# Optimal Pricing in a Rented 5G Infrastructure Scenario with Sticky Customers
## Appendix B Maximization of Profits  
In this appendix, we derive the equations whose solution allows us to maximize the profits of operators. We consider first the infrastructure owner (i.e., the MNO) and then the renters (MVNOs). Profit maximization will be sought by zeroing the derivative of profit. For the infrastructure owner, after recalling Equation (15) and applying the product rule for derivatives, we have  
\[ \begin{split}\frac{\partial\Pi_{\text{MNO}}}{\partial p_{\text{ MNO}}}&=ne^{-z\overline{p}}\left\{-\frac{z}{1+N_{R}}\bigg{[}e^{\xi\frac{P_{ \text{MNO}}}{P_{\text{MNO}}}}(p_{\text{MNO}}-\gamma B)-(1+\alpha)(C_{0}+mB)+ \gamma B\bigg{]}\right\}\\ &+ne^{-z\overline{p}}\bigg{[}e^{\xi\frac{P_{\text{MNO}}}{P_{ \text{MNO}}}}\bigg{[}1-\xi\frac{P_{\text{MNO}}}{\overline{p}_{\text{MNO}}}+ \gamma B\frac{\xi}{\overline{p}_{\text{MNO}}}\bigg{]}\bigg{\}}=0,\end{split}  \tag{18}\]  
which leads to the following equation  
\[e^{\xi\frac{P_{\text{MNO}}}{P_{\text{MNO}}}}\bigg{[}1-(p_{\text{MNO}}-\gamma B) \bigg{(}\frac{z}{1+N_{R}}+\frac{\xi}{\overline{p}_{\text{M}VNO}}\bigg{)} \bigg{]}=\frac{z}{1+N_{R}}[\gamma B-(1+\alpha)(C_{0}+mB)] \tag{19}\]  
Unfortunately, Equation (19) is a transcendental equation that cannot be solved exactly for \(p_{\text{MNO}}\), and we have to resort to a numerical solution. As to the profit of the generic \(i\)-th renter, we can use the following expression of the profit:  
\[ \begin{split}\Pi^{(i)}_{\text{MVNO}}&=n^{(i)}_{\text{ MVNO}}p^{(i)}_{\text{MVNO}}-n^{(i)}_{\text{MVNO}}\frac{C^{(i)}_{\text{MVNO}}}{n^{(i)}_{ \text{MVNO}}}\\ &=n^{(i)}_{\text{MVNO}}\Bigg{(}p^{(i)}_{\text{MVNO}}-\frac{C^{(i) }_{\text{MVNO}}}{n^{(i)}_{\text{MVNO}}}\Bigg{)}\\ &=nf_{\text{tot}}f^{(i)}_{\text{MVNO}}\Bigg{(}p^{(i)}_{\text{MVNO} }-\frac{C^{(i)}_{\text{MVNO}}}{n^{(i)}_{\text{MVNO}}}\Bigg{)}\end{split}  \tag{12}\]  
Because we cannot act on the overall basin dimension \(n\), we can consider the following quantity to be maximized  
\[\Pi^{(i)}_{\text{MVNO}}\propto f_{\text{tot}}f^{(i)}_{\text{MVNO}}\Bigg{(}p^{( i)}_{\text{MVNO}}-\frac{C^{(i)}_{\text{MVNO}}}{n^{(i)}_{\text{MVNO}}}\Bigg{)} \tag{13}\]  
After recalling Equation (2), we also note that the following quantity does not depend on the strategic variable \(p^{(i)}_{\text{MVNO}}\)  
\[\frac{C^{(i)}_{\text{MVNO}}}{n^{(i)}_{\text{MVNO}}}=(\eta_{i}\alpha+\beta_{i} )(C_{0}+mB)+\gamma B. \tag{14}\]  
The derivative to be zeroed is then  
\[ \begin{split}\frac{\partial\Pi^{(i)}_{\text{MVNO}}}{\partial p ^{(i)}_{\text{MVNO}}}&\propto\frac{\partial(f_{\text{tot}}f^{(i )}_{\text{MVNO}})}{\partial p^{(i)}_{\text{MVNO}}}\Bigg{(}p^{(i)}_{\text{MVNO} }-\frac{C^{(i)}_{\text{MVNO}}}{n^{(i)}_{\text{MVNO}}}\Bigg{)}+f_{\text{tot}} f^{(i)}_{\text{MVNO}}\\ &=\Bigg{[}-\frac{z}{1+N_{R}}f_{\text{tot}}f^{(i)}_{\text{MVNO}}+f _{\text{tot}}\frac{\partial f^{(i)}_{\text{MVNO}}}{\partial p^{(i)}_{\text{MVNO }}}\Bigg{]}\Bigg{(}p^{(i)}_{\text{MVNO}}-\frac{C^{(i)}_{\text{MVNO}}}{n^{(i)} _{\text{MVNO}}}\Bigg{)}+f_{\text{tot}}f^{(i)}_{\text{MVNO}}\\ &=f_{\text{tot}}\Bigg{\{}\Bigg{[}-\frac{z}{1+N_{R}}f^{(i)}_{ \text{MVNO}}+\frac{\partial f^{(i)}_{\text{MVNO}}}{\partial p^{(i)}_{\text{MVNO }}}\Bigg{]}\Bigg{(}p^{(i)}_{\text{MVNO}}-\frac{C^{(i)}_{\text{MVNO}}}{n^{(i)} _{\text{MVNO}}}\Bigg{)}+f^{(i)}_{\text{MVNO}}\Bigg{\}}.\end{split}  \tag{15}\]  
Because the term \(f_{\text{tot}}\) cannot be brought down to zero, the equation to solve is finally  
\[\Bigg{[}-\frac{z}{1+N_{R}}f^{(i)}_{\text{MVNO}}+\frac{\partial f^{(i)}_{\text {MVNO}}}{\partial p^{(i)}_{\text{MVNO}}}\Bigg{]}\Bigg{(}p^{(i)}_{\text{MVNO} }-\frac{C^{(i)}_{\text{MVNO}}}{n^{(i)}_{\text{MVNO}}}\Bigg{)}+f^{(i)}_{\text{ MVNO}}=0, \tag{16}\]  
which can be put in the following form after a few algebraic passages:  
\[