gjm comments on What can total utilitarians learn from empirical estimates of the value of a statistical life? - Less Wrong Discussion
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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.
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.
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.