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# The Weitzman model

While uncertainty and risk are discussed in both the Nordhaus and Stern models, they take the form of certainty equivalence. Essentially, the way to model risk with a certainty equivalence approach is to add more “cost” to account for more risk. For example, in a model of a project evaluation, if a project has no risk compared with another risk, the investor in the project with risk will demand a higher rate of return on investment. However, the essence of the model does not change. The relationship between the temperature change and CO2 rise is not without uncertainty. This uncertainty is illustrated in Figure 6.4.

Figure 6.4 shows the following. It shows that a 3°C temperature rise and a 550 ppm CO2 level happen with a probability of 0.99 (99 percent), a rise of 5°C and 650 ppm CO2 level happen with a probability of 0.58 (58 percent), and so on. Thus, there is a great deal ofuncertainty associated with any change in temperature that corresponds to a given level of CO2 concentration.

Martin Weitzman proposed the following theoretical argument when there is uncertainty about different future states of the world: if catastrophic events have higher probabilities than standard exponentially decaying probability distributions, then the standard economic models (of calculating costs and benefits) cannot be applied.[1] This result has become known as the “Weitzman’s Dismal Theorem.” This is the consequence of the so-called fat right tail of a probability distribution and catastrophic losses. Weitzman’s own summary of the result and its relevance for climate change was:

The burden of proof in climate-change cost-benefit analysis (CBA) is presumptively upon whoever calculates expected discounted utilities without considering that structural uncertainty might matter more than discounting or pure risk. Such a middle-of-the-distribution modeler should be prepared to explain why the bad fat tail of the posterior-predictive probability distribution function is not empirically relevant and does not play a very significant—perhaps even decisive—role in climate-change CBA.26

To understand the significance of events of large magnitudes, we can consider the example of the distributions of large earthquakes. It is well known that the magnitude of large earthquakes has a fat tail (a power law distribution). Thus, there were four earthquakes in recorded history that were of the magnitude equal to or greater than the 2011 Japanese earthquake that resulted in a devastating tsunami. With a probability distribution with a fat tail, such events are not that rare. Ifwe were to apply the same event and pretend that the distribution is normal, then it would be equivalent to running into a person 30 feet tall.[2] Weitzman’s approach has been criticized by some economists—notably by Nordhaus.[3] More recently, a new approach has been suggested to reconcile Nordhaus’s model with Weitzman’s.[4]

Weitzman, “On Modeling and Interpreting the Economics.”

Figure 6.4 Uncertainty in the relationship between rise of CO2 and rise of temperature

• [1] Martin Weitzman, “On Modeling and Interpreting the Economics of Catastrophic ClimateChange” (2009) 91 Review of Economics and Statistics 1.
• [2] William Nordhaus, “Economic Policy in the Face of Severe Tail Events” (2012) 14 Journal ofPublic Economic Theory 197.
• [3] Nordhaus, “Economic Policy in the Face of Severe Tail Events.”
• [4] Masako Ikefuji etal., “Weitzman meets Nordhaus: Expected Utility and Catastrophic Risk in aStochastic Economy-Climate Model” Discussion Paper No. 825, The Institute of Social and Economic Research Osaka University, Osaka, Japan (December 2011) (accessed March 15, 2013).

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