Calculating the Elasticity of Demand

Elasticity of demand is equal to the percentage change of quantity demanded divided by percentage change in price. In this video, we go over specific terminology and notation, including how to use the midpoint formula. We apply elasticity of demand to the war on drugs, and more broadly to the prohibition of a good when it has an elastic demand.

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- [Alex] In our first lecture
on the elasticity of demand,

we explain the intuitive meaning
of elasticity.

It measures the responsiveness
of the quantity demanded

to a change in price.

More responsive means more elastic.

In this lecture, we're going
to show how to create

a numeric measure of elasticity.

How to calculate with some data
on prices and quantities,

what the elasticity is over a range
of the demand curve.

So here's a more precise definition
of elasticity.

The elasticity of demand
is the percentage change

in quantity demanded divided
by the percentage change in price.

So let's write it like this.
We have some notation here.

The elasticity of demand is equal
to the percentage change in.

Delta is the symbol for change in,
so this is the percentage change

in the quantity demanded
divided by the percentage change

in the price.

That's the elasticity of demand.
Let's give an example or two.

So, if the price of oil increases
by 10% and over a period

of several years the quantity
demanded falls by 5%,

then the long run elasticity
of demand for oil is what?

Well, elasticity
is the percentage change

and the quantity demanded.

That's -5% divided
by the percentage change

in the price.

That's 10%.

So the elasticity of demand
is -5% divided by 10%, or -0.5.

Elasticities of demand
are always negative

because when price goes up,
the quantity demanded goes down.

When price goes down,
the quantity demanded goes up.

So we often drop the negative sign
and write that the elasticity

of demand is 0.5.

Here's some more important notation.

If the absolute value
of the elasticity of demand

is less than one,
just like the example

we just gave for oil, we say
that the demand curve is inelastic.

Elasticity of demand less than one,
the demand curve is inelastic.

If the elasticity demand
is greater than one,

we say the demand curve is elastic.

And if elasticity of demand
is equal to one,

that is the knife point case,
then the demand curve

is unit elastic.

These terms are going to come back,
so just keep them in mind.

Inelastic: less than one.
Elastic: greater than one.

So we know that elasticity
is the percentage change

in quantity divided
by the percentage change in price,

how do we calculate
the percentage change in something?

This is not so hard,
but it could be a little bit tricky

for the following reason.

Let's suppose you're driving down
the highway at 100 miles per hour.

I don't recommend this,
but let's just imagine

that you are.

You're going 100 miles per hour,
and now you increase speed by 50%.

How fast are you going?
150 miles per hour, right?

Okay, so now you're going
150 miles per hour.

Suppose you decrease speed by 50%.
Now, how fast are you going?

75 miles per hour, right?

So how is it that you can
increase speed by 50%

and then decrease by 50%
and not be back

to where you started?

Well the answer is,
is that intuitively,

we have changed the base
by which we are calculating

the percentage change.

And we don't want to have
this inconsistency

when we calculate elasticity.

We want people to get
the same elasticity

whether they're calculating
from the lower base

or from the higher base.

So, because of that, we're going
to use the Midpoint Formula.

So, the elasticity of demand,
percentage change in quantity

divided by the percentage
change in price,

that's the change in quantity
divided by the average quantity

times 100.

That will give us the percentage
change divided by

the change in price
divided by the average price.

Again, that times 100.

Notice, since we've actually got
100 on top and 100 on the bottom,

those 100s we can actually
cancel out.

Let's expand this
just a little bit more.

The change in quantity.
What is the change in quantity?

Well, let's suppose
we have two quantities.

Let's call them after and before.

It doesn't matter which one
we call after or which one before.

So, we're going to then expand this
to the change in quantity.

That's Q after minus Q before
divided by the average,

Q after plus Q before,
divided by two,

divided by the change in price,
P after minus P before,

divided by the average price,
b after plus b before,

divide by two.

So that's a little bit of a mouthful,
but everything, I think,

is fairly simple.

Just remember change in quantity
divided by the average quantity

and you should always be able
to calculate this.

Let's give an example.

Okay, here's an example
of a type of problem

you might see on a quiz
or a mid term.

At the initial price of $10,
the quantity demanded is 100.

When the price rises to $20,
the quantity demanded

falls to 90.

What is the elasticity is,
what is the elasticity over

this range of the demand curve?

Well, we always want
to begin by writing down

what we know -- our formula.

The elasticity of demand
is the percentage change

in quantity divided
by the percentage change in price.

Now, let's remember
to just expand that.

That's Delta Q over the average Q
all divided by Delta P

over the average P.

Now, we just start
to fill things in.

So our quantity after, okay,
after the change is 90.

Our quantity before that was 100.

So on the top,
the percentage change

in quantity is 90 minus 100
divided by 90 plus 100, over two.

That is the average quantity.

And then on the bottom,
and the only trick here

is always write it
in the same order,

so if you put the 90 here,
then make sure you put the 20,

the number the price
which is associated

with that quantity started off
the same way.

So, always just keep it
in the same order.

So on the bottom, then,
we have the quantity --

the price after -- which is 20
minus the price before,

which is 10, divided
by the average price.

And now, just, it's numerics.

You plug in the numbers
and what you get is the elasticity

of demand is equal to -0.158,
approximately.

We can always drop
the negative sign

because these things,
elasticity of demands,

are always negative.

So it's equal to 0.158.

So does this make the elasticity
of demand over this range

elastic or inelastic?

Inelastic, right?

The elasticity of demand
we've just calculated

is less than one,
so that makes this one inelastic.

There you go.

We need to cover one more
important point

about the elasticity of demand,
and that is its relationship

to total revenue.

So a firm's revenues
are very simply equal

to price times quantity sold.

Revenue is equal
to price times quantity.

Now, elasticity, it's all about
the relationship

between price and quantity,

and so it's also going
to have implications for revenue.

Let's give some intuition
for the relationship

between the elasticity
and total revenue.

So revenue is price times quantity.

Now suppose the price goes up
by a lot and then quantity demanded

goes down, just by a little bit.

What then is going to be
the responsive revenue?

Well, if price is going up
by a lot and quantity

is going down just by a little bit,
then revenue

is also going to be going up.

Now, what kind of demand curve
do we call that, when price goes up

by a lot and quantity falls
by just a little bit?

We call that
an inelastic demand curve.

So, what this little thought
experiment tells us

is that when you have
an inelastic demand curve,

when price goes up
revenue is also going to go up,

and of course, vice versa.

Let's take a look
at this with a graph.

So here's our initial demand curve,
a very inelastic demand curve,

at a price of $10, the quantity
demanded is 100 units,

so revenue is 1,000.

Notice that we can show revenue
in the graph

by price times quantity.

Now, just looking at the graph,
look at what happens

when the price goes up to 20.

Well, the quantity goes down
by just a little bit,

in this case to 90,
but revenues go up to 1,800.

So you can just see,
by sketching the little graph,

what happens to revenues
when price goes up

when you have
an inelastic demand curve.

And again, vice versa.

Let's take a look
about what happens

when you have
an elastic demand curve.

So let's do the same kind
of little thought experiment,

revenue is price times quantity.

Suppose price goes up
by a modest amount

and quantity goes down
by a lot.

Well, if price is going up
by a little bit and quantity

is going down by a lot,
then revenue must also be falling.

And what type of demand curve
is it when price goes up

by a little bit,
quantity falls by a lot?

What type of demand curve is that?

That's an elastic demand curve.

So, revenues fall as price rises
with an elastic demand curve.

And again, let's show that.

If you're ever confused
and you can't quite remember,

just draw the graph.

I can never remember, myself,
but I always draw

these little graphs.

So, draw a really flatter,
elastic demand curve.

In this case, at a price of $10,
the quantity demanded is 250 units.

So revenues is 2,500.

And see what happens,
when price goes up,

price goes up to $20,
quantity demanded falls to 50,

so revenue falls to 1,000.

And again, you can just compare

the sizes of these
revenue rectangles

to see which way
the relationship goes.

And of course, this also implies,
going from $20, the price of $20

to the price of $10,
revenues increase.

So with an elastic demand curve,
when price goes down,

revenues go up.

So here's a summary
of these relationships.

When the elasticity of demand
is less than one,

that's an inelastic demand curve
and price and revenue

move together.

When one goes up,
the other goes up.

When one goes down,
the other goes down.

If the elasticity of demand
is greater than one,

that's an elastic demand curve
and price and revenue move

in opposite directions.

And could you guess what happens
if the elasticity of demand

is equal to one --
if you have a unit elastic curve?

Well then, when the price changes,
revenue stays the same.

Now, if you have to, again,
memorize these,

but it's really much better
to just sketch some graphs.

I never remember them,
as I've said myself,

I never remember
these relationships,

but I can always sketch
an inelastic graph

and then with a few changes
in price, I can see

whether the revenue rectangles
are getting bigger or smaller

and so I'll be able to recompute

all of these relationships
pretty easily.

Here's a quick practice question.

The elasticity of demand for eggs
has been estimated to be 0.1.

If egg producers raise their prices
by 10%, what will happen

to their total revenues? Increase?
Decrease? Or it won't change?

Okay, how should we
approach this problem?

If the elasticity of demand is 0.1,
what type of demand curve?

Inelastic demand.

Now, what's the relationship
between an inelastic demand curve?

When price goes up,
what happens to revenue?

If you're not sure,
if you don't remember,

draw some graphs.

Draw an inelastic,
draw an elastic, figure it out.

Okay, let's see. What happens?
Revenue increases, right?

If you have an inelastic
demand curve and price goes up,

revenue goes up as well.

Here's an application.

Why is the war on drugs
so hard to win?

Well, drugs are typically
going to have

a fairly inelastic demand curve.

What that means
is that when enforcement actions

raise the price of drugs,
make it more costly to get drugs,

raising the price,
that means the total revenue

for the drug dealers goes up.

So check out this graph.

Here is the price
with no prohibition,

here's our demand curve,
our inelastic demand curve.

What prohibition does,
is it raises the cost

of supplying the good.

But that raises the price,
which is what it's supposed to do,

and that does reduce
the quantity demanded of the drug.

But it also has the effect
of increasing seller revenues.

And seller revenues may be
where many of the problems

of drug prohibition come from.

It's the seller revenues
which drive the violence,

which drive the guns,
which make it look good

to be a drug dealer,
which encourage people

to become drug dealers,
and so forth.

So there's a real difficulty
with prohibition,

with prohibiting a good,
especially when it has

an inelastic demand.

Here's another application
of elasticity of demand

and how it can be used
to understand our world.

This is a quotation from 2012
from NPRs food blog "The Salt."

"You've all heard a lot

about this year's devastating
drought in the Midwest, right?

US Department of Agriculture
announced last Friday

that the average US cornfield
this year will yield less per acre

than it has since 1995.

Soybean yields are down, too.

So you think that farmers
who grow these crops

must be really hurting.

And that's certainly the impression
you get from media reports.

But how's this,
for a surprising fact?

On average, corn growers
actually will rake in

a record amount of cash
from their harvest this year."

So can you explain this secret side
of the drought?

I'm not going to answer
this question.

This is exactly the type
of question you might receive

on an exam.

But you should be able
to answer it by now,

with a few sketches
on a piece of paper.

And in particular, what I want you
to answer is,

what type of demand curve,
for corn, would make exactly

this type of outcome
perfectly understandable?

Not a secret or surprise,
but perfectly understandable.

Okay, that's the elasticity
of demand.

Next time we'll be taking up
the elasticity of supply,

and we'll be able to move
through that material much quicker

because it covers
many similar concepts.

Thanks.

- [Narrator] If you want
to test yourself,

click Practice Questions,
or if you're ready to move on,

just click Next Video.

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