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 the semi-parametric econometric framework of Olley and Pakes (1996). We address the simultaneity of inputs and outputs while allowing for the possibility of one unobserved productivity term for each product, all of which may be freely correlated with inputs and outputs. We 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 coefficient 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 We develop two tests for the appropriateness of approximating multi-product production with a collection of single-product production functions. ∗We thank several very helpful referees, Andrew Bernard, Jan De Loecker, Swati Dhingra, Penni Goldberg, John Haltiwanger, Marc Muendler, Ariel Pakes, Peter Schott, Chad Syverson, Dan Trefler, and seminar participants at the 2014 NBB Conference, Riksbank, the 11th International Conference, Monash University, Mannheim, EITI2015, IIOC2015, ATW2015, Rice University, Texas A&M, DIEW2015, NOITS2015, Uppsala, the final COMPNET conference, ESWC2015, LSE, Hong Kong University, UCD, Lund, Toronto, UCLouvain, Chicago Fed, RIDGE, AEA2017, Bundesbank, SED2017, Oxford and the NBER IO Winter meeting 2020. Petrin, Smeets and Warzynski thank the National Bank of Belgium for its financial support. The authors are also extremely grateful for the data support provided by the NBB Statistical department. All presented results respect the all confidentiality restrictions associated with the data sources. The views expressed are those of the authors and do not necessarily reflect the views of the NBB. All errors are ours. †Corresponding author email : fwa@asb.dk