Y, and S and the dynamics are the same form as
those in Becker and Murphy (1988), while all solutions here are dependent on24

the optimal inventory τ ∗.

Pre-inventory problem
Next, suppose the consumer makes a consumption plan at time v ∈ (0, τ ].
The consumer’s problem is then given byvv

Ψ(A , S , v) = max(τ,Yt ,Ct )Z τe−rt U (Yt , Ct , St )dt + Γ(Aτ , S τ , τ ).

(A.5)

v

Z t


e−rw dw Ct − Yt ,s.t. Ȧt = rAt − ert P0 + z

(A.6)

0

Ṡt = Ct − σSt .
Rt
Here, P0 + z 0 e−rw dw is the discounted cost at time 0 for consuming per unit
Rt
Ct at time t, thus ert (P0 + z 0 e−rw dw) is the current value of per uint cost attime t. Then the Lagrangian function is

L{Yt ,Ct ,τ} =Z τh′e−rt U (Yt , Ct , St ) − λt (Ȧt − rAt + ert P0 Ct
v
Z t+ ert Ct z
e−rw dw + Yt )0i
− µt (Ṡt − Ct + σSt ) dt′

+ Γ(Aτ , S τ , τ ).

The first-order conditions are′

Uy (Yt , Ct , St ) =λt ert ,Uc (Yt , Ct , St ) =Pt λt ert − ert µt ,

′

λt
′

µt

=
=

′

′

′

λ0 e−rt ,Z ∞e−(r+σ)w+σt Us (Yt , Ct , St )dw,
t

25

(A.7)

and the first order condition with respect to τ is
i
i
h µ˙′
h λ˙′
′
′
−λτ − τ′ Aτ − rAτ + Yτ + Cτ P0 erτ + Cτ z(erτ − 1)/r − µτ − ′τ Sτ − Cτ + σSτ
λτ
µτ
i
h µ˙
i
h λ˙
= −λτ − τ Aτ − rAτ + Yτ + Cτ Pτ − µτ − τ Sτ − Cτ + σSτ .
λτµτValue-matching and smooth-pasting conditions
Set v = τ and impose the value-matching condition, Ψ(Aτ , S τ , τ ) = Γ(Aτ , S τ , τ ),′and the smooth-pasting conditions, Ψ(Aτ , S τ , τ )A = Γ(Aτ , S τ , τ )A , µτ − = µτ + .
Because the time discount rate is assumed to be equal to the interest rate, thegrowth rate of λ is zero and λ equals a constant, for t ∈ (0, ∞]. Hence, thefollowing conditions are obtained.
hPt − P0 i1/r
τ ∗ = ln 1 +,P0 + z/r(A.8)′

Uy = λ = λ = ΨA = ΓA , for t ∈ (0, ∞],′′Uc = λ(P0 ert + z(ert − 1)/r) − µt , for t ∈ (0, τ ],′′= λPt − µt , for t ∈ (τ, ∞],
Z ∞
′′
µt =
e−(r+σ)(w−t) Us dw, for t ∈ (0, ∞].

(A.9)(A.10)(A.11)(A.12)t

In addition, because P0 , Pt , z and r are given, the τ ∗ can be derived by (A.8).
Furthermore, if the utility function is specified, the relation between Y and C
can be derived by (A.9), (A.10), (A.11) and (A.12), Yt = f(Ct , St ) for t ∈ (0, τ ]and Yt = g(Ct , St ) for t ∈ (τ, ∞]. The budget function at time 0 is

P0

Z τ

Z τ

Z τ
 Z t

−rw
e
dw dt +
e−rt Yt dt
Ct z

Ct dt +00
Z 0∞+e−rt (Pt Ct + Yt )dt = A0 .
τ

26

0

(A.13)

Substituting Ct for Yt in (A.13), then the budget function becomes

P0

i1/r
h
Z ln 1+ Pt −P0
P0 +z/rhi1/r
Z ln 1+ Pt −P0
P0 +z/rCt dt +

0

+

h

Z ln

P −P0
1+ P t+z/r
0

i1/r

0

 Z t

Ct z
e−rw dw dt
0e−rt f(Ct , St )dt

0

Z ∞
h
+

P −P

0
ln 1+ P t+z/r
0

i1/r e−rt (Pt Ct + g(Ct , St ))dt = A0 .

(A.14)(A.14) is a complicate function of Ct . By incorporating the (A.3) and (A.6) into
(A.14), if the consumption set Ct∗ can be solved, the Yt∗ can also be solved.20
The consumption set (Ct∗ , Yt∗ ) for t ∈ (0, ∞] depend on the optimal inventory
period τ ∗. During the inventory period, the inventory affects consumption by
two effects. The first is the income effect caused by the total money saved byinventory. The second is the price effect caused by the inventory cost whichchanges with time. After the inventories are used up, there are still two effects
on the optimal consumption set (Ct∗ , Yt∗ ), the income effect and the price effectwhich is caused by the changed consumption capital St . The optimal purchaseat time zero is
Z τ∗
0Ct∗dt,(A.15)and nothing for t ∈ (0, τ ∗].

Appendix B: Data
Consumer Price Index, Statistics Bureau Ministry of Public Management,
Home Affairs, Posts and Telecommunications, Japan. Annual Report on theConsumer Price Index, 1951-2003.
20 A.14 is too complicate to obtain a closed form solution of C .
t27

Consumer Price Index of Cigarettes, Statistics Bureau Ministry of Public Management, Home Affairs, Posts and Telecommunications, Japan. “Subgroup Index for Japan”. Annual Report on the Consumer Price Index, 19512003.
Nominal Worker Household Disposable Income, Economic Planning
Agency, Government of Japan. Annual Report of Family Income and Expenditure, 1951-2003.
Nominal Retail Cigarette Price, Statistics Bureau Ministry of Public
Management, Home Affairs, Posts and Telecommunications, Japan. “Nationwide Uniform Prices or Charges” Monthly Report on the Retail Price Survey,
2003.
Normal Cigarette Price, Nominal Retail Cigarette Price (2003) times
consumer price index of cigarettes divided by the index (2003).
Per Capital Worker Household Cigarette Purchases, Per worker
household total cigarette consumption expenditure divided by per household
population.
Real Household Disposable Income, Nominal Household Disposable
Income divided by the consumer price index.

28

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Models with an Application to the Demand for Fish.” Review of Economic
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[2] Auld, M. Christopher and Grootendorst, Paul. “An empirical analysis ofmilk addiction.” Journal of Health Economics 2004, 23(6), pp.1117-1133.
[3] Becker, Gary S., Grossman, Michael, and Murphy, Kevin M. “An Empirical
Analysis of Cigarette Addiction”. American Economic Review, June 1994,
84(3), pp.396-418.
[4] Becker, Gary S. and Murphy, Kevin M. “A Theory of Rational Addiction”.
Journal of Political Economy, August 1988, 96(4), pp.675-700.
[5] Boyer, Marcel. “A Habit Forming Optimal Growth Model”. International
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[6] Cameron, Samuel. “Nicotine addiction and cigarette consumption:apsycho-economic model.” Journal of Economic Behavior and Organization,
March 2000, 41(3), pp.211-219.
[7] Chaloupka, Frank J. “Rational Addictive Behaviour and Cigarette Smoking”. Journal of Political Economy, August 1991, 99(4), pp.722-42.
[8] Elster, Jon and Skog, Ole-J∅rgen. Getting Hooked: Rationality and Addiction. Cambridge University Press 1999.
[9] Feenstra, Robert C., and Shapiro, Matthew D. “High-Frequency Substitution and the Measurement of Price Indexes”. NBER Working Paper Series,
2001, No.8176.

29

[10] Gruber, Jonathan and Köszegi, Botond. “Is Addiction “Rational”? Theory
and Evidence.” NBER Working Paper No.7507, 2000.
[11] Gruber, Jonathan and Köszegi, Botond. “Is Addiction “Rational”? Theory
and Evidence.” Quarterly Journal of Economics, November 2001, 116(4),pp.1261-1303.
[12] Hausman, Jerry A. “Specification Tests in Econometrics”. Econometrica,
November 1978, 46(6), pp.1251-71.
[13] Hendel, Igal, and Nevo, Aviv. “Sales and Consumer Inventory”. Institute ofBusiness and Economic Research, Economics Department Working Papers,
University of California, Berkeley, 2001.
[14] Horioka, Charles Yuji. “Stimulating Personal Consumption by Taxation.”
Nihon Keizai Shimbun, 18 January 2002, page 29.
[15] Horioka, Charles Yuji. “To Fix Economy, End Consumption Tax.” The
Daily Yomiuri, 19 February 2002, page 9.
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History of Japanese Tobacco Monopoly), Vol.1-6, 1963-1990.
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Willard G. “Taxation, Regulation, and Addiction: A Demand Function for
Cigarettes based on Time-series Evidence.” Journal of Health Economics,
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[18] Kingston, Geoffrey H. “Efficient Timing of Retirement.” Review of Economic Dynamics, October 2000, 3(4), pp.831-840.
[19] Ryder, Harl E. and Heal, Geoffrey M. “Optimum Growth with Intertemporal Dependent Preference”. Review of Economic Studies, January 1973,
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30

[20] Samwick, Andrew A. “New evidence on pensions, social security, and thetiming of retirement.” Journal of Public Economics, November 1998, 3(4),pp.207-236.
[21] Statistics Bureau and Statistics Center, Government of Japan. Consumer
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31

Figure 1 Inventory for tax increase in t+1

Inventory△Store1(t)△Hoarding(t)△Store2(t)△Store3(t)

t

t+1

t+2

t+3

Source: drawn by author

32

t+n

Figure 2 Daily purchase before and after tax increase

350.00300.00

April
May
June
July
AugustSeptember

200.00150.00100.00

50.00

28

25

22

19

16

13

10

7

4

0.00
1

Daiy purchase ( yen in 2003)

250.00Date
Source: author's calculation based on Reportof Family Income and Expenditure 2003

33

2.7

2

2.65
1.5

2.6

1

2.55
2.5

0.5

2.45

0

2.4

Source: author's calculation based on Annual
Report of Family Income and Expenditure, 20022003

34

Price (100yen per pack)

2.75

2.5

Aug-02
Sep-02Oct-02
Nov-02
Dec-02Jan-03
Feb-03Mar-03
Apr-03May-03Jun-03
Jul-03Aug-03
Sep-03Purchase (p acks per capita)

Figure 3 Monthly cigarette purchase before and after tax increase in 2003

Purchase(packs per
capita)Real price(100 yen per
pack)

300

500

250

400

200

300

150

200

100

100

50

0

0

Source: Annual Report on the Consumer
Price Index, Monthly Report on the Retail
Price Survey

35

Nominal price per pack (yen)600

Jan-54
NovSepJul-65
MayMar-73
Jan-77
NovSepJul-88
MayMar-96
Jan-00Real price per pack (yen), CPI

Figure 4 Real and nominal cigarette price Jan. 1954 – Sep. 2003

Real price ofcigarettes
CPI
Nominal price ofcigarettes

9
7

0.2

6

0

5

-0.2

4

-0.4

3
-0.6

2
1

-0.8

0

-1

Source: Annual Report of Family Income andExpenditure Survey

36

Rate of tax increase

0.4

8

Jan-54
Jul-57
Jan-61
Jul-64
Jan-68
Jul-71Jan-75
Jul-78Jan-82
Jul-85Jan-89Jul-92Jan-96
Jul-99
Jan-03

Cigarette puchase s (packs per
capita, worker household, adjusted
by X-12 ARIMA)

Figure 5 Per capita cigarette purchase versus rate of tax increase Jan. 1954 – Sep. 2003

Monthlypurchase(packspercapita)Rate oftax
increase

Table 1 Daily purchase before and after tax increase
before tax increase

after tax increasepurchasepurchase/averagepurchasepurchase/average

8 days

49.34

1.32

10.92

0.29

7 days

97.35

2.60

22.16

0.59

6 days

79.28

2.12

12.54

0.34

5 days102.87

2.75

24.18

0.65

4 days113.81

3.04

16.55

0.44

3 days

75.20

2.01

13.24

0.35

2 days125.26

3.35

15.64

0.42

1 day

332.90

8.89

28.38

0.76

Source: author’s calculation based on Report of Family Income and Expenditure 2003

37

Table 2 Summary and difference test for the daily purchase before and after the tax increase (from1 April, 2003 to 31 March, 2004) (tax increase was enforced on 1 July, 2003)

Summaries
Variable (packs perperiodcapita)daily purchase 1

daily purchase 2

daily purchase 3

daily purchase 4

from

1

April

to

20

June,

2003
from 21 June to 31 September,2003
from 1 October to 31 December,2003
from 1 January to 31 March,
2004

Obs

Mean Std. Dev. Min

Max

81

0.046

0.008

0.032 0.072

102

0.046

0.044

0.013 0.418

92

0.040

0.008

0.024 0.060

91

0.038

0.008

0.021 0.071

Difference tests
Hypothesis
Ho: daily purchase 1= daily purchase 2;
Ha:daily purchase 1> daily purchase 2
Ho: daily purchase 2= daily purchase 3;
Ha:daily purchase 2> daily purchase 3
Ho: daily purchase 3= daily purchase 4;
Ha:daily purchase 3> daily purchase 4

t-statistics

p-value

0.008

0.497

1.264

0.104

1.583

0.058

Source: author's calculation based on Report of Family Income and Expenditure 2003-2004

38

Table 3 Monthly purchase frequency and expenditure before and after tax increase
Frequency (times per 100 family,

Expenditure

one month)(yen per family)

Jan-0399

1,098

Feb-0396

1,040

Mar-03103

1,142

Apr-03101

1,107

May-03112

1,119

Jun-03110

1,884Jul-0379786

Aug-0399

1,089

Sep-0396

1,024

Source: author’s calculation based on Report of Family Income and Expenditure 2003

39

Table 4 Summary statistics: Jan. 1954 - Sep. 2003

Variable

Mean

Std.Dev

Max.

Min.

Purchaseｔ

1.475

0.460

5.277

0.769

Priceｔ

2.912

0.898

4.959

1.715

Yｔ

945.854345.2101439.970

273.486

ΔYｔ

1.721

26.054108.564-134.337

Source: author’s calculation

40

Table 5 Hoarding size versus the rate of tax increase

Events of tax Purchase (2 months Purchase (one month Hoarding

Rate of tax

increaseincrease

before tax increase)

before tax increase

1

1.472

2.453

0.981

0.188

2

2.447

5.277

2.83

0.491

3

2.153

2.988

0.835

0.211

4

1.914

2.772

0.858

0.119

5

1.71

2.484

0.774

0.111

6

1.464

2.138

0.674

0.078

7

1.224

2.154

0.93

0.08

Average

1.769

2.895

1.126

0.183

Source: author’s calculation based on Report of Family Income andExpenditure 1954 - 2003

41

Table 6 Tests of unit roots (ADF test and Phillips-perron test): Jan. 1954-Sep. 2003
ADF test

Variableconstant time lag length

Phillips-pherron test
test statistics

constant time lag length

test statistics

Purchaseｔ

yes

yes

4

-19.236(***)

yes

yes

5

-6.840(***)

Priceｔnono4-2.363(**)

no

no

5

-2.341(**)

Yｔ

yes

yes

4

-0.174

yes

yes

5

-2.059

ΔYｔ

yes

yes

4

-17.807(***)

yes

yes

5

-57.206(***)

Note: ***: significant at 1% level; **: significant at 5% level; *: significant at 10% level42

Table 7 Estimation results without distinction between purchase and consumption(do not consider optimal inventory)
Independent
Variable

OLS

2SLS

Coefficientt-statistic

Coefficientt-statistic

Constant

2.467 (***)

12.7387.587 (***)

13.731

Purchaseｔ-1

0.235 (***)

6.194-0.264 (***)-3.484Purchaseｔ+1

0.177 (***)

4.394-0.572 (***)-5.800Priceｔ-0.415 (***)-12.163-1.268 (***)-13.542

ΔYｔ

0.000

-1.1810.000-0.206Time-0.001 (***)-10.445-0.004 (***)-12.836

Adjust R-square

0.787

0.539

OID ｒatio

92.508

Wu ratio

100.510Observations

593

593

Note: ***: significant at 1% level; **: significant at 5% level; *: significant at 10% level;
The instruments of 2SLS Column: two lags and two leads of price, hoarding dummy,
store dummy and other explanatory variables; The critical 5% value for Chi-square
distribution with 4 degrees of freedom (OID test) is 9.488. The critical 5% value for
Chi-square distribution with 6 degrees of freedom (Wu test) is 12.592.

43

Table 8 Estimation results with distinction between purchase and consumption(consider optimal inventory, rational addiction model with optimal inventory)
Independent
Variable

OLS

2SLS

Coefficientt-statistic

Coefficientt-statistic

Constant

0.277 (***)

3.511

0.663 (*)

1.722

Purchaseｔ-1

0.469 (***)

13.6090.492 (***)

3.898

Purchaseｔ+1

0.463 (***)

13.306

0.343 (***)

2.834

Priceｔ-0.046 (***)-3.371-0.110 (*)-1.712ΔYｔ

0.000-0.3340.000-0.578

Hoardingｔ+1-2.546 (***)-10.656-1.841 (**)

-2.546

Hoardingｔ

7.222 (***)

43.7496.868 (***)

17.196Store_1ｔ-4.898 (***)-16.986-5.044 (***)-5.366Store_2ｔ0.307 (**)

2.235

0.317

1.223

Store_3ｔ

0.165

1.308

0.207

1.336

Store_3ｔ-1

0.056

0.449

0.099

0.755

Time

0.000 (***)-2.9490.000 (*)

-1.6852

４θ β<1(***)(-3.810)(***)(-1.924)

φ1<1

(***)

(-5.146)(***)(-3.047)

φ1>1

(***)

(3.533)(*)(1.351)

short-run ε-0.414 (***)(-4.464)-0.489 (**)(-2.147)

long-run ε-1.325 (***)(-8.278)-1.311 (***)(-19.074)

Adjust R-square

0.975

0.974

OID ratio

19.162

Wu ratio

23.934

Observations

593

586

Note: ***: significant at 1% level; **: significant at 5% level; *: significant at 10% level;
The instruments of 2SLS Column: four lags and one lead of price, seven lags of ΔY,
three leads of hoarding dummy, five lags of store dummy and other explanatory
variables; The critical 5% value for Chi-square distribution with 14 degrees of freedom(OID test) is 23.685. The critical 5% value for Chi-square distribution with 12 degrees
of freedom (Wu test) is 21.026.

44