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# Utility maximization: equalizing marginal utility per dollar

## Video transcript

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