Doing Better With Less: Implementing Peak-load Pricingfor Managing Residential Water Demand∗

Arnaud Reynaud†

May 12, 2010

Preliminary work. All comments welcome.

AbstractThe first objective of this paper is to determine under which conditions peak pricing canbe an effective tool for managing residential water demand under limited water availabilityconditions. To answer this question, we first develop a theoretical model allowing to assessthe impact of peak pricing on consumer’s surplus and on water consumption. The mainresult of this model is that, whatever the level of the price elasticity of water demandsduring the peak and the off-peak periods, it is always possible to find a system of peakan off-peak prices such as moving from a single price towards seasonal differentiated pricesresults in an increase in the aggregate consumer surplus and a decrease in the aggregatewater consumption. The second part of the paper provides an empirical application of thismodel on French data. We first estimate seasonal residential water demands, using data ona sample of local communities located in South-West of France. This allows us to check ifthe determinants of residential water demand in France vary according to the period of theyear. Based on these econometric estimations, we simulate the impact on consumer surplusof moving toward peak-load pricing. In the French case, peak prices do not result neitherin a substantial increase in the aggregate consumer surplus nor in a significant reductionof the aggregate water consumption. This result is due to the fact that the price elasticitydifference between the peak and the off-peak period is almost negligible in the French case.

Keywords : Peak-load pricing, Residential water demand, Renewable resource, Scarcity.

JEL’s codes : Q2, Q5, L9.∗This work is a part of the projects APPEAU funded by the program "Agriculture and Développement

Durable" from the Agence Nationale pour la Recherche (ANR) and EAUSAGE-quant funded by the program"Eaux et Territoires" and by the research program "Pour et Sur le Développement Regional". The authorsgratefully acknowledge support from the French Ministry of Environment, from the CNRS-EDD and from theMidi-Pyrénées region. We are also grateful to the Adour-Garonne Water Agency for providing data.

†TSE (LERNA-INRA), University of Toulouse 1, Manufacture des Tabacs - Bât.F, 21 allée de Brienne, F-31000Toulouse, e-mail: areynaud@toulouse.inra.fr. Tel: (33)5.61.12.85.12. Fax: (33)5.61.12.85.20.

Introduction

A number of countries have recently moved from an historical water policy based on supply man-agement toward more economic tools aiming at managing water demands1. This is in line witha global trend in favor of incentive-based instruments compared to the more classical “commandand control” approach.

Among the various economic instruments proposed for managing water demands (marginalcost pricing, block rate pricing, tradeable permits, etc.), it has been advocated that peak-loadpricing could be an efficient tools for managing water use conflicts that usually occur at somespecific seasons. There are two main motivations for implementing peak-load water pricing. First,the social value of water is likely to vary according the period of the year. In particular, it is likelythat the social value is higher during the summer where the high competition across water usersmight result in scarcity rents. A second argument form implementing peak load pricing is relatedto the share of outdoor usages into the peak water consumption (usually in summer). Outdoorwater uses have been found much more price reactive than indoor water consumption, essentiallybecause the surplus derived from those uses is likely to be low. This clearly constitutes a secondmotivation for implementing peak pricing. Several studies have estimated seasonal residentialwater demand. For most of these them, the residential water demand in summer is more sensitiveto price changes (Howe (1982), Howe and Linaweaver (1967), Griffin and Chang (1991), Renzetti(1992) or Lyman (1992)). Those empirical result suggest that peak-load water pricing may bean effective tool for managing water demands.

If there is now a consensus on the importance of efficiency criteria in water pricing determi-nation, it is surprising to note that only a few recent studies have evaluated the pricing of waterutilities.2 It is also interesting to see that, while the economics literature is quite consistent inits criticisms of water pricing by water utilities, only few studies have estimated the magnitudeof the gains to be expected from moving to efficient prices. Moreover, estimates of welfare gainsvary from one study to another. Swallow and Marin (1988) show that moving toward efficientprices will result in an increase of welfare within 2% of the actual surplus. Garcia and Reynaud(2004) evaluate the pricing of French water utilities. They show that moving towards efficientprices does not result in important direct welfare effects (less than 1% on average). Last, Ren-zetti (1992) addresses the issue of moving toward a seasonally differentiated pricing. Renzetti(1992) finds that a move to seasonally differentiated pricing raises the aggregated surplus byapproximately 4%.

The first objective of this paper is then to determine under which conditions peak pricingcan be an effective tool for managing water demand in the summer season. To answer thisquestion, we develop a theoretical model allowing to assess the impact peak pricing on consumer’ssurplus and on water consumption. Assessing the impact of peak pricing on consumer surplus isimportant since acceptability by users of new pricing schemes could me a relevant implementationissue. Assessing the impact of peak pricing on the water consumption is also important sincewater utilities wishing to implement those types of pricing may face water scarcity problems.Hence, they may operate under conditions of limited water availability. The main result of ourtheoretical model is that, whatever the level of the price elasticity of water demands during the

1In France, each water user must paid to the Basin Agency for every cubic meter some charges aiming atindicating the social cost of water. The Basin Agency Fees include abstraction and pollution charges. Abstractioncharges vary by location of user and type of resource (ground water or surface water). Pollution charges are basedon pollution emitted by users. Another example of recent move toward more economic-oriented tool is Brazil.Hence, following the approval of the Federal Water Law of January 1997, the Brazilian water management systemhas been going through a wide-ranging reform. Among the several institutional and policy innovations promotedby the law, one of the most prominent is the introduction of quality- and quantity-related water charges into theregulatory framework.

2See Kim (1995) and Renzetti (1999) for water pricing analysis in North America.

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peak and the off-peak periods, it is always possible to find water prices such as moving from asingle price for both periods towards seasonal differentiated prices results in an increase in theaggregate consumer surplus and a decrease in the aggregate water consumption. The second partof the paper provides an empirical application of this model on French data. We first estimateseasonal residential water demands, using data on a sample of local communities located inSouth-West of France. This allows us to check if the determinants of residential water demandin France vary according to the period of the year. Based on these econometric estimations,we simulate the impact on consumer surplus of moving toward peak-load pricing. In the Frenchcase, peak prices do not result neither in a substantial increase in the aggregate consumer surplusnor in a significant reduction of the aggregate water consumption. This result is due to the factthat the price elasticity difference between the peak and the off-peak period is almost negligiblein the French case.

We develop our analysis according to the following structure. In the next section, we brieflyreview the literature on peak-load pricing. In section 2, we present the theoretical model. Sec-tion 3 provides an empirical application of this model to the French case. In this empiricalsection, we specify the econometric model allowing to estimate residential peak and off-peakwater demand functions. We finally conclude by some policy implications of our work.

1 Estimating consumer demands under peak pricing

Historically, estimates of consumer demands facing peak prices have dealt with the electricitysector, see Parks and Weitzel (1984) or Caves et al. (1984). We review here the main issuesrelated to this literature and then we move to the literature focused on peak prices in the watersector. Before that, we briefly present the standard economic model of consumer demand underpeak prices.

1.1 The standard economic model of consumer demand under peak prices

Here we will use either the terms peak prices or time-of-use prices (TOU) for denoting a price ofan economic good varying according to the time period considered. For simplicity reasons andsince most of the literature on peak pricing has focused on the electricity sector, we assume thatthe good for which peak prices are implemented is electricity. Following Parks and Weitzel (1984),we consider a consumer that derives his/her utility from electricity consumption at various timeperiods and from the consumption of other goods:

u = U∗(x, z; a) (1)

where x = (xi, . . . , xr) is a vector of the quantities of electricity consumed during r time intervals,z = (zi, . . . , zs) is a vector of non-electricity goods and services and a is a vector of exogenousvariables (household characteristics, weather and environmental variables, etc.).

Typically, it is assumed that a consumer preferences can be represented by a more specificutility function:

u = U(e(x), z; a) (2)

where the subfunction e(x) is homogeneous of degree one in the elements of x. This subfunctioncan be viewed as a an aggregate quantity index for all electricity used throughout time periods.The utility function in Equation (2) is thus homotheticity separable in x. Homothetic separabilityis imposed largely for reasons of analytical convenience and is common in the empirical literatureon peak-load pricing, see Parks and Weitzel (1984) or Caves et al. (1984) or Bergstrom andMacKie-Mason (1991). In particular, it allows two solve the consumer problem in two stages.

The first stage involves the allocation of total expenditure between electricity and the non-electricity goods and services. This decision depends on total expenditure, on an index of the price

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of electricity (an aggregation of the prices at different times), and on the prices of all other goodsand services. The second stage of this budgeting process involves the allocation of total electricityexpenditure to consumption at various times of use on the basis of the different TOU prices.Notice that the homothetic separability of Equation (2) is a necessary and sufficient condition forthe validity of a decentralized two-stage budgeting approach to the electricity demand problem.The two-stage budgeting framework gives rise to conditional demand functions involving onlyelectricity prices and total electricity expenditure. Then, the conditional price elasticities givethe effect of a change in the electricity price at a given period on the electricity demand at anyperiod, holding total electricity expenditure fixed. In general a change in the price of electricityat a particular time will also change the allocation of total expenditure between electricity andother goods and services. Such effect is encompassed into the unconditional price elasticities.

1.2 Empirical estimates of electric demands facing peak prices

There is an extensive empirical literature on peak pricing in the electricity sector. We limit ouranalysis to the literature focused on price elasticity of the demand in peak and off-peak periods.

Filippini (1995) estimates the residential demand for electricity by time-of-day in Switzer-land. The estimated short-run own-price elasticities are -0.60 during the peak period and -0.79during the off-peak period. These elasticities demonstrate a high responsiveness of electricityconsumption to changes in prices. Moreover, positive values of the cross-price elasticities showthat peak and off-peak electricity are substitutes.3 Ham et al. (1997) estimate elasticities forsmall commercial establishments, finding a range from -0.038 to -0.050 for off-peak own priceelasticities and -0.069 to -0.091 for peak own price elasticities. They find that substitution be-tween peak and off-peak does not differ significantly from zero for this group of users. Boisvertet al. (2004) estimate substitution elasticities for peak-off-peak price differentials of 0.2 to 0.27,and calculate that 75% of the change in electricity usage is due to shifting load. This implies thatthe own price elasticity of electricity is about one quarter of the substitution elasticity. The onlyauthors to address real-time elasticities explicitly are Patrick and Wolak (2001). They give adetailed empirical analysis of real-time demand response for 5 industrial sectors in the UK. Theyfind fairly low price elasticities, ranging from virtually zero to -0.05 for four of the five sectors.The water supply industry exhibits a wider range, with elasticities between zero and -0.27. Forthree out of five sectors, Patrick and Wolak (2001) find the absolute value of the elasticity tobe relatively high at peak hours, late in the afternoon. As prices are higher during peak hours,this finding may support a more or less linear demand relationship. Looking at hour-to-hourpatterns of own and cross price elasticities, Patrick and Wolak (2001) conclude that “most ofthe substitutability in electricity consumption within the day comes from substitution acrossadjacent load periods”.

1.3 Rational for peak prices in the water sector

A first motivation for implementing peak prices is that the social value of water is likely to varyaccording the period of the year. In particular, it likely that the social value is higher duringthe summer where the high competition across water users might result in scarcity rents. Asa result, it makes economic sense to charge more at these times since the opportunity cost ofwater consumption is higher. For instance, the marginal benefit of instream water flow in riversis typically higher during periods of low flow like summer (Loomis (1998)), justifying a higherprice for residential water consumption and other uses where they compete with instream flow.

3In the time-of-day electricity literature, most estimates of the substitution elasticity are quite low. The rangeof 0.10 to 0.14 reported in Caves, Christensen, and Herriges (1984) is then quite typical. It is however importantto notice that, in the long run, when users can adjust their appliance holdings, the substitution elasticity may bemuch greater in absolute value.

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Second, several studies have estimated seasonal residential water demand. For most of thesethem, the residential water demand in summer is more sensitive to price changes (Howe (1982),Howe and Linaweaver (1967), Griffin and Chang (1991), Renzetti (1992) or Lyman (1992)). Forexample, Griffin and Chang (1991) report a summer price sensitivity that can be as great as 30percent more than winter price responses. Lyman (1992) find peak demand more elastic thanoff-peak demand and shows that the elasticity of peak-demand affects off-peak demand whenconsumers purchase more water-efficient appliances. Lyman (1992) estimates peak elasticity ofabout -1.35 compared with an inelastic off-peak elasticity of -0.44. He also find cross-price effectsbetween peak and off-peak periods. This effect is similar to an income effect where peak chargesaffect water use in the off-peak period. For example, peak charges could cause people to buy waterefficient durable goods like dishwashers or washing machines that cut off-peak water use. With allelse constant, Lyman (1992) finds that the long-run effect of a variable influencing demand will be24.5 percent greater in the peak than in the off-peak period. Last, Renzetti (1992) reports a pricesensitivity of the demand higher for summer than for winter. Furthermore, elasticities reportedare quite close in value to the estimates of summer and winter price elasticities reported by Howeand Linaweaver (1967). The higher price sensitivity of residential water demand in summer isrelated to the share of outdoor usages into the peak water consumption.4 Outdoor water useshave been found much more price reactive than indoor water consumption, essentially becausethe surplus derived from those water uses is likely to be low. This clearly constitutes a secondmotivation for implementing peak pricing.

A third argument is related to the literature on the impact of capacity cost on public utilitypricing. The general result from this literature is that the off-peak price should reflect consumermarginal operating costs contrary to the the peak price that should also include the marginalcapacity costs, the rational being that it is the peak consumption that press against capacity(see the seminal paper Boiteux (1949)).5

1.4 Empirical estimates of residential water demands facing peak prices

Residential demand consists of two components, indoor usage and outdoor usage. The demandfunction can be estimated using variables that reflect outside characteristics, such as irrigablearea per dwelling unit (Howe and Linaweaver (1967)), garden size (Nieswiadomy and Molina(1989), Lyman (1992) and Hewitt and Hanemann (1995)), sprinkler system (Lyman (1992)), orpool ownership (Dandy et al., 1997). Renwick and Archibald (1998) have proposed a modelof household water demand that explicitly incorporates measures of domestic and landscapeirrigation technologies. Outdoor use is usually assumed to exhibit higher price sensitivity (Howeand Linaweaver (1967), Foster and Beattie (1979) and Renwick and Green (2000)).

Indoor and outdoor usage differences might require seasonal-demand and peak-load pricing.Several studies have found winter demand less sensitive to price changes (Carver and Boland,1980; Howe, 1982; Griffin and Chang, 1990; Renzetti, 1992 and Dandy et al., 1997). Lyman(1992) has found that peak demand more elastic than off-peak demand and that the elasticityof peak-demand affects off-peak demand when consumers purchase more water-efficient appli-ances. In general, seasonal differences have been found under lagged-response specifications,Nieswiadomy and Molina (1989). According to Lyman (1992), the peak period price elasticityis more than twice the off-peak elasticity. Lyman (1992) has estimated the peak elasticity to

4Several authors have analyzed residential water peak consumption by introducing into demand functionsvariables such as irrigable area per dwelling unit (Howe and Linaweaver (1967)), garden size (Nieswiadomy andMolina (1989), Lyman (1992) and Hewitt and Hanemann (1995)) or sprinkler system (Lyman (1992))

5The literature on peak load pricing essentially emerged in response to problems faced by most public utilities,such as electricity supply industry and telecommunications, whose products are economically non-storable anddemand is time varying. These characteristics tend to result in non-uniform utilization of capacity over time.Peak load pricing offers then a way to reduce the demand in peak load (peak clipping).

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about -1.35 compared with an inelastic offpeak elasticity of -0.44. Thus, peak prices are moreelastic than off-peak. Lyman (1992) has also found cross-price effects between peak and off-peakperiods. This effect was similar to an income effect where peak charges affect water use in theoff-peak period. For example, peak charges could cause people to buy water efficient durablegoods like dishwashers or washing machines that cut off-peak water use. With all else constant,Lyman (1992) has found that the long-run effect of a variable influencing demand will be 24.5percent greater in the peak versus off-peak period. Lyman (1992) concludes that although theliterature on conservation pricing focuses on block price schemes, utilities should consider peakand off-peak pricing.

Seasonal pricing, a form of peak pricing, is also an effective method of using marginal costto price water. Griffin and Chang (1991) found that summer residential demand is more priceresponsive than winter demand. Consequently, price can be a more effective allocative tool in thesummer than winter. Summer price sensitivity can be as great as 30 percent more than winterprice responses. Finally Renzetti (1992) reports that the average values of the demand elasticitiesfor the summer and winter demand equations. It is interesting to note that the absolute value ofthe average summer elasticity exceeds the average winter price elasticity. Furthermore, althoughthe point estimates of the price elasticities are not statistically significant, they are quite close invalue to the estimates of summer and winter price elasticities reported by Howe and Linaweaver(1967).

2 Simple analytics of peak prices

In this section, we develop a theoretical model allowing to assess the impact of peak pricingon consumer’s surplus and on water consumption. Assessing the impact of peak pricing onconsumer surplus is important since acceptability by users of new pricing schemes could me arelevant implementation issue. Assessing the impact of peak pricing on the water consumption isalso important since water utilities wishing to implement those types of pricing may face scarcityproblems. Hence, they may operate under conditions of limited water availability.

2.1 The theoretical model

Time is divided in two periods: a peak period and an off-peak period. Those periods are indexedby k, k ∈ {p, o}. We denote by dp and do the duration (in days) of each period. Following theprevious literature on residential water demand, we assume that the water demand function forthe peak period can be approximated by a log-log form6:

ln Yp = Cp +∑

l

αlp · ln Xlp + βp · ln pp (3)

where Cp is a constant, Xlp represents a vector characterizing the residential water users duringthe peak period, pp is the peak period unit water price and βp is the price elasticity of the peakwater demand which is assumed to be negative, βp < 0. In equation (3), we implicitly assumethat the water demand in the peak period is not affected by the off-peak water price. The mainargument supporting this assumption is that, on the short-term, substitution capacities betweenthe peak and the off-peak periods are likely to be limited since they may require investments in

6See for instance Nauges and Thomas (2000) for an example of log-log specification of the residential waterdemand for France. Dalhuisen et al. (2003) have conducted a meta-analysis on residential water demand basedon 51 published articles. They report that 20 articles provide estimates of price and income elasticities based ona logarithmic demand function (either semi or double logarithmic).

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storage facilities which are typically based on long-term plans.7 The water demand function forthe peak period can be rewritten as:

Yp(pp) = Ap · pβpp (4)

where exp Ap = Cp +∑

l αlp · lnXlp. In Equation (4), Yp(pp) gives the daily water consumptionof the representative residential water user during the peak period if the peak water price is pp.

We also assume that he water demand function for the off-peak period can be approximatedby a log-log form:

ln Yo = Co +∑

l

αlo · ln Xlo + βo · ln po (5)

where Co is a constant, Xlo represents a vector characterizing the residential water users duringthe off-peak period, po is the off-peak period unit water price and βo is the price elasticity of thepeak water demand which is assumed to be negative, βo < 0. Taking the logarithm and usingobvious notations, the water demand function for the off-peak period can be rewritten as:

Yo(po) = Ao · pβoo (6)

In Equation (6), Yo(po) gives the daily water consumption of the representative residential wateruser during the off-peak period if the off-peak water price is po.

Notice that at this stage of the analysis, we only assume that the price elasticity of residentialwater demands is negative both for the peak and the off-peak periods. We however do not makeany assumption concerning the relative price elasticity of the residential water demand duringthe peak and the off-peak periods. Then, price elasticity could be higher or lower during thepeak season.

Initially, there is no peak price implemented by the water utility. In other words, the price ofwater is identical for both periods and is denoted by p0. The water utility wishes to implementpeak prices. Let δ denote the price change rate for the peak period:

pp = p0 · (1 + δ) (7)

with δ ∈] − 1,∞[. If δ is positive [negative] then the implementation of peak pricing implies anincrease [decrease] of the water price during the peak season. We denote by μ the price changerate for the off-peak period:

po = p0 · (1 + μ) (8)

with μ ∈]− 1,∞[. If μ is positive [negative] then the implementation of peak pricing implies anincrease [decrease] of the water price during the off-peak season.

The problem that is addressed by the water utility is the following: Is it possible to find asystem of price changes (δ, μ) such as the total consumer surplus is at least equal to the initialsurplus and the total water consumption is lower than the initial water consumption? The firstconstraint corresponds to an acceptability constraint of the new water pricing by residentialwater users: the residential water users must benefit from this new pricing scheme. The secondconstraint corresponds to a water availability constraint. The water available for residentialwater users is limited by the initial water endowment of the water utility. Hence, the objectiveof the utility is to try to increase the total consumer surplus of residential water users withoutincreasing the total residential water consumption.

7Moreover, in the empirical application, we are able to observe water consumption for the peak and the off-peak periods but, since not peak pricing is currently implemented, the price is the same for both periods. Hence,water substitution between the peak and the off-peak periods cannot be inferred from our data.

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2.2 Implementing peak prices with a consumer’s surplus gain

In this section, we derive the residential water consumer surplus change following the implemen-tation of peak prices. We denote by ΔCSp(δ) the daily consumer surplus change when the rateof the peak price change is δ. Similarly, the daily consumer surplus change when the rate of theoff-peak price change is μ is denoted by ΔCSo(μ). Then exploiting the definition of consumer’ssurplus, and thus calculating the area under the Marshallian demand curve between the old andnew price, the change in daily consumer surplus for the peak period may be written (see Strandand Walker (2003)):

ΔCSp(δ) =1

1 + βpppYp(pp)

[(p0

pp

)1+βp − 1]. (9)

Using Equation (4), this expression can be rewritten as:

ΔCSp(δ) =Ap · p1+βp

0

1 + βp

[1 − (1 + δ)1+βp

]. (10)

Using a similar approach, the change in daily consumer surplus for the off-peak period is:

ΔCSo(μ) =Ao · p1+βo

0

1 + βo

[1 − (1 + μ)1+βo

]. (11)

The peak pricing will result in no change in the aggregate consumer surplus if and only ifthe following condition holds:

dp · ΔCSp(δ) + do · ΔCSo(μ) = 0 (12)

which states that the loss [gain] in consumer surplus during the peak period must be exactlycompensated by the gain [loss] in consumer surplus during the off-peak period. Plugging Equation(10)-(11) into condition (12) gives after some manipulations:

1 − (1 + δ)1+βp

1 − (1 + μ)1+βo= −A0

Ap· do

dp· 1 + βp

1 + βo· pβo−βp

0 (13)

for μ �= 0. This condition implicitly define μ as a function of δ. We denote by μ(δ) this implicitfunction. Differentiating totally condition (13) with respect to μ and δ gives (see Appendix A):

dμ

dδ= −Ap

Ao· dp

do· pβp−βo

0 · (1 + δ)βp

(1 + μ)βo< 0 (14)

with μ �= 0. This result is intuitive: an increase of the price during the peak period must becompensated by a decrease in the price during the off-peak period in order to maintain the levelof the aggregate consumer surplus.

2.3 Implementing peak prices with limited water resources

We wish to explore now the impact of implementing peak pricing of the aggregate water con-sumption. In particular, we wish to characterize the set of price changes (δ, μ) such that the totalwater consumption is not modified. We denote by ΔYp(δ) the change in daily water consumptionwhen the rate of the peak price change is δ. Similarly, the change in daily water consumptionwhen the rate of the off-peak price change is μ is denoted by ΔYo(μ).

By definition, we have:ΔYp(δ) = Yp(pp) − Yp(p0) (15)

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which gives using Equations (4) and (7):

ΔYp(δ) = −Ap · pβp

0 · [1 − (1 + δ)βp ]. (16)

Using a similar approach, the change in daily water consumption for the off-peak period is:

ΔYo(μ) = −Ao · pβo0 · [1 − (1 + μ)βo]. (17)

The peak pricing will result in no change in the aggregate water consumption if and only ifthe following condition holds:

dp · ΔYp(δ) + do · ΔYo(μ) = 0 (18)

which states that the decrease [increase] in water consumption during the peak period must beexactly compensated by the increase [decrease] in water consumption during the off-peak period.Plugging Equations (16)-(17) into Equation (18) gives after simplification:

1 − (1 + δ)βp

1 − (1 + μ)βo= −A0

Ap· do

dp· pβo−βp

0 (19)

with μ �= 0. This condition implicitly define μ as a function of δ. We denote by μ(δ) this implicitfunction. Differentiating totally condition (19) with respect to μ and δ gives (see Appendix A):

dμ

dδ= −Ap

Ao· dp

do· pβp−βo

0 · βp

βo· (1 + δ)βp−1

(1 + μ)βo−1< 0 (20)

This result is intuitive: decreasing the price during the peak period results in an increase inthe peak water consumption. This increase in water consumption must be compensated byan increase in the water price during the off-peak period in order to maintain the level of theaggregate water consumption constant.

2.4 Implementing peak prices with surplus gain and limited water resources

We wish now to characterize the set of price changes (δ, μ) such that neither the total waterconsumption is modified nor the aggregate consumer surplus. In other words, we wish to identifyμ(δ), μ(δ) in the (δ × μ) space.

First, notice that the system of Equations (13)-(19) as an obvious solution. If (δ, μ) = (0, 0)then neither the total water consumption nor the aggregate consumer surplus are modified. Itfollows that necessarily μ(δ) and μ(δ) have an intersection at that point. Second, as demonstratedpreviously both μ(δ) and μ(δ) are strictly decreasing with δ. Third, the condition :

dμ

dδ<

dμ

dδ(21)

gives using Equations (14)-(20):1 + δ

1 + μ>

βp

β0. (22)

The slope of functions μ(δ) is lower than the slope of μ(δ) if and only if:

μ < δ · βo

βp+ (

βo

βp− 1). (23)

Two cases must be distinguished according to the levels of the price elasticity of the peak andthe off-peak demand.

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2.4.1 Case 1: Peak water demand is more reactive to the water price, βp < β0 < 0

First we consider the case where the peak water demand is more reactive to the water pricethan the off-peak water demand, βp < β0 < 0. We can then represent in the (δ, μ) space,the function μ(δ) implicitly defined by Equation (13) characterizing all price changes such thatthe aggregate consumer surplus is not modified (ΔCS(δ, μ) = 0), the function μ(δ) implicitlydefined by Equation (19) characterizing all price changes such that the aggregate consumer waterconsumption is not modified (ΔY (δ, μ) = 0) and the linear function defined by Equation (23)characterizing all price changes such that dμ

dδ < dμdδ .

Figure 1: Zoning in the (δ, μ) space when βp < βo < 0

Figure 1 presents the zoning in (δ, μ) (see Appendix A for a more formal derivation of theshape of the three functions represented on this figure). The frontier μ(δ) such that ΔCS(δ, μ) =0 is given by the green plain curve. Above this frontier, all (δ, μ) correspond to a decrease inthe aggregate consumer surplus. The frontier μ(δ) such that ΔY (δ, μ) = 0 is given by the reddotted curve. Above this frontier, all (δ, μ) correspond to a decrease in the aggregate waterconsumption.

Four zones can be identified in the (δ, μ) space. Zone 1 is of main interest for the waterutility since it corresponds to a situation where implementing peak pricing results in an increaseof the aggregate consumer’s surplus and a decrease in the aggregate water consumption. Since,in this zone the peak price is higher than the initial one, consumer’s surplus during the peakperiod decrease. However, such a decrease is more than compensated by the increase of thesurplus during the off-peak period. Prices in zone 1 allow to reallocate some water from the peakperiod toward the off-peak period with an increase in the aggregate consumer surplus. Whenprices belong to zone 2, the aggregate water consumption still decreased, compared to the initialsituation. However, in that zone the social surplus decreases. Zone 3 corresponds to all (δ, μ)located below frontiers ΔY (δ, μ) = 0 and ΔCS(δ, μ) = 0. In that zone, the decrease in theaggregate water consumption is only made possible by a reduction of the aggregate consumerwelfare. For prices belonging to zone 4, the peak pricing induces an increase of the aggregateconsumer welfare by also an increase in the aggregate water consumption.

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Table 1: Zoning in the (δ, μ) space when βp < β0 < 0Zone ΔCS(δ, μ) ΔY (δ, μ) Conditions on (δ, μ)

Zone 1 + − 1−(1+δ)1+βp

1−(1+μ)1+βo> −A0

Ap· do

pp· 1+βp

1+βo· pβo−βp

0

1−(1+δ)βp

1−(1+μ)βo< −A0

Ap· do

pp· pβo−βp

0

Zone 2 − − 1−(1+δ)1+βp

1−(1+μ)1+βo> −A0

Ap· do

pp· 1+βp

1+βo· pβo−βp

0

1−(1+δ)βp

1−(1+μ)βo> −A0

Ap· do

pp· pβo−βp

0

Zone 3 + + 1−(1+δ)1+βp

1−(1+μ)1+βo< −A0

Ap· do

pp· 1+βp

1+βo· pβo−βp

0

1−(1+δ)βp

1−(1+μ)βo< −A0

Ap· do

pp· pβo−βp

0

Zone 4 − + 1−(1+δ)1+βp

1−(1+μ)1+βo< −A0

Ap· do

pp· 1+βp

1+βo· pβo−βp

0

1−(1+δ)βp

1−(1+μ)βo> −A0

Ap· do

pp· pβo−βp

0

To conclude, when the peak water demand is more reactive to the water price, βp < β0 < 0,there exists a zone in the (δ, μ) space such that implementing peak pricing results in increasingthe aggregate consumer surplus (acceptability constraint) and a reduction of the aggregate waterconsumption (water availability constraint). In that zone, the water price will be higher duringthe peak period. The aggregate consumer increase in made possible by reallocating water fromthe peak season toward the off-peak season through a higher peak price.

2.4.2 Case 2: Peak water demand is less reactive to the water price, βo < βp < 0

Next, we consider the case where the peak water demand is less reactive to the water price thanthe off-peak water demand, βo < βp < 0. Once again, we can then represent in the (δ, μ) space,the function μ(δ) implicitly defined by Equation (13) characterizing all price changes such thatthe aggregate consumer surplus is not modified (ΔCS(δ, μ) = 0), the function μ(δ) implicitlydefined by Equation (19) characterizing all price changes such that the aggregate consumer waterconsumption is not modified (ΔY (δ, μ) = 0) and the linear function defined by Equation (23)characterizing all price changes such that dμ

dδ < dμdδ . The main difference compared to case 1 is

that the slope of line defined by dμdδ = dμ

dδ is greater than one. Moreover, for δ = 0, this conditionis satisfied for a negative value of μ.

Figure 2 presents the zoning in (δ, μ) (see Appendix A for a more formal derivation of theshape of the three functions represented on this figure). Once again, the frontier μ(δ) such thatΔCS(δ, μ) = 0 is given by the green plain curve. Above this frontier, all (δ, μ) correspond toa decrease in the aggregate consumer surplus. The frontier μ(δ) such that ΔY (δ, μ) = 0 isgiven by the red dotted curve. Above this frontier, all (δ, μ) correspond to a decrease in theaggregate water consumption. As previously, four zones can be identified in the (δ, μ) space withcharacteristics in terms of change in aggregated consumer surplus and water consumption similarto case 1. When the peak water demand is less reactive to the water price, βo < βp < 0, therestill exists a zone in the (δ, μ) space such that implementing peak pricing results in increasingthe aggregate consumer surplus (acceptability constraint) and a reduction of the aggregate waterconsumption (water availability constraint).

2.4.3 Summary

The following proposition summarizes the main findings of the theoretical model presented inthe previous paragraphs.

11

Figure 2: Zoning in the (δ, μ) space when βo < βp < 0

Proposition 1 Whatever the level of the price elasticity of water demands during the peak andthe off-peak periods, there exist water prices such as moving from a single price for both periodstowards seasonal differentiated prices results in an increase in the aggregate consumer surplusand a decrease in the aggregate water consumption.

The intuition of this proposition is straightforward. Since the marginal value of water is notnecessarily the same in the peak and in the off-peak periods, using the same water price may beinefficient.

2.5 Second-best water prices

In the previous paragraph, we have shown that it is always possible to find a system of differenti-ated prices such as the aggregate consumer surplus increases (compared to the initial situation)and the aggregate water consumption decreases (compared to the initial situation). The nextquestion of interest is to characterize the pricing schemes that will be chosen according to thespecific objectives of the water utility.

2.5.1 Maximizing the aggregate surplus under limited water availability

First, the water utility may be constrained by the total water availability, that is by the quantityof water that may be consumed during the peak and the off-peak periods. A natural objectivefor the water utility is such a context would be to choose the peak and the off-peak prices suchas the aggregate consumer surplus is maximized under the water availability constraint. This

12

is a second-best optimum, in the sense that we impose on the water utility a water availabilityconstraint.

The second-best pricing scheme is then the solution of the following optimization program:

maxδ,μ

dp · ΔCSp(δ) + do · ΔCSo(μ)

s.t. dp · ΔYp(δ) + do · ΔYo(μ) = 0μ ≥ μ

δ ≥ δ

Assuming an interior solution, the second-best pricing scheme is the solution of the followingsystem of equations:

μ = δ · βo

βp+ (

βo

βp− 1) (24)

1 − (1 + δ)βp

1 − (1 + μ)βo= −A0

Ap· do

pp· pβo−βp

0 . (25)

Equation (24) corresponds to set of (δ, μ) such that:

dμ

dδ=

dμ

dδ(26)

whereas equation (25) corresponds to set of (δ, μ) such that the aggregate water consumption isnot modified by peak pricing. The second-best price scheme is then given by point A in Figures1 and 2.

2.5.2 Minimizing the aggregate water consumption without consumer surplus loss

Another possible way to define second-best differentiated prices in to determine the price schemesuch as the aggregate water consumption is minimized under the constraint that the consumersurplus is not modified. Such an objective for the water utility may fit with a situation in whichwater could be used by competitive users having a high valuation. The pricing should be thenacceptable by current water users (no change in aggregate surplus) and should try to minimizethe water consumption.

The second-best pricing scheme is then the solution of the following optimization program:

minδ,μ

dp · ΔYp(δ) + do · ΔYo(μ)

s.t. dp · ΔCSp(δ) + do · ΔCSo(μ) = 0μ ≥ μ

δ ≥ δ

Assuming an interior solution, the second-best pricing scheme is the solution of the followingsystem of equations:

μ = δ · βo

βp+ (

βo

βp− 1) (27)

1 − (1 + δ)1+βp

1 − (1 + μ)1+βo= −A0

Ap· do

pp· 1 + βp

1 + βo· pβo−βp

0 (28)

13

Equation (27) corresponds to set of (δ, μ) such that:

dμ

dδ=

dμ

dδ(29)

whereas equation (28) corresponds to set of (δ, μ) such that the consumer surplus is not modifiedby peak pricing. The second-best price scheme is then given by point B in Figures 1 and 2.

3 Empirical application to residential water in France

In this section we use the theoretical model presented in the previous section in order to em-pirically assess the impact on residential aggregate consumer surplus and on aggregate waterconsumption of moving toward seasonal differentiated pricing in France. The theoretical modelrequires first to estimate residential water demand functions, both for peak and for off-peakperiods. In the next paragraph, we briefly survey the literature related to the specification ofthe residential water demand. We then move to the econometric model and to the estimationof peak and off-peak residential water demand in France. In the last paragraph, we analyze theimpact of implementing peak prices.

3.1 Motivations for implementing peak prices in France

Globally, water can be viewed as an abundant natural resource in France. However, the increasedpressure on water sources has resulted in an increase of scarcity problems at the local level and atsome specific periods of the year. First, according to the French “Comité Sécheresse” the numberof French Departments having implemented water use restrictions each year over the period1998-02 was 25 (over 91). It has been multiplied by 2 over the period 2003-04 to reach morethan 70 in 2005. These figures indicates that local conflicts for water use are more and moreeffective in France. Second, considering only 2005, 9 French Departments have implementedquantity restrictions of water uses in May. This number has increased to 50 in July and hasreached a maximum of 71 in August. In September, 66 Departments were still restricting wateruses but this number dropped to less than 30 in October. These figures indicate that the issueof intra-annual water allocation is important in term of water policy in France.

3.2 Specifying a residential water demand

Water demand functions have been estimated since the 1950s. For most studies, the primaryobjective was to derive a measure of the price elasticity of water demand. This may be whythe two main econometric issues that have been intensively investigated in the applied literatureare the adequate representation of the price in the demand (marginal versus average pricing)and the implications of block rate pricing. Howe and Linaweaver (1967) have argued earlythat residential consumers are more likely to react to the average price than to the marginalprice. Later, Shin (1985) introduced a ’perceived’ price which is in fact a combination of themarginal and the average prices. As under a block rate pricing, it is difficult to determine theprice specification that should be used for estimating the demand function. Most of the modelsemploy a combination of the marginal price and a ’difference’ variable which aims to capturethe income effect imposed by the block rate structure (Nordin (1976)). But under a block ratetariff, the price is endogenous and specific estimation techniques, such as instrumental variables(two-stage least square), are required. However as suggested by Hewitt and Hanemann (1995),the correct specification of the water demand under multiple-block tariffs requires a two-stagemodel where the block is first selected by the consumer and then the quantity is chosen in acontinuous way within that block.

14

Numerous estimates of price and income elasticities for residential water demand are nowavailable and there is a large consensus among researchers that the residential water demandis inelastic, but not perfectly. Most of the published studies report short-term price elasticitiesvarying from -0.3 to -0.1. However long-term residential water demand appears to be more pricesensitive, see Nauges and Thomas (2000) among others. The estimate of income elasticitiesreveals a more substantial range of values going from 0 to more than 2. In their meta-analysis,Dalhuisen et al. (2003) report that “the distribution of income elasticities has a mean of 0.46”and that “water demand appears to be inelastic in terms of income changes”.

An important issue is to specify a functional form that will be used for estimating the waterdemand function. In their meta-analysis based on 51 published articles, Dalhuisen et al. (2003)report that half of these papers have used a linear specification (lin-lin form) and that 20 articlesprovide estimates of price and income elasticities based on a logarithmic demand function (eithersemi or double logarithmic). In order to easily compare our estimates with those from the previ-ous literature, we will consider a log-log specification. Following Equation (3) of the theoreticalmodel, we denote by Y the water consumption in cubic meters per day of the representativehousehold in a given local community. The water demand model in log-log form can be writtenas follow:

ln Y = f(ln p, ln X|β) (30)

where p represents the price of water, I the representative household’s income and X a vectorof exogenous variables influencing water demand (e.g., household’s income, climate variables,demographic characteristics of the representative household). In Equation (30), β is a vector ofparameters to be estimated.

3.3 Econometric model

We assume that each year is divided into two periods, namely a peak periods (corresponding tothe summer) and an off-peak period (corresponding to the rest of the year). Those two periodsare indexed by k, k ∈ {p, o}. We denote by dp and do the duration (in days) of each period.Let Yitk denote the water consumption per capita (in cubic meter per day) for the representativehousehold in the local community i at year t for period k. The water demand can be written ina very general way as:

ln Yitk = fk(ln pitk, ln Xit, ln Zitk |βk) + ηitk (31)

where pitk represents the unit water price that may vary between the peak and the off-peakperiod, Xit is a vector of exogenous variables influencing water demand which do not vary withthe season (household income, housing characteristics, availability of groundwater, etc.), Zitk isa vector of exogenous variables influencing water demand which do vary with the season (climatevariables) and ηitk represents the error term.

Specifying a demand function for each season requires a few comments. First, it correspondsto the view that in a given local community the residential water consumers during the summerseason are not necessarily the same as those consuming water during the rest of the year. Second,the summer season may correspond to some specific water uses that are not observed during therest of the year (water for swimming-pool or for gardening). It follows that we cannot excludea priori the fact that the demand functions may depend upon the season. Notice that seasonaldependent demand functions have also been estimated for example by Renzetti (1992). Third,we implicitly assume that the water demand in the peak season only depend upon the water priceof that season. It other words, the cross-price elasticitiy of peak and off-peak water demandsis assumed to be equal to zero. As it will be discussed later, there are currently no peak pricesimplemented for residential water demand in France. It follows that the cross-price elasticitiy ofpeak and off-peak water demands cannot be identified and must be set to zero.

15

As it is standard in panel data analysis, we decompose the error term as ηitk = αi+εitk whereαik is i.i.d N(0, σ2

αk) and εitk is i.i.d N(0, σ2

εk). The term αik is a community-specific effect and

epsilonitk is the usual error term. The equation to be estimated becomes:

lnYitk = fk(ln pitk, ln Xit, ln Zitk |βk) + αik + εitk (32)

The community-specific effect creates a non-standard covariance structure. The GeneralisedLeast Squares (GLS) estimator is then efficient and improves over the Ordinary Least Squares(OLS) estimator as long as the fixed effects are non correlated with the regressors. The interpre-tation of estimates with panel data crucially depends on the nature of the community-specificeffect. We first need to test the presence of the community-specific term, αik in equation (32).We use a Breusch/Pagan LM test (BP-LM) to discriminate between the pooled model and therandom effect model. The basic idea is that, in the model described by (32), if σ2

αk= 0, ran-

dom effects are not needed. Next, we need to investigate the potential correlation between thecommunity-specific term and the regressors. If some form of correlation is present in the sample,the random effect estimator will be inconsistent. A Hausmann specification test can be used toanswer this question.

3.4 Data

Data used for estimating the residential water demands come from various sources includingpopulation census and water agency files based on local communities declarations.8 The datasetcovers 3 years, 1998, 2001 and 2004. The data are aggregated at the local community level (338local communities located in the Midi-Pyrénées region in Southwest of France). Due to a fewmissing observations, the final dataset is finally made of 904 observations at the local communitylevel. We briefly describe the main variables used in the econometric application.

Residential water consumption is reported by distinguishing the summer period (from July1st to October 31) from the rest of the year (January 1st to June 30 and November 1st toDecember 31).9 In the remaining of the paper, the peak season will correspond to the summerseason and the off-peak season to the rest of the year.

In Table 2, we report the average residential consumption in cubic meter per day and perhousehold (m3/day/household) both for peak and the off-peak periods. As it can be seen, thedaily consumption is greater in the peak period (summer) than in the off-peak period (rest ofthe year), the difference being on average 0.08 cubic meter per day (+18%). Moreover, thisdifference is statistically different from zero. This may indicate that peak water consumptioninclude specific water uses (outdoor uses for instance).

The price variable considered in the demand function corresponds to the unit price (includingtaxes) per cubic meter for an annual water consumption equal to 120 cubic meter.10 This priceis available from the Agence de l’Eau-IFEN-SCEES survey conducted in 1999, 2001 and 2004.The unit water price has increased from 1.94 euro/m3 in 1999 to 2.29 euro/m3 in 2004. Since,the unit price is obtained by dividing the total bill by the water consumption, it may suffer

8A more exhaustive of the various surveys used for the empirical application may be found in Appendix B.9In fact, the French water agency measures residential withdrawals for both periods. The ratio of the summer

residential withdrawals to the rest of the year residential withdrawals as been applied to the effective residentialconsumption reported by the IFEN-SCESS-Agence de l’eau surveys. We implicitly assume that the summer/restof the year ratio is the same for withdrawals and for effective consumption.

10The specification of the price in the demand function has resulted in a debate over the use of the marginalor average price, see Taylor, McKean, and Young (2004). Howe and Linaweaver (1967) argued that residentialconsumers are more likely to react to the average rather than to the marginal price. Later, Shin (1985) introduceda ’perceived’ price which is in fact a combination of the marginal and the average prices. Another approach hasbeen followed by Chicoine, Deller, and Ramamurthy (1986) who estimate a demand function using the marginalprice. Because the econometric results have been mixed, there is still now no clear consensus on the correctspecification.

16

Table 2: Summary statistics for the main variablesVariable Definition Mean Std. Dev. Min. Max.Yp daily peak water consumption (m3/day) 0.533 0.238 0.04 2.75Yo daily off-peak water consumption (m3/day) 0.453 0.183 0.034 2.239I household income (euros) 21945 4990 11661 41777p water unit price (euros per m3) 2.12 0.887 0.175 5.718ucmf num. of consumption units per household 1.652 0.11 1.32 1.977persmf num. of persons per household 2.458 0.26 1.71 3.203vulne dummy for nitrate vulnerable area 0.477 0.5 0 1codtour dummy if tourist area 0.095 0.294 0 1densite population density (100 inhabitants/km2) 2.026 4.016 0.012 31.028vol_napp_s share of water from groundwater sources 0.263 0.393 0 1vol_napc_s share of water from groundwater sources 0.02 0.117 0 1vol_surf_s share of water from surface sources 0.717 0.402 0 1sais_s share of seasonal population 0.034 0.153 0 2.155tnp summer daily minimal temperature (◦C) 14.02 0.882 12.302 15.431txp summer daily maximal temperature (◦C) 25.694 1.646 22.157 27.739rgp summer daily solar radiation (j/cm2) 1914 122 1689 2168etpp summer Penman’s PET (mm/day) 4.193 0.521 3.175 5.164rainp summer daily rain (mm/day) 1.729 0.633 0.656 2.972tno fall daily minimal temperature (◦C) 4.642 0.893 3.239 6.278txo fall daily maximal temperature (◦C) 13.13 1.064 9.855 14.204rgo fall daily solar radiation (j/cm2) 673 55 551 754etpo fall Penman’s PET (mm/day) 1.143 0.172 0.716 1.47raino fall daily rain (mm/day) 2.013 0.861 0.652 3.851domtot ratio of domestic to other water users 0.939 0.11 0.333 1resprin share of permanent housing 0.791 0.164 0.167 0.983collective share of collective housing 0.823 0.303 0.143 3.579

from an endogenity problem. This endogeneity issue will be adressed first by regressing the unitprice variable on a set of exogenous variables (i.e. instrumental variables which include all thedeterminants the price and a set of technical variables characterizing the water service). Then,the instrumentalised price can be used as an exogenous variable in the water demand equation.

Household income at the local community level has been provided by the French NationalInstitute of Statistic and Economic Studies (INSEE). The income variable I, is the medianhousehold fiscal income defined at the local community level. It varies from 20895 euro/householdon average in 1999 to 23067 euro/household in 2005.

Some variables aiming at capturing some heterogenity across local communities have been alsointroduced. Such variables include the population density, the share of non permanent dwellingsto the total number of dwelling and a dummy variable equal to 1 if the local community is locatedin a tourist area. One may expect that local communities located in a tourist area may presentsome specific water consumption patterns, especially during the summer season.

Residential water consumption may also be affected by the water availability and by qualityconsiderations. The variable qmauv representing the share of rivers classified as bad qualitywithin a given local community aims at capturing such an effect.

Climatic data have been provided by Meteo-France and by the French Institute of Agro-nomic Research (Agroclim unit). Both institutions manage a set of meteorological measurementstations. Climatic information available on daily basis over long periods include minimum tem-perature (tn), maximum temperature (tx), solar radiation (rg), Penman’s potential evapotran-

17

Table 3: Summary statistics for year 1998, 2001 and 2004Variable 1998 2001 2004Yp 0.521 0.528 0.546Yo 0.443 0.446 0.467I 20895 21655 23067P 1.948 2.085 2.294tempp 25.937 24.891 26.164

spiration (etp) and rain (rain). Local communities have been matched with a meteorologicalstation based on a proximity criterion.

In Table 3, we present some statistics for the main variables of interest according to the year.As it can be seen, daily water consumption per household has increased over time both for thepeak and the off-peak seasons. Both the unit water price and the household fiscal income followsimilar increasing trend. The driest and the hottest year of our sample is year is 2004.

3.5 Estimation of residential demand functions

Both the peak and the off-peak demands have been estimated using the sample of 904 observa-tions of residential water consumption at the French local community level. Estimation of theresidential water demands are presented into the two next tables. Before discussing the resultsof these estimates, we must conduct some specification tests.

Table 4: Estimation results (peak period)Variable Coefficient (Std. Err.)

lprixtot -0.154∗∗ (0.031)lrevenu 0.123 (0.087)uc_mf -0.139 (0.171)vol_napp_s -0.081† (0.042)vol_napc_s 0.146 (0.136)tx_E 0.016† (0.009)pluie_P 0.027 (0.026)densite -0.014∗∗ (0.004)vulne -0.079∗ (0.037)qmau -1.297∗∗ (0.493)sais_s 0.190† (0.104)Intercept -1.927∗ (0.780)

N 904Log-likelihood .χ2

(11) 78.61

Specification tests The two following tables give the estimate of the demand function for peakand the off-peak seasons. Based on these estimates, we first proceed to some specification tests.First, in both cases the BP-LM statistics exceeds the tabulated chi-squared value at 1 percent:we reject the null hypothesis (σ2

α = 0). The GLS estimator is more appropriate than a OLSestimate on the pooled model. In other words, there are local-community specific effects in thedata. Second, as the Hausman statistics are lower in both cases than the tabulated chi-squared

18

Table 5: Estimation results (off-peak period)Variable Coefficient (Std. Err.)

lprixtot -0.118∗∗ (0.029)lrevenu 0.114 (0.084)uc_mf -0.078 (0.164)vol_napp_s -0.079∗ (0.040)vol_napc_s 0.120 (0.130)tx_E 0.013 (0.009)pluie_P -0.011 (0.025)densite -0.009∗ (0.004)vulne -0.028 (0.036)qmau -1.185∗ (0.469)sais_s 0.090 (0.099)Intercept -1.965∗∗ (0.749)

N 904Log-likelihood .χ2

(11) 44.665

value at 1 percent, we keep the null hypothesis: the local-community effect is not correlated tothe exogenous variables and GLS estimators are unbiased and efficient.

Residential water demand for the peak period We turn now to the main socioeconomicdeterminants of the residential water demand for the peak period. First, the price variableis a significant determinant of residential water consumption for the peak period. Hence, theprice elasticity is −0.15 and significantly different from zero. This level of price sensitivity isin line with the numerous estimates of price elasticities for residential water demand. Hence,There is a large consensus among researchers that the residential water demand is inelastic, butnot perfectly. Most of the published studies report short-term price elasticities varying from-0.3 to -0.1. Second, the income variable appears to be not significant. Third, accessibilityto groundwater results in a lower water consumption during the peak period. It is likely thatsome outdoor water uses during summer may be satisfied thanks to groundwater. Fourth, thequality seems to be a significant determinant of the residential water demand. Being locatedin an area vulnerable to pesticides or facing a raw water with low quality induce a lower waterconsumption. Last, as expected, the higher is the share of the seasonal population the higherwill be the residential water consumption in the peak period.

Residential water demand for the off-peak period We turn now to the main socioeco-nomic determinants of the residential water demand for off-peak period. First, the price variableis a significant determinant of residential water consumption for the off-peak period. Hence, theprice elasticity is −0.12 and significantly different from zero. The other determinants (quality ofraw water, share of the seasonal population, density population) are similar to the peak periodcase.

A comparison of peak and off-peak water demands The summer water demand appearsto be slightly more reactive to price changes than the rest of the year water demand. This is in linewith several studies having estimated seasonal residential water demand. Hence, for most of thesethem, the residential water demand in summer is more sensitive to price changes (Howe (1982),

19

Figure 3: Water demand functions for peak and off-peak seasons (at the mean sample)

.4.5

.6.7

.8

0 1 2 3price

offpeak peak

Howe and Linaweaver (1967), Griffin and Chang (1991), Renzetti (1992) or Lyman (1992)). Forexample, Griffin and Chang (1991) report a summer price sensitivity that can be as great as 30percent more than winter price responses. Lyman (1992) find peak demand more elastic thanoff-peak demand and shows that the elasticity of peak-demand affects off-peak demand whenconsumers purchase more water-efficient appliances. Lyman (1992) estimates peak elasticity ofabout -1.35 compared with an inelastic off-peak elasticity of -0.44. He also find cross-price effectsbetween peak and off-peak periods. This effect is similar to an income effect where peak chargesaffect water use in the off-peak period. For example, peak charges could cause people to buy waterefficient durable goods like dishwashers or washing machines that cut off-peak water use. With allelse constant, Lyman (1992) finds that the long-run effect of a variable influencing demand will be24.5 percent greater in the peak than in the off-peak period. Last, Renzetti (1992) reports a pricesensitivity of the demand higher for summer than for winter. Furthermore, elasticities reportedare quite close in value to the estimates of summer and winter price elasticities reported by Howeand Linaweaver (1967). The higher price sensitivity of residential water demand in summer isrelated to the share of outdoor usages into the peak water consumption.11 Outdoor water useshave been found much more price reactive than indoor water consumption, essentially becausethe surplus derived from those water uses is likely to be low. This clearly constitutes a secondmotivation for implementing peak pricing. In Figure 3, we have represented the summer andthe rest of the year water demand function at the mean sample as a function of the unit waterprice. Remember that for 90% of our observations, the unit water price lies between 0.46 eurosper cubic meter and 1.99 euros per cubic meter. On this relevant price interval, the estimated

11Several authors have analyzed residential water peak consumption by introducing into demand functionsvariables such as irrigable area per dwelling unit (Howe and Linaweaver (1967)), garden size (Nieswiadomy andMolina (1989), Lyman (1992) and Hewitt and Hanemann (1995)) or sprinkler system (Lyman (1992))

20

water consumption per day is greater in the summer season than in the rest of the year season.

3.6 Implementing peak prices

We consider now a water utility wishing to implement peak prices. Using the peak and theoff-peak water demand function estimates, we now assess the impact on consumer surplus andon water consumption of moving toward such a type of pricing. All the following analysis hasbeen conducted at the mean of our sample.

3.6.1 Zoning in the (δ, μ) space

We first characterize the set of price changes (δ, μ) such that neither the total water consumptionis modified nor the aggregate consumer surplus. This will allow use to identified the four possiblezones as discussed in the theoretical part of the paper.

Since, the peak water demand is more reactive to the water price, βp < β0 < 0, then therelevant situation is the one described in case 1 in the previous section. In Figure 4, we have rep-resented in the (δ, μ) space, the function μ(δ) implicitly defined by Equation (13) characterizingall price changes such that the aggregate consumer surplus is not modified (ΔCS(δ, μ) = 0), thefunction μ(δ) implicitly defined by Equation (19) characterizing all price changes such that theaggregate consumer water consumption is not modified (ΔY (δ, μ) = 0) and the linear functiondefined by Equation (23) characterizing all price changes such that dμ

dδ = dμdδ .

Figure 4: Zoning in the (δ, μ) space in the French case

( , ) 0Y( , ) 0CS

ˆd dd d

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

A

B

Figure 4 calls for a few remarks. First, significant changes both in terms of aggregate con-sumer surplus or in terms of aggregate water consumption can only be achieved through impor-tant price variations. This result is naturally a consequence of the low price elasticities of peak

21

and off-peak water demand functions. Second, as it can be seen, the set of (δ, μ) such differenti-ated prices result both in increasing the aggregate consumer surplus and in decreasing the totalwater consumption (zone 1 in Figure 4) is very limited since curves such that ΔCS(δ, μ) = 0and ΔY (δ, μ) = 0 almost coincide. This is clearly related to the fact that the difference in priceelasticity during the peak and the off-peak period is almost negligible. Since price elasticity aresimilar, the consumer gains to be expected from transferring water consumption from the peakto the off-peak period are small.

3.6.2 Second-best peak pricing

In this last paragraph, we finally characterize the second-best pricing schemes that could beimplemented by the water utility.

As we have seen in the theoretical part of the paper, a first possible way to define thesecond-best differentiated prices is to determine the price scheme such as the aggregate consumersurplus is maximized under the water availability constraint. The second-best pricing scheme,solution of this water utility optimization program, is given by A in Figure 4. Another second-best pricing scheme is the one such that the aggregate water consumption is minimized underwithout consumer surplus reduction. In that case, the second-best pricing scheme is given by Bin Figure 4.

Table 6: Comparison of second-best price schemes with respect to the initial situationVariable A BPeak price change +16.8% +17.9%Peak water consumption change −2.4% −2.5%Off-peak price change -11.3% −10.5%Off-peak water consumption change 1.4% +1.3%Change is total surplus (in euro/year) +2.82 0Change in total water consumption 0 -11.3%

Let us consider first the point A that is the pricing scheme which maximizes the aggregateconsumer surplus without modifying the aggregated water consumption. Solving this programwith our estimates of the peak and the off-peak residential water demands result in δ = 0.168and μ = −0.113. The peak price must be increased by +16.8% compared to the initial waterprice whereas the off-peak price must be decreased by −11.3%. In that case, the decrease inwater consumption during the peak period is exactly compensated by the increase during the offpeak period. Unfortunately, such a pricing scheme does not result in a substantial increase inthe aggregate consumer surplus. The aggregate consumer surplus increases by 2.82 euros whichis negligible compared to the average water bill, 372 euros. Two explanations may be advocatedfor explaining this result. First, as peak and off-peak water demand functions are very inelastic,changes in water prices only result in small modifications of water consumption. Second thedifference in price elasticity during the peak and the off-peak period is almost negligible. Ourresult contrast with the previous literature having measured the impact on the welfare of movingtoward more efficient water prices. Hence, Swallow and Marin (1988) show that moving towardefficient prices will result in an increase of welfare within 2% of the actual surplus and Renzetti(1992) finds that a move to seasonally differentiated pricing raises the aggregated surplus byapproximately 4%. Our result is however in line with Garcia and Reynaud (2004) who show ona sample of French water utilities located in the Bordeaux area that moving towards marginalcost pricing does not result in important direct welfare effects (less than 1% on average).

Next, we consider the point B that is the pricing scheme which minimizes the aggregatewater consumption without the aggregated surplus. Solving this program with our estimates of

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the peak and the off-peak residential water demands result in δ = 0.179 and μ = −0.105. Thepeak price must be increased by 17.9% compared to the initial water price whereas the off-peakprice must be decreased by 10.5%. In that case, the loss of consumer surplus during the peakperiod is exactly compensated by the increase during the off peak period. Notice that such apricing scheme results in a substantial aggregate water consumption reduction, (−11.3%).

4 Conclusion

In this paper, we have determined under which conditions peak pricing can be an effective toolfor managing water demand in the summer season. To answer this question, we have developeda theoretical model allowing to assess the impact of implementing peak pricing on consumer’ssurplus and on water consumption. The main result of this model is that, whatever the levelof the price elasticity of water demands during the peak and the off-peak periods, it is alwayspossible to find water prices such as moving from a single price for both periods towards seasonaldifferentiated prices results in an increase in the aggregate consumer surplus and a decrease inthe aggregate water consumption.

The second part of the paper has provided an empirical analysis of this issue based onresidential water consumption in France. We have first estimated seasonal residential waterdemands, using data on a sample of local communities located in South-West of France. Thishas allowed us to check if the determinants of residential water demand in France vary accordingto the period of the year. Based on these econometric estimation, we have simulated the impacton consumer surplus of moving toward peak-load pricing. The empirical results show that theset of peak and off-peak prices inducing both an increase in the aggregate consumer surplus anda decrease in the total water consumption is very limited. This result is clearly related to the factthat the difference in estimated price elasticity during the peak and the off-peak period is almostnegligible. Since price elasticity are similar, the consumer gains to be expected from transferringwater consumption from the peak to the off-peak period are small. As a result, second-best peakand off-peak prices do not result is significant increase of the objective function of the waterutility. This is not to say that moving towards seasonal differentiated prices is not of interestin general for a social planner since most of the published empirical estimates report substantialdifferences in price elasticities between peak and off-peak periods.

There are several extensions of this framework that could be of interest. The first one is tomake the water demand for a given period depend upon the water price for other periods. Asecond extension would be to introduce a water utility cost function in order to evaluate theimpact of peak pricing not only on the consumer surplus but also on the social welfare. We letthese possible extensions for future researches.

References

Bergstrom, T., and J. MacKie-Mason (1991): “Some simple analytics of peak-load pricing,”The Rand Journal of Economics, 22(2), 241–249.

Boiteux, M. (1949): “La tarification des demandes en pointe: Application de la théorie de lavente au coût marginal,” Western Journal of Agricultural Economics, pp. 321–340.

Caves, D., L. Christensen, P. Schoech, and W. Hendricks (1984): “A comparison of Dif-ferent Methofologies in a Case Study of Residential Time-of-Use Electricity Pricing,” Journalof Econometrics, 26, 7–34.

23

Chicoine, D., S. Deller, and G. Ramamurthy (1986): “Water Demand Estimation UnderBlock Rate Pricing: A Simultaneous Equation Approach,” Water Resources Research, 22(6),859–863.

Dalhuisen, J., R. Florax, H. deGroot, and P. Nijkamp (2003): “Price and IncomeElasticities of Residential Water Demand: A Meta-Analysis,” Land Economics, 79, 292–308.

Filippini, M. (1995): “Swiss residential demand for electricity by time-of-use,” Resource andEnergy Economics, 17(3), 281–290.

Foster, J. H., and B. Beattie (1979): “Urban Residential Demand for Water in the UnitedStates,” Land Economics, 55(1), 43–58.

Garcia, S., and A. Reynaud (2004): “Estimating the Benefits of Efficient Water Pricing inFrance,” Resource and Energy Economics, 26(1), 1–25.

Griffin, R., and C. Chang (1991): “Seasonality in Community Water Demand,” WesternJournal of Agricultural Economics, 16, 207–217.

Ham, J., D. Mountain, and M. L. Chan (1997): “Time-of-use prices and electricity demand:allowing for selection bias in experimental data,” Rand Journal of Economics, 28, 113–141.

Hewitt, J., and W. Hanemann (1995): “A Discrete/Continuous Choice Approach to Resi-dential Water Demand under Block Rate Pricing,” Land Economics, 71(2), 173–192.

Howe, C. (1982): “The Impact of Price on Residential Water Demand: Some New Insights,”Water Resources Research, 18(4), 713–716.

Howe, C., and F. Linaweaver (1967): “The Impact of Price on Residential Water Demandand Its Relation to System Design and Price Structure,” Water Resources Research, 3(1),13–32.

Kim, Y. (1995): “Marginal Cost and Second-Best Pricing for Water,” Review of Industrial Or-ganization, 10, 323–338.

Loomis, J. (1998): “Public’s Values for Maintaining Instream Flow: Economic Techniques andDollar Values,” ournal of the American Water Resources Association, 34(5), 1007–1014.

Lyman, R. (1992): “Peak and Off-Peak Residential Water Demand,” Water Resources Research,28(9), 2159–2167.

Nauges, C., and A. Thomas (2000): “Privately-operated Water Utilities, Municipal PriceNegotiation, and Estimation of Residential Water Demand: The Case of France,” Land Eco-nomics, 76(1), 68–85.

Nieswiadomy, M., and D. Molina (1989): “Comparing Residential Water Demand Estimatesunder Decreasing and Increasing Block Rates Using Household Data,” Land Economics, 65(3),281–289.

Nordin, J. (1976): “A Proposed Modification of Taylor’s Demand Analysis: Comment,” TheBell Journal of Economics, 7, 719–721.

Parks, R., and D. Weitzel (1984): “Measuring the consumer welfare effects of time-differentiated electricity prices,” Journal of Econometrics, 26, 35–64.

Patrick, R., and F. Wolak (2001): “Estimating the Customer-Level Demand for ElectricityUnder Real-Time Market Prices,” NBER Working Papers 8213.

24

Renwick, M., and S. Archibald (1998): “Demand Side Management Policies for ResidentialWater Use: Who Bears the Conservation Burden,” Land Economics, 74(3), 343–359.

Renwick, M., and R. Green (2000): “Do Residential Water Demand Side Management Poli-cies Measure Up? An Analysis of Eight California Water Agencies.,” Journal of EnvironmentalEconomics and Management, 40, 37–55.

Renzetti, S. (1992): “Evaluating the Welfare Effects of Reforming Municipal Water Prices,”Journal of Environmental Economics and Management, 22, 147–192.

Renzetti, S. (1999): “Municipal Water Supply and Sewage Treatment Costs, Prices, and Dis-tortions,” Canadian Journal of Economics, 32(3), 688–704.

Shin, J. (1985): “Perception of Price when Price Information is Costly: Evidence from Residen-tial Electricity Demand,” The Review of Economic Studies, 67(4), 591–598.

Strand, J., and I. Walker (2003): “The value of water connections in CentralAmerican cities: A revealed preference study,” Discussion paper, University of Oslo,http://folk.uio.no/jostrand/watervaluepaper.pdf.

Swallow, S., and C. Marin (1988): “Long-run Price Inflexibility and Efficiency Loss forMunicipal Water Supply,” Journal of Environmental Economics and Management, 15, 233–247.

Taylor, R., J. McKean, and R. Young (2004): “Alternate Price Specifications for EstimatingResidential Water Demand with Fixed Fees,” Land Economics, 80(3), 463–475.

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A Zoning in the (δ, μ) space

In this Appendix, we characterize the set of price changes (δ, μ) such that neither the total waterconsumption is modified nor the aggregate consumer surplus.

A.1 No change in the aggregate water consumption

The peak pricing will result in no change in the aggregate water consumption if and only ifcondition (18) holds:

dp · ΔYp(δ) + do · ΔYo(μ) = 0. (A.1)

Lemma 1 There exists a minimal negative value both for δ and μ such that Equation (A.1) maybe satisfied.

We now demonstrate that Equation (A.1) may only holds if δ and μ are respectively greaterthan δ and μ with δ < 0 and μ < 0. Hence, we can rewrite (A.1) as:

dp · Yp((1 + δ)p0) + do · Yo((1 + μ)p0) = dp · Yp(p0) + do · Yo(p0) (A.2)

which states that aggregate water consumption with peak prices must be exactly equal to waterconsumption without. If μ is negative, then the water consumption during the off-peak periodwill increase since the off-peak price elasticity is assume to be negative. The maximal off-peakprice decrease, denote by μ, is given by:

do · Yo((1 + μ)p0) = dp · Yp(p0) + do · Yo(p0) (A.3)

which is compatible with Equation (A.6) when δ → +∞ since in such a case the water con-sumption during the peak period goes toward zero. Using the log-log specification of the waterdemands gives after simplification:

μ =[1 +

dp

do· Ap

Ao· pβp−βo

0

]1/βo − 1 (A.4)

Using a similar approach allows us to characterize the maximal peak price decrease, denote byδ:

δ =[1 +

do

dp· Ao

Ap· pβo−βp

0

]1/βp − 1 (A.5)

It follows that condition (A.1) may only be satisfied for δ ∈]δ,+∞] and for μ ∈]μ,+∞].

Lemma 2 Equation (A.1) implicitly defines μ as a decreasing function of δ over the relevantinterval δ ∈]δ,+∞].

First, we can notice that (δ, μ) = (0, 0) is an obvious solution to condition (A.1). Now, weconsider a case where μ �= 0 and δ �= 0. Plugging Equations (16)-(17) into Equation (A.1) givesafter simplification:

1 − (1 + δ)βp

1 − (1 + μ)βo= −A0

Ap· do

pp· pβo−βp

0 (A.6)

with μ �= 0. This condition implicitly define μ as a function of δ denoted by μ(δ). Differentiatingtotally condition (19) with respect to μ and δ gives:

dμ

dδ= −Ap

Ao· dp

do· pβp−βo

0 · βp

βo· (1 + δ)βp−1

(1 + μ)βo−1< 0 (A.7)

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This result is intuitive: decreasing the price during the peak period results in an increase inthe peak water consumption. This increase in water consumption must be compensated byan increase in the water price during the off-peak period in order to maintain the level of theaggregate water consumption constant. Moreover, it can be easily shown that:

limδ→δ+

μ(δ) = +∞ and limδ→+∞

μ(δ) = μ (A.8)

As a results of these two Lemma, the frontier μ(δ) has the shape depicted in Figure 1 and2. In particular, if μ > μ(δ), then the implementation of the peak pricing results in a reductionof the aggregate water consumption. On contrary, if μ ≤ μ(δ), then the implementation of thepeak pricing results in an increase in the aggregate water consumption.

A.2 No change in the aggregate consumer surplus

The peak pricing will result in no change in the aggregate consumer surplus if and only if Equation(12) holds:

dp · ΔCSp(δ) + do · ΔCSo(μ) = 0 (A.9)

which states that the loss [gain] in consumer surplus during the peak period must be exactlycompensated by the gain [loss] in consumer surplus during the off-peak period.

Lemma 3 Equation (A.9) implicitly defines μ as a decreasing function of δ over the intervalδ ∈] − 1,+∞].

First, we can notice that (δ, μ) = (0, 0) is an obvious solution to condition (A.9). Now, weconsider a case where μ �= 0 and δ �= 0. Plugging Equations (10)-(11) into Equation (A.1) givesafter simplification:

1 − (1 + δ)1+βp

1 − (1 + μ)1+βo= −A0

Ap· do

pp· 1 + βp

1 + βo· pβo−βp

0 (A.10)

for μ �= 0. This condition implicitly define μ as a function of δ. We denote by μ(δ) this implicitfunction. Differentiating totally condition (A.10) with respect to μ and δ gives:

dμ

dδ= −Ap

Ao· dp

do· pβp−βo

0 · (1 + δ)βp

(1 + μ)βo< 0 (A.11)

with μ �= 0. This result is intuitive: an increase of the price during the peak period must becompensated by a decrease in the price during the off-peak period in order to maintain the levelof the aggregate consumer surplus.

Moreover, it can be easily shown that:

limδ→−1+

μ(δ) = +∞ and limδ→+∞

μ(δ) = −1 (A.12)

As a result of Lemma 3, the frontier μ(δ) has the shape depicted in Figure 1 and 2. Inparticular, if μ > μ(δ), then the implementation of the peak pricing results in a reduction ofthe aggregate consumer surplus. On contrary, if μ ≤ μ(δ), then the implementation of the peakpricing results in an increase in the aggregate consumer surplus.

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B Source of data

B.1 Water related information

B.1.1 IFEN-SCEES surveys

The first source of water related information is the water surveys conducted by the FrenchInstitute for the Environment (IFEN) and the Ministry of Agriculture’s Department of Statistics(SCEES) in 1998, 2001 and 2004. Those surveys defined at the local community level (morethan 5,000 local communities in 2001) are representative at the French national level. Thesampling is exhaustive for all local communities with more than 10,000 inhabitants. The samplingrate decreases with the local community size (1/20 for local community with less than 400inhabitants). These surveys include data at the local community level on:

− type of organization chosen by the local community (private versus public, type of delega-tion contract, inter-communal agreement, etc.);

− cost of the water service (investments realized by the water service, labor cost);

− technical characteristics of the water service (network length, pumping capacity, numberof residential or industrial users, water treatment facilities, etc.);

− number of customers and annual water consumption by type (industrial, residential, mu-nicipality);

− and price of the water service (average domestic water price, type of pricing schemes im-plemented, etc.).

B.1.2 The Agence de l’Eau Adour-Garonne withdrawal database

The Agence de l’Eau Adour-Garonne withdrawal database (AEAG database)reports for eachwithdrawal point the volume of water pumped during a given year and during the summerseason (June 1st to October 31) fro year 1999 to 2005. The type of water source is also reportedby pumping point (ground water or surface water). Information reported is this database isexhaustive for the Midi-Pyrénées region and has been aggregated at the local community level.

B.1.3 Summer and rest of the year water consumption

The residential water consumption for the summer and the rest of the year seasons has beencomputed by applying the ratio of summer residential withdrawals to the rest of the year res-idential withdrawals (AEAG database) to the residential water consumption reported in theIFEN-SCEES surveys. The implicit assumption is that the ratio of summer to rest of the yearresidential withdrawals does not significantly differ from the ratio of summer to rest of the yearresidential consumption.

B.2 Socio-economic information

B.2.1 INSEE-DGI fiscal income databases

Information related to household income comes from the INSEE-DGI databases “Revenus fiscauxdes ménages - France métropolitaine”. This information based on exhaustive fiscal declarationsof French households is available at the local community level from years 2000 to 2005. Forthe missing year (1999), a simple interpolation using 2000 and 2001 data has been realized.Information available at the local community level include the median household fiscal income,

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the average number of persons per fiscal household and the average number of consumption units(following the OECD definition) per fiscal household.

B.2.2 French census

The 1999 census describes the characteristics of French’s people and dwellings. The census ofpopulation provides the population and dwelling counts for each local community. The censusalso provides information about French’s demographic, social and economic characteristics.

B.2.3 INSEE tourist capacity databases

Information related to the touristic capacity comes from the INSEE “Capacité des communesen hébergement touristique - France métropolitaine”. For each year, these databases providethe number of beds in hotels or in camping sites summer residential water consumption perhousehold.

B.3 Climate information

Climatic data have been provided by Meteo-France and by the French Institute of Agronomic Re-search (Agroclim unit). Both institutions manage a set of meteorological measurement stations.Climatic information available on daily basis over long periods include minimum temperature(tn), maximum temperature (tx), solar radiation (rg), Penman’s potential evapotranspiration(etp) and rain (pluie) for 29 meteorological measurement stations. Local communities have beenmatched with a meteorological station based on a proximity criterion.

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C Definition of variables

Table C.1: Definition of variablesVariable Definition Type Source

Water consumptionY p residential summer daily water consumption CS AEAG & IFEN-SCEES

(m3/day/user)Y o residential r.o.y daily water consumption CS AEAG & IFEN-SCEES

(m3/day/user)Water pricep water unit price (euro/m3) CS IFEN-SCEESSocioeconomic variablesI household median income (euro/household/year) CS INSEE-DGIucmf average consumption units per household CS INSEE-DGIpersmf average number of persons per household CS INSEE-DGIdens_01 population density in 2001 (100 inhabitant/km2) 1999 INSEEresid_01 share of non permanent dwellings CS INSEEcodtour_01 dummy if tourist area 1999 INSEEsais_p ratio of the number of hotel rooms and camping CS INSEE

places to 1999 total populationTechnical characteristicsrvodvot_01 share of water sold to domestic users CS IFEN-SCEESvulne_01 dummy for nitrate vulnerable area 1999 IFEN-SCEESqmauv_01 share of rivers classified as bad quality 1999 IFEN-SCEESsurf_tot share of water from surface sources CS AEAG

Climate variabletnp summer daily minimal temperature (◦C) CS METEOtxp summer daily maximal temperature (◦C) CS METEOrgp summer daily solar radiation (◦C) CS METEOetpp summer Penman’s PET (mm/day) CS METEOrainp summer daily rain (mm/day) CS METEOtno r.o.y daily minimal temperature (◦C) CS METEOtxo r.o.y daily maximal temperature (◦C) CS METEOrgo r.o.y daily solar radiation (◦C) CS METEOetpo r.o.y Penman’s PET (mm/day) CS METEOraino r.o.y daily rain (mm/day) CS METEO

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