Logically Consistent Market Share Models

Abstract

A logically consistent market share model should predict market shares that are between zero and one, and sum to one. Few authors have worried about this type of problem in empirical studies, mainly because of the usual interest in a particular brand. It is then implicitly assumed that if predicted market share for that brand is MS,, the other firms combined will get (1 MS,). This point may not be as obvious as it seems. If we were to estimate the market share response functions for the other brands as well, we would often find that the sum of the market shares is not one. The problem becomes apparent particularly when the response functions for all brands or for a group of brands are estimated simultaneously. An article by Neil E. Beckwith in a recent issue of JMR [2] reports an application of Zellner's joint generalized least squares method (joint GLS) [13] to the estimation of linear market share response functions of various competing brands. Beckwith's article illustrates an interesting way of obtaining more efficient estimators, but at the same time, raises the issue of logical consistency more clearly than in other applications. This article will address the problems which arise when market response models are sum-constrained. Reference will be made to Beckwith's study because it is one of the few examples where response functions of various brands have been estimated simultaneously. We will first derive restrictions on the explanatory variables and on the parameters which are implied by a sum constraint on the dependent variable. Beckwith's study will serve as an illustration. The constraint in his work relates to the sum of the individual firms' market shares. Market share models are such that few distributions can describe the behavior of the disturbances. This is discussed in the second section. In the third section we reach the conclusion that for market share functions to be logically consistent their functional form should almost invariably be intrinsically nonlinear.' It would seem intuitively reasonable to require logical consistency as a criterion for judging a model's appropriateness. However, logical consistency leads to more complicated market share functions, and necessitates more sophisticated estimation techniques. So there will be a trade-off between requiring logical consistency and model simplicity. This point will be examined in the final section.