Journal of Hydrology 408 (2011) 226–234

Contents lists available at SciVerse ScienceDirect

Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

Estimating the own-price elasticity of demand for irrigation water in the Musicatchment of India

Brian Davidson a,⇑, Petra Hellegers b

a University of Melbourne, Parkville, Victoria 3010, Australiab Wageningen UR, LEI, P.O. Box 29703, 2502 LS The Hague, The Netherlands

a r t i c l e i n f o s u m m a r y

Article history:Received 1 November 2010Received in revised form 26 May 2011Accepted 31 July 2011Available online 9 August 2011This manuscript was handled by GeoffSyme, Editor-in-Chief

Keywords:Derived demand for irrigationOwn-price elasticity of demand for waterResidual MethodMarginal value product of waterKrishna BasinIndia

0022-1694/$ - see front matter Crown Copyright � 2doi:10.1016/j.jhydrol.2011.07.044

⇑ Corresponding author. Address: Department ofGeography, University of Melbourne, Parkville, Victor8344 8633; fax: +61 3 8344 4665.

E-mail address: b.davidson@unimelb.edu.au (B. Da

As irrigation water is an input into a production process, its demand must be ‘derived’. According to the-ory, a derived demand schedule should be downward sloping and dependent on the outputs producedfrom it, the prices of other inputs and the price of the water itself. Problems arise when an attempt ismade to estimate the demand for irrigation water and the resulting own-price elasticity of demand, asthe uses to which water is put are spatially, temporarily and geographically diverse. Because water isnot generally freely traded, what normally passes for an estimate of the own-price elasticity of demandfor irrigation water is usually a well argued assumption or an estimate that is derived from a simulationmodel of a hypothesized producer. Such approaches tend to provide an inadequate explanation of what isan extremely complex and important relationship. An adequate explanation of the relationship betweenthe price and the quantity demanded of water should be one that not only accords with the theoreticalexpectations, but also accounts for the diversity of products produced from water (which includes themanagement practices of farmers), the seasons in which it is used and over the region within which itis used. The objective in this article is to present a method of estimating the demand curve for irrigationwater. The method uses actual field data which is collated using the Residual Method to determine thevalue of the marginal product of water deployed over a wide range of crops, seasons and regions. Thesevalues of the marginal products, all which must lie of the input demand schedule for water, are thenordered from the highest value to the lowest. Then, the amount of irrigation water used for each product,in each season and in each region is cumulatively summed over the range of uses according to the orderof the values of the marginal products. This data, once ordered, is then used to econometrically estimatethe demand schedule from which the own-price elasticity of demand for irrigation water can be derived.To illustrate the method, the values of the marginal product of water deployed in the Musi catchment inIndia are used to determine an own-price elasticity of demand for irrigation water which has some posi-tive value to producers of approximately �0.64. For water that is most highly valued, the elasticity wasfound to be highly elastic at �2.12, while less valued water used in agriculture was far more inelastic at�0.44. Finally, for almost 36% of water deployed in the catchment the elasticity was logically determinedto be perfectly elastic.

Crown Copyright � 2011 Published by Elsevier B.V. All rights reserved.

1. Introduction

The input demand schedule for irrigation water is a reflectionand embodiment of the factors that affect farmers’ decisions topay for and use that resource. In an aggregate sense, say over acatchment, estimating the demand for irrigation water and its con-comitant own-price elasticities is a complex task. Even within afairly contained catchment irrigated agriculture can be spread overa massive area, using water in different seasons to produce a wide

011 Published by Elsevier B.V. All

Resource Management andia 3010, Australia. Tel.: +61 3

vidson).

array of crops. For a farmer, irrigation is essentially a solution fordealing with the problem of inadequate or unreliable rainfall. Irri-gation can be defined as the movement of water in time and spacewhich results in the movement in agriculture in time and space.Even more compelling is the observation that farmers faced withmore reliable supplies of water may choose to produce crops thatcould not normally be grown in a region due to the lack of water, orwhich have an uneven spatial and temporal distribution of water.Thus, providing irrigation water ultimately results in the geo-graphic and occupational mobility of the products that are pro-duced from it. In addition, reliable supplies of water affect theinvestment decision behavior of farmers. Farmers’ precautionarystrategies to protect against the possibility of catastrophic loss inthe event of unreliable supply may lead them to select less risky

rights reserved.

B. Davidson, P. Hellegers / Journal of Hydrology 408 (2011) 226–234 227

but less profitable crops, under-use fertilizers, shift household la-bor to less profitable off-farm activities, and avoid investment inproduction assets and improved technology. The point is thatfarmers approach the use of irrigation water in a multitude ofways, changing what they grow to produce a greater diversity ofcrops using methods that are appropriate to the task, given the cir-cumstances in which they find themselves.

This complexity of purpose and task can be compared to theprocess and rigor of estimating the own-price elasticity of demandfor irrigation water (i.e. the responsiveness of the quantity de-manded to a change in its price). The methods currently used arefar from sophisticated and do not reflect or account for the diver-sity of decisions that could be made or the conditions that existin a catchment. Ignoring those studies where an elasticity is justassumed (a fairly inelastic one in the short run and a more elasticone in the longer run), the normal approach is to estimate a mar-ginal value from a mathematical programming model of a typicalirrigation farm (for recent reviews of many studies into the de-mand for irrigation water see Bontemps and Couture, 2002; Appelset al., 2004; Young, 2005; Scheierling et al., 2006; Griffin, 2006;Bell et al., 2007). It should be noted that analysts are forced touse these approaches because no reliable market based data existsupon which a range of water quantities and prices could be used toestimate the demand schedule. Thus, it could be argued that theability of analysts to estimate an elasticity for irrigation water isdependent on their perception of how water is used, rather thanwhat is actually occurring.

These programming approaches do not expressly reflect therichness associated with using irrigation water. For instance, ithas been observed that as irrigation water becomes increasinglylimited farmers tend to not to produce certain crops in some preor-dained order (see Vinot, 2009). Molle et al. (2009) examined howfarmers adapted to water scarcity and unreliable water availabilityin six different river basins of Asia and the Middle East. They sug-gested farmers undertake several types of local adjustments includ-ing supply augmentation, conservation, reallocating entitlement toa particular crop and to ‘‘bend’’ predicted allocation patterns bytampering with infrastructures or resorting to sociopolitical net-works. Institutional innovations with irrigation water also occur.In some way, all these actions and responses affect the returnsfarmers get from deploying water and how they react to its price.

The demand for irrigation water is a complex issue, which re-quires more than just an assumption about what its elasticitymight be, or one derived from a hypothesized farm within a catch-ment, regardless of how well it is modeled and simulated. How-ever, it should be noted that obtaining estimates of the own-price elasticity of demand for irrigation water is not an easy task.Ward and Michelsen (2002), Davidson (2004), Young (2005), Grif-fin (2006) and Hanemann (2005) amongst many others have com-mented on the difficulties of dealing with a product that is notactively traded in a market (thus reducing the possibilities of usingan inductive approach) and one that is riddled with market fail-ures. Adding to this is the complexity that irrigation water itself,the products produced from it and the factors that are combinedwith it are either taxed and/or subsidized heavily by governments.

What is required to obtain a better estimate of the own-priceelasticity of irrigation water is better information on the value waterprovides to the production process. This information will need to beobtained using a deductive approach (as water is not traded in amarket) and should ideally be derived from the range of activitiescurrently being undertaken in the catchment in question. Young(2005) reviews these methods and suggests that the ResidualMethod of deducing the value marginal product of water used toproduce an individual crop, grown in a particular region and duringa particular season, can be obtained. According to Salvatore (2004)theoretically each individual crop, season and regional value

marginal products of water should lie on the input demand schedulefor the resource. So, combining this with some knowledge of thequantities of water supplied to each crop, in each region and duringeach season, it should be possible to estimate the input demandschedule for water and its own-price elasticity of demand.

2. Objectives

The aim in this article is to present and apply a method of esti-mating the own-price elasticity of demand for irrigation water in acatchment. This approach needs to accord with the theoreticalnecessities of being a derived demand, i.e. negative and accountfor the prices of outputs and other inputs. But ideally, it must moreadequately reflect its richness of the spatial, temporal, geographi-cal and occupational diversity of irrigation water use. To be usefulto policy makers this estimate should be calculated on the socialdemand curve, not the private one that exists in a taxed or subsi-dized environment.

It should be noted from the outset that the aim in developingthis method is not to replace the other more accepted methods.Rather it is to complement the other methods, in order to developa better understanding of what is occurring.

The method developed in this paper relies on determining thevalue marginal product of water used to produce a wide range ofcrops, grown over different seasons and regions (which weredetermined using the Residual Method outlined in Young, 2005and as applied in Hellegers and Davidson, 2010). These values ofthe marginal products for irrigation water used in each season,for each crop in each region theoretically lie on the input demandschedule for irrigation water, if it is assumed that constant returnsto scale exist and that irrigators maximize profits (Salvatore, 2004).So in order to understand the demand relationship it is necessaryto find the quantity of water deployed in each crop, during eachseason and in each region that corresponds to each value marginalproduct. Thus, the loci of these points should represent the inputdemand for irrigation water and from this relationship, it shouldbe possible to determine the own-price elasticity of demand forirrigation water in a catchment.

This method is applied to the Musi catchment in India, wherethe values of the marginal products of nine different crops grownover two seasons in five different regions are already known (seeDavidson et al., 2009; Hellegers and Davidson, 2010). In obtainingthese estimates the influences of taxes and subsidies have been ex-cluded. The amount of water allocated to each crop, in each seasonand in each region is know (see George et al., 2011). In order to fur-ther investigate the demand for water, the data for the Musi catch-ment can be segregated into three distinct groups in order todetermine how farmers who produce different crops may react dif-ferently to a change in the price of water. In addition, the impactsthat arise from a change in the prices of outputs (the factor Helleg-ers and Davidson found to have the greatest impact on water val-ues and an element that will result in horizontal shifts in demand)can also be investigated.

3. Theoretical considerations

In this study of interest is the estimation of a derived demandschedule for irrigation water and the concomitant own-price elas-ticity that can be derived from it. Varian (1987, p. 4) outlines how ademand schedule should be thought of in what he terms a ‘reser-vation price’. That is, the highest price that a given person will ac-cept and still purchases the good. In this study the reservationprice is revealed by assessing the value of the marginal productfarmers receive from using the good irrigation water alone in theproduction of different crops in different regions and seasons.

228 B. Davidson, P. Hellegers / Journal of Hydrology 408 (2011) 226–234

Salvatore (2004, p. 298) suggests that to determine the deriveddemand for an input it is necessary to assume that the firm will usethe amount of each input in such a way that maximizes its profits.To do this the firm must produce the optimal level of output withthe least cost combination of inputs. The firm will employ each in-put as long as the extra income from the sale of the output (knownas the marginal revenue product or MRPi) produced by the input islarger than the extra cost of the input. The marginal revenue prod-uct (MRPi) of each input is equal to the marginal product of eachinput (MPi) multiplied by the marginal revenue of the output(MRx), and in a perfectly competitive environment the marginalrevenue of the output should equal the price of the output (Px).The value of the marginal product of each input (VMPi) must there-fore equal the marginal revenue product of the input in this situa-tion, which in turn is equal to the marginal product of the inputmultiplied by the price of the output, or

VMPi ¼MPiPx ¼MRPi ð1Þ

As more units of the input are added the marginal product ofthe input must eventually decline and Salvatore (2004, p. 298) arg-ures that this declining portion of the marginal revenue productschedule is the firms demand schedule for the input in question.

However, this would only be the case if the firm was assumed toemploy only one variable input (i). The use of multiple variable in-puts means that any individual marginal revenue product curvecould shift as the price of any variable input changed. A similarproblem occurs when attempting to move from the firm level tothe market demand for the input. If the price of an input changes,all firms in the market will react, producing a different quantity ofthe output and having what is known as an external effect on thefirm and the price in the market. Both these impacts would meanthat the derived demand for any input would become the loci ofpoints on different marginal revenue product curves, where thevalue of the marginal product lies on that input demand curve(Salvatore, 2004, pp. 299–300).

For each individual input used, the value of the marginal prod-uct for the last unit employed must lie on the market derived de-mand schedule for it, provided that it is assumed that a perfectlycompetitive environment exists and that the firms operate underthe assumption that constant economies to scale occur. Thus, inthis study if the value of the marginal product can be calculatedon the value of the last unit of water used, it provides one pointon the overall input demand schedule for water, a point that corre-sponds to the demand for water for that output. By calculating thevalue of the marginal products of water used for different outputs(in different locations and seasons as well), all which theoreticallyshould lie on the demand curve for the input, it should be possibleto derive the input demand for water.

As water is just one input into a production process, the de-mand for it should also accord with that of an input demand. Insuch cases the quantity of the input demanded has a number of un-ique theoretical aspects. First, the relationship between the priceand quantity of water demanded must be a negative (downwardsloping) one. Second, the price of the output produced from the in-put water, the price of other inputs and the price of substitutes towater are hypothesized to be important in determining the inputdemand for a good as suggested above (Varian, 1987).

The own-price elasticity of demand is an important variable asit summarizes the responsiveness (or sensitivity) of the quantitydemanded to a change in its price.

ed ¼ ðDQ w=Q wÞ=ðDPw=PwÞ ð2Þ

where ed,i is the own-price elasticity of demand for irrigation water;Qw is the quantity of input w; and Pw is the value of the marginalproduct of the input w.

There are numerous methods available to estimate the own-price elasticity of demand for a good, none of which are really ame-nable to the case of irrigation water. Most involve the collection ofmarket prices and quantities for a good, or a range of goods, andthen estimate the relationship between the two. In this case thedemand for the good could be expected to be a function of its price,where the slope (the coefficient associated with the price) is nega-tive and significantly different to zero and the intercept term islarge and positive. To estimate the own-price elasticity of demandthe slope coefficient of the demand curve (b) is multiplied by theratio of the quantity demanded to the price at any point alongthe curve.

ed ¼ b � Qw=Pw ð3Þ

More typically, and the method used in this study, transformingthe quantities demanded prices into logs and estimating the rela-tionship by regressing one on the other will yield an estimate ofthe elasticity (see Tomek and Robinson (2003, p. 361), for an ex-panded discussion on this practice).

4. Previous estimates

Many researchers have used mathematical programming mod-els to estimate the elasticities of irrigation water whereby a hypo-thetical farm is assumed to exist and the model is simulated withdifferent constraints to determine the marginal value, or the sha-dow price, of water (see Appels et al., 2004; Scheierling et al.,2006). Others have attempted econometric evaluations and evenfield experimentation.

Scheierling et al. (2006) reviewed a total of 53 studies of esti-mates of the own-price elasticity of demand for irrigation water.They reported estimates ranged from �0.002 to �1.97, with amean of �0.51 and a median value of �0.22. In total, 11 mathemat-ical programming studies were assessed and 21 estimates ob-tained, 4 econometric studies were reviewed with 22 estimatesbeing obtained and 3 obtained estimates from field experimentstudies to obtain 10 estimates of the own-price elasticity of de-mand for irrigation water. One might suggest that any elasticityobtained should fall between the ranges found by Scheierlinget al. (2006) and it should not fall far from their estimated meanof �0.5.

However, large differences could well occur. For instance, in anAustralian study of short run elasticity of demand for irrigationwater across crops, Bell et al. (2007) found elasticities to rangefrom �0.8 for the fruit and vegetable industry groups to �1.9 forsugar. These estimates are less elastic (lower in absolute terms)than others reported in other Australian agriculture (see Appelset al., 2004). These estimates were derived at the farm level usingpanel data of the profit functions of individual producers and thewater prices used were those that farmers pay for water.

These highly synthesized methods (programming models andMeta techniques) are inadequate as they do not necessarily allowfor the diversity of options open to a range of farmers and enter-prises, spread over a wide area. Scheierling et al. (2006) compareestimates across studies which are based in different regions, whileBell et al. (2007) make comparisons across commodities in differ-ent catchments. The modeled farm approach described in Appelset al. (2004) does neither of these things. It is also assumed in theseapproaches that all farmers will act in the same manner and in arational way, with perfect knowledge and foresight, actions whichare hypothesized by the analyst alone to exist. Vinot (2009), Molleet al. (2009) and Hellegers and Davidson (2010) all explain whythis might not be the case. They suggest that farmers plant differ-ent crops for a variety of reasons, some for a financial return andothers for sustenance. This implies that if water is restricted in

B. Davidson, P. Hellegers / Journal of Hydrology 408 (2011) 226–234 229

the region, then the crop with the lowest value is probably not theone sacrificed because of a lack of water alone.

To put this argument another way, it could be suggested thatthe estimates of the own-price elasticity of demand that have beenderived from programming models are dependent on assumptionsbeyond those normally associated with profit maximization andthe economic efficiency of an input. They are based on additionalassumptions imposed by the analysts on what they perceive asthe users actions with respect to different circumstances. Whatare required are estimates that adequately reflect the circum-stances and situations the users of an input face. This can only bederived from an assessment of the outcome from a productive pro-cess, where the values of the marginal products of water are de-rived from the actions of users in the field. Such an approach isdetailed in the next section, which relies on calculating the valueof the marginal products and using that in accordance with theassumptions specified in Section 3.

5. Methods

To estimate the demand function for irrigation water, it is nec-essary to obtain an estimate of the marginal value farmers place onirrigation water. This will vary according to the use farmers put thewater to, the output prices received for the chosen use, the waywater is applied to individual crops and how it is combined withother inputs. Thus, a range of values for different crops in differentregions and seasons will need to be obtained. Taking this approachalso yields a set of estimates that can be used to conduct an ordin-ary least squares regression analysis over the obtained values ofthe total demand for irrigation water.

The approach used in this article to determine the disaggregat-ed value of the marginal products of irrigation water used in agri-culture across crops, regions and seasons is the Residual Method,described by Young (2005), and detailed in Davidson et al. (2009)and Hellegers and Davidson (2010). While this method has its lim-itations (in particular that it tends to overestimate the value ofwater), both Young (2005) and Griffin (2006) attest to the useful-ness and reliability of the approach. This method relies on the va-lue to a producer from producing a good (its price by its quantity)being equal to the summation of the quantity of each input re-quired to produce it multiplied by each inputs corresponding mar-ginal value of the product:

YiPi ¼ RQj;iVMPj;i ð4Þ

where Yi is the quantity of output of crop i (kg ha�1); Pi is the pricereceived for crop i (Rs kg�1); Qj,i is the input j used to produce crop i;and VMPj,i is the value of the marginal product input j used to pro-duce crop i.

Expanding the equation to a two input model (v and w), andassuming that the value of the marginal product of one input (v)is known but the value of the marginal product of water (w) isnot, yields:

YiPi ¼ Qv;iVMPv;i þ Q w;iVMPw;i ð5Þ

where Qv,i is the known quantity of input v required to produce cropi (kg ha�1); VMPv,i is the known value of the marginal product of in-put v (Rs kg�1); Qw,i is the known quantity of input w used to pro-duce crop i (m3 ha�1); VMPw,i is the unknown value of themarginal product of input w (Rs m�3); and all other variables areas defined above.

The value of the marginal product of the unknown input (w) canbe found by rearranging Eq. (5) so it is specified as a function of theprice multiplied by quantity of the output, less the sum of thequantities of all known inputs multiplied by the values of the mar-

ginal products of all known inputs, all divided by the quantity ofthe unknown input. In other words:

VMPw;j ¼ ðYiPi � Qv;iVMPv ;iÞ=Q w;i ð6Þ

where all variables are as described above.Young (2005, p. 61) describes the solution to Eq. (6) as the ‘va-

lue of water’ or the ‘net return to water’ per unit of water for thecrop in question. But in doing so, he assumes that the producerchooses inputs in such a way that the values of the marginal prod-ucts equals the price of the input. This assumption allows for thesubstitution of the values of the marginal products for input pricesand permits the value of the marginal product for water to be equalto the unknown price. It is, in the parlance of economics, the ‘resid-ual value’. If the values of the marginal products can be determinedfor different crops (i) in different zones (k) and in different seasons(t) the number of observations required to estimate the input de-mand schedule for water in a catchment can be increased, as allshould lie on it.

Of interest in this study is not the marginal value of water usedin a particular individual crop, but the marginal values betweendifferent crops, in different regions and seasons. Each individualcalculated value of the marginal product is assumed to lie on theinput demand schedule for water (see Section 3) and as such pro-vides a point which, when combined with the values of the mar-ginal product of water used in other crops, seasons and regionswithin a catchment, can be used to estimate the whole input de-mand curve. It should be remembered that the aim in this studyis not to provide an estimate of the own-price elasticity of demandfor irrigation water used in a particular crop, in a single season at aknown location. Rather it is to provide an estimate of the own-price elasticity of demand for irrigation water over a range of crops,seasons and locations within a catchment.

The price in this study is what farmers derive in value fromdeploying water for different crops in different zones and seasons.Thus, the demand schedule is the loci of values of the marginalproducts farmers derive from deploying water in different uses.These can be interpreted to be, as Varian suggests, the reservationprice for irrigation water used in different uses, seasons and uses.In other words, the implicit assumption is that farmers view thedemand for water from highest value to lowest and thus by cumu-latively summing the quantities employed in this way will revealthe input demand relationships for water.

To calculate the input demand schedule it is necessary to orderand collate the various values of the marginal products of waterused to produce different crops in different regions and in differentseasons within a catchment according to the amount of irrigationwater deployed in each use. To view the sort of reservation pricediscussed by Varian (1987) the crops are ordered from highestvalue of the marginal product to the lowest and then the amountof water used for each subsequently valued crop is cumulativelysummed according to this order. In doing so, it should be possibleto trace out the input demand for irrigation water in thecatchment.

Usually an elasticity estimate is independent of the units cho-sen, as the measure of the percentage change in quantity dividedby the percentage change in price. However, in this case the esti-mated elasticity is not independent of the degree of aggregationused to order the data. The simple act of splitting the quantity datainto half and ascribing the same marginal values to it will changethe estimate of the elasticity. This is a serious limitation of the ap-proach, which cannot be easily resolved. The best approach to solv-ing this problem is to split the quantity data up into as small a setof units as possible, in order to obtain as many observations as pos-sible. Doing so will result in an estimate that is more elastic.

Fig. 1. An example of the approach used to estimate the demand schedule for irrigation water.

Table 1The values of irrigation water for individual crops in five different zones by seasonand weighted average values in the Musi catchment 2001–02 (Rs m�3). Source:Hellegers and Davidson (2010)

Crop Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Average

Kharif (summer)Rice 0 0.11 0.30 0.42 0.01 0.20Vegetables 106.19 106.07 106.15 106.5 106.36 106.20Chilli 4.55 4.52 4.67 4.83 5.01 4.62Fruits 24.26 24.20 24.37 24.59 24.68 24.39Ground nuts 0 0 0 0 0 0Maize 22.27 22.01 21.82 22.47 21.83 22.26Cotton 0 2.82 2.96 3.17 0.22 1.96Other 0 0 0 0 0 0Average 5.38 2.82 2.05 2.45 2.02 3.31

Rabi (winter)Rice 0 0 0.04 0.16 0.08 0.06Vegetables 21.53 21.50 21.68 21.83 22.03 21.61Chilli 1.21 1.19 1.38 1.50 1.75 1.29Gram 15.50 15.49 15.66 15.78 16.03 15.65Ground nuts 3.75 3.73 3.89 4.04 4.25 3.92Maize 3.17 3.14 3.32 3.46 3.67 3.31Other 0 0 0 0 0 0Average 1.99 0.43 0.64 0.70 1.13 1.11Average 3.38 1.35 1.22 1.55 1.46 2.04

Table 2Total quantities of irrigation water applied by crop and zone (MCM). Source: Hellegersand Davidson (2010)

Crop Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Total

Kharif (summer)Rice 157.14 69.11 76.21 186.91 47.39 536.75Vegetables 5.31 0.19 0.15 0.17 0.36 6.18Chilli 1.55 1.06 0.44 0.54 0.22 3.81Fruits 16.14 7.70 3.71 13.54 2.76 43.86Ground nuts 0.13 0.33 0.14 1.04 0.02 1.65Maize 5.62 0.23 1.27 2.89 0.11 10.12Cotton 15.14 11.41 10.54 13.36 3.93 54.38Other 0.99 0.62 0.28 1.65 0.07 3.60Total 202.03 90.65 92.73 220.10 54.85 660.35

Rabi (winter)Rice 250.35 133.49 116.40 205.59 88.39 794.22Vegetables 24.50 0.80 1.21 0.64 4.36 31.51Chilli 3.19 1.10 0.51 1.18 0.10 6.07Gram 0.49 0.57 0.40 1.09 0.01 2.57Ground nuts 2.79 4.62 1.51 11.98 0.18 21.09Maize 9.45 5.77 12.24 15.00 0.11 42.56Other 0.20 0.00 0.00 0.22 0.00 0.42Total 290.97 146.35 132.27 235.70 93.15 898.45Total 493.00 237.00 225.00 455.80 148.00 1558.80

230 B. Davidson, P. Hellegers / Journal of Hydrology 408 (2011) 226–234

These values of the marginal product and the correspondingcumulatively summed quantities of water deployed become thebasis for the econometric estimation of the input demand for irri-gation water and subsequently the estimation of its own-priceelasticity of demand. These data can then be separated into cropswhere the irrigation water is most highly valued and less highlyvalued. In this approach, it is assumed that farmers act rationally,wanting to use the first available water to produce the most valu-able crop and so on until all water is expended.

To illustrate this approach, the highest values of the marginalproducts farmers place on water in any system are often for vege-tables (at say Rs. 0.5/m3), fruits (at Rs. 0.4/m3) then to grains (at Rs.0.2/m3) and then to ground nuts (at Rs. 0.1/m3). These values areVarian’s (1987) ‘reservation prices’. Then, if it is assumed for illus-trative purposes that 10 MCM (million cubic metres) is used forproducing vegetables, another 20 MCM to produce fruit and a fur-ther 100 MCM is used to produce grains and ground nuts, each. Thepattern of rational water use would be to use the first 10 MCM toproduce vegetables, the next 20 MCM to produce fruit, the next100 MCM to produce grains and the final 100 MCM to produce

ground nuts. This implies an ordered schedule of demand for waterwhere the first 10 MCM is valued at Rs. 0.5/m3, 30 MCM is valuedat Rs. 0.4/m3, 130 MCM is valued at Rs. 0.2/m3 and 230 MCM is val-ued at Rs. 0.1/m3. This would be a stepped equation, which couldbe smoothed (see Fig. 1). This is the input demand schedule forwater in that region, given the observed use of water on thosecrops. From this demand schedule it would be possible to estimatethe slope and intercept of a function and from that, estimate theown-price elasticity of demand for irrigation water in the region.Ideally, the curve could be estimated in log form to enable a directestimate the own-price elasticity of demand for water as suggestedby Tomek and Robinson (2003) and specified in Eqs. (2) and (3).

6. Analysis and results

Data on values and the quantities of water deployed in the Musicatchment of India were derived from Davidson et al. (2009) andHellegers and Davidson (2010). In this study the values the mar-ginal products of individual crops grown in different parts of thecatchment in different seasons were estimated. The social prices(those that account for and exclude the impacts of taxes and

00.51

1.52

2.53

3.54

4.55

0 200 400 600 800 1000 1200 1400 1600Cumulative irrigaiton water applied (MCM)

Ave

rage

val

ue (R

s./m

3)

Fig. 2. Ordering of the crops from highest average value (Rs m�3) to the lowest and the associated cumulative water use (MCM). Source: Hellegers, P., Davidson, B., 2010.Determining the disaggregated economic value of irrigation water in the Musi catchment in India. Agricultural Water Management 97 (5), pp. 933–938.

B. Davidson, P. Hellegers / Journal of Hydrology 408 (2011) 226–234 231

subsidies) were used. In estimating these values, Hellegers andDavidson accounted for and measured the impacts changes inthe prices of outputs and other inputs. These estimates were sub-jected to a set of sensitivity tests, where it was found that the out-put prices affected values most. If the output and other input priceswere altered, the values of the marginal products for each use ofwater would change. The cross price elasticities of water and otherinputs could be calculated by altering the input prices within Hel-legers and Davidson’s analysis, something that is beyond the scopeof this paper. By not doing so, only the short run elasticities of thedemand for water are calculated in this study. Ordering all thecrops produced in all the zones (specified in Table 1) from highestto the lowest and by cumulatively summing the associated quanti-ties of water (specified in Table 2) according to the order ofdescending value, is shown in Fig. 2.

It should be noted that these values of the marginal products ofwater would be at the upper limit of any estimate as a limitation ofthe Residual Method is that any input not included in the analysisis attributable to the input under consideration (see Young, 2005;Hellegers and Davidson, 2010). In this case the estimated valuesare relatively low. In total there are 75 observations (eight cropsin Kharif or summer season and seven crops in Rabi or winter sea-son in each of the five zones of the Musi catchment, as defined byGeorge et al. (2011). For reasons of exposition, the crops whichhave a value of the marginal products of more than Rs. 5/m3 (26observations) are not shown in Fig. 2. The crops not shown repre-sent only 6% (or 94 MCM) of the total allocated quantity of water(of 1559 MCM). There are 19 observations that have a value lessthan zero. In these cases, the production of the particular cropwas undertaken at a loss, not a surprising result given that theinfluences of subsidies were removed from the analysis (seeDavidson et al., 2009). Further, it should be noted that making aloss is a perfectly reasonable outcome from this market process.As it is impossible to obtain a logarithm of anything but a positivenumber, these 19 observations were excluded from the economet-ric analysis. However, as the water is used to produce output, theiremployment has an economic interpretation. For these 19 observa-tions the elasticity was logically determined to be perfectly elastic,as they are unresponsive to a change in price and are ordered alongthe quantity axis. They represent 36% (or 561 MCM) of the totalallocated quantity of water in the catchment.

To directly estimate the own-price elasticity of demand forwater applied of the 56 usable observations, which represent a sig-nificant proportion (approximately 64%) of the water used in thecatchment, the logarithmic values of quantities and average valueswere obtained and regressed against one another using the ordin-ary least squares estimator in Microsoft’s Excel program. A loga-rithmic functional form was chosen after a visual inspection ofthe data (see Fig. 2) revealed that it would provide the best fit

for the estimated equation and because it is the most direct meth-od of obtaining an estimate to the own-price elasticity for irriga-tion water. In addition, a logarithmic function form is consistentwith the idea that farmers would value the small amount of waterused in high value crops disproportionally higher than a largequantity of water used on less valuable crops. The implication forusing this type of functional form is that the elasticity is constantover the whole range of data estimated.

Using all the data available for all crops across all zones duringall seasons in the Musi catchment (i.e. all 56 usable observations),the following regression equation was estimated:

ln Q w ¼ 5:494� 0:640 ln Pw

ð35:33Þ ð�10:67Þ n ¼ 56 F ¼ 113:76 R2 ¼ 0:62 ð7Þ

where Qw is the cumulative ordered quantity of water employed inproducing each individual; Pw is the value of the marginal productof water used in that product; and the figures in brackets are theStudents-t statistics.

It would appear that the aggregate own-price elasticity of de-mand for irrigated water in the Musi catchment is equal to�0.64. In other words for every 1% increase in the value of the mar-ginal products of water, the quantity demanded would decline by0.64%. The parameter estimates are correctly signed (negativeand a necessity as water is an input demand). In addition, withan adjusted R2 of 0.62, it would appear that the relationship is ade-quately explained. The Students-t statistic of �10.67 on the valueof the marginal product coefficient would suggest that the esti-mate is significantly different from zero.

To assess whether the elasticities might well be different forcrops in which irrigation water is either highly valued or less val-ued (designated by the subscripts h and l respectively), the 56 use-able observations were segregated into two somewhat equal parts.It should be noted that a Spline regression technique could havebeen used at this point in the analysis to estimate linear slopesfor different ranges of the independent variables. Using a Splinetechnique could be considered to be more statistically efficient;however such an approach would not conform with the a priori be-liefs revealed in a visual inspection of the data.

Crops that were found to value irrigation water greater thanRs. 15/m3, (25 observations representing only 6% of the water de-ployed), were used to estimate the following regression equation:

ln Q h ¼ 10:482� 2:122 ln Ph

ð12:66Þ ð�8:78Þ n ¼ 25 F ¼ 77:02 R2 ¼ 0:77 ð8Þ

The remaining 31 observations, all having an estimated positivevalue of Rs. 5.01/m3 or less, and accounting for 58% of the waterused, were employed to estimate the following regression equation:

232 B. Davidson, P. Hellegers / Journal of Hydrology 408 (2011) 226–234

ln Q l ¼ 5:439� 0:444 ln Pl

ð154:61Þ ð�20:75Þ n ¼ 31 F ¼ 430:41 R2 ¼ 0:94 ð9Þ

As was the case when all the 56 positive observations were as-sessed (designated as w and presented in Eq. (7)), in the equationsfor the higher valued crops (designated as h and presented in Eq.(8)) and the lower valued crops (designated by l and presented inEq. (9)), the estimated coefficients on value were found to be sig-nificantly statistically different from zero and correctly signed.The estimated equations are significant and the goodness of fit sta-tistic reveals even better estimates that the aggregate estimate seeEq. (7)). Most importantly, all three estimates of the elasticities,and what can be implied from the data excluded from the process,are all different.

According to the theory the demand for an input is sensitive tochanges in output and other input prices. Changing these elementsresults in parallel shifts in the demand equation, rather thanchanges in the elasticity estimates, something that would requirere-estimating the values of the marginal products, which is beyondthe scope of this study. Thus, what changes is the amount of irriga-tion water that falls in each of the three elasticity ranges. Hellegersand Davidson (2010) reported the estimated values were most sen-sitive to changes in output prices. They concluded that a 30%reduction in output prices resulted in 87% of irrigation water beingfound to be in the perfectly elastic range and valued at zero, while10% was in the elastic range and only 3% in the inelastic range. Thiscan be compared with the situation presented above where 64%,36% and 6%, were found to be in the perfectly elastic, elastic andinelastic ranges (respectively). If output prices increase by 30%then the amount of water in the perfectly elastic and elastic rangeschange down to 10% and up to 84%, respectively, while the inelasticrange remains unchanged at 6%.

The estimates of the own-price elasticities of demand for irri-gation water in the Musi catchment yield an interesting set ofcircumstances. If taken over the whole range of data (all 75observations) three different estimates can be derived. First, forthe 25 crops where water is valued most highly, an elastic esti-mate, of �2.1 was estimated. Second, for the next 31 crops,where water is less valued, the estimate was found to be fairlyinelastic at �0.44. Third, combining these two sets of observa-tions together (56 in all) to obtain the overall own-price elastic-ity of irrigation water, an inelastic �0.64 estimate was obtained.The estimated elasticities are in the range of those found byScheierling et al. (2006). Finally, it was logically asserted thatfor approximately 36% of the water used in the Musi catchment,that which was found to have a residual value of equal to or lessthan zero, the own-price elasticity of demand was perfectlyelastic.

7. Limitations of the approach

The results obtained in this study can be questioned on a num-ber of broad fronts, any of which relate to the assumptions speci-fied above. The most important relates to the well knownproblems of obtaining values using the Residual Method (thatthe figure includes everything that cannot be accounted for else-where). This leads to the suggestion that the value figure obtainedis too high. Yet what should be noted is that the estimates of thevalues of the marginal products used in the analysis are not veryhigh.

Further, the variability in the resultant values need to be ques-tioned, as some are abnormally high and others are very low ornegative. Barton and Taron (2010) compared the value of the mar-ginal products of water obtained using trading prices with surveysof value and with that obtained from the Residual Method in the

Tungabhadra River in India. They concluded that the ResidualMethod showed greatest variation. This led them to conclude thatthe Residual Method is not as useful and that the resultant esti-mates of demand could be misleading. While Hellegers andDavidson (2010) mount a vigorous defense of their results, thesecriticisms reach into the heart of the approach that is applied inthis case. That is, is it better to take a highly disaggregated ap-proach to assessing the demand for irrigation water, as pursuedin this study, or to take the hypothesized simulated approach usedby others? The point being that Barton and Taron made their judg-ment on finding a single estimate, something that in a catchmentwith many different characteristics, would appear to be illogical.

What cannot be ignored in this study are the numerousassumptions that have been made that force the values of the mar-ginal product of water used to produce individual products, in aparticular season and region. In particular, it was assumed thatperfect competition existed and that farmers produced under a re-gime of constant returns to scale. Both these assumptions arehighly dubious.

Calculating a price elasticity of demand is not independent ofthe institutions which control the allocation of water, as thoseinstitutions may well prevent water travelling from a place whereit is not highly valued to one where it is. Not accounting for alltransactions costs in reallocating water will have the same effect.Institutional rules break down the assumptions of perfect compe-tition, just as decreasing and increasing returns to scale do, result-ing in the values of the marginal products not lying on the inputdemand schedule for water. While in this study an attempt wasmade to use estimates of the values of the marginal products thatexcluded subsidies and taxes, a fact that resulted in a number ofthe estimated values of the marginal products being negative,not all institution influences can be accounted for.

Within an assumption of a perfectly competitive environmentare two elements that have an important bearing on the findings.First, profit maximization is assumed. This leads to the secondassumption that producers would order requirements from highestto lowest value. There is nothing to suggest that farmers operateunder these beliefs, especially in developed countries, as issues ofsubsistence intrude. Third, given this, it could be argued that farm-ers attempt to maximize their utility, rather than their profits.

Another limitation identified with this approach is that the esti-mated elasticities would appear to be dependent on the method ofaggregating the quantity data together. By cumulatively summingthe quantities in a finer graded scale tends to make the estimatemore elastic. There is little that can be done to overcome this lim-itation, other than to be aware of it. In this study the quantitieswere segregated according to the region, crop and season in whichthey were grown, implying that the value of the marginal productfor water in each is perfectly elastic. As a consequence, the demandfunction would be stepped in shape, not smoothed, as was sug-gested in this study. Further insights into this problem can begained from studying the literature on labor economics, a field thatconcentrates on input values and use.

In this study the logarithmic functional form was implied andused to estimate the elasticities. It should be noted that any esti-mate is dependent on the chosen functional form. The one usedin this analysis was chosen from visual inspection of the dataand for convenience. However, such a choice implies that the elas-ticity is constant over the whole range of the observed data. Thismight not be the case and given that in segregating the data intohigh and low value groups resulted in two very different estimates.

Finally, it could be implied that some degree of endogeneitymay exist in the estimation procedure. Endogeneity may occur be-cause in deriving the value of the marginal products using theResidual Method relies on dividing each estimate by the quantityof water (see Eq. (6)) and that this quantity now appears on both

B. Davidson, P. Hellegers / Journal of Hydrology 408 (2011) 226–234 233

sides of Eq. (7). In reality this is not a serious concern as the quan-tity used in the estimation procedure is the cumulative sum ofwater used, not the actual sum. If, in cases where great degreesof aggregation are not possible and few observations exist with asmall total quantity of water deployed, it may be advised to under-take a statistical test for endogeniety and to adjust for it.

The results reported in this study must be interpreted with care.They all allude to further research that could be undertaken in thisfield. In particular, there is a need to work on the impacts thatmotivate farmers to produce a crop for sustenance over profitand a more comprehensive study of the institutional constraintsto moving water between uses, regions and seasons. Despite theselimitations, which it should be noted exist in many studies on theinput demand for water, the estimates obtained in this study areconsidered superior to others as at least they reflect the choicesfarmers directly make, rather than those that analysts might thinkthat they make. They could be used in conjunction with other esti-mates derived using other approaches, in order to improve theunderstanding of the demand for water.

8. Discussion and conclusions

It has been argued throughout this study that those who need toinvestigate the problems of valuing water and/or of estimating theown price elasticity of demand for water need to attempt the taskon an individual crop basis, in an individual region over differentseasons. Doing so reveals a highly variable result across crops in asingle year. If the value of water and its resulting elasticity is deter-mined in aggregate, which is usually the response from other ap-proaches, the precision needed to understand the changes thatwill result from in supply can be lost. These aggregate estimatesare (according to Scheierling et al. (2006)) usually based on a linearor quadratic programming models in which the region/farm is as-sumed to maximize profits from the use of a set quantity of waterprovided at a given price. The process of optimization embodied inthese models forces the value of water to be equal across crops, asthat is the variable that is simulated to change. In these other studiesit is suggested that the limiting variable is water, so applying waterto each crop so that the marginal values from each are equal, forces aresult that there would only be a single average elasticity value ofwater, regardless of the crop produced. This can only be correct inthe highly theoretical world of optimization models, as farmers donot view it this way and change the way they allocate water.

Underlying all these other approaches to estimating the de-mand of water is similar data that has been employed in this study.It is all about discerning the value of water by taking the costs bythe quantities of the inputs from the price of the output by theyield produced. The modeling philosophy is different in this study,as the aim is to disaggregate water by the output resulting in theestimation of the different responses (revealed in the differentelasticities). In the other studies the differing values of water usedin different crops are, by assumption, made to equate.

From the sensitivity tests conducted by Davidson et al. (2009), itcan be concluded that the estimates of the values is highly sensi-tive to changes in commodity prices. It could be argued that watervalues (and what farmers are willing to pay for it) are determinedmainly by commodity prices. So can output or commodity pricesact as a proxy for water values and by implication is the own-priceelasticity of demand for water used in a particular crop in a partic-ular region at a specified time, equal to that of the crop in ques-tion? This is an area that requires further investigation. Otherfactors come into play, especially those at the farm level. Thus,the type of estimates obtained by Bell et al. (2007), where moreelastic estimates of water prices elasticities were obtained fromcrop profit functions, could be questioned.

By ordering the data in the way described above over a range ofcrops grown over a wide area and in different seasons, it was pos-sible to obtain an estimate of the own-price elasticity of demandfor irrigation water in the Musi catchment of approximately�0.64 in aggregate. This estimate is well within the ranges speci-fied in Scheierling et al. (2006) and not that different to the averagethey obtained of �0.51. There are no ‘simulated values’ from mod-els of ‘hypothetical’ farms that optimize water use in the approachspecified above. Rather the observations of values of the marginalproducts for water used, and the own-price elasticities that are de-rived from them, are based on observed situations of water useacross a region over a wide variety of crops grown in different sea-sons. Thus, they can be considered to be superior, as they reflectthe actions of those who use the resource.

However, a wider question still exists: Do farmers faced withwater shortages act rationally cutting the crop that pays the lowestvalue per unit of water. In many cases in the developed world thismay well be the case. Conversely, in India and the rest of the thirdworld where a high proportion of crops are produced and con-sumed on a subsistence basis, this approach can be questioned.Growing cash and higher valued crops is inherently more riskywhen water is in short supply. Producing a crop that you know willfeed you and your household for another year is a more rationalapproach. The other approaches to estimating demand for irriga-tion water, especially the programming model approach, only takeaccount of the financial returns to the crop as a whole. In realityother concerns apart from financial ones, like crop rotation andbeing self sufficient, also play a role and these also need to beinvestigated.

Acknowledgements

This research was funded by the Australian Centre for Interna-tional Agricultural Research. The authors are grateful to the com-ments received from two anonymous reviewers and as a resulthave altered their approach significantly. Despite their best efforts,any remaining errors are the sole responsibility of the authors.

References

Appels, D., Douglas, R., Dwyer, G., 2004. Responsiveness of Water Demand: A Focuson the Southern Murray Darling Basin. A Productivity Commission StaffWorking Paper, Melbourne.

Barton, D., Taron, A., 2010. Valuing Irrigation Water using Survey-Based Methods inthe Tungabhadra River Basin, India’, Irrigation and Drainage Systems, pp. 1–13,<http://dx.doi.org/10.1007/s10795-010-9098-2> (accessed 1.10.10).

Bell, R., Gali, J., Gretton, P., Redmond, I., 2007. The responsiveness of Australian farmperformance to changes in irrigation water use and trade. In: Presented to the51st Annual Conference of the Australian Agricultural and Resource EconomicsSociety, Canberra February.

Bontemps, C., Couture, S., 2002. Irrigation water demand for the decision maker.Environ. Dev. Econ. 7 (4), 643–657.

Davidson, B., 2004. The problems of analysing markets for irrigation water,submitted to Australian economic review. In: A Contributed Paper to the 48thConference of the Australian Agricultural and Resource Economics Society,Melbourne, February.

Davidson, B., Hellegers, P.J.G.J., Samad, M., 2009. Assessing the Economic Impact ofRedistributing Water within a Catchment: A Case Study in the Krishna Basin inIndia, IWMI Research Report 139, Colombo.

George, B., Malano, H., Davidson, B., Hellegers, P., Bharati, L., Sylvain, M., 2011. Anintegrated hydro-economic modelling framework to evaluate water allocationstrategies I: model development. Agric. Water Manage. 98 (5), 733–746.

Griffin, R., 2006. Water Resource Economics: The Analysis of Scarcity, Policies, andProjects. MIT Press, Cambridge, Mass.

Hanemann, W.M., 2005. The Economic Conception of Water, Working Paper 10005,Department of Agricultural and Resource Economics and Policy, University ofCalifornia, Berkley.

Hellegers, P.J.G.J., Davidson, B., 2010. Determining the disaggregated economicvalue of irrigation water in the Musi catchment in India. Agric. Water Manage.97 (6), 933–938.

Molle, F., Vinot, J.P., Lannerstad, M., Hoogesteger, J., 2009. Villains or heroes?Farmers’ adjustments to water scarcity. Irrig. Drain. doi:10.1002/ird.500.

Salvatore, D., 2004. Theory and Problems of Microeconomic Theory, third ed.McGraw Hill, New Delhi.

234 B. Davidson, P. Hellegers / Journal of Hydrology 408 (2011) 226–234

Scheierling, S.M., Loomis, J.B., Young, R.A., 2006. Irrigation water demand: a meta-analysis of price elasticities. Water Resour. Res. 42, 1–9.

Tomek, W., Robinson, K., 2003. Agricultural Product Prices, fourth ed. CornellUniversity Press, Ithica, NY.

Varian, H., 1987. Intermediate Micro-economics. A Modern Approach. W.W. Nortonand Co., New York.

Vinot, J.P., 2009. Rural dynamics and new challenges in the Indian water sector: thetrajectory of the Krishna Basin, South India. In: Molle, Francois, Wester, P. (Eds.),

River Basin Trajectories: Societies, Environments and Development.Wallingford, UK: CABI; Colombo, Sri Lanka: International Water ManagementInstitute (IWMI), pp. 214–237.

Ward, F., Michelsen, A., 2002. The economic value of water in agriculture: conceptsand policy applications. Water Policy 4 (5), 423–446.

Young, R., 2005. Determining the Economic Value of Water: Concepts and Methods,Resource for the Future, Washington, D.C.