Thursday, November 23, 2006

Speed Limits and Economy

I recently participated in a debate about speed limits. Even if none of the participants were really heavy-duty-speeders, most of them nevertheless strongly opposed to current rather low speed limits (30 or 40 km/h on small streets, 50 on main streets in Helsinki), arguing, for example, it would hamper the national economy. Some debaters even argued that people should be allowed to drive their cars at any speed the driver consider safe. To me, a father of a child with whom I walk between home and day care over seven zebra crossings twice every day, this simply disgusted me.

So, let's make some hyper-optimistic assumptions: a driver drives, say, three kilometers every day in an area where speed limits have been reduced from 40 to 30 km/h, how much will it cost the state? Nothing because most likely this wouldn't increase salaries and consequently tax revenues, but theoretically it could cost the driver up to 1.5 minutes every day, or assuming we count only working days, some 330 minutes during the year. That's five and a half hours of taxable work time. Assuming a relatively well-paid (4000 e/month) driver that would be below 150 euros per year, or below 15 million euros for the approximately 100 000 cars suffering from the reduced speed limit.

Of course that's only theory. In practice driving speed is usually limited by other cars, traffic lights, and taking corners. In fact statistics show that average driving speed dropped only by 1.5 km/h, cutting the cost to yearly drivers to only two million euros.

Ok, even two million euros is money, and spending it has to be justified. So let's look at the issue from the victim's point.

If a car hits a person, the probability of death is 5%, 15% or 40% for speeds 30, 40, and 50 km/h, respectively. The severity of non-fatal injuries and likelihood of a hit is also reduced.

What then is the cost of a death to the society? If the victim is a 30-year old adult, the state loses all its investments in his education and future tax revenues. A typical university degree costs the state approximately a quarter of a million euros, the basic education roughly equally much, and missed taxes 20000 euros per year until the victims retirement. So far the loss is one million euros, and we haven't yet accounted for the economic support for the grieving family. Even if statistics are still scarce, the reduction in driving speeds did have the desired effect: in 1999 to 2003 on average seven pedestrians were killed each year, but in 2005 only two.

The five million euro saving in the reduced deaths already counters for the drivers' lost leasure time, but we haven't yet accounted for non-fatal injuries. Unfortunately I don't have good data on them, but let it be noted that during 1999-2003 approximately 150 non-fatal injuries from car-pedestrian accidents were reported each year. A large fraction of them were probably tissue damages and bone fractures that maybe required only a few surgeries and some months' time to recover, but some of them were also tetraplegics whose life-time medical bills to the society easily reach a million - in addition to the one million in their abrupted careers.

In this treatment I've deliberately concentrated on the economic justification of current reduced speed limits, and that is quite strong. Injuries will cause physical pain, death and even moreso paralysis will cause emotional losses I can't formulate in numbers. I'll only say that it definitely is not the drivers' prerogative to define the level of risk his potential victims should to accept.

We, and the society we live in, pay heavily for speed. And I haven't yet treated all aspects of speed.

Thursday, November 16, 2006

Anti-Fibonacci Sequences and Rings of Saturn

Assume a process which repeatedly marks all of positive integers which belong to a Fibonacci sequence which start with the lowest unmarked x and x+c. For example, if c is one, the first Fibonacci sequence marks 1, 2, 3, 5, 8, 13, 21, etc. The process would start the second Fibonacci sequence at 6 and 7, marking thereafter 13, 20, 33, ..., the third one would mark 9, 10, 19, 29, etc. The unmarked numbers 4, 18, 22, 28, 32, 47, 54, 72, etc. are then what I call the anti-Fibonacci numbers for the given c = 1. For c = 2 the anti-Fibonacci numbers would be 5, 8, 9, 24, 34, 38, 45, 50, etc.

One might assume that anti-Fibonacci number generation is a sufficiently complicated process to be essentially pseudo-random. My back-of-the-envelope calculation went as follows: Let p(i) be the probability that i is anti-Fibonacci. A single existing Fibonacci sequence grows sparser with a rate of phi/i where phi is the base of the golden ratio (sqrt(5) + 1) / 2 with which the Fibonacci values grow, but with a probability of p(i)^2 the i and i+c starts a new Fibonacci sequence and p(i) gets a nudge of 2/i upwards. Putting these in equilibrium and solving for p gives sqrt(sqrt(5) + 1) / 2, or roughly 0.899. I knew I had made some lofty approximations in the derivation above, especially in assuming that the density of multiple Fibonacci sequences would decrease as rapidly as that of one sequence, so I wrote a little program that computes some 100 million first anti-Fibonacci numbers. I was somewhat surprized to see that p is only approximately 0.856 for c = 1. For c = 2 the program gave 0.872, which as again surprizing because I didn't expect c to affect p. But when my computer spat out what looks like interference patterns my jaw really dropped.

Having seen the non-randomness of the average of p over all i and observed how it depends on c, a change in coordinates lead me to wonder if conversely the average of p over c's depends on i. The answer is yes. Below I have a sample of p(i) estimated by c up to ten thousand.

The moral of the story? Whenever playing with number theory, staple your jaw so it won't keep falling off constantly.

Thursday, November 02, 2006

Price of a Parking Place

I recently overheard a powerful curse of how costly parking is in the center of Helsinki. At its highest normal street parking costs three euros per hour, but evenings, nights and weekends are free. So, should a parking place be in 100% utilization, and should everybody really pay their parking, it would give a revenue of a little above 700 euros per month, or 50 to 60 euros per square meter per month. This sounds high in comparison to the average appartment rents, typically well over 15 euros per square meter per month, in the same area. But appartments are typically stacked six to nine floors on top of each other, so consequently one can gather up to twice the amount in rent than in street parking for each unit of ground area. The cost of heating, maintenance and repairs is perhaps a fifth of the rent. The cost of construction is harder to estimate, but in the long run - we live in an eighty year old building - it too becomes relatively small. So, in summary, renting a parking place doesn't really seem costlier than renting an appartment.