hi so today we begin our first lecture
hi so today we begin our first lecture

on the solo model this is a workhorse
on the solo model this is a workhorse

model in economics not just for
model in economics not just for

understanding growth but also in further
understanding growth but also in further

developments for understanding business
developments for understanding business

cycles the model was developed by Robert
cycles the model was developed by Robert

Solow and he won the Nobel price in 1987
Solow and he won the Nobel price in 1987

largely for his work on this model in
largely for his work on this model in

the in this lecture we're going to begin
the in this lecture we're going to begin

with the super simple solo model which
with the super simple solo model which

is the version that Tyler and I
is the version that Tyler and I

developed in our textbook modern
developed in our textbook modern

principles so if you want to follow
principles so if you want to follow

along I get a hold of that textbook solo
along I get a hold of that textbook solo

model begins with the production
model begins with the production

function which is simply a mathematical
function which is simply a mathematical

model describing how output is produced
model describing how output is produced

so we're going to write output Y as a
so we're going to write output Y as a

function of physical capital K human
function of physical capital K human

capital which is going to write as e
capital which is going to write as e

times L you can think of that as
times L you can think of that as

education times the number of labourers
education times the number of labourers

ideas which we're going to write a later
ideas which we're going to write a later

on will also interpret this a factor as
on will also interpret this a factor as

productivity so we're going to write
productivity so we're going to write

output is equal to a function of ideas
output is equal to a function of ideas

physical capital and human capital now
physical capital and human capital now

to begin we're going to assume that AE
to begin we're going to assume that AE

and L are constant so there's no
and L are constant so there's no

population growth there's no idea growth
population growth there's no idea growth

there's no education growth this is nice
there's no education growth this is nice

because it means that we can then write
because it means that we can then write

output is simply a function of capital
output is simply a function of capital

now what kind of function is this what
now what kind of function is this what

kind of properties do we want this
kind of properties do we want this

function have basically there are two
function have basically there are two

clearly we want more capital to produce
clearly we want more capital to produce

more output a positive function but we
more output a positive function but we

want it to do so at a diminishing rate
want it to do so at a diminishing rate

that is each addition to capital should
that is each addition to capital should

generate smaller additions to output the
generate smaller additions to output the

more capital we have already I'll
more capital we have already I'll

explain that a bit more in a second a
explain that a bit more in a second a

nice simple function which has these
nice simple function which has these

properties is a square root function so
properties is a square root function so

we're going to write output is equal to
we're going to write output is equal to

the square root of capital for example
the square root of capital for example

if there are 16 units of capital then
if there are 16 units of capital then

output is 4 let's describe this function
output is 4 let's describe this function

in more detail
in more detail

okay so here is our production function
okay so here is our production function

we have capital along the horizontal
we have capital along the horizontal

axis and we have output along the
axis and we have output along the

vertical axis and notice that at a
vertical axis and notice that at a

capital stock of 100 you get an output
capital stock of 100 you get an output

of 10 the square root of that at a
of 10 the square root of that at a

capital stock of 400 you get an output
capital stock of 400 you get an output

of 20 okay notice also that this graph
of 20 okay notice also that this graph

has diminishing returns let me show you
has diminishing returns let me show you

what I mean
what I mean

let's suppose this is a farm a wheat
let's suppose this is a farm a wheat

farm okay and let's imagine that a
farm okay and let's imagine that a

tractor is 100 units of capital this is
tractor is 100 units of capital this is

arbitrary your first tractor okay that
arbitrary your first tractor okay that

helps you to produce 10 units of output
helps you to produce 10 units of output

your first tractor your first tractor
your first tractor your first tractor

helps you to produce 10 units of output
helps you to produce 10 units of output

the second tractor not quite as useful
the second tractor not quite as useful

as the first tractor because you've
as the first tractor because you've

already got one tractor running so the
already got one tractor running so the

second tractor can't run it quite as
second tractor can't run it quite as

often you only get to about 44.1 units
often you only get to about 44.1 units

of output from your second tractor what
of output from your second tractor what

about the third tractor well the third
about the third tractor well the third

tractor you can only use let's say when
tractor you can only use let's say when

the first two tractors break down so on
the first two tractors break down so on

average your third tractor produces even
average your third tractor produces even

less than your second tractor did in
less than your second tractor did in

fact adding a third tractor now gets you
fact adding a third tractor now gets you

only about 33.1 units of output 3.2
only about 33.1 units of output 3.2

something like that okay so this is our
something like that okay so this is our

production function it has diminishing
production function it has diminishing

returns to capital
returns to capital

we've only just begun to develop the
we've only just begun to develop the

model but already there are some hints
model but already there are some hints

that some ideas will be developing in
that some ideas will be developing in

future lectures in particular due to
future lectures in particular due to

diminishing returns to capital countries
diminishing returns to capital countries

with small capital stocks should grow
with small capital stocks should grow

rapidly so you take a country where the
rapidly so you take a country where the

capital stock is really low and that
capital stock is really low and that

means that the productivity of capital
means that the productivity of capital

ought to be high so even a little bit of
ought to be high so even a little bit of

investment ought to generate you a big
investment ought to generate you a big

increase in output high growth rate so
increase in output high growth rate so

China may be one example of this so in
China may be one example of this so in

China they are almost literally
China they are almost literally

literally in fact adopting the first
literally in fact adopting the first

tractors those first tractors are
tractors those first tractors are

increasing agricultural output
increasing agricultural output

tremendously which is a high growth rate
tremendously which is a high growth rate

in fact across all kinds of sectors of
in fact across all kinds of sectors of

the economy China is adding its first
the economy China is adding its first

units of capital and that is growing the
units of capital and that is growing the

economy rapidly what this does also
economy rapidly what this does also

suggest however is that China is gonna
suggest however is that China is gonna

slow down as capital accumulates now if
slow down as capital accumulates now if

if it's the case that countries with
if it's the case that countries with

small capital stocks should grow rapidly
small capital stocks should grow rapidly

well why don't all poor countries grow
well why don't all poor countries grow

rapidly if you look around the world
rapidly if you look around the world

it's Justin simply not the case that
it's Justin simply not the case that

pork run countries on average grow
pork run countries on average grow

faster than rich countries in fact the
faster than rich countries in fact the

reason why they're poor is often that
reason why they're poor is often that

their growth rate is low so why aren't
their growth rate is low so why aren't

all poor countries growing rapidly
all poor countries growing rapidly

well what China did is it adopted more
well what China did is it adopted more

capitalist institutions it freed up its
capitalist institutions it freed up its

property markets it improved incentives
property markets it improved incentives

and you know after the death of Mao in
and you know after the death of Mao in

particular and it was only after it
particular and it was only after it

improved its institutions that it was
improved its institutions that it was

able to take advantage of the high
able to take advantage of the high

productivity of capital and to grow
productivity of capital and to grow

rapidly so here we've got a hint again
rapidly so here we've got a hint again

at something else we're gonna be talking
at something else we're gonna be talking

more about it conditional convergence
more about it conditional convergence

the countries can grow rapidly towards
the countries can grow rapidly towards

when we might call their natural GDP per
when we might call their natural GDP per

capita but we might call the the GDP per
capita but we might call the the GDP per

capita which is sort of fundamentalist
capita which is sort of fundamentalist

comes out of their fundamental quality
comes out of their fundamental quality

of their institutions the fundamental
of their institutions the fundamental

savings rate and so forth things we'll
savings rate and so forth things we'll

be talking more about later okay
be talking more about later okay

here's another interesting application
here's another interesting application

bombing a country can increase its
bombing a country can increase its

growth rate let's look at some data so
growth rate let's look at some data so

this is data from Germany and Japan you
this is data from Germany and Japan you

know
know

States let's start with a 1950 to 1960
States let's start with a 1950 to 1960

the decade after World War two well what
the decade after World War two well what

do we see well we see Germany growing at
do we see well we see Germany growing at

at 6.6 percent growth rate Japan at a
at 6.6 percent growth rate Japan at a

six point eight percent growth rate
six point eight percent growth rate

meanwhile the United States is only
meanwhile the United States is only

growing in 1.2 percent what is going on
growing in 1.2 percent what is going on

how is it that the losers of World War
how is it that the losers of World War

two are growing more rapidly than the
two are growing more rapidly than the

winner - a lot of people this just
winner - a lot of people this just

didn't seem right but the actually the
didn't seem right but the actually the

explanation is pretty simple the
explanation is pretty simple the

explanation is that huge amounts of the
explanation is that huge amounts of the

capital stock in Germany and Japan were
capital stock in Germany and Japan were

destroyed so Germany and Japan had a low
destroyed so Germany and Japan had a low

capital stock but their fundamental
capital stock but their fundamental

convergence
convergence

GDP per capita rate the rate towards
GDP per capita rate the rate towards

which their fundamentals was leading
which their fundamentals was leading

them was high so the their capital stock
them was high so the their capital stock

indeed was very productive they were
indeed was very productive they were

able to invest a lot and growth rates
able to invest a lot and growth rates

were very high
were very high

however as capital accumulated their
however as capital accumulated their

growth rates fell so germany felt at
growth rates fell so germany felt at

1.9% 1980 to 1990 japan to 3.4 percent
1.9% 1980 to 1990 japan to 3.4 percent

and of course in the next decade it
and of course in the next decade it

slowed down even more United States a
slowed down even more United States a

2.3 percent this also suggests another
2.3 percent this also suggests another

idea we're going to be talking more
idea we're going to be talking more

about in future lectures you know if
about in future lectures you know if

these high rates 6.6 percent Germany and
these high rates 6.6 percent Germany and

Japan and so forth China's 10% rate if
Japan and so forth China's 10% rate if

these high rates are all about catching
these high rates are all about catching

up all about really high productivity
up all about really high productivity

capital where you're catching up to your
capital where you're catching up to your

natural level of GDP per capita you know
natural level of GDP per capita you know

what is determining this rate where
what is determining this rate where

you're at you're sort of natural GDP per
you're at you're sort of natural GDP per

capita rate what is determining growth
capita rate what is determining growth

on the cutting edge we'll be talking
on the cutting edge we'll be talking

more about that when we come to talk
more about that when we come to talk

about ideas in a future lecture okay
about ideas in a future lecture okay

let's get back to developing the model
let's get back to developing the model

so we've talked about how capital
so we've talked about how capital

produces output and now we want to say
produces output and now we want to say

well what do you do with that output and
well what do you do with that output and

we're gonna split it into two basic
we're gonna split it into two basic

things you can invest the output or you
things you can invest the output or you

can consume the output so we have a very
can consume the output so we have a very

simple one good model here think about
simple one good model here think about

potatoes you know
potatoes you know

you produce potatoes that's your output
you produce potatoes that's your output

you can invest you can take some of
you can invest you can take some of

those potatoes and plant them back into
those potatoes and plant them back into

the ground that's an investment of
the ground that's an investment of

potatoes in order to produce more
potatoes in order to produce more

potatoes in the next period or you can
potatoes in the next period or you can

consume the potatoes so those are the
consume the potatoes so those are the

two things we're going to let output do
two things we're going to let output do

so we're going to assume that people
so we're going to assume that people

take a constant fraction of output and
take a constant fraction of output and

save and invest that constant fraction
save and invest that constant fraction

so here is our investment curve the
so here is our investment curve the

green line investment in our case is
green line investment in our case is

going to be equal to 30% of output so a
going to be equal to 30% of output so a

savings rate of 30% in other words our
savings rate of 30% in other words our

investment curve is equal to 0.3 times y
investment curve is equal to 0.3 times y

so we take here as our output curve and
so we take here as our output curve and

we multiply that by 0.3 that gives us
we multiply that by 0.3 that gives us

our investment curve in other words
our investment curve in other words

investment is equal to 0.3 times the
investment is equal to 0.3 times the

square root of K let's take a given
square root of K let's take a given

capital stock so if we have a capital
capital stock so if we have a capital

stock of 100 okay then that means that
stock of 100 okay then that means that

output is 10 right square root of that
output is 10 right square root of that

and of that 10 units of output three
and of that 10 units of output three

units are going to be invested and seven
units are going to be invested and seven

units are going to be consumed pretty
units are going to be consumed pretty

simple next thing that we need to add to
simple next thing that we need to add to

our model the next fact we need to take
our model the next fact we need to take

into account is that capital depreciates
into account is that capital depreciates

so those tractors begin to rust they
so those tractors begin to rust they

begin to fall apart think about a car
begin to fall apart think about a car

you bring home a new car it runs
you bring home a new car it runs

perfectly but after a while it needs
perfectly but after a while it needs

more maintenance it needs more oil
more maintenance it needs more oil

checks the brakes start to fail things
checks the brakes start to fail things

start to fail you have to invest money
start to fail you have to invest money

just to keep the car running the way it
just to keep the car running the way it

was when you bought it as new factories
was when you bought it as new factories

begin to depreciate begin to wear and
begin to depreciate begin to wear and

tear on the factories water system sewer
tear on the factories water system sewer

systems they begin to corrode they begin
systems they begin to corrode they begin

to fall apart over time and you need to
to fall apart over time and you need to

invest just to keep you at the same
invest just to keep you at the same

place that you were before so this is
place that you were before so this is

depreciation this is capital
depreciation this is capital

depreciation wear and tear
depreciation wear and tear

here's how we add depreciation to our
here's how we add depreciation to our

model so depreciation we're going to
model so depreciation we're going to

write as just a constant fraction of the
write as just a constant fraction of the

capital stock so we'll write that every
capital stock so we'll write that every

hundred units of capital that you have
hundred units of capital that you have

two units depreciate every period so a
two units depreciate every period so a

depreciation rate of 2% this means that
depreciation rate of 2% this means that

every 100 units of capital that you have
every 100 units of capital that you have

two of them wear out every period now
two of them wear out every period now

it's going to turn out to be very
it's going to turn out to be very

important the relationship between the
important the relationship between the

investment function in green and the
investment function in green and the

depreciation function here in sienna in
depreciation function here in sienna in

fact we want to focus in on this because
fact we want to focus in on this because

this is the whole model a lot of the
this is the whole model a lot of the

models going to be driven by this so
models going to be driven by this so

let's focus in on this area we're gonna
let's focus in on this area we're gonna

blow up these two curves so we can see
blow up these two curves so we can see

what's going on a little bit more
what's going on a little bit more

clearly okay here's our investment curve
clearly okay here's our investment curve

and our depreciation curve exactly how
and our depreciation curve exactly how

we had before I've just blown them up so
we had before I've just blown them up so

they're easier to see now here is the
they're easier to see now here is the

key to the model when investment is
key to the model when investment is

bigger than depreciation okay when you
bigger than depreciation okay when you

are investing more capital than is
are investing more capital than is

depreciating every period then your
depreciating every period then your

capital stock must be growing
capital stock must be growing

similarly when depreciation is bigger
similarly when depreciation is bigger

than investment okay when more capital
than investment okay when more capital

is depreciating every period than you
is depreciating every period than you

are investing then your capital stock is
are investing then your capital stock is

getting smaller so the capital stock is
getting smaller so the capital stock is

growing anywhere in this green area over
growing anywhere in this green area over

here anywhere in this green area the
here anywhere in this green area the

capital stock is gridding is getting
capital stock is gridding is getting

smaller anywhere in this red area so the
smaller anywhere in this red area so the

capital stock is shrinking well if it's
capital stock is shrinking well if it's

capital stock is growing over here and
capital stock is growing over here and

shrinking over here then the one place
shrinking over here then the one place

where the capital stock is constant is
where the capital stock is constant is

where the investment rate is equal to
where the investment rate is equal to

where the investment is equal to
where the investment is equal to

depreciation this point is the
depreciation this point is the

steady-state where investment is equal
steady-state where investment is equal

to depreciation the capital stock is
to depreciation the capital stock is

holding constant it's neither growing
holding constant it's neither growing

nor shrinking the capital stock is
nor shrinking the capital stock is

constant when you are investing every
constant when you are investing every

period just enough to replace the
period just enough to replace the

capital which he has depreciated that's
capital which he has depreciated that's

the steady state
the steady state

when you are investing just enough to
when you are investing just enough to

replace the capital which has
replace the capital which has

depreciated okay so now we've brought
depreciated okay so now we've brought

the output curve back in so we can see
the output curve back in so we can see

the whole model in one diagram let's
the whole model in one diagram let's

remember that the steady state is when
remember that the steady state is when

investment is equal to depreciation that
investment is equal to depreciation that

turns out to be at a capital stock where
turns out to be at a capital stock where

the capital stock is 225 right here okay
the capital stock is 225 right here okay

the investment equal to the depreciation
the investment equal to the depreciation

that's 4.5 okay that's given right here
that's 4.5 okay that's given right here

all right then the output steady state
all right then the output steady state

output okay is equal to 15 that is when
output okay is equal to 15 that is when

the capital stock is 225 the square root
the capital stock is 225 the square root

of that is 15 we can think about this by
of that is 15 we can think about this by

the way as GDP per capita
the way as GDP per capita

so solo model is telling us a couple of
so solo model is telling us a couple of

things first of all it's a model of GDP
things first of all it's a model of GDP

per capita tells us we haven't gone into
per capita tells us we haven't gone into

this in great detail but you can see
this in great detail but you can see

it's gonna have something to do with the
it's gonna have something to do with the

savings rate if the savings rate is
savings rate if the savings rate is

gonna be higher you can see this curve
gonna be higher you can see this curve

is going to shift up okay we're going to
is going to shift up okay we're going to

get a bigger steady state output all
get a bigger steady state output all

right we'll talk more about that in a
right we'll talk more about that in a

future lecture you can also see that
future lecture you can also see that

this tells us something about the growth
this tells us something about the growth

rate that you're gonna grow when you're
rate that you're gonna grow when you're

below your steady state capital stock
below your steady state capital stock

finally it tells us that that growth is
finally it tells us that that growth is

going to taper off that growth is gonna
going to taper off that growth is gonna

slow down you're eventually going to
slow down you're eventually going to

come to a point where the investment is
come to a point where the investment is

equal to the depreciation that is your
equal to the depreciation that is your

capital stock is gonna grow so large so
capital stock is gonna grow so large so

I've said at some point all of your
I've said at some point all of your

investment is gonna be taken up just
investment is gonna be taken up just

maintaining the capital stock so you're
maintaining the capital stock so you're

gonna build so many aqueducts think of
gonna build so many aqueducts think of

Rome so many sewers so many water
Rome so many sewers so many water

supplies so many factories that every
supplies so many factories that every

period the depreciation on those bits of
period the depreciation on those bits of

capital is going to be so large and it's
capital is going to be so large and it's

gonna take up all of your investment and
gonna take up all of your investment and

that's when growth stops so this model
that's when growth stops so this model

is telling us something which it first
is telling us something which it first

is a little bit surprising it's telling
is a little bit surprising it's telling

us that capital growth alone capital
us that capital growth alone capital

deepening cannot be responsible for
deepening cannot be responsible for

long-run growth at some point your
long-run growth at some point your

capital stock will be so large
capital stock will be so large

it's all going to be all of your
it's all going to be all of your

investment is going to be consumed by
investment is going to be consumed by

making up for depreciation so this model
making up for depreciation so this model

is telling us we're gonna have to look
is telling us we're gonna have to look

somewhere else
somewhere else

to understand long-run growth and of
to understand long-run growth and of

course when we get there we're going to
course when we get there we're going to

be talking about ideas we're gonna put
be talking about ideas we're gonna put

the model through its paces in the next
the model through its paces in the next

few lectures Thanks
few lectures Thanks