Gerry's World

A glimpse into my life

April 15, 2022

Background

I’m planning a day-trip and trying to weight my travel options. I want to include the following costs:

Cost of lost productivity

This is also pretty easy to calculate. Since this is happening on a weekend and my weekends are usually just me sleeping most of the day anyway, I valued my time at a measly 5 / hour. I also give multipliers for how “productive” I could be on a bus vs plane. Cost of death This is an interesting one. First I calculate chance of death, then the expected value of days of my life lost, then convert to dollars. Chance of death First calculate the chance of death. This is easy to do at a first-order approximation - passenger cars have ~1.5 deaths / billion vehicle miles according to NHTSA. Buses are purportedly 50 times safer than passenger cars, and planes 750 times safer. More accurate calculations are tricky for a number of reasons: • actually a lot of the deaths due to cars are e.g. pedestrian deaths, not passenger deaths • car safety ratings matter • my driving abilities matter - I give a 1.5 multiplier since I consider myself a below-average driver • time of day matters • geographical location matters (e.g. easy highway vs busy metro city) • non-fatal injury is not accounted but does matter In any case, I just use the first-pass approximation. Expected value of days lost We could analyze this 2 ways: 1. what is the cost of going on the trip in the first place 2. assuming I am definitely going on the trip, what’s the cost of death for different options? I’m pretty sure these both turn out to be very nearly equivalent. Let me explain why. Justification of equivalence First, option 2 is easy to calculate: just multiply the (probability of death) by the (sum of all future joys) to figure out how much joy you’ve lost by exposing yourself to death risk. $$\boxed{\text{Cost due to chance of death} = (\text{probability of death}) \cdot (\text{sum of all future joys})} \tag{1}\label{eq:cost_to_chance_of_death}$$ Option 1 is more interesting. The formula should be something like: $E[\text{no trip}] - E[\text{yes trip}]$ where $$E[\cdot]$$ denotes the expected value. “no trip” means I never went on the trip, while “yes trip” means I did go on the trip, and took a given mode of transportation. Expanding out $$E[\text{yes trip}]$$, $E[\text{yes trip}] = P(\text{not dying})V(\text{not dying}) + P(\text{dying})V(\text{dying})$ where $$P(\cdot)$$ denotes the probability of an event occuring, and $$V(\cdot)$$ denotes the value/joy derived from that event occurring. Both “not dying” and “dying” are assuming we did go on the trip. Note that obviously $$V(\text{dying}) = 0$$ since we will never achieve any joy ever again by dying. $V(\text{not dying}) = V(\text{no trip}) + V(\text{the trip itself})$ here, $$V(\text{the trip itself})$$ denotes the joy I derive from going on the trip. Then, \begin{align*} E[\text{yes trip}] &= [1-P(\text{dying})][V(\text{no trip}) + V(\text{the trip itself})] + P(\text{dying})\cdot 0 \\ &= 1\cdot V(\text{no trip}) - P(\text{dying}) V(\text{no trip}) + (1-P(\text{dying})) V(\text{the trip itself}) \\ &= E[\text{no trip}] - P(\text{dying}) V(\text{no trip}) + (1-P(\text{dying})) V(\text{the trip itself}) \\ &\approx E[\text{no trip}] - P(\text{dying}) V(\text{no trip}) + V(\text{the trip itself}) \end{align*} So the cost ( cost = -value ) of going on the trip is: $E[\text{no trip}] - E[\text{yes trip}] = P(\text{dying}) V(\text{no trip}) - V(\text{the trip itself})$ To do a quick check to see if the trip is worth it in the first place, we calculate $$P(\text{dying}) V(\text{no trip})$$ and see if it’s reasonable. To compare different transportation methods, the $$V(\text{the trip itself})$$ term will cancel and we just compare $$P(\text{dying}) V(\text{no trip})$$ for the different methods. So in other words, the value of we need to calculate is $$P(\text{dying}) V(\text{no trip})$$, or in other words, (probability of death) $$\cdot$$ (sum of all future joys), which is exactly the same as Eq. \eqref{eq:cost_to_chance_of_death} in option 2. (QED) Evaluation In section Chance of death we already calculated $$P(\text{dying})$$ so now we just need to calculate $$V(\text{no trip}) := (\text{sum of all future joys})$$. This depends heavily on both the probability of death vs age curve (e.g. “# of lives” column of an Actuarial Life Table) and also the utility of a year of life at a given age (presumably younger years are more valuable than older years). I just make a 0th order approximation that when you multiply these two together and average over all future years, 1 year is approximately worth 0.5 “units” where I define a “unit” to be the value of one year at my current age. My logic is that when multiplying together the death curve with the joy/utility curve, it will be something like a line that starts at 1 right now and ends at 0 at age 75, so the average will be 0.5. At age 25 and assuming life expectancy 75, that’s $(50 \text{ years}) \cdot (0.5 \text{ joy-years/year}) \cdot (365 \text{ days / year}) = 9125 \text{ joy-days}$ Finally, to convert to dollars, I make the rough approximation that the value of 1 day is worth400. This is based on the fact that I would work for about $50-100/hour for 40 hours / week; and, as a rational being, that is “worth it” to me, so that must be a reasonable estimate of my day’s worth. That makes$2000-4000 / week = $286-571 / day, so I say roughly$400 is reasonable. Note that if I thought my day was worth more, I would work fewer hours; and if I thought my day was worth less, I would work more hours. There’s some subtlety here that I’m missing but in any case, I think it’s a reasonable estimate.

Then I arrive at the figure,

$(\text{sum of all future joys}) \approx (9125 \text{ joy-days}) \cdot (400 / \text{joy-day}) = 3.65 \text{ million}$

Then the cost of traveling due to chance of death according to Eq. \eqref{eq:cost_to_chance_of_death} is

$\boxed{P(\text{dying}) \cdot (3.65 \text{ million})}$

Final Cost Tables

Transportation

Direct Cost Productivity cost Chance of Death Expected cost of lost joy-days $-equivalent joy-day cost Car$254.82 $63.33 0.0000171 0.1560375$62.42
Rental Car $174.60$63.33 0.00002565 0.23405625 $93.62 Bus$80.00 $23.75 0.000000342 0.00312075$1.25
Hotel $87.50 Final Options Direct Cost Cost (including cost of death) Cost (including cost of productivity + death) Rental + Sleep in car$174.60 $268.22$331.56
Car + Sleep in car $254.82$317.24 $380.57 Bus + AirBnB$148.50 $149.75$173.50