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# Mathy version of MPC and multiplier (optional)

## Video transcript

in this video I'm going to work through the exact same scenario that we saw in the last video but it'll be a little bit more mathy and the reason why I'm going to make it a little bit more matte these so that you see it's the same ideas it's going to have a little bit more cryptic notation but it allows us to generalize the ideas that we saw in the last video so let's just assume instead of saying that the marginal propensity to consume in our little island is 0.6 let's just call it a general let's just say our marginal propensity to consume is C and what we want to do is we want to figure out given some initial initial change in expenditure and this guy's change in expenditure will be this guy's change in income what is going to how is that going to end is that that cycles around and round due to the multiplier effect what is going to be the total change in our GDP so this is what we care about we care about we care about our total our total change in GDP and the GDP Y could be viewed as what expenditure or it could be viewed as income depending on how you think about things so let's say this guy instead of saying that he's going to spend all of a sudden $1000 let's just call his and his incremental change in expenditure let's just call that Delta Y nought and Delta just means change in and Y we could view this as aggregate expenditure and I'm putting this little 0 here so this is our first iteration that this is the first time that we're doing this one of these deltas and then as we go keep doing them we're going to have y1 y2 y3 and so on and so forth so if we think about the total change in GDP you're definitely going to have this in the last example this was$1000 this guy's \$1,000 expenditures disguised a thousand dollar income so then you have Delta Y nought but then we saw that this guy he has his marginal propensity to consume is si so he's going to spend of the income he gets he's going to spend si times that so he's now going to do Delta y1 this is the next kind of incremental bump in our GDP we're seeing and that's going to be equal to C times what he just got and so are now after one after doing the 0th iteration in the first iteration our total change is going to be is going to be C actually let me write it this way times Delta y1 and Delta y1 this is just the same thing as C times Delta why not so it's kind of fancy notation but seeing something fairly basic the exact same thing that we said in the last video now this guy all of a sudden above and beyond what he spent in that 0th iteration he's now getting Delta y1 he has a marginal propensity to consume we're just assuming of C so now he's going to spend C times that so he's now going to make an expenditure of I'll do this all just in the same color he's now going to do Delta y2 which is equal to C times Delta y1 and so now we have Delta y2 this up this new incremental bump and they're getting smaller and smaller and smaller but we can go an infinite number of times and just to remember what this is Delta y2 is the same thing as C times Delta y1 Delta y1 is the same thing as as C times Delta Y naught and so this thing right over here is the same this whole thing can be written as C squared times Delta Delta Y naught this right over here is C times Delta Y naught C times Delta iy naught and this of course is equal to Plus this is just Delta Y naught and we can just keep going if this guy this guy will then get this amount and he'll spend C times that to the farmer and so if we had a y3 it would just amount to C times this which is B C to the third times Delta Y naught and we could keep going on and on and on an infinite number of time but each of these terms are going to get smaller and smaller because we're going to assume we're going to assume in order for this to actually work we're going to assume that C is between 0 & 1 C is less than 1 obviously when someone gets new income and thinking of the simple case someone's not going to be able to spend more their marginally marginal propensity to consume we assume it's going to they can't spend more than they just got and in general they're not going to spend the whole thing so we're going to assume that it is less than 1 and so this is exactly the same idea that we did in the last one but now it is general and we can simplify this a little bit mathematically so this is all equal to our Delta Y or total bump in GDP due to that initial due to that initial spark and if we factor that initial mark out we have the Delta Y naught Delta Y naught actually let me do that a different color so the math becomes clear we have the Delta Y naught Delta Y naught Delta y naught Delta Y naught when I say not I'm talking about that zero subscript if we factor that out we get our total bump in GDP our total bump in whether you want to view this output expenditure income is equal to we're going to factor that out the Delta Y naught times and then we're just left with we are just left you factor for the change in Y naught here you get one and then over here plus C plus C squared plus C to the third and you go on and on and on and the last video I told you that this right over here is going to simplify to 1 minus 1 over 1 minus C so this is equal to this part over here is equal to 1 over 1 minus C now you might have not been satisfied and since this is a more mathy video it's a good place to actually show you that it would sum up to 1 1 over 1 minus C so not to do introduce too many variables but let's just call this thing X so let's just say I say that X X is equal to this whole thing right over here it's equal to 1 plus C plus C squared plus C to the third so on and so forth now let's imagine what we would get if we multiply x times C so what happens if I multiply it I'll do this in a different color what happens if I multiply C times X well then each of these terms I can multiply by C so 1 times C is C C times C is C squared C squared times C is C to the third the C to the third times C is C to the fourth so on and so forth now what happens if I subtract this from that if I subtract the left-hand sides I get X minus C X so I get X minus C X on the left hand side I'll do that in that pink color - where did it go I actually think I changed the color on my so I'll just write X minus C X and that's going to be equal to if you subtract this stuff from that stuff over there this you'll have a c-minus see they'll cancel out let me do this in yellow C squared minus C squared that'll cancel out C the third minus is there that'll cancel out so every term other than one is going to cancel out everything is going to cancel out you're just going to be left with a 1 here which is a pretty neat trick in my in my mind and then we can factor out the X right over here so you get x times 1 minus C is equal to 1 and then you divide both sides by 1 minus C you get X is equal to 1 over 1 minus C and X was exactly this thing right over here so this thing is equal to 1 over 1 minus C so this right here we just showed you exactly what we were kind of we we told you in the last video that the total bump in GDP this right over here you could view this as a total bump in GDP is going to be equal to that initial bump in GDP which we called Delta Y naught that was that initial spending that that farmer did for the and the builders initial income that the total bump is going to be equal to that initial bump times this expression which we view is the multiplier and this is e this is the multiplier right here is a function of the marginal propensity to consume so this right over here this right over here let me label it all actually we just rewrite it so the total bump the total bump in our aggregate expenditure or output come or output or income is going to be equal to the initial bump the initial bump times the multiplier which is ends up being a function of our marginal propensity to consume so this right over here is our multiplier multiplier plier and this right over here is you could view that as our initial bump and just to make sure that it works out from what we saw in the last video in the last video our marginal propensity to consume was 0.6 so 6 C was C was 0.6 and our initial bump our initial expenditure was equal to 1000 and if you put point 6 in here you will get 2.5 and so you get the exact same multiplier and you get the exact same total bump in GDP as we got in the last video at least now we have a little general and you're a little bit hopefully a little bit more comfortable with some of this notation that I'm using and unfortunately you'll see different notation in almost every economic textbook I just want to make sure that this makes reasonable sense to you