This is a tricky session. And it is a tricky session in a way because it is one of the most important ones, because it talks about the temperature anomaly predicted by DICE. And on the other hand, it could either be an extremely short one in which I just show you one graph, or an extremely long and messy one in which I go through loads and loads of slides with equations, etc. I have decided to do something in between. So it will be on the short side. I have put all the equations on the slides and I have tried to be as clear in the derivation as possible. But I will try to convey to you the intuition behind all these equations, and what they mean for this key variable, which is the evolution of the temperature normally over time. So the DICE model models the temperature revolution using what is called a two-box system. We won't go into details. I'm just going to outline in the simplest terms, the one-box system. Ultimately, what I would like to obtain is something like equation three here, that tells me the change of the heat content. The Q, more precise, should be the entropy of the system over time, which is equal to the forcing, which is, remember, the difference between energy, energy in and energy out minus the terms, lambda dt. Lambda is called the feedback term. And how do I get actually from equation three to something that I can use more readily to determine the temperature? I simply say that my Q, my energy, is equal to the heat capacity, which is how much of the energy changes when the temperature changes, times how much the temperature changes when time changes. So if you look at the equation one, you can see that simply by integrating this equation here, you can immediately work out what the time behavior would be for temperature. And the key element in this equation here is lambda, which is the feedback term. Intuitively, where does the feedback term come from? The most important source of feedback is the increase in vapor pressure as the temperature increases. Now the vapor pressure increases by about 70% per increase in degree kelvin. Warming oceans evaporate more water and a warmer atmosphere can accommodate more water vapor, which is the most important greenhouse gas. This feedback amplifies the warming effect of CO2, of methane, etc. By the way, when you're reading the slice, CH43, it is not a new compound. 3 is just sending it to the footnote. So this feedback term is a large part of a multiplier effect on climate sensitivity. Now, in the DICE model, there is something which is more complicated, which is a two-box model, in which you measure both the upper and the lower part of the atmosphere. I won't go in a detailed discussion, but I have put a detailed derivation of all the terms, because I very much encourage you to look up the DICE model which has been made public, and it is a great, great thing. So Professor Nordhaus, Yale University, has made the DICE model publicly available. And it is thanks to his, in a way, academic generosity that I've been able to reproduce it myself and started it myself. And it is not difficult to code it yourself. It is a few 100 lines of code. And in that case, if you want to do that, you have to understand all the terms. So if you want to do these super useful exercise, that is why I explained in the slides exactly where all the various coefficients come from. However, for the purpose of this discussion, we can remain at a more qualitative level. So if I have a integrated assessment model such as DICE, and I calculate the temperature anomaly over time, this is one typical profile along an optimized path. So I start from today's temperature anomaly which is between half a degree and one degree. And the 100 is 100 years after today. So as I speak, it is in the first part of 2100, so let's call it 2120. And as you can see, according to simulation, we will be reaching the maximum in temperature anomaly, at least along this optimized path, around a bit more than 100 years from now. And we can also look at the temperature anomaly by the end of the century which is at 80. Look 80 on the x axis and it is about 3.5 degrees. Now this sounds pretty high. We are a long way away from the 2 degree or 1.5 centigrade degree scenario. How come, is it impossible to achieve such a scenario? In the case of a DICE model, it would be possible to remain within a 1.5 or 2 centigrade. However, given the assumptions in the DICE model, it is not optimal to do so. And one of the key elements of a DOCE model in obtaining this conclusion, in the DICE model, there are no tail events. In the DICE model, there is no possibility of abrupt climate change and of damages beyond what is in the run of the mill, business as usual, damage function. The damage function in the DICE model makes the damage depend quadratically on the increase in temperature. This is consistent with the best scientific evidence that we have at the moment for the range that everything is going without abrupt changes. So this has to be taken with a pinch of salt. Perhaps one interesting exercise would be to see to what extent does the optimal temperature anomaly change if we insert in the DICE scenarios, the possibility of abrupt climate changes. And the second ingredient, if you want, in the DICE model that is responsible for these optimal temperature anomaly path is the fact that almost with certainty, we are going to become richer, and richer, and richer, in the next 100 years and beyond that. And then, the logic of utility function that says, given a certain fixed sacrifice, it will cost me much less. It will create a much smaller decrease in utility if I make this sacrifice later on then earlier on. And therefore, this makes the abatement schedule is not as steep as what, for instance, Western Review would recommend. So explaining this temperature anomaly path this way comes from two essential features. The first one is the absence of abrupt climate changes. And the second is the fact that human society is expected, and I hope Professor Nordhaus is right, to be richer, and richer, and richer, as we go on, irrespective and despite of the damage inflicted by climate change. As I said, I hope it's right, and the evidence of the last 100 years makes us hope that that might be the case. However, as I discussed elsewhere in other sessions, it is not a dead cert, it is not effective nature, we perhaps shouldn't rely on it as strongly as sometimes we do. [MUSIC]