Mathy version of MPC and multiplier (optional)발음듣기
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Mathy version of MPC and multiplier (optional)
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In this video I'm going to work through the exact same scenario that we saw in the last video but it will be a little bit more mathy.발음듣기
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The reason why I'm going to make it a little bit more mathy is so that you see it's a same idea it's just going to have a little bit more cryptic notation but it allows us to generalize the ideas that we saw in the last video.발음듣기
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Let's just assume, instead of saying that the marginal propensity to consume in our little island is .6, let's just say our marginal propensity to consume is C.발음듣기
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What we want to do is we want to figure out, given some initial change in expenditure and this guy's change in expenditure will be this guy's change in income.발음듣기
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Y could be viewed as expenditure or it could be viewed as income depending on how you think about things.발음듣기
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Let's say this guy, instead of saying that's he's going to spend all in the thousand dollars, let's just call his incremental change in expenditure, let's just call that delta Y nought.발음듣기
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This is the next incremental bump in our GDP we're seeing and that's going to be equal to C times what he just got.발음듣기
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Now, after doing the zero iteration in the first iteration our total change is going to be ...발음듣기
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Actually let me write it this way, times delta Y1, and delta Y1, this is just the same thing as C x delta Y nought.발음듣기
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It's fancy notation but it's just saying something fairly basic, The exact same thing that we said in the last video.발음듣기
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Now this guy, all of a sudden, above and beyond what he spent in that zeroth iteration, he's now getting delta Y1.발음듣기
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He's now going to make an expenditure of, I'll do this just in the same color, he's now going to do delta Y2 which is equal to C x delta Y1.발음듣기
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Now we have delta Y2, this new incremental bump and they're getting smaller and smaller and smaller but we can go an infinite number of times.발음듣기
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If this guy would then get this amount and he'll spent C times that to the farmer and so if we had a Y3, it would just amount to C times this which is C^3 x delta Y nought and we could keep going on and on and on an infinite number of time but each of these terms are going to smaller and smaller because we're going to assume, in order for this to actually work, we're going to assume that C is between 0 and 1.발음듣기
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Obviously, when someone gets new income and thinking of the simple case, someone is not going to spend more, the marginal propensity to consume, they can't spend more than they just got.발음듣기
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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.발음듣기
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Whether you want to do this output, expenditure or income, is equal to, we're going to factor that out, the delta Y nought times, and then we're just left with, you factor the change in Y nought here you get 1 and then over here, + C + C^2 + C^3 and you go on and on and on.발음듣기
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In the last video I told you that this right over here is going to simplify to 1 over 1 - C.발음듣기
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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 over 1 - C.발음듣기
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I'll just write X - CX and that's going to be equal to, if you subtract this stuff from that stuff over there, you have a C - C, they'll cancel out.발음듣기
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Everything is going to cancel out and you're just going to be left with the 1 here which is a pretty neat trick in my mind.발음듣기
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This right here, we just showed you exactly what we told you in the last video that the total bump in GDP, this right over here, you could view this as the total bump in GDP is going to be equal to that initial bump in GDP which we called delta Y nought.발음듣기
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That was that initial spending that that farmer did and the builders initial income, that the total bump is going to be equal to that initial bump times this expression which we view as the multiplier.발음듣기
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The total bump in our aggregate expenditure or output or income is going to be equal to the initial bump times the multiplier which ends up being a function of our marginal propensity to consume.발음듣기
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This right over here is our multiplier and this right over here is, you could view that as our initial bump.발음듣기
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If you put .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.발음듣기
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At least now we have a little general and you're hopefully a little bit more comfortable with some of these notation that I'm using.발음듣기
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