Generalized Linear Consumption Function

Generalizing a linear consumption function as a function of aggregate income

Male: In the last video,
we began our exploration

of what a consumption function is.

It's a fairly straightforward idea.

It's a function that describes how

aggregate income can drive
aggregate consumption.

We started with a fairly
simple model of this,

a fairly simple consumption function.

It was a linear one.

You had some base level of consumption,

regardless of aggregate income,

and then you had some level of consumption

that was essentially induced by having

some disposable income.

When we plotted this linear
model, we got a line.

We got a line right over here.

I pointed out in the last video

this does not have to be the only way

that a consumption
function can be described.

You might use some fancier
mathematical tools.

Maybe you can construct
a consumption function.

You have an argument.

You would argue that
the marginal propensity

to consume is higher at lower levels

of disposable income and
that it kind of tapers out

as disposable income, as aggregate

disposable income goes up.

You might think that maybe you should have

a fancier consumption function

that when you graph it
would look like this

and then you would have to use things

fancier than just what
we used right over here.

What I want to do in this video

is focus more on a linear model.

The reason why I'm going to focus on

a linear model is because,
one, it's simpler.

It'll be easier to manipulate.

It's also the model that tends to be used

right when people are starting to digest

things like consumption functions

and building on them to learn about things

like, and we'll do this in a few videos,

the Keynesian Cross.

What I'm going to do is,
I'm going to do two things.

I'm going to generalize this linear

consumption function,
and I'm going to make it

a function not just of disposal income,

not just of aggregate disposable income,

which is what we did in the last video,

but as a function of
income, of aggregate income.

Then we will plot that generalized one

based on the variables.

It's really going to be the same thing.

We're just not going to use these numbers.

We're going to use
variables in their place.

Let's give ourselves a
linear consumption function.

We can say that aggregate consumption

where we're going to have some base level

of consumption no matter
what, even if people

have no aggregate income,
they need to survive.

They need food on the table.

Maybe they'll have to dig
in savings somehow to do it.

So, some base level of consumption.

I'll call that lower case c sub zero.

Or lowercase c with a subscript
of zero right over there.

That's the base level
of aggregate consumption

or it's sometimes referred
to as autonomous consumption.

This is autonomous
consumption because people

will do it on their own, or in aggregate

they will do it on their
own, even if they have

no aggregate income.

Then we will have the part that is due,

directly due, to having
some aggregate income.

We call that the induced consumption,

because you can view it as being induced

by having some aggregate income.

Above and beyond what the
base level of consumption,

people are going to consume some fraction

of their disposable income.

So we'll say disposable income.

They're not going to consume all

of their disposable income.

They might save some of it.

So they're going to consume the fraction

that's essentially their
marginal propensity to consume.

This right over here, I'll
do that in this orange color.

Marginal propensity to consume.

Hopefully this makes intuitive sense.

This says, look, if this was 100,

people are going to
consume 100 no matter what,

100 billion whatever
your unit of currency is.

Now, if their marginal propensity

to consume is, let's say, it is 1/3.

You have now above and beyond this

people have disposable
income of let's say 900,

this is saying that
they want to consume 1/3

of that disposable income they're getting.

That is, if you give them 900 of extra

disposable income, they're propensity

to consume that incremental income,

they're going to consume 1/3 of it.

So this would be 1/3, so it would be 900.

Let me give an example.

If you had a situation,
you could have a situation,

where c-nought is equal to 100.

If you have disposable
income is equal to 900,

and c1 is equal to 1/3, or we could say

0.333 repeating forever, c1 is 1/3.

Then this makes sense.

On their own people
would consume this much,

but now they have this disposable income.

Their marginal propensity to consume

if you give them 900 extra of income,

they're going to consume 1/3 of that.

So then you're going to
have, your consumption

is going to be equal to, for
this case right over here,

your consumption is going to
be 100 plus 1/3 times 900.

So your consumption in this situation,

your induced consumption, 1/3 times 900,

would be 300, maybe it's
in billions of dollars,

300 billion dollars.

Then your autonomous
consumption would be 100.

They would add up to 400.

Once again, this is autonomous
and this is induced.

Autonomous, this right over
here is induced consumption.

Now, I did write it in general terms.

I'm using variables here
instead of, or constants, really

instead of using the numbers
we saw in the last example.

But I also said that I would express

aggregate consumption
as a function not just

of disposable income
but of aggregate income;

not just of aggregate disposable income

but aggregate income.

The relationship is fairly simple between

disposable income and overall income.

We saw over here, in
aggregate, you have income,

but the government in
most modern economies

takes some fraction of that out for taxes.

What's left over is disposable income.

Just a reminder, income in aggregate,

aggregate income is the same thing as

aggregate expenditures,

which is the same thing
as aggregate output.

This right over here is GDP.

So this right over here is, let me do this

in a color, I've used
almost all my colors.

This is equal to GDP.

Disposable income is essentially GDP,

or you could say aggregate
income, minus taxes.

I'm going to do the taxes
in a different color.

Minus taxes.

So we can express disposable income

as aggregate income, this right over here

is the same thing as
aggregate income minus taxes.

We could rewrite our
whole thing over again.

Aggregate consumption is equal
to autonomous consumption

plus the marginal propensity to consume

times aggregate income,
which is the same thing

as GDP, times aggregate
income minus taxes.

We fully generalized
our consumption function

and now we've written it as a
function of aggregate income,

not just aggregate disposable income.

To make you comfortable
that this is still a line

if we were to plot it as a function

of aggregate income instead
of disposable income,

let me manipulate this thing a little bit.

We could distribute c1, which is our

marginal propensity to consume, and we get

aggregate consumption is equal
to autonomous consumption

and then we're going to distribute this,

plus c, so we're going to multiply it

times both of these terms,

plus our marginal propensity to consume

times aggregate income,

and then minus our marginal
propensity to consume

times our taxes.

Since we want it as a
function of aggregate income,

everything else here is really a constant.

We're assuming that those
aren't going to change.

Those are constant variables.

What we could do is we could rewrite this

in a form that you're
probably familiar with.

Back in algebra class
you probably remember

you can write it in the form y=mx+b where

x is the independent variable,

y is the dependent variable.

If you were to plot this,

on the horizontal axis is your x axis,

your vertical axis is your y axis.

This right over here
would have a y intercept,

or your vertical axis intercept
of b, right over there.

Then it would be a line with slope m.

If you were to take your
rise divided by your run,

or how much you move up
when you move to the right

a certain amount, that gives you your m.

Slope is equal to m.

The same analogy is here.

We can rewrite this in that form,

where our dependent
variable is no longer y.

Our dependent variable
is aggregate consumption.

Our independent variable is
not x, it is aggregate income.

So let's write it in that form.

We can write it as dependent variable, c,

which we'll plot on the vertical axis,

is equal to the marginal
propensity to consume

times aggregate income,

I'll do that purple color,

times aggregate income,

plus autonomous consumption,

minus marginal propensity
to consume times taxes.

It looks all complicated, but you just

have to realize that this
part right over here,

this is all a constant.

It is analogous to the
b if you were to write

things in kind of
traditional slope intercept

form right over here.

When we plot the line, if you have no

aggregate income, this is what your

consumption is going to be.

Let me draw that.

Once again, our dependent
variable is aggregate consumption.

Our independent variable
in this is no longer

disposable income like
we did in the last video.

It is now aggregate income.

If there's no aggregate income,

this is the independent
variable right over here,

if there's no aggregate income,

then your consumption is just going to be

this value right over here.

So your consumption is just going to be

that value right over
there, which is c-nought

minus c1 times t.

Then as you have larger values of

aggregate income, c1, that fraction of it,

is what's going to contribute
to the induced consumption.

What you essentially
have is this is the slope

of our line, this right
over here is our slope.

Just to kind of draw the analogy,

if you were to say y
is equal to mx plus b.

Actually, maybe I'll write it like this.

If you were to write c is equal to m ...

and I don't want to
confuse you if this m and b

seem completely foreign.

It comes from kind of
a traditional algebra

grounding in slope and y intercept.

If I were to say c is equal to my plus b,

this is the slope.

This is our vertical or our
dependent variable intercept

right over here.

That's where we intercept the dependent

variable axis.

And this is our slope.

It's our marginal propensity to consume.

Our line will look something like this,

where the slope is equal to the marginal

propensity to consume,
which is equal to c1.

If people all of a sudden are more likely

to spend a larger
fraction of their income,

then the marginal propensity to consume

would be higher and our
slope would be higher.

We would have a line that looks like that.

We always assume that the marginal

propensity to consume will be less than 1.

So we'll never have a slope of 1.

We'll also never have a negative slope

because we assume that this is positive.

If people are more likely
to save than consume

when they have extra
income, then this line

might look something like that.

It might have a lower slope.