so in our third lecture on the solo
model we're going to take the model to
data we're going to see whether the
model is consistent with the basic facts
of economic growth let's take a
look so from our previous lectures we
know that in the soal model GDP per
capita is a positive function of savings
or investment positive function of human
capital and a negative function of
population growth so from this point of
view the secret to development is quite
simple is to follow what bill easly is
called the Victorian values work hard
and save study hard and don't have too
much sex that is limit
childbearing so let's see for some data
whether this these predictions hold
up so what we're showing here is the
relationship between real GDP per capita
in 2005 where we're thinking about this
is sort of the steady state level of GDP
per capita and the mean or average
investment rate over the previous 35
years and what we're showing is that
countries which invested more over those
35 years they ended up with a higher
level of GDP per capita so quite
consistent with the solo model by the
way I've taken the data here from David
Wild's book uh economic growth this is a
textbook uh it's very good you may want
to take a look at
that our second bit of data looks at
real GDP per capita against the average
years of schooling and again we see a
pretty nice close tight relationship
more years of schooling more human
capital greater GDP per capita now of
course the causality here could be going
the other way perhaps it's just in rich
countries that people you know have a
lot of schooling because schooling can
be fun at least for some of us not for
all of us so that's possible too but
nevertheless the basic facts are
consistent with the solo
model so here we're showing the um
average or mean population growth rate
over the previous 35 years again real
GDP per capita on the vertical axis and
we see as the solo model predicts a
negative relationship between the
population growth rate and real GDP per
capita now again here the causality is
not necessarily all running from the
population growth rate to real GDP per
capita perhaps in poor countries they
have more children perhaps they have
more children because that's one of the
only ways in which they can save for
retirement that is have somebody there
have a child there to take care of them
so there's different interpretations
nevertheless um the prediction is
consistent with the solo model looking
pretty good for the solo
model the solo model however offers more
than just qualitative predictions uh by
putting a little bit more structure on
the model we can actually get some
quantitative predictions that's really
what we want to do so let's take a
closer look so let's remember that the
key to the solo model is that you reach
the steady state when investment is
equal to
depreciation now let's take the
equations for investment and
depreciation and set that up so that
means that taking the investment here
and depreciation here the steady state
is found when this equation is true now
let's remember that the Y here this is
is output per person okay that is equal
to a times the capital labor ratio to
the uh Power of alpha so let's just
substitute that in okay right there
substitute for y we get this so here is
our equation for the steady state so
let's work a little bit with our
equation and see whether we can get some
more quantitative predictions out of
that okay here's the same uh equation
that we had on the previous page just to
remind us now let's work with this
equation a little bit let's divide both
sides by K to the power of alpha so if
we do that we get this all right now
what we want to do is isolate K so let's
take both sides of this equation and uh
take them to the power of 1 over 1 minus
Alpha okay that's going to get rid of
this term here so if we do that and then
we uh move K over here we just rearrange
put this stuff stuff over here what we
get is this equation and lo and behold
this is the steady state level of K
steady state level of capital or the
capital excuse me the steady state
capital labor ratio and I've marked it
with a star just indicating this is the
steady state level and what we've done
here is we've expressed the steady state
level of capital in terms of these
exogenous factors in terms of the
savings rate the depreciation rate the
population growth rate the technology
term uh a and this term Alpha which is a
term on the production function it's
going to turn out to be important and
we're be going to be talking about Alpha
a little bit more so for now however uh
we have isolated the steady state level
of
capital so from our previous slide we
know that the steady state level of
capital is given by this expression we
also remember that output per worker is
equal to a Time the capital labor ratio
uh to the power of this Alpha term and
so substituting if we substitute this in
for here the capital labor ratio here
then we'll get the steady state level of
output per worker pretty straightforward
the only possible complication is
there's an a out here there's also an a
within the parentheses here and when you
combine them and bring them outside the
parenthesis this is what what you get so
this is the level of steady state per
worker given by the solo model it
depends upon in particular these
fundamental factors here so the next
thing we're going to want to ask is
suppose we want to compare do two
different countries the solo model says
any differences in output per worker
must be due to the differences in these
fundamental factors so let's ask suppose
that the savings rate were different in
two different countries how much of the
differences in output per worker which
we observe in the world can be explained
by reasonably sized differences in these
fundamental factors we're going to focus
on the savings rate but we could do
similar exercises for the other ones as
well so that's the question we're going
to ask
next this looks slightly imposing but
it's really not we're going to compare
two different countries by looking at
the ratio of their gdps per capita so
from the pr previous slide we know that
the steady state GDP per capita for the
United States is given by this
expression here according to the solo
model and the steady state ratio of GDP
per capita for let's say Zambia or
Zimbabwe it's given by again according
to the solo model by this expression
here now let's suppose that everything
in these two countries was the same
except for the savings rate if that's
the case then a bunch of these terms are
going to drop out and you'll see that
the difference in GDP per capita is
explained by the differences in the
savings rate so in particular let's
suppose that this ratio of savings rate
suppose the ratio was four that is
suppose that the United States saved
four times as much as Zambia in
particular let's imagine this is not the
case but let's imagine that the United
States saved 20% of its output and
Zambia saved 5% of its output so the
savings rates differed by a factor of
four if that was the case how large a
difference in GDP per capita could we
explain based upon a difference in
savings rates difference in the ratio of
four well that's turns out is going to
depend upon this Alpha over 1 minus
Alpha term and that's what we're going
to be looking at next so the question
once again is suppose that savings rates
differed by a ratio of four how much
could that ratio of four
explain in terms of ratios of GDP per
capita so what this graph tells us is
that if Alpha is small then a difference
in savings rates of four a ratio of four
that's only going to be able to explain
a fairly small difference in the income
ratios cross
countries on the other hand if Alpha is
large if Alpha is over here okay then
we're going we're going to be able to
explain a large differences large
difference in the income ratios by a
fairly modest difference a reasonable
difference in the savings rate so if
Alpha is big and the savings rate
differs by a factor of four then we're
going to be able to explain differences
in GDP per capita across countries of 30
30 times difference a four times savings
rate difference is going to translate
into a 30 times difference in GDP per
capita if Alpha is big but if Alpha is
small a four times difference of uh
savings rates is only going to be able
to explain a small fraction of the
difference in GDP per capita across
countries so what is Alpha and how big
is
it so how big is Alpha well in order to
get a grip on this equation we need some
way of connecting Alpha with something
which we can measure in the real world
and to do that we need to make some
assumptions we're going to assume uh
reasonably competitive markets and we're
going to assume that the bulk of ideas
we don't have to pay for this so you
know
pasteurization uh Newton's Laws these
are all available to us sort of free of
charge uh as it were of course you still
have to pay for Capital and you still
have to pay for labor educated labor
which understands these issues um but
you don't have to pay for the ideas
themselves that's uh a in other words is
a public is a public good okay both of
these assumptions are not bad not
perfectly true of course but they're
going to get us quite quite a way so
they're worthwhile making these
assumptions even if they're not
perfectly true if we do this if we make
these assumptions then in particular the
wage will be equal to the marginal
product of Labor okay with competitive
markets the wage is going to be equal to
the marginal product of labor right and
the marginal product of labor is the
derivative of output with respect to L
which is this expression right here okay
now let's take the wage and multiply It
by L okay that gives us the total
payments to labor if we then divide by y
this gives us the fraction of output
which which goes to labor or labor share
in national income okay so this is labor
share in national income now in terms of
our expression it's just this and you
can see there's a bunch of things we can
divide okay cancel out and what we have
is this nice expression whoa turns out
that labor share of national income is
equal to one minus alpha or in other
words we can find this number in the
National accounts so if you go to the
national accounts what you can find is
that Labor's share of income is about
2/3 you know wages and labor
compensation that covers the bulk of GDP
that covers the bulk of national income
okay closely related after you take into
account depreciation and so forth so in
other words if 1 minus Alpha is equal to
2/3 Alpha is equal to
1/3 okay let's go back and look at our
diagram okay here's our diagram again
and uhoh if Alpha is equal to 1/3 then
we're only going to be able to explain
an income ratio of two with the savings
ratio of four in other words if Alpha is
equal to 1/3 the solo model is going to
have real difficulty explaining large
income ratios even with you know pretty
large size savings ratios you know any
reasonably size savings ratio is not
going to be able to explain the income
ratios we OB OB erve in the real world
when Alpha is equal to
1/3 all right well Manu RoR and while
famous paper they said well they made
two points they said first if Alpha is
equal to 6 then the solo model actually
does pretty well at explaining
differences in income across countries
moreover they said an alpha equal to0 6
that's actually not too bad because
Labor's
share isn't really about pay payments to
Pure labor the payment to Labor's share
also represents a payment to human
capital and human capital education
takes time to produce so human capital
really ought to be considered a type of
capital hence the name and if you think
about human capital as capital then
Capital share which is Alpha it's not at
all unreasonable then to think that
Alpha is equal to
06 however this is not the end of the
story for a couple of reasons one
objection I have is that if it were
really the case that the differences in
income ratios was explained by savings
ratios and also by human capital you
know why isn't everyone Rich then you
know could it really be the case that
what is holding back poor countries is a
failure to save that seems really hard
to believe like if you went to a poor
country and you said if only you saved
you know a little bit more than you're
saving now not that much more but if
only you saved more then within a
generation you would be much much richer
than you are now I think they would do
it or to put it differently a forn a
would work much better than it actually
does if all it took to generate growth
was a difference in savings rate then we
could push their savings even
temporarily higher that would increase
their income and then it could very well
uh snowball could very well keep going
so uh I don't think the differen is
relas savings rate there's also a couple
of other objections even when Alpha is
equal to 6 and we'll deal with those
next the solo model also has
implications for interest rates and
skilled wages and thus for Capital and
labor flows and here again we're going
to see that it's solo model is really
not capable of explaining some of the
basic facts in particular if poor
countries are poor because they lack
capital and skilled layor then the
return on Capital and skilled labor
should be much higher in poor countries
but is this true no it's not true so if
anything what we observe is that capital
and skilled labor flow from poor to rich
countries so think about a uh doctor
physician physicians in Africa would
like to come practice in the United
States where their income would be much
higher we don't see a lot of physicians
in the United States moving to Africa in
order to make a higher income the same
thing is true with uh Capital we don't
see massively High interest rates in
poorer countries and we don't see
Capital flowing from rich countries to
poor countries you know unfortunately or
in fact what we see quite often is rich
people in poor countries want to take
their Capital abroad want to invest in
richer countries so all of these things
suggest that a higher Alpha is not not
going to solve all the problems and it
suggest that the solo model taken as a
model
of uh capital investment and capital
investment as a source of growth is not
going to be able to explain all of the
big differences in income ratios which
we observe in the real world to get us
really there we're going to have to add
some things to the model we're going to
have to take into account the
possibility of differences in
productivity across Nations and that's
something will look at in the next video
thanks