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Consumption function with income dependent taxes | Macroeconomics | Khan Academy - YouTube
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In the last video where we generalized
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the linear consumption function.
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I said that the tax, the
total amount of taxes,
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the aggregate taxes are constant,
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all of these were constants right here.
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You can merge them into a constant
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that ended up being our
independent variable intercept
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right over here.
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YouTube user nilsor1337
asks a very interesting
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and good question.
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"Aren't taxes in some way a
function of aggregate income?
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"In most modern economies
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"people pay a percentage of their income.
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"In general, the tax base
grows as aggregate income
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"or as GDP grows.
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"Is it appropriate to make this constant?"
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The simple answer is it
depends on how carefully
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you want to model it.
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In some cases you might just say,
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"Well, let's just assume
that this is a bulk tax.
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"We're just trying to
understand one aspect of it."
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You will see that in
some economics courses
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or some economics textbooks.
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The other way is you
could actually model it
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a little bit more realistic.
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You could say, "Hey, taxes really are
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"a function of aggregate income."
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We could say that T really
is going to be equal
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to some tax rate.
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I'll write that as a lower
case t times aggregate income.
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In a place like the U.S.,
this might be close to
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the 30% of aggregate income or 20%.
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Whatever it might be or aggregate income
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is what is going to go for taxes.
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If you do it this way,
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and you substitute back to this
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you could actually get an
expression for consumption
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in terms of aggregate income
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that takes into consideration
the idea that taxes
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are function of aggregate income.
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Just to do that algebraically,
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we can rewrite this expression up here.
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You have aggregate consumption =
my marginal propensity to consume
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times aggregate income +
autonomous consumption,
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the amount that would be
consumed no matter what.
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Minus the marginal propensity to consume,
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shows up again.
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Instead of writing T right over here,
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I'm going to write lower case t x Y,
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tax rate times aggregate income.
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Times the tax rate times aggregate income.
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I just took this, instead
of writing upper case T,
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I wrote lower case t
times aggregate income
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and they should be the same thing.
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But now we've expressed t as a
function of aggregate income.
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Now we can merge both of these,
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these are something
times aggregate income.
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We can combine those 2 terms.
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This one and this one write over here.
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If we factor our a
common factor of c1 x Y,
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we get, let me write it this way.
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Actually, let me just combine them first
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so that the algebra doesn't confuse you.
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We get C = c1 x Y.
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Marginal propensity to
consume times aggregate income
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and I'm going to write this one.
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Minus the marginal propensity
to consume times ...
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I'll switch the order here.
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Well, let me not switch the order,
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times the tax rate,
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not just the aggregate total tax value
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but the actual tax rate
times aggregate income.
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That's those 2 terms there
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and then we're just left with
the autonomous consumption.
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So, plus the autonomous consumption.
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Over here, we have a common factor.
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We can factor out the c1 and the Y,
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or essentially the marginal
propensity to consume
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and the aggregate income.
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This is just algebraic
manipulation right over here.
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We get aggregate consumption is equal to,
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let's see, we could write this c1(1 - t)Y.
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You can multiply this out to verify.
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If you multiply it all out
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then the 1st term is c1(1)Y
is this right over here
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and c1(-t)Y is this term right over here.
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Then you're left with your
autonomous consumption.
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This actually makes a lot of sense
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because when you write it like this,
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when you write it like this
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you could look at this
term right over here.
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What is this term right over here?
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Well, (1 - t)Y, if the tax rate is 30%
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then this 1 - 30% is going to be 70%.
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70% x aggregate income,
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that's essentially what
people get in their pockets.
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This whole term right over here
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is essentially disposable income.
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Disposable income right over here.
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We could actually,
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if we wanted to write this
as some other variable
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we could just put that
variable right over there
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and say it's disposable income
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and then it actually becomes a very simple
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thing to graph.
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We could graph this 2 different ways.
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If we wanted to write a
function of aggregate income
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we would graph it like this.
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Now, when we express it this way,
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taxes as a function of aggregate income
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now our vertical intercept.
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This is aggregate consumption.
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Our vertical intercept is
this term right over here.
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That is C [not]
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and our slope is all of this business.
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The slope of our line
is going to be C1(1 - t)
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and this right over here,
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the independent variable
is aggregate income.
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Another option, we could
set some other variable
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to what we could say disposable income.
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Let me call it Y disposable = (1 - t)Y
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then we could write this.
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It's essentially equal to this
business right over there.
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Then we could rewrite the
consumption function as
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aggregate consumption =
marginal propensity to consume
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times disposable income + sum
level of autonomous consumption.
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plus sum level of autonomous consumption.
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This actually takes us back to the basics.
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This takes us back to our
very original situation here
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where we had some autonomous consumption
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plus our marginal propensity to consume
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times disposable income.
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If we wanted to plot it this way
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as a function of disposable income,
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not aggregate income
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then it would look like this.
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This is consumption, and now
this is an aggregate income,
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this is disposable income
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which is the same thing as (1 - t)Y.
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Now, still our vertical
intercept is C [not]
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and our line slope is the
marginal propensity to consume.
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This is C1 just like that.
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All of these are completely
valid consumption functions
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and I thank nilsor1337
for bringing up a topic
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that actually was a
cause of confusion for me
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because it really does depend.
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Because I thought the way, he or she,
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originally thought about the problem.
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Well, taxes are a function
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and a lot of econ books tend
to treat this as a constant.
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That is actually just
an assumption they make
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to often simplify the calculations.
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If they don't want to make that assumption
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you can still show that
it is a linear function,
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that aggregate consumption
is still a linear function
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of aggregate income.
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