Let's see what we get when we apply this formula to the case of a power utility function. Remember, I have to calculate my derivative of the utility with respect to consumption. I've written here in one and two, my first derivative and my second derivative. Therefore I have the Gamma there, I can write as a ratio of minus the second derivative over the first derivative times consumption. Obviously, Gamma is one measure of the curvature of the utility function, but there is more being linked to the curvature of utility function. Of course, Gamma is related to the same time, degree of risk aversion. But this is not what we are interested in because there is no risk in my economy. So this is not the interpretation that I am after. There is another interpretation of Gamma, and this is what we are going to develop in the following slides. First of all, I'm showing here different utility functions for increasing value of Gamma. As I increase Gamma, it goes up to 10 here, it becomes more and more curved and we all know that this implies risk aversion, but as I've said, this is not what I'm after. We've seen that the ratio of U_1 and U-naught is a key quantity in our old setup. What does it mean if this ratio were equal to one? It would mean that we care about changing future consumption tomorrow just as much as we care about changing consumption today. That is what would happen if that ratio were equal to one. But what is this quantity equal to in our simple model? To answer the question, consider as usual the separable utility function and look at the ratio. Remember, U_0 is a derivative of the total utility with respect to our consumption today, which is just the derivative of the chosen power utility function with respect to consumption today. Similarly for U_1, but with the addition of Beta, which takes into account of discounting. Now, I have here the derivative of my utility function for level of consumption C_1. I can do a power expansion as for any other function. I can say this function here, don't get confused by the fact that the function is a derivative itself, it is a function and I'm looking at the first-order change. The first-order change in a function is its first derivative. The first-order change in the first derivative is the second derivative. I am expressing here the function first derivative of the consumption evaluated at C_1 equal first derivative of consumption at C_0 plus second derivative times change in consumption. Again, this is nothing different than saying f of x plus dx equal f of x plus f prime times dx. If the growth in consumption is set equal to the initial consumption times the rate of growth of the economy, so if I write that Delta C is consumption times 0 times g, substituting, I have an expression for the derivative of my utility function for the level of consumption C_1, which is given by the last terms in equation 7, which is first derivative evaluated for consumption C-naught plus second derivative evaluated for consumption C-naught times Delta C, which is C_0g. At this point, I just substitute the terms and I obtain that my ratio, U_1 over U-naught is equal to Beta times 1 minus Gamma g. What does this show me? This show me that with a power utility function, the coefficient Gamma plays a double role. It controls the risk aversion, and everybody is familiar with this, but it also controls the rate of intertemporal substitution of marginal utility. This comes from the thing that's on the left-hand side here. I have exactly the marginal rate of substitution of intertemporal utility. Why are we focusing so much on this double role of a coefficient? Because this coefficient Gamma is a tiny bit like a short blanket. If we try to establish its magnitude by seeing how real people react to real or hypothetical bets, we obtain one number, the risk aversion bit. When we find it this way, we find some puzzling values. It is much too small, what we estimate, in order to explain, for instance, equity returns, and this is the well-known equity premium puzzle. On the other hand, is much too big in order to explain the level of typical real rates and there is the short rate puzzles. Should we worry? Should we care? Well, perhaps we should. If we want to use the Ramsey equation in a normative manner, what does it mean in an normative manner? In such a way they should tell us what we should do, then we should take this parameter seriously. This is both what Lord Stern and Professor Nordhaus did, and as we saw, they got completely different results. We will explore their results in much greater detail, when we look at integrated assessment models. But we have already seen that this leads to very different recommendations. One says, act now aggressively, Lord Stern, and the other one's, wait for when we are richer and/or smarter. Well, to get out of this conundrum limbo, let's look at the components of the Ramsey equation ourselves more closely. Here is the equation again, as a reminder. Finally, I'm going to talk about this impatience term, Beta in some detail. Now, typically, we call Beta an impatience style. When it comes to your own consumption today and your own consumption tomorrow or next year or in 10 years time, but it's still you, nobody is better placed than you in order to assess which trade off you should make. A slice of cake today, slice of cake tomorrow. How much bigger the future slice of cake should be for you to be indifferent with respect to having a slice of cake today? Nobody can answer this question better than you can. However, when we're dealing with climate change, we are no longer really dealing with your own slice of cake, but with a slice of cake will be eaten by your children and by your grandchildren and by their grandchildren. Therefore, this brings into play really, really different types of consideration, and a consideration which bring into account damage which will be suffered by future generation, so this parameter Beta becomes entwined with how much we care for future generations. Clearly as we put it this way, we are immediately entering an ethical domain and there has been enormous amount of discussion in philosophical terms, in political terms, even in religious terms. Many philosophers and some economists even claim that it does not make any sense, it shows 'a lack of imagination' to ask for a bigger slice of cake tomorrow, even if the decision involve the same person, but a fore theory even more so, if it entails another person. For what it is worth, very little, my own personal view is that when we are dealing with the welfare of future generations, these impatience coefficients, Delta, should be small, but not zero. I will give some arguments in future sessions to argue why I feel that this is the case. Whatever my inclinations, is there some consensus when people are asked at least about their own impatience? Well, in reality, there is no convergence amongst experts towards an agreed or unique rate of impatience. There have been academic papers. Frederick, Loewenstein, and O'Donoghue here, have conducted a meta-analysis. Meta-analysis means looking at many other studies done by other scholars in the literature and they found estimates of the rate of impatience. They find wildly different estimates varying from zero percent to 70 percent. Therefore, this is not really the way to go if you want to get a handle on this. When it comes to intergenerational discounting, the consensus amongst most, but by no means all economists, is that the pure impatience bit, Delta should be small but not zero. Incidentally, Lord Stern, as we shall see in his report, uses for Delta the very low value of 0.1 percent per annum, which is justified rather arbitrarily by the assumption that there is a 0.1 chance that we might disappear per year, that we might disappear perhaps because of an asteroid hits the Earth. I call this the dancing-on-the-Titanic reasoning, because it seems to be why making sacrifices when we might all die tomorrow anyhow? This 0.1 percent per annum, to me, it looks a bit like a fig leaf to avoid putting a flatter number of zero, but is really neither here nor there. When we put all the pieces together, what we get for the correct day rate which future generation should be discounted, as is depicted in this histogram here to which a function has been fitted about answers given by hundreds of professional economists. We seem to be in a bit of a limbo, can we say something a bit more precise and helpful? For once, a degree of consensus has been achieved between what two economists, Weitzmann and Gollier say about this topic. We turn to this important issue in the next sessions.
