I love the first sentence of this abstract. A paper on what climate models can tell us. http://web.mit.edu/rpindyck/www/Papers/Climate-Change-Policy-What-Do-the-Models-Tell-Us.pdf … (viaIntrigued I followed the link to the paper, whose title is "Climate Change Policy: What do models tell us?" and the first sentence is "Not very much." What caught my attention beyond the simple humor was it was written by Robert Pindyck who has had an influence on my academic career. In the late 1990s I was working for a US oil company struggling in an environment of $15/bbl - orange juice was more valuable at the time. Representatives from a well known consultancy proposed employing a new technique to value the company's undeveloped resources that would enhance the firm's balance sheet, the technique was known as "Real Options", and employed ideas from stochastic control to assess the value of real assets. During the course of the presentation the engineering director asked the questions "So when would it be optimal to develop the fields" and the consultant replied "Either immediately or just before expiry of the exploration license". This made no sense to the managers and I was asked to explain why the highly paid consultant came up with such as seemingly dumb (and actually dumb) answer and the standard text at the time was Pindyck's book. While Pindyck moved swiftly on from Real Options theory I became interested in the underlying mathematics and my PhD was substantially on how the structure of the payoff, discounting and driving stochastic process determine the structure of the optimal control strategy.~~@~~RogerPielkeJr)

Because I had this association with Pindyck I read the meat of the paper. What I discovered was that economic assessment of climate change comes down to calculating a "social cost of carbon" and this typically involves building "integrated assessment models" of the future impact of current green-house gas emissions, and in doing this the IAMs "discount" the future "cost" of climate change to give it a current value.

This is bread and butter to a financial mathematician: how do we value an asset with an uncertain value in the future, and I found Pindyck's discussion interesting because it touched on a number of topics I ponder on

We can begin by asking what is the “correct” value for the rate of time preference, (delta)? This parameter is crucial because the effects of climate change occur over very long time horizons (50 to 200 years), so a value of (delta) above 2 percent would make it hard to justify even a very moderate abatement policy. Financial data reflecting investor behavior and macroeconomic data reflecting consumer and firm behavior suggest that is in the range of 2 to 5 percent. While a rate in this range might reflect the preferences of investors and consumers, should it also reflect intergenerational preferences and thus apply to time horizons greater than 50 years? Some economists (e.g., Stern (2008) and Heal (2009)) have argued that on ethical grounds (delta) should be zero for such horizons, i.e., that it is unethical to discount the welfare of future generations relative to our own welfare. But why is it unethical? Putting aside their personal views, economists have little to say about that question. I would argue that the rate of time preference is a policy parameter, i.e., it reflects the choices of policy makers, who might or might not believe (or care) that their policy decisions reflect the values of voters. As a policy parameter, the rate of time preference might be positive, zero, or even negative. The problem is that if we can’t pin down (delta) , an IAM can’t tell us much; any given IAM will give a wide range of values for the SCC, depending on the chosen value of (delta) .Firstly I am interested in how choices made by modelers determine the answer: a good mathematician is one who instinctively knows how to formulate a problem so that it is tractable, for example exponential and power utilities are often preferred to log utility for the reason that they are more likely to result in closed form solutions. But more resonant to me was the discussion of the ethics of the discount rate (the "time preference", delta). In Dixit and Pindyck the advice is to determine the discount rate based on the Capital Asset Pricing Model (CAPM), essentially the discount rate is a linear function of the standard deviation of asset returns: high discount rates (bad) are associated with very uncertain returns (bad, also .

The roots of CAPM are in Markowitz's theory which trades off returns for uncertainty represented by the mathematical operator "variance". Markowitz's choice of variance was novel, but not unique. Simultaneously a less well known British mathematician, A. D. Roy, came up with p the same idea, but Roy's background is revealing in why variance (uncertainty) was regarded as a "bad thing". Roy had originally entered Cambridge to study mathematics just before the start of the Second World War. His studies were interrupted by his service in the Royal Artillery, and he fought at the Battle of Imphal in Burma - Britain's Stalingrad, contracting jaundice and being invalided out with what would now be regarded as post traumatic stress disorder. In his Econometrica (Markowitz published the same year in Journal of Finance) article titled

*Safety First and the Holding of Assets*Roy explicitly observed, in language that is poignant coming from someone who had been besieged in a Burmese town for about four months in 1944, that

Decisions taken in practice are less concerned with whether a little more of this or of that will yield the largest net increase in satisfaction than with avoiding known rocks of uncertain position or with deploying forces so that, if there is an ambush round the next corner, total disaster is avoided.The question of portfolio selection was one of balancing the risks of disaster against the opportunities for reward. This approach is now canaonical but I do not think it is particualry rational: opportunity (and threats) lie in uncertainty; this is why the poor gamble.

In this contemporary paper Pindyck does not explicitly recommend, as others do, that the discount rate should be set by the 'market', though he does quote the market rate. Rather, in rejecting the 'ethical' position that the discount rate should be set to zero, he suggests the rate is a "policy parameter" that can be set arbitrarily by policy makers. In making this observation Pindyck is presenting himself as what Roger Pielke Jnr defines as a "Pure Scientist", someone who stands above the murky process of policy decisions, and in doing so presents climate science as something of a forlorn exercise:

IAMs are of little or no value for evaluating alternative climate change policies and estimating the SCC. On the contrary, an IAM-based analysis suggests a level of knowledge and precision that is nonexistent, and allows the modeler to obtain almost any desired result because key inputs can be chosen arbitrarily.while the climate economist "Pure Scientist" Pindyck describes the "discount rate" is the "rate of time preference", the "Science Arbiter" Nicholas Stern, descibes it as

simply (sic) the proportionate rate of fall of the value of the numeraire used in the policy evaluation.

...number of general conclusions follow immediately from these basic definitions. First,[discount rate] depend on a given reference path for future growth in consumption and will be different for different paths. Second, the discount rate will vary over time. Third, with uncertainty, there will be a different discount rate for each possible sequence of outcomes. Fourth, there will be a different discount rate for different choices of numeraire. In imperfect economies, the social value of a unit of private consumption may be different from the social value of a unit of private investment, which may be different from the social value of a unit of public investment. And the rates of changes of these values may be different too.

A further key element for understanding discount rates is the notion of optimality of investments and decisions. For each capital good, if resources can be allocated without constraint between consuming the good in question and its use in accumulation, we have, for that good, the result that the social rate of return on investment (the marginal productivity of this type of good at shadow prices), ...And it goes on, while people complain about mathematicians obscurifying economic intuition! This text is part of a section entitled Ethics and gives a "positive" justification for a conclusion that I hope to arrive at in an simpler and quicker manner than Stern.

I believe financial economics emerged as an expression of virtue ethics in the face of uncertainty, specifically it is about establishing equality between what is given and what is returned in a reciprocal relationship. In this context I explain interest rates using Poisson's 'Law of Rare Events'.

Possion worked on many of the topics
pioneered by the great Revolutionary physicists, Laplace and Lagrange, and ventured into physics, working
on heat, electricity and magnetism. In respect to probability, and
keeping to Laplace’s division between physical and social sciences,
in 1837 Poisson wrote

*Recherches sur la probabilité des jugements en matiére criminelle et en matiére civile*(‘Research on the Probability of Judgements in Criminal and Civil Matters’). Despite its title, most of Possion’s book was a development of ‘probability calculus’, and according to the historian of probability, Ivo Schneider, after its publication “there was hardly anything left that could justify a young mathematician from taking up probability theory” [__2__, p 203].
The heart of

*Recherches*was a single chapter on determining the probability of someone being convicted in a court, by a majority of twelve jurors, each of whom is “is subject to a given probability of not being wrong” and taking into account the police’s assessment of the accused’s guilt [__2__, p 196]. In order to answer this problem, Poisson needed to understand what has become known as the ‘Law of Rare Events’, in contrast to the Law of Large Numbers [__1__]. Poisson’s starting point was the Binomial Model, based on two possible outcomes such as the toss of a coin, or the establishment of innocence or guilt. De Moivre had considered what would happen as the number of steps in the ‘random walk’ of the Binomial Model became very large, with the probability of a success being about half. Poisson considered what would happen if, as the number of steps increased, the chance of a success decreased simultaneously, so that it became very small.
On this basis, Poisson worked out that if the rate of a rare events
occurring, the number of wins per round, was

*lambda*, then the chance of there being*k*wins in*n*rounds was given by
The Poisson
distribution. Apart from being one of the key models in probability,
along with the Binomial and the Normal, the Poisson distribution has
an important financial interpretation.

Consider a someone lending a sum of money,

using the Law of Rare
Events, we can ignore the second expression, since it is zero, and
for the first, we have that, the probability of no defaults is given
when *L*. The lender is concerned that the borrower does not default, which is hopefully a rare event, and will eventually pay back the loan. Say the banker assesses that the borrower will default at a rate of*lambda**defaults a day, and the loan will last**T*days. The banker might also assume that they will get all their money back, providing the borrower makes no defaults in the*T*days, and nothing if the borrower makes one or more defaults. On this basis the lender’s mathematical expectation of the value of the loan is*k*= 0,

so

The lender is handing
over

*L*with the expectation of only getting
back. To make the
initial loan amount equal the expected repayment, the lender needs to
inflate the repayment by

to make it equal to the
loan amount,

James Bernoulli had identified the number

*e*in 1683 by considering how a bank account grew and as the time between interest payments became infinitesimally small, and in the process solved the fundamental differential equation of science
Notice that neither the
number

*e*, the fundamental differential equation nor the Poission distribution came out of the physical sciences, they all emerged out of the analysis of social sciences and humanities.
This argument explains why a lender charges interest at a rate of

*lambda**a day: the lender is not charging for the use of money, they are equating what they lend with what they expect to get back, just a Pierre Jean Olivi had argued in the thirteenth century [*__3__, p 19], a fact that is crystal clear through the use of Poisson's mathematics, but had escaped philosophers since Aristotle.
I think this mathematical explanation is simpler than Stern's discursive one, but what does it imply for the discount rate employed in climate science? In my formulation the discount rate is determined by the chance that a borrower might default, and it makes the expected amount returned equal to the amount taken. In the case of climate science, the default probability is the chance that humanity is not present to benefit from sacrifices made today, i.e the discount rate is related to the extinction of human beings. Stern, eventually, makes the same point but via a rather precipitous route that eventually justifies it on the basis that other great men have recommended it in the past.

But that is not really the point of this post, the point is as follows. Pindyck adheres to the fact value dichotomy, but in doing so ends up arguing, as far as I can see, that science is impotent to answer the important problems of society. I think an issue related to this position is the belief that "Pure Science" is about Truth, and anything that cannot establish "truth" is not a suitable topic of "Scientific" debate.

I take the Pragmatic position: Truth does not exist. In the face of this fact science is about establishing consensus through communicative action. In this setting science must be sincere (that is it OED 4 "Characterized by the absence of all dissimulation or pretence; honest, straightforward:"). It is at this point I diverge with Stern who I think ties himself in knots (in my humble opinion) because he does not wish to stand up and declare we cannot separate what is from what ought to be.

The popular alternative to Stern's advocacy of a low discount rate and Pindyck's rejection of modelling is to choose a 'market rate'. This is more disingenuous than either Stern's or Pindyck's arguments because while Stern struggles to integrate ethics into the argument, 'market forces' advocates claim to be objective, ignoring the fact that choosing money as the determinant of value is, in itself, highly subjective. Fairness is a more universal concept than money.

In the Pragmatic context, mathematics is not the determiner of truth, but a rhetorical device that assists discourse. This view goes back to Fibonacci's

*Liber Abaci*, which can be seen as being successful because it gave merchants the tools, in Arabic/Hindu numbers and the concept of the algorithm, to communicate the best way of solving commercial problems. This is at the heart of my objection to what I see as Pindyck's approach, he sees the models as useless because they do not provide an accurate answer, I see the models as useful if they are regarded as the focii of discussions about what is important in climate science.

### References

[1] I. J. Good. Some statistical applications
of Poisson’s work.

*Statistical Science*, 1(2):157—170, 1986.
[2] I. Schneider. The Probability Calculus in
the Nineteenth Century. In L. Kruger, L. J. Daston, and
M. Heidelberger, editors,

*The Probabilistic Revolution: Volume 1: Ideas in**History*. MIT Press, 1987.
[3] E. D. Sylla. Commercial arithmetic,
theology and the intellectual foundations of Jacob Bernoulli’s

*Art of Conjecturing*. In G. Poitras, editor,*Pioneers of Financial Economics:**contributions prior to Irving Fisher*, pages 11—45. Edward Elgar, 2006.
If only mathematics had been used as a focus of discussions about what is important in finance, prior to 2008.

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