<p>My conclusion based on reading his paper is that the utility function of an individual or society can't be of the CRRA form presented in the paper. That would indeed lead to scenarios where you could not get out of the bed in the morning, as minimizing the tiniest probability of an enormous risk would warrant infinite sum of money.</p>

<p>I will attempt to explain the mathematics of the paper. First the utility as a function of consumption is of the form:</p>

<p>$$ U(c) = -c^{1-a}, \space a&gt;1$$</p>

<p>This is also called a constant relative risk aversion utility function. Constant relative risk aversion with regards to consumption means that a person prefers a bundle of 1 utils over uncertain bundles with expected value of 1 utils. The same applies to 2 utils to the same degree proportionally. Note that the utility is $-\infty$ when consumption reaches 0. </p>

<p>Now, if you want to calculate the exact utility that a person receives you simply multiply the probability of each bundle with the utility it grants:</p>

<p>$$P_1*U(C_1)+P_2*U(C_2)... = \sum_{n=1}^{\ b}P_n U(C_n)$$</p>

<p>Where P denotes a probability of a given bundle and U(c) is the utility.</p>

<p>Finally, consider a case where there is a possibility of a bundle that has consumption value of 0, with probability greater than zero:</p>

<p>$$ P_x U(c_x) = -\infty $$</p>

<p>It does not matter what the probabilities and utilities of other bundles are as you can't recover from minus infinite utility.</p>

<p>The same principle applies to continuous probability distributions, so just replace the sum sign with an integral and consider case where it doesn't reach 0 probability when c=0.</p>

<p>You may be interested in readin Nordhaus's response:
<a href="http://aida.wss.yale.edu/~nordhaus/homepage/documents/weitz_011609.pdf" rel="nofollow">http://aida.wss.yale.edu/~nordhaus/homepage/documents/weitz_011609.pdf</a> (By the way, he uses the exact same case scenario as I did).</p>

CC BY-SA 3.0
2016-01-05T08:01:53.110Z
Thank you for your answer.
I have read Nordhaus' critique, which is most interesting, as well as Weitzman's answer to those critiques: http://heep.hks.harvard.edu/files/heep/files/dp11_weitzman.pdf
In the end, Weitzman argues (refer to pages 15-16 of the above doc) that asteroids do not qualify for the dismal theorem as their is too much uncertainty about them, nor is there any evidence today that an asteroid killer is going toward us in a foreseeable future. In the opposite, we are very much aware that climate change is happening.
Hector
CC BY-SA 3.0
2016-01-05T11:24:00.687Z
@Hector His refutation doesn't address the underlying math at all, so it's not very credible. Anyone can come up with scenarios with disastrous outcomes that are rare, just because climate change is observed doesn't change that. In the framework, there is of course also a general problem with just assigning welfare functions to people and then forcing them to do things for their own good, this leads to a contradiction (but is again a other topic).
Dole
CC BY-SA 3.0
2016-01-05T15:39:47.643Z
Understood. To conclude on this topic, do you share the opinion that Weitzman points out a meaningful problem (how can we appropriately account for climate change) but doesn't come up with a plausible alternative or do you believe that the 'fat tailness' of climate change outcomes does not require a different treatment than regular events?
Hector
CC BY-SA 3.0
2016-01-05T16:03:45.410Z
@Hector In general he is of course right that fat tailed risks have more potential for damage in dollar terms than non-fat tailed ones (by definition). That damage is expounded if you consider consistently risk averse individuals. I would not necessarily say that what we consider regular events aren't fat tailed, for example a common criticism to VAR analysis is that stock market return/income is fat tailed... So preparing for such risks should be common place in the world of finance.
Dole
CC BY-SA 3.0
2016-01-04T22:30:41.953Z
2016-01-04T22:30:41.953Z
Dole
<p>The so-called "dismal theorem" asserts that we do not appropriately account for catastrophic scenarios which have very small probability of occurrence. It has been studied in details by Martin's Weitzman, notably in his article <em>"<a href="http://scholar.harvard.edu/files/weitzman/files/additivedamages.pdf" rel="nofollow noreferrer">Additive Damages, Fat-Tailed Climate Dynamics, and Uncertain Discounting</a>"</em>.</p>

<p>The articles of Martin Weitzman rely on a large dose of mathematics, and my skills do not allow me to get everything, nor of course to question the reasoning and deductions of Weitzman. His conclusion is broadly that "The take-away message here is that reasonable attempts to constrict bad-tail fatness can leave us with uncomfortably big numbers" (p19)</p>

<p>I would like to know if there are actually any practical implications from this dismal theorem. In particular concerning climate change. I thought so until @Dole <a href="https://economics.stackexchange.com/questions/10050/is-the-international-economic-system-too-chaotic-to-work-towards-carbon-emissi">pointed out</a> that the dismal theorem could also be used to justify trillions of dollars in investment in a anti-asteroid defense system. I would appreciate any insight on the conditions to apply this theorem, if it can ever be used. Any relevant literature would also help me.</p>

<environmental-economics><externalities><climate-change>
Has the "dismal theorem" practical implications?
Hector
# Has the "dismal theorem" practical implications?

(Asked by: username_0 on 2016-01-04)

The so-called "dismal theorem" asserts that we do not appropriately account for catastrophic scenarios which have very small probability of occurrence. It has been studied in details by Martin's Weitzman, notably in his article *"[Additive Damages, Fat-Tailed Climate Dynamics, and Uncertain Discounting](http://scholar.harvard.edu/files/weitzman/files/additivedamages.pdf)"*.

The articles of Martin Weitzman rely on a large dose of mathematics, and my skills do not allow me to get everything, nor of course to question the reasoning and deductions of Weitzman. His conclusion is broadly that "The take-away message here is that reasonable attempts to constrict bad-tail fatness can leave us with uncomfortably big numbers" (p19)

I would like to know if there are actually any practical implications from this dismal theorem. In particular concerning climate change. I thought so until @username_1 [pointed out](https://economics.stackexchange.com/questions/10050/is-the-international-economic-system-too-chaotic-to-work-towards-carbon-emissi) that the dismal theorem could also be used to justify trillions of dollars in investment in a anti-asteroid defense system. I would appreciate any insight on the conditions to apply this theorem, if it can ever be used. Any relevant literature would also help me.

**Tags:** environmental-economics, externalities, climate-change


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## 1 answers

### Answer by username_1 on 2016-01-04. Score: 3 | Accepted answer

My conclusion based on reading his paper is that the utility function of an individual or society can't be of the CRRA form presented in the paper. That would indeed lead to scenarios where you could not get out of the bed in the morning, as minimizing the tiniest probability of an enormous risk would warrant infinite sum of money.

I will attempt to explain the mathematics of the paper. First the utility as a function of consumption is of the form:

$$ U(c) = -c^{1-a}, \space a>1$$

This is also called a constant relative risk aversion utility function. Constant relative risk aversion with regards to consumption means that a person prefers a bundle of 1 utils over uncertain bundles with expected value of 1 utils. The same applies to 2 utils to the same degree proportionally. Note that the utility is $-\infty$ when consumption reaches 0.

Now, if you want to calculate the exact utility that a person receives you simply multiply the probability of each bundle with the utility it grants:

$$P\_1\*U(C\_1)+P\_2\*U(C\_2)... = \sum\_{n=1}^{\ b}P\_n U(C\_n)$$

Where P denotes a probability of a given bundle and U(c) is the utility.

Finally, consider a case where there is a possibility of a bundle that has consumption value of 0, with probability greater than zero:

$$ P\_x U(c\_x) = -\infty $$

It does not matter what the probabilities and utilities of other bundles are as you can't recover from minus infinite utility.

The same principle applies to continuous probability distributions, so just replace the sum sign with an integral and consider case where it doesn't reach 0 probability when c=0.

You may be interested in readin Nordhaus's response:
<http://aida.wss.yale.edu/~nordhaus/homepage/documents/weitz_011609.pdf> (By the way, he uses the exact same case scenario as I did).

> **Comments:**
> **username_0**: Thank you for your answer. I have read Nordhaus' critique, which is most interesting, as well as Weitzman's answer to those critiques: http://heep.hks.harvard.edu/files/heep/files/dp11_weitzman.pdf In the end, Weitzman argues (refer to pages 15-16 of the above doc) that asteroids do not qualify for the dismal theorem as their is too much uncertainty about them, nor is there any evidence today that an asteroid killer is going toward us in a foreseeable future. In the opposite, we are very much aware that climate change is happening.  
> **username_1**: @username_0 His refutation doesn't address the underlying math at all, so it's not very credible. Anyone can come up with scenarios with disastrous outcomes that are rare, just because climate change is observed doesn't change that. In the framework, there is of course also a general problem with just assigning welfare functions to people and then forcing them to do things for their own good, this leads to a contradiction (but is again a other topic).  
> **username_0**: Understood. To conclude on this topic, do you share the opinion that Weitzman points out a meaningful problem (how can we appropriately account for climate change) but doesn't come up with a plausible alternative or do you believe that the 'fat tailness' of climate change outcomes does not require a different treatment than regular events?  
> **username_1**: @username_0 In general he is of course right that fat tailed risks have more potential for damage in dollar terms than non-fat tailed ones (by definition). That damage is expounded if you consider consistently risk averse individuals. I would not necessarily say that what we consider regular events aren't fat tailed, for example a common criticism to VAR analysis is that stock market return/income is fat tailed... So preparing for such risks should be common place in the world of finance.  

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