A discrete/continuous choice model on a nonconvex budget set

ABSTRACT

Decreasing block rate pricing is a nonlinear price system often used for public utility services. Residential gas services in Japan and the United Kingdom are provided under this price schedule. The discrete/continuous choice approach is used to analyze the demand under decreasing block rate pricing. However, the nonlinearity problem, which has not been examined in previous studies, arises because a consumer’s budget set (a set of affordable consumption amounts) is nonconvex, and hence, the resulting model includes highly nonlinear functions. To address this problem, we propose a feasible, efficient method of demand estimation on the nonconvex budget. The advantages of our method are as follows: (i) the construction of an 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 on the nonconvex budget set. The proposed method is applied to estimate the Japanese residential gas demand function and evaluate the effect of price schedule changes as a policy experiment.

KEYWORDS:

• Bayesian analysis
• discrete/continuous choice approach
• Hermite–Hadamard integral inequality
• nonconvex budget set
• residential gas demand

JEL CLASSIFICATION:

• C11
• C24
• D12

Acknowledgment

The authors are grateful to the Editor, the Associate Editor, and the anonymous referee for their valuable suggestions that substantially improved the manuscript. The authors also thank K. Fukumoto (Gakushuin University), M. Fukushige (Osaka University), K. Kakamu (Kobe University), K. Oya (Osaka University), A. C. Harvey (University of Cambridge), M. Kitahara (Osaka City University), N. Terui (Tohoku University), and H. K. van Dijk (Erasmus Universiteit Rotterdam) for their helpful comments.

Notes

The discrete/continuous choice approach is named because it simultaneously considers the discrete and continuous choices. In the block rate pricing case, consumers choose both the block and the consumption amount, which are discrete and continuous, respectively. This approach has also been used to examine a wide range of topics including housing (Lee and Trost, 1978; King, 1980), transportation (Mannering and Winston, 1985; Hensher and Milthorpe, 1987; de Jong, 1990; West, 2004), labor supply (Burtless and Hausman, 1978; Burtless and Moffitt, 1985), electricity demand (Herriges and King, 1994), and water demand (Hewitt and Hanemann, 1995; Olmstead et al., 2007; Miyawaki et al., 2014). When the budget set is convex, Miyawaki et al. (2014) proposed the appropriate estimation method. However, as we will see later, the nonconvex budget case (including the decreasing block rate pricing case) is much more difficult to estimate model parameters properly.

We face the nonlinearity under decreasing block rate pricing, though we do not under increasing block rate pricing (see, e.g., Moffitt, 1986).

Recently, Szabó (2009) proposed the maximum likelihood estimation method for general block rate pricing where the linear demand function is assumed. Szabó (2009) imposed a condition that the direct utility function is quasiconcave. This condition aims to guarantee that the underlying preference relation be strictly convex, that is, the preference relation be well-behaved. However, as stated in Hurwicz and Uzawa (1971), two more conditions (the nonnegative demand condition and the separability condition) are required for the underlying preference relation to be strictly convex. These additional conditions often make it difficult to numerically maximize the likelihood function. See Miyawaki et al. (2014) for the detailed discussion on this issue.

This functional form implicitly assumes that P_k > 0 and Q_k > 0 for all k. Under decreasing block rate pricing, these assumptions are equivalent to P_K > 0 and Q_K > 0.

As pointed out in Hausman (1985), our approach that involves deciding the demand function first and deriving its corresponding indirect utility function has two advantages: (i) we can flexibly choose the functional form of the demand function based on the dataset, and (ii) the stochastic specification becomes convenient.

With the dataset that will be used in Section 6, we conducted the estimation of the gas demand function normalizing the variance of heterogeneity to one. The results are affected by this normalization. In particular, the posterior mean of β₁ is estimated to be −0.094 and its 95% credible interval is (−0.31, − 0.003).

By using this data generating process, it is straightforward to conduct the simulation study. Given true parameter values, we generate the gas demand. Then, given the generated gas demand as the dataset, the efficient MCMC simulator that will be described in Section 4 is applied to draw samples from the posterior distribution. We conducted the simulation study and found that true parameter values were recovered. However, due to the page limitation, we omit the details.

See, for example, Niculescu and Persson (2003) for a proof. Niculescu and Persson (2003) also noted that the first (or last) inequality can define the convex function itself.

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⁹See, for example, Chapter 2 of Hardy et al. (1952) for a proof of the power-mean inequality. This equivalence also uses the fact that f(x) = x^β₁ (β₁ ∈ [l₁, 0]) is decreasing as x(> 0) increases.

Because a₁≷0 for all k≶j, a₁/(P_ik − P_ij) > 0 for any k and j (k ≠ j).

We also conducted the analysis without the prior truncations for both β_j (j = 1, 2). Posterior means and standard deviations for elasticity parameters are −0.83 (0.28) for the price elasticity (β₁) and 0.27 (0.046) for the income elasticity (β₂). Their 95% credible intervals are [−1.38, − 0.28] for β₁ and [0.17, 0.35] for β₂. Thus, price and income elasticities are highly credible to be negative and positive, respectively, in the sense that their 95% credible interval does not include zero. Furthermore, income elasticity is highly credible to be less than one. Other obtained results are very similar to those obtained with priors (34) specified above, which are omitted due to the page limitation. Thus, we conclude that the results given below are not sensitive to the prior truncation.

We also conducted the Bayesian inference with the truncated prior for δ, and obtained results are mostly similar with those found in Table 6. However, the prior truncation affects the marginal posterior distribution for δ₄. The posterior mean for δ₄ is 0.086 with its 95% credible interval [0.016, 0.17].

Because our data are reduced from 1, 250 to 473, such a large data reduction would influence the obtained results. To examine the effect from this reduction, we gathered households whose dependent variables are missing but whose explanatory variables are not. The number of households then became 759. Under the same MCMC setting, we estimated the residential gas demand function. Missing dependent variables were imputed within the MCMC simulation by using the data augmentation method. Obtained results are quite similar to those given in Table 6 and shown in Fig. 9, and we omit the details. Thus, the data reduction in explanatory variables does not influence the estimates of model parameters.

Suppose x₁ > x₀ > 0. Then, because (l = 0, 1) is decreasing (increasing) with respect to x_l if θ < (>)0, the numerator if θ≶0. Therefore, D(x₁, x₀; θ) > 0 if x₁ > x₀ > 0. Similarly, D(x₁, x₀; θ) <0 if x₀ > x₁ > 0.