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Calculating the Elasticity of Demand - YouTube
Channel: Marginal Revolution University
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- [Alex] In our first lecture
on the elasticity of demand,
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we explain the intuitive meaning
of elasticity.
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It measures the responsiveness
of the quantity demanded
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to a change in price.
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More responsive means more elastic.
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In this lecture, we're going
to show how to create
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a numeric measure of elasticity.
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How to calculate with some data
on prices and quantities,
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what the elasticity is over a range
of the demand curve.
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So here's a more precise definition
of elasticity.
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The elasticity of demand
is the percentage change
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in quantity demanded divided
by the percentage change in price.
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So let's write it like this.
We have some notation here.
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The elasticity of demand is equal
to the percentage "change in".
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Delta is the symbol for change in,
so this is the percentage change
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in the quantity demanded
divided by the percentage change
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in the price.
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That's the elasticity of demand.
Let's give an example or two.
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So, if the price of oil increases
by 10% and over a period
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of several years the quantity
demanded falls by 5%,
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then the long run elasticity
of demand for oil is what?
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Well, elasticity
is the percentage change
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and the quantity demanded.
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That's -5% divided
by the percentage change
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in the price.
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That's 10%.
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So the elasticity of demand
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is -5% divided by 10%, or -0.5.
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Elasticities of demand
are always negative
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because when price goes up,
the quantity demanded goes down.
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When price goes down,
the quantity demanded goes up.
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So we often drop the negative sign
and write that the elasticity
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of demand is 0.5.
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Here's some more important notation.
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If the absolute value
of the elasticity of demand
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is less than one,
just like the example
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we just gave for oil, we say
that the demand curve is inelastic.
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Elasticity of demand less than one,
the demand curve is inelastic.
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If the elasticity of demand
is greater than one,
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we say the demand curve is elastic.
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And if elasticity of demand
is equal to one,
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that is the knife point case,
then the demand curve
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is unit elastic.
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These terms are going to come back,
so just keep them in mind.
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Inelastic: less than one.
Elastic: greater than one.
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So we know that elasticity
is the percentage change
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in quantity divided
by the percentage change in price,
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how do we calculate
the percentage change in something?
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This is not so hard,
but it could be a little bit tricky
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for the following reason.
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Let's suppose you're driving down
the highway at 100 miles per hour.
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I don't recommend this,
but let's just imagine
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that you are.
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You're going 100 miles per hour,
and now you increase speed by 50%.
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How fast are you going?
150 miles per hour, right?
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Okay, so now you're going
150 miles per hour.
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Suppose you decrease speed by 50%.
Now, how fast are you going?
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75 miles per hour, right?
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So how is it that you can
increase speed by 50%
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and then decrease by 50%
and not be back
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to where you started?
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Well the answer is,
is that intuitively,
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we have changed the base
by which we are calculating
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the percentage change.
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And we don't want to have
this inconsistency
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when we calculate elasticity.
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We want people to get
the same elasticity
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whether they're calculating
from the lower base
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or from the higher base.
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So, because of that, we're going
to use the Midpoint Formula.
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So, the elasticity of demand,
percentage change in quantity
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divided by the percentage
change in price,
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that's the change in quantity
divided by the average quantity
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times 100.
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That will give us the percentage
change divided by
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the change in price
divided by the average price.
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Again, that times 100.
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Notice, since we've actually got
100 on top and 100 on the bottom,
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those 100s we can actually
cancel out.
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Let's expand this
just a little bit more.
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The change in quantity.
What is the change in quantity?
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Well, let's suppose
we have two quantities.
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Let's call them after and before.
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It doesn't matter which one
we call after or which one before.
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So, we're going to then expand this
to the change in quantity.
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That's Q after minus Q before
divided by the average,
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Q after plus Q before,
divided by two,
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divided by the change in price,
P after minus P before,
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divided by the average price,
b after plus b before,
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divide by two.
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So that's a little bit of a mouthful,
but everything, I think,
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is fairly simple.
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Just remember change in quantity
divided by the average quantity
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and you should always be able
to calculate this.
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Let's give an example.
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Okay, here's an example
of a type of problem
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you might see on a quiz
or a mid term.
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At the initial price of $10,
the quantity demanded is 100.
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When the price rises to $20,
the quantity demanded
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falls to 90.
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What is the elasticity is,
what is the elasticity over
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this range of the demand curve?
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Well, we always want
to begin by writing down
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what we know -- our formula.
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The elasticity of demand
is the percentage change
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in quantity divided
by the percentage change in price.
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Now, let's remember
to just expand that.
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That's Delta Q over the average Q
all divided by Delta P
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over the average P.
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Now, we just start
to fill things in.
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So our quantity after, okay,
after the change is 90.
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Our quantity before that was 100.
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So on the top,
the percentage change
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in quantity is 90 minus 100
divided by 90 plus 100, over two.
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That is the average quantity.
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And then on the bottom,
and the only trick here
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is always write it
in the same order,
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so if you put the 90 here,
then make sure you put the 20,
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the number the price
which is associated
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with that quantity started off
the same way.
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So, always just keep it
in the same order.
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So on the bottom, then,
we have the quantity --
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the price after -- which is 20
minus the price before,
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which is 10, divided
by the average price.
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And now, just, it's numerics.
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You plug in the numbers
and what you get is the elasticity
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of demand is equal to -0.158,
approximately.
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We can always drop
the negative sign
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because these things,
elasticity of demands,
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are always negative.
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So it's equal to 0.158.
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So does this make the elasticity
of demand over this range
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elastic or inelastic?
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Inelastic, right?
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The elasticity of demand
we've just calculated
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is less than one,
so that makes this one inelastic.
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There you go.
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We need to cover one more
important point
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about the elasticity of demand,
and that is its relationship
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to total revenue.
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So a firm's revenues
are very simply equal
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to price times quantity sold.
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Revenue is equal
to price times quantity.
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Now, elasticity, it's all about
the relationship
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between price and quantity,
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and so it's also going
to have implications for revenue.
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Let's give some intuition
for the relationship
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between the elasticity
and total revenue.
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So revenue is price times quantity.
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Now suppose the price goes up
by a lot and then quantity demanded
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goes down, just by a little bit.
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What then is going to be
the response of revenue?
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Well, if price is going up
by a lot and quantity
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is going down just by a little bit,
then revenue
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is also going to be going up.
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Now, what kind of demand curve
do we call that, when price goes up
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by a lot and quantity falls
by just a little bit?
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We call that
an inelastic demand curve.
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So, what this little thought
experiment tells us
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is that when you have
an inelastic demand curve,
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when price goes up
revenue is also going to go up,
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and of course, vice versa.
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Let's take a look
at this with a graph.
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So here's our initial demand curve,
a very inelastic demand curve,
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at a price of $10, the quantity
demanded is 100 units,
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so revenue is 1,000.
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Notice that we can show revenue
in the graph
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by price times quantity.
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Now, just looking at the graph,
look at what happens
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when the price goes up to 20.
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Well, the quantity demanded
goes down by just a little bit,
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in this case to 90,
but revenues go up to 1,800.
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So you can just see,
by sketching the little graph,
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what happens to revenues
when price goes up
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when you have
an inelastic demand curve.
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And again, vice versa.
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Let's take a look
about what happens
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when you have
an elastic demand curve.
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So let's do the same kind
of little thought experiment,
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revenue is price times quantity.
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Suppose price goes up
by a modest amount
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and quantity goes down
by a lot.
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Well, if price is going up
by a little bit and quantity
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is going down by a lot,
then revenue must also be falling.
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And what type of demand curve
is it when price goes up
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by a little bit,
quantity falls by a lot?
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What type of demand curve is that?
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That's an elastic demand curve.
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So, revenues fall as price rises
with an elastic demand curve.
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And again, let's show that.
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If you're ever confused
and you can't quite remember,
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just draw the graph.
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I can never remember, myself,
but I always draw
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these little graphs.
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So, draw a really flatter,
elastic demand curve.
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In this case, at a price of $10,
the quantity demanded is 250 units.
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So revenues is 2,500.
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And see what happens,
when price goes up,
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price goes up to $20,
quantity demanded falls to 50,
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so revenue falls to 1,000.
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And again, you can just compare
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the sizes of these
revenue rectangles
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to see which way
the relationship goes.
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And of course, this also implies,
going from $20, the price of $20
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to a price of $10,
revenues increase.
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So with an elastic demand curve,
when price goes down,
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revenues go up.
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So here's a summary
of these relationships.
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When the elasticity of demand
is less than one,
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that's an inelastic demand curve
and price and revenue
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move together.
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When one goes up,
the other goes up.
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When one goes down,
the other goes down.
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If the elasticity of demand
is greater than one,
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that's an elastic demand curve
and price and revenue move
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in opposite directions.
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And could you guess what happens
if the elasticity of demand
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is equal to one --
if you have a unit elastic curve?
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Well then, when the price changes,
revenue stays the same.
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Now, if you have to, again,
memorize these,
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but it's really much better
to just sketch some graphs.
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I never remember them,
as I've said myself,
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I never remember
these relationships,
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but I can always sketch
an inelastic graph
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and then with a few changes
in price, I can see
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whether the revenue rectangles
are getting bigger or smaller
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and so I'll be able to recompute
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all of these relationships
pretty easily.
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Here's a quick practice question.
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The elasticity of demand for eggs
has been estimated to be 0.1.
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If egg producers raise their prices
by 10%, what will happen
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to their total revenues? Increase?
Decrease? Or it won't change?
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Okay, how should we
approach this problem?
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If the elasticity of demand is 0.1,
what type of demand curve?
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Inelastic demand.
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Now, what's the relationship
between an inelastic demand curve?
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When price goes up,
what happens to revenue?
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If you're not sure,
if you don't remember,
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draw some graphs.
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Draw an inelastic,
draw an elastic, figure it out.
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Okay, let's see. What happens?
Revenue increases, right?
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If you have an inelastic
demand curve and price goes up,
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revenue goes up as well.
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Here's an application.
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Why is the war on drugs
so hard to win?
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Well, drugs are typically
going to have
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a fairly inelastic demand curve.
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What that means
is that when enforcement actions
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raise the price of drugs,
make it more costly to get drugs,
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raising the price,
that means the total revenue
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for the drug dealers goes up.
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So check out this graph.
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Here is the price
with no prohibition,
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here's our demand curve,
our inelastic demand curve.
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What prohibition does,
is it raises the cost
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of supplying the good.
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But that raises the price,
which is what it's supposed to do,
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and that does reduce
the quantity demanded of the drug.
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But it also has the effect
of increasing seller revenues.
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And seller revenues may be
where many of the problems
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of drug prohibition come from.
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It's the seller revenues
which drive the violence,
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which drive the guns,
which make it look good
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to be a drug dealer,
which encourage people
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to become drug dealers,
and so forth.
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So there's a real difficulty
with prohibition,
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with prohibiting a good,
especially when it has
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an inelastic demand.
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Here's another application
of elasticity of demand
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and how it can be used
to understand our world.
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This is a quotation from 2012
from NPRs food blog "The Salt."
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"You've all heard a lot
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about this year's devastating
drought in the Midwest, right?
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US Department of Agriculture
announced last Friday
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that the average US cornfield
this year will yield less per acre
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than it has since 1995.
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Soybean yields are down, too.
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So you think that farmers
who grow these crops
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must be really hurting.
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And that's certainly the impression
you get from media reports.
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But how's this,
for a surprising fact?
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On average, corn growers
actually will rake in
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a record amount of cash
from their harvest this year."
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So can you explain this secret side
of the drought?
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I'm not going to answer
this question.
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This is exactly the type
of question you might receive
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on an exam.
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But you should be able
to answer it by now,
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with a few sketches
on a piece of paper.
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And in particular, what I want you
to answer is,
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what type of demand curve,
for corn, would make exactly
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this type of outcome
perfectly understandable?
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Not a secret or surprise,
but perfectly understandable.
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Okay, that's the elasticity
of demand.
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Next time we'll be taking up
the elasticity of supply,
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and we'll be able to move
through that material much quicker
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because it covers
many similar concepts.
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Thanks.
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- [Narrator] If you want
to test yourself,
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click Practice Questions,
or if you're ready to move on,
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just click Next Video.
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