hi so today we begin our first lecture
on the solo 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 price in 1987
largely for his work on this model in
the in this lecture we're going to begin
with the super simple solo model which
is the version that Tyler and I
developed in our textbook modern
principles so if you want to follow
along I get a hold of that textbook solo
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 is going to write as e
times L you can think of that as
education times the number of labourers
ideas which we're going to write a later
on 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 AE
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 properties do we want this
function 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 a 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 we have 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 okay and let's imagine that a
tractor is 100 units of capital this is
arbitrary your first tractor okay that
helps you to produce 10 units of output
your first tractor your first tractor
helps you to produce 10 units of output
the second tractor not quite as useful
as the first tractor because you've
already got one tractor running so the
second tractor can't run it quite as
often you only get to about 44.1 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 33.1 units of output 3.2
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
that some ideas will 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 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 gonna
slow down as capital accumulates now if
if it's the case that countries with
small capital stocks should grow rapidly
well why don't all poor countries grow
rapidly if you look around the world
it's Justin simply not the case that
pork run countries on average grow
faster than rich countries in fact the
reason why 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 it adopted more
capitalist institutions it freed up its
property markets it improved incentives
and 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 again
at something else we're gonna be talking
more about it conditional convergence
the countries can grow rapidly towards
when we might call their natural GDP per
capita but we might call the the GDP per
capita which is sort of fundamentalist
comes out of their fundamental quality
of their institutions the fundamental
savings rate and so forth things we'll
be talking more about 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 and Japan you
know
States let's start with a 1950 to 1960
the decade after World War two well what
do we see well we see Germany growing at
at 6.6 percent growth rate Japan at a
six point eight percent growth rate
meanwhile the United States is only
growing in 1.2 percent what is going on
how is it that the losers of World War
two are growing more rapidly than the
winner - a lot of people this just
didn't seem right but the 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 was leading
them was high so the 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 felt at
1.9% 1980 to 1990 japan to 3.4 percent
and of course in the next decade it
slowed down even more United States a
2.3 percent this also suggests another
idea we're going to be talking more
about in future lectures you know if
these high rates 6.6 percent Germany and
Japan and so forth China's 10% 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 you're sort of natural GDP per
capita rate what is determining growth
on the cutting edge we'll be talking
more about that when we come to 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 and now we want to say
well what do you do with that output and
we're gonna split it into two basic
things you can invest the output or you
can consume the output so we have a very
simple one 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
so 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% of output so a
savings rate of 30% 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 okay then that means that
output is 10 right square root of that
and of that 10 units of output three
units are going to be invested and seven
units are going to be consumed pretty
simple next thing that 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 begin to wear and
tear on the factories water system sewer
systems they 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 so depreciation we're going to
write as just a constant fraction of the
capital stock so we'll write that every
hundred units of capital that you have
two units depreciate every period so a
depreciation rate of 2% this means that
every 100 units of capital that you have
two of them wear out every period now
it's going to turn out to be very
important the relationship between the
investment function in green and the
depreciation function here in sienna in
fact we want to focus in on this because
this is the whole model a lot of the
models going to be driven by this so
let's focus in on this area we're gonna
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 how
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 okay 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 okay 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 gridding is getting
smaller anywhere in this red area so the
capital stock is shrinking well if it's
capital stock is growing over here and
shrinking over here then the one place
where the capital stock is constant is
where the investment rate is equal to
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 he has depreciated that's
the steady state
when you are investing just enough to
replace the capital which has
depreciated okay so 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 at a capital stock where
the capital stock is 225 right here okay
the investment equal to the depreciation
that's 4.5 okay that's given right here
all right then the output steady state
output okay 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 solo model is telling us a couple of
things first of all it's a model of GDP
per capita tells us we haven't gone into
this in great detail but you can see
it's gonna have something to do with the
savings rate if the savings rate is
gonna be higher you can see this curve
is going to shift up okay we're going to
get a bigger steady state output all
right 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 gonna 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 gonna
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 gonna grow so large so
I've said at some point all of your
investment is gonna be taken up just
maintaining the capital stock so you're
gonna 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 and it's
gonna take up all of your investment and
that's when growth stops so this model
is telling us something which it 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
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 gonna 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 gonna put
the model through its paces in the next
few lectures Thanks