In this session, we're going to derive the fundamental Ramsey equation. We're going to determine the social discount factor. Are we going to shoot on what, it depends in a way? There is a key set of results for all the approach that we are doing in this part of the course. And let's remind ourselves briefly in the previous sessions we obtained that the marginal productivity of capital, the derivative of my function F of K. Must be equal to the marginal rate of substitution. It's in current in future consumption. U sub note over U sub one where the subsequent remember being derivative. They don't mean at time zero and Taiwan. And intuitively this means remember that I am happy to forego consumption today. Just to the extent when the future increased consumption, compensate me for the sacrifice that I have made today. Okay, and then we showed other things. We showed that the welfare preserving rate of return on capital must equal many things. The rate of return on the production technology, technology, the rate of default free borrowing. And the internal rate of return on any project in the economy, and therefore a project remains a good project or a bad project, irrespective of how it is financed. So our goal is to find the appropriate discount rate to discount future benefits of damages and to find their present value. Well, this seems easy because we have three quantities and they all should be identical. And therefore, if we can at least estimate one. I'm done, well, let's see if it is really that easy. So let me try first by using the rate that firms used to discount future cash flows. Typically, firms use for the purpose the weighted average cost of capital. Unfortunately, weighted average cost of capital are very high, 10-15%. Well, this rate is very high because it contains a risk premium. The risk premium is there because the rates refer to very risky projects, in the sense that the returns are positively correlated on how well the economy as a whole is doing so. The project we looked at in our economy where such as they assure returns, no correlation with consumption. Therefore, the weighted average cost of capital is not the right number to use, pity because there was a market available rate. Can we do something else? Yes, perhaps we can look, didn't you tell me Ricardo that also the rate of return on riskless debt had to be equal to all these rates. So why don't we use that one? Why don't we use the risk of return on Treasury bonds? Well, the problem is that first of all, they have to be really riskless. And probably the closest thing to riskless is US Treasuries and or perhaps German bunds. But even here we have some problems. The really liquid ones are nominal bonds, not real bond's nominal bonds are not riskless. Our default free but not riskless because they are exposed to inflation risk. So they contain they embed a uncertainty due to inflation and real rate tips in America Link Ear's in Britain and another types in Europe are not particularly liquid. And they do not have the same depth of market from which we can extract this rate as nominal would have. And in any case, when it comes to the discount factors that I need. I need really, really long data discount factors because many of the damages will occur 100 years and more in the future. And we do not have a riskless Treasury bonds for those very long maturities recently. Amazingly, long Treasurys have been issued and I seem to remember there has been a 50-year Austrian Treasury bonds and Mexico. Yes, Mexico has issued a 100-year bond. But I wouldn't with due respect really call Mexican issued bonds as a default free bond. So also, the long rates on Treasury bonds are not the right ones to use to discount future damages from climate change. So what can we do that? Well, we are faced with the trickiest over three rates that we saw in the previous lecture. That is to say, the welfare preserving greater return on capital. But this is challenging because there is no real world market for these rates. Therefore, we have to derive it from first principles, which means from a reasonable utility function from a reasonable parameter. Ization of the utility functions and from assumptions about future economic growth. As we shall see this is not an easy task, but before moving on to this task, let me comment again. Also, the promise that we can extract these rates from market rates is probably likely to disappoint us very severely. Especially as this time of speaking as I'm talking now, prices of all kinds of assets have been severely distorted by quantitative easing all over the world. If you look at real rates in the UK. Real rates for very long maturity as you can impute it from the prices of Link Ear's of inflation linked bonds are negative minus one minus 2%. It is difficult to believe that these are the correct rates and by the way, if these were the correct rates to use for the purpose attend. It would have dramatic consequences because it would mean that we should care about future generations more than we care for us. So let's be very careful when we say, market rates are always the best that we can use. That implies a lot of assumptions about market efficiency that are not necessarily satisfied, especially in at the present moment in time. But therefore, let's go back to the other assumption. So to the other route, getting this discount factor from first principles and we shall see it is tricky but it is not as impossible in my scene. So in order to make progress, we're going to make going to make again the assumption that the utility function is separable. We could do something more complicated with nazi productivity functions, but functions. But let's just get a flavor for the approach. So I have to date economy. So I have the big you utility is a sum of utility today plus discounted utility at time. One as these equation shows. Next question, which separable utility shall we choose? A common and popular choice is the CRRA. The constant relative risk aversion class. This class of function is characterized by the fact that as we grow richer or poorer, investors are equally risk averse about losing the same fraction of their wealth. So Bill Gates and myself have exactly the same dislike, not for losing £1000 they would hurt me immensely more than Bill Gates, but for losing the same fraction of our wealth. And a popular member of this class of utility function is the power utility function that have written here. It is consumption one minus gamma divided by one minus gamma with a minus one on top which is rather irrelevant. And that is used when gamma is different from one. But the reason putting than minus one is that you can check that S gamma goes to one. Then that function goes to the logarithmic utility function that we have already used. So the log arrhythmic utility function that we looked at in the previous session is a special case of this. C Constant relative risk aversion will come is equal to one. So in the appendix I will derive that the equation we obtained in the last session can be generalist to this very similar result. So my welfare preserving rate of return, the rate of return that I should use for discounting, is equal to the sum of impatience. And again I'll go over this term in a minute. Plus the growth in the economy, which we have already seen times gamma. If gamma is one, if I have a utility function is exactly what I obtained before. But in general, gamma should be bigger than one. And typically economists like numbers in the order of 1.5 to 2.5 in that, in that region. There is a Ramsey equation which is the foundation of mainstream economic analysis of climate change intervention. And we use it explicitly when we look at NPV approaches and we use it implicitly when we look at integrated assessment models. Make no mistake, when you're using, not losing. When you're using integrated assessment models, you see explicitly the delta term, but you're also implicitly using Gamma. And why is this equation is so important? Well, this equation tells us that the discount rate we are after is related to the impatience coefficient. The certain growth in the economy and the coefficient gamma that we show in the appendix is equal to a particular ratio of derivatives with the utility functions. But we have not interpreted yet, this is what we're going to do next. And to get this interpretation, let's do closely at power utility functions