This summer we visited Berlin and enjoyed its excellent public transportation system. Unlike the controlled access system prevalent in the US, Berlin’s (and Prague’s for that matter) system is an honor system. You can buy a ticket at any time and when you board a train or a bus you stamp the ticket with the date and time the trip started. Compliance is enforced by periodic inspections and the fine for riding without a validated ticket (Schwarzfahren) is a modest €60. During our 8 days in Berlin we saw inspectors once and did not get inspected ourselves.
Given the ticket price, is the combination of the fine size and inspection frequency sensible? That is, could BVG (the transit authority) save money by changing the fine amount and/or number of inspectors it has to hire?
To answer this question we must know how people decide whether to buy a ticket or risk being fined. Many factors come into play here. A purely rational and amoral consumer would buy the ticket if the product of the inspection probability and fine amount is larger than the ticket price. This way, not buying tickets costs more in the long run. However, there is a strong national pride and the concept of Beitragen (contributing) which keeps Schwarzfahren low (just a few percent according to this article in German). Being caught without a valid ticket may also result not just in a fine but also in a court summons which presumably has a higher deterrent power.
Let’s explore the effect of the frequency of inspections (and the associated cost) and the fine amount on the total amount of money collected. I will define the single ride ticket cost to be 1 and the fine amount to be . To keep the formalism simple, I will neglect the reality that the fine for repeat offenders is higher. This fact reduces Schwarzfahren but makes calculations more complicated. In the long run, the amount of the fine for repeat offenses is the important quantity.
I will assume for simplicity that the population of transit rides consists of two distinct pools. The conscientious riders always buy the ticket (population size ). The rest of the riders (population size ) are rational fare evaders and buy tickets when the inspection probability per ride is greater than . This assumption requires the ability to accurately measure the frequency of inspections either from personal observations or from those of friends and family. The last ingredient in the recipe for computing the revenue of the system is the cost of inspections where is a new parameter which can be thought of as the cost per inspection.
Within our simplistic model, given the fine amount , if the inspection probability is , the evaders do not buy tickets and if the potential evaders always buy tickets. Because the cost of inspections is linear with , there are two local maxima of the revenue. Either in which case the revenue is simply (i.e. there are no inspectors and the evaders ride free), or in which case the evaders buy the tickets the total revenue is . Therefore, if the cost parameter is greater than the product of the fine amount and the fraction of fare avoiders
it does not make sense to inspect. Another way to look at this equation is that the fine amount has to be at least the cost per inspection divided by the fraction of evaders to justify inspections.
Of course the situation in real life is different. Most people are opportunistic evaders. Each person is characterized by a risk aversion parameter which balances the financial benefit of evasion with the emotional toll of being caught. Because of this fact, some level of inspection is always necessary. The level itself depends on the distribution of the risk aversion parameter in the population and the recovery rate of the fines as a function of the fine amount. But that’s for another post.