We're reaching an important part of the discussion. Let's remind ourselves a tiny bit of what we obtained. We are trying to get the correct discount rate, and we will obtain the Ramsey's equation that says, the discount rate is equal to any ''impatience'' term, putting impatience in double-quotes, plus Gamma times the rate of growth in the economy. We have discussed that impatient is not really impatient, but it is more or how much a sense of fairness towards future generations. We can describe Gamma as our aversion to feast and famine. Feast and famine, I mean our dislike for having a very uneven stream of consumption. Finally, G was the rate of growth in the economy. But looking at it this way, it is a tricky problem. It is not easy to come to this, but we can split the problem in two parts. One is a factual part, which is the rate of growth in the economy. Terribly difficult to ascertain, but it's list as a fact of nature. Another part which has to do with the way we feel, and in a way only we can say how we feel. One is the way we feel about future generations, and the other one is how much we dislike unevenness of inequality in consumption. I will be showing in future sessions that we can engage in mental experiments that at least help us in bounding these values of dislike for inequality, lack of evenness of consumption, and impatience. This is what we're doing at the moment, but it is undeniable that there is a relatively big range of values for the discount factor which are plausible. What are we to do in the presence of these uncertainty? It is exactly what we're going to do in this session here, and the key results is exactly the fact that we are uncertain as an impact or should have an impact on our decision. Last comment before we get started, even the factual piece of information, the rate of growth of the economy is far from being a constant of nature. After all, throughout the history of mankind, stretching back all the way before the invention of agriculture, we have had virtually zero economic growth until roughly the Industrial Revolution. Growth in population and growth if in economic condition, growth in product per person has been virtually zero throughout human history. From 1750, we began to have some growth and this growth has had a period of exceptional increase in the [inaudible]. The post-war years when we got three percent real in many parts of the world. Now, very few people today would think the three percent real is a reasonable rate of growth to extrapolate into the future, but most economists think that 1.5 percent real rate of growth is a reasonable assumption for projecting economic growth in the future. Well, I don't know if it is, but let us not forget that it is absolutely exceptional in the history of mankind, and it's not that we have understood perfectly how the economy grows. Therefore, we've understood, yes, we know why at this point in time these things became to be different and therefore we had such a growth. As we shall see, there is a cultural factor, the total factual of production that basically reflects our ignorance. The fact that even the factual bit is highly uncertain should be kept in mind in this type of analysis. All of this is to say, there is a reasonable range, let's say for this discount factor from, let's call it one percent and five percent that is a reasonable range of uncertainty. What do we do in the presence of this uncertainty? Well, one way to look at this is a very stylized model which has been created by a Harvard Professor, Vitamin. He says, "Suppose that there are n possible scenarios for how the future might unfold. We don't know which scenario will prevail, but we do know the probability of each of these scenarios." In each of these scenarios, there is one time-dependent discount factor. Perhaps each scenario is associated with the different growth in the economy, and that is how we have a different discount factor in all of these scenarios. I have the possibility of 20 percent of growth at 2 percent and therefore at one time-dependent discount factor, etc. For each scenario, the discount factor today will be close, or rather for small t, will be close to the discount factor today. However, in each scenario, it will approach some other limit depending on the scenario is indicated by r_j star as time goes to infinity. Well, this means that as time goes to infinity, the discount factor will be r_j star with probability P_j. I can define an overall discount factor out to time t, so d is valid for every time t for each scenario by saying that for each scenario is equal to that quantity a_j, which is e to the minus, and then I have the integral of my discount factor over time. Now there is another thing that I can do. That a_j, if you want, is an average discount factor. Scenario specific r to time t. R can do something else. I can define an expected discount factor by multiplying each of the scenario-dependent discount factors by their probabilities. That is the quantity A of t in equation 2. What is the meaning of A of t? Well, suppose that an investment choice with consequences in the future must be made now before we know what the relevant scenario will be. The investment is small relatively to the overall size of the economy, and the uncertainty is uncorrelated with the state of the world. If this is true, an extra dollar of investment at time t is worth A of t expected present dollars. The quantity A of t can be interpreted as the average or expected discount factor. Important to be very clear about how this little model works. When the investment choice is made, we do not know which state prevails. But one second after the choice has been made is a tiny bit like a quantum mechanical collapse of the wave function, I immediately collapse on one of these particular states. Afterward, I know for sure in which state of the world we are, and therefore at that point in time, I am in a rice economy with perfect certainty, etc. In that state, everything is known and deterministic, and we are therefore in the rice economy setup that we have already started. The model is super simple. Some people uncharitable might say simplistic, but it's tractable. What do we get from this? Well, it's not difficult to show that I can define what is called the certainty equivalent instantaneous discount rate, denoted by big R of t, forget about the intermediate equality, look at the last term. The last term says that it is the sum over the discount factors of w_j. Those w_js are not quite the probabilities. Intuitively, one might have expected the probabilities there, but they are what is given in Equation 4, which are the adjusted distorted probabilities. These distorted probabilities are distorted, if you want, by these exponents of integrals here. The certainty-equivalent far-distant-future discount rate is defined as the limit of equation 3 as time goes to infinity. The key question is, what is the value of R star, this certainty-equivalent far-distant-future discount factor, and what does R star depend on? Well, this is easy now to see. Define the lowest possible far-distant-future discount rate. Of all the discount rates that we have, pick out the lowest plausible one. Professor Weitzman at this point proves a big result. He shows that the certainty-equivalent far-distant-future discount rate must equal the lowest possible far-distant-future discount rate, so that R star is equal to R star minimum, which is the minimum of all the plausible discount factors. Remember, R star is the far distant one. Therefore, the interest rate that in the present of uncertainty about discount factors we should use for the distant future should be the lowest possible limiting value. The intuition is pretty easy. The key observation is that all of these results is obtained because we average over discount factors. Over time, the impact of higher discount rates diminishes, because when I take the exponential e^ minus rt and r is big, as I push t to infinity, the discount factors with a bigger R will die off much more quickly than the discount factors with a smaller R. This leaves the field, so to speak, to the lowest discount rate. The logic of NPV then says that an investment Delta should be made today if it yields a benefit at a future time t, such that the discounted value of that benefit, e^ minus rt times t, is greater or equal to the cost today. What does this imply about the optimal form of investment for a very long-term project, such as improving the impact of climate warming when I have uncertainty about the discount factor? Let's face it, uncertainty about discount factor is almost unavoidable. What we have obtained implies that uncertainty about future discount rates provides a strong generic rationale for using certain equivalent social discount factors that decline over time. We should therefore act according to what is optimal in the lowest plausible discount rate case. Therefore, in the presence of a type of uncertainty modeled in this simple approach, the lowest plausible discount factor is the one that we should use for the very distant future. This has implications when it comes to assessing the desirability of investing more today or later on. It basically puts more emphasis on those optimal parts that are obtained with lower plausible discount factors. I conclude this by saying, this is definitely true, but let us not forget that the discount factor that we use should nonetheless remain plausible. It's not that anything I can dream up with a tiny probability is still meaningful is going to help in my decision-making. Once I narrow to a reasonable range of discount factors, these type of results pushes my choice towards the lower end of the plausible discount factors. As we shall see, this has big consequences both in net present value approaches and also when we look at the integrated assessment models that we will be treating in a few sessions from now.