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Mathy version of MPC and multiplier (optional)
Generalizing what we did in the last video with more math. Created by Sal Khan.
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You said that you can't spend more than you got, where does debt fit into this? Because that's a whole bunch of people spending money they don't have.(3 votes)
Keep in mind that MPC is just a silly little model that might occasionally be helpful for you to think about economic scenarios. Don't take it too seriously. With that in mind, this model breaks down if there is a universal MPC greater than 1, so it's not allowed. In the MPC world, taking on debt would lead to infinite GDP.
I can't stress enough that this is just a silly little model to help you think about things. :)(8 votes)
Hi,
At5:59ish, Sal simplifies the expression (by making a variable X and then multiplying by C followed by substracting by X and X's value, and finally isolating X). My question is how does he know to do that, does such a technic have a name (so I can google it) and how can one become good at doing such simplifications of seemingly complex expressions.
Thank you.(3 votes)- It's a convergent geometric series. There are other videos on it here. You can also see http://en.wikipedia.org/wiki/Geometric_series#Proof_of_convergence and related info on similar pages.(6 votes)
It is determined that $500 billion net fiscal infusion is needed to prevent a recession. The multiplier is 4. How much money should be injected into the economy?(3 votes)
If I understand this question correctly, I believe you can think of it as:
The "net fiscal infusion needed to prevent a recession" = The amount of money "injected into the economy" x the multiplier (which is the marginal propensity to consume). Substituting the values from this question into the formula:
$500,000,000,000 = X(4); where $500,000,000,000 is the desired amount of fiscal infusion, X = the money needed to be injected into the economy to achieve the desired fiscal infusion, and 4 = the multiplier.
Solving for the equation:
$500,000,000,000/4 = X
$125,000,000,000 = X
The amount needed to be injected into the economy to achieve a $500,000,000 fiscal infusion is $125,000,000 if the multiplier is 4.(1 vote)
The equation you use seems to only take into count money and not labor and resources... if those don't change would it not lead to inflation?
If the farmer choose to cut his hours of operation thus having more money and less labor/resources wouldn't that also lead to a negative outcome?(3 votes)- Yes, this model only takes nominal money into account. There are a number of questions on that in the last video as well - would be great if Sal made another to explain that.
The farmer is first to know there is new found money, so the builder is initially tricked into thinking he's getting old high-valued money. In reality he's getting newly devalued inflated money. Eventually he'll find out and they'll both raise prices.(1 vote)
- This formula assumes that the multiplier (K) is: 1 < K < infinity. This is the formula my textbooks give, as well. However, report by CBO on ARRA has multipliers below 1. Moreover, in a lot of debates on fiscal stimulus I've noted that some experts claim a multiplier to be negative. This does not fit in with what the formula proposes. So, I wonder if anyone could reference or explain how a multiplier is actually being calculated (in relatively simple, structured language, please)?
The only thought I've come up with is the offsetting effect of the 'crowding-out' effect. However, there is no formula for it. So, the two can not be combined to calculate the actual multiplier effect.
Thanks in advance for the guidance provided!(1 vote)- I've never heard anyone suggest that in reality MPC could be negative. It's hard to imagine a realistic scenario where gaining income would cause someone to spend less in total than before. Perhaps provide a citation.(4 votes)
How does the multiplying by c and then subtracting the bottom equasion from the top equasion that Sal does from5:55make sense? It seems so random to me. Is anyone able to explain just this specific step in the math here? the rest I understand well.(1 vote)
He's manipulating the equation to get it into a simple, usable form that still represents the exact same outcome. Think of it as repacking a suitcase to save space.(4 votes)
I don't seem to get the part at6:57.
Following:
x =1+c+c^2+c^3+.......+c^n
-c.x=-[c+c^2+c^3.........c^(n+1)] ----> here
------------------------------------------ Wouldn't it actually be??
x-c.x=1-c^(n+1)
Thus, x(1-c)=1-c^(n+1)
x=[1-c^(n+1)]/(1-c)
I know for huge n numbers c^(n+1) tends to zero, but this here is a more precise way of computing the sum isn't it?
Am I making some mistake?(2 votes)- It seems like you wonder why Sal got rid of that -c^(n+1) part. The first thing you have to realize is that we're calculating an infinite sum, so n = ∞. That results in -c^(∞+1). We now entered the very complicated math that deals with infinity. If you've had some basic calculus you might have heard of limits. We can use limits to say that when n gets larger and larger -c^(n+1) gets closer and closer to 0. So when n = ∞ we can say that -c^(n+1) = 0. This gets writen down like this: https://gyazo.com/7373b014acd9e11a1347f8ed96703f51 It might be hard to get it at first, but this is how mathematicians work with infinity.
It can also get backed up with a more practical argument. Let's say ∆y0 = 100, c = 0.6 and n = 20. Sal's function will return 250 and yours will return 249.99 (rounded to two decimals). The error in Sal's formula is only a cent and when people want to do this calculation they only care about a rough answer. A cent more or less isn't going to change the decision you make, so Sal's function is an easier way to come to the same conclusion.
I hope this cleared up a few things.(2 votes)
- Along with a marginal propensity to consume people need a marginal propensity to save money. The saved money will be used at a later time in the economy.(2 votes)
- Where the 60% of money go?(1 vote)
60% of the extra income is spent again. The other 40% is saved.(2 votes)
If there is a mathy version of MPC and multiplier is there a different version also?(1 vote)
5:59
Video transcript
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. 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. 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. 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. That cycle is round and round due to the multiplier effect. What is going to be the
total change in our GDP? This is what we care about, we care about our total change in the GDP. Y could be viewed as expenditure or it could be viewed as income depending on how you think about things. 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. Delta just means change in, and Y, we could view this
as aggregate expenditure. I'm putting this little zero here. This is our first iteration, this is the first time that
we're doing one these deltas. Then as we keep doing them
we're going to have Y1, Y2, Y3 and so on and so forth. 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 is $1000 expenditures, this guy is a $1000 income. Then you have delta Y nought. Then we saw that this guy, his marginal propensity to consume is C. He's going to spend of the income he gets, he's going to spend C times that. He's now going to do Delta Y1. 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. Now, after doing the zero
iteration in the first iteration our total change is going to be ... 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. It's fancy notation but it's just saying
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 zeroth iteration, he's now getting delta Y1. He has a marginal propensity to consume, we're just assuming of C. Now he is going to spend C times that. 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. 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. Just to remember what this is, delta Y2 is the same
thing as C x delta Y1. Delta Y1 is the same thing
as C x delta Y nought. So this thing right over here, this whole thing could be written as C^2 x delta Y nought. This right over here C x delta Y nought. This of course is equal to,
this is just delta Y nought. We can just keep going. 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. 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. In general they're not going
to spend the whole thing. So we're going to assume
that it is less than 1. 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. This is all equal to our
delta Y, our total bump in GDP due to that initial spark. If we factor that initial spark out with the delta Y nought. Actually, let me do
that in different color just so the math becomes clear. We have the delta Y
nought, delta Y nought, delta Y nought, delta Y nought. When I say nought, I'm talking
about that zero subscript. If we factor that our, we
get our total bump in GDP. 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. In the last video I told you
that this right over here is going to simplify to 1 over 1 - C. This is equal to, this
part over here is equal to 1 over 1 - 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 over 1 - C. Not to introduce too many variables, but let's just call this thing X. Let's just say that X is
equal to this whole thing right over here. It's equal 1 + C + C^2 + C^3, so on and so forth. Now let's imagine what we would get if we multiply X x C. What happens if I multiply, and I'll do this in a different color. What happens if I multiply C x X? Well then, each of these terms I can multiply by C. 1 x C is C, C x C is C^2, C^2 x C is C^3, C^3 x C is C^4, so on and so forth. Now, what happens if I
subtract this from that? If I subtract the left hand sides I get X - CX on the left hand side. I'll do that in that pink color. Where did it go? Actually I think I changed
the color on my ... 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. Let me do this in yellow. C^2 - C^2, that will cancel out. C^3 - C^3, that would cancel out. Every term other than 1
is going to cancel out. 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. Then we can factor out
the X right over here. You get X x 1 - C = 1 and then you divide both sides by 1 - C, you get X = 1 over 1 - C. X was exactly this thing right over here. This thing is equal to 1 over 1 - C. 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. 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. This is the multiplier right here is a function of the marginal
propensity to consume. This right over here, let me label it all. Actually, let me just rewrite it. 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. This right over here is our multiplier and this right over here is, you could view that as our initial bump. 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 .6. C was 0.6 and our initial bump, our initial expenditure
was equal to 1,000. 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. 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. Unfortunately, you'll
see different notation almost every economics textbook. I just want to make sure that this makes reasonable sense to you.