SEARCH

SEARCH BY CITATION

Keywords:

  • Stochastic dynamic programming;
  • water pricing;
  • adoption of efficient irrigation technology

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. REFERENCES

In this section, we structure a farm-based model, where a farmer is assumed to maximize an instantaneous profit function by choosing the amount of water to apply to his crops, and the area of land to be irrigated with efficient irrigation technologies. Suppose there are two types of irrigation technologies: conventional irrigation (Furrow) and efficient irrigation technology (Sprinkler or drip). The technologies are denoted as F and H for conventional and efficient irrigation technology. For each irrigation technology m, irrigation effectiveness is denoted by inline image, where (inline image) and inline image. Suppose the area covered under effective technology at time t is inline image. The rate of change in area covered with efficient irrigation technology can be shown as

  • display math(1)

where inline image and h is the amount of new area brought under new irrigation technology. inline image represents the depreciation rate of new technology. Consider the total applied irrigation water (gross) as w. The net effective water applied for farm i can be represented as

  • math image(2)

where the area of land has been normalized as one hectare and thus the aggregate area of efficient irrigation area (H) is less than one.1

Consider benefit inline image as a function of effective water, inline image. We assume that the benefit function of water is concave in inline image with inline image and inline image.2 Again, we assume that the cost of installing efficient irrigation technology in per unit of area is constant, and is denoted by inline image. We consider water flow, W, to be stochastic, and follows geometric Brownian motion

  • display math(3)

where z represents a Wiener process and inline image denotes the variance rate in the water flow.3

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,

  • display math(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:

  • display math(5)

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

  • display math(6)

We write the associated Bellman equation as

  • display math(7)

As the water supply, W, is stochastic, using Itô's Lemma, we obtain

  • display math

and by substituting inline image, inline image, inline image, and inline image we have,

  • math image(8)

After applying the differential operator inline image, and considering the assumptions of Wiener process where inline image, equation (8) can be written as

  • math image(9)

Hence, we can rewrite the Bellman equation as

  • math image(10)

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

  • math image(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 (inline image).

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

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

  • math image(12)

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

  • math image(13)

Using Itô's Lemma we have

  • math image(14)

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

  • math image(15)

Substituting (15) in (13), we have

  • display math(16)

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

  • display math(17)

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

  • display math(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.

Now differentiating the Bellman equation (10) with respect to the state variable H, for the optimal values of the control variables w and h, we get

  • math image(19)

As the first two terms in parentheses of the above equation are zero from the first-order conditions (11) and inline image, we have

  • display math(20)

Substituting inline image in (20) and setting inline image, we get

  • display math(21)

Equation (21) implies that at equilibrium the benefit of increasing aggregate investment in efficient irrigation is equal to opportunity cost of capital inline image.

Now, substituting inline image from (21) and differentiating equation (17), we have

  • math image(22)

where

  • math image(23)

Rearranging the above equation, we have

  • math image(24)

We find the denominator of the above fraction is positive (see Appendix Appendix). The numerator is negative only if the marginal benefit of water consumption is convex which means inline image. In the latter case, we find inline image indicating a negative relationship between aggregate investment in efficient irrigation and variance in water supply. It suggests that if the marginal benefit is convex (inline image), then a reduction in optimal consumption will have a higher effect on benefit than an increase in consumption; and in such case relationship between investment in efficient irrigation and variance in water supply will be negative.7 Under such circumstances, farmers may buy water rather than invest in costly irrigation technology. Hence, presence of flexible water price cannot guarantee an increase in the adoption rate of efficient irrigation technology under increasing uncertainty in water supply.8

However, for a concave marginal benefit function (inline image), 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 inline image.

The irreversibility of investment can also be considered in the model as costly disinvestment or costly reversibility. Considering the cost of disinvestment, we reformulate the total cost function of efficient irrigation technology as

  • display math(25)

The above function has the following interpretation: if inline image the farmer is investing at a unit cost of inline image, if inline image the farmer is investing at a unit cost of inline image.

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

  • display math(26)

Then following the calculations made earlier in this section, equation (24) can be rewritten as

  • math image(27)
  • math image(28)

In the above equation, we find that with a convex marginal benefit function, if inline image, then the denominator of the equation is positive but with lower value than the case under investment (inline image). It implies that if there is a cost of disinvestment, then the aggregate investment decreases more with the increase in variance of water supply. If the cost of disinvestment, inline image, increases further or investment becomes irreversible, then the duration of “inaction period ” increases, and the aggregate investment decreases more with the increase in the variance of water supply.

3. Investment in efficient irrigation and water storage

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. 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 inline image proportion of water. The water constraint can be rewritten as

  • math image(29)

The stock of water denoted by S can be expressed as

  • display math(30)

Otherwise

  • math image(31)

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

  • display math(32)

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

The net benefit can be rewritten as

  • display math(33)

where inline image is the cost of investment.

We can rewrite the Bellman equation as

  • display math(34)

where ψ is the Lagrange multiplier for the constraint inline image.

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

  • math image(35)
  • display math(36)
  • display math(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 (inline image), 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 inline image and inline image, is equal to the price used to value increments to their stock, inline image and inline image, respectively.

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

  • display math(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

  • math image(39)

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

  • display math(40)

Now using Ito's Lemma we have

  • display math(41)

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

  • display math(42)

Substituting (42) in (40), we obtain

  • display math(43)

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

  • display math(44)

As the first three terms in the parentheses are zero from the first-order conditions (35) and inline image, we obtain

  • display math(45)

Substituting inline image in (45) and setting inline image, we have

  • display math(46)

From Appendix Appendix, we know

  • math image(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 (inline image). While deriving the above equation (Appendix C), we also consider a linear price function of water (inline image) for analytical simplicity.

Now, partially differentiating the latter equation with respect to σ2 and K, we obtain

  • display math(48)

Rearranging the terms in the above equation, we obtain

  • math image(49)

As storage water will weaken the relationship between changes in w and W, the term inline image will be negative. Hence, the denominator of the above fraction becomes negative. If we assume that water consumption increases with increase in water supply under scarcity condition, then inline image; and hence inline image will be positive. In this case, the optimal capital stock for investment in water storage increases with the increase in variance in water supply.

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

  • math image(50)

In the above function, the denominator is positive (see Appendix Appendix). If inline image, then the sign of inline image depends on the curvature of the marginal benefit function of water. If the marginal benefit function of water is convex, inline image, then we find a positive sign of inline image. 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 inline image. 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 (inline image) between investment in efficient irrigation and reservoir, respectively. However, the relationship becomes weaker with increase in variance of water supply as inline image will be negative given inline image.

The degree of complementarity between investments in irrigation technology and storage reservoir, as reflected by inline image, increases with the decrease in cost of irrigation technology. This is evident; as inline image decreases, the denominator of the fraction of equation (50) also decreases, and the magnitude of inline image increases.

The adoption of irrigation technology is also influenced by the cost of water storage. If we differentiate the equation (47) with respect to inline image and H after substituting inline image (from (C7)), we obtain

  • math image(51)

In the above equation as the denominator is positive (from Appendix Appendix), it is quite evident that inline image; and the investment in efficient irrigation technology will decrease with the increase in the cost of storage.

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, inline image with inline image, inline image. 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

  • display math(52)

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

  • display math(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, inline image, 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. REFERENCES

In this section with the help of numerical analysis, we attempt to evaluate how farmers will invest in efficient irrigation technologies with different costs of irrigation technology and water storage and under different levels of uncertainty in water supply. We also examine the role of flexible water price in the adoption of efficient irrigation. The baseline parameter values, shown in Table 1, are intended to be representative of actual values. In the numerical analysis, we have assumed that the instantaneous utility function of the farmer faces constant relative risk aversion inline image, where θ is the degree of risk aversion, and inline image is the intertemporal substitution elasticity between consumption of water in any two periods. The latter measures the willingness to substitute water consumption between different periods. The smaller θ (the larger inline image), the more willing is the farmer to substitute water consumption over time.

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.

image

Figure 1. Expected time path for the adoption of efficient irrigation technology.

Download figure to PowerPoint

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 analysis
ParametersValues
Depreciation rate (δ) per year0.1
Cost of Investment in irrigation technology(dollars) per hainline image973 (sprinkler)800 (drip)
Price of water (dollars) 1000 cubic meter50
 60 (high price)
 40(low price)
Discount rate0.1
Efficiency of Furrow Irrigation (inline image)0.55
Efficiency of Sprinkler Irrigation (inline image)0.8
Construction cost of reservoir (dollars) per 1000 cubic meter inline image400
Nonlinear cost function of water storage (dollars)inline imageinline image thousand cubic meter
Proportion of water consumed (β)0.8
Net Benefit (NB) in (dollars)inline image
Benefit from water(inline image);inline image
 inline image
inline imagePrice of agricultural crops
θConstant relative risk aversion parameter
H (Area covered under efficient irrigation technology)inline image
K (Capital stock for storage reservoir)inline image
p (price of water)inline image
W (water supply)inline image
inline image(variance in water supply)0.15
W0(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.

image

Figure 2. Expected time path for the adoption of efficient irrigation technologies under fixed and flexible water pricing schemes.

Download figure to PowerPoint

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.

image

Figure 3. Expected time path for the adoption of efficient irrigation technologies under different levels of variance in water supply.

Download figure to PowerPoint

5. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. 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.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. REFERENCES

The authors would like to thank two anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

Appendix A

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. REFERENCES

Rearranging the first-order condition (11), we obtain inline image.

Using Itô's Lemma and considering inline image, we obtain

  • display math(A1)

Consider that along the optimal path inline image. Using Itô's Lemma, we obtain

  • display math(A2)

Substituting inline image, inline image, inline image, and inline image, in the above equation, we obtain

  • display math(A3)

The expression for inline image could be simplified if we neglect the higher order terms order in inline image, then inline image, where inline image.

Substituting, inline image in (A1), applying the differential operator inline image, and considering inline image, we obtain

  • display math(A4)

In the long-run equilibrium, the expected change in water consumption is zero. Setting inline image in (A4), we obtain

  • display math(A5)

Appendix B

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. REFERENCES

From (23), we have

  • display math(B1)

As inline image and inline image, the sign of the above expression depends mainly on the sign of the term (inline image).

  • display math(B2)

In the above expression, υ is the constant elasticity of marginal utility with respect to consumption and exceeds unity (inline image) (Hicks, [1965], Evans, [2004], [2005]). Hence given the positive sign of inline image we obtain inline image. Thus inline image.

Again, as inline image, we have inline image as a result,

inline image.

Appendix C

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. REFERENCES

Suppose inline image. From equation (A1) and inline image, we have

  • display math(C1)

Setting inline image, as the expected value of variation in water consumption is zero in the sense of convergence to a long-run (steady-state) distribution for w after considering the existence of a stochastic equilibrium; and assuming a linear price function of water (inline image) for analytical simplicity, we obtain

  • display math(C2)

If we take the differential operator to inline image, we have

  • display math(C3)

Substituting (38) and (43) in (C3), results in

  • display math(C4)

As from first-order condition (35) we have inline image, equation (C4) can be written as

  • display math(C5)

Equating (C2) and (C5) we have

  • display math(C6)

Considering a nonlinear cost function of storage of water, inline image with inline image, inline image, and substituting (38) and (52), we obtain

  • display math(C7)

In this case, equating (C2) and (C7) we obtain

  • display math(C8)
ENDNOTES
  1. 1

    We have normalized the area of land for analytical simplicity.

  2. 2

    For details on concave benefit function, please refer to Ambec [2008]

  3. 3

    Following Fisher and Rubio, and Bhaduri et al., we assume that water supply follows a geometric Brownian motion (Fisher and Santiago [1997], Bhaduri et al. [2011]). The assumption of Geometric Brownian motion is based on the log normal distribution of water flow. Such assumption allows us to evaluate the effect of water variability on decision outcomes. The log normal distribution has been successfully used in the past to model the skewed distribution of annual flows in regions of high interannual variability (Waylen and Zorn [1998]). From a hydrological perspective, the contribution of drift component in the total change of water flow is negligible (Fisher and Santiago [1997]). Hence, we have excluded the deterministic drift component.

  4. 4

    Please note that we are not considering here any access to additional supply to meet this excess demand. The process of demand adjustment implies the process through which demand responds (increase or decrease) to the price.

  5. 5

    A long run steady state indicates stability of system (here water economy).

  6. 6

    Considering a linear price function of water, we have inline image.

  7. 7

    Many studies have considered convex marginal benefit, and justify the “prudent” nature of an individual under this assumption (Pindyck [1982], Deaton [1991], Weikard et al. [2006]). In the context of water management, the studies by Baylis and Vercammen, Fisher and Rubio, and Bhaduri et al. have also considered the effect of convex marginal benefit function on current water consumption (Fisher and Santiago [1997], Kathy and Vercammen [2008], Bhaduri et al. [2011]).

  8. 8

    Under flexible water price regime inline image and inline image will be more positive compared to the case of fixed water price where inline image. As a result inline image will be less negative under flexible water price.

  9. 9

    Here the aggregate adoption of efficient irrigation will decrease if the existing irrigation technology is depreciating and the representative farmer is no longer making new investments in irrigation technology.

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Investment in efficient irrigation and water storage
  6. 4. Numerical analysis
  7. 5. Conclusion
  8. Acknowledgments
  9. Appendix A
  10. Appendix B
  11. Appendix C
  12. REFERENCES
  • U. Amarasinghe, A. Bhaduri, O.P. Singh, A. Ojha, and B.K. Anand [2008], Cost and Benefits of Intermediate Water Storage Structures: Case Study of Diggies in Rajasthan, India, in Proceedings of the CGIAR Challenge Program on Water and Food, Addis Ababa, Ethiopia November 1014, 2003, pp. 79–82.
  • S. Ambec [2008],Sharing a Resource with Concave Benefits, Social Choice Welfare 31, 113.
  • B.C. Bates, Z.W. Kundzewicz, S. Wu, J., and P. Palutikof [2008], Climate Change and Water, IPCC Secretariat, Geneva, 210 pp.
  • F.J. Batz, K.J. Peters, and W. Janssen [1991],The Influence of Technology Characteristics on the Rate and Speed of Adoption, Agric. Econ. 21, 121130.
  • A. Bhaduri, U. Manna, E. Barbier, and J. Liebe [2011],Climate Change and Cooperation in Transboundary Water Sharing: An Application of Stochastic Stackelberg Differential Games in Volta River Basin, Nat. Res. Model. 24, 409444.
  • F.M. Caswell [1989],The Adoption Low Volume Irrigation Technology as a Water Conservation Tool, Water Int. 14, 1926.
  • M. Caswell [1991], Irrigation Technology Adoption Decisions: Empirical Evidence, in (A. Dinar and D. Zilberman, eds.), The Economics and Management of Water and Drainage in Agriculture, Kluwer Academic Press, Boston.
  • M.J. Carey and D. Zilberman [2002],A Model of Investment under Uncertainty: Modern Irrigation Technology and Emerging Markets in Water, Am. J. Agric. Econ. 84, 171183.
    Direct Link:
  • A. Deaton [1991],Savings and Liquidity Constraints, Econometrica, 59(5), 12211248.
  • D. Evans [2004],The Elevated Status of the Elasticity of Marginal Utility of Consumption, Appl. Econ. Lett. 11(7), 443447.
  • D.J. Evans [2005],The Elasticity of Marginal Utility of Consumption: Estimates for 20 OECD Countries, Fiscal Stud. 26(2), 197224.
  • G. Feder, R.E. Just, and D. Zilberman [1985],Adoption of Agricultural Innovations in Developing Countries: A Survey, Econ. Dev. Cult. Change 33, 255298.
  • K. Federick and D. Major [2002], Climate Change and Water Resources, The Management of Water Resource, Edward Elgar, New York.
  • A.C. Fisher and J.R. Santiago [1997],Adjusting to Climate Change: Implications of Increased Variability and Asymmetric Adjustment Costs for Investment in Water Reserves, J. Environ. Econ. Manage. 34(3), 207227.
  • L.R. Gardner [1985],The Potential for Water Markets in Idaho, Idaho Econ. Forecast 7, 2434.
  • J.J. Harou, M. Pulido-Velazquez, D.E. Rosenberg, J. Medellin-Azuara, J.R. Lund and R.E. Howitt [2009],Hydro-Economic Models: Concepts, Design, Applications, and Future Prospects, J. Hydrol. 375, 627643.
  • J.R. Hicks [1965], Capital and Growth, Vol. 343, Oxford University Press, Oxford.
  • R. Jensen [1982],Adoption and Diffusion of an Innovation of Uncertain Profitability, J. Econ. Theory 27, 182192.
  • C.R. Johansson, Y. Tsur, T.L. Roe, R. Doukkali, and A. Dinar [2002],Pricing Irrigation Water: A Review of Theory and Practice, Water Pol. 4, 173199.
  • R.E. Just and D. Zilberman [1983],Stochastic Structure, Farm Size and Technology Adoption in Developing Agriculture, Oxf. Econ. Papers 35, 307328.
  • B. Kathy and J. Vercammen [2008], Environmental Policy Choice with Learning, The Selected Works of Kathy Baylis. Available at http://works.bepress.com/kathy_baylis/16 (Accessed 01/03/2013).
  • K. Knapp and L. Olson [1995],The Economics of Conjunctive Groundwater Management with Stochastic Surface Supplies, J. Environ. Econ. Manage. 28, 340356.
  • P. Koundouri, C. Nauges, and V. Tzouvelekas [2006],Technology Adoption under Production Uncertainty: Theory and Application, Am. J. Agric. Econ. 88, 657670.
    Direct Link:
  • L. Oberkircher and A.K. Hornidge [2011],Water Is Life: Farmer Rationales and Water Saving in Khorezm, Uzbekistan: A Lifeworld Analysis, Rural Sociol. Soc. 76, 394421.
  • R. Pindyck [1982],Adjustment Costs, Demand Uncertainty, and the Behavior of the Firm, Am. Econ. Rev. 72, 415427.
  • R. Ram and S. Athalye [2009],Drought Resilience in Agriculture: The Role of Technological Options, Land Use Dynamics, and Risk Perception, Nat. Res. Model. 22, 437462.
  • M. Rosegrant and R.S. Meinzen-Dick [1996],Water Resources in the Asia Pacific Region, Asia Pacific Econ. Lit. 10, 3253
  • A. Saha, A.H. Love, and R. Schwart [1994],Adoption of Emerging Technologies Under Output Uncertainty, Am. J. Agric. Econ. 76, 386846.
  • R.K. Sampath [1992],Issues in Irrigation Pricing in Developing Countries, World Dev. 20, 967977.
  • E.C. Schuck, W.M. Frasier, R.S. Webb, L.J. Ellingson, and W.J. Umberger [2005],Adoption of More Technically Efficient System as a Drought Response, Water Res. Dev. 21, 651662.
  • D. Sekler, U. Amarsinghe, D. Molden, R. deSilva, and R. Barker [1998], World Water Demand and Supply 1990–2025: Scenarios and Issues, Research Report 19, International Water Management Institute, Colombo, Sri Lanka.
  • F. Shah, D. Zilberman, and U. Chakravorty [1995],Technology Adoption in the Presence of an Exhaustible Resource: The Case of Groundwater Extraction, Am. J. Agric. Econ. 77, 291299.
  • B.C. Shaliba and D.B. Bush [1987], Water Markets in Theory and Practice: Market Transfers, Water Values, and Public Policy, Westview Press, Boulder, CO.
  • D. Sunding and D. Zilberman [2001], The Agricultural Innovation Process: Research and Technology Adoption in a Changing Agricultural Sector, in (B.L. Gardner and G. Rausser eds.), Handbook of Agricultural Economics, Elsevier, New York.
  • Y. Tsur and T. Graham-Tomasi [1991],The Buffer Value of Groundwater with Stochastic Surface Water Supplies, J. Environ. Econ. Manage. 21, 201224.
  • Y. Tsur, M. Sternberg, and E. Hochman [1990],Dynamic Modeling of Innovation Process Adoption with Risk Aversion and Learning, Oxf. Econ. Papers 42, 336355.
  • R. Waylen and M.R. Zorn [1998],Prediction of Mean Annual Flows in North and Central Florida, J. Am. Water Res. Assoc. 34, 149157.
  • H.-P. Weikard, M. Finus, and J.C. Altamirano-Cabrera [2006],The Impact of Surplus Sharing on the Stability of International Climate Coalitions, Oxf. Econ. Papers 58, 209232.
  • D. Yaron, A. Dinar, and H. Voet [1992],Innovations on Family Farms: The Nazareth Region in Israel, Am. J. Agric. Econ. 74, 361370.
  • D. Zilberman, A. Dinar, N. MacDougall, M. Khanna, C. Brown, and F. Castilo [1995], Individual and Institutional Responses to Drought: The Case of California's Agriculture, University of California Berkeley, Working paper.