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# Cross-price elasticity ofÂ demand

## Video transcript

So far, we've been
focused on the elasticity of demand for only one good. We've thought about how
changes in the price of that good affect
changes in its quantity. Now what we're going to explore
is how we can go across goods. So we're going to talk about
the cross elasticity of demand. And there's multiple different
scenarios we could think about, but it's really thinking about
how a price change in one good might affect the quantity
demanded in another good. And to see an example of this,
think about two airlines-- two competing airlines-- maybe
it's the same exact route going at the exact same time, maybe
between New York and London. So airline one, right
over here-- airline two, very competitive,
price right over here is $1,000 for a round trip. Quantity demanded
is 200 tickets, let's say, in a given week. Airline two, price is
$1,000 for the round trip, and the quantity demanded
is 200 tickets as well. Now let's think about
what will happen. What will happen if
airline one raises its price from $1,000 to $1,100? In fact, we could
even do something less dramatic than that,
to $1,050-- so a relatively small increase in price. And remember, when we think
about the percentage price increase, when we're thinking
about elasticities in general, we don't just say,
OK, $50 on top of $1,000, that's a
5% price increase. That's what we would do
in everyday thinking. If you said you went
from $1,000 to $1,050, you would say that's a $50
increase on a base of $1,000 or that is a 5% increase. But when you think
about elasticities, because we want to have the same
percent change between-- if you go from $1,000 to $1,050,
or if you go from $1,050 down to 1,000-- we actually use
the average point as a base. So the percent change
in this scenario-- let me write it right over here. So our percent change--
and I'll write it in quotes, because
it's a little bit different than what you do
in traditional mathematics when you think about
percent changes-- is you had a 50 change in price. Your price went up by
50, and on our base we will use 1,025, which is
the average of 1,000 and 1,050. And so that gives us a
change of 50 divided by 1,025 is equal to, let's
say, roughly 4.9%. So this is approximately 4.9%,
we'll say, "increase" in price, although we're going to put
that increase in quotes, because we're using
it on the average. And we do that so that if we
said it was 1,050 to 1,000, it would still be a 4.9%
decrease using this same idea-- using the midpoint as the base. Now, when that happens--
Everyone today, they use these
travel sites where you can compare
prices-- If these really are the exact same route, going
from the exact same airport to the exact same other
airport in London, leaving at the exact
same time, everyone is going to gravitate
to this one now, because it's only $1,000--
even just to save $50. Why would they ride
on this airline? So this quantity demand
is going to go to 0. And this quantity demanded
is going to go to 400. And we're not going to think
about the actual capacity of the planes and all that. We're going to have a
very simple model here. So what was the percent
change in quantity for airline two right over here? Well, once again, our change
in quantity is 200, not 400. We went from 200 to 400. So we gained 200. And our base, we want to use
the average of 200 and 400, which is 300. And so this is
approximately 67%. So we have, all of a
sudden, our cross elasticity of demand for airline
two's tickets, relative to a1's price. And we get the percent
change in the quantity demanded for a2's tickets, which
is 67% over the percent change, not in a2's price change,
but in a1's price change. That's why we call
it cross elasticity. We're going from
one good to another. So let's just say, for
simplicity, roughly 5%. And so you do the math. So if you have 67% divided by
5%, you get to roughly 13.4. So this is approximately 13.4. So you have a very high
cross elasticity of demand. In fact, if you even
increase this, maybe by $5, you might have had
the same effect. And so you would have had
a very large number here. And that situation right here,
for this cross elasticity of demand-- it's
because these things are near perfect substitutes. The way that we set up
this problem, we said, well, people don't care
which one they take. They're just going to
go for the cheapest one. And so when you have
near substitutes, or nearly perfect
substitutes, for each other, like this example right here,
the cross elasticity of demand approaches infinity. It gets higher and
higher and higher. In theory, if these are really,
really, really identical, even if you raise this a
penny, people will say, well, why would I waste a penny? I would just use airline two. And so this number would be
even lower right over here. And so this thing might
approach infinity. And notice this was a positive. When we just did regular
price elasticity of demand, the only way that you
would increase quantity for a traditional goods
was by lowering price. But here, we raise price on a
substitute competitive product, and we raise the demand
for airline two's product, which actually
made a lot of sense. So it wasn't a
negative relationship. It's actually a positive
value right over here. But you could have
things in other-- you could have that
negative relationship using cross elasticity of demand. This is an example
of a substitute. We could think about the
example of a complement. So what if we're
talking about e-books? So let's say I have
some type of an e-book, and the current quantity
demanded in a given week is 1,000. And let's say that the price
of an e-reader that you would need for my
e-book is $100. But let's say that
price of the e-reader goes down from $100 to $80. So you had a $20
decrease in price. Well, what's going to happen to
my e-book, assuming its price does not change? Well, then the quantity demanded
for my e-book will go up. So let's say the quantity
demanded for my e-book goes up by 100, because more
people are going to be able to afford
this, or they're going to have money
left over when they buy this to
buy more e-books. And so I don't even know what
the price for my e-book is, but at a given price point, the
quantity demanded will go up. And so this goes to 1,100. And so I'll leave it
to you to calculate this price elasticity of demand. But you will see that you will
actually get a negative value, like we're used to
seeing for regular price elasticity of demand. And when you do
calculate it, remember, you want to do your
percent price change in e-book quantity over percent
change in e-reader price. And the other thing
you have to remember, you don't just take
negative 20 over 100. You take negative 20 over
the average of these two, when you're thinking of it
in the elasticity context. So this right over
here-- actually, maybe we'll just
work it through. Pause it, and try
to do it yourself. So this value right over
here is negative 20 over 90-- the average of those two--
and this value right over here is going to be plus 100 over
the average of these two. So the average of
those two is 1,050. And so this is 100
divided by 1,050, which gets you to about 0.95. So about 9 and 1/2% change in
quantity demanded for my book. And then this
denominator right here is negative 20 divided by 90. So you get a drop of 22%. And so if you divide the
numerator by the denominator, you get 0.952 divided
by negative 0.22222-- I'll just put couple
of 2's there-- and you get a negative 0.43. So this is equal
to negative 0.43. And this makes sense. If you lower the
price of an e-reader-- this complement product,
a product that goes along with my e-book-- it
increases the demand. So just like you get with
price elasticity of demand, you get a negative
value over here. And what about completely
two unrelated products? So let's say that
I have basketballs, and the price of basketballs
goes from, let's say, $20 to $30. What's going to
happen to my e-book? Well, my e-book's
not going to change. It's going to stay at $1,000. So my percent change in the
quantity demanded of my e-book is going to be 0
in this example. So we're going to
have 0, when we want to do this cross
elasticity of demand, over my percent
change in basketballs, which would be 30 over 25. So whatever that
is-- 30 over 25 would be 10 over 25-- which
is a 40% increase. So that would be 0 over
40%, which equals 0. So for unrelated
products, products where the price of
change in one of them does not affect the quantity
demanded in the other, it makes complete
sense that you have a 0 cross elasticity of demand. If they're complements, you
would have a negative cross elasticity of demand. And if they're substitutes,
you would have a positive one. And the closer the substitutes
they are, the more positive your cross elasticity of
demand is going to be.