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Mathy version of MPC and multiplier (optional) | Macroeconomics | Khan Academy - YouTube
Channel: Khan Academy
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In this video I'm going to
work through the exact same
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scenario that we saw in the last video
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but it will be a little bit more mathy.
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The reason why I'm going to make it
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a little bit more mathy
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is so that you see it's a same idea
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it's just going to have a
little bit more cryptic notation
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but it allows us to generalize
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the ideas that we saw in the last video.
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Let's just assume, instead of saying
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that the marginal propensity to consume
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in our little island is .6,
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let's just say our marginal
propensity to consume is C.
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What we want to do is
we want to figure out,
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given some initial change in expenditure
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and this guy's change in expenditure
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will be this guy's change in income.
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That cycle is round and round
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due to the multiplier effect.
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What is going to be the
total change in our GDP?
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This is what we care about,
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we care about our total change in the GDP.
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Y could be viewed as expenditure
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or it could be viewed as income
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depending on how you think about things.
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Let's say this guy, instead
of saying that's he's going to
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spend all in the thousand dollars,
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let's just call his incremental
change in expenditure,
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let's just call that delta Y nought.
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Delta just means change in,
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and Y, we could view this
as aggregate expenditure.
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I'm putting this little zero here.
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This is our first iteration,
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this is the first time that
we're doing one these deltas.
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Then as we keep doing them
we're going to have Y1,
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Y2, Y3 and so on and so forth.
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If we think about the total change in GDP,
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you're definitely going to have this.
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In the last example this was $1000.
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This guy is $1000 expenditures,
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this guy is a $1000 income.
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Then you have delta Y nought.
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Then we saw that this guy,
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his marginal propensity to consume is C.
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He's going to spend of the income he gets,
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he's going to spend C times that.
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He's now going to do Delta Y1.
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This is the next incremental
bump in our GDP we're seeing
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and that's going to be equal
to C times what he just got.
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Now, after doing the zero
iteration in the first iteration
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our total change is going to be ...
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Actually let me write it this way,
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times delta Y1, and delta Y1,
this is just the same thing
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as C x delta Y nought.
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It's fancy notation
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but it's just saying
something fairly basic,
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The exact same thing that
we said in the last video.
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Now this guy, all of a sudden,
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above and beyond what he spent
in that zeroth iteration,
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he's now getting delta Y1.
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He has a marginal propensity to consume,
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we're just assuming of C.
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Now he is going to spend C times that.
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He's now going to make an expenditure of,
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I'll do this just in the same color,
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he's now going to do delta Y2
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which is equal to C x delta Y1.
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Now we have delta Y2,
this new incremental bump
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and they're getting smaller
and smaller and smaller
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but we can go an infinite number of times.
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Just to remember what this is,
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delta Y2 is the same
thing as C x delta Y1.
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Delta Y1 is the same thing
as C x delta Y nought.
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So this thing right over here,
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this whole thing could be written as
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C^2 x delta Y nought.
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This right over here C x delta Y nought.
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This of course is equal to,
this is just delta Y nought.
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We can just keep going.
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If this guy would then get this amount
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and he'll spent C times that to the farmer
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and so if we had a Y3,
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it would just amount to C times this
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which is C^3 x delta Y nought
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and we could keep going on and on and on
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an infinite number of time
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but each of these terms are
going to smaller and smaller
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because we're going to assume,
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in order for this to actually work,
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we're going to assume
that C is between 0 and 1.
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Obviously, when someone gets new income
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and thinking of the simple case,
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someone is not going to spend more,
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the marginal propensity to consume,
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they can't spend more than they just got.
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In general they're not going
to spend the whole thing.
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So we're going to assume
that it is less than 1.
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This is exactly the same idea
that we did in the last one
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but now it is general
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and we can simplify this a
little bit mathematically.
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This is all equal to our
delta Y, our total bump in GDP
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due to that initial spark.
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If we factor that initial spark out
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with the delta Y nought.
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Actually, let me do
that in different color
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just so the math becomes clear.
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We have the delta Y
nought, delta Y nought,
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delta Y nought, delta Y nought.
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When I say nought, I'm talking
about that zero subscript.
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If we factor that our, we
get our total bump in GDP.
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Whether you want to do
this output, expenditure
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or income, is equal to, we're
going to factor that out,
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the delta Y nought times,
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and then we're just left with,
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you factor the change in
Y nought here you get 1
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and then over here, + C + C^2 + C^3
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and you go on and on and on.
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In the last video I told you
that this right over here
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is going to simplify to 1 over 1 - C.
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This is equal to, this
part over here is equal to
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1 over 1 - C.
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Now, you might have not been satisfied
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and since this is a more mathy video
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it's a good place to actually show you
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that it would sum up to 1 over 1 - C.
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Not to introduce too many variables,
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but let's just call this thing X.
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Let's just say that X is
equal to this whole thing
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right over here.
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It's equal 1 + C + C^2 + C^3,
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so on and so forth.
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Now let's imagine what we would get
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if we multiply X x C.
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What happens if I multiply,
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and I'll do this in a different color.
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What happens if I multiply C x X?
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Well then, each of these terms
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I can multiply by C.
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1 x C is C, C x C is C^2, C^2 x C is C^3,
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C^3 x C is C^4, so on and so forth.
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Now, what happens if I
subtract this from that?
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If I subtract the left hand sides
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I get X - CX on the left hand side.
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I'll do that in that pink color.
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Where did it go?
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Actually I think I changed
the color on my ...
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I'll just write X - CX and
that's going to be equal to,
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if you subtract this stuff
from that stuff over there,
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you have a C - C, they'll cancel out.
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Let me do this in yellow.
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C^2 - C^2, that will cancel out.
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C^3 - C^3, that would cancel out.
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Every term other than 1
is going to cancel out.
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Everything is going to cancel out
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and you're just going to
be left with the 1 here
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which is a pretty neat trick in my mind.
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Then we can factor out
the X right over here.
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You get X x 1 - C = 1
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and then you divide both sides by 1 - C,
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you get X = 1 over 1 - C.
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X was exactly this thing right over here.
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This thing is equal to 1 over 1 - C.
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This right here, we just showed you
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exactly what we told you in the last video
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that the total bump in GDP,
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this right over here,
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you could view this as
the total bump in GDP
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is going to be equal to
that initial bump in GDP
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which we called delta Y nought.
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That was that initial spending
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that that farmer did and
the builders initial income,
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that the total bump is going to be equal
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to that initial bump times this expression
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which we view as the multiplier.
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This is the multiplier right here
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is a function of the marginal
propensity to consume.
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This right over here, let me label it all.
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Actually, let me just rewrite it.
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The total bump in our
aggregate expenditure
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or output or income
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is going to be equal to the initial bump
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times the multiplier which
ends up being a function
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of our marginal propensity to consume.
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This right over here is our multiplier
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and this right over here is,
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you could view that as our initial bump.
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Just to make sure that it works out
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from what we saw in the last video.
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In the last video our
marginal propensity to consume
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was .6.
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C was 0.6 and our initial bump,
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our initial expenditure
was equal to 1,000.
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If you put .6 in here you will get 2.5
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and so you get the exact same multiplier
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and you get the exact
same total bump in GDP
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as we got in the last video.
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At least now we have a little general
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and you're hopefully a
little bit more comfortable
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with some of these
notation that I'm using.
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Unfortunately, you'll
see different notation
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almost every economics textbook.
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I just want to make sure
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that this makes reasonable sense to you.
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