Irrigated agriculture constitutes approximately 70% of global
freshwater consumption. While the nearly 260 million hectares of
irrigated land worldwide currently provide 40% of the global food
supply, future expansion and intensification is likely necessary to meet
a predicted 40% to 45% increase in food demand by the year 2025 (United
Nations Environment Program 1999), implying additional stress on a
scarce natural resource. (1) Irrigated agriculture is also a major
source of groundwater nitrate pollution. Violations in the maximum
allowable levels of nitrates in drinking water are reported in every
European county, while African nitrate loads in some suburban
groundwater wells are six to eight times World Health Organization
acceptable levels. A survey of nearly 200,000 U.S. water sampling
records found that more than 2 million people drank water exceeding
federal nitrate standards, and nearly 52% of the community water wells
and 57% of the domestic wells are considered nitrate contaminated (Nolan
et al. 1998). In California, nitrates are responsible for more well
closures than any other chemical, and 10% to 15% of the water supply
wells violate federal standards (Bianchi and Harter 2002). (2)
In response to the potential health threats from nitrates in
groundwater, a variety of regulations on irrigated agriculture have been
proposed and implemented, including limits on fertilizer usage and
nitrate concentrations in groundwater (Shortle and Abler 2001). Research
addressing the groundwater nitrate problem has often focused on policies
targeting the nitrogen input (e.g., Choi and Feinerman 1995; Nkonya and
Featherstone 2000); understandably so given that annual fertilizer use,
which has been estimated to add 7 billion pounds more nitrogen than is
taken up by the plants on the field, has increased since the 1990s
(National Research Council 1993; USDA 2005). Yet as highlighted in
research by Helfand and House (1995) and Larson, Helfand, and House
(1996) in a static field-level analysis of lettuce production, the
complementarity between applied water rates and nitrate pollution is
such that a second-best approach consisting of a water surcharge is only
marginally less efficient than an emissions charge, albeit substantially
more efficient than a nitrogen input charge. An added benefit from a
water surcharge is a reduction in applied water, an underpriced,
oversubsidized resource subject to a substantial literature of its own
(Caswell, Lichtenberg, and Zilberman 1990). The complementarity between
water, a scarce natural resource, and nitrates, an environmental quality
problem, demonstrates the need to consider water and nutrient management
policies jointly, as stressed in Lee (1998), and the potential for
cross-policy effects, as shown in Weinberg and Kling (1996) for water
markets and drainage policy.
Dynamic analysis of water and nitrogen inputs, crop yield, and
nitrate emissions is quite limited (Segarra et al. 1989; Vickner et al.
1998). Furthermore, an issue of longstanding concern in the agronomic,
soil science, and agricultural engineering literatures is field-level
spatial variability in soil and irrigation system parameters (Nielsen,
Biggar, and Erh 1973; Seginer 1978). While this variability has several
consequences, the main implication is that irrigation water is typically
distributed nonuniformly over a field with consequent impact on water
infiltration, soil/plant processes, crop yields, deep percolation flows,
and nitrogen leaching. Although this topic has seen only modest
attention by agricultural economists, it is invariably critical when
considered. In particular, Berck and Helfand (1990) show that
von-Liebig-type production functions at the plant-level integrate to
smooth nonlinear functions at the field level. Feinerman, Letey, and
Vaux (1983) show theoretically that spatial variability typically
increases profit-maximizing applied water rates, while Letey, Vaux, and
Feinerman (1984) demonstrate that optimum water applications under
spatial variability can differ by factors of two or more compared to
uniform applications and more closely correspond to observed behavior.
Similarly, Dinar, Letey, and Knapp (1985) establish that field-level
spatial variability is critical to accurately analyzing salinity and
drainage problems associated with irrigated agriculture. Finally,
Larson, Helfand, and House (1996) express caution in policy instrument
choice without more research on the variance of nitrate leaching due to
field-level heterogeneity, while Chiao and Gillingham (1989) incorporate
nonuniformity for applied phosphorous in dry land production.
Within the water-nitrogen economics literature, the only study to
incorporate dynamic spatial variability is Vickner et al. (1998) with
spatial variability defined as the fraction of a field under- or
overirrigated relative to a water requirement. They conclude that
ignoring irrigation application variability understates nitrate
abatement policies. Their model of nonuniform irrigation differs from
models typical of the irrigation economics literature, and results in
some 95% of land area uniformly overirrigated and hence represented by a
single parameter. (3) Somewhat contrary to Helfand and House (1995) and
Larson, Helfand, and House (1996), they find that nitrogen control is a
preferable second-best strategy to controlling applied water. Further
analysis of this problem therefore seems crucial to natural resource
usage and the environment in irrigated agriculture. (4)
This article further explores spatial heterogeneity, dynamic
optimization, and nitrate emissions in irrigated agriculture with
attention toward water-nitrogen complementarity and possible
cross-policy effects. A spatial dynamic model of water and nitrogen
management is developed with endogenous water and nitrogen applications
and interseasonal nitrogen carryover. This model extends the irrigation
and nitrogen economics literature by characterizing water infiltration
with a spatial density function over the field. A major task is
estimation of a plant-level model for yield, carryover, and emissions,
where the function must exhibit appropriate global properties to account
for water infiltration above and below mean levels. To this end, data
from an unusually rich field trial are used to estimate a production
function system exhibiting thresholds, plateau maximums, and input
substitution.
Fundamental properties of the dynamic system are investigated,
including decision rules, spatial moments, and evolution of the soil
nitrogen spatial density function. A key finding is rapid convergence to
a steady-state under a wide variety of initial conditions, which is
significant for regional policy analysis as it simplifies needed
computations and data. Specification tests are conducted for spatial
variability and dynamic optimization; consistent with previous
literature, spatial variability is fundamental for water scarcity and
environmental quality degradation in irrigated agriculture. The effects
of a range of water and nitrogen emission prices are evaluated also.
There is a significant policy-relevant response from water and nitrogen
management alone, even while crop and irrigation system are fixed. As in
Johnson, Adams, and Perry (1991), the implication is that significant
resource conservation and environmental quality improvement is possible
at relatively low cost to agricultural productivity, at least starting
from current conditions. The results also exhibit large cross-policy
effects complementing Larson, Helfand, and House (1996) and Weinberg and
Kling (1996).
Bioeconomics of Field-Scale Crop Growth and Management
Spatial dynamics of field-level water and nitrogen management is
analyzed. Water is distributed nonuniformly over the field in response
to soil heterogeneity and/or nonuniform irrigation systems, implying
spatially variable water uptake and nitrogen uptake and emissions. (5)
Interseasonal carryover dynamics for soil nitrogen are also considered.
With variability in the various driving factors, soil nitrogen also
exhibits heterogeneity over time even with initial soil nitrogen
uniformity. Field-scale crop yield and emissions in each period are an
integration over the field; hence, spatially variable water infiltration
directly impacts current crop yield and nitrogen emissions, and
indirectly affects future levels by inducing soil nitrogen variability.
The importance of spatial variability and dynamics in water and nitrogen
management is analyzed along with input pricing policies for water
conservation and water quality at the field level.
Letting r denote the discount rate and T the planning horizon, the
present value of net benefits to land and management ($/ha) is
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where t is time [years], [[bar.y].sub.t] = field-scale crop yield
[Mg/ha], [[bar.w].sub.t] = field-average applied water depth [cm],
[[bar.n].sub.at] = applied nitrogen [kg/ha], and [[bar.n].sub.et] =
nitrogen emissions/leaching [kg/ha]. Parameters are [p.sub.y],
[p.sub.w], and [p.sub.n] as the prices of crop [$/Mg], water [$/ha/cm],
and nitrogen [$/kg], respectively; [kappa] is nonwater and nonnitrogen
production costs associated with the cropping system [$/ha], and
[p.sub.e] is nitrogen leaching cost [$/kg].
Spatial variability is a dynamic extension of the static model
proposed by Seginer (1978) and used subsequently in Feinerman, Letey,
and Vaux (1983), Dinar, Letey, and Knapp (1985), and Berck and Helfand
(1990). The key concept is a water infiltration coefficient giving the
fraction of field-average water depth infiltrating at a point in the
field. At a particular point in the field, the amount of water
infiltrating into the root zone at time t is [w.sub.t]([beta]) =
[beta][[bar.w].sub.t] where [beta] [member of] [0, [infinity]] is the
water infiltration coefficient. [beta] is distributed over the field
according to a spatial density function, f([beta]), with mean E[[beta]]
= 1, and standard deviation SD[[beta]] that depends on the type of
irrigation system.
Field-level relationships for yield and nitrogen emissions are:
(2) [[bar.t].sub.t] = [[integral].sup.[infinity].sub.0]
[y.sub.t]([beta])f([beta])d[beta] [[bar.n].sub.et] =
[[integral].sup.[infinity].sub.0] [n.sub.et]([beta])f([beta])d[beta]
where [y.sub.t]([beta]) and [n.sub.et]([beta]) are plant-level
yield [Mg/ha] and nitrogen emissions [kg/ha], respectively. Thus
field-level crop yield and nitrogen emissions are plant-level yield and
emissions integrated over the field according to the spatial density
function for water infiltration. Plant-level production functions for
yield and nitrogen emissions are [y.sub.t]([beta]) =
[g.sub.y][n.sub.t]([beta]), [w.sub.t]([beta]), [n.sub.at]([beta])] and
[n.sub.et]([beta]) = [g.sub.e][[n.sub.t][beta]), [w.sub.t]([beta]),
[n.sub.at]([beta])], respectively, where [n.sub.t] is inorganic soil
nitrogen [kg/ha] at the beginning of period t, and [n.sub.at] is applied
nitrogen. At the plant-level, crop yield and nitrogen emissions,
specified as leaching below the rootzone, depend on initial soil
nitrogen, water infiltration, and nitrogen applications at points in the
field characterized by [beta].
Soil nitrogen dynamics or carryover dynamics (Segarra et al. 1989)
for a given [beta] are
(3) [n.sub.t+1]([beta]) = [g.sub.n][[n.sub.t]([beta]),
[w.sub.t]([beta]), [n.sub.at]([beta])]
indicating dependence on the same variables as plant-level crop
yield and nitrogen emissions. Initial soil nitrogen in period 1 is
assumed constant across the field [[n.sub.1]([beta]) = [[bar.n].sub.1]],
and nitrogen is applied uniformly across the field [[n.sub.at]([beta]) =
[[bar.n].sub.at]], the latter assumption following from the use of
mechanical/chemical fertilizer applications consistent with irrigated
agriculture. The model can be modified to make plant-level fertilizer
applications proportional to infiltrated irrigation water; however, this
is not pursued here. For computational tractability in the dynamic
optimization model, the spatial density support is discretized into a
series of intervals, each with a specified [beta] value and representing
a fraction of the field as computed from the spatial density function. A
useful interpretation is that the field is divided into a finite number
of plots each with a specified [beta] value and area. (6)
Control variables are field-level applied water [[bar.w].sub.t] and
nitrogen [[bar.n].sub.at], and state variables are nitrogen carryover
for each of the discrete grid intervals for the [beta] infiltration
coefficients. The dynamic optimization problem is solved using the GAMS
CONOPT nonlinear optimization procedure. To eliminate endpoint effects,
the optimization routine is implemented as a running horizon problem in
which a sequence of finite-horizon optimization problems are solved with
a thirty-year time horizon, each starting from the states resulting from
the first period of the previous solution and retaining only the first
period results from each for the final solution.
Economic Data and Crop-Water-Nitrogen Production Function
The empirical application is corn production in Yolo County,
California with a traditional (furrow one-half mile) irrigation system.
Maximum corn yield is 12.02 Mg/ha, with a price of $102.02
[[Mg.sup.-1]]. Production costs include costs such as seed, land
preparation, and machinery but do not include those associated with
water, nitrogen fertilizer, land and management, and cash overhead (UCCE
2004). Irrigation system data are from University of California
Committee of Consultants (UCCC 1988). Combined, amortized nonwater
production costs are $673 [ha.sup.-1], baseline nitrogen fertilizer
costs are $0.59 [kg.sup.-1], and baseline water costs are $0.64 [[ha
cm].sup.-1]. We assume a discount rate of 5% with all economic data
inflation-adjusted to 2003 dollars.
The infiltration coefficients [beta] are distributed lognormally
over the field with E[[beta]] = 1 for mass balance. The baseline results
assume a Christensen Uniformity Coefficient (CUC) of 0.77, where CUC is
a widely used measure of nonuniformity in the irrigation engineering
literature, and calculated as 1 - [[integral].sup.[infinity].sub.0]
[absolute value of [beta] - 1] f ([beta])d[beta]. SD[[beta]] was
estimated so that the CUC = 0.77 under the lognormal [beta]
distribution. This distribution is discretized into 11 possible [beta]
values, each with an associated fraction of the field computed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where f is the
lognormal density and [[DELTA].sub.i], i = 1,11, is a partition of [0,
[infinity]] containing the discrete [beta] values. This model can be
interpreted as 11 subareas of the field, each characterized by a [beta]
value, constituting a specified fraction of the field, and with an
associated soil nitrogen state variable.
A classic work on water-nitrogen production functions is Hexum and
Heady (1978). Although they investigate several functional forms, they
settle on polynomials (including fractional powers) as a useful
functional form. Ackello-Ogutu, Paris, and Williams (1985), among
others, point out that polynomials generally do not fit qualitative
agronomic theory and evidence: they have a point maximum instead of a
plateau maximum, allow more substitution than is warranted by the data,
and imply excessive input usage. Moreover, von-Liebig functions
demonstrate superior data fit relative to polynomials and other
traditional smooth production functions (Ackello-Ogutu, Paris, and
Williams 1985; Grimm, Paris, and Williams 1987; Paris 1992). However, a
recent sophisticated statistical analysis by Berck, Geoghegan, and Stohs
(2000) rejects both the von-Liebig formulation as well as the
non-substitution hypothesis. Taken together, these results leave open
the appropriate form for plant-level production functions.
[FIGURE 1 OMITTED]
Additional concerns arise at the field-level with spatial
variability. As outlined in Lanzer and Paris (1981; figure 1), a general
conceptual model of yield production functions exhibits convex-concave
behavior initially, followed by a yield plateau and then possibly a
yield decline. In the uniform case, only the concave portion is
economically relevant, hence functional forms constituting local
approximations (e.g., Taylor series approximations via polynomials) may
be reasonable as the optimization model can appropriately bound the
inputs. In the spatial case, though, some parts of the field likely
receive input levels in the convex (increasing returns to scale)
portion, while other parts receive excess input levels leading to yield
declines. Consequently, functions with desirable global properties and
data fit are necessary, raising additional issues to those debated in
the literature. Polynomials with any reasonable order and von-Liebig
functions are unlikely to perform well globally even if they are
reasonable locally.
To overcome some of these difficulties, we develop a production
function system specified by several component functions representing
the major flows and processes in the plant-water-soil system. One reason
for the system approach rather than the approach used in much of the
literature (e.g., Johnson, Adams, and Perry 1991; Vickner et al. 1998)
is that a system approach can capture yield-depressing effects
associated with excess water infiltration in a logical fashion while
still allowing individual component functions to be estimated with
classical properties. We also utilize functional forms that exhibit
convex-concave behavior and plateau maximums. These functional forms
effectively place upper and lower bounds on the levels for individual
variables. In combination with multiplicative functions such as
Mitscherlich-Baule (Paris 1992), this system allows for input
substitution consistent with Berck, Geoghegan, and Stohs (2000), yet
subject to limits consistent with the classic findings of Paris and
others.
The plant-level production system was estimated for corn using an
unusually rich data set from Tanji et al. (1979) (see also Pang, Letey,
and Wu 1997a, b). The experimental data consist of two years of corn
field trials at a University of California-Davis site from October 1974
through September 1976. The trials measure the effects of nitrogen and
water applications rates on yields, nitrogen uptake, inorganic soil
nitrogen levels, nitrate emissions, and organic nitrogen mineralization.
The experiment provides data beyond that typically used in the
agricultural production economics literature (e.g., Hexum and Heady 1978
as used in Berck, Geoghegan, and Stohs 2000), and is key to the analysis
as it allows estimation and testing of the system model without
resorting to hidden variables and speculative functional forms. It
should be emphasized that while these field experiments were performed
in the mid 1970s, recent articles in the soil science literature still
calibrate to this data (Pang, Letey, and Wu 1997a,b).
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 converge