"Bayesian Estimation of Demand Functions under Block-Rate Pricing"

This article proposes a Bayesian estimation of demand functions under block-rate pricing by focusing on increasing block-rate pricing. This is the first study that explicitly considers the separability condition which has been ignored in previous literature. Under this pricing structure, the price changes when consumption exceeds a certain threshold and the consumer faces a utility maximization problem subject to a piecewise-linear budget constraint. Solving this maximization problem leads to a statistical model in which model parameters are strongly restricted by the separability condition. In this article, by taking a hierarchical Bayesian approach, we implement a Markov chain Monte Carlo simulation to properly estimate the demand function. We find, however, that the convergence of the distribution of simulated samples to the posterior distribution is slow, requiring an additional scale transformation step for parameters to the Gibbs sampler. These proposed methods are then applied to estimate the Japanese residential water demand function.

"Discrete/Continuous Choice Model of the Residential Gas Demand on the Nonconvex Budget Set"

The discrete/continuous choice approach is often used to analyze the demand for public utility services under block rate pricing, which is a nonlinear price system. Although a consumer's budget set is convex under increasing block rate pricing, a consumer's budget set is nonconvex under decreasing block rate pricing as is the case with the gas supply in Japan and the United Kingdom. The nonlinearity problem, which has not been examined in previous studies, arises under nonconvex budget sets in which the indirect utility function corresponding to the demand function becomes highly nonlinear. To address this problem, this article proposes a feasible, efficient method of demand on the nonconvex budget set and implements a case study using household-level data on Japanese residential gas consumption. The advantages of our method are as follows: (i) the construction of an efficient Markov chain Monte Carlo algorithm with an efficient blanket based on the Hermite-Hadamard integral inequality and the power-mean inequality, (ii) the explicit consideration of the (highly nonlinear) separability condition, which often makes numerical likelihood maximization difficult, and (iii) the introduction of normal disturbance into the discrete/continuous choice model.

"Bayesian Estimation of Demand Functions under Block Rate Pricing"

This article proposes a Bayesian estimation method of demand functions under block rate pricing, focusing on increasing one, where we first considered the separability condition explicitly which has been ignored in the previous literature. Under this pricing structure, price changes when consumption exceeds a certain threshold and the consumer faces a utility maximization problem subject to a piecewise-linear budget constraint. Solving this maximization problem leads to a statistical model that includes many inequalities, such as the so-called separability condition. Because of them, it is virtually impractical to numerically maximize the likelihood function. Thus, taking a hierarchical Bayesian approach, we implement a Markov chain Monte Carlo simulation to properly estimate the demand function. We find, however, that the convergence of the distribution of simulated samples to the posterior distribution is slow, requiring an additional scale transformation step for parameters to the Gibbs sampler. These proposed methods are applied to estimate the Japanese residential water demand function.

"Panel Data Analysis of Japanese Residential Water Demand Using a Discrete/Continuous Choice Approach"

Block rate pricing is often applied to income taxation, telecommunication services, and brand marketing in addition to its best-known application in public utility services. Under block rate pricing, consumers face piecewise-linear budget constraints. A discrete/ continuous choice approach is usually used to account for piecewise-linear budget constraints for demand and price endogeneity. A recent study proposed a methodology to incorporate a separability condition that previous studies ignore, by implementing a Markov chain Monte Carlo simulation based on a hierarchical Bayesian approach. To extend this approach to panel data, our study proposes a Bayesian hierarchical model incorporating the random and fixed individual effects. In both models, the price and income elasticities are estimated to be negative and positive, respectively. Further, the number of members and the number of rooms per household have positive relationship to the residential water demand when we apply the model with random individual effects, while they do not in the model with fixed individual effects.

Panel Data Analysis of Japanese Residential Water Demand Using a Discrete/Continuous Choice Approach

Block rate pricing is often applied to income taxation, telecommunication services, and brand marketing in addition to its best-known application in public utility services. Under block rate pricing, consumers face piecewise-linear budget constraints. A discrete/continuous choice approach is usually used to account for piecewise-linear budget constraints for demand and price endogeneity. A recent study proposed a methodology to incorporate a separability condition that previous studies ignore, by implementing a Markov chain Monte Carlo simulation based on a hierarchical Bayesian approach. To extend this approach to panel data, our study proposes a Bayesian hierarchical model incorporating the individual effect. The random coefficients model result shows that the price and income elasticities are estimated to be negative and positive, respectively, and the coefficients of the number of members and the number of rooms per household are estimated to be positive. Furthermore, the AR(1) error component model suggests that the Japanese residential water demand does not have serial correlation.Block rate pricing, Bayesian analysis, Panel data, residential water demand