Spatial dynamics of water and nitrogen management in irrigated agriculture.

(9) Analyses can implicitly include spatial variability via an irrigation efficiency coefficient (Caswell, Litchenberg, and Zilberman 1990). However, the plant-level production functions used here are highly nonlinear, and integration over even LRP production functions implies nonlinear field-level production functions (Berck and Helfand 1990). This approach appears as a linear approximation and a priori could only be expected to hold over a limited range of input values. Johnson, Adams, and Perry (1991) include subfields in a crop-simulator model.

(10) Kennedy (1986) and Segarra et al. (1989) evaluate dynamic optimization versus annual optimization for nitrogen application, but not for irrigation or spatial variability.

(11) Even just the optimal steady-state in this model is likely still too complex for direct inclusion in a regional programming model with many crops, irrigation systems, and land quality types. As an alternative, this field-level model with a given crop and irrigation system can be run over a range of water, nitrogen, and emission prices, and a regression model fit to the resulting optimal steady-state values for applied water, applied nitrogen, crop yield, and nitrogen emissions. These estimated production functions can be included in the regional programming model; this would be a computationally feasible system for a large number of activities.

K. Knapp and K. Schwabe are professor and associate professor in the Department of Environmental Sciences, University of California, Riverside.

The authors would like to thank the editor Stephen Swallow, anonymous reviewers, and participants at the annual meetings of the WAEA and AAEA for comments leading to a substantially improved manuscript. Senior authorship is shared equally. Table 1. Optimal Steady-State Values Under Alternative Discount Rates

Applied Applied Soil Discount Water Nitrogen Nitrogen Rate [[bar.w].sub.ss] [[bar.n].sub.a,ss] [[bar.n].sub.ss] r (%) (cm) (kg/ha) (kg/ha) 0 87.5 224 161.4 5 87.9 221 158.9 10 88.2 219 156.8 15 88.5 217 155.2 20 88.7 215 153.8

Nitrogen Discount Emissions Yield Annual Rate [[bar.n].sub.e,ss] [[bar.y].sub.ss] NB r (%) (kg/ha) (tons/ha) ($/ha) 0 36.2 10.08 168.30 5 36 10.07 168.27 10 35.8 10.06 168.19 15 35.7 10.04 168.09 20 35.6 10.03 167.99 Note: PV-optimization with spatial variability and base parameter values as specified in the text. Variables of the form [X.sub.ss] denote steady-state values. Table 2. Optimal Steady-State Values by Behavioral Regime (PP versus PV) and Spatial Heterogeneity (Uniform versus Spatial)

Applied Applied Soil

Water Nitrogen Nitrogen

[[bar.w].sub.ss] [[bar.n].sub.a,ss] [[bar.n].sub.ss]

(cm) (kg/ha) (kg/ha) PV-Spatial 88 221 159 PP-Spatial 91 190 138 PV-Uniform 63 208 173 PP-Uniform 65 171 137

Nitrogen Nitrogen

Emissions Uptake

[[bar.n].sub.e,ss] [[bar.n].sub.u,ss]

(kg/ha) (kg/ha) PV-Spatial 36 231 PP-Spatial 33 213 PV-Uniform 6 237 PP-Uniform 7.3 212

Yield Annual

[[bar.y].sub.ss] NB

(tons/ha) ($/ha) PV-Spatial 10.08 168 PP-Spatial 9.9 163 PV-Uniform 10.1 194 PP-Uniform 9.8 186 Note: Behavioral regimes are Present Value Optimization (PV) and Period-by-Period Optimization (PP). Discount rate = 5% and baseline parameter values, including [p.sub.w] = $.64/ (ha-cm) and [p.sub.e] = 0. Table 3. Water Price Effects on Optimal Steady-State Values

Applied Water Applied Nitrogen Price Water [[bar.n] Soil Nitroge [p.sub.w] [[bar.w].sub.ss] .sub.a,ss] [[bar.n].sub. ($/(ha-cm)) (cm) (kg/ha) a,ss] (kg/ha) 0.64 88 221 158.80 (Current/Baseline) 0.71 86.1 221.1 160.7 (10% Increase) 0.77 73.4 216.8 162.2 (20% Increase) 0.83 72.3 216.2 169.1 (30% Increase) 0.90 71.4 215.7 169.9 (40% Increase) 0.96 70.5 215.2 170.6 (50% Increase) Water Nitrogen Price Yield [[bar.y] Emissions [p.sub.w] .sub.ss] [[bar.n].sub ($/(ha-cm)) (tons/ha) .e,ss] (kg/ha) 0.64 10.10 36 (Current/Baseline) 0.71 10.10 34.9 (10% Increase) 0.77 9.94 25.2 (20% Increase) 0.83 9.93 24.0 (30% Increase) 0.90 9.92 23.0 (40% Increase) 0.96 9.91 22.2 (50% Increase) Water Price [p.sub.w] WTP for Water Grower Profit ($/(ha-cm)) ($/ha) ($/ha) 0.64 227.01 168.27 (Current/Baseline) 0.71 226.95 166.34 (10% Increase) 0.77 213.17 156.80 (20% Increase) 0.83 212.50 152.34 (30% Increase) 0.90 211.78 147.81 (40% Increase) 0.96 211.05 143.37 (50% Increase) Note: PV-optimization with spatial variability and other parameter values at baseline values as specified in the text. WTP for Water = revenue less costs not including water charge. Grower Profit = Annual NB less water charge. Table 4. Optimal Steady-State Values Under a Nitrogen Emissions Charge

Applied Applied N Emissions Water Nitrogen Charge [[bar.w].sub.ss] [[bar.n].sub. [p.sub.e] ($/kg) (cm) a,ss] (kg/ha) 0.00 88 221 0.20 72 213.2 0.40 69.7 209.7 0.60 68 207.1 0.80 66.2 205 1.00 63.5 203

Soil N Emissions Nitrogen Yield Charge [[bar.n].sub.ss] [[bar.y].sub.ss] [p.sub.e] ($/kg) (kg/ha) (tons/ha) 0.00 158.8 10.1 0.20 166.7 9.908 0.40 166.8 9.866 0.60 166.7 9.831 0.80 166.4 9.8 1.00 168.2 9.75

Nitrogen N Emissions Emissions Annual Grower Charge [[bar.n].sub NB Profit [p.sub.e] ($/kg) .e,ss] (kg/ha) ($/ha) ($/ha) 0.00 36 168.27 168.27 0.20 23.5 166.59 161.89 0.40 20.9 165.83 157.47 0.60 19.1 164.87 153.41 0.80 17.4 164.09 150.17 1.00 15.2 161.89 146.69 Note: The emission charge [p.sub.e] = 0 represents the baseline case. Annual NB =revenue less costs not including N charge. Grower Profit= Annual NB less N charge.