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PROFESSOR: Hi, and welcome back
to the 14.01 problem solving
videos.
Today, we're going to work on
Fall 2010, problem set one,
problem number four.
And in this problem,
we're going to be
working with elasticities.
But instead of starting
with a demand function,
and starting with
a supply function,
and calculating the elasticity
given those functions,
we're going to be given
the elasticity of demand
and the elasticity of supply.
And we're going to
have to back out
what the demand functions
and the supply functions
should have looked like.
So we're basically
just working in reverse
from what we did in lecture.
Let's go ahead and read the
full problem up through part A.
You have been asked to
analyze the market for steel.
From public sources,
you are able to find
that last year's price
for steel was $20 per ton.
At this price, 100 million tons
were sold on the world market.
From trade association
data, you are
able to obtain estimates for
their own price elasticities
of demand and supply on the
world markets as negative 0.25
for demand and 0.5 for supply.
Assume the steel has
linear demand and supply
curves throughout.
Part A asks us to solve for the
equations of demand and supply
in this market, and to sketch
the demand and supply curves.
So looking at the formal
definition of elasticity
of demand and
elasticity of supply,
we basically are going to have
three different parts to it.
We have the derivative of
either demand or supply
function with respect to P, in
this case the own price of P,
or the price of steel.
And we also have the
equilibrium price,
or any price at the point on
the curve, and a quantity.
In this case, it's going to
be the equilibrium quantity.
So basically, what
we have now is
we are given-- for the
elasticity of demand,
we're given three variables.
We're given the
price, the quantity,
and the elasticity of demand.
And that means the only
thing that we don't know
is the derivative of the
demand curve with respect to P.
So if we can isolate
this derivative,
then we can integrate
the number that we're
able to solve through for.
And then we can solve
out for what our demand
curve is going to look like.
So let's go ahead and walk
through that process together.
Substituting in for the
elasticity of demand P and Q,
we're gonna have this equation.
And the one thing that
I want you to notice
is since the derivative of the
demand curve with respect to P
is negative 0.25, in this
case, we know that it's linear.
But just because it's
linear at the point
where price is 20
and quantity is
100, that doesn't
necessarily mean it's
gonna be linear throughout.
So it's useful to know that
at any point on this line,
it's always going to
have the derivative equal
to negative 0.25.
So that's useful.
We know we can integrate
and have the correct answer.
Solving for dQD dP, we're
gonna have negative 1.25.
And we're just going to
integrate this with respect
to P. And after we
integrate, we're
going to be left
with a constant.
In this case, we're going
to call the constant a.
This is how much the
demand curve has actually
shifted up to begin
with, shifted up or down.
And to solve for a,
all we have to do
is we can just plug back in for
the $20 and the 100 quantity,
and we can solve through for
what a is going to be equal.
When you solve through
plugging in Q and P,
you're going to find
that a is equal to 125.
So we're gonna have that our
final demand function is gonna
be negative 1.25P plus 125.
Now we can go through
this exact same process
with the elasticity
of supply now.
And all we have to do now is
use the number 0.5 instead,
and we can solve through.
We can integrate.
And then we're going to
solve for the other constant
to, again, get our supply curve.
Substituting in the
information we have,
we're going to be left
with this equation.
And we're gonna go
ahead and isolate
the derivative that we have.
And when we integrate
again, we have
to remember that we are going
to have a constant that we're
gonna have to solve for.
And I'm gonna just
call this constant c.
Again, plug in the price of 20
and the quantity equal to 100
and you're gonna find that the
supply curve is gonna be equal
to 2.5P plus 50.
And we can do a quick sketch
of this on our axes here.
We're just gonna go ahead and
draw our upward-sloping supply
curve, our downward-sloping
demand curve.
And we're gonna mark
the equilibrium point
and label the equilibrium
quantities and the equilibrium
prices, as well.
And before we move on to the
second part of this problem,
we can pause here.
And we can think about what
did the elasticities that we
started with actually mean.
Well, if we were to look at
this point of intersection
at the equilibrium
of the demand curve,
we're looking at the
percentage change
at this point in quantity per
percentage change in price.
So we're basically just
saying, for that tiny change,
an infinitesimally small change
at this point for the demand
curve, how much does quantity
change, percentage-wise,
relative to price?
And that's also
what we're looking
at with the supply curve.
So when you're
given a elasticity,
if you have an
elasticity of supply,
it makes sense that it's gonna
be positive, in this case 0.5,
because when price
increases, suppliers
are willing to supply more.
And it makes sense that the
demand elasticity that we're
given is negative,
or negative 0.25,
because when price
begins to increase,
the consumers are gonna
want less of the product.
Now, the second
part of this problem
is going to give us new
elasticities of demand
and supply.
And I'm gonna just quickly run
through the actual calculation,
because it's gonna be the
same as our calculation
that we just did.
And instead, we're gonna
think about possible causes
for the shifts that we see
in the supply and the demand
curve.
Part B says, suppose
that you discover
that the current price
of steel is $15 per ton
and the current level of
worldwide sales of steel
is 150 million tons.
The most recent
elasticity estimates
from the trade
association this year
are negative 0.125 for
demand and 0.25 for supply.
Describe the change in the
supply and the demand curves
over the past year
using your diagram
from part A. What sort of
events might explain the change?
Now, I've given us the
information for this part
of the problem on this board.
And you'll notice that our
inputs, or our variables,
have changed now.
The price has dropped from $20.
Now it's gonna be down to $15.
You're gonna notice that
the quantity has actually
increased from 100 up to 150.
And our elasticities of demand
and elasticities of supply
have changed, because the price
and the quantity are different,
and we're at a different
point on our graph.
The process we're gonna
do to solve for our demand
curves and our supply curves are
going to be exactly identical.
And when you follow
the same process--
I'll just do the
first step up here--
you're gonna substitute in
for the information that's
given in the problem.
And all you're going
to do is you're,
again, gonna solve through
for the derivative.
You're gonna integrate.
And you're gonna
find the constants.
After you do that
entire process,
you're gonna find that
the demand curve is
given by this equation.
And you're gonna find
that the supply curve is
given by this new equation.
Now, if we look
at this new demand
curve and this new
supply curve, we'll
actually notice that the
slope, with respect to P,
is going to be identical
in both of the cases
that we solved for, both the
beginning case and the case
in the end of the problem.
The only thing that's shifted
between our quantities demanded
and our quantities
supplied, or the curves,
is there's been a shift.
And the shift for
the demand curve--
it went from an intercept of 125
now to an intercept of 168.75.
So our demand curve is
shifting up and out.
So we can represent this
shift in demand like this.
Notice that the slope is
going to be exactly identical.
I'm going to write a
small db for part B.
And then we can do the
same sort of interpretation
for our supply curve.
Looking at our supply curve,
the intercepts, now, is 112.5.
But before, it was only at 50.
And what this means, this
means that the supply curve
is going to shift in and down.
And so my graph with
the equilibrium price
that I've drawn--
it's a little bit off,
but what you should
see-- you should
see that the new equilibrium
price has fallen.
In this case, it's fallen to 15.
And the equilibrium quantity
has increased from 100 to 150.
So since we had both a shift in
supply and a shift in demand,
necessarily we see that
quantity is going to increase.
But if the demand curve
had shifted way up here,
we could see that price
could have increased.
So the effect on the price
in this market is ambiguous.
We can say that, necessarily,
the effect on quantity
is going to be clearly
towards an increase.
So to wrap up this
problem, we saw
that changes in
elasticities can also
represent changes in the
underlying demand and supply
functions.
Let's wrap up by just
thinking about what
could have caused the demand
shift that we've seen.
And what could have caused
the supply shift that we saw?
Now, there are a couple of ideas
that we can have for demand.
The first idea that
we could have is we
could just have had an increase
in the income of a consumer.
If a consumer has
more income, then they
might be willing to
spend more on steel.
A second idea that
we have, we could
have that the price of a
substitute-- perhaps you're
considering building a bridge
out of iron instead of steel--
if the price of the
substitute has increased,
then perhaps the
consumers are going
to be willing to pay more to
get the steel since the iron
is more expensive.
A third possible idea
is that the number
of goods that you need to
make from steel is increasing.
So if you suddenly find
new uses for steel,
then the price that you're
willing to pay at any given
point is going to be higher.
Basically, to affect
the demand curve,
you have to think
about why would people
be more willing to pay
more for a fixed quantity.
And I just listed off a
couple of ideas there.
We can also think about
reasons about why the supply
curve could be shifting in.
In this case, why
is it-- why are
sellers willing to offer a
cheaper price at any fixed
quantity?
And one idea that we
could have for this
is just that there are
more firms in this market.
If this market isn't perfectly
competitive to start off with,
then increasing
the number of firms
is gonna increase
competition, and the producers
are gonna have to
drop their prices.
A second idea for why
we've seen the supply curve
shift out and down could be the
fact that input price for steel
has dropped.
Perhaps the way of manufacturing
or getting the raw material
is cheaper because the machine
they're using to get the steel
is cheaper.
Basically, when you're
thinking about the shift that's
making it cheaper for
suppliers to produce the good,
all you need to
think about is what
could make it so that they're
more willing to produce
at a lower price.
So again, with this
problem, we went
through working with
elasticities and demands.
We've seen that we can go from
a demand curve or supply curve
to elasticities, or we can go
from elasticities to demands.
And then, once we've had the
supply and the demand curves,
we looked at how do we
interpret the shifts and shocks?
And we looked at
possible explanations
for those shift and shocks.
I hope you found
this problem helpful.
