Pricing Decisions Under Demand Uncertainty: A Bayesian Mixture Model Approach
Authors: Kalyanam Kirthi 
Published: October 2008

The advent of optical scanning devices and decreases in the cost of computing power have made it possible to assemble databases with sales and marketing mix information in an accurate and timely manner. These databases enable the estimation of demand functions and pricing/promotion decisions in “real” time. Commercial suppliers of marketing research like A. C. Nielsen and IRI are embedding estimated demand functions in promotion planning and pricing tools for brand managers and retailers. This explosion in the estimation and use of demand functions makes it timely and appropriate to re-examine several fundamental issues. In particular, demand functions are latent theoretical constructs whose exact parametric form is unknown. Estimates of price elasticities, profit maximizing prices, inter-brand competition and other policy implications are conditional on the parametric form assumed in estimation. In practice, many forms may be found that are not only theoretically plausible but also consistent with the data. The different forms could suggest different profit maximizing prices leaving it unclear as to what is the appropriate pricing action. Specification tests may lack the power to resolve this uncertainty, particularly for non-nested comparisons. Also, the structure of these tests does not permit seamless integration of estimation, specification analysis and optimal pricing into a unified framework. As an alternative to the existing approaches, I propose a Bayesian mixture model (BMM) that draws on Bayesian estimation, inference, and decision theory, thereby providing a unified framework. The BMM approach consists of input, estimation, diagnostic and optimal pricing modules. In the input module, alternate parametric models of demand are specified along with priors. Utility structures representing the decision maker's attitude towards risk can be explicitly specified. In the estimation module, the inputs are combined with data to compute parameter estimates and posterior probabilities for the models. The diagnostic module involves testing the statistical assumptions underlying the models. In the optimal pricing module the estimates and posterior probabilities are combined with the utility structure to arrive at optimal pricing decisions. Formalizing demand uncertainty in this manner has many important payoffs. While the classical approaches emphasize choosing a demand specification, the BMM approach emphasizes constructing an objective function that represents a mixture of the specifications. Hence, pricing decisions can be arrived at even when there is no consensus among the different parametric specifications. The pricing decisions will reflect parametric demand uncertainty, and hence be more robust than those based on a single demand model. The BMM approach was empirically evaluated using store level scanner data. The decision context was the determination of equilibrium wholesale prices in a noncooperative game between several leading national brands. Retail demand was parametrized as semilog and doublelog with diffuse priors for the models and the parameters. Wholesale demand functions were derived by incorporating the retailers' pricing behavior in the retail demand function. Utility functions reflecting risk averse and risk neutral decision makers were specified. The diagnostic module confirms that face validity measures, residual analysis, classical tests or holdout predictions were unable to resolve the uncertainty about the parametric form and by implication the uncertainty with regard to pricing decisions. In contrast, the posterior probabilities were more conclusive and favored the specification that predicted better in a holdout analysis. However, across the brands, they lacked a systematic pattern of updating towards any one specification. Also, none of the priors updated to zero or one, and there was considerable residual uncertainty about the parametric specification. Despite the residual uncertainty, the BMM approach was able to determine the equilibrium wholesale prices. As expected, specifications influence the BMM pricing solutions in accordance with their posterior probabilities which act as weights. In addition, differing attitudes towards risk lead to considerable divergence in the pricing actions of the risk averse and the risk neutral decision maker. Finally, results from a Monte Carlo experiment suggest that the BMM approach performs well in terms of recovering potential improvements in profits.