This illustration of the deductive method through the use of residual imputation (Young, 2005) focuses on farm level production when one new input, irrigation water, is added. A more general understanding of the role of irrigation water in agricultural production can be obtained by incorporating a broader range of specific technologies (e.g. fertilizer use and irrigation technology). A crop budget typically is prepared for every crop and technology combination, with each serving as the basis for an activity in a mathematical programming (optimization) problem.<sup>23</sup> When solved for a variety of water supply levels, model solutions can provide cropping mix, production, and irrigation technology use for each water supply level. Alternatively, different water prices can be used to generate cropping and production patterns.<sup>24</sup> Early examples of a linear programming models employing this approach include Hartman and Whittlesey (1961), and Moore and Hedges (1963) who model farms of various sizes with 54 production activities representing combinations of alternative crops, irrigation treatments, and soil grades. Yaron (1967) introduces activities representing flexible irrigation practices. Heady _et al_ (1973) demonstrated the use of linear programming to address such fundamental issues as the use of water resources in meeting future economic development needs, and the ability to substitute between water, land, and other inputs in agricultural production. The study encompassed over 200 land regions, and 51 water supply regions in the United States. Heady _et al_ . demonstrated the possibility of modeling the impact of factors including water prices, population, farming technology, exports, and agricultural policies on regional and national agricultural production.

These early linear programming studies illustrate the use of deductive methods based on the residual imputation of the difference between gross crop revenue and opportunity costs of all other inputs. Typically, constant output prices are used, raising questions of applicability particularly during periods of acute water stress or potential output disruptions. Allowing crop prices to depend on output can be accommodated, and is demonstrated in the quadratic programming model used by Howitt _et al_ . (1980) in estimating agricultural production and short-run irrigation water use in 14 regions of California.<sup>25</sup>

> 23 Each crop budget is implicitly assumed to be at the production frontier and thus technically efficient.  Moreover, the fixed proportion of inputs in the crop budget is presumed to represent allocatively efficient input usage.

> 24 With regard to residual imputation, it needs to be noted that it relies by its nature on finding meaning in the difference between project benefits (usually revenues) and costs (typically dominated by crop production costs).  Because benefits and costs are uncertain, and the difference between benefits and costs is typically small compared to their level, estimates of the difference (the residual) have a much higher level of uncertainty.  It is easy to neglect this underlying factor when interpreting the very detailed and extensive output from typical mathematical programming models.

> 25 A particular challenge across mathematical programming approaches is the problem of calibration.  Naïve mathematical programming models rarely result in solutions which reflect observed crop distributions or technology choices under representative water supplies and output prices.  This is typically ascribed to a myriad of producer constraints and hidden costs which are not evident to the modeler.  This problem is often handled with a series of ad hoc constraints on crop acreages. Another response, proposed and implemented by Howitt (1995), is to recognize the existence of unobserved costs and

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