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.