So a brief reminder of what we're doing, where we're going and the path to there. So ultimately, what we want to establish is we want to see when we should act, we should invest a lot now, later at intermediate times whatever and then we want to decide which abatement strategies are going to be more effective. And we've seen the NPV approach is the tool, the main tool we're using at the moment for this task. And why am I saying at the moment? Because later on we'll be talking about integrated assessment models which are similar but use a different methodologies, for the moment let's stay on message with NPV. And as we have discussed so far, a key element of the NPV approach is finding the correct discount rate. So the purpose of this session here is to show that on the basis of relatively simple no arbitrage arguments, there are several rates that we can determine. That should all be equal, at least at the optimum and in a perfect economy, and therefore, if we can establish any one of these rates, well, that's good enough because that is what we need for our social discount factor. It is a tiny bit complicated in reality that it sounds in theory, but let's go one step at a time. So more specifically, let's remind ourselves that what we have obtained. We have obtained the result that in a simple rice economy, remember a rice economy is an economy where rice is both consumption and capital. So, as I was saying, we have obtained the results that in a simple rice economy, the marginal productivity of capital, which is a derivative of a perfectly known function, f(k). Remember f(k) is a function that translates a grain of rice today into one or more grains of rice in the future. So the multiple productivity of capital, the first derivative of f(k), has to be equal to the marginal rate of utility substitution. And remember there was a ratio of this two derivatives, the ratio of the derivative of the total utility with respect to the consumption today, over the derivative of the total utility with respect to the consumption at time one. And then we also saw that if that total utility can be expressed in a separable form that had a particularly simple expression. So this is the key results so far. So in this session we're going to learn how to express these conditions in terms of various rates, a borrowing rate, a rate of return on capital, a rate of return on investment. And the reason for doing this is to get different different handle on how to calculate this discount factor, which is so key. And clearly once we have this quantity here we can use it for the NPV approach, not just for what I just said. But also for the levelized cost of energy approach that we saw in few sessions back, remember when we were comparing different sources of production of energy, not necessarily renewable, it could be any source of production of energy. And we saw that we were using the NPV approach that also requires the social, not the social in this case, in that case just the discount factor, okay. And we're going to take over from where we left and we're going to make some little changes of variables in order to express all our results in a more palatable form. Remember that we were in a two date one period economy. The two days were T0 and T1 and the one period is the one period that spends the period from T0 to T1. Tomorrow, it could be 24 hours time, could be five years time, it doesn't matter. It is just a arbitrary future point in time. And remember also that my first derivative F prime of kappa, kappa is the investment in rice is the extra rice, which is obtained by investing one additional unit of rice at time T0, okay. Now, instead of investing one single unit of rice at the beginning in a lump investment at the start, let me spread this investment at a constant rate from T0 to T1. And then I can ask the question, what is this rate of investment from T0 to T1? What would this rate have to be in order to have a time T1, the additional amount of rice, F prime of K. So this means as this expression shows here that F prime of K which is my marginal productivity of capital, should be equal to the reinvestment if you want of my rice over time exponents or rho k times the time, the length of the time interval take logarithms and sold for rho k. And that rho k is my welfare preserving rate of return on capital. What does it mean welfare preserving? It is exactly because remember we set this equality here by saying, I am indifferent between giving up a bit of consumption today in order to have a bit more consumption in the future. It was just that sweet spot. Okay, now here's an interesting and simple manipulation, to make things simple, let's assume that we are working with the logarithmic utility function, particularly simple because when I take derivatives of log(c), I just get 1/c. So in that case, first of all I'm assuming that my utility function is separable. So I am writing the total the big utility function as the sum of utility today plus discounted utility at time one. And furthermore, I'm saying that utility at any the smaller u at any point in time is a logarithmic function before when I take a derivative, remember the subscript o and the subscript 1 implying mean, signify derivative with respect of consumption at time zero and the time one, that's what we did in the last session. So ratio of big U sub 0 and big U sub 1 becomes C1 over C0. They invert the order simply because in when I take the logarithm I get 1/c. So if I assume that the rate of growth of consumption is equal to the real rate of growth of the economy, then I have the ratio C1 over C naught is my big Rg. Big R, is a gross rate, so it's a number as 1.03 not a number like 3%. To make things simple, take T1-T naught =1. And use the equation that I obtained before to obtain that my marginal productivity of capital is to be one of the beta C1 or C naught. Take logarithm of both sides and remember the logarithms of 1+x is almost equal to x when x is smaller to obtain a very nice and interesting result. And the interesting results, it tells me that my rate at time looking for rho k, is equal to delta plus Rg. What does it mean? Well, delta is my impatience term. I will revisit this term in a short while but let's just keep the place name in patience for the moment and Rg is my rate of growth in the economy. So in the certainty case when there is no uncertainty about the investment, about anything the welfare preserving rate of return on capital, rho k should be equal to the rate of growth of the economy plus our impatience rate. We will see that this is very similar is a tiny bit simplified to the celebrated Ramsey equation and it is simpler because we specified one particular type of utility function which is a logarithmic one which perhaps is not the most realistic. But nonetheless we get a very simple and clear result. So the welfare preserving greater return on capital is just given by the sum of impatience and rate of growth of the economy. Okay, what else could I do? Well, I could set up a debt market so I set up a frictionless default free credit market. And in this market anybody can borrow each units of rice at time T0 and repay absolutely for sure with no default. Exponents rho t units of rice at time T1 per unit of rice borrowed. So in this rice economy rho without any subscript is the interest rates. Of course every agent still has his rice technology. So it then turns k units of rice today into f(k) units of rice in the future. And therefore a very simple arbitrage argument shows that the borrowing rate must be equal to rho k that we have just determined which in turn is equal to the marginal productivity of capital. Otherwise there would be an arbitrage. I won't go into the details but is intuitive because if it were, let's say more profitable to borrow money, I would borrow the money, obtain more, then I could obtain with my productive, investment and thereby creating arbitrage because there is no uncertainty. Okay, can we do something more? Yes we can, suppose that we create an investment opportunity in our rice economy, an investment opportunity could be a new rice paddock irrigation system. More interesting and more realistically moving away from the rice economy. If you're thinking in terms of climate change or of society in general, it could be investing in education, in infrastructure, in climate change control etc. At this stage, the only important feature about this investment project is that it's return is certain. So if each citizen invest units, a certain amount of rice the project in this project, in this irrigation project, the project we yield surely in exactly exponents are T units of rice at time T per unit that has been put into this investment. Now, we can answer a really important question how are we going to finance the investment project, doesn't make a difference. Well, if you think about it, I've shown here in this slide, there are three ways of financing it. I could consume today the same amount of rice, but I could plant less rice, what we don't plant we invest in the irrigation technology. Otherwise I could consume less rice today. Plant the same amount of rice and what I don't eat today invest in the irrigation technology, or perhaps I can just borrow rice. I'm going to eat less rice in the future because I have to repay the debt. Does how we finance the sure project make a difference. Now, if I have if I call r the internal rate of return on the project. Simple reasoning against again, based on no arbitrage shows that all these different rates are equal. And why is it so important that all this different rates should be equal? Because ultimately I want to determine the rate that I'm going to use in the social discount factor and hopefully perhaps having three ways of getting at this different number might be helpful. For the moment what we have obtained is simply we reached the conclusion that in order to decide whether our irrigation project is worthwhile. So out of metaphor, whether we should invest in climate change, it doesn't matter whether we fund this by consuming less rice, planting less rice or borrowing rice a good project remains a good project and same for a bad one, irrespective of how we finance it. So this also imply that in order and this we are getting really to the heart of the matter. In order to decide how much we should spend on the project today, we should just focus on the net some of the costs and benefits discounted at the appropriate rate where the rate can be any of these three rates which are all identical. When it comes to climate change control. This set of relationships are absolutely fundamental because in mainstream academic thinking, they are the framework on the basis of which we decide how much we should invest at different points in time. And at this point let me elaborate a bit on some features that we will explore further in the future. According to this line of thinking, if we are confident that we're going to be richer tomorrow, we should postpone possibly a large part of the investments to make it when it hurts less because as we are richer in the future, our pain from investment, our decreasing utility will be smaller. This is in a nutshell part of Professor Nordhaus view that we will discuss at great length later on. This is a logical conclusion, but we should think critically about the assumption that we have made in order to reach this conclusion and how well they apply to the case of tackling climate change. One big big assumption is that we know the damages and we know my production function for sure. So again, how what use are we going to make of the identities that we have established just now? Ultimately, we want to choose how society should invest to tackle the problems such as climate change. And we can do this by borrowing by consuming less, by investing less in other activities such as education, health care, etc. One of the big messages are trying to convey is that these is a big deal. It is not something at the margin we have to make sacrifices. And since these investments are not marginal, are not small. We want to choose how to deploy our finite resources in the most efficient way. So the approach that we have developed may suggest, for instance that if abating temperature changes, cost a lot today, perhaps solar panel may be very expensive today. Then we may not maximize our total utility by investing heavily, say, in solar panels today or in negative emission technologies today. Alternatively, if we give a very high negative present value to future damages, then we don't need a super efficient solar panel or we don't need a super cheap net negative emissions technology to decide they should pay its cost of installation today. Or perhaps you should do a mix, invest a bit now and a bit later. In any case economic theory, I suggest that we should do so optimally otherwise, we are leaving money on the table. So in mainstream economic thinking, these equations give us a rational way to decide the optimal level of investment today. That is theoretically appealing. However, in order to see whether this approach gives us practical guidance, there are many gaps that we have to fill in and so far we have kept on saying we're going to use this in order to determine the social discount factor, but we haven't done it yet. This is going to be our next step. And above all, our results so far are very general, but they are empty because we have not specified or parameterized the utility function. In the little example I have made, I have assumed a utility function equal to the logarithmic, in which case I have assumed a particular exponent gamma I'll explain what that is, but I have not justified any of this. So in the next lecture, we're going to see how we can arrive at the social discount factor. Meaning we're going to find the rate at which we should discount future damages in order to obtain their present value then and only then, are we going to be able to make a decision rationally, according to mainstream thinking about how much and when we should invest. [MUSIC]