# Optimal Pricing in a Rented 5G Infrastructure Scenario with Sticky Customers
## 4 Operators' Costs, Revenues, and Profits
### The Demand Function  
In this section, we describe the demand function, which provides us with the number of customers joining one of the operators. We denote the overall number of potential customers (the basin) as \(n\). We need to obtain the number of actual customers of each operator. We obtain that value from the overall number of potential customers through a chain of factors. First, we obtain the number of actual customers through the factor \(f_{\text{tot}}\) (which can be obviously defined as the ratio of the number of actual customers to the number of potential customers); by actual customers, we mean customers joining either the MNO or one of the MVNOs. Then, we subdivide the actual customers among the operators through an additional factor \(f_{x}\), where \(x\) may take one of the following values: MNO for the infrastructure owner (the MNO) or \(r_{i}\) for the \(i\)-th renter (MVNO). The number of customers joining the operator \(x\) is finally given by the following composition  
\[n_{x}=n\cdot f_{\text{tot}}\cdot f_{x}. \tag{3}\]  
We can now see how to derive both factors \(f_{\text{tot}}\) and \(f_{x}\). We start with the proportion \(f_{\text{tot}}\) of actual customers. Here, we postulate that it is an exponential function of the average price \(\overline{p}\), which is simply the arithmetic average of the prices imposed by the MNO (\(p_{\text{MNO}}\)) and the MVNOs (\(p_{\text{MVNO}}^{(i)}\) for the \(i\)-th MVNO):  
\[\overline{p}=\frac{p_{\text{MNO}}+\sum_{i=1}^{N_{R}}p_{\text{MVNO}}^{(i)}}{1+ N_{R}}. \tag{4}\]  
It is to be noted that MVNOs can impose different prices in our model. As to the fraction of actually subscribing customers, we employ an exponential demand function. The exponential function is employed in many contexts [48] and has been adopted for communications services, e.g., in [49; 50; 51]. In our case, the price is considered as the average price offered by operators:  
\[f_{\text{tot}}=e^{-z\overline{p}} \tag{5}\]  
In this equation, the coefficient \(z>0\) is the absolute value of the quasi-elasticity of the overall demand function with respect to the average price. In fact, the quasi-elasticity with respect to the average price \(\overline{p}\) (we employ the quasi-elasticity instead of the elasticity because the demand function outputs a value in the \((0,1]\) range) is  
\[\epsilon=\frac{1}{f_{\text{tot}}}\frac{\partial f_{\text{tot}}}{\partial \overline{p}}=e^{z\overline{p}}e^{-z\overline{p}}(-z)=-z. \tag{6}\]We can now consider the function \(f_{x}\), which determines how customers are divided between the operators. We call this function the repartition function. As hinted in the Introduction, customers may have reasons for selecting either operator other than a lower price. The reasons may range from the preferred operator holding a more established brand to its having a better reputation. In fact, [PERSON] and [PERSON] have shown that customer loyalty implies an increased likelihood of repurchasing services from the same provider and lower price sensitivity [52]. In turn, both [PERSON] et al. and [PERSON] et al. have shown that customer satisfaction is a key determinant of loyalty [53; 54]. Because [PERSON] et al. identified service value and corporate image as antecedents of customer satisfaction [55], we can easily conclude that those two factors at least may drive customers' preferences in addition to sheer pricing. The repartition function must then consider price as a relative variable rather than as on that determining customers' preferences, where the latter choice would lead the customer to always choose the operator proposing the lowest price. In this paper, we call a customer who prefers operator A even though operator B is offering lower prices a sticky customer, because they tend to choose an operator despite lower-priced offers by other operators. Stickiness is a phenomenon different from, yet similar to, lock-in, where customers are bound to an operator for technological or contractual reasons.