### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Investment in efficient irrigation and water storage
- 4. Numerical analysis
- 5. Conclusion
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- REFERENCES

In the framework of a stochastic dynamic programming model, the paper investigates the impact of water supply uncertainty and storage at farm level on adoption of efficient irrigation technologies under a flexible water price regime. We find that even a flexible water pricing cannot guarantee higher adoption of efficient irrigation technology in all cases. Results of the paper indicate that if a farmer invests in water storage capacity, then the value of efficient usage of water increases, and the rate of adoption of efficient irrigation technology will be higher. It establishes a complementarity relationship between investments in storage capacity and adoption of efficient irrigation technology. The relationship becomes stronger with increasing variance in water supply. In a situation without any option to store water at the farm level, we find a negative relationship between investment in efficient irrigation technology and water variability. However, numerical analysis results suggest that a risk averse farmer may invest more in efficient irrigation only if the variance in water supply is very high.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Investment in efficient irrigation and water storage
- 4. Numerical analysis
- 5. Conclusion
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- REFERENCES

In the past, water policy schemes focused more on the development of irrigation infrastructures for expansion of irrigated area. However, these expansionary policies are not sufficient as demand for water continues to increase (Sampath [1992], Rosegrant and Meinzen-Dick [1996]). Global climate change may put further pressure on the existing hydrological systems with increasing water demand as the variability of water supply is expected to change (Federick and Major [2002], Bates et al. [2008]). Coping with the effects of climate change on water requires stronger demand management measures to enhance efficient usage of water (Harou et al. [2009]). In the agricultural sector, which typically uses up to 70% of water resources, conservation and efficient usage are critical to sustainable water management and climate change adaptation. Many have proposed the use of modern irrigation technologies as one of several possible solutions to tackle water scarcity and environmental degradation. Seckler et al., for instance, showed that improvements in irrigation efficiency alone may meet half of the projected increase in water demand by 2025 (Sekler et al. [1998]).

It is pertinent to understand the factors that influence a farmer's decision to adopt irrigation technologies over time. Several studies have recognized risk as the most important factor determining the adoption of efficient irrigation technology among other socioeconomic, structural, and demographic variables (Jensen [1982], Just and Zilberman [1983], Tsur et al. [1990], Saha et al. [1994], Koundouri et al. [2006]). Uncertainty associated with the adoption of technology in agriculture stems from different sources: perceived riskiness of future farm yield after technology adoption, price uncertainty related to farming itself, or production uncertainty due to a possible water shortage. Most of the studies empirically investigated the effect of perception of such risk elements in a farmer's decision on adoption of irrigation technology in a static framework of binary choice models. A dynamic analytical framework, however, will be more appropriate to explain a farmer's decision on the timing of adoption of technology.

Among the literature on stochastic water resource management in a dynamic framework (Tsur and Graham-Tomasi [1991], Knapp and Olson [1995], Bhaduri et al. [2011]), the study by Fisher and Rubio is particularly noteworthy. It finds that the optimal investment in water reservoir increases as a result of an increase in the variance rate of precipitation (Fisher and Santiago, [1997]). However, there is limited literature from the perspective of the theory of investment that examines the effect of stochastic water resource on investment in efficient irrigation technologies. This paper attempts to fill the research gap by investigating the impact of water supply uncertainty on investment in efficient irrigation technologies in the framework of a stochastic dynamic programming model.

We also explore if investment in water storage capacity at a farm level could induce farmers to adopt efficient technology under variable water supply. The water storage at a farm level often helps to mitigate the effects of scarce and unreliable water supply. Such kind of storage facilities, often called “*Diggi,*” are common under the Indira Gandhi Nehar Pariyojna (IGNP) project in Rajasthan, India, where the farmers store the surface water, and then use the water out from the *Diggi* to irrigate crops through field channels.*Diggi* addresses the reliability issue in the water management of the region, where unreliable water supply associated with rigid schedules of water delivery is a major constraint for the farmers. Such kind of water storage also encourages the farmers to adopt efficient irrigation technology that improves the water-use efficiency, as well as crop yield (Amarasinghe et al. [2008]).

This study is also motivated by a case study in Uzbekistan in Central Asia where unsustainable use of irrigation water has led to the Aral Sea crisis. A study by Oberkircher and Hornidge explores how farmers in the region perceive water and its management, and what motivates them to conserve irrigated water (Oberkircher and Hornidge [2011]). They find that farmers lack incentive to conserve water, as storage of saved water is not possible due to the centralized irrigation system. If intermediate water storage is a possibility, then it could create new rationales that facilitate water conservation. Motivated by these two case studies, we explore if there exists complementarity between investments in efficient irrigation and water storage capacity.

It is often argued that water pricing could promote water-use flexibility and establish a recognized water value, and thus provide incentives for more efficient water usage (Shaliba and Bush [1987]). Shah, Zilberman, and Chakraborty argue that it may be optimal to increase water price to encourage quicker adoption of water conservation technologies (Shah et al. [1995]). On the other hand, Carey and Zilberman find that water markets can delay the adoption of modern irrigation technologies for farms with scarce water supplies (Carey and Zilberman [2002]). They predict that farms will not invest in modern technologies unless the expected present value of investment exceeds the cost by a potentially large hurdle rate. Ranjan and Athalye also find water pricing to be least important in influencing adoption of new technologies if it does not capture the true opportunity cost of water (Ram and Athalye [2009]). In this paper, we assume a flexible water price regime, where the water price depends on the excess demand of water. We investigate whether such water pricing alone can guarantee higher adoption of efficient irrigation technologies, given the uncertainty in water supply stemming from climate change. Our theoretical results indicate that flexible water pricing cannot guarantee higher adoption of efficient irrigation technology under increasing variance in water supply. However, the negative impact of uncertainty on investment in efficient irrigation technologies will be less under flexible water pricing. We find if there is an opportunity to invest in water storage capacity, then the farmers will be motivated to save water. Consequently, the relationship between aggregate investment in efficient irrigation technology and uncertainty in water supply might be positive.

The remainder of the paper is structured in the following way. In the following section, the model assumptions are outlined, and we determine the optimal investment in irrigation technology under uncertainty. In the third section, we reformulate the model with the option of a water storage reservoir. In Section 'Numerical analysis', we present the numerical analysis results. Finally, we present the key results in the Conclusion section.

### 2. Model

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Investment in efficient irrigation and water storage
- 4. Numerical analysis
- 5. Conclusion
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- REFERENCES

The price of water is denoted by *p*. The price of water evolves over time and is a function of the aggregate excess demand of water,

- (4)

where *k* is a constant. We also assume that there are *N* homogeneous farmers, who can influence water demand and the price changes are governed by the relative strength of demand given variable water supply. For simplicity, we have assumed that the rate of price change (with respect to time) at any moment is always directly proportional to the excess demand prevailing at that moment. The adjustment of price is affected not through market clearance in every period but through a process of demand adjustment given variable water supply.4

The net benefit function of water can be represented as follows:

- (5)

We maximize the expected present value of net benefit with respect to *w* and *h*

- (6)

We write the associated Bellman equation as

- (7)

We maximize the above equation with respect to *w* and *h*, and obtain the following first-order conditions

- (11)

The above equation suggests that at the margin, the marginal benefit of water equals the marginal cost of water. The marginal cost of water includes the price of water (*p*) and the scarcity value of water reflected in the change in the price of water ().

Also, the marginal value of efficient irrigation, given by , is equal to the cost of adopting efficient irrigation, .

Differentiating the Bellman equation (10) with respect to the state variable *p*, and given the optimal values of the control variables *w* and *h*, we obtain

- (12)

Note that the first two terms in parentheses of the above equation are zero given the first-order conditions (11). Hence

- (13)

Using Itô's Lemma we have

- (14)

We apply the differential operator to the above equation; and considering that , we obtain

- (15)

Substituting (15) in (13), we have

- (16)

Assuming that a stochastic equilibrium exists, the expected value of variation in water consumption is zero () in the sense of convergence to a long-run (steady-state) distribution for *w*.5 Given such assumption, from Appendix Appendix, we have

- (17)

After substituting from first-order condition (11) and , the above expression (17) can be written as6:

- (18)

Equation (18) implies that at equilibrium the marginal benefit of water consumption equals the marginal cost, where the latter includes a term with variance rate of water.

However, for a concave marginal benefit function (), a reduction in optimal water consumption will have a smaller impact on benefit. Hence the farmer will invest more in efficient irrigation with increase in uncertainty in water supply and .

Using such cost function, the net benefit function of water can be represented as follows:

- (26)

### 3. Investment in efficient irrigation and water storage

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Investment in efficient irrigation and water storage
- 4. Numerical analysis
- 5. Conclusion
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- REFERENCES

In this section, we evaluate if there is an opportunity to invest in water storage, then how investment in efficient irrigation will change with variance in water flow.

We assume that the farmer will buy water and use β proportion of water while storing proportion of water. The water constraint can be rewritten as

- (29)

The stock of water denoted by *S* can be expressed as

- (30)

Otherwise

- (31)

where α is a factor that transforms storage capacity to capital stock. The capital stock, *K*, changes over time according to

- (32)

where *I* and represent investment and it's rate of depreciation, respectively.

The net benefit can be rewritten as

- (33)

where is the cost of investment.

We can rewrite the Bellman equation as

- (34)

where ψ is the Lagrange multiplier for the constraint .

We maximize the above equation with respect to *w*, *h*, and *I*. The first-order conditions are

- (35)

- (36)

- (37)

The first-order condition (35) suggests that the marginal value of water for consumption is equal to the marginal cost of water. The marginal cost includes the price of water, scarcity value of water reflected by the change in the price of water (), and the marginal value of water storage. Further, other first-order conditions suggest that for investment, the marginal value of capital in efficient irrigation and for storage capacity, as given by and , is equal to the price used to value increments to their stock, and , respectively.

Similarly as in Section 'Model', differentiating the Bellman equation (34) with respect to the state variable *p*, results in

- (38)

Again, differentiating the Bellman equation (34) with respect to the state variable *S*, for the optimal values of the control variables *w*, *h*, and *I*, we have

- (39)

Note that the first three terms in the parentheses are zero. Hence

- (40)

Now using Ito's Lemma we have

- (41)

From Appendix Appendix, we know

- (47)

The above equation has been derived after assuming the existence of an equilibrium, where the expected value of variation in water consumption is zero in the long run (). While deriving the above equation (Appendix C), we also consider a linear price function of water () for analytical simplicity.

Now, partially differentiating the latter equation with respect to σ^{2} and *K*, we obtain

- (48)

Rearranging the terms in the above equation, we obtain

- (49)

Differentiating the equation (47) with respect to *K* and *H* after substituting (from equation (21)), we can derive a relationship between investment in storage reservoir and efficient irrigation technology

- (50)

In the above function, the denominator is positive (see Appendix Appendix). If , then the sign of depends on the curvature of the marginal benefit function of water. If the marginal benefit function of water is convex, , then we find a positive sign of . It suggests that there exists a complementary relationship between the investment in efficient irrigation and water storage. The relationship also becomes stronger with increasing variance . Here, water storage gives an opportunity to the farmers to save more water, and increasing variance of water supply can induce the farmers to invest more in efficient irrigation. On the other hand, in the case of concave marginal benefit function of water, we find a substitutable relationship () between investment in efficient irrigation and reservoir, respectively. However, the relationship becomes weaker with increase in variance of water supply as will be negative given .

We have assumed so far, the cost of water storage to be linear. However, the storage cost function could be nonlinear as cost of water storage increases with storage either due to higher leakage or evaporation. We attempt to analyze the effects on investment given such a nonlinear cost function. We assume that the cost function is convex with respect to the storage of water, with , . After differentiating the Bellman equation (34) with respect to the state variable *S*, for the optimal values of the control variables *w*, *h*, and *I*, we can rewrite equation (43) as

- (52)

Using the above equation in Appendix Appendix, we obtain after rearranging

- (53)

The above equation implies that at equilibrium the marginal benefit of water consumption equals the marginal cost of water withdrawal for storage and irrigation. The marginal cost of water includes an additional term, , reflecting the nonlinear marginal cost of water storage. It implies that if the overall marginal cost increases further, then it will reduce the amount of water withdrawal for storage. Moreover, given the complementarity relationship between investment in water storage and irrigation technology, such a cost function might also influence the adoption of efficient technology. This is further explored in the following section.

### 4. Numerical analysis

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Investment in efficient irrigation and water storage
- 4. Numerical analysis
- 5. Conclusion
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- REFERENCES

The results in Figure 1 suggest that a relatively risk averse farmer, with a degree of risk aversion of 0.4, may invest in efficient irrigation technologies up to 32% of the total area with an irrigation technology costing $973 per ha. Considering a relatively less costly irrigation technology, for instance drip irrigation with a cost of $800 per ha, we find that the adoption rate to increase by 12% more. However, we find that the decrease in the cost of technology does not influence the initial inaction period of 2 years.

Our results also suggest that if the same farmer has the opportunity to invest in storage capacity, then the expected benefit from using efficient irrigation technologies increases and the adoption of irrigation technology will reach 52%.9 However, in the initial years the rate of adoption of efficient irrigation will be lower due to investment in storage capacity, and afterwards it will increase at a faster rate.

We have also illustrated the case where the cost of storage water is nonlinear and depends on the storage. The tradeoff between increase in storage costs and technology adoption is evident from our simulation results. There is an initial delay in the adoption of technology, and a slow initial increase in the rate of adoption under the assumption of such nonlinear cost. However, the rate of increase in adoption will be be higher only at a later stage.

Table 1. Parameters for numerical analysisParameters | Values |
---|

Depreciation rate (δ) per year | 0.1 |

Cost of Investment in irrigation technology(dollars) per ha | 973 (sprinkler)800 (drip) |

Price of water (dollars) 1000 cubic meter | 50 |

| 60 (high price) |

| 40(low price) |

Discount rate | 0.1 |

Efficiency of Furrow Irrigation () | 0.55 |

Efficiency of Sprinkler Irrigation () | 0.8 |

Construction cost of reservoir (dollars) per 1000 cubic meter | 400 |

Nonlinear cost function of water storage (dollars) | thousand cubic meter |

Proportion of water consumed (β) | 0.8 |

Net Benefit (NB) in (dollars) | |

Benefit from water(); | |

| |

| Price of agricultural crops |

θ | Constant relative risk aversion parameter |

*H* (Area covered under efficient irrigation technology) | |

*K* (Capital stock for storage reservoir) | |

*p* (price of water) | |

*W* (water supply) | |

(variance in water supply) | 0.15 |

*W*_{0}(Initial water supply) | 50 thousand cubic meter |

*w* (amount of water purchased 1000 cubic meter) | Choice variable |

*I* (investment in storage reservoir) | Choice variable |

*h* (investment in efficient irrigation technology) | Choice variable |

Figure 2 compares the expected time paths for adoption of efficient irrigation technology under flexible and different fixed water pricing schemes. We find that a flexible price of water plays a significant role in inducing the adoption of efficient irrigation technology. If the price of water is fixed at $50 per 1000 cubic meter, then the rate of adoption is much slower and the aggregate adoption of efficient irrigation technology will be lower compared to that under flexible water pricing scheme. We have also compared the effects with regards to adoption of irrigation technology under several fixed prices. With a higher fixed price of $60 per 1000 cubic meter, we find that the adoption rate to increase to 22% as the opportunity cost of water rises; and with a lower water price of $40 per 1000 cubic meter, the adoption rate to be lower at only 10%. Interestingly, the increase in water price above $50 per 1000 cubic meter increases the adoption rate much more than the increase in water price from the $40 mark. It implies that the water price above $50 per 1000 cubic meter is more sensitive to the adoption of irrigation technology.

We also examine how the expected time path for adoption rate of modern irrigation technology will change with the change in water supply volatility. Results, as illustrated in Figure 3, show how aggregate investment in irrigation technology changes with increase in variance of water. The result is consistent with the theoretical finding of a negative relationship between investment and the variance of water supply. However, we find that if the risk aversion is high or intertemporal elasticity is low, then the higher variance or uncertainty will produce an increase in the certainty equivalent of the return to investment in efficient irrigation, and as a result farmers may invest more. But in such a case, we find that the increase in variance of water must be sufficiently high for the aggregate investment to be higher. This result also justifies the historical patterns of modern irrigation technology adoption as evident from the studies of Schuck et al. [2005], Zilberman et al. [1995], and Caswell [1991], which have shown that adoption of the drip irrigation increase dramatically when the frequency of droughts is very high.

### 5. Conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Investment in efficient irrigation and water storage
- 4. Numerical analysis
- 5. Conclusion
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- REFERENCES

In many countries, water prices are fixed and determined administratively, reflecting neither the supply cost nor the scarcity value. Moreover, when the scarcity value of water is increasing, it could be inappropriate to insulate the water economy from market forces. In this paper, we investigate how to get the water prices right so that it can induce the farmers to adopt efficient irrigation technology.

It is also crucial that an analytical framework that explores different adoption decisions of farmers also addresses the time pattern of dynamic factors such as water variability and water storage that may affect a farmer's decision. It is very important to learn the effects of the dynamic process of such variables that will help explain why individuals choose different adoption rates. In a stochastic dynamic framework, we explore whether increasing variability in water supply can induce the farmers to conserve water through the adoption of efficient irrigation technologies.

In this paper we find that a flexible water price, which depends on stochastic water demand as well as aggregate demand, can increase the adoption rate of efficient irrigation technology by more than 20% for a risk averse farmer. We also find that if farmers invest in water storage capacity, then the value of efficient irrigation increases, and the rate of adoption will be higher. In our theoretical model, we found a complementarity relationship between investment in storage capacity and efficient irrigation technology. The relationship becomes stronger with increasing variance in water supply. The relationship holds if the marginal benefit function is convex.

We also consider a nonlinear storage cost function as cost of water storage increases with storage either due to higher leakage or evaporation. We find that the marginal cost of water withdrawal for storage increases further; and may reduce the adoption of irrigation technology given the complementary relationship between investment in irrigation technology and water storage.

In a situation without any option to store water at the farm level, we find that the value of the investment in efficient irrigation technology will not be sufficient to guarantee higher investment under uncertainty. However, a risk averse farmer may invest more in efficient irrigation only if the variance in water supply is very high.

In sum, the finding of the paper is consistent with the general view that flexible water pricing is a valid alternative for increasing the efficiency of water usage. However, even a flexible water pricing cannot guarantee higher adoption of efficient irrigation technology under increasing variance of water supply. If farmers also have the option to invest in storage capacity, then the adoption rate will be significantly higher.

However, the majority of the farmers in transition and developing countries are resource-poor small holders who may find it difficult economically to adopt efficient irrigation technology. Hence the approach must be sensitive to match the farmers unique characteristics of low capital availability. Moreover, flexible water price also requires suitable institutional arrangements, for instance metering of water for implementation as well as monitoring. It might lead to additional cost.

Variation in water supply stemming from climate change may also induce the farmers to cultivate alternative crops that require less water or high valued crops to adopt efficient irrigation technologies. In this paper, we have ignored such cropping pattern change for analytical simplicity. Future work can focus on these aspects.