MPC and multiplier

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?