Hi. In this set of lectures we're talking about economic growth. And what we want to do is we want to understand why is it that some countries are rich and some countries are poor. So to get our bearings on how growth works, we've gotta start with a much simpler model. So economic growth models are gonna have a lot going on. Gonna have labor, they're gonna have physical capital, gonna have depreciation rates and saving rates and all sorts of stuff. So to just sort of get us to understand the basics of growth, we're gonna start out with a much simpler case. And we're gonna start by talking about just compounding. So you put money in the bank, and we talk about the rate at which that grows. From then we are gonna talk about then countries growth to make the GDP growing and we're gonna see why different growth rates are so important cuz we're gonna see that growth is sort of exponential, so we're gonna talk about a very simple sort of exponential growth rate. We just keep putting money in the bank from that we're going to learn a really cool trick called the Rule of 72. The Rule of 72 will tell us how quickly our money will double or how quickly GDP will double. So there if you think about it, what's the difference between an eight percent growth rate and a four percent growth rate and I think it's twice as much. We'll actually see from the Rule of 72, that it's even more than that. Doubling the growth rate has really significant effects. Okay, so let's get started, so let's start with just sort of, like, you know. Basic accounting 101, you know, you put some money in the bank. So suppose you've got X dollars and you put it in the bank at R percent interest. How much do you get? Let's suppose you get 100 dollars and you put it in the bank. And let's suppose you get five percent interest. Well, what you're gonna get at the end of the next year is 105 dollars. Right? Because the general formula for this thing is, you just take X times one plus the interest rate. Right? So, that's the general formula. So you put 100 dollars in the bank at five percent, I'll get 105 back. Now if I put that 105 back in the bank I'll get 105 time times. One plus.05 and that's gonna be 105 plus 5.25, which is 110.25. So, in the two years I'll have 110.25. Now, if I kept this money in the bank for ten years, right, I'd just have 100 times one plus.05. [sound] Raised to the tenth power, right? Because they just keep multiplying this by 1.05, times 1.05, times 1.05, and that's what I get. So, that's - you what them to think about if I buy a certificate of deposit for some money, the bank say okay, I'm going to put this thousand dollars in for six years at five percent. They'll tell you, okay, well then you're going to get a thousand dollars times 1.05 raised to the sixth power. That's how much you'll get back at the end of the six years. You want to do that same thing -- the same very, very simple thing with GDP. Now with GDP, what we're going to do is that instead of setting X to be the amount of money we put in the bank, that's going to be per capita GDP. If there's a GDP right now of G, and we have R percent growth, then next year we'll get one Plus R. And in ten years, we'll have one plus R raised to the tenth power. Now, why does that matter so much? Why do we care so much about this R? Why do politicians always talk about it? Why do bankers always talk about it? Why do we care so much about growth rates? Well, to see why, let's look at two cases. Let's look at a sort of low growth case, the country that has a two percent growth rate. And a high growth case, a country that has a six percent growth rate. Let's start'em out both in year zero, with everybody making $1,000. So per capita income's $1,000. Well, what happens after [inaudible] the first year, the first country goes up by twenty%, so it's 1020, and the other countries at 1060. Now, you could say, big deal. $40 more per person. That's not a huge difference. Well let's go ahead ten years. In ten years, if I use that formula, the people in the first country are making $1200 apiece. The people in the second country are making $1800 apiece. So now, they're 50 percent better off and if I go ahead 35 years, right, so really maybe one generation, maybe a generation and a half, the first country has now doubled. So they're now at 2,000. The second country's at 7600. They've gone up 3.8 times. So now, they're almost -- or 7.6 times, I'm sorry -- so they're almost four times better off. Let's suppose I go ahead 100 years, move ahead a century. One country plugs along at two percent growth. The other country plugs along at six percent growth, right? The first country's now making $7,000 per person. The second country, right? People are making $339,000 per person. Right? So that's like forty five times as much. So in a hundred year period, this two percent versus six percent difference just becomes enormous, and that's because this growth is exponential, right? Cuz that's one plus our raise to the power of T, and so if R is bigger, you get a huge increase. So here's the Rule of 72. And this explains sort of why what was going on was going on like that. The Rule of 72 says divide the growth rate into 72. And the answer you get, will give you the number of years it takes to double, right? So let's suppose that our growth rate is two percent, right? If our growth rate is two percent I take seventy two, divide it by two and I get thirty six, that means it will take about thirty six years to double. Let's go back to our graph, go back to our graph you see it took thirty five years to double, so pretty close right? What if I had six percent. Well 72 divide by twelve I'm sorry, divided by six means it's gonna be twelve years to double. So what that means is, in this first period there's two%. It's gonna take me 36 years to double. At six percent, no it takes me twelve years to double. Which means that this country will double three times. Which is two times two times two. Which is eight. So its GDP will be eight times its original GDP, in the time it takes the first country to double. And if we go back and look at our data, sure enough after 35 years it's effectively eight times as big. So you see the rule of 72 isn't exactly right. Like, it took only 35 years to double. And this isn't quite at eight thousand. But it's really accurate. So for low interest rates, it tends to, you know underestimate -- overestimate the number of years and for high interest rates it tends to overestimate the number of years. But at eight percent, nine percent it works just about perfectly. So the rule of 72, right, again, which is really cool -- so it's just, take your growth rate and divide it into 72, that tells you how long it's gonna take to double. So the move from two percent to six percent isn't just a four percent increase in the growth rate. Right? It's a dividing by three of the time to double. So it means that every twelve years your country's gonna double its well being, its GDP. Whereas in the first case at two percent it's gonna take 36 years. That's why this... people focus so much on growth rates and that's why we wanna look at models that explain where growth comes from. Let's go back and look at the United States. Remember we're hanging out at about three, four percent. Well, to think, what's the difference between three, four percent. Well four percent is 72 divided by four, which means every eighteen years we'll double. And three percent is 72 divided by three. Which means every 24 years we'll double. Well, what would you rather do? Double every eighteen years, or double every [laughs] 24 years? Clearly you'd rather double every eighteen years. So that's why we care a lot about boosting that growth rate. Even from something like three percent to four percent. Because that means we are going to increase our well being much, much faster. Okay, whew, exit. There's a lot going on here, right? We've talked about this sort of you know, simple interest rate thing where we've got X right, [sound] times one plus R, raised to the t power. Well this is sort of a cheat here because what we've done is I've just assumed for the interest of simplification that the growth is happening just once a year. So it's like once a year we do growth rates. When I was a kid there was a commercial on television for a bank and they talked about how some banks only gave interest once a year and it said, this new bank, we give interest every second of the day. So what we do is, instead of saying, okay we're going to give you at the end of one year, X times one plus R, we're going to give you interest, let's just say, suppose first we're going to give you interest every say, so we're going to give you one plus R over 365. So we're going to reduce the rate, divide it by 365. We're going to give it to you 365 times. So we're going to compute your interest daily and then they said, we'll do even better than that, we're going to do it, we can even do it hourly, so we'll do it over, we're going to divide the interest rate by the number of hours in the day and then we'll compute this interest every hour. And then we can even do it every second and so on and so on. And they show this guy at a calculator that's pluging away, right, [laugh], computing these interest rates and I thought how are they doing that? Well the way they did it, is that they just used math. It turns out if you have this formula, and this should be an infinity here. If you have the number of periods go to infinity, right so if you're doing it infinitely fast. Then what happens is, this formula, this one plus the interest rate over N, raised to the power NT, just becomes E to the RT. And remember E was that number Euler's constant, which is 2.71828. So why do we do this? Why do we do all this math? Reason we are doing all this math is basically you could think of the growth rate, instead of thinking of this formula -- you know X times, one plus R to the T -- you can do something simpler. You can just use E to the R T, where E is this Euler's constant, this 2.71828. So what you can do is if you think about growth as just sort of occurring continuously, then this nice simple formula will give you sort of the rate at which things are gonna grow, and that's why it's called exponential growth. Because the rate at which you grow is exponential in this function, in this number e. So it's E raised to the exponent R T. So why is that so important? The reason it's so important is, let's go back, remember we talked about linear functions, in the previous set of lectures, right? So remember linear function looks like this. So that would mean that growth would sort of go ten right, eleven twelve thirteen fourteen and so on, right. Exponential growth goes like this, it zooms up. Even faster than something like X squared. So what that means that if you grow at a ten percent rate, right, you're not just going to go ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, right. In fact, if you grow at a ten percent rate, in seven years. Right. Remember the rule of 72. 72 divided by ten is equal to seven. In seven years, what you're gonna do, is you're going to double. So you'll be twice as well off as you were before. [sound]. What have we learned? We've learned that if we get a growth rate of, say, let's say three percent, four percent, five percent. Right, that, that can lead to significantly better, you know higher GDP down the road, than if the growth rate is just a little bit smaller. And the reason why is because we've got this exponential growth. The world is not, the outcome is not going to be sort of linear in these growth rates. It's going to be exponential over time. So what you'd like to do is sustain a higher growth rate. So what we want to do next is construct some models of growth where we can see how this economic growth depend on things like saving, depreciation and technology. Okay, thank you.