Positive Mathematical Programming to model regional or basin-wide implications of producer adoption of practices emerging from plot-based research.

: A method for calibrating models of agricultural production and resource use presented 1 by Howitt [1] for policy analysis is proposed to leverage multidisciplinary agricultural research at 2 the National Center for Alluvial Aquifer Research (NCAAR). An illustrative example for Sunﬂower 3 County, MS is presented to show how plot-level research can be extended to draw systemic region 4 or basin wide implications. A hypothetical improvement in yields for dryland soybean varieties is 5 incorporated to the model and shown to have a positive impact on aquifer outcomes and producer 6 proﬁts. The example illustrates that a change in one practice-crop combination can have system-wide 7 impacts as evidenced by the change in acreages for all crops and practices.


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The National Center for Alluvial Aquifer Research (NCAAR) was created to conduct research 14 aimed at developing novel irrigation and agricultural water management technologies to improve 15 water productivity, decrease irrigation water withdrawal from, and increase the groundwater  practices which must change, and a wide range of socio-economic classes of producers who must 27 all adopt new practices. This paper presents a methodology that can bridge the inter-disciplinary 28 obstacles to translate plot and field level research results to regional or basin-wide potential outcomes 29 that incorporate implicit producer behavior with minimal data requirements: Positive Mathematical  Table Elevation Value High : 344.878 to estimate the elasticities of the derived demand for water [15,16] or policies are adopted (see figure 3).

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Next, we describe the type of disciplinary research that can be fed into a PMP model to draw 118 aquifer and policy implication insights. would be expected at a regional or basin level.

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Economists model producer behavior primarily as pursuing a business objective: maximizing 129 profits or delivering a level of output at the minimum cost. Despite a multitude of other objectives, 130 including cultural ones, the assumption of profit maximization is used because it predicts economic 131 behavior reasonably well, particularly at some level of aggregation [28]. The decision regarding how 132 input use, such as irrigation water, is determined "at the margin", meaning the decision is made based 133 on whether the treatment is expected to return a higher benefit than the cost of applying it. Figure   134 4 illustrates the concept with respect to water use: apply irrigation water until the benefit of the last

Partial budget analysis
Economic data and preliminary analyses

Conserved water
Applied irrigation water (w, mm) Crop yield above rainfed (Y, tons ha −1 ) Example of a nonlinear-plateau yield response to irrigation Full yield Maximum profit Increasing response Plateau agronomic management practices such as row spacing, cover crops, conservation tillage, and skip row 147 irrigation are prime candidates.

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The irrigation technologies that are available to the producers in the LMRB for increasing furrow  the ability of the PMP methodology to model micro-economic behavior capable of reproducing the 181 activity levels at the county level of aggregation is well suited to bridge the interdisciplinary and data 182 availability barriers to basin-wide implications of agricultural experimental outcomes (see Figure 3).

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The PMP-based dynamic simulation process is to: The first step consists in using observed data to obtain the shadow prices on land use acres by 196 solving the following problem for the observed period: s.t. ∑ j x rj ≤ A r = ∑ j a rj ∀r; (2) a rj − ≤ x rj ≤ a rj + ∀r, j; The Lagrangean and first order conditions for the problem for each region at the initial state are: µ j a j + − x j = 0, ∀j; for which the solutions x * j would be very close to the observed levels a j by construction.

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For the second step, a cost function C(w rj , x rj ; α rj , γ rj , δ rj ) to replace c rj in equation (1) is estimated 211 to incorporate additional desired features-i.e., water use, w j . Additionally, we would be interested in 212 calibrating a crop yield function Y j (·) that captures the crop's response to irrigation water application 213 (or other inputs of interest) such that Y j (w rj ) = y rj at the observed levels in the initial period.

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A function that captures crop yield response to irrigation water applied can be specified as proposed by Martin et al.
[31] and calibrated to reflect observed yields and water use [14,18]: where Ymrj is the minimum crop yield before irrigation water is applied; Y f rj is the fully-watered 215 yield; GIR rj is the crop's gross irrigation water requirement to achieve fully watered yield (given 216 observed seasonal weather); and IE rj is the irrigation application efficiency. This function is estimated 217 to reflect the initial observed levels of yield and water use.

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The arguments for the function Y j (w rj ) is the first instance in which results from the plot or 219 field level research can be introduced. Practices that affect minimum yields (for example dryland), where wo rj is the initially observed rate of irrigation water application per acre. At the initial observation levels, the function collapses to C(wo rj , x rj ; α rj , γ rj , δ rj ) = α rj + 0.5γ rj x rj = co rj .
The nonlinear program is now expressed as follows for the calibration problem: and first order conditions: The third step consists in combining the conditions from the two previous steps to match the initial observed levels of the variables of interest. From equations (5) and (12) we obtain: and equation 10 is a second equality which can be used to solve for the two calibrating parameters 226 (α j , γ j ) since the value of the shadow prices (λ, µ j ) where obtained from the original program. The 227 solutions are: The remaining calibrating parameter, δ j , can be found from equation (9) and first order condition (13) by taking the derivative of the yield response function Y j (w j ) specified in equation (8) : The fourth step consists in preparing the cost function to adjust based on updated aquifer status. In this case, the pumping lift affects the pumping costs at time t[18]: where θ et is the price per unit of energy source e; TDH t is total dynamic head at time t; and EF t is 229 energy efficiency of source e. TDH is the sum of pumping lift L t , which depends on aquifer levels 230 at the end of period t − 1; and pumping head which converts the irrigation system pressurization 231 requirement to feet of additional lift.

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The resulting cost function takes the following form: A similar approach can be followed to study the effect of changing costs of other inputs or resources.

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The final step consists in simulating the effects over time by the following aquifer equation of motion: where R is the rate of net natural recharge of the aquifer and A s is the area in the region that overlays 234 the aquifer times the aquifer specific yield. This aquifer formulation can be interpreted as a "localized"  To setup the model, we start with baseline information available from publicly accessible sources.

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County-level parameters are summarized in tables 1 and 2. It fully overlies an acute depression of the MRVAA water table 1 that has drawn concern from producers as well as federal and state agencies The dynamic simulation is run under the two scenarios over 20 years. The "calibrated" scenario 279 is the modified program that includes the ability to update the status of the aquifer which affects 280 pumping lifts over time which in turn affects costs. The "shock" scenario is also modified to update 281 pumping lift but also incorporates an improvement in the level of dryland soybean yields (affecting 282 minimum yield as well). The area is referred to colloquially, and by USGS [32] as the "cone of depression;" a potentially confusing misnomer as a cone of depression occurs at any well actively pumping.  As expected, dryland soybean acreage and profitability increase with the shock. This result is The other important extension of the analysis is with respect to the aggregate results that allow to 297 draw insights at regional or basin-wide scales. research to produce deeper insights on the effects and repercussions experimental plot or field level 313 research can have on regional or basing wide producer welfare and natural resource conditions. The

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We present a clear step-by-step guide to implement the methodology employing straight-forward 319 mathematical optimization techniques and including ways in which the programs can be modified to 320 incorporate unobserved features of interest. The application of this methodology would make highly 321 disciplinary research more relevant across disciplines and to various stakeholders who could more 322 easily assess the implications of the agricultural experimental practices proposed and the eventual 323 technology transfer as producers adopt them.

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A caveat of PMP is that the resulting programs, by design, try to produce allocations that mimic 325 as much as possible those observed in the initial period on which the program is calibrated. But as 326 evidenced by the hypothetical case presented, the directions of change are readily identified.

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Soil Sciences, Mississippi State University, for plot-level research images.

Conflicts of Interest:
The authors declare no conflict of interest.

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The following abbreviations are used in this manuscript: