 So, as you can see, this is a joint paper today with Nogu Kiotaki and Shenzhen Zhang, who is a colleague of mine at the LSE. The title is Credits Horizons, and the question that we want to think about is, along with a without a deep pocket needing to raise funds, typically turns to the financial markets and offers future revenues. And it's true that the number of years worth of future revenues that the financial market is willing to lend against is typically between three to four and a half years. Now, we interpret that to mean the first three to four and a half years. That is to say, the credit horizon is quite short, even though the project, the underlying investment project, could be long-term. So, we would like to explain that and then use the explanation to ask the more macroeconomic question, might it be the case that a drop in long-term real interest rates leads to secular stagnation? And that question is that a lot of prompted by the experience of, I suppose, Japan and then countries in Southern Europe, and is becoming more prevalent around the world, that low interest rates don't seem to be delivering high growth. So, here's one slide to give the sense of our approach. As the government mentioned, human capital is centre stage here. Human capital of the entrepreneurs, or more generally, key workers. I'm thinking of the chief scientists in the research and development department. I'm going to call them engineers when I reach the model. Their human capital is essential for two things, constructing and then maintaining the production facilities. Now, their wage, I use the word wage. Incidentally, can you see my slides? Yes. You can, so you can see me on the slides. We cannot see. Now we can see. Okay, thank you. I'm particularly keen that you should see the slides. I think seeing me is a luxury too far. Okay, so I'm going to keep going. Please interrupt me if you can't see the slides. Now, the wage of these key workers, the entrepreneur or engineer, reflects their marginal contribution to productivity. And that is a contribution they make over the long horizon. That's the nature of contributing to productivity. That it has an effect not just in the immediate future, but it has a slow effect into the more distant future. I use the word wage in inverted commons here because this is not confused with the wages of perhaps more regular workers. Now, the human capital is inalienable, as the government mentioned. We put that right at the heart of it all. That's the one departure from Aroda Burr. What that means is that at the time of investment, an engineer or entrepreneur cannot credibly promise to work for less than that in suing wage. And what that means is that, and this next step is a little opaque at this point, I free admit, and I hope it'll become clearer when I'm done the modeling, that the implication of that is that the fund raise capacity of entrepreneurs is governed largely by their near-term revenues, as I said at the beginning. And now a fall in long-term interest rates means, or may mean, that funds, the fund raising capacity of an entrepreneur at the time of investment, may not expand as much as her investment cost, which has a longer duration. So paradoxically, her investment may be stifled. Now, normally we think of low interest rates as being a good thing for people who are raising money, but here it's time to turn out to be possibly a bad thing, and then when you've got it there's generally a different model that turns out to be stifled too. So here is a smaller company that I thought was real introspect. There is something that I don't want to talk about that makes me run. Maybe it's me that should have done that. Okay, the only thing is that the assumption we're investing in, which has been to speed date, a contender of our agents for the content we've got back there later, which is left to never ask. Now, each, I'm sorry, could you possibly ask me a few questions, can we, can you hear me? Yes, yes we can. Thank you. At each date, an engineer, and I'm going to prefer to, I can jointly produce two things, art and tools, and her inputs are two things as well, goods at normal. So at the beginning of a period of putting in goods, buying a building, the cost of the building is going to be actually that's important. And so the price of X plus Q, as she will produce at the end of the period, plant, or I call plant, which is a short hand for the equipment in some of the building. And then quite distinct from that, she will produce what I call a tool, or in here, a tool that's specific to Emma. Now, that's a short hand for an orientation of her human capital, increasing the capacity that she has to work in the future. So there are a few outputs. One of them, the tool is inalienable and can be competitive or sold, but the other can, and in fact that's the way Emma raises funds. She sells the plant to a saviour that's called him Sam. Remember these agents who drove during the course of their life, but at the moment Sam is a saviour and Emma is an engineer. And we want some wobbling reasons that the match between the plant and the engineer is not specific thereafter. So Sam, the plant owner, will be free to hire any Emma for maintaining his equipment subsequently, and indeed Emma, our Emma, is free to work at any saviour down the line. A little bit at your age, W. And the central part of Ara de Bra is that Emma cannot commit in advance to work for us and you. If she could, then she dealt in effect to raise more funds and increase the scale of her investment. Now, coming to Sam at each stage, he has planted productivity Zed. I should have said on the previous slide here that the starting productivity of any unit of plants at the time that it's first produced and installed is one. But Sam can choose the evolution of the productivity of a plant. So start one day where he starts the day with plant with productivity Zed. He can hire any number of tools that is saying any number of Emma's to produce goods and to maintain plants. So here again, we've got a constant return to scale correction function. The input is a unit of plant with productivity Zed. Any number of tools could be zero, H-grinding could be zero. Generates output goods and notice the output is not affected by the input of H. It's just that Zed produces the output according to this linear correction function. The role of H, the tool, is to maintain or possibly augment the productivity for next periods. Zed prime is next periods of productivity. And that's a combination of today's function. Zed prime is a function of today's productivity and the number of tools that Sam chooses to employ. Everything, the plant, the tools, even the building, we assume depreciates by fact lambda each period. Well, the subject of buildings rather oddly at the end of the buildings are supplied by farmers. That's just in order to keep the national accounting in the domestic economy clean and simple. They have the use of building themselves. One building generates F goods each period, as I said, depreciates by fact lambda. And that is that the international price of buildings will be Q, as promised. It's F divided by R minus lambda. Now, sadly, a plant owner has an option to stop. So the value of the plant, with productivity Zed measured at the end of the period, it's capital V on the left-hand side, is the discounted value of what Q is the failed price of buildings, or, that's if he shuts down, in other words, or he continues and then tomorrow he'll earn the revenue, A Z, minus the amount of weight he has to go for to the H. And then the day after tomorrow's productivity, or the end of tomorrow night's evenings, won't be this little complex production function inside Zed. So that's the prime tomorrow's productivity with appropriate depreciation lambda. Now, although this Bellman equation, as you call it, is pretty clean to read, it is actually surprisingly, we found, tricky to analyze. And it turns out that the plant owner, Sam, has a clear dichotomy between planning to shut down in the short term, in a finite period, capital T, mean it could mean in terms of capital T is his choice, or to completely differently to continue forever. And those choices would be different sequences of choices of H on his part. He could, if he's going to shut down in the medium term, then he won't bother to invest so much in terms of maintaining productivity, and productivity will slowly decline. I mean, in the literature, that's sometimes called the walk in there. On the other hand, he could choose to grow, sorry, not to grow, to continue forever, and thought about might be an auxiliary decision, shall I improve that productivity? Now, the relevance of all this to Emma, remember the entrepreneur who's trying to raise funds from Sam by selling the plant, is that Sam pays for V of one, remember one is the initial productivity of the plant, and B is the amount that Emma can borrow per unit. That's her, what we call borrowing capacity. Now, today's talk is just a short one, so I'm going to focus on, as it were, one half of the private space, a half in which, in fact, no Sam's choose to shut down. The other half is particularly rich and interesting, but I don't have time to talk about it today. Now, the feature of equilibrium is that no plant then stops, and because we've set things up, that there really is a one-to-one ratio, H per plant, the number equilibrium statement there, the number of units of plant, sorry, units of tools per plant is one, and given the nature of the product, the conduct of this function for generating Z prime tomorrow, it means that Z stays at one, it starts at one, it stays at one forever, and therefore, the output per unit of plant is in PA every period. Okay. Now, this slide is complicated. I won't go into the minutiae, but I do want to get away some of the important in duration. The first equation here is Sam's choice of how many is the first order condition for Sam's optimal choice of H, which remember, in equilibrium will be one in order to clear the market for labor, but W in other words is in equilibrium, number which induces Sam to choose H equal to one. The right-hand side, so that's Sam's model of cost of labor. The right-hand side is the marginal product of labor, which is crucially a forward-looking animal. Why? Because any tool or emmer that Sam employs today will enhance tomorrow's productivity, and that'll have an immediate impact on tomorrow's output, but it will also have an indication for what makes it today, Monday. It'll have an indication for Wednesday's output and Thursday's output and so forth. So the range, critically, is a function of R, and let me put my cards on the table, as R drops, and that's the thought experiment I'm going to have in mind, that will push the way forward. Now, meantime, B, remember, that's what Sam is willing to pay, or we have on, is what Sam is willing to pay for a unit of plunge at the beginning, let's say on Saturday night, is what is left of A after Sam has paid the wages of the employees or the engineers that have been maintaining it. Now, at any point, the productivity is the cumulative contribution of workers who've been employed or tools that have been employed to that rate. Now, the word cumulative is crucial, because as time goes by, and we go from today, Monday to Friday, Saturday, et cetera, the fraction of the cherry, the cake, that is left for Sam after he's netted out the cumulative wages that he's paid to the various engineers en route, Monday, Tuesday, Wednesday, the size of the cherry that's left for Sam shrinks. In other words, starting from the moment that he buys the plant, Sam anticipates that his residual care of revenues, that is, once he's contracted wages, will be problem. Another way to think about that is this curved line on the horizontal lines if we've got time. This curved line is Sam's share of output, which is of height A over here. It's Sam's share of output going to him after he's netted out wages, and that share of output drops. And ultimately, most of the productivity out here towards the high values of T, most of the productivity really can be attributed to the cumulative contribution of, as it were, generations of embers that have been maintaining the plant. You might think, like I say, and one other little thing to glean from the diagram is F, which is the opportunity cost of the building. Not just beyond a certain point, Sam appears to be losing money. His share at downward slope and curve dips below the opportunity cost. But that's mirage because at any moment, the wages that have contributed to the output at that time are for the by-large sum. And so the crucial position that Sam faces as to whether or not to continue is his ongoing flow of revenue of wages have been subtracted, which is A minus W, is it bigger than F? If it is, he should continue. So although the red area is there, in truth, Sam does not want to shut down that period. Do you want, he wants to carry on forever in the of the plant space, uninterested. And then this effect can be strong enough to overcome any rising net worth to stifle investment in growth. So this rather stylized equation is taken from a sort of generic equation one sees in any macro-finance model, where on the left-hand side you've got investment. And on the right-hand side, you've got the savings rate later. And then this critical ratio on the top line of net worth of the people who are doing investing, which is that day's worth of embers, their net worth, which they plow into investment. The total cost of investment is X plus Q, as I said earlier. They're borrowing P. Now, in the fourth experiment I'm interested in, which is suppose there's a permanent production in the real interest rate, that pushes Q up, big time, because Q is that this long-run discounted value of all the buildings. What it doesn't do, though, is to push up B commensurately, because B is the A by B short-term revenues that Sam is anticipating getting hands on. And therefore, the denominator of this ratio can go up when interest rates fall. In fact, because of leverage here, it could go up by quite a bit and more than any possible increase in the net worth. The literature on macro-finance is heavily concentrated on the enumerator. Our paper today is one which I guess we could call the denominator. All right, so to close. The upshot is that within my chosen part of the parameter space, there's a subset of it, namely particularly where lambda and r are not too far from 1, where a reduction in the real interest rate causes growth to go down. And we see that as the parable, possibly to describe what's been happening in Japan for the last 30 years or a plus, and what's been happening, let's say, in southern Europe since the advent of the euro. In southern Europe, when the euro was introduced in the early 2000s, the markets probably anticipated that real interest rates were going to stay low permanently in the new way. Q, the price of buildings, shot up. And our thesis today is that B, the borrowing capacity of investors, does not shoot up as much, proportionately speaking. And at the end of it all, this denominator that I'm focusing on goes up, and the overall ratio goes down. And in fact, we find that everybody, this is really, I think, quite striking, everybody in the investing economy is worse off. It's clear that the savers are worse off because the interest rate's gone down. But so too are the engineers. Remember, people take turns to be one or the other. The reason why the engineers are worse off is because their leverage rate of return that's issued here is going to go down on account of this more dramatic issue compared to this. So just the last thing to show you is the diagram. Here's a thought experiment where at age five, among the horizontal, there's an unexpected fall in the interest rate from two and a half to one and a half percent, which is assumed to persist. So this is a model without any aggregate uncertainty. It starts with a single shock. Now, the immediate effect is that Q shoots up. And the value of investment, which of course includes the value of the buildings, really shoots up. Investment itself can actually shoot up a little because of the net worth effect that Nero and I put on. I think they're like 97. Okay, so investment shoots up. Consumption goes up for the similar sorts of reasons that there's a hike in net worth, which causes people to eat more, so to speak. Output goes up too. So it all looks good. This looks like a moon. But actually, it has within it the seeds of stagnation. The real, the good aspect, that is to say, the engine for growth, which is tool and humanity plan, are in a short order shrinking on account of that denominator effect that I told you about. And that persists into the long run. So although there's a boom, a temporary boom, in the long run, there is stagnation. So output assumes a lower growth of service investments, so does consumption ultimately. And in welfare terms, as I say, everybody can be worse off. So turning back to Japan and Eastern and Southern Europe, at the time it looked like a boom when interest rates fell. But actually, with the benefit of the hindsight, it looks more like a form of stagnation. Okay, well, thanks so much, Sean, for this wonderful talk. And once again, we are very privileged to have you and this shows our laureates today for great talks.