What I want to do in this video
is introduce you to the idea of a budget line.
Actually, probably isn't a new idea.
It's a derivative idea of what you've seen
and often in an introductory algebra course
where A, you've gotten a certain amount of money
and you can spend it on a certain combination of goods.
What are all the different possibilities
that you can actually buy?
That's really what a budget line is.
Let's say that you have an income
and I'll do it both in the abstract
and the concrete.
I'll do it variables
and then I'll also do it with actual numbers.
Lets say your income, your income in a month is Y
and lets say that you spend all of your money.
Your income is equal to your expenditures.
Assuming in our little model here
that you're not going to be saving any money.
To show how overly simplified we can make a model
we are going to only assume
that you can spend on two different goods
and that's so that we can actually plot
all the combinations on a two dimensional surface
like the screen over here.
Obviously, most people buy many more
or they at least are choosing between many,
many more than two goods.
But let's say you can choose between 2 goods
and let's just take goods that we've been doing
using in recent videos.
That 2 goods that you buy are either chocolate or fruit.
You could buy chocolate by the bar
or fruit by the pound.
What are going to be your expenditures
assuming you spend it all on chocolate and fruit?
Well, there's going to be the amount
that you spend on chocolate
will be the price of chocolate
times the quantity of chocolate you buy
which is the number of bars.
And then the amount you spent on fruit
will be the price of fruit per pound
times the quantity of fruit.
For example, if Y = $20 a month
and the price, actually we'll plot this in a second,
the price of chocolate is equal to $1 per bar
and the price of fruit is equal to $2 per pound.
I think these were the prices I used
in a per pound of fruit.
Then all of a sudden, you would know what this is,
you would know what this is and this is.
You know what the Ps are and the Y
and then you could actually graph one of these
quantities relative to the other.
What we can do is, and let's do that,
we can graph the quantity of 1 relative to the other.
Why don't we put the quantity of chocolate
on this axis over here and let's put the quantity of fruit
on this axis over here.
First, if we wanted to graph it I like to put it,
since I've put quantity of chocolate
on the vertical axis here,
I'd like to solve this equation for quantity of chocolate
as a function of quantity fruit
and it should make it pretty straight forward to graph.
Let's try that out.
First, I'm just going to rewrite this
without expenditures in between.
We have our income,
our income Y = price of chocolate
times the quantity of chocolate
plus the price of fruit times the quantity of fruit.
Now, I want to solve for the quantity of chocolate.
Let me make that orange
so we know that this is this one right over here.
If I want to solve for that,
the best way I could isolate it one side of this equation.
Let me get rid of this this yellow part right over here
and the best way to do that
is to subtract it from both sides.
Let's subtract the price of fruit
times the quantity of fruit
and I could substitute the numbers in first
and that might actually make it a little bit
easier to understand
but I like to keep it general first.
You see, you don't have to just use with these numbers
you could just see the general result here.
I'm going to subtract it from the left hand side
and the right hand side
and the whole point is to get rid of it
from the right hand side.
This cancels out, the left hand side
becomes your income minus the price of fruit
times the quantity of fruit.
This is going to be equal to your right hand side
which is just the price of chocolate
times the quantity of chocolate.
Now if we want to solve for the quantity of chocolate
we just divide both sides by the price of chocolate
and then you get it, and I'll flip the sides.
You get the quantity of chocolates,
is going to be equal to your income,
your income divided by the price of chocolate
minus the price of fruit times the quantity of fruit
all of that over the price of chocolate.
All over that over the price of chocolate.
We can actually substitute these numbers in here
and then we can actually plot what essentially
this budget line will look like.
In our situation, 20, Y = 20,
the price of chocolate is equal to 1.
Price of chocolate is equal to 1.
This term right over here,
$20 per month divided by $1 per bar
which would actually give you 20 bars per month
if you work out the units.
This term right over here just simplifies to 20.
This is actually an interesting term,
your income, your income in dollars divided by the price
of an actual good or service.
You could view this term right over here
as your real income.
The reason why it's called real income
is it's actually pegging what your earnings
to what you can buy.
It's pegging it to a certain real goods,
it's not tied to some abstract quantity like money
which always has a changing buying power.
What you could buy for $20 in 2010 is very different
than what you could buy for $20 in 1940.
Here, when you divide your income,
divide it a by a price of some good
it's really telling you your income in terms of that good.
You could view your income as $20 per month
or you could view your income
if you wanted your income in chocolate bars.
You could say my income is,
I could buy 20 chocolate bars each month.
So I could say, my income 20 chocolate bars per month.
They would be equivalent to you
assuming that you could sell the chocolate bars
for the same price you could buy it
and that's somewhat of an assumption.
But you could say I have the equivalent income
of 20 bars a month.
You could have also done it in fruit.
I have the equivalent income of 20 divided by 2,
10 pounds of fruit a month.
It's trying your income to real things,
not the abstract quantity like money.
Anyway, this is going to be equal to,
let me write it over here.
My quantity of chocolate
is going to be equal to this term right over here as 20.
If you wanted to do the units,
it would be 20 bars per month
and you could do a little bit of dimensional analysis
to come up with that.
You could treat the units just like numbers
and see how the cancel out.
20 bars per month minus the price of fruit
divided by the price of chocolate.
$2 per pound of fruit.
The price of fruit is going to be $2
and I actually want to look at the units
because that's interesting.
Let me write it here.
The price of fruit is equal to $2 per pound.
Let me write it this way.
$2 per pound of fruit,
I'll show you how the units cancel out.
Then we're dividing that by the price of chocolate.
Dividing it by the price of chocolate
which is equal to $1 per bar of chocolate.
Now, obviously the math is fairly straight forward.
We just get 2, but the units are a little bit interesting.
You have a dollar and the numerator of the numerator
and a dollar, the numerator of the denominator,
those will cancel out.
You could actually view this as,
this is going to be the same thing
just to look at the units.
This is going to be,
this is the same thing as the numerator times the inverse
times the reciprocal of the denominator right over here.
You could say $2 per pound times,
the reciprocal of 1 is just 1,
times 1 bar per dollar.
Then the dollars cancel out
and you are left with 2 bars per pound of fruit.
What we've actually done over here,
this term right over here,
it gives us bars of chocolate per pound of fruit.
It simplifies to 2 bars of chocolate per pound of fruit.
It's actually giving you the opportunity cost
of a pound of fruit.
It's saying hey, you could buy a pound of fruit
but you'd be giving up 2 bars of chocolate.
Because the price, you could get 2 bars of chocolate
for every pound of fruit.
You could view this as the relative price,
this right over here is the relative price
of fruit in this example.
It's telling you the opportunity cost,
it's telling you how much fruit cost
in terms of chocolate bars.
Regardless, that number is fairly straight forward,
it was just a 2.
Minus 2 times the quantity of fruit.
This is fairly straight forward to plot.
If the quantity of fruit it 0,
our quantity of chocolate is 20.
This is going to be 20 over here.
This is 20 and this is going to be 10.
This is 15, this is 5.
This is a point on our budget line right over there.
There is multiple ways that you could think about this.
One way you could say is if you buy no chocolate,
if the quantity of chocolate is 0,
what is going to be the quantity of fruit?
Then you could solve this or you could just say,
"Look, if I have $20 a month
"then I'm going to spend it all on fruit.
"I can buy 10 pounds of fruit."
So to say that this right over here is 10.
Let's say this right over here is 10, this is 5,
so this is also on our budget line
and every point in between
is going to be on our budget line.
Every point in between is going to be on our budget line.
Another way you could have done this
and this comes straight out of kind of your typical
algebra 1 course.
You could say, in this case,
if you view this as the Y axis,
you say your Y interceptor,
you say, "My chocolate quantity interceptor is 20
"and then my slop is negative 2.
"My slope is negative 2."
For every extra pound of fruit I buy
I have to give up 2 pounds of chocolate.
You could also view this as the opportunity cost of fruit.
You see this slope as we go forward,
if we buy one more pound quantity of fruit
we're giving up 2 bars of chocolate.
One statement I did just make,
I said every point on this line is a possibility
and I can only say that if we assume
that both of these goods are divisible goods
which means we can buy arbitrarily small amounts of it,
that we could buy 10th of a bar of chocolate
on average especially.
Or we could buy 100th of a pound of fruit.
If they weren't divisible, they're indivisible
then only the whole quantities
would be the possibility points.
We'll just assume they're divisible,
especially even if the store only sells
indivisible bars of chocolate.
If you buy one bar of chocolate every 4 months,
on average you're buying .25 bars of chocolate per month.
Even that, on average, almost anything,
almost anything here is divisible.
This line right over here shows
all of the combinations we can buy.
All of the combinations
of the divisible goods we could buy
if we spend all of our money.
That right over there is our budget line.
That is our budget line.
And any combination out here is unaffordable.
We don't have enough money for that.
Any combination down here is affordable.
Actually, we would end up with extra money
if we're below the budget line.
This isn't all that different than what we saw
with the production possibilities frontier.
Remember, we had a curve that really showed all of the
if we were producing 2 goods,
what combinations of goods we could produce.
Anything on that curve
for the productions possibility frontier was efficient.
Anything outside of it was unattainable
and anything inside was attainable but inefficient.