4/30/15

By Kevin DeLuca

ThoughtBurner

*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.

*THE GOVERNMENT’S PROBLEM*

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:

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 *N*s, 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:

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*):

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?).

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.

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.

*EMPRICAL ESTIMATES: SPEEDING FINES IN TRAVIS COUNTY*

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.

_______________________________________________________________________

[i] http://onlinemanuals.txdot.gov/txdotmanuals/szn/szn.pdf

[ii]https://research.stlouisfed.org/wp/2006/2006-048.pdf

[iii]http://safety.fhwa.dot.gov/speedmgt/ref_mats/fhwasa1304/Resources3/40%20-%20Accidents%20on%20Main%20Rural%20Highways%20Related%20to%20Speed,%20Driver,%20and%20Vehicle.pdf

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

[v]http://www.rmiia.org/auto/traffic_safety/Cost_of_crashes.asp (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.)

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

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

[viii]https://research.stlouisfed.org/wp/2006/2006-048.pdf

[ix]http://www.foxbusiness.com/personal-finance/2011/06/17/heres-how-many-car-accidents-youll-have/

________________________________________________________________________

## One thought on “Thought: Welfare-Maximizing Speeding Fines”