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## Utility maximization with indifference curves

Current time:0:00Total duration:10:52

# Indifference curves and marginal rate of substitution

## Video transcript

- [Narrator] In this video,
we're going to explore the idea of an indifference curve. Indifference. Indifference curve. And what it is, is it
describes all of the points, all of the combinations of
things to which I am indifferent. In the past, we've thought
about maximizing total utility. Now, we're going to talk
about all of the combinations that essentially give us
the same total utility. So, let's draw a graph that tells us all of the different
combinations of two goods to which we are indifferent and like I've, we've mentioned before, we're focusing on two goods,
because if we did three goods we would have to do it
in three dimensions, and four goods would get very abstract. So, let's say in this
axis, the vertical axis, this is going to be the quantity and we'll stay with the chocolate
and the fruit trade-off. Those are the only two
things that we consume. So, this is going to be the
quantity of chocolate in bars and in the horizontal axis,
this is going to be the quantity of fruit and this is going
to be in pounds of fruit. And this will go, see
this is 10, this is 20 this is 10, and this is 20 and this would be 15, 5, 5, and then 15. And let's say, let's say that
right now, at some point, I am consuming 5 pounds of fruit per month and 15 bars of chocolate per month. So, that would put me right there. And if someone were to ask,
"Sal, how would you feel, how would you feel if instead
of that, instead of that, I were to give you, let's say, 10 bars of chocolate and 7
pounds, and 7 pounds of fruit?" And I would say, you know
what, I'm indifferent. I wouldn't care whether I have,
so this right over here is, I wouldn't care whether I
have 15 bars of chocolate and 5 pounds of fruit or
whether I have 10 bars of chocolate and 7 pounds of fruit. I am indifferent between these two. I have introspected on what I like and what I derive benefit
and satisfaction out of, and I get the same total
utility out of either of these, out of either of these points. So, both of these are on
the same indifference curve and in general, I can plot all
of the different combinations that give me the exact same total utility and it might look something like this. Let me try to draw it
as neatly as possible. I'll do it in magenta. It might look something like this and then keep going all
the way down like that. So, any point on this
curve right over here, I'm indifferent relative
to my current predicament of 15 bars and 5 pounds of chocolate. So, that is my indifference curve. Indifference. Indifference curve. Now, let's think about, so, obviously, if I go all over here, 20 pounds of fruit, and I don't know, that looks
about 2 bars of chocolate, to me, the same utility
based on my preferences, as where I started off with. So, someone just swapped everything out. I would just kind of, you know, shrug my shoulders and
say yeah, no big deal. I wouldn't be happy. I wouldn't be sad. I am indifferent. Now, what about points down here? What about a point like this? Well, that is clearly not
preferable because, for example, that point I just showed,
I can show a point on the indifference curve
where I am better off. For example, that point that I just did, that's 5 pounds of fruit and
about 5 bars of chocolate, but assuming that the marginal
benefit of more chocolate is positive, and the way I've drawn this, or the assumption is that it is, then, I'm obviously getting more benefit if I'm get even more chocolate per month. And so, anything down here,
below the indifference curve, is not preferred. Not preferred. Preferred. And, using the same exact
logic, anything out here, anything out here, well, that would be good
because we're neutral between all of these points on the curve but this green point right over here, I have the same number of
bars as a point on the curve, but I have a lot more pounds of fruit. It looks like I have 11
or 12 pounds of fruit. So, assuming that I'm
getting marginal benefits from those incremental pounds of fruit, and we will make that assumption, then, this right over
here, anything out here, is going to be preferred. So, this whole area is
going to be preferred to everything on the curve. Preferred. And the whole area down
here, is, obviously, we've not preferred to anything on the curve. And let me, just to show
you this, not those points. So, all of this, and let me do that in a different color actually,
'cause our curve is purple, everything in blue is not preferred. Now, the last thing I want
to think about in this video is what the slope of this
indifference curve tells us. When I talk about the slope,
and this is really kind of an idea out of Calculus, 'cause we're used to thinking
about slopes of lines. So, if you give me a line like that, the slope is how much does
my vertical axis change for every change in my horizontal axis? So, in a typical algebra class,
that axis is your Y axis. That is your X axis. And when we think about
slope, we say, okay, when I have a certain change
in Y when I change in X by 1. So, we have something like this. So, when I change, I get
a certain change in Y, the triangle means change
in, delta, change in Y, when I get a certain change in X. And delta Y, the change in Y, over change in X is equal to the slope. But this is when it's a line
and the slope isn't changing. At any point on this line,
if I do the same ratio between the change in
Y and the change in X, I'm going to get the same value. On a curve like this, the
slope is constantly changing. So, what we really do, to
figure out the slope exactly at a point, you can imagine,
it's really the slope of the tangent line at that point. A line that would just
touch at that point. So, for example, let's say
that I draw a tangent line, I am going to draw my best attempt at drawing a tangent line
and I'll do it in pink. Let's say I have a tangent line right from our starting
predicament, just like that. And it looks something like that. It looks something like that. And so, right where we are
now, exactly at this point, you know, if we veer away, it seems like our slope is changing. Matter of fact, it definitely is changing. It's becoming less steep as
we go forward to the right. It's becoming more steep
as we go to the left. But right there, the slope of the tangent line looks right like that or you can view that as the
instantaneous slope right there. And we can measure the
slope of the tangent line. We can say, look, if we want an extra, let's see, this looks like about, if we want an extra 2 pounds of fruit, how many bars are we
going to have to give up? How many bars are we
going to have to give up? Well, it looks like we're
going to have to give up, based on the slope right over there, looks like we're going to
have to give up 5 bars. So, this is 5 and this is 2. So, what is your change
in, what is the slope here? The slope here, is going
to be your change in bars, and I should actually say this is a negative right over there, it's going to be your change in bars, your change in chocolate bars,
over your change in fruit. Over your change in fruit. And in this situation, it is -5 bars for every 2 fruit that you get. So, bars per fruit. Or you can say this is equal
to -2.5 bars per fruit. So, it's essentially saying,
exactly at that point, how are you willing to
trade off bars for fruit? Exactly at that point, it's gonna change, as things change along this curve. But it's saying, exactly where
you're sitting right now, you would be indifferent but it's only as you just slightly move
for an extra drop of fruit, an extra ounce of fruit,
not even a whole pound, you'd be willing to trade
off 2 1/2 bars per fruit. And what this says, so
you're willing to give up, since it's negative,
you're giving up 2 1/2 bars of chocolate for every pound of fruit. Now, it's going to be different. Once you have a lot
more fruit, you're going to be much less willing to
give up bars of chocolate. Over here, you have a lot of
bars and not a lot of fruit. So, you're willing to give
up a lot of bars for fruit. Over here, if we go over here, the slope looks a little bit different. Over here, it is much flatter. So let me draw it in a
color we haven't used yet. So, over here, the tangent
line looks something like this. It looks something like this. And let's say, when you
calculate it, in order to get, I don't know, this looks
about 5 pounds of fruit, in order to get 5 pounds
of fruit, you are going to have to give up 2 bars. So, once again, the slope is
the change in the vertical axis over the change in the horizontal axis. So, over here, at this
point, your change in bars over your change in fruit, is going to be, well, you're going to give
up 2 bars, for every 5 fruit. Bars per fruit. So, this right over here, is -0.4. I'll say B for F. So over here, you're willing
to give up much fewer bars for every incremental fruit. Up here, you were willing
to give many bars away for every fruit and that makes sense. Over here, you had a lot of chocolate bars and not a lot of fruit. So, you were willing to give
up more bars for your fruit. And over here, you have many fewer bars so you're much more resistant
to giving up bars for fruit. But this number, how
many bars you're willing to give up for an incremental
fruit at any point here, or you could view it as a slope
of the indifference curve, or the slope of a tangent line at that point of the indifference curve, this, right over here is called our marginal rate of substitution. Marginal rate of substitution. It's a very fancy word
but all it's really saying is how much you're willing to
give up of the vertical axis for an increment of the horizontal axis. Right at that point, and it
changes, as soon as you move, because this is a curve,
it changes a little bit, but right at that point, for
a super super small amount, how many bars are you
willing to give up for fruit? And obviously, it changes as we go along this indifference curve.