# 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