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## Price elasticity of demand

# Price elasticity of demand using the midpoint method

## Video transcript

What we're going to
think about in this video is elasticity of
demand-- tis-sit-tity, elasticity of demand. And what this is,
is a measure of how does the quantity demanded
change given a change in price? Or how does a change in price
impact the quantity demanded? So change in price--
impact quantity-- want to be careful
here-- quantity demanded. When you talk about
demand, you're talking about the whole curve. Quantity demanded is a specific
quantity-- quantity demanded. And the way that we, as
economist-- I'm not really an economist, but since
we're doing economics, we could pretend
to be economists. The way that
economists measure this is they measure it as a
percent change in quantity over a percent-- over the
percent change in price. And the reason why they do
this, as opposed to just, say, change in quantity
over change in price, is because if you did change in
quantity over change in price you would have a number that's
specific to the units you're using. So it would depend on
whether you're doing quantity in terms of per hour, or
per week, or per year. And so you would have different
numbers based on the time frame, or the units,
that you might use. But when you use a percentage
it is a unitless number. Because the percentage--
you're taking a change in some quantity,
divided by that quantity. So the units themselves
actually cancel out. And the reason why it's
called elasticity-- this might make some sense
to you-- or the reason why I like to think
it's called elasticity, is I imagine something
that's the elastic. Like a elastic band
or a rubber band. And in the rubber
band, if you pull it, depending if something--
so let's say this one is inelastic. So if you pull, you're not
going to able to pull it much. It's going to be fairly stiff. It's not going to stretch a lot. While something is
elastic-- if something is elastic for a given
amount of force-- so this is for a given
amount of force-- you're not able to pull it much. And if something
is elastic, maybe for the same amount
of force, you're going to be able
to pull it a lot. So this right over
here is elastic. And so the analogy, maybe,
might make a little bit sense-- relative to applied
price and demand. Something is elastic-- so
let me write this down. So let me write, very elastic. If a given change in
price-- given price change you have-- and we'll talk about
percentages in a little bit. But a given change in price,
you have a large change in demand-- so large
percentage change. And let me just speak
in terms of percentage. Given a percentage
change in P, you end up having a large
percentage change in Q. That would be very elastic. So you could imagine
the P is like the force, and the Q, the
quantity demanded, is how far the thing
can get stretched apart. And that's why we would
call this very elastic. Just like a very
elastic rubber band. And if something is very
inelastic, if given a percent change in P, you have a
small percent change in Q. So just like a rubber band--
for a given amount of force, if you're not able to
pull it much at all, then it's inelastic. If you're able to pull
a lot, it's elastic. Same thing with
price and quantity. For a given change in price, if
the percent quantity demanded changes a lot-- very elastic. If it doesn't change a
lot-- very inelastic. Now, with that out
of the way, let's actually calculate
the elasticity for multiple points along this
demand curve right over here. And I think that will give
us a bit better grounding. Especially because there are a
little slightly-- I would call them unusual ways of calculating
the percent change in quantity and the percent
change of price-- just so that we get
the same number when we have a positive
change in price. And the same as we get
the negative change in price-- or a negative and a
positive-- or a drop in price and an increase in price. So let me give myself
some real estate over here because I want to do
some actual mathematics. And actually all of this we will
be reviewing in what I'm about do, and it will give me some
real estate to work with. So let me clear all of that. And let me clear is
that right over here. And what I'm going
to do is I'm going to calculate the
elasticity of demand at several points along this
demand curve right over here. And so the first one,
I will do it at point A to point B. So let me
make another column right over here--
elasticity of demand. And actually,
we're going to have one column that's
elasticity of demand. So it's a big E with a little
subscript D. And the other one, I'll just take its
absolute value. Because, depending
on-- sometimes people like to just
think of the number, which will tend to
be a negative number. And sometimes,
people like to look at the absolute value of it. So we'll look at both and
see what it actually means. So let's say our price
drops from point A to point B. So from
point A to point B we have a $1-- a negative
$1 change in price. And we have a positive--
so this is a negative $1 change in price. And we have a
positive $2-- sorry-- a positive two burger per hour
change in quantity demanded. So what is the elasticity
of demand there? So let's write it over here. I'll do it in A's color. So the elasticity
of demand, remember, it's the percent
change in quantity. So percent change in
quantity-- I'll rewrite it. It's the percent
change in quantity over percent change in price. And so we have-- what's our
percent change in quantity? So it's going to be the change
in quantity over some base quantity. So our change in
quantity is two. So it's going to be
equal to 2 over-- now in traditional terms-- and this
is what I want to, kind of, clarify-- is a little bit
unusual in how we do it. But we do it, so that we get
the same elasticity of demand whether we go from
A to B or B to A. Or essentially, we get
the same elasticity of demand along this
whole part of the curve. Instead of just
dividing the change in quantity divided by
our starting point, what I want to do is
I'm going to divide the change in quantity divided
by the average of our starting and our ending, points. So that's going to be 2
over-- and I'll actually do the math explicitly. Actually, no, let's
just think about it. What's the average
between 2 and 4. Well, the average is
just going to be 3. That's the average of 2 and 4. Let me write it down
to, just so it's clear. That right over here
is 2 plus 4 over 2. That's how you get 3. That's how you would
calculate the average. So that is our
proportionate change. And then you want to
multiply by 100-- times 100-- to actually get a percentage. And then, what is
our change in price? Well we're going to do the
same thing, or the percent change in price. Our change in price
is negative 1. It is negative 1 over-- and
once again, we don't just do negative 1 divided
by 9, we do it over the average of 8 and 9. And the average
of 8 and 9 is 8.5. And then multiply by 100
to get your percentage. Now, these 100s,
obviously, cancel out. These 100s cancel out. And so we are going to be
left with-- when you divide by a fraction,
it's the same thing as multiplying by its inverse. So we're going to get 2/3
times negative 8.5 over 1-- or times negative 8.5. I'll get out our
calculator and it is-- well, multiply
2 times negative 8.5, and then divided by 3, which
gives us negative 5.6667. It's really negative 5 2/3. So I'll just write
it negative-- I'll round it-- it's negative 5.67. So this is approximately
equal to negative 5.67. So right over here
it's negative 5.67. And this absolute value
is, obviously, just 5.67. And I'll leave it to you
to verify, for yourself, that you'll get the same
elasticity of demand using this technique--
where you use the average as your
base in the percentage. Going from 9 to 8 as
going-- going from 9 to 8 in price as going
from 8 to 9 in price. Which is different than if you
used the 9 as the base or the 8 as the base. So this right here
is the elasticity of demand-- not just at point
A. You can, kind of, view it is the average
elasticity of demand over this little part of
the curve, which is really a line in this example--
over this part of the arc. So we'll write that
part right over here. I'll write the absolute value. The absolute value of our
elasticity of demand is 5.67. Now let's do the other two
sections right over here. So let's think
about what happens when we go from C to D. So our
elasticity of demand there. So from C to D we have a
change in quantity, once again, of plus 2. And our change in price,
once again, is negative 1. But we'll see, even though
that the change in the quantity over-- the change of
quantity is the same, and the change in
price is the same, we're going to have a
different elasticity of demand, because we have different
starting points. Our starting points and
our ending points for price are lower and our starting
and ending points for quantity are higher. So it will actually
change the percentage. So let's see what we get. So our percent change
in quantity-- we have a change in quantity of 2. And then our average
quantity is 9 plus 11, which is 20,
divided by 2 is 10. All of that over
percent change in price. So we have-- let me scroll down
a little bit-- negative one divided by the average price. So negative 1 is
the change in price. And we want to divide
that by the average price. Well, $5.50 plus $4.50 is
$10-- divided by 2 is $5.00. So the average is $5.00. And we can multiply
the numerator by 100 and the
denominator by 100, but that won't change
anything, because we could just divide both by 100. And so this is
equal to 2 over 10, times-- dividing by a
fraction is the same thing as multiplying by its inverse--
times negative 5 over 1. And this is just because 2 over
10 is the same thing as 1/5. 1/5 times negative 5 over
1-- it is negative 1. So this right over here. So our elasticity of demand
right over here is negative 1. Or it's absolute value is 1. So the absolute value of
the elasticity of demand, right over here, is equal to 1. Now let's just do
one more section, and maybe, the next video
we can think a little bit about what it's telling us. So let's do this last
section over here, just for some practice. I encourage you to pause
it and try it yourself. And so we're going to
think about this section right over here. So once again, our change
in quantity is plus 2, and our change in
price is negative 1. And our elasticity
of demand-- change in quantity-- 2 over average
quantity, which is 17. Change in price is negative
1 over average price-- 1 plus 2 divided by 2 is $1.50. Or $1.50 is right in between
these two-- divided by $1.50. We don't have to multiply the
numerator and the denominator by 100 because those
just cancel out. So we get 2 over 17,
times negative-- well, we could just write this
as negative $1.50 over 1. And this is equal
to-- getting our-- getting our calculator back out. So this is equal to--
I'll just write-- well, it's really just going to be
negative 3 over 17, right? 2 times negative $1.50
is negative 3 over 17. So negative 3 divided by 17
is equal to, I'll just say, negative 0.18. So here it is, negative 0.18,
and its absolute value is 0.18. So the elasticity of
demand over here is 0.18. And I'll leave you there,
and in the next video we'll think about these
results a little bit.