## 4 Methodology and Data  
### Calculation of Total Factor Productivity for Yangtze River Inland Ports  
Total factor productivity (TFP) is a crucial indicator for measuring port productivity, reflecting the rate of change in total output relative to total inputs [3,40,41]. TFP aggregates multiple input (\(x\)) and output (\(y\)) indicators to measure or decompose changes in port productivity over time or among different ports [42]. In this regard, the Malmquist productivity index (MPI) is an effective tool for measuring changes in port TFP between two time periods and decomposing the sources of efficiency change [43]. The Malmquist TFP index measures TFP changes by comparing the distance of each data point relative to a common technology, with its main advantage being that it does not rely on input and output prices or related market clearing assumptions [40,42]. While retaining the advantages of the traditional Malmquist productivity index, we have adopted the more advanced global Malmquist productivity index as a tool for measuring changes in the productivity of Yangtze River inland ports. This index, developed by [PERSON] and [PERSON] [33], not only meets the circularity assumption but also avoids the infeasibility of mixed-period models that may arise in the traditional Malmquist productivity index [44,45]. Specifically, the formula for measuring TFP changes using the global Malmquist productivity index is as follows:  
\[TFPCH=\frac{TFP_{i}^{t+1}}{TFP_{i}^{t}}=M_{i}^{G}(y_{i,t+1},x_{i,t+1},y_{i,t}, x_{i,t})=\frac{D_{i}^{G}(y_{i,t+1},x_{i,t+1})}{D_{i}^{G}(y_{i,t},x_{i,t})} \tag{4}\]  
where variables \(x\) and \(y\) respectively represent the input vector and the output vector of the port. \(D_{i}^{G}\) is the output-oriented distance function based on global reference technology. \(t\) and \(t+1\) represent two time periods. \(M_{i}^{G}\) can be used to measure the TFP change in port \(i\) from period \(t\) to \(t+1\). \(M_{i}^{G}>1\) indicates an increase in TFP for Yangtze River inland port \(i\), while \(M_{i}^{G}=1\) and \(M_{i}^{G}<1\) respectively denote no change and a decrease in TFP for Yangtze River inland port \(i\). Under the VRS (variable returns to scale) condition, the global MPI can be decomposed in three directions using the method of [PERSON] [46], with the decomposition process as follows:  
\[ \begin{split}& M_{i}^{G}(y_{i,t+1},x_{i,t+1},y_{i,t},x_{i,t})= \frac{D_{i}^{G}\big{(}y_{i,t+1},x_{i,t+1}\big{)}\times SE\big{(}y_{i,t+1},x_{i, t+1}\big{)}}{D_{i}^{G}(y_{i,t},x_{i})\times SE\big{(}y_{i,t},x_{i,t}\big{)}}\\ &=\frac{D_{i}^{I+1}\big{(}y_{i,t+1},\ x_{i,t+1}\big{)}}{D_{i}^{G} \big{(}y_{i,t},x_{i,t}\big{)}}\times\Bigg{[}\frac{D_{i}^{G}\big{(}y_{i,t+1},x_{ i,t+1}\big{)}}{D_{i}^{G}\big{(}y_{i,t+1},x_{i,t+1}\big{)}}\Bigg{]}\times\frac{ SE\big{(}y_{i,t+1},\ x_{i,t+1}\big{)}}{SE\big{(}y_{i,t},x_{i,t}\big{)}}\\ &=TEC_{i,(t,t+1)}\times\Bigg{[}\frac{\text{BPC}_{i,j+1}^{G} \big{(}y_{i,t+1},x_{i,t+1}\big{)}}{\text{BPC}_{i}^{G}\big{(}y_{i,t},x_{i,t} \big{)}}\Bigg{]}\times SEC_{i,(t,t+1)}=TEC_{i,(t,t+1)}\times BPC_{i,(t,t+1)} \times SEC_{i,(t,t+1)}\end{split}  \tag{5}\]  
To explore the sources of productivity changes in Yangtze River ports, as demonstrated in Equation (5), we decomposed port productivity changes under the assumption of variable returns to scale (VRS) into three parts: pure technical efficiency change (TEC), best practice change (BPC), and scale efficiency changes (SECs) [45]. Specifically, a TEC value greater than 1 indicates an improvement in the port's technical efficiency, whereas a value less than 1 suggests a decline in technical efficiency. The BPC value reflects how closely port technology aligns with global technology development, where a BPC value greater than 1 indicates that the port's technology has become closer to the global technology level during the comparison period, and a value less than 1 indicates that the port's technology has moved away from the global technology level.