**Nov 28 JDN 2459547**

**Risk compensation. **It’s one of those simple but counter-intuitive ideas that economists love, and it has been a major consideration in regulatory policy since the 1970s.

The idea is this: The risk we face in our actions is partly under our control. It requires effort to reduce risk, and effort is costly. So when an external source, such as a government regulation, reduces our risk, we will *compensate *by reducing the effort we expend, and thus our risk will decrease less, or maybe not at all. Indeed, perhaps we’ll even overcompensate and make our risk worse!

It’s often used as an argument against various kinds of safety efforts: Airbags will make people drive worse! Masks will make people go out and get infected!

The basic theory here is sound: Effort to reduce risk is costly, and people try to reduce costly things.

Indeed, it’s theoretically possible that risk compensation could yield the exact same risk, or even *more *risk than before—or at least, I wasn’t able to prove that for any possible risk profile and cost function it *couldn’t *happen.

But I wasn’t able to find any actual risk profiles or cost functions that would yield this result, even for a quite general form. Here, let me show you.

Let’s say there’s some possible harm H. There is also some probability that it will occur, which you can mitigate with some choice *x. *For simplicity let’s say that it’s one-to-one, so that your risk of H occurring is precisely 1-x. Since probabilities must be between 0 and 1, thus so must *x.*

Reducing that risk costs effort. I won’t say much about that cost, except to call it c(x) and assume the following:

(1) It is *increasing*: More effort reduces risk more and costs more than less effort.

(2) It is *convex: *Reducing risk from a high level to a low level (e.g. 0.9 to 0.8) costs less than reducing it from a low level to an even lower level (e.g. 0.2 to 0.1).

These both seem like eminently plausible—indeed, nigh-unassailable—assumptions. And they result in the following total expected cost (the opposite of your **expected utility**):

(1-x)H + c(x)

Now let’s suppose there’s some policy which will reduce your risk by a factor *r,* which must be between 0 and 1. Your cost then becomes:

r(1-x)H + c(x)

Minimizing this yields the following result:

rH = c'(x)

where c'(x) is the derivative of c(x). Since c(x) is increasing and convex, c'(x) is positive and increasing.

Thus, if I make *r *smaller—an external source of less risk—then I will reduce the optimal choice of *x. *This is risk compensation.

But have I reduced or increased the amount of *risk*?

The total risk is r(1-x); since *r *decreased and so did *x, *it’s not clear whether this went up or down. Indeed, it’s theoretically possible to have cost functions that would make it go up—but I’ve never seen one.

For instance, suppose we assume that *c(x) = ax*^{b}, where *a *and *b *are constants. This seems like a pretty general form, doesn’t it? To maintain the assumption that c(x) is increasing and convex, I need *a >* 0 and* b >* 1. (If 0 < b < 1, you get a function that’s increasing but concave. If b=1, you get a linear function and some weird corner solutions where you either expend no effort at all or all possible effort.)

Then I’m trying to minimize:

r(1-x)H + ax^{b}

This results in a closed-form solution for *x*:

x = (rH/ab)^(1/(b-1))

Since b>1, 1/(b-1) > 0.

Thus, the optimal choice of *x *is increasing in *rH *and decreasing in *ab.* That is, reducing the harm H or the overall risk *r *will make me put in less effort, while reducing the cost of effort (via either *a *or *b*) will make me put in more effort. These all make sense.

Can I ever increase the overall risk by reducing *r? *Let’s see.

My total risk r(1-x) is therefore:

r(1-x) = r[1-(rH/ab)^(1/(b-1))]

Can making *r *smaller ever make this larger?

Well, let’s compare it against the case when *r=1*. We want to see if there’s a case where it’s actually larger.

r[1-(rH/ab)^(1/(b-1))] > [1-(H/ab)^(1/(b-1))]

r – r^(1/(b-1)) (H/ab)^(1/(b-1)) > 1 – (H/ab)^(1/(b-1))

For this to be true, we would need *r > *1, which would mean we didn’t reduce risk at all. Thus, reducing risk externally reduces total risk even after compensation.

Now, to be fair, this isn’t a *fully *general model. I had to assume some specific functional forms. But I didn’t assume *much, *did I?

Indeed, there *is *a fully general argument that externally reduced risk will never *harm *you. It’s quite simple.

There are three states to consider: In state A, you have your original level of risk and your original level of effort to reduce it. In state B, you have an externally reduced level of risk and your original level of effort. In state C, you have an externally reduced level of risk, and you compensate by reducing your effort.

Which states make you better off?

Well, clearly state B is better than state A: You get reduced risk at no cost to you.

Furthermore, state C must be better than state B: You voluntarily chose to risk-compensate precisely because it made you better off.

Therefore, as long as your preferences are rational, state C is better than state A.

Externally reduced risk will never make you worse off.

QED. That’s it. That’s the whole proof.

But I’m a behavioral economist, am I not? What if people *aren’t *being rational? Perhaps there’s some behavioral bias that causes people to overcompensate for reduced risks. That’s ultimately an empirical question.

So, what does the empirical data say? Risk compensation is almost never a serious problem in the real world. Measures designed to increase safety, lo and behold, actually increase safety. Removing safety regulations, astonishingly enough, makes people less safe and worse off.

If we ever do find a case where risk compensation is very large, then I guess we can remove *that *safety measure, or find some way to get people to stop overcompensating. But in the real world this has basically never happened.

It’s still a fair question whether any given safety measure is worth the cost: Implementing regulations can be expensive, after all. And while many people would like to think that “no amount of money is worth a human life”, nobody does—or should, or even *can—*act like that in the real world. You wouldn’t drive to work or get out of bed in the morning if you honestly believed that.

If it would cost $4 billion to save one expected life, *it’s definitely not worth it. *Indeed, you should still be able to see that even if you don’t think lives can be compared with other things—because $4 billion could save *an awful lot of lives *if you spent it more efficiently. (Probablyover a* million*, in fact, as current estimates of the marginal cost to save one life are about $2,300.) Inefficient safety interventions don’t just cost money—they prevent us from doing other, more efficient safety interventions.

And as for airbags and wearing masks to prevent COVID? Yes, definitely 100% worth it, as both interventions have already saved tens if not hundreds of thousands of lives.

It would be interesting (and might broaden your audience) if this were presented in Youtube with animated graphics to illustrate your algebraic reasoning.

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To answer the question about increasing the absolute risk, mathematically this is just a question of letting the cost function grow quickly enough.

If we consider r as a function of x in the equilibrium, i.e. r(x)=c'(x)/H then the risk is (1-x)c'(x)/H.

We want this expression to increase as x decreases, which means it should have a negative derivative somewhere.

This condition gives ((1-x)c”(x)-c'(x))/H < 0, or simplified (1-x)c''(x)b then there can be an area where a decrease in r increases the risk. For example take a=1, b=2 and H=5/4. Then the minimum total expected cost is at x=5/8, with a risk of 3/8=0,375. With r=9/10 however the minimum total expected cost is at x=9/16, with a risk of (9/10)*(7/16)=63/160=0.39375.

Generally of course there are many cost functions which grow faster than log(1/(1-x)) and therefore always increase the risk for a decreasing a, on particularly simple one would be (1-sqrt(1-x)).

I hope this makes sense and that I didn’t misunderstand the model and assumptions.

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To answer the question about increasing the absolute risk, mathematically this is just a question of letting the cost function grow quickly enough.

If we consider r as a function of x in the equilibrium, i.e. r(x)=c'(x)/H then the risk is (1-x)c'(x)/H.

We want this expression to increase as x decreases, which means it should have a negative derivative somewhere.

This condition gives ((1-x)c”(x)-c'(x))/H < 0, or simplified (1-x)c''(x)<c'(x).

So to find the cost function of constant risk we can solve this as a differential equation with equality, which gives c(x)=a*log(1/(1-x)) (for an arbitrary positive a). Note that this function is increasing, convex, with c(0)=0 and c(1)=infinity. The risk under this cost function should always be a/H.

By the same reasoning I don't think your argument for the polynomial cost function is correct. It seems to me that if b<1+b*x then there can be an area where a decrease in r increases the risk. For example take a=1, b=2 and H=5/4. Then the minimum total expected cost is at x=5/8, with a risk of 3/8=0,375. With r=9/10 however the minimum total expected cost is at x=9/16, with a risk of (9/10)*(7/16)=63/160=0.39375.

Generally of course there are many cost functions which grow faster than log(1/(1-x)) and therefore always increase the risk for a decreasing a, on particularly simple one would be (1-sqrt(1-x)).

I hope this makes sense and that I didn't misunderstand the model and assumptions.

(Disregard my other comment, having a less than and greater than cut out half of the text…)

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