Tax multiplier, MPC, and MPS | AP Macroeconomics | Khan Academy

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 going to review what we learned about the multiplier and the marginal propensity to consume. But we're going to 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 0 and 1, 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 a thousand dollars, you would spend six tenths of it; you would spend 600.

And so using this idea, and I'm going to 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? We talked about this in the previous video. The builder is going to say, "I have x dollars."

We're going to 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 a thousand dollars 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'd be the marginal propensity to consume squared. And then this just keeps going on and on and on.

And so he's going to get this amount. What's he going to spend? Well, he's going to spend that times the marginal propensity to consume again, so it's going to be that. We could just keep going on and on and on, but if we want to 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. You might recognize from your algebra class or maybe your precalculus 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 1 over 1 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 one dollar initially on the builder, well, it's going to be one dollar 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 just going to 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 is being taken away from the farmer. So what's going to be? What's the farmer going to do? Well, from the farmer's point of view, 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? What's going to 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, times the multiplier times 1 over 1 minus the marginal propensity to consume. Or if you want to 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 1 minus the marginal propensity to consume.

And so this expression, which seems all fancy and technical, but all it's doing is saying, "All right, 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.

Some of you might have seen something where the denominator looks different; you might see something called a 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 going to spend 30 cents of it.

Well, how much am I going to save? Well, if I spend 30 cents, I'm going to 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. Tax multiplier that looks like this, where they write the negative of the marginal propensity to consume over instead of 1 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, npc; 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.