本次英国代写主要为工业经济学的long take-home assignment
1.考虑一个古尔诺寡头垄断，其中三个rms i = 1； 2; 3.所有Örms
具有相同的恒定边际成本c =1。逆需求
市场的功能由P = 9 Q给出，其中P是市场
价格，而Q =
P3
i = 1 qi是总输出。
（a）解决游戏的纳什均衡，包括Örmout-
看跌期权，市场价格，总产出和有效收益（提示：
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
如图1所示，它们产生的容量q1，q2和q3等于
古诺博弈的纳什均衡量的特征
（a）部分。在第二阶段，他们同时决定他们的
价格p1，p2和p3。每个Örm出售的边际成本
容量为0。不可能卖出超过容量的商品。这
Örmi的剩余需求为
Di（pi; p i）=
8
<
：
9 pi Pj = i
如果pi> pj对于所有j = i，则为qj
9点
3
如果对于所有j = i而言pi = pj
如果pi <pj对于所有j = i，则为9 pi
：
（请注意，此处我们假设有效/并行配给是
层）。证明这是第二阶段的纳什均衡
每个Örm收取市场清算价格p的子游戏
=
9 q1 q2 q3。 [20]
2.考虑一次无限重复的Bertrand寡头博弈，其中包括
计数因子<1。单位生产成本为常数c = 0：2，并且对于所有n> 2个有效值均相同。没有固定费用。
描述可以促进默契合谋的“触发”策略形式
定价方面的问题。确定采取此类策略的条件
在以下每种情况下都可以维持垄断价格：
（a）每个时期的市场需求为D（p）= 1 p。 （计算[15]
垄断价格，并在回答中明确指出。）
（b）在每个时期结束时，市场不复存在，概率为[15]
能力 。
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
作为静态价格竞争游戏，类似于线性城市模型。
两个Örms同时选择价格p1和p2。派生
两个Örm的需求函数为：D1（p1; p2）= S
2个
+ p2 p1
2吨
和D2（p1; p2）= S
2个
+ p1 p2
2吨
，其中S> 0，参数t> 0
测量产品区分度。两个Örm都有
生产的恒定边际成本c> 0。
（a）推导该游戏的纳什均衡，包括价格，
两个Örms的输出和利润。 [10]
（b）从需求函数中，qi = Di（pi; pj）= S
2个
+ pj pi
2吨
，得出
剩余逆需求函数：pi = Pi（qi; pj）（算出
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
就像垄断者面临着剩余的逆需求，而
最优齐（等于边际收入和边际成本）
或pi使Pi（qi; pj）= pi> c，即Örmi具有市场支配力。 [10]
（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]
the monopoly price and proÖt explicitly in your answer.)
(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 !