The inconsistency of common scale estimators when output prices are unobserved and endogenous

This paper explores the inconsistency of common scale estimators when output is proxied by deflated sales, based on a common output deflator across firms. The problem arises when firms operate in an imperfectly competitive environment and prices differ between them. In particular, we show that this problem reveals itself as a downward bias in the scale estimates obtained from production function regressions, under a variety of assumptions about the pattern of technology, demand and factor price shocks. The result also holds for scale estimates obtained from cost functions. The analysis is carried one step further by adding a model of product demand. Within this augmented model we examine the probability limit of the scale estimate obtained from an ordinary production function regression. This analysis reveals that the OLS estimate will be biased towards a value below one, and how this bias is affected by the magnitude of the parameters and the amount of variation in the various shocks. We have included an empirical section which illustrates the issues. The empirical analysis presents a tentative approach to solve the problem discussed in the theoretical part of this paper.