ESTIMATION OF THE DEMAND FOR RESIDENTIAL WATER IN A STONE- GEARY FORM AND THE CHOICE OF THE PRICE VARIABLE MARIE-ESTELLE BINET. Associate Professor

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1 ESTIMATION OF THE DEMAND FOR RESIDENTIAL WATER IN A STONE- GEARY FORM AND THE CHOICE OF THE PRICE VARIABLE MARIE-ESTELLE BINET Associate Professor arie-estelle.binet@univ-rennes1.fr CREM (UMR 611 CNRS), University of Rennes 1, France University of Rennes 1, Faculty of Econoics 7 lace Hoche, CS Rennes cedex, France Abstract. Based on a detailed survey data (concerning the French overseas deartent La Reunion), we investigate the rice secification issue. We estiate the residential water deand function, derived fro a Stone-Geary utility function, using Shin s ercetion rice. As the corresonding araeter is estiated to be around 0.6 and different fro one, we show that the rice erceived by the consuer is between the average and the arginal rice. We conclude that consuers, facing an increasing block ricing, underestiate the true (arginal) rice and therefore consue too uch water. JEL code: Q5, D1, C51. Key-Words: Residential water deand, erceived rice, Stone-Geary. 1

2 1. INTRODUCTION A large eirical literature ais at obtaining consistent values of rice elasticity of deand to analyze the relevance of a ricing olicy to anage water consution. Indeed, estiation of residential water deand has been a ajor issue in environental econoics as roved by the nuber of recent surveys available in the literature (Arbuès et al (003), Dalhuisen et al (003), Worthington and Hoffan (008), Nauges and Whittington (010)). But to roote efficient water use, water authorities need to know what rice variable consuers are resonding to. However, the water tariff structure is often colex with increasing or decreasing rices and fixed charges. Therefore, one iortant issue in the literature is to rovide an adeuate secification of the rice variable in residential water deand odels. The discussion generally focuses on the adeuacy of using the average or the arginal rice. A erfectly infored consuer should react to arginal rice (in addition to Nordin s difference variable). But in case of rice illusion or incolete inforation, the consuer can react to other rice indicator as the average rice. To analyze residential electricity consution, also characterized by the resence of a block ricing structure, Shin (1985) develoed a rice ercetion variable, which is a cobination of arginal and average rices. Indeed, the choice of the rice variable that consuers react to reains an eirical issue. Howe and Linaweaver (1967) are the first to have discussed and coared average and arginal rices for water deand secification. In a first ste, the rice which rovides the best fit is resued to be the rice erceived by consuers, Foster and Beatie (1981). Next, to our knowledge, two other ethodologies have been ileented to exlicitly coare rice variables. Using hyothesis tests on an augented water deand function including arginal rices and two fors of the Nordin s difference variable, Oaluch (198) derived the first test of whether consuers resond to arginal or average rice when faced with block rates. But although Oaluch s odel is creative, it does not erit the consuer to react to a function or both arginal and average rices. Next, Shin (1985) roosed to coare arginal and average electricity rices by testing the value of a rice ercetion araeter. Nieswiadoy and Molina (1991) used this ethodology to deal with residential water consution in a ties series aroach, under a decreasing then increasing block rate schedule. In this context, this aer is an original contribution on residential water deand literature for

3 at least two reasons. Firstly, it is based on a detailed survey data including bills collected on a reresentative sale of households in one overseas French region, La Reunion. This is a uniue icro level data set which allows an exlicit odeling of tariff schee. The second originality of this aer is to aly Shin s ethodology using a Stone-Geary functional for in estiating residential water deand in a cross-sectional aroach. One attractive feature of this secification is that total water consution can be broken down into two constituent arts. The first is the incoressible art, which can be interreted as a iniu level of water consution. The second coonent of total water consution is the variable art which deends on incoe and rice levels. Next, following the translating ethod develoed by Pollack and Wales (1981), one way to roose a better eirical secification and to reduce unexlained variation in exenditure behavior is to ostulate that the araeter easuring the iniu water consution is a function of the characteristics of the households. Using GMM estiator that accounts for both endogeneity of the rice variable, and non linear coefficients, we estiate the water deand function using the Shin s ercetion rice. Our ain finding is that the erceived rice araeter is between zero and one. Then, we conclude that consuers react to a rice which is between the average rice and the arginal rice. Therefore, in an increasing block ricing, consuers underestiate the true (arginal) rice and are consuing too uch water. This aer roceeds as follows. In section, the odel secification is resented. The second section describes the data used in the eirical work. Section 3 exaines and discusses the results. Section 4 concludes.. WATER DEMAND SPECIFICATION First, we describe the ricing odeling. We then resent the water deand secification and describe the translating ethod..1 Price easureent In Reunion island, the ricing of residential water consists of several (between 3 and 4) increasing consution blocks. To silify the resentation, if the rate schedule is assued to consist of two blocks and the fixed charge, the consuer s budget constraint can be written 3

4 as following: (1) F 1 k1 ( k1) x X I Where denotes residential water consution easured at the household level. 1 and are resectively the first and the second block rices. k 1 is the highest consution level in block 1. x and X are resectively an index rice and the corresonding consution of other rivate goods. I is the household incoe. F is the fixed art. To deal with block rate structure colexity, we often consider a variable insired by Nordin (1976). Indeed, Nordin (1976) clais that consuers react not only to arginal rice, but also to the change in the intra-arginal rices The Nordin s difference variable D is eual here to the difference between the bill would have been if the water uantity was consued at the arginal rice less total bill. This variable, as the fixed art, easures an incoe effect iosed by the tariff structure: () D ) 0 ( 1 k1 if we consider an increasing consution block. Then, the budget constraint can be rewritten as: ( 3) x X I F D Therefore, if consuers are erfectly infored, they should react to arginal rice and to adjusted incoe: I F D. However, if consuers are ierfectly infored, they ay k k resond to average rice: AP ( ) We confor to Taylor et al (004) who advice to urge the fixed charges fro the average rice as it creates a bias in the estiated rice elasticity. Next, Shin (1985) defines the erceived rice as the rice to which the consuer resonds. Facing a colex tariff structure, the consuer ay have a artial knowledge of it, and the erceived rice ay include one fraction only of the difference variable (with total bill), and D thus ay lie between arginal and average rices as: AP. As the choice of the rice variable is an eirical uestion, Shin (1985) built a rice ercetion variable as a function of arginal and average rices and a rice ercetion 4

5 araeter k to be estiated: (4) * AP P ( ) k We can see that the ratio AP catures the effect of the intra-arginal effects on rice ercetion. If k is eual to zero, we conclude that consuers resond to arginal rice whereas if k is eual to one, they resond to average rice. If 0 k 1, then, the consuer s * erceived rice is between average and arginal rices. If k 1, we have AP. k was estiated by Shin (1985) and Van Helden, Leeflang and Sterken (1987) for electricity deand and by Nieswiadoy and Molina (1991) and by Nieswiadoy and Cobb (1993) for water deand in a log linear secification. The results of this test gave ixed results. Shin (1985) found a araeter eual to 1.007, with the null hyothesis k 1 not rejected. Shin concluded that the erceived rice is average rice. Nieswiadoy and Molina (1991) obtained a non significant araeter under increasing water consution block and conclude that the consuers resond to arginal rices. But the reverse result was found in Nieswiadoy and Cobb (1993). But they included the fixed charge in the average rice which biased the estiate of k toward unity and therefore invalidates the Shin test according to et Taylor et al (004).. Residential water secification in a Stone-Geary for We secify the consuer s utility function in the following Stone-Geary for: U, x) ln ( ) (1 ) ln ( x ). In the literature, to our knowledge five other ( x articles have used such a secification to analyze residential water consution, Dhararatna and Harris (010), Binet et al (009), Madhoo (009), Martinez-Esineira and Nauges (004) and Gaudin et al (001). The assutions of this function are strong searability, the adding-u restriction, ositive arginal budget share and and x x. x is often considered as a subsistence level of consution. can be regarded as a 5

6 iniu water consution. Then, residential water deand function is derived fro the following rogra MAX U ( x, ) subject to y x x : (5) ( y x x ) Where is the residential water rice variable erceived by consuer. This secification is a (silified) linear exenditure syste (LES) if we consider only two goods (water consution and other consution goods). This secification has the advantage of being coatible with theory while exlicitly odelling a iniu level of consution, irresective of the rice of the consued good or the consuer s incoe. The consuer first urchases the iniu level of each good, and the left-over incoe then is allocated in a fixed roortion to the deand for residential water. Since residential water consution usually is constrained by the size of household and water using caital stock, the LES secification is articularly well suited to account for these features. Without satial variation for x, we suose that the inial value of other consution goods can be incororated in araeter : (6) ( y ) Next, the rice ercetion variable is substituted into (6) to obtain the following secification to be estiated: (7) ( y AP ( ) AP ( ) k k ) Where k is the rice ercetion araeter, is the arginal rice, AP the average rice and is the error ter.. Translating ethod Pollack and Wales (1981) develoed the translating ethod in a deand secification to allow subsistence araeters to deend on deograhic variables. Martinez-Esineira and 6

7 Nauges (004) and Gaudin et al (001) also ileented the translating ethod to estiate the deand for residential water in a LES secification. Using the sae data set, Binet et al (009) have reviously described in the following syste of euations (8-10): (8) 1Nc Garden 3 Swiing ool Firstly, outdoor water uses is suosed to be one deterinant of the incoressible water consution. Indeed, the resence of a garden is an iortant factor as 77% of households leave in a house with a garden in La Reunion island against 30% in etroolitan France, in 004. Only 5 % of household have a swiing ool in our sale but the resence of a swiing ool can exlain the huge consution levels observed. The resence of a garden or a rivate swiing ool is defined here as a dichotoous variable that takes the value of one if the faily has a garden or a swiing ool, zero otherwise. We would have liked to use data giving the size of the garden and the volue of the swiing ool, but our data set gives only inforation on the resence or not of these euients. Following Carlevaro et al (007), the size of the oulation of water users Nc can be differentiated by distinguishing the nuber of child and the nuber of eloyed adults (NEA) fro the rest of the faily (Nuber of Uneloyed Adults): ( 9) Nc NUA 1 NEA CHILD If we consider the needs of daily life, intended for the whole of the faily, we exect that the water consution will increase with the household size. But, we exect a lower iact of the nuber of eloyed adults (NEA), as they send ore tie outside hoe than the other ebers of the faily. As a great roortion of households has a vegetable garden, euation (10) enable us to include the cliatic effects on consution: ( 10) 0 1 Weather Weather is easured by the ercentage of non rainy days over the billing eriod. 1, 0, 1, 1 and are the araeters to be estiated (with and k). Euations (8-10) are substituted in the water deand function (7) to obtain one euation with non linearity in 7

8 araeters. 3. HOUSEHOLD SURVEY We first offer a brief descrition of the survey and then introduce the data selected for eirical analysis. 3.1 Household survey A stratified rando survey was financed by the Regional Directorate for the Environent (DIREN) and conducted in 004 on a sale of 000 households. The survey designers retained a stratified saling of 000 households according to unicialities, i.e. 1% of total household oulation, see Carlevaro et al (007). The uestionnaire included 5 uestions concerning: Socio econoic household characteristics (sex, householder age and occuation, faily incoe and size, nuber of working eole and child, and if they are tenants or roerty owners). Housing characteristics (individual faily house or aartent, age, nuber of roos and altitude). Water consution euient (swiing ool, washing achine, dishwasher, garden or vegetable garden ownershi). Consution habits (washing freuency, business activity at hoe). Carried out by telehone, this survey was coleted with a ailing to 1000 volunteer households, intended to collect the volue of water consution dislayed in the last three bills (covering one year), later by ost. Finally, we collected the volue of water consution recorded on 173 households, including 449 useful water bills (fro 1 to 3 water bills er household). Since the billing eriod varies across unicialities, consution data were converted to a daily consution er household (in liter). Corresonding daily weather indicators as reciitation, teerature were taken fro Meteo France Agency. 8

9 3. Descrition of the variables a) Price easureent Data on water rices and consution blocks length were available fro DIREN. The rice variable includes sewage rices. We subtract the fixed charge fro the total bill to coute the average rice, to avoid bias estiates, see Taylor, McKean and Young (004) for a discussion. By urging the fixed ortion of the bill, the average variable rice includes only the ortion which varies with consution. b) Incoe The DIREN survey, conducted in 004 on a sale of 000 households, recorded household incoe level 1 as an ordered ualitative variable, naely as belonging to one of the following five incoe intervals (in Euros/onth): [0;750], [750;1500], [1500;3000], [3000;4500] and [4500;. + ]. To irove the exlanatory ower of estiates, Carlevaro et al (007) decided to easure household incoe levels as a uantity. Therefore, they develoed an iutation ethod based on an econoetric odel describing the observed ualitative inforation on household incoe according to an ordered olychotoous econoetric odel, where the unobserved household incoe level is secified as a latent variable. Then, they assue that the for of the ersonal incoe distribution is influenced by ordinal easure of household standard of living, stated by households on a four level scale, naely «The household hasn t enough oney to live. It cannot ake ends eet», «The household has just enough to live, but by aking great sacrifices», «The household has enough oney to live without aking great sacrifices», «The household akes no sacrifices, in any case nothing iortant». They finally obtained 0 different incoe values. Next, as discussed before, we use adjusted incoe, euals to incoe less fixed charges lus the difference variable to allow for incoe effects of these variables. c) Cliatic variables Cliatic effects can be introduced in different ways. Here, we follow Matinez-Esineira (00) using the ercentage of non rainy days over the billing eriod. Indeed, Carlevaro et al (007) have showed that users resond ore to the occurrence of rainfall than to its aount. 1 Households were asked to take into account all their incoe sources, including welfare benefits, roerty incoe, 9

10 Therefore, we use daily observations recorded by Meteo France Agency (about one hundred stations set u in La Reunion). These geograhical distributed observations allow us to coute the nuber of days without rainfall for each bill collected according to the observations recorded at the closest weather station. d) Euients and other variables collected We exeriented other exlanatory variables, articular those easuring billing freuency, housing characteristics, consution habits or the resence of a dishwasher or a washing achine. But the araeters associated with all these variables turned out to be highly insignificant (robability greater than 30%), robably owing to their ierfect easure. As variables as the stock of aliances esecially hel to analyze long run reactions to a shock rice, indeed, that uestion reaches beyond the scoe of this article. The set of available variables, easured in 004, and suary statistics for all variables are resented in Table 1: Table 1. Data set descrition, 449 water bills in 004 Descrition of variables Mean Min Max Daily water consution er household (liters) Monthly household incoe (Euros) Average rice without fixed charge (Euros for one liter) Marginal rice (Euros for one liter) Nuber of child in a faily Nuber of eloyed adult in a faily Nuber of uneloyed adult in a faily Percentage of days without rainfall (weather variable) In our sale, average residential water consution is 675 liters er household and er day (which corresond to 53 liter er caita in average). Coutelier and Le Jeannic F. (007) confir the relevance of our estiate as they easured water consution to be eual to 69 liters er inhabitant in La reunion, which is siilar to highest consution levels in OECD. For exale, average daily er caita water use in the UE-15 countries ranges fro 115 liters in Belgiu to 65 liters in Sain. In La reunion, er caita residential water consution is about 60 % greater than in etroolitan France, in average. Likewise, distinctive features like 10

11 water rices, which are very low in Le Reunion (less than one Euro er 3 in average), the way of living (a high roortion of households live in a house with a garden), or cliatic effects, ay contribute to exlain those high water consution levels, see Binet et al (009) for further results. 4. EMPIRICAL RESULTS Estiation strategy is develoed in 4.1. Next, eirical results are reorted in Estiation strategy One roble, which ust be addressed with increasing block rate, is siultaneity as consuers select the uantity of water and the rice siultaneously. We therefore choose to ileent a GMM (Generalized ethod of oents) estiator using an aroriate set of instruents. In the sirit of Hausan and Wise (1976), rices associated with fixed levels of consution (the three first uartiles of the distribution) are used as instruents for arginal and average rices. An instruental variable ust satisfy two reuireents. It ust be correlated with the endogenous variable and be orthogonal to the error ter. We therefore used the Bound et al (1995) test to select relevant instruents and erfored the over-identifying restrictions Hansen-Sargan test to choose valid instruents. As the Breush-Pagan test reveals the resence of heteroscedasticity, robust standard errors are couted in a heteroscedastic-consistent atrix. Once the endogeneity in the rice variable has been urged via IV or GMM estiators, the rice coefficient acuired the exected sign whereas it was ositive with sile OLS techniue. 11

12 4. Results Eirical results obtained for two nested secifications are listed in Table : Table. Estiation results obtained with GMM, Shin s rice ercetion ethodology. Secification all coefficients significant coefficients Coefficient estiates Value (robability) Value (robability) (arginal budget share) (0.071)* (0.000)*** (0.000)*** (0.000)*** (uneloyed adult) (0.17) (garden, coon effect) (0.4) (garden with weather effect) (0.014)** (0.08)* (child) (0.54) (eloyed adult) (Swiing ool) k (ercetion rice araeter) 3 93 (0.65) 0.66 (0.005)*** (0.000)*** Mean value of inial water consution 480 (71%) 84 (4%) Overidentifying restrictions test (-value) 0.011** Adjusted R² Nuber of observations Test: H : 1 Rejection Rejection 0 k Significance level: *** for 1%, ** for 5% and * for 10%. The first odel includes all the araeters (see the second row). In the last row, insignificant values of araeters obtained in reliinary estiations finally have been set eual to 0 1

13 using an iterative rocedure. In both secifications, the value of the consution of other rivate goods is fixed to be 0 to obtain consistent estiates. Hansen test indicates that the validity of instruents is not rejected in the final secification only. Our first oint of interest is the value of Shin s rice ercetion araeter. We obtain robust and significant values for it, eual to 0.66 in the first secification, then to 0.56 in the final one. In both cases the null hyothesis k 1 is rejected, which eans that the consuer s erceived rice is between average and arginal rices. As the water bill reresents a sall ercentage of incoe, consuers are robably not well infored about the ricing structure. Therefore, the consuer ay underestiate the real rice and consue too uch water. Next, the iniu residential water consution level ˆ can be couted for each bill using estiates araeters and euations (8-10). Finally, average values are around 71 ercent of total er household consution with the colete secification yet around 4 % only if we consider the final odel. In the final secification, the resence of a garden has a ositive but insignificant iact uon unicial sending, which exlains these different values for incoressible water consution. We conclude that, at least, 4% of residential water consution will not be sensitive to a rice increase. To delve further into the interretation of the incoressible er household water consution, we can analyze other results obtained. We observe a ositive and significant iact of household size on the iniu water consution: an increase of one uneloyed adult will result in an increase of 178 liters er day er household. This effect is weaker if we retain all the coefficients (resectively +165). Next, one additional child will generate an increase of around 60 liters er day in household water consution in the final secification. But the nuber of eloyed adults has no significant iact on residential water consution ( 0 1 ). 4 CONCLUSION Using one uniue cross-sectional dataset covering a reresentative sale of 000 households in 004, we estiate the residential water deand function using a Stone-Geary for. The translated Stone-Geary secification for water deand and the focus on the erceived water 13

14 rice araeter to be estiated, are uniue to the current aer. Using a non linear GMM ethod with rices that the household would face at different fixed levels of water consution as instruents, we obtain the following eirical results. First, the evidence fro La Reunion island suggests that the erceived rice to which consuers resond is lower than arginal rice. Therefore, consuers ay consue too uch water. Second, as our results show that, at least, 60 % of consution should react to rice variations, which is not negligible, the choice of the aroriate rice secification reains a relevant issue. REFERENCES Arbuès F., Valienas M.A and Martinez-Esieneira R. (003),"Estiation of residential water deand: a state-of-art review", Journal of Socio-Econoics, 3, Binet M.E, Carlevaro F. and M. Paul (009), "La deande d eau otable à La Réunion : estiation à artir de données d enuête", unublished anuscrit. Bound J., Jaeger D. and Baker R. (1995), Probles with instruental variable estiation when the correlation between the instruents and the endogeneous exlanatory variable is weak, Journal of the Aerican Statistical Association, 90, Carlevaro F., Schlesser C., Binet M.E, Durand S. and M. Paul (007), "Econoetric odeling and analysis of residential water deand based on unbalanced data", Alied Econoetrics (Pricladniaya Ekonoetrica), n 4 (8), , 007, Market DS, Moscow. Coutelier and Le Jeannic F. (007), La facture d eau doestiue en 004, 177 Euros ar ersonne et ar an», Ifen, le 4 ages, vol. 117, ars Dalhuisen J., Florax R. De Groot H.and Nijka P. (003), «Price and incoe elasticities of residential water deand: a eta analysis, Land Econoics, vol. 79, issue, Dhararatna D. and Harris E. (010), Estiating residential water deand using the Stoine- Geary functional for: the case of Sri Lanka, Discussion aer, 46/10, Deartent of Econoics, Monash University. Foster H.W and Beatie B.R (1981), On the secification of rice in studies of consuer deand under block rice scheduling, Land Econoics, issue 57, Gaudin S., R. Griffin and Sickes R. (001), Deand secification for unicial water 14

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