Spatial dynamics of water and nitrogen management in irrigated agriculture.

Details of the estimated system are in the Appendix. The estimated functions fit the data extremely well ([R.sup.2] [greater than or equal to] 0.78) and have appropriate global properties. The composite plant-level production functions for yield, emissions, and carryover as functions of soil nitrogen, applied water, and applied nitrogen are constructed from this system and illustrated in figure 1. While generally consistent with prior irrigation economics research, the results can differ with respect to nitrogen and water interactions. In figure la, for example, excessive water application rates at low soil nitrogen levels decrease yields as the additional water leaches nitrogen out of the soil. As more nitrogen is leached out of the soil with excessive water application (figure lb), less is then available as carryover into the next period (figure lc). Knapp and Schwabe (2007) contain additional discussion and graphs of the estimated production function system.

[FIGURE 2 OMITTED]

Dynamics of the Spatially Variable Field

With spatially variable water infiltration and nitrogen carryover dynamics, the field constitutes a relatively complex dynamic system. In this section, computational experiments are used to characterize the dynamic system with the base water price and a zero nitrogen emissions price. Figure 2 partially characterizes the optimal decision rule giving applied water and nitrogen as a function of soil nitrogen. In the figure, soil nitrogen is constant across the field; thus this is only a partial characterization as soil nitrogen in each of the grid cells can take on a range of nonuniform values in principle. As illustrated, water applications are reasonably constant across the range of values; applied nitrogen generally declines linearly to a threshold, after which it is zero.

Time series of the spatial means of the state and control variables were computed starting from (uniform) initial soil nitrogen [n.sub.i0] of 50 kg/ha and 350 kg/ha. The time paths converge to a steady-state independent of the initial conditions, and convergence is rapid (<10 years) even with initial conditions relatively far from the steady-state. This property was found in all of the empirical specifications reported in the article, and similar rapid convergence has been reported in the salinity economics literature (Dinar and Knapp 1986; Knapp 1992; Letey and Knapp 1995). While growers in an actual operating environment need to evaluate and respond to initial conditions in the field, the significant implication for modeling and policy analysis is that one can reasonably focus on the optimal steady-state, thereby lessening the data and computational burden at farm and regional spatial scales. This is important as it would be virtually impossible to estimate and solve full dynamic systems for all fields at larger spatial scales.

[FIGURE 3 OMITTED]

Temporal evolution of the spatial density function for soil nitrogen provides a more detailed view. A piece-wise linear cumulative distribution function (CDF) for soil nitrogen in each year was computed from model output on soil nitrogen state variables and their associated fractional areas. This CDF shows the fraction of the field having soil nitrogen levels less than or equal to a specified value; spatial density functions are estimated from the CDF by finite-differences over an appropriate grid. (7) Figure 3 depicts the spatial density function for soil nitrogen for several years. Consistent with the results for first moments, this spatial density function is relatively invariant after approximately eight years indicating a steady-state for the entire system. Note that the observed rapid convergence to a limiting density function does not imply that the problem can be modeled as a static system; the equations of motion are necessary to compute the steady-state while formal dynamic optimization procedures are required to compute the dynamically optimal steady-state. (8)

Current corn prices are substantially higher than the price assumed in our analysis, a price change largely due to recent increases in the demand for corn to produce ethanol. Knapp and Schwabe (2007) provide a sensitively analysis over a range of corn prices.

Holding other prices fixed, a 50% increase in corn price to the current market rate of approximately $153/Mg results in over a 60% increase in nitrogen emissions. Such a large potential increase in nitrogen emissions can exacerbate greatly an already existing nitrogen pollution problem. However, other factors, such as the price of nitrogen, which is surging lately as well due to fuel price increases, will likely regulate some of the behavioral response to the ethanol-generated price changes.

A key parameter in any dynamic analysis is the discount rate. Risk-free interest rates and rates of return on agricultural and general assets typically are fairly low (4% to 5%) in developed countries. Sustainability concerns, however, have spawned a large literature on discounting as an appropriate criterion in view of intergenerational equity over long horizons. At the other end of the spectrum, capital scarcity in developing countries can imply larger discount rates. Table 1 explores alternate discount rates and optimal management. Higher discount rates tend to increase water applications and decrease nitrogen applications slightly resulting in reduced nitrogen supply and crop yields. These results are consistent with the observation that increased discount rates imply reduced concern for the future. The results also suggest declining nitrogen emissions with increased interest rates. In general, though, the quantitative changes are modest and indicate relatively little sensitivity to discount rates.

Spatial Variability and Dynamic Optimization: Specification Tests

This section evaluates the significance of spatial variability and dynamic optimization. That is, do analyses require consideration of these factors, or can they safely be neglected? As before, base prices, a 5% interest rate, and a zero nitrogen emission price ([p.sub.e] = 0) are considered. Table 2 contrasts steady-state values for water and nitrogen management with and without spatial variability under alternate assumptions on optimal behavior, either present value (PV) or period-by-period (PP) optimization. PP optimization selects input levels in each time period to maximize profits in that period conditioned on the states entering that period; states for the next period are calculated from the equations of motion and selected input levels. In contrast to PV optimization, the impacts of current decisions on future periods are ignored under PP optimization.

Introducing spatial variability can have very significant impacts. As shown in table 2, applied nitrogen rates ([n.sub.a]) increase by a modest 6% under PV-Spatial relative to PV-Uniform. However, consistent with previous literature, optimal steady-state applied water rates increase substantially by nearly 40%. Feinerman, Letey, and Vaux (1983) demonstrate that the latter effect arises because, after a threshold level, spatial variability increases the marginal product of water resulting in increased optimal applied water. Crop yields are not affected by uniformity in the water-nitrogen empirical example here; however, profits are somewhat lower under nonuniform irrigation reflecting the increased costs of greater inputs.

The most striking effect is on nitrate emission rates. As table 2 demonstrates, a level of spatial variability associated with a traditional irrigation system results in optimal steady-state nitrate emissions six times greater than that predicted in the uniform scenario. This is primarily due to the applied water effect noted above, that is, the desire to maintain adequate moisture levels in all parts of the field results in over-irrigating some parts of the field; consequently, higher levels of deep percolation result with subsequent increases in nitrate emissions. Such results also explain the higher applied nitrogen levels as compensation for the reduced soil nitrogen levels.

The implications of these observations for agricultural natural resource and environmental policy are compelling. As noted above, computed water application rates are much closer to observed values than those from uniform calculations (e.g., those reported in Hexum and Heady (1978)). More novel here, the results also raise the hypothesis that observed nutrient loadings to environmental media may be much more due to field-level spatial variability than to lack of emission prices. Certainly it would not be possible to understand current nitrate loadings from irrigated agriculture--and by extension grower policy response--without accounting for spatial variability. Similar results can be expected for other agricultural chemicals in irrigated agriculture as well.