Calculation and Visualization of Dynamic Price Elasticities

Abstract

ABSTRACTApplying the Hierarchical Bayesian Regression model to weekly aggregated sales history data from 92 retail stores located around Tokyo, I calculated price elasticities by item, week, and store. These elasticities are more stable than figures calculated using the Hierarchical Regression or Bayesian Regression models. Furthermore, using Google Earth, I visualized the calculated price elasticities of these 92 stores over 67 weeks, providing a better understanding about heterogeneity of price elasticities across time and space.Keywords: Price Elasticity, Hierarchical Bayesian Regression Model, Markov Chain Monte Carlo (MCMC) method, Google EarthINTRODUCTIONCurrently consumers in Japan are more price-conscious, especially when purchasing commodity goods in supermarkets. In response to this, many retailers cut shelf price to gain business. It is however necessary to optimize the price and/or promotions in addition to cutting prices in order to maintain profits and sustain growth. Because consumer behavior depends on location, competitors, and other factors, store customization is a very important consideration (Levy and Weitz, 2011).Chintagunta, Dube, and Singh (2008) measured the effects of price discrimination by a supermarket chain in Chicago, and proposed optimal stores pricing. In contrast, Dobson and Waterson (2008) proposed uniform pricing rather than optimal pricing specific to each store.Price elasticities by store and time provide important, fundamental information for price customization. A popular way to calculate price elasticity by store and week is the estimation of parameters using a regression model, that is, a model with parameters depending on store and week, using daily sales data aggregated by store. Because of limited storage capacity, however, some retailers retain only weekly historical sales data. However, estimating parameters using a regression model and input by store and week using stores' weekly sales data, could be unstable (Blattberg & George, 1991; Montgomery, 1997).Therefore, in this study, I applied the MCMC method using Gibbs' sampling to estimate the parameters of the Hierarchical Bayesian Regression model and calculated weekly price elasticities by store. I also dynamically visualized weekly price elasticities by store, using Google Earth.BAYESIAN UPDATING USING MCMC METHOD Calculation of Posterior ExpectationGiven parameter q and observed data y= (y1,L yn ), and using the definition of conditional probability, Bayes' theorem holds: Pr(qy)= Pr(q)Pr(yq)Pr(y), where Pr(q) is a prior distribution of parameter q including all a priori information, Pr(y q) is a likelihood function of the observed data given the parameter q, and Pr(qy) is a posterior distribution of y . The expectation of posterior distribution Pr(qy)is called a Bayesian estimator. The advantages of Bayesian estimation are the ability to use prior information, parameter updating, and so on.To estimate (calculate the expectation of) posterior distribution we must solve the integral stated above, which is usually very difficult. If solving the integral is difficult or impossible, then conjugating prior, asymptotic expansion and Monte-Carlo integration are used to estimate the posterior distribution.Monte-Carlo Method and MCMC MethodIf the random sample following the posterior distribution available, we can approximate the expectation of posterior distribution by the Monte-Carlo aN q(n) N integral n=1 . By the law of large numbers, the Monte-Carlo integral is a valid {q(1),q(2),L,q(N)} is approximation of expectation (Tienery & Kadane, 1986). However, when random sampling of the posterior distribution is not available and if the parameter is high-dimensional, random sampling is very inefficient. In this case, using a sample following Markov chains derived from posterior distribution, we calculate the posterior expectation using the Monte-Carlo integral. …