1. Introduction
The power law is ubiquitously used in the fields of physics, earth, and planetary science, biology, finance [
1], as well as in urban planning. A city is a complex system with bottom-up growth characteristics. The scale distribution of cities requires appropriate structures to obtain proper urban populations and numbers of towns. The power law plays a significant role in characterizing the laws of urban development. Thus, the exploration of the effects on the power law of the urban scale deepens research about the law of urban self-organization, improves current urban-scale system and balances development of regional cities.
Since Auerbach firstly applied the power rate to the study of urban scales, the impact studies of the distribution law of urban scales have been continuously enriched [
2]. Brakman [
3] introduced a negative feedback mechanism model to explore the impact of economic factors, such as transportation and congestion costs, economical levels of industrial activities and returned to scale on the power law at the urban scale. The increase in industrialization and the decrease in transportation costs would lead to faster growth and agglomeration in big cities’ population, with a corresponding increase in the power index. The diversity of economic scales among industries resulted in the diversity of industrial locations, which was transformed into a diversity of urban scale through the spatial coordination of industries, and then resulting in a certain difference in the power index [
4]. Additionally, regional household registration policies, population size, transportation level, and urban output also had effects on the power index [
5,
6,
7]. For example, Devadoss et al. [
8] explored the power law of Indian rural and urban areas and found that the difference in regional higher power indices stems from larger population sizes. Empirical studies of Chinese cities had also found that a better spatial allocation of economic resources and a higher degree of population agglomeration led to a higher power index [
9,
10,
11]. Therefore, policies, economic and social aspects can have an impact on the power law of urban scale. Moreover, the impact on the power law at urban scale is generally defined from the macro perspective, while research from the perspective of micro decision-making behavior is yet to be further analyzed.
The relationship between interactive decision-making and the dynamic changes in land use is complex, which determines the possibility of regional land development and distribution characteristics at the urban scale. Ligmann-Zielinska [
12] showed that, in the same area, various types of decision-makers had different competitions and choices for land. Bacon et al. [
13] simulated the decision-making process of land managers through BN models, and Chen et al. [
14] constructed an RLC-REP model to determine the land use selection preferences of residential areas for use in the decision-making process. Parker et al. [
15] also found that interactions among residential agents would trigger developers to pursue the area, followed by an increase in the development of land. Furthermore, the developers were more likely to choose downtown sites due to the reputation effect since they had higher probabilities to develop [
16]. In addition, the government at multiple levels played huge roles in decision-making in urban land management. The demarcation of urban construction boundaries would not only affect the psychological value and preference of developers but promote the occurrence of out-of-boundary development for impacts on the urban scales [
17]. The increased uncertainty of government tenure [
18] and regional competition for land use [
19] also influenced the decision-making behavior in land development. As a result, the combination of diverse tactics gave the gradual exploration of land use a variety of chances depending on how well the “personal preferences” of individual decision-makers were met [
20]. That is, land developments will be affected by micro decision-makers, who influence and lead to uncertainty in the laws of urban development.
Meanwhile, different decision-makers were not completely rational, and their cognitive systems were limited [
21,
22]. Due to the subjective judgments of utility and probability of uncertain exogenous factors, subjects would behave differently [
23]. However, judging was risky. While meeting uncertain risks, decision-makers had varied risk attitudes, which influence their decisions [
24]. Their aversion tendencies in risk attitudes intensify when they regard the risks of benefit-taking was less than their responsibility-taking [
25]. For example, when decision-makers faced a strong livelihood risk, their attitudes generally showed a tendency toward avoidance [
26], which finally affected their decisions. Some scholars had constructed the attitude utility functions (AUFs) to quantify risk attitudes. They pointed out that land development was closely related to risk attitudes by evaluating the spatial impact of land use decisions [
12]. Based on the previous research, Lu et al. [
27] explored the influencing factors of urban expansion patterns and found that reckless decision-makers were more likely to lead the urban sprawl. Han et al. [
28] explored the formation of urban settlements through analyzing the decision-makers’ risk perceptions and found that it was highly correlated with the power law. That was shown that the scilicet risk attitude had a certain influence on the power law. Although several studies revealed that risk attitudes influence the distribution of urban scales, these studies were based on the overall regional rule. Moreover, Wegmann et al. [
29] researched households in urbanized areas with different incomes. They revealed that people with more assets tend to have lower risk aversion, and risk attitudes vary with spatial distribution of asset owners. However, the study was limited to the relationship between assets and owners and did not discuss from the perspective of urban scale.
In summary, the power law has been widely applied to the study of the internal influence mechanism of urban scale worldwide. However, the existing studies have not deeply considered the impact of regional differences in internal risk attitudes, particularly from the perspective of micro-decision-making behaviors. Since regional differences in risk attitudes of various decision-makers affect regional urban development, this study sets regional differences in risk attitudes (while using computer simulation technology), yields to explore the influence of the power law at urban scale. Added to that, the study contributes to deepening the interpretive study at the urban scale distribution and promotes regional sustainable development.
In the following sections,
Section 2 shows the theoretical presentation of the power law and the risk decision.
Section 3 delivers the system modeling.
Section 4 reveals the simulation results of the modeling and the discussions will be generated in
Section 5. Finally,
Section 6 summarizes the conclusion and proposes suggestions for future research.
3. Model Construction
3.1. Basic Ideas
This paper is based on Netlogo and sets the simulation space to be a homogeneous plane, then divides it equally into several areas while applying two rules (regional quantitative differences in the same attitudes and mixed multi-attitudes), to represent regional different risk reference simulation scenarios. In the model, this study set the same time steps to record different development forms. Then count the settlement scale of them and rank it. Afterward, through the linear regression, the results are recorded to find the changing law of the urban scale’s power law. Finally, a comparative analysis is conducted with the distribution law of urban scale in the real world, and the impact of regional differences in risk attitudes is studied.
3.2. Computer Simulation Design
In the section, the step of the system modeling will be proposed:
- (1)
The plane space of the study is made up of 200 rows by 200 columns, using the concept of cellular automata, making each development unit a cell, which indicated the smallest land development unit. There are 32,400 cells, making every development area with 60 rows by 60 columns through nine equal area divisions. In addition, each area has the same possibilities to be developed. The cells have two states of development: developed and undeveloped, assigned 0 or 1, respectively. Moreover, the neighbors’ effect and scale-mixing modes’ attraction affect the development of one cell. Each cell’s developing possibilities are determined by a potential value as:
where
Q is the initial potential value of the cell,
N is the agglomeration strength coefficient,
S is the number of developed cells in the neighborhood, and
T denotes the largest neighbor’s settlement scale of the nearby target cell. The product of
S and
T is the hybrid mode attraction. If one cell has three developed cells in its neighbors, and the largest neighbors’ settlement scale is eight; next, the potential values are
(
Figure 3). The development principle follows the location of cells with higher development potential value has a higher probability to be selected.
- (2)
According to the risk attitude to have the new potential of land development, the attitude utility functions (AUFs) are numerically approximated as:
where
y is the utility based on risk attitudes,
α is a curving coefficient driving the shape of the AUF, can measure the different effects on the development potential caused by the same risk attitude.
In this study, α is equal to 1.2 and 1.4 in the above approximation, and x is the original value.
In terms of regional division, the largest number of the prefecture-level city in different provinces is 21, and the others are around 10 in China. Thus, it is equivalent to reality to divide the area into nine equal spaces. In addition, nine areas are more convenient to simulate, and the rules of model design include two scenarios (
Table 1):
The first one is the regional quantitative differences in the same attitudes: setting one-third, two-thirds, and all of the plane space as seeking or aversion attitudes in turn. The increasing percentage can be interpreted as the intensity of a shift in a particular attitude. The other area is created with a neutral attitude based on the environment, which is completely logical and has no risk effect. Therefore, the same attitudes apply to this circumstance.
The second one is the mixed multi-attitudes: it includes four types of settings: main risk-seeking, main risk-neutral, main risk-averse, and equality. In the first three settings, the ratio of main risk attitudes with the other two is 5:2:2, while the last one is 3:3:3.
- (3)
All the settings are choosing 15 as the time step, in this circumstance, and 15% of the cells have been developed in the whole space. At this time, the settlement form is clearer and can present obvious urban hierarchical features. Thus, it has a better representation effect and avoids the phenomenon of large settlements connecting with large settlements caused by the increase of time steps. In addition, each group takes the mean value five times, and finally counts the scale of the development of land in the region.
5. Discussion
The exponent of the power law is basically kept around −1 when explaining the internal laws of cities [
40], which means the scale system is more reasonable. However, this research shows that the slope gradually approaches or breaks −1, which is related to the high possibility of exploitation brought by agglomeration. In addition, the overall
R2 of the model is greater than 0.940, which indicates that it is close to the real world. Meanwhile, decision-making subjects with different risk preferences have a different tendency to distinct areas. The more favorable geographical conditions are [
41], the higher the level of economic development [
42] and the degree of cultural openness will be [
43], and the more strongly the risk-taking subjects pursue profit in advantageous areas will be [
44], for example, the southeast coast of China. Therefore, for exploring the laws of the real world under the influences of risk attitudes, this study considers six provinces in China as targeted regions: Shandong, Zhejiang, Hunan, Jiangxi, Gansu, and Guizhou. Data were mainly obtained from the “China Urban Construction Statistical Yearbook” [
45]. We took the prefecture-level cities as the research unit and selected the urban built-up area data from 2012 to 2020 are generated for statistical analysis. Referring to
Table 6, the slope and the primacy of different regions showed their respective size differences and changing trends, which reflect the power distribution characteristics within a region.
The absolute value of the slope in Shandong and Zhejiang Provinces is at a relatively high level, reaching almost 0.98. This phenomenon is similar to the averse model, which reflects the result that the seeking attitude causes a large slope value. But the primacy ratio is not larger, which is contrary to the simulation results. Since the Reform and Opening-up In China, these two provinces have a rapid economic development as large economic provinces in the coastal areas. Especially the priority development in their large cities, such as Jinan in Shandong and Hangzhou in Zhejiang, which have gathered many brands, resources, and industries. Since they have large differences in urban scales from regional small cities such as Liaocheng, Zaozhuang, Quzhou, and Lishui. In addition, the open environment is more likely to attract entrepreneurs and developers with active thinking, who will have a higher level of awareness and a reckless attitude toward performance and profitability [
22]. Therefore, policy innovation and land use development are more likely to be promoted with easier breakthroughs in operational boundary constraints. Moreover, Shandong province has two major economic construction cores cities: Qingdao and Jinan. The variation of built-up area of between two cities is only 35.49 m
2 in 2020. Neither of the scales of the two cities showed its potential as a capital city in the province. Therefore, the overall primacy ratio is low. The change flatter magnitude of the change represents indicates, because the urban planners in Zhejiang Province paid more attention to the development of small and medium cities, and the primacy ratio is also relatively small.
The slope and the primacy ratios of Hunan, Jiangxi Province are at the medium level, which is similar to the mainly neutral model. Data show that the monopoly position of the large city is not strong. The slope is lower than the whole southeast coastal area, which proves the impact of mixed various risk attitudes. On the one hand, the unique geographical location enables the two provinces to accept the influence of the eastern region and stimulates the potential for innovation and reform. On the other hand, the two provinces undertake the influence of the conservative culture of the western region, and the overall risk level is relatively neutral. Changsha-Zhuzhou-Xiangtan is the “one city” integration in Hunan Province, and its urban space development is far greater than that of other cities, while that of Western Hunan is the lowest [
46]. The implementation of the central China development policy has led to an increase in the indicators of construction land, based on the government’s “rational choice”, it has a tendency towards the dominant area [
43]. The introduction of the dot axis model is also manifested in more risky land development. Thus, the larger gaps between large, medium, and small cities have led to a significant increase in slope, which is consistent with the impact of reckless and averse attitudes in this model. The Jiangxi Province regional built-up areas in the north and south urban areas can be nearly seven times different in urban scales, which manifested as the “polarization effect”. The inclination of policy indicators attracts foreign investment and industrial agglomeration. The increase of investment of various reckless subjects occurs in advantageous areas, such as Gannan, and Yingtan industrial clusters. Though they are small-scale cities, there still have the attractive areas which have large-scale areas in the area.
As for Gansu and Guizhou Provinces, they are located inland with a low economic level. The phenomenon of the two provinces is similar to the mainly averse model, thus, the overall slope is at a relatively low level, but the change range is clear with a larger primacy ratio. Lanzhou is the development core in Gansu Province, with a Gross Domestic Product (GDP) of 283.7 billion (10 times that of the lowest city). The size of regional built-up areas of the city is 23 times different from Longnan Province. In addition, the urban scale development in areas such as Tianshui, Wuwei, and Pinglh havelarge differences. There is no radiation effect of a large metropolis. The built-up areas within the region of Guizhou Province are at most about 7 times different in urban scales. The GDP of Guiyang City reached 404 billion. It can be seen that the rapid development of the primary cities has led to large gaps in urban scales with other areas in these two provinces, where gather more reckless subjects. Furthermore, these two provinces are deeply affected by several factors such as regional historical and geographical environment, farming culture, resources, and environmental pressure. This situation fundamentally shaped the averse preferences of different agents such as urban managers and corporate investors [
43], particularly for small cities. Thus, the two provinces have a few large urban-scale cities, affected by aversion and the attitude of seeking.
Therefore, the various change characteristics of slope and primacy ratio among regions are mainly due to the differences in agglomeration benefits and varying degrees of risk attitudes affect. Core cities have higher economic levels and the better resource configuration. They have larger urban scales since they have clear labor force agglomeration, the policy preference, and the cultural openness. The power law at the urban scale also resembles to the phenomenon of the simulation scenario: the large settlements are concentrated but hold fewer numbers; the small settlements are scattered but with a larger number. When a dominant area has greater land development intensity, the impact of the seeking attitude is stronger. The underdeveloped areas that are less attractive can highlight differences in urban scales between large and small cities, which is similar to the mixed multi-attitudes. Added to that, the regional differences in risk attitudes render the urban scale having an agglomeration and a dispersion effects. The different degrees of influence make the power law change, which complies with the laws of the real world. We deduce that the social and economic functions undertaken by cities in the regional urban system are closely related to their urban scales. The development of urban lands in different regions in China is affected by risks. The advancement of reasonable spillover of risk-taking subjects benefits in promoting the layout of productivity and enterprise enrichment among regions, provinces and cities to form reasonable urban scale systems.
6. Conclusions and Prospects
To explore the influence of the differences in risk attitudes within regions in China, this study considers the increase of the returns and the AUF through computer simulation technologies to analyze the changing trends of urban settlements. As a result, the R2 values are all around 0.97, indicating that the regional risk difference setting still conforms to the power phenomenon. Based on the above, we concluded that:
- (1)
The seeking attitude in the scenarios of the same attitudes with regional quantitative differences and mixed multi-attitudes has a greater impact on the scales and levels of settlements. Precisely, the increased numbers of seeking attitude settings indicate the obvious phenomenon of large and small numbers of settlements. It is more prominent in the second scenario with the characteristics of significant agglomeration gains and rapid growth. The overall primacy ratio of the seeking model is larger in the first scenario. The agglomeration effect of the primary city plays a crucial role. However, in the second scenario, as the degree of aversion increases, the influence of risk attitudes becomes strong, and the absolute value of the slope tends to increase synchronously.
- (2)
Empirical studies of six provinces in China have found that the size and number of urban scales are highly related to the law of situational settlement size. The results show a phenomenon of large-scale cities holding small numbers, while the opposite occurs in small-scale cities. Areas with higher levels of economic development with opened policies correspond to higher slopes, which is relevant to the influences of seeking attitudes. Culturally conservative and resource-stressed areas are similar to the lower slopes of the effects of aversion attitudes; the excessive area remains at a moderate level. The primacy ratio is more similar to the mixed attitude scenario, with some volatility in each region. Economies play more role in core areas than the backward regions.
- (3)
Risk attitude exists in any decision. The uncertain development of land use leads to the phenomenon of uneven resource allocation and slow urbanization development at the urban scale during urban development. In the past, the concept of “Garden City Theory” in the urban system regards multiple regions as a whole. The ideas of centralization, metropolis, and border town system constantly emphasize integrated urban development. When the centralized provinces (districts) are formulated through policymaking, the polarization center in the region can be guided; the development preferences of risk-taking subjects for small cities can be stimulated; moreover, the “diffusion effect” and “ trickle-down effect” in economically developed areas will be strengthened for promoting the balanced development of regions. As for the decentralized provinces (districts), it will be beneficial to promote the development of regional economies, and cultural exchanges through improving urban economic activities’ degree of agglomeration, and guiding the agglomeration of population and capital.
This study aims at enriching the research on the influence of the power law on an urban scale. It provides a reference for new urbanization construction from the perspective of risk attitude control, in order to promote a complete urban structural system. However, in the procedure of the introduction in risk attitudes, this study does not consider the game mechanism among the government, the developers, or the residents. The simulated scenarios are also based on idealized states, without considering the heterogeneity of the real world, which requires further research.