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Problem Set #2
Micro II – Spring Term 2010
SUGGESTED SOLUTIONS
1. (Third Degree Price Discrimination) Suppose that for some good, the marginal cost of
production is $5. Demand comes from two sources, senior citizens and the rest of the
population. Senior citizen demand is given by the inverse demand function
Pseniors = 15 – Q/500. Demand by the rest of the population is given by the inverse demand
function Prest = 20 – Q/2000.
a. If a monopolist facing these two demands must set a single price for the good, what price
would it optimally set? What would be its profit?
For setting single price, total demand curve has to be derived. Q = Qs + QR
Qs = (15 – Ps )500
Qs = 7500 – 500Ps
QR = 40000 – 2000PR
Since Q = QS + QR and Ps = PR
Q = 7500 – 500P + 40000 – 2000P
Q = 47500 – 2500P
P as a function of Q  P = 47500/2500 – Q/2500 = 19 – Q/2500
MR = 19 – 2Q/2500 = MC = 5
Q = 17500 and putting this in P = 19 – 17500/2500 = $12
Profits = (P – MC) * Q = (12 – 5)*17500 = $1,22,500
b. Suppose this monopolist could set two prices, one for seniors and another for regulars.
What prices maximize profit and what is that level of profit?
The monopolist can set different prices by engaging in third-degree price discrimination.
MRS = 15 – Qs/250 and MRR = 20 – QR/1000 and MC = 5
1. Total output should be divided between groups so that MR for each group is equal
2. Total output is chosen so that MR for each group of consumers is equal to the MC
of production
MR for each group will be equated to MC
15 – Qs /250 = 5 or QS = 2500
20 – QR/1000 = 5 or QR = 15000
PS = 15 – 2500/500 = $20
PR = 20 – 15000/2000 = $12.5
Profits = 20 * 2500 + 12.5 * 15000 – 17500 * 5 = $1,50,000
NOTE: DO CHECK IF THE TWO CONDITIONS ARE BEING FOLLOWED
2. (Third Degree Price Discrimination) Suppose a firm sells to students and others at a single
price of $10 per unit. At this price, it sells 10,000 units in total; 2,000 to students and 8,000
to the others (non-students). At the price of $10, demand by students has elasticity -3, while
demand by the others has elasticity -1.5.
a. Suppose the firm decided to raise the price it charges non-students by $0.10. At the same
time, it will lower the price facing students. If it wants to lower the price charged to students
so that it sells (approximately) the same 10,000 units (so that the decrease in demand by nonstudents is balanced by an increase in demand by students), by how much should it decrease
its price to students? What will be the new price charged to students? Hint: Use the formula for
elasticity of demand.
ED = ∆Q/∆P * P/Q; P1 = Students and P2 = Non-students
3 = ∆Q/∆P * 10/2000  ∆Q/∆P = 600
Q = a – 600P  a = 2000 + 6000 = 8000
Q1 = 8000 – 600P1
1.5 = ∆Q/∆P * 10/8000  ∆Q/∆P = 1200
Q = a – 1200 P  a = 8000 + 12000 = 20000
Q2 = 20000 – 1200P2
If P2 = 10.1  Q2 = 20000 – 1200 * 10.1 = 7880
10000 – 7880 = 2120 = Q1
P1 = 8000/600 – 2120/600 = $9.8
b. What will be the approximate impact on the firm’s profits if it simultaneously raises its
price to non-students to $10.10 and lowers the price charged to students as calculated in part
a? Hint: Use MR = P(1+1/ed)
The output should be divided such that MR1 = MR2
MR1 = 10.1 ( 1 – 1/1.5) = 10.1 * 1/3 = 3.367
MR2 = 9.8 ( 1 – 1/3) = 9.8 * 2/3 = 6.53
Since the current output distribution does not equate MR1 to MR2, it can be concluded that
profits will be lower. Prices are not optimal
c. Note that we say nothing in this problem about MC. Why don’t you need to know this to
answer part b?
 Equating MR1 and MR2 gives the following relationship that must hold for prices
 The higher price will be charged to consumer with the lower demand elasticity
P1 ( 1  1 E 2 )

P2 ( 1  1 E1 )
3. You are selling two goods to a market consisting of three customers with reservation
prices as follows:
Consumer
A
B
C
Good 1
10
40
70
Good 2
70
40
10
a. Graph the reservations prices. Are demands negatively or positively correlated? Are they
perfectly correlated?
b. Suppose the MC1 = MC2 =20. Compute the optimal prices and profits for: i) selling the
goods individually, ii) pure bundling, iii) mixed bundling.
i) 70 and 70  140 – 40 = 100
ii) 80*3 – 20*6 = 120
iii) 69.95*2 + 80 – 20*4 = 139.9, Therefore go for mixed bundling
c. In part b, which was the most profitable strategy? Why?
Since the MC are significant, mixed bundling should be a more profitable technique than
others
d. Suppose MC1 = MC2 = 0. Would you change your pricing strategy (as compared to your
answer in part b)? Why or why not?
i) P1 = P2 = 40
4 * 40 = $160
ii) Same as in a(ii) except that MC =0 so profits are $240
iii) 69.95*2 + 80 = $219.9, Therefore go for Pure Bundling. This was predicted because MC
=0
4. Sal’s satellite company broadcasts TV to subscribers in Los Angeles and New York. The
demand functions for each of these two groups are
QNY = 60 – 0.25PNY
QLA = 100 – 0.50PLA
where Q is in thousands of subscriptions per year and P is the subscription price per year. The
cost of providing Q units of service is given by
C = 1,000 + 40Q
where Q = QNY + QLA.
a.
What are the profit-maximizing prices and quantities for the New York and Los
Angeles markets?
We know that a monopolist with two markets should pick quantities in each
market so that the marginal revenues in both markets are equal to one another
and equal to marginal cost. Marginal cost is $40 (the slope of the total cost
curve). To determine marginal revenues in each market, we first solve for
price as a function of quantity:
PNY = 240 - 4QNY and
PLA = 200 - 2QLA.
Since the marginal revenue curve has twice the slope of the demand curve, the
marginal revenue curves for the respective markets are:
MRNY = 240 - 8QNY and
MRLA = 200 - 4QLA.
Set each marginal revenue equal to marginal cost, and determine the profitmaximizing quantity in each submarket:
40 = 240 - 8QNY, or QNY = 25 and
40 = 200 - 4QLA, or QLA = 40.
Determine the price in each submarket by substituting the profit-maximizing
quantity into the respective demand equation:
PNY = 240 - (4)(25) = $140 and
PLA = 200 - (2)(40) = $120.
1. Total output should be divided between groups so that MR for each group is equal
TRUE: QNY = 25, MRNY = 40 and QLA = 40, MRLA = 40, MRLA = MRNY
2. Total output is chosen so that MR for each group of consumers is equal to the MC
of production: TRUE: MRLA = MCT and MRNY = MCT
b.
As a consequence of a new satellite that the Pentagon recently deployed, people in
Los Angeles receive Sal’s New York broadcasts, and people in New York receive
Sal’s Los Angeles broadcasts. As a result, anyone in New York or Los Angeles can
receive Sal’s broadcasts by subscribing in either city. Thus Sal can charge only a
single price. What price should he charge, and what quantities will he sell in New
York and Los Angeles?
Given this new satellite, Sal can no longer separate the two markets, so he now
needs to consider the total demand function, which is the horizontal
summation of the LA and NY demand functions. Above a price of 200 (the
vertical intercept of the demand function for Los Angeles viewers), the total
demand is just the New York demand function, whereas below a price of 200,
we add the two demands:
QT = 60 – 0.25P + 100 – 0.50P, or QT = 160 – 0.75P.
Rewriting the demand function results in
P
160
1

Q.
0.75 0.75
2
Now total revenue = PQ = (213.3 – 1.3Q)Q, or 213.3Q – 1.3Q , and therefore,
MR = 213.3 – 2.6Q.
Setting marginal revenue equal to marginal cost to determine the profitmaximizing quantity:
213.3 – 2.6Q = 40, or Q = 65.
Substitute the profit-maximizing quantity into the demand equation to
determine price:
65 = 160 – 0.75P, or P = $126.67.
Although a price of $126.67 is charged in both markets, different quantities are
purchased in each market.
QNY  60  0.25126.67  28.3 and
QLA  100  0.50126.67  36.7.
Together, 65 units are purchased at a price of $126.67 each.
c.
In which of the above situations, (a) or (b), is Sal better off? In terms of consumer
surplus, which situation do people in New York prefer and which do people in
Los Angeles prefer? Why?
Sal is better off in the situation with the highest profit. Under the market
condition in 4a, profit is equal to:
 = QNYPNY + QLAPLA - (1,000 + 40(QNY + QLA)), or
 = (25)($140) + (40)($120) - (1,000 + 40(25 + 40)) = $4,700.
Under the market conditions in 4b, profit is equal to:
 = QTP - (1,000 + 40QT), or
 = (126.67)(65) - (1,000 + (40)(65)) = $4633.33.
Therefore, Sal is better off when the two markets are separated.
Consumer surplus is the area under the demand curve above price. Under the
market conditions in 4a, consumer surpluses in New York and Los Angeles are:
CSNY = (0.5)(240 - 140)(25) = $1250 and
CSLA = (0.5)(200 - 120)(40) = $1600.
Under the market conditions in 4b the respective consumer surpluses are:
CSNY = (0.5)(240 – 126.67)(28.3) = $1603.67 and
CSLA = (0.5)(200 – 126.67)(36.7) = $1345.67.
The New Yorkers prefer 4b because the equilibrium price is $126.67 instead of
$140, thus giving them a higher consumer surplus. The customers in Los
Angeles prefer 4a because the equilibrium price is $120 instead of $126.67.
5. A cable TV company offers, in addition to its basic service, two products: a Sports
Channel (Product 1) and a Movie Channel (Product 2). Subscribers to the basic service can
subscribe to these additional services individually at the monthly prices P1 and P2,
respectively, or they can buy the two as a bundle for the price PB, where PB < P1 + P2. (They
can also forego the additional services and simply buy the basic service.) The company’s
marginal cost for these additional services is zero. Through market research, the cable
company has estimated the reservation prices for these two services for a representative
group of consumers in the company’s service area. These reservation prices are plotted (as
x’s) in Figure 11.16, as are the prices P1, P2, and PB that the cable company is currently
charging. The graph is divided into regions, I, II, III, and IV.
Figure 11.16
a.
Which products, if any, will be purchased by the consumers in region I? In
region II? In region III? In region IV? Explain briefly.
Product 1 = sports channel. Product 2 = movie channel.
Region
Purchase
Reservation Prices
I
nothing
r1 < P1, r2 < P2, r1 + r2 < PB
II
sports channel
r1 > P1, r2 < PB - P1
III
movie channel
r2 > P2, r1 < PB - P2
IV
both channels
r1 > PB - P2, r2 > PB - P1, r1 + r2 > PB
To see why consumers in regions II and III do not buy the bundle, reason as follows: For
region II, r1 > P1, so the consumer will buy product 1. If she bought the bundle, she would
pay an additional PB - P1. Since her reservation price for product 2 is less than PB - P1, she
will choose only to buy product 1. Similar reasoning applies to region III.
Consumers in region I purchase nothing because the sum of their reservation values are less
than the bundling price and each reservation value is lower than the respective price.
In region IV the sum of the reservation values for the consumers are higher than the bundle
price, so these consumers would rather purchase the bundle than nothing. To see why the
consumers in this region cannot do better than purchase either of the products separately,
reason as follows: since r1 > PB - P2 the consumer is better off purchasing both products than
just product 2, likewise since r2 > PB - P1, the consumer is better off purchasing both products
rather than just product 1.