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Hi. In this set of lectures we're talking about economic growth. And what we want to

Â do is we want to understand why is it that some countries are rich and some countries

Â are poor. So to get our bearings on how growth works, we've gotta start with a

Â much simpler model. So economic growth models are gonna have a lot going on.

Â Gonna have labor, they're gonna have physical capital, gonna have depreciation

Â rates and saving rates and all sorts of stuff. So to just sort of get us to

Â understand the basics of growth, we're gonna start out with a much simpler case.

Â And we're gonna start by talking about just compounding. So you put money in the

Â bank, and we talk about the rate at which that grows. From then we are gonna talk

Â about then countries growth to make the GDP growing and we're gonna see why

Â different growth rates are so important cuz we're gonna see that growth is sort of

Â exponential, so we're gonna talk about a very simple sort of exponential growth

Â rate. We just keep putting money in the bank from that we're going to learn a

Â really cool trick called the Rule of 72. The Rule of 72 will tell us how quickly

Â our money will double or how quickly GDP will double. So there if you think about

Â it, what's the difference between an eight percent growth rate and a four percent

Â growth rate and I think it's twice as much. We'll actually see from the Rule of

Â 72, that it's even more than that. Doubling the growth rate has really

Â significant effects. Okay, so let's get started, so let's start with just sort of,

Â like, you know. Basic accounting 101, you know, you put some money in the bank. So

Â suppose you've got X dollars and you put it in the bank at R percent interest. How

Â much do you get? Let's suppose you get 100 dollars and you put it in the bank. And

Â let's suppose you get five percent interest. Well, what you're gonna get at

Â the end of the next year is 105 dollars. Right? Because the general formula for

Â this thing is, you just take X times one plus the interest rate. Right? So, that's

Â the general formula. So you put 100 dollars in the bank at five percent, I'll

Â get 105 back. Now if I put that 105 back in the bank I'll get 105 time times. One

Â plus.05 and that's gonna be 105 plus 5.25, which is 110.25. So, in the two years I'll

Â have 110.25. Now, if I kept this money in the bank for ten years, right, I'd just

Â have 100 times one plus.05. [sound] Raised to the tenth power, right? Because they

Â just keep multiplying this by 1.05, times 1.05, times 1.05, and that's what I get.

Â So, that's - you what them to think about if I buy a certificate of deposit for some

Â money, the bank say okay, I'm going to put this thousand dollars in for six years at

Â five percent. They'll tell you, okay, well then you're going to get a thousand

Â dollars times 1.05 raised to the sixth power. That's how much you'll get back at

Â the end of the six years. You want to do that same thing -- the same very, very

Â simple thing with GDP. Now with GDP, what we're going to do is that instead of setting X to be the

Â amount of money we put in the bank, that's going to be per capita GDP. If there's a

Â GDP right now of G, and we have R percent growth, then next year we'll get one Plus

Â R. And in ten years, we'll have one plus R raised to the tenth power. Now, why does

Â that matter so much? Why do we care so much about this R? Why do politicians

Â always talk about it? Why do bankers always talk about it? Why do we care so

Â much about growth rates? Well, to see why, let's look at two cases. Let's look at a

Â sort of low growth case, the country that has a two percent growth rate. And a high

Â growth case, a country that has a six percent growth rate. Let's start'em out

Â both in year zero, with everybody making $1,000. So per capita income's $1,000.

Â Well, what happens after [inaudible] the first year, the first country goes up by

Â twenty%, so it's 1020, and the other countries at 1060. Now, you could say, big

Â deal. $40 more per person. That's not a huge difference. Well let's go ahead ten

Â years. In ten years, if I use that formula, the people in the first country

Â are making $1200 apiece. The people in the second country are making $1800 apiece.

Â So now, they're 50 percent better off and if I go ahead 35 years, right, so really

Â maybe one generation, maybe a generation and a half, the first country has now doubled. So

Â they're now at 2,000. The second country's at 7600. They've gone up 3.8 times. So now, they're

Â almost -- or 7.6 times, I'm sorry -- so they're almost four times better off.

Â Let's suppose I go ahead 100 years, move ahead a century. One country plugs along

Â at two percent growth. The other country plugs along at six percent growth, right?

Â The first country's now making $7,000 per person. The second country, right? People

Â are making $339,000 per person. Right? So that's like forty five times as much. So

Â in a hundred year period, this two percent versus six percent difference just becomes

Â enormous, and that's because this growth is exponential, right? Cuz that's one plus

Â our raise to the power of T, and so if R is bigger, you get a huge increase. So

Â here's the Rule of 72. And this explains sort of why what was going on was

Â going on like that. The Rule of 72 says divide the growth rate into 72. And the

Â answer you get, will give you the number of years it takes to double, right? So

Â let's suppose that our growth rate is two percent, right? If our growth rate is two

Â percent I take seventy two, divide it by two and I get thirty six, that means it

Â will take about thirty six years to double. Let's go back to our graph, go

Â back to our graph you see it took thirty five years to double, so pretty close

Â right? What if I had six percent. Well 72 divide by twelve I'm sorry, divided by six

Â means it's gonna be twelve years to double. So what that means is, in this

Â first period there's two%. It's gonna take me 36 years to double. At six percent, no

Â it takes me twelve years to double. Which means that this country will double three

Â times. Which is two times two times two. Which is eight. So its GDP will be eight

Â times its original GDP, in the time it takes the first country to double. And if

Â we go back and look at our data, sure enough after 35 years it's effectively

Â eight times as big. So you see the rule of 72 isn't exactly right. Like, it took only

Â 35 years to double. And this isn't quite at eight thousand. But it's really

Â accurate. So for low interest rates, it tends to, you know underestimate --

Â overestimate the number of years and for high interest rates it tends to

Â overestimate the number of years. But at eight percent, nine percent it works just

Â about perfectly. So the rule of 72, right, again, which is really cool -- so it's just,

Â take your growth rate and divide it into 72, that tells you how long it's gonna

Â take to double. So the move from two percent to six percent isn't just a four

Â percent increase in the growth rate. Right? It's a dividing by three of the

Â time to double. So it means that every twelve years your country's gonna double

Â its well being, its GDP. Whereas in the first case at two percent it's gonna take

Â 36 years. That's why this... people focus so much on growth rates and that's why we

Â wanna look at models that explain where growth comes from. Let's go back and look

Â at the United States. Remember we're hanging out at about three, four percent. Well, to think,

Â what's the difference between three, four percent. Well four percent is 72 divided

Â by four, which means every eighteen years we'll double. And three percent is 72

Â divided by three. Which means every 24 years we'll double. Well, what would you

Â rather do? Double every eighteen years, or double every [laughs] 24 years? Clearly you'd

Â rather double every eighteen years. So that's why we care a lot about boosting

Â that growth rate. Even from something like three percent to four percent.

Â Because that means we are going to increase our well being much, much faster.

Â Okay, whew, exit. There's a lot going on here, right? We've talked about this sort

Â of you know, simple interest rate thing where we've got X right, [sound] times one

Â plus R, raised to the t power. Well this is sort of a cheat here because what

Â we've done is I've just assumed for the interest of simplification that the growth

Â is happening just once a year. So it's like once a year we do growth rates. When I was a kid

Â there was a commercial on television for a bank and they talked about how some banks

Â only gave interest once a year and it said, this new bank, we give interest every

Â second of the day. So what we do is, instead of saying, okay we're going to

Â give you at the end of one year, X times one plus R, we're going to give you

Â interest, let's just say, suppose first we're going to give you interest every

Â say, so we're going to give you one plus R over 365. So we're going to reduce the

Â rate, divide it by 365. We're going to give it to you 365 times. So we're going

Â to compute your interest daily and then they said, we'll do even better than that,

Â we're going to do it, we can even do it hourly, so we'll do it over, we're going

Â to divide the interest rate by the number of hours in the day and then we'll

Â compute this interest every hour. And then we can even do it every second and so on

Â and so on. And they show this guy at a calculator that's pluging away, right,

Â [laugh], computing these interest rates and I thought how are they doing that? Well

Â the way they did it, is that they just used math. It turns out if you have this

Â formula, and this should be an infinity here. If you have the number of periods go

Â to infinity, right so if you're doing it infinitely fast. Then what happens is,

Â this formula, this one plus the interest rate over N, raised to the power NT, just

Â becomes E to the RT. And remember E was that number Euler's constant, which is 2.71828.

Â So why do we do this? Why do we do all this math? Reason we are doing all this

Â math is basically you could think of the growth rate, instead of thinking of this

Â formula -- you know X times, one plus R to the T -- you can do something simpler. You

Â can just use E to the R T, where E is this Euler's constant, this 2.71828. So what

Â you can do is if you think about growth as just sort of occurring continuously, then

Â this nice simple formula will give you sort of the rate at which things are gonna

Â grow, and that's why it's called exponential growth. Because the rate at

Â which you grow is exponential in this function, in this number e. So it's E

Â raised to the exponent R T. So why is that so important? The reason it's so important is, let's

Â go back, remember we talked about linear functions, in the previous set of

Â lectures, right? So remember linear function looks like this. So that would

Â mean that growth would sort of go ten right, eleven twelve thirteen fourteen and

Â so on, right. Exponential growth goes like this, it zooms up. Even faster than

Â something like X squared. So what that means that if you grow at a ten percent

Â rate, right, you're not just going to go ten, eleven, twelve, thirteen, fourteen,

Â fifteen, sixteen, right. In fact, if you grow at a ten percent rate, in seven years. Right.

Â Remember the rule of 72. 72 divided by ten is equal to seven. In seven years, what

Â you're gonna do, is you're going to double. So you'll be twice as well off as

Â you were before. [sound]. What have we learned? We've learned that if we get a

Â growth rate of, say, let's say three percent, four percent, five percent.

Â Right, that, that can lead to significantly better, you know higher GDP

Â down the road, than if the growth rate is just a little bit smaller. And the reason

Â why is because we've got this exponential growth. The world is not, the outcome is

Â not going to be sort of linear in these growth rates. It's going to be exponential

Â over time. So what you'd like to do is sustain a higher growth rate. So what we

Â want to do next is construct some models of growth where we can see how this

Â economic growth depend on things like saving, depreciation and technology. Okay,

Â thank you.

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