ESTIMATION OF THE DEMAND FOR RESIDENTIAL WATER IN A STONEGEARY FORM AND THE CHOICE OF THE PRICE VARIABLE
MARIE-ESTELLE BINET
Associate Professor
[email protected]
CREM (UMR 6211 CNRS), University of Rennes 1, France
University of Rennes 1, Faculty of Economics
7 place Hoche, CS 86514
35065 Rennes cedex, France
Abstract. Based on a detailed survey data (concerning the French overseas department La
Reunion), we investigate the price specification issue. We estimate the residential water
demand function, derived from a Stone-Geary utility function, using Shin’s perception price.
As the corresponding parameter is estimated to be around 0.6 and different from one, we
show that the price perceived by the consumer is between the average and the marginal price.
We conclude that consumers, facing an increasing block pricing, underestimate the true
(marginal) price and therefore consume too much water.
JEL code: Q25, D12, C51.
Key-Words: Residential water demand, perceived price, Stone-Geary.
1
1. INTRODUCTION
A large empirical literature aims at obtaining consistent values of price elasticity of demand to
analyze the relevance of a pricing policy to manage water consumption. Indeed, estimation of
residential water demand has been a major issue in environmental economics as proved by the
number of recent surveys available in the literature (Arbuès et al (2003), Dalhuisen et al
(2003), Worthington and Hoffman (2008), Nauges and Whittington (2010)). But to promote
efficient water use, water authorities need to know what price variable consumers are
responding to.
However, the water tariff structure is often complex with increasing or decreasing prices and
fixed charges. Therefore, one important issue in the literature is to provide an adequate
specification of the price variable in residential water demand models. The discussion
generally focuses on the adequacy of using the average or the marginal price. A perfectly
informed consumer should react to marginal price (in addition to Nordin’s difference
variable). But in case of price illusion or incomplete information, the consumer can react to
other price indicator as the average price. To analyze residential electricity consumption, also
characterized by the presence of a block pricing structure, Shin (1985) developed a price
perception variable, which is a combination of marginal and average prices.
Indeed, the choice of the price variable that consumers react to remains an empirical issue.
Howe and Linaweaver (1967) are the first to have discussed and compared average and
marginal prices for water demand specification. In a first step, the price which provides the
best fit is presumed to be the price perceived by consumers, Foster and Beatie (1981). Next, to
our knowledge, two other methodologies have been implemented to explicitly compare price
variables. Using hypothesis tests on an augmented water demand function including marginal
prices and two forms of the Nordin’s difference variable, Opaluch (1982) derived the first test
of whether consumers respond to marginal or average price when faced with block rates. But
although Opaluch’s model is creative, it does not permit the consumer to react to a function
or both marginal and average prices. Next, Shin (1985) proposed to compare marginal and
average electricity prices by testing the value of a price perception parameter. Nieswiadomy
and Molina (1991) used this methodology to deal with residential water consumption in a
times series approach, under a decreasing then increasing block rate schedule.
In this context, this paper is an original contribution on residential water demand literature for
2
at least two reasons. Firstly, it is based on a detailed survey data including bills collected on a
representative sample of households in one overseas French region, La Reunion. This is a
unique micro level data set which allows an explicit modeling of tariff scheme. The second
originality of this paper is to apply Shin’s methodology using a Stone-Geary functional form
in estimating residential water demand in a cross-sectional approach. One attractive feature of
this specification is that total water consumption can be broken down into two constituent
parts. The first is the incompressible part, which can be interpreted as a minimum level of
water consumption. The second component of total water consumption is the variable part
which depends on income and price levels.
Next, following the translating method developed by Pollack and Wales (1981), one way to
propose a better empirical specification and to reduce unexplained variation in expenditure
behavior is to postulate that the parameter measuring the minimum water consumption is a
function of the characteristics of the households. Using GMM estimator that accounts for both
endogeneity of the price variable, and non linear coefficients, we estimate the water demand
function using the Shin’s perception price. Our main finding is that the perceived price
parameter is between zero and one. Then, we conclude that consumers react to a price which
is between the average price and the marginal price. Therefore, in an increasing block pricing,
consumers underestimate the true (marginal) price and are consuming too much water.
This paper proceeds as follows. In section 2, the model specification is presented. The second
section describes the data used in the empirical work. Section 3 examines and discusses the
results. Section 4 concludes.
2. WATER DEMAND SPECIFICATION
First, we describe the pricing modeling. We then present the water demand specification and
describe the translating method.
2.1 Price measurement
In Reunion island, the pricing of residential water consists of several (between 3 and 4)
increasing consumption blocks. To simplify the presentation, if the rate schedule is assumed
to consist of two blocks and the fixed charge, the consumer’s budget constraint can be written
3
as following:
(1) F
k
1 1
2
(q k1 )
px X
I
Where q denotes residential water consumption measured at the household level.
1
and
2
are respectively the first and the second block prices. k1 is the highest consumption level in
block 1. p x and X are respectively an index price and the corresponding consumption of other
private goods. I is the household income. F is the fixed part.
To deal with block rate structure complexity, we often consider a variable inspired by Nordin
(1976). Indeed, Nordin (1976) claims that consumers react not only to marginal price, but
also to the change in the intra-marginal prices The Nordin’s difference variable D is equal
here to the difference between the bill would have been if the water quantity was consumed at
the marginal price less total bill. This variable, as the fixed part, measures an income effect
imposed by the tariff structure:
(2) D
(
2
1
) k1
0 if we consider an increasing consumption block.
Then, the budget constraint can be rewritten as:
(3)
2
q
px X
I
F
D
Therefore, if consumers are perfectly informed, they should react to marginal price
adjusted income: I
F
2
and to
D . However, if consumers are imperfectly informed, they may
respond to average price: AP
k
1 1
2
q
(q k1 )
. We conform to Taylor et al (2004) who
advice to purge the fixed charges from the average price as it creates a bias in the estimated
price elasticity.
Next, Shin (1985) defines the perceived price as the price to which the consumer responds.
Facing a complex tariff structure, the consumer may have a partial knowledge of it, and the
perceived price may include one fraction only of the difference variable (with total bill), and
thus may lie between marginal and average prices as: AP
2
D
.
q
As the choice of the price variable is an empirical question, Shin (1985) built a price
perception variable as a function of marginal and average prices and a price perception
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parameter k to be estimated:
(4) P *
2
(
AP
)k
2
We can see that the ratio
AP
captures the effect of the intra-marginal effects on price
2
perception. If k is equal to zero, we conclude that consumers respond to marginal price
whereas if k is equal to one, they respond to average price. If 0
perceived price is between average and marginal prices. If k
k
2
1 , then, the consumer’s
1 , we have p *
2
AP.
k was estimated by Shin (1985) and Van Helden, Leeflang and Sterken (1987) for electricity
demand and by Nieswiadomy and Molina (1991) and by Nieswiadomy and Cobb (1993) for
water demand in a log linear specification. The results of this test gave mixed results. Shin
(1985) found a parameter equal to 1.007, with the null hypothesis k
1 not rejected. Shin
concluded that the perceived price is average price. Nieswiadomy and Molina (1991) obtained
a non significant parameter under increasing water consumption block and conclude that the
consumers respond to marginal prices. But the reverse result was found in Nieswiadomy and
Cobb (1993).
But they included the fixed charge in the average price which biased the
estimate of k toward unity and therefore invalidates the Shin test according to et Taylor et al
(2004).
2.2 Residential water specification in a Stone-Geary form
We specify the consumer’s utility function in the following Stone-Geary form:
U m (q, x)
ln (q mq ) (1
) ln ( x mx ). In the literature, to our knowledge five other
articles have used such a specification to analyze residential water consumption, Dharmaratna
and Harris (2010), Binet et al (2009), Madhoo (2009), Martinez-Espineira and Nauges (2004)
and Gaudin et al (2001).
The assumptions of this function are strong separability, the adding-up restriction, positive
marginal budget share
and q
mq and x
mx .
m x is often considered as a subsistence level of consumption. mq can be regarded as a
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minimum water consumption. Then, residential water demand function is derived from the
following program MAX U m ( x, q) subject to y m
(5) q
( ym
mq
p q mq
pq q
px x :
p x mx )
pq
Where p q is the residential water price variable perceived by consumer.
This specification is a (simplified) linear expenditure system (LES) if we consider only two
goods (water consumption and other consumption goods). This specification has the
advantage of being compatible with theory while explicitly modelling a minimum level of
consumption, irrespective of the price of the consumed good or the consumer’s income. The
consumer first purchases the minimum level of each good, and the left-over income then is
allocated in a fixed proportion
to the demand for residential water. Since residential water
consumption usually is constrained by the size of household and water using capital stock, the
LES specification is particularly well suited to account for these features.
Without spatial variation for p x , we suppose that the minimal value of other consumption
:
goods can be incorporated in parameter
(6)
q
( ym
mq
p q mq
)
pq
Next, the price perception variable is substituted into (6) to obtain the following specification
to be estimated:
( ym
(7)
2
q
mq
mq
2
(
AP
)k
)
2
2
(
AP
)k
2
Where k is the price perception parameter,
2
is the marginal price, AP the average price and
is the error term.
2.2 Translating method
Pollack and Wales (1981) developed the translating method in a demand specification to
allow subsistence parameters to depend on demographic variables. Martinez-Espineira and
6
Nauges (2004) and Gaudin et al (2001) also implemented the translating method to estimate
the demand for residential water in a LES specification. Using the same data set, Binet et al
(2009) have previously described mq in the following system of equations (8-10):
(8) m q
1
Nc
2
Garden
3
Swimming pool
Firstly, outdoor water uses is supposed to be one determinant of the incompressible water
consumption. Indeed, the presence of a garden is an important factor as 77% of households
leave in a house with a garden in La Reunion island against 30% in metropolitan France, in
2004. Only 5 % of household have a swimming pool in our sample but the presence of a
swimming pool can explain the huge consumption levels observed. The presence of a garden
or a private swimming pool is defined here as a dichotomous variable that takes the value of
one if the family has a garden or a swimming pool, zero otherwise. We would have liked to
use data giving the size of the garden and the volume of the swimming pool, but our data set
gives only information on the presence or not of these equipments.
Following Carlevaro et al (2007), the size of the population of water users N c can be differentiated by
distinguishing the number of child and the number of employed adults (NEA) from the rest of the
family (Number of Unemployed Adults):
(9) N c
NUA
1
NEA
2
CHILD
If we consider the needs of daily life, intended for the whole of the family, we expect that the
water consumption will increase with the household size. But, we expect a lower impact of
the number of employed adults (NEA), as they spend more time outside home than the other
members of the family.
As a great proportion of households has a vegetable garden, equation (10) enable us to include
the climatic effects on consumption:
(10)
2
20
21
Weather
Weather is measured by the percentage of non rainy days over the billing period.
1,
20 ,
21 , 1 and
2 are
the parameters to be estimated (with
and k) . Equations (8-10) are
substituted in the water demand function (7) to obtain one equation with non linearity in
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parameters.
3. HOUSEHOLD SURVEY
We first offer a brief description of the survey and then introduce the data selected for
empirical analysis.
3.1 Household survey
A stratified random survey was financed by the Regional Directorate for the Environment
(DIREN) and conducted in 2004 on a sample of 2000 households. The survey designers
retained a stratified sampling of 2000 households according to municipalities, i.e. 1% of total
household population, see Carlevaro et al (2007).
The questionnaire included 25 questions concerning:
Socio economic household characteristics (sex, householder age and occupation,
family income and size, number of working people and child, and if they are tenants or
property owners).
Housing characteristics (individual family house or apartment, age, number of rooms
and altitude).
Water consumption equipment (swimming pool, washing machine, dishwasher,
garden or vegetable garden ownership).
Consumption habits (washing frequency, business activity at home).
Carried out by telephone, this survey was completed with a mailing to 1000 volunteer
households, intended to collect the volume of water consumption displayed in the last three
bills (covering one year), later by post.
Finally, we collected the volume of water consumption recorded on 173 households,
including 449 useful water bills (from 1 to 3 water bills per household). Since the billing
period varies across municipalities, consumption data were converted to a daily consumption
per household (in liter). Corresponding daily weather indicators as precipitation, temperature
were taken from Meteo France Agency.
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3.2 Description of the variables
a) Price measurement
Data on water prices and consumption blocks’ length were available from DIREN. The price
variable includes sewage prices. We subtract the fixed charge from the total bill to compute
the average price, to avoid bias estimates, see Taylor, McKean and Young (2004) for a
discussion. By purging the fixed portion of the bill, the average variable price includes only
the portion which varies with consumption.
b) Income
The DIREN survey, conducted in 2004 on a sample of 2000 households, recorded household
income level1 as an ordered qualitative variable, namely as belonging to one of the following
five income intervals (in Euros/month): [0;750], [750;1500], [1500;3000], [3000;4500] and
[4500;. + ].
To improve the explanatory power of estimates, Carlevaro et al (2007) decided to measure
household income levels as a quantity. Therefore, they developed an imputation method based
on an econometric model describing the observed qualitative information on household
income according to an ordered polychotomous econometric model, where the unobserved
household income level is specified as a latent variable. Then, they assume that the form of
the personal income distribution is influenced by ordinal measure of household standard of
living, stated by households on a four level scale, namely «The household hasn’t enough
money to live. It cannot make ends meet», «The household has just enough to live, but by
making great sacrifices», «The household has enough money to live without making great
sacrifices», «The household makes no sacrifices, in any case nothing important». They
finally obtained 20 different income values. Next, as discussed before, we use adjusted
income, equals to income less fixed charges plus the difference variable to allow for income
effects of these variables.
c) Climatic variables
Climatic effects can be introduced in different ways. Here, we follow Matinez-Espineira
(2002) using the percentage of non rainy days over the billing period. Indeed, Carlevaro et al
(2007) have showed that users respond more to the occurrence of rainfall than to its amount.
1
Households were asked to take into account all their income sources, including welfare benefits, property
income,…
9
Therefore, we use daily observations recorded by Meteo France Agency (about one hundred stations
set up in La Reunion). These geographical distributed observations allow us to compute the number
of days without rainfall for each bill collected according to the observations recorded at the closest
weather station.
d) Equipments and other variables collected
We experimented other explanatory variables, particular those measuring billing frequency,
housing characteristics, consumption habits or the presence of a dishwasher or a washing
machine. But the parameters associated with all these variables turned out to be highly
insignificant (probability greater than 30%), probably owing to their imperfect measure. As
variables as the stock of appliances especially help to analyze long run reactions to a shock
price, indeed, that question reaches beyond the scope of this article.
The set of available variables, measured in 2004, and summary statistics for all variables are
presented in Table 1:
Table 1. Data set description, 449 water bills in 2004
Mean
Min
Max
675
102
5204
2096
426
7374
0.00089
0.00001
0.00218
0.0011
0.00013
0.0027
Number of child in a family
0.91
0
4
Number of employed adult in a family
0.98
0
3
Number of unemployed adult in a family
1.29
0
5
58
0
100
Description of variables
Daily water consumption per household
(liters)
Monthly household income (Euros)
Average price without fixed charge (Euros
for one liter)
Marginal price (Euros for one liter)
Percentage
of
days
without
rainfall
(weather variable)
In our sample, average residential water consumption is 675 liters per household and per day
(which correspond to 253 liter per capita in average). Coutelier and Le Jeannic F. (2007)
confirm the relevance of our estimate as they measured water consumption to be equal to 269
liters per inhabitant in La reunion, which is similar to highest consumption levels in OECD.
For example, average daily per capita water use in the UE-15 countries ranges from 115 liters
in Belgium to 265 liters in Spain. In La reunion, per capita residential water consumption is
about 60 % greater than in metropolitan France, in average. Likewise, distinctive features like
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water prices, which are very low in Le Reunion (less than one Euro per m3 in average), the
way of living (a high proportion of households live in a house with a garden), or climatic
effects, may contribute to explain those high water consumption levels, see Binet et al (2009)
for further results.
4. EMPIRICAL RESULTS
Estimation strategy is developed in 4.1. Next, empirical results are reported in 4.2.
4.1 Estimation strategy
One problem, which must be addressed with increasing block rate, is simultaneity as
consumers select the quantity of water and the price simultaneously. We therefore choose to
implement a GMM (Generalized method of moments) estimator using an appropriate set of
instruments2.
In the spirit of Hausman and Wise (1976), prices associated with fixed levels of consumption
(the three first quartiles of the distribution) are used as instruments for marginal and average
prices. An instrumental variable must satisfy two requirements. It must be correlated with the
endogenous variable and be orthogonal to the error term. We therefore used the Bound et al
(1995) test to select relevant instruments and performed the over-identifying restrictions
Hansen-Sargan test to choose valid instruments.
As the Breush-Pagan test reveals the presence of heteroscedasticity, robust standard errors are
computed in a heteroscedastic-consistent matrix.
2
Once the endogeneity in the price variable has been purged via IV or GMM estimators, the price coefficient
acquired the expected sign whereas it was positive with simple OLS technique.
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4.2 Results
Empirical results obtained for two nested specifications are listed in Table 2:
Table 2. Estimation results obtained with GMM, Shin’s price perception methodology.
Specification
all coefficients
significant coefficients
Coefficient estimates
Value
Value
(probability)
(probability)
0.0017
0.0048
(0.071)*
(0.000)***
165
178
(0.000)***
(0.000)***
130
0
(marginal budget share)
1
(unemployed adult)
20
(garden, common effect)
(0.17)
156
21
(garden with weather effect)
2
(child)
1
(employed adult)
(0.42)
0.53
0.32
(0.014)**
(0.08)*
0.21
0
(0.54)
93
3
(Swimming pool)
0
0
(0.65)
k
0.66
0.56
(perception price parameter)
(0.005)***
(0.000)***
Mean value of minimal water consumption
480 (71%)
284 (42%)
Overidentifying restrictions test (p-value)
0.011**
0.135
Adjusted R²
0.20
0.08
Number of observations
449
449
Rejection
Rejection
Test: H 0 : k
1
Significance level: *** for 1%, ** for 5% and * for 10%.
The first model includes all the parameters (see the second row). In the last row, insignificant
values of parameters obtained in preliminary estimations finally have been set equal to 0
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using an iterative procedure. In both specifications, the value of the consumption of other
private goods is fixed to be 0 to obtain consistent estimates. Hansen test indicates that the
validity of instruments is not rejected in the final specification only.
Our first point of interest is the value of Shin’s price perception parameter. We obtain robust
and significant values for it, equal to 0.66 in the first specification, then to 0.56 in the final
one. In both cases the null hypothesis k
1 is rejected, which means that the consumer’s
perceived price is between average and marginal prices. As the water bill represents a small
percentage of income, consumers are probably not well informed about the pricing structure.
Therefore, the consumer may underestimate the real price and consume too much water.
Next, the minimum residential water consumption level m̂q can be computed for each bill
using estimates parameters and equations (8-10). Finally, average values are around 71
percent of total per household consumption with the complete specification yet around 42 %
only if we consider the final model. In the final specification, the presence of a garden has a
positive but insignificant impact upon municipal spending, which explains these different
values for incompressible water consumption. We conclude that, at least, 42% of residential
water consumption will not be sensitive to a price increase.
To delve further into the interpretation of the incompressible per household water
consumption, we can analyze other results obtained. We observe a positive and significant
impact of household size on the minimum water consumption: an increase of one unemployed
adult will result in an increase of 178 liters per day per household. This effect is weaker if we
retain all the coefficients (respectively +165). Next, one additional child will generate an
increase of around 60 liters per day in household water consumption in the final specification.
But the number of employed adults has no significant impact on residential water
consumption (
4
1
0 ).
CONCLUSION
Using one unique cross-sectional dataset covering a representative sample of 2000 households
in 2004, we estimate the residential water demand function using a Stone-Geary form. The
translated Stone-Geary specification for water demand and the focus on the perceived water
13
price parameter to be estimated, are unique to the current paper. Using a non linear GMM
method with prices that the household would face at different fixed levels of water
consumption as instruments, we obtain the following empirical results. First, the evidence
from La Reunion island suggests that the perceived price to which consumers respond is
lower than marginal price. Therefore, consumers may consume too much water. Second, as
our results show that, at least, 60 % of consumption should react to price variations, which is
not negligible, the choice of the appropriate price specification remains a relevant issue.
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