馃攳
Generalized linear consumption function | Macroeconomics | Khan Academy - YouTube
Channel: Khan Academy
[0]
Male: In the last video,
we began our exploration
[2]
of what a consumption function is.
[4]
It's a fairly straightforward idea.
[6]
It's a function that describes how
[8]
aggregate income can drive
aggregate consumption.
[12]
We started with a fairly
simple model of this,
[14]
a fairly simple consumption function.
[16]
It was a linear one.
[17]
You had some base level of consumption,
[19]
regardless of aggregate income,
[21]
and then you had some level of consumption
[22]
that was essentially induced by having
[25]
some disposable income.
[27]
When we plotted this linear
model, we got a line.
[30]
We got a line right over here.
[32]
I pointed out in the last video
[33]
this does not have to be the only way
[35]
that a consumption
function can be described.
[37]
You might use some fancier
mathematical tools.
[39]
Maybe you can construct
a consumption function.
[42]
You have an argument.
[43]
You would argue that
the marginal propensity
[45]
to consume is higher at lower levels
[47]
of disposable income and
that it kind of tapers out
[50]
as disposable income, as aggregate
[52]
disposable income goes up.
[54]
You might think that maybe you should have
[55]
a fancier consumption function
[57]
that when you graph it
would look like this
[59]
and then you would have to use things
[61]
fancier than just what
we used right over here.
[63]
What I want to do in this video
[65]
is focus more on a linear model.
[68]
The reason why I'm going to focus on
[70]
a linear model is because,
one, it's simpler.
[72]
It'll be easier to manipulate.
[74]
It's also the model that tends to be used
[76]
right when people are starting to digest
[78]
things like consumption functions
[80]
and building on them to learn about things
[84]
like, and we'll do this in a few videos,
[85]
the Keynesian Cross.
[87]
What I'm going to do is,
I'm going to do two things.
[89]
I'm going to generalize this linear
[91]
consumption function,
and I'm going to make it
[92]
a function not just of disposal income,
[95]
not just of aggregate disposable income,
[98]
which is what we did in the last video,
[99]
but as a function of
income, of aggregate income.
[103]
Then we will plot that generalized one
[105]
based on the variables.
[106]
It's really going to be the same thing.
[108]
We're just not going to use these numbers.
[109]
We're going to use
variables in their place.
[111]
Let's give ourselves a
linear consumption function.
[114]
We can say that aggregate consumption
[118]
where we're going to have some base level
[120]
of consumption no matter
what, even if people
[122]
have no aggregate income,
they need to survive.
[124]
They need food on the table.
[126]
Maybe they'll have to dig
in savings somehow to do it.
[128]
So, some base level of consumption.
[130]
I'll call that lower case c sub zero.
[133]
Or lowercase c with a subscript
of zero right over there.
[137]
That's the base level
of aggregate consumption
[140]
or it's sometimes referred
to as autonomous consumption.
[143]
This is autonomous
consumption because people
[150]
will do it on their own, or in aggregate
[152]
they will do it on their
own, even if they have
[154]
no aggregate income.
[156]
Then we will have the part that is due,
[159]
directly due, to having
some aggregate income.
[162]
We call that the induced consumption,
[164]
because you can view it as being induced
[166]
by having some aggregate income.
[168]
Above and beyond what the
base level of consumption,
[171]
people are going to consume some fraction
[174]
of their disposable income.
[176]
So we'll say disposable income.
[184]
They're not going to consume all
[185]
of their disposable income.
[186]
They might save some of it.
[187]
So they're going to consume the fraction
[189]
that's essentially their
marginal propensity to consume.
[192]
This right over here, I'll
do that in this orange color.
[197]
Marginal propensity to consume.
[202]
Hopefully this makes intuitive sense.
[203]
This says, look, if this was 100,
[205]
people are going to
consume 100 no matter what,
[207]
100 billion whatever
your unit of currency is.
[210]
Now, if their marginal propensity
[211]
to consume is, let's say, it is 1/3.
[214]
You have now above and beyond this
[217]
people have disposable
income of let's say 900,
[220]
this is saying that
they want to consume 1/3
[223]
of that disposable income they're getting.
[226]
That is, if you give them 900 of extra
[228]
disposable income, they're propensity
[230]
to consume that incremental income,
[232]
they're going to consume 1/3 of it.
[234]
So this would be 1/3, so it would be 900.
[236]
Let me give an example.
[237]
If you had a situation,
you could have a situation,
[240]
where c-nought is equal to 100.
[245]
If you have disposable
income is equal to 900,
[255]
and c1 is equal to 1/3, or we could say
[258]
0.333 repeating forever, c1 is 1/3.
[261]
Then this makes sense.
[263]
On their own people
would consume this much,
[265]
but now they have this disposable income.
[267]
Their marginal propensity to consume
[269]
if you give them 900 extra of income,
[270]
they're going to consume 1/3 of that.
[273]
So then you're going to
have, your consumption
[275]
is going to be equal to, for
this case right over here,
[278]
your consumption is going to
be 100 plus 1/3 times 900.
[287]
So your consumption in this situation,
[289]
your induced consumption, 1/3 times 900,
[293]
would be 300, maybe it's
in billions of dollars,
[296]
300 billion dollars.
[298]
Then your autonomous
consumption would be 100.
[300]
They would add up to 400.
[304]
Once again, this is autonomous
and this is induced.
[307]
Autonomous, this right over
here is induced consumption.
[316]
Now, I did write it in general terms.
[317]
I'm using variables here
instead of, or constants, really
[321]
instead of using the numbers
we saw in the last example.
[325]
But I also said that I would express
[327]
aggregate consumption
as a function not just
[330]
of disposable income
but of aggregate income;
[333]
not just of aggregate disposable income
[335]
but aggregate income.
[337]
The relationship is fairly simple between
[339]
disposable income and overall income.
[342]
We saw over here, in
aggregate, you have income,
[346]
but the government in
most modern economies
[347]
takes some fraction of that out for taxes.
[350]
What's left over is disposable income.
[353]
Just a reminder, income in aggregate,
[357]
aggregate income is the same thing as
[358]
aggregate expenditures,
[359]
which is the same thing
as aggregate output.
[361]
This right over here is GDP.
[364]
So this right over here is, let me do this
[366]
in a color, I've used
almost all my colors.
[368]
This is equal to GDP.
[371]
Disposable income is essentially GDP,
[374]
or you could say aggregate
income, minus taxes.
[380]
I'm going to do the taxes
in a different color.
[382]
Minus taxes.
[386]
So we can express disposable income
[387]
as aggregate income, this right over here
[392]
is the same thing as
aggregate income minus taxes.
[396]
We could rewrite our
whole thing over again.
[398]
Aggregate consumption is equal
to autonomous consumption
[403]
plus the marginal propensity to consume
[406]
times aggregate income,
which is the same thing
[411]
as GDP, times aggregate
income minus taxes.
[418]
We fully generalized
our consumption function
[421]
and now we've written it as a
function of aggregate income,
[424]
not just aggregate disposable income.
[426]
To make you comfortable
that this is still a line
[429]
if we were to plot it as a function
[431]
of aggregate income instead
of disposable income,
[434]
let me manipulate this thing a little bit.
[436]
We could distribute c1, which is our
[439]
marginal propensity to consume, and we get
[441]
aggregate consumption is equal
to autonomous consumption
[447]
and then we're going to distribute this,
[448]
plus c, so we're going to multiply it
[451]
times both of these terms,
[452]
plus our marginal propensity to consume
[454]
times aggregate income,
[458]
and then minus our marginal
propensity to consume
[461]
times our taxes.
[466]
Since we want it as a
function of aggregate income,
[470]
everything else here is really a constant.
[472]
We're assuming that those
aren't going to change.
[474]
Those are constant variables.
[475]
What we could do is we could rewrite this
[477]
in a form that you're
probably familiar with.
[479]
Back in algebra class
you probably remember
[481]
you can write it in the form y=mx+b where
[487]
x is the independent variable,
[489]
y is the dependent variable.
[491]
If you were to plot this,
[492]
on the horizontal axis is your x axis,
[497]
your vertical axis is your y axis.
[500]
This right over here
would have a y intercept,
[502]
or your vertical axis intercept
of b, right over there.
[506]
Then it would be a line with slope m.
[509]
If you were to take your
rise divided by your run,
[512]
or how much you move up
when you move to the right
[514]
a certain amount, that gives you your m.
[518]
Slope is equal to m.
[520]
The same analogy is here.
[521]
We can rewrite this in that form,
[523]
where our dependent
variable is no longer y.
[526]
Our dependent variable
is aggregate consumption.
[528]
Our independent variable is
not x, it is aggregate income.
[533]
So let's write it in that form.
[534]
We can write it as dependent variable, c,
[537]
which we'll plot on the vertical axis,
[539]
is equal to the marginal
propensity to consume
[543]
times aggregate income,
[547]
I'll do that purple color,
[548]
times aggregate income,
[549]
plus autonomous consumption,
[555]
minus marginal propensity
to consume times taxes.
[562]
It looks all complicated, but you just
[563]
have to realize that this
part right over here,
[567]
this is all a constant.
[569]
It is analogous to the
b if you were to write
[574]
things in kind of
traditional slope intercept
[577]
form right over here.
[578]
When we plot the line, if you have no
[580]
aggregate income, this is what your
[584]
consumption is going to be.
[585]
Let me draw that.
[588]
Once again, our dependent
variable is aggregate consumption.
[594]
Our independent variable
in this is no longer
[597]
disposable income like
we did in the last video.
[599]
It is now aggregate income.
[604]
If there's no aggregate income,
[606]
this is the independent
variable right over here,
[608]
if there's no aggregate income,
[609]
then your consumption is just going to be
[611]
this value right over here.
[612]
So your consumption is just going to be
[614]
that value right over
there, which is c-nought
[617]
minus c1 times t.
[620]
Then as you have larger values of
[625]
aggregate income, c1, that fraction of it,
[628]
is what's going to contribute
to the induced consumption.
[632]
What you essentially
have is this is the slope
[634]
of our line, this right
over here is our slope.
[636]
Just to kind of draw the analogy,
[638]
if you were to say y
is equal to mx plus b.
[642]
Actually, maybe I'll write it like this.
[643]
If you were to write c is equal to m ...
[649]
and I don't want to
confuse you if this m and b
[651]
seem completely foreign.
[653]
It comes from kind of
a traditional algebra
[655]
grounding in slope and y intercept.
[658]
If I were to say c is equal to my plus b,
[664]
this is the slope.
[665]
This is our vertical or our
dependent variable intercept
[670]
right over here.
[671]
That's where we intercept the dependent
[673]
variable axis.
[674]
And this is our slope.
[676]
It's our marginal propensity to consume.
[678]
Our line will look something like this,
[681]
where the slope is equal to the marginal
[685]
propensity to consume,
which is equal to c1.
[688]
If people all of a sudden are more likely
[690]
to spend a larger
fraction of their income,
[692]
then the marginal propensity to consume
[696]
would be higher and our
slope would be higher.
[698]
We would have a line that looks like that.
[700]
We always assume that the marginal
[701]
propensity to consume will be less than 1.
[704]
So we'll never have a slope of 1.
[705]
We'll also never have a negative slope
[707]
because we assume that this is positive.
[709]
If people are more likely
to save than consume
[711]
when they have extra
income, then this line
[714]
might look something like that.
[716]
It might have a lower slope.
Most Recent Videos:
You can go back to the homepage right here: Homepage