Hi. So today we begin our first lecture
on the Solow Model. This is a workhorse model in economics,
not just for understanding growth, but also in further developments
for understanding business cycles. The model was developed by Robert Solow and he won the Nobel Prize in 1987,
largely for his work on this model. In this lecture, we're going to begin
with the super simple Solow Model, which is the version that Tyler and I developed in our textbook
"Modern Principles". So if you want to follow along,
I'd get a hold of that textbook. The Solow Model begins with
the production function, which is simply a mathematical model
describing how output is produced. So we're going to write output 'Y'
as a function of physical capital 'K', human capital, which we're
going to write as 'e times L'. You can think of that as education
times the number of laborers. Ideas, which we're going to write as 'A'. Later on, we will also interpret
this 'A' factor as productivity. So we're going to write output is
equal to a function of ideas, physical capital and human capital. Now to begin, we're going to assume
that 'A', 'e', and 'L' are constant. So there's no population growth,
there's no idea growth, there's no education growth. This is nice because it means
that we can then write: output is simply a function of capital. Now, what kind of function is this? What kind of property do we want
this function to have? Basically, there are two. Clearly, we want more capital to produce
more output, a positive function. But we want it to do so
at a diminishing rate. That is, each addition to capital
should generate smaller additions to output,
the more capital we have already. I'll explain that a bit more in a second. A nice simple function
which has these properties is the square root function. So we're going to write: output is
equal to the square root of capital. For example, if there are 16 units
of capital, then output is 4. Let's describe this function
in more detail. Okay, so here is our production function. We have capital along the horizontal axis
and output along the vertical axis. And notice that at a capital stock of 100,
you get an output of 10, the square root of that. At a capital stock of 400,
you get an output of 20. Okay, notice also that this graph
has diminishing returns. Let me show you what I mean. Let's suppose this is a farm,
a wheat farm, and let's imagine that a tractor is
100 units of capital, this is arbitrary. Your first tractor, that helps you
to produce 10 units of output. Your first tractor helps you
to produce 10 units of output. The second tractor's not quite
as useful as the first tractor because you've already
got one tractor running. So the second tractor you
can't run quite as often. You only get about 4 or 4.1 units
of output from your second tractor. What about the third tractor? Well, the third tractor you can only use, let's say, when the first two
tractors break down. So on average, your third tractor produces
even less than your second tractor did. In fact, adding a third tractor now gets
you only about 3 or 3.1 units of output, 3.2 or something like that. Okay, so this is our production function. It has diminishing returns to capital. We've only just begun to develop the model but already there are some hints and some ideas we'll be developing
in future lectures. In particular, due to
diminishing returns to capital, countries with small capital stocks
should grow rapidly. So you take a country where
the capital stock is really low and that means that the productivity
of capital ought to be high. So even a little bit of investment
ought to generate you a big increase in output,
a high growth rate. So China may be one example of this. So, in China they are almost literally,
literally in fact, adopting the first tractors. Those first tractors are increasing
agricultural output tremendously, which is a high growth rate. In fact, across all kinds of
sectors of the economy, China is adding its first units of capital and that is growing the economy rapidly. What this does also suggest, however, is that China is going to slow down
as capital accumulates. Now, if it's the case that countries with small
capital stocks should grow rapidly, why don't all poor countries grow rapidly? If you look around the world,
it's just simply not the case that poor countries on average
grow faster than rich countries. In fact, the reason they're poor
is often that their growth rate is low. So why aren't all poor countries
growing rapidly? Well, what China did is
adopt more capitalist institutions. It freed up its property markets. It improved incentives, you know,
after the death of Mao, in particular. And it was only after
it improved its institutions, that it was able to take advantage of the high productivity of capital
and to grow rapidly. So, here we've got a hint
at something else we're going to be talking more about,
conditional convergence. The countries can grow rapidly towards what we might call
their natural GDP per capita, what we might call the GDP per capita,
which is sort of fundamentalist, comes out of the fundamental
quality of their institutions, the fundamental savings rate,
and so forth. Things we'll be talking about more later. Okay, here's another
interesting application. Bombing a country,
can increase its growth rate. Let's look at some data. So this is data from Germany,
Japan and the United States. Let's start with 1950 to 1960,
the decade after World War II. What do we see? Well, we see Germany growing
at a 6.6 percent growth rate, Japan at a 6.8 percent growth rate. Meanwhile, the United States
is only growing at 1.2 percent. What is going on? How is it that the losers of World War II
are growing more rapidly than the winner? To a lot of people
this just didn't seem right. But actually, the explanation
is pretty simple. The explanation is that huge amounts of the capital stock
in Germany and Japan were destroyed. So Germany and Japan
had a low capital stock but their fundamental
convergence GDP per capita rate, the rate towards which their fundamentals
were leading them was high. So their capital stock
indeed was very productive. They were able to invest a lot
and growth rates were very high. However, as capital accumulated
their growth rates fell. So Germany fell to
1.9 percent from 1989 to 1990, Japan to 3.4 percent, and of course, in the next decade
it slowed down even more, the United States at 2.3 percent. This also suggests another idea we're going to be talking
more about in future lectures. You see if these high rates, 6.6 percent
in Germany and Japan and so forth, China's 10 percent rate, if these high rates
are all about catching up, all about really high
productivity capital, where you're catching up to
your natural level of GDP per capita. You know, what is determining this rate, where you're at your sort of
natural GDP per capita rate? What is determining growth
on the cutting edge? We'll be talking more about that when
we talk about ideas in a future lecture. Okay, let's get back to
developing the model. So, we've talked about
how capital produces output. Now we want to say, well,
what do you do with that output? We're going to split it
into two basic things. You can invest the output,
or you can consume the output. So, we have a very simple
and good model here. Think about potatoes, you know you
produce potatoes, that's your output. You can invest. You can take some of those potatoes
and plant them back into the ground. That's an investment of potatoes in order to produce
more potatoes in the next period, or you can consume the potatoes. So those are the two things
we're going to let output do. We're going to assume that people
take a constant fraction of output and save and invest
that constant fraction. So here is our investment curve,
the green line. Investment, in our case, is going to be
equal to 30 percent of output, so a savings rate of 30 percent. In other words, our investment curve
is equal to 0.3 times "Y". So we take here as our output curve
and we multiply that by 0.3. That gives us our investment curve. In other words, investment is equal to
0.3 times the square root of "K". Let's take a given capital stock. So if we have a capital stock of 100, that means that output is 10,
the square root of that. And of those 10 units of output,
three units are going to be invested and seven units are going to be consumed. Pretty simple. The next thing we need
to add to our model, the next fact we need
to take into account, is that capital depreciates. So those tractors begin to rust,
they begin to fall apart. Think about a car. You bring home a new car. It runs perfectly but after a while
it needs more maintenance. it needs more oil checks. The brakes start to fail. Things start to fail. You have to invest money just to keep the car running
the way it was when you bought it as new. Factories begin to depreciate. There is wear and tear on the factories. Water systems, sewer systems
begin to corrode. They begin to fall apart over time. And you need to invest just to keep you
at the same place that you were before. So this is depreciation, this is
capital depreciation, wear and tear. Here's how we add
depreciation to our model. Depreciation, we're going to write as just
a constant fraction of the capital stock. So we'll write that every
100 units of capital you have, two units depreciate every period, so a depreciation rate of 2%. This means that
every 100 units of capital, you have two of them
wear out every period. Now, what's going to turn out
to be very important, is the relationship between
the investment function, in green, and the depreciation function
here in siena. In fact, we want to focus in on this
because this is the whole model. A lot of the model
will be driven by this. Let's focus in on this area. We're going to blow up these two curves so we can see what's going on
a little bit more clearly. Okay, here's our investment curve
and our depreciation curve, exactly as we had before. I've just blown them up
so they're easier to see. Now, here is the key to the model. When investment is bigger
than depreciation and when you are investing more capital
than is depreciating every period, then your capital stock must be growing. Similarly, when depreciation
is bigger than investment, when more capital is depreciating
every period than you are investing, then your capital stock
is getting smaller. So the capital stock is growing
anywhere in this green area over here, anywhere in this green area. The capital stock is getting smaller
anywhere in this red area. So the capital stock is shrinking. If capital stock is growing over here
and shrinking over here then the one place where
the capital stock is constant is where the investment
is equal to depreciation. This point is the steady state,
where investment is equal to depreciation. The capital stock is holding constant. It's neither growing nor shrinking. The capital stock is constant
when you are investing, every period, just enough to replace
the capital which has depreciated. That's the steady state, when you are investing just enough to
replace the capital which has depreciated. Okay, now we've brought
the output curve back in, so we can see the whole model
in one diagram. Let's remember that the steady state is
when investment is equal to depreciation. That turns out to be a capital stock, where the capital stock is 225,
right here. The investment equal to
the depreciation is 4.5. That's given right here. Then the output,
the steady-state output, is equal to 15. That is when the capital stock is 225. The square root of that is 15. We can think about this,
by the way, as GDP per capita. So, the Solow Model is
telling us a couple of things. First of all, it's a model of
GDP per capita. It tells us, we haven't
gone into this in great detail, but you can see it will have something
to do with the savings rate. The savings rate is going to be higher. You can see this curve
is going to shift up. We're going to get
a bigger steady-state output. We'll talk more about that
in a future lecture. You can also see that this tells us
something about the growth rate, that you're going to grow when you're
below your steady-state capital stock. Finally, it tells us that
that growth is going to taper off, that growth is going to slow down. You're eventually going to come to a point where the investment is
equal to the depreciation. That is, your capital stock
is going to grow so large, such that at some point
all of your investment is is going to be taken up
just maintaining the capital stock. So, you're going to build
so many aqueducts, think of Rome, so many sewers, so many
water supplies, so many factories that every period the depreciation on those bits of capital
is going to be so large that it's going to take up
all of your investment. And that's when growth stops. So this model is telling us something,
which at first is a little bit surprising. It's telling us that capital growth alone,
capital deepening, cannot be responsible for long-run growth. At some point your capital stock will be
so large that it's all going to be, all of your investment is going to be
consumed by making up for depreciation. So this model is telling us we're going
to have to look somewhere else to understand long-run growth. And of course, when we get there,
we're going to be talking about ideas. We're going to put the model through
its paces in the next few lectures. Thanks.