Edited to add: As Salim suggests in comments, the entire problem is that I assumed an implausible value for wealth (which should be interpreted as lifetime consumption). With a more plausible number, everything makes sense. Mea culpa for not realizing this right away. I will leave this post up as a monument to my rashness, but have inserted boldfaced comments in appropriate places to update for my new understanding.
This is bugging me. It’s a perfectly simple exercise in valuing lives for the purposes of cost-benefit analysis. I would not hesitate to assign it to my undergraduates. But it leads me to a very unsettling and unexpected place, and I want to know how to avoid that place.
It’s also a little geeky, so I hope someone geeky will answer — ideally, someone geeky who thinks about this stuff for a living.
Start here: You’re a trapeze artist who currently works without a net. There’s a small probability p that you’ll fall someday, and if you fall you’ll die. You have the opportunity to buy a net that is sure to save you. What are you willing to pay for that net?
Well, let’s take U to be your utility function and W your existing wealth. If you don’t buy the net, your expected utility is
p U(death)+(1-p) U(W)
But we can simplify this by adding a constant to your utility function so that U(death)=0. So if you don’t buy the net, your expected utility is just
(1-p )U(W)
If you do buy a net at price C, then you’re sure to live, with utility
U(W-C) = U(W) – C U′ (W)
where the equal sign means “approximately equal” and the approximation is justified by the assumption that the probability of falling (p) is small, so your willingness to pay (C) is presumably also small.
Equating these two expected utilities gives me C = p U(W)/U′ (W). If we set V = U(W)/U′ (W), then C = pV. That is, you’re willing to pay pV to protect yourself from a p-chance of death. This justifies calling V the “value of your life” and using this value in cost-benefit calculatios regarding public projects that have some small chance of saving your life (guard rails, fire protection, etc.)
So far, so good, I think. But now let’s see what happens when we posit a particular utility function.
I will posit U(W) = log (W), which is a perfectly standard choice for this sort of toy exercise, though actual real-world people are probably a bit more risk-averse than this. Except I can’t just leave it at U(W) = log(W), because my analysis requires me to add some constant T to make the utility of death equal to zero.
So let’s take E to be the income-equivalent of death; that is, living with E dollars is exactly as attractive as not living at all. Then I have to choose T so that log(E) + T = 0. In other words, T = -log(E).
Now I know that, with your current wealth equal to W, the value of your life is U(W)/U'(W) = W log(W/E) .
Now as a youngish but promising trapeze artist, you’ve probably got some modest savings, so lets make your current wealth W=50,000 (with everything measured in dollars). (Edited to add: This was the source of all the difficulty. W represents something like lifetime consumption, so 50,000 is a ridiculously small number. Let’s go with 5 million instead.) Then here is the value of your life, as a function of E, the income-equivalent of death.
If E = .0001 (that is, if dying seems just as attractive to you as living with your wealth equal one-one-hundredth of a penny), then the value of your life is $1 million. (Edited to add: This should actually be E= 4.1 million dollars, which is considerably more than one-one-hundredth of a penny.)
If E = 6.92 x 10-82, then the value of your life is $10 million. (Edited to add: This should be E = $677,000 which might be a plausible figure.)
If E = 1.29 x 10-864, then the value of your life is $100 million. (Edited to add: This should be E equal to about one cent, which is of course implausible, but that’s fine, because a $100 million value of life is also implausible.)
Edited to add: I won’t continue to edit the details in the rest of this post, but I think this is all straightened out now. Thanks to those who chimed in, and sorry to have taken your time on this!
Now I am extremely skeptical that you, I, or anyone else is capable of envisioning the difference between living on 10-82 dollars and living on 10-864 dollars. Yet the decision of whether to value your life at $10 million or at $100 million hinges entirely on which of these seems more to you to be the utility-equivalent of death.
There is some purely theoretical level at which this is no problem. It is possible that you’d rather die than live on 10-864 dollars and would rather live on 10-863 dollars than die. But I am extremely skeptical of any real-world cost-benefit analysis that hinges on this distinction.
(And this is the range in which we have to be worried, since empirical estimates of the value of life tend to come in somewhere around $10 million.)
If I make you less risk-averse — say with a relative risk aversion coefficient of 4 — almost the entire problem disappears. But the tiny part that remains is still plenty disturbing. Then I get:
If E = .007 (that is, about 2/3 of a penny), the value of your life is $1 million.
If E = .003 (about 1/3 of a penny), your life is worth $10 million.
If E = .0015 (a sixth of a penny), your life is worth $100 million.
So we need to tell the folks in accounting to value your life at either $1 million or $100 million, depending on where you draw the suicide line between having two thirds of a penny and having one sixth of a penny.
This is nuts, right? And how squeamish should it make me about the whole value-of-life literature? And what, if anything, am I missing?
I can’t follow the math, so maybe you address this, but one problem/paradox I see in a situation like this is that the artist derives utility from the net on the day they fall, but probably derives dis-utility on the days she doesn’t. Put another way, there’s utility in successfully performing without the net…
Daniel Hill: I do not address that. I assume that the only downside of having a net is that you have to pay for it. I could, of course, tell a more complex (and maybe more realistic) story along the lines of your suggestion, but it wouldn’t solve my problem. In order to focus on the key problem without distractions, I want to look at the simplest possible example in which the problem arises.
This seems related to the fact that adding a constant term to the utility function–which does not matter for consumption optimization–does in fact matter for the value of life because the value of life depends explicitly on the flow of utility. This was first noted by Rosen (1988), who pointed out some interesting implications of this dependence, and shows up e.g. as the “subsistence consumption” parameter in Murphy and Topel (2006).
Some thoughts.
1. This seems like it’s going to depend a lot on the choice of utility function. Maybe log is a choice that’s useful in a lot of senarios but not in this one. How do these numbers change if we use something else?
2. Expanding on that log only works for positive wealth and has a weird assymptote at zero. If you’re $500 in debt (is that W=-500??) then what’s your utility? In this model it seems worse than death, so I’m leaning to it being a “bad” choice of utility function. By which I mean a choice which isn’t appropriate to this situation. Disclaimer I’m not at all confident I understand how W is defined here (future time adjusted earnings or current money in the bank?) so I could be completely off base.
Jonathan Kariv: Toward the bottom of the post, I report figures for a constant relative rate of risk aversion equal to 4, which means a utility function of (up to a multiplicative constant that doesn’t matter) $-x^(-3)$. (More generally, rate of risk aversion equal to sigma means a utility function of $(1/(1-sigma) )(x^(1-sigma))$, unless sigma =1, in which case it means Log(x).
The industry standard in investigations like this is to start by assuming a constant rate of risk aversion, and empirical studies suggest that that rate should be somewhere between 1 and 10, and probably somewhere between 3 and 5, which is why I used 4 in the example. The results there are about 10^(82) times less disturbing than the results from the log function, but still, I think, plenty disturbing.
Julian: Thank you. This gives me a promising angle to investigate.
What is W? Financial wealth, or even net worth doesn’t seem like a great basis for a utility function. Why not NPV of future consumption or something like that? Future consumption of 0 seems plausibly equivalent to death. For one, it would actually lead to death.
On second thought, NPV of future consumption runs into its own problems. I’d rather consume $50,000 per year for 50 years than consume $2.5 million, or even $25 million, in one year and then die. We need a model that accounts for the fact that consumption in a given time period exhibits diminishing marginal utility, while consumption across more time periods either does not, or has a different, less aggressive diminishing schedule. Log within time periods vs square root across time periods, or something like that.
Steve #5:The class of families of functions you describe seem to assume W>0? How does this model treat someone with $50,000 in the bank and a $50,001 loan to be repaid?
Surely the problem is just that log W is a really silly utility function near zero.
I don’t think it is very different to have 1 cent to having a dollar. Compared to the difference between 1000 and 100 000 dollars.
I think Leo is right that log (and similar) utility functions lose their descriptive value near zero. But that leaves another problem: if we stipulate that the consumption-equivalence of death is much larger than a penny – say $1,000 – then the value of your life is only $196,000.
This arises from the time-scaling of the problem. If we frame the problem in terms of annual consumption, then $50,000 and $1,000 seem like reasonable values. But if we multiply by 10 to get decadal consumption (with 0 discounting), we get a value of life 10 times larger. [Recall that V = W*ln(W/E), so 10W*ln(10W/10E)=10V]
This is occurring because the way Steve has posed the problem makes V “the value of living long enough to consume W”, not “the value of life”. If your net present value of all future consumption is just $50,000, you are poor indeed! Keeping discount rates at zero, the value of 30 such life-years is about $6 million, which is close to the range used in policy.
So I think Steve’s problem is solved in three ways:
0) He should frame problems as U(consumption), not U(wealth)
1) He needs a more realistic subsistence level of consumption
2) He needs to be explicit about time
Salim (and others): point taken re (2), but as for the subsistence level, I don’t get to choose that freely, because this is the unique subsistence level that yields a value of life that matches standard empirical estimates.
Off to an appt but will play with alternative formulations in a few hours.
I recently watched the movie Free Solo. You could do a whole new post on the calculation, if any, that occurred in the head of the climber.
It’s almost as if he’s indifferent to the possibility of death, especially if it’s not being filmed. The film includes a list of free climbers who have died on these climbs, to which he says “What did you expect?” Indeed.
I don’t mean to veer off the central question at hand here. But I’ve wondered whether there is useful data in this regard that might be derived from life insurance policies, or perhaps settlements from wrongful death lawsuits.
#14. I believe this sort of data is where this comes from:
“And this is the range in which we have to be worried, since empirical estimates of the value of life tend to come in somewhere around $10 million.”