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50:20
Lecture 14: Saving, Capital Accumulation, and Output
MIT OpenCourseWare
·
May 11, 2026
Open on YouTube
Transcript
0:16
I couldn't connect but
0:18
so the Fed just hiked by 25 basis
0:20
points.
0:22
And
0:23
as people expected, you know, this is
0:25
the way that it works when there's lots
0:27
of uncertainty essentially
0:29
the Fed starts communicating
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0:31
what's going to do
0:33
and the communication was still very
0:34
clear that
0:36
that 25 basis points was
0:39
to be expected and and apparently I was
0:41
reading this right now. It was released
0:42
at
0:43
3 minutes ago, 4 minutes ago.
0:45
Um
0:47
they also said that that further hikes
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0:50
are no longer guaranteed. So remember
0:52
that we saw that expected
0:55
hikes sort of we saw several several
0:57
expected hikes for the next few months
0:59
before
1:01
the SVB mess and right after it we sort
1:05
of saw the whole thing declining and and
1:07
at least the minutes are consistent with
1:09
that. Um so there we are. So not big
1:13
uncertainty I mean the markets are
1:14
rallying or something like that at least
1:16
for the next 10 minutes or so but uh
1:18
we shall see.
1:20
Anyway, so but today we we're going to
1:23
really start
1:24
I'm going to I'm going to show you sort
1:26
of the first model of economic growth.
1:28
Uh
1:29
And
1:31
before I do that, who knows who that
1:33
person is?
1:37
No? No clue?
1:41
He
1:42
actually he's Robert Solow. He was an
1:46
He's an emeritus professor at MIT.
1:48
Together with Paul Samuelson essentially
1:50
he's responsible for building the
1:52
economics department at MIT. And he won
1:54
the Nobel Prize in 1987.
1:57
I was a student then here. Uh and
2:00
and
2:02
for his work primarily for his work on
2:05
economic growth. And so what we're going
2:08
to do in the next two three lectures are
2:10
essentially things that Bob Solow
2:13
developed many many years ago.
2:17
The basic mechanism, you know, remember
2:20
that we had this Keynesian cross before
2:21
where we have this multiplier in the
2:23
goods market and aggregate demand
2:25
feeding into income and so on and so
2:26
forth. That was sort of the start
2:28
mechanism in in in short-run macro.
2:31
In long-run macro growth theory
2:34
this is sort of the the key mechanism
2:37
and and you can think of it as the
2:38
following. At any point in time
2:41
an economy has, you know,
2:43
factors of production primarily labor
2:45
and capital.
2:47
That capital stock, labor is more or
2:49
less fixed or depends on population
2:50
growth, things that are sort of
2:52
difficult to to control or they're not
2:54
really that endogenous to to economics.
2:57
Not at least in the current times. Many
3:00
centuries ago yes they were. We had this
3:02
Malthusian theories in which you know
3:04
population growth determined determined
3:06
growth because
3:08
food is scarcity and stuff like that but
3:11
that's no longer the case fortunately.
3:13
Uh for in most parts of the world. So
3:16
but what can what can change over time
3:19
and quite a bit and it depends on
3:21
economic decisions is the capital stock.
3:23
But at any point in time there is
3:25
certain capital stocks which combine
3:27
with labor give you some certain output.
3:29
Output is income.
3:31
Part of that income will be saved as
3:33
we're seeing
3:34
and that those savings will be used for
3:37
investment. Okay?
3:39
But investment is nothing else than
3:41
capital accumulation.
3:43
So
3:44
this income will lead to saving which
3:47
will fund investment which will change
3:49
the stock of capital will feed into
3:50
capital stock that will feed into income
3:52
and so on. All this is happening very
3:54
slowly because the capital stock
3:56
accumulates slowly. I mean
3:59
but but but this is what is happening
4:01
and so all the models we're going to
4:03
look at certainly the model we're going
4:04
to look at in this lecture is all about
4:06
this mechanism. Okay?
4:10
So let's remember what we did in the
4:12
previous lecture. We
4:15
uh
4:16
and I'm going to assume that population
4:18
is constant. I'm going to relax that at
4:19
the very end but assume that the
4:20
population is constant and equal to n
4:22
and remember we're not worrying about
4:24
unemployment and stuff like that here.
4:26
Um so output per capita or per person
4:31
uh is y over n and we remember we had an
4:35
production function f of k and n then
4:38
because of constant returns to scale we
4:40
could divide by n on both sides
4:43
everything and we ended up with this
4:46
relationship. So output per person is
4:50
equal to an is is a is an increasing
4:52
function of capital per person. It's an
4:54
increasing function of capital per
4:55
person but it's also concave
4:58
function
4:59
of capital per person. Why is it
5:01
concave?
5:02
That is why is it increasing at a
5:04
smaller pace?
5:13
Uh yeah.
5:17
Decreasing
5:18
marginal product of capital exactly.
5:21
You know, for fixed amount of labor the
5:23
more capital you put in in into
5:25
production well
5:27
output keeps expanding but by less and
5:29
less because it has less and less labor
5:31
to work with each each unit of capital.
5:34
Perfect. That's very important. Uh then
5:37
let's we're going to work in closed
5:38
economy. I haven't opened it. I'm going
5:40
to do that after
5:42
uh
5:43
quiz two.
5:44
Um
5:45
So and I'm going to assume also no
5:47
public deficits so g equal to t capital
5:50
T.
5:51
And in that case then we know that uh
5:54
private investment private savings equal
5:57
to private investment. Okay? That's
5:59
that's the way we derive the IS curve.
6:01
Um so that's that's that's not new.
6:05
I'm going to modify a little bit what we
6:07
did in the short run.
6:08
Uh um and I'm going to assume that that
6:11
savings is proportional to income. So
6:14
savings little s times y.
6:17
Notice that that this is
6:20
is different from what we did in the
6:22
short run. In the short run remember we
6:23
had a c0 floating around. We had a
6:25
constant in the consumption function. So
6:28
savings which was equal to income minus
6:30
consumption also had a constant floating
6:32
around.
6:33
Now that that constant was important in
6:35
the short-run model because you were
6:37
approximating for a bunch of things that
6:39
are not related to short-term income.
6:41
Wealth you know, the price of houses,
6:44
stuff like that. We put all that in that
6:48
constant there.
6:50
When you think about the long run though
6:52
uh most of those things that we excluded
6:54
there asset prices, stuff like that tend
6:57
to scale with output as well. So so this
7:00
is this are inconsistent on the surface
7:03
but if you were to fully work out what
7:05
is behind the c0 in in the consumption
7:07
function then
7:09
this is not a bad approximation. They're
7:11
not that inconsistent because you
7:13
endogenize things that over the long run
7:15
scale with income. I mean, you know,
7:18
wealth tends to rise with income and all
7:20
these things tend to move together. At
7:22
not at high frequency, you can have all
7:23
sort of fluctuations but over the long
7:25
run they tend to scale up together. So
7:29
that's going to be our saving function.
7:31
So that means that we know in
7:34
equilibrium
7:35
this is not investment function. We know
7:37
that in equilibrium investment will be
7:40
equal to uh it will be proportional to
7:43
income.
7:44
Okay? So remember what we were going
7:46
through the box. We had at the top of
7:49
the box we had capital that led to
7:51
output. We're doing everything in terms
7:53
per capita. That less led to saving
7:57
and that
7:58
funded investment. Okay? So that's
8:02
that's what we have. So
8:05
this growth model is really about
8:08
uh these three functional forms and then
8:10
a dynamic equation for the stock of
8:12
capital.
8:13
So
8:15
the evolution of the stock of capital
8:18
capital will increase because of
8:19
investment.
8:21
Uh that's what investment is. It's an
8:22
increase in the stock of capital.
8:24
Uh but but it will also decrease
8:28
as a result of depreciation. I mean
8:30
things do break up, you know.
8:32
Uh once in a while. And so
8:36
and different type of capital have
8:37
different depreciation rates. Equipment
8:39
depreciate much faster than structures
8:41
and buildings and so on. But we're going
8:44
to not going to make those distinctions
8:45
here. But you see this tells you the
8:47
capital stock at t plus one is equal to
8:50
the capital stock we had before minus
8:52
what is depreciated of that stock of
8:54
capital plus any new investment we do
8:57
today.
8:58
Okay?
9:00
In per worker terms and remember that
9:02
for now I'm keeping population growth
9:04
constant
9:05
equal to zero. Not not population growth
9:07
constant. Yeah, constant but equal to
9:08
zero. So population is constant.
9:12
I can divide this both sides by n, you
9:14
know, and I get that capital per worker
9:18
uh per worker or per person
9:21
is equal to this expression here. I did
9:24
two things here. I divided by n and I
9:26
replaced replaced this I function this
9:30
investment for savings. Okay? Because I
9:33
know in equilibrium they have to be
9:35
equal. So I have that.
9:37
Um I can rewrite this, you know, just
9:40
subtract kt over n on both sides and
9:43
then you get at the change in capital
9:45
per person is
9:47
is an increasing function of savings
9:50
and decreasing of depreciation. Okay? So
9:53
the last step
9:55
that is important in this model is to So
9:58
here I have essentially a difference
10:00
equation for capital, but we have an
10:01
output per capita on the right-hand
10:03
side. But it turns out that that I know
10:06
that output per capita per person I said
10:09
per capita per person the same thing
10:11
per worker it's the same thing in this
10:13
part of the course.
10:14
Uh so this
10:16
is equal to uh
10:19
is a function is an increasing and
10:20
concave function of capital
10:23
per person. Okay?
10:25
So this is I would say is the sort of
10:27
fundamental equation of the Solow growth
10:29
model.
10:30
It says the change in the stock of
10:32
capital
10:33
increases
10:35
uh with uh
10:38
with investment of course and decreases
10:41
with depreciation. And both of these
10:44
expressions here are increasing
10:46
functions of the
10:48
stock of capital per person. Okay?
10:51
So let's let's try to understand what is
10:53
in here.
10:54
So
10:57
why
10:58
uh
10:59
so this is linear obviously because
11:01
depreciation is linear. You you say say
11:04
you you lose 5% of your stock of capital
11:06
every year because it breaks down.
11:09
Obviously, the more capital per person
11:11
you have, the more units of capital
11:13
you're going to lose. This is in units
11:15
of capital per person. If you have a
11:17
larger stock of capital, you're going to
11:18
lose 5% of of a larger number is a
11:21
larger number. So this and this is
11:23
proportional is linear.
11:25
Now this one remember this comes here
11:27
from the saving function and this term
11:30
here is equal to income per person.
11:33
Uh
11:33
now suppose that that you start in a
11:36
situation where the capital stock is
11:38
relatively low and this
11:40
is positive.
11:42
What does it mean that this is positive?
11:44
I mean the implication of this being
11:45
positive is that the stock stock of
11:47
capital per person will be growing.
11:50
But what does it mean that this is
11:51
positive in words?
11:56
I mean if you have a stock of capital,
11:58
there are things that reduce the stock
12:00
of capital and there are things that
12:01
increase the stock of capital.
12:03
This is the thing that increases the
12:05
stock of capital that's the thing that
12:06
reduces the stock of capital.
12:08
So
12:10
if this
12:13
is greater than that what is that What
12:15
does that mean? It's that means this is
12:17
positive, but in words
12:19
what is happening?
12:31
Let me simplify it. This is remember
12:33
this is just investment per person.
12:39
Well, this just says
12:41
that
12:42
this economy in this economy there is
12:44
more investment than destruction of
12:46
capital due to depreciation.
12:49
Okay? That's what this means.
12:51
This is investment
12:53
and and this is positive means that the
12:55
investment that which is a function of
12:57
saving the saving rate and stuff like
12:59
that uh is a function of the funding
13:01
available for investment
13:03
is equal to the funding available for
13:04
investment. Uh
13:07
uh
13:07
if this is positive, well this is
13:09
greater than the stock of capital.
13:12
Another way of saying it you need a
13:14
minimum level of investment in an
13:15
economy to maintain the stock of
13:17
capital.
13:20
The minimum level of investment that you
13:21
need to maintain the stock of capital is
13:24
equal to the depreciation. So 10 machine
13:26
breaks
13:27
you need to invest at least 10 machines
13:30
in order to maintain the stock of
13:31
capital constant. Okay?
13:34
Now if this is positive, it means you're
13:36
investing more than the machines that
13:37
are breaking down.
13:39
Now suppose you start in a situation
13:41
where that's the case.
13:42
So that means the stock of capital is
13:44
growing.
13:45
I suppose I ask you the next period do
13:48
you think that gap will be larger or
13:49
smaller
13:51
than it used to be?
14:01
Yeah, actually that's not a great
14:03
question.
14:04
Well
14:05
because I'm not doing it in the right
14:07
units for that.
14:10
Let me ask you a
14:12
variation of that question. Suppose we
14:14
keep going.
14:16
After a while, do you think that number
14:20
will get larger or smaller? After a
14:23
after let it run for a little for for
14:25
quite a while.
14:26
Do you think that number will So
14:28
remember I said we start with some stock
14:30
of capital. This is positive.
14:32
If this is positive, it means that the
14:33
capital stock is growing. That means
14:35
this guy is growing and that guy is
14:37
growing. And they're growing equally.
14:41
But after a while, do you think this
14:42
number will get smaller or bigger?
14:45
After a long while just to make sure
14:47
that my approximation is not bad here.
14:57
Exactly. It's going to get smaller
14:59
because
15:00
this guy keeps growing linearly
15:03
with the stock of capital and this one
15:04
is not. It's concave, you know?
15:06
At some point this income sort of you
15:08
need to put a lot of capital for for
15:10
income to keep rising and therefore for
15:12
saving to keep rising and therefore for
15:14
investment to keep rising. And at some
15:16
point yes it won't be able to
15:19
uh
15:20
to really grow. I mean you're going to
15:21
be using all your investment really to
15:22
maintain the stock of capital.
15:24
That's sort of the logic
15:26
of the Solow model.
15:29
And it's all in this diagram. So this is
15:32
diagram you should really really
15:33
understand well and control it and play
15:36
with it and all that. It's the
15:38
equivalent to your IS-LM model in in in
15:42
the first part of the course.
15:44
So look at what you have here.
15:46
So I'm going to plot output per worker
15:51
per worker per person against capital
15:53
per worker here.
15:55
And so
15:58
this red line here
16:01
is just
16:02
the depreciation. Okay? This
16:05
term here.
16:07
And that's is a linear function of the
16:10
capital per worker. Okay? That's what it
16:12
is.
16:15
Uh
16:15
the blue line here
16:18
is output per worker
16:20
which as we said is a concave function
16:22
of K over N. Remember I showed you that
16:24
production function last in the last in
16:26
the previous lecture.
16:28
There you are.
16:29
Okay?
16:30
What is the green line?
16:32
Is investment per worker which is equal
16:34
to saving per worker and saving per
16:36
worker is little s the saving rate times
16:40
uh output. So it's little s which is a
16:43
number like 0.1 if if if we're talking
16:46
about the US and you know 0.4 if we're
16:49
talking about Singapore it varies a lot
16:51
across countries. But but uh
16:54
but so this this green line here is
16:57
nothing else than this blue line
16:59
multiplied by a number that is less than
17:00
one. That's the reason it's lower. Okay?
17:06
Okay, good. So the point I was
17:08
describing before is was a point like
17:10
this.
17:12
Remember?
17:13
Uh the point that I was describing
17:15
suppose the economy starts in a point
17:17
like this one K0 over N.
17:20
Well
17:21
and I want to understand the dynamics of
17:23
this economy. How will it grow over
17:25
time?
17:26
So
17:27
what you have see here is that that
17:31
uh
17:32
at this level of capital per worker
17:36
investment is greater than
17:38
uh
17:39
than depreciation.
17:41
So that's exactly a situation where this
17:43
is positive.
17:45
Okay?
17:46
That distance here
17:49
is that.
17:52
Okay?
17:54
And the reason I sort of
17:55
say I'm not going to do any local
17:57
analysis because we could have a started
17:58
with a K over zero over here and then
18:01
that number is growing, but it's growing
18:03
if you were to normalize by the stock of
18:05
capital is is is declining. That's
18:07
that's that but I didn't want to do that
18:08
then. But now that's what I So let's
18:11
look at this case. You're you're in a
18:12
situation where this is positive. If
18:14
this is positive
18:16
it means the capital stock per worker is
18:18
growing. So you're moving to the right.
18:21
In the next period you're going to be
18:23
here.
18:24
That
18:25
that means the capital stock keeps
18:26
growing
18:27
but by a smaller steps.
18:30
Eventually
18:33
uh
18:33
the investment is entirely used
18:36
for uh
18:39
recovering from the depreciation of
18:40
capital. So covering the depreciation of
18:42
capital. And that point the capital
18:45
stock stock stops growing. We call that
18:48
a steady state stationary state. We stop
18:52
Okay? So that's the steady state of this
18:54
model.
18:55
That means
18:56
this economy regardless of where is I do
18:59
analysis from the other side. Suppose
19:00
you start from a situation like this.
19:02
You start with a lot of capital.
19:04
Okay? Well, if you start with a lot of
19:06
capital in this economy
19:08
what happens
19:10
when here?
19:12
Well, what happens here is that the
19:13
investment you're putting to the ground
19:15
in this economy is less than what you
19:17
need to maintain the stock of capital
19:19
which is depreciation.
19:21
And that means the stock of capital will
19:23
be shrinking over time.
19:25
Okay? You're moving that way.
19:27
So regardless of where you start in this
19:30
economy if I I ask you the question 100
19:33
years from now, where are you?
19:35
You I tell you tell me I don't need to
19:38
know where you start from. I know that
19:39
we're going to end up around there.
19:41
You can either you start from here, you
19:43
go there
19:44
from here you go there and so on. That's
19:46
the reason we call this a steady state.
19:47
This is where you converge in the long
19:49
run. Okay?
19:55
Now,
19:56
this is already interesting because it
19:57
tells you
19:59
you know, at this mo- at this point
20:01
here, the economy was growing. You know,
20:04
the capital stock was growing and and
20:06
and and the and output was growing. You
20:08
see, the capital is if you start from
20:10
here,
20:11
the capital stock is growing, well,
20:13
output is also growing.
20:15
Okay? You're moving up there.
20:17
Okay? So, you had growth.
20:20
That kind of growth we call transitional
20:22
growth.
20:23
You know, it goes from one point to
20:26
another point. It's not a permanent
20:28
growth. It's transitional growth.
20:30
It's the fact that you were away from
20:32
your steady state and then you're going
20:34
converging towards your steady state.
20:37
A lot of the growth we observe and the
20:39
difference of growth we observe across
20:40
countries, remember I showed you the
20:41
downward sloping curves and all that,
20:44
is as a result of that. Poorer economies
20:47
tend to have lower capital
20:49
uh
20:50
capital labor capital employment ratios,
20:53
capital population ratios, and therefore
20:55
they they tend to grow faster because
20:57
they're catching up with their steady
20:58
state.
21:00
Very advanced economies that have been
21:01
more or less in the same place for a
21:03
long time are moving around there.
21:05
So, there's less catching up growth.
21:08
And that's the main responsible for the
21:10
the downward sloping curve I showed you
21:12
within OECD countries and even broader
21:14
than that. Africa was a little of a
21:16
problem there.
21:18
Okay.
21:20
So, that's This is an important model.
21:23
Okay for you.
21:24
Important diagram.
21:26
Let's let's play a little with it. So,
21:28
suppose that, you know, at the time,
21:30
this is a very simple model, but
21:32
at the time,
21:34
the the view was that uh
21:36
well,
21:37
what really supports growth is saving.
21:40
So, economies that save a lot
21:42
grow a lot. And this sort of sort of
21:44
makes sense here because
21:46
investment, which is what leads to
21:48
capital accumulation, is entirely funded
21:50
by savings. It makes sense.
21:52
You have more saving, you should grow
21:54
more.
21:55
Okay, so let's This is something we can
21:58
do an experiment. Suppose you start at
22:00
at at a steady state, if you will.
22:03
And now we increase the saving rate.
22:06
What moves?
22:09
Which curve This is the kind of things
22:10
you should know when you work with this
22:12
model.
22:13
If I change the saving rate, which curve
22:16
moves
22:17
in this model?
22:19
Let me go one by one.
22:20
Does the red line move?
22:24
No, has nothing to do with savings.
22:26
That's to do with depreciation. If I
22:27
move the depreciation rate, that curve
22:29
will move, but not
22:32
Will the production function move?
22:35
No. So, the blue line cannot move.
22:37
All that will move is the green line
22:39
because the green line is the saving
22:41
rate times the
22:43
the the blue line. So, if I increase the
22:46
saving rate, I'm going to move the green
22:47
line up.
22:49
Okay? And that's what we have here.
22:52
So, you see what happens is you start
22:54
for for This was a steady state for this
22:57
saving rate in this economy.
22:59
Now, all of the sudden this economy
23:00
starts saving more.
23:02
What happens then?
23:05
This tells you very much the story of
23:06
Asia, the Asian miracle
23:09
of the '60s, '70s, and so on is very
23:11
much something like that.
23:14
A little more complicated, but, you
23:16
know,
23:18
a big part of what explains sort of the
23:20
fast growth of Asia
23:22
uh
23:23
during that period
23:25
uh is that something like that happened.
23:28
Now, why the saving rate increases and
23:29
so on, that's all very interesting and
23:30
so on, but but it's not what I want to
23:32
discuss today.
23:34
So,
23:35
but what happens here then? So, what
23:37
happens See, this economy was in a
23:38
steady state, so there was no growth. It
23:40
was growing at zero in steady state, you
23:42
know?
23:43
Because
23:44
this says in a steady state output per
23:47
per worker per remains constant and
23:49
since we have no population work and
23:51
growth, then that means output is not
23:52
growing either.
23:54
Okay? The only way you can have that
23:55
ratio constant with the denominator not
23:58
moving is that the numerator is not
23:59
moving either. Okay?
24:02
Okay, good.
24:03
So, now
24:05
boom, all of the sudden we get a higher
24:07
saving rate. So, what happens now?
24:11
What reacts?
24:14
So, the saving rates go up. It's a
24:16
closed economy, it means the investment
24:18
rate will go up.
24:19
Okay?
24:21
What happens now?
24:29
What does that gap tell you?
24:35
Now, you have a positive gap there,
24:37
which means you're investing more than
24:39
the the what you need in order to
24:41
maintain the stock of capital at the
24:42
previous steady state.
24:45
So, that means the stock of capital is
24:46
going to start growing to the right.
24:48
It's going to start growing.
24:49
Okay?
24:50
And as the stock of capital grows, then
24:53
output per capita also grows.
24:56
And this will keep happening until you
24:58
reach the new steady state.
25:02
So, a higher saving rate, so important
25:05
conclusion there. This This as simple as
25:07
it is
25:08
proves something.
25:10
Uh
25:11
that, you know, the conventional wisdom
25:13
that a higher saving rate would give you
25:15
sustained growth, higher growth,
25:18
isn't really true.
25:19
And not certainly not in this model.
25:21
Eventually, you'll go back to growth
25:23
equal to zero.
25:24
Okay? When you reach a new steady state,
25:26
you're going to be also growing at zero.
25:29
Okay?
25:31
What is true, though,
25:33
is that you get what again what is
25:34
called transitional growth. It goes Oh,
25:38
here you're going to start growing very
25:39
fast, in fact. Okay? And then you're
25:42
going to keep growing at a low slow
25:44
lower pace until you go back to zero,
25:45
but you're going to get lots of growth
25:47
in the transition
25:48
as a result of that. And it turns out in
25:50
the data when you're looking at 20 30
25:52
years of data, it's difficult to uh
25:56
disentangle sort of very permanent rates
25:58
of growth versus transitional rate of
26:00
growth.
26:01
This is one of the things that has
26:02
concerned China quite a bit, you know,
26:04
they have been they grow very very fast.
26:05
They have been growing very very fast
26:07
for a long time, but it's very clear
26:09
it's becoming harder and harder for them
26:10
to grow at the type of rate of growth
26:12
that they had in the
26:14
20 years ago.
26:15
Okay? They had rates of growth of 15% or
26:17
so.
26:18
They had very high They had a very low
26:21
initial capital
26:22
population ratio,
26:24
big population, little capital, and
26:26
enormous saving rates.
26:29
So, so they grew very very fast.
26:32
They had like the green line very close
26:33
to to the blue line, the capital stock
26:36
very low, so they grew very very fast.
26:39
But they have been growing very fast for
26:40
a very long period of time, so now it's
26:42
getting a lot harder because they're
26:43
getting closer and closer to their
26:45
steady state. That's the issue. Okay.
26:47
There are other sources of growth, and
26:49
that's what we're going to talk about in
26:50
the next lecture,
26:52
but but this This is something called
26:54
the easy part of growth.
26:56
It's sort of running out in China.
27:03
Okay.
27:09
And it has to run out
27:11
in all developed economies for quite a
27:13
while.
27:17
Um
27:18
good.
27:22
Is this clear?
27:23
It's important. I mean, a question like
27:25
that is guaranteed in your quiz.
27:28
It's 81.
27:30
What happens if the saving rate does
27:31
something?
27:33
So,
27:34
so so this is a plot over time or um
27:38
so, this is a case in which you were in
27:40
a steady state and at time T the saving
27:42
rate goes up.
27:44
S1 greater than S0 jump.
27:47
Then output cannot jump.
27:50
So, the saving rate goes up, but output
27:52
can- cannot jump at day zero. Why?
27:54
Why is it that output doesn't jump
27:55
immediately to a new steady state?
28:02
You know,
28:02
this is the
28:04
I'm I'm saying
28:05
this is what will happen to output.
28:07
You're going to start growing very fast
28:08
early on, and then you keep growing,
28:10
keep growing at a slower and lower pace
28:12
because of decreasing returns to
28:13
capital,
28:15
uh and eventually you'll converge to a
28:16
new steady state with with a rate of
28:19
growth equal to zero, well, like the one
28:21
you had before this savings shock.
28:25
And the question I'm asking now is, why
28:26
doesn't out- Why does output have to do
28:28
this? Why Why doesn't it just jump?
28:35
What would What is the only variable
28:36
that could make it jump?
28:41
Well, you need to look at the production
28:42
function.
28:44
The production function is a function of
28:45
K over N. N is fixed. The only thing
28:47
that can make it jump is if the capital
28:48
stock jumps.
28:50
But the capital stock's not jumping.
28:52
That's a stock.
28:53
And in order to accumulate a larger
28:54
stock of the new steady state, you're
28:56
going to go through a lot of flows.
28:58
That's investment. You know, every year
29:00
you're going to be adding a little more
29:01
to the stock of capital on net or or or
29:03
or
29:03
That's the way you grow. You It's not
29:05
that all of the sudden
29:06
your stock of capital jumps.
29:10
That's very much because this is a
29:12
closed economy. If you're in an open
29:14
economy, the capital stock can move a
29:15
lot faster in a transition because you
29:18
can borrow from abroad. You don't need
29:20
to fund it all with domestic
29:22
sources. And in fact, that's what
29:24
typically happens
29:26
in in in emerging markets and so on is
29:28
they typically borrow for a long time.
29:31
Problem is that they tend to consume it
29:32
rather than invest it, and that's the
29:33
reason you end up in financial crisis
29:35
and so on. But but but in principle,
29:37
things could go much faster if you have
29:39
an open economy and and you have capital
29:41
inflows into your country. But that
29:43
you'll we'll talk about more about that
29:46
five or six six lectures from Anyways,
29:48
but this is what happens when I'm
29:50
increasing the saving rate. So, yes, it
29:52
affects the rate of growth of the
29:54
economy during the transition,
29:56
uh but but not in the long run. Now,
29:58
this transition can be very long.
30:00
Okay?
30:02
Now, what about consumption? So, so
30:06
uh uh
30:07
invariably, and there's no way around
30:09
that, if if
30:11
uh
30:11
given a technology and so on, if the
30:14
saving rate goes up, then output per
30:16
worker will go up.
30:18
Okay?
30:19
The question is the next question is
30:21
what happens to consumption per worker?
30:22
Does consumption per worker go up
30:25
or not?
30:27
You are inclined to say, well, I mean,
30:29
it makes sense that it goes up because
30:32
uh
30:32
we have more income, no? The saving rate
30:34
is little s times y, then consumption is
30:38
1 minus little s times y. So, income
30:41
goes up, consumption should go up.
30:48
And and yes, that's a dominant source,
30:52
but it's not all the story because
30:54
remember I I what I told you.
31:03
So, consumption here is going to be
31:04
equal to
31:05
1 minus little s
31:08
times y, so consumption
31:11
per person will be
31:13
that.
31:15
Remember that what is increasing y over
31:17
n there, so what is making this guy go
31:20
up, which will lead to an increase in
31:22
consumption over n,
31:24
is that this
31:25
guy went up.
31:28
And that's a force in the opposite
31:30
direction.
31:32
Okay?
31:33
So, in fact, that was one of the debates
31:36
with the
31:38
East Asian mirror Southeast Asian
31:39
miracle
31:40
is that it was fueled by lots of
31:42
savings. So, people say, okay, that's
31:44
wonderful. Your output growth is very
31:45
fast, but consumption growth is not so
31:47
fast. And at some point, it may be
31:49
hurting you. I think that they were
31:51
right though for other reasons, but
31:53
but
31:55
but that's that picture makes a point.
31:58
You know, so if if if your saving rate
32:00
to start with, this is a general lesson.
32:02
If the saving rate is
32:04
you start with is very very low,
32:07
then an increase in the saving rate will
32:08
lead to a strong increase in consumption
32:10
because this change is a small relative
32:12
to the big bang you get on output.
32:14
Because if you have low saving rate,
32:16
that also means that the
32:19
the capital stock is very low.
32:21
And if the capital stock is very low, f
32:24
prime is is very big. You know, this is
32:26
a concave function and you're in in the
32:27
steep part of the function.
32:29
Later on, if saving is very high, you're
32:32
going to tend to have capital stock very
32:34
high, and then first of all,
32:37
more capital won't increase output per
32:39
worker a lot because
32:41
because of decreasing returns,
32:43
and and this is a big number. So, it
32:45
starts dominating. And that's what you
32:46
see here.
32:47
This economy as increases saving rate,
32:50
uh consumption per worker rises, but at
32:53
some point, it reaches a a maximum, and
32:55
then it starts declining.
32:56
I mean, think of the limit. If you save
32:58
100% of your income,
33:01
you don't consume anything. No matter
33:03
how much is your output, if your saving
33:05
rate is 100%, then you're not going to
33:07
consume anything.
33:09
If you have no income, no saving rate,
33:12
no savings, no income, no capital stock,
33:15
no income, you're not going to consume
33:16
anything either. Okay? So, you at least
33:19
you know these two points. And since you
33:21
know there are some positive points in
33:23
the in the middle,
33:24
uh you know that the curve is going to
33:25
tend to have that that kind of change.
33:27
It's not going to be it's going to be
33:28
non-monotonic.
33:30
And that's the way
33:34
So, let me just
33:36
play with a little a few numbers. This
33:37
is
33:40
Yeah, let me play with a few numbers.
33:41
It's not that crazy.
33:43
Uh suppose you have a a production
33:45
function that gives equal weight to
33:47
capital and workers. So, this production
33:49
function.
33:51
That's a production function of constant
33:53
return to scale.
33:55
It better be because that's what we're
33:57
doing, but
33:58
what do you think?
34:01
Yes, no.
34:03
The sum of the exponents is one. So,
34:06
it's k to the 1/2 n to the 1/2. The sum
34:09
of the exponents is one, so you know
34:11
that
34:12
it's proportional to the scaling factor.
34:14
So,
34:15
we're going to use
34:17
as a scaling as before n, so
34:20
um
34:21
so we have this.
34:23
Okay?
34:24
This is a this is a
34:25
f of little f of k over n is the square
34:29
root of k over n.
34:31
Okay?
34:32
Minus delta k over n. So, all that I'm
34:34
doing is I'm plugging in that function.
34:37
Uh
34:39
So, here only
34:41
I'm replacing all these functions by
34:44
by a
34:45
a specific example, one in which this is
34:46
a square root of k over n.
34:49
Okay? That's a concave function, square
34:52
root.
34:56
Good.
34:57
Now, do it as an exercise. If you solve
35:00
for the steady state, how do you solve
35:01
for the steady state? Well, set this
35:03
equal to zero.
35:04
That will give you the steady state.
35:06
No?
35:07
If the steady state is when the capital
35:09
is not growing anymore, it's when this
35:11
is equal to zero.
35:13
When this is equal to zero, I can solve
35:14
for the steady state level of k over n,
35:17
no, from here.
35:19
This equal to zero, I can solve for k
35:21
over n, and I'm going to call that the
35:22
steady state. k star.
35:25
We typically use the stars for the
35:26
steady states in growth theory.
35:29
Okay?
35:29
Well, the answer to this is is k
35:33
uh the steady state stock of capital per
35:35
per person is the saving rate over delta
35:39
squared. That's what it is.
35:41
Output
35:43
uh per person, which is the square root
35:45
of k over n, is therefore the square
35:47
root of s over delta squared, so it's s
35:50
over delta.
35:52
Okay?
35:53
So, in this particular model, in the
35:55
long run, output per worker doubles when
35:58
the saving rate doubles. Okay? If I
36:00
double the saving rate, then output per
36:03
worker will double.
36:07
Notice that the stock of capital is
36:09
is going to grow a lot more
36:11
in the
36:12
when you increase the saving rate.
36:14
Okay?
36:15
It's square.
36:19
So, in that economy,
36:21
if you do increase the saving rate from
36:23
10 to 20%,
36:25
this is the way it goes.
36:27
Okay?
36:28
So, uh remember, 10 to 20% that means
36:31
that the the new steady state output per
36:33
worker will be twice what it was in the
36:35
previous steady state.
36:36
Okay? So, you go from one to two.
36:39
But it takes a long time.
36:42
And the numbers are not crazy. 50 years
36:44
takes you to go to the new steady state.
36:48
Okay? So, so that's sort of the time
36:50
frame we're talking about. So, it is
36:52
true that the saving rate will not
36:53
change the long run rate of growth
36:56
absent other mechanisms.
37:00
But you can grow faster than your
37:02
average, your steady state level for
37:04
quite quite some time. Okay? And and
37:06
again, a lot of that of the Asian
37:09
miracle has been of that kind.
37:13
This is what I was telling you of China
37:15
before, no? Well, yeah, you you can grow
37:17
very fast, especially if you have saving
37:19
rate much higher than 20%, I mean, 50%
37:22
or so.
37:23
But but but the rate of growth will have
37:25
a tendency to decline. Absent some other
37:28
miracle, there are a lot of the reasons
37:29
why we have all these fight about
37:30
technology and so on.
37:33
It has to do with cuz that's the main
37:35
mechanism you alternative mechanism to
37:37
grow.
37:38
It's technology. Okay? We're going to
37:40
talk about that in the next lecture. But
37:42
but this force, which is what I'm saying
37:44
the force, the easy part of growth, it's
37:47
very difficult to fight this pattern.
37:49
Okay?
37:53
So, here you have numbers
37:55
uh for the steady states.
37:57
So, if the saving rate is zero,
37:59
obviously, everything is zero.
38:01
No way around.
38:03
Uh if the saving rate is 0.1, 10%, then
38:06
in this model, capital per worker is
38:08
one, output per worker is one.
38:10
Consumption per worker didn't go from
38:12
zero to one. Why? Because you were
38:13
saving something. So, it's zero is 1
38:15
minus 0.1, which is the saving rate.
38:18
Suppose you double the saving rate.
38:20
Well, we know that we're going to double
38:22
output per worker in this economy. We
38:23
said that we're going to go from one to
38:25
two.
38:26
The capital stock is going to have to
38:27
grow a lot more to double the amount of
38:29
output.
38:31
Why is that? Decreasing returns.
38:34
To double output, you're going to have
38:35
to much more than double capital
38:37
because, you know, you need you're going
38:39
to be fighting decreasing returns.
38:42
What about uh consumption? Well, it
38:44
won't double because you're doing this
38:46
out of increasing the saving rate. So,
38:48
you get the two minus now 0.2, not 0.1.
38:51
Okay?
38:52
Minus 0.2 times two. So, you get 1.6.
38:56
And so on.
38:58
And
38:59
the higher you go with your saving rate,
39:01
uh
39:02
the harder it gets for capital to bring
39:04
along uh
39:06
uh
39:07
um
39:09
output per capita,
39:11
and the more the drag on consumption
39:12
because you need to be saving a lot in
39:14
order to maintain this high stock of
39:17
capital that you're having. Okay? You
39:18
have a very large stock of capital, that
39:20
means you need to save a lot just for
39:23
the sake of maintaining that stock of
39:25
capital. And so
39:28
little is left for
39:30
extra
39:31
output per capita. And so, you see that
39:33
here in this particular for this
39:35
particular model, when the saving rate
39:37
exceeds 0.5,
39:39
then
39:40
uh Uh, output obviously keeps rising
39:42
when you increase the saving rate, but
39:44
but output starts declining. So, that's
39:46
your
39:47
in the declining part.
39:48
And if you get to one, of course,
39:50
there's no consumption. So, that's a
39:52
that's a curve that we trace.
39:59
Okay.
40:03
Is everything clear? Now, I'm going to
40:05
That's a basic solo model, and that's a
40:07
model that again you need to control
40:10
completely. Okay.
40:12
All that I'm going to do now is very
40:14
simple. I'm going to just
40:16
modify a little bit this model
40:19
to uh add population growth.
40:22
Okay.
40:23
So, what happens
40:24
By the way,
40:26
for for centuries population growth has
40:28
been one of the main In this model,
40:32
we concluded that output per worker
40:35
was not growing.
40:37
What we're going to conclude in a second
40:40
is that
40:41
output per worker will not grow if
40:43
population is growing.
40:45
But that means that output is growing.
40:48
If population is growing and output per
40:50
worker is not growing, it's constant,
40:52
that means output is also growing. And
40:54
for a long time,
40:56
growth
40:58
of output, not of output per worker, was
41:01
driven by large population growth. And
41:04
sometimes you get big migration flows
41:06
into a country that leads sort of to
41:07
growth and so on.
41:09
Now, big parts of the world
41:11
have negative population growth. So, now
41:13
we're going through a cycle in which is
41:15
things are going the the other way
41:17
around in in in many large parts of the
41:20
world. I mean, this true in almost all
41:22
of continental Europe,
41:24
uh certainly in Japan, I said South
41:26
Korea, China,
41:29
and even some places Latin America.
41:31
Okay. So, the drug actually is is
41:34
against that.
41:36
Uh we don't have the natural force for
41:38
growth that we had for for many many
41:40
years.
41:42
So, let me let me introduce population
41:44
growth. So, assume now that that
41:46
population rather than being constant
41:47
growth growth at the rate gn, which
41:50
could be positive or negative. I'm going
41:51
to do the example for the pos a positive
41:54
uh population growth example.
41:56
So, there's no equation that changes in
41:58
the sense that
42:00
this is still true. It's still true that
42:02
investment equal to saving. It's still
42:05
true that that uh output is equal to
42:09
output per worker. Output is equal to f
42:12
of k and n,
42:14
and so on and so forth.
42:16
The the thing that
42:17
is a little trickier is that that, you
42:20
know,
42:22
in this model,
42:24
if I don't normalize things for
42:27
if I if I you know, in this case here
42:30
where population was not growing,
42:32
I could have just eliminated this n.
42:34
It's a constant, and I would have done
42:36
everything in in in capital in the space
42:37
of capital here and output here. Would
42:39
have been the same, just scaled by a
42:41
number, a constant n.
42:44
When I have population growth, I'm not
42:46
indifferent between doing one way or the
42:48
other.
42:49
Because if I don't have if I don't if I
42:51
do it in the space of k and y,
42:54
and population is growing, then all
42:56
these curves are moving.
42:58
So, it's a very unfriendly diagram
42:59
because my curves are all moving. As n
43:01
is moving, everything is moving. So, the
43:03
the trick in all these growth models,
43:05
and it's going to be even more important
43:06
in the next lecture, is to find the
43:08
right scaling of capital so there is a
43:11
steady state. So, you your curves are
43:13
not moving around as population grows.
43:16
It's very easy to find the scaling
43:18
factor. It's population.
43:20
Okay. So,
43:22
that's what I'm going to do.
43:25
But remember, what is different here is
43:27
So, I want to what I'm saying here I
43:29
want to get all my variables as scaled
43:32
by population at some point in time.
43:33
That's what I want to do.
43:35
Because I know I practice enough with
43:37
these things that's going to give me a
43:38
steady state. Okay.
43:40
Uh
43:41
um Now, what is trickier relative to
43:43
what I showed you before is that before
43:45
I just divided by n both sides and and I
43:48
was home.
43:49
Now, I can't really do that. Okay. Let
43:51
me divide by n t plus one both sides.
43:55
So, that's nice. I get my capital per
43:58
worker at t plus one.
44:01
But there's certain things that are not
44:02
as nice.
44:03
What I have on the right-hand side is
44:05
not what I really want. I don't want
44:06
capital over
44:08
population next period.
44:11
Now, my steady state's going to be in
44:12
the space of
44:14
capital over population at the same
44:17
time. That's my steady state.
44:20
So, this is not so nice.
44:23
So, what I have to do is I want to
44:24
convert this the right-hand side in
44:26
something that is of the kind of things
44:27
that I want to have.
44:29
So, what I'm going to do is divide and
44:31
multiply each of these sides by nt over
44:33
nt plus one.
44:35
So, sorry. I'm going to divide and
44:37
multiply each of these by nt.
44:40
Okay.
44:41
So,
44:42
multiply by nt, divide by nt. So, I'm
44:45
multiplying by one.
44:46
Well, and and then I can rearrange the
44:48
terms in this way. So, I get what I
44:50
want, which is capital per
44:52
uh person at time t, all at time t, but
44:55
then I get this ratio here.
44:58
Okay. And I can do the same for this
44:59
expression here.
45:01
Now, what is that ratio?
45:06
Population
45:07
today divided by population tomorrow.
45:15
Well,
45:16
it's one
45:17
over one plus the rate of growth of
45:19
population.
45:21
nt plus one is equal to nt times one
45:23
plus gn.
45:25
That's the rate of growth
45:26
of population.
45:41
Okay.
45:43
So, so what I have here
45:46
is one over one plus g
45:48
gn. Now, gn is not a big number.
45:52
So, one over one plus gn,
45:55
one over one plus gn is approximately
45:57
equal to minus gn.
46:00
Okay. So, one over one plus gn, gn is
46:02
very close to zero, is approximately
46:05
equal to minus gn. Okay.
46:08
So, that's the reason this guy became
46:11
that guy, approximately that guy.
46:14
I can do the same here, but it turns out
46:16
that
46:17
the term there's an extra term here,
46:19
therefore
46:20
uh which is equal to
46:22
s times gn
46:25
times yt over nt.
46:27
Well, that's second order. That's the
46:29
reason I'm going to drop it. Okay. It's
46:30
a saving rate, which is sorry it's it's
46:33
a it's a small number times
46:35
uh
46:36
a rate of population growth, which is a
46:38
number like, you know, 0.01 or something
46:40
like that. So, that's a small number.
46:41
So, I'm dropping it.
46:43
That's a bigger approximation than that
46:44
one, actually, but I'm going to do it.
46:46
Everything becomes a lot simpler, but
46:48
So, this is an approximation.
46:50
Okay. I'm just dropping second-order
46:51
terms.
46:54
And once I have that, I have the system
46:56
I want because now I have
46:58
a a system for the evolution of the of
47:00
the
47:02
capital per
47:04
po- per worker. Okay.
47:07
Or per person.
47:10
And if you see, it looks exactly as we
47:12
had before. Remember, this is exactly
47:13
what we had before.
47:16
s f k over We used to have n not sub-
47:18
subscript t. Now, it's k over nt.
47:22
But what is different
47:24
is that now, rather than having only the
47:26
depreciation rate here, we have the
47:27
depreciation rate plus the rate of
47:29
growth of population.
47:32
Why do you think we have the rate of
47:33
growth of population there?
47:36
Remember the the the economics
47:39
behind this expression before.
47:42
It was
47:44
This is what adds to capital.
47:47
To capital per worker.
47:49
This is what you need to maintain. What
47:52
takes away from capital.
47:54
Okay.
47:55
Now,
47:56
it's what takes away from
47:58
given we're doing everything in the
47:59
space of capital per worker, that takes
48:01
away from capital Oh, that's a typo.
48:04
There's a t there.
48:05
Okay.
48:07
t
48:11
Okay.
48:12
So, why do you think I have this gn
48:14
here?
48:16
Well, I have only one minute, so I don't
48:17
have time to.
48:19
Because if I want to maintain a stock of
48:21
capital
48:23
per worker,
48:25
and workers
48:26
are growing,
48:28
then I need to be growing the capital
48:29
stock. Even if I had no depreciation, if
48:31
I want to maintain the capital per
48:33
worker constant, and workers are
48:35
growing,
48:37
then I need to grow the stock of
48:38
capital.
48:39
So, in order to maintain the capital I
48:41
still need to spend what I used to spend
48:43
for depreciation of the capital stock.
48:46
But if I want to maintain the the
48:48
capital per worker constant, then I'm
48:50
going to need more investment.
48:52
Okay.
48:53
Just to make make up for that that extra
48:56
component.
48:58
So, now, set ga equal to zero. That's
49:01
Your diagram is exactly as before in
49:03
this space. Set a equal to one and
49:05
constant.
49:06
But this
49:07
line, the red line here, will have delta
49:10
plus gn. Okay. So, it rotates up.
49:17
So, you can play here and see what
49:18
happens if there's change in population
49:20
growth,
49:21
and so on and so forth.
49:23
It's going to be counterintuitive
49:24
initially because you see, if I increase
49:26
population growth, this curve will
49:28
rotate up,
49:29
and then it would appear as if that
49:31
leads to negative growth.
49:35
But you don't get negative growth. In
49:37
this diagram, you do get that
49:40
Y over over N will decline.
49:43
But that doesn't mean that you get
49:44
negative growth. It just means that
49:47
output is not growing as fast as
49:48
population.
49:50
But but both are growing. Just the
49:52
population is growing faster than
49:53
output. I'll I'll I'll start from that.
49:56
Uh
49:57
oh, I think it's after your break. So,
49:59
you're going to have forgotten
50:00
everything by then. So, I'll do a review
50:02
of this and then and then we
50:04
Okay. Have a Have a nice break.
— end of transcript —
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