Estimating Residential Water DemandUnder Block Rate Pricing: A Nonparametric Approach

Céline NaugesLEERNA-INRA, Toulouse and University College London1

Richard Blundell2

University College London and Institute for Fiscal Studies

Abstract

In this paper we use exogenous variation in the block structure and tari� ratesto nonparametrically estimate the price and income elasticities of demand for waterunder nonlinear pricing. The data was collected at the household level on water price,water consumption and household characteristics. Cypriot households in di�erentareas were charged a di�erent gross price for water units and also a di�erent blockstructure. We present results that show important biases in parametric structuralmodels and more reduced form approaches.

1This research was completed while the �rst author was a fellow at the Department of Economics atUniversity College London. The European Commission is gratefully acknowledged for the �nancial sup-port provided (Grant HPMFCT-2000�00536). The authors also wish to thank Phoebe Koundouri forprovision of the data.2Address for correspondence: University College London, Department of Economics, Gower Street, LON-DON WC1E 6BT, UK. Tel: 44 (0)20 7679 5863, Fax: 44 (0)20 7916 2773, e-mail: r.blundell@ucl.ac.uk.

1

1 Introduction

Nonlinear tari�s are common in the pricing structure of utilities. In particular, in the

electricity and water supply industries where there exist substantial �xed costs in pro-

duction. The impact of such nonlinear pricing on consumer demand and on consumer

welfare will depend on the precise nature of income and substitution elasticities. However

with nonlinear pricing the marginal price is endogenous to consumer demand and the un-

biased estimation of price and income e�ects is di�cult. This problem is found elsewhere

in empirical microeconomics, in particular in the study of labour supply with progressive

piece-wise linear income taxation (see, among others, Burtless and Hausman 1978 or Mof-

�tt 1986, 1990). The standard approach to estimation of preferences is to use maximum

likelihood, however this is sensitive to assumptions on the form of preferences and the dis-

tribution of unobservable tastes. Instead one can adopt a semiparametric approach that

exploits the structural features of the optimal choice problem without imposing strong

distributional and preference restrictions. Blomquist and Newey (2002) have recently pro-

posed a nonparametric approach for the labour supply problem and we adapt it here for

the study of nonlinear pricing.

Nonlinear tari�s are now more and more frequently applied by water authorities

and the choice of the relevant price to include in the demand function is still a source

of controversy. The most common approach, following the work of Taylor (1975) and

Nordin (1976), is to include in the demand function, in addition to the marginal price

of water, the di�erence variable (de�ned below) in order to take into account the in-

tramarginal rate structure. This speci�cation has been combined to the use of either

least squares or instrumental variables techniques to estimate water demand functions by

1

Billings and Agthe (1980) and Nieswiadomy and Molina (1989) among others. Hewitt

and Hanemann (1995) apply maximum-likelihood to the estimation of residential water

demand. Despite the di�erence in terms of the price variables included in the demand

function and the econometric techniques employed, almost all studies agree on the in-

elasticity of residential demand (price-elasticity is found signi�cant but less than one in

absolute value) whatever the region considered (United States or Europe) and the type

of data used (cross-sectional, time-series or panel data).

Our application is to a nonlinear pricing `experiment' in Cyprus. In this data, house-

holds in di�erent areas were charged a di�erent gross price for water units and also a

di�erent block structure. The gross price was determined by local conditions (water

availability and quality, density of population, delivery costs etc.) and we will assume

this is exogenous for preferences over water demand. The marginal price could take three

values for any consumer. The block structure is a simple piecewise linear convex pricing

rule which we describe in detail below. The rule changed across consumers according to

the area in which they were resident, but the marginal price paid will depend on the actual

block chosen which depends on the quantity consumed. This exogenous variation across

households in the block structure and gross price is a signi�cant advantage in recovering

consistent estimates of the underlying preference parameters.

We begin the paper with a brief outline of nonlinear tari�s and consumer demand. We

then go on to outline the maximum likelihood approach and the nonparametric alternative.

Section 3 presents the data which is collected at the household level on water price, water

consumption and household characteristics. Section 4 presents our results where we show

the biases that can occur when simple methods or strong distributional assumptions are

adopted in estimation.

2

2 Nonlinear Tari� Structure

2.1 Theoretical Issues

It is well known, see Hall (1973) for example, that piecewise-linear budget constraints due

to multiple-block pricing raise issues of model speci�cation and econometric estimation

because, contrary to traditional consumer demand analysis, the demand function for a

good facing block rate pricing is typically nonlinear, nondi�erentiable and often includes

discrete jumps.

Let us consider the simplest case of a nonlinear tari� structure with two blocks.1

Suppose that a consumer with income I maximizes a strictly quasi-concave utility function

U(q, z) where q could represent water and z a composite good. Price of z is normalized to

1 and water is sold under a two-block rate tari� that can be increasing or decreasing. Let

us call pj, j = 1, 2 the water price in the jth block and l1 the limit of the �rst block. The

budget constraint is de�ned by two linear segments and the budget set can be described

by the following conditions:

I =

p1q + z q ≤ l1

p1l1 + p2(q − l1) + z q > l1

(1)

or equivalently:I = p1q + z q ≤ l1

I + (p2 − p1)l1 = p2q + z q > l1.(2)

The term y = I + (p2 − p1)l1 is the virtual income whereas D = (p2 − p1)l1 is known

as the di�erence variable, see for example Nordin (1976). The latter is de�ned as the dif-

ference between the bill if all units had been charged at the marginal rate and consumer's

actual payment. It follows that the di�erence variable is positive under increasing block1Extension to a multiple block rate tari� is straightforward.

3

rates and negative under decreasing ones. This variable is thus often interpreted as an

implicit income tax under a decreasing block pricing schedule and as an implicit income

subsidy under increasing block rates (see Figure 1 for a representation of the budget set

under nonlinear tari�s).

[Insert Figure 1 here]

So multi-block tari�s generate budget sets that di�er in two ways from the `conven-

tional' budget set:2 the budget constraint is clearly nonlinear and the budget set may be

nonconvex. Demand curves cannot be derived analytically and be obtained directly as

close-form expressions from �rst-order conditions of utility maximization. Moreover, the

resulting demand functions may be kinked (point of nondi�erentiability) or discontinuous

with jumps from one segment of the budget constraint to another.3 As it is frequently the

case with highly nonlinear models, maximum likelihood has been an attractive estimation

method because it generally provides estimates with the desirable large-sample properties

of consistency, asymptotic normality and asymptotic e�ciency. However, maximization

of the likelihood function (see Blundell and MaCurdy 1999, for a review) requires dis-

tributional assumption and is subject to speci�cation error when these assumptions are

invalid.2The budget set that derives from the `traditional' assumption that consumers face a constant price

which is independent of the quantity demanded.3Nondi�erentiability comes from the fact that several consumers may be at a kink with di�erent

marginal willingnesses to pay. Moreover, small changes in price or income may create discontinuitiesbecause consumers may switch from one segment or kink to another. In the case of a nonconvex budgetset, the nonlinearity of the demand function is enhanced by the fact that it may be multi-valued. It canbe the case when an indi�erence curve is simultaneously tangent to the piecewise-linear budget constraintat two or more points.

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2.2 The Maximum-Likelihood Approach

This approach is based on the full representation of the agent's behaviour. The choice

of the block (discrete choice) and the choice of the level of consumption within this

particular block (continuous choice) are modelled. This two-step choice model leads to

distinguish between conditional and unconditional demand functions. The conditional

demand function is the function that states the quantity of water used conditionally to

the choice made by the user to be inside one particular block or at a kink. In a two-block

setting, conditional demand is de�ned as:

q =

q∗(p1, y1) if q < l1l1 if q = l1q∗(p2, y2) if q > l1

(3)

where (p1, p2) and (y1, y2) are respectively the prices and virtual incomes in block 1 and 2.

The agent then determines the block of consumption maximising a global utility function

that depends on conditional indirect utility functions. The unconditional Marshallian

demand is �nally obtained as the combination of the discrete and continuous choices:

q =

q∗(p1, y1) if q∗(p1, y1) < l1l1 if q∗(p2, y2) ≤ l1 ≤ q∗(p1, y1)q∗(p2, y2) if q∗(p2, y2) > l1.

(4)

See Burtless and Hausman (1978), Hausman (1985), Mo�tt (1986, 1990) and Hewitt and

Hanemann (1995) for a more detailed derivation of the unconditional demand function.

When coming to the econometric speci�cation, it is common practice since Burtless

and Hausman's (1978) paper to consider two sources of error: a �rst error term that

accounts for heterogeneity in preferences (called from now on ε) and an optimization error

5

(η) which accounts for the discrepancy between desired and observed level of consumption.

More precisely, the �rst error term will allow for a distribution of preferences in the

population or in other words, we allow households to have di�erent parameters in their

utility and demand functions.4 The household makes a utility maximisation decision to

consume a particular level of water q but, because of unexpected variations in water used

(due to meter errors or leaks for example), the observed consumption level may di�er.

So, the econometric model reads:

q =

q∗(p1, y1) + ε + η if ε < l1 − q∗(p1, y1)l1 + η if l1 − q∗(p1, y1) ≤ ε ≤ l1 − q∗(p2, y2)q∗(p2, y2) + ε + η if ε > l1 − q∗(p2, y2).

(5)

The likelihood for one observation can be derived from this last expression, under the

assumption of normality and homoskedasticity of the error terms.

However, the use of maximum-likelihood can appear di�cult in some cases, mainly be-

cause the likelihood function is not globally concave. Some of the problems encountered in

previous studies using maximum-likelihood to estimate piecewise-linear-constraint mod-

els are listed in Mo�tt (1986). Among others, this author mentions the sensitivity of

the maximum-likelihood estimates to various forms of speci�cation error or the normality

assumption that may be a poor approximation in small samples.

2.3 A Nonparametric Approach

Blomquist and Newey (2002) propose a nonparametric approach to estimate demand

functions under nonlinear budget sets. The basic idea is to think of the choice (e.g. hours4This speci�cation will allow both parameters to be identi�ed.

6

of work or cubic meters of water in the present case) as being a function of the entire

budget set that will be characterized by the intercept and slope of each segment and

the limits of each block. They argue that this approach has the advantage of being less

sensitive to the functional form as well as being not technically challenging (it requires

the use of least squares only). The suggested method is to estimate the conditional mean

of water used given the budget set:

E[qi|xi] = q̄(xi) (6)

where qi is the quantity of water consumed by household i (i = 1, . . . , n) and xi represents

his budget set. It is important to note that, in this case, distributional assumptions on

the errors are not required. As mentioned before, the budget set is fully described by the

slopes of each segment (p1, p2, . . ., pJ) (J ≥ 2), their intercepts or virtual incomes (y1, y2,

. . .,yJ), and the limits of each block (l1, l2, . . ., lJ−1). Blomquist and Newey show that, if

the budget set is convex,5 the dimensionality of the problem can be reduced if we assume

the following:

(A1) desired consumption for a linear budget set is given by q∗ = π(y, p, v)where v is unobserved and represents individual heterogeneity,

(A2) v is statistically independent of the budget set,(A3) π(y, p, v) is strictly increasing in v and,(A4) the probability of no consumption is zero.

Under these assumptions, the conditional mean of consumption can be written as

q̄(x) = π̄(yJ , pJ) +J−1∑j=1

[µ(yj, pj, lj)− µ(yj+1, pj+1, lj)] (7)

where

µ(y, p, l) =

∫ π−1(y,p,l)

−∞[π(y, p, v)− l]g(v)dv,

5From now on, we will consider the case of a convex budget set only.

7

g(v) being the density of v. The conditional mean of water consumption is thus decom-

posed as the sum of the average consumption for a linear budget set, π̄(yJ , pJ), and a

second term that corrects for the nonlinearity.6

One approach to implement this method is to use series as a nonparametric estimation

technique to estimate model (7). In our case, a series estimator will be the predicted value

from a regression on some approximating functions for p and y. Let x = (y1, p1, . . . , yJ , pJ)

be the vector of virtual incomes and prices. Let wK(x) = (w1K(x), . . . , wKK(x))′ be

a vector of K approximating functions. For data (xi, qi), (i = 1, . . . , n), let W =

(wK(x1), . . . , wK(xn))′ and Q = (q1, . . . , qn)′. A series estimator of q̄(x) is given by

q̂(x) = wK(x)′β̂ with β̂ = (W ′W )−W ′Q (8)

where − denotes any symmetric generalized inverse. We will consider in the application

power series as approximating functions (spline would have been another possible tech-

nique). Power series are formed by choosing the elements of wK(x) to be products of

powers of the individual components of x. They are easy to compute and have good

approximation rates for smooth functions, although they may be sensitive to outliers in

x, and can be highly collinear (Hausman and Newey 1995).

The �rst term in equation (7), π̄(yJ , pJ), will be approximated by products of powers of

yJ and pJ . Approximating series for the second term,∑J−1

j=1 [µ(yj, pj, lj)−µ(yj+1, pj+1, lj)],

will use di�erences of powers of yj, pj (j = 1, . . . , J) and lj (j = 1, . . . , J − 1). Series

estimator are sensitive to the number of terms K chosen in the approximating function.

In the empirical application, we will choose the number of terms to minimize the following6As pointed out by Blomquist and Newey, when π(y, p, l) is linear in v, this correction term is exactly

analogous to the Heckman (1979) correction in the sample selection model.

8

cross-validation criterion:

CV =1

n

n∑i=1

(qi − q̂(x−i)

)2 (9)

where q̂(x−i) is the least squares coe�cient computed from all the observations except

the ith.

3 Dataset Description

The sample considered in this paper has been built from a 1997 survey of 2,700 randomly

selected Cypriot households who have been questioned about their family characteristics

(size, occupation), housing (size, urban/rural environment, equipment) and their water

expenses in 1997. The households surveyed belong to di�erent water authorities areas,

each having its own tari� structure. Accurate information on prices charged in each zone

was obtained and was then used to recover water consumption of each household. We

select from this sample the 1,686 households who face a three-block rate tari�.7 All tari�

structures (in the 36 water authorities) are increasing so the convexity of the budget set

is guaranteed. Another advantage of this dataset is the variation in the shapes of the

budget sets, not common in labour taxation analyses.

We report some descriptive statistics in Table 1.

[Insert Table 1 here]

In this table the �xed charge corresponds to a fee paid by households whatever their

level of consumption is.8 In addition to this �xed fee consumers are charged for each cubic

meter used. From now on, marginal price is the price of the last unit of consumption and7We assume that this type of selection will not create any estimation bias.8This fee should be designed in order to cover �xed costs associated with water delivery such as

maintenance of meters or issuing of bills.

9

average price is the ratio of total water bill over total water consumption. p1, p2 and p3

are respectively the unit prices in the �rst,9 second and third block. The limits of the �rst

and the second block are denoted by l1 and l2. y1, y2 and y3 are virtual incomes (reported

income + di�erence variable) for users respectively in block 1, 2 and 3.

We observe a large span of water bills ranging from a total monetary amount of 9

to 360 CYP/year. Households in the sample use an average of 115 m3/year which is

in the range of the average domestic consumption in most European countries (OECD,

1999). Consumption varies between 40 and 546 m3 a year. Apart from measurement

errors, extremely low and high observed consumptions could be explained respectively by

the reliance on other sources of water supply (e.g. wells) and the presence of leakages.

Surveyed households di�er also in terms of income (from 61 to 177,929 CYP/year), size

(from one to sixteen members in the household), residence size (from 15 to 600 squared

meters) and equipment in the house. If almost 100% of the households surveyed are

supplied with electric current and hot water and own a fridge and a washing machine,

a reduced proportion of houses are equipped with central heating (32% of the sample),

air conditioning (26%) and dish washer (33%). The average price charged for one cubic

meter is respectively 0.51, 0.70 and 0.88 in block 1, 2 and 3. Prices vary from one area

to the other, ranging from 0.20 to 1.10 CYP/m3. On average, the marginal price charged

to residential users is 0.81 CYP/m3 (this corresponds to 1.20 US Dollars). This price is

in the range of the water prices charged in southern European countries and in general

lower than prices faced by residential users in northern Europe.10 We �nally note that the

three virtual incomes (y1, y2 and y3) corresponding to the three blocks are almost equal.9The �rst block corresponds to a low level of consumption.

10A comparative study of residential water prices made by the OECD (1999) gives the following �gures:Belgium (2.06 US Dollars per cubic meter in 1997); France (3.11 USD in 1996); Great-Britain and Wales(3.11 USD in 1998); Italy (0.84 USD in 1996); Spain (1.07 USD in 1994); Sweden (2.60 USD in 1998).

10

This follows from the relative small share of water expenses in overall income (0.006 on

average in the sample).

We report in Table 2 the repartition of households between the three blocks and some

average values for households' characteristics in each block. The share of the population

in each block is: 9% in the �rst block, 21% in the intermediate block and 70% in block 3.

The number of members in the household is in average greater for those in block 3. Those

households are also characterized by a bigger average size of their residence and a higher

average income (gross income as well as income per adult). Equipment in central heating,

washing-machine and air-conditioning is also more frequent in the third block. Finally,

the location in urban or rural areas does not seem to be linked with the choice of the

block.

[insert Table 2 here]

4 Estimation Results

4.1 Nonparametric Estimation

In this section we implement the nonparametric series estimator. In addition to price and

income we include as demand shifters the number of household members (SIZE), the area

of the house as measured in squared meters (SQRMT) and a dummy for washing-machine

ownership (WM). These three variables are added linearly to the model.

Tables 3 and 4 give the results of the series estimation. At each step we include addi-

tional variables (see column 2 of Table 3) and we compute three goodness-of-�t criteria:

the cross-validation criterion (see section 2.3 for its de�nition), the Fisher-statistic (F-

stat) which tests for the signi�cance of extra terms and the Mean Sum of Squared Errors

(MSSE). For notational simplicity, in the presentation of our results, we write lm4yrps

11

in place of∑

j lmj (yrjp

sj − yr

j+1psj+1) and so on. We stop our addition of terms in the series

at step 7 where the parameter estimates and the various criteria stabilize. In Table 4,

the mean price [resp. income] elasticities are reported. Elasticities have been computed

for each household at the price and income of the last block. Standard errors have been

computed using the Delta method (Kmenta, 1986).

[Insert Tables 3 and 4 here]

The Fisher-statistic and the Mean Sum of Squared Errors (MSSE) decrease as new

variables are added in the model and as one would expect the standard error associated

with the price elasticity is increasing with the number of terms. However, the �nal

estimate of the price elasticity is �.31 which seems plausible and is in the range of estimates

provided by previous European studies (even if very few household-level data have yet

been used). Income elasticity is estimated at .36.

4.2 Alternative Estimators

We present in this section the results of parametric techniques `usually' applied to the

estimation of water demand. We follow here the common approach: the demand equation

is �tted through a log-log linear relationship between water used by the household on

the left-hand-side and price, income and other demographics on the right-hand-side. To

make things comparable with the nonparametric model, we include as extra explanatory

variables squared price and income as well as the cross-product of these two variables.

In each case household consumption is written as a function of the price and virtual

income corresponding to the block matching observed water use. As mentioned in the

introductory section the nonlinearity in prices is then accounted for either by including

12

in the model a di�erence variable or, which is equivalent, by considering virtual income

(income plus di�erence variable) instead of reported income.

[Insert Table 5 here]

We present in Table 5 the results of Ordinary Least Squares (OLS, henceforth), Two

Stage Least Squares (2SLS) and Maximum-Likelihood (ML) estimates. For 2SLS estima-

tion we instrument marginal price and di�erence by the prices of each block, the �xed

charge, the income and the three demand shifters SIZE, SQRMT and WM. The predicted

values for the marginal price and the di�erence variable are then used in the second stage

in which the demand function is estimated. In all models consumption, prices and incomes

have been transformed into logarithms. For the ML estimation (see derivation of the log-

likelihood in Appendix), we use the 2SLS estimates as starting values. Convergence was

achieved using the method proposed by Berndt et al. (1974).11

Most of the variables are signi�cant in the three models and have the expected signs.

Price (in level, squares and through the cross-product with income) is found highly signi�-

cant whatever the econometric method is, but the estimated coe�cients are quite di�erent

from one model to another. This discrepancy is an evidence for price endogeneity. The

OLS method clearly produces biased estimates as the endogeneity of price is not corrected

for. Income (in level and in squares) is not signi�cant in any of the models, it only comes

out through its interaction with price. The positive impact of the household size, the

house surface and the ownership of durables on the expected level of water used is con-

�rmed through the three models, the magnitude of these e�ects being also quite similar

from one model to another. The results in the ML column also show that more of the11It is a scoring method that uses the cross-product of the matrix of �rst derivatives to estimate the

Hessian matrix.

13

unexplained variability is due to the inability to characterize heterogeneity in preferences

(σ̂η < σ̂ε). This could be explained by the omission of variables such as age and occupa-

tion of the members of the household, education level of the head of the household, use

of water for sprinkling etc. that were not available in the database but are often proved

to be determinants of water consumption. Ideally the observation of those households on

successive time periods would allow us to explicitely account for unobserved heterogeneity.

As the consistency of the ML approach largely relies upon the assumption made on

the error term, we propose to test whether disturbances are normally distributed if the

disturbances are replaced by the residuals. We follow the methodology of Jarque and

Bera (1980) and use the critical values tabulated by Deb and Sefton (1996). The test

statistic is:

LMN = N( 1

24(b2 − 3)2 +

1

6(√

(b1)2)

(10)

where√

(b1) is the sample skewness statistic and b2 is the sample kurtosis. This statistic

is found equal to 487.92 when using residuals after ML estimation from our data. This

value is far above the critical value at the 5% level of signi�cance so we reject the null of

normality of the error term, making di�cult to rely upon ML estimates.

We report in Table 6 the estimated elasticities obtained from the parametric and

nonparametric estimators. To make things comparable with results obtained from the

nonparametric approach, we compute price and income elasticities for each household

at the price and income of the last block. We report the mean, median, 25% and 75%

quantiles of the distribution of price and income elasticities.

[Insert Table 6 here]

14

The preferred nonparametric estimators show an important di�erence with parametric

maximum likelihood and much simpler (but inconsistent) estimation approaches. As one

would expect they also show a reduction in precision although the estimated elasticities

remain signi�cant. On our sample, we �nd that the traditional parametric approaches

(OLS, 2SLS and ML) lead to an over-estimation of the price elasticity of demand and

to an under-estimation of income elasticity. The mean price elasticity derived from the

nonparametric approach (step 7) is found equal to −0.31, which is lower, in absolute

value, than the elasticities computed from the parametric models (from −0.46 using 2SLS

to −0.38 using OLS). Using the �exible nonparametric approach, 50% of the households

in the sample are found to have a price elasticity between −0.42 and −0.19 whereas the

dispersion is larger when using the ML technique (−0.94 is the 25% quantile and −0.04

is the 75% quantile). Income elasticity is found highly biased in parametric models:

income elasticity is estimated at .10 whatever parametric technique (OLS, 2SLS or ML)

is used while it is estimated at 0.36 in the nonparametric model. In the latter, 50% of the

households in the sample exhibit an income elasticity ranging from 0.21 to 0.49.

When the normality assumption is rejected as in the present study, we have shown

that ML can produce biased estimates. The �exible nonparametric approach is thus an

easy way to avoid assumptions about error terms and demand functional form. The

nonparametric approach also provides some insights on the way prices and income a�ect

demand. In our case the best nonparametric model includes more complex interactions

of prices and income than the common and quite restricted demand equations speci�ed

when parametric models are used.

15

5 Summary and Conclusions

This paper has used variation in the block structure for water pricing across households

in Cyprus to derive nonparametric estimates of price and income elasticities. In this data

households in di�erent areas were charged a di�erent gross price for water units and also

a di�erent block structure. The gross price was determined by local water speci�city and

the block structure is a simple piecewise linear convex pricing rule which we describe

in detail. The rule changed across consumers according to the area in which they were

resident, but the marginal price paid depended on the actual block chosen which depends

on the quantity consumed. This `quasi-experimental' variation across households in the

block structure and gross price were shown to provide a signi�cant advantage in recovering

consistent estimates of the underlying preference parameters.

The nonparametric approach as proposed by Blomquist and Newey (2002) provides

an easy way to avoid assumptions both on the functional form and on the distribution of

the error terms. As shown in the present study, if these assumptions are not correct then

the estimates derived from parametric methods such as the preferred ML can be severely

biased. Thus, as we generally do not know the exact form of the relationship between

demand, price and income, the nonparametric should always be the approach to consider.

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Kmenta, J. (1986): Elements of Econometrics. New York: Macmillan Publishing Com-pany.

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(1990): �The Econometrics of Kinked Budget Constraints,� Journal of EconomicPerspectives, 4(2), 119�139.

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17

OECD (1999): �Household Water Pricing in OECD countries,� Paris: OECD.

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18

Appendix: derivation of the log-likelihood

We present the derivation of the log-likelihood function in a three-blocks setting, which isthe case corresponding to our particular dataset. A log-linear form is chosen for the con-ditional demand. Denoting by Z a vector containing the constant and some demographicvariables, by (p1, p2, p3) and (y1, y2, y3) respectively the prices and virtual incomes in thethree blocks, we have the following econometric model:

ln(q) =

Zδ + α ln(p1) + µ ln(y1) + ε + η if ε < ln(l1)− Zδ − α ln(p1)− µ ln(y1)ln(l1) + η if ln(l1)− Zδ − α ln(p1)− µ ln(y1) ≤ ε

and ε ≤ ln(l1)− Zδ − α ln(p2)− µ ln(y2)Zδ + α ln(p2) + µ ln(y2) + ε + η if ln(l1)− Zδ − α ln(p2)− µ ln(y2) < ε

and ε < ln(l2)− Zδ − α ln(p2)− µ ln(y2)ln(l2) + η if ln(l2)− Zδ − α ln(p2)− µ ln(y2) ≤ ε

and ε ≤ ln(l2)− Zδ − α ln(p3)− µ ln(y3)Zδ + α ln(p3) + µ ln(y3) + ε + η if ε > ln(l2)− Zδ − α ln(p3)− µ ln(y3).

(δ, α, µ) are unknown parameters. Assuming that ε and η are independent normal vari-ables with zero mean and respective variances σ2

ε and σ2η, that gν,ε(ν, ε) with ν = ε + η is

a binormal distribution, the corresponding log-likelihood is:

ln L =∑

i ln{

exp(−w21/2)

σνΦ(r1) +

exp(−u21/2)

ση[Φ(t2)− Φ(t1)] +

exp(−w22/2)

σν[Φ(r3)− Φ(r2)]

+exp(−u2

2/2)

ση[Φ(t4)− Φ(t3)] +

exp(−w23/2)

σν[1− Φ(r4)]

}where

wk =[ln(q)− Zδ − α ln(pk)− µ ln(yk)

]/σν

u1 = [ln(q)− ln(l1)]/ση

u2 = [ln(q)− ln(l2)]/ση

t1 = [ln(l1)− Zδ − α ln(p1)− µ ln(y1)]/σε

t2 = [ln(l1)− Zδ − α ln(p2)− µ ln(y2)]/σε

t3 = [ln(l2)− Zδ − α ln(p2)− µ ln(y2)]/σε

t4 = [ln(l2)− Zδ − α ln(p3)− µ ln(y3)]/σε

r1 = (t1 − ρw1)/√

(1− ρ2)

r2 = (t2 − ρw2)/√

(1− ρ2)

r3 = (t3 − ρw2)/√

(1− ρ2)

r4 = (t3 − ρw3)/√

(1− ρ2)σν =

√σ2

ε + σ2η

ρ = σε/σν

with ρ the correlation coe�cient between ε and ν.

19

Tables and Figures

Table 1: Descriptive statisticsVariable Mean Std Dev Minimum Maximum

water bill (CYP/year) 59.17 43.57 9.00 360.00consumption (m3/year) 115.22 51.08 40.00 546.43gross income (CYP/year) 15,108.95 12,388.17 61.00 177,929.00average price (CYP/m3) 0.51 0.24 0.11 1.01marginal price (CYP/m3) 0.81 0.30 0.22 1.10�xed charge (CYP/year) 3.06 1.66 0.00 6.00p1 (CYP/m3) 0.51 0.20 0.22 0.70p2 (CYP/m3) 0.70 0.22 0.34 0.90p3 (CYP/m3) 0.88 0.24 0.46 1.10y1 (CYP/year) 15,095 12,388 47 177,911y2 (CYP/year) 15,098 12,388 49 177,913y3 (CYP/year) 15,105 12,388 52 177,917l1 (m3) 71.08 29.11 50.00 120.00l2 (m3) 90.84 42.10 60.00 160.00CH∗ 0.32 0.47 0.00 1.00WM∗ 0.94 0.24 0.00 1.00DWM∗ 0.33 0.47 0.00 1.00FR∗ 0.99 0.08 0.00 1.00HOTWTR∗ 0.97 0.17 0.00 1.00AC∗ 0.26 0.44 0.00 1.00ELECTR∗ 0.99 0.08 0.00 1.00URBAN∗ 0.87 0.34 0.00 1.00SIZE 4.26 2.43 1.00 16.00ROOMS 5.47 1.55 1.00 15.00SQRMT 144.95 62.35 15.00 600.00Notes: CYP is for Cyprus pounds (1 CYP is currently around 1.50 US Dollars).∗ indicates a dummy variable.CH: equal to one if central heating, WM: equal to one if washing machine,DWM: equal to one if dish washer, FR: equal to one if fridge,HOTWTR: equal to one if hot water, AC: equal to one if air-conditioning,ELECTR: equal to one if electricity, URBAN: equal to one if urban area,SIZE: size of household, ROOMS: number of rooms,SQRMT: size of the house in squared meters.

Table 2: Households' characteristics in each blockBlock 1 Block 2 Block 3

frequency 159 (9.4%) 352 (20.9%) 1,175 (69.7%)

gross income 11,111 12,644 16,388income/adult 4,135 4,236 4,983CH 0.16 0.17 0.39WM 0.81 0.93 0.96AC 0.19 0.24 0.27URBAN 0.82 0.93 0.86SIZE 3.42 3.89 4.49SQRMT 115.10 127.26 154.29

Table 3: Series estimation - Goodness of �t criteriaStep Additional Cross F-stat(1) MSSE(2)

variables Validation

1 constant, p3, y3 2,009 103.582 (�) 1,999.932 4p, 4y 1,892 94.135 (0.00) 1,880.953 l4y, l4p 1,894 73.672 (0.17) 1,879.264 y2

3, p23 1,893 60.747 (0.12) 1,876.72

5 4y2, 4p2 1,899 51.686 (0.20) 1,875.356 p3y3 1,899 48.301 (0.07) 1,872.727 4yp 1,900 45.203 (0.21) 1,872.051,686 observations.(1) We report in parentheses the p-value of the F-statistic for the test ofsigni�cance of the extra terms.(2) Mean Sum of Squared Errors.

Table 4: Series estimation - Estimation of elasticitiesStep Additional Price Std Err. T-stat Income Std Err. T-stat

variables elasticity elasticity

1 constant, p3, y3 �0.7941 0.0354 �22.4504 0.0789 0.0129 6.09872 4p, 4y �0.2928 0.0600 �4.8775 0.3644 0.0670 5.43713 l4y, l4p �0.3736 0.1204 �3.1041 0.3673 0.1661 2.21064 y2

3, p23 �0.2377 0.1350 �1.7604 0.3520 0.1733 2.0317

5 4y2, 4p2 �0.2990 0.1355 �2.2076 0.3472 0.1760 1.97226 p3y3 �0.2851 0.1363 �2.0922 0.3496 0.1780 1.96357 4yp �0.3137 0.1379 �2.2754 0.3581 0.1798 1.99161,686 observations.

Table 5: Parametric estimationsOLS 2SLS ML

Est Coef Std Err(1) Est Coef Std Err(1) Est Coef Std Err

constant 3.1714 0.5057 3.6268 0.4570 3.3063 0.8039price �1.5430 0.1856 �1.4389 0.2722 �1.9397 0.3497income 0.1156 0.1115 �0.0012 0.1019 0.0553 0.1764price x price �0.3949 0.0451 �0.0437 0.0925 0.7425 0.0789income x income 0.0008 0.0062 0.0069 0.0057 0.0041 0.0096price x income 0.0990 0.0183 0.1027 0.0252 0.1876 0.0365household size 0.0248 0.0042 0.0257 0.0039 0.0256 0.0044residence area 0.0003 0.0002 0.0006 0.0002 0.0007 0.0002washing machine 0.1266 0.0422 0.1284 0.0323 0.0785 0.0463

σε + ση 0.1293 � 0.1122 � � �σε � � � � 0.3031 0.0133ση � � � � 0.1900 0.0130

Mean LL 0.7735LM test of normality(2) 487.92 (0.000)1,686 observations.(1) robust standard errors.(2) critical values from Deb and Sefton (1996), p-value in brackets.

Table 6: Comparison of elasticity measuresOLS 2SLS ML NP-step 7

Price elasticitymean �0.3827 �0.4565 �0.4522 �0.3137median �0.5475 �0.4472 �0.2909 �0.314825% quantile �0.6570 �0.5015 �0.9372 �0.417475% quantile �0.0990 �0.3984 �0.0383 -0.1947(Std error) (0.0280) (0.0315) (0.0378) (0.1379)

Income elasticitymean 0.1005 0.0976 0.0994 0.3581median 0.1201 0.1125 0.1218 0.303025% quantile 0.0711 0.0557 0.0327 0.205575% quantile 0.1396 0.1351 0.1508 0.4856(Std error) (0.0138) (0.0128) (0.0149) (0.1798)

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