Spatial dynamics of water and nitrogen management in irrigated agriculture.

These results demonstrate substantial nitrogen emission reduction with minimal impact on agricultural productivity or social net benefits. For an emissions price of $1, emissions are reduced by 58%, while yield and social net benefits decline by 3% and 13%, respectively. While additional control will eventually become increasingly expensive, these results are broadly consistent with the findings for other pollutants in which substantial reductions from uncontrolled levels can be achieved at relatively low costs (Tietenberg 2006). These results again demonstrate rather large and possibly surprising cross-policy effects, namely that nitrate emission pricing engenders a large drop in irrigation water, consistent with the earlier hypothesis that field-scale spatial variability is a major determinant of pollutant loadings.

The emissions charge induces efficient management for the associated level of N-emissions, and if the charge equals the marginal damages, then full social efficiency is achieved. Emission reductions also can be induced by input-side instruments. As an example, Tables 3 and 4 show that a water price of $.83/ha-cm leads to almost identical results as the nitrogen emissions charge of $.20/kg. In general, efficient input-side policy requires charges on all pollution generating inputs (Griffin and Bromley 1982), implying a surcharge for both water and nitrogen applications. The efficient input charges can be computed using the shadow values associated with the equations of motion, but this is not pursued here due to space limitations. A closely related topic is equity effects on grower profits, which generally depend on the selected policy instruments. For instance, in the example given the emission charge is somewhat more favorable to the grower than the water charge. Again, we do not pursue this topic in detail as a full analysis needs to account for rebates or tiered pricing that influence equity even for a given choice of instruments, as well as entry/exit considerations.

Conclusions

The article develops a spatial dynamic optimization model of field-scale water and nitrogen management. The model incorporates spatial variability consistent with the agronomic and irrigation engineering literature, includes nitrogen carryover dynamics, and estimates a plant-level production function system exhibiting substitution consistent with Berck, Geoghegan, and Stohs (2000) while subject to limits as implied by Paris (1992). Qualitative dynamics exhibited by the model indicate a relatively rapid convergence to the optimal steady-state independent of initial conditions. This finding has potentially significant implications for quantitative policy analysis. If dynamics and optimization are important and transition time-scales long, then accurate regional policy analysis requires specifying initial conditions for all fields and solving a very large optimization problem, a heroic task from a data and computational perspective. The results here suggest that the essentials of the problem are well-captured by the dynamically optimal steady-state, a computationally and informationally much more tractable problem. (11)

Spatial variability is fundamental to resource scarcity and environmental quality in irrigated agriculture. While spatial variability does not imply large changes in nitrogen applications, it does have very large effects on water applications and nitrogen emissions such that overlooking spatial variability leads to erroneous results. The results demonstrate that input demand, pollutant loadings, and grower response are much larger than would be predicted from a uniform model. The extent to which simplifications used in the agricultural production economics literature are an acceptable approximations, and over what range, is an open question requiring further investigation. The model developed here can be used as testbed for this purpose.

Dynamic optimization versus static (period-by-period) optimization also was tested. Static optimization implies lower nitrogen application rates and higher water application rates than PV-optimality. Higher water applications leach additional nitrogen out of the soil leaving less carryover for future periods and, consequently, less nitrogen uptake and lower yields. While the quantitative loss from static optimization is not large in percentage terms, it can still translate into significant farm-level losses. Emission effects, meanwhile, are ambiguous as the static optimization procedure reduces applied nitrogen but increases applied water.

Water conservation and nitrate pollution control policies are evaluated as well. While estimated water demand is inelastic, water price increases well within estimated values consequent to a variety of possible policy reforms can result in policy-relevant quantity reductions. For example, a 20% water price increase from the base level here still leaves the price considerably less than the true shadow value facing California agriculture as calculated in other studies; this price increase, though, induces water reductions, which if scaled to all of California, would imply almost a two-thirds increase for urban uses. In the quantity dimension and given the crop and water prices considered, establishing a needed 10% to 20% agricultural water transfer rate to support urban growth and environmental restoration goals in California over the next several decades can be achieved with an annual loss of $15/ha or less in agricultural net benefits. Of course, equity consequences for growers would depend on specific policy mechanisms and instruments.

Similar findings hold for nitrate pollution control. The results suggest that efficient emission reductions are achieved primarily through reduced applied water relative to nitrogen fertilizer, a direct result of spatial variability. As with water, and starting from baseline conditions, significant reductions in nitrate emissions are obtained with relatively modest consequences for agricultural production. In particular, a $1/kg emission charge that induces a 55% emissions reduction incurs only a 6% loss in agricultural net benefits. This result holds starting from no regulation and for the crop and water prices considered here. Eventually, though, nitrate regulation becomes increasingly expensive as standards are tightened. Note that the water and nitrate results follow from crop management solely; irrigation systems and crop choice as stressed in previous work are not considered. Consideration of these strategies strengthens the results as additional compliance methods further reduce the already low costs found here.

An unanticipated finding of this research is a very strong cross-policy effect: water management implies strong reductions in nitrogen emissions, while emissions management implies large reductions in applied water. These results follow from the observation that nitrogen is transported through the rootzone via water flows, and the latter are larger than might be anticipated to maintain adequate moisture levels in all portions of the field. This complements Weinberg and Kling (1996) who find strong cross-policy effects for regional water and drainage management, and Larson, Helfand, and House (1996) who illustrate, both theoretically and empirically, the complementary relationship of water and nitrate pollution. Interestingly, the results differ from Vickner et al. (1998) who find that nitrogen management is more efficient than water management implying lower cross-over effects on water conservation.

The findings in this article suggest that nitrogen management in irrigated agriculture is as much water management as it is nitrogen input policy. In particular, the role of field-scale water infiltration variability appears crucial; it does not seem possible to either understand existing levels of resource demand/environmental loadings, or to accurately model and predict growers' policy response, without consideration of this phenomena. It can be readily hypothesized that this is likely the case for other nutrients and agri-chemicals in irrigated agriculture as well.

Appendix

Plant-Level Production Function System

The plant-level production function system consists of six component functions representing the major soil/plant processes and fluxes. After estimation, this system specifies composite functions giving crop yield, nitrogen emissions, and carryover dynamics as functions of initial (inorganic) soil nitrogen and applied water and nitrogen at a point within the field as characterized by [beta]. Integration over [beta] then determines field-scale production relations.

Corn yield [y.sub.t] with maximum potential yield [y.sup.max] [Mg/ha] is

(A.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, [w.sub.t]([beta]) is infiltrated water [cm], [n.sub.ut]([beta]) is plant nitrogen uptake [kg/ha], and [w.sub.50] and [nu.sub.75] are scaling coefficients for infiltrated water and nitrogen uptake implying 50% and 75% maximum crop yields, respectively (these allow parsimonious function estimation and representation). The parameters to be estimated are [y.sup.max], [w.sub.50], [[PHI].sub.yw], [nu.sub.75], and [[PHI].sub.yu]. In equation (A.1), crop yield is convex-concave in the individual inputs with a plateau maximum at [y.sup.max]; the multiplicative form allows a degree of input substitution.

Plant nitrogen uptake [n.sub.ut]([beta]) with maximum potential plant uptake [n.sup.max.sub.u]is

(A.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]