In this session, we're going to look at the role played by the population and the population growth in the DICE model. In particular, we're going to look at how the population growth is modeled and what the potential shortcomings of this way of modeling population might be. Let's refresh our memory why population is so important in the DICE model. Remember, in the DICE model, population equal labor force, everybody works, so that is the first important thing to keep in mind. Therefore, the population is a direct input via the Cobb Douglas function that we've seen to the production function. Growth in population therefore, automatically means growth in output. The growth is less than linear, but it is growth in output nonetheless. Here, we have to be very careful about what we're maximizing. As we shall see in the last slide of this session here, the DICE model maximizes over the total utility, whereby total utility, it means that if I have one billion people, I'm going to add up the utility of one billion people that everything else being equal is going to be greater than the population of eight billion people, [inaudible], everything else being equal. Now, is this a logical thing to do? This is a hotly debated point amongst economists and one might argue that what really matters is not the more the merrier approach, but it is the utility understood as happiness per person. We will look again at this in the last slide, but I wanted to stress this point at the beginning. There is another important point that I would like to stress. We made a distinction in a previous session between endogenous and exogenous variables. In the DICE model, population is completely exogenous. The growth in population is assigned from the outside, and by the way, with no uncertainty. But it's not the uncertainty bits, that I am stressing here. The fact that it is exogenous has some implications. Now, the population according to the DICE model, grows over time. Growing population means increase in output, and you might say, increase in output will automatically give rise to higher utility because greater output, greater consumption, greater utility. Not so fast and not so simple, because by increasing output, I'm also increasing emissions, and increasing emissions creates damage and therefore there is a reduction in future consumption given by this increase in emissions, given by increase in output, given by increase in population. Arguably, there is a weak point here in making the population growth completely exogenous. We will see later on whether we can do something about it and whether it is reasonable to do something about it. More precisely, the population is assumed to grow in the DICE model but at a declining rate, so the starting point to this function is calibrated in order to obtain today's population. The population in 2100 is assumed to be 10.5 billion. This estimate comes from an assessment made by the United Nations, and as usual, there is lots of referencing in the DICE model and it is a very reasonable assumption. Therefore, having the starting point and having the 2100 population, given a functional form with two parameters, I can fit a function that goes through these two points. The initial rate of growth to the population is roughly what we are experiencing at the moment, which is about 13 percent. These are the exact equations that describe how populations grows. We said that the population at time's t is equal to the population at time's t minus 1 times a growth factor, 1 plus g. Now, the g is not constant, as you can see from the second equation, but the g rate of increase of the population decreases over time. This is what the population growth looks like, as I said, starting from the value today and leveling off, basically after 2200 there is no further growth in population, and this is what my growth rate looks like, going progressively down to zero. Now, the growth in population is, as I said, exogenous and deterministic. Whether the economy grows a lot or a little, this is no direct effect on the population. Not everybody agrees. Richard Toll puts it very starkly, I'll read his quote here. "The assumption that population growth is independent of per capita income is at odds with everything we know about fertility and mortality." But there is a very strong statement, and indeed it may be true for some countries, for instance, Hong Kong, since the 1970s, there has been a huge economic growth and the birth rate has fallen by half. However, if we look across the planet, this relationship, while it is true, it is a negative relationship, but it is not very strong. However, there could be a case to be made to endogenize population growth, so to say, along the spots where I have greater growth, I will have a slower rate of increase of a population and therefore, it is not sensible to draw a regression line in this graph here, and to say, let's make the growth of a population depend along each path on the output that the economy is experiencing at each point in time. Now we come to the last point that I wanted to discuss. The total utility is made up of a sum of what are called in DICE, the period utility, so the utility in each period of time. The period utility takes its input, consumption per person. This is key, it takes in consumption per person. Then this utility is discounted and multiplied by the total population. Notice in this equation here, that I have multiplied the last term in red, I'm multiplying by the population at time Ti. Now this raises a question, is really the sum of all these quantities, of these period utilities, the most relevant quantity to maximize, given that an ever increasing population has been posited? The arguments here is, since the future generations are assumed to be one-and-a-half times bigger, rather the future population is assumed to be one-and-a-half times bigger. This by construction, everything else being equal, means that the happiness of the future is going to be one-and-a-half times greater than today's happiness. On average every body, not only on average, but also in distribution, everybody is as rich and as poor as today. This is a debatable assumption. I find more palatable the assumption to say, what really matters is not the total utility, so its not the utility that I obtained, multiplying in each period by the population in that period, but not including this term. When you do that, there is some variation in the output of DICE, but I'm happy to say that the variations are not massive. That is what I wanted to bring to your attention when it comes to understanding how the DICE model works.