Keynesian cross and the multiplier | Macroeconomics | Khan Academy

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In the last video, we saw
how the Keynesian Cross

could help us visualize an increase in

government spending
which was a shift in our

aggregate planned expenditure
line right over here

and we saw how the
actual change, the actual

increase in output if you take all the

assumptions that we
took in this, the actual

change in output and
aggregate income was larger

than the change in government spending.

You might say okay,
Keynesian thinking, this

is very left wing, this is the

government's growing larger right here.

I'm more conservative.
I'm not a believer in

Keynesian thinking.

The reality is you actually might be.

Whether you're on the right or the left,

although Keynesian economics tends to be

poo-pooed more by the
right and embraced more

by the left, most of the
mainstream right policies,

especially in the US,
have actually been very

Keynesian.

They just haven't been
by manipulating this

variable right over here.

For example, when people
talk about expanding

the economy by lowering taxes, they are a

Keynesian when they say
that because if we were

to rewind and we go back to our original

function so if we don't
do this, if we go back to

just having our G here,
we're now back on this

orange line, our original
planned expenditure,

you could, based on this
model right over here,

also shift it up by lowering taxes.

If you change your taxes to be taxes minus

some delta in taxes, the
reason why this is going

to shift the whole curve
up is because you're

multiplying this whole thing by a negative

number, by negative C1.

C1, your marginal
propensity to consume, we're

assuming is positive.

There's a negative out here.

When you multiply it
by a negative, when you

multiply a decrease by
a negative, this is a

negative change in taxes,
then this whole thing

is going to shift up again.

You would actually shift up.

You would actually shift
up in this case and

depending on what the
actual magnitude of the

change in taxes are,
but you would actually

shift up and the amount
that you would shift up -

I don't want to make my graph to messy so

this is our new aggregate
planned expenditures -

but the amount you
would move up is by this

coefficient down here, C1, -C1 x -delta T.

You're change, the amount
that you would move up,

is -C1 x -delta T, if we assume delta T is

positive and so you
actually have a C1, delta T.

The negatives cancel out
so that's actually how

much it would actually move up.

It's also Keynesian when you say if we

increase taxes that will
lower aggregate output

because if you increase
taxes, now all of a

sudden this is a positive,
this is a positive

and then you would shift the curve by that

much.

You would actually
shift the curve down and

then you would get to a
lower equilibrium GDP.

This really isn't a difference between

right leaning fiscal
policy or left leaning

fiscal policy and
everything I've talked about

so far at the end of the
last video and this video

really has been fiscal policy.

This has been the spending
lever of fiscal policy

and this right over here
has been the taxing lever

of fiscal policy.

If you believe either of those can effect

aggregate output, then you are essentially

subscribing to the Keynesian model.

Now one thing that I did
touch on a little bit

in the last video is
whatever our change is,

however much we shift
this aggregate planned

expenditure curve, the
change in our output

actually was some multiple of that.

What I want to do now is
show you mathematically

that it actually all works
out that the multiple is

actually the multiplier.

If we go back to our
original and this will just

get a little bit mathy
right over here so I'm

just going to rewrite it all.

We have our planned
expenditure, just to redig

our minds into the actual expression, the

planned expenditure is
equal to the marginal

propensity to consume
times aggregate income

and then you're going to have all of this

business right over here.

We're just going to go
with the original one,

not what I changed.

All this business, let's just call this B.

That will just make it
simple for us to manipulate

this so let's just call
of this business right

over here B.

We could substitute that back in later.

We know that an economy is in equilibrium

when planned expenditures
is equal to output.

That is an economy in
equilibrium so let's set this.

Let's set planned expenditures equal to

aggregate output, which
is the same thing as

aggregate expenditures, the same thing as

aggregate income.

We can just solve for
our equilibrium income.

We can just solve for it.

You get Y=C1xY+B, this
is going to look very

familiar to you in a second.

Subtract C1xY from both sides.

Y-C1Y, that's the left-hand side now.

On the right-hand side,
obviously if we subtract

C1Y, it's going to go away
and that is equal to B.

Then we can factor out
the aggregate income from

this, so Yx1-C1=B and
then we divide both sides

by 1-C1 and we get, that cancels out.

I'll write it right over here.

We get, a little bit of
a drum roll, aggregate

income, our equilibrium, aggregate income,

aggregate output.

GDP is going to be equal to 1/1-C1xB.

Remember B was all this business up here.

Now what is this?

You might remember this
or if you haven't seen

the video, you might
want to watch the video

on the multiplier.

This C1 right over here is our marginal

propensity to consume.

1 minus our marginal propensity to consume

is actually - And I
don't think I've actually

referred to it before which
let me rewrite it here

just so that you know the
term - so C1 is equal to

our marginal propensity to consume.

For example, if this is
30% or 0.3, that means

for every incremental dollar of disposable

income I get, I want to spend $.30 of it.

Now 1-C1, you could view
this as your marginal

propensity to save.

If I'm going to spend
30%, that means I'm going

to save 70%.

This is just saying
I'm going to save 1-C1.

If I'm spending 30% of that incremental

disposable dollar, then I'm
going to save 70% of it.

This whole thing, this is the marginal

propensity to consume.

This entire denominator
is the marginal propensity

to save and then one over
that, so 1/1-C1 which

is the the same thing
as 1/marginal propensity

to save, that is the multiplier.

We saw that a few videos ago.

If you take this infinite
geometric series,

if we just think through
how money spends, if I

spend some money on some
good or service, the

person who has that
money as income is going

to spend some fraction
of it based on their

marginal propensity to
consume and we're assuming

that it's constant
throughout the economy at all

income levels for this
model right over here.

Then they'll spend some
of it and then the person

that they spend it on,
they're going to spend

some fraction.

When you keep adding all
that infinite series up,

you actually get this
multiplier right over here.

This is equal to our multiplier.

For example, if B gets
shifted up by any amount,

let's say B gets shifted
up and it could get

shifted up by changes in any of this stuff

right over here.

Net exports can change,
planned investments

can change, could be shifted up or down.

The impact on GDP is
going to be whatever that

shift is times the multiplier.

We saw it before.

If, for example, if C1=0.6, that means for

every incremental disposable
dollar, people will

spend 60% of it.

That means that the
marginal propensity to save

is equal to 40%.

They're going to save
40% of any incremental

disposable dollar and
then the multiplier is

going to be one over
that, is going to be 1/0.4

which is the same thing
as one over two-fifths,

which is the same thing
as five-halves, which

is the same thing as 2.5.

For example, in this
situation, we just saw that

Y, the equilibrium Y is
going to be 2.5 times

whatever all of this other business is.

If we change B by, let's
say, $1 billion and

maybe if we increase B by $1 billion.

We might increase B by
$1 billion by increasing

government spending by $1
billion or maybe having

this whole term including
this negative right

over here become less
negative by $1 billion.

Maybe we have planned
investment increase by

$1 billion and that could
actually be done a little

bit with tax policy too
by letting companies

maybe depreciate their assets faster.

If we could increase net
exports by $1 billion.

Essentially any way that we
increase B by $1 billion,

that'll increase GDP by
$2.5 billion, 2.5 times

our change in B.

We can write this down this way.

Our change in Y is going
to be 2.5 times our

change in B.

Another way to think
about it when you write

the expression like
this, if you said Y is a

function of B, then you
would say look the slope

is 2.5, so change in Y over change in B is

equal to 2.5, but I just
wanted to right this

to show you that this isn't some magical

voodoo that we're doing.

This is what we looked at
visually when we looked

at the Keynesian Cross.

This is really just describing the same

multiplier effect that
we saw in previous videos

and where we actually derived
the actual multiplier.