Thought: Welfare-Maximizing Speeding Fines


By Kevin DeLuca


Opportunity Cost of Reading this ThoughtBurner post: $3.31 – about 15.1 minutes

Before you read this, make sure you’ve read my earlier post about speeding and the driver’s maximization problem. This is the second of a two-part ThoughtBurner series, and this post will look at the issue of speeding from the perspective of a city planner or benevolent government.

Imagine that you are working for a city government, and the local police chief wants to know how much the monetary penalty for speeding should be. The reason there are speed limits (allegedly) is so that people will not drive dangerously. Clearly, then, as a government official you would want people to follow the speed limits. This would probably be best overall, since it is bad for ‘society’ if a lot of people are driving dangerously.

But your purpose isn’t to completely stop people from speeding; it is simply to deter them from speeding. Or, in other words, you want to create incentives that will make them follow the speed limit more often. If you really wanted to make sure people didn’t speed ever, you could just make the fine for speeding ridiculously high – $100 million if you get caught. The expected costs would be way too high for any rational person to speed. But, clearly, taking every penny from someone who drives a little too fast is a net negative for society, so you want to avoid that scenario.

To make things even more confusing, your government actually makes money if it gives out more speeding tickets. Even though speed limits and speeding fines are meant to stop people from speeding, the government will lose revenue if that were to happen. The fines from speeding tickets can be spent on other useful things, for example fixing roads, and this benefits society as well.

So, it’s a tricky balance to find. People benefit from speeding, but speeding also imposes costs on society. The government benefits financially if more speeding tickets are given out, but if more people are speeding then roads are more dangerous. The government’s problem, and more specifically your problem as the city planner, is to figure out how expensive these speeding ticket fines ought to be given all the benefits and costs of speeding.


Taking the perspective of the government, we can try to find the optimal level of speeding ticket fines. The benevolent government can be thought of as an optimizing agent itself, who wants to maximize the welfare of society.

In a simple model, there are three things that the government needs to take into consideration when deciding how expensive fines should be:

  • The speeding ticket revenue that can pay for other public goods
  • The cost of enforcing speed limits
  • The cost of accidents caused by speeding

The government can also control three things:

  • The actual speed limits (which affects the cost-benefit analysis of speeding individuals)
  • How expensive speeding tickets are (revenue for public goods)
  • The amount of enforcement (cost of enforcement)

Accident costs are outside of government control – they will affect the optimizing solution but the government can do nothing about them (except indirectly, as we will soon see).

For simplicity, I will make a few additional assumptions for easier calculations. First, even though the government has control over many factors, we will suppose that the government only focuses on choosing optimal fine levels. Notice that this is the easiest policy route to take – the process of changing the cost of a speeding ticket is basically costless itself, so any necessary adjustments would be relatively easy to make. Increasing enforcement is expensive, hard to do, and probably unpopular. Also, there has already been a lot of research done by road building people on how best to set speed limits[i], so I will assume that the government has already set speed limits at an optimal level.

By altering the fines, the government can affect drivers in two ways. The government can either stop people from speeding completely (turn more people into Punctual Perrys), or change the how speedy people speed (change the value of s*). These two strategies are illustrated below:

RaiseBase copy


RaiseSlope copy

The first figure happens when the government raises the base fine for speeding – in Travis County, this would mean raising the $105 base fine. The second figure happens if the government changes how quickly the fine increases (the derivative of fines with respect to speeding) – in Travis County, this would mean raising the $10 per 1 mph-over-the-limit penalty.

We will also assume that the government solves this problem on a year-to-year basis, so it considers costs and benefits over the course of a year (rather than on a single day or over an entire century). This is a realistic assumption because governments usually make budget decisions annually[ii].

Every year then, the government should try to solve what I call the government’s optimization problem. In words, the government’s objective is to maximize the benefits to society minus the costs to society, by choosing speeding fines. We can write this as:


Don’t worry if it looks complicated. It’s not, it will all be ok! R(p,N,F) is just the amount of revenue brought in by speeding tickets. C(e) is just the fixed costs of enforcing some “e” amount. And the last term, A(r(F),N,c), represents the societal costs of accidents.

The revenue function is straightforward to calculate. It is just the number of tickets multiplied by the cost of a ticket. We can express the number of tickets each year as the probability of getting a ticket, p, multiplied by the number of people in the city, N. If we let F be the average cost of a speeding ticket, we have an expression for revenue:


Since we are focusing on setting fines, we are also assuming that enforcement costs remain the same. The government doesn’t increase the number of police officers, or the amount of time they spend trying to catch people. They still have to pay all of the costs of enforcement, but they are fixed costs, which will not affect our optimizing solutions. This means that C(e) is fixed and we know that changing F will not affect these fixed costs.

Next, while the government cannot control how much an accident costs on average, it can reduce the total costs of accidents by decreasing how many people get into accidents. By increasing speeding fines, the government can deter people from speeding, which might lead to a reduction in accidents. In this sense, the accident rate can be thought of as a function of speeding fines; r(F).

Last, the societal cost of accidents is really just the number of accidents multiplied by the average cost of an accident. If we express the number of accidents as the accident rate, r(F), times the number of people, N, then we just need to multiply this by the average cost of an accident, which we will call c, and we’ll have our expression for the societal cost of accidents:


Let us rewrite the government’s problem, substituting in the expressions for ticket revenue and accident costs:


In order to solve this maximization problem, we just take the first derivative of the expression with respect to speeding fines, F, and set it equal to zero. The fixed costs of enforcement drop out since they are not functions of F, and we are left with:


So obviously, as I’m sure all of you guessed from the beginning, p times N has to equal dr over dF times N times c. Duh, haha, who didn’t see that coming! Now we’ve solved the government’s maximization problem! [smirk emoji about here]

Our first order conditions actually describe something pretty simple: that the marginal increase in ticket revenue must equal the marginal increase in accident costs given some increase in speeding fines. Ok, so it’s not that simple. Let’s make it easier to deal with.

We already know how the constants p and c are defined, so the only confusing term left is dr over dF, which is how the accident rate changes when speeding fines change. It doesn’t have an intuitive interpretation at this point, because raising fines doesn’t actually change the accident rate directly. But, we do know that speed affects accident rates, especially speeding. Observe the magic of mathematics:


So, the change in the accident rate with respect to fines (dr over dF) is really just a function of how the accident rate changes as speed changes (dr over ds), adjusted (divided) by how fines change as speed changes (dF over ds). This is much better for us (well, the government) because some smart people have already figured out a lot about how speed affects accident rates, which means that I (or government officials) have much less work to do.

If we plug this back into the equation above, cancel out the Ns, and solve for the change in traffic fines with respect to speed:


Now, we have discovered the condition that needs to hold in order for the government to be acting optimally. It says that if the rate at which the fines increase is equal to the rate at which accidents increase multiplied by the cost of accidents, divided by the probability of getting caught, then the government is maximizing net social benefits. As long as this condition is met, the government will be solving its optimization problem.

Luckily for the government, they have direct control over all aspects of speeding fines, including their rate of change. For example, we knew from part 1 that Travis County had set the rate at which the fine increases as speed increases equal to a constant, $10 ($10 more fine for every 1 mph over the speed limit). Now the only question that remains is: how does the accident rate increase as speed increases?

According to an oldish traffic report done by David Solomon at the Department of Commerce[iii], the accident rate is better thought of as a function of how much you deviate from the average speed, not your actual speed. Figure 7 taken from the paper shows it well:

Screen Shot 2015-04-23 at 6.26.46 PM

This is essentially a graph of the shape of dr over ds. It shows how many accidents occurred at different deviations from the average speed (over a given distance, 100-million vehicle miles). The lowest rate seems to be at or slightly above the average speed, and the fitted line increases exponentially in both directions as the deviation from the average speed increases. Notice that the y-axis is in log scale, so the increases are even bigger than they appear visually. In the paper, they claim that this general pattern holds regardless of what the average speed actually is, though the end behavior changes a bit at really low or really high average speeds.

Intuitively, I think this makes sense – if you are going 60 mph in a 40 zone, I’d imagine you’d be way more likely to cause an accident than if you were going 60 mph in a 60 zone. What was initially surprising to me is how the rate of getting into an accident by driving too slow is actually just as high and sometimes higher than the rate of getting into an accident by driving to fast (more about this later).

If we assume that most people travel at a speed close to the speed limit when they drive, then we can use this information to assess the risk of speeding violations independent of the actual speed. Travis County’s penalties for speeding violations already sort of do this – you pay the same fine for going 5 mph over the speed limit regardless of the actual speed limit. This risk can then be used in our conditions above to calculate the optimal speeding fines.

So, in order for the government to set their speeding fines to the correct levels, they will need to make sure that the fines account for the fact that the accident rate does not increase constantly with speed. Rather than being a straight upward sloping line (as it is in Travis County), the optimal fine schedule should increase more as drivers’ speed-over-the-limit increases. It will look just like how the accident rate changes according to speed (multiplied by the constant c over p):

OptimalVsActual copy

With these sorts of fines, your fine would increase more as drivers sped more. 5 mph over the speed limit might be $155, but 10 mph over the limit wouldn’t be $205 – it would be much higher (maybe like $305). And then 20 mph would be waaaaaay higher, maybe like $700. This would accurately reflect the fact that as you deviate more from the average speed, the rate at which you get into accidents increases very quickly. The government, acting optimally and wanting to prevent accidents, should therefore also make the rate at which fines increase also increase very quickly. In this way, the government basically makes people more precisely “pay” for the increased danger to society that they create via their increased expected accident rate.

Notice that, in the theoretical fines in the above graph, the optimal fines are sometimes increasing faster and sometime increasing slower than the old, constantly increasing fines. This, in additional to the fact that we do not know the slope of driver’s utility curves U(s), means that we cannot know in advance how the new optimal fines will change the solutions to drivers’ optimization problems. It could cause some people to speed faster while others speed slower.

If we take the government’s optimization a step further, we can actually devise what I will call a “negative speed limit” that accounts for the increased risk of auto accidents at slower-than-average speeds. If the government is really all about optimizing, they should also penalize people who make roads more dangerous by driving extremely slowly – give out speeding tickets for slow speeds (slowing tickets?).

NegativeSLNoSafe copy

While it probably wouldn’t ever catch on politically, if the government justifies upper speed limits by claiming that it makes roads safer, then it’s no different to set a lower speed limit for the same reason. Since driving at an exact speed is probably too strict an enforcement rule, the government could set a window of safe driving speeds for each road, and then give out speeding tickets to people who drive at speeds outside the safe zone. For example, on the highway the rule could be something like: “Drive between 55 and 75 mph”, and then going too fast or too slow could result in a ticket. There could also be exceptions for slow/heavy traffic situations. In other situations, the government might just have an upper speed limit – “Drive between 5 and 25 mph” is effectively just an upper speed limit.

NegativeSL copy

I don’t really think that people or police would really be down for this though. I’m also suspicious that maybe what’s really happening is: people who drive quickly are more likely to get into accidents with people who are driving slowly. This would mean that really the danger is fast drivers, and the victims are disproportionately slow drivers, so it looks like driving slowly is dangerous (it is, but because people would be more likely to hit you, not because you “cause” more accidents).

Assuming that the government doesn’t want to optimize with a negative speed limit, we can still use our theoretical model to test whether the current speeding fines in Travis County are optimal or not, which is what I’ll do next.


When people decide whether to speed or not, it ultimately depends on their own preferences (whether they are Punctual Perrys or Lackadaisical Lucys). We now know that changing the base fine will turn some speeders into non-speeders by raising the expected costs of speeding at any speed. But, we can’t figure out the optimal base fine without knowing specifically how all people react to changes in the base fine, and how these changes affect ticket revenue and the costs of accidents.

Instead of speculating, I will simply say that, at this moment, I cannot assess whether Travis County’s base fine of $105 is set at optimal levels or not. But, based on the models that have been developed in these posts, if we assume that $105 is optimal (or even just that it is fixed) we can devise what government optimal speeding fines would look like.

As I showed in my last post, the probability of getting caught speeding is so low that people who are acting optimally and who decide to speed should almost completely ignore speed limits. The expected cost of speeding is so low that as long as they gain any value from speeding (well, more than $0.002 worth) they should increase their speed.

This, to me, suggests that the rate at which speeding fines are increasing – the additional $10 per 1 mph over that you pay if you get caught – is far too low if the actual intention is to stop people from driving dangerously (i.e. reduce the actual speed at which most people speed). But rather than wondering if that is true, we now know the conditions to check whether the government is acting optimally. Taking our government optimizing conditions, let’s just plug in the actual observed values into the expression:


We know dF over ds is equal to 10 ($10 extra fine)[iv]; c is average accident costs which, according to this website[v], are at least $3,144; p is the probability of getting caught in a year, 0.206[vi]; and dr over ds is how the accident rate changes as speed changes. If you plug in everything except for dr over ds, you get:


Since we know that dr over ds is not constant (it changes as speed changes), we already know that these are not perfectly optimal fines. But is this result at all close? Maybe they aren’t perfect, but instead the government just approximated in order to make the fines easier to understand. In that case, given the current fines of Travis County, the rate at which accidents increase would need to be close to about 0.0007 per mph faster a driver speeds.

I don’t have the actual data used in the Solomon study, so instead I’ll just use this cool ability I have where I point my face at the graphs and use these optic sensors in my head to send a signal to my brain which then comes up with numbers that I can use to calculate close approximations of actual data. The results in table form are shown below:


On average, the number of accidents just about doubles (a little bit more than doubles, actually) for every 5 mph more a person is driving away from the average speed. More specifically, the number of accidents increased 108.54% on average for every 5 mph faster and 113.39% on average for every 5 mph slower (I excluded extremely slow deviations).

However, these are changes in the number of accidents over some given distance (100-million vehicle miles), not the change in number of accidents for some given number of drivers. Before we can compare these numbers, we have to get the unit of change to be accident rates per driver, per year (because our probability, p, is chance of a driver getting caught per year).

Luckily, we have the information to do this. We can turn all of the accident rates above into accidents per driver by figuring out how many drivers it would take to drive those 100-million vehicle miles. From the 2009 National Household Travel Survey[vii], Table 3 shows that drivers drive 28.97 miles per day on average. Then, over the course of a year, a single average driver drives 10574.05 miles total. Divide 100-million by this number, and we get that it takes about 9458 drivers to drive 100-million miles in one year. If we divide the average number of accidents at each speed by the number of average drivers it would take to drive that distance, we can approximate the accident rate (for a given number of drivers) at each speed deviation. Below are the results:


For the rest of the analysis, I leave out the places where the accident rate is greater than one, since the approximation obviously doesn’t work well there. If you look at the “Change” column, you can see that once you get past 10 mph, the change in the accident rate is always greater than 5*0.0007 = 0.0035 (which is what the change should be if Travis County were setting the fines optimally i.e. if dr over ds actually equaled 0.0007). If we approximate the changes as a linearly increasing function of speed (OLS), we get that the accident rate increases by about 0.0113 for each mph over the average. Notice that this is much higher than 0.0007 (16 times higher, actually). The plot below should help you visualize how close these approximations are to the optimal.


The Travis County optimally assumed accident rates (given their fines) are close to zero, which, as you can see, means that the conditions for them to be acting optimally are far from both the actual accident rate and the crude linear optimal approximation of the accident rate (except for maybe at low-speed deviations). With this evidence, I think it is safe to say that the Travis County speeding fines are not optimal. For many speeding violations, the fines will be too low to account for the increased risk of accidents associated with speeding.

So what should the fines be? Like I said before, I don’t know the optimal base fine, but if we want to optimally account for the increased risk of those who do speed we can describe how the fines should change as your speed increases. We just use the optimizing conditions from the theoretical model:


We have estimates for c, p, and we can use the “Change” column in the previous table as our dr over ds in the equation. In the table below, I have calculated the optimal changes in speeding fines and the resulting fine schedule, assuming that the base fine of $105 remains the same:


Weirdly enough, the actual speeding ticket cost at 15 mph is about where the optimal fines and the actual fines intersect (highlighted above). Actual speeding ticket costs for people going more than 15 mph over the limit, however, are much lower than the optimal fines. This is a result of Travis County’s (implicit) assumption that driving 5 mph faster always increases the risk of accident by the same amount. But, as we’ve seen from the data, the increase in accident rate depends on how fast you are already speeding, and it increases very quickly. For example, changing your speed from 20 mph over to 25 mph over almost triples your risk of getting into an accident, so the optimal fine for speeding at 25 mph over is almost triple the fine at 20 mph. Focusing on only the positive speed deviations, we can compare the optimal fines to the actual Travis County fines:


While the Travis County fines are (relatively) close approximations at low speed deviations, they are not at all close to the optimal fines at high-speed deviations (anything over 15 mph). Why does this matter? Because it means that Travis County is not accounting for the danger of speeders to society at exactly the speeds where speeders are most likely to actually cause accidents. It also means that it is overcharging speeders at low-speed deviations, where speeding is least likely to cause damage.

While it may technically be optimal to set the fine for going 30 mph over the limit at more than $6,000, it may not be possible (politically). But who really needs to speed by 30 mph? Shouldn’t we want to deter that person from doing that? A $6,000 fine would certainly accomplish that.

One last thing to consider: it is not clear that Travis County is actually trying to act optimally in the way we described. It might be that Travis County is trying to maximize revenue, rather than maximize revenue minus societal costs of accidents. This would have implications for what the government thinks is “optimal”, and it might mean that Travis County would want to keep high-speed fines lower so that more people speed and get caught, leading to more ticket revenue. There is also literature to suggest that local governments actually use speeding tickets as a way to make up for lost tax money during recessions[viii]. I mentioned these alternative objectives just to point out that other models might better describe how Travis County set its speeding fines. Or it might just be that the fines were made up off the top of someone’s head (a likely scenario, I think).

Besides helping the government, I also hope this helps drivers who are considering whether speeding is worth it. I don’t believe people include the cost of getting into an accident when they choose how speedy they should speed, and this is probably not a big deal for them usually – the average driver gets into an accident every 18 years[ix], so the probability is really low per trip (about 0.00005). But, the risk of having an accident increases extremely rapidly as you speed more and more. At low speeds you’re probably ok not including expected accident costs, but at the upper end you might want to consider the increased risk of crashing into someone.






[v] (note: this is the cost of automobile damage from an accident, and doesn’t include the costs of personal injuries or death. I didn’t include these costs because of the many complicated factors that go into the process of estimating the true “value” of a life and of injuries. These cost estimates will be “low” then, in the sense that they will tell us the lower bound estimates of the optimal fines.)






PDF Version


Thought: At What Speed Should You Speed?


By Kevin DeLuca


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).


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:


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:


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?


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:


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:


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:


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


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


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


And substituting in gives:


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:


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


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 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.










 PDF Version

Thought: Countries With Names That Sound Really Free Aren’t Actually Free


By Boyd Garriott


Opportunity Cost of Reading this ThoughtBurner post: $0.73 – about 3.3 minutes

What are some buzzwords to indicate that a country is free? A democracy? A government by the people? How about a republic?  If these are indeed the case, then it would seem that the freest country on the planet would have to be none other than the Democratic People’s Republic of Korea. For those that are less geographically inclined, that’s North Korea. For those that are even less geographically inclined, that’s the bad Korea.

So what gives; why does the most unfree country have such a free-sounding name? North Korea isn’t alone in being a country notoriously bad on civil rights with a very “free-sounding” official name. Think Democratic Republic of the Congo or People’s Republic of China. Could it be that countries are compensating for their definitive lack of freedom with a “free-sounding” name? A normal person would probably take this a face value and have a laugh at the irony, but we’re not normal people at ThoughtBurner, so I’ve done some statistical analysis to figure out whether countries with free-sounding names are actually more oppressive.

First off, to figure out how free or oppressive a country is, I used a dataset[i] from Freedom House of around 200 countries that are rated based on their political rights and civil liberties. I downloaded the 2015 dataset, aggregated those two numbers and then converted the result to a scale between 0 and 100 with 0 being not free at all (think North Korea) and 100 being totally free (think USA). This allows us to calculate percentage changes in freedom.

Second, to figure out how “free-sounding” a country name is, I used the formal names of countries (as opposed to the short names; think Democratic People’s Republic of Korea vs. North Korea) which are listed on Wikipedia[ii]. I then gave countries a point for every term they used that seemed to endorse freedom: any variations of “Republic”, “Democracy”, or “People’s”. A score of 0 (Canada) indicates that a country’s name makes no endorsement of freedom while a score of 3 (back to our friends in North Korea) indicates that a country’s name sounds like the preamble to the Constitution.

I then regressed these two numbers, and I found some interesting results. For every “freedom-endorsing” term in a country’s title, its citizens can expect to be 14% less free at a statistically significant level. To give you an idea of what that means, check out this chart below:


That’s right; as a country gets freer in name, it gets less free in reality. The average freedom for a country without any free-sounding descriptors is 69%, better than the world average of 61%. However, the average freedom for a country with three free-sounding descriptors – including the Democratic People’s Republic of Korea – is an appalling 11%. As a matter of fact, every country with that many free-sounding descriptors is classified as “not free” by Freedom House.

Even countries with just “Republic” in the name are, on average, 10% less free than countries without free-sounding descriptors. By the time that jumps to something like “Democratic Republic”, we’re talking 17% less freedom!

If that wasn’t ironic enough, consider this: countries with any variation of “king” or “kingdom” in their name are actually, on average, more free than countries with any freedom-sounding descriptor. That’s right: countries that explicitly endorse monarchy in their names are freer than those that explicitly endorse freedom. Granted, many countries with “king” in their name are modern European democracies like the United Kingdom and the Kingdom of Demark, so that explains some of the irony.

There are three important lessons to take from this.

  1. North Korea is the bad Korea (again, this one is more of a reminder for the less geographically inclined).
  2. Countries that sound really free usually aren’t that free.
  3. Yep, countries compensate for being terrible places by having nice-sounding names.

Wonky Stuff:

Regression of Freedom Score on Number of Freedom Descriptors


Important Statistics


Further Explanation of the Methodology

I came to the conclusion to use the freedom descriptors that I did after quite a bit of thought. First off, the words “Republic”, “Democracy”, and “People’s” are pretty bold adjectives that describe a form of government that represents the needs of free citizens. It should also be noted that I used the search string “Democr” to get any name that described itself as “Democratic” as well. Similarly, I searched the string “King” which also included “Kingdom”.

Next, I actually put quite a bit of thought into choosing my freedom descriptors, so here’s some explanation on other contenders that I didn’t count. “United” seems to be an obvious contender, but it’s not a word that actually describes a free society; citizens under the rule of a dictator are “united”, but they certainly aren’t free. “State” also came up pretty often in the dataset, but a state is just a sovereign territory that doesn’t make any claim as to the type of government it employs. On similar grounds, I rejected using “Principality” or “Commonwealth”. “Federal” came up, but that describes multiple states under a central entity – nothing about freedom.

I used country’s official English names because… well… I don’t speak like a hundred languages.

Lastly, to be clear about the graph, the labels on the bottom are illustrative but still accurate. The true labels would be “0 Freedom Descriptors”, “1 Freedom Descriptors”, etc. However, I took the liberty of putting common country titles that illustrated the amount of “Freedom Descriptors” they corresponded to. For example, “Democratic Republic” is usually what a country with freedom descriptors looks like, but there are exceptions such as the People’s Republic of China. There are two freedom descriptors, but it doesn’t fit neatly into the graph. In the end, however, I think it presents the information fairly.

Boyd Garriott is ThoughtBurner’s Chief Contributor. Boyd received his undergraduate degree in economics from the University of Texas at San Antonio. He currently lives in Washington D.C. and will be attending Harvard Law School in the fall.






PDF Version