optimal crop choice, irrigation allocation, and the impact of contract farming

Optimal Crop Choice, Irrigation Allocation, and theImpact of Contract Farming

Woonghee Tim HuhOperations and Logistics Division, Sauder School of Businesss University of British Columbia, Vancouver, British Columbia V6T-1Z2,

Canada, [email protected]

Upmanu LallColumbia Water Center and The Department of Earth and Environmental Engineering, Columbia University, New York,

New York 10027, USA, [email protected]

T he changing climate and concerns over food security are prompting a new look at the supply chain reliability ofproducts derived from agriculture, and the potential role of contract farming as a mechanism to address climate and

price risk while contributing toward crop diversification and water use efficiency is also emerging. In this study, the deci-sion problem of a farmer associated with allocating his land among different crops with varying water requirements isconsidered, given that a subset of the crops may be associated with a forward contract that is being offered by a buyer.The problem includes a decision to acquire a certain amount of irrigation water capacity prior to the season and to allo-cate this capacity as irrigation water to be applied during the season to each of the crops selected. Rainfall in the growingseason and the market price of each crop at the end of the season are considered to be random variables. Two stochasticprogramming models are developed to consider facets of this problem and to understand how contracts that reduce mar-ket price uncertainty from the problem may change the farmer’s decision. The structural properties of these models arediscussed, and selected implications are illustrated through an application to data from the Ganganagar district in Rajas-than, India.

Key words: crop planning; contract farming; irrigation; agriculture; IndiaHistory: Received: October 2010; Accepted: July 2012 by George Shanthikumar, after 2 revisions.

1. Introduction

1.1. MotivationFood security is a key global social goal. According toa comprehensive report by the United Nations Foodand Agriculture Organization (2002), which assessesthe long-term development of world food and agri-culture for the next three decades, the worldwidedemand for agriculture products is projected to growat a modest rate of 1.5% per year, and global shortagesare unlikely to occur provided that appropriate inter-ventions exist. However, serious supply shortages aswell as land scarcity may persist at regional and locallevels in conjunction with chronic or episodic (e.g.,drought) water scarcity.According to the United Nations Food and Agricul-

ture Organization (2003), the land held for agricul-tural use during the past five decades has expandedby a mere 12% to 11% of the global land surface whilearable land per capita has in fact declined by 40%.Such a decline has been compensated for by anincrease in cropping intensities and crop yieldgrowth. For example, the world average grain yielddoubled to 2.4 tonnes per hectare from 1962 and 1996.

This increase is expected to account for 80% of thefuture increases in crop production in developingcountries. The ability to increase production yield isclosely related to water, as the potential to improveyield is restricted in non-irrigated agriculture (whichdepends entirely upon rainfall and accounts for 60%of the production in developing countries). Irrigatedagriculture, which covers only 20% of the total arablearea, is responsible for 40% of the production. Whileirrigation is important for agricultural production, itaccounts for a large portion (approximately 70%) ofthe global water usage taken from rivers and ground-water. For example, in India, agriculture constitutesat least 90% of water usage (Kumar et al. 2005). Theimportance of irrigation and its impact on water willcontinue to assume a greater role since the areaequipped for irrigation is projected to increase by 20%by the year 2030, through an estimated investment of25–50 billion US dollars per year.Water is routinely cited as a critical resource limita-

tion and a persistent crisis for many developing coun-tries, particularly India and China (Briscoe 1999,Dinar and Subramanian 1997, Kinzelbach et al. 2004).As illustrated in Brown and Lall (2006), average

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Vol. 22, No. 5, September–October 2013, pp. 1126–1143 DOI 10.1111/poms.12007ISSN 1059-1478|EISSN 1937-5956|13|2205|1126 © 2013 Production and Operations Management Society

annual rainfall, inter-annual variability in rainfall,and intra-annual variability in rainfall are highly cor-related with the per capita national GDP (the excep-tions being countries that have oil or other resourcesthat stimulate their economy, e.g., Libya and Nigeria).The developing and poor countries are typically theones with low to medium average annual rainfall,and high variability in that rainfall. This sensitivity toclimate and water availability has the greatest impactin agriculture, which is the dominant water user.Since a large proportion of the population in many

countries (e.g., 70% in India) is rural and engaged inagriculture, crop failure assistance and rural liveli-hood support programs may constitute significantnational expenses, even if they are relatively ineffec-tive (Jha et al. 2007). Access to free (or nearly so) reli-able water or energy sources to pump groundwaterbecomes a major political issue, and meeting thisdemand translates into significant subsidies andexpenses for both water and energy, while impactingthe other uses of these resources negatively (Monari2002). To meet national food security goals in thishighly variable climate, price supports for grains areoffered and procurement targets are set. Since cropfailure is financially protected and a firm price isoffered in the event of crop success, farmers’ cropplanning decisions are determined by non-marketforces. This, combined with the provision of freewater, maintains suboptimal water usage strategiesand limits the possible investment in efficient waterinfrastructure development and maintenance (Jhaet al. 2007, Monari 2002).The Green Revolution led to significant gains in

crop yield through the application of fertilizer, irriga-tion, and improvements in the genetic strains of manycrops. However, there has been little improvementin Asia in the last decade and very little change inAfrica. The limited effectiveness of government inthese programs is often cited. There is potential for adramatic increase in crop yield per unit of water forexisting crops through investments in appropriateirrigation methods and technology. However, mostdeveloping country farmers are caught in a povertytrap. They may not have the capital to invest in irriga-tion technology (according to the Food and Agricul-ture Organization 2003, costs can range from US$1000to US$10,000 per hectare, not counting the develop-ment of a water storage or source) and may not haveguaranteed access to the source of irrigation water.Consequently, when farmers are faced with uncertainrainfall, the ability to transition to a more effectivetechnology is not there. If private capital is effectivelyinvested, it could help stabilize the incomes of bothfarmers and corporations. This is the premise of con-tract farming, where corporations could engage in for-ward contracts with farmers and also supply needed

inputs in exchange for a quality controlled productavailable in a timely manner. A more sophisticatedapproach could lead to dynamic crop choices thatwould consider the global marketplace and climatechange as key variables (Bravo-Ureta and Pinheiro1993, Hamdy et al. 2003). Greater private participa-tion in crop selection and water supply could intro-duce local farmers to options that increase revenue,save water, and provide access to the global market-place. Exploring how to achieve this transition tohigher efficiency and productivity can provide anopportunity for sustainable agricultural developmentand water management. This transition requires anunderstanding of how farmers make decisions as wellas what changes the farmers are able to adopt.The primary focus of this study is to provide a

simple and stylized model for the farmers’ decisionmaking process for crop planning and water manage-ment in the contract farming context. This model isused to develop insights into the interplay among cropselection and allocation, irrigation decisions, and con-tract farming. It is also hoped that our model will serveas a reference point for future empirical fieldwork aswell as a prototype for buildingmore detailedmodels.

1.2. Summary of Our ResultsWe consider the problem of a farmer who decideswhich crops to plant and how to allocate a fixedamount of land to the crops when a forward contractmay or may not be present. These decisions are madewhile rainfall and crop prices are still uncertain. Thefarmer also decides the amount of irrigated water tobe applied to each crop, which may occur before orafter the rainfall amount becomes known. We formu-late our problems as stochastic optimization prob-lems, where uncertainties arise from rainfall and cropprices. We identify concavity properties, which notonly enable us to understand the underlying proper-ties of the farmer’s problems and the relationshipamong several parameters, but also position ourmodel of the farmer as a useful building block formore complicated models.We also report the results of a computational inves-

tigation. Based on historical data in one of the districtsin India, we consider the farmer’s problem where thepossible crop choices are rice and sugarcane. Gener-ally, rice requires more water than sugarcane, but thenonlinearities in the yield rates of sugarcane and ricewith respect to water lead to an interesting trade-offthat depends on the water usage. Our results showthat, as the expected price of sugarcane increases(e.g., due to the increase in the global demand forbiofuels), not only does the overall net profit increasebut more land is allocated to sugarcane; however, theirrigation capacity may increase or decrease depend-ing on its cost. For a risk-averse farmer, as the degree

Huh and Lall: Crop Choice, Irrigation Allocation, and Contract FarmingProduction and Operations Management 22(5), pp. 1126–1143, © 2013 Production and Operations Management Society 1127

of risk aversion decreases (which occurs as the farmergains easier access to capital or as independent farm-ers merge to become a larger cooperative or corpora-tion), the irrigation capacity decreases due to thefarmer’s reluctance to make an investment up front.In the case of contract farming, the farmer’s decisionsas well as his water usage are sensitive to the contractand cost parameters.

1.3. Literature Review and Our ContributionThere is a vast literature on the application of opera-tions research and operations management to cropplanning and other related problems in agriculture.Many of these studies are reviewed in Glen (1987)and Lowe and Preckel (2004), and the intention of thissection is to highlight a limited number of relatedstudies. A large portion of the literature is devoted todeveloping linear programming or other mathematicalprogramming formulations for specific problemsinvolving farm land allocation, crop rotation, harvesting,etc. (e.g., Audsley 1985, Cocks 1968, Heady 1954,Jolayemi and Olaomi 1978, Rae 1971). The primaryfocus of the aforementioned studies is to demonstratethe viability of mathematical programming tools inefficiently solving a range of proposed problems,whereas, in comparison, our objective also includesobtaining the structural characterization of the opti-mization problem.Yield uncertainty is a common characteristic in

agriculture, and several models have been developedto mitigate the risk of mismatch between supply anddemand. Jones et al. (2001, 2002, 2003) study seedcorn production and show the benefit of the secondproduction opportunity. Kazaz (2004) and Allen andSchuster (2004) study the capacity and productionquantity decisions under yield uncertainty for oliveoil production and the Concord grape harvest, respec-tively. Recently, Kazaz and Webster (2011) examinedhow yield uncertainty affects the optimal decisions ofa food processing company. In the manufacturingand production literature, the randomness in yield istypically represented as an exogenous random vari-able (e.g., Yano and Lee 1995), and the source ofrandomness is usually not investigated. In the agricul-tural setting, where water availability is the mainsource of yield uncertainty, it is reasonable to build astochastic optimization model in which the recourseaction depends on rainfall and irrigation. In theMaatman et al. (2002) study of the Central Plateau ofBurkina Faso in West Africa, stochastic recourse deci-sions depend on the observed rainfall values, butthere exists no control to affect the yield of crops. Incontrast, we endogenize the stochastic yield rate byallowing the farmer to possibly increase water supplythrough irrigation (recourse decision). While thereexist several papers that demonstrate the dependency

of the crop yield rate on multiple factors or investigatethe control problem of dynamically managing severalfactors to increase the yield rate, their results are spe-cific to the specific site under investigation, and fur-thermore they do not consider decisions at a higherscope such as crop planning limitations imposed byresource availability (Drummond et al. 2004, Li et al.2004, Liu et al. 2001, Shani et al. 2004). To our knowl-edge, we are unable to identify any existing model foroptimal crop planning that explicitly incorporates irri-gation decisions under rainfall uncertainty.The model in this study is general enough to study

contract farming in a single framework. Under con-tract farming, the farmer enters into a contract with abuyer to sell the harvest before the beginning of theselling season. While there are several types of con-tracts used in contract farming, one of the most popularcontracts is the forward contract, which specifies boththe quantity and price in the future transaction.Under contract farming, the buyer can potentiallybenefit from the reduced risk of purchase price andsupply availability, and the farmer can also benefitfrom access to financial resources and technology(Rehber 1998). Yet, it is not clear whether and how thecurrent practice of contract farming benefits all theparties involved. Existing research on contract farm-ing is mostly empirical or case-based (e.g., Runstenand Key 1996, Warning and Key 2002). There are alimited number of recent analytic models for contractsin the agriculture business (e.g., Burer et al. 2008, Heand Zhang 2008), but these models do not address thecontract farming problem directly.A well-known concept in operations management

similar to contract farming is “contract manufactur-ing.” The motivating rationale for such an arrange-ment is core competency, where the originalequipment manufacturer (buyer) focuses on productinnovation and the contract manufacturer (supplier)focuses on capacity utilization through pooling(Plambeck and Taylor 2005, Ulku et al. 2007). Con-tract farming differs from contract manufacturingin several aspects. The primary purpose of the buyerin contract farming is to ensure quantity and pricestability of the supply instead of mitigating the riskassociated with investment. Whereas the contractmanufacturer achieves the pooling effect by supply-ing to multiple buyers, the contract farmer supplies toonly one buyer, who procures from a large number offarmers in each season. The random yield effect has amore dramatic effect in the agricultural setting, andthe implementation of contract farming can poten-tially infringe upon more public policy issues than itsmanufacturing counterpart.In this study, we propose a stochastic programming

model for studying a farmer’s crop planning problemthat explicitly accounts for rainfall uncertainty and

Huh and Lall: Crop Choice, Irrigation Allocation, and Contract Farming1128 Production and Operations Management 22(5), pp. 1126–1143, © 2013 Production and Operations Management Society

irrigation decisions. Our model enables us to under-stand the relationship among several parameters ofthe model and their impact on one of the most impor-tant but often neglected resources, namely, water. Italso provides a unified and parsimonious frameworkfor studying the impact of risk aversion and contractfarming. Our study can be viewed as one of thesteps toward extending the operations literature tothe globalization of agriculture, following the call ofresearch by Lowe and Preckel (2004). As Schweigmanet al. (1990) noted, agriculture is one of the areaswhere operations research can make significant con-tributions in developing countries. From a technicalperspective, the concavity properties that we identifyfor our formulations do not follow from standard con-vexity preservation results, and we overcome this dif-ficulty by using an approach based on the firstprinciples.

2. Basic Model: With Static IrrigationControl (Model S)

We assume that the crop allocation and irrigationcapacity decisions are made at the beginning, beforeany of the uncertainties are resolved, and marketprices for crops are realized at the end after all thedecisions are made. For the modeling of the irrigationusage decision (the quantity and allocation of irri-gated water) and rainfall realization, the most realisticmodeling approach would be a multi-period dynamiccontrol of irrigated water that depends on the evolu-tion of rainfall during the season; however, such anapproach would be computationally intensive andmay not produce additional insights, and thus weintroduce simplifications. In this section, we intro-duce the case where the irrigation usage decision pre-cedes rainfall realization, which we refer to as theModel with Static Irrigation Control, or simply Model S.The other case where this decision follows rainfallrealization is referred to as the Model with ResponsiveIrrigation Control or Model R, and it is discussed insection 4.

2.1. DescriptionWe consider the decision of the farmer, who has con-trol over a fixed area of agricultural land, and thisarea cannot be increased or decreased during the

planning horizon. Without loss of generality, the totalarea of land is 1 unit. We assume that the parametersof a contract, if it exists, are already determined, andthey are exogenous to our model and known to thefarmer. There are N types of potential crops, indexedby i = 1,…,N. We assume that the following sequenceof events occurs—see Figure 1 for an illustration.(Below, “A” represents action and “O” representsobservation.)

(A) At the beginning of the year, the farmerplants a subset of potential crops for theupcoming farming season. Let ui representthe amount land allocated to crop i. Thus,u1 þ �� þ uN ¼ 1. In addition, he may decideto invest in the irrigation project, which pro-vides additional water for his farmland dur-ing the season. Let v be the irrigationcapacity, which is the maximum amount ofwater available for irrigation, and this capac-ity v is also a part of the farmer’s decisions.1

(If the farmer does not invest in the irrigationproject, then v = 0.) Furthermore, the farmeralso makes the decisions regarding how theirrigated water will be used. His allocationshould satisfy the capacity constraint thatthe total amount of irrigated water cannotexceed v. Let wi � 0 denote the total amountof water irrigated to crop i. Then, w1 þ ��þwN � v. If the farmer distributes wi amountof water unevenly in the farmland of crop i,it can be shown to be suboptimal, andthus we assume without loss of generality aneven distribution of water within each croparea.

(O) Let R � 0 denote the total amount of rainfallin a season, which we assume to fall evenly inthe land. The value of the exogenous randomvariable R is realized, and we denote the real-ized value by r. (While the farmer observesthe realized rainfall r, we assume that he can-not block rainfall from a portion of his land ordivert it to another portion.) The yield amountof each crop is given by the amount of landallocated to the crop, ui, multiplied by theyield rate. The yield rate of each crop, ci,depends on the total amount of water input,which is the sum of the rainfall and water

Figure 1 Sequence of Events for Model S (Static Irrigation Control)


irrigation. The actual market prices for cropsare also realized, and the actual payoff to thefarmer is based on these prices as well as thecontract parameters, if applicable.

We now describe the components of the farmer’spayoff function (objective function). Our model isflexible enough to accommodate contract farming andthe farmer’s risk aversion.CROP YIELD RATE. We use ciðyÞ to denote the yield

rate of crop i, which represents the quantity of crop iharvested from a unit area of farmland, given that thetotal amount of water input is y. The total amount ofwater y is the sum of (i) the rainfall and (ii) the irri-gated water. We model the yield rate as a function ofthe total amount of input water only. (In the case ofwheat, it has been identified that the key drivers forthe yield rate are water and fertilizers usages (Li et al.2004). We exclude the fertilizer decision in our model,which is a reasonable modeling assumption if theoptimal amount of fertilizer can be dispatched rela-tively easily since its availability is less constrained bystochastic elements of the system (e.g., rainfall), or iffertilizer is not available at all. The dominant effect ofwater on the yield has also been recognized by theempirical model of Kumar and Khepar (1980), whohave developed the relationship between water inputand production yield in India, and also by the opera-tions research model of Schweigman et al. (1990),who have employed statistical regression for studyingthe rainfall data in Tanzania.As the amount of water input increases, it is reason-

able to expect that the yield rate increases initially;however, an excessive amount of water is not desir-able. We model that ci is a concave function with afinite maximum (e.g., Kumar and Khepar 1980). Forsimplicity of exposition, we assume that the maxi-mizer of ci is unique and positive, and ci is strictlyincreasing between 0 and the maximizer. (In reality,the yield rate will be zero unless the total water inputexceeds a certain minimum threshold quantity, inwhich case, the yield rate function fails to be concave.However, unless there is an extreme drought or flood,it is reasonable to expect that this function for the cropof our interest is concave within a reasonable domainof water input.)Suppose that the amount of irrigated water, wi, is

distributed in the area of ui. Then, the amount ofirrigated water normalized for per unit area iszi ¼ wi=ui. Since r represents the amount of rainfallper unit area, the total amount of water a unit areawould receive is r þ wi=ui, and the yield rate of thisarea is a function of this quantity:

ciðrþ wi=uiÞ:

The realized amount of crop i is given by theamount of land allocated to crop i multiplied by theyield rate, which is

xi ¼ ui � ciðrþ wi=uiÞ ¼ ui � ciðrþ ziÞ:IRRIGATION COST. We model the cost of irrigation

with the following two components: capacity invest-ment and usage. The capacity investment cost is theupfront cost spent to ensure the availability of irri-gated water. It may include, for example, drilling awell or building a water reservoir or acquiring waterrights (where the cost is annualized). We assume thatthe investment cost is proportional to the irrigationcapacity v. The usage cost is associated with theactual amount of water irrigated to the land andincludes the cost of electricity for pumping waterfrom a well or the transportation cost for water fromthe reservoir to the farmland. We also assume that theusage cost is linear in the amount of irrigated water.Thus, the total cost of irrigation, C, is given by thesum of j1 � v and j2 � ðw1 þ � � � þ wNÞ, wherej1; j2 � 0. Since wi ¼ ui � zi, it follows that this costcan be written as

C ¼ Cðv; z1; . . .; zNÞ ¼ j1 � vþ j2 �XNi¼1

ui � zi:

(Note that C depends on ðu1; . . .; uNÞ, but we sup-press such dependence in our notation for brevity.)NET REVENUE AND CONTRACT FARMING. Let x ¼

ðx1; . . .; xNÞT denote the total yield vector, wherexi ¼ ui � ciðrþ ziÞ and zi ¼ wi=ui. Let P ¼ ðP1; . . .;PNÞTdenote the vector of nonnegative random variables,each corresponding to the market price of crops.These random variables, P1; . . .;PN , may be corre-lated. Then, the farmer’s revenue without any con-tract is given by

P � x ¼XNi¼1

Pi � xi:

Note that the farmer’s revenue is separable in z.Furthermore, it is also linear in this case.Now, we show how we can incorporate contract

farming into our model. We consider the commonlyused forward buy/sell contract, where the farmer andthe buyer commit to both the quantity and the priceat the beginning of the planning horizon. LetfðpFi ; qFi Þ : i ¼ 1; . . .;Ng represent the family of for-ward contracts. (In our model, we allow the possibilityof multiple contracts, one for each crop, though inpractice a farmer typically commits to at most onecontract. If qFi ¼ 0, it indicates the absence of aforward contract for crop i.) The specification of thesecontracts is exogenous to our model, and consequently


neither the contract price nor the quantity are consid-ered explicitly as decision variables. This situationcan be changed where we consider a more sophisti-cated model of the interaction between the farmerand the contract offerer. For now, we investigate theeffects of changes in these variables parametrically. Ifthe farmer produces exactly the quantity specified bythe contract, then he earns pFi � qFi from crop i. If thefarmer produces more than the forward contractamount, then he sells any excess amount of the cropto the market at the spot price; thus, the additionalrevenue from the market for crop i is Pi � ½xi � qFi �þwhere xi is the production quantity of crop i. Other-wise, if the farmer does not produce enough quantityto satisfy the forward contract, we assume that hemust pay the buyer a penalty cost proportional to theshortage amount. (The shortage penalty possibly rep-resents additional costs such as emergency activitiesand transportation and acquisition of supply by thecontract offerer at spot market prices.) Letb ¼ ðb1; . . .; bNÞ be the vector of per-unit shortagepenalty costs. Furthermore, we allow b to depend onP and require only that the penalty cost satisfiesb � P (for example, bi ¼ Pi þ fi where fi � 0 orbi ¼ fi � Pi where fi � 1). This constraint of b basedon P rather than on pF addresses the situation wherethe farmer may default on the contract if the marketprice were higher than the contract price. Then, thefarmer’s net revenue is given by

R ¼ RðxÞ ¼ pF � qF þ P � x� qF� �þ�b � qF � x

� �þ;

where qF ¼ ðqF1 ; . . .; qFNÞ and pF ¼ ðpF1 ; . . .; pFNÞ.The following result shows that the net revenue

function is a well-behaved function of x, even withthe presence of the contract.

PROPOSITION 1. For any P, RðxÞ is separable, nonde-creasing and concave in x.

RISK AVERSION AND PAYOFF FUNCTION. Recall that Cdenotes the irrigation cost andR denotes the net reve-nue from crop yields. We then define the farmer’sprofit by computing R minus C. To incorporate thefarmer’s aversion to the stochasticity in revenue, wetake the approach of evaluating the expected utility ofprofit, where the utility function is a single-dimen-sional function U : < ! < that is both increasing andconcave. The farmer’s objective is to maximize hisexpected utility, which we refer to as his payoff:

max P ¼ E UðR � CÞ½ �;

where the expectation is taken over rainfall R andcrop prices P. Note that Π is a function of u ¼ðu1; . . .; uNÞ, v and w ¼ ðw1; . . .;wNÞ. If U is an iden-

tity function, then Π is simply E½R � C�, and werefer to this special case as the risk neutral case.

2.2. Stochastic Programming FormulationWe now present a mathematical programming formula-tion of the stochastic optimization problem. In thedescription of section 2.1, the farmer’s decisions are thetriplet of u, v, and w. It is convenient to replacew ¼ ðw1; . . .;wNÞ with z ¼ ðz1; . . .; zNÞ using the rela-tionship zi ¼ wi=ui. None of these decisions depends onthe realizations ofR orP. The formulation is given below:

(SP-S) maxu;v;z

ER;P

�UðRðu1 �c1ðz1þRÞ; . . .;uN �cNðzNþRÞÞ�Cðv;z1; . . .;zNÞÞ

�s. t.

XNi¼1

ui �zi�v u1þu2þ��þuN ¼ 1

v�0;ui�0;zi�0 for each i¼ 1; . . .;N:

2.3. AnalysisWe now describe the properties of (SP-S) that are use-ful in developing an algorithm for finding an optimalsolution for this formulation. Since both the objectivefunction and the feasible region of (SP-S) fail to beconvex, such problems are in general analyticallydifficult to solve. However, we can show that ourproblem can be written as a sequence of convex pro-gramming formulations, and consequently it becomeseasy to solve.WATER DISTRIBUTION WITHIN A SINGLE CROP. We first

comment on how irrigated water should be distrib-uted within the given crop area. If wi amount ofirrigated water is available for crop i, should it be dis-tributed evenly or concentrated in a smaller portionof the crop area? It can be argued that it is optimal todistribute wi amount of irrigated water evenlythroughout the crop area of size ui. This retroactivelyjustifies the even distribution assumption that we hadalready introduced.OPTIMAL IRRIGATION CAPACITY AND USAGE DECISION.

While the decision variables in (SP-S) consist of (u,v,z), we restrict our attention to a smaller problem ofoptimizing over (v,z) while holding u fixed. Supposeu1 þ u2 þ � � � þ uN ¼ 1. Note that in any optimalsolution, v ¼ w1 þ �� þ wN ¼ u1z1 þ �� þ uNzN mustalways be satisfied (since both irrigation capacity andallocation decisions are made at the same time, andthere is no reason to reserve irrigation capacity that isnot used in Model S). Let Π(u,v,z) denote the objectivefunction of (SP-S). Define

(SP-Sa) /ðuÞ ¼ maxz

P u;XNi¼1

uizi; z

!

s. t. zi � 0 for each i ¼ 1; . . .;N;


which is an optimization problem with respect to zfor fixed u. The following results follow from stan-dard results of convex analysis.

THEOREM 2. (SP-Sa) is a concave function maximizationproblem with nonnegativity constraints with respect toz ¼ ðz1; . . .; zNÞ.

There exist many efficient algorithms to solveconcave function maximization problems with linearconstraints. If the farmer is risk neutral, then we canfurther characterize the optimal solution. Since thenet revenue function R is separable, it can be writtenas Rðx1; . . .; xNÞ ¼ R1ðx1Þ þ � � � þ RNðxNÞ for certainsingle-variable functions fR1; . . .;RNg. By the concav-ity and monotonicity of R, each Ri is nondecreasingand concave. Thus, ER;Pi

½Riðui � ciðzi þ RÞÞ� is also con-cave in zi. The following result shows that the optimalsolution to (SP-Sa) can be found by optimizing overeach component separately.

PROPOSITION 3. In the risk neural case, an optimalsolution z to (SP-Sa) is given by, for each i = 1,…,N,

@

@ziER;Pi

½Riðui � ciðzi þ RÞÞ� ¼ ðj1 þ j2Þ � ui:

OPTIMAL CROP ALLOCATION DECISION. In our discus-sion of (SP-Sa), we have discussed the problem offinding the optimal decisions of z for given u, where/(u) denotes the optimal value of (SP-Sa). We nowconsider the problem of optimizing / over u, i.e.,

(SP-Sb) maxu

/ðuÞ s. t.XNi¼1

ui ¼ 1

ui � 0 for each i ¼ 1; . . .;N:

Our objective is to show the concavity of u, which isuseful in finding the optimal solution for (SP-Sb). Infact, we will show the convexity result for everysample path of r. We note that this result does notfollow from the standard convexity result of convexprograms (e.g., Rockafellar 1970, Chapter 29, Boydand Vandenberghe 2004, Chapter 5) since the objectivefunction contains a non-convex term ui � ciðzi þ RÞ.The main idea in the proof of Theorem 4 is to carefullyconstruct a z vector for amidpoint of a pair of arbitrarychoices of u vectors, such that the constructed z vectoris feasible and yields sufficiently high objective value.

THEOREM 4. The optimal value of (SP-Sb) is concave inthe first stage decisions ðu1; . . .; uNÞ.

Thus, in this section, we have shown that while(SP-S) for Model S is not a convex problem, it can be

written as a sequence of convex problems that can besolved easily.

3. Numerical Results

In this section, we present numerical results based onempirical data. In section 3.1, we present data and ourmodel parameters. In sections 3.2 and 3.3, we analyzethe risk neutral and risk averse cases, and then studythe impact of contract farming in section 3.4. Thediscussion in this section is based on Model S ofsection 2.

3.1. DataThe numerical computation of this study is basedon data from the Ganganagar district, in the state ofRajasthan in northwestern India, where the majorityof the population depends on agriculture. Thisregion has witnessed a transformation of the land-scape in the last century as the Gang canal hasbrought excess water from Punjab and HimachalPradesh to a region with comparatively low annualrainfall (around 300 mm). We focus on this regionbecause of the relative importance of irrigation inthis region and also because the Rajasthan state gov-ernment has encouraged contract farming. Wheatand cotton are the major crops grown in this dis-trict, but significant amounts of rice and sugarcaneare also grown.We take historical prices for rice between years

1961 and 1987 from an agricultural data set fromthe World Bank (Sanghi et al. 1998). (This databasecontains various statistics pertaining to Indian agri-cultural, climatological, edaphic and geographicalvariables.) The data set does not include the sugar-cane prices but includes the sugar prices, and wetake scalar multiples of sugarcane prices to be sugarprices. (The multiplier is chosen under the assump-tions that 100 kg of sugarcane is required to pro-duce 8.5 kg of sugar and that 60% of the cost ofsugar is due to the cost of sugarcane. These assump-tions are based on and consistent with varioussources such as India Infoline 2002, Gill 2000, TheHindu 2006, Multi Commodity Exchange of IndiaLtd 2006, and Sri Lanka Sugarcane Research Insti-tute 2005.) The prices are adjusted to the 1987 levelusing the annual Indian inflation rates for agricul-tural commodities, available from the World Develop-ment Indicators 2007, which is the World Bank’sannual compilation of data concerning development.The annual rainfall data in this region is based onthe historical data between the years 1951 and 2002(52 data points), available from the Indian Meteoro-logical Department (2007). The empirical data ofthe inflation-adjusted prices and rainfall have theaverage values of


E½Price� ¼ 470:1;E½Psugarcane� ¼ 24:09; and E½R�¼ 263:2;

and their histograms are shown in Figures 2 and 3.Furthermore, we observe that both the prices andrainfall are significantly variable, as indicated byfollowing standard deviations: r½Price� ¼ 206:9,r½Psugarcane� ¼ 8:06, and r[R] = 94.71. In addition,data show that the prices of two crops are positivelycorrelated whereas price and rainfall are negativelycorrelated; more precisely, the correlation coeffi-cients are q½Price;Psugarcane� ¼ 0:44, q½Price;R� ¼�0:20, and q½Psugarcane;R� ¼ �0:31.For our computation, since the random variables

are correlated, we use the k-nearest neighbor boot-strap method (Lall and Sharma 1996) to generate sam-ples of ðPrice;Psugarcane;RÞ. For R, we use 27 years ofdata between 1961 and 1987 (for which the corre-sponding price data are available). Then, for eachvalue of R, we choose samples of ðPrice;PsugarcaneÞ thatcorrespond to similar values of R. More precisely, letfðptrice;ptsugarcane;rtÞ : t ¼ 1; . . .;27g denote the historical

data. Let (t) denote the sorted index for rt’s, that is,rð1Þ � . . .� rð27Þ. We sample R uniformly fromfrð1Þ; . . .; rð27Þg. For any rðtÞ, we choose fromðpðt�2Þ; pðt�1Þ; pðtÞ; pðtþ1Þ; pðtþ2ÞÞ with probability propor-tional to ð13 ; 12 ; 1; 12 ; 13Þ. (We make necessary truncationsif t is either too small or too large.)We model the yield rates as a function of water

input using the water-production model estimated byKumar and Khepar (1980); in particular,

criceðyÞ ¼ 5:9384� 0:035206yþ 2:412043y0:5

csugarcaneðyÞ ¼ �11:5441þ 2:92837y� 0:0027y2;

where the input variable y is the total amount ofwater in millimeters per hectare, and the output c(y)is given in quintal per hectare. (One quintal isequivalent to 100 kg or 0.1 t.) Each of these func-tions is concave. For rice, crice is maximized at1173.48 mm/ha achieving the output value of 47.25qt/ha, and for sugarcane, csugarcane is maximized at542.30 mm/ha achieving the value of 782.50 qt/ha(see Figure 4). Note that, in general, rice requires agreater amount of water than sugarcane does, butthe yield rate of sugarcane is more sensitive to waterinput than that of rice. Furthermore, the amount ofwater input available from rainfall is insufficient forthe maximum yield of both crops.For the irrigation cost, we vary the irrigation capac-

ity investment parameter j1 in the range of 2–20rupees per millimeter-hectare. (This range wasderived based on the Food and Agriculture Organiza-tion 2003 estimate of the infrastructure cost ofUS$1000 to US$10,000/ha, using the annualizedinvestment cost rate of 15% under the assumptionthat the estimate was based on the capacity of 1000mm. Inflation and exchange rate data are taken fromInternational Monetary Fund and Yahoo finance.) For

Figure 3 The Histogram of the Rainfall Distributions (mm): 1951–2002. Based on 52 Observations in Total

(a)

(b)

Figure 2 The Histograms of the Inflation-Adjusted Price Distributions(rupees/qt): 1961–1987. Based on 27 Observations in Totalfor Each


simplicity, we fix j2 ¼ 0, that is, irrigation usage iscostless.For the utility function, we use the following

increasing concave function for the utility function:UðzÞ ¼ ½1� expð�kzÞ�=k where k > 0. Since UðzÞ ! zas k ↓ 0, we also define UðzÞ ¼ z if k = 0. We note thatthis utility function displays the “constant absoluterisk aversion” property, that is, �U00ðzÞ=U0ðzÞ is a con-stant.

3.2. Without Contracts: Risk Neutral CaseWe first consider the payoff function for the risk-neu-tral farmer in the absence of any forward contract.Here, the expected revenue is given by RðxÞ ¼E½Price� � xrice þ E½Psugarcane� � xsugarcane where xrice andxsugarcane represent the production quantities of riceand sugarcane, respectively. In this section, we fix theprice distribution of rice, but multiply the distributionof sugarcane by various scalars to study the effect ofprice changes in sugarcane. We also vary the costparameter j1 for irrigation investment.The computational results are summarized in

Figure 5. In all three figures, the horizonal axisrepresents a range of values of j1, and each line is

indexed by a multiplicative adjustment factor for theprice of sugarcane, Psugarcane, to study the impact ofchanges in the relative price. (Thus, the adjustmentfactor of 1.0 represents no modification.) In Figure 5a,we observe that the payoff objective function expect-edly decreases in the cost parameters j1 and increasesin the expected selling price of sugarcane, E½Psugarcane�.Figure 5b shows the allocation of land between the

two crops. Without any price adjustment, the cropallocation is always to plant rice (i.e., urice ¼ 1), for all

0 200 400 600 800 1000 1200 140025

30

35

40

45

50Yield Rate as a function of Water Input: Rice

Water Input (mm)

Yie

ld R

ate

(qui

ntal

/ha)

100 200 300 400 500 600 700 800200

300

400

500

600

700

800Yield Rate as a function of Water Input: Sugarcane

Water Input (mm)

Yie

ld R

ate

(qui

ntal

/ha)

(a)

(b)

Figure 4 Yield Rates (qt/ha) as a Function of Water Input (mm/ha)

0 5 10 15 201.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4x 10

4Expected Utility vs Kappa1: Risk Neutral Case

Kappa1 (Rs.)

Pay

off (

Rs.

)1.01.11.21.3

0 5 10 15 200

0.2

0.4

0.6

0.8

1Land Allocated to Rice vs Kappa1: Risk Neutral Case

Kappa1 (Rs.)

Fra

ctio

n A

lloca

ted

to R

ice

1.01.11.21.3

0 5 10 15 200

100

200

300

400

500

600

700Irrigation Capacity vs Kappa1: Risk Neutral Case

Kappa1 (Rs )

Irrig

atoi

n C

apac

ity (

mm

)

1.01.11.21.3

(a)

(b)

(c)

Figure 5 Without Contract: Risk Neutral Case (Model S). LegendRepresents the Multiplicative Adjustment for the Price of Sug-arcane Psugarcane. (a) Payoff Function Π vs. j1. (b) Crop-LandAllocated to Rice urice vs. j1. (c) Irrigation Capacity m vs. j1.


values of j1 tested. However, when the price of sugar-cane is 10% higher, as j1 increases, the crop allocationswitches from rice to sugarcane since sugarcanerequires less water than rice (crice is maximized at1173.48 mm/ha whereas csugarcane is maximized at542.30 mm/ha). The allocation, however, eventuallyswitches back to rice, which is less sensitive to waterinput than sugarcane. When the price of sugarcane ismuch higher (20% or higher), it is more profitable toplant sugarcane only.We consider the effect on the water resource in

Figure 5c, which shows an unsurprising result thatmore water is used when the cost of irrigation isaffordable. The comparison of all rice and allsugarcane (corresponding to the adjustment factors of1.0 and 1.2, respectively) shows that the sensitivity ofthe irrigation capacity on its cost parameter j1 ishigher in rice than in sugarcane. This observation canbe explained by the fact that the yield curve of sugar-cane is comparatively more sensitive, implying thatbringing the total water input to its maximizerbecomes crucial. A few implications of our findingsare (i) the crop-by-crop land allocation, with disregardto the irrigation cost, is not necessarily a good indica-tor for estimating water usage and (ii) reducing thefarmer’s cost of irrigating water (e.g., through subsidy)has quite a substantial impact on the water usage.

3.3. Without Contracts: Risk AversionTo study the effect of risk aversion, we test variousvalues of the parameter k, a proxy for the degree ofrisk aversity. From the definition of the utility func-tion U, a higher value of k implies lower payoff, whichis shown in Figure 6a. This figure also shows that theexpected payoff decreases in the irrigation costparameter j1, as expected.Figure 6b shows that while the risk-neutral farmer

(k = 0) plants rice only, some mix of sugarcanebecomes more preferable as the farmer becomes morerisk averse (k ¼ 0:50 � 10�4 or 0:75 � 10�4). Thisobservation can be explained by the fact that diversifi-cation is generally a good strategy against risk (as infinancial assets). Moreover, we note that the farmercan easily control the yield rate of sugarcane by satis-fying its modest water requirement; for rice, however,it becomes too expensive to prepare enough irrigationfor the maximum yield of rice and thus its yield rateis subject to the outcome of the rainfall. This explainswhy the rice is the more risky crop, which the riskaverse farmer likes to avoid.In Figure 6c, we see that irrigation capacity tends to

decrease in the risk aversion parameter k as well asthe irrigation cost j1, as the farmer becomes reluctantto spend an upfront investment that may or may notbecome useful. It has an important policy implicationsince the initiatives to reduce the irrigation cost to the

farmer (e.g., through government subsidy) and todecrease the farmer’s risk aversion (e.g., through cropinsurance, corporatization of farming and easy accessto capital), albeit politically appealing, are likely toresult in the higher usage of irrigation if appropriateinterventions fail to exist.In summary, we see that, in the absence of any con-

tract, risk aversion of the farmer leads to crop diversi-fication. It also discourages the farmer from investingin irrigation capacity since the risk-averse farmer

0 5 10 15 200.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

4 Payoff vs Kappa1: Risk Averse Case

Kappa1 (Rs.)

Pay

off (

Rs.

)0.000.250.500.75

0 5 10 15 20

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Land Allocated to Rice vs Kappa1: Risk Averse Case

Kappa1 (Rs.)

Fra

ctio

n A

lloca

ted

to R

ice

0.000.250.500.75

0 5 10 15 200

100

200

300

400

500

600

700Irrigation Capacity vs Kappa1: Risk Averse Case

Kappa1 (Rs.)

Irrig

atoi

n C

apac

ity (

mm

)

0.000.250.500.75

(a)

(b)

(c)

Figure 6 Without Contract: Risk Averse Case (Model S). Legend Rep-resents the Risk Aversion Parameter c in 10�4 (a) PayoffFunction Π vs. j1. (b) Crop-Land Allocated to Rice urice vs.j1. (c) Irrigation Capacity m vs. j1.


would like to avoid the up‐front cost; thus, measuresto reduce the risk aversion of the farmer may increaseoverall water usage.

3.4. With Forward ContractsWe now consider the impact of contract farming byincorporating a forward contract into the computa-tional analysis. We assume that there is no contractfor rice and that the forward contract for sugarcane isgiven by two parameters qFsugarcane and pFsugarcane. Wefix the penalty cost at 150% of the market price, thatis, bsugarcane ¼ 1:5 � Psugarcane. If xrice and xsugarcanerepresent the total yield of rice and sugarcane, respec-tively, then the profit function is given by

R� C ¼ Price � xrice þ pFsugarcane � qFsugarcaneþ Psugarcane � ½xsugarcane � qFsugarcane�þ

� bsugarcane � ½qFsugarcane � xsugarcane�þ:

In the definition of the utility function U, we usek ¼ 0:25 � 10�4.We study the impact of the contract parameters by

varying pFsugarcane in the range of 80% and 160% ofE½Psugarcane� and also varying qFsugarcane between 0 and1000. The value of qFsugarcane ¼ 0 corresponds to thecase without any contract. Figure 7 summarizesthe case of j1 ¼ 2, corresponding to a low value ofthe irrigation cost, and Figure 8 summarizes the caseof j1 ¼ 10, a moderate value of the irrigation cost.We observe that the payoff to the farmer depends

on the contract parameters. For a fixed value offorward quantity qFsugarcane, the expected payoff inFigures 7a and 8a is unsurprisingly higher if theforward price pFsugarcane is higher. It is interesting tonote that when the forward price is the same as theexpected market price, there is no benefit of the con-tract to the farmer regardless of the quantity pFsugarcane.The contract provides the farmer with price certainty,which the risk-averse farmer prefers. At the sametime, this exposes the farmer to a greater penalty incase that the yield is not sufficiently high enough tocover the quantity under contract, and thus yielduncertainty becomes a dominating consideration rela-tive to price uncertainty under contract farming. Italso forces the farmer to reduce the amount of ricegrown. This trade-off may not always be favorable tothe farmer, who therefore requires a premium on theforward contract price compared with the expectedmarket price in order to compensate the farmer forthe quantity risk associated with sugarcane and theopportunity cost associated with rice.In addition, these figures show that, for the fixed

forward price, the payoff initially increases as theforward quantity qFsugarcane increases, but it eventuallydecreases as the farmer is unable to satisfy the

contract quantity and becomes penalized. The payoffis a concave function of the forward quantity. Thisimplies that if the farmer is unsure about the benefitof contract farming, it is safer for him to experimentwith it on a smaller scale—with a smaller amount ofthe contract quantity.The land allocation decisions are shown in Figures

7b and 8b, which show that the land is to be exclu-

0 200 400 600 800 10000.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

4Payoff vs Cane Contract Quantity: Kappa1=2

Sugarcane Forward Contract Quantity

Pay

off (

Rs.

) 0.81.01.21.41.6

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1Land Allocation to Rice vs Cane Contract Quantity: Kappa1=2


Fra

ctio

n A

lloca

ted

to R

ice

0.81.01.21.41.6

0 200 400 600 800 1000250

300

350

400

450

500

550

600

650

700Irrigation Capacity vs Cane Contract Quantity: Kappa1=2


Irrig

atoi

n C

apac

ity (

mm

)

0.81.01.21.41.6

(a)

(b)

(c)

Figure 7 Contract Farming: j1 ¼ 2 (Model S). Legend Represents theSugarcane Forward Price as a Multiple of the Expected Mar-ket Price, That Is, pFsugarcane/E [Psugarcane]. c ¼ 0:25 � 10�4.Note that qFsugarcane ¼ 0 Corresponds to No Contract Farming.(a) Payoff Function Π vs. qFsugarcane. (b) Crop-Land Allocated toRice urice vs. qFsugarcane. (c) Irrigation Capacity m vs. qFsugarcane


sively used for rice when there is no contract, butmore and more land is used for sugarcane as the for-ward contract requires an increased amount of sugar-cane. This monotonicity result is again intuitive. Weobserve that when the contract quantity qFsugarcane issmall, the area of land allocated to sugarcane is pro-portional to qFsugarcane; in other words, we do not seeany aggregation or pooling effect on this area. The

pooling effect on aggregate quantities usually existsbecause of the reduction of variability from the sum-mation of random variables, but in our case, such areduction does not exist since the yield rate in onepart of the land is perfectly correlated with the yieldin another part. When the contract quantity is suffi-ciently high, the entire land is dedicated to sugarcane.We note, however, that the farmer’s allocation deci-

sion is insensitive to the forward price; this phenome-non can be explained by the fact that the value ofpFsugarcane does not affect the optimal decisions (sincepFsugarcane appears as a constant in the expression ofthe profit Π, and the utility function that we use hasthe CARA property). Since pFsugarcane, however, affectsthe farmer’s payoff Π, an appropriate choice of thiscontract parameter will be important in order toentice the farmer to accept this contract in the firstplace. We deduce that the forward contract quantityis the single-most important predictor for the amountof land in growing sugarcane, provided that the cur-rent relative prices between rice and sugarcaneremain similar.Figures 7c and 8c illustrate how the irrigation

capacity or quantity depends on the forward quantity.We use the terms “capacity” and “quantity” inter-changeably since these two quantities (v andw1 þ � � � þ wN) are the same under Model S. We firstnote that the vertical axis in these two figures aredifferent, and the irrigation capacity is much higherwhen the associated price j1 is lower. It confirms anintuitive result that water usage decreases as a func-tion of the irrigation cost, and thus subsidizing thiscost, for example, in the form of subsidized electricity,has an adverse effect on water consumption.There has been emerging debate regarding whether

contract farming contributes to or mitigates watershortage. Some have argued that contract farmingincreases water usage when farmers shift to cropswith high water requirement (Singh 2004), and othershave argued that contract farming may potentiallyreduce water usage through better technology(Sindhu 2010). Figures 7c and 8c further address thequestion of whether contract farming is beneficial orharmful to water usage. Increasing qFsugarcane has twopossibly opposing effects on the irrigation capacityplanning decision: (i) Since more land is allocated tosugarcane as opposed to rice, and sugarcane requiresless water than rice, it follows that the need for irriga-tion decreases if irrigation is not expensive and (ii)since the contract specifies a higher quantity of sugar-cane, more irrigation is required to increase the yieldrate of sugarcane. We observe that when irrigation isinexpensive (j1 ¼ 2), the first effect becomes domi-nant, and the commitment on sugarcane reduces theoverall water usage. However, when the irrigationcost is moderate (j1 ¼ 10), the first effect is not

0 200 400 600 800 10000.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4Payoff vs Cane Contract Quantity: Kappa1=10


Pay

off (

Rs.

) 0.81.01.21.41.6

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1Land Allocation to Rice vs Cane Contract Quantity: Kappa1=10


Fra

ctio

n A

lloca

ted

to R

ice

0.81.01.21.41.6

0 200 400 600 800 1000170

180

190

200

210

220


S F d C Q i

Irrig

atoi

n C

apac

ity (

mm

)

0.81.01.21.41.6

(a)

(b)

(c)

Figure 8 Contract Farming: j1 ¼ 10 (Model S). Legend Representsthe Sugarcane Forward Price as a Multiple of the ExpectedMarket Price, That Is, pFsugarcane/E [Psugarcane]. c ¼ 0:25�10�4: Note that qFsugarcane ¼ 0 Corresponds to No contractFarming. (a) Payoff Function Π vs. qFsugarcane. (b) Crop-LandAllocated to Rice urice vs. qFsugarcane. (c) Irrigation Capacity m

vs. qFsugarcane.


significant and the second effect is more pronounced;since sugarcane is more sensitive to water input thanis rice, the amount of irrigation for sugarcane is lesssensitive to price compared to rice. (We have alsoconducted computations with larger values of j1 andthe results are qualitatively similar to the case ofj1 ¼ 10.) What we can conclude from these observa-tions is that one cannot say whether contract farmingunilaterally increases or decreases water usagesince the impact of contract farming to the waterusage depends on both irrigation cost and contractparameters.Summarizing the findings of this section, we note

that whether the farmer prefers contract farming tono contract farming depends on the contract details.The contract price should be set higher than theexpected market price, accounting for the penalty thatthe farmer needs to pay in case of shortage. Theamount of land dedicated to the contract cropdepends on the contract quantity, but appears to beinsensitive to the contract price. While the impact ofcontract farming on water usage depends on contractparameters, any subsidy given to the irrigation costincreases water usage.

4. Extension: With ResponsiveIrrigation Control (Model R)

In this section, we study the sensitivity of our analy-sis and findings with respect to how we modeledirrigation. In particular, while it was assumed in sec-tion 2 that irrigation decisions are made before rain-fall is realized, we now consider the case where thisdecision is made after the farmer obtains a perfectrainfall forecast (see Figure 9). Thus, the farmerknows the exact quantity of rainfall before he startsmaking decisions on how the irrigated water willbe used. The development and use of long-rangeclimate forecasts is a very active area (e.g., Li et al.2008, Shukla and Mooley 1987, Shukla 1998, forempirical evidence and discussion on how certainaspects of the climate can be predicted). Rather thanconsidering the specific attributes of such forecasts,here we consider the limiting case where a perfectforecast may be available. We refer to this model asthe Model with Responsive Irrigation Control, or ModelR. We formulate this model as a two-stage stochastic

optimization problem, where the recourse decisionin the second stage depends on the realization ofrainfall. Our formulation approach and the emphasison rainfall are consistent with the multi-stage modelof Maatman et al. (2002) This section is motivated byan observation that a more realistic model of irriga-tion should be more dynamic, taking into accountthe evolution of both rainfall and irrigation through-out the season, but such a model would be more dif-ficult to analyze. Since irrigation precedes rainfall inModel S, we consider an alternative model whereirrigation follows rainfall and investigate whetherthe results of earlier sections continue to hold in thiscase.

4.1. DescriptionWe now present the description of Model R byemphasizing the difference of this model from ModelS. The sequence of events are given as follows.

(1A) At the beginning of the year, the farmermakes a crop allocation decision ðu1; . . .; uNÞ,where u1 þ � � � þ uN ¼ 1, and the irrigationcapacity decision v� 0.

(1O) The value of rainfall R is realized, and wedenote the realized value by r.

(2A) The farmer makes the irrigation water usagedecision ðw1ðrÞ; . . .;wNðrÞÞ where w1ðrÞþ� � � þ wNðrÞ � v. (While this decision alsodepends on ðu1; . . .; uNÞ and v, we suppressthis dependence in our notation for exposi-tional simplicity.)

(2O) The market prices of crops are realized.

The definition of the payoff function in Model R issimilar to the corresponding definition in Model S,except that the irrigation water usage vectorðw1; . . .;wNÞ needs to be replaced by ðw1ðrÞ; . . .;wNðrÞÞ,which is a recourse decision made after the observa-tion of the rainfall quantity r. For Model R, it is conve-nient to introduce yiðrÞ, which is defined byr þ wiðrÞ=ui. Then, the yield rate of crop i is ciðyiðrÞÞ,and the total yield of crop i is xi ¼ ui � ciðyiðrÞÞ. Wecan also write the irrigation cost in terms ofðy1ðrÞ; . . .; yNðrÞÞ:

C ¼ Cðv; y1; . . .; yNÞ ¼ j1 � v� j2 � rþ j2 �XNi¼1

ui � yiðrÞ;

Figure 9 Sequence of Events for Model R (Responsive Irrigation Control)


which follows from wiðrÞ ¼ ui � ðyiðrÞ � rÞ andu1 þ � � � þ uN ¼ 1.

4.2. Two-Stage Stochastic ProgrammingFormulationIn presenting the mathematical formulation for ModelR, it is convenient to use yðrÞ ¼ ðy1ðrÞ; . . .; yNðrÞÞinstead ofw ¼ ðw1ðrÞ; . . .;wNðrÞÞ. Note that for fixed rand ui, yiðrÞ ¼ r þ wiðrÞ=ui is a one-to-one transfor-mation of wiðrÞ. In the following formulation, y(r) isthe second stage decision vector (after the rainfallrealization r) while u ¼ ðu1; . . .; uNÞ and v are thefirst-stage decisions.

(SP-R) maxu;v

ER;P½UðRðu1 � c1ðy1ðRÞÞ; . . .;uN � cNðyNðRÞÞÞ � Cðv; y1ðRÞ; . . .; yNðRÞÞÞ�

s. t.XNi¼1

ui � ðyiðrÞ � rÞ� v for each

r in the support of R

u1 þ u2 þ � � � þ uN ¼ 1

v� 0; ui � 0; yiðrÞ� r for each i ¼ 1; . . .;N:

4.3. Analysis: An OverviewWhile we have presented a two-stage stochasticprogramming formulation for the farmer’s decisionsin section 4.2, we now provide a sketch of how tosolve this model efficiently. The details of this anal-ysis can be found in Appendix B. We first considerthe second stage problem, followed by an analysisof the first stage problem. We show that the prob-lem at each stage can be written as a convexprogram.OPTIMAL IRRIGATION USAGE DECISION (SECOND STAGE).

In the second stage, the farmer decides the recoursedecision y(r). At this time, the first stage decisions, uand v, have already been made, and the realized rain-fall r has been observed. Let Π(u,v,y(r)) denote theobjective function of (SP-R) condition on the rainfall r,that is,

Pðu; v;yðrÞÞ ¼ EP½UðRðu1 � c1ðy1ðrÞÞ; . . .;uN � cNðyNðrÞÞÞ� Cðv; y1ðrÞ; . . .; yNðrÞÞÞ�;

where we recall

Cðv;y1ðrÞ; . . .;yNðrÞÞ ¼ j1 � v� j2 � rþ j2 �XNi¼1

ui � yiðrÞ" #

:

Then, the second stage problem can be written asfollows:

(SP-Ra) /ðu;vÞ ¼maxyðrÞ

Pðu;v;yðrÞÞ

s. t.XNi¼1

ui � yiðrÞ� rþ v

yiðrÞ� r for each i¼ 1; . . .;N:

It can be shown that the above problem is a convexprogramming problem with linear constraints (Theo-rem 5 in the Appendix). In the special case of the riskneutral farmer, the first order condition provides aeasy-to-understand characterization of the optimalirrigation policy that equalizes the marginal benefit ofirrigation for each crop (Proposition 6).OPTIMAL CROP ALLOCATION AND IRRIGATION CAPACITY

DECISIONS (FIRST STAGE). In our discussion of (SP-Ra),we discussed how to solve the second stage decisionsy(r) while treating the first stage decisions as fixed.For fixed (u,v), recall that u(u,v) is the optimal valueof (SP-R) given that the second stage decisions aremade optimally. In this section, we study the problemof making the optimal first-stage decisions, whichconsist of the crop-land allocation u ¼ ðu1; . . .; uNÞand the irrigation capacity v, that is,

(SP-Rb) maxu;v

/ðu; vÞXNi¼1

ui ¼ 1; v� 1

��( )

:

We can show that the objective function that wemaximize in (SP-Rb) is jointly concave in the decisionvector (u,v) (Theorem 8). This structural result is use-ful in deciding globally optimal crop-land allocationand irrigation capacity investment Model R. Weprove this concavity result by first showing the jointconcavity of u—note that this result does not followimmediately from standard results and requires acareful construction of the recourse vector.

4.4. Numerical ResultsRecall that in section 3 we have reported computationresults and insights based on Model S. A questionthat naturally arises at the end of section 3 is whetherthe results are artifacts of a particular irrigation modelthat we adopted or whether they are robust withrespect to how irrigation is modeled. To answer thisquestion, we have conducted a similar set of experi-ments for Model R, a more dynamic and alternativemodel of irrigation.To summarize our findings, the results that we

obtain for Model R are similar to those reported forModel S. In particular, we generated plots that areanalogous to 5–8, and the corresponding plots arevery close. Each of the three subplots in each figure,especially the total amount of payoff to the farmer,does not seem to be affected by whether irrigationusage decision is a recourse action or not. This implies


that the value of flexibility in this recourse action doesnot have a first-order impact on the payoff. Further-more, this observation reinforces the insights (that wehave identified in section 3) to be robust and insensi-tive to how we modeled irrigation—since a more real-istic dynamic version of irrigation usage lies betweenthe two extremes that we modeled (Models S and R).One issue that we further investigate for Model R is

the distinction between the irrigation capacity andirrigation usage and how it impacts irrigation deci-sions. Since Model R is dynamic, it is possible toreserve a certain level of capacity v, but the amountthe farmer uses wrice þ wsugarcane, may in fact be smal-ler than the capacity, depending on rainfall. (In con-trast, under the static model of Model S, all irrigationdecisions are made before rainfall realization and theirrigation capacity equals usage.) Figure 10a showsthe irrigation capacity for Model R when the irriga-tion capacity cost is low (j1 ¼ 2). This plot has a simi-lar shape as the one for Model S (see Figure 7c), butwe observe that the irrigation capacity for Model R ishigher compared to Model S, which can be explained

by the fact that not all the capacity needs to be used.Furthermore, this difference in irrigation capacitybetween two models is greater when the forward con-tract quantity qFsugarcane is higher. This is due to the factthat sugarcane has a yield rate that is more sensitiveto water input than rice, and therefore there is agreater value of flexibility associated with waterusage recourse action. These findings are consistentwith the moderate value of j1 (Figure 11a).Since the irrigation usage may not be the same as

irrigation capacity, we consider how much of theirrigation capacity is actually used. This is shown inFigures 10 and 1. When qFsugarcane is small, most of theirrigation is used, but when qFsugarcane is high, there is asizeable gap. This again is related to the water sensi-tivity of sugarcane. With high values of qFsugarcane, theland is entirely allocated to sugarcane, and the irriga-tion usage decision is made to bring the total waterinput up to the optimal quantity only since excesswater input is harmful. However, with low values ofqFsugarcane, the farmer also grows rice, and any excesswater can be used for rice since it requires a large

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Figure 10 Contract Farming: j1 ¼ 2 (Model R). Legend Representsthe Sugarcane Forward Price as a Multiple of the ExpectedMarket Price, That Is, pFsugarcane/E [Psugarcane]. c ¼ 0:25�10�4. Note that qFsugarcane ¼ 0 Corresponds to No ContractFarming. (a) Irrigation Capacity m vs. qFsugarcane. (b) ExpectedIrrigation Usage xrice þ xsugarcane vs. qFsugarcane.

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Figure 11 Contract Farming: j1 ¼ 10 (Model R). Legend Representsthe Sugarcane Forward Price as a Multiple of the ExpectedMarket Price, That Is, pFsugarcane/E [Psugarcane]. c ¼ 0:25�10�4. Note that qFsugarcane ¼ 0 Corresponds to No ContractFarming. (a) Irrigation Capacity m vs. qFsugarcane. (b) ExpectedIrrigation Usage xrice þ xsugarcane vs. qFsugarcane.


amount of water but is not as sensitive to water input.This observation leads to a useful implication for apolicymaker or anyone else interested in reducing theamount of water usage through contract farming insugarcane. To this end, it is preferable to set up a con-tract that specifies the forward contract quantityqFsugarcane to be high enough to induce the farmer togrow sugarcane exclusively. Otherwise, if the farmergrows both sugarcane and rice, the farmer mayincrease his investment in irrigation in order to con-trol irrigation for water-sensitive sugarcane, and anyresidual excess capacity will be used for rice, therebyincreasing total irrigation usage. This occurs since thewater usage cost j2 is low compared to the watercapacity cost j1, which is the case of the Indian sub-continent context. We note that the policymaker’spreference for a large contract is in contrast to thefarmer’s preference for a smaller contract that we dis-cussed in section 3.In this section, we have investigated whether a dif-

ferent model of irrigation would change our earlierfindings, and it turns out that irrigation flexibilitydoes not introduce substantial changes to our results.We observe that when the irrigation usage decision ismade after the rainfall is realized, not all capacity thatwas planned for will be actually used. When the waterusage cost is relatively inexpensive, however, we seea tendency that if the irrigation capacity planned for awater-sensitive crop is not used for this crop, it mayend up being used for a less water-sensitive crop.

5. Concluding Remarks

In this study, we have developed a simple and styl-ized model for the farmer’s decision under the rainfalland price uncertainty that explicitly incorporates theirrigation investment and usage decisions. We haveidentified several structural properties (e.g., the con-cavity of objective functions) that not only producemanagerial insights but also allow the numerical com-putation for the optimal solution to be uncomplicated.We have studied the relationship between crop plan-ning and irrigation management through numericalanalyses based on historical agricultural data fromthe Ganganagar district of India and illustrated thatcontract farming can be used to increase the profit ofthe farmer as well as to encourage the responsibleusage of water, though it is dependent on the contractand cost parameters.In particular, we have shown that the farmer under

forward contracts is, and this means shielded fromprice uncertainty, but is exposed to a greater risk ofquantity or yield uncertainty, Because of this trade-off, the contract price may need to reflect a premiumon the market price for the farmer to accept the con-tract. The farmer’s payoff is concave in the contract

quantity, and this means the farmer may start the con-tract farming relationship with a small contract to testits benefit. With regard to water usage, contract quan-tity seems to be the single most important factor in itsimpact on water usage, but whether this impact ispositive or negative depends on cost parameters.Therefore, the impact of contract on water cannot besimply stated—although a carefully designed contractin certain settings can reduce water usage. Finally,water usage may increase with diversified farming inwhich the farmer grows multiple crops, and thisincrease is due to the increased value of irrigationcapacity when it is flexible—the irrigation capacitycan be used for any subset of crops. Thus, with diver-sified farming, one can obtain a better solution byallowing the irrigation cost to depend not only on thecapacity investment but also on actual water usage.There are several interesting research thrusts

related to this study. These are outlined below andreflect possible directions in our current researchwithin this area.First, we need to consider the buyer’s problem and

its interface with the farmer’s problem. In this study,our focus was primarily on the farmer since under-standing his objectives and behavior is a prerequisiteto what kind of contract the buyer can offer, but thefarmer’s perspective has not been satisfactorily stud-ied in the literature. Potentially, the buyer will have abasket of crop needs and superior information rela-tive to the farmer as to potential market prices, rain-fall forecasts, and technologies. Also, the buyer mayintend to contract with potentially hundreds or thou-sands of farmers in diverse geographical areas to meetthe target product demands. The asymmetry in infor-mation and in capital resources could translate intopredatory contract price setting. On the other hand,the local market prices potentially provide a balanc-ing mechanism. If the buyer procures a large fractionof the local product, then he will change local marketprices, possibly making an adverse impact on partici-pation or default incentives of the farmer. There isalso opportunity for cooperative interaction betweenthe farmer and the buyer. Both face climate andmarket price risks, but in somewhat different ways.Geographical diversification of procurement couldreduce the buyer’s climate risk, but increase transac-tion costs. The farmer may not have the capital or thefinancial resources to afford crop insurance or to pur-chase irrigation equipment or fertilizer to the optimalproduction level. The buyer could then negotiate interms of optimizing his expense across geographicalrisk pooling or building sustainable capacity andgoodwill with specific farmer or farmer groupsthrough the needed capital. The insurance case isinteresting because the geographical risk pooling(perhaps to a limited degree) would still provide the


buyer with the ability to get reinsurance cover reflect-ing the reduced risk, as opposed to the retail insur-ance policy the farmer is constrained to buy.Second, the opportunity to reduce the stochastic

elements of the problem could come either throughinvestment (perhaps social or corporate) in forecast-ing systems (climate or commodity prices) and thesupporting monitoring programs or through the opti-mization of water storage, irrigation, and the fertilizermix. Crop diversification is also a social and individ-ual goal that possibly enters as a natural linear con-straint into such a formulation either from a policyperspective for the corporation or government oreven for the farmer considering subsistence as well asrevenue goals. The models as formulated could beused to explore the trade-offs in risk reduction by dif-ferent means with a parametric evaluation of the cropdiversification and budgetary constraints. In this case,one would also need to consider objective functionsor goals that reflect elements of risk that are not cap-tured well by the expected value analysis consideredin this study.Third, in a setting such as India, the government

offers a variety of subsidies for water inputs, providessupport prices for certain crops deemed to be in theinterest of national security, and provides relief fromflood and drought losses. While many of these pro-gram elements are politically expedient, some areneeded for risk management (though not necessarilyin the most optimal sense) of rural livelihoods, andthe net effect on environmental and water resourcesustainability is perceived as negative due to theresulting farmer decisions to unsustainably pumpgroundwater to grow rice or other crops in regionsand seasons that are climatologically unsuitable forsuch crops. Here, the question is whether the privatesector, through contract farming, can effectively com-pete with or help re-shape government programs topromote more sustainable water, crop, and rural live-lihood outcomes at the national scale. This requiresmulti-scale and multi-agent modeling in the sameframework as initiated here, and a broader set ofquestions needs to be considered.For the deployment of our model in a field setting

(our current work focuses on India), the underlyingassumptions of the model need to be calibrated forthe weather and soil conditions of the specific regionunder study, and furthermore, additional featuresand constraints should be added to the model (e.g.,the trade-off between water use and fertilizer applica-tion that are generally well studied and hence wereimplicitly parameterized in the current application).Regarding contract farming, we have illustrated itspotential benefits using forward contracts assumingthat the details of the contracts are exogenously given,and we are currently investigating the negotiation

process of determining specific parameters of the for-ward contract and other types of contracts as well asthe possibility of contract enforcement and renegotia-tion (Plambeck and Taylor 2007). Overall, we envisionthat the increased role and participation of privatecorporations, in collaboration with policymakers andfarmers, can address and resolve the problems ofpublic interest (namely, agriculture and water), pro-vided that competing and incongruent objectives areproperly balanced and managed with appropriateincentives.

Acknowledgments

This research was a part of a Pepsico Foundation fundedproject entitled “Improving Rural Water and LivelihoodOutcomes in India, China, Africa and Brazil.” We wouldalso thank the referee board for their helpful comments.

Note

1In this study, the amount of water is measured not interms of volume, but rather in terms of the “height” whenit is spread over the unit area. We use this convention tobe consistent with the unit of rainfall, which is also mea-sured in height.

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Supporting InformationAdditional Supporting Information may be found in theonline version of this article:

Appendix A.Appendix B.


optimal crop choice, irrigation allocation, and the impact of contract farming

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