In looking at the cost-effectiveness of health interventions in fantasy communities we have shown that the infinite lifespan of elves creates analytical problems, and other commenters have suggested that the cost-effectiveness of clerical interventions to reduce infant mortality should be balanced against the need for clerics to go to war. Well, Professor John Quiggin at the Crooked Timber blog recently broached the issue of doing a benefit-cost analysis of US military spending, and has found that the US defense department has killed a million Americans since 2001. His benefit-cost analysis is really just an exercise in peskiness, though it does have a valid underlying point, and I think actually you could show with a simple cost-effectiveness analysis that the wars of the last 10 years have, under quite reasonable assumptions, not been a cost-effective use of American money. Of course, we don’t make judgments about military spending on cost-effectiveness or cost-benefit grounds.

In comments at Crooked Timber[1], I listed a few examples of how US Defense Department money could be better spent, and one of those examples was vaccination. Obviously, disease eradication would be a very good use of this money, because of its long-term implications, but in thinking about the cost-effectiveness (or cost-benefit) of this particular intervention, I think we can see another clear example of how these purely economic approaches to important policy debates just don’t work. So, here I’m going to look at this in a little more detail, and give some examples of how we can come to outrageous policy conclusions through looking at things through a purely econometric lens. I think I came to this way of thinking by considering the cost-effectiveness of interventions in elven communities, and ultimately it’s relevant to the debate on global warming, because a common denialist tactic is to demand that AGW abatement strategies be assessed entirely in terms of cost-benefit analyses, which are very hard to do and, as one can see from the comments thread at Crooked Timber, are anathema to supporters of the military establishment. As we can see here, they also break down in quite viable real-life circumstances.

The Problem of Disease Eradication

So, you’re the US president in 2001, and you’re reading a book on goats to some schoolkids, and as happens in this situation, you have to make a snap decision about how to spend 200 billion US $ over the next 10 years. You could spend it going to war with a small nation that harbours terrorists; let’s suppose that if you don’t your country will be subject to one 9/11 -style attack every year for the next 20 years (until OBL dies). If you do, you’ll commit your own and the next administration to spending 200 billion US $. Is this a good use of your money? 200 billion US $ to save about 50,000 US lives over 20 years, minus the casualties (wikipedia tells me it’s about 5000). So you get a net benefit of 45,000 lives, or 4,444,444  US $ per life – this actually comes under the US government’s 5 million US$-per-life-saved threshold, so it’s a viable use of your money. But one of your alternatives is to spend the money on eradicating HIV using a vaccine that was recently developed, and it has been shown that by spending 200 billion US$ over 10 years you could eliminate HIV from the face of the earth. You don’t care about the face of the earth, but you need to eradicate it everywhere to make Americans safe from it. Should you ignore the terrorist attacks and spend the money?

For a standard cost-effectiveness analysis you would calculate the incremental benefit (in lives saved) from this vaccine compared to the war on terror. Lives saved in the future are discounted at a fixed rate (usually about 3% a year) and decline in value over the term of the intervention. But the problem with this calculation for disease eradication (specifically) is that the term of the intervention is infinite. All future lives saved forever go into the calculation. The actual formula for this calculation is the integral over (the negative exponent of (discount rate*time t)) multiplied by (lives saved at time t)[2]. Usually we model a policy over 20 or 30 years, giving a finite result; but in this case we need to model the benefit over all future time, and the integral of any bounded function multiplied by the negative exponent, over an infinite range, is infinite. So even with furious discounting we get an infinite benefit from eradicating any disease. Not only does this make comparing disease eradication decisions – e.g. smallpox vs. HIV – impossible, but it makes comparing disease eradication to any other policy objective impossible, and it tells us – quite reasonably, I should say – that we should bend all our health care resources to this task.

In this case, the president of the USA should decide not to go to war because 20 September 11ths are a small price to pay for the eradication of HIV. Eventually Osama bin Laden will give up[3]; HIV won’t. But the stupidity of this decision doesn’t end here. If it costs a trillion dollars to eradicate HIV, the president would be better off defunding his army and paying the price than not; and if Mexico were to invade, killing a million Americans, the infinite benefit of having eradicated HIV would still outweigh the loss.

Now, one argument against this logic is that you shouldn’t include the yet-unborn in a policy evaluation; yet this is standard practice. For example, in considering the cost-effectiveness of different interventions to reduce HIV transmission, we might run a model on the 15-64 year old population, and when we do this we allow for maturity into and out of the population; if we run the model for more than 15 years we are implicitly allowing the yet-unborn into the model. Furthermore, you could surely argue that modeling disease eradication without including the unborn devalues the whole concept – what is disease eradication except a project to protect the unborn generations of the future?

So we can’t use econometric analyses by themselves to assess the value of interventions, because a perfectly reasonable economic analysis of a valid healthcare goal throws up an impossible contradiction. The world expects – with significant help from Bill Gates, I might add – to eliminate polio by 2015 and with the recent announcement of a vaccine for malaria you can bet that the international health movement will turn its gaze on that bastard protozoan next. And there is no economic argument you can mount against spending money on it – even if the cost is everything you own.

Implications for the Global Warming Debate

A common argument mounted by “hard-headed realists” and AGW deniers is that money spent on AGW mitigation needs to be justified by a solid cost-benefit analysis, because the alternative is to spend this money on targeting real problems now, especially in third world countries (often also the countries most vulnerable to AGW’s worst effects). Money spent on infant mortality now, they argue, is far better than money spent on AGW mitigation in the future – even if you accept that the negative effects of AGW are a certainty. This is a particularly powerful argument since we don’t have solid evidence for exactly how bad the effects of AGW will be, and we know that the future benefits of reducing infant mortality now are huge. This economic defense will usually also depend on discount rates – we’re much more interested in lives saved now than in the future, and AGW mitigation’s effects will be felt in the future, not now. Exactly what the relative benefits of mitigation will be are very sensitive to discount rates.

In this case, though, one can argue: well, let’s spend the entire defense department’s money on eradicating HIV. If we test everyone in Africa every 6 months – surely possible with the full funding of the US military on the case – and treat them immediately (or, hey, just treat everyone in Africa with anti-HIV drugs for the next 30 years – let’s put them in the water!) then we can eliminate HIV, and save an infinite number of lives. It’s guaranteed on both cost-benefit and cost-effectiveness grounds, with the added benefit that you don’t need to quibble over the discount rate – it’s guaranteed to be cost-effective under any finite discount rate. The natural argument against this will be that someone might invade America. But we can say in response to this, “uh uh! Precautionary principle! You don’t know how bad that invasion will be or even if it will happen.” If the precautionary principle doesn’t apply to the putative risks of AGW, why should it apply to defense? Or rather, if we need to attach a monetary value to the future risks of AGW, why not attach one to the future invasion of the USA? And when we do, it will be of lower value than the benefits from elimination of HIV, even if the entire population is wiped out during the invasion.

Which brings us back to the simple problem that we can’t assess any policy in isolation using only the econometric tools at our disposal. Everyone understands this, of course, which is why people on the Crooked Timber thread are bridling at Professor Quiggin’s analysis. They attach additional, non-economic considerations to these problems. But one of the rear-guard actions of the anti-AGW movement is to demand that we use exclusively economic methods for assessing the value of AGW mitigation – and it was in response to this fiction that the Stern review was commissioned. I think it needs to be recognized that these econometric tools offer false clarity, and only apply within a very limited framework, that of limited improvements in a limited temporal framework (pacemakers vs. aspirins, essentially). Defense, disease elimination, and AGW mitigation lie outside that framework. This should be abundantly clear to anyone who has tried to do a cost-effectiveness calculation of the relative merits of slavery and genocide for elven communities. It’s just a shame that most economists haven’t bent their mind to these truly important questions; fortunately, we at the C&C University are here to help with the more profound philosophical questions. No, don’t thank me, we do it for free. Or, alternatively, pick apart the argument in the comments … I’m eager to hear how a valid mathematical framework can be constructed for the analysis of disease eradication goals, because it’s relevant to my work…


Actually while I was watching a band in Kichijoji at 3am last night I realized that my interpretation of the formula for total effectiveness in the disease eradication was wrong[5]. Ultimately, the benefits that accrue from disease eradication are approximately (1/(discount rate))*average number of lives saved in any year. So for a discount rate of 3% and 1,000,000 lives saved per year from (e.g. ) eradicating malaria you would get a total benefit of about 33 million. It’s not infinite but it’s very very large. So the general argument holds, but it is possible to compare disease eradication programs. Note that there’s an argument that can be made for a lower discount rate in the case of disease eradication (it is all about saving future generations, not the current generation) and even a small change in the discount rate makes a big difference to the outcome. Also, under certain conditions (exponential population growth bigger than the discount rate) the benefits of disease eradication are infinite; I think most people expect the population to stabilize at 7 billion though so this doesn’t apply on earth.

fn1: for historical reasons I comment there as sg

fn2: or something similar

fn3: Actually it’s an interesting question, isn’t it? If you ignore a terrorist who is incapable of waging a conventional war on you, refuse to give into his demands, mount a purely law-enforcement operation to prevent his worst excesses, and wait him out, how long will it be before he just gives up and goes away? How long can OBL recruit people for if killing Americans leads to … nothing? And if after a few years the US said quietly to the Taliban, “we’ll give you a billion a year in aid if you get rid of him,” how long would it be before he had no safe bases?

fn4: I find this very interesting. A few years ago it was getting hard to find doctors in the west who would perform circumcisions on babies; ten years ago doctors were equivocal on the issue and there has been a long-standing community opposition to circumcision for non-medical reasons; yet now we’re recommending it (and funding it!) en masse in African countries. I wonder how Americans would have felt if, in 1987, Nelson Mandela or Robert Mugabe had come to the USA and suggested that the solution to their growing HIV problem was to circumcise all adult gay men?

fn5: I did this calculation only recently, so I really should have got this right from the start…