Download Potential midterm exam questions, David Figlio

Practice problem set 2, Professor David Figlio
Fall 2004
(1) This morning, I found a book about abnormal psychology being sold on Amazon.com
for $96.25, and at the same time sold on the Amazon.uk site for the equivalent of $45.
(This is not an international student edition or anything like that. It is the same book,
published by the same publisher, in the same format.) What do you think is going on
here?
(2) Suppose that for some item, the marginal cost of production is $5. Demand comes
from two sources, students and the rest of the population. Student demand is given by the
demand curve Q=7500-500P. Demand by the rest of the population is given by
Q=40000-2000P. Assume that fixed costs are zero.
(a) Suppose that the monopolist can set two prices, one for students and another for the
rest of the population. What prices maximize profit and what is the level of profit?
(b) Describe the conditions under which your answer to (a) could occur.
(c) Suppose that the monopolist can’t charge different prices. What single price would
the firm charge then? How much lower are the profits in this case than in the case of
question (a)?
(3) Suppose that you find that people who live in the east and people who live in the west
have different demands for your product. There are an equal number of people living in
the east as people living in the west. You estimate that EACH person living in the east
has demand for your product q=80-2P and EACH person living in the west has demand
for your product q=160-2P. Marginal cost is constant at 20 per unit.
(a) Suppose that you can BLOCK PRICE and sell DIFFERENT BLOCKS AT
DIFFERENT PRICES to the east and the west. What is the BLOCK SIZE and BLOCK
PRICE that you will charge customers in the EAST?
(b) Suppose that you must sell the SAME BLOCK AT THE SAME PRICE in both the
east and the west. What is the BLOCK SIZE and BLOCK PRICE that you will charge
customers? [Hint: This is difficult. Think about all of the possible block prices that you
could charge, and which types of people would buy at each price. Which earns you the
highest profits?]
Sample solutions
(1) Amazon is engaging in price discrimination. It is taking advantage of its presumption
that students in the United States and Britain have different demands for this textbook—
or textbooks in general. (Why exactly these differences exist is not material for this
class.) Amazon can price discriminate when it can (1) identify multiple groups of
consumers with different willingness to pay for the product; (2) identify their different
demands for the product; and (3) prevent transfer from one group to another. Shipping
costs between Britain and the United States are sufficiently high to effectively prevent
resale. (Note that this is one major reason that Amazon.uk does not ship to U.S.
addresses!!)
(2) This question is a standard price discrimination question. Note that because the
marginal cost of production is constant ($5), you can solve for the profit-maximizing
prices and quantities in the two markets separately. If marginal costs were NOT constant,
you would have to solve for the profit-maximizing prices and quantities as a system of
equations. I’ll do an example of that in a moment. But first, back to the problem at hand:
(a) Invert the student demand curve to get Ps=15 – (1/500)Qs so MRs=15 – (1/250)Qs.
Set equal to marginal cost of 5 to find Qs=2500. Plug into the student demand curve to
find Ps=10. Invert the other demand curve to get Pr=20 – (1/2000)Qr. So MRr=20 –
(1/1000)Qr. Set equal to marginal cost of 5 to get Qr=15000. Plug into the other demand
curve to find Pr=12.5. Profit from students is (10-5) x (2500) = 12500. Profit from the
rest of the market is (12.5 – 5) x (15000) = 112500. So total profit = 125000.
(b) You can price discriminate so long as you can distinguish students from non-students
and prevent transfer from one group to the other.
(c) In this case, you have to think about the overall market as a single demand curve. So
aggregate the demand curves: Q=7500-500P+40000-2000P=47500-2500P. Invert the
demand curve to get P=19-(1/2500)Q. [Note that this is only the equation for the portion
of the demand curve where BOTH groups are consuming. You will want to check
this…look for my bracketed note later in this paragraph.] So MR [for this portion of the
demand curve!] = 19 – (1/1250)Q. Set equal to MC=5, and solve for Q=17500. The
corresponding price (from the demand curve) is 12. [OK…now you can look to see if
this is the right demand curve to look at. Will both groups buy at least something at a
price of 12? Yes—so this is the right demand curve to look at. Otherwise, you’d only
look at the high valuation group!!]
Profits here are (12 – 5) x (17500) = 122500. So profits are 2500 lower here than in the
case of (a).
OK. NOW suppose that costs were increasing, rather than constant. The only difference
here is that you have to solve the two MR=MC equations simultaneously. So suppose
that MC=2+(1/500)Q. Rewrite MC=2+(1/500)Qs+(1/500)Qr. Then write MRs=MC and
MRr=MC. The two equations would be:
For students: 15 – (1/250)Qs = 2+(1/500)Qs+(1/500)Qr
For the rest: 20 – (1/1000)Qr = 2+(1/500)Qs+(1/500)Qr
Solve the system of two equations and two unknowns to find Qs=214.2857 and
Qr=5857.143. The price you charge students would be 14.57, and the price charged to
the rest is 17.07.
(3) (a) To find the block size, find the place where the demand curve intersects marginal
costs: so rewrite East demand curve as p=40 – (1/2)q, and set equal to MC=20 to solve
for q=40. The block price is the area beneath the demand curve up to that quantity,
which you calculate to be 1200.
(b) You might be tempted here to combine the two demand curves, because this “looks”
like a price discrimination question. Resist this temptation, because it’s not!
First, look at the block price that you would charge to westerners. Follow the same
procedure as in (a) to find that you’d charge 6000 for a block size of 120 to westerners.
So what do you do? If you sell only to westerners, you’ll want to charge a block price of
6000 for a block size of 120. Your profit per westerner is 3600.
To sell to both westerners and easterners, you have to charge the block price that
easterners would be willing to pay: a price of 1200 for a block size of 40. You earn
profits of 400 per easterner with this block pricing scheme. Now, westerners would be
willing to buy three blocks apiece at these prices [see below for details], so you’ll sell one
block to every easterner and three to every westerner. If there is one easterner and one
westerner, you’ll earn profits of 400+3x400=1600. This is worse than the 3600 profit if
you only sold to the one westerner!
[I am not going to ask you to do this on the exam…I’ll only go as far as asking for an
optimal block price. But I wanted you to think a little about this type of comparison…]
[If you’re still reading: Now…how did I know that westerners would buy three blocks?
Well, it’s clear that they’ll be willing to buy the first block. So, after that, their demand is
p=60-(1/2)q…so they’ll be willing to buy a second block, and you’ll see the same thing
for the third block. But after the third block, the demand is p=20-(1/2)q…so you
wouldn’t go past three.]