Thought: At What Speed Should You Speed?

4/16/15

By Kevin DeLuca

ThoughtBurner

Opportunity Cost of Reading this ThoughtBurner post: $2.28 – about 10.4 minutes

While I’m sure that many of you readers are outstanding citizens who would never ever even dream about ever breaking the law ever, I know that some of you are natural-born rebels and straight‑up gangsters that look at the list of minor traffic violations[i] and say, “Nah, Imma do me.” Speeding ensues.

Most people will not drive faster than they feel comfortable driving, but the prevalence of speeding tickets suggests that often times people’s maximum comfortable driving speed is above the set speed limits. According to the info graphic on this webpage[ii], 20.6% of all drivers will get a speeding ticket over the course of a year, costing them an average of $152 per ticket. And about 41 million people get speeding tickets each year cumulatively bringing in over $6 billion in revenue from fines.

If people are making an appropriate cost-benefit analysis when they decide to speed, then these facts mean that the time saved from speeding over the course of a year is worth at least $6 billion dollars. But I don’t think that most people are actually doing any calculations before they make their decision to speed, so there is a large potential for inefficiency – I suspect people are speeding in a non-optimal way.

There are costs and benefits to speeding, and the prevalence and price of speeding tickets are non-trivial, so ThoughtBurner is here to help you out. In Speeding Part 1, I will attempt to answer the question: at what speed over the speed limit should you, the driver, speed? In Speeding Part 2, I will help the government out by helping them decide how expensive a speeding ticket really should be (sorry everyone).

THE DRIVER’S PROBLEM

So, you want drive somewhere and you’re wondering if speeding there to save some time is worth risking getting a speeding ticket. How do you decide? Before you can make an informed decision, you need to know a few things:

  • The time you will save by speeding
  • The (subjective) value of the time you will save by speeding
  • The probability that you will get caught speeding
  • The cost you will have to pay if you get caught speeding

The time you will save by speeding is fairly easy to calculate; distance divided by (speeding) speed, minus the distance divided by (non-speeding) original speed. The trick is, for each mile per hour over the speed limit you travel, you don’t save the same amount of time. An easy example: say it takes you 10 minutes to get somewhere going 20 mph. If you go 40 mph, you’ll get there twice as fast – in 5 minutes, which means you’ll save 5 minutes of your time. If you go 60 mph, you’ll get there three times as fast – in 3.33 repeating (of course) minutes, which means you’ll save 6.66 repeating (of course) minutes of your time. The first 20 mph over the limit saves you 5 minutes, but the next 20 mph over the limit only saves you an additional 1.67 minutes.  If you do some calculations, you can see that the higher the speed limit, the less valuable speeding is (in terms of time saved). But in general, we can show the relationship graphically, holding distance traveled constant:

Time Saved vs Speed

The subjective value of the time you will save by speeding is a bit trickier. It probably depends on a lot of different things – like how busy of a person you are, how late you are already running, how much you subjectively don’t like driving, etc. – and so I can’t know the actual level of your subjective value of the time you will save. But, I think it is safe to assume that the subjective value of time saved, or the utility of time saved – U(s) – is diminishing. For example, imagine that you are running 5 minutes late and you are deciding whether to speed to save 5 minutes or not speed (save 0 minutes). Those 5 minutes could be very valuable. Now imagine that you are 5 minutes late and you are deciding whether to speed to save 35 minutes or 30 minutes (yeah, you’re going, like, super-fast). When you are already saving a lot of time from speeding, e.g. 30 minutes, then saving an additional 5 minutes isn’t really worth much. Even if the utility from time saved wasn’t diminishing, actual time saved is diminishing as you go faster and faster, so the value of time saved will be diminishing as well. We can represent this graphically too:

Utility vs Speed

The probability of getting caught is another very tricky number to estimate. I spent a lot of time thinking about more sophisticated models, where the probability of getting caught is a function of the distance you are traveling, the speed at which you are going, and other factors, but I think that for our purposes the best thing for us to do is try to estimate the probability of getting caught per trip. I will provide actual estimates later, but for now let us call this constant probability of getting caught “p”. We can expect, given that only 1/5th of drivers actually get speeding tickets every year, that p is probably small.

Last, we need to know the penalty for getting caught. I will consider the case where there is a base fee for speeding, plus a fine that increases depending on how fast over the speed limit you were traveling, as is the case in Travis County (Austin, Texas)[iii]. This is not actually super common – I will consider alternate fine schedules later, but this particular case leads to an easier solution. The cost of the speeding fine, then, is linearly related to how fast you are speeding:

Fine vs Speed

If multiply the graph above by p, we get a graph of the expected costs of speeding; the amount you have to pay if you get caught weighed by the chance that you actually get caught.

Expected Costs vs Speed

We can combine all this information to solve what I will call the driver’s maximization problem, which is: maximize the benefits from speeding minus the expected costs of speeding. More precisely, the problem is:

DriversProb

Where U(s) is the driver’s subjective value of time saved (Figure 2) and E[C(s)] is the expected cost of speeding (Figure 4). Notice that the driver is choosing at what speed to drive which, in this model, will determine the value of both U(s) and C(s).

I will now consider two possible types of drivers: Punctual Perry and Lackadaisical Lucy.

  • Case 1: Punctual Perry

Punctual Perry doesn’t like missing out on anything. So, he always plans ahead and makes sure to leave early whenever he has to drive anywhere. Speeding and saving time doesn’t really give him much value, since he’s never really rushed for time. When Punctual Perry plots his driver maximization problem, it looks like this:

Punctual Perry

Notice that Punctual Perry never subjectively values his time saved more than the value of the expected cost of a speeding ticket for any given speed. This is because Punctual Perry is true to his name (punctual); he doesn’t need to save time since he’s always on time, so saving more time isn’t very valuable to him. He is more worried about the expected cost of the hypothetical ticket than saving a few extra minutes.

We don’t even have to do any math (yay!) to see what the solution to Punctual Perry’s driver maximization is: don’t speed. The expected costs are always higher than his benefits, so any amount of speeding leads to negative values for the driver maximization problem. If he doesn’t speed, there are no benefits but also no chance of getting caught speeding, and since zero value is better than negative value, Punctual Perry just never speeds.

  • Case 2: Lackadaisical Lucy

Lackadaisical Lucy is a more interesting case. She is typically late to things, which means that time saved by driving a little faster is more valuable to her compared to time saved by Punctual Perry. When Lackadaisical Lucy plots her curves for the driver maximization problem, it looks like this:

Lackadaisical Lucy

For Lackadaisical Lucy, there are speeds at which the value of the time saved is greater than the expected costs at that speed. If Lackadaisical Lucy chooses the right speed, she can maximize the benefits from speeding conditional on expected costs. But what speed is the right speed? It is the speed at which the distance between U(s) and E[C(s)] is the biggest.

Refer back to the driver’s maximization problem. In order to maximize, Lackadaisical Lucy can simply take the derivative of the driver’s problem with respect to speed and set it equal to zero. Doing so results in:

FOCs

In words, this means that Lackadaisical Lucy should choose a speed, s*, where the additional benefit of speeding a little more is equal to the additional cost of speeding a little bit more (marginal benefit equals marginal costs). Graphically, this speed is shown on the graph at the point where the slope of Lackadaisical Lucy’s utility curve is equal to the slope of the expected cost curve:

Lackadaisical Lucy FOCs

The speed where the slopes are equal is the optimal speed that Lackadaisical Lucy should drive in order to maximize her utility. By choosing speed s*, Lackadaisical Lucy is maximizing the difference between the benefits of speeding and the expected cost of speeding, for some given distance. This is good, because Lackadaisical Lucy is a rational human being who wants to maximize her utility.

In summary, this model predicts that some people (or all people in some circumstances) will decide not to speed when the benefits from speeding never exceed the expected costs (Punctual Perry). And people who decide to speed (Lackadaisical Lucy) should drive at a speed where the marginal expected cost of speeding is equal to the marginal subjective benefit of speeding.

So, if you find yourself being a Punctual Perry, then not speeding is the right choice. But what if you are being a Lackadaisical Lucy? How do you know how fast to speed? What is the actual marginal cost of speeding?

EMPIRICAL ESTIMATES

Using the simple model developed above, I will now provide some empirical estimates to help you all solve your own driver maximization problems.

I’m guessing that, in real life, many people are Lackadaisical Lucys in the sense that there is a point where they value saving time more than the expected cost of speeding, though not necessarily just because they are always late. For example, they could just hate driving a lot so that speeding to drive less is worth the risk. Regardless of their reason, these types of drivers can use the model developed above to determine their own individualized solution to the driver maximization problem whenever they want to go somewhere.

The subjective value of time saved is all about you guys, and it could vary pretty widely across individuals (Punctual Perrys vs. Lackadaisical Lucys). Also, remember that the amount of time saved – and therefore your utility gained from it – depends on the distance you are traveling. But the probability of getting caught and the costs of traffic fines faced by everyone are the same, so I’ll first focus my attention on providing some guesses of these values.

Start with the statistics from above that says 20.6% of all drivers get a speeding ticket each year. The most recent estimates from the 2009 National Household Travel Survey[iv] put the average daily number of vehicle trips per driver at about 3 a day (see Table 3). This means that each driver makes (3*365) 1095 car trips every year. There is a 20.6% chance that at least one of those trips will result in a speeding ticket. Which also means that there is a 79.4% chance that none of those trips will result in a speeding ticket. If we let p equal the probability of getting a speeding ticket per trip, then it follows that (1 – p) is the probability of not getting a speeding ticket. Then:

ProbEstimate1

The left hand side is the probability of not getting a ticket 1095 trips in a row, and the right hand side is the observed proportion of people who don’t get a speeding ticket each year. Solving for p gives:

ProbEstimate2

That is, the probability of getting a ticket per car ride is one minus the 1095th root of 0.794, which comes out to be about 0.0002. While this may seem really low, it is.

People drive a lot, and considering that drivers also probably have strategies for avoiding speeding tickets (e.g. don’t speed on certain roads where cops hang out), it is not that surprising to me that the probability of getting caught is so low. You actually have to be pretty unlucky to get a speeding ticket.

Using the fine schedule from Travis County, we know exactly how much the speeding ticket will cost you at any given speed over the speed limit. It is a $105 base fine, plus $10 per mph over the speed limit you get caught speeding[v]. Mathematically, this means that:

Cost1

Adjust this by our newfound predicted probability p, and you get:

Cost2

Remember that the maximizing condition is when marginal benefits are equal to marginal costs:

FOC2

Taking the derivative of E[C(s)] with respect to s:

CostDerivative

And substituting in gives:

FOC3

This means that, if you speed, you should speed at s* where the marginal benefit of speeding is equal to 2 tenths of a penny. Which is almost nothing. So, your optimizing speed will be very close to the speed at which you will no longer gain any benefits from increasing your speed. For example, based on my own previous estimates of opportunity costs[vi], the value of $0.002 is approximately equal to the value of 0.54 seconds of leisure time.

Imagine how happy you would be if I told you that you would have an extra 0.54 seconds today to do whatever you want! If you would be at least that happy by speeding a little faster, then you should do it.

The implication of this is that people acting optimally (in Travis County) should basically just completely disregard speed limits and drive at the fastest speed they feel comfortable driving – well, marginally below it. Only go 4.9999 mph over the limit vs. 5 mph over the limit.

The results initially surprised me. People who aren’t making any optimizing calculations are also probably getting really close to choosing the correct speed the model says they should speed. Since the probability of getting a speeding ticket per trip is so low, the expected costs are also very low and the marginal cost of increasing your speed is even lower (close to zero).  So just keep increasing your speed until the marginal benefit is close to zero as well. Graphically, it would look something like this:

Empirical Graph

Basically, yeah you were most likely already doing it right. Speed at the max speed you are comfortable driving, because the probability of getting caught is so low per trip that the marginal expected cost of speeding is less than half a penny. You were already acting optimally! Wow, brilliant.

Even if you change some of the assumptions that would lead to higher estimates of p, the results are essentially the same. For example, implicit in our estimate of p above is that people speed on every trip they make throughout the year and have a non-zero chance of getting a speeding ticket. However, it seems likely that this is not the case – people don’t always speed no matter what. So, let’s be extremely generous and assume that people only speed one days-worth of trips (3 trips) per month. Then we would have:

CostEstimate2

And, substituting in the new p with our optimizing conditions give:

CostEstimate3

Which means that you should speed until the marginal value of speeding is worth only $0.064 in time saved which, again based on my own previous estimates of opportunity costs, is about equal to the value of 17.5 seconds of leisure time. Again, imagine how happy you would be if I told you that you would have an extra 17.5 seconds today to do whatever you want! If speeding a little faster makes you at least that happy, you should do it (with the above specification).

In general, the strategy of just speeding at whatever speed you want is probably a very close approximation to the optimizing strategy. I originally thought I would end this post by telling everyone to speed less, but instead I think that in the spirit of ThoughtBurner’s mission I have to encourage you all to ignore speed limits. What have I done.

THE GOVERNMENT’S PROBLEM

The purpose of speed limits and speeding fines is to deter people from speeding. In this model, drivers acting optimally will basically ignore speed limits. This makes me wonder whether the City of Austin has set traffic fines high enough to deter speeding. For example, other states have speeding fines of up to $1,000[vii]. Perhaps there are other incentives that determine the schedule of speeding fines, such as revenue collection.

So, imagine now that you are a city government planner and you are tasked with determining how expensive speeding fines will be. How do you know you have chosen the right penalty amount? If drivers are basically ignoring speed limits right now because their expected costs are so low, then in order to get them to stop speeding either you need to increase the probability of getting a speeding ticket (hard way) or increase speeding fines (easy way). This will be the problem I will solve in Speeding – Part 2.

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[i] https://www.tdcj.state.tx.us/divisions/hr/hr-policy/pd-27a.pdf

[ii] https://www.thezebra.com/insurance-news/315/speeding-ticket-facts/

[iii] https://www.traviscountytx.gov/justices-of-peace/jp1/court-costs

[iv] http://nhts.ornl.gov/2009/pub/stt.pdf

[v] https://www.traviscountytx.gov/justices-of-peace/jp1/court-costs

[vi] https://thoughtburner.wordpress.com/2015/02/26/thought-the-value-of-reading-a-blog-post/

[vii] https://www.thezebra.com/insurance-news/315/speeding-ticket-facts/

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