MPC and multiplier

going with my habit of overly simplified economy let's that imagine 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 it has mr. farmer 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 yes a lot more maybe this is the Builder right over here so this is mr. 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 marginal propensity propensity to consume consume is when I'll put that in parentheses it's often referred to as M PC that is equal to so you could either say 60 percent or it's 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 a sudden gets an extra dollar he's going to spend another 60 cents on other things and the only person to really spend it with his the farmer if the farmer gets another dollar he's going to spend 60% of that or 60 cents 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 there they were all living happily they 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 it's got a little bit of their agreed-upon currency maybe the agreed-upon currency in this island is the dollar they've maybe got a stash when they when they're shipwrecked on this or whatever so the agreed-upon currency is actually the dollar and the farmer discovers that he's got that he wants to he discovers a big pile of dollars in is in his in his in his sock and he says well I'm going to spend $1,000 on I need to do some repairs to my buildings so let's say so we have this and we have this kind of increase in spending that's going on so the farmer says hey I'm going to spend I'm going to spend $1,000 and I'm going to give it to the Builder now the Builder says well you own gee I've I've just gotten $1,000 I have a marginal propensity to consume of 60 percent or 0.6 I'm going to spend 60% of that so he's going to spend so he's going to spend and the only person he can spend it with is the farmer he's going to spend 60% 60% times $1,000 which is equal to $600 well now the farmer says well I just I got above and beyond the thousand that I just spent somehow you know the economy seems to be picking up the Builder just spent $600 more on me than I 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 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 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 dollars well now the builder says well you know I got that initial thousand dollars I spent 600 that but now I got another 360 dollars and I have a marginal propensity to consume of 0.6 so I'm going to press pend 60 percent of that so above and beyond this bending he also spends 60 percent of this right over here and 60 percent 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 zero point six times zero point six times $1000 now this number right over here I don't know what this is is a 60 percent of three hundred and sixty dollars I don't get a calculator out to figure out what that is exactly so let's say that let's say that I have so 0.6 we can actually say point six to the third power point six or let's just let's just write that or point six to the third and then I'm going to multiply that times 1000 gives us two hundred and sixteen dollars so this guy so this right over here gives us two hundred and sixteen dollars this guy says hey I got another two hundred sixteen dollars I'm going to spend sixteen percent of that I think you see where this is going and sixty percent 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 zero point six times zero point six times zero point six so you're gonna have zero point six times zero point six to the third power that's going to be zero point six to the fourth power times 1000 times 1000 which is whatever 60 percent of 216 is and I'll just calculate it so that's so times 0.6 gives us one hundred thirty dollars is going to get one hundred twenty nine point six dollars now this guy the biller says oh I got another under twenty nine point six dollars I'm gonna spend 60% of that and it goes on and on and on so given this let's think about how much from that incremental that incremental increase of spending of a thousand dollars how much total new production and spending happened in this economy so the way to think about that so the total total total and we could view it either way remember you can view kind of the GDP you could do 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 total output measured in our agreed-upon currency which is let's say dollars this is now going to be it was this original thousand that the farmer spent for the Builder so it's going to be that original thousand plus this first right over here this point six times a thousand that the Builder spend at six hundred dollars so that's 0.6 times 1,000 plus then we had this time the farmer said I'm going to spend sixty percent of that so that was 0.6 squared times one thousand plus zero point six squared times 1,000 and then this guy said oh I'm going to spend 60% of that now that I got that point six squared times 1,000 so that he's going to take 60% of that and spend it and that gave us a point six to the third power times 1,000 plus zero point six to the third power times 1,000 and then we and then the last one we did it would keep going on and on forever theoretically is you're going to have the zero point six is you're going to have plus zero point six to the fourth power times 1,000 and then this would keep going on and on forever we could then it would be plus zero point six to the fifth power times 1,000 plus zero point six to the sixth power keep going on and on forever and the one way and then me one of the fascinating things about mathematics and maybe the next video I will 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 one 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 from that that's kind of sparked by that original $1000 we can factor out the thousand I'll do this in a new color so we can factor out the thousand and we are left with well if you factor out a thousand there you get one plus zero point six plus zero point six squared plus zero point six to the third power plus zero point six 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 this right here simplifies 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 this would be equal to 1 over 1 over 0.4 and zero-point-four is 2/5 so this is equal to 1 over 2/5 which is equal to 5 halves so your total output your total output is going to be equal to is going to be equal to 1,000 times times 5 halves or this is the same thing as equal to one thousand times two and a half which is equal to 2500 so there's two interesting ideas that are going here is 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 and that that sixty percent it keeps getting multiplied and going through the economy you essentially have this multiplier effect that that thousand got spent some fraction of that gets spent and some fraction of that gets spent and so what we ended up doing is that thousand that first thousand dollars got x 2.5 got 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 that drives the multiplier and all the multipliers 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