## 4 The Impacts of Parking Capacity and Parking Fees
### Parking Pricing Scheme  
Furthermore, we investigate the influence of parking prices on the total travel cost to understand how the parking manager can improve social welfare by adjusting parking prices. To show the impacts of parking fees on the total travel cost, we take the first-order partial derivatives of total travel cost \(\pi\) in Equation (10) with respect to \(p_{1}\) and \(p_{2}\), respectively. The results of \(\frac{\partial\pi}{\partial p_{1}}\) and \(\frac{\partial\pi}{\partial p_{2}}\) are included in Table 1 in Appendix B. Obviously, the parking prices charged at the central cluster and the peripheral cluster impose opposite impacts on the total travel cost. Thus, we focus on the analysis regarding the parking price \(p_{1}\) for simplicity.  
Case 5 represents a special case since the conditions \(\eta^{a}\Delta v=(\eta^{h}-\beta)\Delta w\) and \(\eta^{a}\Delta v=\Delta p\) are very strict and may hardly be observed in practice. In this case, both AV and HV commuters are indifferent with the two parking clusters, which leads the equilibrium problem to the classic morning commute problem with a single cluster. Thus, we only focus on Cases 1-4 in the following analysis. By excluding Case 5, we obtain the tendency of the total travel cost against the parking price at the central cluster in Proposition 4.  
**Proposition 4**.: _The impacts of parking prices at two clusters on the total travel cost can be categorized into the following scenarios._  
1. _For Case 1 with_ \(K_{1}\geq N\) _and Case 2 and Case 3-2 with_ \(K_{1}\geq N^{h}\)_, the total travel cost does not vary with_ \(p_{1}\)_._
2. _For Case 1 with_ \(K_{1}<N^{h}\)_, Case 3-1 with_ \(K_{1}<N^{h}+y\)_, and Case 3-2 with_ \(K_{1}<N^{h}\)_, the total travel cost may increase or decrease with_ \(p_{1}\)_, depending on the specific level of_ \(K_{1}\)_._
3. _Otherwise, the total travel cost does not decrease with_ \(p_{1}\)_._  
Proposition 4 shows that the system performance can be improved by adjusting the parking prices at the central. In particular, for cases in Scenario (c) as stated in Proposition 4, the total travel cost becomes higher when the parking price at the closer cluster \(p_{1}\) is higher. Meantime, the total travel cost will decrease with the parking price at the farther cluster \(p_{2}\). In other words, increasing \(p_{2}\) helps to improve social welfare while increasing \(p_{1}\) will harm system performance in most cases. To achieve a better system performance, it is required to set the parking price at the closer cluster at a lower level or set the parking price at the farther cluster at a higher level. For Scenario (a), the total travel cost does not vary with \(p_{1}\) or \(p_{2}\). So, we could not reduce the total travel cost by adjusting the prices. The reason is that the potential parking demand for the central cluster can be met. In other words, the capacity of the central cluster is sufficient to accommodate all potential users. For Scenario \(b\), the total travel cost may increase with \(p_{1}\) if \(K_{1}\) is low, but decreases with \(p_{1}\) if \(K_{1}\) is high. The joint effects of parking prices \(\pi\) and parking capacity \(K_{1}\) on the total travel cost are demonstrated in Section 6.2. Proposition 4 provides insights for public parking managers in order to better regulate parking pricing schemes and improve system performance.  
The findings provided in Proposition 4 describe the tendency of the total travel cost for each equilibrium case. It is still unclear that how the total travel cost differs among different equilibrium cases. Therefore, we further investigate how the minimized total travel cost can be achieved through the optimal design of parking prices in the next section.