1.考虑一个古尔诺寡头垄断，其中三个rms i = 1； 2; 3.所有Örms

P3
i = 1 qi是总输出。
（a）解决游戏的纳什均衡，包括Örmout-

NE是对称的。 [20]
（b）现在假设这三个Örm玩一个2阶段的游戏。在舞台上

（a）部分。在第二阶段，他们同时决定他们的

Örmi的剩余需求为
Di（pi; p i）=
8
<

9 pi Pj = i

9点
3

（请注意，此处我们假设有效/并行配给是

=
9 q1 q2 q3。 [20]
2.考虑一次无限重复的Bertrand寡头博弈，其中包括

（a）每个时期的市场需求为D（p）= 1 p。 （计算[15]

（b）在每个时期结束时，市场不复存在，概率为[15]

3.考虑一个双头垄断市场，其中两个公司出售差异化产品，
ucts，这是不完美的替代品。市场可以建模

2个
+ p2 p1
2吨

2个
+ p1 p2
2吨
，其中S> 0，参数t> 0

（a）推导该游戏的纳什均衡，包括价格，

（b）从需求函数中，qi = Di（pi; pj）= S
2个
+ pj pi
2吨
，得出

Pi（qi; pj））。证明对于t> 0，Pi（qi; pj）是向下倾斜的，
IE。，
@Pi（qi; pj）
@qi
<0。以给定的pj 0来争论，Örmi

（c）计算均衡价格和利润的极限为t！

1. Consider a Cournot oligopoly with three Örms i = 1; 2; 3. All Örms
have the same constant marginal cost c = 1. The inverse demand
function of the market is given by P = 9 Q, where P is the market
price, and Q =
P3
i=1 qi is the aggregate output.
(a) Solve for the Nash equilibrium of the game including Örm out-
puts, market price, aggregate output, and Örm proÖts (Hint: the
NE is symmetric). [20]
(b) Now suppose these three Örms play a 2-stage game. In stage
1, they produce capacities  q1,  q2 and  q3, which are equal to the
Nash equilibrium quantities of the Cournot game characterised
by part (a). In stage 2, they simultaneously decide on their
prices p1, p2 and p3. The marginal cost for each Örm to sell up
to capacity is 0. It is impossible to sell more than capacity. The
residual demand for Örm i is
Di (pi; p i) =
8
<
:
9 pi Pj=i
 qj if pi > pj for all j = i
9 pi
3
if pi = pj for all j = i
9 pi if pi < pj for all j = i
:
(Note, here we assume that the e¢ cient/parallel rationing ap-
plies). Prove that it is a Nash equilibrium of the second stage
subgame that each Örm charges the market clearing price p
=
9  q1  q2  q3. [20]
2. Consider an inÖnitely repeated Bertrand oligopoly game with dis-
count factor  < 1. The unit cost of production is a constantc = 0:2 and the same for all n > 2 Örms. There are no Öxed costs.
Describe a formof ìtriggerîstrategies that can facilitate tacit collu-
sion in pricing. Determine the condition under which such strategies
can sustain the monopoly price in each of the following cases:
(a) The market demand in each period is D(p) = 1 p. (Calculate [15]

(b) At the end of each period, the market ceases to exist with prob- [15]
ability .
3. Consider a duopoly market, where two Örms sell di§erentiated prod-
ucts, which are imperfect substitutes. The market can be modelled
as a static price competition game, similar to a linear city model.
The two Örms choose prices p1 and p2 simultaneously. The derived
demand functions for the two Örms are: D1 (p1; p2) = S
2
+ p2 p1
2t
and D2 (p1; p2) = S
2
+ p1 p2
2t
, where S > 0 and the parameter t > 0
measures the degree of product di§erentiation. Both Örms have
constant marginal cost c > 0 for production.
(a) Derive the Nash equilibrium of this game, including the prices,
outputs and proÖts of the two Örms. [10]
(b) From the demand functions, qi = Di (pi; pj) = S
2
+ pj pi
2t
, derive
the residual inverse demand functions: pi = Pi(qi; pj) (work out
Pi(qi; pj)). Show that for t > 0, Pi(qi; pj) is downward-sloping,
i.e.,
@Pi(qi;pj)
@qi
< 0. Argue that, taking pj  0 as given, Örm i
is like a monopolist facing a residual inverse demand, and the
optimal qi (which equates marginal revenue and marginal cost)
or pi makes Pi(qi; pj) = pi > c, i.e., Örm i has market power. [10]
(c) Calculate the limits of the equilibrium prices and proÖts as t !