# Optimal Pricing in a Rented 5G Infrastructure Scenario with Sticky Customers
## 4 Operators' Costs, Revenues, and Profits
### Cost Models  
The structure of costs is different for the MNO offering its infrastructure for rent and the MVNOs renting it. The expectations for the MVNOs are to have a slimmer set of costs (see the considerations on the strategic advantage of MVNOs in [36]). These lower costs allow their business to be profitable though they cannot expect to set prices as high as those of MNOs. In fact, the latter category is perceived as more reliable or having a stronger brand image, as suggested in [37]. Hence, the structure of costs is essential to set prices for both types of operators. In this section, we describe the cost models separately for the MNO and the MVNOs. In describing the models, we rely on the distinction between CAPEX (capital expenditures) and OPEX (operational expenditures). CAPEXs include all the expenses made by the company into fixed, long-duration assets. In telecommunications, these include the expenses incurred to purchase, e.g., routers, optical fibres, antennas, and towers. On the other hand, OPEXs include the recurring expenses incurred during day-to-day activities and, more generally, all the expenses that are consumed within a year. OPEXs include consumables and renting fees. The distinction between CAPEX and OPEX is well established in the telco sector, as adopted, e.g., by [38] for LTE (long-term evolution) networks, Ref. [39] for access networks, and by [40; 41; 42] for the core network. However, CAPEX and OPEX refer to different timescales, whereas OPEXs exist during a one-year horizon, CAPEXs extend their effect over several years. In order to arrive at a common time frame reference in the budget equation for the operators, we distribute CAPEXs over the lifetime of the assets. The lifetime can be generally deemed as dictated by the accounting rules of the company for their amortization and recording in ledgers. Here we assume that such distribution over time has been accomplished through a straight-line method (i.e., a uniform distribution of costs over the asset's lifetime) [43] so that we can obtain a yearly CAPEX to be added to the OPEX to arrive at the overall expenses over a year. Hereafter, we make the following assumptions to define our cost model:  
* The CAPEXs of the MNO are linearly related to the bandwidth;
* OPEXs are proportional to the CAPEXs;
* The CAPEXs and OPEXs of MVNOs are a fraction of those pertaining to the MNO;
* Renting is priced linearly in the bandwidth so that the renting fee is expressed per unit bandwidth. The choice of a linear model for the relationship between CAPEX and bandwidth is based on the data provided by [44]. The assumption of proportionality between CAPEX and OPEX allows us to consider CAPEX through a single proportionality factor. Though in principle OPEX could be determined through a detailed bottom-up approach (see, e.g., the procedure outlined by [45]), OPEXs appear to be proportional to CAPEX in many real cases. For example, Ref. [45] estimate OPEX costs for equipment installation to amount to 30% of the CAPEX costs, and their final estimation for the OPEX/CAPEX ratio is 75%. Though the exact value of the ratio may depend on the operator's business model and the specific network, considering OPEX to be proportional to CAPEX is a common assumption, adopted, e.g., in [46] for virtual networks, where the OPEX/CAPEX ratio appears to depend on the virtual network function utilization rate, or in [47] for data centres, where the OPEX/CAPEX ratio is seen to move in the 31-61% range. We can now describe the overall cost function for the MNO and the MVNOs. For the MNO we have  
\[C_{\text{MNO}}=(n_{\text{MNO}}+\sum_{i=1}^{N_{R}}n_{\text{MVNO}}^{(i)})(C_{0}+mB)(1+ \alpha), \tag{1}\]  
where \(n_{\text{MNO}}\) and \(n_{\text{MVNO}}^{(i)}\) are, respectively, the number of customers of the MNO and the number of customers of the \(i\)-th MVNO, \(C_{0}\) is the zero-consumption fee in the linear cost model, \(B\) is the bandwidth, \(m\) is the marginal cost of bandwidth, and \(\alpha\) is the OPEX/CAPEX ratio.