So we have said we've seen for net present value approaches and in particular for the determination of a social discount factor. Now we're going to get on with it and we're trying to create a little model from which the social discount factor will drop out as a natural output. So the little model is based on what is called the rice economy and I'll explain why it is called the rice economy. It is a strange economy in which the only good is rice and we make super simplified model, but that's okay. Super simplified models are ubiquitous in science, not just in economics. If you look at physics, you're supposed to be the hardest science par excellence, you have lots of frictionless planes, and point balls, etc. So one shouldn't be ashamed for making super simple models. They allow you to see things clearly, and if you want to add bells and whistles, you can always do it at a later time. So we're going to introduce the rice economy, just characterized by having some sweeping assumptions. And let's keep in mind where we're trying to go. Ultimately, we want to find answers to the questions, when should we act? How much should we act at different points in time? And which abatement strategies are most attractive at any point in time? So if you think about it, there are no questions that are more important than these. And since the topic is very meaty, we're going to break it into digestible bits. So in these sessions, we're going to look at the, what I call the first-order conditions. So how much consumption it is optimal to give up today for a future benefit? And then in the next session, we're going to look at various rate, the investment rate, the borrowing rate. And what we're going to do in the next section is to invoke in no arbitrage argument to show that at the optimum and in a perfectly working economy with rational agents, etc., all these different rates are the same. And why do we care about having so many different rates? Because the social discount rate is the key things we're trying to get to and therefore having different routes of estimating this is or might be helpful. So, Before we get started, just by way of reassurance, the analytical tools we are going to use, they're not made up ad hoc and they have not been invented for this particular problem. The first-order condition is ubiquitous in economics and it is at the heart of asset pricing, growth theory, etc. And the derivation of the equivalence of the different rates rests on the condition of no arbitrage, which is also almost ubiquitous. In finance, you find it in option pricing, you find it in corporate finance, you finance it in the arbitrage pricing theory, etc. By itself having an illustrious pedigree does not guarantee that the theory is valid, but at least we're not making things up as we go along. So I promised that I would talk about a rice economy and this is tongue in cheek, this is the ultimate rice economy. What do we mean by this? Well, as I said we're going to introduce sweeping assumptions. We are in a two-times, one-period economy. So the two times are t0 and t1, two times one, a single period between the two. There is one single good in the economy that can be either consumed today or invested. Investing means planting the rice, they will produce more rice in the future. In economics, one is rice economies and trees economies. In trees economies, they also produce products in the future, apples, pears or whatever, but you can't eat them today. So rice is different because you can either eat it today, hopefully after cooking it, or you can plant it and you get more rice in the future. So if instead of being consumed, k units of rice are planted, we know for sure that they will produce f(k) units of rice in the future with certainty. Therefore, rice is also capital. So we know how productive rice is. The function f(k) is fully deterministic. And we know that if we plant k units of rice, we're going to get f(k) units of rice in the future for sure. When I say in the future, I don't have to specify when, because I only have two times in my economy, t0 and t1. So saying in the future, it is a time t1. So what else do we know about this production function? We know that it's concave, which means that if I plant more, and more, and more rice, I get less and less extra future rice. So I will get more rice, but the rate of increase of my rice production will decrease. And it's also obvious that if I plant no rice, I get no rice in the future. I'm eating everything today, but I will get nothing, I will starve in the future. Also it makes sense, but it is not necessary to require that for a small initial investment, f of epsilon should be greater than epsilon. Which means at the origin, giving up some rice today must yield a bit more rice tomorrow, otherwise, why should I not consume the rise in the first place. Anyhow, this is not necessary, but these are the assumptions that we need about our production function. And these assumptions are summarized in these two graphs here. The blue line shows my production technology. And on the x-axis, I have a unit of rice planted today. And on the y-axis, I have the amount of rice produced in the future. And the orange line is the marginal productivity or capital, which gives me the derivative of the production function. Notice our derivative becomes very, very sharp at the beginning, which means that at the beginning it really pays a lot. It gives me a lot of extra rice, giving up a bit of rice today. So the marginal productivity of capital in the limit goes towards infinity as I approach 0. Then we need some assumptions about preferences. We have seen the assumption about the production functions. What about our preferences? Well, let's keep things uper simple. Everybody has the same preferences which are described by a utility function, which is a big utility of consumption today and consumption at time t1, a big U utility. And I say big U because we may or we may not separate this big utility into two sub utilities, the utility today and the utility in the future. This has to be discounted, but for the moment, I'll just keep it general. So just a function today, a function that gives me the utility as a function of consumption today and tomorrow. And our task is to maximize this utility function by varying consumption and investment over the two dates. So we want to determine how much we should invest today, which means give up consumption today, in order to obtain the maximum overall happiness. And we also make the assumption that the utility function is concave. So as I consume more and more rice, I am happier and happier, I never get indigestion. But I am less and less happy as I consume more and more rice. And automatically this means that we are risk-averse, but actually risk aversion doesn't enter the discussion because everything is deterministic in this economy here. Nonetheless, there is a substitution of utility across time and this will be extremely important. Now we can also make another assumption which is very, very handy, but it's not innocuous. And this assumption is the utility function is separable. What does it mean? If we look back at the expression we have used before, there was a big U as a function of C0 and C1. We say I can split this total utility as the sum of two components. And the two components are the utility today and the discounted utility from consuming in the future. And by the way, these are utility functions of a particular class. Power utility function, which is a class of constant relative risk aversion utility function. I will explain what that means. And you can see these utility functions for different levels of risk aversion. And you might say, well, this function look very different. Yes, they look different, but they have one important feature. If I look at the change in utility for the same percentage change in consumption, it is always the same. My utility does not decrease the same if I lose $100 when I'm rich and when I'm poor. But it decrease by the same amount if I lose 10% of my wealth irrespective of whether I am poor or I am rich. That is a feature of this class of utility functions. As I said, making separable utilities, employing separate utilities is a very handy, but it is not an innocuous assumption. In particular, if we work with a separable utility functions, the coefficient of risk aversion which describes how I dislike a risky better today, also controls the rate of intertemporal substitution of utility. How much I am willing to change my consumption pattern in order to avoid feast and famine. So given two consumption patterns which have consumed a lot today and be a pauper tomorrow or vice versa. Live under a bridge today and live in a mansion at time t1. You might probably prefer an intermediate outcome which is live in the terraced house both at time t0 and at time t1. This is what I mean when I say avoiding feast and famine, we're going to discuss this later. And finally, we need assumptions about how much rice different people have and about how much they like rice. So as I said, it is super simple. Everybody has the same initial amount of rice, there are no issues of inequality in the super simple model. And since everybody has the same amount of rice, and everybody has the same preferences, and the same endowment, the choices made by Johnny are identical to the choices made by Jenny, for any Johnny and Jenny. So therefore, if optimally one individual decides to invest k grain of rice today, she and everybody will also decide to invest k units of rice today. [MUSIC]