continuoussimultaneousgamesproblemset answers

8/13/2019 ContinuousSimultaneousGamesProblemSet Answers

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*I will only solve the questions that I think are more relevant for exams.

1.Competitive rent seeking.

Two contestants are fighting for a prize that worth V. The conflict technology function whichdecides their chance of winning is:

1

1 2

m

m m

FP

F F

where P is the probability that contestant 1 will win,1Fis the fighting effort of contestant 1,

2F is the fighting effort of contestant 2 and m is the economies of scale in conflictmeasuring the advantage that a contestant with higher fighting effort has over a contestant

with lower fighting effort. Contestant 1 faces a constant marginal cost of1

Cwhen investing

in fighting effort and contestant 2 faces a constant marginal cost of2

C when investing infighting effort. Solve for the Nash equilibrium of the simultaneous game.

2.Bertrand (price) and Cournot (quantity) competition.The linear demand demand functions faced by two firms are given as

1 1 1 1 2y a b p cp

2 2 2 2 1y a b p cp Assuming zero costs of production, show that quantities are alwayslower and prices higher inCournot competition than in Bertrand competition.Answer:Bertrand competition:

1

1 1 1 1 1 1 1 2

11 1 1 2

1

1 21

1

max

2 0

2

pp y p a b p cp

a b p cpp

a cpp

b


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2

2 2 2 2 2 2 2 1

2 2 2 2 1

2

2 12

2

max

2 0

2

pp y p a b p cp

a b p cpp

a cpp

b

1 2

11

2 12

2

2 1 22

2 2 1

2 12 2

2 2 1 2 1

2

1 2 1 1 22

1 2 1 2

1 2 122

1 2

2

2

2 2 2

2 2 2 2 2

2 4

4 4

2

4

a cpp

b

a cpp

b

a a cpcp

b b b

a ac c cp p

b b b b b

b a ca b b cp

b b b b

b a cap

b b c


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1 21

1

2 12

2

1 21 1

1 1 2 1 2

21 2 2 1 2

1

1 2 1 2

1 2 212

1 2

2

2

2 2 2 2 2

2 4

4 4

2

4

a cpp

b

a cpp

b

a ac c cp p

b b b b b

a b ca b b cp

b b b b

a b cap

b b c


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1 2 212

1 2

1 2 122

1 2

1 1 1 1 2

2 2 2 2 1

1 2 2 1 2 11 1 1 2 2

1 2 1 2

2

1 1 2 1 1 2 2 1 2 1

1 2

1 2

1 2 1 1 2 22 2 2 2 2

1 2 1 2

2

4

2

4

2 24 4

4 2 2

4

2 24 4

a b cap

b b c

b a cap

b b c

y a b p cp

y a b p cp

a b ca b a cay a b cb b c b b c

a b b c b a b ca c b a cay

b b c

b a ca a b cay a b cb b c b b c

22 1 2 2 1 2 1 1 2 22 2

1 2

4 2 2

4

a b b c b b a ca c a b cay

b b c

Cournot competition:

2 2 2 2 1y a b p cp


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1 1 1 1 2

1 1 2

11

2 2 2 2 1

2 2 12

2

1 1 22 2 2 2

1

1 2 2 1 1 2 2 1 1 2

2

1 2 2 1 1 2 2 1 1 2

21 2 2 2 2 1 1 1 2 1

2

1 2 2 2 1 1 1 2 1

2 12

y a b p cp

a y cp

p b

y a b p cp

a y cpp

b

a y cpy a b p cb

b y a b b b p c a y cp

b y a b b b p ca cy c p

b b p c p a b ca b y cy

b b c p a b ca b y cy

a b cp

1 1 2 1

2

1 2

a b y cy

b b c


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2 2 11 1 1 1

2

2 1 1 2 1 2 1 2 2 1

2

2 1 1 2 1 2 1 2 2 1

2

1 2 1 1 1 2 2 2 1 2

2

1 2 1 1 2 2 2 1 2

1 2 2 2 1 21 2

1 2

a y cpy a b p c

b

b y a b b b p c a y cp

b y a b b b p ca cy c p

b b p c p a b ca b y cy

b b c p a b ca b y cy

a b ca b y cyp

b b c


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1

2

1 1 2 1 2 1 2 1 1 22

1 2

11 2 2 2 1 22

1 1 2

1 2 2 21

2

1max

12 0

2

ya b y ca y b y cy y

b b c

a b ca b y cyy b b c

a b ca cyy

b

2

2

2 2 1 2 1 2 1 2 1 22

1 2

2

2 1 1 1 2 12

2 1 2

2 1 1 1

2

1

1max

12 0

2

ya b y ca y b y cy y

b b c

a b ca b y cyy b b c

a b ca cy yb


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1 2 2 21

2

2 1 1 12

1

2 1 11 2

1 1

2 1 1 1 2 2 22

1 1 2

2 1 1 1 2 22 2

1 1 2 1 2

22 1 1 1 2 2 1 2

2

1 1 2 1 2

2

2

2 2

2 2 2

2 2 2 2 2

4

2 2 2 4

a b ca cyy

b

a b ca cyy

b

a b ca cy y

b b

a b ca a b ca cyc yb b b

a b ca a b cac c cy y

b b b b b

a b ca a b ca b b cc yb b b b b

22 1 1 2 1 2 2 1 2

2

1 2 1 2 1 2

2 1 1 2 1 2 2

22

1 2

2 4

2 2 2 2 4

24

a b ca b a b ca b b ccy

b b b b b b

a b ca b c a b ca yb b c


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2 1 1 12

1

1 2 22 1

2 2

1 2 2 2 1 1 11

2 2 1

1 2 2 2 1 1 11

2 2 1 2 1

21 2 2 1 2 1 1 1 2

1

1 2 2 1 1 2

1 2 2 1

2

2 2

2 2 2

2 2 2 2 2

2 4

4 2 2 4

2

a b ca cyy

b

a b ca cy y

b b

a b ca a b ca cycy

b b b

a b ca a b ca cyc cy

b b b b b

a b ca b a b ca b b ccy

b b b b b b

a b ca b c a

2 1 112

1 24

b cay

b b c


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1, 1,

2

1 1 2 1 1 2 2 1 2 1

21 2

1 2 2 1 2 1 1

2

1 2

2

1 1 2 1 1 2 2 1 2 1

2

1 2

1 2 2 1 2 1 1

2

1 2

2 2

1 1 2 1 1 2 1 2 1 2 1

2

1 2

4 2 2

4

2

4

4 2 2

42

4

4 4 2

4

B Cy y

a b b c b a b ca c b a ca

b b c

a b ca b c a b ca

b b c

a b b c b a b ca c b a ca

b b ca b ca b c a b ca

b b c

a b b c b a b b ca b a c c a

b b c

1 2 2 1 1

2

1 2

2

1

2

1 2

2

4

04

b ca c a b ca

b b c

c a

b b c

3.Bertrand competition.Two firms, named 1 and 2, each produce a homogeneous good whosemarket demand at a

price of p is 1-p. Firm 1s average cost is constant andequal to 0; firm 2s average cost is alsoconstant and equal to c>0. Thetwo firms compete by simultaneously setting prices for thegood. Firm 2sells to the entire market (and firm 1 sells nothing) if and only if its priceis lowerthan firm 1s price; otherwise, firm 1 sells to the entire market andfirm 2 sells nothing. a.What would firm 1 charge if it is a monopolist?

b.Let p be any price between 0 and the minimum of c and 1/2. Explainwhy there is anequilibrium in which both firms set a price of p.


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c. Can there be any other equilibria for the Bertrand competition market?

Answers:1.If firm 1 is monopoly, he maximizes p(1-p) which gives p=1/2.

2.Assuming c


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Pareto optimality requires that social welfare or surplus, which is sum of consumer andproducer surplus, be maximized. That is, to maximized area under demand curve andmarginal costs curves. Therefore, P=5=MC1=MC2 ,qb=5, qa=185.

b.Monopoly sets MR=MC=5. Qa=195/2, qb=5. P=195/4.

c.Solving firm as problem gives:95-qa-0.5qb=0Solving firm bs problem gives:100-0.5qa-2qb=0.Therefore, qb=30 and qa=80.

5.Cournot competition and perfect competition.Consider a homogeneous product market where demand function is

1

100N

iP q

iq is the output level of firm i and there are N identical firms in themarket. All the firmsfaces a constant and marginal costs of C. Supposethe firms move simultaneously setting theiroutput level taking the outputlevels of other firms as given.a.Solve for price, quantity and profit of firm i when N = 1.

b.

Solve for price, quantity and profit of firm i when N = 2.c.Solve for price, quantity and profit of firm i as a function of N.d.Suppose there is free entry and exit in the market. Solve for the perfectcompetition price,quantity and N.Answer:


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1

1

1 2 3

100

100

100 ... ... 0

100 1 0

100

1

100

100 1

1100

1

100 1

N

i

N

j i j

j

j N

j

j

P q

q c q

c q q q q qq

c N q

cq

N

c

N c qN

N Nc q

N

N

c qN

6.The inverse demand function, production function and inverse labor demand function of anisolated economy with only one homogenous product and a production function involvingonly one input (labor) are

P = 100 QQ = 2L


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W = 4Li.Solve for the equilibrium where the monopoly firm in this economy is also a monopsony inthe labor market.ii.

If the government is to impose a minimum wage of 80, how will this change the equilibriumin i.iii.Solve for the social optimal level of production and employment.Answer:a.

2

max

100 22

100 2

100 4 0

25

12.5

75

50

QPQ WL

QQ Q Q

Q Q

QQ

Q

L

P

W

Alternatively


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2

max

100 2 2 4

200 8

200 16 0

12.5

25

50

75

LPQ WL

L L L L

L L

LQ

L

Q

W

P

b.

Now marginal cost of labor is 80 for L=20.

So set marginal cost of labor equal to marginal revenue product of labor

80=200-8L

8L=120

L=15

Q=30

W=80

P=70

c.


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100 2

100

3

200

3

100

6

400

6

P MC

Q Q

Q

P

L

W

Alternatively

100 2 2 2

200

6

1003

400

3

200

3

L

L

VMP W

P MP W

L L

L

Q

W

P


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7.The demand function in a particular industry is of the form

2

100

q p

where q is the quantity demanded and p is the price. The technology in themarket is such thateach unit can be produced at the constant cost of $5.i.If the industry is supplied by perfectly competitive firms, what is theequilibrium price andindustry output? What is the price elasticity of demandfaced by a single firm at this price ifthe firm increases its price?ii.If the industry is supplied by a monopolist, what is the equilibrium priceand industry output?

What is the price elasticity of demand faced by themonopolist at this price if thefirmincreases its price?iii.If the industry is supplied by two identical oligopolies engaged inCournotcompetition, whatis the equilibrium price and industry output?iv.If the industry is supplied by two identical oligopolies engaged in Betrandcompetition, whatis the equilibrium price and industry output? Ifthere are n Bertrand oligopolies, what is theequilibrium price and industryoutput?

8.

Choosing firms location.

In a product market with a constant regulated price P, two firms simultaneously make

decision on where to locate on their product space (in a linear city): 1 2, 0,1l l .Consumers are uniformly distributed on 0,1 . Each consumer at most buys one unit fromone of the firm. From product i, consumer x derives utility:

i iv x r x l P .

Assuming1 2

l l , the indifferent consumer is located at

^ 1 2

2

l lx

.

Find the Nash equilibriums.


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9.

Agency cost

1 and 2 jointly operate a firm that produces Y which sells in a perfectly competitive

market at price P. The production function is:

1 2Y AX X

1s share of revenue is and 2s share of revenue is . 1 incurs a constant marginal

cost of 1c when contributing to the common effort and 2 incurs a constant marginal cost

of 2c when contributing to the common effort.

a.

Find the production efforts and profits under Nash equilibrium.

b.

Find the socially optimal production efforts and profits.

10.

Tragedy of Commons

The Straits of Malacca is teeming with fishes. A fishing ship will bring in gross revenue

of 1000-5S where S is the number of fishing ships operating in the straits. It costs 10 per

ship to operate and there is free entry and exit.

a.

Find the Nash equilibrium number of ships.

b.

Find the socially optimal number of ships.

11.

Cavalry warfare

State 1 owns a heavy cavalry force. State 2 owns a light cavalry force. The two states fight

for world hegemony. The two armies engage each other on a plain. , 0,1x y . 1

minimizes 2

x y . 2 minimizes 2

x y a . Solve for the Nash equilibriums.

12.

Offense vsDefense

State 1 is on the strategic offensive. State 2 is defending herself. Both State 1 and 2 incur a

constant marginal cost of C when fielding a military unit. They fight over a fixed prize of

value V. The conflict technology function is of the form:

1

1 1

FP

F F


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denotes the advantage of being on the offensive. Solve for the Nash equilibriums of military

expenditures given the parameter of V, C and .

continuoussimultaneousgamesproblemset answers

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