Matrix Inversion of Demand Systems Reconsidered

This paper shows that there are serious problems with matrix inversion of firm-level demand systems. When firms sell differentiated, imperfect substitutes, matrix inversion does not control for the nature of the product when the number of firms is permitted to expand beyond duopoly. Under these conditions, matrix inversion imparts a bias is in favor of quantity competition relative to price competition in models explicitly designed to compare them and this can be significant.An alternative method for analytically inverting log-linear or linear demand systems involving any number of firms under any degree of product differentiation among symmetric firms is proposed. It is shown that this alternative does not suffer from the issues tied to matrix inversion in the context of linear and log-linear demand systems. The use of matrix inversion (or any other method that does not properly control for the nature of the product) in comparative studies of price vs. quantity competition involving more than two firms producing imperfect substitutes should be reconsidered