Problem set 4:
Monopoly and Duopoly
Lecturer: Johan Stennek
Teaching Assistant: ConnyWollbrant
Class date: Friday 3rd December
1. Microware develops and sells software, among them the word-processing
program Sentence. This program has terrific quality. The basic features are
better designed than in any alternative software on the market, and there are
several more advanced features not found in any other product. The costs of
developing the program have already been taken and the cost of producing a
DVD with a copy of the software is c which is rather small.
There are two types of consumers. Business people have a high valuation of
quality; they are willing to
pay H for the basic features and h for the advanced
features. They receive utility uH  H  h  p if they buy a license of Sentence at
price p . Students havea low valuation of quality; they are willing to pay L  H


for basic features and l  h for the advanced features. They only receive utility

uL  L  l  p if they buy. Any consumer not buying the software
 receives zero
utility (this is the utility received if using the malfunctioning freeware that


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anyone can download from the Internet). There are equally many business
people as students in the population.
1. Show that the monopolist will set p  H  h if c  2L  H  2l  h.
Assume this condition to be fulfilled.
Microware suddenly realizes that
 they can offer
 a student version of Sentence,
with the advanced features removed. This does not affect the cost of producing
the software.
2. Show that Microware can charge at most ps  L for the student version, if
the students are to prefer buying the student version to not buying the
program at all. What is this condition usually called?

3. Show that Microware can charge at most p  L  h for the full version, in
order to discourage the business people from buying the student version.
What is this condition usually called?

4. What trade-off does Microware face, when deciding whether to market a
student version in addition to the full-featured version?
5. Show that Microware will indeed market the student version if
c  2L  H . Assume this condition to be fulfilled.
6. Assume that the cost of a DVD is so small that it can be approximated to

zero, c  0 . Then, the conditions assumed so far can be written 2L  H and
h 2l 2L  H . Try to give these conditions an economic interpretation.

 buy the full7. Finally show that the business people actually prefer to

feature version to not buying at all and that the students prefer the
student version to the full-feature version. What are these conditions
usually called?
8. How about selling the full-feature version to both groups? What is the
trade-off?
2. Consider a homogenous goods market with n firms competing a la Cournot.
Let qi be firm i’s quantity and Q  i 1 qi the aggregate quantity. The price is
n
given by the inverse demand function, which is linear and given by p    Q .
2
Firm i’s marginal cost is constant and denoted ci . Let c  1n  ci be the average
i1
n
cost in the market.
1. Compute the equilibrium price.
2. How does the equilibrium price depend on the distribution of costs?
3. Compute firm i’s quantity.
4. Under what conditions will firm i produce a positive output?
5. Keeping the number of firms in the market fixed, what determines firm i’s
market share si  qi Q ?
6. Are production costs minimized in the market?
3. This exercise investigates whether production costs are minimized in a
Bertrand oligopoly market.
Consider a homogenous goods market with 2 firms competing a la Bertrand. Let
pi be firm i’s price. Assume that firm 1’s cost is cL and that firm 2’s cost is higher
cH  cL . Market demand is given by D  p  . If the firms charge the same price, all
consumers buy from firm 1.
1. Compute the Nash equilibrium.
2. Is there any market power and, if so, what determines market power?
3. Are production costs minimized in the market?
4. Explain the difference to Cournot.
4. Consider a differentiated goods market with n firms competing a la Cournot.
The consumer’s utility is quadratic and given by
n
n
n
i 1
i 1
i 1
U    qi  12  qi2    qi  q j  y
j i
where qi denotes the quantity of firm i’s product that the consumer consumes,
y is the quantity consumed of an outside good (also serving as a numeraire) and
 0,1 is the degree of substitution between the q-goods. Let pi be firm i’s
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price. All firms have the same constant marginal cost c, which is assumed to be
lower than the demand intercept, i.e. c   . Firm i’s inverse demand is then
given by:
pi    qi    q j .
j i
1. Compute the equilibrium prices.
2. What determines the strength of competition? Hint: How does price
depend on the number of firms and the degree of substitutability?
5. Consider a differentiated goods market with n firms competing a la Bertrand.
The consumer’s utility is quadratic and given by
n
n
n
i 1
i 1
i 1
U    qi  12  qi2    qi  q j  y
j i
where qi denotes the quantity of firm i’s product that the consumer consumes,
y is the quantity consumed of an outside good (also serving as a numeraire) and
 0,1 is the degree of substitution between the q-goods. Let pi be firm i’s
price. One may then show that firm i’s demand is linear and given by
 1

qi  A  Bpi  G  n1
pj 



j i
where
1   
0
1  n  2   n  1 2
1  n  2 
B
0
1  n  2   n  1 2
.
n  1

G
0
1  n  2   n  1 2
A
All firms have the same constant marginal cost c, which is assumed to be lower
than the demand intercept, i.e. c   .
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1. Compute the equilibrium prices (considering A, B and G as parameters).
You may assume the equilibrium to be symmetric, i.e. that all firms charge
the same price ( pi  p ) in equilibrium.
2. Compute the equilibrium price as a function of the “deep parameters,” i.e.
,,n .
3. What determines the strength of competition (market power) in the
market? Hint:
a. How does price depend on the number of firms? What happens to
price if the number of firms is one, two or tends to infinity?
b. How does price depend on the degree of substitution?What
happens to price if goods are unrelated   0 of almost perfect
substitutes  1?
6. Compare the Cournot equilibrium price and the Bertrand equilibrium price,
when firms produce differentiated products but have the same marginal cost.
Which is higher?
7. Consider Hotelling’s model of horizontal product differentiation, as discussed
in the lecture. Consumers live on a street of length 1. Two stores sell identical
goods, but at different locations. The consumers are uniformly distributed along
the entire street. All consumers have unit demand and are willing to pay v for
one unit of the good.
There are two differences compared to the model discussed in the lecture.
First, we assume that the firms are not necessarily located at the end points of
the street, but may have their stores somewhere between the end-points. One is
located at l and the other located at h  l .
0
x
l
h
1
Second, we assume that the consumers have quadratic transportation costs.
Thus, a consumer living at x visiting the store located at l would have to pay the
transportation cost t l  x  . The same consumer would have to pay t h  x  if
2
2
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visiting the store at h . Quadratic transportation costs implies that the extra cost
of travelling one extra meter is increasing in the distance.1
The rules of the game is that first the firms simultaneously choose their
locations, second they simultaneously set their prices and finally the consumers
decide whether they want to buy the good and, if so, from which firm. We solve
the model backwards.
1. Find the location of the indifferent consumer, z .
2. Find the two demand functions Dl and Dh.
3. How is demand affected if the two stores are located close to each other?
4. Find the equilibrium prices Pl and Ph
5. Firm l ’s equilibrium profit can be written
l l,h  Pl l,h  cDl Pl l,h, Ph l,h,l,h. Use the envelope theorem to
find an expression for dl dl .
6. Show that firm l will choose l  0 (when assuming that l  h ).
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The reason for using a quadratic cost function is that the linear cost function
gives rise to some technical problems when the firms are not located at the endpoints of the street. If a firm would lower its price to the point where it attracts
all the consumers located between the two firms, it would also attract all the
consumers on the other side of the rival. As a result, the firms demand and profit
functions would be discontinuous and under some conditions an equilibrium in
pure strategies fail to exist.
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