As I said, the DICE model is probably the most famous and the best known of the integrated assessment models. Therefore, we're going to look at it in some detail. Also, I have re-implemented the DICE model in its entirety, and therefore, I can discuss many of the results that it produces having done the calculations myself and possibly making some variations on the inputs, etc. Therefore, let's try to understand what type of IAM DICE is. What type of integrated assessment model DICE is. Well, conceptually, in this conceptual framework, the DICE model fits in in the general class of neoclassical growth models. We shall see briefly what this means. We want to understand how climate damage fits into these frameworks, and we want to understand the form of the output produced by the DICE model. As I said, the DICE model belongs to the class of neoclassical models of growth theory that was pioneered by Solow in the 1970s. What is the idea in this class of models? Well, agents invest in capital, invest in education, invest in technology, and so and so. To do all these investments, they reduce consumption today, but capital, education, technology will increase their future consumption. There is a trade-off between how much I should reduce consumption today by investing in capital education technology in order to consume more in the future. By the way, I'm presenting this, it is clear that what I in simplified terms call impatience is going to play a key role. You can call it impatience or when it is not your own generation, let's call it regard for future generations, just to use a simple term. This is going to come from an interplay from sacrifices today against benefits in the future. The key insight is that the increase in the future, possibly expected consumption, must be big enough to compensate for today's sacrifice. How does all this fit with climate change? Well, the DICE model extends this framework by introducing the natural capital of the climate system. Now, the concentration of CO2 in the atmosphere can be seen as a negative natural capital. Therefore, emission reductions plays the role of an investment that increases natural capital or if you want, decreases the deterioration of natural capital. If we invest in emission reduction today, we consume less today, however, is not that we consume more in the future, but we reduce the reduction of consumption in the future. They would be caused by an unbridled growth of climate damage. As in the standard class of models, we have to model the gross output, what the economy produces. The factors that go into the economy are labor, capital, and the total factor of production that I will discuss in a moment. Let's begin with labor and capital. The way they enter to produce the output of the economy is via Cobb-Douglas production function. What does it mean? Well, we see it in this equation here, it says that the y gross, which is the gross output, is given by let's leave for one second TFP. It is the population to the one minus gamma times the capital to the gamma. With gamma is a number between zero and one. What are the properties of these Cobb-Douglas production function? By the way, population and capital are the two factors of production. If we increase capital, we have more production. If we increase the labor force in the DICE model, everybody works, so increasing the population is the same thing as increasing the labor force. If I increase the population and I increase the capital, I increase the production. But there is another term, which is the total factor of production. Now, this term has been introduced in economic growth theory because when we try to account for observed past growth in output, in terms of growth in capital and growth in population, we don't get a full explanation of the growth in output. Therefore, there is this mysterious term, which is the total factor of production that I'm going to give it an interpretation in a minute, but in a rather uncharitable way, it can be described as a fudge factor or a correction factor or something that makes the right-hand side equal to the left-hand side. I should also say that in some formulation, the total factor of production is multiplied together with the population before taking the power one to minus Gamma. In some version, it is total factor of production times population to the one minus Gamma. It doesn't make much difference because if you reshuffle terms, one can be expressed in the function of the other. As I said, a Cobb-Douglas production function implies that there is more output for more labor and more capital. It also implies that if I keep on adding more and more labor, the increasing output is less than linear. Similarly, if I keep on putting more and more capital, the increasing output is less than linear. There is a decreasing rate of increase in output as capital and labor increases. There is another interesting feature. If I increase both labor and capital by the same fraction, the output increases exactly by the same fraction. The mathematical feature is called that the function is homogeneous of degree one. It is very easy to see because in the previous formula that we have just seen, if I multiply both the population and the capital by a constant C, I will have C one to the minus Gamma times C to the Gamma, and therefore one minus Gamma C to C to the Gamma just becomes C. Very easy to see. Let's go back for a moment to this total factor of production, which is a catch-all terms. What does it capture? Well, it capture the technology, the quality of institutions, the education of the labor force, all those factors which are very important, essential, the level of corruption. All these factors which are super important in determining how productive the economy will be, but they cannot be directly described in terms of capital or labor. The fact of a total factor of production has been growing since the beginning of the Industrial Revolution is fundamentally the reason why Malthus law does not apply and why we have been exceptionally in the history of mankind, observing an increased economic growth per person and in end at the same time an increased population since 1750. We have observed that since the start of the Industrial Revolution, the total factor of production has increased. How we've partly observed is too strong an expression. We observe the increase in total output and we say, we cannot account for this total output by increasing capital and increasing labor alone. They explain something but not all of it, and therefore, there must be another factor, this total factor of production, that as a residual has been growing. Therefore the assumption is made in DICE that the total factor of production is going to increase over time. DICE assumes that the rate of increase of a total factor of production is going to decrease over time. It is very, very important to stress that in DICE, the growth in the total factor of production is the main reason why we're going to be so much richer in the future per person. Now, why am I stressing this point so much? Well, the intuition should be familiar to you from what we have seen when we were talking about the social discount factor. Suppose let's say I want to decide when to make a sacrifice. As we have explained, investing in climate change abating initiatives is a sacrifice because it diverts consumption from fun to abating climate change. Now, given a fixed amount of investment for consumption, it makes a lot of sense to pay this price, to make the sacrifice when I'm richer than when I'm poorer, because the same amount will cause a smaller decrease in utility when I'm richer than when I'm poorer. The fact that in the DICE model, we are going to be richer in the future virtually with certainty, I wouldn't say that it bakes some of the answer in the question. But really is a key ingredient in determining the solution that we're going to find. We're going to discuss later. Let's keep these in mind. For the moment, important thing is, according to the DICE model, we are going to be much richer in the future, mainly because the total factor of production is expected to grow over time, albeit at a declining rate. This is indeed, this picture shows the time behavior over a very extended period of time because the DICE model spends 500 years, and it is how the total factor of productivity increases at a slightly declining rate over time from today to after 2,500.