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GREG HUTKO: Hi, and welcome
back to the 14.01

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problem-solving videos.

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Today I'm going to do Fall
2010 Problem Set 7,

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Problem Number 2.

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00:00:32,925 --> 00:00:35,600
And we're going to work through
both parts A and B to

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begin with, so I'm
going to read the

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00:00:37,120 --> 00:00:39,190
beginning part of the problem.

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00:00:39,190 --> 00:00:42,630
Suppose Napster's considering
selling music via email.

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There are two types of users,
students and non-students.

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Each non-student has an inverse
demand function of 200

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minus x and each student has an
inverse demand function of

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00:00:53,840 --> 00:00:57,580
160 minus x, where x is the
number of songs delivered by

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email, p is measured in cents.

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The marginal cost to Napster of
sending an additional song

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00:01:06,360 --> 00:01:08,690
via email is 0.

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Suppose Napster can identify all
users as either students

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or non-students.

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If Napster offers a fixed number
of songs per year to

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each person, what is the profit
maximizing level of

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00:01:19,210 --> 00:01:22,500
songs offered to a student
and a non-student.

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In other words, what is the
equilibrium level of output

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for each type of person under
first degree price

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discrimination?

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Part B is going to ask us, given
the equilibrium level of

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outputs that we've calculated,
what is the dollar price

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00:01:37,220 --> 00:01:42,250
charged to students and
non-students per year?

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00:01:42,250 --> 00:01:44,630
Now what this problem is really
asking us about, it's

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asking us about a situation
in economics

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called a two-part tariff.

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00:01:49,260 --> 00:01:52,220
Now in some cases, when you have
two groups of consumers

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00:01:52,220 --> 00:01:55,350
that have differing demand
functions, a monopolist can

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00:01:55,350 --> 00:01:58,790
actually capture the
vast majority of

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00:01:58,790 --> 00:01:59,730
the consumer surplus.

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00:01:59,730 --> 00:02:02,660
Or in most cases, all the
consumer surplus if he can

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00:02:02,660 --> 00:02:05,580
differentiate between
the two groups.

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00:02:05,580 --> 00:02:08,860
So in this case, Napster is
probably going to set up a

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situation where they can
differentiate between students

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00:02:11,560 --> 00:02:12,710
and non-students.

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00:02:12,710 --> 00:02:15,550
So if they can discriminate by a
student by having them enter

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00:02:15,550 --> 00:02:18,700
their school ID or their school
email address, then

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they'd be able to do the
situation that we're doing for

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00:02:21,050 --> 00:02:22,860
parts A and B.

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00:02:22,860 --> 00:02:26,140
The two parts of a two-part
tariff, the first part is an

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access fee that's equal to
the consumer surplus.

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00:02:29,580 --> 00:02:32,960
So you can say to a consumer,
sure, you can have this set

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bundle of songs, this
set number of songs.

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But you're going to have to
pay all of the potential

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00:02:39,620 --> 00:02:41,590
benefit that it would
bring you.

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00:02:41,590 --> 00:02:45,190
So if it's going to bring you
160 units of benefit, we're

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00:02:45,190 --> 00:02:47,240
going to take that from you and
we're going to give it to

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the producers instead.

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The second part, in addition
to the access fee, the

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producer has to decide the
price per unit when the

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consumer is consuming
the bundle.

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And they also have to decide
how many songs to bundle.

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The way the producer makes this
decision is by setting

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the price per unit equal to the
marginal cost. So they're

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basically going to say, I'm
going to throw in as many

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songs into this bundle, get you
using as many of the songs

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as possible, or listening to
the songs as possible.

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But I'm going to only thrown
in as many songs until it

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starts costing me more than it
could potentially bring me by

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capturing your consumer
surplus.

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00:03:28,960 --> 00:03:31,220
So we're going to set the price
per unit equal to the

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00:03:31,220 --> 00:03:35,260
marginal cost equal to
0 in this problem.

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Now looking at our graph, I've
tossed up the demand curves

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00:03:38,780 --> 00:03:41,230
for the student and
the non-students.

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And what's going to happen
in this problem is in the

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equilibrium case, our supply
curve is actually straight

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along the x-axis because our
marginal cost is equal to 0.

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So if this was a competitive
equilibrium, the producer's

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surplus would be equal to 0.

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But what we're going to do
in this case, or what the

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producers are going to do in
this case, is they're going to

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say I can bundle for the
students 160 songs, so that

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the quantity produced is here.

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00:04:08,760 --> 00:04:12,080
I'm going to charge you the
marginal cost equal to 0 for

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each song you listen to.

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But then I'm going to come back
and I'm going to make you

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pay all of your consumer surplus
each year to download

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those 160 songs.

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So again, I went to
the marginal cost,

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which is right at 0.

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Then I set the price
where it intersects

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with the demand curve.

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00:04:38,850 --> 00:04:40,560
And then again, I said that
we're going to charge an

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access fee equal to that area.

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The same thing is going to
happen to the non-students.

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In addition to charging the
consumer surplus for A, we're

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00:04:50,640 --> 00:04:53,840
going to make a larger bundle
with 200 songs, so that we can

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capture this additional consumer
surplus out here.

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00:04:57,020 --> 00:05:01,323
And I'm going to break this area
up into two pieces called

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00:05:01,323 --> 00:05:03,990
B and C. And you'll
see why I'm doing

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00:05:03,990 --> 00:05:06,550
that in just a second.

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00:05:06,550 --> 00:05:09,460
So for part A, how big are we
going to make the bundles for

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00:05:09,460 --> 00:05:11,550
the students and the
non-students?

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00:05:11,550 --> 00:05:18,910
The bundle for the students is
going to be equal to 160.

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00:05:18,910 --> 00:05:24,130
And the bundle for the
non-students is going to be

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equal to 200.

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00:05:27,540 --> 00:05:31,300
Now for part B, we know that
we're going to charge a price

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00:05:31,300 --> 00:05:33,310
per unit equal to 0.

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We just need to calculate what
the access fee for each of

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00:05:35,690 --> 00:05:37,520
these groups is going to be.

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00:05:37,520 --> 00:05:42,670
And so the access fee for the
group of students who have a

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00:05:42,670 --> 00:05:46,950
song bundle of 160 is just
going to be the area of

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triangle A. So their fee is
going to be consumer surplus

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00:05:53,810 --> 00:05:58,480
of students, which is equal to
A, which is going to be equal

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00:05:58,480 --> 00:06:06,850
to 1/2 times 160 squared.

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00:06:06,850 --> 00:06:10,320
So we're going to charge the
consumers a fee that's going

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00:06:10,320 --> 00:06:14,740
to be equal to 12,800.

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00:06:14,740 --> 00:06:18,400
And a common mistake on this
problem would be to just have

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00:06:18,400 --> 00:06:20,840
this answer and put
dollars here.

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00:06:20,840 --> 00:06:22,750
But if you look back on the
problem it said that all

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prices are given in cents.

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00:06:24,840 --> 00:06:33,380
So we know that the fee
for the students

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is going to be $128.

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00:06:37,170 --> 00:06:39,430
And now we know that the fee for
the non-students is going

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to be higher because their
consumer surplus is greater

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00:06:41,950 --> 00:06:45,040
when they have a
bundle of 200.

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00:06:45,040 --> 00:06:46,890
So the fee for the non-students
is just going to

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00:06:46,890 --> 00:06:50,030
be the consumer surplus
of the non-students.

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00:06:50,030 --> 00:06:58,520
And that's going to be the area
of A plus B plus C. Which

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00:06:58,520 --> 00:07:02,530
again, that's going to
just be equal to

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00:07:02,530 --> 00:07:06,690
1/2 times 200 squared.

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00:07:06,690 --> 00:07:11,600
So in this case, our fee for the
non-students is going to

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be equal to $200.

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So when the monopolist has a way
of telling who's a student

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and who's a non-student, then
they can charge the

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00:07:22,100 --> 00:07:25,230
non-students a higher price
because their demand curve is

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00:07:25,230 --> 00:07:27,760
further up and they're willing
to pay for more

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00:07:27,760 --> 00:07:30,330
songs within a bundle.

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00:07:30,330 --> 00:07:35,480
Part B is going to ask us, or
part C, moving on to part C.

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00:07:35,480 --> 00:07:39,340
It's going to ask us, what's
the maximum price that

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00:07:39,340 --> 00:07:43,560
students are willing to
pay for this bundle?

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00:07:43,560 --> 00:07:46,980
So for the 160 song bundle,
we're asked, what's the

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00:07:46,980 --> 00:07:49,410
maximum price students
are willing to pay?

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00:07:49,410 --> 00:07:51,850
Well, if we charge them a
price higher than their

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00:07:51,850 --> 00:07:55,730
consumer surplus of $128, then
they're just going to drop out

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00:07:55,730 --> 00:07:56,240
of the market.

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00:07:56,240 --> 00:07:58,830
They're going to say, I'm not
going to even buy this bundle.

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00:07:58,830 --> 00:08:01,580
You're taking away all of
the benefit and then

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00:08:01,580 --> 00:08:03,900
an extra one penny.

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00:08:03,900 --> 00:08:06,140
It's worth it for me to just go
someplace else and take my

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00:08:06,140 --> 00:08:07,220
money elsewhere.

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00:08:07,220 --> 00:08:10,600
So the answer to part C, the
maximum price you can charge a

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00:08:10,600 --> 00:08:15,530
consumer in a student for the
160 song bundle is just the

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00:08:15,530 --> 00:08:21,310
$128 that we calculated
before.

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00:08:21,310 --> 00:08:23,810
Part D brings up an interesting
implication,

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00:08:23,810 --> 00:08:27,930
however, for the non-students
who can look over and see the

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00:08:27,930 --> 00:08:30,160
maximum price that we're
charging the students for

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00:08:30,160 --> 00:08:31,900
their bundle.

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00:08:31,900 --> 00:08:35,750
Part D says, what is the gross
consumer surplus that

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non-students enjoy if they
consume 160 songs per year at

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00:08:39,830 --> 00:08:41,370
the price from C?

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00:08:41,370 --> 00:08:42,859
What is the net consumer
surplus?

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00:08:45,550 --> 00:08:50,330
So gross consumer surplus is
just going to be the situation

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00:08:50,330 --> 00:08:51,790
that we had before.

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00:08:51,790 --> 00:08:55,540
In this case, we're looking
at this point here.

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00:08:55,540 --> 00:08:57,770
We're dealing with
the non-students.

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00:08:57,770 --> 00:09:01,420
They're going to now be
consuming the 160 song bundle.

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00:09:01,420 --> 00:09:04,970
The consumer surplus for this
group of students is going to

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00:09:04,970 --> 00:09:13,530
be both A and B. So working with
part C, we know that the

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00:09:13,530 --> 00:09:19,780
gross consumer surplus for the
non-students when they're

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00:09:19,780 --> 00:09:27,250
consuming 160 songs is going
to be the area of A plus B.

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00:09:27,250 --> 00:09:30,740
And we know that the net
consumer surplus, they're

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00:09:30,740 --> 00:09:34,670
going to be charged a fee
equal to A. So the net

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00:09:34,670 --> 00:09:44,160
consumer surplus is just going
to be what we calculate for

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00:09:44,160 --> 00:09:48,940
the gross consumer surplus
minus the fee

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00:09:48,940 --> 00:09:52,670
the area of A, 128.

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00:09:52,670 --> 00:09:55,450
So even though the non-students
will get no

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00:09:55,450 --> 00:09:58,820
consumer surplus in net if
they go with the 200 song

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00:09:58,820 --> 00:10:02,970
bundle, maybe if they go with
the 160 song bundle, they'll

180
00:10:02,970 --> 00:10:06,910
be left over with some
consumer surplus.

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00:10:06,910 --> 00:10:10,290
Now to calculate the area of A
plus B, all we're going to do

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00:10:10,290 --> 00:10:15,300
is we're going to say that we
know the area of A plus B plus

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00:10:15,300 --> 00:10:19,320
C. Now we're just going to
subtract out the area of C.

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00:10:19,320 --> 00:10:21,580
And if we were to go back to
our demand curve for the

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00:10:21,580 --> 00:10:25,700
non-students, plugging in the
quantity of 160 into that

186
00:10:25,700 --> 00:10:29,120
demand curve, we would find
that it intersects

187
00:10:29,120 --> 00:10:33,490
this axis at 40.

188
00:10:33,490 --> 00:10:39,660
So the area of A plus B plus C
is going to be equal to 200.

189
00:10:39,660 --> 00:10:46,320
And then the area of C is going
to be equal to 1/2 times

190
00:10:46,320 --> 00:10:49,630
40 squared.

191
00:10:49,630 --> 00:10:53,670
And so that means that the gross
consumer surplus for a

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00:10:53,670 --> 00:11:02,080
non-student is going to
be equal to $192.

193
00:11:02,080 --> 00:11:05,240
So that's the area of A
plus B in our diagram.

194
00:11:05,240 --> 00:11:07,670
Now to find the net consumer
surplus, we're just going to

195
00:11:07,670 --> 00:11:14,900
subtract out 128.

196
00:11:14,900 --> 00:11:19,860
And we're going to find that the
net consumer surplus for a

197
00:11:19,860 --> 00:11:23,290
non-student who's consuming
160 songs is going

198
00:11:23,290 --> 00:11:35,780
to be equal to $64.

199
00:11:35,780 --> 00:11:41,030
So if the monopolist was still
to price the same prices that

200
00:11:41,030 --> 00:11:43,700
they had before, if they were
to make the 160 song bundle

201
00:11:43,700 --> 00:11:50,780
$128 and the 200 song bundle
$200, in that case, all of the

202
00:11:50,780 --> 00:11:54,010
non-students would look at their
two choices between the

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00:11:54,010 --> 00:11:57,120
bundles and they'd say, even
though this non-student bundle

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00:11:57,120 --> 00:12:00,610
is targeted towards me, I'm
better off in net consumer

205
00:12:00,610 --> 00:12:02,460
surplus if I go with
the bundle that's

206
00:12:02,460 --> 00:12:04,780
targeted toward students.

207
00:12:04,780 --> 00:12:06,660
But now the monopolists
aren't going to be

208
00:12:06,660 --> 00:12:07,800
stupid in this situation.

209
00:12:07,800 --> 00:12:09,890
They're going to go back and
they're going to think, well,

210
00:12:09,890 --> 00:12:12,260
if you're going to go and buy
the non-student bundle, then

211
00:12:12,260 --> 00:12:14,970
what I have to do is I
have to make sure--

212
00:12:14,970 --> 00:12:17,200
if you're going to go back and
buy the student bundle, I have

213
00:12:17,200 --> 00:12:19,780
to make sure that the
non-student bundle gives you a

214
00:12:19,780 --> 00:12:23,210
higher net consumer surplus, so
that you're still willing

215
00:12:23,210 --> 00:12:27,420
to buy the non-student bundle
with 200 songs.

216
00:12:27,420 --> 00:12:31,160
Part B asks us, what is the
maximum price Napster can

217
00:12:31,160 --> 00:12:36,530
charge for 200 songs per year
if it offers 160 songs per

218
00:12:36,530 --> 00:12:42,470
year at the highest price that
students are willing to pay?

219
00:12:42,470 --> 00:12:47,130
And so simply put, what they
can charge for the 200 song

220
00:12:47,130 --> 00:12:54,490
bundle is they can charge for
an area of A and C. So we're

221
00:12:54,490 --> 00:12:56,490
going to subtract out.

222
00:12:56,490 --> 00:12:58,970
We have to leave the
non-students with at least

223
00:12:58,970 --> 00:13:06,340
enough consumer surplus
for the area of B.

224
00:13:06,340 --> 00:13:09,640
We know from our previous
problems, we know that the

225
00:13:09,640 --> 00:13:28,450
area of A is just going to be
equal to 1/2 times 160 squared

226
00:13:28,450 --> 00:13:31,720
plus 1/2 times 40 squared.

227
00:13:31,720 --> 00:13:35,620
This is the area of A and this
is the area of C. So we can

228
00:13:35,620 --> 00:13:42,360
find that the maximum fee
is going to be $136.

229
00:13:42,360 --> 00:13:45,090
The other way of calculating
this is to just say, well, I

230
00:13:45,090 --> 00:13:50,520
know that the non-students are
starting off with 200 hours of

231
00:13:50,520 --> 00:13:51,940
consumer surplus.

232
00:13:51,940 --> 00:13:56,040
I know that I need to leave
them with at least $64 of

233
00:13:56,040 --> 00:13:57,460
consumer surplus.

234
00:13:57,460 --> 00:13:59,700
So you can just think
intuitively.

235
00:13:59,700 --> 00:14:21,550
You know the maximum price
is going to be $136.

236
00:14:21,550 --> 00:14:24,560
And so what we're really going
to look at in this situation

237
00:14:24,560 --> 00:14:26,560
where they can't discriminate
between the students and the

238
00:14:26,560 --> 00:14:29,560
non-students, this is the price
that they're going to be

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able to charge for the
200 song bundle.

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00:14:32,690 --> 00:14:36,480
For the 160 song bundle, they
can charge the maximum price

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00:14:36,480 --> 00:14:37,810
the students are
willing to pay.

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00:14:51,550 --> 00:14:55,170
And if we were in a scenario
where there were only two

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00:14:55,170 --> 00:14:59,630
customers in this market, one
non-student and one student,

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00:14:59,630 --> 00:15:02,710
then we could calculate the
overall profits for the

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00:15:02,710 --> 00:15:08,130
monopolist. And so in this
scenario, the overall profits

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00:15:08,130 --> 00:15:13,250
are going to be $264.

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00:15:13,250 --> 00:15:15,700
The last part of this problem
is going to give you another

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00:15:15,700 --> 00:15:18,510
scenario that the monopolist
might consider.

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00:15:18,510 --> 00:15:20,620
The last part of these problems
is going to ask you

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00:15:20,620 --> 00:15:23,550
to look at, what happens if
instead of designing a student

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00:15:23,550 --> 00:15:28,540
bundle that has 160 songs, what
happens if we design a

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00:15:28,540 --> 00:15:36,610
student bundle that instead is
only going to have 140 songs?

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00:15:36,610 --> 00:15:38,790
And again, when you're designing
this bundle, you're

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00:15:38,790 --> 00:15:41,820
going to have to plan the prices
so that only students

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00:15:41,820 --> 00:15:45,600
will want the 140 song bundle
and only non-students are

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00:15:45,600 --> 00:15:47,690
going to want the
200 song bundle.

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00:15:47,690 --> 00:15:50,430
And the process is going
to be the same.

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00:15:50,430 --> 00:15:52,430
So we're going to stop
here on this problem.

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00:15:52,430 --> 00:15:55,650
And just to summarize what we
looked at here, we looked at

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the consequences and the
implications of having a

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00:15:58,340 --> 00:16:00,660
price-discriminating monopolist.
When you can

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00:16:00,660 --> 00:16:03,180
actually discriminate, you can
capture all the consumer

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00:16:03,180 --> 00:16:05,900
surplus and you can charge
different prices to different

264
00:16:05,900 --> 00:16:07,490
groups in the market.

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00:16:07,490 --> 00:16:09,260
But when you can't discriminate,
what you have to

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00:16:09,260 --> 00:16:11,520
do is you have to make packages,
bundle them and

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00:16:11,520 --> 00:16:14,320
price them, so that the
consumers will discriminate

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00:16:14,320 --> 00:16:17,990
themselves by picking the bundle
that gives them the

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00:16:17,990 --> 00:16:20,220
highest net consumer surplus.

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00:16:20,220 --> 00:16:21,980
I hope you found this part
of the problem helpful.

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00:16:21,980 --> 00:16:24,140
And go ahead and continue
through with part I and check

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00:16:24,140 --> 00:16:25,390
your answers against
the solution.