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```					University of Essex
Department of Economics                                           Spring term 2010-11

EC 202 Spring term Problem Set 2

1.   Malcolm Smith is thinking of opening a club at Essex University. There are two
types of students interested in this club: 20 punks and 20 hippies. They have
different demand curves for attending the club: demand of each hippy is
q = 120 − 10 p , demand of each punk is q = 120 − 2 p , where q is the number of
time a person goes to the club per year. The marginal cost of admitting one more
student is zero.
Malcolm cannot price discriminate. Thus, he has two choices. He can set an
admission price that is the same for both types. Alternatively, he can charge each
person an annual fee to join the club, and allow anyone with a club card in free.

(a)   What is the profit-maximising output and price if a uniform price
is charged?
(b)   If the fixed fee is £600, how many punks and hippies will join?
Will Malcolm be better off with this policy?
(c)   Discuss other fixed fees that could be offered and how that
would affect his profits.

(a) If price = p, then total revenue is:

R = 20 [ p q H ] + 20 [ p q P ]

= 20 [ p (120 − 10 p ) + p (120 − 2 p ) ]

[
R = 20 240 p − 12 p 2           ]
Maximise revenue by choosing p:

dR
= 240 p − 24 p = 0
dp
p = 10

R = 20 [2400 − 1200 ]

R = 24000
To find the profit maximising price without using calculus, combine the
two sets of demand functions into an overall demand:

Q = 20(120 − 10 p) + 20(120 − 2 p) = 4800 − 240 p .

Inverse demand: p = 20 − (1 / 240)Q .

Then MR = P + Q (ΔP/ ΔQ) = 20 − (1 / 240)Q − (1 / 240)Q = 20 − (1 / 120)Q .

Setting MR = MC = 0, Q = 2400.

Therefore p = 20 − (1 / 240)2400 = 10.

(b) For F = 600, we have:

1
CS P =    × 60 × 120 = 3600 > 600
2
1
CS H   = × 12 × 120 = 720 > 600
2
Both groups will buy the membership.

Since p = 0, each student will choose q = 120.
Hence total revenue TR = 600 x 40 = 24000.

TR is the same in both cases.

(c) If entry fee = 800, all the punks will buy, but the hippies will not, so:

TR = 20 x 800 = 16000.

If the fee is set so high that only punks buy, the best option would then be to
set membership fee at 3600 to extract all the consumer surplus of the punks.
In this case:

TR = 20 x 3600 = 72000.

What is the best pricing scheme that ensures that both hippies and punks buy?
It is setting F = 720, i.e. equal to the CS of a hippy. Then:

TR= 720 x 40 = 28800.
2.   A firm can price discriminate between two markets. Demand in the first market
is Q1 = 80 – 2P1 and demand in the second market is Q2 = 120 – 3P2. The firm
produces all of its output in a single plant. The cost function of this plant is TC =
10 + Q2, where Q is the firm’s total output. Hence MC = 2Q.

(a) What is the marginal revenue curve of the firm in each of the two
markets?
(b) If the firm chooses Q1 and Q2 to maximise its total profits, how much will
it sell in each of the two markets? Check whether the price is indeed lower
in the market with the more elastic demand.
(c) Suppose that revenues earned in market 1 are free of any tax but revenues
earned in market 2 are taxed at a rate of 25%. What would be the profit-
maximising price in each market now?

a. MR1 = P1 + Q1 (ΔP1/ΔQ1) = 40 – (Q1/2) + Q1(–1/2) = 40 – Q1.
Similarly, MR2 = P2 + Q2 (ΔP2/ΔQ2) = 40 – (Q2/3) + Q2(–1/3) =
40 – 2Q2/3.

b. MR1 = MR2 ↔ 40 – Q1 = 40 – 2Q2/3 ↔ Q1 = 2Q2/3.
Hence Q = Q1 + Q2 = 5Q2/3 and MC = 2Q = 10Q2/3.

We can then write MR2 = MC as 40 – 2Q2/3 = 10Q2/3 → Q2* = 10 and
Q1* = 20/3.

Therefore, P1* = 40 – (10/3) = 110/3 = 36.67 and P2* = 40 – 10/3 =
110/3.

Hence Є1 = P1/Q1 (–2) = –11 and Є2 = P2/Q2 (–3) = –11.
Even though the demand functions have different slopes, the firm
charges the same price in both markets and, at the profit-maximising
prices, the elasticities are the same in the two markets.

c. The tax implies that MR2 = (0.75)(40 – 2Q2/3) = 30 – (Q2/2).

MR1 = MR2 ↔ Q1 = 10 + Q2/2. Hence Q = Q1 + Q2 = 10 + 3Q2/2 and
MC = 2Q = 20 + 3Q2.

MR2 = MC can be written as 30 – Q2/2 = 20 + 3Q2, so that Q2* = 20/7,
which is lower than in part b.

In market 1 we have Q1* = 10 + 10/7 = 80/7, which is higher than in
part b.

The corresponding prices are now P1* = 40 – 40/7 = 240/7 = 34.29 and
P2* = 40 – 20/21 = 820/21 = 39.04.
3.   A property developer is selling new villas in Spain. The cost of building each
villa is 100,000 euros. There are two types of potential buyers. Type 1 buyers
are willing to pay up to 250,000 – 10,000M, where M is the number of months
that they would have to wait before taking possession of the villa. Type 2 buyers
are willing to pay up to 200,000 – 1,000M. Let us assume for simplicity that the
developer could supply all villas immediately if he/she wanted. Assume further
that there are 20 impatient (i.e. type one) buyers and 20 patient (i.e. type two)
different types of buyers apart. Moreover, administrative fees are high enough to

(a) Suppose first that M = 0. What is the price that would maximise the
developer’s profit? How much profit does the developer make?

(b) Suppose now that the developer can choose M as well as the price he/she
asks for the villa. The developer offers two options. In the first option,
the villa costs 190,000 and is available in 6 months. In the second option,
the villa costs 240,000 and is available now. Which option would a type
one buyer choose? Which option would a type 2 buyer choose? How
much profit does the developer make?

(c) Can the developer do even better than in part b.? How? What is the best
possible combination of offers that the developer can select?

a. Only two prices make sense. The first possible solution is to charge
250,000, foregoing the business from type 2 buyers but extracting all the
surplus of type 1 buyers. This yields a profit of (250,000 – 100,000) x 20
= 3,000,000.

The second solution is to sell to both types and set a price equal to
200,000, which is equal to the CS of type 2 buyers (and lower than the CS
of type 1 buyers). The corresponding profit is (200,000 – 100,000) x 40 =
4,000,000. Clearly, this is the preferred option.

b. Let us first consider the choice of type 1 buyers. From option 1 they get a
surplus equal to 250,000 – 190,000 – 60,000 = 0. From option 2, they get
250,000 – 240,000 = 10,000 > 0. Hence type 1 buyers decide to purchase
the houses that are available immediately for a price of 240,000. This
yields a profit of (240,000 – 100,000) x 20 = 2,800,000 for the developer.

Now consider type 2 buyers. From option 1, they get 200,000 – 190,000 –
6,000 = 4,000. Option 2 is just too expensive. Hence type 2 buyers buy
houses for 190,000 and wait six months. This yields a profit of (190,000 –
100,000) x 20 = 1,800,000.
Thanks to this dual offer, the developer now makes a profit of 4,600,000.
By degrading the quality of the good offered to the patient (type 2) buyers,
(i.e., by including a wait of 6 months), the monopolist has managed to
separate the two groups of buyers. The developer makes less profit on
patient buyers than in part a. but it can now charge a much higher price to

c. Define as (P1, M1) the price/waiting time package aimed at impatient (type
1) buyers and (P2, M2) the package aimed at patient (type 2) buyers.

The principles are as follows.
Firstly, it does not make sense to make the impatient buyers wait since it
simply reduces their willingness to pay. (Recall that the developer could
supply all villas immediately if he/she wanted.) Hence M1 = 0.

It also does not make sense to degrade the quality of the good offered to
the patient buyers (i.e. make them wait) more than what is just enough to
patient buyers. This point is important: the reason for setting M2 > 0 here
is to separate the two groups.
Hence we must have 250,000 – P1 = 250,000 – P2 – 10,000M2. The RHS
of this equation is the surplus of the type 1 buyers if they choose the
package (P1, M1). The LHS of this equation is the surplus of the type 1
buyers if they choose the package (P2, M2). The equation is saying that the
type 1 buyers must be just indifferent between the two packages – in
which case, it is assumed, they will choose (P1, M1).

Finally, we want to extract all possible surplus from patient buyers, i.e.
200,000 – P2 – 1,000M2 = 0 and, if possible, extract all possible surplus
from impatient buyers as well, i.e. P1 = 250,000.
Note that this level of P1 ensures that patient buyers will not buy the

Combining these equalities, we have P1* = 250,000, M1* = 0,
P2 + 1,000 M2 = 200,000 and P2 + 10,000 M2 = P1 = 250,000.
Solving the last two equations for the two unknowns, we get M2* = 50/9
and P2* = 200,000 – 50,000/9 = 194,444.

Offering these two options yields profits of (94,444 x 20) + (150,000 x 20)
= 4,888,880

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