main groups: single-factor productivity measures, total factor productivity (TFP) indices, and frontier models.

Single-factor (also called partial) productivity ratios or indices, relating output to only one input, are easy to calculate.<sup>5</sup> Yet they have a number of limitations, including that they are average, not marginal products; and because they are affected by the intensity of use of the excluded inputs, they give an incomplete picture of the underlying drivers of productivity change—especially when used in isolation. These issues were pointed out early on in the production economics literature.<sup>6</sup> Nevertheless, single-factor productivity ratios in the form of “crop per drop” continue to be widely used in the irrigation literature.

In contrast, the economics literature on agricultural productivity and efficiency mainly employs TFP indices and frontier models—not least because it is concerned with the inclusion not just of water, but also the other inputs (see Coelli _et al_ ., 2005, for an excellent introduction to this literature). TFP indices, the second group of methods, are particularly concerned with the incorporation of all inputs of the production process.<sup>7</sup> TFP indices compare a single output or an aggregate output index to an aggregate input index. Different ways of aggregation lead to different TFP indices, with Laspeyre, Paasche, Fisher, Törnqvist and Eltetö-Köves-Szulc indices being commonly used (Latruffe, 2010). These indices require both quantity and price information for the outputs and inputs included, and implicitly assume that all firms are efficient (which is unlikely to be true); hence, TFP changes over time are attributed to technological change.<sup>8</sup> The third group of methods is frontier models, which measure efficiency as a potential input reduction or potential output expansion, relative to a reference “best practice” or efficient frontier, constructed from observed inputs and their output realizations. Techniques for defining the frontier can be classified into parametric and non-parametric methods. Parametric methods rely on specifying a production frontier and estimating its parameters econometrically. There are two types: deterministic frontier analysis that assumes that any deviation from the frontier is due to inefficiency, and stochastic

> 5  The early partial productivity studies in the agricultural sector focused on output per unit of land and labor, often in intercountry comparisons (see, for example, Hayami and Inagi, 1969).

> 6 For example, Heady and Dillon (1961) in an inter-country comparison of production function estimates from farm samples emphasized that “the resultant average [computed as the mean output divided by the mean input of a resource] includes the product returns of all inputs, not simply the product return attributable to the single resource” (p. 590). Furthermore, a singlefactor productivity measure does not account for the possibility of either factor substitution or output substitution (Latruffe, 2010).

> 7 Because of the difficulty of capturing all of the inputs and outputs that interact in the production process, many studies refer to them as multifactor productivity (MFP), rather than TFP, indices.

> 8  Besides using index numbers, TFP changes can also be calculated from econometric estimates of a production function or frontier.

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