This article uses a logarithmic growth model to analyze the correlation between the national economy and real estate. It reveals the Granger causality between the national economy and the real estate economy. The results show a long-term equilibrium relationship and a two-way Granger causality between real estate prices and economic growth. Excessive growth in real estate prices will create bubbles and will also drive economic growth backward.

#### Keywords

- Logarithmic growth function
- Economic development
- Real estate
- Logistic model

#### MSC 2010

- 33B30

Economic change is one of the important driving factors of urbanization. And urbanization is a social consequence of economic development. The relationship between the level of urbanization and economic development is a common research topic in multiple disciplines such as geography and economics [1]. For Western countries, there are more studies around the 1950s. In the 1980s, the focus of attention shifted to the urbanization and economic development of developing countries. China is currently in a period of rapid urbanization. Industrialization is proceeding rapidly, and the industrial structure is gradually adjusted. It is time to study the relationship between urbanization and economic development. As early as the 1980s, Chinese scholars began to explore the relationship between urbanization and the level of economic development. In recent years, research results from various angles have been reported continuously.

The main function of science is to explain and predict, and mathematical modeling is an important way to realize the two basic functions of science. The main purpose of explanation is to reveal causality, while prediction (including prediction) is to judge unknown results. A simple and convenient way to deal with many research methods is to find a suitable measure. Reveal the numerical relationship between these two measures and express it with mathematical equations. This results in a concise and easy-to-understand mathematical model. The model is just the right simplification of the real world [2].

Around the 1980s, many scholars tried to establish a relationship model between urbanization and economic development. The growth of urbanization level has a fixed upper limit, while the level of economic development has no clear upper limit [3]. Therefore, the growth rate of urbanization corresponding to economic development will eventually become smaller and smaller. In this way, if the per capita economic output is the independent variable, the urbanization level is the dependent variable. Establishing the functional relationship in this way can get a convex curve in the second half of the arc. Many functions can give this kind of curve. The content includes logarithmic function, power exponential function, hyperbolic function, two-type exponential function, logistic function, etc. This article intends to systematically summarize three mathematical models describing the relationship between urbanization and the level of economic development. Reveal the hidden and previously unknown dynamic mechanism behind it. At the same time, it clarifies the application methods, scope of application, interpretation, and prediction effects of different models.

Measurement is the basic connection between mathematical modeling and empirical analysis. The measurement is easier to understand and establishes a mathematical model. To this end, it is first necessary to explain a few basic measures of urbanization and economic development [4]. The commonly used measure to reflect in a region is the level of urbanization:

In formula (1), L represents the level of urbanization.

Formula (2)

The above formula represents a hyperbola with special parameters. In addition, the rate of change of urbanization level can also be used to reflect the rate of urbanization. The absolute speed formula is:

Correspondingly, the relative speed can be expressed as:

The speed of urbanization can be expressed in the form of differentiation.

There are two simplest measures of the economic development level of a region. The first is the per capita gross regional product, that is, the per capita GDP. The second is per capita national income, referred to as per capita income. GDP and national income are both related and different. National income can only be obtained after depreciation, indirect taxes, transfer payments, and government subsidies in GDP are converted [6]. The average statistical analysis is carried out based on the per capita GDP and per capita income of all countries globally. The results show that there is an allometric relationship between the two:

The scaled index is slightly greater than 1. Since the scale index of formula (7) is very close to 1, the relationship between per capita income and per capita GDP is approximately proportional. Therefore, if a certain mathematical equation is satisfied between the urbanization level and per capita income, then per capita GDP is used to replace per capita income [7]. The functional form of the equation remains unchanged.

The first mathematical model of the relationship between the level of urbanization and the level of economic development is the logarithmic model. Usually expressed as:

In formula (8), x represents per capita output value or per capita income. L represents the level of urbanization. A and b are parameters greater than 0. There are many studies on this model at home and abroad, mainly based on cross-sectional data from countries worldwide to describe the relationship between the proportion of urban population and per capita output value.

The second mathematical model of the relationship between the level of urbanization and the level of economic development is the power index relationship model. It is usually expressed as

In formula (9), c and d are parameters greater than zero. Other symbols are the same as formula (8). Take the logarithm of equation (9) to obtain the dual logarithmic, linear relationship

The parameter

This paper proposes the Logistic model to examine the relationship between China's per capita output value x and urban-rural ratio O to obtain an exponential function:

In formula (11), C and k are parameters. We substitute equation (11) into equation (3) to get the Logistic function immediately:

Formula (12)

Formula (13)

Formula (14), ^{2} = 0.878 (Figure 1). The above formula can be equivalently expressed as

Furthermore, the model parameters for 2012 and 2017~2019 can be calculated. The model parameters from 2013 to 2016 were obtained by interpolation (Table 1). The results show that the model scale coefficient C's estimated value decreases year by year. In contrast, the estimated value of the parameter k reflects the rate of change increases year by year [9]. We use the same data to test the single logarithmic and double logarithmic models, and the goodness of fit is relatively low (see Table 1). Generally speaking, the cross-sectional data of various provinces in China are most suitable to be described by the Logistic function, followed by the logarithmic function, and then the power exponential function.

The model parameters of the relationship between urban-rural ratio and GDP per capita in various regions of China (2012~2020).

Logarithm | Single logarithm | Double logarithm | |||
---|---|---|---|---|---|

Model parameters | C | k | R | R | R |

2012 | 0.255 | 1.078 | 20.851 | 20.818 | 20.776 |

2013 | 0.296 | 0.837 | - | - | - |

2014 | 0.307 | 0.738 | - | - | - |

2015 | 0.304 | 0.681 | - | - | - |

2016 | 0.301 | 0.63 | - | - | - |

2017 | 0.311 | 0.618 | 0.898 | 0.849 | 0.84 |

2018 | 0.321 | 0.542 | 0.899 | 0.847 | 0.844 |

2019 | 0.329 | 0.467 | 0.899 | 0.847 | 0.846 |

2020 | 0.319 | 0.424 | 0.878 | 0.833 | 0.801 |

The mathematical model of the human geography system is non-unique. Different system evolution conditions lead to different models. Different models reflect different dynamic mechanisms. The key lies in the purpose of these models. The purpose of modeling urbanization and economic development is to explain theoretically and forecast in practice (Table 2). However, to effectively use these models to carry out interpretation and prediction work, it is necessary to reveal the meaning of the parameters of the model and the dynamic mechanism behind the model.

The main uses of the urbanization-economic development level relationship model.

Modeling purpose | Details |
---|---|

explain | (1) Reveal the causal relationship between urbanization and economic development (who decides who) |

(2) Determine whether urbanization and industrial development are in harmony (whether urbanization lags behind industrialization) | |

(3) Understand the dynamic mechanism of urbanization and economic development (what are the control variables) | |

Predict | (1) Given the per capita income of a region, estimate the level of urbanization in the region |

(2) Given the urbanization level of a region, estimate the per capita income of the region |

First, consider the single logarithmic model. Take the derivative of equation (2) to get the rate of change of urbanization level corresponding to per capita output value:

This shows that for every unit increase in per capita output value, urbanization increases by

In formula (17), _{0} is a constant about the level of urbanization. The logarithmic model can only be established when the level of urbanization changes linearly and the per capita output value increases exponentially. What the power exponent model reflects is an allometric growth relationship. Take the derivative of equation (9) to obtain the rate of change of the urban population proportion corresponding to the level of economic development:

From this, the allometric coefficient is obtained:

The allometric growth coefficient in geography and biology is also the elasticity coefficient in economics. 1) The rate of change of the urbanization level is directly proportional to the urbanization level. The level of urbanization and the level of economic development jointly control the rate of change. 2) The relative growth rate of urbanization level to the relative growth rate of per capita output value is a constant. This is the basic meaning of allometric growth, and this constant is the allometric coefficient

In formula (20),

Due to the parameter

In formula (22), _{0} is a constant about the per capita output value. The first formula implies that the level of urbanization is a logistic curve in the time direction. The second formula shows that the per capita output value increases linearly. This means that the logistic growth process of the urbanization level in the time direction is the premise of the logistic change of the corresponding economic level.

In addition to empirical statistical standards, the model selection also has theoretical symmetry standards. If a mathematical model has invariance under a certain transformation, it has a kind of symmetry. The more symmetrical the model, the more widely applicable it is. The inverse function of the logarithmic model has translation invariance. The power exponential model has scale invariance. And the Logistic model has translation invariance. They are all symmetrical models [12]. Therefore, we can express the main points of the model comparative analysis as shown in Table 3.

Comparison of the characteristics of the three urbanization-economic development level relationship models.

Model | Logarithmic model | Power Exponent Model | Logistic model |
---|---|---|---|

relation | Single logarithm | Double logarithm | Logarithm |

Variation characteristics | From fast to slow, fast first, then slow | From fast to slow, fast first, then slow | Fast in the middle and slow at both ends |

Law of change | Characteristic scale | Featureless scale | Characteristic scale |

Rate of the change control variable | Per capita output value (economic variable) | Per capita output value and urbanization level (two variables) | Urbanization level (urban variable) |

Kinetic characteristics | The level of urbanization changes linearly, and the per capita output value increases exponentially | Both the level of urbanization and per capita output value have increased exponentially | Logistic growth of urbanization level, linear growth of per capita output value |

symmetry | Translational symmetry | Scale symmetry | Translational symmetry |

1) The logarithmic model is suitable for areas dominated by economic variables, and it reveals a single logarithmic, linear relationship. Suppose this model represents the situation of a region, the rate of change in the level of urbanization corresponding to the level of economic development changes from fast to slow. With the development of the social economy, the impact of per capita output value on urbanization is weaker.

2) The power index model is suitable for areas where economic variables and urban variables are balanced. What it reveals is a double logarithmic, linear relationship. If this model represents the situation of a region, the rate of change of urbanization level corresponding to the level of economic development also changes from fast to slow. The rate of change is slower than the logarithmic model, and the change curve has no characteristic scale. With the development of the social economy, the impact of per capita output value on urbanization is weaker.

3) Logistic model is suitable for areas dominated by urban variables. What it reveals is a logarithmic, linear relationship. Suppose this model represents the situation of a region. In that case, the rate of change of the urbanization level corresponding to the level of economic development is slow at both ends and fast at the middle. The change curve has characteristic scales. The impact of per capita output value on urbanization in the initial stage is relatively weak and gradually strengthened.

The purpose of modeling the relationship between urbanization and the economic development level of the three models is the same. When the objects described by these algorithms have different background conditions, the model structure reflects different dynamic characteristics. The exponential model and the power exponential model are the results of predecessors, and the Logistic model is the innovation of this article. The analysis and comparison of the dynamic mechanism of different models is also the main innovation of the article.

#### The main uses of the urbanization-economic development level relationship model.

Modeling purpose | Details |
---|---|

explain | (1) Reveal the causal relationship between urbanization and economic development (who decides who) |

(2) Determine whether urbanization and industrial development are in harmony (whether urbanization lags behind industrialization) | |

(3) Understand the dynamic mechanism of urbanization and economic development (what are the control variables) | |

Predict | (1) Given the per capita income of a region, estimate the level of urbanization in the region |

(2) Given the urbanization level of a region, estimate the per capita income of the region |

#### The model parameters of the relationship between urban-rural ratio and GDP per capita in various regions of China (2012~2020).

Logarithm | Single logarithm | Double logarithm | |||
---|---|---|---|---|---|

Model parameters | C | k | R | R | R |

2012 | 0.255 | 1.078 | 20.851 | 20.818 | 20.776 |

2013 | 0.296 | 0.837 | - | - | - |

2014 | 0.307 | 0.738 | - | - | - |

2015 | 0.304 | 0.681 | - | - | - |

2016 | 0.301 | 0.63 | - | - | - |

2017 | 0.311 | 0.618 | 0.898 | 0.849 | 0.84 |

2018 | 0.321 | 0.542 | 0.899 | 0.847 | 0.844 |

2019 | 0.329 | 0.467 | 0.899 | 0.847 | 0.846 |

2020 | 0.319 | 0.424 | 0.878 | 0.833 | 0.801 |

#### Comparison of the characteristics of the three urbanization-economic development level relationship models.

Model | Logarithmic model | Power Exponent Model | Logistic model |
---|---|---|---|

relation | Single logarithm | Double logarithm | Logarithm |

Variation characteristics | From fast to slow, fast first, then slow | From fast to slow, fast first, then slow | Fast in the middle and slow at both ends |

Law of change | Characteristic scale | Featureless scale | Characteristic scale |

Rate of the change control variable | Per capita output value (economic variable) | Per capita output value and urbanization level (two variables) | Urbanization level (urban variable) |

Kinetic characteristics | The level of urbanization changes linearly, and the per capita output value increases exponentially | Both the level of urbanization and per capita output value have increased exponentially | Logistic growth of urbanization level, linear growth of per capita output value |

symmetry | Translational symmetry | Scale symmetry | Translational symmetry |

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