- [Instructor] So in this video we're going to revisit
another super simple economy that only has
a farmer and a builder on an island.
And we're gonna review what we learned about the
multiplier and the marginal propensity to consume,
but we're gonna do it a little bit more abstractly.
So just as a bit of a review, the marginal propensity
to consume, this is a number usually between zero and one
that says if you were to get an extra dollar,
how much of that would you spend?
And so your marginal propensity to consume
if it is 0.25, that means that if you were to get a dollar
you would spend 25 cents of it.
If your marginal propensity to consume is 0.6,
that means that if you were to get $1,000,
you would spent six tenths of it,
you would spend $600.
And so using this idea, and I'm gonna keep
it abstract, let's just say that we start
with the farmer, she spends X dollars on products
or services from the builder.
So X dollars goes to the builder,
now what's the builder going to do?
And we talked about this in the previous video.
The builder's gonna say, I have X dollars,
we're gonna assume everyone on this island
has the same marginal propensity to consume,
he's now going to spend some of it.
And he only has one place to spend it,
it's a very simple economy, what's he going to spend?
Well he's going to spend X times the marginal
propensity to consume.
If this seems very abstract, think about it.
If X was $1,000, and if the marginal propensity
to consume is let's say 25 hundredths,
or 0.25, now he's going to spend $1,000
times 0.25, which would be $250.
But now all of a sudden, the farmer has this much money,
and what's she going to do?
Well she's going to spend this times
the marginal propensity to consume.
So she is now going to spend X times the marginal
propensity to consume, which is how much
she got from the builder, and how much of
that is she going to spend?
Well that times the marginal propensity
to consume again.
So it would be the marginal propensity to consume squared.
And then this just keeps going on and on and on.
And so he's gonna get this amount,
what's he gonna spend?
Well he's gonna spend that times the marginal
propensity to consume again, so it's going to be that,
and we could just keep going on and on and on,
but if we wanna figure out how much increased output
is there in this economy because of that initial
expenditure of X dollars, well you would
just sum everything up.
It would be X plus X times the marginal propensity
to consume plus X times the marginal propensity
to consume squared, goes on and on and on.
If we want, we can factor out an X,
so X times one plus marginal propensity to consume
plus marginal propensity to consume squared
on and on and on, and you might recognize
from your algebra class, or maybe your pre-calculus
class that this is an infinite geometric series
that we cover in other videos, and you can actually
sum this up, it's one of the cooler things
in mathematics.
This can be summed up as being equal to one
over one minus the marginal propensity to consume.
And so you have that initial expenditure of X,
but the total amount of output is X times this expression,
and so that's why this expression right over here
is known as the multiplier.
If the farmer spent $1 initially on the builder,
well it's going to be $1 times this expression,
in terms of how much increased output there is
in the simplified economy.
So this is the multiplier.
Now let's do something interesting
and something new that we hadn't done before.
Let's imagine that somehow some magic government
shows up on this island and decides to tax the farmer.
Just says, hey, we're gonna take some money
from this farmer for whatever it might be,
and so this magic government, the amount
that they take away, we will call that delta T,
you could sometimes do that as your change in taxes.
So this is positive, that means that the government
is getting money, and money's being taken away
from the farmer.
So what's gonna be, what's the farmer going to do?
Well, from the farmer's point of view,
they, she now has the delta T less dollars,
and so that's going to affect her consumption.
It's going to decrease her consumption.
How much is it going to decrease her consumption by?
Well it's gonna be the amount she has to give
to the government, times her marginal propensity
to consume.
Times the marginal propensity to consume.
Why does this make sense?
Well in the situation where if I give the farmer a dollar,
let's say the marginal propensity to consume
is 0.5, if I gave her a dollar, she would spend 50 cents.
If I took a dollar away from her,
she would spend 50 cents less.
The amount I take away, times the marginal propensity
to consume.
But if this is the effect on the farmer,
what's going to be the total effect on output?
Well, this is the situation where essentially X
is equal to that.
If I take away delta T from the farmer,
that's essentially saying X is equal to this thing
right over here.
So X is equal to this thing right over here,
and so what's going to be the impact?
Well, the total output is going to be equal
to X, which is negative delta T times the
marginal propensity to consume,
that part right over here, that is that,
times this, time the multiplier.
Times one over one minus the marginal propensity
to consume.
Or if you wanna express this multiplier
as something multiplying by that increase in taxes,
this would be equal to delta T times
the negative of your marginal propensity to consume
over one minus the marginal propensity to consume.
And so this expression, what seems all fancy
and technical, but all it's doing is saying,
alright look, if you take money away from the farmer,
this would be the effect on her initial spend,
and so then you get this multiplier on the taxes,
and this negative is because if you have a positive
taxation right over here, it would have a
decrease in output because of the multiplier.
And some of you might have seen something
where the denominator looks different,
you might see something called an MPS,
or marginal propensity to save.
And that just comes out of the simple idea
that if I have, let's say that I have a marginal
propensity to consume of 0.3,
well that means for every dollar I get,
I'm gonna spend 30 cents of it.
Well how much am I going to save?
Well if I spend 30 cents, I'm gonna save the rest,
so my marginal propensity to save in this situation
would be 70 cents.
Or another way of thinking about it,
your marginal propensity to consume
plus your marginal propensity to save
is equal to one, or another way of thinking about it,
subtract the marginal propensity to consume
from both sides, your marginal propensity to save
is equal to one minus your marginal propensity to consume.
And so that's why this thing is the exact same
thing as this thing, and so sometimes,
you might see a formula for the tax multiplier,
let me write this down.
Tax multiplier, that looks like this,
where they write the negative of the marginal
propensity to consume over, instead of one minus
the marginal propensity to consume,
they write the marginal propensity to save.
And when you're learning this in economics,
it all seems so cryptic, where did this come from,
MPS, MPC, people try to memorize these things,
but as you can see, it's coming out of reasonably
straightforward algebra.
Economists sometimes have a knack for
making straightforward algebra seem a little bit
more complicated than it needs to.
In the next video, I'll do some worked examples
so that we can actually apply these formulas
to see, and see that it makes sense.