frontier analysis (SFA) that also allow for statistical noise.<sup>9</sup> Non-parametric methods, on the other hand, use mathematical programming techniques to construct piece-wise a surface (or frontier) over the outputinput space and then calculate the level of inefficiency as the distance to the frontier. The most popular method to do this is data envelopment analysis (DEA) (Latruffe, 2010).<sup>10</sup> In addition to these three groups of methods, there is a fourth group that is not much discussed in the agricultural productivity and efficiency literature but constitutes an important part of the agricultural and irrigation water economics literature. Following Young (2005), this group can be termed deductive, and includes various alternatives such as the residual imputation method, mathematical programming, hydroeconomic models, and computable general equilibrium (CGE) models. In contrast to the parametric methods, which employ inductive logic (in this case econometric procedures) to infer generalizations from individual observations, deductive methods apply constructed models comprising a set of behavioral postulates (such as profit maximization) and assumptions to reason from general premises to particular conclusions. They are based on the specification of a physical production function and assume that firms operate on the frontier. Because of their ability of modeling different scenarios, deductive methods are useful for policy analysis and project planning.

The literature related to agricultural water productivity and efficiency is extensive, yet fragmented among disciplines. This paper reports on the findings of a survey of the agricultural productivity and efficiency literature with regard to the explicit inclusion of water aspects in productivity and efficiency measurements, with the aim of contributing to the discussion on how to assess and possibly improve water productivity in the agricultural sector. The focus is on studies that apply one of the four groups of methods and make an effort to incorporate water quantity aspects, i.e., single-factor productivity measures, TFP indices, frontier models, and deductive methods. Water quality aspects and other agricultural activities besides irrigated crop production are not included. To our knowledge, this is the first attempt to undertake a review and analysis of this type. Given the extensive literature on agricultural productivity and efficiency, the survey does not attempt to be exhaustive. A key finding is that most studies presented in the agricultural productivity and efficiency literature either incorporate field- and basin-level aspects but focus only on a single input (water), or they apply a multi-factor approach but do not tackle the basin-level. It seems that no study on agricultural water productivity has yet presented an approach accounting for both multiple inputs and basin-level issues. However, deductive

> 9 This is modeled through a composed error structure, with a one-sided component measuring inefficiency and a two-sided symmetric term capturing statistical noise.

> 10 DEA and SFA can also be used to obtain the distance measures required for the Malmquist TFP index, which provides a decomposition into technological and efficiency change.

8