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PROFESSOR: Hi, welcome
back to the 14.01
problem solving videos.
Today we're going to work
on p-set 1 problem
number 3 from Fall 2010.
And we're going to work
through all four
parts of this problem.
But to start off I'm just going
to read through part A.
Consider the market
for apple juice.
In this market the supply curve
is given by quantity
supplied equals 10 pj minus 5
pa, and the demand curve is
given by quantity demanded
equals 100 minus 15 pj plus 10
pt, where j denotes apple juice,
a denotes apples, and t
denotes tea.
Part A asks us to assume
that pa is fixed at $1
and pt equals 5.
We need to calculate the
equilibrium price and quantity
in the apple juice market.
So to start off this problem,
I wrote down both the supply
and the demand functions.
But before we get started with
the algebra, I wanted to come
over to this graph and I
wanted to think about
conceptually what we're
going to be doing.
When we solve for an equilibrium
price and the
equilibrium quantity, all we're
doing is we're finding
the point at which the quantity
supplied and the
quantity demanded is equal.
Looking at the graph, that point
is going to be right
here where the two
curves intersect.
So this will be q star, our
equilibrium quantity, and this
will be p star, our
equilibrium price.
So for part A we're just solving
for the equilibrium
price and quantity.
And they try to trip you up on
this problem by throwing in
the price of apples and
the price of tea.
But since they tell us what
these prices are initially,
we're just going to plug
these into our supply
and our demand functions.
And once we do that, we'll have
isolated the pj variable
and the q variable so we'll
be able to solve
through for this problem.
So starting off with part A.
We're going to go ahead and
we're going to set the quantity
supplied equal to the
quantity demanded.
And so for my supply function,
I've already plugged in pa.
And after plugging in pa I found
that 10 pj minus 5 is
the supply curve.
And the demand curve is equal
to 150 minus 15 pj.
Solving out for pj I find
that the equilibrium
price is equal to 6.2.
And since I know this is an
equilibrium price I'm going to
go ahead and I'm going to
label this with a star.
Solving through for the
equilibrium quantity all we
have to do is we have to take
this equilibrium price we
found and plug it back into
either the supply curve or the
demand curve.
I'm going to go ahead and I'm
going to plug it into the
supply function.
And that lets us solve for the
equilibrium quantity denoted
with the star.
And that in this case is 57.
So looking at our graph, the
equilibrium price and the
equilibrium quantity, we
can now label them.
We can label the price,
6.2, and the
equilibrium quantity 57.
Let's go ahead and move on to
part B. Part B is going to be
the exact same scenario as we
started off with in part A,
only what we're going to do now
is we're going to shift
the supply curve by changing
the price of apples.
Part B states, suppose that a
poor harvest season raises the
price of apples to
pa equals 2.
Find the new equilibrium price
and quantity of apple juice
and draw a graph to illustrate
the answer.
Now what's happening in this
scenario is that the demand
curve is completely
unaffected.
The only thing that's
changing is our
supply curve is shifting.
So when we look at our supply
curve we have to think about
conceptually what do
apples represent.
Well they're an input
for the suppliers.
It's something they have to use
to make the apple juice.
And if the price of apples is
increasing then we intuitively
know that this quantity, or this
q star that they produce
before, it's going to be
more expensive for
them to produce it.
And, in fact, it's going to
be more expensive for the
suppliers to produce
any given quantity.
So this means that the supply
curve for part B is shifting
up and to the left.
I'm going to denote this by
labeling our new supply curve
sb for Part B.
So for Part B all we're going
to do is we're going to plug
in this new pa price.
And then we're going to
do the same thing.
We're going to set the quantity
supplied and the
quantity demanded equal.
When we solve through for a new
supply curve we find that
10pj minus 10 is our
new supply curve.
And we know that are demand
curve is going to be exactly
the same as the scenario that we
started off with in Part A.
Solving through for the new
equilibrium price we find that
pj is going to be
equal to 6.4.
I'm going to label this new
equilibrium price with a B.
And then we can take this
equilibrium price, we can plug
it back into either our
new supply curve
or our demand curve.
And we can find that the
equilibrium quantity is going
to equal 54.
And when we look at our graph we
can think about what these
quantities and these new
prices looks like.
We can see that the quantity for
part B clearly should have
shifted down, and it did
drop from 57 to 54.
And we can see that the price
is higher than what
we started off with.
It rose from 6.2
now up to 6.4.
So we can see that our intuition
that the supply
curve would shift up because
it's more expensive for
suppliers makes sense according
to both our algebra
and to our graph.
Let's go ahead and try part C.
Part C is going to be another
shift, but what's happening now
is this new shift is going
to affect the demand curve
not the supply curve.
Part C states, suppose that pa
equals 1, but the price of tea
drops to pt equals 3.
Find the new equilibrium price
and quantity of apple juice.
So for this scenario now, our
pa is back to 1, like it
started off with in part one.
But now t is dropped
from five to three.
Now before we start, let's think
about what t actually
represents to the consumers
in this market.
If I'm a consumer and I'm
debating what I want to drink
in the morning, one of my other
choices might be tea.
Now the price of tea drops,
then maybe I'll be less
willing to pay as much for the
apple juice because they can
just go out and get
the cheap tea now.
Looking at our graph, if the
consumer is less willing to
pay as much for each quantity
of apple juice as they were
before, this scenario means
that the demand curve has
shifted down.
So, for example, the equilibrium
quantity that we
started off with an point A,
they used to be willing to pay
this much, now they would only
be willing to pay this much
for the same quantity because
tea is cheaper.
Now we're going to be using the
same supply curve that we
started off with before.
So in this scenario we're
looking to set supply and
demand equal and find this
new equilibrium point.
When we go through and we
actually plug in the new pa
and the new pt, we're going to
find that the supply and the
demand functions.
When we set them equal we're
going to set 10 pj minus 5
equal to 130 minus 15 pj.
Again we're going to solve
through for pj.
And we're going to find that our
pjc for part C is going to
be equal to 5.4.
And we're going to find that the
new equilibrium quantity,
again either plugging back
into our supply curve or
demand curve, is going
to be equal to 49.
And when we think back to our
original part A, we found that
the price was 6.2.
Now the price dropped
down to 54 or 5.4.
And what we see here, is that
we did see a price decrease,
or we could have predicted
a price decrease
according to our graph.
And we also find out the
quantity dropped from part A
in 57 now the 49.
And according to our graph, we
would have predicted that
quantity would have
decreased as well.
So the algebra and our
graph both match up.
Let's go ahead and move on to
part D. Now we're going to
look at the effect of a
government intervention on the
market for apple juice.
Part D says, suppose that pa
equals 1 and pt equals 5 and
there's a price ceiling
on apple juice of
pj star equals 5.
What is the excess demand for
apple juice as a result?
Draw a graph to illustrate
your answer.
So now we're back to the
original scenario we started
with in part A. We have the
price of apples is equal to 1.
And we're back to the price
of tea equal to five.
And now the only thing that's
different in this scenario for
part D is that the government
said that these suppliers
can't charge any price
higher than 5.
So let's go ahead and let's
think about what this looks
like conceptually on a graph.
In this scenario we started
off with an
equilibrium price of 6.2.
If we were to have a case
where the government was
saying you could charge only as
much as 7 in the market for
apple juice, the suppliers
wouldn't even be affected.
Because they would say, that
doesn't affect us, we're
charging 6.2.
That's too high.
We're not going to have to
change anything we're doing.
But in this scenario the
government is saying the most
you can charge is 5.
Now at the price of 5, the
quantity supplied, or the
intersection of the price of
5 with the supply curve, is
going to lead to a qs that's
different than the quantity
that's demanded.
And more specifically we're
going to find that too many
people are demanding the product
and there's not enough
being supplied in the market.
This is what we're referring
to when we talk
about excess demand.
We're talking about the space
between the quantity that's
supplied and the quantity
that's demanded.
And that's what we're going
to be solving for.
So by plugging in the price cap
of 5 into both our supply
curve and our demand curve,
we'll be able to find the
difference between the amount
that's supplied and the amount
that's demanded.
So plugging into our supply
curve we can find that the
quantity supplied is going
to be equal to 45.
And plugging into our demand
curve we can find that the
quantity demanded is going
to be equal to 75.
Now one of the biggest problems
that students run
into on this problem is they
think they're done when they
reach this point.
All right, I found the quantity
supplied, I found the
quantity demanded, I know
they're different.
I'm done.
But really what you need to find
is exactly how different
the quantity supplied and the
quantity demanded is.
You need to find out how
many consumers are
left out of the market.
So our excess demand is the
difference between the
quantity supplied and the
quantity demanded.
And that's going to be
30 in this case.
So just to review what we did
in this problem, we started
off by setting a quantity
supplied and a quantity
demanded equal to solve
for a basic
equilibrium price and quantity.
We then looked at shifts in
supply and demand and looked
at how that affected the price
and quantity in a market.
And finally, we ended with part
D with looking at the
effect of a government
intervention.
How does a price cap affect
how many people can get a
product compared to how many
people want a product.
