Several studies (i.e., Fan; Ahmad and Bravo-Ureta; Wu; Kalirajan, Obwona and Zhao; Kalirajan and Shand; Giannakas, Tran and Tzouvelekas; Giannakas, Schoney and Tzouvelekas) have attempted to explain and identify the sources of output growth in agriculture. By using a parametric production frontier approach, they have attributed output growth to factor accumulation (input growth), technical change, and changes in technical inefficiency. Factor accumulation refers to movements along a path on or beneath the production frontier, technical change is associated with shifts in the production frontier, and changes in technical inefficiency are related to movements towards or away from the production frontier. Implicit in this framework, initiated by Nishimizu and Page, are the assumptions of constant returns to scale and of allocative efficiency. Consequently, changes in total factor productivity (TFP) have been attributed to only two sources: technical change and changes in technical inefficiency. This however restricts unnecessarily the analysis of the potential sources of output growth.
Despite this limitation of previous studies, the parametric production frontier approach has two other shortcomings. First, it is unable to accommodate multi-output technologies, which are quite common in agricultural production. It is well known that inappropriate and unnecessary aggregation of outputs (and inputs) often results in misrepresentation of the structure of production, which may also affect the degree of technical efficiency. Second, even if input prices data are available, the effects of scale economies and of allocative inefficiency on TFP changes cannot be separated from each other (Bauer; Lovell). Indeed, the scale effect can only be identified if input allocative efficiency is presumed, and in this case there is no need for input price data. In contrast, the effect of input allocative inefficiency cannot be identified even if the assumption of constant returns to scale is maintained. Thus, within the parametric production frontier approach, TFP changes may at most be attributed to changes in technical inefficiency, technical change, and the scale effect. However, under the assumption of expected profit maximization, the parametric production frontier approach has the advantage of single-equation estimation and of requiring only input and output quantity data.
On the other hand, cost frontiers can satisfactorily deal with decomposing TFP changes in the presence of multi-output technologies, input allocative inefficiency and non-constant returns to scale (Bauer). As long as panel data are available, this can be achieved by estimating a system of equations consisting of the cost frontier and the derived demand (or cost share) equations, which allows firm-specific and time-varying technical and allocative inefficiencies to be separated from each other (Kumbhakar and Lovell, 2000, pp. 166-75). Clearly, this is a more complicated econometric problem than single-equation estimation, and also requires firm specific data on input prices. Notice however that the effects of technical and allocative inefficiency cannot be identified separately if the cost frontier is estimated with a single-equation procedure.
The objective of this paper is to propose a tractable approach for analyzing the sources of TFP changes (i.e., technical change, changes in technical and allocative inefficiency, and the scale effect) in a multi-output setting, while retaining the single- equation nature of the econometric procedure used to estimate the parameters of the underlying technology. The proposed approach relies on Bauer’s cost function based decomposition of TFP changes and the duality between input distance and cost functions. Specifically, the cost function (i.e., dual representation) is used for the theoretical decomposition of TFP changes whereas all necessary information for quantifying the sources of TFP changes are recovered from an econometrically estimated input distance function (i.e., primal representation). Thus, instead of using a system-wise procedure to estimate a cost frontier, all necessary information for decomposing TFP changes within a cost function approach can be recovered from an input distance function, which also fully describes the production technology.
In this context, the input distance function could be seen as an alternative that overcomes the shortcomings of production frontiers while retains the advantages of a single-equation estimation. By definition, the input distance function can easily accommodate multi-output technologies and thus has an obvious advantage over production frontiers. In addition, estimates of the input-oriented measure of technical inefficiency may be directly obtained from the estimated input distance function (Fare and Lovell). On the other hand, by using the duality between input distance and cost functions (e.g., Fare and Primont), it can be shown that the effects of scale economies and of allocative inefficiency on TFP changes can be separated from each other. The only assumption required to measure allocative efficiency from an input distance function is that one observed price equals the cost-minimizing price at the observed input mix (Fare and Grosskopf). However there is an endogeneity problem with input quantities in the single-equation estimation if the assumption of cost minimization is maintained. This problem may be solved by using an instrumental variable estimation procedure.