Parking Law

In Chile, on February 15 (correct me if I’m wrong), the parking law was enacted, which among other things requires parking companies to:

• Compensate customers who are victims of theft.
• Prohibit fines for lost tickets.
• Choose one of the following two pricing methods:
  1. Charge per minute actually used.
  2. Charge per segment rounded down, where the first segment cannot be less than 30 minutes and subsequent segments cannot be less than 10 minutes.

The result of this regulation was that parking lots significantly increased their final price to the user. In this article, we will analyze why.

Available Information

The SERNAC conducted a review of price changes, which is presented in the following table, where it can be seen that on average parking costs increased by 7.14%. On the other hand, EMOL informs us that 75% of usage is between 20 and 120 minutes.

Before the law came into effect, most facilities charged for every 30 minutes rounded up.

Introductory Analysis

We will work under the assumption that any variation in prices is because companies want to maintain the expected value of the revenue received before the regulation.

We will begin the analysis with a graph where the horizontal axis represents the elapsed time of a parking user (the customer), while the vertical axis represents the amount of time charged to the customer for the following 4 cases:

1. Standard pricing before the law came into effect: charge every 30 minutes rounded up.
2. Continuous per-minute pricing.
3. Pricing with an initial window of 30 minutes and charges every 10 minutes, without rounding up.
4. Pricing with an initial window of 30 minutes and charges every 30 minutes, without rounding up (I include this case in case there is a technological barrier to charging every 10 minutes).

Clearly, in the 3 simulations representing the new law (lilac, blue, and green), effectively fewer minutes are charged than in the previous standard situation (red). This suggests that it is indeed necessary to raise prices to maintain the revenue needed to pay salaries, loans, etc.

Assumptions

Given that under the new law fewer minutes can be charged than before, we will run a simulation to see how much parking lots need to increase in each case to offset the lost revenue.

We will start by calibrating the simulation, assuming that parking times follow a Poisson distribution shifted on the X-axis (meaning no car stays 0 minutes), which will be calibrated with the information provided by EMOL: "75% of usage is between 20 and 120 minutes."

With the above assumptions, and choosing the second minimum, the average stay time is 113 minutes and no one stays less than 7.3 minutes. The model was calibrated with the following R code (which is sensitive to the starting point, but the results always yield similar parameters):

# x lambda
objetivo = function(x){
  ( ppois(120,x) - ppois(20,x) - 0.75 )^2
}
plot(sapply(1:400, objetivo), type="l")
sol = optimise(objetivo,c(50,150))
print(sol)

Minima of function:

Simulation

With the parameters obtained above, we will run a simulation with 100,000 cases and apply the 4 pricing cases mentioned, then compare the average number of minutes charged.

On average, under the new law, fewer minutes will be billed than before. According to the simulation, the minutes billed for each case and the corresponding adjustment factor to keep revenue constant are:

Immediately we can conclude that companies must increase the price between 13% and 31% to maintain their revenue. However, this extra amount is not paid equally by everyone. In the following graph, we see on the X-axis the length of stay and on the Y-axis the amount of minutes paid adjusted by the required increase. Users who stay a short time will pay less than before, while users who stay a long time will feel the change.

Conclusions

Basically, we have three important conclusions:

1. A price increase was to be expected, ranging between 13% and 31%.
2. Given that on average parking lots increased their price by only 7.1%, parking lots will see a decrease in revenue, not considering their new security obligations.
3. If you run short errands (less than an hour), you will benefit from this law.