JOHN BROOME: (00:10) I'm going to be talking at a rather high level of abstraction about economic evaluations, the most common method of economic evaluation which is a two-step procedure. Most economists working in practice to value policies, events in a country or across the world, do it in two steps. The first thing they do is aggregate all the values that there are in the country (or the world) at each particular time to reach an overall value at a time. So they evaluate the world, say, in 2025 and also the world in 2026 and so on. So they arrive at a value at each time. And the second step is to aggregate those values across time to arrive at an overall evaluation. And generally, most typically, when they're aggregating across time, they apply discounting. So they often attach less importance to values that come later than to values that come earlier. That's the second step. 

(01:35) But actually, I'm going to concentrate on the first step of this two-step procedure. I'm going to concentrate on the question of whether the world even has such a thing as a value at each time, the property that together, if you take the value at this time and put it together with the values at other times, those values together determine the overall value of the world or of the country. That idea is called separability, strictly weak separability by times. The idea is that each world or the country has a value at each time which can be added up with other values in order to arrive at the overall value of the world or of the country. That’s separability and I'm going to be talking about a problem that arises with separability. And of course, if there is a problem with that, then there's going to be a greater problem with discounting because you can really only do discounting of the normal sort, if you have in the first place, values at each time that you can discount. So this question is prior to the question of whether we should discount with times.

(02:55) I’ll start with an example of this difficulty. There is a value that I think we mostly recognize. We think that in at least one respect in the last couple of centuries, the world has been getting better. And that respect is that it's been giving people longer lives. People get to live longer now than they used to a couple of hundred years ago and we think actually, that is, that’s one of the greatest benefits that economic development brings to us. However, that's not a benefit that we can readily recognize in this standard two-step procedure that I described. So that's a serious problem. This important value doesn't fit into the two-step procedure, at any rate without some extra work. And I'm going to describe why not and then I'll talk about ways of getting round this difficulty. And that will, it turns out, raise another interesting question, which is, how do we take account of the size of the population within economic valuations? That question turns out actually to be inseparable from issues to do with temporal separability and consequently, with discounting the future. 

(04:19) Now, let me just pause for a moment because I'm slightly anxious about one thing. I don't seem to have a chat showing on my screen. And that's fine so long as the sound works all right. But if it doesn't… I don't think I've got any way of finding out whether it doesn’t.

SVEN: (04:44) It's all fine, John.

JOHN BROOME: (04:46) Sorry. 

SVEN: (04:49) It's all fine.

JOHN BROOME: (04:50) The chat button is not available. Now I'm sharing my screen. But look, it's like Andreas said… I just heard. Can you interrupt me if there's a problem with the sound. That's the only thing I'm worried about.

ANDREAS: (05:01) Yes, we definitely will do. Don't worry. So far, everybody’s saying they can hear you very clearly.

JOHN BROOME: (05:04) I’ll just keep going then. I can see everything is going okay. 

ANDREAS: (05:07) Alright. 

JOHN BROOME: (05:09) So let me explain this difficulty of valuing longevity, the increase in people’s… the value of the lengths of people's lives and why that's a problem for separability. 

(05:23) So first of all, let me remind you of a well-recognized empirical fact about the increase in longevity. It doesn't happen just by itself. There are further consequences. What happens in the course of what's called the demographic transition, which is when people get to live much longer lives. The first thing that happens is that the death rate declines and then some decades later, the birth rate declines. And in the meantime, the country's total population increases because people die less quickly before they start being born less quickly. So you end up with a bigger population. Now the value that I'm concentrating on, the value of longevity, is certainly not the value of the increase in population, whatever that may be, but the value of the increase in longevity. So in order to focus on that value, I'm going to deal with a very stylized version of the demographic transition, in which I take away the consequent effect on the country's population. I'm going to fix it so that the population of the country remains the same. This is what would happen if the death rate and the birth rate happened at the same time. So I'm going to compare a world before the transition with a world after the transition when the population, which is to say the population at each time, is the same. And to keep things simple, I'm going to assume that the standard of living is not affected by this change. So as people get to live longer, they continue to have lives of the same quality. And then I can give you an illustration of this demographic transition in a sort of diagram.

(07:11) Here it is.

(07:13) So I've got two halves to this diagram, one before the transition, the one after the transition, how the world is before, how it is after. I've got time in the horizontal direction here and people in the vertical direction. Before the transition, I've got each person living for a couple of times. This one for instance, gets born here, lives a life here and then dies. So if you look across, each person lives for two years at a standard of w per year. Over here after the transition, everybody is living for four years. So there's been this increase in longevity and I've arranged it so that the population at every time in both cases is two people. Here there are two people alive at every time and over here, there are two people alive at every time too. So the increasing population has been canceled out and we've got everybody living longer. 

(08:17) Now if we try to assess these two situations comparatively, in a separable fashion, we won't get the right conclusion because if you look at the world before transition time by time, separately, looking at one time at a time, at each time we've got just two people living. Look along these vertical lines. You've got two people living and they've each got a wellbeing of w. And that's exactly the same over here, that each time you've got two people living and they've both got a wellbeing of w. So if we assume that we can make a valuation of these worlds separately, time by time, we will have to think that they have the same value. But that's what we don't think or we mostly don't think they have the same value. We think it's better to have people living longer. So a separable valuation misses the crucial point of the value of longevity. 

(09:20) We can define separability algebraically. And I just put this here in case you want to do it. To keep it simple, I am assuming that time is finite and the number of possible people is finite. And in that case, we can assume that the value of the world, V, is given by these w's for each person and for each time. There's a w11 here, which is w for the first person and the first time. This is w21, which is for the second person the first time. This is w12 which is for the first person the second time. So these w's are all the first person ones. These are the second person ones. I took a w because these things stand for the person's wellbeing at times when she's alive, but at times when she's not alive, w is some sort of arbitrary… It's assigned some sort of arbitrary value, which indicates that she doesn't exist. I generally write it as an Ω when looking at a time when the particular person doesn't exist. So in practice, if things are going for a long time, there are going to be a lot of Ω’s in this vector here. There's going to be Ω’s at all the times for each person when she doesn't exist. And there may be people who never exist at all in the particular world we're thinking about. Their lives will all be Ω’s. But when they live, w is their wellbeing. So this vector captures both whether a person exists and how well off she is when she does exist. And we will say that the value function – I’ve written value as a function of all these things – the value function is weakly separable by times if and only if it takes a particular form. 

(11:21) And here is the form. This says that the value of the world at the first time, which depends on everybody's w's at that time, their wellbeings and whether or not they exist. The value of the world at the second time, their wellbeings and whether or not they exist, and so on, right through to the last time. And if you can do the valuation by first valuing the times – that's  the value of the first time, that's the value of the second time – and then putting them together in some way to get an overall valuation. That is the assumption of separability by times and this is the function that in the first instance, gives no value to longevity, as I just explained. So from now on, I'm going to drop this word weakly. Separability by times – by that I mean weak separability by times. 

(12:20) Let me say a bit more about just what is this value of longevity that can't be registered by a separable function.

(12:29) There's certainly one feature of longevity that can be registered. A separable function can register the extra wellbeing that comes to a person if her life is extended. If a person gets to live longer than she otherwise would have done, then she'll go through a period of years and during that period of years, extra period of years, she will have some wellbeing and that wellbeing can be taken account of perfectly well in a separable function because that extra wellbeing all takes place at some particular time and therefore can be counted at that particular time in a separable function. And so that value, the extra wellbeing a person gets that is registered, can be registered in a separable function. 

(13:21) But notice that you can get extra wellbeing in the world in two different ways. You can get it by extending the life of a person that we've already got, in which case, it will be registered in a separable function. Or you can get it in a very different way, which is by creating a new person who would live at later times and she will have wellbeing and that will be new wellbeing brought into the world and it will also be registered by a separable function. So those two ways of adding wellbeing in a separable function are going to have the same effect. They're going to be valued in the same way. 

(14:02) But most people think intuitively that the second way of adding wellbeing to the world is less good than the first way. We think, most of us we think, think that at any rate within limits, it's better to extend the lives of people that there are rather than create new lives, even if the people who lived those new lives would be getting all the same wellbeing that the people whose lives were extended would have had. That means that in effect, we think that if you've got a given quantity of wellbeing to be enjoyed in the world, it's better to have that wellbeing divided among fewer lives. We prefer the wellbeing to be given to fewer people rather than having a profusion of people, each of whom has a smaller share of that wellbeing. And that value is the one that's not recognized in a separable function. A temporally separable value function can't recognize that second value that I described, the value that appears when we think it's better to extend a life rather than to have a new person living with the same amount of wellbeing. 

(15:21) That value – which is what I should call the value of longevity simply – that value you can only identify when you're looking at one of these grids that I've illustrated already, when you're looking across time, so you're comparing one time with another and looking in the grid to see whether the same person is alive at one time as the other. And that's just the sort of thing that you can't do in a separable value faction because it doesn't allow the comparison between different times that's involved. So the two-step procedure, which assumes separability of time, can't recognize this value and it's therefore mistaken unless it can be fixed, somehow. 

(16:11) Here's a note for economists. I've been talking about separability of time of people's wellbeings and whether or not they exist – those w's. As a matter of fact, economists when they assume separability of times, are normally doing it for consumption, what people are consuming at the different times, the material goods that they're consuming. And they often assume implicitly that consumption is separable by time. It would be just conceivable to have consumption separable by time, even if wellbeing is not separable by time. But although it's just conceivable, I can't think of any way in which that would really, really possibly happen. So it's virtually certain that separability of consumption by time implies separability of wellbeing by time and that's why I'm concentrating on the separability of wellbeing. 

(17:13) So here's a summary of what I've said so far. Temporal values are generally aggregated with a discount factor. The issue of separability is prior to the issue of discounting because discounting doesn't make sense without separability. But there is a serious difficulty with separability. It doesn't recognize the value of longevity and that's a problem. And it's particularly going to be a problem for climate policies and other long-term policies, policies whose effects extend over a very long stretch of time because that's going to influence how long people live and also the development of the population of the world over those times. And in the case of climate change, we know that one of the biggest harms it's going to do is to shorten people's lives. Climate change kills people, a lot of people.

(18:06) So remember, we're stuck if we want to do discounting and we're stuck even if we are doubtful about separability, unless we can find a way around this problem. Can we recognize the value of longevity within a separable function by some sort of a fix? And I'm now going to go on to fixes we might try. 

(18:28) I've got one further note to make about separability before we get to that. That is, there is another important value which a separable function can't recognize and that's the value of equality between people's lifetime wellbeings. Look at these two grids here. In the first one, we've got, as I said here, lifetime inequality. Again, I've got time going horizontally and people going vertically. And in this diagram here, I've got different sorts of people living alternatively. Here, I've got a person living at level 1 for two periods. Here, a person living at level 2 for two periods. Here, one living at level 1 for two periods and so on. So this population is unequal in terms of the lifetime wellbeing of people. There are people with better lifetime wellbeing and people with less good lifetime wellbeing. Over here, there's no lifetime inequality. Everybody starts at level 1 and ends up at level 2. But on the other hand, there is temporal inequality because every time there is an old person living at level 2 and a young person living at level 1. So there's inequality between the old and the young, and that obtains at every time. So we've got temporal inequality here but no lifetime inequality. Here, we've got both. 

(20:03) But make the comparison between these two time by time looking at one period at a time. In every period here, you've got a person living at level 1 and a person living at level 2. It's so here and it's so here. There's always a person at level 1 and a person at level 2. And that's exactly the same over here. There's always a person at level 1 and a person at level 2. So if we evaluate these two worlds separably, we're not going to be able to give more value to this one on the right than the one on the left. But that is intuitively incorrect because there is a sort of inequality over here, which there is not over here. And this is an important sort of inequality, lifetime inequality. So that's a further objection to separability. It doesn't recognize the value of lifetime equality. 

(21:03) So how are we going to fix these problems? But I'm going to concentrate on the longevity one and try out a couple of fixes to see how they work. The first is one that I identify with Iwao Hirose. He suggested that we should fix it this way. We should recognize that when a person dies, she suffers a loss. She loses the rest of her life and that's a loss that she suffers when she dies at the time of her death, that's to say. So we should subtract that loss from the value of the world for the person at the time when she dies. So now we can compare the world before transition and after transition, taking account of this loss which people suffer when they die. This is how Hirose recommends we do it. 

(22:12) So let's take the ‘before transition’ world where everybody lives for a couple of years at level w. Well, potentially it turns out, because we could live for longer, potentially, they're losing two years of life when they die. So this is a bad thing that happens to them and Hirose suggests we should take that badness off at the time that they die. So the way we should value the world for this person at this time, is her wellbeing at that time, but we take away the 2w because she then dies and that's a bad thing. So this is a picture of the values that we should be putting together at each time. Whereas, after the transition, this bad thing doesn't happen. People live for their full potential life of four years. So this is how we should do the value according to Hirose. And you can see now that we can recognize by this means that this is a better world than this one because at each period, we value things separably period by period. At each period, if we do it by addition, we get a value of zero here. Whereas each period, if we do it by addition, we get a value of 2w here. So this is the better world. That's the idea and it gives you a value of longevity that can be dated.

(23:39) But I think he's incorrect because it involves double counting. Remember, I've already said that if you extend a person's life, one value that we can recognize separably is all the extra good that she would get in her life, that she gets in her life during the extended time that she lives. And that's already taken account of in the separable function. I mean, let's go back to now, go back to the two worlds that I have just illustrated. 

(24:20) These ones where these people live for two years and die, these people live for four years. So compare those worlds now but without the 2w’s here. Here we have everybody living for two years, here we have everybody living for four years. And you can see that we get extra value here because we've got people living these extra years. In fact the value in a separable value function of this distribution is 4w in every year measured separably, whereas here, it's 2w in every year, measured separably. So that's already taken account of.

(25:09) In fact, we can see that this double counting can lead to the wrong conclusion by comparing these worlds here. So I want to compare the value of worlds where everybody potentially lives for two periods but in one of them, actually, people die after two periods. They die, whereas in the other world they do continue to live for the remaining two periods but their life is miserable during that time. So they live for two good years then they live for two bad years. And I'm now going to set my scale of wellbeing so that zero, 0w, represents a standard of life which is such that it would be better to die rather than continue living at a wellbeing of less than zero. So these minuses here – which are levels of wellbeing in the later periods of these people's lives – these levels of life, which is such that it would be better for these people had they died earlier after two years. 

(26:29 ) Now, how do we compare these in a separable valuation? Well, do it year by year. Here we get, if we do it simply by addition again, we get 2w –  2w = 0. Here in each year we get 2w – ½w = 1½ w. So here each year the value is positive. Here each year the value is zero. So this is, on the separable valuation, a better world. 

(27:01) But it isn't a better world. For each person, it would be better if she died young if she dies over here rather than continue to live in these conditions of suffering, which were imposed on her. So the Hirose method, I think, is not satisfactory. 

(27:25) This is a third idea, but that's because I didn't bother to mention the first one. This is the next idea that I'm going to be talking about. And that's to say that there's a ‘critical level’ or a ‘neutral level’ and to say that the existence of a life adds to the value of the world only if the total wellbeing of that life exceeds the total wellbeing in the life. So what this is saying in effect, is that from the point of view of the universe when it's valuing the world and deciding whether the world is good or not, each time a person gets born, born into this world, it actually is a bad thing. It's a hit against the value of the world, having a new person and that hit can only be overcome by having the person live a long enough and good enough life that her total wellbeing during the life is greater than this negative value which her creation brings into the universe.

(28:34) Here is the value function for this theory. What we do is we look at each person. Here is a person i. Look at her lifetime wellbeing, which I've got down as wi and subtract from it the neutral level n. So this is positive if her lifetime wellbeing is greater than the neutral level, otherwise it's negative, and then we add up across everybody. So somebody whose life doesn't get up as far as the neutral level is a bad thing. Her existence is a bad thing so far as the universe is concerned. If she gets above the neutral level, it's a good thing. 

(29:14) Now this formula recognizes the value of longevity so long as your life is good because the longer you go on living, the more w goes up and the more likely it is to get greater than n and then once it is greater than n, it gets progressively more and more greater than n and adds to the value of the world. So this ‘neutral level’ theory recognizes the value of longevity but it's not, at least not simply, separable by times. We can get it to be separable by times if we take these n values (the negative values that I've been talking about), if we take those n values and somehow or other allocate them to times, and if we do that, then we'll be able to count those in a temporally separable value function by picking them up at the time that they've been allocated to. And we can do that. 

(30:12) So here is a way of doing it. We could just allocate the n’s, the bad features of the existence of a person to the time when a person dies. So each time a person dies, we would just take away the and  if we did that, we would have managed to get this negative neutral level into the separable function. It will now appear at each time. We will succeed in showing that this world where people live longer than this world here, is a better world. You can see… Look! If you look at each time here, at each time you get a value of 2w – n. Over here, you get a value of 2w – n  in some years, but in other years you get a value of two 2w. So this is a better world than this one. We can recognize the value of longevity in this world here and we can do that without violating separability. We won't get into the trouble that Hirose got into because we're taking this n away from everybody. Everybody gets an n subtracted from her wellbeing at the last time that she lives. This sort of very vaguely represents, it corresponds to the way that many economists do evaluations because they often think that death is a bad thing that happens to people and they typically assume that death is equally bad for everybody, so they subtract it from the value of the world at the time when people die. This vaguely resembles economic practice [inaudible 32:00]. That's one way of doing it. 

(32:05) I actually prefer a different way. I think that the better time to locate the badness of the neutral level is at the time of creation, when a person is creating rather than when she dies. That's because this then allows us to recognize another value that mostly we intuitively do recognize, which is quite hard to fit into a good value theory. But treating the hit of creation as something that occurs at the time of creation allows us to recognize this. And now I'm going to explain what this value is and how we can recognize it. 

(32:54) Plausibly the creation of a person is something that happens gradually. People don't suddenly spring into existence as fully created people at an instant. It takes some time before a person comes into existence after the beginning of the creation process. And once we recognize that and once we think of this neutral level as a hit, it should be allocated to the time when a person is created, we can spread that hit over the extended period of the person's creation and that gives the solution to this other problem of life and death that I mentioned. 

(33:44) Intuitively, a lot of people think that the death of an infant, a very young human being, is less bad than the death of a young adult human being. Although both are no doubt bad events, if a very young child dies, a very young baby dies, that's not so bad as the death of somebody, say, a 20 year old. Whereas naively, you might well think we should count it as worse because the infant loses an extra 20 years of life that the young adult has already had. So naively, you might think it should be more bad for a child to die. But many of us think, no, the death of an infant is less bad. How can that be? 

(34:34) Well, the answer that I can offer you is that an infant is not yet fully a person. A person takes time and we can allocate the negative value of existence, the n, you can spread it over the time during which it comes into existence and you will see what the effect of that is. So suppose that in a good long life, there are four periods. But the first two periods are the time over which the person comes into existence. So if you were valuing a person who dies in a ripe old age, this is how you would do it. During the first two periods and I'm assuming that her standard of life is still with but she's not completely a person, she's only acquiring personhood during this time. So I'm taking away half of the neutral level during each of those periods. And then she's getting two periods of life after that. So that's how things are for a person who lives all her life, her full life. This is how things are for a person who dies when she just becomes a person. So she's gone through the creation and now she's a person. This is what she has in her life. And here is a person who dies in infancy before she gets to be a person. Now compare these two. This is going to be worse than this, provided n by 2, n/2 > w. If this negative value, half the negative value is greater than the wellbeing that happens during this second period of creation, this is going to come out less good than this. 

(36:31) And that is a crude example of my theory of why the death of a young adult is more of a bad thing than the death of an infant. It just supports my idea that we should think of the neutral level is a hit that the universe takes from someone's creation at the time she's created rather than at the time she dies. 

(36:59) So what's happened so far? Well, we found a way to give value to longevity and do it in a way that's separable across times and it also has the merit of answering this other question about the value of life and death. Of course, it's not at all like the common practice of the valuation that economists carry out, but at least it’s separable. So it looks as if it might be possible to incorporate this into a theory that allows for some kind of discounting. I'm not going to try to do that, but it's given us separability at any rate. But I think perhaps the most important lesson that I want you to draw from what's happened so far, is that in order to do separability, which is to say in order to even make sense of discounting during periods when people's lives and existence are in question, we had to think about the ethics of population. I was using a theory from the ethics of population, the neutral level theory. That's where this theory comes from. And that is incorporated in the value function in order to make it possible to give value to longevity within a separable function. We can't think about discounting, we can't think about separability without thinking about population ethics, which immediately makes everything a lot more difficult because population ethics is extremely difficult.

(38:40) But actually, it should have been clear from the beginning to any economist, that you can't think about discounting without thinking about population because the standard theory of discounting within economics has a theory of population ethics built into it. Maybe you haven't noticed that, but I'm going to explain it in a moment. But the standard Ramsey formula – which is the formula for the discount rate which is universally applied in the economics theory of discounting – the standard Ramsey formula depends on a particular theory of population ethics, and not a terribly plausible one. And I've already explained that discounting using the standard Ramsey formula can give no value to longevity but it has this further implication, this further strong implication for a particular theory on population ethics. And I'm just going to show you what that theory is. 

(39:46) Here's the Ramsey formula. Now as I say, I have no idea who all of you are, who I assume are still listening to this event. I don't know how many of you might have any familiarity with economics and how many of you don't. If you don't like formulae of the sort that they use in economics, I'm afraid this is not going to be very nice for you for the next few slides. You might want to turn off your attention. But if you do, I'm giving you the Ramsey formula in a very simple form and I'll talk about where it can be derived from. 

(40:26) This is the Ramsey formula. This is the formula for the discounting of commodities. It tells you the rate at which the value of future commodities – by that I mean material goods like food and bicycles and things – the rate at which the value of commodities as we look through the future diminishes, the rate at which we should give less value to future commodities. And the formula tells us that this rate is made up of two components. The first is this thing ρ, which stands for what's generally called the ‘pure’ rate of discount. It's the rate of discount on value itself, how much we discount future value compared with present value. It's pure. If you think that there is a positive ρ, so you attach a positive pure rate, you use a positive pure rate of discounting in your formula. That means you think that good things that happen in the future are less valuable than exactly the same good things that happen now. That's what ρ is about. So that's a pure rate of discount. 

(41:43) And then there's this extra component, which is a product of two things. g is easy. That's the growth rate of consumption per period. So this is what you might call an economic parameter. This is how fast the economy is growing per person. It's not the increase in GDP, it's the increase in GDP per person. This η is something known as the elasticity of value with respect to individual consumption. It's a measure of how much somebody’s consuming extra goods adds to the value of the world. I'm actually, in a moment, going to assume that the way it adds to the value of the world is by adding to her wellbeing and we take account of that in the value of the world. So people consume goods and we're assuming that it's this consumption that makes their lives worthwhile. And η tells you how much increasing your consumption of goods increases the value of your life. I should say, not your whole life, the value of your life in a particular time. It’s how much it increases your temporal wellbeing. How much increasing your consumption increases your temporal wellbeing. It's a measure of that. It's called an elasticity. That's just an economist’s [inaudible 43:12]. That's the formula.

(43:16) This is a general value formula from which it can be derived. So have a look at this. V is the value of the world. So this is a value function for the value of the world. It's an integral over time and it's a separable value function because what you're doing is you're valuing the world at each time and then you're integrating across time. So you're integrating across time here and what you're integrating is the value that's assigned to the world at each time. Here, that is. Start over here. That's total consumption C. This is the number of people (n). So this fraction (C(t)/n(t)) is consumption per person. This is per capita consumption here and v is the value that derives from per capita consumption. So, if we assume that everybody is consuming the same, then they will each be consuming per capita consumption and this v tells you the value that is derived from that, how much good is delivered to the world by people consuming this. So that's what the v is. This is the individual value that there is. Now this is a crucial thing in the formula. This is then multiplied by n, which is the number of people who are alive at time t. So you take the value of each person and you multiply it by the number of people. So this is the total value in the world which is the value belonging to each person multiplied by the number of people and δ here is the discount factor. δ, which will decline over time in the standard formula, is the relative value that you apply to later values compared to earlier values. This is the world valued at each time and if it's later in time, δ is less, so it gets less value. So we treat the value of later things as less and less the later in history. This formula, the Ramsey formula can be derived from this value function. Now, I'm not going to try and do it. 

(45:42) Though there is actually a derivation on this page. I'm just telling you, you can derive it that way. And that value function can be thought of as the foundation of the Ramsey formula. 

(45:56) The Ramsey formula can be derived from this value function here. This is the value function that underlies the Ramsey formula. Now, I haven't proved that. What I did on the previous page that you didn't look at is that the Ramsey formula can be derived from this value function. I didn't prove that this is the only value function from which the Ramsey formula can be derived and I haven't done that. I should, really, but I haven't. But I'm pretty sure that there's no significantly different value function that will give you the Ramsey formula. And that means that the Ramsey formula embeds the theory that's inside this value function. This value function represents the theory of value that's implied by the Ramsey formula. 

(46:53) So now we need to ask, how do we make sense of that? How can we understand this value function? Well, I've already made the simplifying assumption implicitly, the simplifying assumption that everybody is consuming the same. That is implicit in here. That's why we attach a value to the consumption behavior. Everybody is consuming consumption per head. And I've also actually mentioned that I'm going to assume that this is each person's wellbeing. So what we're valuing is the wellbeing that people have at each time, which they derive from consuming consumption at each time. So those are a couple of simplifying assumptions. Let's make those and given that let's see how we can make sense of that formula. 

(47:42) For the moment, ignore the δ. So we'll not try doing any pure discounting on it. 

(47:47) So look at this formula without the δ. Suppose that's just 1. What is this formula? Well it says… So what you do is you look at everybody's wellbeing at a time, you take the total of people's wellbeings at that time by taking each person's wellbeing and multiplying by the number of people and then you add up across times. So it's saying we're taking the total of wellbeings at every time and adding those across time. And that function is what can be called total complete utilitarianism. It’s total utilitarianism because it's adding up across everybody's lifetime wellbeing. But as well as that, it's assuming that each person's lifetime wellbeing is the total of the wellbeing that she has during her life. So to get the value, you first of all add up across lives, or you can do it this way, you add up across lives and then you add up across people. Each life is simply the total of wellbeing… has a value that’s simply the total wellbeing of each and the world has a value which is simply the total of everybody's lifetime wellbeings. So that's the theory. It's total complete utilitarianism. 

(49:05) Now, there are severe objections to total complete utilitarianism. In fact, they all (all the ones I know of) stem from the one that I've been talking about up to now. It's the assumption of separability at times. Separability at times prevents us – done in a crude way rather than the subtle ways that I described later – it prevents us from making any distinction in value between adding wellbeing to the world, put into the life of somebody we’ve already got and adding wellbeing to the world by adding a new person. I said very early on that most of us intuitively think that wellbeing is a good thing, that the wellbeing is a good thing, and furthermore, it's better to have wellbeing added into an life we've already got, rather than to have a new person living a bad life. So it's ruling out the neutral level theory that I've just described because neutral level theory does give more value to extending a life than to creating a new life. It also incidentally, for any of you who are economists, it rules out average utilitarianism, which is implicitly at least very popular in economics. So total complete utilitarianism is, at the very least, a controversial theory, but it's implicit in the standard theory of discounting in economics. 

(50:39) If we put the discount factor back, we get a discounted total complete utilitarianism. We have some further doubts that we can add against it. 

(50:51) So here's the conclusion that I want to draw. That even before we start thinking about discount rates, we have to think seriously about the ethics of population, which is difficult. And indeed, our standard theory of discounting within economics is committed to one particular theory of population ethics, which I have to say I suspect Ramsey arrived at without really thinking very much about it. So it's a theory that economists find themselves committed to without really having applied the thought that they need to. And other theories are not even consistent with theories of population ethics, are not even consistent with discounting as [inaudible 51:38]. So discounting and population ethics do not sit well together and at least you have to think about population ethics if you want to do discounting. 

(51:51) And that's the end.