The following content is provided under a Creative
Commons license.
Your support will help MIT OpenCourseWare continue to
offer high-quality educational resources for free.
To make a donation or view additional materials from
hundreds of MIT courses, visit MIT OpenCourseWare at
ocw.mit.edu.
GREG HUTKO: Hi, and welcome back to the 14.01
problem-solving videos.
Today I'm going to do Fall 2010 Problem Set 7,
Problem Number 2.
And we're going to work through both parts A and B to
begin with, so I'm going to read the
beginning part of the problem.
Suppose Napster's considering selling music via email.
There are two types of users, students and non-students.
Each non-student has an inverse demand function of 200
minus x and each student has an inverse demand function of
160 minus x, where x is the number of songs delivered by
email, p is measured in cents.
The marginal cost to Napster of sending an additional song
via email is 0.
Suppose Napster can identify all users as either students
or non-students.
If Napster offers a fixed number of songs per year to
each person, what is the profit maximizing level of
songs offered to a student and a non-student.
In other words, what is the equilibrium level of output
for each type of person under first degree price
discrimination?
Part B is going to ask us, given the equilibrium level of
outputs that we've calculated, what is the dollar price
charged to students and non-students per year?
Now what this problem is really asking us about, it's
asking us about a situation in economics
called a two-part tariff.
Now in some cases, when you have two groups of consumers
that have differing demand functions, a monopolist can
actually capture the vast majority of
the consumer surplus.
Or in most cases, all the consumer surplus if he can
differentiate between the two groups.
So in this case, Napster is probably going to set up a
situation where they can differentiate between students
and non-students.
So if they can discriminate by a student by having them enter
their school ID or their school email address, then
they'd be able to do the situation that we're doing for
parts A and B.
The two parts of a two-part tariff, the first part is an
access fee that's equal to the consumer surplus.
So you can say to a consumer, sure, you can have this set
bundle of songs, this set number of songs.
But you're going to have to pay all of the potential
benefit that it would bring you.
So if it's going to bring you 160 units of benefit, we're
going to take that from you and we're going to give it to
the producers instead.
The second part, in addition to the access fee, the
producer has to decide the price per unit when the
consumer is consuming the bundle.
And they also have to decide how many songs to bundle.
The way the producer makes this decision is by setting
the price per unit equal to the marginal cost. So they're
basically going to say, I'm going to throw in as many
songs into this bundle, get you using as many of the songs
as possible, or listening to the songs as possible.
But I'm going to only thrown in as many songs until it
starts costing me more than it could potentially bring me by
capturing your consumer surplus.
So we're going to set the price per unit equal to the
marginal cost equal to 0 in this problem.
Now looking at our graph, I've tossed up the demand curves
for the student and the non-students.
And what's going to happen in this problem is in the
equilibrium case, our supply curve is actually straight
along the x-axis because our marginal cost is equal to 0.
So if this was a competitive equilibrium, the producer's
surplus would be equal to 0.
But what we're going to do in this case, or what the
producers are going to do in this case, is they're going to
say I can bundle for the students 160 songs, so that
the quantity produced is here.
I'm going to charge you the marginal cost equal to 0 for
each song you listen to.
But then I'm going to come back and I'm going to make you
pay all of your consumer surplus each year to download
those 160 songs.
So again, I went to the marginal cost,
which is right at 0.
Then I set the price where it intersects
with the demand curve.
And then again, I said that we're going to charge an
access fee equal to that area.
The same thing is going to happen to the non-students.
In addition to charging the consumer surplus for A, we're
going to make a larger bundle with 200 songs, so that we can
capture this additional consumer surplus out here.
And I'm going to break this area up into two pieces called
B and C. And you'll see why I'm doing
that in just a second.
So for part A, how big are we going to make the bundles for
the students and the non-students?
The bundle for the students is going to be equal to 160.
And the bundle for the non-students is going to be
equal to 200.
Now for part B, we know that we're going to charge a price
per unit equal to 0.
We just need to calculate what the access fee for each of
these groups is going to be.
And so the access fee for the group of students who have a
song bundle of 160 is just going to be the area of
triangle A. So their fee is going to be consumer surplus
of students, which is equal to A, which is going to be equal
to 1/2 times 160 squared.
So we're going to charge the consumers a fee that's going
to be equal to 12,800.
And a common mistake on this problem would be to just have
this answer and put dollars here.
But if you look back on the problem it said that all
prices are given in cents.
So we know that the fee for the students
is going to be $128.
And now we know that the fee for the non-students is going
to be higher because their consumer surplus is greater
when they have a bundle of 200.
So the fee for the non-students is just going to
be the consumer surplus of the non-students.
And that's going to be the area of A plus B plus C. Which
again, that's going to just be equal to
1/2 times 200 squared.
So in this case, our fee for the non-students is going to
be equal to $200.
So when the monopolist has a way of telling who's a student
and who's a non-student, then they can charge the
non-students a higher price because their demand curve is
further up and they're willing to pay for more
songs within a bundle.
Part B is going to ask us, or part C, moving on to part C.
It's going to ask us, what's the maximum price that
students are willing to pay for this bundle?
So for the 160 song bundle, we're asked, what's the
maximum price students are willing to pay?
Well, if we charge them a price higher than their
consumer surplus of $128, then they're just going to drop out
of the market.
They're going to say, I'm not going to even buy this bundle.
You're taking away all of the benefit and then
an extra one penny.
It's worth it for me to just go someplace else and take my
money elsewhere.
So the answer to part C, the maximum price you can charge a
consumer in a student for the 160 song bundle is just the
$128 that we calculated before.
Part D brings up an interesting implication,
however, for the non-students who can look over and see the
maximum price that we're charging the students for
their bundle.
Part D says, what is the gross consumer surplus that
non-students enjoy if they consume 160 songs per year at
the price from C?
What is the net consumer surplus?
So gross consumer surplus is just going to be the situation
that we had before.
In this case, we're looking at this point here.
We're dealing with the non-students.
They're going to now be consuming the 160 song bundle.
The consumer surplus for this group of students is going to
be both A and B. So working with part C, we know that the
gross consumer surplus for the non-students when they're
consuming 160 songs is going to be the area of A plus B.
And we know that the net consumer surplus, they're
going to be charged a fee equal to A. So the net
consumer surplus is just going to be what we calculate for
the gross consumer surplus minus the fee
the area of A, 128.
So even though the non-students will get no
consumer surplus in net if they go with the 200 song
bundle, maybe if they go with the 160 song bundle, they'll
be left over with some consumer surplus.
Now to calculate the area of A plus B, all we're going to do
is we're going to say that we know the area of A plus B plus
C. Now we're just going to subtract out the area of C.
And if we were to go back to our demand curve for the
non-students, plugging in the quantity of 160 into that
demand curve, we would find that it intersects
this axis at 40.
So the area of A plus B plus C is going to be equal to 200.
And then the area of C is going to be equal to 1/2 times
40 squared.
And so that means that the gross consumer surplus for a
non-student is going to be equal to $192.
So that's the area of A plus B in our diagram.
Now to find the net consumer surplus, we're just going to
subtract out 128.
And we're going to find that the net consumer surplus for a
non-student who's consuming 160 songs is going
to be equal to $64.
So if the monopolist was still to price the same prices that
they had before, if they were to make the 160 song bundle
$128 and the 200 song bundle $200, in that case, all of the
non-students would look at their two choices between the
bundles and they'd say, even though this non-student bundle
is targeted towards me, I'm better off in net consumer
surplus if I go with the bundle that's
targeted toward students.
But now the monopolists aren't going to be
stupid in this situation.
They're going to go back and they're going to think, well,
if you're going to go and buy the non-student bundle, then
what I have to do is I have to make sure--
if you're going to go back and buy the student bundle, I have
to make sure that the non-student bundle gives you a
higher net consumer surplus, so that you're still willing
to buy the non-student bundle with 200 songs.
Part B asks us, what is the maximum price Napster can
charge for 200 songs per year if it offers 160 songs per
year at the highest price that students are willing to pay?
And so simply put, what they can charge for the 200 song
bundle is they can charge for an area of A and C. So we're
going to subtract out.
We have to leave the non-students with at least
enough consumer surplus for the area of B.
We know from our previous problems, we know that the
area of A is just going to be equal to 1/2 times 160 squared
plus 1/2 times 40 squared.
This is the area of A and this is the area of C. So we can
find that the maximum fee is going to be $136.
The other way of calculating this is to just say, well, I
know that the non-students are starting off with 200 hours of
consumer surplus.
I know that I need to leave them with at least $64 of
consumer surplus.
So you can just think intuitively.
You know the maximum price is going to be $136.
And so what we're really going to look at in this situation
where they can't discriminate between the students and the
non-students, this is the price that they're going to be
able to charge for the 200 song bundle.
For the 160 song bundle, they can charge the maximum price
the students are willing to pay.
And if we were in a scenario where there were only two
customers in this market, one non-student and one student,
then we could calculate the overall profits for the
monopolist. And so in this scenario, the overall profits
are going to be $264.
The last part of this problem is going to give you another
scenario that the monopolist might consider.
The last part of these problems is going to ask you
to look at, what happens if instead of designing a student
bundle that has 160 songs, what happens if we design a
student bundle that instead is only going to have 140 songs?
And again, when you're designing this bundle, you're
going to have to plan the prices so that only students
will want the 140 song bundle and only non-students are
going to want the 200 song bundle.
And the process is going to be the same.
So we're going to stop here on this problem.
And just to summarize what we looked at here, we looked at
the consequences and the implications of having a
price-discriminating monopolist. When you can
actually discriminate, you can capture all the consumer
surplus and you can charge different prices to different
groups in the market.
But when you can't discriminate, what you have to
do is you have to make packages, bundle them and
price them, so that the consumers will discriminate
themselves by picking the bundle that gives them the
highest net consumer surplus.
I hope you found this part of the problem helpful.
And go ahead and continue through with part I and check
your answers against the solution.