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# Tax multiplier, MPC, and MPS

The spending multiplier and tax multiplier will cause a $1 change in spending or taxes to lead to further changes in AD and aggregate output. The spending multiplier is always 1 greater than the tax multiplier because with taxes some of the initial impact of the tax is saved, which is not true of the spending multiplier.

## Want to join the conversation?

- At5:09, why does the X is equal to negative delta T multiplied by MPC?(27 votes)
- I had the same question. I watched a few videos on the tax multiplier and I believe the reason it's negative is because of the way it affects aggregate demand in an economy. When taxes go down, the aggregate demand curve will increase (people will be consuming more with the extra money not going towards taxes); and vise versa, when taxes go up, the aggregate demand curve will decrease. Like in this example, taxes were introduced, and the effect on the net GDP in the economy will be negative. One can calculate the estimated effect on an economy due to tax increases or decreases using this tax multiplier formula. If you want GDP to go up, decrease taxes and use this formula to find out how much the aggregate demand will be shifted; it will be positive, as there will be another negative sign within delta t due to taxes going down.

I hope this helps clear it up a bit!(8 votes)

- At5:05, how is x = -ΔT.MPC ?

In my opinion shouldn't it be x - ΔT. MPC? because the tax he gave is deducted from the amount he spent. For eg - Earlier if I gave 1000$ but now I can to pay $920 as 80$ is paid in taxes.

I didn't get the reasoning/logic sal gave. Please, help me understand.(23 votes)- Not sure if it's correct but this is my approach; the additional $1000 x is hypothetical. Before the $1000 was in the picture, there was economic stability, so she didn't have to make a "choice" whether to spend or save (maybe she was using her stable in a fixed manner perhaps buying more fertilizer, hiring labor, etc.)

But the MPC is applied to her new "extra" income. Now she can make a "choice", and we the economists have observed that given a choice, the farmer would spend a fraction (MPC) of the new income.

When the government decides to levy tax, they are deducting regardless of economic stability or "extra income". The farmer may or may not have an extra $1000 when the taxes are levied.(1 vote)

- Something is wrong about this video. Please check it.In my opinion, the formula must be Δx = -ΔT.MPC. Even if it is not, can you explain it more clearly?(5 votes)
- As 5 people have mentioned here at Khan Academy, and another person in YouTube comments, at5:05the formula

X = -Δt * MPC

shouldn't be allowed.

In the video it is established that the variable X is a positive number, as well as the MPC being a positive number between 0 and 1.

I'm not aware of the correct formula in economics, but it's not possible to combine a negative factor (-Δt) and a positive factor (MPC) to create a positive product (X). I may be entirely wrong and Δt is negative if the government is taxing negative dollars, but taxing a negative amount in generally not the case. Δt is not explained in the video, nor is the formula

X = -Δt * MPC.

I believe it would benefit users to remove this video for the time being, seeing as a large part of it is unclear.

Nevertheless, I am inspired by Khan Academy's affect on the world and education as a whole. To whomever is reading this, thank you for your contribution and have a great day.(4 votes) - This video does not make much sense, it says x = -ΔT * MPC, but that would not make sense, she is paying the builder negative dollars? What about the money she actually has, this only counts the decrease caused by the tax. Secondly, ΔT will not be the same for every transaction, how come there was no mention of a Tax Rate or something, so that you can establish what ΔT really means in each context?

Please, Please Khan Academy, please remake this one.(4 votes) - This video is terrible. It just confuses you talking about geometric series and some obscure beating around the bush equation. Dislike.(3 votes)
- Let's say I spend 30% of my income (and save 70%). If the government taxes me more, not only I will have less income but I will think that's better to save more. So the more Government tax people, the more their MPC gets smaller. If after taxes and my regular expenses (household, food, etc..) I am left with 1$ I'll probably not spend 30% but just keep it because.. you never know(3 votes)
- That's a valid point. When the government taxes individuals more, it reduces their disposable income and consequently their ability to save or spend. This can lead to a decrease in the marginal propensity to consume (MPC), which is the percentage of additional income that people typically spend.

If your income after taxes and regular expenses leaves you with only $1, it is understandable that you might choose to save that amount rather than spend 30% of it. People tend to prioritize saving for unexpected expenses or future needs, especially when their disposable income is limited.

Overall, an increase in government taxes can potentially lead to a decrease in consumption and a higher propensity to save, which impacts the economy's overall spending levels.(1 vote)

- Isn't X=delta t? Because as Khan explains, we could think it that way: if one get same amount of money as the tax, then x/1-c is the total amount contributed to the output, and as the money is gone, one would not spend that amount of money, then shouldn't the x here equals to delta t instead of delta t times MPC?(3 votes)
- x is how much her consumption has increased (decreased if negative). if you give that person a dollar, their spending will increase by 1*MPC. Now, if you take away a dollar from that person, their spending will DECREASE by 1*MPC, or increase by -1*MPC. Now, the amount you take away from them is the amount of the tax dollars (multiply -1 in -1*MPC by Δt). so their spending will increase by -Δt*MPC.(1 vote)

- why tax multiplyer is like = -mpc/mps

what is the algebra behind it?

why dont we write it like mps/-mpc?(1 vote) - Good to know that there really is MPS!(1 vote)

## Video transcript

- [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, 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.