# Model for Determining Parking Demand Using Simulation-Based Pricing
## 3 Results and Data Analysis
### Mathematical Model of Demand
#### 3.1.1 Natural Demand for Parking Spaces  
If demand \(Q\) depends on the price \(P\) according to some unknown function \(Q=Q(P)\), we inferred from the data collected, shown in Figure 2, a proportional dependency:  
\[\frac{dQ}{dP}=-\lambda\cdot Q,\ \ \lambda>0 \tag{7}\]  
Differential Equation (7) indicates that the decline in demand is proportional to the current demand \(Q\): with an increase in price, the reduction in demand will be greater the higher the current demand is. The differential equation is solved by separating variables, where the right-hand side is negative, corresponding to the observed natural decline in demand with a positive price change in a parking area that has sufficient available spaces number (1). When solving, the variables are treated equally: \(Q(P)=Q\).  
\[\frac{dQ}{Q}=-\lambda\cdot dP\]  
\[\int\frac{dQ}{Q}=-\int\lambda\cdot dP\]  
\[lnQ=-\lambda P+C,\]  
where \(C\) is an arbitrary constant. Furthermore:  
\[e^{lnQ}=e^{-\lambda P+C}\]  
\[Q=e^{-\lambda P}\cdot e^{C}\]  
\[Q=e^{-\lambda P}\cdot N,\]  
where \(e=2.71828\) is the base of the natural logarithm, and \(N=e^{C}\) is again some arbitrary but positive constant. The unknown function \(Q=Q(P)\) is now given by the following explicit formula:  
\[Q(P)=N\cdot e^{-\lambda P}\,\ \ \ \lambda>0 \tag{8}\]  
The parameters \(N\) and \(\lambda\) are determined from two known pairs of prices and corresponding demands taken from the collected data:  
* \((P_{v},Q_{v}=Q(P_{v}))\)--higher price and (lower) demand at the higher price
* \((P_{m},\ Q_{m}=Q(P_{m}))\)--lower price and (higher) demand at the lower price.  
The parameter \(\lambda\) is calculated by substituting the price-demand pairs into Formula (8), followed by solving the system of exponential equations:  
\[Q_{v}=N\cdot e^{-\lambda\cdot P_{v}}\]  
\[Q_{m}=N\cdot e^{-\lambda\cdot P_{m}}\]  
By dividing the equations, we get the following:  
\[\frac{Q_{v}}{Q_{m}}=e^{-\lambda\cdot(P_{v}-P_{m})}\]  
\[ln\frac{Q_{v}}{Q_{m}}=-\lambda\cdot(P_{v}-P_{m}),\]which ultimately gives the parameter \(\lambda\) a positive value; since the logarithm is negative as the demand at the higher price is naturally less than the demand at the lower price by (1):  
\[\lambda=\frac{-1}{P_{v}-P_{m}}\cdot ln\frac{Q_{v}}{Q_{m}}. \tag{9}\]  
From (8), it follows that for the price-demand pair at the higher price we have \(Q_{v}=N\cdot e^{-\lambda\cdot P_{v}}\), so by substituting \(\lambda\) from (9), we obtain the value of parameter \(N\) with a positive argument in the exponential function. The same value of \(N\) would be obtained using the pair with the lower price and higher demand.  
\[N=Q_{v}\cdot e^{\lambda\cdot P_{v}}=Q_{v}\cdot e^{\frac{P_{v}}{P_{m}-P_{v}}\cdot ln \frac{Q_{v}}{Q_{m}}} \tag{10}\]  
The parameter \(N\) gives the number of users in the case that the service is free and is therefore referred to in this work as the gravitational demand, representing all potential users of the garage. The normal demand is calculated using the final formula:  
\[Q(P)=Q_{v}\cdot e^{\frac{P_{v}}{P_{m}-P_{v}}\cdot ln\frac{Q_{v}}{Q_{m}}\cdot(1 +P)}. \tag{11}\]  
Calculating the elasticity coefficient by (1), and using (8), we get the following:  
\[\eta(P)=\frac{P\cdot N\cdot(-\lambda)\cdot e^{-\lambda\cdot P}}{N\cdot e^{- \lambda\cdot P}}=-\lambda\cdot P. \tag{12}\]  
With \(\lambda\) from (9), Equation (12) indeed results in negative elasticity, and the decline in demand is greater the higher the price at which the increase occurs.