I couldn't connect but
so the Fed just hiked by 25 basis
points.
And
as people expected, you know, this is
the way that it works when there's lots
of uncertainty essentially
the Fed starts communicating
what's going to do
and the communication was still very
clear that
that 25 basis points was
to be expected and and apparently I was
reading this right now. It was released
at
3 minutes ago, 4 minutes ago.
Um
they also said that that further hikes
are no longer guaranteed. So remember
that we saw that expected
hikes sort of we saw several several
expected hikes for the next few months
before
the SVB mess and right after it we sort
of saw the whole thing declining and and
at least the minutes are consistent with
that. Um so there we are. So not big
uncertainty I mean the markets are
rallying or something like that at least
for the next 10 minutes or so but uh
we shall see.
Anyway, so but today we we're going to
really start
I'm going to I'm going to show you sort
of the first model of economic growth.
Uh
And
before I do that, who knows who that
person is?
No? No clue?
He
actually he's Robert Solow. He was an
He's an emeritus professor at MIT.
Together with Paul Samuelson essentially
he's responsible for building the
economics department at MIT. And he won
the Nobel Prize in 1987.
I was a student then here. Uh and
and
for his work primarily for his work on
economic growth. And so what we're going
to do in the next two three lectures are
essentially things that Bob Solow
developed many many years ago.
The basic mechanism, you know, remember
that we had this Keynesian cross before
where we have this multiplier in the
goods market and aggregate demand
feeding into income and so on and so
forth. That was sort of the start
mechanism in in in short-run macro.
In long-run macro growth theory
this is sort of the the key mechanism
and and you can think of it as the
following. At any point in time
an economy has, you know,
factors of production primarily labor
and capital.
That capital stock, labor is more or
less fixed or depends on population
growth, things that are sort of
difficult to to control or they're not
really that endogenous to to economics.
Not at least in the current times. Many
centuries ago yes they were. We had this
Malthusian theories in which you know
population growth determined determined
growth because
food is scarcity and stuff like that but
that's no longer the case fortunately.
Uh for in most parts of the world. So
but what can what can change over time
and quite a bit and it depends on
economic decisions is the capital stock.
But at any point in time there is
certain capital stocks which combine
with labor give you some certain output.
Output is income.
Part of that income will be saved as
we're seeing
and that those savings will be used for
investment. Okay?
But investment is nothing else than
capital accumulation.
So
this income will lead to saving which
will fund investment which will change
the stock of capital will feed into
capital stock that will feed into income
and so on. All this is happening very
slowly because the capital stock
accumulates slowly. I mean
but but but this is what is happening
and so all the models we're going to
look at certainly the model we're going
to look at in this lecture is all about
this mechanism. Okay?
So let's remember what we did in the
previous lecture. We
uh
and I'm going to assume that population
is constant. I'm going to relax that at
the very end but assume that the
population is constant and equal to n
and remember we're not worrying about
unemployment and stuff like that here.
Um so output per capita or per person
uh is y over n and we remember we had an
production function f of k and n then
because of constant returns to scale we
could divide by n on both sides
everything and we ended up with this
relationship. So output per person is
equal to an is is a is an increasing
function of capital per person. It's an
increasing function of capital per
person but it's also concave
function
of capital per person. Why is it
concave?
That is why is it increasing at a
smaller pace?
Uh yeah.
Decreasing
marginal product of capital exactly.
You know, for fixed amount of labor the
more capital you put in in into
production well
output keeps expanding but by less and
less because it has less and less labor
to work with each each unit of capital.
Perfect. That's very important. Uh then
let's we're going to work in closed
economy. I haven't opened it. I'm going
to do that after
uh
quiz two.
Um
So and I'm going to assume also no
public deficits so g equal to t capital
T.
And in that case then we know that uh
private investment private savings equal
to private investment. Okay? That's
that's the way we derive the IS curve.
Um so that's that's that's not new.
I'm going to modify a little bit what we
did in the short run.
Uh um and I'm going to assume that that
savings is proportional to income. So
savings little s times y.
Notice that that this is
is different from what we did in the
short run. In the short run remember we
had a c0 floating around. We had a
constant in the consumption function. So
savings which was equal to income minus
consumption also had a constant floating
around.
Now that that constant was important in
the short-run model because you were
approximating for a bunch of things that
are not related to short-term income.
Wealth you know, the price of houses,
stuff like that. We put all that in that
constant there.
When you think about the long run though
uh most of those things that we excluded
there asset prices, stuff like that tend
to scale with output as well. So so this
is this are inconsistent on the surface
but if you were to fully work out what
is behind the c0 in in the consumption
function then
this is not a bad approximation. They're
not that inconsistent because you
endogenize things that over the long run
scale with income. I mean, you know,
wealth tends to rise with income and all
these things tend to move together. At
not at high frequency, you can have all
sort of fluctuations but over the long
run they tend to scale up together. So
that's going to be our saving function.
So that means that we know in
equilibrium
this is not investment function. We know
that in equilibrium investment will be
equal to uh it will be proportional to
income.
Okay? So remember what we were going
through the box. We had at the top of
the box we had capital that led to
output. We're doing everything in terms
per capita. That less led to saving
and that
funded investment. Okay? So that's
that's what we have. So
this growth model is really about
uh these three functional forms and then
a dynamic equation for the stock of
capital.
So
the evolution of the stock of capital
capital will increase because of
investment.
Uh that's what investment is. It's an
increase in the stock of capital.
Uh but but it will also decrease
as a result of depreciation. I mean
things do break up, you know.
Uh once in a while. And so
and different type of capital have
different depreciation rates. Equipment
depreciate much faster than structures
and buildings and so on. But we're going
to not going to make those distinctions
here. But you see this tells you the
capital stock at t plus one is equal to
the capital stock we had before minus
what is depreciated of that stock of
capital plus any new investment we do
today.
Okay?
In per worker terms and remember that
for now I'm keeping population growth
constant
equal to zero. Not not population growth
constant. Yeah, constant but equal to
zero. So population is constant.
I can divide this both sides by n, you
know, and I get that capital per worker
uh per worker or per person
is equal to this expression here. I did
two things here. I divided by n and I
replaced replaced this I function this
investment for savings. Okay? Because I
know in equilibrium they have to be
equal. So I have that.
Um I can rewrite this, you know, just
subtract kt over n on both sides and
then you get at the change in capital
per person is
is an increasing function of savings
and decreasing of depreciation. Okay? So
the last step
that is important in this model is to So
here I have essentially a difference
equation for capital, but we have an
output per capita on the right-hand
side. But it turns out that that I know
that output per capita per person I said
per capita per person the same thing
per worker it's the same thing in this
part of the course.
Uh so this
is equal to uh
is a function is an increasing and
concave function of capital
per person. Okay?
So this is I would say is the sort of
fundamental equation of the Solow growth
model.
It says the change in the stock of
capital
increases
uh with uh
with investment of course and decreases
with depreciation. And both of these
expressions here are increasing
functions of the
stock of capital per person. Okay?
So let's let's try to understand what is
in here.
So
why
uh
so this is linear obviously because
depreciation is linear. You you say say
you you lose 5% of your stock of capital
every year because it breaks down.
Obviously, the more capital per person
you have, the more units of capital
you're going to lose. This is in units
of capital per person. If you have a
larger stock of capital, you're going to
lose 5% of of a larger number is a
larger number. So this and this is
proportional is linear.
Now this one remember this comes here
from the saving function and this term
here is equal to income per person.
Uh
now suppose that that you start in a
situation where the capital stock is
relatively low and this
is positive.
What does it mean that this is positive?
I mean the implication of this being
positive is that the stock stock of
capital per person will be growing.
But what does it mean that this is
positive in words?
I mean if you have a stock of capital,
there are things that reduce the stock
of capital and there are things that
increase the stock of capital.
This is the thing that increases the
stock of capital that's the thing that
reduces the stock of capital.
So
if this
is greater than that what is that What
does that mean? It's that means this is
positive, but in words
what is happening?
Let me simplify it. This is remember
this is just investment per person.
Well, this just says
that
this economy in this economy there is
more investment than destruction of
capital due to depreciation.
Okay? That's what this means.
This is investment
and and this is positive means that the
investment that which is a function of
saving the saving rate and stuff like
that uh is a function of the funding
available for investment
is equal to the funding available for
investment. Uh
uh
if this is positive, well this is
greater than the stock of capital.
Another way of saying it you need a
minimum level of investment in an
economy to maintain the stock of
capital.
The minimum level of investment that you
need to maintain the stock of capital is
equal to the depreciation. So 10 machine
breaks
you need to invest at least 10 machines
in order to maintain the stock of
capital constant. Okay?
Now if this is positive, it means you're
investing more than the machines that
are breaking down.
Now suppose you start in a situation
where that's the case.
So that means the stock of capital is
growing.
I suppose I ask you the next period do
you think that gap will be larger or
smaller
than it used to be?
Yeah, actually that's not a great
question.
Well
because I'm not doing it in the right
units for that.
Let me ask you a
variation of that question. Suppose we
keep going.
After a while, do you think that number
will get larger or smaller? After a
after let it run for a little for for
quite a while.
Do you think that number will So
remember I said we start with some stock
of capital. This is positive.
If this is positive, it means that the
capital stock is growing. That means
this guy is growing and that guy is
growing. And they're growing equally.
But after a while, do you think this
number will get smaller or bigger?
After a long while just to make sure
that my approximation is not bad here.
Exactly. It's going to get smaller
because
this guy keeps growing linearly
with the stock of capital and this one
is not. It's concave, you know?
At some point this income sort of you
need to put a lot of capital for for
income to keep rising and therefore for
saving to keep rising and therefore for
investment to keep rising. And at some
point yes it won't be able to
uh
to really grow. I mean you're going to
be using all your investment really to
maintain the stock of capital.
That's sort of the logic
of the Solow model.
And it's all in this diagram. So this is
diagram you should really really
understand well and control it and play
with it and all that. It's the
equivalent to your IS-LM model in in in
the first part of the course.
So look at what you have here.
So I'm going to plot output per worker
per worker per person against capital
per worker here.
And so
this red line here
is just
the depreciation. Okay? This
term here.
And that's is a linear function of the
capital per worker. Okay? That's what it
is.
Uh
the blue line here
is output per worker
which as we said is a concave function
of K over N. Remember I showed you that
production function last in the last in
the previous lecture.
There you are.
Okay?
What is the green line?
Is investment per worker which is equal
to saving per worker and saving per
worker is little s the saving rate times
uh output. So it's little s which is a
number like 0.1 if if if we're talking
about the US and you know 0.4 if we're
talking about Singapore it varies a lot
across countries. But but uh
but so this this green line here is
nothing else than this blue line
multiplied by a number that is less than
one. That's the reason it's lower. Okay?
Okay, good. So the point I was
describing before is was a point like
this.
Remember?
Uh the point that I was describing
suppose the economy starts in a point
like this one K0 over N.
Well
and I want to understand the dynamics of
this economy. How will it grow over
time?
So
what you have see here is that that
uh
at this level of capital per worker
investment is greater than
uh
than depreciation.
So that's exactly a situation where this
is positive.
Okay?
That distance here
is that.
Okay?
And the reason I sort of
say I'm not going to do any local
analysis because we could have a started
with a K over zero over here and then
that number is growing, but it's growing
if you were to normalize by the stock of
capital is is is declining. That's
that's that but I didn't want to do that
then. But now that's what I So let's
look at this case. You're you're in a
situation where this is positive. If
this is positive
it means the capital stock per worker is
growing. So you're moving to the right.
In the next period you're going to be
here.
That
that means the capital stock keeps
growing
but by a smaller steps.
Eventually
uh
the investment is entirely used
for uh
recovering from the depreciation of
capital. So covering the depreciation of
capital. And that point the capital
stock stock stops growing. We call that
a steady state stationary state. We stop
Okay? So that's the steady state of this
model.
That means
this economy regardless of where is I do
analysis from the other side. Suppose
you start from a situation like this.
You start with a lot of capital.
Okay? Well, if you start with a lot of
capital in this economy
what happens
when here?
Well, what happens here is that the
investment you're putting to the ground
in this economy is less than what you
need to maintain the stock of capital
which is depreciation.
And that means the stock of capital will
be shrinking over time.
Okay? You're moving that way.
So regardless of where you start in this
economy if I I ask you the question 100
years from now, where are you?
You I tell you tell me I don't need to
know where you start from. I know that
we're going to end up around there.
You can either you start from here, you
go there
from here you go there and so on. That's
the reason we call this a steady state.
This is where you converge in the long
run. Okay?
Now,
this is already interesting because it
tells you
you know, at this mo- at this point
here, the economy was growing. You know,
the capital stock was growing and and
and and the and output was growing. You
see, the capital is if you start from
here,
the capital stock is growing, well,
output is also growing.
Okay? You're moving up there.
Okay? So, you had growth.
That kind of growth we call transitional
growth.
You know, it goes from one point to
another point. It's not a permanent
growth. It's transitional growth.
It's the fact that you were away from
your steady state and then you're going
converging towards your steady state.
A lot of the growth we observe and the
difference of growth we observe across
countries, remember I showed you the
downward sloping curves and all that,
is as a result of that. Poorer economies
tend to have lower capital
uh
capital labor capital employment ratios,
capital population ratios, and therefore
they they tend to grow faster because
they're catching up with their steady
state.
Very advanced economies that have been
more or less in the same place for a
long time are moving around there.
So, there's less catching up growth.
And that's the main responsible for the
the downward sloping curve I showed you
within OECD countries and even broader
than that. Africa was a little of a
problem there.
Okay.
So, that's This is an important model.
Okay for you.
Important diagram.
Let's let's play a little with it. So,
suppose that, you know, at the time,
this is a very simple model, but
at the time,
the the view was that uh
well,
what really supports growth is saving.
So, economies that save a lot
grow a lot. And this sort of sort of
makes sense here because
investment, which is what leads to
capital accumulation, is entirely funded
by savings. It makes sense.
You have more saving, you should grow
more.
Okay, so let's This is something we can
do an experiment. Suppose you start at
at at a steady state, if you will.
And now we increase the saving rate.
What moves?
Which curve This is the kind of things
you should know when you work with this
model.
If I change the saving rate, which curve
moves
in this model?
Let me go one by one.
Does the red line move?
No, has nothing to do with savings.
That's to do with depreciation. If I
move the depreciation rate, that curve
will move, but not
Will the production function move?
No. So, the blue line cannot move.
All that will move is the green line
because the green line is the saving
rate times the
the the blue line. So, if I increase the
saving rate, I'm going to move the green
line up.
Okay? And that's what we have here.
So, you see what happens is you start
for for This was a steady state for this
saving rate in this economy.
Now, all of the sudden this economy
starts saving more.
What happens then?
This tells you very much the story of
Asia, the Asian miracle
of the '60s, '70s, and so on is very
much something like that.
A little more complicated, but, you
know,
a big part of what explains sort of the
fast growth of Asia
uh
during that period
uh is that something like that happened.
Now, why the saving rate increases and
so on, that's all very interesting and
so on, but but it's not what I want to
discuss today.
So,
but what happens here then? So, what
happens See, this economy was in a
steady state, so there was no growth. It
was growing at zero in steady state, you
know?
Because
this says in a steady state output per
per worker per remains constant and
since we have no population work and
growth, then that means output is not
growing either.
Okay? The only way you can have that
ratio constant with the denominator not
moving is that the numerator is not
moving either. Okay?
Okay, good.
So, now
boom, all of the sudden we get a higher
saving rate. So, what happens now?
What reacts?
So, the saving rates go up. It's a
closed economy, it means the investment
rate will go up.
Okay?
What happens now?
What does that gap tell you?
Now, you have a positive gap there,
which means you're investing more than
the the what you need in order to
maintain the stock of capital at the
previous steady state.
So, that means the stock of capital is
going to start growing to the right.
It's going to start growing.
Okay?
And as the stock of capital grows, then
output per capita also grows.
And this will keep happening until you
reach the new steady state.
So, a higher saving rate, so important
conclusion there. This This as simple as
it is
proves something.
Uh
that, you know, the conventional wisdom
that a higher saving rate would give you
sustained growth, higher growth,
isn't really true.
And not certainly not in this model.
Eventually, you'll go back to growth
equal to zero.
Okay? When you reach a new steady state,
you're going to be also growing at zero.
Okay?
What is true, though,
is that you get what again what is
called transitional growth. It goes Oh,
here you're going to start growing very
fast, in fact. Okay? And then you're
going to keep growing at a low slow
lower pace until you go back to zero,
but you're going to get lots of growth
in the transition
as a result of that. And it turns out in
the data when you're looking at 20 30
years of data, it's difficult to uh
disentangle sort of very permanent rates
of growth versus transitional rate of
growth.
This is one of the things that has
concerned China quite a bit, you know,
they have been they grow very very fast.
They have been growing very very fast
for a long time, but it's very clear
it's becoming harder and harder for them
to grow at the type of rate of growth
that they had in the
20 years ago.
Okay? They had rates of growth of 15% or
so.
They had very high They had a very low
initial capital
population ratio,
big population, little capital, and
enormous saving rates.
So, so they grew very very fast.
They had like the green line very close
to to the blue line, the capital stock
very low, so they grew very very fast.
But they have been growing very fast for
a very long period of time, so now it's
getting a lot harder because they're
getting closer and closer to their
steady state. That's the issue. Okay.
There are other sources of growth, and
that's what we're going to talk about in
the next lecture,
but but this This is something called
the easy part of growth.
It's sort of running out in China.
Okay.
And it has to run out
in all developed economies for quite a
while.
Um
good.
Is this clear?
It's important. I mean, a question like
that is guaranteed in your quiz.
It's 81.
What happens if the saving rate does
something?
So,
so so this is a plot over time or um
so, this is a case in which you were in
a steady state and at time T the saving
rate goes up.
S1 greater than S0 jump.
Then output cannot jump.
So, the saving rate goes up, but output
can- cannot jump at day zero. Why?
Why is it that output doesn't jump
immediately to a new steady state?
You know,
this is the
I'm I'm saying
this is what will happen to output.
You're going to start growing very fast
early on, and then you keep growing,
keep growing at a slower and lower pace
because of decreasing returns to
capital,
uh and eventually you'll converge to a
new steady state with with a rate of
growth equal to zero, well, like the one
you had before this savings shock.
And the question I'm asking now is, why
doesn't out- Why does output have to do
this? Why Why doesn't it just jump?
What would What is the only variable
that could make it jump?
Well, you need to look at the production
function.
The production function is a function of
K over N. N is fixed. The only thing
that can make it jump is if the capital
stock jumps.
But the capital stock's not jumping.
That's a stock.
And in order to accumulate a larger
stock of the new steady state, you're
going to go through a lot of flows.
That's investment. You know, every year
you're going to be adding a little more
to the stock of capital on net or or or
or
That's the way you grow. You It's not
that all of the sudden
your stock of capital jumps.
That's very much because this is a
closed economy. If you're in an open
economy, the capital stock can move a
lot faster in a transition because you
can borrow from abroad. You don't need
to fund it all with domestic
sources. And in fact, that's what
typically happens
in in in emerging markets and so on is
they typically borrow for a long time.
Problem is that they tend to consume it
rather than invest it, and that's the
reason you end up in financial crisis
and so on. But but but in principle,
things could go much faster if you have
an open economy and and you have capital
inflows into your country. But that
you'll we'll talk about more about that
five or six six lectures from Anyways,
but this is what happens when I'm
increasing the saving rate. So, yes, it
affects the rate of growth of the
economy during the transition,
uh but but not in the long run. Now,
this transition can be very long.
Okay?
Now, what about consumption? So, so
uh uh
invariably, and there's no way around
that, if if
uh
given a technology and so on, if the
saving rate goes up, then output per
worker will go up.
Okay?
The question is the next question is
what happens to consumption per worker?
Does consumption per worker go up
or not?
You are inclined to say, well, I mean,
it makes sense that it goes up because
uh
we have more income, no? The saving rate
is little s times y, then consumption is
1 minus little s times y. So, income
goes up, consumption should go up.
And and yes, that's a dominant source,
but it's not all the story because
remember I I what I told you.
So, consumption here is going to be
equal to
1 minus little s
times y, so consumption
per person will be
that.
Remember that what is increasing y over
n there, so what is making this guy go
up, which will lead to an increase in
consumption over n,
is that this
guy went up.
And that's a force in the opposite
direction.
Okay?
So, in fact, that was one of the debates
with the
East Asian mirror Southeast Asian
miracle
is that it was fueled by lots of
savings. So, people say, okay, that's
wonderful. Your output growth is very
fast, but consumption growth is not so
fast. And at some point, it may be
hurting you. I think that they were
right though for other reasons, but
but
but that's that picture makes a point.
You know, so if if if your saving rate
to start with, this is a general lesson.
If the saving rate is
you start with is very very low,
then an increase in the saving rate will
lead to a strong increase in consumption
because this change is a small relative
to the big bang you get on output.
Because if you have low saving rate,
that also means that the
the capital stock is very low.
And if the capital stock is very low, f
prime is is very big. You know, this is
a concave function and you're in in the
steep part of the function.
Later on, if saving is very high, you're
going to tend to have capital stock very
high, and then first of all,
more capital won't increase output per
worker a lot because
because of decreasing returns,
and and this is a big number. So, it
starts dominating. And that's what you
see here.
This economy as increases saving rate,
uh consumption per worker rises, but at
some point, it reaches a a maximum, and
then it starts declining.
I mean, think of the limit. If you save
100% of your income,
you don't consume anything. No matter
how much is your output, if your saving
rate is 100%, then you're not going to
consume anything.
If you have no income, no saving rate,
no savings, no income, no capital stock,
no income, you're not going to consume
anything either. Okay? So, you at least
you know these two points. And since you
know there are some positive points in
the in the middle,
uh you know that the curve is going to
tend to have that that kind of change.
It's not going to be it's going to be
non-monotonic.
And that's the way
So, let me just
play with a little a few numbers. This
is
Yeah, let me play with a few numbers.
It's not that crazy.
Uh suppose you have a a production
function that gives equal weight to
capital and workers. So, this production
function.
That's a production function of constant
return to scale.
It better be because that's what we're
doing, but
what do you think?
Yes, no.
The sum of the exponents is one. So,
it's k to the 1/2 n to the 1/2. The sum
of the exponents is one, so you know
that
it's proportional to the scaling factor.
So,
we're going to use
as a scaling as before n, so
um
so we have this.
Okay?
This is a this is a
f of little f of k over n is the square
root of k over n.
Okay?
Minus delta k over n. So, all that I'm
doing is I'm plugging in that function.
Uh
So, here only
I'm replacing all these functions by
by a
a specific example, one in which this is
a square root of k over n.
Okay? That's a concave function, square
root.
Good.
Now, do it as an exercise. If you solve
for the steady state, how do you solve
for the steady state? Well, set this
equal to zero.
That will give you the steady state.
No?
If the steady state is when the capital
is not growing anymore, it's when this
is equal to zero.
When this is equal to zero, I can solve
for the steady state level of k over n,
no, from here.
This equal to zero, I can solve for k
over n, and I'm going to call that the
steady state. k star.
We typically use the stars for the
steady states in growth theory.
Okay?
Well, the answer to this is is k
uh the steady state stock of capital per
per person is the saving rate over delta
squared. That's what it is.
Output
uh per person, which is the square root
of k over n, is therefore the square
root of s over delta squared, so it's s
over delta.
Okay?
So, in this particular model, in the
long run, output per worker doubles when
the saving rate doubles. Okay? If I
double the saving rate, then output per
worker will double.
Notice that the stock of capital is
is going to grow a lot more
in the
when you increase the saving rate.
Okay?
It's square.
So, in that economy,
if you do increase the saving rate from
10 to 20%,
this is the way it goes.
Okay?
So, uh remember, 10 to 20% that means
that the the new steady state output per
worker will be twice what it was in the
previous steady state.
Okay? So, you go from one to two.
But it takes a long time.
And the numbers are not crazy. 50 years
takes you to go to the new steady state.
Okay? So, so that's sort of the time
frame we're talking about. So, it is
true that the saving rate will not
change the long run rate of growth
absent other mechanisms.
But you can grow faster than your
average, your steady state level for
quite quite some time. Okay? And and
again, a lot of that of the Asian
miracle has been of that kind.
This is what I was telling you of China
before, no? Well, yeah, you you can grow
very fast, especially if you have saving
rate much higher than 20%, I mean, 50%
or so.
But but but the rate of growth will have
a tendency to decline. Absent some other
miracle, there are a lot of the reasons
why we have all these fight about
technology and so on.
It has to do with cuz that's the main
mechanism you alternative mechanism to
grow.
It's technology. Okay? We're going to
talk about that in the next lecture. But
but this force, which is what I'm saying
the force, the easy part of growth, it's
very difficult to fight this pattern.
Okay?
So, here you have numbers
uh for the steady states.
So, if the saving rate is zero,
obviously, everything is zero.
No way around.
Uh if the saving rate is 0.1, 10%, then
in this model, capital per worker is
one, output per worker is one.
Consumption per worker didn't go from
zero to one. Why? Because you were
saving something. So, it's zero is 1
minus 0.1, which is the saving rate.
Suppose you double the saving rate.
Well, we know that we're going to double
output per worker in this economy. We
said that we're going to go from one to
two.
The capital stock is going to have to
grow a lot more to double the amount of
output.
Why is that? Decreasing returns.
To double output, you're going to have
to much more than double capital
because, you know, you need you're going
to be fighting decreasing returns.
What about uh consumption? Well, it
won't double because you're doing this
out of increasing the saving rate. So,
you get the two minus now 0.2, not 0.1.
Okay?
Minus 0.2 times two. So, you get 1.6.
And so on.
And
the higher you go with your saving rate,
uh
the harder it gets for capital to bring
along uh
uh
um
output per capita,
and the more the drag on consumption
because you need to be saving a lot in
order to maintain this high stock of
capital that you're having. Okay? You
have a very large stock of capital, that
means you need to save a lot just for
the sake of maintaining that stock of
capital. And so
little is left for
extra
output per capita. And so, you see that
here in this particular for this
particular model, when the saving rate
exceeds 0.5,
then
uh Uh, output obviously keeps rising
when you increase the saving rate, but
but output starts declining. So, that's
your
in the declining part.
And if you get to one, of course,
there's no consumption. So, that's a
that's a curve that we trace.
Okay.
Is everything clear? Now, I'm going to
That's a basic solo model, and that's a
model that again you need to control
completely. Okay.
All that I'm going to do now is very
simple. I'm going to just
modify a little bit this model
to uh add population growth.
Okay.
So, what happens
By the way,
for for centuries population growth has
been one of the main In this model,
we concluded that output per worker
was not growing.
What we're going to conclude in a second
is that
output per worker will not grow if
population is growing.
But that means that output is growing.
If population is growing and output per
worker is not growing, it's constant,
that means output is also growing. And
for a long time,
growth
of output, not of output per worker, was
driven by large population growth. And
sometimes you get big migration flows
into a country that leads sort of to
growth and so on.
Now, big parts of the world
have negative population growth. So, now
we're going through a cycle in which is
things are going the the other way
around in in in many large parts of the
world. I mean, this true in almost all
of continental Europe,
uh certainly in Japan, I said South
Korea, China,
and even some places Latin America.
Okay. So, the drug actually is is
against that.
Uh we don't have the natural force for
growth that we had for for many many
years.
So, let me let me introduce population
growth. So, assume now that that
population rather than being constant
growth growth at the rate gn, which
could be positive or negative. I'm going
to do the example for the pos a positive
uh population growth example.
So, there's no equation that changes in
the sense that
this is still true. It's still true that
investment equal to saving. It's still
true that that uh output is equal to
output per worker. Output is equal to f
of k and n,
and so on and so forth.
The the thing that
is a little trickier is that that, you
know,
in this model,
if I don't normalize things for
if I if I you know, in this case here
where population was not growing,
I could have just eliminated this n.
It's a constant, and I would have done
everything in in in capital in the space
of capital here and output here. Would
have been the same, just scaled by a
number, a constant n.
When I have population growth, I'm not
indifferent between doing one way or the
other.
Because if I don't have if I don't if I
do it in the space of k and y,
and population is growing, then all
these curves are moving.
So, it's a very unfriendly diagram
because my curves are all moving. As n
is moving, everything is moving. So, the
the trick in all these growth models,
and it's going to be even more important
in the next lecture, is to find the
right scaling of capital so there is a
steady state. So, you your curves are
not moving around as population grows.
It's very easy to find the scaling
factor. It's population.
Okay. So,
that's what I'm going to do.
But remember, what is different here is
So, I want to what I'm saying here I
want to get all my variables as scaled
by population at some point in time.
That's what I want to do.
Because I know I practice enough with
these things that's going to give me a
steady state. Okay.
Uh
um Now, what is trickier relative to
what I showed you before is that before
I just divided by n both sides and and I
was home.
Now, I can't really do that. Okay. Let
me divide by n t plus one both sides.
So, that's nice. I get my capital per
worker at t plus one.
But there's certain things that are not
as nice.
What I have on the right-hand side is
not what I really want. I don't want
capital over
population next period.
Now, my steady state's going to be in
the space of
capital over population at the same
time. That's my steady state.
So, this is not so nice.
So, what I have to do is I want to
convert this the right-hand side in
something that is of the kind of things
that I want to have.
So, what I'm going to do is divide and
multiply each of these sides by nt over
nt plus one.
So, sorry. I'm going to divide and
multiply each of these by nt.
Okay.
So,
multiply by nt, divide by nt. So, I'm
multiplying by one.
Well, and and then I can rearrange the
terms in this way. So, I get what I
want, which is capital per
uh person at time t, all at time t, but
then I get this ratio here.
Okay. And I can do the same for this
expression here.
Now, what is that ratio?
Population
today divided by population tomorrow.
Well,
it's one
over one plus the rate of growth of
population.
nt plus one is equal to nt times one
plus gn.
That's the rate of growth
of population.
Okay.
So, so what I have here
is one over one plus g
gn. Now, gn is not a big number.
So, one over one plus gn,
one over one plus gn is approximately
equal to minus gn.
Okay. So, one over one plus gn, gn is
very close to zero, is approximately
equal to minus gn. Okay.
So, that's the reason this guy became
that guy, approximately that guy.
I can do the same here, but it turns out
that
the term there's an extra term here,
therefore
uh which is equal to
s times gn
times yt over nt.
Well, that's second order. That's the
reason I'm going to drop it. Okay. It's
a saving rate, which is sorry it's it's
a it's a small number times
uh
a rate of population growth, which is a
number like, you know, 0.01 or something
like that. So, that's a small number.
So, I'm dropping it.
That's a bigger approximation than that
one, actually, but I'm going to do it.
Everything becomes a lot simpler, but
So, this is an approximation.
Okay. I'm just dropping second-order
terms.
And once I have that, I have the system
I want because now I have
a a system for the evolution of the of
the
capital per
po- per worker. Okay.
Or per person.
And if you see, it looks exactly as we
had before. Remember, this is exactly
what we had before.
s f k over We used to have n not sub-
subscript t. Now, it's k over nt.
But what is different
is that now, rather than having only the
depreciation rate here, we have the
depreciation rate plus the rate of
growth of population.
Why do you think we have the rate of
growth of population there?
Remember the the the economics
behind this expression before.
It was
This is what adds to capital.
To capital per worker.
This is what you need to maintain. What
takes away from capital.
Okay.
Now,
it's what takes away from
given we're doing everything in the
space of capital per worker, that takes
away from capital Oh, that's a typo.
There's a t there.
Okay.
t
Okay.
So, why do you think I have this gn
here?
Well, I have only one minute, so I don't
have time to.
Because if I want to maintain a stock of
capital
per worker,
and workers
are growing,
then I need to be growing the capital
stock. Even if I had no depreciation, if
I want to maintain the capital per
worker constant, and workers are
growing,
then I need to grow the stock of
capital.
So, in order to maintain the capital I
still need to spend what I used to spend
for depreciation of the capital stock.
But if I want to maintain the the
capital per worker constant, then I'm
going to need more investment.
Okay.
Just to make make up for that that extra
component.
So, now, set ga equal to zero. That's
Your diagram is exactly as before in
this space. Set a equal to one and
constant.
But this
line, the red line here, will have delta
plus gn. Okay. So, it rotates up.
So, you can play here and see what
happens if there's change in population
growth,
and so on and so forth.
It's going to be counterintuitive
initially because you see, if I increase
population growth, this curve will
rotate up,
and then it would appear as if that
leads to negative growth.
But you don't get negative growth. In
this diagram, you do get that
Y over over N will decline.
But that doesn't mean that you get
negative growth. It just means that
output is not growing as fast as
population.
But but both are growing. Just the
population is growing faster than
output. I'll I'll I'll start from that.
Uh
oh, I think it's after your break. So,
you're going to have forgotten
everything by then. So, I'll do a review
of this and then and then we
Okay. Have a Have a nice break.