For cost-benefit analysis, the usual ballpark figure for the value of a life is about $10,000,000. But I keep hearing it suggested that when it comes to fighting a disease like Covid-19, which mostly kills the elderly, this value is too high. In other words, an old life is worth less than a young life.
I don’t see it.
People seem to have the intuition that ten years of remaining life are more precious than one year of remaining life. That’s fine, but here’s a counter-intuition: An additional dollar is more precious when you can spend it at the rate of a dime a year for ten years than when you’ve got to spend it all at once — for example, if your time is running out. (This is because of diminishing marginal utility of consumption within any given year). So being old means that both your life and your dollars have become less precious. Because we measure the value of life in terms of dollars, what matters is the ratio between preciousness-of-life and preciousness-of-dollars (or more precisely preciousness-of-dollars at the margin). If getting old means that the numerator and the denominator both shrink, it’s not so clear which way the ratio moves.
Instead of fighting over intuitions, let us calculate:
Suppose you’re a young person with 2 years to live and 2N dollars in the bank, which you plan to consume evenly over your lifetime, that is at the rate of $N per year. I’ll write your utility as
Suppose also that you’re willing to forgo approximately pX dollars to avoid a small probability p of immediate death. Then (by definition!) X is the value of your life. (The reasons why this is the right definition are well known and have been discussed on this blog before. I won’t review them here.) This means that
= U(N,N) – (pX/2)U1(N,N) – (pX/2)U2(N,N)
(where the last equal sign should be read as “approximately equal” and the Ui are partial derivatives).
Because you’ve optimized, U2(N,N) = U1(N,N), so we can write
Now suppose you’re an old person, having lived half your life and spent half your endowment. With only one year left to live, you can spend $N now and $0 next year, so your utility is
If the value of your life is Y, then
so that
Now that we have expressions for the value of a young person’s life (X) and the value of an old person’s life (Y), we want to compare them to see which is bigger. This of course depends on the utility function $U$. A natural first guess is Cobb-Douglas: U(A,B)=AαBβ for some constants α and β . This won’t work here because we want to be able to insert B=0, so the natural fix is to add some constant K to A and B:
(Footnote: To be consistent with my earlier assumption that young people spread consumption equally over their lifetimes, I should take α=β , but everything still works, modulo a little fiddling, with α and β arbitrary.)
With this definition, the results of our first and second calculations show that X=Y. That is, an old life is worth exactly as much as a young one
Could you get a different result if you used a different utility function? Sure. There are enough utility functions out there that with a little cherry-picking you can get any result you want. But when you’re not cherry-picking, Cobb-Douglas is a good starting point—which leaves you with the conclusion that old and young lives are equally valuable. Of course there might be a good non-cherry-picking reason to depart from Cobb-Douglas, but if so, that reason needs to be articulated.
If you’re going to skip that step and just pull things out of the air like “of course old lives are worth less than young ones”, then you might as well skip the cost-benefit analysis entirely and just pull your desired conclusion out of the air instead. But if prefer to think about this stuff, then it’s worth thinking about it carefully. This post is Step One. Who wants to propose Step Two?
Suppose that you’re compelled to play a round of Russian roulette at some point in your life. You can play it at age 20, or age 80. Which do you choose?
Sam raises a good point, but I read Steven’s analysis to be:
[a] What would you pay to avoid playing Russian roulette aged 20;
[b] What would you pay to avoid playing Russian roulette aged 80;
It is not clear that $[a] is different, let alone higher, than [b].
Steve,
your calculation leaves the typically much reduced quality of life due to ill health in old age out of consideration, and this is an element that you can’t ignore.
In Italy, the average Covid-19 victim lost 3 years of life.
Basically, the large majority of the victims is already with one foot in the grave.
#1. That depends entirely on whether you are 20 or 80. I suggest you would pick the one you are not.
Another measure of the value of a life would be the value to everyone else. What is the cost to the world if a given person dies? Not the cost to them, nor to their nearest and dearest, but to the world in general. That value is the value of what they will produce in the remainder of their lives, whether that’s lifting potatoes, writing, or inventing hyperdrive.
The elderly, from this point of view, produce almost nothing. The frail elderly are an ongoing cost due to the care that they need.
A new born baby has a whole life of production ahead of them, but if it dies, it is easily replaced. Only nine months of resources have been expended on them. So the cost of neonatal death is small.
As the baby grows up through childhood, more and more resources are invested, against the expectation of a lifetime of productivity. The value of their future minus the cost of their past reaches a maximum as they reach adulthood and begin producing. That is the point where their death incurs the greatest cost, and therefore where it is worth the greatest amount of resources to prevent.
Thereafter, depending on the course of their life, their value — the cost of their premature death — may go up and down, but until we defeat aging there comes a point where that value declines towards zero and below.
Whether that is an appropriate way to value lives when shortage of resources makes decisions necessary is another question. But it does seem to correspond fairly well with how upset people are when someone dies. It is sad for a newborn, but one gets on with life. It is sad for anyone over 90, but expected: one says it was their time. It is seen as the greatest tragedy at around the beginning years of adulthood, when the investment in their upbringing has reached its maximum, and the productive life that that expenditure was all for is lost. And depending on the talents revealed by that age, the greater their future promise, the greater the loss is felt.
What is politically optimal and what is morally optimal are not necessarily the same. Under current institutions, the Congressmen who decide our policies also take into account future elections. They care about votes and votes don’t come in terms of fractions of years remaining. They are one person one vote. Moreover, old people vote more frequently than young people. Thus, all else equal, our political institutions would lead to policies that disproportionately favor the elderly’s interest. Of course, not all old people will vote narrowly in their self-interest for more protection from the virus, but politicians may anticipate that it may move the needle, which is sufficient.
The initial assumption is clearly not correct – we don’t start with a sum in the bank and spend it evenly over our life. It does not seem to be trivial whether this makes any difference, but I am sure this has been discussed before, so I am happy to go along with it. I like the shortening it to 2 years so we can more or less ignore discount rates. Very useful.
This is the value of a statistical rather than an individual life. We accept that the value of a life is priceless in some circumstances – slavery is not permitted, for example. Yet for policy we must decide how to trade one thing against another in general terms. The value of a life is necessary to do this. This is what people pay on average for a small reduction in risk. The value will depend on the wealth of the country, because when discussing policy we are asking what that that country should spend to save a statistical life. We are not actually putting a monetary value on individuals within that country. This of course raises all sorts of ethical questions when we are dealing internationally.
An alternative approach is the Quality Adjusted Life Year (QUALY). A study on dialysis calculated the value of one year of quality life to be $129,000 in 2009. Median age in the US is 38 years and life expectancy is 78, so if the average expected life has 40 years to go, and we round down to $100,000 to compensate for reduced quality as we age, we get an average value remaining of $4M. This is ball-park similar to the $10 million often used for the value of life. This is somewhat reassuring that there is some consistency within these approaches. The idea of the QUALY has problems, because if you lose a finger, say, your life becomes worth less as the quality is reduced. Most people find this somewhat troubling.
The QUALY approach says that less should be spent to save an old life than a young life. SL’s analysis here suggest that an old life is worth the same as a young one. This is intriguing.
The value of each life is what each person would pay to avoid a small chance of immediate death. The calculation shows that for two people with exactly equal lifetime resources, they would pay the same when they are old as they would when they were young. They have less money, but less time to spend it in, so on balance they spend the same to avoid the risk. This does not sound unreasonable when it is put like that.
Out of interest, the image is of Stanislav Spáčil, at 17 and 102. He died in 2015 at the age of 107.
It is interesting that you have mentioned that the two values are equal but you have not mentioned what the value is. If I did my mathematics correctly, it is (A + K) / alpha. Which means that the value of a life is wealth divided by life expectancy. Which means that the rich should be saved at the expense of the poor.
Do you still stand by your analysis or would you want to redo it?
Nimesh: No, the result of the calculation is that the value of a life is independent of remaining life expectancy.
As for the rest, you seem surprised by the conclusion that safety is a normal good. But there is ample empirical evidence for that. More luxurious cars tend to have more safety equipment, for starters.
If we follow this line of logic, doesn’t it suggest that the value of extending a life by one day (or one minute or one microsecond) is the same as extending a life by a few decades? If so, then in the extreme it shows that the reasoning must be going wrong somewhere. It would suggest that all the value of a life is compressed in its very last instant.
Do you reach a different conclusion if you use willingness to accept rather than willingness to pay? How much would someone have to pay your heirs for you to accept a certain death today? Might an 80 year old who feels they are nearing the end of their life accept a smaller sum than a 30 year old?
I think Steve’s analysis here is missing an important point. (Alternative hypothesis – I have fundamentally misunderstood what the “value of life” calculation has been meant to demonstrate all along. I freely admit this could be the issue, and if so, someone please do feel free to correct me.)
Steve suggests that when we say something like “the statistical value of a life is ten million dollars” that we ought to consider the marginal value of those dollars to the person being saved, with respect to their remaining lifespan. But, why should we consider that? I have never understood the statistical value of life as being meant to measure how much value someone places on their own life continuing. I always took it to be an attempt to measure the social value that is derived from saving a life. When we “measure the value of life in terms of dollars,” we’re not measuring how much those dollars are worth to the specific person who might be saved, or how they personally would value spending those dollars at particular rates on particular timelines. We’re measuring the social value that is gained by saving their life, against the alternative uses of those ten million dollars (or more precisely, by the resources represented by those dollars). By those lights, the question of how much additional life can be expected to be gained seems like a very important factor.
A near-Hansonian analysis.
How much would you pay for a new car? The same make and model with 100,000 miles on it? 200,000 miles?
Which society works best: one in which everyone who reaches 80 is executed, one in which everyone who reaches 40 is executed, or one in which everyone who reaches 10 is executed?
KevinDC (#12):
Steve suggests that when we say something like “the statistical value of a life is ten million dollars” that we ought to consider the marginal value of those dollars to the person being saved, with respect to their remaining lifespan. But, why should we consider that?
Because the whole point of cost-benefit analysis is to respect people’s preferences.
If you are facing a 1% chance of death, and if I can eliminate that chance of a death at a cost of $100,000, then the relevant question is: Which would you prefer, the safety or the cash? And if the latter, then it’s a policy mistake to give you the safety.
If we’ve decided not to respect people’s preferences, then of course the “value of life” calculation, along with many other things, becomes irrelevant.
J Storrs Hall (#13): I’m not sure I see the relevance of your analogy. The point of cost-benefit analysis is to make choices that respect the preferences of those who are affected. But cars don’t have preferences.
Mike Bravo (#2): Yes, that’s the point exactly. Thank you.
Jim (#11):
If we follow this line of logic, doesn’t it suggest that the value of extending a life by one day (or one minute or one microsecond) is the same as extending a life by a few decades?
Yes, but I’m not sure why you find this problematic. If you’ve got a few seconds to live, what are you going to spend your money on?
As for willingness-to-pay versus willingness-to-accept, there is no difference for small probabilities. You could argue that COVID-19 has a 1% chance to kill you and 1% is not small, but that makes the mistake of treating policy as an event, not a process. At the point when you lay down a blanket policy for how you’ll behave in pandemics, pandemics are rare events.
Steve (#17)
I am not an economist, so these are closer to the class of ‘gut reactions’ rather than well-theorised objections.
Two problems strike me with the line of logic in the extreme situation.
1. It seems to me that it values all intervening years at zero. If the value of my life at 30 = some constant K and the value of my life at 80 = K too, then the value of my 50 years in between is K-K = 0, no?
2. You ask “If you’ve got a few seconds to live, what are you going to spend your money on?” Plenty! Inheritance. Either for my heirs or if the state wants to tax my estate at 100% then it’s a social inheritance for other people. Hey, I’ve got 10 seconds left to live, will I give all my possessions for an extra 10 seconds? No – I’ll leave those possessions for someone who’ll get more out of it.
In terms of willingness-to-pay vs. willingness-to-accept, the reason I raised it was the individual situation of poor people. I’ve seen in the comments section on Andrew Gelman’s blog the example of a homeless person who can only scratch together $3. Their WTP to save their life (even to avoid certain death) cannot be any higher than the $3 they have available, but their WTA even a fractional risk of death could well be higher.
I see that Harold (#7) raises the potential for using QALYs. These are also used in cost benefit analyses, and do not imply a constant value of life. So that suggests a way of using the method that you feel respects people’s preferences (#15) whilst avoiding some of these odd-seeming conclusions.
Jim:
1. It seems to me that it values all intervening years at zero. If the value of my life at 30 = some constant K and the value of my life at 80 = K too, then the value of my 50 years in between is K-K = 0, no?
I have no idea what “the value of my 50 years in between” means. Are you referring to your willingness to pay to avoid some probability of spending those years in a coma? If so, I don’t see why you’d expect to be able to compute this value from the kind of arithmetic you’re doing.
2. You ask “If you’ve got a few seconds to live, what are you going to spend your money on?” Plenty! Inheritance.
Yes, and the calculations here ignore that. The next step is to add inheritance to the model and see where it leads. It might or might not lead to where you expect it to.
I’ve seen in the comments section on Andrew Gelman’s blog the example of a homeless person who can only scratch together $3. Their WTP to save their life (even to avoid certain death) cannot be any higher than the $3 they have available, but their WTA even a fractional risk of death could well be higher.
But this is quite irrelevant both to the way economists do calculate the value of life and the way economists should calculate the value of life. Willingness to pay to avoid a small probability of death should be essentially equal to willingness to pay to accept that probability, which is all we need.
I see that Harold (#7) raises the potential for using QALYs. These are also used in cost benefit analyses, and do not imply a constant value of life. So that suggests a way of using the method that you feel respects people’s preferences (#15) whilst avoiding some of these odd-seeming conclusions.
Only if you’ve got a good argument that ties the QALYs to willingness-to-pay (or accept).
Steve #19
OK – I’m definitely out of my depth here – thanks for engaging anyway!
That caveat well and truly made and meant – quickly searching [qalys willingness to pay] in Google Scholar seems to show that it’s at least not absurd to tie QALYs to willingness-to-pay. From the titles, lots of the papers on the first couple of pages look like they’re trying to establish peoples WTP for a QALY or similar.
Here is one that looks relevant, based on the Abstract, and comes up with WTP numbers for a QALY in a bunch of countries (e.g., $62,000 for the USA):
https://onlinelibrary.wiley.com/doi/abs/10.1002/hec.1481
Jim: Yes, that looks like a good starting point. Thanks for posting it.
Hello Steve –
You said “Because the whole point of cost-benefit analysis is to respect people’s preferences.”
I’m not sure I agree with this. When you want to respect people’s preferences, cost-benefit analysis is a good way to do that. But I’m not sure it follow from this that respecting people’s preferences is the “whole point” of cost-benefit analysis. And, perhaps of greater interest, I’m not sure I see how this is consistent with your previous writings on the topic.
I’ve been reading your work for years, and following this blog since the beginning (although like most of the blogs I follow, I generally avoid leaving comments). What I find striking about that comment is that in many of your previous works, you’ve used cost benefit analysis to argue that we shouldn’t accede to people’s personal preferences, when such preferences were shown to fare poorly with cost benefit analysis from a social (rather than individual) point of view.
(The below paragraph is me working from memory, so I may have some specifics wrong, but I think the basic point is accurately represented.)
For example, in The Armchair Economist, you argued that if you were to see a dollar blowing down the street and towards a sewer drain to be lost forever, your personal preference would be to pick it up. You’d expend three cents worth of effort and gain a dollar’s worth of benefit. But you also argued that if you let the dollar go, then the money supply would decrease, which would make other people’s dollars worth slightly more, giving society in general a total dollar in additional purchasing power. And since you’d gain a net 97 cents of benefit by picking up a dollar, which is smaller that the social gain of one dollar by letting it go, a good use of cost benefit analysis would be to prevent you from retrieving the dollar for yourself, despite your personal preferences and despite how you’d benefit in your own personal cost benefit analysis.
An individual might value the years from 80 to 82 at exactly the same dollar amount they value the years from 20 to 80. (I’m not entirely convinced that’s true from what you’ve argued above, but I’ll take it for the sake of argument.) Even so, that doesn’t mean the social benefit is the same in providing a 20 year old an extra 60 years of life as it is to provide an 80 year old an extra 2 years of life. If I decide to spend a million dollars, and I could use it to give 60 extra years of life to 10 young people or 2 years of extra life to 10 old people, it very well may be the case that both groups equally prefer to receive that benefit. But it still seems obvious to me that much greater social benefit is gained by saving the younger group. (Assuming away unusual circumstances like one of the young would be the next Hitler or one of the elderly was six months away from curing cancer.) Given that you’re previously argued on many occasions that individual preferences should be overruled (or at least, not actively provided for!) when they fail a social cost/benefit analysis, your current position seems inconsistent to me. What might I be missing?
Sl. #10. In your reply you say “Nimesh: No, the result of the calculation is that the value of a life is independent of remaining life expectancy.”
Yet the comment says the analysis shows the rich should be saved at the expense of the poor, nothing to do with life expectancy.
However, it seems reasonable that the analysis would say the life of the rich is worth more than the poor, since it is based on that persons willingness to pay. This is why the value of a statistical life is more in the US than it is in Bangladesh. Your analysis is not looking at this, since it is comparing two people with the same wealth over their lives in order to isolate the age factor.
The point, I think, is that policy applies to the whole of a specific population, and policy makers have to have some basis for thier choices. Tehe economic approach is to base this on preferences, which requires some sort of averaging process – hence the VSL. If it cost $2M to build a safety rail that would save 1 unidentified future life, it is worth it. It if costs $20M it is not, by this analysis. We do not know if that person would be a pauper or a millionaire since they are statistical. As soon as we ask how much should we pay to save this or that individual, the whole basis is out of the window.
Another factor that is implied in some of this discussion is trading a young life against an old one, rather than both lives against, say, a smaller military, cheaper education or more tax. In normal policy making, we can decide to spend as much as required to save both, and to value them equally. With Covid, we may be presented with different choices. If healthcare is overwhelmed it is not about public spending, but allocating the resources that are already there as it is not possible to increase them in time. We may be presented with two people with idendical SVL, but one could contribute more to society. Is it OK to use this criterion in this circumstance, since in a sense existing policy has already failed?
I think we find the concept of the statistical life more acceptable if the value obtained is representative of the entire population the policy will affect. So a policy on handrails will be publically acceptable as long as it is applied evenly within the juristriction. I think that if a city Government started calculating VSL based on districts and spending on safety measures accordingly there would be disagreements.
This paper looks at adults and children, and find the VSL for children is higher than for adults.
https://www.sciencedirect.com/science/article/abs/pii/S0928765518303014
They say “However, the bulk of VSL research has focussed on estimating average values rather than taking account of the potential variation in VSL across groups.” My feeling is that this is deliberate and politically necessary. However, if it were possible to divide people into groups and use the VSL for that group when allocating resources, would that be the best economic approach? After all, the VSL is just a way of estimating preferences, and if we could know the value better, we would be closer to the real preference. Thus a policy that would tend to prevent the death of a rich person would have more spent on it than a policy that would tend to save a poor person. I know this often happens now, but should we enshrine this in policy making as a matter of sound economic principles ?
There are other methods besides preferences we could use to make the same choices. But if we are going to use preferences, this analysis shows at least as a first estimation, that we do not have a good reason to think the old should be valued less than the young.
One question that seems to be posed is should we use preference to decide these policies? Or should we value a life not by what the individual thinks they are worth, but what society thinks it can get from them?
There are some things about the VSL that make us uncomfortable, but there are also things about the State picking who to save and who to let die also makes us uncomfortable.
Governments that have applied the “value to society” rule have often been viewed poorly.
“No, the result of the calculation is that the value of a life is independent of remaining life expectancy.
As for the rest, you seem surprised by the conclusion that safety is a normal good. But there is ample empirical evidence for that. More luxurious cars tend to have more safety equipment, for starters.”
I don’t think you answered my questions. My first question was as per your analysis, what is the value of a life – is it (A + K) / alpha? If it is, does your analysis imply that a rich person’s life is more valuable than that of a poor person?
Regarding your claim that value of life is independent of life expectancy, all you have shown is that a person values his life the same at any age – provided that his wealth divided by life expectancy is the same. It is not clear to me that this translates to saying that society should value all lives equally. Since your model does not even value rich and poor lives equally, I doubt it can allow us to reach the conclusion that all lives are equal.
Nirmesh:
I don’t understand much of what you’ve written. I’ll try to respond as best I can.
what is the value of a life – is it (A + K) / alpha?
Yes.
does your analysis imply that a rich person’s life is more valuable than that of a poor person?
Yes. If it didn’t imply this, there would of course be something very wrong.
Regarding your claim that value of life is independent of life expectancy, all you have shown is that a person values his life the same at any age – provided that his wealth divided by life expectancy is the same.
Here’s where we get to the parts I don’t understand. To say that the value of life is independent of life expectancy is the same thing as saying that a person value his life the same at any age, so I don’t get the “all you have shown” bit.
– provided that his wealth divided by life expectancy is the same.
Here I’m totally confused. Why are you dividing wealth by life expectancy?
It is not clear to me that this translates to saying that society should value all lives equally. Since your model does not even value rich and poor lives equally, I doubt it can allow us to reach the conclusion that all lives are equal.
But that conclusion is manifestly false, so we wouldn’t want a model that reaches that conclusion. Do rich and poor people spend equally on car safety equipment? On vitamins and nutrition? On elective medical care? On home defibrillators? On exercise classes? It’s hard to imagine a model that answers these questions in the affirmative and is still actually useful for policy analysis.
Also, is it standard practice to add constants to a Cobb Douglas utility function or it significantly changes the key attributes of the function?
The discussion between Nirmesh and SL overlaps with my comment.
“Do rich and poor people spend equally on car safety equipment? On vitamins and nutrition? On elective medical care? On home defibrillators? On exercise classes? It’s hard to imagine a model that answers these questions in the affirmative and is still actually useful for policy analysis.”
I think SL is saying that to be efficient, policy should mirror the willingness to pay of the individuals. However, my instinctive, or moral objection is that it is wrong to value lives differently. So lets think it through.
Say we wanted a value of a statistical life (VSL) that was more targeted to give more efficient policy. We split the population into (say) above and below median wealth and conclude that the VSL for these two groups was $5M for the below median and $15 million for above median wealth. We divide localities into either majority wealthy or majority poor, and apply a policy that we should spend 3 times more in majority wealthy areas to save a statistical life. This initially sounds very wrong, but is it?
This is all about cost/benefit analysis, so we must also calculate the cost of saving a life for the two populations. Rich areas already have safety measures. The individuals within them already take vitamins, have home defibrilators and medical care. It may be that by spending $2M in the poor area we could save 3 lives, whereas spending $15M in the rich area would fail to save a single life. Policy would then direct spending more to the poor.
An example – say rolling out a defibrilator to every street corner. Over all USA, say this cost $11M to save a life. With current single SVL it is not cost effective, in part because defibs in wealthy areas go unused due to home defibs. If we divide into rich and poor areas, then cost per statistical life in poor areas is say $4M, but in wealthy areas is $16M. Same number of defibrilators, but fewer lives saved in wealthy areas. Policy to install defibrilators in poor areas is approved.
Harold (#27):
I think SL is saying that to be efficient, policy should mirror the willingness to pay of the individuals.
Yes, exactly.
However, my instinctive, or moral objection is that it is wrong to value lives differently.
I doubt this is true. Do you have an instinctive moral objection when you see relatively poor people choosing to spend money on car repairs or child care or perhaps a Netflix subscription when they could instead be spending every dollar they’ve got on health care? If you don’t have that instinctive moral objection when people make these choices for themselves, why would you have an instinctive moral objection when policymakers choose to respect those choices?
An example – say rolling out a defibrilator to every street corner. Over all USA, say this cost $11M to save a life. With current single SVL it is not cost effective, in part because defibs in wealthy areas go unused due to home defibs. If we divide into rich and poor areas, then cost per statistical life in poor areas is say $4M, but in wealthy areas is $16M. Same number of defibrilators, but fewer lives saved in wealthy areas. Policy to install defibrilators in poor areas is approved.
I can’t make any sense out of this. People who buy defibrillators are well aware that they might never be used (in fact they hope they will never be used!) but they still want them. If unused defibrillators are wasteful, then so is buying insurance which is almost never used. But people want insurance.
If you spend $1000 to buy a defibrillator for a person who is rich enough to consider a defibrillator a priority, you’ve done little harm. If you spend $1000 to buy a defibrillator for somebody who would have far preferred $1000 worth of help with the groceries or car repairs, you’ve made a mistake — unless you seriously do have an instinctive moral objection to poor people caring about groceries and car repairs, which again, I very much doubt.
KevinDC (#22): It’s inefficient to pick up hundred dollar bills in the street because the cost of picking them up (three cents worth of effort) exceeds the social benefit (which is zero). Why, then, do people pick up those hundred dollar bills? Because the private benefit is $100. The social problem is caused by the divergence between social and private benefits.
But I don’t see how you’re applying this to the valuation of lives. For one reason or another, I’ve decided to spend $1000 to help you. At this point, what’s relevant is how can I help you the most? If you’re rich and your basic needs are mostly taken care of, your preference might be that I fund a policy that will increase the chance that you’ll live another year. If you’re poor and struggling, your preference might be help with the groceries or child care or car repairs. Efficient policy respects those preferences, unless there are offsetting externalities.
In the case of the hundred dollar bill, the externality is central to the argument. I don’t see any analogue to that externality here.
“I can’t make any sense out of this. ” Yes, reading it back I can see why. I am not suggesting we fund defibrillators for all street corners. I was not very coherent. I will try to construct an argument. I see your argument that it makes sense to use the life-saving resources for the rich because the poor need other things more, but this is slightly outside that argument.
Assumption: Currently, for public spending, a cost benefit analysis is done. For a policy who’s purpose is to save lives, if the benefits outweigh the costs, then the policy is enacted. The assumed benefit is the SVL.
1) Current policy mostly uses one value for all statistical lives in the US of about $10M.
2) This means that due to the cost benefit analysis, the US would spend as much to save a poor person as a rich person.
3) This is inefficient because the life of the rich person is worth more than the life of a poor person.
4) We cannot get a figure for every life for policy purposes, but we could do a bit better than one value for everyone. We could divide the population into 2 groups, rich and poor and assign a SVL to each of these groups. *If* policy could be directed separately towards these groups, we would spend more to save a rich life than a poor life. Considering benefits only,this implies more public life-saving resources going to the rich.
5) However, now we have divided the benefits we must also divide the costs involved. It probably costs less to save a poor life than a rich life, because the rich already have lots of risk reducing stuff.
6) As we are talking about targeting the policy, we may end up directing more life-saving resources to the poor. It is the ratio of costs to benefits, not the benefits alone that is important.
Harold: Yes, your point is much clearer now.
But I think the main thing that remains is just this: If we’re going to spend $1000 per person, rich people are a lot more likely than poor people to want their $1000 spent on defibrillators. So if we’re trying to help rich people, it might make sense to provide defibrillators, but if we’re trying to help poor people, we probably want to provide something else.
But we’re off on a tangent. The question is not “Are poor lives worth less than rich lives?”. This is an uninteresting question because all thoughtful people agree on the answer. The question that might be interesting is “Are old lives worth less than young lives?”. To focus on that question, we want to hold wealth constant.
Would you rather live in a world in which Adam Smith had died at 7, or at 67? Add in the contributions of Aristotle, Homer, Plutarch, Fibonacci, Aquinas, Luther, Pascal, Leibniz, Newton, Maxwell, Ricardo, Fermi, Einstein, Darwin, and Milton Friedman, possibly using a Cobb-Douglas function.
I believe there is a confusion between the value of reducing the risk of death for the person and the value of reducing the risk of death of a person ~for society~. Surely the dollar value that Bill Gates puts on reducing his risk of death is much greater than of a poor 60 year old man. But that does not mean that society, or government, should spend that much more money on saving Gates’ life over the other man. If all total lives are equally valuable for society, and if people typically prefer to die later rather than sooner, than comparing QALYS instead of lives makes perfect sense.
SL: “People seem to have the intuition that ten years of remaining life are more precious than one year of remaining life.”
Re-phrase: Assuming a life expectancy of 80, society values the
loss of ‘life time’ for killing a twenty year old, far greater
than murder of a seventy year old. (philosophically, this transforms homicide into a crime of theft)
I’ll hypothesize that this implied belief is already operational, in the judicial system. That is, judges impose sentences proportionally severe to the age of the victims (measured by mandatory time until parole).
This would make a nice research project, or PhD dissertation, for a sociologist.
For those motivated to see this debate on stage, I refer you to … where do you go for coverage of philosophy, society, morality? … yep, Twilight Zone: “The self-improvement of Salvador Ross”
Your analysis is about policy choices on how much we should spend to save lives.
If we have limited ventilators and no way to get more for now, we may need to choose between saving two individuals, say an old or a young one or a rich and a poor one.
Do you think this is effectively the same question as the policy one?
1) I have the same concern as some others about personal vs social utility (basically it seems that for most policy decisions, it’s the latter that is more relevant), but I might have missed some great response so I won’t beat that one to death.
2) Accepting the personal utility perspective then: Why do you consider the multiplicative Cobb-Douglas function at all appropriate for this situation? IANAE, but I gather from wikipedia that C-D was developed to model e.g. productivity as a function of LaborInput and CapitalInput; this seems at least plausible, since
S1: an incremental dollar of labor increases the incremental production resulting from each incremental dollar of Capital, and vice versa.
But I can’t see how S1 applies at all to our value-of-life discussion. So I guess my “non-cherry-picking reason to depart from Cobb-Douglas” is that Cobb-Douglas implicitly assumes that the aggregate effect on personal utility of the amounts spent in successive years is multiplicative, without any basis for doing so.
Arch1 (#37) There is no compelling reason to choose Cobb-Douglas. Another option is CES, which definitely does imply that the old life is worth less than the young life. But of course there’s no compelling reason to choose CES either.
So my point is not that we should accept the consequences of one particular utility function or another, but that one very natural choice of utility function yields a conclusion that is at odds with what many people are saying. Therefore to support their conclusions, they need to rule out that utility function. You can do that by cherry-picking an alternative (which seems unsatisfying) or you can do it by a priori ruling out any function that doesnt give you the conclusion you were hoping for (equally unsatisfying) or you can do it for some good reason. But I haven’t heard anyone suggest what that good reason might be.
Harold (#36):
Your analysis is about policy choices on how much we should spend to save lives.
If we have limited ventilators and no way to get more for now, we may need to choose between saving two individuals, say an old or a young one or a rich and a poor one.
Do you think this is effectively the same question as the policy one?
No.
Policy is a process, not an event. A policy consists of a contingency plan for what you’ll do in a variety of possible future circumstances, not an ad hoc decision about what you’ll do at the moment.
If you’re going to spend $1000 on something that will benefit a rich person, he might want you to spend it by installing more ventilators in the rich people’s hospital in case he ever needs one. If you’re going to spend $1000 on something that will benefit a bunch of poor people, they might prefer that you spend it on providing childcare. This raises two policy questions: First, how much should you spend on the rich and how much on the poor. Second, given what you’re spending, how much of that should go to ventilators? Given our plausible assumptions, you’ll probably want to provide ventilators for the rich but not the poor.
Now comes the pandemic and poor people are dying, so you want to take some of those ventilators from the rich and give them to the poor. But then your policy is: “Give poor people childcare instead of ventilators, and then give them ventilators anyway”. This is not a coherent policy.
You can construct a coherent policy that guarantees ventilators to the poor, but then you’re up against the fact that they don’t want that policy.
Of course you might endorse a policy along the lines of “make poor people so rich that they’ll want ventilators”. Then we can have an argument about whether this is feasible and whether it’s desirable. That’s a real policy argument. But I don’t see “should we make policies and then act as if we hadn’t made those policies?” as anything like a legitimate policy question.
Steve (#38): Sorry for not being clear enough about what I think is a good reason to rule out Cobb-Douglas for this scenario.
The reason is that it makes obviously invalid (even internally inconsistent) assumptions in this context.
Example: Suppose my experienced utility over a 1 year period increases as the square root of my spending. How much would my utility increase if my spending rate increased by 10x for a year? Ans: It would increase by a factor of 3.16, right?
And if we ask the same question for a 2-year period, the answer (at least to first order) would be the same, because we’re talking about two 1-year periods in succession, right?
But Cobb-Douglas (with each exponent = 0.5, and the additive parameters set to 0 for simplicity) says the answer for the 2-year period is an increase by a multiplicative factor of 3.16*3.16 = 10. And for a 4-year period it would be 100, 1000 for a 6-year period, etc. Alternatively, if we decide to analyze at monthly rather than annual granularity, then instead of a factor of 3.16 increase for a 1-year period, Cobb-Douglas tells us the factor becomes 1 million for the same 1-year period, when nothing at all has changed about the situation being modeled.
The reason for these bizarre and internally inconsistent results is that unlike factory production (in which labor and capital expenditures can each act as ‘force multipliers’ for the other), my spending amounts in successive time periods have no such interaction wrt their impact on my overall utility.
arch1 (#40): Thank you, your point is clearer now.
You are right that if I measure time more finely and try to keep the same Cobb-Douglas exponents, I get nonsense. But this only means that when I adjust the time period, I have to adjust the exponents accordingly. I can say that you live two years, with annual consumption of x and y, and apply Cobb-Douglas with exponents alpha and beta — or I can say you live two half-years, with annual consumption of x/2, x/2, y/2, y/2, and use exponents alpha/2, alpha/2, beta/2, beta/2 . This gives me effectively the same utility function (up to a multiplicative constant that doesn’t matter).
As to the rest, I don’t see why you’d think that consumption in various periods can’t interact the way capital and labor do. If I visit Hawaii this year, and then go back next year, this year’s experience will surely impact next year’s, either because I’ve learned which islands I like the best or because I’ve already seen the best of everything and am now down to the second best of everything. My enjoyment of everything I do today is heavily colored by all of the things I’ve done in the past.
I am thinking of a situation where the system is overwhelmed and there are not enough ventilators to go around. We would not be taking them from the rich since they never actually paid for them, since they don’t exist. Some people who have paid for healthcare coverage that usually would include ventilators will not get them.
This reminds me of the hypothetical where if a rich person would pay $1000 to punch someone, and that person would accept $500 to be punched, social welfare is increase by the punch even if the money is not paid.
In this case we have rich people who would pay more for the ventilator, so social welfare is increased by giving it to the rich person even if they don’t actually make the payment.
Is this a reasonable comparison?
Steve (#40): Thanks respectively to your Hawaii vacations and adjustable exponents, I withdraw my rash assertions that Cobb-Douglas is obviously invalid and internally inconsistent as applied to this scenario.
I still don’t see any evidence or reasoning as to why it *does* apply to this scenario, so I’m not sure why you are effectively considering it to be the default (I’m sure you wouldn’t use it just because it’s mathematically tractable, offers 3N adjustable parameters, is a popular choice in other contexts, and leads in this case to an edgy contrarian conclusion:-).
Maybe Cobb-Douglas for economists is like the normal distribution for statisticians, though, in that there is theoretical justification for applying it to a broader range of situations than one might at first expect?
Based on this analysis it seems unclear whether life at a young age is more valuable than life at an old age — to the same individual.
Based on past analyses I’ve seen from you (and others), society would generally judge the life of the young more valuable than the life of the old.
I guess it gets a little confusing at the beginning when it’s not clear who or what is doing the judging.
arch1 (#43): I have a well documented lifetime habit of always testing first-draft models with Cobb-Douglas utility first (and CES second), so I don’t think I can be fairly accused of choosing it because it leads to what you call an “edgy contrarian conclusion”, but I don’t think I can defend it on any theoretical grounds of the sort you’re (reasonably) asking for.
But that’s not the point. The point is: Many people say that X must be true. I’ve found a reasonable assumption under which X can not be true. Therefore there must be some gap in whatever argument leads to the false conclusion that X must be true. (X in this instance is “young lives are more valuable than old lives”, but the logic of course is quite general).
What reasonable assumptions would make X true? I’d say that the burden of settling that questions falls on the people who claimed that X must be true in the first place. Mr. A says “X is always true” and offers no proof. Mr. B (or in this case Mr. L) says “Actually, X is true under some assumptions and false under others” and does offer a proof. I’d say at this point that Mr. B (or L has made an actual contribution to the conversation and now it’s Mr A’s turn.
Scott H (#44): I guess it gets a little confusing at the beginning when it’s not clear who or what is doing the judging.
But it is clear. The value of your life is defined in the content of the blog post (and of course in many other places) as that dollar amount X such that you would be willing to pay pX to avoid a small probability p of death. So it’s you who are doing the judging.
I think this is an interesting take. If I understand it correctly, what you’re basically saying it‘s not necessarily clear that being older drives a lesser $ value placed on life by said old person vs this same person when younger b/c older people likely value $ less because they don’t have as much time to spread their consumption out. They also value their life less because there simply isn’t as much left to be had, but since they value $ less you’d have to pay them more than you otherwise would have at a younger age to give it up, all else equal, and it’s not clear that just doesn’t make up for the lost value in life to get to the same dollar nominal $ as someone younger. Interesting perspective if I’m understanding correctly.
We obviously can’t *really* do this, but I think an interesting experiment would be to take, say, healthy older people and health younger people and see what it would take to make them trade their lives.
Like, for example, would a random 60-year-old be willing not only to give up all his current wealth, plus take on lots of debt to switch lives with a healthy 22-year-old, say, in his current social class and roughly “all else equal.” If so, how much exactly would he be willing take on? On the flip side, the young person I imagine… man that would be a tough trade. But of course this could happen uh my imagination across multiple ages, so whereas the market might not exist for a 70-year-old swapping with a 20-year-old despite having $billions to give, there might would be a robust market for swaps between, say, 35-year-olds and, day, 21-year-olds. I wonder if we could ask people what they’d be willing to give to switch since obviously this market can only exist in our country imaginations.
Let’s see . . . I am 82 years old. I would not pay as much to save my life now than I would have when I was 30, assuming the same wealth at both ages.
#48,
I think the intuition here is that money isn’t as worth much when you’re old so you may be like “screw it, this money isn’t as worth as much to me since I can’t really spend it like I could when I was younger, therefore I’m willing to give more of it up.”
#47.
“The self-improvement of Salvador Ross” was mentioned by Richard D above – you can get the radio version on youtube.
SPOILER
Ross found he was able to exchange any condition he had for someone else. He traded a broken hand for a cough, his hair for someones baldness. He was able to sell 40 years of his life for a very large sum to a rich old man, then buy the years back cheaply from a series of relatively poor 19 year olds, each selling him 1 year.
#48 It is dangerous to extrapolate from reducing a small chance of death to a avoiding certainty of death. People will give almost everything they have to avoid certain death. They do make different and more revealing choices to avoid a small risk. Perhaps the question you should ask yourself is would you pay less now for safety features in your car than you would have at age 30?
I found this interesting:
https://www.oecd.org/officialdocuments/publicdisplaydocumentpdf/?cote=ENV/EPOC/WPNEP(2010)9/FINAL&doclanguage=en
They found little correlation between age and VSL, in line with SL analysis above, but they do note that some authors had found an inverted U shape – indicating less value to the young and old.
They conclude “In summary, our analysis appears to suggest that the value of statistical life primarily depends on individual wealth, but also – and this is problematic – on technical considerations with regard to how this value is calculated. In any theoretical scenario, the value of the risk reduction proposed during the
contingent valuation should be independent of VSL. But this is not what we observe. It even appears that in rich countries great value is attributed to life, in part because average willingness to pay is high (reflecting high levels of individual wealth), but also because there is limited scope for reducing risk, mechanically yielding a large ratio.”
I guess this is what I was getting at with some of my comments above. Both value and the cost of life-saving is less for poorer people.
Dear Steven
I don’t know if you have the patience to answer this question, but let me try.
There seems to me that there is an inconsistency in the economics literature. Health economists use QALYs (quality-adjusted life years), and the value per each additional year of life is something like USD 150.000.
SVL (value of a statistical life), which corresponds to the definition in your post, is something like 10 million.
Given a disease that affects mainly the old people, these different values will have wildly different implications; therefore, they cannot be both correct.
You favor SVL, I guess. But in that case, why are health economists using such different criteria? Is this, somehow, related to probabilities? In the case of medication versus no medication, evaluated in terms of QALYs, one is talking about reasonably certain outcomes. At the same time, when we estimate SVL we deal will small probabilities. Does this explain the different approaches?
Can you provide any insight?
Someone beat me to it but Step Two must be intertemporal i.e. caring about future generations. Otherwise it seems we’ve proven that the old enjoy cash bonfires much more than the young. Anecdotally I’ve seen many comments along the lines of “I’m old or otherwise high risk and I’m willing to isolate myself and take my chances, but my kids need to work”. I have yet to see anyone who’s lost their job (i.e. the other high-risk group) argue that the critics of blanket lockdown policies are being ridiculous.
It almost feels like “willingness to pay” breaks down when we’re measuring it with reference to a ‘unit of value’ that two parties value very differently? Maybe this is always the case (and I’ve always struggled with how economics reconciles this with *ability* to pay, although this post was helpful in that regard…certainly the equality argument is a fascinating (even beautiful) result I had not considered).
I’ll just leave this thought Larry Lindsey recently penned on the matter: “Each life lost is a tragedy. But “science” would likely not measure success in terms of lives lost – all lives are ultimately lost – but in terms of maximizing life expectancy. While all deaths are tragic, would decision analytics weight the death of a 93-year-old with a heart condition and diabetes the same as that of a 45-year-old mother of three who contracted breast cancer? Would you?” Note he is referring to the fact that “flattening the curve” was *never* intended to minimize lives lost, and the lockdowns have had opportunity costs in terms of other healthcare.
“and I’ve always struggled with how economics reconciles this with *ability* to pay.”
Simple – it does not, nor does it attempt to do so. See my comment #50 “In summary, our analysis appears to suggest that the value of statistical life primarily depends on individual wealth,”
“Note he is referring to the fact that “flattening the curve” was *never* intended to minimize lives lost,”
It was certainly about reducing the number. if too many people present at once the hospitals will no be able to treat them all.
If we never get a vaccine or a treatment but natural immunity is permanent, ultimately I guess we will end up with about 60% of people getting the disease and having natural immunity. There will be a death rate associated with this and we will not be able to reduce this. Eventually a certain number will die. That death rate could be a lot higher if we all get it at once, as will the death rate from other factors. It will be lower if we develop treatments or vaccines. Let us hope that natural immunity is permanent, or at least long lived.
i would say ‘swamping the hospitals’ was the risk scenario not the expected scenario. reaching a threshold level of natural immunity (if possible) protects the vulnerable as well. there are also the casualties from recession and deferred regular medical care to consider (I see Australian researchers fear many more suicides than covid deaths, and the US has shed millions of health care jobs). the political incentive is always to focus on what is most visible now
This is a topic that appeals to the motion, which is probably why there is so little discussion of the math. And, no, I’m not tempted to discuss the math, either. While it seems obvious that old lives would be worth less (many of us reach an age where we’d happily pay to quickly die, my grandfather willingly gave up eating because he did not want to live any longer), I think this is more because we see age as a proxy for health (and also for remaining lifetime). If I had the vigor I had in my twenties, it would no longer seem as obvious that another year of life would be worth less; but if I were no longer able to walk, struggled even to think (I am already no longer as good at memorizing lyrics as I once was!), I might be quite happy to say goodbye.
Emotion, that is!