# Dependence of Parking Pricing on Land Use and Time of Day

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Data and Study Area

Variable | Level | Percentage (%) | Variable | Level | Percentage (%) |
---|---|---|---|---|---|

Parking period
^{a} | peak period | 56.71 | Vehicle ownership | Private
^{c} | 14.46 |

non-peak period | 43.29 | none-private | 85.54 | ||

Parking duration (hour) | ≤3 | 17.29 | Number of passengers | 0 | 78.32 |

3–12 | 35.30 | 1–3 | 21.40 | ||

>12 | 47.40 | >3 | 0.27 | ||

Trip purpose | work | 53.47 | Walking time (min) | ≤2 | 47.23 |

maintenance | 16.83 | 2–5 | 46.66 | ||

entertainment | 21.78 | 5–10 | 5.64 | ||

other | 7.92 | >10 | 0.47 | ||

Parking rate (yuan/h ^{b}) | 0 | 60.96 | Parking location | temporary stop | 7.07 |

0–1 | 0.48 | on-street parking | 3.38 | ||

1–3 | 1.90 | illegal parking | 8.81 | ||

3–5 | 3.48 | off-street parking | 9.44 | ||

5–10 | 20.20 | residential zone parking | 50.63 | ||

>10 | 12.97 | Employer-sponsored parking | 20.45 | ||

other | 0.21 |

^{a}Peak hours refer to parking starts between 7 AM and 9 AM or between 5 PM and 7 PM, and off-peak hours refer to parking starts at the rest of the day;

^{b}The yuan is a Chinese monetary unit. One yuan is approximately equal to 0.16 U.S. dollars.

^{c}Private refers to the private car, while none-private means the employer-provided car.

^{2}(square kilometers) and had a permanent registered population of 21.52 million. It consisted of 16 urban and suburban districts and two rural districts. These 18 administrative districts were divided into four functional areas: The Core Districts of Capital Function, Extended Areas for Urban Function, New Districts of Urban Development, and Ecological Preservation Development Districts (Figure 1).

## 4. Two-Level Parking Model

#### 4.1. Model Construction

Level-1: Game between government and car users | Players | Government | Car Users |

Strategies | To establish parking rates according to the overall performance of the parking system to maximize social benefit | To choose parking strategy according to parking cost and benefit to maximize personal benefit | |

Actions | To determine parking rate p for different parking locations and starting times | To choose parking strategy s_{i}: parking inside/outside a business zone and parking during peak/off-peak hours | |

Utility function | ${\pi}_{g}({S}^{*}(p),p)={\displaystyle \sum _{i=1}^{n}{\pi}_{i}+{\displaystyle \sum _{i=1}^{n}{p}_{i}\times {t}_{i}}}$ | ${\pi}_{i}({s}_{i}(p),{s}_{-i}(p),p)={\pi}_{i}^{0}-{C}_{i}$ | |

Expected outcome | To obtain the optimal parking rate p^{*} | To park according to the most satisfying strategy S* (P) | |

Level-2: Game among car users | Players | Car users | |

Strategies | To choose the optimal parking strategy under the influence of the parking rate to maximize individual parking utility | ||

Actions | Car user i chooses parking strategy s_{i} according to parking rate p, whereas the strategies of the others are s_{−i} = ( s_{1},…, s_{i-1}, s_{i+1}, …, s_{n}) | ||

Utility function | ${\pi}_{i}({s}_{i}(p),{s}_{-i}(p),p)={\pi}_{i}^{0}-{C}_{i}$ |

_{g}is the utility of the government, which is calculated as the sum of the total parking utility of all of the car users and the parking pricing income. The parking benefit of non-car users is not represented in the government’s utility function because non-car users do not have a parking demand and do not need to pay a parking fee either. Superficially, their parking benefit in the parking system is zero. However, being a kind of public resource, parking facilities or the income collected from a parking system should be shared by not only car users but also non-car users. The usual practice is to utilize the parking pricing income ($\sum _{i=1}^{n}{p}_{i}\times {t}_{i}$) collected from the car users to support investment in public projects, particularly transport investment [22,23]. This fact demonstrates that the government represents the benefit of the public, i.e., not only of car users but also of non-car users. Thus, we can also state that π

_{g}represents a consideration of the benefit of non-car users.

_{bz&p}is the rate to park in business zones during peak hours, p

_{bz&n-p}is the rate to park in business zones during non-peak hours, p

_{n-bz&p}denotes the rate to park outside business zones during peak hours, and p

_{n-bz&n-p}designates the rate to park outside business zones during non-peak hours. For most of the public parking facilities in Beijing, the current parking rate is 5–10 yuan/h. Considering the rapidly increasing commodity prices in China, we set the basic parking rate for parking outside business zones and during off-peak hours as 10 yuan/h: p

_{n-bz&n-p}= 10 yuan/h. The values of p

_{bz&p}, p

_{bz&n-p}and p

_{n-bz&p}are calculated using the parking model.

_{i}is car user i’s parking utility, i = (1,…, n), and n is the total number of car users in the parking system.

_{i}is car user i’s parking duration. According to the results of a correlation analysis of t

_{i}and other variables, we select parking rate p

_{i}as the independent variable and establish a linear regression model for t

_{i}[28,29]:

Variable | Coef. | Standard Error | t-Stat. | Sig. |
---|---|---|---|---|

a | 5.283 | 0.521 | 10.140 | 0.000 |

b | −0.046 | 0.013 | −3.538 | 0.000 |

_{i}refers to car user i’s parking cost for each instance of parking, which consists of two parts: the actual parking cost and the time cost of parking activities. Specifically, C

_{i}= p

_{i}× t

_{i}+ vot × t

_{i}’, where t

_{i}is the parking duration, t

_{i}’ is the time consumed by parking activities and vot is the car user’s value of time. According to statistical information, the annual average wages of staff and workers in Beijing in 2014 was 64,116 yuan [30]. Thus, the average hourly wage can be calculated as 64116/(250 × 8) = 32.06 yuan/h because generally the total number of annual working days in China is 250. The average hourly wage (32.06 yuan/h = 0.53 yuan/min) is consequently adopted as the general value of vot [32]. Based on the general value of vot, a different specific value of vot is determined for the different types of time consumed by parking-related activities (Table 4) according to the travel survey data in Beijing and the estimation results presented by related previous studies [32,33,34,35]. The economic conditions, residents’ consumption concept in China, and our previous experiences on travel behavior analysis were also involved in the estimation of the values of vot. The value of walking time was set to be less than that of cruising and parking operation times (see Table 4) because driving is more costly compared with walking (because of fuel consumption). The statistical data in China indicate that most travelers prefer to conduct a walking trip (especially a short-distance walking trip) than to drive a car in order to save money. In addition, the anxiety a traveler feels when waiting for a bus ora train and the congestion during getting on/off a bus or subway may be the reason for the relatively high value of the waiting and on/off times for public transport (see Table 4).

_{i}’) is defined as the sum of walking time from the parking location to the destination and the cruising time to search for available parking location (for simplicity, the cost of fuel consumption is taken as a part of the parking time). By assuming that different car users choose different parking decisions, we examine the basic time consumed in parking activities. When car user i chooses the same parking strategies as the other car users, a congestion cost, which is defined as 5.00 min, has to be concluded in his or her parking time (t

_{i}’).

_{i}’ is calculated according to different parking locations and parking starting times. According to statistics on Beijing’s business zones, the average radius of such a zone is approximately 1 km. The average riding time for a trip by bus or subway from a parking location outside a business zone to a travel destination in the business zone was calculated as 8.00 min for parking during peak hours and 6.00 min for parking during off-peak hours based on the Beijing parking survey data [36,37,38,39]. In addition, according to the parking survey data, the average walking time (from the parking location to the destination) after parking in a business zone is approximately 3.14 min, and the average time required to walk from outside a business zone to a travel destination in the business zone was set at 8.00 min [40,41,42]. In addition, the cruising time was set at 8.00 min, 6.00 min, 2.00 min, and 0 min for the bz&p parking strategy (i.e., business zones and peak hours), the bz&n-p strategy (i.e., business zones and non-peak hours), the n-bz&p strategy (i.e., non-business zones and peak hours), and the n-bz&n-p strategy (i.e., non-business zones and non-peak hours), respectively. The parking operation time was defined as 3.00 min for any parking location and parking starting time. An additional cost of parking during off-peak hours was defined, which was set at 15.00 min based on the survey data. This cost represents the time loss of parking during non-peak hours because the driver would have arrived at the destination during peak hours. The time consumed and the value of time for the different parking activities of the four parking strategies is shown in Table 4.

_{i}’) for different parking strategies can be obtained from Table 4. In contrast, when car user i chooses the same parking strategies as the other car users, a congestion cost, which is defined as 5.00 min, must be included in the user’s parking time (t

_{i}’). The vot of congestion time was set to be 0.65 yuan/min. Thus, t

_{i}’ for car user i can be calculated. The results are shown in Table 5.

**Table 4.**Time consumed and the value of time for the different parking activities of the four parking strategies.

The Composition of t_{i}’ | vot (yuan/min) | Car User
i’s Parking Strategy--s_{i} (p) | |||
---|---|---|---|---|---|

bz&p ^{a} | bz&n-p | ||||

t_{i}’ (min) | vot × t_{i}’ (yuan) | t_{i}’ (min) | vot × t_{i}’ (yuan) | ||

Cruising time | 0.65 | 8.00 | 5.20 | 6.00 | 3.90 |

Parking operation time | 0.55 | 3.00 | 1.65 | 3.00 | 1.65 |

Walking time after parking | 0.45 | 3.14 | 1.41 | 3.14 | 1.41 |

Loss of parking before peak hours | 0.45 | 0.00 | 0.00 | 15.00 | 6.75 |

Riding time by bus or subway | 0.50 | 0.00 | 0.00 | 0.00 | 0.00 |

Waiting time and on/off time for bus or subway | 0.60 | 0.00 | 0.00 | 0.00 | 0.00 |

Total value | - | 14.14 | 8.26 | 27.14 | 13.71 |

Cruising time | 0.65 | 2.00 | 1.30 | 0.00 | 0.00 |

Parking operation time | 0.55 | 3.00 | 1.65 | 3.00 | 1.65 |

Walking time after parking | 0.45 | 8.00 | 3.60 | 8.00 | 3.60 |

Loss of parking before peak hours | 0.45 | 0.00 | 0.00 | 15.00 | 6.75 |

Riding time by bus or subway | 0.50 | 8.00 | 4.00 | 6.00 | 3.00 |

Waiting time and on/off time for bus or subway | 0.60 | 5.00 | 3.00 | 3.00 | 1.80 |

Total value | - | 26.00 | 13.55 | 35.00 | 16.80 |

^{a}bz&p means parking in the business zone during peak hours, bz&n-p denotes parking in the business zone during non-peak hours, n-bz&p signifies parking outside the business zone during peak hours, and n-bz&n-p represents parking outside the business zone during non-peak hours.

Car User
i’s Parking Strategy: s_{i} (p) | The Other Car Users’ Parking Strategy: s_{−i} (p) | vot × t_{i}’ | Congestion Cost | Total Value of vot × t_{i}’ for s_{i} (p) |
---|---|---|---|---|

bz&p ^{a} | Not same as
s_{i} (p) | 8.26 | 0.00 | 8.26 |

bz&n-p | 13.71 | 0.00 | 13.71 | |

n-bz&p | 13.55 | 0.00 | 13.55 | |

n-bz&n-p | 16.80 | 0.00 | 16.80 | |

bz&p | Same as
s_{i} (p) | 8.26 | 3.25 | 11.51 |

bz&n-p | 13.71 | 3.25 | 16.96 | |

n-bz&p | 13.55 | 3.25 | 16.80 | |

n-bz&n-p | 16.80 | 3.25 | 20.05 |

^{a}bz&p means parking in the business zone during peak hours, bz&n-p denotes parking in the business zone during non-peak hours, n-bz&p signifies parking outside the business zone during peak hours, and n-bz&n-p represents parking outside the business zone during non-peak hours.

#### 4.2. Model Solution

#### 4.2.1. Nash Equilibrium

_{i}according to p, while the other car users’ strategies are s

_{−}

_{i}= (s

_{1},…, s

_{i-1}, s

_{i+1}, …, s

_{n}) accordingly (s

_{i}, s

_{−}

_{i}∈ S

_{i}). S

_{i}denotes the set of available parking strategy choices, i.e., S

_{i}={parking in the business zones during peak hours (bz&p), parking in the business zones during non-peak hours (bz&n-p), parking outside the business zones during peak hours (n-bz&p), parking outside the business zones during non-peak hours (n-bz&n-p)}. We also assumed that parking duration (t

_{i}) is selected by car users considering the parking rate (p

_{i}) with respect to the specific parking strategy, which he or she chooses accordingly. Therefore, the value of t

_{i}can be calculated using Equation (1).

_{−}

_{i}

^{*}, car user i’s optimal strategy is,

^{*}(p) = (s

_{i}

^{*}(p), s

_{−}

_{i}

^{*}(p)) are the optimal strategies for all of the car users, i.e., the Nash equilibrium solutions of the Level-2 game.

^{*}is the optimal parking rate for the government, then p

^{*}can be determined as follows:

^{*}is the Nash equilibrium of the Level-1 game.

#### 4.2.2. Solution of Level-2 Game

_{i}(Equation (1)) and the value of vot × t

_{i}’ (Table 4), we obtain:

_{i}for each value of vot × t

_{i}’, respectively (different parking strategy corresponds to different value of vot × t

_{i}’, as shown in Table 4. In addition, two sets of vot × t

_{i}’ were defined according to car user –i’s two strategies, i.e., being same as car user i’s strategy vs. being distinct with car user i’s strategy (Table 5). Therefore, two sets (i.e., totally four solutions) of p

_{i}were obtained for each parking strategy, respectively. By deleting the solutions, which are too large (i.e., greater than 100 yuan/h) to be used as parking rate, we obtain the ranges of p

_{bz&p}, p

_{bz&n-p}and p

_{n-bz&p}. The results are shown as follows:

- p
_{bz&p}∈ (0, 12.41], (12.41, 13.20] (yuan/h) - p
_{bz&n-p}∈ (0, 11.12], (11.12, 11.88] (yuan/h) - p
_{n-bz&p}∈ (0, 11.15], (11.15, 11.92] (yuan/h)

Parking Strategies | Parking Rate (yuan/h) | Parking Duration (t_{i}) (min) | ||
---|---|---|---|---|

p1 ^{c} | p2 ^{c} | t1 | t2 | |

bz&p ^{a} | 13.20 | 12.41 | 4.68 | 4.71 |

bz&n-p | 11.88 | 11.12 | 4.74 | 4.77 |

n-bz&p | 11.92 | 11.15 | 4.73 | 4.77 |

n-bz&n-p | - | - | 4.82 ^{b} | 4.82 ^{b} |

^{a}bz&p means parking in the business zone during peak hours, bz&n-p denotes parking in the business zone during non-peak hours, n-bz&p signifies parking outside the business zone during peak hours, and n-bz&n-p represents parking outside the business zone during non-peak hours.

^{b}The value of t

_{i}for parking outside the business zone during non-peak hours (i.e., n-bz&n-p) is calculated based on p

_{n-bz&n-p}= 10 yuan/hour.

^{c}p1 refers the upper limit value of the value ranges of parking rate calculated using the value of vot × t

_{i}’ when car user –i’s strategy is as same as car user i’s strategy. p2 refers the upper limit value of the value ranges of parking rate calculated using the value of vot × t

_{i}’ when car user –i’s strategy is distinct with car user i’s strategy (Table 5).

_{bz&p}, p

_{bz&n-p}and p

_{n-bz&p}in other words, is chosen, respectively. Then the upper limit value of each range of parking rate is selected and taken as its potential values. The values of p

_{i}for different parking strategies are shown in Table 6. Accordingly, the values of t

_{i}for different parking rate values are calculated using Equation (1). The results are shown in Table 6.

_{bz&p}, p

_{bz&n-p}, and p

_{n-bz&p}, the payoffs (i.e., the parking utilities) of car user i and −i in the Level-2 game are calculated. The results are shown in Table 7.

**Table 7.**Parking utilities and optimal parking strategies for different parking strategies in the Level-2 model.

Parking Rate (yuan/h) | Car User i parking Strategy | Car User −i Parking Strategy | |||||
---|---|---|---|---|---|---|---|

s_{−i}(p) = bz&p | s_{−i}(p) = bz&n-p | s_{−i}(p) = n-bz&p | s_{−i}(p) = n-bz&n-p | ||||

p1 ^{a} | p_{bz&p} | 13.20 | s_{i}(p) = bz&p | (−3.25, −3.25) | (0, 0) | (0, 0) | (0, 5.00) |

p_{bz&n-p} | 11.88 | s_{i}(p) = bz&n-p | (0, 0) | (−3.25, −3.25) | (0, 0) | (0, 5.00) | |

p_{n-bz&p} | 11.92 | s_{i}(p) = n-bz&p | (0, 0) | (0, 0) | (−3.25, −3.25) | (0, 5.00) | |

p_{n-bz&n-p} | 10 | s_{i}(p) = n-bz&n-p | (5.00, 0) | (5.00, 0) | (5.00, 0) | (1.75,1.75) ^{*} | |

p2 ^{a} | p_{bz&p} | 12.41 | s_{i}(p) = bz&p | (0, 0) | (3.25, 3.25) | (3.25, 3.25) | (3.25, 5.00) ^{*} |

p_{bz&n-p} | 11.12 | s_{i}(p) = bz&n-p | (3.25, 3.25) | (0, 0) | (3.25, 3.25) | (3.25, 5.00) ^{*} | |

p_{n-bz&p} | 11.15 | s_{i}(p) = n-bz&p | (3.25, 3.25) | (3.25, 3.25) | (0, 0) | (3.25, 5.00) ^{*} | |

p_{n-bz&n-p} | 10 | s_{i}(p) = n-bz&n-p | (5.00, 3.25) ^{*} | (5.00, 3.25) ^{*} | (5.00, 3.25) ^{*} | (1.75,1.75) |

^{a}p1 refers to the upper limit value of the value ranges of parking rate calculated using the value of vot × t

_{i}’ when car user –i’s strategy is as same as car user i’s strategy. p2 refers to the upper limit value of the value ranges of parking rate calculated using the value of vot × t

_{i}’ when car user –i’s strategy is distinct with car user i’s strategy (Table 5). The solutions with

^{*}are the Nash equilibrium solutions. The utilities with negative value are not considered in choosing the optimal solution. The game matrix is composed of different players’ utilities (yuan).

_{i}(p) = n-bz&n-p, which equals to 5.00, is larger than that of other strategies (3.25, 0 and 3.25 for s

_{i}(p) for bz&p, bz&n-p and n-bz&p, respectively), s

_{i}(p) = n-bz&n-p is chosen as the optimal strategy. We underline 5.00 to mark it. The underlining indicates that s

_{i}(p) = n-bz&n-p is the dominant strategy when s

_{−}

_{i}(p) = bz&p. Similarly, 5.00, 5.00 ,and 1.75 are underlined for s

_{i}(p) under the condition that s

_{−}

_{i}(p) is bz&n-p, n-bz&p and n-bz&n-p, respectively.

_{−}

_{i}(p) = n-bz&n-p is the dominant strategy when s

_{i}(p) is bz&p, bz&n-p, n-bz&p, and n-bz&n-p, respectively. Thus, the underlined mixed strategies (1.75, 1.75)

^{*}represent the Nash equilibrium choices of this level of game.

#### 4.2.3. Solution of Level-1 Game

^{*}) (Table 8).

P (yuan/h) | Game Strategy | Car User i’s Utility π_{i} (yuan) | Car User −i’s Utility π_{−i} (yuan) | The Government’s Utility π_{g} ^{a} (yuan) | ||
---|---|---|---|---|---|---|

s_{i}(p) | s_{−i}(p) | |||||

p1 ^{b} | p_{bz&p} = 13.20 | n-bz&n-p | n-bz&n-p | 1.75 | 1.75 | 101.5 |

p_{bz&n-p} = 11.88 | ||||||

p_{n-bz&p} = 11.92 | ||||||

p2 ^{*} | p_{bz&p} = 12.41 | bz&p ^{*} | n-bz&n-p ^{*} | 3.25 | 5.00 | 114.94 ^{*} |

p_{bz&n-p} = 11.12 | bz&n-p | n-bz&n-p | 3.25 | 5.00 | 109.49 | |

p_{n-bz&p} = 11.15 | n-bz&p | n-bz&n-p | 3.25 | 5.00 | 109.65 | |

p2 ^{*} | p_{bz&p} = 12.41 | n-bz&n-p ^{*} | bz&p ^{*} | 5.00 | 3.25 | 114.94 ^{*} |

p_{bz&n-p} = 11.12 | n-bz&n-p | bz&n-p | 5.00 | 3.25 | 109.49 | |

p_{n-bz&p} = 11.15 | n-bz&n-p | n-bz&p | 5.00 | 3.25 | 109.65 |

^{*}are the Nash equilibrium solutions.

**π**

^{a}_{g}is calculated with Equation (5).

^{b}p1 refers to the upper limit value of the value ranges of parking rate calculated using the value of vot × t

_{i}’ when car user –i’s strategy is as same as car user i’s strategy. p2 refers to the upper limit value of the value ranges of parking rate calculated using the value of vot × t

_{i}’ when car user –i’s strategy is distinct with car user i’s strategy (Table 5).

#### 4.2.4. Optimal Solution

_{bz&p}is 12.41 yuan/h, p

_{bz&n-p}is 11.12 yuan/h, p

_{n-bz&p}is 11.15 yuan/h, and p

_{n-bz&n-p}is 10 yuan/h. Under this condition, the parking strategy of car user i and –i is that s

_{i}(p) = bz&p when s

_{−i}(p) = n-bz&n-p or s

_{i}(p) = n-bz&n-p when s

_{−i}(p) = bz&p. These outcomes indicate that car users prefer a parking strategy opposite to that of the other car users to avoid the congestion loss.

## 5. Parking Utility Analysis

_{i}(p) = bz&p when s

_{−i}(p) = n-bz&n-p or s

_{i}(p) = n-bz&n-p when s

_{−i}(p) = bz&p. Therefore, to simplify the process of parking utility analysis, we consider only the scenario of parking in business zones during peak hours (bz&p) vs. parking outside business zones during non-peak hours (n-bz&n-p).

_{i}) for bz&p vs. n-bz&n-p are calculated with p

_{bz&p}= 12.41 yuan/h and p

_{n-bz&n-p}= 10 yuan/h. The results are shown in Figure 4.

_{bz}= 10 yuan/h), parking strategy bz&p after adjusting the parking rate (p

_{bz&p}= 12.41 yuan/h), and parking strategy n-bz&n-p (p

_{n-bz&n-p}= 10 yuan/h), respectively.

^{*}= 0.237, of the two curves of parking utilities. At point r

^{*}, the utilities of the bz&p and n-bz&n-p parking strategy are equal. That is, car users will choose to park outside the business zone during non-peak hours when the ratio of parking in the business zone during peak hours to the total number of parkers reaches 0.237. This outcome reflects the objective of the government, which is to guide the car users who usually park in the business zone during peak hours to park outside the business zone during non-peak hours. The positive impacts of the adjustments of the bz&n-p and n-bz&p parking rates can also be explained using a similar method.

## 6. Conclusions

_{i}, which are determined at a general level in the present study, should also be examined at an individual level in further studies.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Zong, F.; He, Y.; Yuan, Y.
Dependence of Parking Pricing on Land Use and Time of Day. *Sustainability* **2015**, *7*, 9587-9607.
https://doi.org/10.3390/su7079587

**AMA Style**

Zong F, He Y, Yuan Y.
Dependence of Parking Pricing on Land Use and Time of Day. *Sustainability*. 2015; 7(7):9587-9607.
https://doi.org/10.3390/su7079587

**Chicago/Turabian Style**

Zong, Fang, Yanan He, and Yixin Yuan.
2015. "Dependence of Parking Pricing on Land Use and Time of Day" *Sustainability* 7, no. 7: 9587-9607.
https://doi.org/10.3390/su7079587