Evaluating Irrigation Water Demand

Evaluating Irrigation Water Demand CHRISTOPHE BONTEMPS and STEPHANE COUTURE 1. Introduction Irrigation is a ig-volume user of water. 1 Irrigated agriculture accounts for a large proportion of water use, especially in many water-scarce areas. Imbalances between water needs and resources are likely to induce conflicts between different categories of users (rural, urban, industrial and oter users). In tese situations, as water becomes scarce, 2 policymakers attempt to induce farmers to reduce water consumption by implementing economic tools suc as quotas or water pricing. Recent reforms on water policies ave been implemented (cf. Dinar, 2000) to make farmers increasingly responsible for water conservation. Te final result would lead to force all water users to pay te water at its value. Under scarcity, understanding te influence of water price on water-concerned activity is a key contribution to policy analysis. Tis knowledge, wic is te starting point to define water pricing, is inerent to te estimation of te water demand. As in most oter regions and countries, farmers in France are carged for water. Te fees are fixed at low levels and do not make farmers responsible for te costs tey impose on water supplies. Moreover te knowledge of farmers water consumption remains unprecise. In tis context, estimating farmers water demand is difficult and several questions are still unanswered. Tere is an important literature assessing ow farmers react to canges in te price of water using eiter econometric models or programming models. Tese two approaces on irrigation water demand estimation use different tecniques, different data and lead to various results. 1.1. STATISTICAL ESTIMATION OF DERIVED DEMAND FOR IRRIGATION WATER Irrigation water demand estimates relying on actual farmer beavior are usally based on cross-sectional water use data (Ogg and Golleon, 1989). Tese estimations ave commonly used dual input demand specifications (Moore and Negri, We would like to tank Pascal Favard for elpful comments on an earlier draft, and for is faitful support. We also would like to tank all te participants at te Symposium on Water Resource Management, in Nicosia (Cyprus) 2000, for critical remarks. P. Pasardes et al. (eds.), Current Issues in te Economics of Water Resource Management: Teory, Applications and Policies, 69 83. 2002 Kluwer Academic Publisers. Printed in te Neterlands. cap2-2a.tex; 6/11/2001; 11:19; p.1

70 C. BONTEMPS AND S. COUTURE 1992; Moore et al., 1994a, 1994b; Hassine and Tomas, 1997). Tey represent farmers as a multicrop production firm taking decisions concerning crop coices, crop-level allocations of land in a long-run setting, and water use quantities in te sort-run. Tese studies suggest tat water irrigations are unresponsive to cange in prices. Te inelasticity of irrigation water demand provided by econometric models may be due to te lack of data on crop-level water use. Moreover tese works do not use an acreage-based model or a fixed allocatable input model of water use tat could better explain sort-run decisions. 1.2. DERIVED DEMANDS FROM PROGRAMMING MODEL Te absence of observations on water consumptions as induced te use of matematical programming approaces for te estimation of te derived demand for water. Demand estimates ave been derived from sadow prices obtained by computer simulations of profit maximizing beavior. Many of tese programming studies use linear programming (Sunway, 1973) or quadratic programming (Howitt et al., 1980). Irrigation demand curve estimates were found to be significatively elastic. Tese approaces conclude tat irrigation water demand is completely inelastic below a tresold price, and elastic beyond (Montginoul and Rieu, 1996; Garrido et al., 1997; Varela-Ortega et al., 1998; Iglesias et al., 1998). Te programming metod studies are based on te following sceme. For a given price, one estimates te quantity of water maximizing te farmer s profit. Variations in water prices induce different levels of optimal water quantities. Te autors use tese informations directly to represent te derived demand for irrigation water. Several assumptions are made concerning te crop yield response function to irrigation water. A major critic is tat tese pre-specified functions may not precisely represent te biological and pysical process of plant growt. Anoter drawback is tat tese models ignore te impact of multiple applications of te water for eac crop and give more empasis on crop pattern sifts. Tis capter aims at proposing a metodology to estimate irrigation water demand using programming metods. Our approac is owever different from te existing works and is based on te evaluation of te farmer s value for water. Since markets for water do not exist in France it is not simple to determine tis value. Te first difference wit te existing works concerns te definition of te farmer s value for water. We define it as te maximum amount of money te farmer would be willing to pay for te use of te resource. Anoter major difference concerns te production function used in te estimation. Inere, we do not specify tat function and use a crop-growt simulation model, wic gives precise results concerning water-yield relations. Te procedure we propose is a two-stage metod composed of a model of farmer beavior description and optimization, followed by an estimation procedure, te latter being cosed to be nonparametric ere. First, using a primal cap2-2a.tex; 6/11/2001; 11:19; p.2

EVALUATING IRRIGATION WATER DEMAND 71 optimization approac as modeling framework, we describe te program of te farmer. Tis one must allocate a limited water supply over an irrigation season in order to maximize is profit evaluated at te arvesting of te crop. We propose a numerical metod for obtaining solutions to tis problem based on an optimization algoritm. Tis algoritm integrates te agronomic model, EPIC-PHASE 3 (Cabelguenne and Debaeke, 1995), an economic model, and an algoritm of searc of te optimum. Tis numerical procedure is used for different weater conditions, and for several total water quantities available for irrigation. It generates a database composed of levels of quotas and te associated maximized profits. Second, using tis database and nonparametric metods, we estimate te maximized profit functions. Te irrigation water demand functions are also estimated by a nonparametric derivation procedure. Tis metodology is applied to estimate water irrigation demand in te Soutwestern of France. Our results sow tat te irrigation water demand functions are decreasing, nonlinear, and strongly depend on weater conditions. Tese functions can be decomposed into several areas. Irrigation water demand is inelastic for low water prices; and becomes more elastic as prices increase. Te price levels at wic te canges in responsiveness to prices appear depend on climates and vary around 0.30 F/m 3 for wet weater conditions up to rougly 1.60 F/m 3 for a dry year. Tese informations are a decisive contribution for defining water pricing policies wen water scarcity becomes a leading issue. Te capter is structured in te following manner. Section 2 describes te metodology for evaluating irrigation water demand function. First, we describe te teoritical metod for calculating demand functions; second, te numerical approac to estimate irrigation water demand is presented. Section 3 develops te empirical specification of te model used to estimate a specific regional-dependent water demand and reports te empirical results. We conclude in Section 4. 2. Metodology 2.1. DEMAND FUNCTION CALCULATIONS Te calculations of te water demand of te farmer are based on te evaluation of te farmer s value for water under water scarcity. Te water demand functions are derived from te dynamic model of te farmer s decisions. 2.1.1. Analytical Model Consider a farmer facing a sequential decision problem of irrigation sceduling on a calendar 1,...,T wit T 1 decision dates. At date 1, te farmer knows te water quota available for te season, Q, te initial water stock in soil, V,andte state of crop biomass, M. At eac decision date, te farmer as to irrigate, 4 using a quantity of water denoted q t. cap2-2a.tex; 6/11/2001; 11:19; p.3

72 C. BONTEMPS AND S. COUTURE Te dynamics of te tree state variables respectively are te following: for t = 1,...,T 1, M t+1 M t = f t (M t,v t ), (1) V t+1 V t = g t (M t,v t,q t ), (2) Q t+1 Q t = q t. (3) Te cange in te level of te biomass at any date (Equation (2)) is a function (f t ) of current date biomass and water stock in soil. Te cange in water stock in soil (Equation (3)) depends on te same state variables and on te decision taken at te current date. Te quota as a simple decreasing dynamic (Equation (3)). Tere exists tecnical constraint on irrigation represented by te following equation: q q t q for q t > 0. (4) Te final date (t = T) corresponds to arvesting wen actual crop yield becomes known. Let Y( ) denote te crop yield function; tat quantity depends only on te final biomass at date T and is denoted Y(M T ). Te profit per ectare of te farmer can be written as: T 1 = r Y(M T ) C FT (c q t + δ t C F ), (5) t=1 were r denotes output price; C FT denotes fixed production costs; c is water price; δ t is a dummy variable taking te value 1 if te farmer irrigates and 0 if not. C F represents fixed costs; tese costs appear since te farmer is facing labour and energy costs for eac watering. Te farmer sequential problem is te following: T 1 Max r Y(M T ) C FT (c q t + δ t C F ) (6) {q t } t=1,...,t 1 t=1 s/c and s/c { Mt+1 M t = f t (M t,v t ), V t+1 V t = g t (M t,v t,q t ), Q t+1 Q t = q t, { 0 if qt = 0, δ t = 1 if q t > 0, q q t q for q t > 0, M t 0, V t 0, Q t 0, M 1 = M, V 1 = V, Q 1 = Q. (7) (8) cap2-2a.tex; 6/11/2001; 11:19; p.4

EVALUATING IRRIGATION WATER DEMAND 73 Te constraints (7) are te main dynamics wile (8) are tecnical, and pysical constraints. Tis problem is purely analytical ere and te functions f t and g t are unknown. We will owever solve it in te next section using an agronomical simulation tool and an optimization procedure. Te solution of tis program is te optimal sequence of decisions: q (Q) ={q t } t=1,...,t 1. (9) Tis sequence strongly depends on te quantity of water available for irrigation, Q. Using (9) and (5), we obtain te optimized profit function depending on Q as follows: T 1 (Q) = r Y (M T ) C FT (c qt (Q) + δ t C F ). (10) t=1 2.1.2. Definition of te Value of Water Under limited water supply, te value of water as an economic good is te amount of money te farmer is willing to pay for it. Like any oter good, water will be used by farmers as long as te benefits from te use of an additional unit of ressource exceed its cost. As water becomes scarce, te farmer value of water becomes greater tan te real water price. In oter words, te marginal profit dπ/dq is greater tan te water carge c and terefore te farmer is willing to consume more water. For a given Q, wedefinete water value as te maximum amount of money te farmer would be ready to pay for te use of one additionnal unit of te ressource. Tis opportunity cost, noted λ(q), is defined as te derivative of te optimized profit function: λ(q) = d (Q) (11) dq evaluated for te given quota. Te knowledge of λ(q) for any value of te quota Q gives te willingness to pay function of te farmer. Tis function, usually noted p(q), is te inverse of te irrigation water derived demand function, noted by Q(p) were p is te irrigation water pricing. Terefore, te irrigation water derived demand function is completely determined once its inverse, te willingness to pay is known. 2.2. DEMAND FUNCTION ESTIMATION In order to estimate demand function, initially it is necessary to ave data relating to quotas and associated maximized profits, and secondly to define an estimation procedure. cap2-2a.tex; 6/11/2001; 11:19; p.5

74 C. BONTEMPS AND S. COUTURE Figure 1. Numerical procedure of resolution. 2.2.1. Data Generation We generate data composed of maximized profits and quotas by solving te sequential decision model described in Section 2.1.1. We will present only a brief description of numerical resolution procedure of tis problem ere; a detailed description of te sequential decision model can be found in Bontemps and Couture (2001) or Couture (2000a). Figure 1 sows te numerical procedure integrating te agronomical model, EPIC-PHASE, an economic model, and an algoritm of searc of te solution. To simplify te presentation of te numerical framework, we consider te case of known weater conditions and for a given quota. Te agronomical model, EPIC-PHASE, was used to generate informations relative to te state variables (previously represented by te transition functions f t and g t ), and to determine crop yield for various irrigation scedules. Given te prices of inputs and output, te economic model uses te yield predictions from te plant-growt simulation model, and evaluates te profit for eac decision sequence. Finally te algoritm cap2-2a.tex; 6/11/2001; 11:19; p.6

EVALUATING IRRIGATION WATER DEMAND 75 of searc identifies te optimal decision pat by examining exaustively te set of all simulated profits. 5 Ten te maximized profit,, is obtained. By repeating tis procedure for different levels of quotas and ten for different weater conditions, we obtain a database on wic we base our estimation procedure. 2.2.2. Nonparametric Estimation In order to estimate te water value, it is necessary to estimate first te optimized profit function and its derivative. We cosed a widely used nonparametric kernel estimator (Pagan and Ulla, 1999), to estimate profit function and demand function. A major advantage of nonparametric metod is tat it allows to estimate an unknown function witout assuming its form. Estimation is only based on observed data and is very powerful. In te past decades several nonparametric estimators ave been developed (Härdle, 1990), tey are all based on a weigted sum of functions of te data. Profit function estimation Using (10), te estimation of te optimized profit function lies on te set of simulated data (Q i, i ) i=1,...,n obtained by te numerical model. Te unknown function, ( ), is ponctually estimated on tese n couples (Q i, i ).Tekernel estimator of profit function evaluated for any value of Q, is a weigted sum of te observed responses Q i, te weigt being a continuous function of observed quantities, Q i, and evaluation point Q. It is defined as: π (Q) = ( ) n i=1 i K Qi Q n i=1 K ( Qi Q ) Q R, (12) were K( ) is a kernel function, continuously differentiable. We use a Gaussian kernel function among existing kernel functions. 6 Note tat π (Q) will inerit all te continuity and differentiability properties of K. Terefore π (Q) is continuous and differentiable. Te bandwidt, noted, determines te degree of smootness of π (Q); its coice will be discussed latter in tis section. Demand function estimation In order to estimate te demand function, we will use te property tat te profit function estimator is differentiable. If te estimate π (Q) properly reflects te profit function, (Q), ten te estimate of te profit function derivative is equal to te derivative of te estimate of te profit function (Härdle, 1990). Terefore a derivation of (12) wit respect to Q will give an estimator of te demand function. 7 cap2-2a.tex; 6/11/2001; 11:19; p.7

76 C. BONTEMPS AND S. COUTURE In oter words, te estimator π / Q( ) of / Q( ) is just te derivative of te estimator π ( ). More precisely: π Q (Q) = ( ) n i=1 i K Qi Q Q ( ) Q R. (13) n i=1 K Qi Q Tis can be rewritten as: π Q (Q) = 1 ( n ( ) ) 2 i=1 K Qi Q ( ( n i 1 ( ) ) ( n ( ) ) Qi Q Qi Q K K i=1 i=1 ( n ( ) ) ( n ( ) )) + i K Qi Q 1 Qi Q K. i=1 i=1 Te estimator π / Q(Q) is also continuously differentiable because it incorporates te kernel function K( ) and its derivative K ( ). Smooting parameter selection Coosing te bandwidt,, is always a crucial problem. If is small, ten we get an interpolation of te data. On te oter and, if is ig, ten te estimator is a constant function tat assigns te sample mean to eac point. Tere exist several approaces to bandwidt selection (Vieu, 1993) using teorical considerations (plug-in metod) or based-data metod (cross-validation metod). A feature of tese approaces is tat te selected bandwidt is not fully adapted, particularly if te number of observations is small. We use as a bencmark te value obtained by cross-validation. Te aim of tis metod is to coose a value for minimizing te cross-validation criterion, defined as a sum of distances between te estimator π ( ) evaluated at Q i and te real data observed i.we denote te bandwidt selected by te cross-validation criterion by. In practice, a refinement consists in using a sligtly smaller bandwidt tan in order to limit oversmooting. Following Härdle (1990), te smooting parameter selected for demand function estimator is te same tat te one coosed for profit function estimator, even if tis argument may be discussed. cap2-2a.tex; 6/11/2001; 11:19; p.8

EVALUATING IRRIGATION WATER DEMAND 77 Table 1. Caption?. Source: ITCF (1998); Micalland (1995) and Couture (2000a). Year Output price Water price Fixed Cost per irrigation Fixed cost r (Francs/Tonne) c (Francs/m 3 ) C F (Francs) C FT (Francs) 1989 1049 0.25 150 2150 1991 1038 0.25 150 2150 1993 778 0.25 150 2150 3. An Application in te Sout-West of France Te procedure for demand function determination was applied using data prevailing in te area of te Sout-West of France. In tis area, agriculture is te largest water consumer wit 2/3 of total water consumption. During low river flow periods tere is a strong competition on water wit urban and industrial uses. In tis area, irrigated agriculture is quite recent and concerns a large part of crops suc as corn. Irrigation needs depend strongly on weater conditions. Irrigation water was drawn from rivers supplied by mountain reservoirs. Te irrigation tools used in te Sout- West of France are generally sprinkler systems. Te reference crop is corn because it remains te main irrigated crop in tis area. 3.1. DATA A first set of data is required by te crop growt simulator model. Tis data set includes weater, soil, tecnical and irrigation practices, and crop data. Te daily weater input file was developed from data collected at te INRA station in Toulouse, for a 14-year series (1983 1996). Te soil caracteristic data were included in te crop growt model. Te soil is clayey and calky. Economic output and input price data are included as a secondary data set, see Table 1. Output prices are farm-level producer prices. Input prices include irrigation variable costs and fixed costs by watering, and oter fixed production costs. Te fixed cost, (C F ), per irrigation includes energy and labour costs. Te fixed production costs, (C FT ), are composed of fertilizer, nitrate, seed, and ail insurance costs. 3.2. ESTIMATION RESULTS We ave allowed te quota of available water for irrigation to vary 8 between 0 and 4500 m 3 per ectare. In order to account for weater variability, we ave run te model over 14 years available, but we ave restricted our attention for tree climates : a dry year (1989), a medium year (1991) and a wet year (1993). cap2-2a.tex; 6/11/2001; 11:19; p.9

78 C. BONTEMPS AND S. COUTURE Figure 2. Profits functions for dry, medium and wet year. cap2-2a.tex; 6/11/2001; 11:19; p.10

EVALUATING IRRIGATION WATER DEMAND 79 Figure 2 sows te estimation of te maximized profit functions. Tese estimations are presented for eac year considered in our study. Note tat tese estimated profit functions ave te same trend; tey increase to reac a peak tat depends on weater conditions, and ten remain constant at a maximum level. Tis trend is due to te yield-water relation; from te agronomic knowledge we know tat plant yield increases as water quantity increases below some maximum value, ten te yield decreases wile increasing water quantities (Hexem and Heady, 1978). Te estimated profit functions ave some similarity wit yieldwater functions except wen water is no more a limiting factor; in tis case, te profit remains constant. Te maximum profits vary wit weater conditions (tey reac rougly 10500, 8400, and 6200 Francs per ectare for dry, medium and wet climates respectively), and are obtained for various quantities depending on te weater (tese quantities are 2900, 1700, and 1350 m 3 per ectare for dry, medium, and wet weaters respectively). Te demand function estimations are directly derived from te estimated profit functions, as sown in Figure 3. Te results of te demand estimates in Figure 4 sow clear differences in water demand for te tree weater conditions. Te drier te weater, te greater te irrigation water demand. However tese curves present te same sape and trends. Tese tree demand functions are decreasing and nonlinear. Water prices being set at te range of 0.6 to 2.61 F/m 3, depending on weater conditions, induce a null water demand. From te water quantity available equal to 2900 m 3 per ectare in dry year, to 1700 m 3 per ectare in average year, and to 1350 m 3 per ectare, it is not in te farmer s interest to irrigate. Te differences between tese maximum quantities can be very important. Note tat tis water quantity increases twofold from te wet year to te dry one. Tese differences are due to precipitations. 9 Water demand will be all te more ig so since precipitations will be low. Anoter interesting feature is tat te estimated water demand functions ave inflexion points and can be decomposed into areas. At low water prices, irrigation water demand seems to be inelastic. On te oter and, demand appears more and more elastic as price increases. Te price levels at wic te canges in price-responsiveness appears depend on weater conditions and are ranging from 0.30 F/m 3 in wet year to 1.60 F/m 3 in dry year. Up to tese prices te variations in water consumptions wit respect to te cange in prices appear quite significant. Tese results are confirmed by existing studies using programming metods in te literature on irrigation water demand (Montginoul and Rieu, 1996; Iglesias et al., 1998; Varela-Ortega et al., 1998). Te knowledge of te price tresolds provides crucial information for te policymaker in order to initiate a water pricing reform. Te response to price signals in water saving strategies depends on climate conditions. For example, an increase of price in te range of 0 to 0.9 F/m 3 may not produce a significative reduction of cap2-2a.tex; 6/11/2001; 11:19; p.11

80 C. BONTEMPS AND S. COUTURE Figure 3. Profit and demand function for te dry year (1989). water consumption according to weater conditions; in a dry year, te consumption of water is not modified wereas under wet year, water demand becomes null. Tis aspect needs to be integrated in defining water pricing. If water savings are te policy objective, water prices need to be set at sufficiently ig levels. 4. Conclusion Our paper presents a metodology for evaluating irrigation water demand based on te evaluation of te farmer s willingness to pay for aving one additional unit of water. Demand functions are obtained troug a sequential making-decision cap2-2a.tex; 6/11/2001; 11:19; p.12

EVALUATING IRRIGATION WATER DEMAND 81 Figure 4. Demand functions for dry, medium and wet year. program. Te estimation procedure is based on data generated by a numerical metod integrating a crop-growt model, an optimization procedure linked to a economic model, and nonparametric metods. Tis metod is applied to used to estimate demand functions for data prevailing in te Sout-West of France. We represent tese functions for tree representative climatic years. We sow tat irrigation water demands strongly depend on weater conditions, but ave te same sape: tey are decreasing and nonlinear. We can decompose eac of tem into two major areas: water demand is inelastic at low water prices and ten becomes responsive to water prices up to some price level. Te price levels at wic water demand appears elastic depend on climates and vary around 0.30 F/m 3 for wet weater conditions up to rougy 1.60 F/m 3 for a dry year. Water policies ave to include tis information for defining pricing scemes taking into account climate. Our metod can also be used in a broader framework were te weater is unknown and te farmer decision process is stocastic. As time goes by, te farmer observes te climate and integrates tis information in is decision process. Te complexity of te resolution procedure is increased, but te metod remains valid and operational. Te demand function under stocastic condition can terefore cap2-2a.tex; 6/11/2001; 11:19; p.13

82 C. BONTEMPS AND S. COUTURE be estimated (Bontemps and Couture, 2001) and te value of te information be quantified during te irrigation season (Couture, 2000a, 2000b). Anoter natural extension concerns te problem of on-farm irrigation sceduling in order to take into account competition for water between crops. Using te proposed metod, on-farm water demand function may be estimated in te same way. Due to te probable complexity of any on-farm model, some advanced numerical tools, suc as genetic algoritm (Goldberg, 1989), may be needed. Notes 1 During summer, irrigated agriculture represents 80% of total water consumption in France. 2 Tis problem as been mentioned by economists a long time ago, te first volume of te American Economic Review in 1911 contains a paper on tis problem (Coman, 1911). 3 Tis model is included as a yield-water response function in order to represent te production function. EPIC-PHASE is a crop growt simulator model. 4 If te farmer does not irrigate, ten q t = 0. 5 Te set of constraints defined by (8) reduces te space of available irrigation scedules. Terefore te problem may be solved using an algoritm of searc on all possible cases. Oterwise, we would ave be obliged to use a global optimization program suc as Genetic Algoritm (Goldberg, 1989), to find te optimal scedule. 6 Estimations based on Epanecnikov kernel sligtly differ from te Gaussian kernel estimator. 7 Note tat te Mack and Müller s estimator (1989) is easier to use for derivation since it as a denominator wic does not depend on te derivative variable (Q ere). Since te derivation calculus are quite obvious in our case, we ave used te classic kernel estimator, but we suggest to use tis estimator for advanced derivation estimation. 8 We ave ran te simulation for only 19 value of quotas, mainly because te computation time for te agronomic model and te optimization procedure was important. 9 Te annual precipitations are 402.5 mm for 1989; 676.5 mm for 1991; 901.5 mm for 1993, in te considered area. References Bontemps, C. and Couture, S. (2001) Dynamics and incertainty in irrigation management, Caiers d économie et sociologie rurales 55 56, to appear. Bontemps, C., Couture, S. and Favard, P. (2001) Is te irrigation water demand really Convex?, Mimeo, INRA. Cabelguenne, M. and Debaeke, P. (1995) Manuel d utilisation du modèle EWQTPR (Epic-Pase temps réel) version 2.13, Ed. Station d Agronomie, Toulouse INRA. Coman, K. (1911) Some unsettled problems of irrigation, American Economic Review I(1), 1 19. Couture, S. (2000a) Aspects dynamiques et aléatoires de la demande en eau d irrigation, P.D. Dissertation, Université de Toulouse I. Couture, S. (2000b) Risques multipériodes, prime de risque et valeur de l information, Mimeo, Cemagref. Dinar, A. (2000) Te Political Economy of Water Pricing Reforms, World Bank. Oxford University Press, New York. cap2-2a.tex; 6/11/2001; 11:19; p.14

EVALUATING IRRIGATION WATER DEMAND 83 Garrido, A., Varela-Ortega, C. and Sumpsi, J.M. (1997) Te interaction of agricultural pricing policies and water districts modernization programs: A question wit unexpected answers, Paper presented at te Eigt Conference of te European Association of Environmental and Resource Economists, Tilburg, Te Neterlands, June 26 28. Goldberg, D.E. (1989) Genetic Algoritm in Searc Optimization and Macine Learning, Addison- Wesley, Redwood, CA. Härdle W. (1990) Applied Nonparametric Regression, Econometric Society Monograps, Cambridge University Press, Cambridge. Hassine, N.B.H. and Tomas, A. (1997) Agricultural production, attitude towards risk, and te demand for irrigation water: Te case of Tunisia, Working paper, Université de Toulouse. Hexem, R.W. and Heady, A.O. (1978) Water Production Functions for Irrigated Agriculture, Te Iowa State University Press, Ames, IA. Howitt, R.E., Watson, W.D. and Adams, R.M. (1980) A reevaluation of price elasticities for irrigation water, Water Resources Researc 16, 623 628. Iglesias, E., Garrido, A., Sumpsi, J. and Varela-Ortega, C. (1998) Water demand elasticity: Implications for water management and water pricing policies, Paper presented at te World Congress of Environmental and Resource Economists, Venice, Italy, June 26 28. ITCF (1998) Données prix céréales, Document de travail, ITCF. Mack, Y. and Müller (1989) Derivative estimation in nonparametric regression wit random predictor variable, Sankya 51 59 72. Micalland, B. (1995) Approce économique de la gestion de la ressoruce en eau pour l usage de l irrigation, P.D. Dissertation, Université de Bordeaux. Montginoul, M. and Rieu, T. (1996) Instruments économiques et gestion de l eau d irrigation en France, La Houille Blance 8, 47 54. Moore, M.R. and Negri, D.H. (1992) A multicrop production model of irrigated agriculture, applied to water allocation policy of te bureau of reclamation, Journal of Agricultural and Resources Economics 17, 30 43. Moore, M.R., Golleon, N.R. and Carrey, M.B. (1994) Alternative models of input allocation in multicrop systems: Irrigation water in te Central Plains, United States, Agricultural Economics 11, 143 158. Moore, M.R., Golleon, N.R. and Carrey, M.B. (1994) Multicrop production decisions in western irrigated agriculture: Te role of water price, American Journal of Agricultural Economics, 359 374. Ogg, C.W. and Golleon, N.R. (1989) Western irrigation response to pumping costs: A water demand analysis using climatic regions, Water Resource Researc 25, 767 773. Pagan, A. and Ulla, A. (1999) Nonparametric Econometrics, Temes in Modern Econometrics, Cambridge University Press, Cambridge. Sunway, C.R. (1973) Derived demand for irrigation water: Te California aqueduct, Soutwestern Journal of Agricultural Economics 5, 195 200. Varela-Ortega, C., Sumpsi, J.M., Garrido, A., Blanco, M. and Iglesias, E. (1998) Water pricing policies, public decision making and farmers response: Implications for water policy, Agricultural Economics 19, 193 202. Vieu, P. (1993) Bandwidt selection for kernel regression: A survey, in W. Härdle and L. Simar (eds.), Computer Intensive Metods in Statistics, Vol. 1, Pysica-Verlag,, pp.. cap2-2a.tex; 6/11/2001; 11:19; p.15