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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- 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 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 minus 5%

divided by the percentage change in the
price. That's 10%. So the elasticity of

demand is minus 5% divided by 10%, or
negative 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 negative 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 as 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
firms 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

that 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 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 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 gun,

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.

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