Appendix - How much should government spend to save a single life?

This is a moral question, not an empirical question, so it's impossible to prove -- or even, really, to produce "evidence" -- that one has found the right answer. I think the best you can do is to:

• come up with a set of rules for answering the question
• verify the rules do not lead to any paradoxes or positions that you feel are so morally repugnant that they would be worse than not trying to answer the question at all
• apply the rules to see what answer you get

Landsburg kicked off a round of controversy with his column arguing that if we're going to spend money to help the poor at all, it makes more sense to spend it on groceries for large numbers of people than for expensive rescue efforts for small numbers of people. (And in his follow-up article published as a companion piece to the book, he argues that in any case there should be a cap on what we are willing to spend to save any particular person.)

Robert Frank argues that Landsburg's reasoning is incorrect, but I think his counter-argument contains a number of flaws itself. It makes more sense to address them out of order, starting with the claim at the end of the editorial:

Mr. Landsburg's argument finesses the important distinction between a "statistical life" and an "identified life." The concepts were introduced by the economist Thomas C. Schelling, who observed the apparent paradox that communities often spend millions of dollars to save the life of a known victim - someone trapped in a mine, for example yet are often unwilling to spend even $200,000 on a highway guardrail that would save an average of one life each year.

This disparity is not economically irrational, Mr. Schelling insisted, because the community values what it is buying so differently in the two cases. It is one thing to risk one's own life in an unlikely automobile accident, but quite another to abandon a known victim in distress.

But, as Frank quotes him, Schelling was making an empirical statement -- people are more willing to spend money to save the life of a identified accident victim, than the life of a hypothetical future traffic accident victim -- not a moral statement. This is not an argument why we should spend money to save the accident victim rather than spending it on guardrails. If what you care about is the number of lives saved, then you should spend the money on guardrails. Otherwise, you're caring more about pandering to people's emotions than about saving the maximum number of lives. Which brings me to an argument that Frank makes earlier in his column:

Had the opportunity presented itself, many would have eagerly contributed to Ms. Habtegiris's care. But organizing an endless series of individual private fund-raisers for such cases is impractical. So, we empower government to step in when the need arises.

Translated into the language of economic efficiency, this seems to say: If we had held a private fundraiser to save the life of a victim in an emergency, usually the funds would come through. Each contributor was making their donation voluntarily, so we have to assume the benefits (to them) of knowing someone's life was saved, exceed the cost to them of the donation, and therefore the transaction is economically efficient. We can assume this would usually be the case, so we just skip the fundraiser stage and let the government raise the money all at once (through taxes) and spend it on rescue efforts.

The problem with this is that it means the correctness of the rescue effort would be proportional to the public's willingness to support it through a (hypothetical) fundraiser. In other words, it's OK to spend more to save the lives of those that more people care about. To the extent that "missing white woman syndrome" is a reality, then the government should in fact spend more to locate missing white women! But the worst disparity inherent in this logic is not between saving white women and black men, it's between saving statistical lives and identified lives as described above. If we let hypothetical fundraisers be our guide, then government would spend millions to rescue accident victims, but no money to build the guardrails that would have prevented those accidents in the first place (because fundraisers to build guardrails don't tug at the heartstrings nearly as much).

And why stop at saving lives? Suppose in the 1940's, a black man and a white woman wanted to get married. A lawmaker assumes -- probably correctly -- that so many people would be offended by the thought, that if it were possible to hold a fundraiser such that the marriage would be blocked if you could reach the $1 million mark (in today's money), at least a million people would donate $1 each. On the other hand, the couple may want to get married, but they might value the prospect of getting married at something less than $1 million. Therefore, if 1 million strangers collectively place a $1 million value on stopping the couple from getting married, and the couple places a value of something less than $1 million on getting married, economic efficiency compels the government to outlaw the marriage! (Landsburg addresses something like this paradox in a column about Terri Schiavo.)

In Landsburg's own attempt to answer the question, he -- correctly, I think -- puts the same value on "statistical" and "identified" lives, and then tries to calculate how much we should be willing to spend to save a life in either category. He argues mathematically for a way to calculate the cap -- for example, if the average person would pay $1 to avoid a 1 in a million chance of dying, then efficiency requires us to spend up to $1 million to avoid one additional statistical death.

In rejecting the logic that identified lives are worth more, he addresses the situation of a person trapped in a dangerous situation and facing a 50% chance of death, and would pay up to $4 million (if they had it) to avoid that danger. He then says "Using a willingness-to-pay criterion, the numbers in the above example are impossible" and gives the mathematical reasoning why. But hang on -- saying these numbers are "impossible" is an empirical question. Does that mean there are no people -- or virtually no people -- who would pay only about $1 to avoid a 1 in a million chance of death, but who wouldn't pay $4 million to avoid a 50% chance of death? I'll bet that of the multi-millionaires in the world, there are many who don't even spend a dollar's worth of their time to fasten a seat belt (which would reduce your chance of death by a lot more than 1 in a million), but many of them would pay $4 million rather than bet their life on a coin flip.

More fundamentally, I don't think you can calculate the value of a "life" by asking either what people would pay to avoid a given risk of death, or what people would demand as compensation to accept a given risk of death. You can observe that someone would take a 1 in a million chance of death for a gain of $1 or more, and divide the gains by the probability of death, and conclude that they value their life at about $1 million. But you could also observe that they would demand a payment of $infinity to accept a 100% chance of death, and therefore that they value their life at $infinity (and economic efficiency would demand that we spend all the world's wealth, if necessary, to save their life). Or perhaps they would demand nothing less than a billion dollars to accept a 50% risk of death. Do we then spend up to $2 billion to save their life?

Since you can plug in different probabilities and get different cash values for a person's life, that would seem to me to discredit the whole method. You might argue that we should go by the 1-in-a-million probability and the resulting $1 million value for a life, because "the other values are absurd". But if the algorithm is flawed in general, it could be just a coincidence that some inputs to the algorithm give reasonable-sounding answers. If you want to accept the algorithm's $1 million output because it sound's "reasonable", but reject the algorithm's $2 billion output because it sounds "unreasonable", then you must be using some other criteria to decide which answers are "reasonable", and if you have that criteria, why bother with the algorithm at all?

So here's my answer to the question of how much society should be willing to spend the life of a single person: As much money as it takes, until the expenditure cramps the economy and the infrastructure so much that it results in the statistical death of some other person somewhere else. And when choosing where to spend money, start with the cheapest lives to save first, and work your way up -- regardless of whether those are "statistical lives" or "identified lives". These criteria do not seem to be inherently self-contradictory, or to lead to any paradoxes or morally repugant conclusions. Unfortunately it gives no guidance as to how to calculate how much money the government would have to spend, before the economy contracts so much that it results in the loss of 1 additional life somewhere in the infrastructure. However, by starting with the cheapest life-saving measures first, it does mean that we should spend money on the guardrails before almost anything else.