Panel Data Analysis of Japanese Residential Water Demand Using a Discrete Continuous Choice ...

30 Pages·2010·0.15 MB·English

CIRJE-F-717
Panel Data Analysis of Japanese Residential Water
Demand Using a Discrete/Continuous Choice
Approach
Koji Miyawaki
National Institute for Environmental Studies
Yasuhiro Omori
University of Tokyo
Akira Hibiki
National Institute for Environmental Studies
February 2010
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Panel Data Analysis of Japanese Residential Water
/
Demand Using a Discrete Continuous Choice
Approach
∗
Koji Miyawakia Yasuhiro Omorib Akira Hibikic
February 17, 2010
a
NationalInstituteforEnvironmentalStudies,Ibaraki305-8506,Japan([email protected]).
b
FacultyofEconomics,UniversityofTokyo,Tokyo113-0033,Japan([email protected]).
Phone:[phone]. Fax:[phone].
c
NationalInstituteforEnvironmentalStudies,Ibaraki305-8506,Japan([email protected]).
Abstract
Blockratepricingisoftenappliedtoincometaxation,telecommunicationservices,
andbrandmarketinginadditiontoitsbest-knownapplicationinpublicutilityservices.
Under block rate pricing, consumers face piecewise-linear budget constraints. A dis-
crete/continuouschoiceapproachisusuallyusedtoaccountforpiecewise-linearbudget
constraintsfordemandandpriceendogeneity. Arecentstudyproposedamethodology
to incorporate a separability condition that previous studies ignore, by implementing a
Markov chain Monte Carlo simulation based on a hierarchical Bayesian approach. To
extend this approach to panel data, our study proposes a Bayesian hierarchical model
incorporatingtheindividualeffect. Therandomcoefficientsmodelresultshowsthatthe
priceandincomeelasticitiesareestimatedtobenegativeandpositive,respectively,and
∗Correspondingauthor: Tel:[phone],E-mail:[email protected]
1
thecoefficientsofthenumberof membersandthenumberof roomsperhouseholdare
estimated to be positive. Furthermore, the AR(1) error component model suggests that
theJapaneseresidentialwaterdemanddoesnothaveserialcorrelation.
Keywords: Blockratepricing,Bayesiananalysis,Paneldata,Residentialwaterdemand.
JELclassification: C11,C23,C24,Q25.
1 Introduction
Blockratepricingusuallyhasbeenappliedtoservicesinpublicutilitysectorssuchaswater,
gas,andelectricity.1 However,blockratepricingisbecomingcommoninareassuchaslocal
and wireless telephone services and brand marketing. Under block rate pricing, unit price
changeswithquantityconsumed. Whenunitpriceincreaseswithquantityconsumed,shown
inFigure1,suchapricescheduleiscalledtheincreasingblockratepricing. Whenunitprice
decreaseswithquantityconsumed,itiscalledthedecreasingblockratepricing. Then,under
block pricing consumers maximize utility by selecting the unit price and the consumption
amount. Thiscircumstanceleadstoautility-maximizationproblemunderapiecewise-linear
budgetconstraint.
As surveyed by Olmstead (2009), there are two types of estimation approaches that deal
with this problem: reduced-form approaches, such as instrumental variables, and struc-
tural approaches. The structural approach solves a consumer’s utility maximization prob-
lem in two steps. A consumer first decides appropriate consumption given each block’s
price, and then selects the block that maximizes consumer utility. This is also called a dis-
crete/continuous choice approach because the block selection is discrete while the amount
consumed is continuous. Its important feature is that the derived model explicitly addresses
therelationshipbetweentheblockchoiceandtheamountconsumedunderblockratepricing.
As discussed in Olmstead (2009), the reduced-form approaches can incorporate only
1Theotherexamplewherethesameratestructureisappliedisaprogressivetaxrateinincometaxsystems.
2
limitedaspectsofthepiecewise-linearbudgetconstraint,whilethestructuralapproachesac-
count just for the particular implications of piecewise-linear budget constraints for demand
as well as price endogeneity. Thus the latter approaches have two main advantages over the
former ones: (1) the structural approaches can produce unbiased and consistent estimates of
parameters of the price and the income2and (2) they are consistent with utility theory. De-
spite these advantages, most previous studies employ reduced-form approaches. Structural
approaches are rare in demand analysis.3 This is because the discrete/continuous choice ap-
proach had been applied only to the simplified block rate price structure—for example, the
numberofblocksisfixedattwo.
Pint (1999); Rietveld et al. (2000); Olmstead, Hanemann, and Stavins (2007); Olmstead
(2009) considered multiple-block pricing—for example, the number of blocks varies from
twotofour. Miyawaki,Omori,andHibiki(2010)proposedamethodologytosolvetheprob-
lemofmulti-tierblockratepricingbyimplementingaMarkovchainMonteCarlo(MCMC)
simulationbasedonahierarchicalBayesianapproach. Theirmethodalsoshowedthatmodel
parameters are subject to the separability condition, or set of linear inequality constraints.
Despite its importance, previous literature generally ignores the condition because the pa-
rameterregionbecomestightlyrestrained,makingnumericalmaximizationofthelikelihood
functiondifficult(Pint,1999;Rietveldetal.,2000;Olmsteadetal.,2007;Olmstead,2009).
To extend Miyawaki et al. (2010) for the panel data analysis, this study proposes a
Bayesian hierarchical model. It incorporates the individual effect to estimate the residential
water demand function under the separability condition using panel data of Japanese house-
holds. Thisisthefirststudythatincorporatestheindividualeffectinthediscrete/continuous
choiceapproach.
2Previous studies suggested that water demand is price inelastic. However, as is suggested in the meta-
analysis(Dalhuisen,Florax,deGroot,andNijkamp,2003),thechoiceoftheapproachmayaffecttheestimates,
sincethewaterdemandispriceinelasticinpreviousstudiesemployingthereducedformapproach,however,
arepriceelasticinthediscrete/continuouschoiceapproach.
3Olmstead (2009) reported that, between 1963 and 2004, there were only three studies on water de-
mand (Hewitt and Hanemann, 1995; Pint, 1999; Rietveld, Rouwendal, and Zwart, 2000) that adopted the
discrete/continuouschoiceapproach.
3
We organize this article as follows. Section 2 describes block rate pricing and reviews
previousstudies. Section3explainsthederivationofthemodel,incorporatingtheindividual
effect to extend Miyawaki et al. (2010) for panel data analysis and proposes its MCMC
estimation method. Section 4 shows the empirical analysis of Japanese residential water
demandusingpaneldata. Section5concludes.
2 Block rate pricing system and literature review
Figure 1 shows the example of a three-tier increasing block rate pricing where Y is the
consumptionofthegoodorservice, P istheunitpriceofY inblockk (k=1;2;3)andY¯ is
k k
the boundary quantity between block k and k+1, i.e., the upper limit of block k. Under this
system,whenconsumptionofY exceedsY¯k theunitpricejumpsfrom Pk to Pk+1.
P
P
3
P
2
P
1
Block 1 Block 2 Block 3
0 Y Y Y
1 2
Figure1: Three-Tierincreasingblockpricestructure.
Ablockratepricingsystemgenerallyhasthefollowingcharacteristics:
1. The total payment is comprised of the fixed charge, FC, and the variable charge, VC,
depending on the volume consumed. A practical example of the fixed charge is a
minimumaccesschargeforwaterandelectricityservices.
2. In a block rate pricing system with K blocks, potential consumption is divided into K
consumptionregions—i.e., K blocks.
4
3. K unitpricescorrespondto K blocks. Thek-th(1≤k≤K)unitpriceinthek-thblock
isappliedtoconsumptionandisconstantwithinthek-thblock.
Nowweconsidertwoconsumptiongoods. Thefirstisagood,suchaswaterorelectricity,
to which block rate pricing is applied. The second is a composite good, to which a single
constant price is applied. We denote Yi and Yi as consumption of the first and the second
a
good by the consumer i. We denote Y¯i as an upper limit (threshold) of the k-th block and
k
Pi as the unit price of the k-th block. The superscript i is attached because we assume
k
that different consumers face different block rate pricings. Then, the budget constraint of
consumer i, who chooses the consumption, Yi, within the k-th block, Y¯i ≤ Yi < Y¯i, is
k k−1 k k
expressedasfollows.
∑k−1( )
PiYi+Yi ≤Qi ≡Ii−FCi− Pi −Pi Y¯i; (1)
k k a k j j+1 j
j=1
where we denote Ii as the income of consumer i. The Qi is called as the virtual income
k
of consumer i, who chooses to consume at the k-th block. We also set Y¯i =0 and Y¯i =∞
0 Ki
without loss of generality. As shown in the above equation, the budget constraint becomes
piecewise-linear(seealsoFigure2).
The discrete/continuous choice approach is a common structural approach that solves
the utility maximization problem under a piecewise-linear budget constraint first proposed
byBurtlessandHausman(1978). Otherstudiesincludeexpenditureswithfoodstamps(Mof-
fitt, 1989), car ownership, and use (de Jong, 1990), electricity demand (Herriges and King,
1994;ReissandWhite,2005),waterdemand(HewittandHanemann,1995;Olmsteadetal.,
2007; Miyawaki et al., 2010), choice of wireless service calling plans (Iyengar, 2004), and
consumer preference regarding multiple product categories and brands (Song and Chinta-
gunta,2007).
Although the discrete/continuous choice approach is based on economic theory and can
beextendedtoallowvariousdependencies,itisdifficulttoestimatethemodel’sparameters.
5
Forexample,Moffitt(1986)pointedoutaproblemwithnon-differentiabilityinthelikelihood
function and also a computational burden. This is why most previous studies estimate the
demand function in a simplified manner in which, for example, all consumers face identical
two-tierblockratepricing.
Severalrecentstudies(Pint,1999;Rietveldetal.,2000;Olmsteadetal.,2007;Olmstead,
2009)considermultiple-blockpricing(thenumberofblocksvariesfromtwotofour). How-
ever,theyignoredtheseparabilitycondition. Miyawakietal.(2010)addressedthisproblem
byimplementinganMCMCsimulationbasedonahierarchicalBayesianapproach,butthey
didnotconsidertheapplicationtothepaneldata.
ThemaincontributionsofourarticlearetoextendMiyawakietal.(2010)toincorporate
theindividualeffectandtotakeahierarchicalBayesianapproachinestimatingtheresidential
waterdemandfunctionundertheseparabilityconditionusingJapanesehouseholdpaneldata.
3 Derivation of the demand function under block rate pric-
ing and application of MCMC
3.1 Derivation of the demand function under block rate pricing
Inthissubsectionweexplainthederivationofthedemandfunctionunderablockratepricing
systemwithinthesettingofSection2.
Suppose consumer i determines the consumption of a good that is subject to Ki-block
rate pricing, Yi, to maximize utility, U(Yi;Yi), under the piecewise-linear budget constraint
a
as is shown in (1). Since the increasing block rate pricing is used for the residential water
supplyin Japan, we suppose Pi <Pi (k=1;:::;Ki−1). Figure 2 illustratesan exampleof
k k+1
autilitymaximizationproblemunderapiecewise-linearbudgetconstraintduetoablockrate
pricing system (the case of three-tier increasing block pricing), where we denote Vi as the
levelofanindifferencecurveandwherethesecondblockisoptimalwithitsoptimaldemand
6
Yi .
opt
Y i
a
Vi
O Y i Y i Y i Y i
1 opt 2
Figure2: Utilitymaximizationproblem: three-blockcase.
Before derivating the demand function under block rate pricing, we consider Ki condi-
tionalutilitymaximizationproblems. Fork=1;:::;Ki,thek-thconditionalproblemisgiven
asfollows:
( )
maxU Yi;Yi s.t. PiYi+Yi ≤Qi: (2)
k a k k a k
Yi;Yi
k a
Theoptimalconditionalconsumption,Yi,isdeterminedasifconsumerifacedasingleprice
k
Pi and given income Qi. With these Ki optimal conditional consumptions, the demand
k k
functionunderincreasingblockratepricingforconsumer iisgivenby

Yi =YY¯kii;; iiffYY¯kii−1 <≤YY¯kii <≤YY¯kii aannddkk==11;;::::::;;KKii;−1: (3)
k k+1 k k
Sinceweapplythelog-linearconditionaldemandmodelusedinpreviousstudies,Equa-
tion(3)isrewrittenas

yik; ify¯i;k−1 <yik <y¯ik andk=1;:::;Ki;
yi =y¯ik; ifyi;k+1 ≤y¯ik ≤yik andk=1;:::;Ki−1; (4)
7
y =(cid:12) p +(cid:12) q ≡ xxx′ (cid:12)(cid:12)(cid:12); (5)
ik 1 ik 2 ik ik
where(y;y ;y¯ ;p ;q )=(logYi;logYi;logY¯i;logPi;logQi), xxx =(p ;q )′,and(cid:12)(cid:12)(cid:12)=((cid:12) ;(cid:12) )′.
i ik ik ik ik k k k k ik ik ik 1 2
As per Miyawaki et al. (2010), we now introduce two unobserved random variables into
∗ ∗
the demand function of the i-th consumer: the heterogeneity, w , and the state variable, s .
i i
Theheterogeneityisastochastictermthatmodelsconsumers’characteristicsandisassumed
to be an additive to the log conditional demand yi. Thus, w∗ is assumed to follow the linear
k i
model
w∗ =zzz′(cid:14)(cid:14)(cid:14)+v; v ∼i.i.d. N(0;(cid:27)2); (6)
i i i i v
where zzz and (cid:14)(cid:14)(cid:14) are d×1 vectors of explanatory variables for the heterogeneity and corre-
i
spondingparameters,respectively,andv isanindependentlyandidenticallydistributeddis-
i
turbance term with a normal distribution of mean 0 and variance (cid:27)2. The state variable, s∗,
v i
isadiscreterandomvariablethatindicateswhichblockispotentiallyoptimalforconsumers.
Then,thebasicmodelforthedemandfunctionunderincreasingtheblockratepricingis
givenbythefollowingequations:
y = xxx′ (cid:12)(cid:12)(cid:12); xxx =(p ;q )′; k=1;:::;Ki; (7)
ik ik ik ik ik
w∗ =zzz′(cid:14)(cid:14)(cid:14)+v; v ∼i.i.d. N(0;(cid:27)2); (8)
i i i i v

s∗i =22kk;−1; iiffww∗i∗i ∈∈RRii;;22kk−=1 =(y¯i(;ky¯−i;1k−−1y−iky;iy¯ki;ky¯−iky−iky)ik) aannddkk==11;;::::::;;KKii;−1; (9)

y∗i =yy¯ik;+w∗i; iiff ss∗i∗ ==22kk−an1daknd=k1;=::1:;;:K::i;−K1i;; (10)
ik i
y =y∗+u; u ∼i.i.d. N(0;(cid:27)2): (11)
i i i i u
The extension of the above model to incorporate the individual effect for parameter (cid:14)(cid:14)(cid:14) in
the panel data with n observations and T time periods, which we call a random coefficients
8
model(RC),isrewrittenasfollows:
( )
yit;k = xxx′it;k(cid:12)(cid:12)(cid:12); xxxit;k = pit;k;qit;k ′; k=1;:::;Ki; (12)
( )
www∗ =ZZZ (cid:14)(cid:14)(cid:14) +vvv; vvv ∼i.i.d. N 000;(cid:27)2III ; (13)
i i i i i T v

s∗it =22kk;−1; iiffww∗i∗itt ∈∈RRiitt;;22kk−a1nadnkd=k=1;1::;::;:K:;iKt−it;1; (14)

y∗it =yy¯iitt;;kk;+w∗i = xxx′it;k(cid:12)(cid:12)(cid:12)+w∗i; iiff ss∗i∗itt ==22kk−an1daknd=k1;=::1:;;:K::it;−K1it;; (15)
y =y∗ +u ; u ∼i.i.d. N(0;(cid:27)2); (16)
it it it it u
where subscripts i and t denote the observation i and time t, respectively, N ((cid:22)(cid:22)(cid:22);ΣΣΣ) denotes
T
a T-variate normal distribution with mean (cid:22)(cid:22)(cid:22) and covariance matrix ΣΣΣ, and III is an identity
matrix.
This RC model extends the basic model in three ways. First, it introduces a linear
structure and a normal error vvv to consumer heterogeneity, www∗ = (w∗ ;:::;w∗ )′. The ZZZ =
i i i1 iT i
(zzz ;:::;zzz )′ and (cid:14)(cid:14)(cid:14) are a T ×d vector of explanatory variables and a d×1 vector of their
i1 iT i
coefficients,respectively. Theheterogeneityintervalsaregivenby
( ) ( )
Rit;2k−1 = y¯it;k−1−xxx′it;k(cid:12)(cid:12)(cid:12); y¯it;k−xxx′it;k(cid:12)(cid:12)(cid:12) ; Rit;2k = y¯it;k−xxx′it;k(cid:12)(cid:12)(cid:12); y¯it;k−xxx′it;k+1(cid:12)(cid:12)(cid:12) : (17)
To capture the individual effect for the coefficient of ZZZ , the (cid:14)(cid:14)(cid:14)’s are assumed to be indepen-
i i
dently and identically distributed random samples from a normal distribution Nd((cid:22)(cid:22)(cid:22)(cid:14)(cid:14)(cid:14);(cid:27)2vΣΣΣ(cid:14)(cid:14)(cid:14))
∗
as in (18). Second, a discrete latent variable, s , is used to indicate potentially optimal de-
it
mand chosen by the i-th consumer at time t. When s∗ is odd (s∗ =2k−1 for k=1;:::;Kit),
it it
the i-th consumer would select the optimal conditional demand. When, on the other hand,
s∗ is even (s∗ = 2k for k =1;:::;Kit−1), one of threshold values would be optimal for the
it it
9