Arnold Kling

DeLong, Dasgupta, and Stern

The Distribution of Consumer S... What I Learned at the Tower Re...

On the question of choosing a discount rate to determine the cost of global warming, Brad DeLong chimes in.



Let me see if I can explain the economics to people who find Greek letters intimidating.

Suppose that Robyn Crusoe lives on a large island with a small fruit forest. If Robyn eats all the fruit very year, she can have 1000 pieces of fruit per year. However, if she foregoes eating some fruit and plants it instead, she will have more fruit in the future. Fruit planting is subject to diminishing returns.

(Diminishing returns might mean, for example, that if she plants 1 unit of fruit, she will get an additional 5 units of fruit per year forever starting 15 years from now. But if she plants 10 units of fruit, she will only have an additional 20 units of fruit starting 15 years from now.)

If Robyn follows an equal-fruit strategy, she will not plant any fruit, and she will eat the same amount every year. At the other extreme, if she follows a total-fruit strategy, she will plant all of her fruit, and eat nothing this year. The first extreme corresponds to a social discount rate of infinity, and the second extreme corresponds to a social discount rate of zero. Most likely, her discount rate will be somewhere in between those extremes, and she will plant some fruit as well as eat some fruit.

Dasgupta argued that the Stern report was based on a discount rate that was much too close to zero. It looks like it is telling Robyn to eat only 25 units of fruit and to plant the other 975. This seems like an absurdly high weight on the total-fruit strategy and a correspondingly low weight on the equal-fruit strategy.

DeLong is saying: but what if there is spontaneous growth of fruit trees, such that they yield 3 percent more each year without Robyn planting any new trees? In that case, even if Robyn leans toward a total-fruit strategy as Stern assumes, there will be so much more fruit in the future that the small weight she gives to equal-fruit will be sufficient to get her to save only 22 percent of her fruit and to eat 78 percent of her fruit. (As an aside, I do not think it would be easy for DeLong to defend an assumption of pure technical progress as high as 3 percent per year. And I suspect that with that much technical progress we could enjoy lots of GDP with relatively low carbon emissions...but no matter)

My concern is with Stern, Dasgupta, or DeLong playing social engineer and picking a social discount rate that deviates from market interest rates. I think you get unreliable conclusions any time you do that.

If as a social engineer you think that real interest rates of 2 to 4 percent are too high, then this is a huge issue, with or without global warming. You are saying that we need to lean toward a total-fruit strategy regardless, and save a lot more for the future. The form of that saving does not matter--it could consist of reduced carbon emissions (as an investment in the environment), but it equally well could consist of investments in human and physical capital.

Even if the Stern report had nothing to do with global warming, its assumption for the social discount rate has radical policy implications. Implicitly, it argues for an all-out effort to reduce private-sector and public-sector consumption and to increase investment instead.

It seems to me that the idea of engaging in social engineering to address global warming is plenty ambitious. To add to it an even more significant effort of social engineering in order to get people to defer a much larger share of consumption than they would otherwise is very brave.

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COMMENTS (12 to date)
asg writes:

This was a really good explanation, thanks.

flash writes:


Maybe I'm missing something here but if you want people to save more wouldn't you want a higher interest rate. Higher interest rates increase the opportunity cost of spending money. Unless your talking about the interest on consumer loans which lower consumption. Sorry, just a little confused.

Arnold Kling writes:

People's saving depends on how they weigh their required return on saving relative to what the market offers them in terms of returns on saving.

When you say that people want a higher interest rate, that means that they will save more if the market offers them a higher return. But I am treating what the market has to offer as given. In that case, the only way to increase saving is to lower what they *require* as a rate of return to match the market.

So, if people are inclined to require a 3 percent rate of return, the social engineer who wants much more saving has to do things to make them require only a 1 percent rate of return.

Hope that clears it up.

flash writes:

So, basically the social engineer wants to persuade people to save more than they usually would, by changing what they will accept as a rate of return.

Michael Sullivan writes:

Arnold K: since Brad D. isn't answering, would you be willing to explain the parameters of the growth in consumption equation given by brad, that appears to be where Dasgupta is getting his information?

I don't get what eta is supposed to be other than in very hand wavy terms or exactly what the "social discount rate" is supposed to be in order for that equation to make sense. I've heard people describe what they are supposed to be, but I can't see why they would be related in the way described by that equation. Either I'm grossly misunderstanding, or there's some critical math that's missing.

I understood the other equation Brad used quite well from his own post, and it looks like you've explained *that* part here again in simpler terms.

Even just a pointer to the name (if there is one) of this consumption growth equation or a specific discipline/text where it is explained in detail would be very helpful. I couldn't find anything on the web in my 5 minute search except for a few people linking to Brad D.

I follow what you say here, but I can't connect it to the equations. It's obvious that saying we should save 97.5% for the future is foolish, but you haven't really made clear where in the argument this breaks down. Dasgupta (and presumably you) are claiming that it's purely in the assumption of the "social discount rate", and Brad is saying there's a problem with the model. It would also be good to have a clear explanation of what the "social discount rate" is (because it's obviously *not* what most people think of as a "discount rate" which is an opportunity cost and should always be roughly equivalent to the social rate of return, no?)

Without a clear picture of these things, it's impossible to judge which of you is correct. I'm guessing that if I was an econ PhD I would have encountered this equation at some point and know where to look, and it's certainly fair that it isn't your (or Brad's) job to teach a course in your blog. But I know that I have the math and basic Econ background to follow an explanation if there's a textbook that covers this stuff. Any pointer would be greatly appreciated.

M. Simon writes:

What is the value of reducing delay in the system?

Suppose for an investment of x fruits you can reduce the delay from 15 years to 5 years to fruit production?

In any case, as in all control systems, shortening inherent delays adds stability to the system.

Or think of an economy as a preditor-prey oscillator coupled in n dimensions (lots of predators competing with lots of prey). Oh yeah. The gain factor and coupling of each oscillator is not fixed.

Any way the period of the oscillation is an inherent function of the delay. The shorter the delay the less the following error and overshoot. Too much overshoot and the system oscillates. Boom is following delay - demand increasing faster than supply. Bust is overshoot.

M. Simon writes:

Don't you also need to define a consumption function as well to make this work?

How much fruit is required for

1. Fat
2. Adequate
3. Barely inadequate

Then you need to describe the relative worth of each position. As well as the time value of moving from one area to another on the consumption scale.

This could get very complicated.

Of course if you make certain assumptiions....

Arnold Kling writes:

There are two reasons to discount future consumption. One is pure time discounting--all of us would prefer a hamburger today than a hamburger next Tuesday. The other is a motivation to equalize consumption, given future economic growth. That is, if we know that the next generation is going to benefit from better technology, we are entitled to consume a bit more today, in order to be "fair." That is the eta parameter.

This all gets rather messy, both philosophically and technically. I'm not sure where the papers are that discuss it.

Tom Grey writes:

Your example seems weak: "eat only 25 units of fruit and to plant the other 975"

The consumption ratio of 77.5% should be eating 775 and planting 225.

Which still seems too much planting. But how much is Brad or whoever advocating? 22%?
With real interest rates of 2-4%, how much is being saved now? The savings rate to interest rate relationship isn't at all clear to me.

Mikael writes:

I would recommend:
"Frederick, Shane. 2006. Valuing future life and future lives: A framework for understanding discounting. Journal of Economic Psychology"

It can be downloaded from his web page:

catquas writes:

Even if market interest rates work for discount rates give the population, it doesn't seem obvious to me that they would work for discount rates spanning generations. Is there some reason that they would?

garhane writes:

This helps this lay reader to get a handle on this discount business, for which I say thanks. But suppose one starts not with one but 10,000 Robinson's each one on a separate island, and they have been steadily planting,watering and eating their fruit in a traditional way for decades. Then one of them becomes a scientist, starts measuring the well based water supply on one island and announces that due to the numbers to which they have grown (asssume asexual repro) the water is declining and will stop in 50 years. Island sites have been used up. There are no continents.

Now what is the basis of decision making that will occur. I do not think it will be based on economics. It seems more likely that people will start with what they learn from science and go from there to work out abstemption or rationing to maintain the water supply as is, or gradually improve it. But surely they must work by dealing with physical reality and such sacrifice as that dictates, not economic reasoning such as this discount reasoning seems to be based on.

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