## 3 Materials and Methods  
### Research Area  
This study selected prefecture-level cities in China as the study area. The choice of spatial scale on the employment problem is very important in the study; prefecture-level cities are the main economic and social units in China. Prefecture-level cities are marked by certain independence and significant levels of difference. Therefore, prefecture-level cities were selected as research subjects.  
Choosing Chinese cities as the study area of the employment multiplier reflects unique advantages. First, there are vast differences among many cities in China. The literature focuses on developed countries with high urbanization levels and low labor-supply elasticity. The formation process of China's urban system in the stage of rapid urbanization has unique advantages for studying the formation and evolution of employment problems. Second, in developed countries, particularly the United States, the manufacturing sector is not the main sector of the economy, but that economy is dominated by financial innovation and other activities. Therefore, the impact of changes in the trading sector on non-tradable sectors is not as significant as imagined. China is a manufacturing power, and the migration of manufacturing to developing countries, which is common in developed countries, is rare in China [22]. The driving role of the manufacturing industry in the urban economy is stronger in developing countries than in the case of China, a developed country in a period of transition. Third, both government and market forces affect urban and regional development [42], which is not covered by the current research. Further study of Chinese cities is of great significance for improving the employment multiplier theory and providing reference for developing countries.  
China is divided into three regions in the following analysis. The eastern region is mainly coastal provinces with the highest density of population and the most advanced urban economy. It is the most attractive region for internal migrants. The central region is also populous, but urban economy and attractiveness of cities in this region are moderate. The western region is sparsely populated and economically backward. Employment multipliers in cities located in different regions are very likely to be different and their influencing factors are expected to have both similarities and differences as well.  
According to the previous analysis, expanding cities were selected as the research unit, so it is necessary to identify and exclude shrinking cities. There are many ways to identify shrinking cities; this study chose the more direct method, which is the reduction of population size, or rate of change. In order to classify the types of urban growth or contraction more carefully, both population growth in municipal districts and population change in municipal districts relative to those of non-municipal districts were investigated. The formula is as follows:  
\[\Delta\text{P}_{\text{s}}\%_{\text{e}}=\frac{\left|\text{P}_{\text{s}(2018)}- \text{P}_{\text{s}(2008)}\right|}{\text{P}_{\text{s}(2008)}}-\text{P}_{\text{a}} \tag{1}\]  
\[\Delta\text{P}_{\text{s}}\%_{\text{e}}=\frac{\left|\text{P}_{\text{ns}(2018)} -\text{P}_{\text{ns}(2008)}\right|}{\text{P}_{\text{ns}(2008)}}-\text{P}_{\text {a}} \tag{2}\]  
\[\text{D}=\Delta\text{P}_{\text{s}}\%_{\text{e}}-\Delta\text{P}_{\text{ss}}\% \tag{3}\]  
where \(\text{P}_{\text{s}}\) is the municipal population, \(\text{P}_{\text{ns}}\) is the population of non-municipal districts, and D is the difference in the rate of population change between municipal and non-municipal districts. Here, the change in the rate of the population of both municipal and non-municipal districts is subtracted from the national average population growth rate for that period, \(\text{P}_{\text{a}}\), to exclude the effects of overall population growth. If D is less than 0, the city is in the contraction stage and was not included in the research scope. As a result, 169 cities were selected for the study (Figure 1).