Some unpleasant markup arithmetic: Production function elasticities and their estimation from production data

The ratio estimator of a firm's markup is the ratio of the output elasticity of a variable input to that input's cost share in revenue. This note raises issues that concern identification and estimation of markups using the ratio estimator. Concerning identification: (i) if the revenue elasticity is used in place of the output elasticity, then the estimand underlying the ratio estimator does not contain any information about the markup; (ii) if any part of the input bundle is either used to influence demand, or is neither fully fixed nor fully flexible, then the estimand underlying the ratio estimator is not equal to the markup. Concerning estimation: (i) even with data on output quantities, it is challenging to obtain consistent estimates of output elasticities when firms have market power; (ii) without data on output quantities, as is typically the case, it is not possible to obtain consistent estimates of output elasticities when firms have market power and markups are heterogeneous. These issues cast doubt over whether anything useful can be learned about heterogeneity or trends in markups, from recent attempts to apply the ratio estimator in settings without output quantity data.