Hi, during this set of lectures. I talked about how, in the past, Japan had, had extremely high levels of growth. Remember in the 1950's to 1970's, early 1970's, you see that Japan has growth rates that hang out in the seven to eight percent range. I also talked about how China is currently experiencing levels of growth that are equally high. In the same sort of range eight to nine percent. And there's a question, can China continue to sustain those levels of growth? But what we want to see, is we want to use our very simple growth model to see why countries like Japan and the post [inaudible] can have incredibly high growth. Why China during this era of industrialization can have incredibly high growth. But why would we, we should be somewhat dubious about the prospects of China being able to maintain these levels of growth, unless, they're able to somehow have massive increases in technological improvement. So we'll see that if you're catching up. To other countries. And it's highly likely that you can have, experience super high growth levels just like the ones we see here. But once you've caught up to other countries it's gonna be hard to maintain those growth levels unless you somehow can support massive amounts of innovation. So let?s see how this works we're going to do it just by doing some math so we use slightly different numbers than we used before with the exact same argument so here what we're going to do is we're going to assume that the savings rate is twenty%. The depreciation is ten percent and we're also going to assume a much larger economy, so we get bigger numbers. So instead of assuming that there's a 100 people, so the square root of that is ten, we're going to assume that there's 10,000 people so that the square root is 100. We're also going to start with a lot more machines, we're going to start with 3.600 machines, as opposed to just six, so this is a bigger economy and it that will allow us to get slightly more reasonable. Numbers when we work through the math. So remember how this works right, what we do is, we compute total output. We see how many new machines then get invested in. We see how many machines get depreciated. We solve for the new number of machines, and after we solve for the new number of machines, then we can solve for the output, next period. So let?s get started here we go 3,600 machines so output is 100 times the square root of 3,600 so that's 100 times 60 so that means output is 6,000 so let's think of that as per-capita. Income of $6,000 dollars. So how much do they need to invest? Well investment is going to be.2 times 6,000 so that's 1200, they're going to invest in 1200 new machines. The depreciation is ten percent of the existing machines of 3600 which is 36, 360 so if we net those out we get 840 new machines. And with those 840 new machines that means the new machines the next period is going to be 4,440 so that's we got the first period. Output is 6,000 and now they've got 4,440 new machines, so let's go to the next period, the next period we're going to start out with 4,440 machines and what we're going to do is then we're going to compute the output in this set there we go the square root of 4,440 is approximately. 67. So we're gonna get 100 X the square root of 4,440, which is 100 X 67, which is 6,700. So investment in new machines is gonna be twenty percent of that, which is gonna be 1,000. 340. And our depreciation is gonna be ten percent of this which is roughly 440. So we're gonna invest in about 900 new machines. So, the new machines next time is going to be 5,340. What we can also do is we can look at, what is our growth rate. So, currently we've got $6,700 per person. Before we had a GDP of 6,000. So that's roughly eleven percent growth. So this is a growth rate not unlike, a little bit higher, but not unlike what we currently see in China. We went from 6,000 to 6,700 and now we've got 5,340 machines. Let?s go ahead in one more period we've now had [laugh] 5,340 machines we want to ask what's output going to be and what's growth going to be this is just some very basic math. So if we take the output as 100 times the square root of 5,340 [inaudible] again that's going to be 100 times turns out again do this your [inaudible] about 73 so that's 7,300 that's going to be GDP per capita. Investment in new machines, is twenty percent of that which is 14-. 160. That's twenty percent of 7300. And depreciation is ten percent which is 530. So we are gonna add 930 new machines. Which is gonna gave us 60 to 70 new machines. What's our growth rate. Our growth rate we had 6700 before now 7300 went up 600 which is about. Eight to nine percent. So what we look at, is, in the previous period, we had eleven percent growth, now we've got nine percent growth. So what we're seeing is consistently high levels of growth, because of the fact that, relative to how much labor you've got, the amount of capital is pretty low. So just by investing and saving, you can maintain fairly high levels of growth. Now let's jump ahead. Let's suppose that we've been accumulating capital. We've gone from 3,600 to 4,400 machines to 5,300 machines to 6,200 machines. And now, we're all the way up to 10,000 machines. We can ask, what's gonna happen to output. And we can ask even what's gonna happen to growth. So, let's again, do the math. The output is gonna be 100 times the square root of 10,000, which is 100. That means, now, per capita GDP, let's just say is 10,000. What is investment gonna be? Investment is gonna be. Twenty percent of that which is two thousand, but what is depreciation going to be. Depreciation is ten percent of ten thousand machines, which is a thousand, so that now we're going to have eleven thousand machines next period. Well if we have eleven thousand machines next period, we can ask what?s growth going to be well that's going to be a hundred times the square root of eleven thousand is what g d p is going to be so that's going to be a hundred times approximately a hundred and five. Which is 10,500. So GDP is going to go from 10,000 to 10,500. But what sort of growth rate is that? Well that's only five percent growth. So what you can see is when we went from 3,000 to 4,000, to 5,000 to 6,000, during that period, the economy is sustaining, you know, ten, eleven, eight, nine percent growth. Really high growth rates. Just like those growth rates we're seeing in China. Once the machine number gets to 10,000, now suddenly the growth rate drops to five percent. But we're still adding net, 1000 machines. Well, let's keep going. Let's go ahead, you know, another 10-20 years. And let's suppose now, the economy's got 22, 500 machines. And let's ask what happens here. What's, what's total output gonna be? Well, that's gonna be 100 times the square root of 22-5. Which is 100 times 150 so now, we've got 15,000 per capita GDP. That's really good. How much are we gonna invest when investment is gonna be twenty percent of that which is 3,000. That's a lot of new machines. But what's depreciation gonna be? It's gonna be ten percent of that which is. Of the 22,000 machines we have. Which is 225. I'm sorry. 2,250 machines. So that means net we add 750 machines. So that gives us 23,250 machines. Well now we can ask how many, what's our output gonna be next period. Well next period our output is gonna be a 100 times the square root of 23,250. Well how much is that? Well, that's 15,000. And 250, ballpark. So now GDP's only gone up 250, and that's really like, just a one to two percent increase. So growth is now in the one per, one to two percent range. So we've seen this economy, let's step all the way back a second. When we started out with 3,600 machines, output was 6,000. Then we had eleven percent growth. Then we had eight to nine percent growth. So this economy's growing at a fairly fast clip. Once it gets to 10,000 machines, though, now GDP only grows at five%. And once it gets to 22,000 machines, now GDP only grows at one to two percent. What's going on? Well number our curve output if you have a fixed technology is going to be concave, so there's this fixed region here where you see ten percent growth. Early on when you don't have enough capital given your amount of labor you can get massive amounts of growth but once you get in this region here growth is going to fall to one to two percent it's going to fall off unless, right remember from our mono- unless you get lots of investment in new technology, so if you look at a country like China. With really high growth rates and ask what's causing that. And you can think, well maybe what's causing it is they don't have enough capital, given how much labor they've got. In the case of China that's true. So they're at this part of the curve. So they're experiencing incredibly high growth. When they get to this part of the curve, then, most likely, you not going to see the same, sort of, high growth rate. It's going to look a lot like, more like the picture in Japan. So for them to sustain high growth, they can't just continue to pour money into capital. What they're going to need to do is they're going to need to change technology they're going to need A to increase, now increases in A necessary to give up eight to nine percent growth just have never been seen before so it would be remarkable if technological advances occurred at such a rate that they can maintain this level of growth once their capital levels get to the same level of other countries. What the model tells us, is we're much more likely to see China follow a path similar to Japan. Where we see high growth rates for a period but as capital accumulates. Those growth rates fall over time. So what have we done? What we've done is kind of fun: taken a very simple growth model, Solo growth model. We've looked at data from China and Japan and seen how China has these incredibly high growth rates now. Japan used to have incredibly high growth rates. And we've seen from our model, that if a country's undercapitalized, that's their levels of capital are low level to the amount of labor they've got, then you can sustain super high growth rates for quite a while as you accumulate capital. But once your capital levels become sort of appropriate for the technology your growth rate is going to fall off and the only way you can sustain that growth rate is through technological innovation. By driving up in solos models that parameter A. This doesn't mean that China's not going to continue to have growth, or that Japan is all growth will end. Or that the United States, or Europe all growth is going to end. What it says is there's two different ways to think about growth. One type of growth, what we're seeing in China now and what we saw in Japan post war. And what we saw in Europe and the United States post war, is growth that occurs through capital accumulation. Another type of growth, is what we're, which is what we see in the United States and Japan and Europe now, but not in China, occurs from technological advances, not from buildup of capital. And as you advance technology and you increase that A term, then it makes sense to. Buy more capital, but different types of capital and that's what drives growth. Okay [laugh] lot going on here lot of fun though right? What we're going to talk about next, final lecture, is why then are some countries poor? This, this seems to make a lot of sense just invest in capital if you invest in capital you get lots of growth you catch up with everybody else why is it still the case that some countries are poor if we understand where growth comes from? Thanks.