Keywords

1 Introduction

Hierarchical central place systems are a part of central place theory, which was originated by Christaller (1933 [1966]) and Lösch (1938, 1944 [1954]). Although they were first developed in the domain of geography and the location of economics, central place systems can also be used to classify financial and administrative systems (Parr 2008). In the theoretical work, these central place structures are typically associated with large metropolitan areas, including capital cities. Such centrality can result from a variety of spatial, historical, and economic factors, but it can also lead to problematic disparities when it comes to well-being and quality of life in rural regions. This paper examines how spatial economic organisation can be used to sustain regional welfare, by coordinating access to goods and services in ways that support local populations and their economic activity.

Regional welfare can be enhanced, at least in part, through the economies of agglomeration. These economies are referred to as externalities, with Baumol and Blinder (2016: 304) having categorised spatially unconstrained economies into those marked by beneficial externalities versus detrimental externalities. The difference between these types of economies turns on incidental costs that are borne by others who receive no compensation for any resulting damage to their well-being. Such costs are positive in cases of beneficial externality, but negative in cases of detrimental externality. Furthermore, Parr (2002) proposes a classification of spatially constrained economies that are external to the firm; he divides these into localisation economies, urbanisation economies, and activity-complex economies.

Although localisation and activity-complex economies are commonly linked with manufacturing industries, urbanisation economies are not necessarily connected with aspects of firms and industries. In addition, urbanisation economies also have detrimental externalities such as pollution and congestion, which are referred to as urbanisation diseconomies. These concepts need to be expanded to analyse issues of regional welfare in general and the welfare of rural areas in particular. As demonstrated by Nakamura (2010), rural areas tend to enjoy less availability of goods and services due to the profit-maximisation priorities of firms, which choose to maximise profits rather than revenues. This paper analyses how goods and services can nonetheless be more securely distributed in these areas, arguing that such well-organised distribution systems may partly contribute to improving regional welfare levels.

The economics of welfare was first formally investigated by Pigou (1932), though the limited scope of his analysis was criticised by Robbins (1938), Little (1957), and others. While successors of welfare economics such as Bergson (1938), Hicks (1939), Kaldor (1939), Scitovsky (1941), and Samuelson (1947) attempted to develop the conceptual framework further, this later work gave rise to theoretical contradictions and other problems (Arrow 1950; Arrow and Scitovsky 1969). Later, in a remarkable expansion of the approach, Sen (1974) managed to extend welfare analysis but without creating methodological difficulties, and Stiglitz et al. (2009) built on this work to provide an index of well-being that informed the better-life index of the Organisation for Economic Co-operation and Development (OECD) (see also Stiglitz et al. 2010). Relatedly, Sustainable Development Goals (SDGs) were proposed by the United Nations; these goals include several aspects of well-being in rural as well as urban areas.Footnote 1 Some of the goals involve quality of life (QOL), and economists have developed various approaches related to this subject of study. For example, Blomquist et al. (1988) and Glaeser et al. (2001) focused on the attractiveness of urban areas when it comes to QOL. Greenwood and Hunt (1989) explored how metropolitan migration has been shaped by considerations of employment and also amenities. For their part, Jensen and Leven (1997) compared and contrasted life in suburbs with life in central cities, and Ifcher and Zarghamee (2011) investigated quantitative metrics for well-being. These studies established that QOL is typically higher in urban areas, although those areas also have negative factors such as congestion and high crime rates.

More recently, Jackson et al. (2012) revealed the importance of cooperative behaviour in society, using the conceptual framework of social quilts and neighbour communication networks. Desmet and Rossi-Hansberg (2013) developed an index of welfare in urban areas that includes governments’ budgetary constraints as well as externalities to amenities in those areas. Jones and Klenow (2016) provided another index of welfare using micro-data for several countries that could argue about existing studies which extend beyond GDP-based measurements.

With these previous studies mainly focusing on the evidence of the agglomeration forces of economic activity, the present paper explores what sort of economic system is required to create optimal regional welfare in rural areas. Here, the argument assumes that rural areas offer benefits that offset negative factors in urban areas, such as a rich environment of natural resources. Better access to goods and services is still required in rural areas, however especially necessary goods and services. Access to the market and inputs were studied by Weber (1909 [1928]) via a location triangle model. While the original study investigated the optimal plant location under the calculus of minimum transportation costs, which encompass a combination of two types of input and a single final product, this model can be generalised for an analysis of the optimal locations for the distribution of goods and services and those locations’ corresponding transportation costs.

The paper is organised as follows. In Sect. 6.2, which draws on ideas from central place theory, a spatial model in a simple framework is introduced to examine regional welfare and rural spatial attributes of the central place system used in the model. Then, alternative spatial organisations, proposed via hypothetical analyses, are introduced in Sect. 6.3, and issues of regional sustainability are considered in Sect. 6.4. Section 6.5 indicates further possible expansions of the analysis, with Sect. 6.6 providing concluding comments.

2 A Spatial Model

In order to use the framework of central place theory to investigate regional welfare in rural areas from the standpoint of the availability of goods and services, it is first necessary to employ established market area analysis originating from Lösch (1944 [1954]) and Mills and Lav (1964) and summarised as follows by Nakamura (2007). First, the total revenue, TR, of a representative profit-maximising firm under a spatial monopoly with a regularly formed circular market area can be stated using the following expression:

$$ \mathrm{TR}=\frac{1}{3} Dt\pi {U}^3\left(a- btU\right), $$
(6.1)

where D (D ≥ 0) = density of demand, t (t ≥ 0) = unit distribution transportation cost, π ≈ 3.14159, U (U ≥ 0) = maximum market area radius, and a and b (a ≥ 0, b ≥ 0) = components of the given demand curve. Here, note that there is no demand beyond the maximum radius of the market area.

Second, the total cost, TC, can be expressed as:

$$ \mathrm{TC}(q)={ckq}^2+{F}_{\mathrm{T}}, $$
(6.2)

where c (c ≥ 0) = a unit cost to use as input for production, k (k ≥ 0) = technological indicators for efficiency of production, q (q ≥ 0) = quantity of output, and F T (F T ≥ 0) = the fixed or terminal cost for the distribution of goods and services. Regarding parameter k, as more efficient production becomes available, the indicator k approaches 0.

In order to solve the optimal market area radius of the representative firm, the marginal revenue and the marginal cost should be found under the spatial monopoly condition. The marginal revenue, MR, is immediately available from Eq. (6.1):

$$ \mathrm{MR}(U)=\frac{1}{3} Dt\pi {U}^2\left(3a-4 btU\right). $$
(6.3)

Meanwhile, the marginal cost can be derived from Eq. (6.2), which is replaced by a function of the market area radius u. The conversion uses the following equation for a circular spatial configuration with an additional variable μ (μ ≥ 0) representing a physical obstacle to accessibility between different locations:

$$ q=\mu \pi {u}^2. $$
(6.4)

To sum up, the marginal cost, MC, as a function of the market area radius becomes:

$$ \mathrm{MC}(u)=\partial \frac{ck^2{\mu}^2{\pi}^2{u}^4}{\partial u}4{u}^3{ck}^2{\mu}^2{\pi}^2. $$
(6.5)

Since U in Eq. (6.3) can be treated as u, the optimal market area radius, u ∗, can be specified by equalising Eq. (6.5) to Eq. (6.3):

$$ {u}^{\ast }=\frac{3 at}{4\left({bt}^2+3{k}^2{\mu}^2\pi c\right)}. $$
(6.6)

Also,

$$ \frac{\partial {u}^{\ast }}{\partial b},\frac{\partial {u}^{\ast }}{\partial t},\frac{\partial {u}^{\ast }}{\partial k},\frac{\partial {u}^{\ast }}{\partial \mu },\frac{\partial {u}^{\ast }}{\partial c}<0\kern1em \mathrm{and}\kern1em \frac{\partial {u}^{\ast }}{\partial a}>0. $$
(6.7)

In this way, producers determine the size of their market area. In other words, there are consumers who are not able to obtain goods and services in question beyond the optimal market area radius. Nakamura (2010) characterises this situation in spatial terms as “consumer exclusion.” Such spatial consumer exclusion may cause households in the affected areas to accept lower availabilities of goods and services than households in large metropolitan areas.

While services necessary to the public can be supplied by local or municipal authorities in the form public services, those authorities’ budget constraints are generally tighter in industrialised countries due to the ageing of the total population, for instance. Under such circumstances, it is necessary to develop a well-organised regional system for the distribution of goods and services. The system can be expected to reduce the level of consumer exclusion and increase regional welfare levels, even when the region in question involves rural areas with small-scale economies.

3 Hypothetical Analysis

This section demonstrates how to avoid creating spatial consumer exclusion, over the longer term, in rural areas. In Fig. 6.1, the origin O 1 is the centre of a large metropolitan area where many goods and services are available—based on the idea of central place theory. The horizontal axis shows the physical distance from the centre O 1. The maximum market area radius is point B, and there are still targeted households at this point. Beyond that point, households are able to obtain their goods and services from other central places, as indicated in Eq. (6.1). The vertical axis depicts price levels. As the distance increases from O 1, these price levels rise. Correspondingly, under the condition of a “freight on board” (f.o.b.) pricing system for market area analysis, actual spending in households increases due to transportation costs.

Fig. 6.1
figure 1

Price and distance of goods and services

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In Fig. 6.1, the price level p m shows the maximum reserve price of households for a representative commodity or service. In that case, the commodity or service which has a price p 1 cannot be bought by households that are located beyond point A. As a result, as Fig. 6.1 illustrates, households located between point A and point B face consumer exclusion unless mill price decreases to p s and transportation costs for distribution also decrease.

An alternative framework is shown in Fig. 6.2. The figure has a centre O 2 which is a local hierarchical central place. The maximum market area radius may be B − A, where AO 2 = BO 2 on an economic plane without any physical obstacles. In this case, a high price p 2 is still acceptable, and the difference p 2 − p 1 can be devoted to the management of the local central place O 2. Management of this sort is needed because O 2 will be in a more inefficient situation, due to the limitations of large-scale economies, than the central place O 1.

Fig. 6.2
figure 2

Local lower hierarchical central place system

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A problem may appear where there is spatial market competition between the centre O 1 and the local centre O 2. As illustrated in Fig. 6.3, this can be observed when the price of O 1 declines to the level p s or, perhaps, together with a reduction of distribution transportation costs from the centre O 1. The actual problem is that the local centre O 2 loses its territory h − A + B − z within the market area radius. If the remaining market area, which is not a regularly formed circular market area, falls below the point where normal profit levels can be managed, a spatial structure of this sort is not sustainable for the distribution of goods and services from the local central place O 2.

Fig. 6.3
figure 3

Hierarchical central place system

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This situation corresponds to a shrinking of the demand curve for O 2 that reduces the value a. Further, applying Eq. (6.7) to this situation shows that the optimal market area radius also decreases on average, due to the altered shape of the circular formation after that shift.

4 Regional Sustainability

The previous section demonstrated that spatial market competition under a hierarchical spatial structure may cause severe problems with regional sustainability when it comes to distributing goods and services. As depicted in Fig. 6.4, where the horizontal axis and the vertical axis, respectively, show the size of market area and the level of operating cost at the local central place O 2, a nonoverlapped market area situation satisfies bv (wider market area radius and less cost), while an overlapped market area gives rise to the area gw (narrower market area radius and more cost) for the local central place O 2.

Fig. 6.4
figure 4

Territorial overlaps

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Here, costs to distribute goods and services from the local centre O 2 become higher when the original market area is partly eroded by the centre O 1, since the economies of a larger scale on the centre O 1 may allow for a further reduction of costs by the narrowing of the market area of the local centre O 2. That pushes up the mill price and transportation costs for distribution from the local centre O 2. Eventually, more spatial consumer exclusion appears over the longer term, if the centre O 1 attempts to engage in spatial competition as a short-run entry deterrence behaviour. In that circumstance, regional coordination systems might be necessary to restore operational efficiency of the local central system O 2.

While online stores can be an alternative method for distributing goods and services, there may be more household utility in actual, face-to-face shopping, as long as maximum information and immediate use are key priorities for a given household. In addition, a competitive explicit price set by an online merchant, p 0 (p 0 ≥ 0), should theoretically satisfy p 0 ≤ p m where p m represents the maximum reserve price of households mentioned earlier. Since online stores generally involve longer waiting periods after purchases and also less complete product information than actual, brick-and-mortar shopping, those aspects of online shopping can be added as implicit costs. Such costs can be expressed by σ(σ ≥ 0), with the actual price then being p 0 + σ. Hence, the sufficient condition for online stores to prevail over physical stores would be p 0 + σ ≤ p m. As long as such firms are not directly related to a given region, when it becomes apparent that the distribution of goods and services to that region is unprofitable, their decision to cease business operations may be reached much faster than the decision of locally oriented firms that use the local central place system O 2.

Highly ranked central places, such as capital cities and financial centres of the country, commonly face heavy spatial concentration. These are reasons why many countries enact decentralisation policies as part of top-down, national land-planning, and policy-making initiatives. However, such measures may not work well in the long run if they involve only cost-saving opportunities such as taxation rewards in rural areas. Above and beyond such opportunities, firms must remain in place once they have migrated to those regions. In other words, it is necessary to provide a secure environment for economic activity over the longer term, to ensure that every economic agent can maintain a satisfactory level of utility or profit.

However, the provision of a secure economic environment of this kind should not be expected under the condition where each individual or firm’s sole aim is to maximise its utility or profit level. Instead, a bottom-up approach is needed to maximise regional welfare. To be concrete, when households purchase goods and services at lower prices, their doing so may naturally expand the market share of the larger central places outside the region. It would be ideal if these market areas would then steadily and reliably have goods and services distributed to them. Otherwise, rural areas will face a severe spatial consumer exclusion.

A simple two-player normal-representative form in Table 6.1 can describe the situation at issue. The table shows that there are two local households, person A and person B. Each player can either choose strategy ‘together’ or ‘individual’. Here, ‘together’ implies that the player purchases items always via the local central place system, while ‘individual’ represents the player who pursues only individual utility maximisation, by purchasing from the lower-price market (O 1). Also, note that A 12 ≈ A 21 > A 11 > A 22> and B 12 ≈ B 21 > B 11 > B 22. In the long run, it is apparent that the combination (A 11, B 11) is the best solution for all, ensuring that regional welfare in terms of availability of goods and services is maximised. However, the players’ actual selection is a different combination, (A 22, B 22), which results in much lower payoffs for all than the combination (A 11, B 11). The reason for this selection derives from other potential patterns—namely, (A 12, B 12) and (A 21, B 21)—since a player who selects ‘individual’ can earn a larger payoff in a non-cooperative game.

Table 6.1 Pay-offs among different selections
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However, a regional system might be able to consider organising a cooperative game rather than a non-cooperative one in this connection. If there is a referee (i.e., regional planners) and if that referee offers the possibility for cooperative behaviour by providing sufficient information (i.e., informing players about the payoff matrix that results from sustaining the local central place system O 2 versus continuing to select an ‘individual’ node), an infinitely repeated game may lead local households to select ‘together’ unless others deviate from the triggering strategy. To minimise any risk of deviation, beneficial externality across the region needs to be arranged by substantially increasing its attractiveness.

5 Further Avenues for Inquiry

The framework outlined in this paper may also be applicable to local firms that obtain inputs either via a local central place system or from somewhere else outside the region. Once all relevant economic agents, including those in the governmental sector, are included in the model framework, the impact of utilising the local central place system on the regional economy can be evaluated by input-output analysis. Thus, in an expanded version of the approach sketched here, a regional econometric input-output model (REIM) can provide regional economic forecasting, in the manner originally established by Israilevich et al. (1997). This analysis can then be connected with community-level central place theory through the economic-based approach proposed by Parr et al. (1975), the lower-hierarchical forecasting model developed by Chalmers et al. (1978), and the economic-based approach, involving linkage between different hierarchical sectors, developed by Mulligan (1979). Likewise, the analysis will be able to address Robinson’s (1997) model of problematic factors in household economies, by attempting to establish a direct link between the input-output model and central place system proposed by Sonis and Hewings (2003). It can factor in, too, the Löschian market area analysis via input-output analysis presented in Sonis (2007).

To sum up, the sustainable local central place would become necessary, and its sustainability guaranteed more securely, if local economic agents had constant access to the local central place system as a result of cooperative behaviour undertaken to maximise regional welfare over the longer term—as opposed to non-cooperative behaviour undertaken to maximise short-run individual utility. In the long run, moreover, increased regional welfare also raises individual utility levels. This pattern can be connected, in turn, with Parr’s (2015) analysis of what he calls ‘regional externalities’.

6 Concluding Comments

This paper has explored an optimal spatial economic system for improving regional welfare by means of the secure distribution of goods and services. Using the approach of market area analysis within the framework of central place theory, the paper indicates that rural areas, where local population and economic activity do not reach a sufficient level of scale, find it difficult to compete for the distribution of goods and services from outside the region. Hence, for them to be sustainable in the long run, it is necessary to enhance beneficial economies by promoting cooperative behaviour within rural regions.