Incorporating Prior Information into a GMM Objective for Mixed Logit Demand Systems
It is well known that random parameters specifications can generate upward sloping demands for a subset of products in the data. Nevo (2001), for example, found 0.7 percent of demands to be upward sloping. Possibly less well known is that demand system estimates can imply margins outside of the theoretical bounds for profit maximization. If such violations are numerous enough, they can confound merger simulation exercises. Using Lerner indices for multiproduct firms playing static Bertrand games, we find that up to 35 percent of implied margins for beer are outside the bounds. We characterize downward sloping demand and the theoretical bounds for profit maximization as prior information and extend the GMM objective function, incorporating inequality moments for product-level own-elasticities and brand-level or product-level Lerner indices. These moments impose a cost when the inequality is violated, and equal zero otherwise. Very few violations remain when an inequality constrained estimator is used. Importantly, the unconstrained GMM objective has multiple minima, while the constrained objective has only one minimum when the product-level constraints are used in our illustration. This is valuable for policy purposes as it enables one to limit attention to a single theoretically consistent model. Inputs to merger simulations are likewise consistent with economic theory, and, as a result, confidence in the output is increased.
In a second innovation, this paper introduces merger simulation for static Stackelberg price competition games. Our illustration uses beer data, a perfect vehicle for introducing Stackelberg games as the economics literature and industry trade press have long considered Anheuser-Busch to be the industry price leader. We find evidence of positive pre-merger price conjectures consistent with beer brands being strategic complements. Allowing the leader to update their conjectures in response to a merger provides dramatically different post-merger price and share changes relative to Bertrand. The Stackelberg conjectures are used as a strategic tool that allows post-merger product repositioning unavailable under Bertrand.
Keywords: Random Coefficients Logit; Inequality Constrained Optimization; Merger Simulation; Stackelberg Pricing