Reviewing the Stern Review, again

by John Q on December 19, 2006

Following the publication of this piece in the NY Times, I’ve had a string of email exchanges with Hal Varian, cc:ing Brad DeLong in the role of interested onlooker. I was surprised by the NY Times article since it included both a correct statement of the way in which Stern treats discounting and income redistribution (roughly speaking a 1 per cent change in income has the same value whenever it is incurred and whoever receives it) with a lot of statements that were either misleading or downright wrong, implying that the near-zero rate of pure time preference in the Stern Review implied a near-zero discount rate for cash flows.

Since Varian is one of the brightest and most technically careful people in the economics profession, I was unsurprised by the correct statement, but very surprised to see errors I’d already refuted when put forward by Arnold Kling, Bjorn Lomborg, Megan McArdle and others. Email revealed that the main problems arose from editorial attempts to ‘simplify’ things for readers, but we still have a lot of disagreements about the justifiability or otherwise of inherent discounting.

In any case, all this has spurred me on to produce my long-promised review of Stern on discounting, at least in draft form. Read, enjoy and criticise.



Chris Edmond 12.19.06 at 10:34 am


A couple of technical grumbles. The consumption Euler equation formula on page 4 is:

r = \rho + \eta \times g

and you refer to \eta as measuring the “elasticity of substitution for consumption”. But with your notation in this equation, the elasticity of substitution for consumption is the *reciprocal* \eta^{-1}. With your notation, consumers want a perfectly flat consumption path when \eta = \infty (an IES of zero) and are indifferent to the temporal pattern of consumption when \eta = 0 (an IES of \infty). Of course in the log utility case these are equal as you rightly point out. I’m being pedantic because a lot of people already seem to be confused about these things and a typo like this doesn’t help.

Also: why do you give the intuition for the elasticity of substitution in terms of “income”: we are talking about preferences for consumption and my willingness to trade consumption over time is quite different from my willingness to trade income because I can do things with my income other than consume. (Or, technically if you prefer: in a dynamic model the envelope theorem does not give the result that my marginal utility of consumption equals the derivative of my value function with respect to income; instead in a dynamic model the marginal utility of consumption each period equals the derivative of my value function with respect to my net asset position that period). Since a lot of people already seem to be confused by this discussion, I don’t think it helps to be careless with the intuition you’re providing.



aaron_m 12.19.06 at 10:58 am

”one per cent of income now has the same value as one per cent of income at any time in the future.”

Am I missing something? I must be because this just seems to be a huge mistake

You say

“Under the projections used in the Stern Review, average world income in 2100 is estimated at about $US 100 000. Using eta =1, a sacrifice of $70 per person (1 per cent of income) today would be justified if (and only if) it increased the income of our great-grandchildren in 2100 by at least $1 000. If this trade-off appears reasonable, then a value of eta =1 is appropriate.”

You highlight that following the logic of eta=1, $1000 from the more rich to produce a $70 gain for less rich has the same moral value as visa versa. In other words if it is reasonable for the less rich to give $70 to the more rich, then the more rich out also to give a $1000 for the sake of raising the income of the less rich 70$. But the more rich should be willing to give much more than 1% of their income to raise the income of the less rich by 1% because of declining marginal utility.

When do transfers from the poor to the rich make sense?
Obviously we do not say that one percent of my income (lets say $20 000/year) has the same value to me as one percent of the income of an extremely poor person. Lets call our poor person Dave, he makes 200/year. One percent of income for both Dave and I represents a little less than four days income, but the loose of four days of income means that I can’t buy a new Ipod while the loose of fours days of income for Dave means that he will starve to death. So it is not the case that if Dave can raise my income by 1% by giving 2 dollars to the Help the Ipodless Programme (HIP) that he does or very nearly does have a moral obligation to do so. Obviously because of the starvation consequences of 2 dollars Dave will never be morally obliged to give this money to HIP. In fact the only way that it makes sense on utilitarian grounds for Dave to make a transfer to me is if, for example, his giving 1 dollar (i.e. 0.5% of his income) will prevent me from loosing a little less than 90 percent (or something like that) of my income.

Obviously the value of 1% of income for Dave and I are not the same. So why these utility values become equal when the poorer person is asked to make a transfer to a richer person in the future.


aaron_m 12.19.06 at 11:19 am

Oops forgot to include that the $70 is one percent of the current average world income, which is $7000.


aaron_m 12.19.06 at 12:12 pm

and sorry about all the typos


radek 12.19.06 at 5:23 pm

That’s a really nice write up John.
And the Ramsey reference you’re looking for is 1928, Economic Journal, Vol. 38, “A Mathematical Theory of Saving”.


Tom T. 12.20.06 at 10:43 am

I don’t understand why the appropriate measure of intertemporal tradeoffs is exclusively the risk-free bond rate? Any investment is an intertemporal tradeoff, and people as a whole (or society, or the market, if you prefer) invest their money in a variety of vehicles, including some percentage in higher-risk items such as junk bonds or equities. Money that is spent on (or income that is foregone due to) abatement of global warming would presumably be invested in a mixed basket of investments at varying rates of return, so why does your analysis assume that only the risk-free bond rate matters?

There’s a typo on page 5. This sentence is missing a word (perhaps “decrease”): “a policy that made income (not the growth rate of income!) by one percentage point from 2000 to 2050….”


Tom T. 12.20.06 at 10:46 am

Stern’s figure that you cite, of $100,000 average income in 2100, is that assuming that we have invested in global-warming abatement in the meantime, or that we have not?


Hal Varian 12.20.06 at 11:14 am

I disagree that there are “errors” in my NY Times piece. The two passages that are in dispute are the following:

“is it really ethical to transfer wealth from someone making $7,000 a year to someone making $94,000 a year?”

You claim that this “suggests a one-for-one transfer, not taking $1 from the poor person and giving around $14 to the rich person.” However, just a few sentences before I made it clear that I was thinking of percentage transfers.
(However, I agree that I could have reminded readers of that point.)

The other passage that you claim is an error is
the statement “a dollar a year over a million years is a million dollars.” This simply indicates that summing a small sum (of anything) over a long time gives you a big number.

I don’t think that either of these passages could remotely be called mistakes.

But coming back to my critique of the Stern Review, the important point is the one I make at the end: he should have done a sensitivity analysis. This would have made the report much more useful.

I’m happy to see that he subsequently did some sensitivity analysis in a follow up piece, which is definitely a step in the right direction.


Matt Kuzma 12.20.06 at 11:28 am

This isn’t a critique as much as an edit. In the last paragraph before the subsection Expected Utility, the first sentence is a fragment:

“The main focus of discussion of the Stern review has been the way in which future costs and benefits.”


conchis 12.20.06 at 8:38 pm

I’m more of a prioritarian than a utilitarian – which, if I understand things correctly, means that (given equivalent assumptions about the shape of utlity functions) I should be willing to assign an eta greater than 1, because I’m less inclined to transfer utility from people in the (relatively poor) present, to people in the (relatively richer) future.

Does it strike anyone else as interesting that the a greater preference for redistribution should lead us to be less enthusiastic about attempts to mitigate global warming? And that this seems to be precisely the opposite of the way people’s opinions correlate in practice?

(Although cutting against this somewhat are considerations of hedonic adaptation and reference effects, which suggest that, even at significantly higher incomes, people in the future aren’t likely to be that much better of in utility terms, and so transfers of utility to them are more likely to be justified at the margins.)


Michael Sullivan 12.20.06 at 8:58 pm

This is a good writeup john, and I finally get what eta is supposed to mean, although I haven’t yet reasoned through why the consumption equation would hold in the model.

This comment will tackle the moral question about deciding on eta as a utilitarian (or semi-utilitarian).

I see two problems with basic utility theory for these questions. The first is that real utility functions don’t have the nice properties that a function based on eta would expect.

The second is that moral questions have another component, and that is compulsion or social pressure. To me social pressure is a significant cost, and compulsion is a very large cost. If I am willing to give up my $10 for some utility maximizing exercise, not because I believe it is best, but only because of social pressure, then the cost to me is really more than the $10. In the case of compulsion (taxes, theft) the cost is *much* greater than $10.

What this means is that there is a spread between the transfers a poor person could morally be asked to make for a rich person and those a rich person could morally be asked (or compelled) to make for a poor person.

In the real world, I would have a lot of difficulty if asked to give up $1000 so that some poor person could have $70, when I could find a poor person on my own and give them $100, leaving me with $900 of the $1000. Or I could give the $1000 to some efficient charitable organization and expect somewhere between $800 and $900 to end up in the hands of those who are no better off than the guy who would have ended up with $70. Only if I were in a universe where there were no other ways to get my money to the person earning only $7000/year than to accept this mephistophelean exchange which increases global utility by only a tiny fraction would I consider social pressure to do so even remotely justified.

Meanwhile I consider social pressure on the poor person to transfer wealth to the richer unjustified in all cases. Why? Because if some poor soul giving up $70 has the potential to earn me $1000, I ought to be able to profitably offer her $100 to do so and get nearly all the benefit while enriching her by $30 instead of taking away her money.

Again, in the world this is these horribly exchanges are the only possible option to increase global utility, we might consider this exchange worthy, but I still think there’s a middle area where the social pressure would not be moral on either side, because the cost of the social pressure is greater than the utility benefit of the exchange.

When you consider compulsion, that middle area becomes very large. I cannot easily conscience compelling anyone to give up money or resources to those who are richer, even if doing so increases global utility significantly. If there is a utility gain, then there is clearly a large nominal money gain as well, and the transaction should go off when the richer person is willing to pay for it and not otherwise.

Similarly, I find it hard to swallow compelling anyone to give up resources even to those who are poorer, if those poor are well above subsistence level. The costs of compulsion are high enough that I regret imposing them for any but a large benefit, and the marginal utility of persons well above subsistence is not improved greatly enough by income in my estimation to justify this.

Therefore I see no paradox here. The utility function is just talking about money and consumption, all else being equal. It’s not considering “everything else”, and everything else is not at all equal when there is a social obligation, or even more so, a legal obligation for transfer of wealth in one case and not in another.

And the problem with using utilitarian considerations on a societal level is that some sort of obligation (usually legal) becomes necessary to enforce any change in the status quo.

As I’ve long maintained, the ends do justify the means, but only if one has accounted for *all* of the ends, which often include a lot of unintended consequences of the means choice.


aaron_m 12.21.06 at 5:41 am

M Sullivan

1. None of your alternative scenarios for wealth transfers entail a criticism. All utilitarians think providing a poor person 100 dollars at a cost of 100 dollars is better than providing 70 dollars at a cost of 1000 dollars. You change the choice situation to reject the moral demands of the example, but changing the choice situation simply is avoiding the moral dilemma at hand and not at criticism at all. What is at issue here is a situation where we have a choice to transfer wealth to the future or not to do this. And this means we need to spend $70 for $1000 in the 2100 or impose a cost of $1000 on the future so that we can keep our $70.

2. You claim that the negative utility from coercion almost always outweighs the benefits. This claim is not defended and I doubt that many would be convinced if you tried. Ultimately to defend your coercion as hugely negative for utility claim you would need to argue that utility is better maximised in conditions of real anarchy (i.e. not state coercion for property rights but not for public goods anarchy, or ordered and thus socially coercive non-state community anarchy).

3. Although there is a lot to criticise about the way a utilitarian view on justice deals with intergenerational distributive justice we should be aware that the normative claim that “a sacrifice of $70 per person (1 per cent of income) today would be justified if (and only if) it increased the income of our great-grandchildren,” by one percent (in 2100 their income will be $100 000) is MUCH MUCH MUCH more demanding than what utilitarianism would require of us for intergenerational distributive justice. Following the view that 1 percent of anybody’s income now or in the future is equal to 1 percent of anybody else’s income now or in the future regardless of how rich they are we DRAMTICALLY weaken the demands for transfers from the rich to the poor and DRAMTICALLY increases the demands from the less rich today to the more rich tomorrow. I do not know of any serious utilitarian that would accept the view that 1% of income or consumption is of equal utility value to a rich person as it is to a poor person.


John Quiggin 12.21.06 at 8:28 am

‘“a dollar a year over a million years is a million dollars.” This simply indicates that summing a small sum (of anything) over a long time gives you a big number.’

It seems to me that this example can only mislead readers into thinking that Stern discounts cash flows in this way, when in fact Stern’s procedure values an infinite stream of $1 payments at about $50.

As a general point about adding up it’s true of all discounting procedures that a million payments, each with present value $1 will be valued at $1 million. Adding up present values is what discounting does. That’s true whether you’re adding up dollars (as in the example), marginal increments to utility (as in Stern) or time-weighted increments to utility (as in Nordhaus). So I can’t see that the example, correctly interpreted, tells us anything about Stern. Incorrectly interpreted, it suggests that Stern is using a zero discount rate, which is wrong.


John Quiggin 12.21.06 at 8:33 am

“Does it strike anyone else as interesting that the a greater preference for redistribution should lead us to be less enthusiastic about attempts to mitigate global warming? And that this seems to be precisely the opposite of the way people’s opinions correlate in practice?”

It has struck me. The real divide is, as Brad DeLong has pointed out, between those who take the ethical imperatives seriously and those whose real viewpoint is “What I have, I hold”.


lemuel pitkin 12.21.06 at 12:35 pm


Good question. Good answer.


Hal Varian 12.21.06 at 2:57 pm

Here’s the first part of the paragraph.

“Given these assumptions it is easy to see where the large numbers come from. Unchecked global warming will certainly make future generations worse off to some degree. If we add up these losses over all time using a zero social discount rate, we get a large sum.”

This simply makes the point that an undiscounted sum of a small number (of utils, dollars or whatever) will be large if the time period is large.

Suppose I had said: “Let us imagine that the cost of global warming was very small — equivalent in utility to $1 per year. Summed over a million years, this would be a million dollars.”

Would you have a problem with that?


Hal Varian 12.21.06 at 3:09 pm

I went back and look at the column again and it seems very clear that I was talking about discounting welfare, not discounting consumption.
I don’t see how one could possibly think otherwise, when reading the entire document in context.


the “social rate of time discount,” the rate used to compare the well-being of future generations to the well-being of those alive

…chooses to weigh all generations’ welfare almost equally

Unchecked global warming will certainly make future generations worse off to some degree. If we add up these losses over all time…

So, should the social discount rate be 0.1 percent, as Sir Nicholas Stern, who led the study, would have it, or 3 percent as Mr. Nordhaus
prefers? There is no definitive answer to this question because it is inherently an ethical judgment that requires comparing the well-being
of different people: those alive today and those alive in 50 or 100 years.

appropriate policy toward global warming depends heavily on how one weighs the costs and benefits it imposes on different generations.


John Quiggin 12.21.06 at 4:18 pm

The term “social rate of time discount” is used in general, and by the Stern Review in particular, to refer to the rate at which the value of future consumption (not the social value of utility) declines. What you are describing is the pure rate of time preference.

To quote the IPCC Third Report , whose approach is followed in the Stern Review

The ethical approach applies the so-called social rate of time discount, which is the sum of the rate of pure time-preference and the rate of increase of welfare derived from higher per capita incomes in the future.

(emphasis added). So, the claim that Stern uses a social discount rate of 0.1 is wrong. The examples tend, in my view, to imply that this incorrect statement is the one being discussed.

As you say, if the example had been expressed in utility terms, it would have been more helpful.


Michael E. Sullivan 12.22.06 at 3:45 pm


your point 1 is fair for the consideration of global warming. In effect we are forced on a choice. Either we sacrifice $70 to save them $1000, or we are in essence asking them to sacrifice $1000 in order to save us $70. And we are effectively compelling them to do so, so the compulsion issue doesn’t really apply.

I’m thinking this stuff through, and was considering all kinds of potential redistribution, not just that of remediation/non-remediation of climate change.

2. We can have many differences about what the cost of compulsion is, but surely you will admit that there is some cost. People tend to want the things they have. Compelling them to give them up costs them utility over and above the lack of the thing. Compelling people to give things up in the absence of very good arguments and some kind of bought-into due process costs *enormous* utility.

On further thought, it turns out that the cost of compulsion is largely irrelevant in terms of climate change, unless one considers the current generation to have an inalienable right to pour CO into the air that overrides the rights of future generations. If you don’t, then it is just as accurate to say that by failing to reduce emissions, we are compelling future generations to give us a subsidy, and to do so at these very inefficient rates that we would decry if demanded of us to support the current poor. So compulsion is irrelevant to the uutility calculations of global warming question.

3. Where do I find this information on what the utilitarian academy considers the appropriate value of eta for intergenerational transfers? You are making claims about what “utilitarianism” requires, but I didn’t know there was a generally agreed definition. It’s true that log utility (eta =1) is considered a very risk-loving posture, but I’d like to know exactly what utility function you think I should be using instead and who endorses it.

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