Hi. So in our second lecture on the Solow Model,
we're going to do some comparative statics.
We're going to see what happens when you change
the saving rate, for example.
We're also going to add in ideas and population growth,
and see what happens when you change those factors as well.
Let's get going.
Let's briefly review the Solow Model.
So, remember that the key to the model is the steady-state,
is when investment is equal to depreciation.
When investment is equal to depreciation,
the economy is neither growing nor shrinking,
and we can read off this graph what the GDP per capita is.
Okay.
What happens now if our savings rate, which is, let's say, 30%,
what happens if the savings rate were to go let's say 40%?
So with an increase in the savings rate,
what happens is the entire investment curve shifts up.
In particular, beginning at the old steady-state rate,
what happens immediately is that you get
investment which is now greater than depreciation.
And because investment is greater than depreciation,
you get capital accumulation.
That capital stock will continue to grow,
will get larger and larger
until we reach the point where investment is equal to depreciation,
until we reach the point where the capital stock is so large
that it depreciates enough each period that you need all of your investment
simply to maintain the capital stock at its new higher level.
Notice that at the new higher level of capital stock,
we have a higher GDP per capita.
Let's now add ideas to the model.
Remember we began with a very general production function
in which output was created by ideas, physical capital and human capital,
and we simplified it to output as a square root of the capital stock.
Let's now do something again very simple,
let's make output as equal to A times the square root of capital.
Notice here that an increase in A
means that you get more increase for the same amount of capital stock.
So that's the sense in which A represents ideas or productivity;
more output from the same capital stock.
This graph looks initially a little bit complicated,
but it's really not, we're going to take it in steps.
What this graph shows us
is what happens when you increase productivity,
what happens when you increase ideas.
So our previous model, we can think of as A as being equal to 1.
Now let's suppose that A goes up to 1.5.
This means because you've got better productivity,
for every unit of capital, you now are getting 50% more output --
from that unit of capital.
So, a couple of things happen.
Let's begin at our old steady-state, 225 units of capital.
The old curves are shown by the dashed line.
Here's our old steady-state.
Suppose that at this point, --
that we have an increase in productivity
which increases A to 1.5.
Well, the first thing that happens is that better ideas,
means you get more output from the same amount of capital we had before,
more output from the same capital stock.
But because we have more output, we also have more investment.
Okay, more output means more investment.
And now investment is going to be greater than depreciation.
Because investment is greater than depreciation,
we get capital accumulation.
Capital accumulation increases GDP, --
increases GDP per person, GDP per capita.
So we get two things going on.
We get the initial effect of better ideas increases output,
that secondary effect then with higher levels of output for a given capital stock
that encourages more capital accumulation.
We're going to continue accumulating capital
until we reach the point where investment is equal to depreciation,
until our capital stock is so large that we're using --
all of our investment just to maintain the capital stock.
Notice, however, that with better ideas, our capital stock increases,
as does the amount of GDP per capita.
We're not going to say a lot in this course about the development of ideas,
about cutting edge growth,
about why the United States grows at, you know, 2.5% instead of 7%,
you know, what determines improvements in technology over time.
Do however take a look at some of the work of Paul Romer,
or you could take a look at my own TED talk,
or my book, "Launching the Innovation Renaissance".
We will come back later
to talk about a somewhat different interpretation of the A term
in our production function, as a productivity term.
We will be asking why is it that some countries
don't seem able to take advantage of all the ideas on the technological frontier,
or why some countries are less able to combine capital and labor
in the most efficient ways,
to get the most output out of their capital and labor as is possible.
So we'll come back and talk about that second interpretation
a little bit later.
Let's add population growth to the model.
This is going to take a little bit of mathematics,
not much, but bear with me.
So we wrote our production function in very general terms,
and then we simplified it to
output is equal to A times the square root of K.
We can also write this as
A times K to .5 or to the one half.
This is exactly the same thing, just a slightly different notation style,
it means exactly the same thing.
When we do that, a natural way to generalize this
might be to add in labor, with the same format,
and as long as we're generalizing, let's do that this way.
At .5, there's nothing special about .5,
it's just an easy square root which is just easy to work with,
let's generalize this to output is equal to A times K to the alpha
times L to the 1 minus alpha.
A good reason for writing the production function in this way,
which is what is called the Cobb-Douglas production function, --
is that when you double capital and labor,
you double output.
So if we're thinking about capital and labor as the only factors of production,
it makes sense that if you double the only things you need to produce,
you have to double output as well,
and this production function has that property.
Okay. It's going to be a little bit easier
if we write this in terms of
GDP per capita or output per person.
Let's assume that every person is a worker.
So we're going to divide both sides by L,
and then we're going to simplify this term and rewrite a few things.
Let's do that.
Here's our production function carried over from the last slide.
We want to simplify this,
so we just use our rules for exponents to divide through with the L,
that gets us L to the negative alpha.
Notice that L to the negative alpha is the same as 1 over L to the alpha.
So we can rewrite this portion as --
output per worker is equal to A
times the capital per worker to the power of alpha.
Now, let's just define --
output per worker as little y
and define capital per worker as little k,
and then we get this new production function:
little y is equal to A k to the alpha.
Now notice that, even though the interpretation is different,
the form of this equation is exactly our super-simple Solow form.
It's exactly,
with the difference being we've got an alpha instead of a square root,
we've generalized it slightly,
but it's exactly the same form as we had before.
So in fact, the Solow Model goes through, exactly as we had before,
except now we're just going to interpret it in terms of
output per worker and capital per worker.
Let's take a look at that.
So here we have the Solow Model again,
except now we're writing it in per worker terms,
so this is output, now explicitly per worker,
this is capital per worker.
Here's our production function, again, in terms of
output per worker and capital per worker.
Our investment function looks the same
except, again, it's in per worker terms,
it's now investment per worker,
and I've generalized a little bit, giving this a gamma --
to denote our savings rate.
The major difference, and it's really not even a major difference,
is our depreciation function.
It's written again in terms of capital per worker,
and then we have this extra term here.
Now, think about capital per worker.
How could you depreciate the amount of capital per worker?
Well, there's really two ways of doing it.
One is, you could depreciate capital
in the sense that we've been talking about before, wear and tear;
capital just wears out.
Okay? So you lose capital, that means
capital per worker depreciates or goes down.
The other way you can depreciate capital per worker, however,
is with more workers.
And we're going to introduce the population growth rate,
or the worker growth rate as N.
So there's two ways of depreciating capital per worker:
you can have less capital with straight depreciation or more workers.
So, even though the terms are a little bit different,
the interpretation's a little bit different,
the flavor of the model's exactly as we have before,
and we can now see, for example, that
if you were to increase the depreciation rate, --
either because capital depreciation goes up;
maybe we're dealing with computers now which depreciate faster than buildings;
or, because population growth rate goes up
then this curve is going to shift, you know, like this,
and you're going to get an equilibrium,
a lower amount of output per worker and capital per worker.
Let's take a look at that using a Mathematica example.
Okay, here's a nice little demonstration project
using Mathematica.
You can find this demonstration at this address up here.
If you get Wolfram's CDF player, sort of like a PDF,
you can actually easily embed this demonstration project on any web page.
So here we begin, here's our steady state, as usual
down with a capital stock of 225
and investment equals to depreciation.
You can adjust this, so let's have the savings rate.
You increase the savings rate, and, as we know,
the curve, the savings curve, shifts up, okay?
You get a higher capital stock,
you get greater GDP per capita;
you can move the savings rate like this.
Here's what happens with better ideas, better technology,
that shifts the curve upwards.
Let's reduce our savings rate down here,
shift the technology curve upwards, the better idea curve, and we get --
greater GDP per capita for this same capital stock,
and because higher GDP per capita increases the incentive to invest,
we get a larger capital stock as well.
Let's move the savings rate back here, --
move this down here.
Here's the depreciation rate.
And you can see with a higher --
depreciation rate, you get lower GDP per capita,
lower capital.
Remember that we can also now interpret the depreciation rate
as capital depreciation plus population growth.
So this tells us that if we're thinking about
GDP per worker and capital per worker,
that greater population growth reduces GDP per capita.
Smaller population growth increases GDP per capita and the capital stock.
Okay.
Again, you can easily embed this onto your web page if you want to.
Last thing we want to talk about is --
let's take this model to data.
How much of the data can the Solow Model explain?
That's what we're going to take up in our next lecture.
Thanks.