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# MPC and multiplier

Video transcript

Going with my habit of
overly simplified economy, let's then imagine an economy
that has only two actors in it. So it has Mr. Farmer
right over here. Do my best to draw
the farmer, maybe he has a mustache of some kind. So it has Mr. Farmer
right over here. He's got a hat on. So that is the farmer
in this economy. And then let's say we
also have a builder. So this economy, they're
producing two things. They're producing
food, and this builder can help maintain stuff. So maybe he has
a lot more, maybe this is the builder
right over here. So this is Mr. Builder. And let's say, for the sake of
what we're going to do here, let's say that for this economy,
it's kind of a constant. If either of these fellows
gets an extra dollar to spend, he's going to spend 60% of it. And so, what I'm going to do
is introduce a formal word that really is just
another way of saying that. In this economy, the marginal
propensity to consume is-- and I'll put that
in parentheses, it's often referred to as
MPC-- that is equal to you could either say 60%
or is equal to 0.6. And all this is saying is that
if someone in this economy somehow finds another
dollar in their pocket, they're going to
spend 0.6 of that. Or they're going to
spend 60% of that. So if you give the builder--
if a builder all of the sudden gets an extra dollar, he's going
to spend another $0.60 on other things. And the person to really
spend it with is the farmer. If the farmer gets
another dollar, he's going to spend 60% of that,
or $0.60, with the builder. Now given this
assumption, let's think about what would
happen in this economy if all of a sudden
one of them decided to increase their
spending a little bit. So we'll assume that they
were all living happily. The economy was kind
of at a steady state. And let's say the
farmer discovers a sock in a drawer that he
didn't realize was there. And it's got a little bit of
their agreed upon currency. Maybe the agreed upon currency
in this island is a dollar. They've maybe got a stash when
their shipwrecked on this, or whatever. So the agreed upon currency
is actually the dollar. And the farmer
discovers that he's got-- he discovers a big
pile of dollars in his sock. And he says, well, I'm
going to spend $1,000. I need to do some
repairs to my buildings. So we have this kind of increase
in spending that's going on. So the farmer says, hey,
I'm going to spend $1,000, and I'm going to give
it to the builder. Now the builder
says, well, you know, gee, I've just gotten $1,000. I have a marginal propensity
to consume of 60%, or 0.6. I'm going to spend 60% of that. So he's going to spend,
and the only person he can spend it
with is the farmer. He's going to spend 60% times
$1,000, which is equal to $600. Well, now the
farmer says, well, I got above and beyond the
$1,000 that I just spent. Somehow the economy
seems to be picking up. The builder just
spent $600 more on me then he would have
otherwise done. He bought that much more food. I have $600 more. I have a marginal propensity
to consume of 0.6 or 60%. So I will spend 60% of
that $600 that I just got. And so it will be
60% of this thing. So it will be 60-- I'll
write it as a decimal-- it'll be 0.6 times this thing,
which is 0.6 times 1,000. Or you could say it is
60% of the $600, which is going to be equal to $360. Well, now the
builder says, well, I got that initial $1,000. I spent $600 of that. But now I've got
another $360, and I have a marginal propensity
to consume of 0.6. So I'm going to
spend 60% of that. So above and beyond
this spending, he also spends 60% of
this right over here. And 60% of this is 0.6
times this whole thing. So he's going to spend
0.6 times this thing-- and I'll write it in green--
times 0.6 times $1,000. Now this number
right over here, I don't know what this
is, is it 60% of $360. I don't know, I could
get a calculator to figure out what
that is exactly. So let's say that I have
0.6-- we could actually say 0.6 to the third power. Or let's just write
that-- 0.6 to the third. And then I'm going to
multiply that times 1,000, gives us $216. So this guy-- so this right
over here gives us $216. This guy says, hey
got another $216, I'm going to spend 60% of that. And I think you see
where this is going. And 60% of that is going to be
0.6 times this whole quantity. So it's going to be--
I'll write it here-- it's going to be 0.6
times this thing, which was already 0.6
times 0.6 times 0.6. So you're going to have 0.6
times 0.6 to the third power. That's going to be 0.6 to the
fourth power times 1,000, which is whatever 60% of 216 is. And I'll just calculated it. So times 0.6 gives us $130,
is going to get $129.60. Now this guy, the builder,
say, I got another $129.60. I'm going to spend 60% of that. And it goes on and on and on. So given this, let's think
about how much from that incremental increase of spending
of $1,000, how much total new production and spending
happened in this economy? So the way to think about
that, so the total-- and we could view it either way. Remember, you could
view kind of the GDP. You could view that as
the aggregate output. You could view that as
the aggregate income, aggregate expenditure. These are all views
because really the economy is a very circular thing. One person's expenditure turns
into another person's income. But we could say
total output here, measured in our agreed upon
currency, which is let's say dollars. This is now going to be,
it was this original $1,000 that the farmer spent
for the builder. So it's going to be
that original $1,000 plus this first, right
over here this 0.6 times 1,000 that the builder
spent, that $600. So that's 0.6 times
1,000 plus-- then we had this time
the farmer said, I'm going to spend 60% of that. So that was 0.6
squared times 1,000. Plus 0.6 squared times 1,000. And then this guy said, oh,
I'm going to spend 60% of that now that I got that 0.6
squared times 1,000. So he's going to take
60% of that and spend it. And that gave us that 0.6 to
the third power times 1,000. Plus 0.6 to the third
power times 1,000. And then the last one we
did, it would keep going on and on forever,
theoretically, is you're going to have plus 0.6
to the fourth power times 1,000. And this would keep
going on and on forever. We could then would be plus 0.6
to the fifth power times 1,000, plus 0.6 to the sixth power. Keep going on and on forever. And one of the fascinating
things about mathematics, and maybe the next
video, I'll reprove this. I've proven this in
multiple playlists, is that you can actually
sum up because this value right over here is less
than 1, this actually ends up being a finite sum. You can actually take
this infinite sum and get a finite number. So just to simplify this,
the total output that's kind of sparked by
that original $1,000, we can factor out the 1,000--
I'll do this in a new color-- so we can factor out the 1,000. And we are left with--
well if we factor of 1,000 there you get 1 plus
0.6 plus 0.6 squared plus 0.6 to the third power
plus 0.6 to the fourth power. And it goes on and on and on. And in the next video, maybe
I'll prove it, just for fun. But this right over here,
it's an infinite sum of a geometric series. And this will actually
simplify to-- I'll do it in the same green
color-- as 1 over 1 minus 0.6. So whatever this number is right
over here, it'll be 1 minus 1 over that. And so in this case, this
would be equal to 1 over 0.4. And 0.4 is 2/5. So this is equal to 1 over
2/5, which is equal to 5/2. So your total output is going
to be equal to 1,000 times 5/2. Or this is the same thing as
equal to 1,000 times 2 and 1/2, which is equal to 2,500. So there's two interesting
ideas that are going here. One is, when people get
a little bit more income, they're going to
spend some of it. And that's where the marginal
propensity to consume is. We're assuming it's linear,
that no matter how much you give them, they're just going
to spend 60% of that. And then given that, that 60%,
it keeps getting multiplied and going through the economy. You essentially have
this multiplier effect, that that 1,000 got spent,
some fraction of that gets spent, then some fraction
of that gets spent. And so what we ended up
doing is that first $1,000 got multiplied by 2.5. And this 2.5 was
completely a function of what the marginal
propensity to consume was. So we have this
relationship here is that whatever the marginal
propensity to consume is, that drives the multiplier. And all the multiplier
is saying is if you spend an extra dollar
in this economy, given people's marginal propensity to
consume, how much will that increase total output?