LESSWRONG

What can total utilitarians learn from empirical estimates of the value of a statistical life?

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ericyu3
Feb 15, 2014
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This post was inspired by Carl Shulman's blog post from last month—if you have time, read that first, since this is basically a response to it. My goal here is to combine

  1. Empirical studies of how much people are willing to pay to reduce their risk of death, and
  2. The "total utilitarian" assumption that potential people are as important as existing people, and the value of an additional person is independent of the number of preexisting people
  3. An additional (quite strong!) assumption that the utility gain from being born and becoming an adult is the same as the utility loss from a premature adult death
and see whether it's more effective for a total utilitarian to improve the incomes of existing people or to increase/decrease the total number of people.

Suppose everyone has identical preferences, and only two variables affect expected utility: their probability of survival  and their income . Since von Neumann–Morgenstern utility functions are invariant under affine transformations, we can define the utility of being dead as 0 and still have one degree of freedom left (two utility functions are equivalent iff they are related by a positive linear transformation). Fixing a reference (minimum) income level , we can always the write the utility function as

,

where  is some function defined on  with . This condition ensures that  is the utility at the minimum income. For instance, if utility from income is logarithmic, we can let . A logarithm with any other base can be turned into  by a linear transformation, so the choice of base doesn't matter.

We can infer  from empirical estimates of the value of a statistical life if we have a hypothesis for the form of —so total utilitarians should pay a lot of attention to these estimates! If you're willing to pay  for a small relative increase in your probability of survival,  (as opposed to an absolute increase ), then your value of life is defined as

.

If your utility from income takes the same form as  and you're rational, then it's also true that

.

In other words, the value of life is the marginal rate of substitution between income and log survival probability. So

and

.

In the case of , we have

.

$6 million is a reasonable estimate (although on the low side) for the value of a statistical life.  is in units of income, so the $6M estimate needs to be translated into an income stream. At an interest rate of 3% over 40 years, this will require payments of ~$257,582 per year. If the $6M estimate was for people making $50,000 a year, then . With  at $300 per year, this gives us . It's just a coincidence that  is so close to 0: slightly different parameters will shift  substantially away from that point. I biased all my parameter estimates (except for the interest rate, which I understand very poorly) so that  would have a downward bias, so if my estimates are wrong  is probably higher.

I'm not going to draw any conclusions about what a total utilitarian should do, since there are many problems with this method of estimation:

  • The value-of-statistical-life studies are from high-income countries, so it's questionable to extrapolate to very low incomes.
  • Utility from income probably isn't logarithmic, since people exhibit relative risk aversion.
  • The value of  depends strongly on the interest rate.
  • I assumed that somebody with $300 per year has the same life expectancy as someone with $50K per year. This isn't as big of a problem as it seems. If they live half as long, you can compare two $300/year people versus one $50K/year person and get a similar result.
The estimate of  is extremely sensitive to the inputs, partly because it's calculated as the difference of two much larger numbers (caused by the value-of-life estimates being from a high income level), partly because I don't know exactly what level of income value-of-life is calculated at, and partly because the relationship between annual income and a lump sum of money depends on the interest rate (I don't know what rates were used for the value-of-life estimates).

Any suggestions on how to make a more robust estimate?

(A big thanks to http://lwlatex.appspot.com/ for helping me format the equations!)
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ericyu3

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gjm
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4 years ago

At least one of us is very confused about pretty much everything here.

Since von Neumann–Morgenstern utility functions are invariant under affine transformations, we can [...]

Not if you're serious about total utilitarianism, which needs to be able to add up utilities and therefore looks quite different (at least when the number of lives can vary) as the constant term varies.

[EDITED to add: The issues below are because the mathematical typesetting got messed up in a way that made + signs disappear; they are not real mistakes and the error has now been fixed in the original post.]

This condition ensures that s is the utility at the minimum income.

I must be misunderstanding. You've written U = psf(y) where the condition in question is f(y0)=0. But this implies U=p.s.0=0 when y=y0. So s is not the utility at minimum income. In fact it looks to me as if s is simply a scaling factor applied to utilities, and as such is perfectly arbitrary.

Then, a little later, you go from VL = s f(y)/f'(y) to s = VL.f'(y) - f(y) but that's completely wrong; it should be s = VL.f'(y)/f(y), which in the log case says s = VL / y log(y/y0); in the case y=y0 this just says that VL=0 whatever s may be, which is not surprising since you deliberately chose to rescale your utilities to make it so ("we can define the utility of being dead as 0").

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ericyu3
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4 years ago

Not if you're serious about total utilitarianism, which needs to be able to add up utilities and therefore looks quite different (at least when the number of lives can vary) as the constant term varies.

Sorry, I was unclear. I meant that the constant term cannot be determined from empirical studies alone, since it doesn't affect decision-making. Estimates of the "value of life" compare the utility change from a small change in income to the utility change from a small change in survival probability, and the point of my post was to extrapolate these to large changes (creating a new person at a very low income level).

The conclusions are unchanged when the utility of death is nonzero, as long as you only look at "changes" in total utility (and not total utility itself, which will be infinite). For example, if the utility of death is fixed at 1 and your utility is fixed at 2, then creating a copy of you would make total utility "2+2+[lots of others]+1+1+1+..." instead of "2+1+[lots of others]+1+1+1+..." and total utility would increase from infinity to infinity+1. Obviously this is ill-defined mathematically (which is why I set death to 0), but you can see that it still makes sense to talk about utility changes.

[math mistakes]

When Vladimir_Nesov changed the images, the plus signs weren't URL-encoded, so they all disappeared. It's supposed to be U = p*(s + f(y)) and VL = (s + f(y))/f'(y).

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gjm
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4 years ago

The two links you give to discussions of "the statistical value of a life" are discussing very different things. Thing One: An extrapolation (from infinitesimal changes to the change from p=0 to p=1) of the dollar-value to a given individual of their survival. Thing Two: An estimate of the dollar-value placed by society on a person's survival.

Thing One (which is what your VL is measuring) is inevitably going to be very sensitive to the person's wealth. Thing Two needn't be, and in fact isn't (most modern societies are willing to go to about as much trouble to save a poor person's life as a rich person's). I think the $6M figure you cite is more a Thing Two than a Thing One.

If we take your calculations at face value, here is what they tell us. We start with a broadly-plausible estimate that in some sense a life is worth about $6M. We suppose that a "typical" life corresponds to an income of about $50k/year. We do some calculations. And we arrive at the conclusion that the life of a very poor person -- someone whose income is your y0 of $300/year -- is worth something on the order of $250. (!!!)

First reaction: This is a reductio ad absurdum: something must be desperately wrong here. Second reaction: Well, maybe not so much; this is not really about assigning different values to rich and poor people's lives, but about how they, in their very different financial situations, convert between utility and money. Third reaction: No, wait, this really is about assigning different values to these people's lives; in particular there is an income level (not very far from y0, in this particular model) at which the utility reaches zero, and no talk of conversion factors will change that.

So I think you either need to bite the bullet and say that very poor people's lives aren't worth saving, or reconsider some assumptions. (Fiddling with the details of the utility function, etc., as in your closing comments, might move the value assigned to a life-at-income-y0 from, say, $250 to, say, $5k, which -- taken as an indication of how desperately important money is to someone so poor, rather than of the absolute value of their life -- is at least semi-reasonable. But it won't do anything to change the fact that someone sufficiently poor will get zero or negative utility.)

The assumption I would suggest revisiting is the one that says, roughly, that death is like merely not-having-lived in terms of utility.

It seems to me entirely possible, and in fact probably right, that (1) quite a lot of people's lives are bad enough that if we were choosing, godlike, between two possible worlds that differ simply in the addition or subtraction of some of those lives, we could reasonably prefer there to be fewer rather than more of them, but also that (2) once one of those lives is there, ending it is a very bad thing. A life just barely bad enough that the person living it considers death an improvement is probably quite a lot worse than a life just barely bad enough that adding another to the world is neutral.

(Of course quality of life isn't the same thing as income, but that's just a matter of the toy model being used here.)

So this would leave us with the following state of affairs: The life of a rather miserably-off person (for which very low income is a kinda-passable proxy) is bad enough that having more such lives in the world doesn't, as such, improve the world. (So they would have U=0 or even U<0.) But, once that life is there, taking it away or failing to save it is still a very bad thing (because of that person's preferences, and the impact on other people). That seems fair enough. But at this point it's worth noting that those value-of-life estimates are all concerned with the value of saving the life, rather than that of having it exist in the first place. Which probably means that there's still something wrong with the calculation.

It's nearly 2am local time so I'll leave my thoughts in that rather half-baked form.

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ericyu3
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4 years ago

Thanks for posting such a detailed response!

It didn’t occur to me to distinguish between Thing One and Thing Two, and you’re right that they’re qualitatively quite different, but it shouldn’t make too much of a difference quantitatively. This is because the Thing Two number is basically derived from Thing One estimates, except that everyone is assumed to have the same value-of-life as a “representative” person. Thing One studies do produce values in the range of $6M.

someone sufficiently poor will get zero or negative utility

In reality, very poor people do try to stay alive, so any model that assigns them negative utility is incorrect - it’s a good sanity check to verify that this isn’t the case. The model I gave in the post suffers from this problem. However, a model where utility becomes utility at low incomes is not necessarily incorrect! Since there’s a minimum income required for survival (actually a minimum consumption level, since other people can give you free stuff, but I’ll ignore the distinction since this is a toy model), very few of the observed poor people will have income smaller than that, since they would quickly die. As long as the zero-utility income level is well below this survival threshold, the model is consistent with the fact that very poor people don’t want to die.

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Vladimir_Nesov
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4 years ago

It's a problem on codecogs.com, and will probably get fixed soon (it's affecting many other posts as well, for example). For the time being, I've replaced the latex renderer in this post with Google Chart Infographics, so it should work now, while codecogs is recovering.

Edit: (Sorry for breaking '+' in formulas, didn't properly test for the potential problem of different renderers requiring different levels of URL encoding.)

Edit (16 Feb): And back to codecogs now. (By the way, when using Google Chart, the URL with 'chart.apis.google.com' is better than one with 'chart.googleapis.com', since the latter has robots.txt preventing crawlers from saving the images.)

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buybuydandavis
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4 years ago

The "total utilitarian" assumption that potential people are as important as existing people

Every sperm is sacred, every sperm is good, every sperm is needed in your neighborhood...

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