A critique Robert Pindyck's model to estimate maximum plausible willingness to pay to prevent global warming

So I've been completely buried with finals and grading and PhD prelim exams and putting my tenure packet together and working with a visiting out-of-town colleague on a old paper we really need to finish up, going to a wonderful Bioenergy Camp in Perry Iowa (muchos gracias to Jim Bushnell), and a little surfing break in Costa Rica....etc..etc...

Despite all this, I've long been meaning to write a critique of Robert Pindyck's seminar at Duke a few  weeks ago.  The talk was about the model he uses to grapple with uncertainty about climate change and its impacts, and what this uncertainty implies about the most we, as a planet, should be willing to pay to curb greenhouse gas emissions.

For the record:  I really like Pindyck's articles and books.  There are few economists who write theory as clearly and elegantly as he does--you can learn both the math and the economics from him.  With Pindyck, there is no "lets use math to obfuscate how shallow my ideas really are."  He's truly one of the best.

So maybe I'm hanging myself by writing this, but despite my general admiration of his work, I think Robert Pindyck is naive when it comes to thinking about uncertainty and climate change. (You can find one version of his paper here).  There is a bit of math in his model so, for the less wonkish, I'll try to keep things relatively intuitive in my critique.  And I will try to present his key ideas fairly.

Also for the record: I don't disagree with Pindyck's broad policy conclusions.  I too would feel extremely wary about spending more than 2% of GDP curbing greenhouse gas emissions.  However, I also think we could bring CO2 emissions close to zero pretty quickly if we spent 2% of GDP in an efficient manner.  But his model doesn't address costs, only maximum willingness to pay to prevent severe global warming.  Fair enough; it's challenging enough to just think seriously about the demand side of curbing CO2 emissions.

Pindyck's model is, I think, mainly a response to Weitzman, who shows how easy it is to justify spending almost all our current income combating global climate change.  Note that Weitzman is not advocating such a policy (nor would I!); he's just trying to sketch out the possibilities, and in doing so, pouring a very cold glass of water on how we economists might like to answer this question.  He basically concludes that cost-benefit analysis, as applied to this problem, is hopeless for determining the optimal amount to spend to curb greenhouse gas emissions.  I don't know that I accept Weitzman's conclusion. Yet.  But I'm feeling cold and wet.

So, in a nutshell, Pindyck takes a pretty neoclassic model of a representative global individual and plugs into that model two kinds of uncertainty:

(1) Uncertainty about the amount temperatures will actually change due to human activity (CO2 emissions);

(2) Uncertainty about the effect of warmer temperatures will have on global economic growth

I like the way he models uncertainty, with simple parametric distribution functions that are easy to manage analytically but also allow for fat tails.

Components of the model that are crucial to the outcome but that are not so transparent are:

(A) The amount of (assumed certain) economic growth that would occur in the absence of climate change;

(B) The nature of the utility function he uses

Factors (A) and (B) are central to the how we balance future potential losses from climate change with current costs of curbing climate change.  This gets at the heart of the so-called 'discount rate' we should use in trading present costs of curbing emissions against future benefits of less warming.

In most models another component of discount rate--a pure rate of time preference--is added to the mix.  But this is usually small and there are fairly persuasive ethical reasons to set it equal to zero.  Pindyck didn't seem to understand these arguments, even though his model embraces them; he seemed to think they have to do with one's individual preference for consuming something today rather than tomorrow, holding all else the same.  The issue is really about whether the utility of someone who is born today is intrinsically worth more than someone born in 50 or 100 years, holding all else the same.  Yes, it matters that people of the future may be much richer (or possibly poorer) and in many ways better off (or possibly worse) than we are today, but that is what (A) and (B) take into account.

For (A), Pindyck assumes a baseline of 2% per-capita growth in consumption worldwide. If one is used to thinking about U.S. growth over the last 100 years this seems pretty reasonable at first blush.  Over the long run, growth has been remarkably steady and just a tad beneath this rate.  For the world as a whole growth may even be a little more rapid.

But in many countries growth has been highly erratic.  Some countries have stagnated or declined for decades or more.  Indeed, even in the U.S., productivity growth was much more rapid 1950-1980 than has been sense.  And individual growth rates vary way, way more than national rates.  Looking more broadly than simple averages, a few lucky ducks experience much more than the average while most experience much less.  This heterogeneity happens to matter a whole lot. Pindyck ignores this heterogeneity and does not acknowledge how important it may be. 

It also matters a lot that he assumes the baseline growth rate is certain.   If he instead modeled the baseline growth rate as a random variable, and assumed that uncertainty interacted in a plausible way with uncertainty about the effect of climate changes on growth (number 2 above)--say, more baseline growth is negatively correlated with impacts from climate change, following from the idea that greater technological advance would make coping with climate change much easier--then I'm confident he would obtain very different and likely much higher willingness to pay to curb warming.

For (B), Pindyck assumes the classic CRRA (that's for Constant Relative Risk Aversion) utility function that many economists use.  This is a nice utility function that is reasonably intuitive and flexible.  But it does have some serious shortcomings. The main shortcoming is that it implies a powerful symmetry between the elasticity of intertemporal substitution and risk aversion that grossly violates real historical investment opportunities. (Sorry, this gets a pretty wonkish.)  To explain the risk premia historically observed on risky assets, we need a very large risk aversion coefficient, which would imply highly inelastic intertemporal substitution.  But highly inelastic intertemporal substitution would imply interest rates much higher and far more volatile that we have observed in history.  Thus, the reality that is our last 100-plus years of financial history strongly violates CRRA utility. 

All of this may seem technical, and it is, but it matters a lot to Pindyck's conclusions. It matters because if Pindyck were to instead use a utility function with high risk aversion and more elastic intertemporal substitution--which is what we observe in real financial data--then his willingness to pay numbers would become much, much larger.  They would become larger still if he were to combine a more realistic utility function with other kinds of uncertainty, like baseline growth, described above.

What utility function should Pindyck use?  One possibilty would be the recursive utility function by Epstein and Zin.  Pindyck made a snyde comment about this utility function in his seminar (he claimed not to have these preferences).  So what about the habit formation model of Cochrane and Cambell (JPE 1999)? Or similarly Ryder and Heal (1973)?  The essential idea is that utility of consumption is relative to expected or recent consumption.  These kinds of utility functionS can be show to roughly fit the facts, unlike CRRA.  But they imply, at times, huge risk aversion and relatively more elastic intertemporal substitution, without the bizarre non-expected-utility recursive structure of Epstein and Zin.

It is not clear to me that these alternative theories are necessarily correct while Pindyck's is wrong.  But the point is that Pindyck is purportedly sketching out the range of possible willingness to pay.  And while he is generous with some of his assumptions, his is clever to compensate generous assumptions in certain respects with much less generous assumptions in other respects.  And relaxing these other assumptions--like baseline growth, certainty of baseline growth, homogeneity in growth, and a utility function that when combined with modest growth pushes marginal values of the future toward zero at an unrealistically rapid rate--could cause his willingness-to-pay numbers to skyrocket.

So while my own subjective judgment would likely put the maximum willingness to pay to prevent global warming in the ballpark of Pindyck's, I think reasonable people could plug numbers into a model not all that different from Pindyck's, but equally if not more realistic, and obtain willingnesses to pay that are much, much larger.  If the point is to place limits on what is plausible, I don't think Pindyck has done so very convincingly.


  1. Hi Michael,

    I haven’t read Pindyck’s paper (and likely won’t), but I am confused by some of your criticisms of his discount rate assumptions.

    You state: “In most models another component of discount rate--a pure rate of time preference--is added to the mix. But this is usually small and there are fairly persuasive ethical reasons to set it equal to zero. Pindyck didn't seem to understand these arguments, even though his model embraces them.”

    Based on this comment, it appears as though you and he are taking different perspectives. I think that you are referring to a social discount rate, which means that you are assuming that a social planner’s WTP is representative of the populations’. It seems that Pindyck is looking at the decisions of economic agents. In this sense, you point may be correct in general but is misplaced in this particular model. All that matters to an individual is her intertemporal rate of substitution. There is no "intrinsic rate of time preference".

    Next: “To explain the risk premia historically observed on risky assets, we need a very large risk aversion coefficient, which would imply highly inelastic intertemporal substitution.”

    Some recent re-evaluations of Mehra’s and Prescott’s results show that the equity premium may not as large as once thought. Some even think that it’s zero (see Falkenbog.blogspot.com for his entertaining opinion and some references). This means that using the CRRA with “standard” levels of risk aversion may not be unrealistic. More to the point, it is the expected level of future risk aversion that is relevant. Although I’ve never seen this claimed, you could probably convince me that the average level of risk aversion has declined in the last 10-15 years.

    All this to say, based on your critique, Pindyck’s conclusions may not be that far off-the-mark. I could be off-base on these comments. Your criticisms are unclear to me however. If I've missed a key argument, please correct me.


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  3. Brandon,

    I suppose I could be a lot more clear. There's a lot of literature here to digest and demystify...

    Pindyck advertises his work as sketching out the most we could plausibly be willing to pay to prevent more than 2C warming. While his model is plausible, I'm arguing that equally plausible models--a few simple, plausible tweaks to Pindyck's model--that would give much lower effective discount rates and much higher willingness to pay.

    There's a lot here and it's not just the equity premium puzzle. And I don't think the puzzle is solved yet. There's a new hypothesis every week, most of which are quickly disproved.

    If risk aversion were even as large as Pindyck assumes on the high end (which is quite low relative to what the data suggest is likely) and the elasticity of substitution were consistent with that level of risk aversion, real interest rates would still be higher and more variable than we observe.

    Pindyck seems to have a answer in mind and is picking his parameters accordingly. It's super easy to make things explode. I don't know about you, but I find that humbling.


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