This article estimates a general linear demand model for three fluid milk brands. It finds perverse signs and complementarity among these substitutes. We develop three constrained models to eliminate perverse signs: a price conduct constraint in the linear model that links private label and Garelick pricing, a generalized Mills address model, and the logit model. Statistical tests indicate that each of the three constrained models produces acceptable substitution patterns and that none is preferred to the other two. On nonstatistical grounds, however, the Mills and price-constrained models perform best. This is the first empirical study of the Mills address model of private-label pricing. Finally, we demonstrate that estimated price-reaction functions and the corresponding total elasticities, which capture only strategic non-collusive interaction in these Nash Bertrand models, provide interesting insight into competition among brands. EconLit citation: L670, L130.] © 2005 Wiley Periodicals, Inc. Agribusiness 21: 149–166, 2005.