A Production Function Model for Aggregate Time‐Series Data

Abstract

A problem in using survey data on yield and input levels in estimating a production function for commercial farmers is that often only averages of individual farm or field input levels are available. Since the average data points should lie below the true production function (as long as it is strictly concave from below), traditional regression models applied directly to the average data will not yield an estimate of the true function unless it is linear (Taylor and Swanson). In this note, a model specification is presented which makes estimation of true production functions from aggregate farmer experiences more likely than under traditional specifications. To estimate this model, time-series data on the variation of input levels as well as the average input levels are needed. While the variation of input levels are often not available for past surveys, such data could be easily made available in future surveys. For the model specification presented in this note to provide meaningful estimates, the necessary conditions are: (a) that the area under study be homogeneous in that it can be represented by a single production function, (b) that there has been a significant change in the distribution of the input levels over the observation period, and (c) that all inputs not included in the analysis are kept constant. The model specification is based on the following identity. The actual (observed) average yield is a summation over all input levels of the product of the unobserved yield at a specific input level and the fraction of farmers applying that level. This identity is substituted into an hypothesized production function which relates the unobserved yields to specific input levels in order to derive an equation which can be estimated using ordinary least squares. The parameters characterizing this regression equation are the parameters of the production function. For simplicity only one input is considered in the derivation.