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Written Exam for the B.Sc. in Economics, Winter 2010/2011

Microeconomics C

Final Exam

February 21, 2011

(2-hour closed book exam)

Please note that the language used in your exam paper must correspond to the language of the titlefor which you registered during exam registration. I.e. if you registered for the English title of thecourse, you must write your exam paper in English. Likewise, if you registered for the Danish titleof the course or if you registered for the English title which was followed by eksamen p dansk inbrackets, you must write your exam paper in Danish.

If you are in doubt about which title you registered for, please see the print of your examregistration from the students self-service system.

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PLEASE ANSWER ALL QUESTIONS.

PLEASE EXPLAIN YOUR ANSWERS.

1. (a) Solve the game below by iterated elimination of strictly dominatedstrategies. Describe each step.

A2 B2 C2 D2A1 7,1 9,2 2,3 4,4B1 4,2 5,1 1,0 3,0C1 3,3 4,3 3,4 5,3

(b) Find all (pure and mixed) Nash equilibria in the following game:

L R

T 1,1 1,0B 0,1 3,2

(c) Consider the game given by the following game tree:

L

R

L R

1

2

4

3

2

3

R

L

1

6

3

0

2

0

4

3

1

2

5

4

RL

i. How many subgames are there in the game (excluding thegame itself)? Write down the strategies for each player.

ii. Find the pure strategy subgame perfect Nash equilibrium.

iii. Find a pure strategy Nash equilibrium that is not subgameperfect.

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2. Two prot maximizing rms (i = 1; 2) produce a homogeneous product.

The quantity produced by rm i is qi 0. The inverse demand functionis

P(Q) = 70Q;

where Q = q1 + q2 (do not worry about negative prices in this exercise).The cost function for rm i is

Ci(qi) = (qi)2; i = 1; 2:

Note that this is the actual cost of the rm, not the marginal cost.

(a) Consider the game where the rms choose their quantities simulta-neously and independently. Write down the prot functions andderive the best response functions for the two rms. Find the(pure strategy) Nash equlibrium.

(b) Suppose now that rm 1 chooses its quantity rst. Firm 2 observesthis quantity and then chooses its own quantity. Find the (purestrategy) subgame perfect Nash equilibrium of this game.

(c) Compare the equilibria from (a) and (b) with respect to the quan-tity that each rm produces. Give an intuitive explanation of theresults.

(d) Suppose the two rms merge. Because of synergies, the cost func-tion for the merged entity is

Cm(qm) = (1 s)(qm)2;

where s 2 (0; 1) is a constant. Intuitively, how does the quantityproduced by the merged entity depend on s? Find the value of ssuch that the quantity produced by the merged entity is the sameas the aggregate quantity in the Nash equilibrium from (a).

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3. Consider the normal-form game G:

L R

T 2,2 4,1B 0,4 3,3

(a) Draw a gure illustrating the set of feasible payos for G.

(b) Consider the innitely repeated game G(1; ) where G is thestage game and 2 (0; 1) is the discount factor. Draw a gureillustrating the set of feasible payos for G that can be achievedas average payos in a subgame perfect Nash equilibrium (SPNE)

of G(1; ) provided that is suciently close to one. Explain.

4. Consider the following signalling game:

R

L

u

0.5d

3,0

[p] L

R

u

u u

d d

d

t1

t2

0.5

2,1

1,2

0,0

0,2

0,1

1,0

2,1[q]

(a) Find a perfect Bayesian equilibrium (PBE) where both types ofthe sender (t1 and t2) send the message L.

(b) Does the PBE you found in (a) satisfy Signalling Requirement 5from Gibbons? Explain your answer.

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