If You've Never Missed a Flight, You're Spending Too Much Time at Airports

Waiting in a basically empty terminal for a flight not scheduled to depart for another hour reminded me of something Dr. Jordan Ellenberg convincingly discussed in his brilliant book How Not to Be Wrong: The Hidden Maths of Everyday Life. Back in October I thought I praised his book slightly too much, only to realize that I had forgotten to discuss this point.

In one of his chapters he discusses the optimal time to arrive at airports before the flight departures. He uses expected utility theory (for all its problems...) to suggest that maybe those hours waiting in the departure area or simply wandering the airport are pretty much wasted and could be avoided by showing up later – which also means that from time to time you may miss a flight or two. This is how Business Insider summarized the point when his book came out:

Figuring out when to go to the airport involves a tradeoff: the earlier you show up, the more likely you are to actually make your flight. But, showing up early also involves a cost in the form of time spent at the airport rather than elsewhere.

In building on a famous quote by Nobel laureate George Stigler, Ellenberg suggests that never missing a flight would suggest that you are at a corner position, too far out one axis and could improve your situation, your utility, by showing up later; in econo-speak moving towards the optimal point of tangecy between the indifference curve and the budget line:

What is always the case is that the best point is somewhere in between the two extremes, which means, in particular, that when you take the optimal time, whatever it is, your chance of missing the plane is not zero. It’s probably pretty small for most people, but it’s not literally zero.

Having nothing better to do, I tried to recount all the times I've been on a flight and how often I have missed them. It's not a pretty read. I counted 84 flights over the last decade and I'm fairly sure I never missed a single one (though Toronto last summer was very very close – running-like-a-maniac-close). On the face of it, it suggests I am spending too much time at airports. 

But hang on a moment, it could also be the case that my utility function includes a very large loss aversion with a high sensitivity to the money I spend on flights; flights do tend to be expensive outlays, and I'm a student after all (privileged, apparently, but nevertheless student). So I tried summing up what I spend on my last ten flights and averaged them out: it turns out I spend around £100/flight (I blame Australia for this high number), a sizeable amount of money for a small-scale student economy.  

Implicit in Ellenberg's (and Stigler's) argument is the opportunity cost of idling away in empty terminals, and minimizing this might justify missing a flight from time to time. It involves an idea of how much money you could have earned instead of wandering the airport shops or people-watching. 

There are a couple of problems here that I believe justify my 84-in-a-row streak and the (conservatively measured) extra hour I waste away waiting for flights to depart:

1. My hourly wage is very low, measured by the marginal value my future income increases from an hour of academic work. This is, after all, the prudent reason to pursue higher education (if we discard the real reason for a second). Besides, the actual wage I do get paid from various jobs during weekends or holidays is also very low, but more importantly tends to be temporally remote from whenever I am travelling; i.e., there isn't really a monetary opportunity cost of showing up too early.
2. Time spent "Idling away" at the airport perhaps isn't so idle work after all, if I manage to produce blog posts or get some reading done while I'm waiting. 
3. Maybe the emotional cost of disrupting plans, paying overprice for a new flight, or just barely make it is substantial enough to tilt the tradeoff in favour of even more time in advance?

Insofar as my hourly wage – my monetary opportunity cost – is low, and my flights average a hundred quid each, the cut-off point for where it would make sense to start arriving later may be far off in the future; 84 observations may not be enough. Moreover, for the same reason it is quite possible that I am currently at the optimal amount of time, even though I have never missed a flight, simply because the sample is too small. Maybe my optimal trade-off is arriving an hour earlier than I would need to, and miss only one flight in every 300. Then, obviously, I may not have reached that point yet. This last point also seems consistent with the above reasons I gave, in econo-speak my utility function may simply be steep enough for the optimal point to be very very close to the corner. 

But, I also think Ellenberg & Stigler etc takes too lightly the non-monetary cost of missing flights. It is not just the price of the flight that has to be factored in, but the additional time loss and the opportunity cost of that. You are commonly travelling for a reason, with a purpose in mind; if that purpose has a time-element to it, missing flights may be hugely expensive. For instance, what scared me so much about almost missing the Toronto flight this summer was that I would have been seriously late for Mises U 2016 – like 6hs or so, and definitely missed the first couple of sessions. A very high price to pay, indeed. 

Yes, guys, your claim sounds really cool and has certain appeal to it, but for most practical reasons it doesn't quite apply. A bit like EUT in general, isn't it?