Kenneth Arrow Makes the Climate Change Math Work

It falls to an uber economist like Nobel Prize winner Ken Arrow to look at climate change and make the economic case for prevention work without resorting to smoke and mirrors. This is a non-trivial accomplishment.

For those who have not followed this aspect of the debate, one Sir Nicholas Stern prepared the so-called Stern Report at the request of the British government. It was the first effort by an economist to analyze the consquences of global warming. His report concluded that action was warranted: failing to take corrective measures would lead to a reduction in world GDP of 5% to 20% (this is not one time, but ongoing) while taking action now would cost 1% of GDP.

That seems clear cut, until you learn that Stern had to fudge the math, using what most would consider to be a ridiculous discount of 0.1%. Why? At least when he prepared his study, most estimates anticipated that Really Bad Outcomes wouldn’t occur till, say, 2100 (of course, that was before the forecast that Arctic summer ice would be a thing of the past by 2013). Any event 50 years or more out at any reasonable discount rate has zero present value. Hence the need to find some way to allow for the fact that our actions might end civilized life down the road (the philosophical issues are discussed in some detail in the report).

Other economists who have looked at the report (this discussion of a seminar among heavyweight economists is typical) tend to have similar reactions: they deplore Stern’s methodology, but nevertheless endorse his conclusion. This seems odd, since economists are supposed to be able to parse problems just like this with math. Is this just a quasi-religious discomfort, that no one wants to be the one who said, “Go ahead, pollute, future generations be damned” because that’s what the models would endorse, or is something missing in how everyone has been looking at the question?

Ken Arrow would say the latter, and he has found the missing element in the calculus: additional consumption does not have a linear value. The story that describes Arrow’s logic is somewhat technical. Consider it intuitively: the more you have, the less valuable incremental stuff is. Arrow doesn’t mention it, but the curve no doubt flattens out once one’s reasonable needs are met and one has a safety buffer (studies of happiness repeatedly confirm this pattern, that once people have reached this level, more goods have little to no impact on happiness. But marketers in advanced economies have gotten very clever at playing on status competitiveness).

Arrow alludes to well established norms regarding the declining marginal value of consumption, and concludes that investing 1% of GDP per year in cutting greenhouse gases in a good investment.

Now we need someone to translate Arrow’s logic into sound bites.

From the Taipei Times (hat tip Mark Thoma):

Last fall, the UK issued a major government report on global climate change directed by Sir Nicholas Stern, a top-flight economist. The Stern Review Report on the Economics of Climate Change amounts to a call to action: It argues that huge future costs of global warming can be avoided by incurring relatively modest cost today.

Critics of the report don’t think serious action to limit carbon dioxide emissions is justified, because there remains substantial uncertainty about the extent of the costs of global climate change, and because these costs will be incurred far in the future.

However, I believe that Stern’s fundamental conclusion is justified: We are much better off reducing carbon dioxide emissions substantially than risking the consequences of failing to act, even if, unlike Stern, one heavily discounts uncertainty and the future.

Two factors differentiate global climate change from other environmental problems.

First, whereas most environmental insults — for example, water pollution, acid rain, or sulfur dioxide emissions — are mitigated promptly or in fairly short order when the source is cleaned up, emissions of carbon dioxide and other trace gases remain in the atmosphere for centuries. So reducing emissions today is very valuable to humanity in the distant future.

Second, the externality is truly global in scale, because greenhouse gases travel around the world in a few days. As a result, the nation-state and its subsidiaries, the typical loci for internalizing externalities, are limited in their remedial capacity. (However, since the US contributes about 25 percent of the world’s carbon dioxide emissions, its own policy could make a large difference.) Thus, global climate change is a public “good” — as defined in economic terms — on an enormous scale.

Cost-benefit analysis is a principal tool for deciding whether altering it through mitigation policy is warranted. Two aspects of that calculation are critical. First, it has to be assumed that individuals prefer to avoid risk. That is, an uncertain outcome is worth less than the average of the outcomes; that is avoiding an uncertain outcome is worth more. Because the possible outcomes of global warming in the absence of mitigation are very uncertain, though surely bad, the uncertain losses should be evaluated as being equivalent to a single loss greater than the expected loss.

The second critical aspect is how one treats future outcomes relative to current ones — an issue that has aroused much attention among philosophers as well as economists. At what rate should future impacts — particularly losses of future consumption — be discounted to the present?

The consumption discount rate should account for the possibility that, as consumption grows, the marginal unit of consumption may be considered to have less social value. This is analogous to the idea of diminishing marginal private utility of private consumption, and is relatively uncontroversial, although researchers disagree on its magnitude.

There is greater disagreement about how much to discount the future simply because it is the future, even if future generations are no better off than us. Whereas the Stern Review follows a tradition among British economists and many philosophers against discounting for pure futurity, most economists take pure time preference as obvious.

However, the case for intervention to keep carbon dioxide levels within bounds (say, aiming to stabilize them at about 550 ppm) is sufficiently strong to be insensitive to this dispute. Consider some numbers from the Stern Review concerning the future benefits of preventing greenhouse gas concentrations from exceeding 550 ppm, as well as the costs of accomplishing this. The benefits are the avoided damages, including both market damages and non-market damages that account for health and ecological impacts.

Following a “business as usual” policy, by 2200, the losses in GNP have an expected value of 13.8 percent, but with a degree of uncertainty that makes the expected loss equivalent to a certain loss of about 20 percent. Since the base rate of economic growth (before calculating the climate change effect) was taken to be 1.3 percent per year, a loss of 20 percent in the year 2200 amounts to reducing the annual growth rate to 1.2 percent. In other words, the benefit of mitigating greenhouse gas emissions can be represented as the increase in the annual growth rate from today to 2200 from 1.2 percent to 1.3 percent.

As for the cost of stabilization, estimates in the Stern Review range from 3.4 percent of GNP to -3.9 percent (since saving energy reduces energy costs, the latter estimate is not as startling as it appears). Let’s assume that costs to prevent additional accumulation of carbon dioxide (and equivalents) come to 1 percent of GNP every year forever, and, in accordance with a fair amount of empirical evidence, that the component of the discount rate attributable to the declining marginal utility of consumption is equal to twice the rate of growth of consumption.

A straightforward calculation shows that mitigation is better than business as usual — that is, the present value of the benefits exceeds the present value of the costs — for any social rate of time preference less than 8.5 percent. No estimate of the pure rate of time preference, even by those who believe in relatively strong discounting of the future, has ever approached 8.5 percent.

These calculations indicate that, even with higher discounting, the Stern Review estimates of future benefits and costs imply that mitigation makes economic sense. These calculations rely on the report’s projected time profiles for benefits and its estimate of annual costs, about which there is much disagreement. Still, I believe there can be little serious argument about the importance of a policy aimed at avoiding major further increases in carbon dioxide emissions.

Print Friendly, PDF & Email


  1. Ronnie Horesh

    I can’t help thinking that the cost-benefit method, in being applied here, has little relevance, or meaning. For one thing, it’s being applied to the so-called costs of mitigation, but the uncertainties extend to the relationship between ghg emissions and climate change. I’d prefer to see climate change tackled directly and in fact targeting climate stability, rather than gas emissions, and subordinating policy to that would make life much easier. I suggest agreeing on a target for climate stability, and contracting out the achievement of a stable climate to the market. This means no prejudging of how best to achieve the goal: mitigation might be necessary, but there will other, possibly more efficient solutions, as yet unthought of by those who would commit us to gas reductions and only gas reductions. (More info on

  2. blueskies

    One of the main problems is that nobody knows how to achieve climate stability. Carbon dioxide is a greenhouse gas, if there is more then all other things being equal temperature will go up. The basic physics is clear. But worldwide climate modeling is not a solved problem. Models in use today do not explain the little ice age, or the medieval warm period. So, if a model does not match the past how do you use it to predict the future? you dont (if you are honest). Thats why the 13.8 number in the post made me cringe, 3 significant digits that is ridiculous. If you increase carbon dioxide you are conducting an experimient, you don’t know what will happen. The argument should be “lets not experiment on climate”. Next should be a discussion of PRACTICAL ways to reduce carbon dioxide. The list here is short, allow lighter cars to be sold (eg smart car), and use as much nuclear power as possible. The safety lobby blocks the first, and the environmental lobby blocks the second. If economists apply math to the issue they are wasting their time, the input values are not known. In the end they will not come up with an argument better than lets not experiment. Interesting that Arrow scores uncertain outputs as undesirable. He should just stop there.

Comments are closed.