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# Constant unit elasticity

Video transcript

We've already talked about
linear demand functions that actually have changing price
elasticity as we go down the curve. And we've shown the extremes. We've shown things that
are perfectly inelastic and things that are
perfectly elastic. What I want to do
in this video-- and it'll be a quick little
video-- is think about can we construct a demand curve, or at
least understand what it looks like, that has a constant
elasticity across the curve? And just for fun, let's make
it a constant elasticity of 1. So it has constant unit
elasticity of demand. So let's think about
how we can create that. And hopefully, it will give
us a bit more intuition on how this elasticity
business even works. So let's draw our axes. So there you go, that is price. And that right
there is quantity. And let me put quantity. Now let's put some
numbers there that'll just help us draw
this demand curve that has unit elasticity at
every point, at every price, in every quantity. So I'm just going to
put some numbers here. So let's say that that
right over there is 10. So that's $10 or
whatever we're doing. And then this is 10 units per
time period, 10 units per week, or 10 units per month,
or whatever else. Now, we want the absolute value
of the elasticity of demand to be equal to 1 at all points. And we're going to assume
that this curve meets the law of demand, which
means as price goes down, quantity demanded goes up. So let's think. This is going to be
a downward sloping, so really we're going to say
that the elasticity of demand is going to be
equal to negative 1. If we have a 1%
decrease in price, we're going to have a
1% increase in quantity, and vice versa. So let's think about it. If we're up here, where
the price is near $10, and maybe where the
quantity is closer to $1, let's think about what
a 10% movement in price would look like, a
10% movement down. It would be roughly
about this size. A 10% movement would
be roughly there. And I'm just trying to get the
general shape of this curve. I'm not going to go into
the deep mathematics or the calculus of it. So that is a 10%
price movement down. And we also want to 10%
quantity movement up. But remember, our
quantity is only at 1. So a 10% quantity movement
up would only be 10% of 1. So if we're moving 10%
in price downwards, this is a 10%
upwards in quantity. So our curve up here would
look something like this. It would actually have
to be quite steep. Now let's think about what the
curve would look over here. Once again, we want
10% for both of them, because we want the price
elasticity of demand to be 1 throughout the curve. So if we go over here, a
10% movement in price-- so let's say we're down here,
where price is close to 1-- a 10% movement in price
is going to be very small. So a 10% movement in price
is going to be like that. It's going to be roughly
a tenth of a movement. So that's a 10%
movement in price. But a 10% movement in
quantity demanded over here, it's going to be much larger. It's going to look
something like that, because the quantity is
approaching 10, so 10% of that is about 1 unit just like that. So at this point
in the graph, it would look something like this. It would flatten out a
good bit, just like that. And then when the price and
the quantity is about the same, so let's say this
point right over here, where the price and the
quantity is about the same-- so let's say that that is
2, this is 3, this is 2, this is 3 right over here--
your percent movements are going to be the same. But since the price and
quantity are the same, the absolute movements are
also going to be the same. So at that point, our
curve should look something like that. It should have a slope of 1. And so if you
connect the dots, you get the general
shape of a demand curve that has a price
elasticity of demand at negative 1
throughout the curve, or whose absolute value of
the price elasticity of demand is 1. So let's just do that. So the curve would
look something like-- I'll just
draw a dotted line; it's easier to do-- so it'll
look something like that. It'll keep getting steeper as
we get the quantity closer to 0, and it'll keep flattening out
as the quantity grows and grows and grows. Anyway, hopefully you
found that interesting.