# Governance and Dynamic Efficiency with Network Structure in the Brazilian Natural Gas Utilities
## 2 Materials and Methods
### Dynamic DEA with Network Structure: A Slacks-Based (DNSBM)
The linking variables are essential for the analysis between periods, as they evaluate whether there are variations in the efficiency of the companies when they are exposed to linking variables between periods [38]. Therefore, we present in Figure 2 the proposed model for the DNSBM in our study for the efficiency of piped gas distributors in Brazil. In this structure proposed by the DNSBM model, we indicate the two divisions, operational technical efficiency and financial efficiency, with their respective variables. The connection between the divisions is made through carry-overs, considering the initial condition \(\mathrm{c_{K}}\left(\mathrm{w=1,\ldots,\mathrm{c_{K}}}\right)\) carry-overs from the period \(\mathrm{t-1}\). It is important to highlight that the Network variable is considered the main variable for analyzing piped gas companies, as it is through the distribution network that gas reaches residences, industries, and commerce. Therefore, the information about the distribution network must be precise, because the data are used to monitor its production capacity and to publicize its efficiency. Figure 2: DEA DNSBM model of a piped gas distribution company for any period \(\mathrm{t}\). Source: Own elaboration. However, the length of the network may not be measured by its different sizes, nor by the different compositions of its construction material, because according to [PERSON], [PERSON], and [PERSON] [39], these can influence the efficiency result of a distributor. According to [PERSON], [PERSON], and [PERSON] [24], there are four factors that affect the efficiency of gas distribution due to its cost: delivery volume, number of customers, network length (pipeline), and supply area (concession). However, in the literature, there is no consensus on how these variables should be treated, as some consider them endogenous, while others consider them exogenous. It is important to highlight that the literature investigating the determinants of efficiency in network industries emphasizes that some contexts or exogenous variables can greatly influence the operational efficiency of gas suppliers [40; 41; 42]. Another point is that piped gas distributors in Brazil do not operate under identical or even similar contextual conditions. In fact, a number of factors can affect their efficiency, such as being publicly or privately owned, the size of the concession area or market, and. in particular, the number of customers as one of the main causes of economies of scale, which consequently influences the gain in efficiency. It is observed that some of the variables used in the DNSBM model are not present in the literature as being used in research in the piped gas segment, which reinforces the importance of our study. This is especially true of the variables used in the DIV EF, which reflect how monetary resources impact the efficiency of the distribution companies. However, creating monetary efficiency measures is problematic. First, it is difficult to obtain a consistent series of values because accounting methods and revaluation policies are not uniform within the companies. Therefore, variations in efficiency may reflect different accounting standards. Secondly, there is still debate on which is the correct approach. Once the model is defined, it is necessary to choose the production scale that adjusts the model in terms of Constant Returns to Scale (CRS) and Variable Returns to Scale (VRS). The methodology used to determine the appropriate returns to scale for this study is based on statistical tests initially developed by [43] and later refined by [44], who presented the paired sample \(t\)-test as a solution. The test was performed with the scores generated by the CRS model and with the scores generated by the VRS model. The test result indicates that the efficiency coefficients in the CRS and VRS models are significantly different at a 5% probability level. Thus. the model chosen in this study was the VRS.