We have established that economists have a metric to
measure responsiveness of quantity demanded to price, it's called elasticity.
The formal definition of elasticity is equal to
the percentage change in quantity over the percentage change in price.
If that ratio, if the absolute value of that ratio is greater than one,
it means that quantity effects dominate price effects.
We sort of thought about that intuitively.
We thought about a product where if that number is greater than one,
it's like, say, relatively flat demand curve.
This looks elastic to us because the impact of
a small change in price can have wide swings in quantity,
meaning that ratio is greater than a one.
See how quantity jumps dramatically just for a small step function down in price.
Alternatively, we could think about a product that had a very steep demand curve,
and this particular product with a steep demand curve,
you can have wild swings in price but hardly any change in quantity.
So, in this case, the denominator dominates the size of the numerator.
So, that ratio in an absolute value sense is less than one.
But it turns out, we have to be a little bit more careful about that.
So, what I'm going do is I'm going to go on a little exercise of analytics here.
Okay. So, let's walk through here.
Say, let's recall that elasticity is equal to
the percentage change in quantity over the percentage change in price.
That could be rewritten as the change in
quantity over quantity divided by the change in price over price.
Simple algebraic manipulation, which is the same thing as writing change in
quantity over change in price times price or quantity.
Nothing fancy there, simple straightforward algebraic manipulation.
Well, now, let's assume that demand curve is linear.
Well, if the demand curve is linear,
then we know several things about this ratio.
If the demand curve is linear about this formula,
this is the inverse of the slope of demand.
If you think about our demand curve,
this is price, this is quantity,
this is our demand curve.
The slope of that curve would be change in price over change in quantity.
Okay. Rise over run.
But in this case, we actually have the inverse of that.
We have change in quantity over change in price,
which is one over the slope. But that's not important.
What's important is that you know for a straight line demand curve,
what a linear function means is that the slope is a constant.
So, if the slope is a constant,
the inverse of the slope is also a constant,
which means that this term is going to be constant.
This is the ratio of price over Q, P over Q.
As you could see along this demand curve,
every point you pick along this demand curve, pick any point,
the ratio of P over Q is going to be falling everywhere along there.
So, what we say is,
along the linear demand curve,
this P over Q ratio changes along the entire demand.
Now, you might say, "Well, why are we doing this?"
Very well, we have decomposed our formula for elasticity into simple algebra,
changing Q over base Q change in P over base P,
which is through algebraic manipulation,
just this two terms.
The first term constant along any linear demand curve.
The second term changes everywhere along the linear demand curve.
Now, what that means is that
every point along a linear demand curve has a different elasticity.
Have to be a little bit more careful,
as long as the demand curve has some slope to it.
You can't have a perfectly vertical or perfectly horizontal.
As long as there's some downward slope, everything changes along.
In fact, we're not going to do it but you can
look up tons of sources that will make the proof for you.
There's a simple proof that says that if you have a straight line demand curve,
we'll put price on the horizontal axis and quantity
on the vertical axis along a straight line demand curve.
Suppose this is the midpoint.
If that's the midpoint,
we could prove that every point in the top half of the demand curve is elastic,
every point in the bottom half of the demand curve is inelastic,
and exactly at this midpoint,
that demand curve is what we called unit elastic.
Now, let's think about this.
We're not going to do this proof but what it says is that,
if you give me a straight line demand curve that has some slope to it,
some downward slope, it's not horizontal,
it's not vertical, it's got some downward slope,
I can prove to you that the exactly half of the points,
the upper half are elastic and the lower half are inelastic.
Now, the reason I'm showing you this is because that's the truth.
But if we were to go back to this picture,
I asked you when you looked at these two graphs,
let's call this market one and let's call this market two, I told you,
you would be correct to think about market two as being
elastic and market one is being inelastic.
Market one looks steep and it feels inelastic.
Market two is flatter and it feels elastic, and that's true.
But then you would say, "Larry,
you just showed us that, in fact,
if those are straight-line demand curves which they looked like,
that's what you're trying to represent,
half of that curve is elastic and half is inelastic."
That's true. But think about our picture.
If you were going to complete this curve out,
you'd have to complete it down here and you'd have to complete way up there.
So, essentially, what I've shown you is really the bottom half of a linear demand curve.
Like on this picture here,
I've shown you the bottom half because the top half is kind of off the edge of the page.
Over here, I said to think of this as elastic and you could say, "Well, yeah.
But Larry, it's linear so half of it would always be elastic and half inelastic.
But the way you've drawn it,
you're really only showing what turns out to be the upper half of
a demand curve that would have to go a long way over
there to find the horizontal intercept."
So, this region would correspond to
the upper half of
the total demand curve that I was trying to represent with this linear picture,
or on this other picture,
it'll correspond to this upper half part which is indeed elastic.
So, elasticity.
Point here is that you should know that if it's a linear demand curve,
exactly half is elastic and half is inelastic.
Yet, when I tell you that for any given demand curve that is inelastic,
you know that in the space you're drawing it, if I say,
"Draw me an inelastic curve," you're really only showing me the bottom half.
It's true that the slope's the same across and it's true that the half of it,
way up off the top of the screen,
would be elastic but this is definitely inelastic.
So, it's okay to think in your head that steep curves are inelastic,
flat curves are elastic.
Larry DeBrockDean Emeritus and Professor of Finance and Professor of Economics
