Economic Benefits from Irrigation-water and Nitrogen-fertilizer Applications

Traditional crop-production functions assume that all of the variable inputs, including irrigation water and nitrogen fertilizer, are fully employed to produce the crop output.Footnote ⁵ If crop production and nonpoint-source pollution are characterized by jointness in output, a crop-production function cannot be estimated (Shumway, Pope, and Nash Reference Shumway, Pope and Nash1983). Therefore, the crop-production process and generation of nonpoint-source pollution are more-appropriately characterized as nonjoint production processes (Kohli Reference Kohli1983, Kim, Schaible, and Daberkow Reference Kim, Schaible and Daberkow2000). And since nitrates are highly soluble in water and are transported into aquifers through return flows of irrigation water, both over-exploitation of ground water for irrigation and deterioration in the quality of ground water from transported nutrients may be managed by adopting irrigation technologies that use water more efficiently.

Aggregate production functions for estimating crop responses across fields or regions with heterogeneity or nonuniformities in the distribution of inputs such as irrigation water and nitrogen fertilizer result in smooth nonlinear functions that are concave with positive marginal products (Berck and Helfand Reference Berck and Helfand1990). Therefore, we assume that the crop-production function is quadratic so that the linear factor-demand functions are tractable mathematically. The crop-production function based on consumptive use of irrigation water, W*(t), and consumptive use of nitrogen fertilizer, n*(t), is then represented by

(1) $$\eqalign{Y\left( {W^\ast\left( t \right),n^\ast\left( t \right)} \right) & = a_0 + a_1W^\ast\left( t \right) - a_2\left( {W^\ast\left( t \right)} \right)^2 + \,a_3n^\ast\left( t \right) \cr & \quad - a_4\left( {n^\ast\left( t \right)} \right)^2 +\, a_5W^\ast\left( t \right)n^\ast\left( t \right)}$$

where Y is output, W*(t) and n*(t) are the consumptive use of irrigation water and nitrates, respectively, and a _i (i = 0, 1, … 5) are positive parameters.

Since the producer's costs for pumping and of fertilization are based on the rates of irrigation water (W) and fertilizer (n) applied, it is more convenient to present the crop-production function (equation 1) in terms of the rates of consumptive-equivalent irrigation water and nitrogen fertilizer applied. Since nitrates are highly soluble in water and are transported to aquifers in irrigation water, we assume that the fertilization efficiency is the same as the irrigation efficiency to reduce complexity.

Let α₀ > 0 be the rate of efficiency of the current irrigation technology such that a linear transformation of the use of irrigation water (IW) is represented by W*(t) = α₀ W ₀(t) where W ₀ is the rate of IW applied using the current irrigation technology. Similarly, the fertilization efficiency associated with the current irrigation technology is represented by n*(t) = α₀[n ₀(t) + f ₀ N(t)] where n ₀(t) is the amount of nitrogen fertilizer (NF) applied using the current irrigation technology, N(t) is the nitrate stock in the aquifer, and f ₀ is a fractional coefficient representing the ratio of IW using the current irrigation technology to the volume of water in the aquifer so that f ₀ N(t) represents the concentration of nitrates in the IW (i.e., the nutrient recycle).Footnote ⁶ The crop-production function (equation 1), which is based on the consumptive-equivalent rate of application of IW and NF, is then rewritten as

(2) $$\eqalign{Y({\rm \alpha} _0W_0(t),{\rm \alpha} _0(n_0(t) + f_0N(t))) =\,& a_0 + a_1{\rm \alpha} _0W_0(t) - a_2{\rm \alpha} _{_0 }^2 [W_0(t)]^2 \cr & + a_3{\rm \alpha} _0[n_0(t) + f_0N(t)] \cr & - a_4{\rm \alpha} _0^2 [n_0(t) + f_0N(t)]^2 \cr & + a_5{\rm \alpha} _0^2 W_0(t)[n_0(t) + f_0N(t)].}$$

The marginal benefits from the consumptive use of IW and NF differ from the marginal benefits associated with the application rates for IW and NF. Let the linear transformation of the marginal benefit from consumptive use of IW be represented by

$$P_{w{^\ast}} = \displaystyle{{P_{w_0}} \over {{\rm \alpha} _0}} $$

where P _w* and P _w0 are the marginal benefits from IW based on the consumptive use and the application rate, respectively. A linear transformation of the marginal benefit from use of NF is represented by

$$P_{n{^\ast}} = \displaystyle{{P_{n_0}} \over {{\rm \alpha} _0}} $$

where P _n* and P _n0 are the marginal benefits from NF based on the consumptive use and the application rate, respectively.

The consumptive-equivalent inverse IW and NF demand functions are then represented byFootnote ⁷

(3) $$\displaystyle{{P_{w_0}} \over {{\rm \alpha} _0}} = P_y {\rm \alpha} _0 \{ [a_1 + a_5 {\rm \alpha} _0 (n_0 \lpar t \rpar + f_0 N\lpar t \rpar)]{\rm} -{\rm} 2a_2 {\rm \alpha} _0 W_0 \lpar t \rpar\} $$

or, equivalently,

(3′) $$P_{w_0} = P_y {\rm \alpha} _0^2 \lcub \lsqb a_1 + a_5 {\rm \alpha} _0 \lpar n_0 \lpar t\rpar + f_0 N\lpar t\rpar \rpar \rsqb - 2a_2 {\rm \alpha} _0 \, W_0 \lpar t\rpar \rcub . $$

Similarly, the inverse NF demand based on consumptive-equivalent NF use is obtained using equation 2 as follows:

(4) $$\displaystyle{{P_{n_0}} \over {{\rm \alpha} _0}} = P_y {\rm \alpha} _0 \, \lcub \lsqb a_3 - 2a_4 {\rm \alpha} _0 f_0 N\lpar t\rpar + a_5 {\rm \alpha} _0 W_0 \lpar t\rpar \rsqb - 2a_4 {\rm \alpha} _0 n_0 \lpar t\rpar \rcub $$

or, equivalently,

(4′) $$P_{n_0} = P_y {\rm \alpha} _0^2 \, \lcub \lsqb a_3 - 2a_4 {\rm \alpha} _0 f_0 N\lpar t\rpar + a_5 {\rm \alpha} _0 W_0 \lpar t\rpar \rsqb - 2a_4 {\rm \alpha} _0 n_0 \lpar t\rpar \rcub.$$

Using equations 3′ and 4′, the economic benefits from IW and NF use are represented in equations 5 and 6, respectively.

(5) $$\eqalign{B(W_0:P_{w_0}) & =\,\, \int\limits_0^{W_0(t)} {P_y} {\rm \alpha} _0^2 \{ [a_1 + a_5{\rm \alpha} _0(n_0(t) + f_0N(t))] - 2a_2{\rm \alpha} _0\,x\} dx \cr & = P_y{\rm \alpha} _0^2 \{ [a_1 + a_5{\rm \alpha} _0(n_0(t) + f_0N(t))]W_0(t) - a_2{\rm \alpha} _0\,W_0^2 (t)\} }$$

(6) $$\eqalign{B(n_0:P_{n_n}) & =\,\, \int\limits_0^{n_0(t)} {P_y} {\rm \alpha} _0^2 \{ [a_3 + a_5{\rm \alpha} _0W_0(t) - 2a_4{\rm \alpha} _0f_0N(t)]\, - 2a_4{\rm \alpha} _0x\} dx \cr & = P_y{\rm \alpha} _0^2 \{ [a_3 + a_5{\rm \alpha} _0W_0(t) - 2a_4{\rm \alpha} _0f_0N(t)]n_0(t) - a_4{\rm \alpha} _0\,n_0^2 (t)\} }$$

As producers adopt improved irrigation systems, the inverse demand curves for IW and NF shift to either the right or the left. To investigate whether adoption of an efficient irrigation technology would lead to increased or decreased use of IW, the differential of equation 3′ with respect to the efficiency rate, α₀, is given in equation 7:

(7) $$\eqalign{\left[ \displaystyle{\partial W_0 \lpar t\rpar \over \partial {\rm \alpha}_0} - \displaystyle{\partial W_0 \lpar t\rpar \over \partial n_0 \lpar t\rpar} \left(\displaystyle{\partial n_0 \lpar t\rpar \over \partial {\rm \alpha}_0} \right) \, \right] &= \displaystyle{1 \over {{\rm \alpha} _0}} \left( \displaystyle{\partial W_0 \lpar t\rpar \over \partial P_{w_0}} \right) \lsqb P_y a_1 {\rm \alpha}_0^2 + P_{w_0} \lpar {\rm \eta} - 3\rpar \rsqb \cr & \gt\!\! = \!\!\lt 0}$$

where

$${\rm \eta} = \displaystyle{{\partial P_{w_0}} \over {\partial {\rm \alpha} _0}} \displaystyle{{{\rm \alpha} _0} \over {P_{w_{_0}}}} $$

is the technology elasticity of the marginal benefit of IW or, equivalently,

(7′) $$\displaystyle\left[{{\partial W_0(t)} \over {\partial {\rm \alpha} _0}} - \displaystyle{{\partial W_0(t)} \over {\partial n_0(t)}}\left(\displaystyle{{\partial n_0(t)} \over {\partial {\rm \alpha} _0}}\right)\,\right] \gt\!\!=\!\!\lt \,0 \quad {\rm if}\quad {\rm \eta} \, \lt\!\!= \!\!\gt \left[ 3 - \displaystyle{{P_{y_{}}a_1{\rm \alpha} _0^2 } \over {P_{w_0}}}\right].$$

The lefthand side of the equality in equation 7′ represents the aggregate marginal change in use of IW as the irrigation efficiency changes. The first and second terms represent the direct and indirect (through a complementary input) marginal change in use of IW. The result from 7′ indicates that the demand curve for IW shifts to the right (left) as irrigation efficiency improves (declines) depending on whether the change in the rate of the marginal benefit of IW when the change in irrigation efficiency is less than (greater than)

$$\left[ 3 - \displaystyle{{P_{y_{}} a_1 {\rm \alpha} _0^2} \over {P_{w_0}}} \right] .$$

Similarly, the differential of equation 4′ with respect to the rate of irrigation efficiency, α₀, is given by

(8) $$\eqalign{\left[ \displaystyle{\partial n_0 \lpar t\rpar \over \partial {\rm \alpha}_0} - \displaystyle{\partial n_0 \lpar t\rpar \over \partial W_0 \lpar t\rpar} \left(\displaystyle{\partial W_0 \lpar t\rpar \over \partial {\rm \alpha}_0} \right) \, \right] &= \displaystyle{1 \over {\rm \alpha}_0} \left(\displaystyle{\partial n_0 \lpar t\rpar \over \partial P_{n_0}} \right) \left[P_y a_3 {\rm \alpha}_0^2 + P_{n_0} \lpar {\rm \varepsilon} - 3\rpar \right] \cr &\gt\!\! = \!\!\lt\,0}$$

where

$${\rm \varepsilon} = \displaystyle{{\partial P_{n_0}} \over {\partial {\rm \alpha} _0}}\displaystyle{{{\rm \alpha} _0} \over {P_{n_{_0 }}}}$$

is the technology elasticity of the marginal benefit of NF. Equation 8 can be rewritten as

(8′) $$\left[ \displaystyle{{\partial n_0 \lpar t\rpar } \over {\partial {\rm \alpha} _0}} - \displaystyle{{\partial n_0 \lpar t\rpar } \over {\partial W_0 \lpar t\rpar }}\left( \displaystyle{{\partial W_0 \lpar t\rpar } \over {\partial {\rm \alpha} _0}} \right) \, \right] \gt\!\!=\!\!\lt\,0 \quad {\rm if}\quad {\rm \varepsilon}\,\lt\!\!=\!\!\gt \, \, \left[ 3 - \displaystyle{{P_{y_{}} a_3 {\rm \alpha} _0^2} \over {P_{n_0}}} \right] .$$

The result in 8′ also indicates that the demand function for NF would shift right (left) as an improved irrigation technology is adopted depending on whether the rate of increase of the marginal benefit of NF is less than (greater than)

$$\left[ 3 - \displaystyle{{P_{y_{}} a_3 {\rm \alpha} _0^2} \over {P_{n_0}}} \right] .$$

The results in equations 7′ and 8′ provide some indication of whether use of IW and NF increases or decreases because of adoption of an improved irrigation technology. However, those equations are derived from the inverse demand functions for IW and NF in equations 3′ and 4′, respectively, and any economic interpretation that is based on factor-demand functions can be somewhat misleading because those functions do not account for the process of adoption of technology. Therefore, we use the optimal economic properties of use of IW and NF to further examine how producers adapt their use of IW and NF when they adopt improved irrigation technologies.

Development and Adoption of Induced Irrigation Technologies

Development

Irrigation enterprises respond to the economic and policy incentives provided to producers.Footnote ⁸ Endogenous technological changes can be assigned to one of two broad, disjointed categories: invention and learning by doing (Romer Reference Romer1990, Reference Romer1994, Young Reference Young1993). We focus on development of new irrigation technologies through deliberate research by government agencies, academics, and irrigation enterprises to resolve regional problems. As a factor of production (IW in this case) becomes increasingly scarce (more costly) and government agencies subsidize the cost of adopting more-efficient irrigation technologies, irrigation enterprises allocate more R&D resources to developing technologies that save or substitute for the scarce resource (Hayami and Ruttan Reference Hayami and Ruttan1985, Kamien and Schwartz Reference Kamien and Schwartz1978, Kim et al. Reference Kim, Uri, Sandretto and Parry1996). We do not distinguish between public and private research as the efforts are often collaborative (Center for Irrigation Technology 2015, Irrigation Association 2015, Irrigation Technology Center 2015).

In our dynamic framework of ground water management, we use a hazard function to explain development of an induced irrigation technology.Footnote ⁹ We assume that the probability of developing technology i at a given point in time (prior to that technical innovation occurring) is an increasing function of (i) the capital stock (K(t | k, g _i )) of irrigation system manufacturers,Footnote ¹⁰ which depends on the cost of energy to pump one acre-foot of ground water per foot of lift (k) and the government subsidy rate to producers to adopt improved irrigation technologies (g _i ), and (ii) the amortized annual unit price of an induced irrigation system (z _i (t)).Footnote ¹¹

Let M _i (t) be the probability that a new more-efficient irrigation technology is developed by time t when M _i (0) = 0 and 0 ≤ M _i (t) ≤ 1. The conditional probability of developing a technical innovation at time t, m _i [z _i (t), K _i (t | k, g _i )], is the probability that development of such an innovation will occur during the next time period, t + Δt, given that it had not been developed at time t. The probability that development of a technical innovation occurring by time t is then represented as

(9) $$M_i\left( t \right) = 1 \hbox{-} {\rm exp}\left[ { \hbox{-} b_im_i\left( {z_i\left( t \right),K_i\left( {t \,\vert\, k,\; g_i} \right)} \right)} \right]t$$

where b _i  = 1/(1 + ϕ) and ϕ is the time elasticity of the hazard rate.Footnote ¹² The probability density function, which is obtained by differentiating equation 9 with respect to time variable t, is then written as a state equation:

(10) $$\displaystyle{{\partial M_i \lpar t\rpar } \over {\partial t}} = m_i \, \lsqb z_i \lpar t\rpar \comma \; K_i \lpar t \,\vert\, k\comma \; g_i \rpar \rsqb \lpar 1 - M_i \lpar t\rpar \rpar $$

where

$$m_i \lpar t = 0\rpar = 0\comma \, {\rm \partial} m_i / {\rm \partial} z_i \; \gt \; 0\comma \; \, {\rm \partial} m_i / {\rm \partial} K_i \; \gt \; 0\comma \; \, t\; \ne \; 0$$

is the probability density function for development of the induced irrigation technology.

Once technical innovation occurs at time t*, the economic and environmental benefits associated with use of IW and NF depend largely on whether the technology is adopted.

Adoption

Once an induced technology is developed, its adoption may be delayed if it does not provide a greater net economic benefit (π _i ) than use of the conventional technology (π₀). Burness and Brill (Reference Burness and Brill2001) and Ashwell and Peterson (Reference Ashwell and Peterson2013) considered models of adoption of an improved irrigation technology by producers without development of a new irrigation technology, and Shah, Zilberman, and Chakravorty (Reference Shah, Zilberman and Chakravorty1995) considered the process of adoption of an irrigation technology without considering capital investments.

Let V _i (θ) be the probability of adoption of the ith induced irrigation technology developed at time θ ≥ t* where V _i (θ = t*) ≥ 0 and 0 ≤ V _i (θ) ≤ 1. A producer's decision to adopt an improved irrigation system depends on how much the cost of pumping ground water is rising (mc(θ)), the amortized annual capital costs associated with adoption and maintenance of the new system (z _i (θ)), the government subsidy rate (g _i ), and the cost of any consultants needed to learn how to operate the new system (E(θ)). Since the agricultural irrigation industry is largely unregulated and lacks independent design standards for irrigation systems (Center for Irrigation Technology 2015), most agricultural irrigators do not have the expertise necessary to determine if the system will perform as claimed. Therefore, agricultural irrigators must be trained to use new systems.

The conditional probability of adopting a new ith irrigation technology at time θ,

$$v_i \, \lsqb E\lpar {\rm \theta} \rpar \comma \; mc\lpar {\rm \theta} \rpar \comma \; \lpar 1 - g_i \rpar z_i \lpar {\rm \theta} \rpar \, \vert \, {\rm \pi} _i \gt {\rm \pi} _0 \rsqb \comma \; $$

is the probability that adoption of such an innovation will occur during the next time period, θ + Δθ, given that the new technology had not been adopted at time θ where π _i is the expected profit resulting from adoption of the technology. We also assume that time of adoption of an innovation is uncertain and that the likelihood of adoption can be expressed as

(11) $$V_{i} \lpar {\rm \theta} \rpar = 1 - \exp \!\lsqb\!- {\rm \sigma} _i v _i \lpar E\lpar {\rm \theta} \rpar \comma \; mc\lpar {\rm \theta} \rpar \comma \; \lpar 1 - g_i \rpar z_i \lpar {\rm \theta} \rpar \,\vert \, {\rm \pi} _i \gt {\rm \pi} _0 \rpar \rsqb {\rm \theta} $$

where σ _i  = 1/(1 + ψ) and ψ is the time elasticity of the hazard rate.

The probability density function, ∂V _i (θ)/∂θ, which is obtained from equation 11, is represented as a state equation in our model as

(12) $$\displaystyle{{\partial V_i({\rm \theta})} \over {\partial {\rm \theta}}} = v_i[E\left({\rm \theta} \right),mc\left({\rm \theta} \right),\left({1 \hbox{-} g_i} \right)z_i\left({\rm \theta} \right) \, \vert \, {\rm \pi} _i \gt {\rm \pi} _0]\left[{1 \hbox{-} V_i\left({\rm \theta} \right)} \right]$$

where v _i (θ = t*) = 0 and

$$\displaystyle{{\partial v_i} \over {\partial E}} \lt 0\comma \; \, \, \displaystyle{{\partial v_i} \over {\partial mc}} \gt 0\comma \; \, \displaystyle{{\partial v_i} \over {\partial z_i}} \, \lt 0\comma \; {\rm} \, {\rm and}\, \, \displaystyle{{\partial v_i} \over {\partial g_i}} \gt 0.$$

Equation 10, associated with development of an induced irrigation technology, and equation 12, associated with adoption of the technology, are then incorporated as state equations into a nested optimal control model of ground water and nitrate stock management.

Dynamic Model of Irrigation and Nitrogen Fertilizer Applications

Applications of NFs by farmers generate both economic benefits, as shown in equation 6, and ground water pollution as an externality. In shallow aquifers underlying highly permeable soils, fertilizer applications and corresponding nitrate leaching can be assumed as simultaneous without loss. In areas where the farm land is above deep aquifers that have low permeability, one can expect that there will be a long time lag between application of the fertilizer and corresponding contamination of the ground water. Estimates of the contamination externality that ignore the time lag may underestimate the user cost associated with stock of nitrates in the aquifer (Kim, Hostetler, and Amacher Reference Kim, Hostetler and Amacher1993, Nkonya and Featherstone Reference Nkonya and Featherstone2000).

Since nitrates are transported to the aquifer in soluble form through IW, we model changes in the stock of nitrates in the aquifer as

(13) $$\eqalign{\dot N\, \lpar t\rpar \approx\,& \lpar 1 - {\rm \alpha} _0 \rpar \lsqb n_0 \lpar t - {\rm \tau} \rpar + f_0 N\lpar t - {\rm \tau} \rpar \rsqb + \lpar 1 - {\rm \alpha} _i \rpar \lsqb n_i \lpar t - {\rm \tau} \rpar \cr& + f_i N\lpar t - {\rm \tau} \rpar \rsqb - \, \lcub f_0 + f_i + {\rm \rho} \rcub \, N\lpar t\rpar} $$

where α₀ and α _i are the rate of irrigation efficiency of the conventional irrigation system and the ith induced irrigation technology, respectively, and α _i (i ≠ 0) > α₀, τ is a time lag;Footnote ¹³ f ₀ N(t) and f _i N(t) are the amounts of nitrate removal from the aquifer through irrigation with the conventional and ith irrigation technology (i.e., nutrient recycle), respectively; and ρ is the net natural discharge rate and denitrification.

Similarly, the hydrologic differential equation is represented by

(14) $$\dot h\lpar t\rpar \approx \left( \displaystyle{{R + \lsqb \lpar 1 - {\rm \alpha} _0 \rpar W_0 \lpar t - {\rm \tau} \rpar - W_0 \lpar t\rpar \rsqb + \lsqb \lpar 1 - {\rm \alpha} _i \rpar W_i \lpar t - {\rm \tau} \rpar - W_i \lpar t\rpar \rsqb } \over {AS}}\right) $$

where h(t) represents how many feet the water table is above sea level, R is the natural recharge rate, A is the number of acres over the aquifer, S is the storativity coefficient, and W ₀ and W _i are the amount of IW applied with the conventional and ith irrigation technologies, respectively.

Since adoption of an induced irrigation technology can occur only after the technology is developed, the expected action-value function (EV) associated with ground water management to be maximized for use of IW and NF under uncertain dates of development and adoption can be represented by a nested action-value function as follows (Kim et al. Reference Kim, Schaible, Lewandrowski and Vasavada2010, Lewandrowski, Kim, and Aillery Reference Lewandrowski, Kim and Aillery2014):Footnote ¹⁴

(15) $$\eqalign{Max\,EV = & \int\limits_0^T e^{ - {\rm \delta} t} \{ (1 - M_i(t))[NB(W_0(t)) + NB(n_0(t))] \cr & + M_i(t)[V_i(t)\{ NB(W_i(t)) + NB(n_i(t)) - (1 - g_i)z_i\} \cr & + (1 - V_i(t))\{ NB(W_0(t)) + NB(n_0(t))\} ]\} dt \cr & = \int\limits_0^T {e^{ - {\rm \delta} t}} \{ [NB(W_0(t)) + NB(n_0(t))] \cr & + \,M_i(t)V_i(t)[NB\,(W_i(t)) \cr & + NB(n_i(t)) - (1 - g_i)z_i - NB(W_0(t)) - NB(n_0(t))]\} dt} $$

where

$$\eqalign{NB(W_0(t)) & = \{ P_y{\rm \alpha} _0^2 [a_1 + a_5{\rm \alpha} _0(n_0(t) + f_0N(t))] - k(SL - h(t))\} \,W_0(t) \cr &\quad - P_y{\rm \alpha} _0^3 a_2W_0^2 (t)\, \cr NB(n_0(t)) & = \{ P_y{\rm \alpha} _0^2 [a_3 + a_5{\rm \alpha} _0W_0(t) - 2a_4{\rm \alpha} _0f_0N(t)] - q\} \,n_0(t) \cr &\quad - P_y{\rm \alpha} _0^3 a_4n_0^2 (t) \cr NB(W_i(t)) & = \{ P_y{\rm \alpha} _i^2 [a_1 + a_5{\rm \alpha} _i(n_i(t) + f_iN(t))] - k(SL - h(t))\} \,W_i(t) \cr &\quad - P_y{\rm \alpha} _i^3 a_2W_i^2 (t) \cr NB(n_i(t)) & = \{ P_y{\rm \alpha} _i^2 [a_3 + a_5{\rm \alpha} _iW_i(t) \cr &\quad - 2a_4{\rm \alpha} _if_iN(t)] - q\} \,n_i(t) - P_y{\rm \alpha} _i^3 a_4n_i^2 (t)} $$

and T is terminal time, δ is the rate of discount, k(SL – h(t)) is the marginal pumping cost per acre where k is a constant pumping cost per acre-foot of ground water per foot of lift, SL is the elevation in feet of land above sea level, and q is the unit price of NF.

The expected action-value function in equation 15, which measures the present value of the expected net benefit from use of IW and NF, assumes that the induced technology is developed with probability M _i (t) and is subsequently adopted with probability V _i (t) while the probability of using the conventional irrigation technology is (1 – V _i (t)).

The dynamic optimization problem of maximizing the expected net social welfare of use of ground water and NF in the presence of development and adoption of an induced irrigation technology with uncertainty is expressed by maximizing the action-value function in equation 15 subject to the state equations, 10, 12, 13, and 14. The current-value Hamiltonian equation is then presented as

(16) $$\eqalign{H & = \{[NB(W_0(t)) + NB(n_0(t))] + M_i (t) V_i(t) [NB(W_i(t)) + NB(n_i(t)) \cr &\quad - (1 - {\rm g}_i) z_i - NB (W_0(t)) - NB(n_0(t))] \} \cr &\quad + \displaystyle{{\rm \lambda}_1(t) \over A \ast S} \left\{ R + [(1 - {\rm \alpha} _0)W_0(t - {\rm \tau} ) \!-\! W_0(t)] + [(1 - {\rm \alpha}_i) W_i (t - {\rm \tau}) - W_i (t)] \right\} \cr &\quad + {\rm \lambda} _2(t) \{(1 - {\rm \alpha}_0) [n_0(t - {\rm \tau}) + f_0 N (t - {\rm \tau})] \cr &\quad + (1 - {\rm \alpha}_i) [n_i(t - {\rm \tau}) + f_i N (t - {\rm \tau})] - [f_0 + f_i + {\rm \rho}] N(t)\} \cr &\quad + {\rm \lambda}_3(t)\,\{ m_i [z_i(t), K_i (t \vert k, g_i)] [1 - M_i(t)]\} \cr &\quad + {\rm \lambda}_4(t) \left\{v_i [E_i ({\rm \tau}), mc ({\rm \tau}), (1 - g_i) z_i ({\rm \tau}) \vert {\rm \pi}_i \gt {\rm \pi} _0]\,\left[1 \hbox{-} V_i \left({\rm \tau} \right) \right] \right\} \cr & \quad \quad \quad \quad h \left( t \right) = h_0 \quad {\rm for}\,\hbox{-} {\rm \tau} \le t \le 0 \cr & \quad \quad \quad \quad N \left( t \right) = {\tilde N}_0 \quad {\rm for}\, \hbox{-} {\rm \tau} \le t \le 0 \cr & \quad \quad \quad \quad M_i \left( t \right) = 0 \quad {\rm for}\, \hbox{-} {\rm \tau} \le t \le 0 \cr & \quad \quad \quad \quad V_i \left( t \right) = 0 \quad {\rm for}\, \hbox{-} {\rm \tau} \le t \le 0}$$

where W ₀(t), W _i (t), n ₀(t), n _i (t), and g _i are control variables; h(t), N(t), M _i (t), and V _i (t) are state variables; and λ₁(t), λ₂(t), λ₃(t), and λ₄(t) are adjoint variables associated with h(t), N(t), M _i (t), and V _i (t), respectively.

The necessary conditions for control variables are presented in equations 17-1 through 19 (the optimality conditions for the state and adjoint variables are presented in Appendix A, which is available from the authors).Footnote ¹⁵

(17-1) $$\eqalign{& \displaystyle{{\partial H} \over {\partial W_0(t)}} + \displaystyle{{\partial H} \over {\partial W_0(t - {\rm \tau} )}}\left \vert {{\left( {t + {\rm \tau} } \right)}} \right. = 0\;\;{\rm for }\;\;0 \le t \lt T - {\rm \tau} , \cr & {\rm which}\,\,{\rm implies}\,\,{\rm that} \cr & (1 - M_i(t)V_i(t))\left[P_{w_0} - k(SL - h(t)) + \displaystyle{{\partial B(n_0(t))} \over {\partial W_0(t)}}\right] = \displaystyle{{{\rm \lambda} _1(t)} \over {A \ast S}} \cr & \quad- \displaystyle{(1 - {\rm \alpha} _0){\rm \lambda} _1(t + {\rm \tau} ) \over A \ast S}}$$

(17-2) $$\eqalign{& \displaystyle{{\partial H} \over {\partial W_0(t)}} = 0\;{\rm for}\;T - {\rm \tau} \le t \le T, \cr & {\rm which}\,\;{\rm implies}\,\;{\rm that} \cr & (1 - M_i(t)V_i(t)) \left[P_{w_0} - k(SL - h(t)) + \displaystyle{{\partial B(n_0(t))} \over {\partial W_0(t)}} \right] = \displaystyle{{{\rm \lambda} _1(t)} \over {A \ast S}}} $$

(17-3) $$\eqalign{& \displaystyle{{\partial H} \over {\partial W_i(t)}} + \displaystyle{{\partial H} \over {\partial W_i(t - {\rm \tau} )}}\left \vert {(t + {\rm \tau} )} = 0\;{\rm for }\;0 \le t \lt T - {\rm \tau} , \right. \cr & {\rm which}\,\;{\rm implies}\,\,{\rm that} \cr & (M_i(t)V_i(t)) \left[P_{w_i} - k(SL - h(t)) + \displaystyle{{\partial B(n_i(t))} \over {\partial W_i(t)}} \right] = \displaystyle{{{\rm \lambda} _1(t)} \over {A \ast S}} \cr & \quad- \displaystyle{{(1 - {\rm \alpha} _i){\rm \lambda} _1(t + {\rm \tau} )} \over {A \ast S}}} $$

(17-4) $$\eqalign{& \displaystyle{{\partial H} \over {\partial W_i \lpar t\rpar }} = 0\;for\; t - {\rm \tau} \le t \le T\comma \; \cr & {\rm which}\,\, {\rm implies}\,\, {\rm that} \cr & \lpar M_i \lpar t\rpar V_i \lpar t\rpar \rpar \left[ P_{w_i} - k\lpar SL - h\lpar t\rpar \rpar + \displaystyle{{\partial B\lpar n_i \lpar t\rpar \rpar } \over {\partial W_i \lpar t\rpar }}\right] = \displaystyle{{{\rm \lambda} _1 \lpar t\rpar } \over {A {\ast} S}}} $$

(18-1) $$\eqalign{& \displaystyle{\partial H \over \partial n_0 \lpar t\rpar} + \displaystyle{\partial H \over \partial n_0 \lpar t + {\rm \tau} \rpar} \left\vert {\lpar t + {\rm \tau} \rpar} \right. = 0 {\rm for} 0 \le t \lt T - {\rm \tau}\comma \; \cr & {\rm which}\,{\rm implies}\,{\rm that} \cr & \lpar 1 - M_i \lpar t \rpar V_i \lpar t\rpar \rpar \left[ P_{n_0} - q + \displaystyle{\partial B \lpar W_0 \lpar t\rpar \rpar \over \partial n_0 \lpar t\rpar} \right] = - \lpar 1 - {\rm \alpha}_0 \rpar {\rm \lambda}_2 \lpar t + {\rm \tau} \rpar}$$

(18-2) $$\eqalign{& \displaystyle{\partial H \over \partial n_0 (t)} = 0\,for\,T - {\rm \tau} \le t \le T, \cr & {\rm which}\,{\rm implies}\,{\rm that} \cr & (1 - M_i (t) V_i (t)) \left[P_{n_0} - q + \displaystyle{\partial B(W_0 (t)) \over \partial n_0 (t)} \right] = 0} $$

(18-3) $$\eqalign{& \displaystyle{\partial H \over \partial n_i \lpar t\rpar} + \displaystyle{\partial H \over \partial n_i \lpar t - {\rm \tau} \rpar} \left\vert {\lpar t + {\rm \tau} \rpar} \right. = 0 {\rm for} 0 \le t \lt T - {\rm \tau}\comma \; \cr & {\rm which}\,{\rm implies}\,{\rm that} \cr & M_i \lpar t \rpar V_i \lpar t\rpar \left[ P_{n_i} - q + \displaystyle{\partial B\lpar W_i \lpar t\rpar \rpar \over \partial n_i \lpar t\rpar} \right] = - \lpar 1 - {\rm \alpha}_i \rpar {\rm \lambda}_2 \lpar t + {\rm \tau} \rpar} $$

(18-4) $$\eqalign{& \displaystyle{{\partial H} \over {\partial n_i \lpar t\rpar }} = 0\, \, for\, \, T - {\rm \tau} \le t \le T\comma \; \cr & {\rm which\;\;implies\;\;that} \cr & \, \, \, M_i \lpar t\rpar V_i \lpar t\rpar \left[ P_{n_i} - q + \displaystyle{{\partial B\lpar W_i \lpar t\rpar \rpar } \over {\partial n_i \lpar t\rpar }}\right] = 0} $$

(19) $$\eqalign{& \displaystyle{\partial H \over \partial g_i} = 0\,for\,0 \le t \le T, \cr & {\rm which}\,{\rm implies}\,{\rm that} \cr & M_i (t)V_i (t)z_i + \left\{[1 - M_i (t)] V_i (t) \left(b_i t \left(\displaystyle{\partial m_i (t) \over \partial g_i} \right)\right)\right. \cr & \left. + M_i (t)[1 - V_i (t)] {\rm \sigma}_i t \left(\displaystyle{\partial v_i (t) \over \partial g_i} \right) \right\} [B(W_i (t)) + B(n_i (t)) \cr &- B(W_0 (t)) - B(n_0 (t))] \cr &= - {\rm \lambda} _3 (t)(1 - M_i (t)) [1 - b_i m_i t] \left(\displaystyle{\partial m_i (t) \over \partial g_i} \right) \cr &- {\rm \lambda}_4 (t)(1 - V_i (t)) [1 - {\rm \sigma} _i v_i t] \left(\displaystyle{\partial v_i (t) \over \partial g_i} \right)}$$

The economic interpretation of optimal conditions presented in equations 17-1 through 19 is better served by first explaining the adjoint variables λ₁(t) through λ₄(t), which are presented in Appendix B (available from the authors). Adjoint variable λ₁(t), which measures the marginal contribution of state variable h(t) to the expected action-value function in equation 15, is positive, as shown in equation B.1. As the level of the water table in feet rises, the pumping cost to producers declines and the net economic benefit increases. Adjoint variable λ₂(t), which measures the marginal contribution of state variable N(t) to the expected action-value function, is negative, as shown in equation B.2. Since the nitrates available in the IW and the NF are perfect substitutes, producers reduce their applications of NF when the nitrates available in ground water increase. Therefore, the economic benefit from use of NF declines when greater amounts of nitrates are available from the IW at no cost. This result indicates that economic and environmental benefits increase as the stock of nitrates in the ground water is reduced by adoption of an improved irrigation technology.

Adjoint variable λ₄(t) in equation B.4, which measures the marginal contribution of state variable V _i (t) to the expected action-value function associated with an increase in the probability of adopting a new improved irrigation technology, is positive. This result, however, requires that adjoint variable λ₃(t) in equation B.3, which measures the marginal contribution of state variable M _i(t) to the expected action-value function associated with an increase in the probability of developing an improved irrigation technology, is positive. Adoption of an improved irrigation technology, therefore, requires that the associated net economic benefit over time be greater than the benefit accrued by use of the conventional irrigation technology. With this knowledge regarding the adjoint variables, we can interpret the optimal conditions presented in equations 17-1 through 19.

Since IW and NF are complementary inputs in crop production, the results in equations 17-1 (conventional) and 17-3 (induced new technology) illustrate that efficient use of ground water for irrigation requires the sum of the expected marginal net economic benefit from use of IW and increase in the amount of complementary NF used to equal the cost associated with use of ground water minus the cost associated with delayed return flows of ground water. The results show, therefore, that models that do not consider development and adoption of an improved irrigation technology overestimate the marginal net economic benefit, and models that do not consider the time lag associated with return flows underestimate the cost of using ground water, which leads to a greater use of ground water than is optimal for irrigation.

Equations 17-2 and 17-4 show how, at the terminal time period, the marginal net economic benefit from use of IW is overstated when development and adoption of an irrigation technology is not considered, causing producers to use more ground water than is optimal for irrigation.

Equations 18-1 (conventional technology) and 18-3 (improved technology) show that the expected net economic benefit from use of NF must equal the user cost when incorporating the time lags associated with return flows and leaching. When development and adoption of an improved irrigation technology are not considered, the marginal net economic benefit from use of NF would be overstated, and the user cost would be overstated when failing to account for the time lag associated with nitrate leaching. Equations 18-2 and 18-4 demonstrate that efficient use of NF in the terminal period under each irrigation technology requires that the marginal benefit from use of NF equals the marginal cost of use.

We find that the economic properties of the optimal government subsidy are more complex than prior studies have considered. Equation 19 shows that, under the optimal government subsidy to encourage producers to adopt an induced irrigation technology, the sum of (i) the expected price of the induced irrigation system, (ii) the expected foregone benefit from failing to develop it, and (iii) the expected foregone benefit from failing to adopt it equals the sum of (i) the user cost from failing to develop the system and (ii) the user cost to adopt a new technology.

Effects of a Cost-share Program on Ground Water Exploitation and Nitrate Stocks

Using the optimal conditions of use of IW and NF in equations 17-1 and 18-4, we evaluate the effects of a cost-share program on use of IW and NF. The government subsidy (g _i ) to encourage producers to adopt better irrigation systems was treated as a control variable in the preceding section (see equation 19). To gain qualitative insight into the properties of optimal decision rules, we must discuss how the subsidy affects the cost to the user associated with nitrate and ground water stocks over time. Therefore, we now treat the government subsidy to producers as an exogenous variable. Though comparative dynamic analyses can be used to evaluate expected changes in the shadow values, λ₁ through λ₄ (Kim et al. Reference Kim, Uri, Sandretto and Parry1996), such an analysis is needlessly complex, and we therefore use a simpler approach to evaluate the effects of a government subsidy.

For convenience, we let the time lag, τ, be zero in equations 17-1 (conventional irrigation) and 17-3 (improved irrigation). The cost of pumping ground water is then represented as

(20) $$\eqalign{ {\rm \lambda} _1 \lpar t\rpar = \, & \displaystyle{{\lpar AS\rpar } \over 2} \left\{ \left[ \displaystyle{{P_{w_0} - k\lpar SL - h\lpar t\rpar \rpar + \displaystyle{{\partial B\lpar n_0 \lpar t\rpar \rpar } \over {\partial W_0 \lpar t\rpar }}} \over {{\rm \alpha} _0}} \right] \right. \cr & + \lpar M_i \lpar t\rpar V_i \lpar t\rpar \rpar \left[ \left( \displaystyle{{P_{w_i} - k\lpar SL - h\lpar t\rpar \rpar + \displaystyle{{\partial B\lpar n_i \lpar t\rpar \rpar } \over {\partial W_i \lpar t\rpar }}} \over {{\rm \alpha} _i}} \right) \right. \cr & - \left. \left. \left( \displaystyle{{P_{w_0} - k\lpar SL - h\lpar t\rpar \rpar + \displaystyle{{\partial B\lpar n_0 \lpar t\rpar \rpar } \over {\partial W_0 \lpar t\rpar }}} \over {{\rm \alpha} _0}} \right) \right] \right\} .} $$

Similarly, the cost associated with nitrate stocks in ground water is obtained from equations 18-1 (traditional irrigation) and 18-3 (improved irrigation) under the assumption that τ = 0:

(21) $$\eqalign{ {\rm \lambda} _2 \lpar t\rpar &= - \left[ \displaystyle{{\left( P_{n_0} - q + \displaystyle{{\partial B\lpar W_0 \lpar t\rpar \rpar } \over {\partial n_0 \lpar t\rpar }}\right) } \over {2\lpar 1 - {\rm \alpha} _0 \rpar }}\right] + \displaystyle{{\lpar M_i \lpar t\rpar V_i \lpar t\rpar \rpar } \over 2}\cr &\quad\left \{ \left( \displaystyle{{P_{n_0} - q + \displaystyle{{\partial B\lpar W_0 \lpar t\rpar \rpar } \over {\partial n_0 \lpar t\rpar }}} \over {\lpar 1 - {\rm \alpha} _0 \rpar }}\, \right) - \left( \, \displaystyle{{P_{n_i} - q + \displaystyle{{\partial B\lpar W_i \lpar t\rpar \rpar } \over {\partial n_i \lpar t\rpar }}} \over {\lpar 1 - {\rm \alpha} _i \rpar }}\right) \right\} .} $$

Inserting equations 9 and 11 for M _i (t) and V _i (t), respectively, into equations 20 and 21 and then differentiating the resulting equations with respect to the rate of the government subsidy, g _i , results in

(22) $$\eqalign{\displaystyle{{\partial {\rm \lambda} _1 \lpar t\rpar } \over {\partial g_i}} & = \displaystyle{{\lpar AS\rpar } \over 2}\left[ \left( \displaystyle{{P_{w_i} - k\lpar SL - h\lpar t\rpar \rpar + \displaystyle{{\partial B\lpar n_i \lpar t\rpar \rpar } \over {\partial W_i \lpar t\rpar }}} \over {{\rm \alpha} _i}} \right) \right. \cr & \,\,\quad \left. - \left( \displaystyle{{P_{w_0} - k\lpar SL - h\lpar t\rpar \rpar + \displaystyle{{\partial B\lpar n_0 \lpar t\rpar \rpar } \over {\partial W_0 \lpar t\rpar }}} \over {{\rm \alpha} _0}} \right) \right] \cr & \,\quad\times \left [ V_i \lpar t\rpar \lpar 1 - M_i \lpar t\rpar \rpar b_i t\, \left ( \displaystyle{{\partial m_i} \over {\partial g_i}} \right) + M_i \lpar t\rpar \lpar 1 - V_i \lpar t\rpar \rpar {\rm \sigma} _i t\, \left( \displaystyle{{\partial v_i} \over {\partial g_i}} \right) \right] \cr& \quad\,\, \gt\!\!=\!\!\lt\,0.} $$

(23) $$\eqalign{ \displaystyle{{\partial {\rm \lambda} _2 \lpar t\rpar } \over {\partial g_i}} & = \displaystyle{1 \over 2}\left[ \left( \displaystyle{{P_{n_0} - q + \displaystyle{{\partial B\lpar W_0 \lpar t\rpar \rpar } \over {\partial n_0 \lpar t\rpar }}} \over {\lpar 1 - {\rm \alpha} _0 \rpar }}\right) - \left( \displaystyle{{P_{n_i} - q + \displaystyle{{\partial B\lpar W_i \lpar t\rpar \rpar } \over {\partial n_i \lpar t\rpar }}} \over {\lpar 1 - {\rm \alpha} _i \rpar }}\right ) \right] \, \cr & \,\quad\times \left[ V_i \lpar t\rpar \lpar 1 - M_i \lpar t\rpar \rpar b_i t\, \left( \displaystyle{{\partial m_i} \over {\partial g_i}} \right) \, \, + \, \, M_i \lpar t\rpar \lpar 1 - V_i \lpar t\rpar \rpar {\rm \sigma} _i t\, \left ( \displaystyle{{\partial v_i} \over {\partial g_i}} \right) \, \right] \cr&\,\,\quad\gt\!\! =\!\!\lt\,0.} $$

The marginal effect of the government subsidy on the cost associated with ground water and the nitrate stock in the aquifer are obtained from equations 22 and 23, respectively, as follows.

(24) $$\displaystyle{{\partial {\rm \lambda} _1 \lpar t\rpar } \over {\partial g_i}} \gt\!\!=\!\!\lt\,0 \quad if \quad \left ( \displaystyle{{{\rm \alpha} _i} \over {{\rm \alpha} _0}} \right) \, \, \lt\!\!=\!\!\gt \left( \displaystyle{{P_{w_i} - k\lpar SL - h\lpar t\rpar \rpar + \displaystyle{{\partial B\lpar n_i \lpar t\rpar \rpar } \over {\partial W_i \lpar t\rpar }}} \over {P_{w_0} - k\lpar SL - h\lpar t\rpar \rpar + \displaystyle{{\partial B\lpar n_0 \lpar t\rpar \rpar } \over {\partial W_0 \lpar t\rpar }}}}\right) $$

(25) $$\displaystyle{{\partial {\rm \lambda} _2 \lpar t\rpar } \over {\partial g_i}} \gt\!\!=\!\!\lt\,0 \quad if \quad \left( \displaystyle{{1 - {\rm \alpha} _i} \over {1 - {\rm \alpha} _0}} \right) \, \gt\!\!=\!\!\lt \left( \displaystyle{{P_{n_i} - q + \displaystyle{{\partial B\lpar W_i \lpar t\rpar \rpar } \over {\partial n_i \lpar t\rpar }}} \over {P_{n_0} - q + \displaystyle{{\partial B\lpar W_0 \lpar t\rpar \rpar } \over {\partial n_0 \lpar t\rpar }}}}\right)$$

Equation 24 shows that the user cost associated with the ground water stock increases (decreases) as the rate of the government subsidy increases when the rate of increase in irrigation efficiency is less than (greater than) the rate of increase in the marginal net economic benefit resulting from adoption of an improved irrigation technology. When the rate of increase in the marginal net economic benefit is greater than the rate of increase in irrigation efficiency, producers use more IW and the user cost associated with the ground water stock increases. Most earlier empirical studies (e.g., Huffaker and Whittlesey Reference Huffaker and Whittlesey2003, Pfeiffer and Lin Reference Pfeiffer and Lin2014) support this outcome, finding that adoption of an induced irrigation technology can lead to a greater use of ground water because some producers switch to high-value crops and/or convert from dryland production to irrigated production. However, when the rate of increase in irrigation efficiency is greater than the rate of increase in the marginal net economic benefit, less water is used relative to an inefficient irrigation system because some producers invest more capital in IW to drive down the marginal net economic benefit. This result is partially supported by Peterson and Ding (Reference Peterson and Ding2005).

Similarly, equation 25 shows that the user cost associated with the nitrate stocks in the aquifer decreases (increases) as the rate of the government subsidy increases when the rate of increase in the marginal net economic benefit from adoption of an improved irrigation technology is greater than the rate of reduction of nitrate leaching. In that case, producers use a greater amount of nitrates so the user cost associated with the nitrate stocks in the ground water declines (becomes more negative) as the subsidy rate increases. This result is consistent with several earlier studies (Shaffer, Halvorson, and Pierce Reference Shaffer, Halvorson and Pierce1991, Wu et al. Reference Wu, Mapp and Bernardo1994). It suggests that a reduction in pollution by adoption of an induced irrigation technology could lead to an increase in the nitrate stocks because of the endogenous technological change. When the rate of reduction in nitrate leaching is greater than the rate of increase in the marginal net economic benefit, producers use less nitrates and the user cost increases (becomes less negative), as demonstrated by Kim, Schaible, and Daberkow (Reference Kim, Schaible and Daberkow2000).

Numerical Illustration

Most previous studies of conjunctive use of ground and surface water at the river-basin level employed mathematical programming models (Huffaker and Whittlesey Reference Huffaker and Whittlesey2003, Ward and Pulido-Velazquez Reference Ward and Pulido-Velazquez2008) or ground water management (Peterson and Ding Reference Peterson and Ding2005) to compare use of IW before and after adoption of an improved irrigation technology. To provide a numerical example that we could compare to our predictions, we obtained data from Peterson and Ding (Reference Peterson and Ding2005), which studied the impact of an improvement in irrigation efficiency achieved by converting from flood to center-pivot and drip irrigation systems on conservation of ground water in western portions of the high plains in Kansas for continuous corn production (see Table 1).

Table 1.

Revenues, Pumping Costs, and Net Benefits by Well Capacity and System Type

^a Corn price of $2.28 per bushel.

^b Acreages for the flood, sprinkler, and drip irrigation systems are based on 160, 126, and 160 acres, respectively.

Source: Peterson and Ding (Reference Peterson and Ding2005).

We find a rate of increase in irrigation efficiency of 1.25 for sprinkler irrigation and 1.67 for drip irrigation, a rate of increase in the marginal (average) net benefit for a 300-gallon-per-minute well capacity of 0.99 for sprinkler irrigation and 1.37 for drip irrigation, and a rate of increase in the marginal net benefit for a 500-gallon-per-minute well capacity of 1.08 for sprinkler irrigation and 1.29 for drip irrigation. These results show that the increase in the rate of irrigation efficiency is greater than the increase in the rate of marginal net benefit for both sprinkler and drip irrigation systems regardless of pumping capacity. Therefore, applications of water would decrease after adoption of an improved irrigation technology. Table 1 shows that water applications increase with sprinkler irrigation and decrease with drip irrigation as expected.

Pfeiffer and Lin (Reference Pfeiffer and Lin2014) also evaluated the effects of converting traditional center-pivot irrigation systems to highly efficient dropped-nozzle center-pivot systems on conservation of ground water in western Kansas. Their empirical results show that the shift to a more-efficient irrigation technology increased extraction of ground water, in part due to a shift to high-value crops that led to a higher marginal net economic benefit from the use of IW.