Theory for Extending Single-Product Production Function Estimation to Multi-Product Settings

We introduce a new methodology for estimating multi-product production functions. It embeds the seminal contributions of Diewert (1973) and Lau (1976) in our extended version of the semi-parametric econometric framework of Olley and Pakes (1996), where we address the simultaneity of inputs and outputs by allowing for the possibility of a possible vector of unobserved ”productivities,” all of which may be freely correlated with inputs and outputs. We show how to use the multi-product production function to recover estimates of firm-product marginal costs using the input and output elasticities by extending Hall’s (1988) single-product result to our multi-product setting using McFadden (1978). We focus on six 6-digit Belgian ”industries” that produce two products, finding all but five of the forty-eight input coefficients are positive and thirty eight are strongly significant. We find outputs are substitutes as the coefficients on ”other good output” is always negative and highly significant. 100% of marginal cost estimates are positive and close to 80% of markups are estimated to be greater than 1. We find very similar results when we move to 4-digit industries, when we use similar multi-product data from France, and when we use the trans-log approximation.