We spent this past weekend in Florida, visiting Kate’s mom and her husband, who moved down there in October. This was a huge hit with the kids, who were very excited to fly on an airplane (four of them, actually, as we changed planes in Baltimore both ways). They also got a big kick out of driving around in a rental car– The Pip chattered happily about “My new car” for a while– which we did a lot of, going to a beach and the Mote Aquarium in Sarasota.
Doing all that driving in a rented SUV and a state with a 70mph speed limit got me thinking about optimum driving speed. Particularly on the trip back to the airport, where I was trying to eke out the last few miles on the single tank of gas I had purchased from Hertz (I managed to roll in with the needle just hitting E, and the rental return person complimented me on getting the most out of that tank).
It’s well known, of course, that the efficiency of a car engine drops as the speed increases– the force of air resistance increases as you go up in speed, probably as something like the square of the speed, so the engine needs to work harder to maintain the same speed, and burns more gas. You can find any number of sites that sanctimoniously lecture you about how much better your car would run if you would just drive at 55mph everywhere– this one even includes a price calculator.
Of course, there’s something fundamentally disingenuous about these calculations, namely that they assume the time you spend driving is free. I could, that calculator informs me, save $467 a year by driving at 60mph instead of 75mph for a 30-mile commute, at the cost of only 6 minutes per day. Which sounds like a great deal, provided I don’t put any value on my time.
But, of course, time behind the wheel is time not doing anything else, and if we’re going to properly calculate an optimum speed, we really ought to put some monetary value on that, as well. As this classic xkcd reminds us.
If we attach a monetary value to the time spent driving, then the optimum speed is not necessarily the speed at which the engine is most efficient. The cost of fuel will be lowest at the maximum efficiency point, of course, but that’s only a part of the equation. You also need to figure in the cost of your time. So, fuel costs will go up as the speed increases, because the engine works less well, but personal time costs will go down, because the higher speed reduces the time in transit, and thus the number of hours you’re “paying” yourself for. At some point, those two trends will cross over, and there will be an optimum speed where the total cost is a minimum.
How does this work out in practice? I originally planned to do this as a post with equations, setting up some toy model for the efficiency, adding the increasing time cost, and doing freshman calculus to find the optimum speed where the cost was minimized. When I went looking for numbers, though, the figures for the efficiency look like a more complicated function than I want to try to figure out, so we’ll do this by an empirical method.
That fuel cost calculator page gives approximate values of the decrease in efficiency for speeds above 55mph, so we’ll use those six points, and calculate the total cost for a 100 mile trip, assuming different hourly rates and gas at $3.50/gallon. If we do that for a rental SUV getting 20mpg (which is probably not a horrible guess for the car we were driving this past weekend), we get the following graph:
As your hourly rate increases from $10/hr to $25/hr, the total cost of the trip increases, because they all take some time, and you pay for that time. If you look closely, though, you can see a trend in the points for each color, particularly at the lower pay rates– at lower speed, the cost is a little higher because of the increased time. At lower pay rates, there’s a minimum value of the total cost, at around 65mph, because the fuel cost eventually overcomes the increased time. You can see this trend slightly better if we normalize all the points to their value at the maximum efficiency point of 55mph:
Here you see that, as a fraction of the initial cost, the total cost drops as you increase the speed, up to a point, then at the low pay rates, goes back up. For higher pay rates, though, the cost of the fuel just isn’t that significant compared to the cost of the time, so the curve flattens out, and then for the highest pay rates, continues to drop. So, according to this calculation, if you value your personal time at $20/hr or more, you should drive at 80mph. If your personal time is worth $17/hr or less, the optimum speed is around 65mph.
Of course, this also depends on the efficiency of your car. If instead of an SUV we were driving a higher-efficiency vehicle with some sort of internal pocket dimension to hold all the crap that’s required when you travel with kids, the normalized cost graph for a 30mpg car looks like this:
So, if your car gets 30mpg at 55mph, and you vale your time at $15/hr or above, you should drive as fast as you can. Time is money, and it’s more money than the gas you burn.
Of course, if I stop there, this not-entirely-serious post is all but guaranteed to get me some comments from tiresome scolds telling me I’m a horrible person for not regarding energy efficiency as the be-all and end-all of everything. And it’s true, I’m a terrible person. I also shower with the lights on, thus wasting electricity.
Of course, those issues are, in principle, something we could capture in this model by increasing the effective price of gas to more appropriately capture the externalities of carbon emissions and pollution, etc. If we bump the price of gas in the toy model up from $3.50/gallon to $4.50/gallon, the result looks like this:
That’s got a minimum cost point at lower speed for all but the highest pay rate. And if you bumped the cost of gas up even more, even the high-salary folks would no doubt stop benefiting from driving at maximum speed. Then it’s all a question of arguing about what the “right” price of gas is, and what the “right” value to put on your time is. Should you be billing your free time at the same rate as your salary? More? Less? And how do we price in the likelihood of getting a speeding ticket at the high end of these graphs?
Those are all interesting questions, but I’ve already devoted more time to getting this out of my head than I ought to, and need to get back to writing my book. Time is word count, and word count is money, too.