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# Optimal point on budget line

Using indifference curves to think about the point on the budget line that maximizes total utility. Created by Sal Khan.

Video transcript

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.