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Disaggregate Hierarchical Decision Huff Model Incorporating Consumer Kaiyu Choices Among Shopping Sites Read chapter Read first chapter

2023 | OriginalPaper | Chapter

A Dynamical Huff Model: Computing the Competitive Equilibrium Distribution of Shop Floor Areas over a City Center Commercial District Using a Fixed-Point Algorithm

Authors : Saburo Saito, Hiroyuki Motomura

Published in: Recent Advances in Modeling and Forecasting Kaiyu

Publisher: Springer Nature Singapore

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Abstract

The equilibrium point of the dynamical retail model for Saga City is computed by Merrill’s method. The dynamical retail model endogenizes the shop floor areas of shopping centers so that its equilibrium point is considered a competitive spatial equilibrium of shop floor areas. The recent revision of the Large-Scale Retail Store Law in Japan, which follows the Japan-US Structural Impediments Initiative, suggests that a new urban commercial policy is required to establish some guidelines to allocate shop floor areas among city retailers from the viewpoint of city formation. The competitive spatial equilibrium of shop floor areas might serve as a guideline. This study compares the actual and the computed equilibrium of shop floor area distribution and suggests further research.

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previous chapter Disaggregate Hierarchical Decision Huff Model Incorporating Consumer Kaiyu Choices Among Shopping Sites
next chapter Kaiyu Markov Model and Evaluation of Retail Spatial Structures
Footnotes
1
The Large-Scale Retail Store Law before the amendment was criticized at the US-Japan Structural Impediments Initiative as a typical structural barrier for the entry by foreign large retailers into the Japanese retail market and was revised by the Japanese Government commitment according to the agreement at US-Japan SII from 1989 to 1990.
 
2
In Eq. (8), putting Wj = Sj; α = μ; cij = logTij; and β = λ as Oi = ciIiPOPi, it becomes Eq. (3), the equation in the previous section. Since only the value of α is related in Theorem 1, its conclusion also holds in the Eqs. (3) and (5) if α is read as μ.
 
3
The equilibrium condition in Eq. (9) is a nonlinear equation. Thus, Newton-Raphson’s method is the first method that can be considered. However, since there is no global convergence, there is no guarantee that it always converges to a solution from an arbitrary initial value. In addition, the method that has been taken in previous studies traces a solution curve by discretizing differential equations. However, if there is no condition of being an asymptotically stable solution, it does not necessarily converge to an equilibrium solution, and even if the solution is unique and asymptotically stable, there is also a problem that it is necessary to concretely specify fj(Wj) representing the reaction rate. Although this tracing method has the advantage that it can describe the process in which the system approaches the equilibrium solution, the fixed-point algorithm is considered superior in directly obtaining the equilibrium solution itself.
 
4
Here, we assume that k is constant in all commercial districts, but it is possible to consider models in which k is different for each commercial district or models in which k is endogenized (Cf. [2, 3]). However, in this case, we have yet to obtain a theoretical result as to whether or not there is still a unique equilibrium solution and whether it is asymptotically stable. Because of this, we would like to consider the case where the value of k differs among different commercial districts as a future research topic.
 
5
While stated like this, the absolute value of the equilibrium shop floor area, of course, will have a significant meaning if the value of k has been specified empirically. However, the specification of the empirical value of k is itself an important research topic. Thus, this research focuses only on the share of the equilibrium shop floor area among the commercial districts or the equilibrium shop floor distribution.
 
6
This survey is a questionnaire survey of citizens’ opinions and behaviors for randomly sampled 3100 citizens distributed and collected by mail. The number of collected samples is 1453 samples.
 
7
The number of valid samples for these question items collected is 918 samples. The question items use the recall method to ask the respondents about their most recent visit to the city center commercial district of Saga City, excluding the purpose of commuting to work or school.
 
8
More precisely, the choice data is about the choice of the entrance commercial district for shopping. This question item was the first home-based survey of consumer Kaiyu behaviors that recorded the whole Kaiyu history of the respondents, i.e., starting from home, visiting several shopping sites, and returning home. Here we use the data about the choice of the entrance commercial district among all commercial districts at the city center of Saga City. In other words, the entrance choice is the respondents’ first visited commercial district on their Kaiyu behaviors.
 
9
In the present study, we aim to obtain a competitive equilibrium shop floor distribution by dealing with a model in which the shop floor efficiency k is constant. Therefore, regarding the competitive equilibrium shop floor distribution, the value of exogenous variables that do not influence this determination can be arbitrary. Since income, consumption propensity, and outflow rate are the same for all residential areas, these values and the value of k will not change the competitive equilibrium shop floor distribution. While we rely on actual statistical data for deciding the values of these exogenous variables to be used, we are not deeply involved in the empirical study on these values.
 
10
The original version of this chapter first appeared in Saito and Motomura [14].
 
Literature
1.
Huff, D. L. (1964). Defining and estimating a trading area. Journal of Marketing, 28(3), 34–38. CrossRef
2.
Birkin, M., & Wilson, A. G. (1989). Some properties of spatial-structural-economic-dynamic models. In: Hauer et al. (Eds.) Urban dynamics and spatial choice behaviour. Theory and decision library (vol. 49, pp. 175–201). Kluwer.
3.
Clarke, G. (1986). Modelling retail centre size and location. Working paper no. 482, School of Geography, University of Leeds.
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Lombardo, S. T., & Rabino, G. A. (1989). Urban structures, dynamic modelling, and clustering. In Urban dynamics and spatial choice behaviour (pp. 203–217). Springer. CrossRef
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Harris, B., & Wilson, A. G. (1978). Equilibrium values and dynamics of attractiveness terms in production-constrained spatial-interaction models. Environment and Planning A, 10(4), 371–388. CrossRef
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Rijk, F. J., & Vorst, A. (1983). Equilibrium points in an urban retail model and their connection with dynamical systems. Regional Science and Urban Economics, 13(3), 383–399. CrossRef
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Kojima, M. (1981). Complementarity and fixed points: An algorithmic approach. Tokyo: Sangyotosho. (in Japanese).
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Scarf, H. E. (1982). The computation of equilibrium prices: An exposition. Handbook of Mathematical Economics, 2, 1007–1061. CrossRef
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Todd, M. J. (1983). Computing fixed points with applications to economic equilibrium models. In: W. F. Lucas (Ed.) Discrete and system models (vol. 3). Modules in applied mathematics. Springer
10.
Saito, S. (1986). Analysis report of Saga citizen opinion survey: City attractiveness and policy demand of Saga City. Saga City Government, Saga City. (in Japanese).
11.
Motomura, H. (1991). Analysis of equilibrium points in dynamical Huff model by fixed point algorithms. Master thesis, Graduate School of Economics, Fukuoka University. (in Japanese).
12.
Saito, S. (1984). A disaggregate hierarchical Huff model with considering consumer’s shop-around choice among commercial districts: Developing SCOPES (Saga commercial policy evaluation system) (in Japanese) planning and public. Management, 13, 73–82. (in Japanese).
13.
Saito, S. (1984). A survey report on the modified Huff model: The development of SCOPES (Saga commercial policy evaluation system). Saga City Government, Saga City. (in Japanese).
14.
Saito, S., & Motomura, H. (1992). Computing competitive equilibrium spatial distribution of retail shopfloor of dynamical retail model using fixed point algorithm. Papers on City Planning, 27, 451–456. (in Japanese).
Metadata
Title
A Dynamical Huff Model: Computing the Competitive Equilibrium Distribution of Shop Floor Areas over a City Center Commercial District Using a Fixed-Point Algorithm
Authors
Saburo Saito
Hiroyuki Motomura
Copyright Year
2023
Publisher
Springer Nature Singapore
Book
Recent Advances in Modeling and Forecasting Kaiyu

Print ISBN: 978-981-9912-40-7

Electronic ISBN: 978-981-9912-41-4

Copyright Year: 2023

DOI
https://doi.org/10.1007/978-981-99-1241-4_2