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Equalizing Marginal Utility per Dollar Spent - YouTube
Channel: Khan Academy
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In the last video,
we thought about how
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we would allocate our $5 between
chocolate bars and fruit.
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And the way we did it,
and it was very rational,
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we thought about how much bang
would we get for each buck.
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And we saw, look, starting
off, our first dollar
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we got a lot of bang
for our buck-- and this
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is really just another way
of saying bang for the buck,
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marginal utility per price.
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So we got a lot of utility
for price starting off
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for that first chocolate bar.
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A little less for the
next chocolate bar,
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but still more than we would
get for a pound of fruit.
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Then more for the
next chocolate bar,
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and only then did
we start buying
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some fruit, buying
some pounds of fruit.
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What do I do in this
video is generalize it.
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I want to think about maybe
a more continuous case
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where we can buy very, very
small increments of each
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of the products.
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It doesn't have to be in
chunks, like chocolate bars.
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And what I'm going
to do is I'm going
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to plot the marginal utility
per price, which is really
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bang for your buck,
on the vertical axis.
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So
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This right over
here on this axis.
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Let's say this is the
marginal utility per price.
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And let's say it also
goes from 0 to 100.
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So that would be 50.
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And the numbers actually
don't matter so much here.
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And then this will
be dollar spent.
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So dollars spent, so your buck.
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So this is bang for your buck
and then this is your buck.
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So this is 1, 2, 3, 4, 5 and 6.
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Now we're going to do
arbitrary products.
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So let's say one product
looks something like this.
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And once again, you
have diminishing utility
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as you get more and
more of that product.
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In the case of fruit,
the more pounds of fruit
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you get the more tired
you get of fruit.
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The less fruit
you need for that,
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or the less you want fruit for
that next incremental pound.
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So let's-- but it
could be anything.
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This is true of most things.
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So this is product A,
could be a service as well.
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So product A, let me
write it this way.
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So this is the marginal
utility for A per price of A.
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And let me get another
product right over here.
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So let's say my other product
looks something like this.
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So this is my marginal utility
for product B per price of B.
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So it's really saying
bang for the buck.
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So just to start
off-- and I won't even
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constrain how much
money we have.
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I just want to think about
how we would spend that money.
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So if I were to spend,
if I had a penny,
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where would I spend a penny.
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And I'm assuming I can buy these
in super small chunks, as small
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as maybe the penny or even
maybe fractions of penny.
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So if I just had
a penny, and I had
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to think about where
am I getting the best
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bang for my buck for that
penny, I'm clearly getting it
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with product A. So I would
spend that penny on product A
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and I would get this much
bang for my buck, which
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would be this entire
part right over here.
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Let me color it in.
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So my first-- I'll
spend it right on A.
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Let me do it in a color
that's more likely to be seen,
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so I'll do it in
this blue color.
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So I'll spend it on
A. My first, in fact,
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where would I spend
my first dollar?
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Well, the whole first dollar
I'm getting a better bang
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for my buck on A. So my first
dollar I will spend on A.
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And the total utility
I will get is actually
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going to be the area
under this curve.
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It's going to be
this whole area.
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It's going to be dollars times
marginal utility with price.
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That would give you, obviously,
the area of this rectangle
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right over here.
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The reason why it wouldn't
be the area of this larger
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rectangle, it would just be
the area under the curve,
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is you're not getting 100
marginal utility per price
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for the entire dollar.
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It's going down the entire time.
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And so your actual
total marginal utility
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is actually just
the area under this.
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And when you take
calculus you'll
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get a better
appreciation for that.
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But let's just think
about, once again, where
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our dollar is going to be spent.
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So actually even if
we've spent already $1,
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our next penny we would still
want to spend on product A,
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because we're still getting
more bang for the buck.
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We're still getting more
bang for the buck all the way
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until right around there.
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Now something
interesting is happening.
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So we've spent about $2.
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We've spend our first
$2 all on product A
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because we're getting
more bang for buck,
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even though that bang was
diminishing every penny or even
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every fraction of a
penny that we spent.
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But now where will we
spend our next penny?
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Well, we could spend
it on product A again.
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But look, we can get about the
same marginal utility spending
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it on product B. So we
could jump right over there,
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spend it on product B. Now where
could we spend our next dollar?
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Well, we get about the
same marginal utility
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whether we spend it on a
little bit more of product B,
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or a little bit more of product
A. So we could do either.
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If we spent a little bit
too much on product A,
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then we could have gotten
more marginal utility spending
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on product B. So
what we would do
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is, once we've gotten to this
threshold right about here,
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we actually are going to spend
every incremental fraction
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of a penny-- we're
actually going
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to want to split between
product A and product
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B. If we spend too much on
one and we go down this curve,
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we could have gotten higher
utility spending on this one.
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If we spend too
much on this one we
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could get higher
utility spending
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on this one right over here.
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So there's a very
interesting phenomenon here.
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Assuming that we
eventually spent enough
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that we buy some of
both, obviously we
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started just buying
product A because it
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had higher utility, at least,
for those first few dollars--
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but assuming that we end up
buying some mix of the two,
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which we do end up spending
if we spend more than $2--
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there's an interesting thing.
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The marginal utility for B, or
the marginal utility for price
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for B that I spent on
that last little increment
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is going to be the same as
the marginal utility per price
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for that last increment of
A. So if this was, if B was,
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I don't know, if it was fruit
and let's say A was chocolate
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but we could buy them in
very, very small increments--
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we're saying for that last
fraction of a pound of fruit
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you're getting the same
marginal utility per price
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as you're getting for that
last fraction of a bar
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or fraction of a
pound of chocolate.
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So there's a general
principle over here.
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And it really just comes from
this very straightforward thing
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that as soon as you can
get better marginal utility
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on the other one, you
start spending there.
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But then they start
to look equal.
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And you would keep dividing
your money between the two.
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And so the general
principle, if you're
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allocating money
between two goods,
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for that last increment--
not across the board,
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just that last increment--
that's why the word marginal
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is so important.
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For that last ounce of
chocolate versus that
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very last ounce of fruit,
the marginal utility
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for price for that last
increment of one good
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will be the same as
the marginal utility
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per price of the second good.
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Now I really want to
emphasize what this is saying.
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This is not saying that the
marginal utility for price
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of the two goods are the same.
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And not even that one is
better than the other.
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This is just saying
as you spend money,
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and let's say you spend
enough money to buy both,
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at some point you're going
to get to a threshold
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where you're neutral
between the two, where
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the marginal
utility for price is
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the same for an incremental of
B versus an incremental of A.
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And at that point you're
juts going to keep switching
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between the two products.
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Because obviously,
if you focus too much
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on this right over here--
let's say you focus,
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let's say at that
point you switch
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and you just start buying
a bunch of product B
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right over here.
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Well, that didn't make sense.
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Because you were
buying product B
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when you could have
actually gotten
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higher marginal utility
buying some of product A.
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And that's the same
reason why you didn't just
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keep going down A, because
you could have gotten
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higher marginal
utility over here.
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This is closer to, I don't
know, 75 while you're only
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getting 70 right over here.
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