Institut f r Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
u
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 1
Problem 1 (Quasi-linear Utility Maximisation)
A consumers preferences are given by the quasi-linear utility function U (x1 ,
Problem Set 8 - Solution
Problem 1 (Horizontal Product Dierentiation - Linear City Model)
We assume prices do not dier too much. Find the marginal consumer just indierent between S1 and
S2 (). All consumers between 0 and will buy at S1, while consumers be
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 8
Problem 1 (Horizontal Product Dierentiation - Linear City Model)
Suppose a village in a mountainous area has only one street. E
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 9
Problem 1 (Stackelberg Game)
Firm 1 and rm 2 are engaged in quantity competition. However, rm 1 is a Stackelberg leader, that i
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 10
Problem 1 (Dynamic Price Competition / Tacit Collusion)
Suppose market demand for a homogeneous good in each period is t = 0,
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 11
Problem 1 (Dynamic Competition)
Inverse market demand for a good is given by p(Q) = 50 Q. The marginal costs of production are
Problem Set 11 - Solution
Problem 1 (Dynamic Competition)
a) Innovation 1 (I1): pm = 37.5 > c = 30 non-drastic
Innovation 2 (I2): pm = 27.5 < c = 30 drastic
b) non-drastic: g = c c (if the fee were higher, rms would prefer to use the old technology).
dras
Problem Set 10 - Solution
Problem 1 (Dynamic Price Competition / Tacit Collusion)
col
col
a) col = (1 + + 2 + .) t , whereas t =
m
2
= 50
col =
50
1
Best deviation: pdev = p
m
dev = t + ( + 2 + .) 0 = 100
Collusion sustainable as long as col dev :
1
50
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 12
Problem 1 (Monopoly threatened by Entry)
Due to the sole access to a patent on a superior production method, producer A has a
Problem Set 12 - Solution
Problem 1 (Monopoly threatened by Entry)
a) WTP (A): m C = 5
WTP (B): C 0 = 4
Z sells innovation to A for xed fee of 5.
A has no interest in actually using Zs innovation but wants to stay a monopolist (patent shelving). Bs
WTP i
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 7
Problem 1 (Contracts as Entry Barriers)
Inverse market demand for a good of which rm A1 is (initially) the only producer is giv
Problem Set 7 - Solution
Problem 1 (Contracts as Entry Barriers)
a) Integrated monopoly: maxq m = p(q) q c(q) = (a c1 q)q
FOC: a c1 2q = 0 q m = ac1
2
m = (q m )2 = ( ac1 )2
4
b) Downstream: maxq B = (a q)q rq = (a r q)q
FOC: a r 2q = 0 q(r) = ar ; B (r)
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 6
Problem 1 (Moral Hazard)
Firm A is a monopolist producing a specialised software at zero production cost (the xed R&D costs can
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 3
Problem 1 (Monopoly)
Inverse market demand for a homogeneous good (good 2) is given by pD (Q) = 1 Q (Q: aggregate
quantity of g
Problem Set 1 - Solution
Problem 1 (Quasi-linear Utility Maximisation)
a) Consumers utility maximisation problem:
max U (x1 , x2 ) s.t. p1 x1 + p2 x2 =
x1 ,x2
b) (general) Tangency condition: budget line and indierence curve have the same slope in an opt
Problem Set 2 - Solution
Problem 1 (Robinson Crusoe Economy)
prot maximisation:
max = p z wz
z
z
FOC:
q=
=
p
2 z
p2
4w2
z=
(p, w) =
p2
2w
!
w = 0 z (p, w) =
=
p2
4w2
p
2w
2
p
w 4w2 =
p2
4w
utility maximisation:
max U (x1 , x2 ) = lnx1 + lnx2 s.t. px2 =
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 2
Problem 1 (Robinson Crusoe Economy)
Consider a one-consumer, one-producer economy with 2 commodities. x1 is the consumption of
Problem Set 4 - Solution
Problem 1 (Perfect Competition vs. Cournot Oligopoly)
a) perfect competition:
max (q) = pq C(q) = pq q 2 1
q
!
FOC: = p 2q = 0 p(q) = 2q
q
q S (p) = p
2
aggregate supply: S(p) = J q S (p) = J
S(p ) = D(p ): J
p
2
p
2
= 50 p
p
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 4
Problem 1 (Perfect Competition vs. Cournot Oligopoly)
Let market demand be given by D(p) = 50 p. The market is supplied by J id
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke,
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 5
Problem 1 (Double Marginalisation with 2 Monopolists)
Inverse market demand for a good of which rm A is the only producer is g
Problem Set 3 - Solution
Problem 1 (Monopoly)
a) Prot maximisation:
max m (q) = pD (q) q C(q) = (1 q)q
q
3
q2
= (1 q)q
2
2
!
FOC: 1 3q = 0
1
q m = 3 , pm = 1 q m = 2 , m = (1 1 )
3
2
1
3
=
1
6
[Note: You would arrive at the same result if you were to ma
Problem Set 5 - Solution
Problem 1 (Double Marginalisation with 2 Monopolists)
Industry structure:
a)
Bs maximisation:
max B = (150 q)q rq = (150 r q)q
q
!
FOC: 150 r 2q = 0
150 r
2
150 + r
p(r) = 150 q(r) =
2
150 r 2
B (r) = (q(r)2 = (
)
2
q(r) =
As max
Problem Set 6 - Solution
Problem 1 (Moral Hazard)
Industry structure:
Moral hazard: B chooses unobservable eort level e which aects As utility. If e were specied in a
contract where A would want to make sure B chooses the optimal e, such contract clause w
Institut fr Wettbewerbspolitik - Prof. Dr. Ulrich Kamecke
Dr. Robert Schmidt, Miyu Lee
Competition Policy - SS 13
Problem Set 13 - Recap
Problem 1 (Monopoly)
A monopolists cost function is C(q) = 0.5cq 2 . It is facing a market demand function of D(p) = a