CIRJE-F-424. Miyawaki, Koji, Yasuhiro Omori and Akira Hibiki, "Bayesian Estimation of Demand Functions under Block Rate Pricing", May 2006; Revised in May 2007.

This article proposes a Bayesian estimation method of demand functions under block rate pricing, focusing on increasing block rate pricing. Block rate pricing is often observed in public utilities, such as water and electricity. Under this price structure, price changes when consumption exceeds a certain threshold, and the demand function is subject to a piecewise-linear budget constraint. We apply the so-called discrete/continuous choice approach to analyze consumer behavior when facing such a price system. Moreover, a separability condition is explicitly considered to obtain proper estimates. The log-linear model is assumed for the conditional demand to illustrate our procedure because it is one of most popular models in the literature. Taking a hierarchical Bayesian approach, we implement a Markov chain Monte Carlo simulation to estimate the demand function. It is, however, found that the convergence of the distribution of simulated samples to the posterior distribution is extremely slow. To improve the mixing properties of the Markov chain, we introduce an additional scale transformation step for parameters to the Gibbs sampler. The model is also extended to allow random coefficients for panel data and spatial correlation to account for consumer heterogeneity in spatial data. Algorithms for these two extensions are also provided. The proposed methods are applied to estimate the Japanese residential water and electricity demand function under increasing block rate pricing.