## 5 Optimal Parking Pricing Scheme  
Based on the equilibrium results derived in the previous sections, we further examine the optimal design of parking prices charged at the central cluster to minimize the total travel cost. Recalling the analysis in Section 2.2, the parking preferences and the occurrence of equilibrium cases depend on \(\Delta p\), \(\Delta v\) and \(\Delta w\). Additionally, we can obtain the transition patterns of equilibrium cases based on the relative magnitudes of \(\Delta p\), \(\eta^{a}\Delta v\) and \((\eta^{h}-\beta)\Delta w\). Here, we investigate the optimal design of parking management policies following the same way. When \(\eta^{a}\Delta v=(\eta^{h}-\beta)\Delta w\), i.e., in Case 5, both AV commuters and HV commuters are indifferent between the closer and the farther parking clusters. The equilibrium problem is reduced to a simplified version with only one parking cluster. In the following, we mainly focus on the scenario under condition \(\eta^{a}\Delta v<(\eta^{h}-\beta)\Delta w\). For the scenario under condition \(\eta^{a}\Delta v>(\eta^{h}-\beta)\Delta w\), we can perform the analysis following similar procedures. The optimal parking pricing scheme for the central cluster can be achieved by exploring the tendency of the total travel cost \(\pi\) when the parking capacity \(K_{1}\) and the difference in parking fees \(\Delta p\) jointly vary. Thus, given fixed \(p_{2}\), \(v_{j}\) and \(w_{j}\), we derive the first-order derivative of the total travel cost \(\pi\) (Equation (10)) with respect to the parking price \(p_{1}\), i.e., \(\frac{\partial\pi}{\partial p_{1}}\). We find that the sign of \(\frac{\partial\pi}{\partial p_{1}}\) is affected by the relation between \(x\) and \(N^{h}-\frac{\alpha^{a}}{\alpha^{h}}N^{h}\), in addition to the ranges of \(K_{1}\) and \(\Delta p\). To facilitate the analysis, we summarize the sign of the first-order derivative \(\frac{\partial\pi}{\partial p_{1}}\) in Tables 4 and 5. The two tables also illustrate the transition of equilibrium cases when \(\Delta p\) changes. From Tables 4 and 5, we can find that the tendency of \(\pi\) relies on the range of \(k_{1}\) and \(\Delta p\). Given the fixed parking price at the peripheral cluster \(p_{2}\), each row in Tables 4 and 5 reflects how the total travel cost \(\pi\) changes with the parking price at the central cluster \(p_{1}\). In this case, we are able to determine the minimum point of the total travel cost within an interval of \(\Delta p\) and an interval of \(K_{1}\). However, given \(K_{1}\), the total travel cost \(\pi\) may be discontinuous at the boundaries of each interval of \(\Delta p\). Thus, we need to combine both the sign of the first-order partial derivative and the level of \(\pi\) at each boundary to obtain the minimized total travel cost. We provide the optimal price at the central cluster \(p_{1}\) when its parking capacity varies in Proposition 5. **Proposition 5** (The optimal parking price \(p_{1}\) when \(\eta^{a}\Delta v<(\eta^{h}-\beta)\Delta w\)). : _At equilibrium, the optimal parking price charged at the central cluster depends on the level of its parking capacity \(K_{1}\), specifically as follows:_  
* _When_ \(K_{1}\in[N^{h}-\frac{a^{a}}{a^{h}}N^{a},N^{h})\)_, the minimized total travel cost is observed in Case 3-3 and the optimal parking price at the central cluster is given by_ \(p_{1}^{*}=p_{2}+(\eta^{h}-\beta)\Delta w\)_._
* _Otherwise, the total travel cost is minimized when_ \(p_{1}\) _approaches_ \(p_{2}\)_._  
Proposition 5 indicates that the optimal parking price at the central cluster should be determined based on its capacity. First, when the parking capacity of the central cluster is at the intermediate level, i.e., \(K_{1}\in[N^{h}-\frac{a^{a}}{a^{h}}N^{a},N^{h})\), the corresponding parking price should be set as \(p_{1}^{*}=p_{2}+(\eta^{h}-\beta)\Delta w\). That is, a higher parking price should be charged from commuters using the central cluster compared with the peripheral one.