Tag Archives: Health

Risk-Aversion Sets Life Value

Many pandemic cost-benefit analyses estimate larger containment benefits than did I, mainly due to larger costs for each life lost. Surprised to see this, I’ve been reviewing the value of life literature. The key question: how much money (or resources) should you, or we, be willing to pay to gain more life? Here are five increasingly sophisticated views:

  1. Infinite – Pay any price for any chance to save any human life.
  2. Value Per Life – $ value per human life saved.
  3. Quality Adjusted Life Year (QALY) – $ value per life year saved, adjusted for quality.
  4. Life Year To Income Ratio – Value ratio between a year of life and a year of income.
  5. Risk Aversion – Life to income ratio comes from elasticity of utility w.r.t. income.

The first view, of infinite value, is the simplest. If you imagine someone putting a gun to your head, you might imagine paying any dollar price to not be shot. There are popular sayings to this effect, and many even call this a fundamental moral norm, punishing those who visibly violate it. For example, a hospital administrator who could save a boy’s life, but at great expense, is seen as evil and deserving of punishment, if he doesn’t save the boy. But he is seen as almost as evil if he does save the boy, but thinks about his choice for a while.

Which shows just how hypocritical and selective our norm enforcement can be, as we all make frequent choices that express a finite values on life. Every time we don’t pay all possible costs to use the absolutely safest products and processes because they cost more in terms of time, money, or quality of output, we reveal that we do not put infinite value on life.

The second view, where we put a specific dollar value on each life, has long been shunned by officials, who deny they do any such thing, even though they in effect do. Juries have awarded big claims against firms that explicitly used value of life calculations to not to adopt safety features, even when they used high values of life. Yet it is easy to show that we can have both more money and save more lives if we are more consistent about the price we pay for lives in the many different death-risk-versus-cost choices that we make.

Studies that estimate the monetary price we are willing to pay to save a life have long shown puzzlingly great variation across individuals and contexts. Perhaps in part because the topic is politically charged. Those who seek to justify higher safety spending, stronger regulations, or larger court damages re medicine, food, environmental, or job accidents tend to want higher estimates, while those who seek to justify less and weaker of such things tend to want lower estimates.

The third view says that the main reason to not die is to gain more years of life. We thus care less about deaths of older and sicker folks, who have shorter remaining lives if they are saved now from death. Older people are often upset to be thus less valued, and Congress put terms into the US ACA (Obamacare) medicine bill forbidding agencies from using life years saved to judge medical treatments. Those disabled and in pain can also be upset to have their life years valued less, due to lower quality, though discounting low-quality years is exactly how the calculus says that it is good to prevent disability and pain, as well as death.

It can make sense to discount life years not only for disability, but also for distance in time. That is, saving you from dying now instead of a year from now can be worth more than saving you from dying 59 years from now, instead of 60 years from now. I haven’t seen studies which estimate how much we actually discount life years with time.

You can’t spend more to prevent death or disability than you have. There is thus a hard upper bound on how much you can be willing to pay for anything, even your life. So if you spend a substantial fraction of what you have for your life, your value of life must at least roughly scale with income, at least at the high or low end of the income spectrum. Which leads us to the fourth view listed above, that if you double your income, you double the monetary value you place on a QALY. Of course we aren’t talking about short-term income, which can vary a lot. More like a lifetime income, or the average long-term incomes of the many associates who may care about someone.

The fact that medical spending as a fraction of income tends to rise with income suggests that richer people place proportionally more value on their life. But in fact meta-analyses of the many studies on value of life seem to suggest that higher income people place proportionally less value on life. Often as low as value of life going as the square root of income.

Back in 1992, Lawrence Summers, then Chief Economist of the World Bank, got into trouble for approving a memo which suggested shipping pollution to poor nations, as lives lost there cost less. People were furious at this “moral premise”. So maybe studies done in poor nations are being slanted by the people there to get high values, to prove that their lives are worth just as much.

Empirical estimates of the value ratio of life relative to income still vary a lot. But a simple theoretical argument suggests that variation in this value is mostly due to variation in risk-aversion. Which is the fifth and last view listed above. Here’s a suggestive little formal model. (If you don’t like math, skip to the last two paragraphs.)

Assume life happens at discrete times t. Between each t and t+1, there is a probability p(et) of not dying, which is increasing in death prevention effort et. (To model time discounting, use δ*p here instead of p.) Thus from time t onward, expected lifespan is Lt = 1 + p(et)*Lt+1. Total value from time t onward is similarly given by Vt = u(ct) + p(et)*Vt+1, where utility u(ct) is increasing in that time’s consumption ct.

Consumption ct and effort et are constrained by budget B, so that ct + etB. If budget B and functions p(e) and u(c) are the same at all times t, then unique interior optimums of e and c are as well, and also L and V. Thus we have L = 1/(1-p), and V = u/(1-p) = u*L.

In this model, the life to income value ratio is the value of increasing Lt from L to L+x, divided by the value of increasing ct from c to c(1+x), for x small and some particular time t. That is:

(dL * dV/dL) / (dc * dV/dc) = xu / (x * c  * du/dc) = [ c * u’(c) / u(c) ]-1.

Which is just the inverse of the elasticity of with respect to c.

That non-linear (concave) shape of the utility function u(c) is also what produces risk-aversion. Note that (relative) risk aversion is usually defined as -c*u”(c)/u’(c), to be invariant under affine transformations of u and c. Here we don’t need such an invariance, as we have a clear zero level of c, the level at which u(c) = 0, so that one is indifferent between death and life with that consumption level.

So in this simple model, the life to income value ratio is just the inverse of the elasticity of the utility function. If elasticity is constant (as with power-law utility), then the life to income ratio is independent of income. A risk-neutral agent puts an equal value on a year of life and a year of income, while an agent with square root utility puts twice as much value on a year of life as a year of income. With no time discounting, the US EPA value of life of $10M corresponds to a life year worth over four times average US income, and thus to a power law utility function where the power is less than one quarter.

This reduction of the value of life to risk aversion (really concavity) helps us understand why the value of life varies so much over individuals and contexts, as we also see puzzlingly large variation and context dependence when we measure risk aversion. I’ll write more on that puzzle soon.

Added 23June: The above model applies directly to the case where, by being alive, one can earn budget B in each time period to spend in that period. This model can also apply to the case where one owns assets A, assets which when invested can grow from A to rA in one time period, and be gambled at fair odds on whether one dies. In this case the above model applies for B = A*(1-p/r).

Added 25June: I think the model gives the same result if we generalize it in the following way: Bt, and pt(et,ct) vary with time, but in a way so that optimal ct = c is constant in time, and dpt/ct = o at the actual values of ct,et.

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Why Not Walking Talks?

Most people see exercise as healthy, and see walking as a reasonably comfortable form of exercise. Some think that they should spend precious exercise time doing something more athletic, and others just can’t find the time to walk. But most seem to enjoy walking and see it as healthy, if only they could find the time.

I’ve been spending a lot of time giving talks lately, mostly on my book. I’m also back to teaching now that summer is over. Usually, these events all happen in a room, where I stand in front while everyone else sits. Sometimes I teach my class out on the grass instead of in a room. And so I wonder: why can’t we give talks while walking outside?

Yes, you’d have to forego visual aids, unless someone works out some pretty fancy tech. And yes, you’d need to pick a walking route that is quiet enough so that the audience could hear the speaker, and so a full-throated speaker won’t bother others along the route. Sometimes the weather isn’t agreeable. The audience would find it harder to see the speaker’s face, and a bigger group would need a louder speaker and more tolerant neighbors. And those who can’t walk might need someone else to push them in a wheelchair.

But none of these seem insurmountable barriers. We already manage to schedule lots of shared activities outdoors. We already have walking talks when guides take groups through battlefields, museums, and other special places. Is it so hard to have talks not focused on the immediate physical surroundings?

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Is US gun control an important issue?

After the shocking massacre in Connecticut it looks like gun control is going to draw a lot of attention from Obama and Congress this year. This got me thinking about how important gun control might be as a political cause. The potential good achieved by focussing on this policy is in large part determined by the damage done by guns in the first place. In that light, does it deserve it?

A natural measure of the importance of the problem is the number of years of healthy life lost due to gun violence. At  the moment there are a bit over 8,000 murders with firearms each year in the US, some two thirds of the total. If we guess that the typical age of death from gun violence is 30, then the average survivor would have enjoyed another 50 years or so of healthy life. Firearm homicides would than lead to the loss of 400,000 years of healthy life each year. We would then have to add health problems among survivors of gun violence. To confirm that these figures are sensible I looked up the World Health Organisation’s Global Burden of Disease, which suggest ‘intentional violence’ as a whole cost the US and Canada about 1,100,000 years of healthy life each year. Two thirds of this would be 650,000 years, a figure which amounts to about 0.8% of the total burden of disease and injury in the US.

Another even larger problem than murder – at least as far as years of healthy life lost – is suicide. Easy access to guns makes suicide attempts more likely to succeed. The US suicide rate is 12 per 100,000; tragically high, though sadly unexceptional by international standards. If the typical suicide victim would have lived another 45 healthy years, this amounts to an annual burden of 1,600,000 each year, roughly the WHO’s figure. [1]  Firearms are used for about half of these suicides, so we’ll say they have a burden of 800,000 years of healthy life, or about 1% of the total burden of disease and injury.

How much could the US hope to reduce these figures? Of course the relationship between the number of guns and violence is contested, and I don’t really want to get drawn into that debate. I will just assume, for the sake of argument, that gun control policies could indeed help reduce violence. For that purpose, let’s imagine it could get firearm violence and suicide down to the average of other OECD countries. [1] Doing so would reduce the gun death rate (and I will assume injuries too) by 80% from ~10 to ~2 per 100,000. This is wildly optimistic given the other drivers of violence and suicide in the US, and the timidity of any likely gun control laws under the Second Amendment. Even if guns did become hard to access, we would expect to see substitution to other weapons. Nonetheless, it offers a useful upper bound.

An 80% drop in firearm deaths and injuries would prevent the loss of 1.15 million years of healthy life each year, or around 1.4 per cent of all the damage done by disease and injury in the US. This falls inconveniently between ‘very little’ and ‘quite a bit’. How can we put this figure in perspective? One option would be to consider how much people claim to value their lives, while another would be to compare it to other available options for saving lives. Here I will use the latter to give some idea of how focussing on gun control compares to other policies or causes that might improve the health of Americans.

How much does it cost to save a life in the US?  The NHS in Britain conveniently uses £30,000 (around $US50,000) for each year of healthy life as the highest price at which a treatment is worth funding. The US has no central body for making these decisions, so no generic ‘marginal cost’ exists. A conclusion of the classic paper, Five-hundred life-saving interventions and their cost-effectiveness, is that the cost of extending lives varies across several orders of magnitude depending on the approach you take. Nonetheless, many interventions in medicine and general safety fell between $5-50,000 for a year of life, at least in the mid-90s. A quick search turns up vaccination of US girls against HPV, which buys a year of healthy life for about $44,000, total knee arthroplasty for $18,300, HIV screening for under $25,000 and flu vaccination at $8,000-52,000. The availability of all of these could be expanded. At a rounded $50,000 figure, the equivalent of 1.15 million years of healthy life could be saved for $57 billion, or 0.38% of US GDP – a significant sum, though under a fifth of long run annual growth. By comparison, the US Federal Government already spends about 24% of US GDP, and all healthcare spending accounts for some 15%. Based on Robin’s work on the inefficacy of much US healthcare spending, redirecting some of that enormous budget to truly life-saving activities would go a long way.

If American activists or voters currently preoccupied with gun control were willing to look farther afield in their desire to prevent unnecessary death, directing government spending to provide bed nets to protect children in developing countries against malaria could save 30,000 kids for a meagre $70 million, or 0.00000046% of GDP. Sadly, the effectiveness and size of US foreign aid is barely discussed.

Of course this health story is not the full picture of the damage done by gun violence. We ought also consider the:

  • Costs incurred in trying to stay safe
  • Costs of caring for the injured
  • Loss of human capital from adults dying
  • Resulting distress and fear
  • Reduced urbanisation as a result of crime (which lowers productivity, among other things).

I would appreciate attempts to quantify these costs but don’t have time to pursue them myself right now. I would note in passing that many other interventions that improve health and safety would also reduce these harms to some extent.

My interpretation of the above is that gun violence is a serious issue in the US. It is not being blown out of proportion like shark attacks or terrorism. At the same time, the impact of guns on US health-span is modest, and lower than many common and avoidable diseases or accidents which fail to inspire a national conversation. Guns have become a hot issue because of their grisly and visible results, as well as fierce identity politics, rather than the absolute scale of the damage they do. If the main goal of gun control advocates were to save lives, their cause would not stand out as low-hanging fruit, especially if they cared about foreigners as well as Americans. Given the host of major problems facing the US, the limited attention of Congress and the White House, and the improbability of achieving a significant reduction in the number of dangerous weapons available, it is not a cause I would jump on.

[1] Some would say that a death by suicide isn’t as bad as a murder, because someone who is preventing from committing suicide probably has a low quality of life. There is some truth to this but I will ignore it, consistent with my desire to define an upper bound.

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Which supplements should a healthy person take?

I have recently been looking into which, if any, nutritional supplements I should start taking. I am in good general health so am looking for supplements that are likely to maintain or improve that health, not cure any particular condition. I have been using three excellent sources for this project, which I can recommend checking out: [1]

For those who want to save time, I will outline my key conclusions here in the hope that doing so will help you. I have decided to start taking:

  • Vitamin D3 (10µg or so a day)
  • Creatine (5g a day)
  • Zinc (30-160mg and Vitamin C (>1g)  each day for the duration of colds.

Tyrosine and potassium are also both cheap and so I will trial them to see if they improve my concentration. I don’t consider them likely to work, but they are at least worth testing. Fluoride mouthwashes also seem a cheap way to reduce the risk of cavities.

Vitamin D has a large evidence base suggesting it significantly lowers ‘all-cause mortality’ and improves both general and bone health. It is especially important now that I am living in the UK, where it is much harder to get Vitamin D from sun exposure.  It is also inexpensive. [2] Basically, it is a no-brainer. The 10µg is twice the daily recommended dietary dose in the UK. For some reason, Gwern is taking a very large 125µg each day. Personally I am tempted to err on the low side due to recent research suggesting too much Vitamin D can raise mortality.

Creatine is best known as a supplement for body-builders, but I am taking it primarily because I hope it will improve my cognition. The evidence to back this is thin, and only finds a significant effect among subgroups like vegetarians, perhaps because they get less creatine from meat consumption. However, the effect size identified was very large, it is cheap and largely safe. I am an almost-vegetarian and lift weights so it is more likely to be worthwhile for me. I will also be able test whether it improves my energy and concentration and stop using it if it doesn’t. This review also finds a range of other worthwhile positive impacts on health.

There is compelling evidence that zinc helps reduce the intensity and duration of colds. As summarised by Cochrane:

Zinc inhibits rhinoviral replication and has been tested in trials for treatment of the common cold. This review identified 15 randomized controlled trials, enrolling 1360 participants of all age groups, comparing zinc with placebo (no zinc). We found that zinc (lozenges or syrup) is beneficial in reducing the duration and severity of the common cold in healthy people, when taken within 24 hours of onset of symptoms.

There are some concerns about side effects, but they do not seem significant in the scheme of things. The tablets can also be obtained cheaply and easily. The appropriate dose is unclear, but studies included in the meta-analysis used between 30-160mg. I will probably choose a figure in the middle of that, and keep some tablets at work and home so I can always take them immediately at the onset of symptoms.

Despite a large number of studies, evidence to back an effect of Vitamin C on colds in the general population is mixed, with positive effects only reliably found on those engaging in extreme exercise. I worry that positive results on such sub-populations could just be the result of data mining, publication bias or other chicanery. Nonetheless, there are no side effects and the tablets are cheap. I consider it worth taking at the onset of colds, even if the probability of any real effect is under a third. Furthermore, effervescent vitamin C tablets are tasty and comforting to drink, and being as conspicuous as they are, may produce a larger than usual placebo effect.

Incidentally, most infection by common colds is caused by surface to surface contact. Using an ethanol handwash after touching shared surfaces, and reducing how often you touch your face with your hands, is likely to significantly reduce their occurrence. If you didn’t already have one, the desire not to get colds is a good selfish reason to wash your hands after using the bathroom. Poor general health is not the problem, as even healthy people who are exposed to the virus are highly likely to become infected.

If I were particularly worried about my blood pressure or cardiovascular health I would start

However, I am young, and consider heart disease to be a problem for the future.

I am keen to hear if I am making mistakes in the above, or missing out on other valuable chances to improve my life. Thanks to Seb Farquhar and Will Crouch for help with this research.

[1] Cochrane’s ‘house effect’ is to frequently find that there is insufficient evidence to draw any conclusion. Where they do make a recommendation, the evidence backing it is likely to be compelling. Gwern’s advice extends to unusual supplements about which there is little other information. Unfortunately, is in based in significant part on personal experiences. While he has tried to do blind and controlled trials  on himself with sufficient sample sizes, I don’t consider one individual’s experiences to be compelling evidence relative to large trials and meta-analyses. He often doesn’t have a statistically significant effect, in part due to small samples. Nonetheless, if the cost of a supplement is low, and it is safe, it can be worth taking even with a low probability of an effect. Snake-Oil Supplements falls somewhere in the middle.

[2] Reasonably cheap sources of: creatine, Vitamin D, Vitamin C and Zinc, tyrosine and potassium. Mouthwashes with over >200ppm of fluoride are widely available, but you should check the label.

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