hi so when our second lecture on the
solo model we're going to do some
comparative statics we're gonna see what
happens when you change the savings 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 solo 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 GDP per capita is
okay what happens now if our savings
rate which is what say 30 percent what
happens if the savings rate were to go
to let's say 40 percent 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 and its new
higher level notice that the new higher
level of the 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 is a square
root of the capital stock let's now do
something again very simple let's make
output is equal to a times the square
root of capital
notice here that an increase in a means
that you get more output for the same
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
gonna 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
one now let's suppose that a goes up to
one point five this means because you
have better productivity for every unit
of capital you're now getting 50% more
output from that unit of capital okay 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 one point five 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 per 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
percent 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 out put 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 grow 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 the point 5 or to the
one-half this is exactly the same thing
just a slightly different notation style
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 0.5 there was
nothing special about 0.5 was just an
easy square root was 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 the nice way a good
reason for writing the production
function in this way which is called a
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 ought to double output as
well and this production function has
that
property ok it's gonna be a little bit
easier if we write this in terms of a
GDP per capita or output per person okay
let's assume that every person is a
worker so we're gonna divide both sides
by L and then we're gonna 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 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 time's the capital
per worker to the power of alpha okay
now let's just define output per worker
as little wide 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 Solo form it's
exactly with the difference being we've
got an alpha instead of a square roots
we've generalized slightly but it's
exactly the same form as we have before
so in fact the solo 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 term 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 given 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 can depreciate capital in the
sense that we've been talking about
before wear and tear capital just
where's 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 interpretations a
little bit different the flavor of the
models exactly as we have before and we
could 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
gonna get in equilibrium a lower amount
of output per worker and capital per
worker
let's take a look of that look at that
using a mathematical 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
webpage 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 get
greater GDP per capita you can move the
savings rate around like this here's
what happens with better ideas better
technology that shifts the curve upwards
let's reduce our savings right down here
shift the technology curve up or it's
the better idea curve and we get
greater GDP per capita for the same
capital stock and because higher GDP per
capita increases the incentive to invest
we get a larger capital stock as well
okay let's move the savings rate back
here mm-hmm this down here here's the
depreciation rate and you can see we had
a higher depreciate depreciation rate
you get at 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 the data how much of the data can
this solar model explain that's what
we're going to take up in our next
lecture Thanks