# Keynesian cross and theÂ multiplier

## Video transcript

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.