Tuesday morning I was woken up at 5AM by a police siren. As I lay awake, I started thinking. I couldn’t be the only one woken up. Should police cars use their sirens at night? It seems like a classic example of diffuse costs and easy-to-see benefits. But is it worth it? Let’s make a quick back-of-the-envelope calculation.

The cost of running a siren at night can be modeled by the following equation (I love Fermi problems!):```
LengthDriven * (2 * SirenDistance) * Density * PctWoken *
DailyIncome * ProductivityLoss
`

where:

• multiply by twice the SirenDistance, how far away on each side of the police car a person can hear the siren, to get the geographic area affected
• multiply by the Density to get the number of people potentially affected
• multiply by the PctWoken to get the number of people who were woken up
• multiply by the average DailyIncome to see how much economic value those people create each day
• multiply by the ProductivityLoss (as a percent) that they experience when groggy to see how much economic value was lost by the siren

This model makes some assumptions. It assumes the police car drives in a straight line, that the density is uniform, that the PctWoken is constant within the SirenDistance and zero outside of it, that everyone works (so ignoring children), that everyone works a day shift, that the ProductivityLoss is independent of DailyIncome, etc. But it seems like a reasonable first step.

Let’s plug in some values:

LengthDriven = 0.25 miles

SirenDistance = 400 feet \approx 0.08 miles

Density = 18,187 people/square mile in SF (from Wikipedia)

PctWoken = 5% of people. I made this up out of nowhere.

DailyIncome = \$45000 yearly per-capita income in SF / 200 work days per year = \$225 per day

ProductivityLoss = 25%, I made this up too

This gives us an economic cost of \$1023 every time a police officer flips a siren on at night.

Even if this only happens once per night in SF, it creates a cost of \$375,000 over the course of the year — equivalent to the salary of about 4 police officers, or about 1% of the SFPD’s budget. Use of sirens also appears to be dangerous. I wonder what the benefits are — how much additional public safety is provided for this cost?

I reached out to the SFPD asking if they have any guidelines about siren use — stay tuned.