Proxy Variable Estimation of Multi-Product Production Functions

Most empirical studies seeking to estimate firm-level production technologies via proxy variable estimators focus on single-output production functions despite that, in practice, the majority of firms produce multiple outputs. Arguably, the primary reason for vast popularity of a single-product model is the systematic lack of data on input usage by products. While one can always employ such a model by defining the firm's "output" as an aggregate of its multiple outputs in the form of deflated total revenue as customarily done in the literature, such a formulation rarely provides an adequate representation of the multi-product firm's technology: (i) it does not allow identification of technological trade-offs between outputs along the firm's production possibilities frontier, and (ii) it unrealistically assumes perfect substitutability of outputs effectively embedded in a linearly additive revenue-based output aggregator which, if misspecified, would produce biased estimates of the firm's production technology and productivity. This paper provides a new methodology for the proxy variable identification of multi-product production technology which not only does not require the access to information beyond that available in most datasets but also successfully identifies the cross-output technological relationship within the firm by not a priori aggregating outputs. We operationalize the model nonparametrically using B-spline sieves and showcase it using data on Norwegian dairy-and-beef farms