A Fixed-Point Approach to Equilibrium Pricing in Differentiated Product Markets.
Morrow, William Ross
AbstractThe use of passenger cars in the U.S. is widely recognized as a primary source of greenhouse gas emissions. Among other concerns related to transportation energy usage, these emissions have motivated policy makers to reform regulatory policies that impact the design and pricing behavior of automotive firms. However many prominent analyses supporting these reforms neglect aspects of imperfect competition that many economists have come to recognize in the automotive industry, particularly Bertrand competition. Understanding the impacts of alternative regulatory policies accounting for imperfectly competitive behavior may improve policy makers' decisions regarding regulatory policy, as well as firms' abilities to profitably respond to the policies chosen. Examining and exploiting this potential first requires further development of the theory of Bertrand competition. This dissertation makes three contributions the application of Bertrand competition in regulated differentiated product markets. We focus on the classes of Logit and Mixed Logit Discrete Choice Random Utility Models of consumer demand and avoid assumptions on the number or type of products offered by different firms. Our first contribution is a proof of the existence of unregulated equilibrium prices for Logit models using a new fixed-point equation equivalent to the first-order necessary condition for equilibrium, minimal assumptions on the utility specification, and mathematical tools from differential topology. This fixed-point equation is then generalized to the class of Mixed Logit models, one of the most flexible and popular empirical forms for representing consumer demand, under a weak hypothesis on the utility specification and mixing distribution. Several numerical approaches based on fixed-point characterizations of stationarity are demonstrated to be efficient and reliable methods for the computation of unregulated equilibrium prices in large-scale and complex differentiated product markets. Finally, we further extend this fixed-point approach to regulated equilibrium pricing problems with regulatory policy forms inspired by those considered for the U.S. automotive industry. Modified fixed-point iterations are derived for a number of regulatory policies with differentiable regulatory costs. A hybrid fixed-point approach is developed for the computation of regulated equilibrium prices under standard-based policies with non-differentiable regulatory costs, such as the Corporate Average Fuel Economy standards currently active in the U.S.
Bertrand CompetitionDifferentiated Product MarketsMixed LogitPoincare-Hopf TheoremGMRES-Newton Hookstep
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