Advanced Microeconomic

7/23/2019 Advanced Microeconomic

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Advanced Microeconomics

Harald Wiese

University of Leipzig

Winter term 2012/2013

Harald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 1 / 70


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What I like about economic theory and microeconomics

Not just talk, but theory with formal models (> sociology)

The core of neoclassical economics: Walras model of perfectcompetition (chapter XIX)

interdependence of markets andprices to bring markets into equilibrium

and also and beyond:decisions under uncertainty (chapters III and IV)detailed analysis of consumer behavior (chapters VI and VII)interactive decisions = game theory (chapters X to XIII)simple concepts like Pareto eciency with wide ramications (chapter

XIV)auction theory (chapters XVII and XVIII)adverse selection (chapter XXII) and hidden action (XXIII)

However: hard work ahead



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Technical stu

3 class meetings

Monday 09:15 - 10:45 in SR 13 (I 274)Wednesday 09:15 - 10:45 in SR 17 (I 374)Friday 09:15 - 10:45 in SR 13 (I 274)

mixing lectures and exercises

written exam 120 min. at two datesmidterm on 07.12.2012 (60 min.)nal in February (60 min.)



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Literature

Harald Wiese: Advanced microeconomics (which is more detailed

than the lecture)Andreu Mas-Colell, Michael D. Whinston, Jerry R. Green:Microeconomic Theory

Hugh Gravell, Ray Rees: Microeconomics

Georey Jehle, Philip Reny: Advanced Microeconomic TheorySamuel Bowles: Microeconomics Behavior, Institutions, andEvolution

Thomas Hnscheid: Schwester Helga



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Course overview

Part A. Basic decision and preference theory:

Decisions in strategic formDecisions in extensive form

Ordinal preference theory

Decisions under risk

Chapter IIDecisions

in strategic form

Decision theory Game theory

Chapter IIIDecisions

in extensive form

Chapter XGames

in strategic form

Chapter XIIGames

in extensive form



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Course overview

Part B. Household theory and theory of the rm:

The household optimumComparative statics and duality theory

Production theory

Cost minimization and prot maximization



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Course overview

Part C. Games and industrial organization:

Games in strategic formPrice and quantity competition

Games in extensive form

Repeated games



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Course overview

Part D. Bargaining theory and Pareto optimality:

Pareto optimality in microeconomicsCooperative game theory



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Course overview

Part E. Bayesian games and mechanism design:

Static Bayesian gamesThe revelation principle and mechanism design



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Course overview

Part F. Perfect competition and competition policy:

General equilibrium theory I: the main resultsGeneral equilibrium theory II: criticism and applications

Introduction to competition policy and regulation



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Course overview

Part G. Contracts and principal-agent theories:

Adverse selectionHidden action



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Part A. Basic decision and preference theory

1 Decisions in strategic (static) form

2 Decisions in extensive (dynamic) form3 Ordinal preference theory

4 Decisions under risk



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Decisions in strategic formOverview

1 Introduction

2 Sets, functions, and real numbers

3 Dominance and best responses

4 Mixed strategies and beliefs

5

Rationalizability



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Introduction

Decision in strategic form:

helps to ease into game theory;concerns one-time (once-and-for-all) decisions where payos dependon:

strategy;

state of the world.



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Introduction

Examplestate of the world

bad weather good weather

strategy

production

of umbrellas100 81

productionof sunshades

64 121



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IntroductionExample 1: production of umbrellas or sunshades

DenitionA decision situation in strategic form is a triple

= (S, W, u) ,

withS (decision makers strategy set),

W (set of states of the world), and

u : S

W

!R (payo function).

= (S, u : S! R) is called a decision situation in strategic form withoutuncertainty.


I d i


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In this example:

S = fumbrella, sunshadeg;W = fbad weather, good weatherg;payo function u given by

u(umbrella, bad weather) = 100,

u(umbrella, good weather) = 81,

u(sunshade, bad weather) = 64,

u(sunshade, good weather) = 121.

Note that:

decision maker can choose only one strategy from S;

only one state of the world from W can actually happen.


I d i


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DenitionDecision situation without uncertainty:

u(s, w1) = u(s, w2)

for all s2 S and all w1, w2 2 W.Trivial case: jWj = 1.


I d i


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IntroductionExample 2: Newcombs problem

Assumptions:

2 boxes:

1000 e in box 1;0 e or 1 mio. e in box 2.

strategies: open box 2,

only, or both boxes;Higher Being put the money into the boxes depending on Hisprediction of your choice;

Higher Beings predictions can be correct (She knows the books youread) or wrong.

Problem

What would you do?


I t d ti


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IntroductionExample 2: Newcombs problem, version 1

prediction:box 2, only

prediction:both boxes

you open

box 2, only

1 000 000 Euros 0 Euro

you openboth boxes

1 001 000 Euros 1 000 Euro


I t d ti


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IntroductionExample 2: Newcombs problem, version 2

predictionis correct

predictionis wrong

you open

box 2, only

1 000 000 Euros 0 Euro

you openboth boxes

1 000 Euro 1 001 000 Euros

We come back to Newcombs problem later ...


Introduction


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IntroductionExample 3: Cournot monopoly

Denition = (S, ), where:

S = [0,) (set of output decisions),

: S! R (payo function) dened by (s) = p(s) s C(s) withp : S! [0,) (inverse demand function);C : S! [0,) (cost function).


Decisions in strategic form


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Decisions in strategic formSets, functions, and real numbers

1 Introduction




5 Rationalizability


Sets


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SetsSet, element, and subset

Denition (set and elements)Set any collection of elements that can be distinguished from eachother. Set can be empty: ?.

Denition (subset)

Let M be a nonempty set. A set N is called a subset of M (N M) ifevery element from N is contained in M. fg are used to indicate sets.M1 = M2 i M1 M2 and M2 M1.

ProblemTrue?

f1, 2g = f2, 1g = f1, 2, 2g

f1, 2, 3

g f1, 2

gHarald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 24 / 70

Sets


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SetsTuples

DenitionLet M be a nonempty set. A tuple on M is an ordered list of elementsfrom M. Elements can appear several times. A tuple consisting of nentries is called n-tuple. () are used to denote tuples.(a1 , ..., an ) = (b1, ..., bm ) if n = m and ai = bi for all i = 1, ..., n.

Problem

True?

(1, 2, 3) = (2, 1, 3)

(1, 2, 2) = (1, 2)


Sets


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SetsCartesian product

DenitionLet M1 and M2 be nonempty sets. The Cartesian product of M1 and M2(M1 M2) is dened by

M1

M2 :=

f(m1, m2) : m1

2M1 , m2

2M2

g.

Example

In a decision situation in strategic form, SW is the set of tuples (s, w).

ProblemLet M := f1, 2, 3g and N := f2, 3g. Find MN and depict this set in atwo-dimensional gure where M is associated with the abscissa (x-axis)and N with the ordinate (y-axis).


Functions


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FunctionsInjective and surjective functions

DenitionLet M and N be nonempty sets.

A function f : M! N associates with every m 2 M an element fromN, denoted by f (m) and called the value of f at m.

SetsM domain (of f)N range (of f)

f (M) := ff (m) : m 2 Mg =[

m2Mff (m)g image (of f).

Injective function if f (m) = f (m0) implies m = m0 for allm, m0 2 M.Surjective function if f (M) = N holds.

Bijective function both injective and surjective.


Functions


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Functionsvalue versus image

mx

( )mf

M

( )Mf

f (M) := ff (m) : m 2 Mg =[

m

2M

ff (m)g


Functions


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FunctionsExercise

ProblemLet M := f1, 2, 3g and N := fa, b, cg.Dene f : M! N by

f (1) = a,

f (2) = a and

f (3) = c.

Is f surjective or injective?


Functions


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FunctionsTwo dierent sorts of arrows

1 f : M

!N (domain is left; range is right).

2 m 7! f (m) on the level of individual elements of M and N.

Example

A quadratic function may be written as

f : R ! R,x 7! x2 .

or in a shorter form: f : x

7!x2

or very short (and in an incorrect manner): f (x) = x2


Functions


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FunctionsInverse function

DenitionLet f : M! N be an injective function. The function f1 : f (M) ! Mdened by

f1 (n) = m , f (m) = n

is called fs inverse function.

Problem

Let M := f1, 2g and N := f6, 7, 8, 9g. Dene f : M! N byf (m) = 10

2m. Dene f s inverse function.


Cardinality


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y

Denition

Let M and N be nonempty sets and let f : M! N be a bijective function.=) M and N have the same cardinality (denoted by jMj = jNj).If a bijective function f : M! f1, 2, ..., ng exists, M is nite and containsn elements. Otherwise M is innite.

ProblemLet M := f1, 2, 3g and N := fa, b, cg. Show jMj = jNj.



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Real numbers


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Theorem

Theorem (cardinality)The setsN,Z, andQ are innite and we have

jNj = jZj = jQj .

However: jQj < jRjand even

jQj < jfx2 R : a x bgjfor any numbers a, b with a < b.


Real numbers


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Intervals

[a, b] : = fx2 R : a x bg ,[a, b) : = fx2 R : a x< bg ,(a, b] : = fx2 R : a < x bg ,(a, b) : = fx2 R : a < x< bg ,

[a,) : = fx2 R : a xg and(, b] : = fx2 R : x bg .

Problem

Given the above denition for intervals, can you nd an alternativeexpression forR?


Convex combination


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Denition

Let x and y be elements ofR. )kx+ (1 k) y, k2 [0, 1]

the convex combination (the linear combination) of x and y.

Problem

0 x+ (1 0) y =?

Problem

Where is 14 1 + 34 2, closer to 1 or closer to 2?

[a, b] = fx2 R : a x bg= fka + (1 k) b : k2 [0, 1]gHarald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 36 / 70

Decisions in strategic form:


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Dominance and best responses

1 Introduction




5 Rationalizability


Weak and strict dominance


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Denition

= (S, W, u) decision situation in strategic form.Strategy s2 S (weakly) dominates strategy s0 2 S i

u(s, w) u(s0, w) holds for all w2 W andu(s, w) > u(s0, w) is true for at least one w2 W.

Strategy s2 S strictly dominates strategy s0 2 S iu(s, w) > u(s0, w) holds for all w2 W.Dominant strategy a strategy that dominates every other strategy(weakly or strictly).

s0 is called (weakly) dominated or strictly dominated.




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Newcombs problem, version 1

prediction:box 2, only

prediction:both boxes

you openbox 2, only 1 000 000 Euros 0 Euro

you openboth boxes

1 001 000 Euros R 1 000 Euro R

Which strategy is best

for the rst state of the world,

for the second state of the world?




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Newcombs problem, version 2

predictionis correct

predictionis wrong

you openbox 2, only 1 000 000 Euros R 0 Euro

you openboth boxes

1 000 Euro 1 001 000 Euros R

Which strategy is best

for the rst state of the world,

for the second state of the world?


Power set


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Denition

Let M be any set. The set of all subsets of M is called the power set of Mand is denoted by 2M.

Examples

M :=f

1, 2, 3g

has the power set

2M = f, f1g , f2g , f3g , f1, 2g , f1, 3g , f2, 3g , f1, 2, 3gg .

Note that the empty set also belongs to the power set of M !

Problem

Can you derive a general rule to calculate the number of elements of anypower set?


arg max


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Example

Consider a rm that tries to maximize its prot by choosing the outputx optimally. The output x is taken from a set X (e.g. the interval [0,))and the prot is a real (Euro) number.

(x) 2 R : prot resulting from the output x,max

x (x) 2 R : maximal prot by choosing x optimally,

argmaxx

(x) X : set of outputs that lead to the maximal prot

Of course,

maxx (x) = (x) for all x from argmax

x (x) .


arg max


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The following cases are possible:

argmaxx (x) contains several elements.argmaxx (x) contains just one element.

argmaxx (x) = .


Best response


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Denition

= (S, W, u) .The function sR : W ! 2S a best-response function (a best response, abest answer) if sR is given by

sR (w) := arg maxs2

Su(s, w) .

Problem

Use best-response functions to characterize s as a dominant strategy.Hint: characterization means that you are to nd a statement that isequivalent to the denition.


Decisions in strategic form:Mi d t t i d b li f


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Mixed strategies and beliefs

1 Introduction




5 Rationalizability


Probability distribution


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Denition

Probability distribution on M : prob : 2M

! [0, 1] withprob() = 0,

prob(A[B) = prob(A) + prob(B) for all A, B2 2M obeyingA\ B = andprob(M) = 1 (summing condition).

Subsets of M are also called events.

Problem

Probability for the eventsA : "the number of pips (spots) is 2",

B : "the number of pips is odd", and

A

[B


Mixed strategy


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Denition

Let S be a nite strategy set. A mixed strategy

is a probabilitydistribution on S :

(s) 0 for all s2 S and s2S

(s) = 1 (summing condition)

(set of mixed strategies) > also stands for sum, as above

(s) = 1 for one s2 S > identify s with (s) > 0 and (s0) > 0 for s6= s0 > properly mixed strategy

Notation:

(s1) ,

(s2) , ...,

sjSj

.


Mixed strategyDecision situation in strategic form with mixed strategies


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Decision situation in strategic form with mixed strategies

Denition

= (S, W, u)

Example

Choosing umbrella with probability 13

and sunshade with probability 23

is amixed strategy.


Belief


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Denition

Let W be a set of states of the world. (set of probability distributions on W);

2 (belief);Notation: (w1) , ..., wjWj .

Example

Bad weather with probability 14 and good weather with probability34


Extending the payo function


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So far,u : S

W

!R

Two extensions:

probability distribution on Wrather than a specic state of the world w2 W> lottery

mixed strategy on Srather than a specic strategy s2 S


Extending the payo functionbeliefs and lotteries


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beliefs and lotteries

state of the worldbad weather, 14 good weather,

34

strategy

productionof umbrellas

100 81

productionof sunshades

64 121

For example,

Lumbrella =

100, 81;

1

4,

3

4


Lottery and expected value


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Denition

A tupleL = [x; p] := [x1 , ..., x ; p1 , ..., p ]

is called a lottery where xj 2 R is the payo accruing with probability pjL (set of simple lotteries).` = 1

L is called a trivial lotteryidentify L = [x; 1] with x.

Denition

Assume a simple L = [x1 , ..., x ; p1, ..., p ] . Its expected value is denotedby E (L) and given by

E (L) =`

j=1

pjxj.


Extending the payo functionmixed strategies


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g

Denition

The payo under and w is dened by

u(, w) := s2S

(s) u(s, w)

Problem

Calculate the expected payo in the umbrella-sunshade decision situationif the rm chooses umbrella with probability 13 and sunshade with

23 .

Dierentiate between w =bad weather and w ="good weather".

Thus, the payo for a mixed strategy is the mean of the payos forthe pure strategies.

How are best pure and best mixed strategies related?


Best pure and best mixed strategies


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Lemma

Best pure and best mixed strategies are related by the following two claims:

Any mixed strategy that puts positive probabilities on best purestrategies, only, is a best strategy.

If a mixed strategy is a best strategy, every pure strategy with

positive probability is a best strategy.


Mixing strategies and states of the worldExercise


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Problem

Consider again the umbrella-sunshade decision situation in which

the rm chooses umbrella with 13 and sunshade with probability23 and

the weather is bad with probability 14 and good with probability34 .

Calculate u

13 ,

23

,

14 ,

34

!


Extending the payo functionsummary


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The payo function u : SW ! R has been extended:beliefs:

u : S! R(s, ) 7! u(s, ) =

w2W (w) u(s, w) = E (Ls)

mixed strategies

u : W ! R(, w) 7! u(, w) =

s2S(s) u(s, w)

both beliefs and mixed strategies

u : ! R(, ) 7! u(, ) =

s2S

w2W(s) (w) u(s, w)


Best-response functions


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Denition

Given

= (S,

W,

u),

there are four best-response functions:

sR,W : W ! 2S, given by sR,W (w) := arg maxs2S

u(s, w) ,

R,W : W ! 2, given by R,W (w) := arg max

2u(, w) ,

sR,

: ! 2S

, given by sR,

() := arg maxs2S u(s, ) , and

R, : ! 2, given by R, () := arg max2

u(, )

sR or R instead of sR,W, etc. (if there is no danger of confusion)

Problem

Complete the sentence: 2 R,W (w) implies(s) = 0 for all ... .


Best-response functionsTheorem


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Theorem

= (S, W, u) . We have

2 ands2sR,() (s) = 1 imply2 R, () and2 R, () implies s2 sR, () for all s2 S with (s) > 0.

These implications continue to hold for W and w rather than and.




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Example

w1 w2s1 4 1s2 1 2

Let := (w1) be the probability of w1. We have s1 2 sR, () in thecase of

4 + (1) 1 1 + (1) 2,i.e., if 14 holds.




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Best-reply function is given by R, : ! 2.For

6= 14 , there is exactly one best strategy,

= 0 ( = (0, 1) = s2) or = 1 ( = (1, 0) = s1).

= 14 implies that every pure strategy (> every mixed strategy) isbest.

R, () =

8 14[0, 1] , = 140, < 14




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4

1 1

1


Best-response functionsExercise


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Problem

Sketch the best-reply function R, for

w1 w2s1 1 3

s2 2 1


Decisions in strategic form:Rationalizability


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1 Introduction

2 Sets, functions, and real numbers3 Dominance and best responses


5 Rationalizability


Rationalizable strategies


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Example

w1 w2s1 4 4s2 1 5s3 5 1

Can a rational decider choose s1 although it is not a best response tow1 and not a best response to w2?

Yes: the rational decider may entertain the belief on W with (w1) = (w2) =

12 .

Given this belief, s1 is a perfectly reasonable strategy.

Problem

Show s1 2 sR,

12 ,

12

!


Rationalizable strategies


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Denition

= (S

,

W,

u).

A mixed strategy 2 is called rationalizable with respect to W if aw2 W exists such that 2 R,W (w).Strategy 2 is called rationalizable with respect to if a belief

2 exists such that

2

R,

(

).


Nobel price 2010

I 2010 th S i Rik b k P i i E i S i i M f


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In 2010, the Sveriges Riksbank Prize in Economic Sciences in Memory ofAlfred Nobel was awarded to the economists

1/3 Peter A. Diamond (Massachusetts Institute of Technology,Cambridge, MA),

1/3 Dale T. Mortensen (Northwestern University, Evanston, IL), and

1/3 Christopher A. Pissarides (London School of Economics and Political

Science)

for their analysis of markets with search frictions.


Further exercises: Problem 1

(a) If strategy s 2 S strictly dominates strategy s 0 2 S and strategy s 0


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(a) If strategy s2 S strictly dominates strategy s0 2 S and strategy s0strictly dominates strategy s00 2 S, is it always true that strategy sstrictly dominates strategy s00?

(b) If strategy s2 S weakly dominates strategy s0 2 S and strategy s0weakly dominates strategy s00 2 S, is it always true that strategy sweakly dominates strategy s00?


Further exercises: Problem 2

Prove the following assertions or give a counter example!


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Prove the following assertions or give a counter-example!

(a) If

2 is rationalizable with respect to W, then is rationalizable

with respect to .

(b) If s2 S is a weakly dominant strategy, then it is rationalizable withrespect to W.

(c) If s

2S is rationalizable with respect to W, then s is a weakly

dominant strategy.


Further exercises: Problem 3 (without gure)

Consider the problem of a monopolist faced with the inverse demand


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Consider the problem of a monopolist faced with the inverse demandfunction p(q) = a b q, in which a can either be high, ah, or low, al.The monopolist produces with constant marginal and average cost c.Assume that ah > al > c and b> 0. Think of the monopolist as settingthe quantity, q, and not the price, p.

(a) Formulate this monopolists problem as a decision problem instrategic form. Determine sR,W!

(b) Assume ah = 6, al = 4, b = 2, c = 1 so that you obtain the plot given

in the gure. Show that any strategy q /2h

alc2b ,

ahc2b

iis dominated

by either sR,W

ah

or sR,W

al

. Show also that no strategy

q2 h

alc2b

,ahc

2b i dominates any other strategy q0 2 halc

2b,

ahc2b i .

(c) Determine all rationalizable strategies with respect to W.(d) Dicult: Determine all rationalizable strategies with respect to .

Hint: Show that the optimal output is a convex combination ofsR,W a

h

and sR,W

al

.Harald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 69 / 70

Further exercises: Problem 3 (gure)

4


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0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

quantity

profit


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Advanced Microeconomic