We exploit a unique data set to estimate the degree of economies of scale in water consumption, controlling for the standard demand factors. We found a linear Engel curve in water consumption: each additional household member consumes the same water quantity regardless of household size, except for a single-person household. Our evidence suggests that the increasing block tariffs (IBT) structure, which is indifferent to household size, has unintended consequences. Large households, which are also likely to be poor given the negative correlation between income and household size, are charged a higher price for water. The degree of economies of scale found here erodes the effectiveness of IBT price structure as a way to introduce an equity consideration. This implication is important in view of the global trend toward the use of IBT.
 The main goal of this paper is to estimate the degree of economies of scale in urban water consumption exploiting a unique data set. We uncover the water demand structure with regard to household size. In particular, we explore what is the additional water consumption, as a result of an additional family member, for various household sizes.
 Water utilities and regulators in many countries are moving toward IBT (increasing block tariff) pricing structure. In this structure two or more prices for water are set, each price pertaining to consumption within a defined block. Efficiency and equity are the two commonly stated justifications for using that pricing structure. There are other arguments made in support of IBT such as discouraging wasteful use and promoting public health but these two intended justifications are commonly mentioned [see Whittington, 1992; Howe, 1998; Boland and Whittington, 2000; Organisation for Economic Co-operation and Development (OECD), 2003].
 Yet, IBT may not be an optimal pricing structure even after taking equity consideration into account [Bös, 1994]. Under certain conditions IBT is an optimal pricing structure. Martin et al  state that IBT is economically efficient and equitable if large consumers have stronger peak usage than small consumers. Boland and Whittington  claim that in practice IBT is likely to promote inefficiency, unfairness and revenue instability in developing countries. However they also suggest that IBT may indeed increase equity but it depends on the size of the first block.
 Assuming that the high rate equals roughly the marginal cost, IBT pricing structure implies that those consumers who are at the lower blocks may pay a price that is below the marginal cost of producing water. From a point of view of resource management, this is clearly inefficient and yet IBT is in place in a growing number of countries. Presumably, promoting equity consideration plays a role in using IBT. The low rate at the initial block ensures that a minimum quantity of water is affordable for poor families. Subsidizing water might not be the most desirable way to solve the efficiency/equity tradeoff but this is beyond the scope of this paper.
 The concept of equity here refers to one prominent dimension which is the incidence of water expenditures across the income distribution, although there are other dimensions of equity in water policy that are not covered here (see OECD  for a detailed discussion of other dimensions of equity). In this paper we provide empirical evidence on the extent of economies of scale in-household water consumption that allows us to examine the consequences of IBT on equity.
 When an IBT pricing structure is used, policymakers have to determine the range of water consumption at which the lowest price applies. This range may be a constant quantity of water or a function of household size. Whether the first block should be in accordance with household size depends on the assumptions underlying the normative model.
 Although there are considerable variations in water-pricing structure among the OECD countries, there is a general move away from fixed price and decreasing block tariff (DBT) pricing structures toward volumetric charging and increasing block tariffs [OECD, 1999]. As Table 1 shows, there has been a significant shift to increasing block tariffs in the United States over the past two decades. Only 4% of American utilities used IBT pricing policies in 1982; 36% did so in 2004. A survey of urban water utilities in Asia found that the majority of utilities in the sample used an IBT pricing structure [Asian Development Bank, 1993].
Table 1. U.S. Residential Public Water Supply Rate Structure, 1982–2004a
 A large variation exists among OECD countries that use IBT pricing policies in the way they treat household size in determining the first block. Typically, in the United States and other OECD countries that use IBT pricing structures, the first block is the same for households of all sizes. This feature of IBT pricing, contrasts with the equity consideration as stated above, if water consumption varies with household size. Large households tend to belong to the lower-income classes. In an IBT pricing scheme that uses the same first block regardless of household size, large households are pushed to pay a higher average price for water. Obviously, the more important economies of scale in water consumption are, the weaker the inequity effect is.
 Spain (Barcelona), Belgium (Flanders), and Greece (Athens), which use IBT pricing policies, take only partly household size into account in setting the price of the first or second block. However, it is hard to explain in terms of economies of scale why the first block is indifferent to household size for households of four individuals or less and increases at a linear rate for larger household sizes, as is the case in Spain (Barcelona). No empirical study known to us shows this pattern of economies of scale in water consumption. Thus insofar as economies of scale exist in water consumption, this pricing policy reflects cross subsidization among households of different sizes in comparison to a price structure that takes household size into account.
 Most households consist of four or fewer members. In Jerusalem, for example, more than 75% of households fall into this size cohort. In a typical developed city, the share of households of four or less is even larger and tends to exceed 90%. In 1999, around 90% of households in the United States consist of four persons or less [United Nations Economic Commission for Europe, 2004]. This share is even higher in European countries. For example, the share of households of four members or less in France, Belgium, and Finland are 92, 93, and 94%, respectively. The fact that the first block is not related to household size for most households may affect the actual incidence of water subsidy.
 Taking household size into account in setting the first block price in an IBT structure is an administrative challenge and may promote larger families to extent it reduces child rearing costs. It imposes a cost on water utilities by forcing them to update their databases continually as people move in and out. Therefore policymakers have to weigh the administrative cost against the equity consideration among other considerations. The tension between these two considerations is affected by the economies of scale in water consumption.
 Metering is an essential part of IBT pricing policy. Two large water companies in England and Wales have withdrawn metering expansion due to stiff opposition occasioned by the possibility of negative effects on low-income households with children. This example highlights the importance of the equity consideration in shaping water-pricing policy.
 These studies estimate the price elasticity while controlling for various demand factors such as spatial variables (e.g., temperature and rain), geographic variables, and demographic variables. A key demographic variable in many of these studies is household size. In those papers that include household size as one of their explanatory variables, it appears in a linear specification. Linear specification, however, ignores the potential of economies of scale in water consumption. This paper aims to fill the gap by estimating the economies of scale of water consumption.
 For this purpose, we use a unique data set composed of disaggregated data on households' water consumption, size, and other characteristics. The data pertain to all households in Jerusalem. The demographic structure of the population of Jerusalem is propitious for the estimation of economies of scale due to its large variation in the size of households.
 The next section briefly reviews the theory of consumer demand that faces piecewise linear budget constraints and the associated econometric issues. Section 3 describes the data. Section 4 presents the estimation results. Section 5 offers policy implications and concludes.
2. Theoretical Background
 Some water uses are family shared consumption (housecleaning, dishwashing, cooking and outdoor consumption) while other uses are private (such as toilets, showers, and drinking). The existence of economics of scale with respect to household size is assumed to be trivial, but the degree is an open empirical question.
2.1. Demand for Urban Water
 Consider a utility-maximizing household with income Y and a piecewise-linear convex budget such as that in Figure 1. For simplicity, assume that the consumer utility comes from water consumption, denoted by w, and from a composite good, c, the price of which is normalized to 1. The consumer faces three increasing price blocks. We define pi as the price of water in the ith block, w1 as the range of the first block, and w2 as the range of the second block. The budget constraint consists of three segments that are described by the following equations:
Where three block tariffs exist, there are two differences. It is by now common to define d2 = (p2 − p1)w1 as a difference for households that face p2. Therefore the virtual income of these households is equal to their actual income plus this difference. For households in the third block, the difference is equal to d3 = (p3 − p2)w2 + (p3 − p1)w1.
 The consumer may be located within one of the three segments or at one of the two kinks. A household's demand for water is determined, among other things, by its taste for water and its size. Since it is natural to expect larger households to have higher demand for water, large households are pushed into higher price segments unless the block range is related to household size. In this regard, the economies of scale are a key factor in shaping the demand for water, which is the focus of this paper.
 Increasing block tariffs generate a nonconventional reaction to price changes. A price change in the first block, for example, affects households in the first block but may have no effect on all other households given that water is not an inferior good (Figure 2). Therefore the magnitude of price elasticity depends on the distribution of households along the water consumption continuum.
2.2. Econometric Issues
 The problem of reverse causality that is arising here due to the simultaneous determination of the price of water (and virtual income) and the choice of water demand can be solved by constructing an instrumental variable. However, this method of 2SLS does not solve the problem of clustering. This is why maximum likelihood has become a standard tool in estimating the demand for water in the case of increasing block tariffs. Demand for water is a combination of choice of block (discrete choice) and choice of the level of water consumption within the block chosen (continuous choice). The following equation describes the demand for water:
 The joint probability includes the probability of continuous choice of water consumption and the conditional probability that the desired level of water consumption, given the existence of choice, lies at a particular kink or block. The use of a maximum likelihood method to maximize the joint probability elicits parameter estimates. The derivation of the log likelihood function that we use in the empirical section of this paper is shown in Appendix A. The maximization of likelihood function solves both endogeneity and the clustering of observations around the kinks.
 In general, maximum likelihood is a better method than two-stage least squares because 2SLS does not account for the potential clustering around the kinks. However, the pricing structure in Israel is such that both w1 and w2 depend positively on the size of lawn, up to a certain limit and on household size for those households that have 5 members or more. Thus all households have two kinks but its location varies considerably across households. As a result the kink location is almost a continuous variable.
 This feature, of course, reduces significantly the likelihood of clustering of households around at a particular kink location and as a result diminishes the advantage of ML technique. More importantly, our data set do not contain enough variation in water prices that reduces further more the reasoning of using ML. Several other problems are covered by Moffitt  such as the likelihood function is not globally concave and may be sensitive to starting values in the computation.
 In our case 2SLS estimation seems a more suitable technique because of the combination of low variation in prices, a large variation in quantities and the large variation in the location of kinks.
 In the empirical section we display the estimated economics of scale in the demand for water based both on ML and 2SLS. The results of ML are presented to show that that estimation technique in our case is not suitable unless an artificial variation in prices is introduced.
 The data used in this study cover all households in Jerusalem for the year 2003 (115,887 households, after excluding 64,720 observations for several reasons (mainly commercial consumers, but also shared meters, household larger than 12 individuals and household metered during part of the year). Our data set comes from three main sources: “Hagihon”, the only water supply company in Jerusalem; the Municipality of Jerusalem; and the Israel Ministry of the Interior. Most of the data originate with the Municipality of Jerusalem and were merged with household water consumption data from “Hagihon” and household size data from the Ministry of the Interior. Table 2 reports descriptive statistics.
Table 2. Descriptive Statistics
Apartment/home size, m2
Lawn size, m2
Share of households at P1
Share of households at P2
Share of households at P3
Number of units in building
Share of households below poverty line
 Most studies on demand for water tend to use samples rather than a whole population. Our database, composed of the almost all households in Jerusalem, allows us to estimate economies of scale in the demand for water. Jerusalem's highly diversified household structure makes this data set very attractive. It contains enough large households to permit precision in estimating the parameters: 22,658 observations of households with six individuals or more, including 692 households with 12 persons (Table 3). The average household size in Jerusalem is 3.65.
Table 3. Household Characteristics by Household Size
Consumption per capita, m3/yr
Apartment/home size, m2
Share of households below poverty line
Lawn size, m2
Number of units in building
Share of households at P1
Share of households at P2
Share of households at P3
Number of observations
 Average per capita water consumption in Jerusalem is 56 m3/year, ranging from 125 m3/year for single-member households to 33 m3/year for households of 12 members (Table 3). The overall average is 7% lower than that in the OECD countries, 60 m3/year (Figure 3).
 One of the limitations of our data is the absence of information on household current income. However, other characteristics in the set, such as apartment (or home) and lawn size, may be viewed as indicators of wealth. A lawn and its size are positively correlated with wealth in countries where the value of land is relatively high. It is particularly true for Jerusalem. However, a lawn may also reflect the taste for water (green view) for a given wealth.
 The number of apartments in the building is likely to be positively correlated with wealth and therefore we choose to use it as one of the wealth indicators. The wealth indicators, coupled with those of residential neighborhood and poverty line, may provide better information about households' permanent income. (In this paper, a household is below the poverty line if it is entitled to municipal tax deduction. This tax deduction is means tested and it is closely related to the formal poverty line in Israel). As Table 2 shows, the average size of an apartment (including private homes) in Jerusalem is 79 m2.
 As noted, the price structure is crucial in choosing the estimation technique. Israel is one of the pioneers in using IBT pricing where that structure was determined thirty years ago and has hardly been changed since then. Throughout the years the number of blocks (three), the range of each block and it relation to household and lawn size are almost identical. Though, the rates have been changed frequently especially during the high inflation period. All municipal authorities in the country use the same pricing policy.
 Jerusalem uses a three-block IBT pricing structure sets by the Israeli parliament. In 2003, the average price in the first block, applying to the first 96 m3, is $1.2/m3 including a sewage surcharge (hereinafter: P1). The price in the second block, for additional consumption up to 84 m3, is $1.5/m3 (hereinafter: P2). The charge for all extra consumption is $1.9/m3 (P3). The price of water in Jerusalem is slightly below the median in the developed countries (Figure 4).
 The pricing structure in Israel has two additional features. Households larger than four persons are entitled to an additional 36 m3 per person per year at a low price. Households with irrigated lawns are allowed an additional 0.6 m3 per m2 per year, up to 300 m3, at a low price (excluding sewage surcharge). The sewage charge is based on total water consumption except for irrigated lawns. A household with irrigated lawn is allowed for an additional quantity of water at a low price and that quantity is exempt from sewage charge regardless of the actual consumption. This additional quantity is granted during the period April–October. Every household has two kink points but each household has its own kink locations depends on household and lawn size. This price structure generates a very large variation in the location of kinks given the continuous nature of lawn size.
 The bimonthly water bill includes a table of the actual marginal rates and consumption quantity at each rate. Each apartment/private home in our data has its own water meter. Almost all consumers have an easy access to water meter but we do not have direct information on whether they read their meter nor their bill. In the empirical part we assume that customers are aware of the marginal prices.
 The use of yearly data raises the question of whether marginal or average price should be used in the estimation. Following Williams , who found that marginal price estimates are more reliable, and Nieswiadomy and Molina , who conclude that urban water consumers respond to marginal price when faced with IBT, we chose to use the marginal price. The marginal price in our study is defined as the highest price paid by the household during the year, even if it applied to one billing period (usually 2 months) only.
4. Estimation Results
 To estimate the extent of economies of scale in water consumption, the central question in this paper, we control for various factors of demand for water that are common in the literature. Those factors include the price of water, wealth indicators (apartment/home size, lawn size, number of apartments in the building, entitlement to municipal tax discounts), and geographical regions within Jerusalem. Obviously, the wealth indicators may capture additional demand factors, such as taste for water in the case of lawn size.
 The dependent variable in our regressions is yearly household water consumption in absolute terms (cubic meters). One expositional advantage of this specification is that the estimators are expressed in units of cubic meters. We also use a log-log specification to test the sensitivity of the scale economies estimates to the particular specification. These regressions appear in Table 4.
The full list of explanatory variables includes 178 regions within Jerusalem, 12 types of municipal tax relief, and number of apartments in the building. For expositional purposes we do not present their estimators. P values are shown in parentheses.
The +d stands for the present value of the difference which is translated into apartment size units by using the average price of an apartment in Jerusalem.
 The way we specify economies of scale in the regression is by using 11 dummy variables for each household size up to 12 members. The omitted variable is a single-person household. This specification permits a nonlinear relationship between water consumption and household size. We are particularly interested in the marginal consumption at each household size. To date, the literature on water demand shows household size in a linear specification. This assumes zero economies of scale.
Table 4 presents the results of three estimations: OLS, 2SLS, and maximum likelihood. As Table 4 shows, in the OLS estimation the additional consumption occasioned by an additional household member is roughly linear for households with more than four members. The economies of scale for households of four persons or less, in contrast, seem to be non negligible.
 However, the sign of marginal price is positive in the OLS model due to the endogeneity that IBT pricing policy in Jerusalem causes. This endogeneity generates a bias not only in the price estimator but also in all other estimators, including those of household size. The interaction between that endogeneity and pricing policy, regarding the additional quantity of water that households receive at a low price for each additional member, exacerbates the bias of the household size estimator. It produces artificial economies of scale for households of four members or less, as mentioned above.
 In 2SLS, we instrument the marginal price and the difference in the first stage by the observed characteristics of households. In the second stage, predicted marginal price and difference are used as explanatory variables in the estimated demand function. The 2SLS method solves the endogeneity problem and tends to remove these biases. Under certain conditions, 2SLS may be an even better technique for describing household behavior than the maximum likelihood method that we present below.
 In the 2SLS estimation, the sign of the marginal price estimator becomes negative, as theory would predict. The absolute effect of the highest price (P3) is greater than the effect of the intermediate price (P2), which is also consistent with the standard theory of demand.
 The sign of the marginal price is negative in the log-log specification as well (Table 5). The calculated price elasticity, based on this specification, is −0.18, falling into the lowest range that appears in the literature.
Table 5. Estimates of Water Demand Model: Log-Log Specificationa
The full list of explanatory variables includes 178 regions within Jerusalem, 12 types of municipal tax relief, and the number of apartments in the building. For expositional purposes, we do not present their estimators. P values are shown in parentheses.
Apartment/home size + d
Below poverty line
Number of observations
 The 2SLS estimation also produces plausible signs and magnitudes for all other estimators. Wealth indicators such as apartment/home and lawn size have positive and quantitatively large coefficients. Though, as noted before the coefficient of lawn may capture also the taste for water.
 A 10 m2 increase in apartment/home size results in additional water consumption of 12 m3. Households below the poverty line consume 13 m3 less than households that share the same characteristics but are over the poverty line.
 The main focus of our paper is on estimating the extent of economies of scale in water consumption. On the basis of the 2SLS results, the economies of scale in water consumption exist only up to two members household (Table 4). Beyond that the marginal water consumption of households follows almost a linear pattern: the additional consumption as a result of an additional member is relatively similar regardless of household size.
 We examine the linear pattern also by running a regression of the marginal water consumption (as a result of additional household member) on household size. The marginal water consumption is computed as the difference between the actual water consumption and the estimated water consumption for a household with the same characteristics but with one member less (based on the 2SLS in Table 4). We found that the marginal water consumption above one person is not (statistically significant) affected by household size.
 Marginal consumption, although by no means constant, fluctuates around 20 m3. The estimator of two persons implies additional consumption of 23 m3 compared with a single-person household. The marginal consumption of very large households is more volatile, ranging from 15 m3 for an 11-member household to 27 m3 for households of nine members.
 Note that the estimated consumption pattern reflects no economies of scale in households larger than two. The determination of the range of the first block in Israel is consistent with infinite economies of scale up to a household of four persons. No additional quantity of water is given at a low price as the household expands. In contrast, for households larger than four persons, the expansion of the first block with household size is in line with zero economies of scale. Notice that each additional member is entitled to 36 m3, much more than the estimated marginal consumption.
 The partial consideration of household size, in determining the range of water consumption at which the low price applies, results in cross subsidization among households of different size. It is also has unintended redistribution consequences due to the interaction between household size and income. For example, the predicted water consumption of a five member household in the 20th percentile is similar to a three member household in the 80th percentile (in the distribution of apartment/home size).
 The share of households that pay a marginal low price follows an inverted U shape (Table 3). Single-member households that pay a low price at the margin are the highest (32%). The share decreases in inverse proportion to household size up to five members. From that point on, the share rises commensurate with household size.
 We also estimated the economies of scale using the maximum likelihood method, as is common in the more recent literature on demand for water. Convergence was achieved by means of the dual quasi-Newton optimization. The marginal price estimator carries the expected negative sign but the magnitude is too large by any standard. It turns out that many of the estimators are not significant and have both an opposite sign and extremely large quantitative effect. In particular, the effect of household size, which is insignificant, follows a peculiar pattern.
 The problem with maximum likelihood estimation in our case is that there is not enough exogenous variation within price segments relative to the extremely large variation in quantities. To show that ML is indeed less suitable in our case we introduce artificial variation in prices. A random noise is added to the actual price at each of the three blocks. This assumes that consumers know the pricing structure but are uncertain about the exact price. The maximum likelihood estimation is based on a subsample of our data using 10,000 randomly selected households. The artificial price variation is generated by adding a randomly selected noise (from a uniform distribution between −4 and 4 and zero expected value) to each of the set of prices. Table 4 (column 4) shows that the estimators of maximum likelihood with enough variation in prices are again plausible. The estimators have both the expected sign and reasonable magnitude. In fact, most of the estimators are similar to those of 2SLS.
 It seems that 2SLS method is least limited in the absence of price variation and relatively small clustering around the kinks. The results of the 2SLS estimation turn out to be much more plausible than those elicited by the maximum likelihood estimation without adding artificial price variation. Therefore we draw our conclusions on the basis of the estimate brought forth by the 2SLS technique.
5. Policy Implications and Conclusions
 We used a unique data set, composed of disaggregated data on water consumption and household size, to estimate the economies of scale. We found that water consumption exhibits almost no economies of scale (with regard to household size) for households of two persons or more.
 Our study suggests that IBT pricing policy used in OECD countries clashes with one of its main stated justifications, the equity consideration. We have shown that IBT pricing structure forces large households to pay higher average prices for water (ceteris paribus). This outcome is inconsistent with the equity consideration because large households tend to belong to the lower income classes.
 One of the main motives for the use of an IBT pricing structure and, in particular, the low price range is to reflect the equity consideration. IBT may not be an optimal way to address equality concern but once that pricing structure is used as in many countries, our evidence implies that ignoring household size is self-defeating in this regard. The relatively low economics of scale in water consumption we found becomes more important in view of the general move away from fixed price and decreasing block tariff (DBT) pricing structures toward volumetric charging and increasing block tariffs [OECD, 1999].
 Taking household size into account in determining the range at which the low price applies is an administrative challenge because the need to continually update databases as people move in and out inflicts a cost on water utilities. Therefore policymakers have to weigh the administrative cost against the equity consideration given that IBT pricing structure is in place. In view of the real increase in household expenditure on water in many OECD countries, the importance of the equity consideration is on the rise.
Appendix A:: Derivation of Log likelihood
 This appendix describes the derivation of the likelihood function in a three-block setting that corresponds to our data set. The discrete continuous choice model is based on the Hausman model and the generalization of Moffitt . A linear form is chosen for the conditional demand.
 The econometric model is as follows:
a vector containing the following variables in addition to a constant: number of persons per household, low-income households that are entitled to discounts on municipal taxes, ownership of a lawn, lawn size, number of tenants per building, and geographical region;
(p1, p2, p3)
prices in the three blocks;
(y1, y2, y3)
virtual apartment sizes in the three blocks;
(δ, α, μ)
 Assumptions are as follows:
and fv, (v, ɛ) is binormal.
 The log likelihood is:
 We thank Ziv Bar-Shira, Yoav Kislev, Tal Otiker, and three anonymous referees for their helpful comments. We thank the participants of department seminar at the Faculty of Agriculture of the Hebrew University of Jerusalem. We are very grateful to “Hagihon,” which provided us with the household water consumption data for this research project, and the Israel Water Commission, which is funding the study.