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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.