Estimating Disaggregate Production Functions: An Application to Northern Mexico

This paper demonstrates a robust method for achieving disaggregation in the estimation of flexible-form farm-level multi-input production functions using minimally-specified data sets. Since our ultimate goal is to address important questions related to the distributional effects of policy changes, we place emphasis on the ability of the model to reproduce the characteristics of the existing production system and to predict the outcomes of these changes at a high level of disaggregation. Achieving this requires the use of farm-level models that are estimated across a wide spectrum of sizes and types, which is often difficult to do with traditional econometric methods, due to limitations of data. The approach to estimating flexible-form production functions used in this paper overcomes these limitations, and also avoids the problems that frequently hinder the application of budget-based representative farm models to these type of analyses namely, that of poor calibration to observed behavior. In our estimation procedure, we use a two-stage approach that first generates a set of observation-specific shadow values for incompletely priced inputs, such as irrigation water or family labor, which are used in the second stage, along with the nominal input prices, to produce estimates of crop-specific production functions using Generalized Maximum Entropy (GME) methods. These functions are able to capture the individual heterogeneity of the local production environment, while still allowing the production function to replicate the input usage and outputs produced in the sample data. Since we are able to generate demand, supply, and substitution elasticities, a wide range of policy responses can be modeled. Our paper demonstrates this methodology through an empirical application to Mexico, drawing from a small set of cross-section data collected in the northern Rio Bravo regions. The estimates show that there is considerable heterogeneity in the behavioral response of farmer households of different sizes, both in terms of the returns to scale, as well as in the elasticities of substitution and derived demands for water. Compared to the aggregate-level estimation, we obtain much more accurate and informative policy response behavior, when shocks are imposed on the model.