So let's just review what we've seen with budget lines.
Let's say I'm making $20 a month.
So my income is $20 per month.
Let's say per month.
The price of chocolate is $1 per bar.
And the price of fruit is $2 per pound.
And we've already done this before,
but I'll just redraw a budget line.
So this axis, let's say this is the quantity of chocolate.
I could have picked it either way.
And that is the quantity of fruit.
If I spend all my money on chocolate,
I could buy 20 bars of chocolate a month.
So that is 20.
This is 10 right over here.
At these prices, if I spent all my money on fruit
I could buy 10 pounds per month.
So this is 10.
So that's 10 pounds per month.
That would be 20.
And so I have a budget line that looks like this.
And the equation of this budget line is going to be-- well,
I could write it like this.
My budget, 20, is going to be equal to the price
of chocolate, which is 1, times the quantity of chocolate.
So this is 1 times the quantity of chocolate,
plus the price of fruit, which is
2 times the quantity of fruit.
And if I want to write this explicitly
in terms of my quantity of chocolate,
since I put that on my vertical axis
and that tends to be the more dependent axis,
I can just subtract 2 times the quantity of fruit
from both sides.
And I can flip them.
And I get my quantity of chocolate
is equal to 20 minus 2 times my quantity of fruit.
And I get this budget line right over there.
We've also looked at the idea of an indifference curve.
So for example, let's say I'm sitting
at some point on my budget line where
I have-- let's say I am consuming 18 bars of chocolate
and 1 pound of fruit.
18-- and you can verify that make sense,
it's going to be $18 plus $2, which is $20.
So let's say I'm at this point on my budget line.
18 bars of chocolate, so this is in bars,
and 1 pound of fruit per month.
So that is 1.
And this is in pounds.
And this is chocolate, and this is fruit right over here.
Well, we know we have this idea of an indifference curve.
There's different combinations of chocolate and fruit
to which we are indifferent, to which
we would get the same exact total utility.
And so we can plot all of those points.
I'll do it in white.
It could look something like this.
I'll do it as a dotted line, it makes it a little bit easier.
So let me draw it like this.
So let's say I'm indifferent between any
of these points, any of those points right over there.
Let me draw it a little bit better.
So between any of these points right over there.
So for example, I could have 18 bars of chocolate
and 1 pound of fruit, or I could have--
let's say that is 4 bars of chocolate
and roughly 8 pounds of fruit.
I'm indifferent.
I get the same exact total utility.
Now, am I maximizing my total utility
at either of those points?
Well, we've already seen that anything
to the top right of our indifference
curve of this white curve right over here-- let me label this.
This is our indifference curve.
Everything to the top right of our indifference curve
is preferable.
We're going to get more total utility.
So let me color that in.
So everything to the top right of our indifference curve
is going to be preferable.
So all of these other points on our budget
line, even a few points below or budget line,
where we would actually save money, are preferable.
So either of these points are not
going to maximize our total utility.
We can maximize or total utility at all of these other points
in between, along our budget line.
So to actually maximize our total utility
what we want to do is find a point on our budget line
that is just tangent, that exactly touches at exactly one
point one of our indifference curves.
We could have an infinite number of indifference curves.
There could be another indifference curve
that looks like that.
There could be another indifferent curve
that looks like that.
All that says is that we are indifferent between any points
on this curve.
And so there is an indifference curve that touches exactly
this budget line, or exactly touches the line at one point.
And so I might have an indifference curve
that looks like this.
Let me do this in a vibrant color, in magenta.
So I could have an indifference curve that looks like this.
And because it's tangent, it touches at exactly one point.
And also the slope of my indifference curve,
which we've learned was the marginal rate
of substitution, is the exact same as the slope of our budget
line right over there, which we learned
earlier was the relative price.
So this right about here is the optimal allocation
on our budget line.
That right here is optimal.
And how do we know it is optimal?
Well, there is no other point on the budget line
that is to the top right.
In fact, every other point on our budget line
is to the bottom left of this indifference curve.
So every other point on our budget line is not preferable.
So remember, everything below an indifference curve--
so all of this shaded area.
Let me actually do it in another color.
Because indifference curve, we are different.
But everything below an indifference curve, so all
of this area in green, is not preferable.
And every other point on the budget line
is not preferable to that point right over there.
Because that's the only point-- or I guess you could say,
every other point on our budget line
is not preferable to the points on the indifference curve.
So they're also not preferable to that point right over there
which actually is on the indifference curve.
Now, let's think about what happens.
Let's think about what happens if the price of fruit
were to go down.
So the price of fruit were to go from $2 to $1 per pound.
So if the price of fruit went from $2 to $1, then
our actual budget line will look different.
Our new budget line.
I'll do it in blue, would look like this.
If we spent all our money on chocolate,
we could buy 20 bars.
If we spent all of our money on fruit at the new price,
we could buy 20 pounds of fruit.
So our new budget line would look something like that.
So that is our new budget line.
So now what would be the optimal allocation
of our dollars or the best combination that we would buy?
Well, we would do the exact same exercise.
We would, assuming that we had data
on all of these indifference curves,
we would find the indifference curve that
is exactly tangent to our new budget line.
So let's say that this point right over here
is exactly tangent to another indifference curve.
So just like that.
So there's another indifference curve that looks like that.
Let me draw it a little bit neater.
So it looks something like that.
And so based on how the price-- if we assume we have access
to these many, many, many, many, many indifference curves,
we can now see based on, all else equal,
how a change in the price of fruit
changed the quantity of fruit we demanded.
Because now our optimal spent is this point on our new budget
line which looks like it's about, well, give or take,
about 10 pounds of fruit.
So all of a sudden, when we were-- so let's
think about just the fruit.
Everything else we're holding equal.
So just the fruit, let's do, when the price was $2,
the quantity demanded was 8 pounds.
And now when the price is $1, the quantity
demanded is 10 pounds.
And so what we're actually doing,
and once again, we're kind of looking at the exact same ideas
from different directions.
Before we looked at it in terms of marginal utility per dollar
and we thought about how you maximize it.
And we were able to change the prices
and then figure out and derive a demand curve from that.
Here we're just looking at it from a slightly different lens,
but they really are all of the same ideas.
But by-- assuming if we had access
to a bunch of indifference curves,
we can see how a change in price changes our budget line.
And how that would change the optimal quantity
we would want of a given product.
So for example, we could keep doing this
and we could plot our new demand curve.
So I could do a demand curve now for fruit.
At least I have two points on that demand curve.
So if this is the price of fruit and this
is the quantity demanded of fruit, when the price is $2,
the quantity demanded is 8.
And when the price is-- actually,
let me do it a little bit different.
When the price is $2-- these aren't to scale--
the quantity demanded is 8.
Actually let me do it here-- is 8.
And these aren't to scale.
But when the price is $1, the quantity demanded is 10.
So $2, 8, the quantity demanded is 10.
And so our demand curve, these are two points on it.
But we could keep changing it up assuming we had access
to a bunch of indifference curves.
We could keep changing it up and eventually plot
our demand curve, that might look something like that.