# Model for Determining Parking Demand Using Simulation-Based Pricing
## 2 Methodology  
The demand and price model developed in this study is grounded in the negative price elasticity of demand and is designed to predict and quantify the effects of parking charges. For a given demand function \(Q=Q(P)\), which calculates the demand for parking spaces \(Q\) depending on the price \(P\), the elasticity coefficient \(\eta(P)\) is defined as the ratio between the relative change in demand \(\frac{dQ(P)}{Q(P)}\) and the relative change in price \(\frac{dP}{P}\):  
\[\eta(P)=\frac{\frac{dQ(P)}{Q(P)}}{\frac{dP}{P}}.\]  
The core hypothesis of this study is that if provided with sufficient parking spaces, then the elasticity \(\eta(P)\) cannot be positive:  
\[\eta(P)=\frac{\frac{dQ(P)}{Q(P)}}{\frac{dP}{P}}=Q^{\prime}(P)\cdot\frac{P}{Q(P )}=\frac{P\cdot Q^{\prime}(P)}{Q(P)}\ \leq\ 0. \tag{1}\]  
A high elasticity coefficient indicates that demand is highly sensitive to price changes, such that a small price increase leads to a substantial decrease in demand. It is essential that when enough parking spaces are available, demand always decreases as price increases. Hence, the elasticity coefficient (1) cannot be positive because of \(Q^{\prime}(P)<0\). The proof is simple, as a sufficient price increase \(P\) will arbitrarily reduce demand \(Q(P)\), as demonstrated in [19]. The model classifies users into categories: short-term users (STU) utilizing hourly tickets under either a day or night tariff; monthly subscribers include Daily Monthly Pass (DMP), Night Monthly Pass (NMP), and Complete Monthly Pass (CMP) users. The model determines optimal prices for these categories via price simulation-based demand forecasting, adjusting category composition to achieve profitable and sustainable garage management. The formalization of the causal relationship between pricing and demand, based on collected and analyzed data, enables parking charge policy to manage user composition and facilitate sustainable garage operation. Vehicle license plates entering and exiting the garage are recorded by the management system, which extracts relevant business data. This method ensures accurate, continuous data collection for robust analysis, but requires careful selection due to the large data volume. The research was conducted in the city of Zagreb, at the Gorica Public Garage (PG), located on the edge of the city center with 353 parking spaces. Data gathered from the garage management system during one weekday, broken down into hourly intervals, served as the basis for the study design. For method demonstration, we used data from one weekday before and after the price change on 1 January 2023. A comparison of parking prices is shown in Figure 1.  
From the accumulation of parked vehicles \((A_{t})\) during hours \(t=1,2,\ldots,24\), in a space with a total of \(M=353\) spots, the occupancy rate (\(Occ_{t}\)) in hour \(t\) is calculated as the ratio of the number of accumulated vehicles at a given time to the total number of spaces in the observed area, typically expressed as a percentage:  
\[Occ_{t}=\frac{A_{t}}{M}. \tag{2}\]  
Based on the data, users of the garage are divided into categories by parking duration. Thus, STUs use the garage at an hourly rate from 08:00 to 18:00 (day rate) or 18:00-08:00 (night rate). Long-term DMP users can access the garage every day from 06:00 to 21:00, NMP users from 21:00 to 07:00 and all day on weekends, while CMP subscribers have access 24/7. The accumulation \((A_{kt})\) is determined for each user category \(k\):  
\[k\;\in\;K=\{STU,\;DMP,\;NMP,\;CMP\}.\]  
Figure 1: Comparison of parking prices in Gorica PG. Total hourly accumulation \(A_{t}\) is then calculated as follows:  
\[A_{t}=\sum_{k\in K}A_{kt}. \tag{3}\]  
Using these characteristics and comparisons, the forecasting model can determine optimal prices in the garage by simulating category-specific prices that forecast demand.