How much should we value statistical lives?

June 9 JDN 2458644

The very concept of putting a dollar value on a human life offends most people. I understand why: It suggests that human lives are fungible, and also seems to imply that killing people is just fine as long as it produces sufficient profit.

In next week’s post I’ll try to assuage some of those fears: Saying that a life is worth say $5 million doesn’t actually mean that it’s justifiable to kill someone as long as it pays you $5 million.

But for now let me say that we really have no choice but to do this. There are a huge number of interventions we could make in the world that all have the same basic form: They could save lives, but they cost money. We need to be able to say when we are justified in spending more money to save more lives, and when we are not.

No, it simply won’t do to say that “money is no object”. Because money isn’t just money—money is human happiness. A willingness to spend unlimited amounts to save even a single life, if it could be coherently implemented at all, would result in, if not complete chaos or deadlock, a joyless, empty world where we all live to be 100 by being contained in protective foam and fed by machines. It may be uncomfortable to ask a question like “How many people should we be willing to let die to let ourselves have Disneyland?”; but if that answer were zero, we should not have Disneyland. The same is true for almost everything in our lives: From automobiles to chocolate, almost any product you buy, any service you consume, has resulted in some person’s death at some point.

And there is an even more urgent reason, in fact: There are many things we are currently not doing that could save many lives for very little money. Targeted foreign aid or donations to top charities could save lives for as little as $1000 each. Foreign aid is so cost-effective that even if the only thing foreign aid had ever accomplished was curing smallpox, it would be twice as cost-effective as the UK National Health Service (which is one of the best healthcare systems in the world). Tighter environmental regulations save an additional life for about $200,000 in compliance cost, which is less than we would have spent in health care costs; the Clean Air Act added about $12 trillion to the US economy over the last 30 years.

Reduced military spending could literally pay us money to save people’s lives—based on the cost of the Afghanistan War, we are currently paying as much as $1 million per person to kill people that we really have very little reason to kill.

Most of the lives we could save are statistical lives: We can’t point to a particular individual who will or will not die because of the decision, but we can do the math and say approximately how many people will or will not die. We know that approximately 11,000 people will die each year if we loosen regulations on mercury pollution; we can’t say who they are, but they’re out there. Human beings have a lot of trouble thinking this way; it’s just not how our brains evolved to work. But when we’re talking about policy on a national or global scale, it’s quite simply the only way to do things. Anything else is talking nonsense.

Standard estimates of the value of a statistical life range from about $4 million to $9 million. These estimates are based on how much people are willing to pay for reductions in risk. So for instance if people would pay $100 to reduce their chances of dying by 0.01%, we divide the former by the latter to say that a life is worth about $1 million.

It’s a weird question: You clearly can’t just multiply like that. How much would you be willing to accept for a 100% chance of death? Presumably there isn’t really such an amount, because you would be dead. So your willingness-to-accept is undefined. And there’s no particular reason for it to be linear below that: Since marginal utility of wealth is decreasing, the amount you would demand for a 50% chance of death is a lot more than 50 times as much as what you would demand for a 1% chance of death.
Say for instance that utility of wealth is logarithmic. Say your currently lifetime wealth is $1 million, and your current utility is about 70 QALY. Then if we measure wealth in thousands of dollars, we have W = 1000 and U = 10 ln W.

How much would you be willing to accept for a 1% chance of death? Your utility when dead is presumably zero, so we are asking for an amount m such that 0.99 U(W+m) = U(W). 0.99 (10 ln (W+m)) = 10 ln (W) means (W+m)^0.99 = W, so m = W^(1/0.99) – W. We started with W = 1000, so m = 72. You would be willing to accept $72,000 for a 1% chance of death. So we would estimate the value of a statistical life at $7.2 million.

How much for a 0.0001% chance of death? W^(1/0.999999)-W = 0.0069. So you would demand $6.90 for such a risk, and we’d estimate your value of a statistical life at $6.9 million. Pretty close, though not the same.

But how much would you be willing to accept for a 50% chance of death? W^(1/0.5) – W = 999,000. That is, $999 million. So if we multiplied that out, we’d say that your value of a statistical life has now risen to a staggering (and ridiculous) $2 billion.

Mathematically, the estimates are more consistent if we use small probabilities—but all this assumes that people actually know their own utility of wealth and calculate it correctly, which is a very unreasonable assumption.

The much bigger problem with this method is that human beings are terrible at dealing with small probabilities. When asked how much they’d be willing to pay to reduce their chances of dying by 0.01%, most people probably have absolutely no idea and may literally just say a random number.

We need to rethink our entire approach for judging such numbers. Honestly we shouldn’t be trying to put a dollar value on a human life; we should be asking about the dollar cost of saving a human life. We should be asking what else we could do with that money. Indeed, for the time being, I think the best thing to do is actually to compare lives to lives: How many lives could we save for this amount of money?

Thus, if we’re considering starting a war that will cost $1 trillion, we need to ask ourselves: How many innocent people would die if we don’t do that? How many will die if we do? And what else could we do with a trillion dollars? If the war is against Nazi Germany, okay, sure; we’re talking about killing millions to save tens of millions. But if it’s against ISIS, or Iran, those numbers don’t come out so great.

If we have a choice between two policies, each of which will cost $10 billion, and one of them will save 1,000 lives while the other will save 100,000, the obvious answer is to pick the second one. Yet this is exactly the world we live in, and we’re not doing that. We are throwing money at military spending and tax cuts (things that many not save any lives at all) and denying it from climate change adaptation, foreign aid, and poverty relief.

Instead of asking whether a given intervention is cost-effective based upon some notion of a dollar value of a human life, we should be asking what the current cost of saving a human life is, and we should devote all available resources into whatever means saves the most lives for the least money. Most likely that means some sort of foreign aid, public health intervention, or poverty relief in Third World countries. It clearly does not mean cutting taxes on billionaires or starting another war in the Middle East.

The injustice of positional goods

July 15 JDN 2458315

At Disneyland, you can now buy a special pass that will let you skip ahead in line. On several airlines including American, Delta, Spirit, and Southwest, you can pay extra to be allowed to board before other passengers (which has been particularly salient for me on the many flights I’ve been taking this summer). This is only an extreme form of a long-standing phenomenon: Since the beginning of commercial ship and train travel, there have been first-class and second-class tickets.

I don’t have any formal survey data on the matter, but just about everyone I have spoken to about such policies is at least vaguely uncomfortable with them, if not totally outraged. The exception is other economists, who typically don’t express any concern whatsoever. “People are willing to pay for this service because they value it,” they say; “so what’s the problem?”

On this one, I think the economists are wrong and everyone else is right. There is something different about this sort of service.

Part of the difference between first-class and second-class is in actual quality of services that actually incur additional costs (I hate to break it to you, but legroom on an aircraft is just such an example; every inch of legroom on each seat is another row of seats they can’t have, which is another $2000 or so they don’t get in revenue on each and every flight). But part of it is something else, something that costs the company literally nothing.

This makes early boarding a clearer example. What are you buying when you pay for early boarding? On most airlines, it’s not even a better seat; your seat is pre-assigned (Southwest is an exception). We could say you are paying for extra time, but that’s not really even true; the plane leaves at the same time for everyone. From your perspective, you are paying for convenience; you get to settle in on the plane, maybe get started working or whatever, before everyone else. Maybe you’d rather wait on the plane than wait in the airport (though frankly I’m not sure why; the airport has restaurants and comfortable restrooms).

What you are really buying is position. Early boarding is a positional good. Every person who gets bumped forward in the queue is someone else who is bumped backward. The net benefit for all customers as a whole is precisely zero, as is the cost for the company to provide it—and yet, it still has a positive price! This is impressive economic alchemy: The airline has managed to take something with zero marginal cost and zero marginal benefit, and still make money off of it. They have transmuted the lead of something costless and worthless into the gold of profit.

They achieve this by pitting customers against one another. In a post awhile back I talked about rent-seeking, such as lobbying and advertising. Usually it’s the corporations doing the rent-seeking, but early boarding and queue-jumping are examples of corporations intentionally generating a circumstances where they can obtain revenue from the rent-seeking of others.

To be fair, there might be some welfare gains to be had from auctioning off order in a queue. Some people have genuinely higher costs of time than others (a cardiac surgeon’s time is particularly important, for example), and an auction could potentially order people who have very high cost of time first.

But this argument is much weaker than it may at first appear, because people also have very different marginal utility of wealth, and indeed I think the correlation between your willingness to pay for time and your total wealth is considerably higher than the correlation between your willingness-to-pay for time and your actual real cost in terms of pain and suffering.

This is a more general problem, as I’ve discussed in previous posts; but I think it’s especially acute in the case of time, because real cost of time doesn’t actually vary all that much between most people. The reason poor people take buses and rich people take limousines isn’t because poor people don’t care about their time; it’s because they can’t afford limousines. A cardiac surgeon and an economist could very well have the same salary and the same willingness-to-pay for time, but people rarely die when an economist turns up an hour late. (It’s not that our work isn’t important—actually a good development economist can save far more lives than any cardiac surgeon—but it’s not nearly so urgent.) Also, consider the fact that teachers and social workers generally contribute a good deal more to society than derivatives traders (and thus, from a social welfare perspective, their time should be considered more valuable), but they are far less likely to pay for first-class seats. In fact, a first-come, first-served method actually seems better than an auction from a social welfare perspective: If your time is really important to you, you’re more likely to go out of your way to check in as soon as you can. That costly signal provides a sorting mechanism which relies directly upon real costs of time, rather than indirectly via monetary willingness-to-pay.

And of course when it comes to Disneyland, this argument utterly fails; I see little reason to think that a cardiac surgeon’s vacation time is substantially more valuable to society. (Don’t get me wrong; surgeons need and deserve vacation time—but if they get too much, their performance actually suffers!) So maybe paying for a place in queue isn’t completely rent-seeking, but it’s pretty close.

That is why paying for positional goods feels unjust to most people: Because it is. Charging a price for positional goods is a means of extracting profit from customers without providing any (net) real service. It’s a way of applying price discrimination without even having much monopoly power. If another airline doesn’t let you pay to skip ahead in the queue, you have a slightly lower expected wait time on that other airline, but any revenue they lose from charging a bit less for economy tickets can be easily made up by charging more for the front of the line.

For example, if the first 10% of the line on airline A is decided by selling spots, while airline B chooses at random, and the average time waiting in line to board is 30 minutes, the expected wait times are as follows. Fly airline A and don’t buy a spot: 16.5 minutes. Fly airline A and buy a spot: 1.5 minutes. Fly airline B: 15 minutes. Those 10% are paying for, on average, 13.5 minutes; but you’re only gaining 1.5 minutes. Of course, there are more people waiting that extra 1.5 minutes than saving those 13.5 minutes (9 times as many, in fact). If the per-minute willingness-to-pay were exactly the same, the airline would break even; but they know of course that the willingness-to-pay of that top 10% is considerably higher than that of most of the bottom 90%. If they have any market power at all (which they generally do, by being the only airline serving certain routes, offering loyalty benefits, etc.), they can squeeze out even more profit.

They may even sometimes go out of their way to make life miserable for those who don’t pay extra, increasing the incentive to pay extra. This requires some market power to pull off, but as I said, they often have that. Most airlines don’t offer power outlets at every seat, for example. This is not a serious question of installation cost or even power consumption. We’re talking about a few hundred dollars on an aircraft that costs hundreds of millions of dollars, or a few kilowatts from a system that can generate over one hundred megawatts (of course most of it is used for propulsion, but adding an alternator that would generate an extra few kilowatts of electrical power would still not be difficult or expensive). This is a way of making life worse for the economy-class passengers so they have a stronger incentive to pay for upgraded tickets.

It’s not always easy to tell what is a positional good: First-class seats are ambiguous, for example. But I think a good heuristic is to ask, “Could everyone benefit from this?” If the answer is “No, even in principle”, then you are definitely dealing with a positional good. Not everyone can be first in line at Disneyland. Not everyone can board the plane first. In theory at least, everyone could be provided the same legroom and meal service as a first-class ticket (it would be expensive, but not impossible), so that is at least in part not a positional good.

The “pay-to-win” effect of some video game downloadable content (DLC) is also a positional good, which we can see by the above heuristic: If everyone pays to have the best gun in the game, there’s no point in having the best gun in the game. This is why gamers are rightfully outraged by “pay-to-win” effects, but typically have no objection to paying for DLC that provides them with extra game content (such as new characters, locations, or missions) or cosmetic upgrades (hats, decorations, and “skins”). Personally I tend to think that most DLC is overpriced, and succeeds at being so due to a kind of monopoly power (Mass Effect DLC doesn’t work on Skyrim or vice-versa) but I certainly don’t object to the basic idea of charging additional money for additional content. The reason we object to “pay-to-win” is not that winning the game is so important; it’s that this business model is so obviously a form of rent extraction. (It’s interesting that gamers in China don’t seem to be as bothered by “pay-to-win” as gamers in the US; this runs counter to the standard narrative that American people are competitive capitalists and Chinese people are collectivist socialists, don’t you think?)

There may be some circumstances in which we have no choice but to allow corporations to charge prices for positional goods—especially if we can’t tell whether we are dealing with a positional good or not. But it would not be very difficult to draft legislation that would at least reduce such business practices: We could simply use my “Could everyone benefit?” heuristic. If a business charges money for something that even in principle they could not possibly provide all of their customers, they are charging a price for a positional good, and should be penalized. The benefits of such a policy would be relatively small, but the costs would be even smaller. If we are really concerned about letting cardiac surgeons board aircraft faster (we should really be concerned about deboarding faster—and especially faster security screening!), we could make such a rule that applies to particular classes of high-urgency professions; we don’t need to allow airlines to extract millions of dollars in rent by pitting their customers against each other.