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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.