Connecting the Keynesian Cross to the IS-Curve

In the last video we
hopefully got the intuition

between how real interest rates

might impact planned investment.

We saw that if real
interest rates went up,

then planned investment went down.

If real interest rates went down,

then planned investment went up.

What we want to do in this video is

take this conclusion right over here,

this hopefully fairly intuitive
conclusion right over here

and apply it to our Keynesian Cross

and think about how real interest rates

would effect overall planned expenditure

and what that would do in a model

like the Keynesian Cross,

what that would do to our
equilibrium real GDPs.

Just as a reminder,

let's just draw our Keynesian Cross first,

or parts of it.

On this axis right over here,

we have expenditures.

This axis right over here,

we have income.

We know, from many videos now,

that an economy is a equilibrium

when income is equal,

when aggregate real income

is equal to aggregate real expenditures.

Circular flow of GDP.

Let's draw …

Let me make a line that's all the points

where Y is equal to expenditures.

Along this 45 degree line right over here.

This is our expenditures.

At this point right over here,

that should be the same value

as what our aggregate income is.

That's part of the Keynesian Cross.

The other part is to actually

plot planned expenditures relative to this

and then see where they intersect.

What the equilibrium for that
planned expenditure line?

I'll write it here as ...

I've written it in the past as planned.

I just wrote out the word.

Planned expenditures.

We could write it as
expenditures planned, like that.

It's equal to our aggregate consumption.

Our aggregate consumption,

we can write it as a function
of disposable income.

Y - T is disposable income.

Aggregat income minus aggregate taxes.

I want to be very clear here.

This is not saying C x Y - T.

This is saying C is a function of Y - T.

Give my a Y - T and I will give you a C.

For the sake of our
Keynesian Cross analysis,

and this is kind of kind
of what you would see

in a traditional intro class,

we assume that we have a
linear consumption function.

We assume that our consumption functions.

C as a function of disposable income.

It might be something like
our autonomous consumption

plus our marginal propensity to consume

times our aggregate income, minus taxes.

This right over here
really is multiplication.

We could distribute this C 1.

This is just saying C
as a function of Y - T.

That's only one part of
planned expenditures.

Above and beyond that,

we have planned investment.

We're talking about the
planned side of things.

Now we know that planned investment ...

In the past we viewed it as a constant,

but now we know it can actually be

a function of real interest rates.

Above and beyond that,

we have government expenditures

and then net exports.

For some given real interest rate,

we can plot this line.

The consumption function right over here

is just a line with a
positive slope that intersects

the vertical axis at some place up here.

It has a positive intersect.

All of these, for given interest rate,

these are all going to be constant.

Our planned expenditures would
look something like this.

It might look something like that.

This is YP.

Let's call this YP_1.

This is the YP we get when we pick …

I'll just write ...

I'll just rewrite the
whole thing over again.

We have our consumption,

which is a function of Y - T,

plus the level of
planned investment at ...

Let's say interest rate R1,

so at some given interest rate,

plus government spending,

plus net exports.

We see ...

We've done this Keynesian Cross analysis

several times now, already.

This is our equilibrium level of GDP.

This is where along our
planned expenditure line,

where income is equal to expenditures,

or output is equal to expenditures.

We are equilibrium right over here.

We're not eating into
inventories in an unplanned way

and we're not building excessive inventory

above and beyond what we had planned.

Now, what I want to think about,

what happens if interest
rates go from R1 to R2?

What happens if interest
rates go from R1 to R2

and in particular let's assume that R2?

Now, we're going have
planned investment at R2

and we're going to assume
that R2 is less than R1.

We're essentially saying,

what happens when interest rates go down.

We already know.

When interest rates go down,

planned investment goes up.

Everything else equal,

if this thing shifts up,

if this term right over
here goes from R ...

if the input into it, if the
real interest rate goes down,

then this whole expression
is going to go up

and so you're going to have an increase.

You're going to have a shifting up

of your planned expenditure

for any level of income.

It might look something like this.

It would look something like this.

This delta right over here, this ...

Let me do it right over here.

This distance right over here

is going to be your change
in planned investment.

It went up because
interest rates went down.

We saw that in the last video.

We saw that we got to a new level,

or we see now that

when you shift that up,

that investment goes up.

Because real interest rate went down,

you get to a new equilibrium point.

That equilibrium point
is a higher level ...

it's a higher level of GDP or income.

We know from previous videos as well,

that this distance right over here

is the same as our multiplier

times the amount that
things got bumped up.

The amount that things got bumped up

was the change in planned investment.

Then, we multiply that
times our multiplier.

Our multiplier is 1 over

the marginal propensity to save,

or 1 over 1- the marginal
propensity to consume.

The marginal propensity to consume ...

We assume it's going to be constant

in order to even be able to do this map.

That's this piece right over there.

That is equal to our C1.

The main theme here,

the real big picture
here as we go on our way

to constructing our ISLM model,

is really that all we're seeing ...

when real interest rates go up,

planned investment goes down.

When interest rates go down ...

which is what we saw in this
example right over here.

Actually, let me write this down.

Y, planned expenditures 2 at
C as a function of Y - D +.

Our new planned investment,
at this lower interest rate,

+ G + net exports.

This is our Y2 right over
here, our planned expenditures.

We saw in this example,

when real interest rates went down,

planned expenditures ...

When real interest rates went down,

planned investment went up.

That made total planned
expenditures go up.

That made total GDP go up.

Now we can have another relationship,

which is really very analogous to this.

Really, by changing this,
we're just shifting this curve.

Then, you have the multiplier
effect on our equilibrium output.

The big takeaway from here is,

if real interest rates go up,

not only does planned investment go down,

that would shift this entire curve down.

Then, that would also cause

our equilibrium real GDP to go down.

It would go down by some multiplier,

by the multiplier of
how much this goes down.

If real interest rates go down,

then planned investment,

because of what we saw in the last video,

goes up.

Then, that would cause ...

That would cause this whole ...

That's what we did in this video.

This curve would shift up.

If this curve shifts up,

our equilibrium GDP is going to be

however much this shifted,
times the multiplier,

so your equilibrium GDP is going to go up.

You really have a very
similar relationship

in terms of just how things move.

We can plot this.

Economist are famous for

not always plotting the
independent variable

the way you would want to.

As we construct our ...

What we're going to see is our IS curve.

It stands for investment savings.

What we're going to do

and we'll talk more
about that in the future.

We plot the convention is to put

real interest rates on the vertical axis

and to put real GDP right over here.

If you want to look at this relationship,

when we have a high real interest rate,

we're going to have a low real GDP.

When we have a low real interest rate,

we're going to have a high GDP.

It's going to make spending go up.

If spending goes up,

you have a multiplier effect.

It makes our equilibrium output go up.

Low interest rate, high real GDP,

so you have a curve that relates.

If you want to relate real
GDP to real interest rates

you get a curve like this,

and it's called the IS curve.

IS comes for investment savings.

We're really more focused
on the I part of it,

the way we analyzed here.

The whole reason, based
on the logic in this video

and the last one as well,

the whole reason why we
have this relationship

is due to real interest
rates impact on investment.

When you have high real interest rates,

you don't have much investment.

Also, you'll be sapping out of GDP.

If you lower interest rates,

then that makes you end up
having a lot more investment,

like we saw in the last video.

That will expand GDP by the multiplier

that we see right over there.