The Treatment of Non-Essential Inputs in a Cobb-Douglas Technology : An Application to Mexican Rural Household-Level Data

The standard approach for fitting a Cobb-Douglas production function to micro data with zero values is to replace those values with "sufficiently small" numbers to facilitate the logarithmic transformation. In general, the estimates obtained are extremely sensitive to the transformation chosen, generating doubts about the use of a specification that assumes that all inputs are essential (as the Cobb-Douglas does) when that is not the case. The author presents an alternative method that allows one to estimate the degree of essentiality of the production inputs while retaining the Cobb-Douglas specification. By using the properties of translatable homothetic functions, he estimates by how much the origin of the input set should be translated to allow the Cobb-Douglas functional form to capture the fact that the data have a positive output even when some of the inputs are not used. To highlight the empirical importance of the approach, he applies it to Mexican farm-level production data that he gathered. Many households did not use family or hired labor in farm production, or had different capital composition (that is, zero value for non-land farm assets). The estimations provide a clear measurement of the degree of essentiality of potentially non-essential inputs. They also indicate the size of the error introduced by the common "trick" of adding a "small" value to zero input values