EC202: Microeconomic Principles 2 2012/13 Handout.pdf · 2013. 1. 7. · EC202 - Extra Game Theory Problems Lent Term 2013 Weekly Course Assignments Due to popular requests I suggest

EC202: Microeconomic Principles 2

2012/13 http://darp.lse.ac.uk/EC202

Office hour

Lecturers Frank Cowell x7277 [email protected]

Francesco Nava x6353 [email protected]

R520

S482

Thu 11:30-12:30

Thu 11:30-12:30 TBC

Class teachers Michel Azulai [email protected]

Kasia Grabowska [email protected]

Luis Martinez [email protected]

Secretary Gisela Ladfico x6674 [email protected]

Lectures Wed 11:00 Hong Kong Theatre [CLM D1] Thu 14:00 Old Theatre

Course Organisation

20 lectures in the first term focusing primarily on price-taking market behaviour by economic agents. Topics include the firm, the consumer, general equilibrium, uncertainty and risk, welfare economics. Where possible the similarity of the economic problems in each topic is exploited in order to highlight the re-use of results.

20 lectures in the second term examining problems of strategic interaction among economic agents. This part of the course introduces fundamental concepts in microeconomic theory for strategic environments. The course begins with an introduction to game theory. Fundamental solution concepts are presented and discussed in detail. Static models of complete and incomplete information and dynamic models of complete information are covered in the course. The subgame perfect Folk theorem and its consequences on repeated interactions are detailed in a few lectures. Some of the most relevant applications of such models of behaviour are also introduced. Particular attention will be devoted to: imperfect competition, adverse selection, signalling, moral hazard, externalities and public goods.

20 weekly classes beginning in week 3. Class material follows the lectures with about a two-week lag.

The course text is F. A. Cowell, Microeconomics: Principles and Analysis (Oxford University Press 2006) [C]. In the second term Most materials covered can be found in the main textbook. Supplementary readings in the second term may be taken from M. J Osborne, An Introduction to Game Theory (Oxford University Press, 2004) [O] and B. Salanié, The Economics of Contracts: A Primer (MIT Press, 2005) [S].

On-line resources are provided at http://darp.lse.ac.uk/EC202. These include answers to weekly class work hand-in exercises and PowerPoint presentation files of lectures.

Lectures, Reading and Classes: First Term

Week Lecture Topics Text Exercises Hand-in Work

1 The firm C 2 NO

2 The firm and the market C 2,3 NO

3 The consumer C 4 2.5 *

4 The consumer and the market C 4,5 2.9, 3.2 *

5 A simple economy C 6 4.2, 4.3, 4.6 NO

6 General equilibrium 1 C 7 4.7, 4.8, 4.11 *

7 General equilibrium 2 C 7 5.3, 5.7, 5.8 *

8 Uncertainty and risk C 8 6.5, 7.3 NO

9 Welfare 1 C 9 7.4, 8.1 *

10 Welfare 2 C 9 8.9, 9.1 *

2

Lectures, Reading and Classes: Second Term

Week Lecture Topics Text Extra Weekly Work Hand-in Work

1 Static Games

Dominance and Nash Equilibrium

C 10.2

C 10.3.1-4

O2.1, O2.6

O2.8-9

9.2 , 9.3, 9.5 NO

2 Mixed Strategy Nash Equilibrium

Oligopoly

C 10.3.5

C 10.4

O4.1-4

O3.1-2

9.6 NO

3 Incomplete Information Games

Bayes Nash Equilibrium

C 10.7

O9.1-3 10.1,10.2, 10.3 *

4 Dynamic Games

Subgame Perfection

C 10.5-6 O5.1-4 10.4, 10.7.1-3, 10.17, O282.1

NO

5 Imperfect Competition

Repeated Games: Introduction

10.7.4-5, 10.12, O183.1-2

*

6 Repeated Games: Folk Theorem

Adverse Selection: Monopoly

C 10.5 O10.1-3, O15.1

S2, S3.1.3

10.13, 10.15.1-2 NO

7 Adverse Selection: Competition

Competitive Insurance Markets

C 11.2 S3.2.1 10.15.3, 10.16, O429.1, O442.1

*

8 Signalling C 11.3 S4, O10.5-6 11.1, 11.2 NO

9 Moral Hazard C 11.4 S5.1-3.5 11.5, 11.6 *

10 Externalities

Public Goods

C 13.4

C 13.6.1-4

O2.8.4, O9.5 11.8,13.5, 13.6 *

Course Requirements

Classes

Classes begin in the third week of the first term and continue into the third term.

Teachers assign specific students to prepare short presentations of the exercises in the text chapters. But all students should make a reasonable attempt in advance of the class. Team up in small groups if you find this helpful, but make sure that you personally understand why the exercise “works.”

Answers to exercises will be posted on the website.

You should also do the mini problems in the text since they are designed to help you with some of the steps involved in the reasoning. Again, feel free to work together on this. (Answers are in the text, Appendix B).

Hand-in Assignments

In the starred weeks you have to hand in written assignments which will be posted on the website

These assignments are of the same scope and difficulty as exam questions.

These have to be your own work. Do not work with others on your hand-in assignments.

Examination

There will be a single three-hour, four-question paper. The 2007-2011 papers (in the Library) can be used as general guidance to the style and level of difficulty. The 2012 paper will follow the format of the 2011 paper:

Question 1 (the compulsory question) is worth 40% of total marks.

Question 1 requires candidates to answer 5 out of 8 parts.

The three other questions are worth 20% each.

EC202 - Extra Game Theory ProblemsLent Term 2013

Weekly Course Assignments

Due to popular requests I suggest a few extra practice game theory problems from Osborne’s manual. You don’tneed to practice on these problems, but you may benefit from them. Some are harder than what I would ask onexam day. Solutions are available on the webpage under "Solutions O".

1. Static Complete Information Games

• 33.1 (Contributing to a Public Good)• 48.1 (Voting)• 34.3 (Choosing a Route)• 80.2 (A Fight)• 114.2 (Games with Mixed Strategy Equilibria)• 114.4 (Swimming with Sharks)• 141.1 (Finding All Mixed Equilibria)

2. Static Incomplete Information Games

• 282.1 (Fighting an Opponent of Unknown Strength)• 282.2 (An Exchange Game)• 282.3 (Adverse Selection)• 291.1 (reporting a crime with an unknown number of witnesses)

3. Dynamic Complete Information Games: NE vs SPE

• 163.1 (Nash Equilibria of Extensive Form Games)

• 163.2 (Voting by Alternating Veto)• 173.2 (Finding Subgame Perfect Equilibria)• 173.4 (Burning a Bridge)• 183.1 (NE of the Ultimatum Game)

• 183.2 (SPE of the Ultimatum Game)

• 189.1 (Stackelberg with Quadratic Costs)

4. Repeated Complete Information Games: Subgame Perfect Folk Theorem

• 428.1 (Strategies in an Infinitely Repeated PD)• 429.1 (Grim Trigger Strategy in a General PD)

• 442.1 (Deviations from Grim Trigger Strategy)

• 443.1 (Delayed modified Grim Trigger Strategy)

• 452.3 (a-b) (Minmax Payoffs)

Hand In —Problem Set 1 Francesco Nava

Microeconomic Principles II —EC202 Lent Term 2013

Each question is worth 25 marks. Please give your answers to your class teacher by Friday of week 3 LT.If you do not hand in at your class, make arrangements with your class teacher about where to bring it.Thank you!

1. Consider the following complete information strategic form game:

1\2 L C R

T 3, 4 1, 3 6, 2M 2, 1 9, 4 0, 2B 2, 2 2, 3 4, 2

(a) Find the pure strategy Nash equilibria.

(b) Are there any strictly dominated strategies if players can only play pure strategies?

(c) Are there any strictly dominated strategies if players can employ a mixed strategies?

(d) Find the mixed strategy Nash Equilibrium.

2. Consider an auction with two buyers participating and a single object for sale. Suppose that eachbuyer knows the values of all the other bidders. Order players so that values decrease, x1 > x2.Consider a 2nd price sealed bid auction. In such auction: all players simultaneously submit a bid bi;the object is awarded to the highest bidder; the winner pays the second highest submitted bid to theauctioneer; the losers pay nothing. Suppose ties are broken in favor of player 1. That is: if b1 = b2then 1 is awarded the object.

(a) Characterize the best response correspondence of each player.

(b) Characterize all the Nash equilibria for a given profile (x1, x2).

(c) Consider the Nash equilibrium in which both players bid their value [ie: bi = xi for i ∈ 1, 2].Is this a dominant strategy equilibrium?



Each question is worth 25 marks. Please give your answers to your class teacher by Friday of week 5 LT.If you do not hand in at your class, make arrangements with your class teacher about where to bring it.Thank you!

1. Consider an economy with two producers competing to supply a market. Suppose that the costfunction of the first firm displays a constant marginal costs, while the second firm displays increasingmarginal costs. In particular assume that:

c1(q1) = q21 + 2q1

c2(q2) = 4q2

Suppose that the inverse demand in this market is linear and satisfies:

p(q) = 10− 2q

Assume that the two firms compete à la Cournot.

(a) Derive the Cournot production levels, profits and the equilibrium price.

(b) Assume that firms form a cartel to sell their output as a monopolist. Derive the cartel productionlevels, profits and the equilibrium price. Compare them to the competitive and Cournot outcomes.

(c) Assume that firms do not account for their market power, but simply equalize marginal coststo prices. Derive the competitive production levels, profits and the equilibrium price. Comparethem to the Cournot outcomes.

2. Four patients have to undergo surgery and rehabilitation in one of two hospitals. Hospital A specializesin surgery. But its elite surgery unit is small. The likelihood of successful surgery, pSA, depends on thenumber of patients treated in the surgery unit, n, as follows:

pSA(n) =

17/20 if n = 115/20 if n = 211/20 if n = 37/20 if n = 4

.

The rehabilitation unit of hospital A is large, but conventional. The likelihood of successful rehabilita-tion pRA = 1/2 is independent of the number of patients treated. Hospital B specializes in rehabilitation.But its elite rehabilitation unit is small. The likelihood of successful rehabilitation, pRB, depends onthe number of patients treated by the unit, n, as follows:

pRB(n) =

1 if n = 1

16/20 if n = 214/20 if n = 312/20 if n = 4

The surgery unit of hospital B is large, but conventional. The likelihood of successful surgery pSB = 1/2is independent of the number of patients treated. Surgery outcomes are independent of rehabilitationoutcomes. A patient’s payoff is 1 if both treatments are successful, and 0 otherwise. A patient isclassified as recovered, only if both treatments are successful. [Hint: Recall that if events A and B areindependent Pr(A ∩B) = Pr(A) Pr(B)].

Microeconomic Principles II F. Nava

(a) First suppose that the existing health regulations require all patients to undergo surgery andrehabilitation at the same hospital. Patients can only choose in which of the two hospital to getboth treatments. Set this problem up as a strategic-form game. Find a Nash equilibrium of thisgame. What is the average probability of recovery among the four patients in this equilibrium?

(b) In an attempt to increase the average recovery probability regulators decide to lift the ban onhaving surgery and rehabilitation at different hospitals. Now patients are free to choose in whichhospital to get either treatment. Set this problem up as a strategic-form game. Find a Nashequilibrium of this game. What is the average probability of recovery among the four patients inthis equilibrium?

(c) Still unsatisfied about the average recovery probability, regulators decide to try a third policy,in which one of the four patients is randomly selected and sent to the two elite units (surgery inA and rehabilitation in B), while the remaining three are sent to the larger conventional units(surgery in B and rehabilitation in A). What is the average probability of recovery among thefour patients with this policy in place?

(d) Compare the average probabilities of recovery under the three regulations. Give an intuitiveexplanation to the observed change in recovery probabilities.

2



Each question is worth 25 marks. Please give your answers to your class teacher by Friday of week 7 LT.If you not to hand in at your class, make arrangements with your class teacher about where to bring it.Thank you!

1. Consider a static game of incomplete information with two players and in which player 2 has twopossible types. Call them type a and type b. Suppose that the probability of player two being of typea is 0.7 and that payoffs are described by the matrix below:

1\2.a L R 1\2.b L RT 4, 2 0, 1 T 0, 1 0, 2M 3, 0 1, 1 M 1, 1 9, 1B 2, 4 3, 3 B 3, 2 4, 1

(a) What are the possible strategies of each player? Is any one of them dominated. [10 marks]

(b) Compute a pure strategy Bayes Nash equilibrium. [10 marks]

(c) Is it a dominant strategy equilibrium? [5 marks]

2. Consider the following extensive form game:

(a) Find the unique Subgame Perfect equilibrium of this game. [8 marks]

(b) Find a pure strategy Nash equilibrium with payoffs (3, 5). [8 marks]

(c) Find a pure strategy Nash equilibrium with payoffs (4, 2). [9 marks]

3. Two firms compete to sell a good. Firm 1 has higher total costs of production than firm 2, but is theStackelberg leader and has to produce before firm 2. Firm 2 can observe the output of firm 1 prior tomaking his output decision. The total costs for the two firms respectively satisfy:

C1(q1) = 3 (q1)2

C2(q2) = (q2)2 + 4q2

Microeconomic Principles II F. Nava

The inverse demand for the output produced by the two firms in this market satisfies:

p(q1 + q2) =

10− 2(q1 + q2) if q1 + q2 ≤ 5

0 if q1 + q2 > 5

Firms choose how much output to produce in order to maximize their profits.

(a) Compute subgame perfect equilibrium output of each firm and the price for this economy. [12marks]

(b) Now alter the previous game to allow both firms to make their output decisions simultaneouslyas in the Cournot model. Compute equilibrium prices and output. [8 marks]

(c) How were prices and output levels affected by such a change? Did the profits of any of the twofirms increase? Explain. [5 marks]

4. Consider the following asymmetric Prisoner’s Dilemma:

1\2 C D

C 3, 4 1, 6D 4, 0 2, 2

(a) Find the minmax values of this game. Consider the infinitely repeated version of this game inwhich all players discount the future at the same rate δ. The following is a "tit for tat" strategy:any player chooses C provided that the other player never chose D; if at any round t a playerchooses D, then the other player chooses D in round t + 1 and continues playing D until theplayer who first chose D reverts to C; if at any round t a player chooses C, then the other playerchooses C in round t+ 1. Write this strategy explicitly. [7 marks]

(b) Find the unique value for the common discount factor δ for which the strategy of part (a) sustainsalways playing C as a SPE of the infinitely repeated game. [9 marks]

(c) Then, consider the following "trigger" strategy: any player chooses C provided that no player everplayed D; otherwise any player chooses D. Write the two incentive constraints that if satisfiedwould make such a strategy a NE. Then, write the two additional incentive constraints that ifsatisfied would make such a strategy a SPE. What is the lowest discount rate for which suchstrategy satisfies all the constraints. [9 marks]

2



Each question is worth 25 marks. Please give your answers to your class teacher by Friday of week 9 LT. Ifyou do not to hand in at your class, make arrangements with your teacher about where to hand in. Thankyou!

1. Consider an economy with a monopolistic electricity supplier. Assume that the costs of producing aunit of electricity are 1$. There are only two goods in this economy namely money, y, and electricity,x. All consumers in this economy are endowed with 100$ in money and no electricity. There aretwo types of buyers in the economy: type H has high value for electricity, while type L does not. Inparticular assume that preferences satisfy:

u(x, y|H) = 8x1/2 + y

u(x, y|L) = (9/2)x1/2 + y

(a) If the monopolist can recognize the type of any individual, find the optimal pricing schedules forboth types. Why is this outcome effi cient?

(b) Suppose that 1/8 of all individuals in the population are of type H. If the monopolist cannotrecognize the type of any individual, find the separating equilibrium optimal pricing schedulesfor both types. Why is the outcome ineffi cient?

(c) What happens to the economy in part (b) if there is perfect competition? What is the uniqueprice for electricity? Who purchases more electricity than in part (b)?

2. Consider Spence’s signalling model. A worker’s type is t ∈ 0, 1. The probability that any worker isof type t = 1 is equal to 2/3, while the probability that t = 0 is equal to 1/3. The productivity of aworker in a job is (t+1)2. Each worker chooses a level of education e ≥ 0. The total cost of obtainingeducation level e is C(e|t) = e2(2− t). The worker’s wage is equal to his expected productivity.

(a) Characterize all pooling perfect Bayesian equilibrium in which both types of workers choose astrictly positive education level.

(b) Find all separating perfect Bayesian equilibria.

(c) Which separating equilibrium survives the intuitive criterion? Is it the one with the lowesteducation level?



Each question is worth 25 marks. Please give your answers to your class teacher by Friday of week 10 LT. Ifyou do not to hand in at your class, make arrangements with your teacher about where to hand in. Thankyou!

1. Consider a “Principal-Agent”model. Suppose that the Agent is a worker who can choose any one oftwo effort levels, e ∈ 1, 2. Two output levels are feasible for the Principal, namely q ∈ 0, 9. Theprobability that the Principal achieves high output depends on the effort of the Agent. In particularassume that Pr(q = 9|e = 1) = 1/6 and Pr(q = 9|e = 2) = 5/6. If w denotes the salary of the worker,the expected payoff of the Principal is:

Π(w|e) = 9 Pr(q = 9|e)− w

The reservation payoff of the worker is 0, while his payoff satisfies:

U(e|w) = 2w − e2

(a) Suppose that the Principal can observe the effort chosen by the worker. Characterize the fullinformation contracts. Do these contracts induce the worker to choose the effi cient effort level?

(b) Suppose that the Principal cannot observe the worker’s effort choice, but only output. Thus, hecan offer only wage contracts w,w which depend on the output produced. If the worker chooseshis effort to maximize expected utility, characterize the incentive compatibility constraint thatthe Principal must satisfy if the worker is to exert high effort e = 2.

(c) Find the optimal wages that a Principal would set in this environment to maximize its profits.Which effort level would be chosen by the Agent in the equilibrium?

2. Consider a Principal-Agent problem with: three exogenous states of nature H,M,L; two effort levelsea, eb; and two output levels distributed as follows as a function of the state of nature and the effortlevel:

H M L

Probability 20% 60% 20%

Output Under ea 25 25 4

Output Under eb 25 4 4

The principal is risk neutral, while the agent has a utility function w1/2, when receiving a wage w,minus the effort cost which is zero if eb is chosen, and 1 otherwise. The agent’s reservation utility is 0.

(a) Derive the optimal wage schedule set by the principal when both effort and output are observable.

(b) Derive the optimal wage schedule set by the principal when only output is observable.

(c) If the principal cannot observe effort, how much would he be willing to pay for a technology that,prior to the beginning of the game, reveals when the state L is realized?

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FrancescoNava

(LondonSch

oolofEco

nomics)

Microeconomic

PrinciplesIIEC202

January2013

7/10

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FrancescoNava

(LondonSch

oolofEco

nomics)

Microeconomic

PrinciplesIIEC202

January2013

8/10

Not

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FrancescoNava

(LondonSch

oolofEco

nomics)

Microeconomic

PrinciplesIIEC202

January2013

9/10

Materials

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inT

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FrancescoNava

(LondonSch

oolofEco

nomics)

Microeconomic

PrinciplesIIEC202

January2013

10/10

Not

es

Not

es

StaticCompleteInformationGames

EC202—LecturesI&II

FrancescoNava

LSE

January2013

FrancescoNava(LSE)


January2013

1/27

Summary

GamesofCompleteInformation:

Definitions:

Game:Players,Actions,Payoffs

Strategy

BestResponse

SolutionConcepts[inpurestrategies]:

DominantStrategyEquilibrium

NashEquilibrium

PropertiesofNashEquilibria:

Non-ExistenceandMultiplicity

Inefficiency

Examples

FrancescoNava(LSE)


January2013

2/27

Notes

Notes

IntroductiontoGames

Anyenvironmentinwhichthechoicesofanindividualaffectthewellbeing

ofotherscanbemodeledasagame.

Whatpinsdownaspecificgame:

Whoparticipatesinagame

[Players]

Thechoicesthatparticipantshave

[Choices]

Thewellbeingofindividuals

[Payoffs]

Theinformationthatindividualshave

[RulesoftheGame]

Thetimingofeventsanddecisions

[RulesoftheGame]

FrancescoNava(LSE)


January2013

3/27

IntroductiontoGames

InmostmodelsdiscussedinMT:

individualdecisionsdidnotaffectthewellbeingofothers

anydependencewouldjusthingefrom

equilibrium

prices

LecturesI,II&IIIdiscusscompleteinformationstrategicform

games.

Insuchenvironments:

Individualsknoweverything,butforthedecisionsmadebyothers

Alldecisionstakeplaceatonce

FrancescoNava(LSE)


January2013

4/27

Notes

Notes

CompleteInformation(StrategicForm)Game

AcompleteinformationgameGconsistsof:

Asetofplayers:

Nofsizen

Anactionsetforeachplayerinthegame:

Aiforplayeri’s

Anactionprofilea=(a1,a2,...,an)picksanactionforeachplayer

Autilitymapforeachplayermappingactionprofilestopayoffs:

u i(a)denotesplayeri’spayoffofactionprofilea

B\G

sm

s5,2

1,2

m0,0

3,5

FrancescoNava(LSE)


January2013

5/27

RepresentingSimoultaneousMoveCompleteInfoGames

StrategicForm

1\2

sm

s5,2

1,2

m0,0

3,5

ExtensiveForm

FrancescoNava(LSE)


January2013

6/27

Notes

Notes

RepresentingSimoultaneousMoveCompleteInfoGames

StrategicForm

1\2

sm

s5,2

1,2

m0,0

3,5

ExtensiveForm

FrancescoNava(LSE)


January2013

7/27

FeasiblePayoffsandEfficiency

StrategicForm

1\2

sm

s5,2

1,2

m0,0

3,5

FeasiblePayoffs

01

23

45

012345

u1

u2

Efficientpayoffsareonthenorth-eastboundary(ingreen)

FrancescoNava(LSE)


January2013

8/27

Notes

Notes

InformationandPureStrategies

Astrategyinagame:

isamapfrom

informationintoactions

itdefinesaplanofactionforaplayer

Inacompleteinformationstrategicformgame:

playershavenoprivateinformation

playersactsimultaneously

Inthiscontextastrategyisanyelementofthesetofactions

Forinstancea(pure)strategyforplayeriissimplya i∈Ai

FrancescoNava(LSE)


January2013

9/27

BestResponses

Defineaprofileofactionschosenbyallplayersotherthanibya −

i:

a −i=(a1,...,ai−1,ai+1,...,an)

Thebestresponsecorrespondenceofplayeriisdefinedby:

b i(a−i)=argmaxa i∈A

iu i(ai,a −

i)foranya −

i

Thusa iisabestresponsetoa −

i—i.e.a i∈b i(a−i)—ifandonlyif:

u i(ai,a −

i)≥u i(a′ i,a −

i)foranya′ i∈Ai

BRidentifiestheoptimalactionforaplayergivenchoicesmadebyothers

Forinstance:

B\G

sm

s5,2

1,2

m0,0

3,5

FrancescoNava(LSE)


January2013

10/27

Notes

Notes

StrictDominance

Strategya istrictlydominatesa′ iif:

u i(ai,a −

i)>u i(a′ i,a −

i)foranya −

i

a iisstrictlydominantifitstrictlydominatesanyothera′ i

a iisstrictlyundominatedifnostrategystrictlydominatesa i

a iisstrictlydominatedifastrategystrictlydominatesa i

InthefollowingexamplesisstrictlydominantforB:

B\G

sm

s5,-2,-

m0,-1,-

FrancescoNava(LSE)


January2013

11/27

WeakDominance

Strategya iweaklydominatesa′ iif:

u i(ai,a −

i)≥u i(a′ i,a −

i)foranya −

i

u i(ai,a −

i)>u i(a′ i,a −

i)forsomea −

i

a iisweaklydominantifitweaklydominatesanyothera′ i

a iisweaklyundominatedifnostrategyweaklydominatesa i

a iisstrictlydominatedifastrategystrictlydominatesa i

InthefollowingexamplesisweaklydominantforB:

B\G

sm

s5,-2,-

m0,-2,-

FrancescoNava(LSE)


January2013

12/27

Notes

Notes

DominanceExamples

Onestrictlyandoneweaklydominatedstrategy:

1\2

LC

RT

-,1

-,2

-,1

B-,0

-,1

-,3

Onestrictlydominantstrategy:

1\2

LC

RT

-,1

-,2

-,3

B-,0

-,1

-,2

Oneweaklydominantstrategy:

1\2

LC

RT

-,1

-,2

-,2

B-,0

-,0

-,1

FrancescoNava(LSE)


January2013

13/27


Definitions(DominantStrategyEquilibriumDSE)

AstrictDominantStrategyequilibrium

ofagameGconsistsofastrategy

profileasuchthatforanya′ −

iandi∈N:

u i(ai,a′ −

i)>u i(a′ i,a′ −

i)foranya′ i∈Ai

ForweakDSEchange>with≥...

aprofileaisaDSEiffa i∈b i(a′ −i)foranya′ −

iandi∈N

Example(Prisoner’sDilemma):

B\S

NC

N5,5

0,6

C6,0

1,1

FrancescoNava(LSE)


January2013

14/27

Notes

Notes

IterativeEliminationofDominatedStrategies

Tofinddominantstrategieseliminatedominatedstrategiesfrom

thegame

Ifnecessaryrepeattheprocesstopossiblyruleoutmorestrategies

Considerthefollowingexample:

1\2

LC

RT

1,0

2,1

3,0

M2,3

3,2

2,1

D0,2

1,2

2,5

⇒

1\2

LC

RT

1,0

2,1

3,0

M2,3

3,2

2,1

D0,2

1,2

2,5

AtthefirstinstanceonlyDisdominatedforplayer1

Nostrategyisdominatedaprioriforplayer2

[Strategiesingreeninthetablearedominatedandthuseliminated]

FrancescoNava(LSE)


January2013

15/27

IterativeEliminationofDominatedStrategies

OnceDhasbeeneliminatedfrom

thegame:

StrategyRisdominatedforplayer2

Nostrategyisdominatedforplayer1

1\2

LC

RT

1,0

2,1

3,0

M2,3

3,2

2,1

D0,2

1,2

2,5

⇒

1\2

LC

RT

1,0

2,1

3,0

M2,3

3,2

2,1

D0,2

1,2

2,5

OnceRhasbeeneliminatedfrom

thegame:

StrategyTisdominatedforplayer1

Afinaliterationyields(M,L)astheonlysurvivingstrategies

FrancescoNava(LSE)


January2013

16/27

Notes

Notes

Dominance:FinalConsiderations

Dominanceisoftenconsideredabenchmarkofrationality:

Rationalplayersneverchoosedominatedstrategies

Commonknowledgeofrationalitymeans:

playersonlyemploystrategiesthatsurviveiterativeelimination

Dominanceisasimpleconceptbuywithimportantlimitations:

Oftenthereisnodominantstrategyevenafteriteration

Itoftenleadstoinefficientoutcomes

Thusaweakernotionofequilibrium

needstobeintroducedtomodel

behaviorespeciallyforrichersetups

FrancescoNava(LSE)


January2013

17/27

NashEquilibrium:Introduction

Dominancewastheappropriatesolutionconceptifplayershadno

informationorbeliefsaboutchoicesmadebyothers

Theweakernotionofequilibrium

thatwillbeintroducedpresumesthat:

playershavecorrectbeliefsaboutchoicesmadebyothers

playerschoicesareoptimalgivensuchbeliefs

theenvironmentiscommonknowledgeamongplayers

Suchmodelallowsfortighterpredictionswhendominancehasnobite

FrancescoNava(LSE)


January2013

18/27

Notes

Notes

NashEquilibrium

Definition(NashEquilibriumNE)

A(purestrategy)Nashequilibrium

ofagameGconsistsofastrategy

profilea=(ai,a −

i)suchthatforanyi∈N:

u i(a)≥u i(a′ i,a −

i)foranya′ i∈Ai

aprofileaisaNEiffa i∈b i(a−i)foranyi∈N

Properties:

Strategyprofilesareindependent

Strategyprofilescommonknowledge

FrancescoNava(LSE)


January2013

19/27

MoreExamples

GamesmayhavemoreNE’s(BattleoftheSexes):

B\G

sm

s5,2

1,2

m0,0

3,5

Nashequilibriamaynotbeefficient(Prisoner’sDilemma):

B\S

NC

N5,5

0,6

C6,0

1,1

PurestrategyNashequilibriamaynotexist(MatchingPennies):

B\G

HT

H0,2

2,0

T2,0

0,2

FrancescoNava(LSE)


January2013

20/27

Notes

Notes

Examples,PropertiesandLimitations

GamesmayhavemoreNE’s(BattleoftheSexes):

B\G

sm

s5,2

1,2

m0,0

3,5

Nashequilibriamaynotbeefficient(Prisoner’sDilemma):

B\S

NC

N5,5

0,6

C6,0

1,1

PurestrategyNashequilibriamaynotexist(MatchingPennies):

B\G

HT

H0,2

2,0

T2,0

0,2

FrancescoNava(LSE)


January2013

21/27

SomeNashEquilibriaareMoreRisky

ConsidertheStagHuntgame: B\G

Stag

Hare

Stag

9,9

0,8

Hare

8,0

8,8

Bestresponsesforthisgameare:

B\G

Stag

Hare

Stag

9,9

0,8

Hare

8,0

8,8

BothplayerschoosingtogoforthestagisNE

SuchNEinvolvesgreaterrisksofmiscoordination

thantheNEinwhichbothgoforthehare

FrancescoNava(LSE)


January2013

22/27

Notes

Notes

ThreePlayerExample

Agamewithmorethan2players:

3L

R1\2T D

AB

1,0,1

1,0,0

0,1,1

1,2,0

AB

0,1,1

0,1,1

1,0,1

2,1,1

TofindallPNEcheckbestreplymaps:

3L

R1\2T D

AB

1,0,1

1,0,0

0,1,1

1,2,0

AB

0,1,1

0,1,1

1,0,1

2,1,1

FrancescoNava(LSE)


January2013

23/27

ThreePlayerExample


3L

R1\2T D

AB

1,0,1

1,0,0

0,1,1

1,2,0

AB

0,1,1

0,1,1

1,0,1

2,1,1

TofindallPNEcheckbestreplymaps:

3L

R1\2T D

AB

1,0,1

1,0,0

0,1,1

1,2,0

AB

0,1,1

0,1,1

1,0,1

2,1,1

FrancescoNava(LSE)


January2013

23/27

Notes

Notes

FirstPriceAuctionExample

Anauctioneersellsoneobjectwiththefollowingrule:

Buyerssimultaneouslysubmitsealedbids(ai)

Highestbidwinstheobject

Winnerpayshisownbid

Tiesarebrokeninfavorofbuyerwithlowestindex

Nbuyersparticipateattheauction

Theirvaluesfortheobjectarev 1>v 2>...>v N

Payoffsofplayeriis:

(vi−a i)I(ai≥maxj6=ia j)

NE:a 1=a 2=v 2

anda i≤v 2

foranyi>2

FrancescoNava(LSE)


January2013

24/27

FirstPriceAuctionAllNashEquilibria

ConsiderthecaseforN=1,2andvaluesv 1>v 2

Thebestresponsemapsforthetwoplayersare:

b 1(a2)=

a 1<a 2

ifa 2>v 1

a 1≤a 2

ifa 2=v 1

a 1=a 2

ifa 2<v 1

b 2(a1)=

a 2≤a 1

ifa 1≥v 2

a 2=a 1+

εifa 1<v 2

TheNashEquilibriaofthegamesatisfy:

b 1=b 2∈[v2,v1]

FrancescoNava(LSE)


January2013

25/27

Notes

Notes

WarofAttritionExample

ConsideragamewithtwocompetitorsN=1,2involvedinafight:

Supposethatthevalueofwinningthefightisv ifori∈N

Competitorschoosehowmuchefforttoputinafighta i∈[0,∞)

Thepayoffofcompetitorigiventheireffortlevelsare:

v iI(ai>a j)+(vi/2)

I(ai=a j)−mini∈1,2a i

Bestresponsefunctionssatisfy:

b i(aj)=

a i>a j

ifa j<v i

a i=0ora i>a j

ifa j=v i

a i=0

ifa j>v i

AllNashEquilibriaofthegamesatisfyoneofthefollowing:

a 1=0anda 2≥v 1

a 2=0anda 1≥v 2

FrancescoNava(LSE)


January2013

26/27

DSEimpliesNE

Fact

Anydominantstrategyequilibrium

isaNashequilibrium

Proof.

IfaisaDSEthena i∈b i(a′ −i)foranya′ −

iandi∈N.

WhichimpliesaisNEsincea i∈b i(a−i)foranyi∈N.

FrancescoNava(LSE)


January2013

27/27

Notes

Notes


EC202—LectureIII:MixedStrategies

FrancescoNava

LSE

January2013

FrancescoNava(LSE)


January2013

1/15

Summary

GamesofCompleteInformation:

Definitions:

MixedStrategy

DominantStrategy

SolutionConcepts:

NashEquilibrium

Examples

NEExistence

FrancescoNava(LSE)


January2013

2/15

Notes

Notes

IntroductiontoMixedStrategies

Aproblematicaspectofthesolutionconceptsdiscussedinthefirsttwo

lectureswasthatequilibriadidnotalwaysexist.

Reasonsforthelackofexistencewere:

Non-convexitiesinthechoicesets

Discontinuitiesofthebestresponsecorrespondences

Todayweintroducemixedstrategieswhichsolvebothproblemsand

guaranteeexistenceofatleastaNashequilibrium.

FrancescoNava(LSE)


January2013

3/15

MixedStrategyDefinition

Considercompleteinformationstaticgame N,

Ai,u i i∈N

Amixedstrategyfori∈NisaprobabilitydistributionoveractionsinAi

Thus

σ iisamixedstrategyif:

σ i(ai)≥0foranya i∈Ai

∑a i∈A

iσi(a i)=1

Intuitively

σ i(ai)istheprobabilitythatplayerichoosestoplaya i

E.G.

σ 1(B)=0.3and

σ 1(C)=0.7isamixedstrategyfor1in:

1\2

BC

B2,0

0,2

C0,1

1,0

FrancescoNava(LSE)


January2013

4/15

Notes

Notes

Simplex

ThesetofpossibleprobabilitydistributionsonasetBis:

calledsimplex

anddenotedby

∆(B)

Suchsetis:

Closed

Bounded

Convex

FrancescoNava(LSE)


January2013

5/15

Payoffsfrom

MixedStrategies

Asinthelastlecturesoftenwedenote:

ff−i=(σ1,...,σi−1,σi+1,...,σN)

Thepayofftoplayerifrom

choosing

σ i∈

∆(Ai)whenothersfollow

σ−iis:

u i(σi,ff−i)=

∑a∈APr(a)u i(a)=

=∑a∈A

∏j∈N

σ j(aj)u i(a)

E.G.Ifplayersfollow

σ 1(B)=

σ 2(B)=0.3inthegame:

1\2

BC

B2,0

0,2

C0,1

1,0

Thepayofftoplayer1is:u 1(σ1,σ2)=(.09)2+(.49)1+(.42)0=0.67

FrancescoNava(LSE)


January2013

6/15

Notes

Notes

BestReplyMaps

Denotethebestreplycorrespondenceofibyb i(ff−i)

Themapisdefinedby:

b i(ff−i)=argmax

σ i∈∆(Ai)u i(σi,ff−i)

Forinstanceconsiderthegame: 1\2

sm

s5,2

1,2

m0,0

3,5

Ifσ 1(s)=1thenany

σ 2(s)∈[0,1]satisfies

σ 2∈b 2(σ1)

Ifσ 1(s)<1thenonly

σ 2(s)=0satisfies

σ 2∈b 2(σ1)

FrancescoNava(LSE)


January2013

7/15

DominatedStrategies

Strategy

σ iweaklydominatesa iif:

u i(σi,a −

i)≥u i(ai,a −

i)foranya −

i

u i(σi,a −

i)>u i(ai,a −

i)forsomea −

i

a iisweaklyundominatedifnostrategyweaklydominatesit

Thisallowsustoruleoutmorestrategiesthanbefore,eg:

1\2

LC

RT

6,6

0,2

0,0

B0,0

0,2

6,6

σ 2(L)=

σ 2(R)=0.5strictlydominatesCsince:

u 2(σ2,a1)=3>u 2(C,a1)=2

FrancescoNava(LSE)


January2013

8/15

Notes

Notes

NashEquilibrium

Definition(NashEquilibriumNE)

ANashequilibrium

ofagameconsistsofastrategyprofileff=(σi,ff−i)

suchthatforanyi∈N:

u i(ff)≥u i(ai,ff−i)foranya i∈Ai

ImplicittothedefinitionofNEarethefollowingassumptions:

Eachagentchooseshismixedstrategyindependentlyofothers

Eachagentknowsandbelieveswhichstrategiestheothersadopt

Eachagentchooseshisstrategytomaximizeexpectedutilitygivenhis

beliefs

FrancescoNava(LSE)


January2013

9/15

NashEquilibriumComputationHelp

Astrategyprofile

σisaNashEquilibriumifandonlyif:

u i(ff)=u i(ai,ff−i)foranya isuchthat

σ i(ai)>0

u i(ff)≥u i(ai,ff−i)foranya isuchthat

σ i(ai)=0

Ifa iisstrictlydominated,then

σ i(ai)=0inanyNashequilibrium

Ifa iisweaklydominated,then

σ i(ai)>0inaNashequilibrium

only

ifanyprofileofactionsa −

iforwhicha iisstrictlyworseoccurswith

zeroprobability

FrancescoNava(LSE)


January2013

10/15

Notes

Notes

Examples

GamesmayhavemoreNEs(BattleoftheSexes):

1\2

sm

s5,2

1,1

m0,0

2,5

Thereare2PNE&amixedNEinwhich

σ 1(s)=5/6&

σ 2(s)=1/6:

u 1(s,σ2)=5σ2(s)+(1−

σ 2(s))=2(1−

σ 2(s))=u 1(m,σ2)

u 2(m,σ1)=

σ 1(s)+5(1−

σ 1(s))=2σ1(s)=u 2(s,σ1)

GamesmayhaveonlymixedNE(MatchingPennies):

B\G

HT

H0,2

2,0

T2,0

0,2

ThereisauniqueNEinwhich

σ 1(H)=

σ 2(H)=1/2:

2(1−

σ i(H))=2σi(H)

FrancescoNava(LSE)


January2013

11/15

MoreExamples

GamesmayhaveacontinuumofNEs:

1\2

LR

CT

4,1

0,2

3,2

D0,2

2,0

1,0

Thegamehas1PNE&acontinuumofNEs,namely:

σ 1(T)=2/3&

σ 2(R)=1−

σ 2(L)−

σ 2(C)

σ 2(L)=1/3−(2

/3)

σ 2(C)for

σ 2(C)∈[0,1

/2 ]

TheseareallNE’ssince:

u 2(L,σ1)=

σ 1(T)+2(1−

σ 1(T))=2σ1(T)=u 2(R,σ1)=u 2(C,σ1)

u 1(T,σ2)=4σ2(L)+3σ2(C)=2σ2(R)+

σ 2(C)=u 1(D,σ2)

FrancescoNava(LSE)


January2013

12/15

Notes

Notes

ThreePlayerExample


3L

R1\2T D

AB

1,0,1

0,1,1

0,1,0

1,0,1

AB

1,1,0

0,0,0

0,0,1

1,1,0

TheNEconditionsrequire:

u 1(T,ff−1)=

σ 2(A)=(1−

σ 2(A))=u 1(D,ff−1)

u 2(A,ff−2)=(1−

σ 1(T))

σ 3(L)+

σ 1(T)(1−

σ 3(L))=

=σ 1(T)σ3(L)+(1−

σ 1(T))(1−

σ 3(L))=u 2(B,ff−2)

u 3(L,ff−3)=

σ 1(T)+(1−

σ 1(T))(1−

σ 2(A))=

=(1−

σ 1(T))

σ 2(A)=u 3(R,ff−3)

Theuniquesolutionrequires:

σ 2(A)=

σ 3(L)=1/2and

σ 1(T)=0

FrancescoNava(LSE)


January2013

13/15

NashEquilibriumExistence

Theorem

(NEExistence)

AnygamewithafinitenumberofactionspossessesaNashequilibrium.

Theorem

(PNEExistence)

Anygamewithconvexandcompactactionsetsandwithcontinuousand

quasi-concavepayofffunctionspossessesapurestrategyNashequilibrium.

Assumptionsinboththeoremsguaranteethatbestresponsemapsare

continuous(upper-hemicontinuous,closedandconvexvalued)onconvex

compactsetsandthusexistence...

FrancescoNava(LSE)


January2013

14/15

Notes

Notes

ContinuousBestResponsesImplyExistence

AgamewithoutPNEandwithasingleNE:

1\2

BC

B2,0

0,2

C0,1

1,0

10.

750.

50.

250

1

0.75 0.

5

0.25 0

s2(B

)

s1(B

)

s2(B

)

s1(B

)

FrancescoNava(LSE)


January2013

15/15

Notes

Notes

OligopolyPricing

EC202—LectureIV

FrancescoNava

LSE

January2013

FrancescoNava(LSE)

OligopolyPricing

January2013

1/13

Summary

ThemodelsofcompetitionpresentedinMTexploredtheconsequenceson

pricesandtradeoftwoextremeassumptions:

PerfectCompetition

[Manysellerssupplyingmanybuyers]

Monopoly

[Onesellersupplyingmanybuyers]

Todayintermediateassumptionisdiscussed:

Oligopoly

[Fewsellerssupplyingmanybuyers]

Twomodelsofcompetitionamongoligopolistsarepresented:

QuantityCompetition

[akaCournotCompetition]

PriceCompetition

[akaBertrandCompetition]

FrancescoNava(LSE)

OligopolyPricing

January2013

2/13

Notes

Notes

ADuopoly

Considerthefollowingeconomy:

TherearetwofirmsN=1,2

Eachfirmi∈Nproducesoutputq iwithacostfunction

c i(qi)mappingquantitiestocosts

Aggregateoutputinthiseconomyisq=q 1+q 2

Bothfirmsfaceanaggregateinversedemandforoutput

p(q)mappingaggregateoutputtoprices

Thepayoffofeachfirmi∈Nisitsprofits:

u i(q1,q2)=p(q)q i−c i(qi)

Profitsdependontheoutputdecisionsofboth

FrancescoNava(LSE)

OligopolyPricing

January2013

3/13

CournotCompetition:Duopoly

Competitionproceedsasfollows:

Allfirmssimultaneouslyselecttheiroutputtomaximizeprofits

Eachfirmtakesasgiventheoutputofitscompetitors

Firmsaccountfortheeffectsoftheiroutputdecisiononprices

Inparticularthedecisionproblemofplayeri∈Nisto:

max q iu i(qi,q j)=max q ip(q i+q j)qi−c i(qi)

[HistoricallythisisthefirstknownexampleofNashEquilibrium—1838]

[Example:VisavsMastercard]

FrancescoNava(LSE)

OligopolyPricing

January2013

4/13

Notes

Notes

CournotCompetition:Equilibrium

Ifstandardconditionsonprimitivesoftheproblemhold:

apurestrategyNashequilibrium

exists

thePNEischaracterizedbytheFOC

Ifso,theproblemofanyproduceri∈Nsatisfies:

∂ui(q i,qj)

∂qi

=p(q i+q j)+

∂p(qi+q j)

∂qi

q i︸

︷︷︸

MarginalRevenue

−∂ci(q i)

∂qi

︸︷︷︸

MarginalCost

≤0ifq i=0

=0ifq i>0

Marginalrevenueaccountsforthedistortioninprices

Pricesdecreaseifinversedemandisdownward-sloping

FOCdefinesthebestresponse(akareactionfunction)ofplayeri:

q i=b i(qj)

FrancescoNava(LSE)

OligopolyPricing

January2013

5/13

CournotExample


p(q)=2−q

c 1(q1)=q2 1

andc 2(q2)=3q2 2

Firmi’sproblemistochooseproductionq igivenchoiceoftheotherq j:

max q i(2−q i−q j)qi−c i(qi)

ThebestreplymapofeachfirmisdeterminedbyFOC:

2−2q1−q 2−2q1=0⇒

q 1=b 1(q2)=(2−q 2)/4

2−2q2−q 1−6q2=0⇒

q 2=b 2(q1)=(2−q 1)/8

CournotEquilibriumoutputsare:

q 1=14

/31andq 2=6/31

Perfectcompetitionoutputsarelarger:

q∗ 1=3/5andq∗ 2=1/5

FrancescoNava(LSE)

OligopolyPricing

January2013

6/13

Notes

Notes

CollusionandCartels

Supposethattheproducerscolludebyformingacartel

Acartelmaximizesthejointprofitsofthetwofirms:

max

q 1,q2p(q)q−c 1(q1)−c 2(q2)

Firstorderoptimalityofthisproblemrequiresforanyi∈N:

p(q)+

∂p(q)

∂qq

︸︷︷

︸MarginalRevenue

−∂ci(q i)

∂qi

︸︷︷︸

MarginalCost

≤0ifq i=0

=0ifq i>0

Aggregateprofitsarehigherinthecartel

Playersaccountforeffectsoftheiroutputchoiceonothers

Buttheprofitsofeachindividualdonotnecessarilyincrease

FrancescoNava(LSE)

OligopolyPricing

January2013

7/13

CollusionExample

Considerthepreviousduopoly,butsupposethatacartelisinplace

Ifso,FOCforthecartelproductionsatisfy:

2−2q1−2q2−2q1=0⇒

q 1=(1−q 2)/2

2−2q2−2q1−6q2=0⇒

q 2=(1−q 1)/4

Carteloutputsare:

q 1=3/7andq 2=1/7

Cournotoutputsarelarger:

q 1=14

/31andq 2=6/31

Cartelprofitsare:

u 1=3/7andu 2=1/7

Cournotprofitsare:

u 1=392/961andu 2=144/961

Totalprofitsarelargerwithacartelinplace,butnotallfirmsmaybenefit

FrancescoNava(LSE)

OligopolyPricing

January2013

8/13

Notes

Notes

Defectionfrom

aCartel

Supposethatthefirmjproducesthecarteloutputq j

Ifso,firmimaybenefitbyproducingmorethanthecarteloutputsince:

b i( qj)>q i

Inthisscenariosustainingacartelmaybehardwithoutoutputmonitoring

Intheexamplethiswasthecaseasfirmspreferredtoincreaseoutput:

b 1(1

/7)=13

/28>3/7

b 2(3

/7)=11

/56>1/7

IfsotheproblemofsustainingthecartelbecomesaPrisoner’sdilemma

FrancescoNava(LSE)

OligopolyPricing

January2013

9/13

BertrandCompetition:Duopoly

Competitionproceedsasfollows:

Allfirmssimultaneouslyquoteapricetomaximizeprofits

Eachfirmtakesasgiventhepricequotedbyitscompetitors

Firmsaccountfortheeffectsoftheirpricingdecisiononsales

Consideraneconomywith:

Twoproducerswithconstantmarginalcostsc

Aggregatedemandforoutputgivenbyq(p)=(b0−p)

/b

Giventhepricesdemandofoutputfrom

firmi∈Nis:

q i(pi,p j)=

q(p i)

ifp i<p j

q(p i)/2ifp i=p j

0ifp i>p j

FrancescoNava(LSE)

OligopolyPricing

January2013

10/13

Notes

Notes

BertrandCompetition:Monopoly

Ifonlyonefirmoperatedinsuchmarketitwouldchoosethepriceto:

max pu(p)=max pq(p)(p−c)

Thusaprofitmaximizingmonopolistwouldsellgoodsataprice:

p=(b0+c)

/2

Withtwoproducerstheproblemofeachfirmbecomes:

max p iu i(pi,p j)=max p iq i(pi,p j)(p i−c)

FrancescoNava(LSE)

OligopolyPricing

January2013

11/13

BertrandCompetition:BestResponses

IntheBertrandmodel,i’soptimalpricingisaimedat“maximizingsales”

Inparticularfirmiwouldsetpricesasfollows(for

εsmall):

Ifp j>p,setp i=pandcaptureallthemarketatthemonopolyprice

Ifp≥p j>c,setp i=p j−

ε,undercutjandcaptureallthemarket

Ifc≥p j,setp i=castherearenobenefitsbypricingbelowMC

Suchlogicrequiresthebestresponseofeachplayertosatisfy:

p i=b i(pj)=

p

ifp j>p

p j−

εifp≥p j>c

cif

c≥p j

ThusintheuniqueNEbothfirmssetp 1=p 2=candperfect

competitionemergeswithjust2firms!

FrancescoNava(LSE)

OligopolyPricing

January2013

12/13

Notes

Notes

CournotvsBertrandCompetition

Bertrandmodelpredictsthatduopolyisenoughtopushdownpricesto

marginalcost(asinperfectcompetition)

Cournotmodelinsteadpredictsthatfewproducersdonotsufficeto

eliminatemarkups(pricesabovemarginalcost)

Inbothmodelsthereareincentivestoformacartelandtochargethe

monopolyprice

Neithermodelisintrinsicallybetter

Accuracyofeithermodeldependsonthefundamentalsoftheeconomy:

Bertrandworksbetterwhencapacityiseasytoadjust

Cournotworksbetterwhencapacityishardtoadjust

FrancescoNava(LSE)

OligopolyPricing

January2013

13/13

Notes

Notes

GamesofIncompleteInformation

EC202LecturesV&VI

FrancescoNava

LSE

January2013

FrancescoNava(LSE)


January2013

1/22

Summary

GamesofIncompleteInformation:

Definitions:

IncompleteInformationGame

InformationStructureandBeliefs

Strategies

BestReplyMap

SolutionConceptsinPureStrategies:


BayesNashEquilibrium

Examples

EXTRA:MixedStrategies&BayesNashEquilibria

FrancescoNava(LSE)


January2013

2/22

Notes

Notes

IncompleteInformation(StrategicForm)

Anincompleteinformationgameconsistsof:

Nthesetofplayersinthegame

Aiplayeri’sactionset

Xiplayeri’ssetofpossiblesignals

Aprofileofsignalsx=(x1,...,xn)isanelement

X=×j∈N

Xj

fadistributionoverthepossiblesignals

u i:A×

X→

Rplayeri’sutilityfunction,u i(a|x)

FrancescoNava(LSE)


January2013

3/22

BayesianGameExample

ConsiderthefollowingBayesiangame:

Player1observesonlyonepossiblesignal:

X1=C

Player2’ssignaltakesoneoftwovalues:

X2=L,R

Probabilitiesaresuchthat:f(C,L)=0.6

Payoffsandactionsetsareasdescribedinthematrix:

1\2.L

y 2z 2

1\2.R

y 2z 2

y 11,2

0,1

y 11,3

0,4

z 10,4

1,3

z 10,1

1,2

FrancescoNava(LSE)


January2013

4/22

Notes

Notes

InformationStructure

Informationstructure:

Xidenotesthesignalasarandom

variable

belongstothesetofpossiblesignals

Xi

x idenotestherealizationoftherandom

variableXi

X−i=(X1,...,X

i−1,X

i+1,...,X

n)denotesaprofile

ofsignalsforallplayersotherthani

PlayeriobservesonlyXi

PlayeriignoresX−i,butknowsf

Withsuchinformationplayeriformsbeliefsregardingtherealizationof

thesignalsoftheotherplayersx −i

FrancescoNava(LSE)


January2013

5/22

BeliefsaboutotherPlayers’Signals[Take1]

Inthiscourseweconsidermodelsinwhichsignalsareindependent:

f(x)=

∏j∈Nf j(xj)

Thisimpliesthatthesignalx iofplayeriisindependentofX−i

Beliefsareaprobabilitydistributionoverthesignalsoftheotherplayers

Anyplayerformsbeliefsaboutthesignalsreceivedbytheotherplayersby

usingBayesRule

IndependenceimpliesthatconditionalobservingXi=x ithebeliefsof

playeriare:

f i(x−i|xi)=

∏j∈N\if j(xj)=f −i(x −i)

FrancescoNava(LSE)


January2013

6/22

Notes

Notes

Extra:BeliefsaboutotherPlayers’Signals[Take2]

Alsointhegeneralcasewithinterdependenceplayersformbeliefsabout

thesignalsreceivedbytheothersbyusingBayesRule

ConditionalobservingXi=x ithebeliefsofplayeriare:

f i(x−i|xi)=Pr(X−i=x −i|X

i=x i)=

=Pr(X−i=x −i∩

Xi=x i)

Pr(Xi=x i)

=

=Pr(X−i=x −i∩

Xi=x i)

∑y −i∈

X−iPr(X−i=y −i∩

Xi=x i)=

=f(x −i,x i)

∑y −i∈

X−if(y −i,x i)

Beliefsareaprobabilitydistributionoverthesignalsoftheotherplayers

FrancescoNava(LSE)


January2013

7/22

Strategies

StrategyProfiles:

Astrategyconsistsofamapfrom

availableinformationtoactions:

αi

:Xi→Ai

Astrategyprofileconsistsofastrategyforeveryplayer:

α(X)=(α1(X1),...,

αN(XN))

Weadopttheusualconvention:

α−i(X−i)=(α1(X1),...,

αi−1(Xi−1),

αi+1(Xi+1),...,

αN(XN))

FrancescoNava(LSE)


January2013

8/22

Notes

Notes

BayesianGameExampleContinued

Considerthefollowinggame:

Player1observesonlyonepossiblesignal:

X1=C

Player2’ssignaltakesoneoftwovalues:

X2=L,R

Probabilitiesaresuchthat:f(C,L)=0.6

Payoffsandactionsetsareasdescribedinthematrix:

1\2.L

y 2z 2

1\2.R

y 2z 2

y 11,2

0,1

y 11,3

0,4

z 10,4

1,3

z 10,1

1,2

Astrategyforplayer1isanelementoftheset

α1∈y1,z1

Astrategyforplayer2isamap

α2

: L,R→y2,z2

Player1cannotactupon2’sprivateinformation

FrancescoNava(LSE)


January2013

9/22


Strategy

αiweaklydominates

α′ iifforanya −iandx∈

X:

u i(αi(x i),a −i|x)≥u i(α′ i(x i),a −i|x)[strictsomewhere]

Strategy

αiisdominantifitdominatesanyotherstrategy

α′ i

Strategy

αiisundominatedifnostrategydominatesit

Definitions(DominantStrategyEquilibriumDSE)

ADominantStrategyequilibrium

ofanincompleteinformationgameisa

strategyprofile

αthatforanyi∈N,x∈

Xanda −i∈A−isatisfies:

u i(αi(x i),a −i|x)≥u i(α′ i(x i),a −i|x)forany

α′ i

:Xi→Ai

I.e.

αiisoptimalindependentlyofwhatothersknowanddo

FrancescoNava(LSE)


January2013

10/22

Notes

Notes

Interim

ExpectedUtilityandBestReplyMaps

TheinterimstageoccursjustafteraplayerknowshissignalXi=x i

ItiswhenstrategiesarechoseninaBayesiangame

Theinterimexpectedutilityofa(pure)strategyprofile

αisdefinedby:

Ui(

α|xi)=

∑X−iui(

α(x)|x)f(x−i|xi)

:Xi→

R

Withsuchnotationinmindnoticethat:

Ui(a i,α−i|xi)=

∑X−iui(a i,α−i(x −i)|x)f(x−i|xi)

Thebestreplycorrespondenceofplayeriisdefinedby:

b i(α−i|xi)=argmaxa i∈A

iUi(a i,α−i|xi)

BRmapsidentifywhichactionsareoptimalgiventhesignalandthe

strategiesfollowedbyothers

FrancescoNava(LSE)


January2013

11/22

PureStrategyBayesNashEquilibrium

Definitions(BayesNashEquilibriumBNE)

ApurestrategyBayesNashequilibrium

ofanincompleteinformation

gameisastrategyprofile

αsuchthatforanyi∈Nandx i∈

Xisatisfies:

Ui(

α|xi)≥Ui(a i,α−i|xi)foranya i∈Ai

BNErequiresinterimoptimality(i.e.doyourbestgivenwhatyouknow)

BNErequires

αi(x i)∈b i(α−i|xi)foranyi∈Nandx i∈

Xi

FrancescoNava(LSE)


January2013

12/22

Notes

Notes

BayesianGameExampleContinued

ConsiderthefollowingBayesiangamewithf(C,L)=0.6:

1\2.L

y 2z 2

1\2.R

y 2z 2

y 11,2

0,1

y 11,3

0,4

z 10,4

1,3

z 10,1

1,2

Thebestreplymapsforbothplayerarecharacterizedby:

b 2(α1|x2)=

y 2if

x 2=L

z 2ifx 2=R

b 1(α2)=

y 1if

α2(L)=y 2

z 1if

α2(L)=z 2

Thegamehasaunique(purestrategy)BNEinwhich:

α1=y 1,

α2(L)=y 2,

α2(R)=z 2

DONOTANALYZEMATRICESSEPARATELY!!!

FrancescoNava(LSE)


January2013

13/22

RelationshipsbetweenEquilibriumConcepts

IfαisaDSEthenitisaBNE.Infactforanyactiona iandsignalx i:

u i(αi(x i),a −i|x)≥u i(ai,a −i|x)∀a−i,x −i⇒

u i(αi(x i),

α−i(x −i)|x)≥u i(ai,

α−i(x −i)|x)∀α−i,x −i⇒

∑X−iui(

α(x)|x)fi(x −i|xi)≥

∑X−iui(a i,α−i(x −i)|x)fi(x −i|xi)∀α−i⇒

Ui(

α|xi)≥Ui(a i,α−i|xi)∀α−i

FrancescoNava(LSE)


January2013

14/22

Notes

Notes

BNEExampleI:Exchange

Abuyerandasellerwanttotradeanobject:

Buyer’svaluefortheobjectis3$

Seller’svalueiseither0$or2$basedonthesignal,

XS=L,H

Buyercanoffereither1$or3$topurchasetheobject

Sellerchoosewhetherornottosell

B\S.L

sale

nosale

B\S.H

sale

nosale

3$0,3

0,0

3$0,3

0,2

1$2,1

0,0

1$2,1

0,2

ThisgameforanypriorfhasaBNEinwhich:

αS(L)=sale,

αS(H)=nosale,

αB=1$

SellingisstrictlydominantforS.L

Offering1$isweaklydominantforthebuyer

FrancescoNava(LSE)


January2013

15/22

BNEExampleII:EntryGame

Considerthefollowingmarketgame:

FirmI(theincumbent)isamonopolistinamarket

FirmE(theentrant)isconsideringwhethertoenterinthemarket

IfEstaysoutofthemarket,Erunsaprofitof1$andIgets8$

IfEenters,Eincursacostof1$andprofitsofbothIandEare3$

Icandeterentrybyinvestingatcost0,2dependingontypeL,H

IfIinvests:I’sprofitincreasesby1ifheisalone,decreasesby1ifhe

competesandE’sprofitdecreasesto0ifheelectstoenter

E\I.L

Invest

NotInvest

E\I.H

Invest

NotInvest

In0,2

3,3

In0,0

3,3

Out

1,9

1,8

Out

1,7

1,8

FrancescoNava(LSE)


January2013

16/22

Notes

Notes

BNEExampleII:EntryGame

Let

πdenotetheprobabilitythatfirmIisoftypeLandnotice:

αI(H)=NotInvestisastrictlydominantstrategyforI.H

Foranyvalueof

π,

αI(L)=NotInvestand

αE=InisBNE:

u I(Not,In|L)=3>2=u I(Invest,In|L)

UE(In,

αI(XI))=3>1=UE(Out,αI(XI))

For

πhighenough,

αI(L)=Investand

αE=OutisalsoBNE:

u I(Invest,Out|L)=9>8=u I(Not,Out|L)

UE(Out,αI(XI))=1>3(1−

π)=UE(In,

αI(XI))

E\I.L

Invest

NotInvest

E\I.H

Invest

NotInvest

In0,2

3,3

In0,0

3,3

Out

1,9

1,8

Out

1,7

1,8

FrancescoNava(LSE)


January2013

17/22

Extra:MixedStrategiesinBayesianGames

StrategyProfiles:

Amixedstrategyisamapfrom

informationtoaprobability

distributionoveractions

Inparticular

σ i(ai|xi)denotestheprobabilitythatichoosesa iifhis

signalisx i

Amixedstrategyprofileconsistsofastrategyforeveryplayer:

σ(X)=(σ1(X1),...,

σ N(XN))

Asusualσ−i(X−i)denotestheprofileofstrategiesofallplayers,buti

Mixedstrategiesareindependent(i.e.

σ icannotdependon

σ j)

FrancescoNava(LSE)


January2013

18/22

Notes

Notes

Extra:Interim

Payoff&BayesNashEquilibrium

Theinterimexpectedpayoffofmixedstrategyprofiles

σand(ai,

σ−i)are:

Ui(

σ|xi)=

∑ X−i

∑ a∈Au i(a|x)

∏ j∈N

σ j(aj|xj)f(x −i|xi)

Ui(a i,σ−i|xi)=

∑ X−i

∑a −

i∈A−iui(a|x)

∏ j6=iσ j(aj|xj)f(x −i|xi)

Definitions(BayesNashEquilibriumBNE)

ABayesNashequilibrium

ofagame

Γisastrategyprofile

σsuchthatfor

anyi∈Nandx i∈

Xisatisfies:

Ui(

σ|xi)≥Ui(a i,σ−i|xi)foranya i∈Ai

BNErequiresinterimoptimality(i.e.doyourbestgivenwhatyouknow)

FrancescoNava(LSE)


January2013

19/22

Extra:ComputingBayesNashEquilibria

TestingforBNEbehavior:

σisBNEifonlyif:

Ui(

σ|xi)=Ui(a i,σ−i|xi)foranya is.t.

σ i(ai|xi)>0

Ui(

σ|xi)≥Ui(a i,σ−i|xi)foranya is.t.

σ i(ai|xi)=0

StrictlydominatedstrategiesareneverchoseninaBNE

Weaklydominatedstrategiesarechosenonlyiftheyaredominated

withprobabilityzeroinequilibrium

ThisconditionscanbeusedtocomputeBNE(seeexamples)

FrancescoNava(LSE)


January2013

20/22

Notes

Notes

Extra:ExampleI

Considerthefollowingexampleforf(1,L)=1/2:

1\2.L

XY

1\2.R

WZ

T1,0

0,1

T0,0

1,1

D0,1

1,0

D1,1

0,0

AllBNEsforthisgamesatisfy:

σ 1(T)=1/2and

σ 2(X|L)=

σ 2(W|R)

SuchgamessatisfyallBNEconditionssince:

U1(T,σ2)=(1

/2)

σ 2(X|L)+(1

/2)(1−

σ 2(W|R))=

=(1

/2)(1−

σ 2(X|L))+(1

/2)

σ 2(W|R)=U1(D,σ2)

u 2(X,σ1|L)=

σ 1(T)=1−

σ 1(T)=u 2(Y,σ1|L)

u 2(W

,σ1|R)=(1−

σ 1(T))=

σ 1(T)=u 2(Z,σ1|R)

FrancescoNava(LSE)


January2013

21/22

Extra:ExampleII

Considerthefollowingexampleforf(1,L)=q≤2/3:

1\2.L

XY

1\2.R

WZ

T0,0

0,2

T2,2

0,1

D2,0

1,1

D0,0

3,2

AllBNEsforthisgamesatisfy

σ 1(T)=2/3and:

σ 2(X|L)=0(dominance)and

σ 2(W|R)=3−2q

5−5q

SuchgamessatisfyallBNEconditionssince:

U1(T,σ2)=2(1−q)

σ 2(W|R)=

=q+3(1−q)(1−

σ 2(W|R))=U1(D,σ2)

u 2(X,σ1|L)=0<2σ1(T)+(1−

σ 1(T))=u 2(Y,σ1|L)

u 2(W

,σ1|R)=2σ1(T)=

σ 1(T)+2(1−

σ 1(T))=u 2(Z,σ1|R)

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January2013

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Notes

Notes

DynamicGames

EC202LecturesVII&VIII

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January2013

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Summary

DynamicGames:

Definitions:

ExtensiveFormGame

InformationSetsandBeliefs

BehavioralStrategy

Subgame

SolutionConcepts:

NashEquilibrium

SubgamePerfectEquilibrium

PerfectBayesianEquilibrium

Examples:ImperfectCompetition

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Notes

Notes

DynamicGames

Allgamesdiscussedinpreviouslectureswerestatic.Thatis:

Asetofplayerstakingdecisionssimultaneously

ornotbeingabletoobservethechoicesmadebyothers

Todaywerelaxsuchassumptionbymodelingthetimingofdecisions

Incommoninstancestherulesofthegameexplicitlydefine:

theorderinwhichplayersmove

theinformationavailabletothem

whentheytaketheirdecisions

AwayofrepresentingsuchdynamicgamesisintheirExtensiveForm

Thefollowingdefinitionsarehelpfultodefinesuchnotion

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BasicGraphTheory

Agraphconsistsofasetofnodesandofasetofbranches

Eachbranchconnectsapairofnodes

Abranchisidentifiedbythetwonodesitconnects

Apathisasetofbranches:

x k,xk+1|k=1,...,m

wherem>1andeveryx kisadifferentnodeofthegraph.

Atreeisagraphinwhichanypairofnodesisconnectedbyexactly

onepath

Arootedtreeisatreeinwhichaspecialnodesdesignatedastheroot

Aterminalnodeisanodeconnectedbyonlyonebranch

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Notes

Notes

AnExtensiveFormGame

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ExtensiveFormGames

Anextensiveformgameisarootedtreetogetherwithfunctionsassigning

labelstonodesandbranchessuchthat:

1.Eachnon-terminalnodehasaplayer-labelinC,1,...,n

1,...,naretheplayersinthegame

NodesassignedtolabelC

arechancenodes

Nodesassignedtolabeli6=Caredecisionnodescontrolledbyi

2.Eachalternativeatachancenodehasalabelspecifyingits

probability:

Chanceprobabilitiesarenonnegativeandaddto1

3.Eachnodecontrolledbyplayeri>0hasasecondlabelspecifyingi’s

informationstate:

Thusnodeslabeledi.sarecontrolledbyiwithinformations

Twonodesbelongtoi.sifficannotdistinguishthem

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Notes

Notes

ExtensiveFormGames

4.Eachalternativeatadecisionnodehasmovelabel:

Iftwonodesx,ybelongtothesameinformationset,forany

alternativeatxtheremustbeexactlyonealternativeatywiththe

samemovelabel

5.Eachterminalnodeyhasalabelthatspecifiesavectorofnnumbers

ui(y) i∈1,...,nsuchthat:

Thenumberu i(y)specifiesthepayofftoiifthegameendsatnodey

6.Allplayershaveperfectrecallofthemovestheychose

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PerfectRecall

Withperfectrecallinformationsets1.2and1.3cannotcoincide:

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Notes

Notes

WithoutPerfectRecall

Withoutperfectrecallassumptionthisispossible:

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PerfectInformation

Anextensiveformgamehasperfectinformationifnotwonodesbelongto

thesameinformationstate

WithPerfectInformation

WithoutPerfectInformation

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Notes

Notes

BehavioralStrategies

Throughoutlet:

S ibethesetinformationstatesofplayeri∈N

Ai.sbetheactionsetofplayeriatinfostates∈S i

Abehavioralstrategyforplayerimapsinformationstatestoprobability

distributionsoveractions

Inparticular

σ i.s(ai.s)istheprobabilitythatplayeriatinformationstages

choosesactiona i.s∈Ai.s

Throughoutdenote:

abehavioralstrategyofplayeriby

σ i=σi.ss∈Si

aprofileofbehavioralstrategyby

σ=σii∈N

thechanceprobabilitiesby

π=π

0.ss∈S0

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ProbabilitiesoverTerminalNodes

Foranyterminalnodeyandanybehavioralstrategyprofile

σ,letP(y|σ)

denotetheprobabilitythatthegameendsatnodey

E.g.inthefollowinggameP(c|σ)=

π0.1(1)σ1.1(R)σ2.2(C)=1/9:

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Notes

Notes

ExpectedPayoffs

IfΩdenotesthesetofendnotes,theexpectedpayoffofplayeriis:

Ui(

σ)=Ui(

σ i,σ−i)=

∑y∈ΩP(y|σ)ui(y)

E.g.inthefollowinggameU1(σ)=4/9andU2(σ)=5/9

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NashEquilibrium

Definition(NashEquilibrium—NE)

ANashEquilibriumofanextensiveformgameisanyprofileofbehavioral

strategiessuchthat:

Ui(

σ)≥Ui(

σ′ i,

σ−i)forany

σ′ i∈×s∈Si∆(Ai.s)

Recallthat

σ′ iisanymappingfrom

informationsetstoprobability

distributionsoveravailableactions

ThedefinitionofNEisasinstrategicformgames

Whatdiffersisthestrategy(behavioral)thatisexpressedateverysingle

decisionstageandnotonprofilesofdecisionsforeveryindividual

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Notes

Notes

NEExample

AgamemayhavemanyNEs:

σ 1(A)=0,

σ 2(C)=0,

σ 3(E)=0

σ 1(A)=

σ 3(E)/(1+

σ 3(E)),

σ 2(C)=1/2,

σ 3(E)∈[0,1]

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NEExampleContinuedI

Profile

σ 1(A)=0,

σ 2(C)=0,

σ 3(E)=0isNEsince:

U1(B,σ−1)=1>0=U1(A,σ−1)

U2(D,σ−2)=0≥0=U2(C,σ−2)

U3(F,σ−3)=2≥2=U3(E,σ−3)

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Notes

Notes

NEExampleContinuedII

Profile

σ 1(A)=

σ 3(E)

(1+

σ 3(E)),

σ 2(C)=

1 2,

σ 3(E)∈[0,1]isNEsince:

U1(A,σ−1)=

σ 2(C)=(1−

σ 2(C))=U1(B,σ−1)

U2(D,σ−2)=

σ 1(A)=(1−

σ 1(A))

σ 3(E)=U2(C,σ−2)

U3(F,σ−3)=(1−

σ 1(A))(2−

σ 2(C))=U3(E,σ−3)

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Ultimatum

Game

NE1:

σ 1.1=[0]

σ 2.1=[A]

NE2:

σ 1.1=[1]

σ 2.2=[A]

σ 2.1=[R]

NE3:

σ 1.1=[2]

σ 2.3=[A]

σ 2.2=[R]...

NE4:

σ 1.1=[3]

σ 2.4=[A]

σ 2.3=[R]...

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Notes

Notes

Ultimatum

Game

Strategy

σ 1.1=[0],

σ 2.1=[A]isNEforany(σ2.2,σ2.3,σ2.4)since:

U1(0,σ2)=4>U1(a1,σ2)foranya 1∈1,2,3

U2(σ2,0)=0=U2(a2,0)foranya 2∈A,R4

Strategy

σ 1.1=[0],

σ 2.2=[A],

σ 2.1=[R]isNEforany(σ2.3,σ2.4)since:

U1(1,σ2)=3>U1(a1,σ2)foranya 1∈0,2,3

U2(σ2,1)=1≥U2(a2,1)foranya 2∈A,R4

Asimilarargumentworksfortheothertwoproposedequilibria

Onlythefirsttwoequilibria,however,involvethreatsthatarecredible,

sinceplayer2wouldneverwanttorefuseandofferworthatleast1$

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SubgamePerfectEquilibrium

Asuccessorofanodexisanodethatcanbereachedfrom

xforan

appropriateprofileofactions

Definition(Subgame)

Asubgameisasubsetofanextensiveformgamesuchthat:

1Itbeginsatasinglenode

2Itcontainsallsuccessors

3Ifagamecontainsaninformationsetwithmultiplenodestheneither

allofthesenodesbelongtothesubsetornonedoes

Definition(SubgamePerfectEquilibrium—SPE)

Asubgameperfectequilibrium

isanyNEsuchthatforeverysubgamethe

restrictionofstrategiestothissubgameisalsoaNEofthesubgame.

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Notes

Notes

NEbutnotSPE

ThefollowinggamehasacontinuumofNEbutonlyoneSPE:

σ 1.1(T)=1and

σ 2.1(L)=1isuniqueSPE

σ 1.1(T)=0and

σ 2.1(L)≤1/2areallNE,butnoneSPE

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NEbutnotSPE

Strategy

σ 1.1(T)=1and

σ 2.1(L)=1istheuniqueSPEsince:

U1.1(T,L)=2>1=U1.1(B,L)

U2.1(L)=3>2=U2.1(R)

Anystrategy

σ 1.1(T)=0and

σ 2.1(L)=q≤1/2isNEsince:

U1(B,σ2)=1≥2q=U1(T,σ2)

U2(L,B)=4=4=U2(R,B)

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Notes

Notes

ComputingSPE—BackwardInduction

DefinitionofSPEisdemandingbecauseitimposesdisciplineonbehavior

eveninsubgamesthatoneexpectsnottobereached

SPEhoweveriseasytocomputeinperfectinformationgames

Backward-inductionalgorithmprovidesasimpleway:

Ateverynodeleadingonlytoterminalnodesplayerspickactionsthat

areoptimalforthem

ifthatnodeisreached

Atallprecedingnodesplayerspickanactionsthatoptimalforthem

ifthatnodeisreachedknowinghowalltheirsuccessorsbehave

Andsoonuntiltherootofthetreeisreached

ApurestrategySPEexistsinanyperfectinformationgame

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BackwardInductionExample

SPE:

σ 1.2=

σ 1.3=[A],

σ 2.1=[t],any

σ 2.2and

σ 1.1=[T]

NEbutnotSPE:

σ 1.2=

σ 1.3=[A],

σ 2.1=[b],any

σ 2.2and

σ 1.1=[B]

AgainNEmaysupportemptythreats

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Notes

Notes

Duopoly:StackelbergCompetition

ImplicittobothCournotandBertrandmodelswastheassumptionthatno

producercouldobserveactionschosenbyothersbeforemakingadecision

IntheStackelbergduopolymodelhowever:

Playerschoosehowmanygoodstosupplytothemarket(asCournot)

Oneplayermovesfirst(theleader)

Whiletheotherplayermovesafterhavingobservedthedecisionof

theleader(thefollower)

Bothplayersaccountforthedistortionsthattheiroutputchoices

haveonequilibrium

prices

ToavoidemptythreatsrestrictattentiontotheSPEofthedynamicgame

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Duopoly:StackelbergCompetition

Thegameissolvedbybackwardinduction:

Considerthesubgameinwhichtheleaderhasproducedq Lunits

Inthissubgame(asinCournot)thedecisionproblemoffolloweristo:

max q Fp(q L+q F)qF−c F(qF)

Solvingsuchproblemdefinesthebestresponsetothefollowerb F(qL)

BySPEtheleadertakesthefollower’sstrategyintoaccountwhen

choosinghisoutput

Thusthedecisionproblemoftheleaderisasfollows:

max q Lp(q L+b F(qL))q L−c L(qL)

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Notes

Notes

StackelbergExample


d(q)=2−q

c F(q)=q2

andc L(q)=3q2

Theproblemsofbothplayersarerespectivelydefinedby:

maxq F(2−q L−q F)qF−c F(qF)

maxq L(2−q L−b F(qL))q L−c L(qL)

OptimalityofeachfirmisdeterminedbyFOC:

2−2qF−q L−2qF=0⇒

q F=b F(qL)=(2−q L)/4

3/2−(3

/2)q L−6qL=0⇒

q L=1/5

StackelbergEquilibriumoutputsare:

q F=9/20andq L=1/5

CournotEquilibriumoutputsare:

q F=14

/31andq L=6/31

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Duopoly:MarketEntry

Considerthefollowinggamebetweentwoproducers

Firm1istheincumbentandfirm2isthepotentialentrant

AssumeP>L>0andM>P>F

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Notes

Notes

Duopoly:MarketEntry

TofindanSPEwithsuccessfuldeterrence,noticethat:

If1doesnotinvest,itpreferstoconcedeifentrytakesplaceasP>F

Thusfirm2preferstoenterif1doesnotinvestasP>L

If1doesinvestitpreferstofightifentrytakesplace,providedthat:

F>P−K

Ifsofirm2preferstostayoutif1hasinvestedasL>0

Thusfirm1preferstoinvestanddeterentryif:

M−K>P

AnSPEexistsinwhichentryiseffectivelydeterredifthecostsatisfies:

M−P>K>P−F

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Extra:DynamicsandUncertainty

Ifanextensiveformgamedoesnotdisplayperfectinformation,subgame

perfectioncannotbeimposedateveryinformationset,butonlyon

subgames

Insuchgamesafurtherequilibrium

refinementmayhelptohighlightthe

relevantequilibriaofthegamebyselectingthosewhichBayesrule

Definition(PerfectBayesianEquilibrium—PBE)

AperfectBayesianequilibrium

ofanextensiveformgameconsistsofa

profileofbehavioralstrategiesandofbeliefsateachinformationsetofthe

gamesuchthat:

1strategiesformanSPEgiventhebeliefs

2beliefsareupdatedusingBayesruleateachinformationsetreached

withpositiveprobability

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Notes

Notes

RepeatedGames

EC202LecturesIX&X

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Summary

RepeatedGames:

Definitions:

FeasiblePayoffs

Minmax

RepeatedGame

StageGame

TriggerStrategy

MainResult:

FolkTheorem

Examples:Prisoner’sDilemma

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Notes

Notes

FeasiblePayoffs

Q:Whatpayoffsarefeasibleinastrategicformgame?

A:Aprofileofpayoffsisfeasibleinastrategicformgameifcanbe

expressedasaweighedaverageofpayoffsinthegame.

Definition(FeasiblePayoffs)

Aprofileofpayoffsw

ii∈Nisfeasibleinastrategicformgame

N, Ai,u i i∈N ift

hereexistsadistributionoverprofilesofactions

πsuchthat:

wi=

∑a∈A

π(a)ui(a)

foranyi∈N

Unfeasiblepayoffscannotbeoutcomesofthegame

Pointsonthenorth-eastboundaryofthefeasiblesetareParetoefficient

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Minmax

Q:What’stheworstpossiblepayoffthataplayercanachieveifhechooses

accordingtohisbestresponsefunction?

A:Theminmaxpayoff.

Definition(Minmax)

The(purestrategy)minmaxpayoffofplayeri∈Ninastrategicform

game N,

Ai,u i i∈N is: u i

=min

a −i∈A−i

max

a i∈A

i

u i(ai,a −i)

Mixedstrategyminmaxpayoffssatisfy:

v i=min

σ−imax σ iu i(σi,

σ−i)

Themixedstrategyminmaxisnothigherthanthepurestrategyminmax.

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Notes

Notes

Example:Prisoner’sDilemma

MinmaxPayoff:(1,1)

FeasiblePayoff:containedinredboundaries

ParetoEfficientPayoffs:(2,2;3,0)and(2,2;0,3)

StageGame

Payoffs

1\2

ws

w2,2

0,3

s3,0

1,1

01

23

0123

u1

u2

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Example:BattleoftheSexes

MinmaxPayoff:(2,2)

FeasiblePayoff:containedinredboundaries

ParetoEfficientPayoffs:(3,3)

StageGame

Payoffs

1\2

ws

w3,3

1,0

s0,1

2,2

01

23

0123

u1

u2

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Notes

Notes

RepeatedGame:Timing

ConsideranystrategicformgameG= N,

Ai,u i i∈N

CallG

thestagegame

Aninfinitelyrepeatedgamedescribesastrategicenvironmentinwhichthe

stagegameisplayedrepeatedlybythesameplayersinfinitelymanytimes

Round1

...Roundt

...

......

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RepeatedGames:PayoffsandDiscounting

Thevaluetoplayeri∈Nofapayoffstreamui(1),ui(2),...,ui(t),...is:

(1−

δ)∑

∞ t=1δt−1u i(t)

Theterm(1−

δ)amountstoasimplenormalization

...andguaranteesthataconstantstreamv,v,...hasvaluev

Futurepayoffsarediscountedatrate

δ

Aninfinitelyrepeatedgamecanbeusedtodescribestrategicenvironments

inwhichthereisnocertaintyofafinalstage

Insuchview

δdescribestheprobabilitythatthegamedoesnotendatthe

nextroundwhichwouldresultinapayoffof0thereafter

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Notes

Notes

RepeatedGames:PerfectInformationandStrategies

Todaywerestrictattentiontoperfectinformationrepeatedgames

Insuchgamesallplayerspriortoeachroundobservetheactionschosenby

allotherplayersatpreviousrounds

Leta(s)=a1(s),...,a n(s) denotetheactionprofileplayedatrounds

Ahistoryofplayuptostagetthusconsistsof:

h(t)=a(1),a(2),...,a(t−1)

Inthiscontextstrategiesmaphistories(ieinformation)toactions:

αi(h(t))∈Ai

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DilemmaFolkTheorem

Prisoner’s

Considertheprisoner’sdilemmadiscussedearlier:

1\2

ws

w2,2

0,3

s3,0

1,1

Tounderstandhowequilibrium

behaviorisaffectedbyrepetition,let’s

showwhy(2,2)isSPEoftheinfinitelyrepeatedprisoner’sdilemma

Folktheoremshowsthatanyfeasiblepayoffthatyieldstobothplayersat

leasttheirminmaxvalueisaSPEoftheinfinitelyrepeatedgameifthe

discountfactorissufficientlyhigh

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Notes

Notes

Grim

TriggerStrategies

[DrawPaths...]

Considerthefollowingstrategy(knownasgrimtriggerstrategy):

a i(t)=

wifeithera(t−1)=(w,w)ort=0

sotherwise

Ifallplayersfollowsuchstrategy,noplayercandeviateandbenefitatany

givenroundprovidedthat

δ≥1/2

Insubgamesfollowinga(t−1)=(w,w)noplayerbenefitsfrom

adeviationif:

(1−

δ )( 3+

δ+

δ2+

δ3+...) =

3−2δ≤2⇔

δ≥1/2

Insubgamesfollowinga(t−1)6=(w,w)noplayerbenefitsfrom

adeviationsince:

(1−

δ )( 0+

δ+

δ2+

δ3+...) =

δ≤1⇔

δ≤1

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FolkTheorem

Theorem

(SPEFolkTheorem)

Inanytwo-personinfinitelyrepeatedgame:

1Foranydiscountfactor

δ,thediscountedaveragepayoffofeach

playerinanySPEisatleasthisminmaxvalueinthestagegame

2Anyfeasiblepayoffprofilethatyieldstoallplayersatleasttheir

minmaxvalueisthediscountedaveragepayoffofaSPEifthe

discountfactor

δissufficientlycloseto1

3IfthestagegamehasaNEinwhicheachplayers’payoffishis

minmaxvalue,thentheinfinitelyrepeatedgamehasaSPEinwhich

everyplayers’discountedaveragepayoffishisminmaxvalue

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Notes

Notes

TestingSPEinRepeatedGames

Definition(One-DeviationProperty)

Astrategysatisfiestheone-deviationpropertyifnoplayercanincreasehis

payoffbychanginghisactionatthestartofanysubgameinwhichheisthe

first-movergivenotherplayers’strategiesandtherestofhisownstrategy

Fact

Astrategyprofileinanextensivegamewithperfectinformationand

infinitehorizonisaSPEifandonlyifitsatisfiestheone-deviationproperty

ThisobservationcanbeusedtotestwhetherastrategyprofileisaSPEof

aninfinitelyrepeatedgameaswedidinthePrisoner’sdilemmaexample

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Example

Topracticelet’sshowwhy(1.5,1.5)isalsoandSPEoftherepeatedPD:

a i(t+1)=

wif[ai(t)=sanda j(t)=wand

a(k)

/∈(s,s),(w,w)

fork<t]

orif[t=0andi=1]

sotherwise

Ifbothfollowsuchstrategy,noplayercandeviateandbenefitatanygiven

roundif

δ≥1/2

Aftera(t−1)=(s,w),(w,s),noplayerbenefitsfrom

adeviationif

δ≥1/2:

1≤(1−

δ )( 3δ+

3δ3+3δ5+...) =

3δ/(1+

δ)

(1−

δ )( 2+

δ+

δ2...) =

2−

δ≤(1−

δ )( 3+

3δ2+3δ4...) =

3/(1+

δ)

Aftera(t−1)=(w,w),(s,s),noplayerbenefitsfrom

adeviationsince:

(1−

δ )( 0+

δ+

δ2+

δ3+...) =

δ≤1⇔

δ≤1

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Notes

Notes

LastExample

Considerthefollowinggame—withminmaxpayoffsof(1,1):

1\2

AB

A0,0

4,1

B1,4

3,3

TwoPNE:(A,B)and(B,A)withpayoffs(1,4)and(4,1)

Alwaysplaying(B,B)isSPEoftherepeatedgamefor

δhighenough

Considerthefollowinggrimtriggerstrategy:

a(t)=

(B,B)ifa(s)=(B,B)foranys<t

ort=0

(B,A)ifa(s)=

(B,B)fors<z

(A,B)fors=z

forz∈0,..,t−1

(A,B)ifa(s)=

(B,B)fors<z

(B,A)fors=z

forz∈0,..,t−1

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NoProfitableDeviation

1\2

AB

A0,0

4,1

B1,4

3,3

Ifallfollowsuchstrategy,noplayercandeviateandbenefitif

δ≥1/3

Followinga(s)=(B,B)for∀s<tnoplayerdeviatessince:

(1−

δ )( 4+

δ+

δ2+

δ3+...) =

4−3δ≤3⇔

δ≥1/3

Followinga(s)=(B,B)for∀s<zanda(z)=(A,B)noonedeviatesas:

(1−

δ )( 0+

δ+

δ2+

δ3+...) =

δ≤1⇔

δ≤1

(1−

δ )( 3+

4δ+4δ2+4δ3+...) =

3+

δ≤4⇔

δ≤1

Followinga(s)=(B,B)for∀s<zanda(z)=(B,A)noonedeviates

forsymmetricreasons.

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Notes

Notes

AdverseSelection

EC202LecturesXI&XII

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AdverseSelection

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Summary

AdverseSelection:

1HiddenCharacteristics

2Uninformedpartymovesfirst

Monopoly:

Onetypeofconsumer

Multipletypesofconsumer

Competition

Definitions:

PoolingEquilibrium

SeparatingEquilibrium

InsuranceMarkets

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January2013

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Notes

Notes

AdverseSelection:MonopolySetup

Twogoodseconomy:xandy

Onefirmproducesgoodxusingy

Constantmarginalcostc

FirmchoosesapricingscheduleP(x),eg:

Uniformprice:

P(x)=px

Two-parttariff:

P(x)=

p 0+p 1x

ifx>0

Multi-parttariff:

P(x)=

p 0+p 1x

if0<x≤z

p 0+p 1z+p 2(x−z)

ifx>z

FrancescoNava(LSE)

AdverseSelection

January2013

3/27

CompleteInformation:Onecosumertype

Beginbylookingatthecompleteinformationbenchmark:

Allconsumersareallidentical

Endowmentsgivenby(ex,ey)=(0,Y)

ForafeescheduleF,thebudgetconstraintofanindividualis:

y(x)=

Y−P(x)ifx>0

Yifx=0

Preferencesoverthetwogoodsaregivenby:

U(x,y(x))=

ψ(x)+y(x)=

Y+

ψ(x)−P(x)ifx>0

Yifx=0

Assume:

ψ(0)=0,

ψx>0,

ψxx<0

FrancescoNava(LSE)

AdverseSelection

January2013

4/27

Notes

Notes

Consumer’sProblem

&ParticipationConstraints

ConsiderthedecisionproblemofaconsumerfacingscheduleP:

Aconsumerpurchasessomegoodxonlyif:

U(x,y)−U(0,Y)=

ψ(x)−P(x)≥0

(PC)

SuchconstraintisknownasParticipationConstraint(PC)

IfPC,holdsaconsumerchoosesx>0inorderto:

maxxU(x,y(x))⇒Px(x)=

ψx(x)

Forp=Px&

ϕ=

ψ−1

x,thedemandassociatedtoPis:

x∗(P)=

ϕ(p(x∗ ))if

ψ(x∗ )−P(x∗ )≥0

0if

ψ(x∗ )−P(x∗ )<0

(FOC)

FrancescoNava(LSE)

AdverseSelection

January2013

5/27

Firms’sDecisionProblem

Givensuchdemandconsiderthedecisionproblemofthefirm:

AfirmchoosesPtomaximizeprofits:

max PP(x)−cx

subjecttoPCandx=x∗(P)

PCmustholdwithequalityatx∗(P)orelsethefirmcouldincrease

profitsbyraisingpricesbyalumpsumuntilPCholds,thus:

P(x∗ )=

ψ(x∗ )

Thefirmcanineffectchoosex∗bychangingP(exploitingFOC)

Usingthesetwofactstheproblemofthemonopolist’sbecomes:

maxx

ψ(x)−cx⇒

ψx(x)=c=p(x)

Theresultingequilibrium

demandisx∗(P)=

ϕ(c)

FrancescoNava(LSE)

AdverseSelection

January2013

6/27

Notes

Notes


Afewmorecommentsonthesolutionofthefirm’sproblem:

Theoptimalpricingscheduleisatwo-parttariff:

P(x)=p 0+p 1xwithp 1=c&p 0

suchthatPCholds

⇒p 0=

ψ(ϕ(c))−p 1

ϕ(c)

Unlikeinthestandardmonopolistproblem,thesolutionofthis

problemisefficientaspricesequalmarginalcosts

Itisstillexploitativehoweverbecausebuyersareleftattheir

reservationutility:

U(x,y)−U(0,Y)=0

Thereareotherwaysofimplementingthesameoutcomesuchasa

takeitorleaveitoffer: [x,P]=[ϕ(c),

ψ(ϕ(c))]

FrancescoNava(LSE)

AdverseSelection

January2013

7/27

CompleteInformation:Multiplecosumertypes

Supposethatconsumershavemultipletypes:

Lett∈L,HdenotethetypeofaconsumerwithH>L

Let

π(t)denotetheproportionoftypestinthepopulation

Themonopolistknowsthetypeofeveryconsumer

Preferencesofaconsumeroftypetare:

U(x,y)=tψ(x)+y=

Y+tψ(x)−P(x)ifx>0

Yifx=0

Setupmeetstheregularityconditionknownassinglecrossing

condition(itrequiresindifferencecurvesoftwotypestocrossonly

once)

Consumerscannotreselltheunitspurchased

FrancescoNava(LSE)

AdverseSelection

January2013

8/27

Notes

Notes

CompleteInformation:Multiplecosumertypes

Thewithmoretypesissimilartothesingletypescenario:

ThefirmpricediscriminatesbothtypesofcostumersP(t)

Theparticipationconstraintoftypetbecomes:

U(x,y|t)−U(0,Y|t)=tψ(x)−P(x|t)≥0

(PC(t))

IfPC(t),holdsaconsumerchoosesx(t)>0inorderto:

maxxU(x,y(x)|t)⇒tψx(x)=Px(x|t)≡p(x|t)

ThedemandbytypetassociatedtoP(t)is:

x∗(P|t)=

ϕ

( p(x∗ (t)|t)

t

) iftψ(x∗ (t))−P(x∗ (t)|t)≥0

0iftψ(x∗ (t))−P(x∗ (t)|t)<0

FrancescoNava(LSE)

AdverseSelection

January2013

9/27


Givensuchdemandconsiderthedecisionproblemofthefirm:

AfirmchoosesPtomaximizeprofits:

maxP

∑tπ(t)[P(x(t)|t)−cx(t)]subjecttoPC(t)andFOC(t)

PC(t)holdswithequalityatx∗(P|t),thusP(x(t)|t)=tψ(x(t))

Thefirmcaneffectivelychoosex(t)bychangingP

Usingthesetwofactstheproblemofthemonopolist’sbecomes:

maxx(t)

∑tπ(t)[tψ(x(t))−cx(t)]⇒tψx(x(t))=c=p(x(t)|t)

Theresultingequilibrium

demandisx∗(t)=

ϕ(c

/t)

Toatypetconsumerthefirmoptimallyoffersatwo-parttariff:

P(x|t)=p 0(t)+p 1xsuchthat

:p 1=c&p 0(t)=tψ(x∗ (t))−cx∗ (t)

FrancescoNava(LSE)

AdverseSelection

January2013

10/27

Notes

Notes

IncompleteInformation:Multiplecosumertype

Ifthefirmcannotrecognizethetwotypesandknowsonly

π(t):

FirmmaystillofferseveralpricingschedulesP(t)...

...butcannotguaranteethattypetpurchasesonlyP(t)

Eachconsumerdecideswhichtypehereportstobe...

...andpaysaccordingtoP(s)ifhereportstobetypes

Thenet-payoffofaconsumeroftypetclaimingtobesis:

V(s|t)=tψ(x∗ (s))−P(x∗ (s)|s)

IfthefirmkeepsofferingthecompleteinformationP(t)...

...bothtypesofconsumerspurchaseP(L)since:

V(L|H)=(H−L)

ψ(x∗ (L))>0=V(H|H)

V(L|L)=0>(L−H)ψ(x∗ (H))=V(H|L)

FrancescoNava(LSE)

AdverseSelection

January2013

11/27

IncompleteInformation:NoPooling

Offeringthesamecontractshoweverisnotoptimalforthefirm:

Considerdecreasingp 0(H)top 0(H)>p 0(L)sothat:

V(H|H)=H

ψ(x∗ (H))−[p0(H)+p 1x∗(H)]=V(L|H)

Suchachangewouldincreasethefirm’sprofitsas:

π(H)p0(H)+

π(L)p0(L)≥p 0(L)

Theorem

(NoPooling)

Itisnotoptimalforthefirmtooffercontractsthatleadconsumerstopool

FrancescoNava(LSE)

AdverseSelection

January2013

12/27

Notes

Notes

IncompleteInfo:Participation&IncentiveConstraints

Thepreviousremarkimpliesthatthefirmwantstosatisfyboth:

Theparticipationconstraintforanytypet∈L,H:

V(t|t)≥0

(PC(t))

Theincentiveconstraintforanytypet6=s∈L,H:

V(t|t)≥V(s|t)

(IC(t))

ThefirmchoosesP(t)andineffectalsox∗(t)byexploitingFOC(t):

Px(x∗ (t)|t)=tψx(x∗ (t))

(FOC(t))

ForP(t)=P(x(t)|t),theproblemofthefirmcanbewrittenas:

maxx(t),P(t)

∑t∈L,Hπ(t)[P(t)−cx(t)]subjectto

V(t|t)≥V(s|t)foranyt∈L,H

V(t|t)≥0foranyt∈L,H

FrancescoNava(LSE)

AdverseSelection

January2013

13/27

IncompleteInformation:OptimalPricing

Priortosolvingtheproblem,noticethat:

PC(L)holdswithequality(otwfirmcanincreaseprofitsraisingP(L)):

V(L|L)=Lψ(x(L))−P(L)=0

IC(H)holdswithequality(otwfirmcanincreaseprofitsraisingP(H)):

V(H|H)=H

ψ(x(H))−P(H)=H

ψ(x(L))−P(L)=V(L|H)

PC(H)isstrict(bytheprevioustwoequalitiesandH>L):

V(H|H)=H

ψ(x(H))−P(H)>0

IC(L)isstrict(bynopoolingtheoremasotwx(H)=x(L)):

V(L|L)=Lψ(x(L))−P(L)>Lψ(x(H))−P(H)=V(H|L)

FrancescoNava(LSE)

AdverseSelection

January2013

14/27

Notes

Notes

IncompleteInformation:OptimalPricing

Thepreviousremarkssimplifythefirm’sproblemto:

max

x(t),P(t)

[ ∑t∈L,Hπ(t)[P(t)−cx(t)]] +λ

V(L|L)+

µ[V(H|H)−V(L|H)]

Firstorderoptimalityrequires: −

π(H)c+

µH

ψx(x(H))=0

(x(H))

−π(L)c+

λLψ

x(x(L))−

µH

ψx(x(L))=0

(x(L))

π(H)−

µ=0

(P(H))

π(L)−

λ+

µ=0

(P(L))

Noticethat

µ=

π(H),

λ=1andthus:

Hψx(x(H))=c

Lψx(x(L))=

c1−(π(H)/

π(L))[(H

/L)−1 ]

P(H)andP(L)arepinneddownbythetwobindingconstraints

FrancescoNava(LSE)

AdverseSelection

January2013

15/27

IncompleteInformation:NoDistortionattheTop

Noticethattheoptimalityconditionsforx(t)requirethat:

MRS xy(H)=MRTxy=c

MRS xy(L)>MRTxy=c

Thisprinciplecarriesovertomoregeneralsetupsandrequires:

Theorem

(NoDistortionattheTop)

Inthesecond-bestpricingoptimum

forthefirmthehighvaluationtypes

areofferedanondistortionary(efficient)contract

Ingeneral(ifthesingle-crossingconditionismet)secondbest-optimum

x SB(t)ifcomparedtofull-informationoptimum

x FB(t)satisfies:

x SB(H)=x FB(H)

x SB(L)<x FB(L)

x SB(L)<x SB(H)

FrancescoNava(LSE)

AdverseSelection

January2013

16/27

Notes

Notes

IncompleteInformation:Competition

Withcompetitionandfreeentryfirmsdonotrunpositiveprofits

OrelseenteringfirmswouldprofitbyofferingP′ (x|t)∈[C(x),P(x|t))

Astheywouldselltoallbuyers=⇒competitionrequiresP(x|t)=C(x)

01

23

4012

x

P(x)

,C(x

)

InblueP(x),inblackC(x),dashedinlightredx(L),indarkredProfits(L),

dashedinlightgreenx(H),indarkredProfits(H)

FrancescoNava(LSE)

AdverseSelection

January2013

17/27

Example:CompetitioninInsuranceMarkets


IndividualshavetwotypesH,L

Thefractionofindividualsoftypetis

πt

Anyindividualcanbehealthyorsick

Theprobabilityoftypetbeingsickis

σ t

Assumethat

σ H>

σ L

TheincomeofanindividualisYifhealthyandY−Kifsick

Lety tdenotetheconsumptionoftypetifhealthy&x tifsick

Preferenceoftypetsatisfy:

σ tu(x t)+(1−

σ t)u(yt)

FrancescoNava(LSE)

AdverseSelection

January2013

18/27

Notes

Notes


Theinsurancemarketiscompetitive(freeentry)

Consumerscanbuyinsurancecoveragez t∈[0,K]...

...ataunitpricep t

[ietotalpremiump tz t]

Iftheydosotheirconsumptioninthetwostatesbecomes:

y t=Y−p tz t

x t=Y−K−p tz t+z t=Y−K+(1−p t)zt

Ifsotheproblemofaconsumerbecomes:

maxz t∈[0,K]σ tu(x t)+(1−

σ t)u(yt)

Thus,FOCwithrespecttoz trequiresfortypet:

σ t(1−p t)u′ (x t)=(1−

σ t)ptu′ (y t)

FrancescoNava(LSE)

AdverseSelection

January2013

19/27


FOCcanbewrittenintermsofMRSas:

u′(xt)

u′(yt)=1−

σ tσ t

p t1−p t

Thusaconsumeroftypetwants:

FullInsurance:

z t=K

ifp t=

σ tUnderInsurance:

z t<K

ifp t>

σ tOverInsurance:

z t>K

ifp t<

σ t

Theprofitsofaninsurancecompanyaregivenby:

∑tπtzt(p t−

σ t)

thusacompanydoesnotrunalossprovidedthatp t≥

σ t

FrancescoNava(LSE)

AdverseSelection

January2013

20/27

Notes

Notes

CompetitioninInsuranceMarkets:FullInfo

Assumethatinsurancecompaniescandistinguishthetwotypes

Ifso,thecompaniessetadifferentpriceforeachtype

Sincethemarketsareperfectlycompetitiveinsurancecompanies:

Offerpricep t=

σ ttotypet∈H,L

Atsuchpricesallconsumersfullyinsure

Andeachfirmmakeszeroprofits

Thusnoentrantcouldbenefitfrom

offeringcompetingpolicies

FrancescoNava(LSE)

AdverseSelection

January2013

21/27

CompetitioninInsuranceMarkets:IncompleteInfo

Ifinsurancecompaniescannotdistinguishthetwotype:

Offeringthecompleteinformationcontractsissuboptimal...

...asallplayersclaimtobeoftypeLtopayp L=

σ L<p H

Thiscannotbeoptimalforafirmsinceitwouldrunaloss:

πHK(pL−

σ H)+

πLK(pL−

σ L)<0

Alternativelyafirmmaynotattempttodistinguishconsumers...

butmayofferapricethatwouldleadtobreakevenifallfullyinsure:

p=

πH

σ H+

πLσ L

Ifso,lowrisktypeLwantstounderinsureas

σ L<pand...

highrisktypeHwantsoverinsuranceas

σ H>pand...

wouldpickz H=K

FrancescoNava(LSE)

AdverseSelection

January2013

22/27

Notes

Notes

CompetitioninInsuranceMarkets:NoPooling

If,however,differenttypesrespondtopasdetailedabove:

AcompanycantelltypesapartasonlytypeHbuysfullinsurance...

Andpreferstoraisepricestothoseindividualsto

σ H

AconsumeroftypeHthuspreferstomimictypeL:

Buyingz Lunits(definedbyFOC(L))atpricep

Ifso,thefirmbenefitsbyofferingapolicy( p,z

) :thatispreferredbytypeLconsumersbutnotbytypeH

itentailsalowerpricep∈(σL,p)andalowerz<z L

todiscouragetypeHfrom

purchaseandtosignalthem

out

moreoversuchpolicyrunsaprofitasp>

σ L

FrancescoNava(LSE)

AdverseSelection

January2013

23/27

CompetitioninInsuranceMarkets:NoPooling

y

x

pD

p0

pF

pP

p0=

(0,0

)

pF=

(K,K

p)

pP=

(zL,

z Lp

)

pD=

(z,z

p)

Theorem

(NoPooling)

Thereisnopoolingequilibrium

inacompetitiveinsurancemarket

FrancescoNava(LSE)

AdverseSelection

January2013

24/27

Notes

Notes

InsuranceMarkets:SeparatingEquilibriamaynotExist

Thusfirmshavetoofferseparatingcontractsifanequilibrium

istoexist:

Consider(pH,zH)=(σH,K)and(pL,zL)=(σL,w)

ForplayersoftypeHtochoose(pH,zH)requiresIC:

u(Y−

σ HK)≥

σ Hu(Y−K+(1−

σ L)w)+(1−

σ H)u(Y−

σ Lw)

SimilarlyplayersoftypeLwouldchoose(pL,zL)since:

σ Lu(Y−K+(1−

σ L)w)+(1−

σ L)u(Y−

σ Lw)≥u(Y−

σ HK)

PROBLEM:if

πLishighenoughbothcontractsaredominated...

...bypoolingcontract(p′ ,z′)=(p+

ε,K−

ε )

Ifsoacompetitiveinsurancemarketmayhavenoequilibrium

Cause:Profitsfrom

eachtypedependdirectlyonhiddeninfo!

FrancescoNava(LSE)

AdverseSelection

January2013

25/27


y

x

p L

p 0

p Pp H

p 0=(

0,0)

p H=(

K,Ks

H)

p L=(

w,w

s L)

p P=(

z',z'p

')

Theorem

(NoEquilibrium)

Noequilibrium

mayexistinacompetitiveinsurancemarket

FrancescoNava(LSE)

AdverseSelection

January2013

26/27

Notes

Notes


Themagentaregion(leftplot)identifiesthepoolingcontractsthatare

profitableifpurchasedbybothtypesandthatareacceptedbybothtypes:

ifsuchregionisnon-empty(leftplot)noequilibrium

exists

iftheregionisempty(rightplot)aseparatingequilibrium

exists

y

x

p L

p 0

p H

y

x

p L

p 0

p H

FrancescoNava(LSE)

AdverseSelection

January2013

27/27

Notes

Notes

Signaling

EC202LecturesXIII&XIV

FrancescoNava

LSE

February2013

FrancescoNava(LSE)

Signaling

February2013

1/16

Summary

Signaling:

1HiddenCharacteristics

2Informedpartymovesfirst

CostlySignals:

EducationalChoice

PoolingEquilibria

SeparatingEquilibria

CostlessSignals

FrancescoNava(LSE)

Signaling

February2013

2/16

Notes

Notes

ASignalingModel

Considerthefollowingeducationalchoicemodel:

Therearetwotypesofworkersg,b

Typethavingprobability

πt

Workerscansignaltheirtypebyacquiringeducation

Differenttypeshavedifferentcoststoacquireeducation

Firmscandistinguishworkersonlybytheireducationand...

...competeonwagestohireworkersgivensuchinformation

Timing:

1Naturedeterminesthetypeofeachworker

2Workersdecidehowmucheducationtoacquire

3Firmssimultaneouslymakewageoffers(Bertrand)

4Workersdecidewhetherornottoacceptanoffer

FrancescoNava(LSE)

Signaling

February2013

3/16

Signaling:EducationalChoiceModel

Inparticularconsiderthefollowingmodel:

Individualscanacquireanylevele∈[0,1]education

Thecostofacquiringleveleeducationfortypetisc(e|t)

Supposethatcostssatisfy:

c(0|t)=0&c e(e|t)>0&c ee(e|t)>0

c e(e|g)<c e(e|b)

Supposethatfirmsofferawageschedulew(e)

Ifsothepayoffofaworkeroftypetwitheducationzis:

u(e|t)=w(e)−c(e|t)

Assumptionsoncostsandpreferenceimplythatthesinglecrossing

conditionismet(ieindifferencecurvescrossonce)

FrancescoNava(LSE)

Signaling

February2013

4/16

Notes

Notes

EducationalChoiceModel:Firms

Inthiseconomyfirmsaremodeledasfollows:

Firmsknowthattheproductivityofaworkeroftypetisf(t)

Iffirmsknewthetypeofaworker,they’dpaytypetexactlyf(t)

Bertrandcompetitionimpliesthatwagesequalproductivity

RecallthatBertrand⇒priceequalsmarginalcost

Initiallyfirmsknowonlytheprobabilitythataworkerisoftypet,

πt

Firmscanobservetheeducationaldecisionsofworkers

Afterworkershavemadetheireducationaldecision,firms:

1updatetheirbeliefsonthebasisofthisnewinformation

2pickawageschedulew(e)thatdependsoneducation

FrancescoNava(LSE)

Signaling

February2013

5/16

EducationalChoiceModel:TypesofEquilibria

Thismodelhastwodifferenttypesofequilibria:

Separatingequilibriainwhichthetwotypesofworker:

1choosedifferenteducationlevels

2arepaiddifferentwages

3prefernottomimictheothertype

Poolingequilibriainwhichthetwotypesofworker:

1choosethesameeducationlevel

2arepaidthesamewage

3prefernottobeseparatedfrom

theothertype

FrancescoNava(LSE)

Signaling

February2013

6/16

Notes

Notes

EducationalChoiceModel:SeparatingEquilibriaI

Inaseparatingequilibrium:

Workersofdifferenttypechoosedifferenteducationlevelse t

Firmsrecognizeeithertypetbyhiseducatione t

Firmspayeachtypeitsmarginalproductivityw(et)=f(t)

[Bertrandcompetition⇒

wagesequalproductivity]

Fortypestorevealthemselvesatwagew(et)itmustbethat:

f(g)−c(e g|g)≥f(b)−c(e b|g)

(IC(g))

f(b)−c(e b|b)≥f(g)−c(e g|b)

(IC(b))

Sinceeducationhasnoeffectonproductivityandbecause

badworkersareidentifiedbytheireducationanyway⇒e b=0

[Inamodelwithmoretypesonlythelowestacquiresnoeducation]

FrancescoNava(LSE)

Signaling

February2013

7/16

EducationalChoiceModel:SeparatingEquilibriaII

Thelastobservation⇒incentiveconstraintscanbewrittenas:

c(e g|g)≤f(g)−f(b)

c(e g|b)≥f(g)−f(b)

ICconditionsrequirethate g∈[e,e]withboundariesdefinedby:

c(e|g)=f(g)−f(b)=c(e|b)

Ifthetwotypesofworkerschoseinequilibrium

educationlevels

e g∈[e,e]&e b=0theex-postbeliefsofafirmare:

πg(e)=

1ife=e g

0ife=e b

Beliefsdon’thavetobepinneddownfore6=e g,eb

FrancescoNava(LSE)

Signaling

February2013

8/16

Notes

Notes

EducationalChoiceModel:SeparatingEquilibriaIII

Toguaranteethatnoworkerchoosese6=e g,ebsupposethat:

w(e)=

f(b)

ife<e g

f(g)ife≥e g

Ifsucharethewagesnogoodworkerpreferstodeviatesince:

f(g)−c(e g|g)≥f(g)−c(e|g)fore>e g

f(g)−c(e g|g)≥f(b)−c(e|g)fore<e g

Moreovernobadworkerpreferstodeviatesince:

f(b)≥f(g)−c(e|b)fore>e g

f(b)≥f(b)−c(e|b)fore<e g

Moregeneraloffequilibrium

wagesschedulesachievethesameresult,

butcomplicatetheanalysisunnecessarily

FrancescoNava(LSE)

Signaling

February2013

9/16

EducationalChoiceModel:SeparatingEquilibriaIV

ThereisamultiplicityofseparatingPerfectBayesianequilibria:

Theyarecharacterizedbytheeducationlevelssatisfying:

e g∈[e,e]&e b=0

Workersreceivetheefficientwage,namelytheirproductivity

Butnoequilibrium

isefficientsincegoodworkersloseresourcesto

signaltheirtypebyinvestingineducation

TheParetodominantequilibrium

istheoneinwhiche g=esincethe

costofacquiringeducationisthelowest

Themultiplicityismainlyduetotheunspecifiedoffequilibrium

beliefs

FrancescoNava(LSE)

Signaling

February2013

10/16

Notes

Notes

EducationalChoiceModel:SeparatingEquilibriaV

Themultiplicitydisappearsforappropriatelychosenbeliefs:

Theintuitivecriterionforoutofequilibrium

beliefssays:

e>e=⇒

πg(e)=1

Thiscriterionisreasonablebadworkersprefere=0toe>e:

f(b)>f(g)−c(e|b)

whichholdsbydefinitionofe

Iffirms’beliefsmeettheintuitivecriteriontheonlyPBEthatsurvives

istheoneinwhiche g=eande b=0

Indeedife g>e,goodworkersprefertoswitchtoesince:

f(g)−c(e g|g)<f(g)−c(e|g)

TheParetodominantequilibrium

istheonlyPBEthatsurvivesthe

intuitivecriterionandinvolvesthelowesteducationlevels

FrancescoNava(LSE)

Signaling

February2013

11/16

EducationalChoiceModel:PoolingEquilibriaI

Inapoolingequilibrium:

Allworkerschoosesameeducationlevelse∗

Firmscannotrecognizeworkersbytheireducatione∗and

payallworkerstheirexpectedproductivity:

w(e∗ )=

πgf(g)+

πbf(b)

Toguaranteethatnoworkerchoosese6=e∗supposethat:

w(e)=

f(b)

ife<e∗

πgf(g)+

πbf(b)

ife≥e∗

Fortypesnottorevealthemselvesatwagew(e∗ )itmustbethat:

w(e∗ )−c(e∗|b)≥f(b)

Suchconditionrequirese∗≤ewhereeisdefinedby:

c(e|b)=

πg[f(g)−f(b)]

FrancescoNava(LSE)

Signaling

February2013

12/16

Notes

Notes

EducationalChoiceModel:PoolingEquilibriaII

ThereisamultiplicityofpoolingPerfectBayesianequilibria:

Theyarecharacterizedbyaneducationlevele∗∈[0,e]

Workerswagesareinefficientanddifferfrom

theirproductivity

Againthemultiplicityisduetotheunspecifiedoffequilibrium

beliefs

Nopoolingequilibrium

meetstheintuitivecriterion

Forconveniencelete+

bedefinedby:

w(e∗ )−c(e∗|b)=f(g)−c(e+|b)

Iffirmsbelievethat

πg(e)=1ife>e+

andsetwagew(e)=f(g)

Thengoodworkerschoosee=e++

εwhilebadonesdonotsince:

f(g)−c(e|g)>w(e∗ )−c(e∗|g)>f(g)−c(e|b)

FrancescoNava(LSE)

Signaling

February2013

13/16

Comparison:SignalingvsAdverseSelection

Theresultsonsignalingdifferfrom

thoseonadverseselectionsince:

Thereisamultiplicityofpoolingequilibria

Thereisamultiplicityofseparatingequilibrium

Theincentiveconstraintneithertypemaybindinapooling

equilibrium

Howevermostresultcoincidewhentheintuitivecriterionisapplied:

Therearenopoolingequilibria

Thereisauniqueseparatingequilibrium

Theincentiveconstraintofthebadtypebinds

Theincentiveconstraintofthegoodtypedoesnotbind

Inefficienciesarisetoprovideincentivestothegoodtype

FrancescoNava(LSE)

Signaling

February2013

14/16

Notes

Notes

CostlessSignalsI

Thetheoryoncostlesssignalsismoreinvolved.Somemoreresultcanbe

understoodfrom

thefollowingexample:

ThereareNrisk-neutralindividuals

Allofthem

canparticipateintheproductionofapublicgood

Inparticularplayerichooseshisefforte i∈0,1

Thepublicgoodisproducedonlyifallexerte i=1

Thecostofexertingeffortc iisprivateinformationand

costsareuniformlydistributedon[0,1]—iePr(c i<b)=b

Thepreferencesofplayeriwithcostc iare—fora<1:

u i(e|ci)=a ∏

j∈Ne j−c ie i

TheuniqueBNEofthisgamehasallplayersexertingnoeffort

Thisfollowssincethereispositiveprobabilitythatc i>aforsomei

FrancescoNava(LSE)

Signaling

February2013

15/16

CostlessSignalsII

Ifasignalingstageisintroducedpriortoeffortdecisionsuchthat:

Eachagentcanannouncehiswillingnesstoexerteffort

InparticulareachagentcansayYes,Notohimexertingeffort

Whensuchsignalingstageisadded,thenthereisaBNEinwhich:

AnyagentannouncesYesifandonlyifc i≤a

Eachagentchoosese i=1ifandonlyifallsaidYes

ThisisaBNEsinceindividualsnolongerriskwastingtheireffort

However,manyotherBNEarepossibleinwhichnousefulinformationis

exchangedatthecommunicationstage.SuchBNEareknownasbabbling

equilibria.

FrancescoNava(LSE)

Signaling

February2013

16/16

Notes

Notes

MoralHazard

EC202LecturesXV&XVI

FrancescoNava

LSE

February2013

FrancescoNava(LSE)

MoralHazard

February2013

1/19

Summary

HiddenActionProblem

aka:

1MoralHazardProblem

2PrincipalAgentModel

Outline

SimplifiedModel:

CompleteInformationBenchmark

HiddenEffort

AgencyCost

GeneralPrincipalAgentModel


HiddenEffort

AgencyCost

FrancescoNava(LSE)

MoralHazard

February2013

2/19

Notes

Notes

Outline:MoralHazardProblem

Thebasicingredientsofamoralhazardmodelareasfollows:

Aprincipalandanagent,areinvolvedinbilateralrelationship

PrincipalwantsAgenttoperformsometask

Agentcanchoosehowmuchefforttodevotetothetask

Theoutcomeofthetaskispinneddownbyamixofeffortandluck

PrincipalcannotobserveeffortandcanonlymotivateAgentby

payinghimbasedontheoutcomeofthetask

Timing:

1Principalchoosesawageschedulewhichdependsonoutcome

2Agentchooseshowmuchefforttodevotethetask

3Agent’seffortandchancedeterminetheoutcome

4Paymentsaremadeaccordingtotheproposedwageschedule

FrancescoNava(LSE)

MoralHazard

February2013

3/19

AsimplePrincipal-AgentModel

Considerthefollowingsimplifiedmodel:

Ataskhastwopossiblemonetaryoutcomes: q,q

withq<q

Agentcanchooseoneoftwoeffortlevels:e,ewithe<e

Theprobabilityofthehighoutputgivenefforteis:

π(e)=Pr(q=q|e)

Assumethat

π(e)<

π(e)—iemoreeffort⇒

betteroutcomes

Principalchoosesawageschedulew

Agentisriskaverseandhispreferencesare:

U(w,e)=E[u(w,e)]

Principalisriskneutralandhispreferencesare:

V(w)=E[q−w]

FrancescoNava(LSE)

MoralHazard

February2013

4/19

Notes

Notes

SimplePrincipal-AgentModel:CompleteInfoI


PrincipalcanobservetheeffortchosenbyAgent

Principalpicksawageschedulew(e)thatdependonAgent’seffort

Agent’sreservationutilityisu—iewhathegetsifheresigns

ThustheparticipationconstraintofAgentis:

U(w(e),e)=u(w(e),e)≥u

BypickingwagesappropriatelyPrincipaldefactochooseseandw

TheproblemofPrincipalthusisto:

maxe,wE[q|e]−

w(e)+

λ[u(w(e),e)−u]

RecallthatE[q|e]=

π(e)q+

π(e)q

FrancescoNava(LSE)

MoralHazard

February2013

5/19

SimplePrincipal-AgentModel:CompleteInfoII

RecalltheproblemofPrincipal:

maxe,wE[q|e]−

w(e)+

λ[u(w(e),e)−u ]

Thelowestwagew(e)thatinduceseffortefrom

Agentis:

u(w(e),e)=u

ThusPrincipalchoosestoinduceefforte∗ifandonlyif:

e∗∈argmax

e∈e,eE[q|e]−

w(e)

Principaltheninducessucheffortchoicebyofferingwages:

w∗ (e)=

w(e)

ife=e∗

w(e)−

εife6=e∗

CompleteinfoimpliesthatFOCforthewagerequiresMC=Price:

1/u w(w,e)=

λ

FrancescoNava(LSE)

MoralHazard

February2013

6/19

Notes

Notes

SimplePrincipal-AgentModel:IncompleteInfoI

NowconsiderthecaseinwhicheffortisunobservableforPrincipal:

SupposethatPrincipalprefersAgenttoexerthighefforte

Principalcanonlyconditionwagew(q)onoutcomeq:

w(q)=

wifq=q

wifq=q

Agent’sparticipationconstraintaterequires:

U(w(q),e)=

π(e)u(w,e)+

π(e)u(w,e)≥u

(PC(e))

Agent’sincentiveconstraintguaranteesthathepickshigheffort:

U(w(q),e)≥U(w(q),e)

(IC)

FrancescoNava(LSE)

MoralHazard

February2013

7/19

SimplePrincipal-AgentModel:IncompleteInfoII

Theproblemofaprincipalwhowantstheagenttoexerteisto:

Maximizehisprofitsbychoosingw(q)subjectto:

1Agent’sparticipationconstraintate

ietheagentpreferstoexerthigheffortthantoresign

2Agent’sincentiveconstraint

ietheagentpreferstoexerthigheffortthanloweffort

ThustheLagrangianofthisproblemis:

max

w,wE[q−w(q)|e]+

λ[U(w(q),e)−u]+

µ[U(w(q),e)−U(w(q),e)]

Recallthat:

U(w(q),e)=

π(e)u(w,e)+

π(e)u(w,e)

E[q−w(q)|e]=

π(e)[q−w]+

π(e)[q−w]

FrancescoNava(LSE)

MoralHazard

February2013

8/19

Notes

Notes

SimplePrincipal-AgentModel:IncompleteInfoIII

WritingoutLagrangianexplicitlythePrincipal’sproblembecomes:

maxw,w

π(e)[q−w]+

π(e)[q−w]+

+λ[π(e)u(w,e)+

π(e)u(w,e)−u]+

+µ[π(e)u(w,e)+

π(e)u(w,e)−

π(e)u(w,e)−

π(e)u(w,e)]

Firstorderconditionsforthisproblemare:

π(e)[−1+

λu w(w,e)+

µu w(w,e)]−

π(e)µu w(w,e)=0

π(e)[−1+

λu w(w,e)+

µu w(w,e)]−

π(e)µu w(w,e)=0

Byrearrangingitispossibletoshowthat:

1Both

µand

λarepositiveifuisincreasingandconcave

2IncentiveConstraintbindssince

µ>0

3Participationconstraintathigheffortbindssince

λ>0

4Wagesw,w

arefoundbysolvingthetwoconstraintsIC&PC

FrancescoNava(LSE)

MoralHazard

February2013

9/19

SimplePrincipal-AgentModel:IncompleteInfoIV

Foranexplicitcharacterizationletubeadditivelyseparableinwande:

u(w,e)=

υ(w)+

η(e)

Firstorderconditionsinthisscenariobecome:

π(e)[−1+

λυw(w)+

µυw(w)]−

π(e)µ

υw(w)=0

π(e)[−1+

λυw(w)+

µυw(w)]−

π(e)µ

υw(w)=0

Solvingwefindthat

λ,µ>0(condition(L)parallelsthecompleteinfo):

λ=

π(e)

υw(w)+

π(e)

υw(w)>0

(L)

µ

[ 1−

π(e)

π(e)

] =π(e)

[ 1 υw(w)−

1υw(w)

] >0

(M)

υ(w)=u+

η(e)

π(e)

π(e)−

π(e)−

η(e)

π(e)

π(e)−

π(e)

υ(w)=u−

η(e)

π(e)

π(e)−

π(e)+

η(e)

π(e)

π(e)−

π(e)

FrancescoNava(LSE)

MoralHazard

February2013

10/19

Notes

Notes

SimplePrincipal-AgentModel:Example

Example:e∈0,1,u(w,e)=w−e2,u=1,

q=4,q=0,

π(1)=3/4,

π(0)=1/4

CompleteInfo:whatarew(0),w(1),e∗?

Wagesw(1)=2andw(0)=1arefoundbyPC(e):

w(1)−1=1andw(0)=1

Optimalefforte∗=1isfoundbycomparingprofits:

(3/4)(4−2)+(1

/4)(−2)=1>(1

/4)(4−1)+(3

/4)(−1)=0

IncompleteInfo:whatarew,w,iffirmwantse∗=1?

Wagesw=5/2andw=1/2arefoundbysolvingPC(1)andIC:

(3/4)(w−1)+(1

/4)(w−1)=1

(3/4)(w−1)+(1

/4)(w−1)=(1

/4)w+(3

/4)w

FrancescoNava(LSE)

MoralHazard

February2013

11/19

Principal-AgentModel

Considerageneralsetupinwhich:

Agentchoosesanyeffortlevele∈[0,1]

Agent’sreservationutilityisstillu

Thestateoftheworld

ωlivesinsomeinterval

Ω

Outputisproducedaccordingtoaproductionfunction

q=q(e,

ω)

Principalisriskneutralorriskaverseandhispreferencesare:

V(w)=E[v(q−w)]


U(w,e)=E[u(w,e)]

PrincipalmovesfirstandtakesAgent’sresponseasgiven

FrancescoNava(LSE)

MoralHazard

February2013

12/19

Notes

Notes

Principal-AgentModel:CompleteInfoI

Let’sbeginbyanalyzingthecompleteinfomodel:

PrincipalcanobserveAgent’sefforteandoutputq...

...thushecaninfer

ωbecauseheknowsq=f(e,

ω)

Agent’sparticipationconstraintremains

U(w,e)≥u

Principalcanchoosewagesthatdependonbotheand

ω...

...thisisequivalenttoPrincipalpickingbotheandw(ω)...

...sincePrincipalcouldchooseawageschedulesuchthat:

w(e,ω)=

w(ω)

ifeisoptimalforPrincipal

w′stU(w′ ,e)<uif

Otherwise

FrancescoNava(LSE)

MoralHazard

February2013

13/19

Principal-AgentModel:CompleteInfoII

ThustheproblemofPrincipalbecomes:

maxe,w(·)E[v(q(e,ω)−w(ω))]+

λ[E(u(w(ω),e))−u]

Firstorderconditionsrequirethat(notev w=v e≡v x):

E[vx(q(e,ω)−w(ω))q e(e,ω)]+

λE(ue(w(ω),e))=0

−v x(q(e,ω)−w(ω))+

λu w(w(ω),e)=0

Combiningthetwoequationsonegetsthat:

E[vx(q(e,ω)−w(ω))q e(e,ω)]=−E[ v x(q(e,ω)−w(ω))u e(w(ω),e)

u w(w(ω),e)

]IfPrincipalisriskneutralthisconditionrequires(efficiency):

MRTe,w=E[qe(e,ω)]=−E[ u e(

w(ω),e)

u w(w(ω),e)

] =MRS e,w

SolvingFOC&PCyieldstheoptimalefforteandwageschedulew(ω)

FrancescoNava(LSE)

MoralHazard

February2013

14/19

Notes

Notes

Principal-AgentModel:IncompleteInfoI

Principalobservesoutputq,butnotefforteandisthusunabletoinfer

ω:

Letf(q|e)denotetheprobabilityofoutputqgivenefforte

LetF(q|e)denotethecumulativedistributionassociatedtof(q|e)

Assumethatf(q|e)satisfies:

1Pdfofoutputhasboundedsupport[ q,q

]2Thesupport[ q,q

] ispubliclyknown

3Thesupport[ q,q

] doesnotdependone

4Ife>e′thenF(q|e)<F(q|e′ )

Defineaproportionateshiftinoutput

βz(q|z)by:

βe(q|e)=f e(q|e)/f(q|e)

SinceF(q|e)=1impliesF e(q|e)=0wegetthat:

E[βe(q|e)]=∫ q q

βe(q|e)f(q|e)dq=F e(q|e)=0

FrancescoNava(LSE)

MoralHazard

February2013

15/19

Principal-AgentModel:IncompleteInfoII

SupposethatPrincipalofferswageschedulew(q)

IfAgentparticipates,hechoosesefforttomaximizehiswellbeing:

max

e∈[0,1]U(w(q),e)=max

e∈[0,1]

∫ q qu(w(q),e)f(q|e)dq

Firstorderconditionrequires:

∫ q q[ue(w(q),e)f(q|e)+u(w(q),e)f e(q|e)]dq=0

⇒E[ue(w(q),e)]+

E[u(w(q),e)

βe(q|e)]=0

Reductioninwellbeingduetoextraeffortisexactlycompensated...

...bytheincreaseinexpectedincomeduehighereffort

PrincipaltakesAgent’sFOCasaconstraintonhisprogram

[aswasthecasewithICinthesimplifiedmodel]

FrancescoNava(LSE)

MoralHazard

February2013

16/19

Notes

Notes

Principal-AgentModel:IncompleteInfoIII

Principalchoosesthewagestomaximizehiswellbeingsubjectto:

1Agent’sparticipationconstraint(PC)

2Agent’sfirstordercondition(FOC)

InparticulartheproblemofPrincipalis:

max

w(q),eV(w(q))+

λ[U(w(q),e)−u]+

µ[∂U(w(q),e)

/∂e]=

max

w(q),eE[v(q−w(q))]+

λ[E[u(w(q),e)]−

u]+

+µ[E[ue(w(q),e)]+

E[u(w(q),e)

βe(q|e)]]

ForconvenienceassumethatAgent’sutilitysatisfiesu we=0

Morethanwithcompleteinfoifeishighsince

βe(q|e)>0

Firstorderconditionsrequire:

−v x(q−w(q))+

λu w(w(q),e)+

µu w(w(q),e)

βe(q|e)=0[w(q)]

E[v(q−w(q))

βe(q|e)]+

µ

[ ∂2E[u(w(q),e)]

∂e2

] =0[e]

FrancescoNava(LSE)

MoralHazard

February2013

17/19

Principal-AgentModel:IncompleteInfoIV

ByrearrangingtermsFOCbecome:

v x(q−w(q))

u w(w(q),e)

=λ+

µβe(q|e)foranyq

E[v(q−w(q))

βe(q|e)]+

µ

[ ∂2E[u(w(q),e)]

∂e2

] =0

Thesecondconditionimplies

µ>0since:

1E[βe(q|e)]=0&v x>0implythat

E[v(q−w(q))

βe(q|e)]>0

2Agent’ssecondorderconditionsimply

∂2E[u(w(q),e)]/

∂e2<0

ThereforeAgent’sFOCholdswithequality

ThefirstconditionrequiresthatPrincipalpaysAgent:

1Morethanwithcompleteinfoifqishighsince

βe(q|e)>0

2Lessthanwithcompleteinfoifqislowsince

βe(q|e)<0

FrancescoNava(LSE)

MoralHazard

February2013

18/19

Notes

Notes

Principal-AgentModel:IncompleteInfoV

MainConclusionswithMoralHazard:

ComparedtocompeteinfoPrincipal:

1paysAgentmorewhenoutputishigh

2paysAgentlesswhenoutputisbad

3nolongerprovidesfullinsurancetoAgentonthevariableoutput

HedoessotoprovideincentivesforAgenttoexerteffort...

...sinceafullyinsuredAgentwouldhavenomotivestoexerteffort

ThisconclusionsrelyontheinformationproblemofPrincipaland

...wouldholdevenifPrincipalwereriskneutral

TheyalwaysholdsolongasAgentisriskaverse

FrancescoNava(LSE)

MoralHazard

February2013

19/19

Notes

Notes

Externalities

EC202LecturesXVII&XVIII

FrancescoNava

LSE

February2013

FrancescoNava(LSE)

Externalities

February2013

1/24

Summary

AcommoncauseofMarketFailuresareExternalities:

1ProductionExternalities

Eg:Pollution(negative)&Research(positive)

2ConsumptionExternalities

Eg:Tobacco(negative)&Deodorant(positive)

CompetitiveOutcomeisnotParetoOptimal

SolutionstotheProblem

Taxes&Subsidies

PrivateSolutions:Reorganization

PrivateSolutions:Pseudo-Markets

CoaseTheorem

(Take1)

FrancescoNava(LSE)

Externalities

February2013

2/24

Notes

Notes

Production&ConsumptionExternalities

Definition(Externality)

Thereisanexternalitywhenanagent’sactionsdirectlyinfluencethe

choicepossibilities(productionsetorconsumptionset)ofanotheragent.

Definition(ConsumptionExernality)

Thereisaconsumptionexternalitywhenanagent’sactionsdirectly

influencetheconsumptionsetofanotheragent.

Definition(ProductionExernality)

Thereisaproductionexternalitywhenanagent’sactionsdirectly

influencetheproductionsetofanotheragent.

ClassicalexamplebyMeade:beekeeperandnearbyorchard,bothincrease

theotheragent’sproductivityandproductionpossibilities.

FrancescoNava(LSE)

Externalities

February2013

3/24

Positive&NegativeExternalities

Definition(PositiveExernality)

Thereisapositiveexternalitywhenanagent’sactionsincreasethe

choicepossibilitiesofanotheragent.

Definition(NegativeExernality)

Thereisanegativeexternalitywhenanagent’sactionsdecreasethe

choicepossibilitiesofanotheragent.

Commoncausesofexternalitiesare:

NetworkingEffects(investmentinassetsthatfacilitatecooperation)

CivicAction(goodnormsofbehaviorthatbenefitothers)

UndefinedOwnershipofResources(excessiveuseofresources)

FrancescoNava(LSE)

Externalities

February2013

4/24

Notes

Notes

ASimpleModelwithExternalitiesI

Topicisdiscussedwithasimpleexampleofproductionexternalities:

Twogoods1,2,twofirmsa,bandoneconsumerc

Good1ispollutingwhilegood2isnot

FirmaissituatedonriverA

ConsumercissituatedonriverB

Firmbissituatedaftertheconfluenceofthetworivers

Firm

aC

onsu

mer

Firm

b

FrancescoNava(LSE)

Externalities

February2013

5/24

ASimpleModelwithExternalitiesII

Firmaproducesgood1usinggood2:

y 1=f(xa 2)

Consumerchaspreferencesforthetwogoodsdefinedby:

U(xc 1,xc 2)

Firmbproducesgood2usinggood1:

y 2=g(xb 1,y1,xc 1)

itsoutputdecreaseswithriverpollutionwhichdependsonthe

quantityofgood1producedandconsumedupstream

Let(e1,e2)denotetheinitialresourcesoftheeconomy

FrancescoNava(LSE)

Externalities

February2013

6/24

Notes

Notes

ParetoOptimum

I

TheParetoOptimaofthiseconomyaresolutionsofthefollowingprogram:

max

xc 1,xc 2,y1,y2,xb 1,xa 2

U(xc 1,xc 2)subjectto

xc 1+xb 1≤e 1+y 1

(λ1)

xc 2+xa 2≤e 2+y 2

(λ2)

y 1≤f(xa 2)

(µ1)

y 2≤g(xb 1,y1,xc 1)

(µ2)

Asproductionconstraintsbindthiscorrespondsto:

max

xc 1,xc 2,xb 1,xa 2


xc 1+xb 1≤e 1+f(xa 2)

(λ1)

xc 2+xa 2≤e 2+g(xb 1,f(xa 2),xc 1)

(λ2)

FrancescoNava(LSE)

Externalities

February2013

7/24

ParetoOptimum

II

TheParetoOptimaofthiseconomyaresolutionsof:

max

xc 1,xc 2,xb 1,xa 2


e 1+f(xa 2)−xc 1−xb 1≥0

[λ1]

e 2+g(xb 1,f(xa 2),xc 1)−xc 2−xa 2≥0

[λ2]

Takingfirstorderconditionswegetthat:

U1−

λ1+

λ2g 3=0

[xc 1]

U2−

λ2=0

[xc 2]

λ2g 1−

λ1=0

[xb 1]

λ1f 1−

λ2+

λ2f 1g 2=0

[xa 2]

Solvingthesystem

ofFOCrequires:

U1

U2+g 3=g 1=1 f 1−g 2

FrancescoNava(LSE)

Externalities

February2013

8/24

Notes

Notes

ParetoOptimum

III

Thusefficiencyinthiseconomyrequires:

U1

U2+g 3=g 1=1 f 1−g 2

(PO)

WhatdoesthePOconditionrequire?

TheLHSistheSocialMRSofconsumerc

Itaccountsfortheexternalityofconsumingxc 1unitsofgood1

TheRHSistheSocialMRToffirma

Itaccountsfortheexternalityofproducingy 1unitsofgood1

ThecentraltermissimplytheMRToffirmb

Ifexternalitiesarepresent,POrequiresMRSandMRTtobeadjustedby

theirsocialvaluetoaccountfortheexternaleffectsofeachagent’s

decisionshaveontherestoftheeconomy(Pigou1920)

FrancescoNava(LSE)

Externalities

February2013

9/24

CompetitiveEquilibriumI

Optimalityinacompetitiveequilibrium

agentsrequires:

U1

U2=g 1=1 f 1

(CE)

WhatdoestheCEconditionrequire?

TheLHSisthePrivateMRSofconsumerc

Itdoesn’taccountfortheexternalityofconsumingxc 1

(over-consumption)

TheRHSisthePrivateMRToffirma

Itdoesn’taccountfortheexternalityofproducingy 1

(over-production)

ThecentraltermisthePrivateMRToffirmb

Ifexternalitiesarepresent,CEisnotPObecauseagentsonlyconsiderfor

theirprivateMRSandMRTandneglectthesocialconsequencesoftheir

behavior

FrancescoNava(LSE)

Externalities

February2013

10/24

Notes

Notes

CompetitiveEquilibriumII

CEconditioncanbederivedbysolvingtheproblemsofthe3players:

max xb 1p 2g(xb 1,y1,xc 1)−p 1xb 1

⇒p 2g 1=p 1

max xa 2p 1f(xa 2)−p 2xa 2

⇒p 1f 1=p 2

max

xc 1,xc 2

U(xc 1,xc 2)stp 1xc 1+p 2xc 2<y

⇒U1

U2=p 1 p 2

CollectingthethreeFOC,onegetsthedesiredcondition:

U1

U2=g 1=1 f 1

(CE)

FrancescoNava(LSE)

Externalities

February2013

11/24

Externalities:Remedies

SeveralremedieshavebeenproposedtofixMarketFailures(CE6=PO)

duetotoexternalities:

Quotas

SubsidiesforDepollution

RightstoPollute

PigovianTaxes

IntegrationofFirms

CompensationMechanisms

Wesaythatamechanism

internalizesanexternalityifitimplementsthe

ParetoOptimum

intheeconomy(ieifCE=PO)

FrancescoNava(LSE)

Externalities

February2013

12/24

Notes

Notes

Quotas

QuotasarethesimplestwaytoimplementPOconsumptionofgood1:

ComputePOconsumptionlevelsofgood1(xc 1, y1)

Forbidfirmafrom

producingmorethany 1

Forbidconsumercfrom

consumingmorethanxc 1

Problemswithquotasare:

ComputingPOrequiresadetailedknowledgeoftheeconomy

It’sanauthoritariansolution

It’sacommonlyusedsolution(thoughinalessbrutalform),eg:

Limitingquantitiesofpollutantsemittedbyfirmsandconsumers

LimitsinCO2emissionsofautomobiles

FrancescoNava(LSE)

Externalities

February2013

13/24

SubsidiesforDepollutionI

Anotherwaytorelaxtheexternalityistosubsidizefirmfordepollution:

Assumethatfirmacaninvestz 2unitsofgood2indepollution

Ifso,itspollutiondropsfrom

y 1toy 1−d(z2)

Theresourceconstraintforgood2consumptionbecomes:

xc 2+xa 2+z 2≤e 2+y 2

Theproductionconstraintforfirmbbecomes:

y 2≤g(xb 1,y1−d(z2),xc 1)

InwhichcasethePOconditionsbecome

U1

U2+g 3=g 1=1 f 1−g 2=1 f 1+1 d 1

(PO)

sinceFOCwithrespecttoPOz 2implythat−g 2d 1(z2)=1

FrancescoNava(LSE)

Externalities

February2013

14/24

Notes

Notes

SubsidiesforDepollutionII

ConsidertheCEofthiseconomyifthegovernmentsubsidizesdepollution:

Lets(z 2)denotethesubsidyofthegovernment

Withthesubsidiesinplacetheprogramoffirmabecomes:

max

xa 2,z2p 1f(xa 2)+s(z 2)−p 2(xa 2+z 2)

Whiletheproblemoffirmbbecomes:

max xb 1p 2g(xb 1,y1−d(z2),xc 1)−p 1xb 1

Governmentinducesthesociallyoptimallevelofdepollutionby

choosings(·)sothats 1(z2)=p 2

However,theCEforthiseconomystillrequires:

U1

U2=g 1=1 f 1

andisthereforenotPO

FrancescoNava(LSE)

Externalities

February2013

15/24

RightstoPolluteI

Thisisthepreferredsolutionbyeconomist(butnotbypolicymakers):

Inparticularconsiderthefollowingremedy:

Firmb(pollutee)sellsrightstopollute

tofirmaandconsumerc(thepolluters)

Itreceivesapricerforanypollutionrightitsellstoconsumerc

Itreceivesapriceqforanypollutionrightitsellstofirma

Ifso,thesolutiontoconsumerc’sproblembecomes:

U1

U2=p 1 p 2+r p 2

Whilethesolutiontofirma’sproblembecomes:

1 f 1=p 1 p 2−q p 2

FrancescoNava(LSE)

Externalities

February2013

16/24

Notes

Notes

RightstoPolluteII

Theproblemoffirmbismorecomplexasitneedstodecide:

onhowmuchoutputtoproduce

onhowmanypollutionrightstoselltothepolluter

Inparticularfirmbsolvesthefollowingprogram:

max

xb 1,y1,xc 1

p 2g(xb 1,y1,xc 1)−p 1xb 1+rxc 1+qy1

Optimalityconditionsforthisprogramrequire:

g 1=p 1 p 2

&−g 2=q p 2

&−g 3=r p 2

SolvingFOCforallthreeplayersimpliesefficiencysince:

U1

U2+g 3=g 1=1 f 1−g 2

FrancescoNava(LSE)

Externalities

February2013

17/24

RightstoPolluteIII

Thecreationofmarketsforrightstopollutetherefore:

implementstheParetoOptimum

requireslessinformationthanquotasasthegovernmentdoesnot

needtoknowpreferencesandtechnologiesofallindividualsandfirms

Furtherconsideration:

Notallindividualspaythesamepriceforthesamerighttopollute

Inourexampleq=ronlyifg(xb 1,y1,xc 1)=G(xb 1,y1+xc 1)

Inourexamplethereisonlyonesupplierandbuyerineachopen

pollutionmarket.Toavoidstrategicconsiderationsitwouldbebetter

ifthereweremore.

Wehavediscusseda"polluterspays"scheme.

Similarargumentsworkif"depollutionrights"marketsareopened

wherepolluteesbuyfrom

polluters.Ofcoursethedistributionof

equilibrium

utilitieswoulddiffer.

FrancescoNava(LSE)

Externalities

February2013

18/24

Notes

Notes

PigovianTaxes

Adifferentwaytosolvetheexternalityproblemisthroughtaxes:

Onecouldtaxproductionofgood1atrateT

Onecouldtaxconsumptionofgood1atratet

WhereT=qandt=r(fromthepreviousremedy)

Suchtaxrateswould:

solvetheexternalityproblemsincePO=CE

requirealotofinformationtobecomputedexactly

ThesetaxlevelsareoftencalledPigoviantaxesinhonorofPigouwhofirst

wroteaboutthem

in1928.

FrancescoNava(LSE)

Externalities

February2013

19/24

IntegrationofFirms

Forconvenienceassumethatconsumercdoesnotpollute:g 3=0.

Anotherremedytotheexternalityproblemistohavebothfirmsmerge.

Inwhichcasethemergedfirmsolvesthefollowingproblem:

max

xb 1,xa 2

p 1f(xa 2)+p 2g(xb 1,f(xa 2))−p 1xb 1−p 2xa 2

ThesolutiontothisproblemimpliesPOsinceFOCrequire:

p 1 p 2=g 1=1 f 1−g 2

Thisisnotthepreferredsolutionsince:

Itdisregardsconsiderationsofmarketpower

Bigfirmsusuallyextracthigherrents

Itdisregardspropertyrights

Firmsmayprefernottomerge

FrancescoNava(LSE)

Externalities

February2013

20/24

Notes

Notes

ACompensationMechanism

CompensatingMechanismshavebeendesignedtointernalizeexternalities

whenever:

thefirmsandconsumersknowallfundamentalsoftheeconomy

whilethegovernmentdoesnot

Suchmechanism

guaranteethatthegovernmentcan:

elicitthePigoviantaxratesfrom

producersandconsumers

setsucheffectivetaxratessothatPO=CE

Thelimitationstothisapproach(suggestedinVarian1994)arethat:

itrequiresalotofknowledgeonthepartofconsumersandproducers

itdoesnobetterthanmarketsforpollutionrights

FrancescoNava(LSE)

Externalities

February2013

21/24

CoaseTheorem

I

Inaseminalpaperof1960,Coasedoubtedthenecessityofany

governmentinterventioninpresenceofexternalities.

Hisargumentproceededasfollows:

Letb(q)denotethebenefittothepolluterofqunitsofpollution

Letc(q)denotethecosttothepolluteeofqunitsofpollution

Assumethatb′>0,b′′<0,c′>0,c′′>0

Efficientpollutionq ∗wouldrequireb′(q∗)=c′(q∗)

Ifpollutionq 0isinefficientitmustbethatb′(q0)<c′(q0)

Ifso,thepolluteecanaskthepolluter:

1toreducepollutionbysomesmallnumber

εtoq 0−

ε2inexchangeofatransfertεwheret∈(b′ (q 0),c′(q0))

Suchtradebenefitsthepollutersincet>b′(q0)

Suchtradealsobenefitsthepolluteesincet<c′(q0)

FrancescoNava(LSE)

Externalities

February2013

22/24

Notes

Notes

CoaseTheorem

II

Thepreviousargumentcanberepeatedsolongasb′<c′

Moreoverasimilarargumentworksforthecaseinwhichb′>c′

ThusCoaseconcludedthatthefollowingresulthadtohold:

Theorem

Ifpropertyrightsareclearlydefinedandtransactioncostsarenegligible,

thepartiesaffectedbyanexternalitysucceedineliminatingany

inefficiencythroughthesimplerecourseofnegotiation.

Thetwoessentialingredientsforhisclaimare:

1Negligibletransactioncosts

2Welldefinedpropertyrights

FrancescoNava(LSE)

Externalities

February2013

23/24

CoaseTheorem

III

LimitationsoftheCoaseTheorem

aredue:

1Non-negligibletransactioncosts

Infacttheresultfails:

1Ifalawyerisneededandifhechargesmorethan

ε(c′−b′)

2Ifinformationaboutcostsandbenefitsisprivate[Myersonetal1983]

2Welldefinedpropertyrights

Infacttheresultfailswhenrightsarenotwelldefined:

1Aswithopenwaterfishing

2Asforpollution

Butseveralexampleshavebeenreportedinwhichsuchbargainingoccurs

Cheung1973showsthatinUSarrangementwithsidepaymentsbetween

beekeepersandorchardsarecommon.

FrancescoNava(LSE)

Externalities

February2013

24/24

Notes

Notes

PublicGoods

EC202LecturesXIX&XX

FrancescoNava

LSE

February2013

FrancescoNava(LSE)

PublicGoods

February2013

1/17

Summary

MarketFailures—PublicGoods:

TheFreeRiderProblem:PO6=CE

PrivatebenefitsaresmallerthanSocialbenefits

ParetoOptimawithpublicgoods(BLScondition)

PrivatecontributionsarenotPO

SubscriptionEquilibrium

SolutionstotheProblem

LindahlEquilibrium

PersonalizedTaxation

PlanningProcedure

TheImportanceoftheFreeRiderProblem

LocalPublicGoods

FrancescoNava(LSE)

PublicGoods

February2013

2/17

Notes

Notes

WhatisaPublicGood?

Definitions(PrivateGoods)

Agoodisrivalifconsumptionbyanagentreducesthepossibilitiesof

consumptionbytheotheragents.

Agoodissubjecttoexclusionifyouhavetopaytoconsumethegood.

Aprivategoodisbothrivalandsubjecttoexclusion.

Definitions(PublicGoods)

Apublicgoodisnonrival.

Apurepublicgoodisbothnonrivalandnotsubjecttoexclusion.

Apurepublicgoodexampleisnationaldefense

Agoodthatissubjecttoexclusion,butnonrivalispatentedresearch

Agoodthatisnotsubjecttoexclusion,butrivalisfreeparking

FrancescoNava(LSE)

PublicGoods

February2013

3/17

TheFreeRiderProblem

Definition(FreeRiderProblem)

Freeridersarethosewhoconsumemorethantheirfairshareofapublic

resource,orbearlessthanafairshareofthecostsofitsproduction.

Freeridingisconsideredtobea"problem"onlywhenitleadstothe

under-productionofapublicgood(andthustoParetoinefficiency),or

whenitleadstotheexcessiveuseofacommonpropertyresource.

ThisproblemwasfirstformalizedbyWicksellin1896

Thoughdiscussionsabouttheunder-provisionofpublicgoodsdate

backtoAdamSmith’s"WealthofNations"in1776

The1stsolutiontotheproblemwasproposedbyLindahlin1919

FrancescoNava(LSE)

PublicGoods

February2013

4/17

Notes

Notes

ASimpleEconomywithPurePublicGoods

Considerthefollowingtwogoodseconomy:

Letndenotethenumberofconsumers(idenotesanyoneofthem)

Letxdenotetheprivategood(anaggregateofallprivategoods)

Letzdenotethepurepublicgood

Goodzisproducedusinggoodxaccordingtoaproductionfunction:

z=f(x)

Theproductionfunctionfcanalsobeexpressedasacostfunctiong:

x=g(z)=f−

1(z)

Endowmentsofthetwogoodsare(ex,ez)=(X,0)

PreferencesofconsumeriaregivenbyUi (x i,zi)

FrancescoNava(LSE)

PublicGoods

February2013

5/17

ParetoOptimalitywithPurePublicGoodsI

Thefeasibilityconstraintfortheprivategoodrequiresthat:

∑n i=1x i≤X−x

Sincegoodzisnonrival,feasibilityonlyrequiresthat:

z i≤zforanyi∈1,...,n

Let

αidenotetheParetoweightontheutilityofplayeri

TheParetooptimaarefoundbysolvingthefollowingproblem:

max

x,x1,...,xn,z,z1,...,zn

∑n i=1αiUi (x i,zi)subjectto

z≤f(x)

∑n i=1x i≤X−x

z i≤zforanyi∈1,...,n

FrancescoNava(LSE)

PublicGoods

February2013

6/17

Notes

Notes

ParetoOptimalitywithPurePublicGoodsII

Eliminatingconstraintsthatcertainlybind,theproblembecomes:

max

x 1,...,xn,z

∑n i=1αiUi (x i,z)subjectto

∑n i=1x i≤X−g(z)

(µ)

Firstorderconditionsrequirethat:

αiUi x(xi,z)=

µ(xi)

∑n i=1αiUi z(xi,z)=

µg′ (z)

(z)

SolvingthetwoequationsyieldthefollowingPOcondition:

∑n i=1Ui z(xi,z)

Ui x(xi,z)=g′ (z)=

1f′(x)

(BLS)

whichisknownastheBowen-Lindhal-Samuelsoncondition

ThusBLSimpliesthatSocialMRSequalsMRT

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PublicGoods

February2013

7/17

SubscriptionEquilibriumI

Consideraneconomyinwhichindividualsdecidehowmuchtocontribute

totheproductionofthepublicgood:

DenotebyXitheresourcesofplayeri

Denotebys itheresourcesthatidevotestotheproductionofz

Theresourceconstraintforplayerirequiresthat:

x i+s i=Xi

Theamountofpublicgoodproducedzhencesatisfies:

z=f( ∑

n i=1s i)

Theproblemofconsumeriistochoosehiscontributions igivens −i:

max

x i,siUi (x i,z)

subjectto

z=f( ∑

n i=1s i)

x i+s i=Xi

FrancescoNava(LSE)

PublicGoods

February2013

8/17

Notes

Notes

SubscriptionEquilibriumII

Sincetheconstraintsbind,theproblemofconsumerisimplifiesto:

max s iUi (Xi−s i,f( ∑

n i=1s i))

Firstorderconditionsthusrequirethat:

Ui z(xi,z)

Ui x(xi,z)=

1f′( ∑

n i=1s i)

ThusFOCimpliesthatPrivateMRSequalsMRT

FOCdiffersfrom

BLSbecauseindividualsonlycareabouttheir

privatebenefitsfrom

investment

Undergeneralconditionssubscriptionequilibrium

impliesthattoo

littlepublicgoodisproduced(iethereisfreeriderproblem)

FrancescoNava(LSE)

PublicGoods

February2013

9/17

LindahlEquilibriumI

In1919Lindahl(aSwedisheconomist)proposedthefollowingsolutionto

theaforementionedfreeriderproblem:

Assumethatpersonalpricescanbeestablished

Letp ibethepricepaidbyconsumeri

Theproducerofthepublicgoodthusreceivesaprice:

p=

∑n i=1p i

andchooseshisoutputgiventhepricetomaximizeprofits:

maxzpz−g(z)

Thusheproducesuntilmarginalcostsareequaltothepricep:

p=g′ (z)

(O1)

FrancescoNava(LSE)

PublicGoods

February2013

10/17

Notes

Notes

LindahlEquilibriumII

Consumerichoosesx i&z itomaximizeutilitygivenhispricep i:

max

x i,ziUi (x i,zi)

subjectto

x i+p iz i=Xi

FOCthereforerequirethatheequalizeshisMRStothepricep i:

Ui z(xi,z i)

Ui x(xi,z i)=p i

(O2)

Marketclearinginthepublicgoodsectorrequiresthattheindividual

demandofanyconsumerbeequaltothesupply:

z i(p)=z(p)

foranyi∈1,...,n

(MC)

FrancescoNava(LSE)

PublicGoods

February2013

11/17

LindahlEquilibriumIII

ConditionsO1andO2addedupforanyitogetherwithMCimplythat:

∑n i=1Ui z(xi,z i)

Ui x(xi,z i)=

∑n i=1p i=p=g′ (z)

whichistheBLSconditionforParetooptimality.

TheadvantageofsuchimplementationisthatitrestoresPO.

Thedisadvantageisthatitrequirestheexistenceofn"micromarkets"in

whichasoleconsumerbuysthepublicgoodathispersonalizedprice.

Thusitishardtoimplementsuchsolutionifconsumersvaluationsforthe

publicgoodareprivateinformation,sinceeveryonehasanincentiveto

underestimatehisdemandinordertopayalowerprice,

FrancescoNava(LSE)

PublicGoods

February2013

12/17

Notes

Notes

PersonalizedTaxation

Ifconsumerscanbetaxedfortheirconsumptionofthepublicgood:

Thebudgetconstraintofanyconsumeribecomes:

x i+t i(zi)=Xi

ConsumersequalizeMRStothemarginalcostofthepublicgood:

Ui z(xi,z i)

Ui x(xi,z i)=t′ i(zi)

SettingtaxratesequaltotheLindahlpricesyieldsPO:

t i(zi)=p iz i

Thisapproachhasthesameadvantagesanddisadvantagesthanthe

Lindahlequilibrium(itworksifthegovernmenthasdetailed

informationaboutthepreferencesinthepopulation)

FrancescoNava(LSE)

PublicGoods

February2013

13/17

PlanningProcedures&PivotMechanisms

Morerecentlyimplementationmechanismshavebeendevisedthatdonot

requiretheregulatortobeinformedaboutthetastesofconsumersand

thatstillrestoreParetooptimality:

1PlanningProcedures(Malinnvaud,Dreze&Poussin1971):

adynamicrevelationmechanism

thatconvergestoPO

planningofficeisuninformedaboutpreferencesintheeconomy

2PivotMechanism

(Vickrey,Clarke,Groves1971)

astaticrevelationmechanism

thatimplementsanyPO

thedesignerisuninformedaboutpreferencesintheeconomy

itimplementsPOasadominantstrategyequilibrium

FrancescoNava(LSE)

PublicGoods

February2013

14/17

Notes

Notes

PropertyofPublicGoods

Itisoftenassertedthatbecauseofthefreeriderproblempublicgoods

shouldbeprovidedbythepublicsector(eg:police,defense,justicesystem)

SuchclaimwasfirstmadebyAdamSmithinthe"WealthofNations"

Severalclassicalauthors(Mill&Samuelson)illustratedtheprinciple

throughthelighthouseexample(apurepublicgood)

In1974Coasecontestedsuchargumentpointingoutthat:

thefreeriderproblemwassimplyaproblemofpositiveexternalities

thusitcouldbesolvedbyprivatebargaining(CoaseTheorem)

Coaseargued:thatBritishlighthousesweretraditionallyaresponsibilityof

aprivatenationalcompanythatperceivedafixedrightwhichwas

dischargedbyanyshiplandinginaBritishport;andthatthiswasnot

detrimentaltoBritishnavalcommercesinceshipownersweremore

consciousofpayingforthisserviceandhadmoreincentivestomonitor

thattheservicewasrendered

FrancescoNava(LSE)

PublicGoods

February2013

15/17

TheImportanceoftheFreeRiderProblem

Inlargeeconomieswithoutpublicgoodsindividualshavenoincentiveto

gamepricesbyalteringtheirdemand(Roberts-Postlewaite1976)

Withpublicgoods,however,individualshaveincentivestounderestimate

theirdemandsinceitaffectstheircontributionsignificantly,butitaffects

theprovisionofthepublicgoodonlymarginally

Severalempiricalstudieshoweverhavepointedoutthattheimportanceof

thefreeriderproblemhasbeenexaggeratedbytheorysinceindividuals

haveatendencytowardshonesty(Bohm1972,Ledyard1995—survey)

Theconclusionsofthesestudiesarethatindividuals:

gameprices,butlessthanintheNEsubscriptiongame

contributelessifthegameisrepeated

contributemoreifallowedtocommunicate

FrancescoNava(LSE)

PublicGoods

February2013

16/17

Notes

Notes

LocalPublicGoods

Definition(LocalPublicGood)

Localpublicgoodsarepublicgoodsthatapplyonlytotheinhabitantsof

aparticulargeographicalarea(garbagecollection,publictransport,parks)

Tieboutwasthefirsttostudytheirtheoryin1956

Themainfeatureofthesemarketsisthatindividualsarefreetodecidein

whichcommunitytolive

Thusindividualswillmoveawayfrom

communitieswithtoofeworpoorly

financedpublicgoods

Tieboutshowedthatthisprocesshadanefficientequilibrium

ifthereare

perfectmobilityandperfectinformation

FrancescoNava(LSE)

PublicGoods

February2013

17/17

Notes

Notes

ChoiceUnderUncertainty

Review

FrancescoNava

LSE

January2013

FrancescoNava(LSE)


January2013

1/17

Summary

ChoiceUnderUncertainty:

StatisticsReview

Definitions:

Lottery&FairLottery

ExpectedUtility

RiskAttitudes(Aversion,Neutrality,Loving)

CertaintyEquivalent

MeasuresofRiskAversion:

RelativeRiskAversion

AbsoluteRiskAversion

Insurance,afirsttake:

ActuariallyFairInsurance

Under-insuranceatUnfairPrices

FrancescoNava(LSE)


January2013

2/17

Notes

Notes

StatisticsReview

Arandom

variableXisavariablethatrecordsthepossibleoutcomesx

ofarandom

event

Anyrandom

variableXischaracterizedby:

thesetofpossibleoutcomesthatcanoccur(X)andby

aprobabilitydistributionoverthepossibleoutcomes(f

:X→[0,1])

Givenanumericalrandom

variableX,f(thatis

X⊆

R):

TheprobabilityofX=xisdenotedbyf(x)

TheexpectedvalueoftheRVXisdenotedanddefinedby:

E(X)=

∑x∈Xxf(x)

ThevarianceoftheRVXisdenotedanddefinedby:

V(X)=

∑x∈X(x−E(x))2f(x)

FrancescoNava(LSE)


January2013

3/17

LotteriesandFairLotteries

AlotteryXisarandom

variableovermonetaryoutcomes

Anylotteryischaracterizedbyanoutcomesetandaprobability

distributionovermonetaryoutcomesX,f

Ingeneralmonetaryoutcomescanalsobenegative

AlotteryXissaidtobefairifE(X)=0

Examples:

X=2,−1 ,f(2)=1/3,f(−1)=2/3isfairsince:

E(X)=2∗(1

/3)−1∗(2

/3)=0

X=2,−1 ,f(2)=1/2,f(−1)=1/2isunfairsince:

E(X)=2∗(1

/2)−1∗(1

/2)=1/2

Suchalotterywouldbefairifanentryfeeof1/2werecharged

FrancescoNava(LSE)


January2013

4/17

Notes

Notes

ExpectedUtility&RiskPreferences

Consideradecisionmakerthathaspreferencesovermonetaryoutcomes

definedbyastrictlyincreasingutilityfunctionu

:X→

R

TheexpectedutilityofalotteryXisdefinedby:

E(u(X))=

∑x∈Xu(x)f(x)

Theexpectedutilitymaydifferfrom

theutilityoftheexpectedvalue!!!

Preferencesoverlotteries:

Anindividualisriskaverseifu′′<0

Anindividualisriskneutralifu′′=0

Anindividualisrisklovingifu′′>0

FrancescoNava(LSE)


January2013

5/17

RiskPreferences

Preferencesoverlotteries:

AriskaverseindividualprefersE(X)tothelotteryX:

E(u(X))<u(E(X))

AriskneutralindividualisindifferentbetweenalotteryXandE(X):

E(u(X))=u(E(X))

ArisklovingindividualprefersalotteryXtoE(X):

E(u(X))>u(E(X))

Example,consider

X=1,9andf(1)=f(9)=1/2:

Ifpreferencesareconcave,sayu(x)=x1

/2,wegetthat:

E(u(X))=2<√5=u(E(X))

Ifpreferenceareconvex,sayu(x)=x2,wegetthat:

E(u(X))=41>25=u(E(X))

FrancescoNava(LSE)


January2013

6/17

Notes

Notes

CertaintyEquivalent

Thecertaintyequivalentx CEofalotteryXisdefinedby:

u(x CE)=E(u(X))⇔

x CE=u−

1(E(u(X)))

Thecertaintyequivalentistheamountofmoneyx CEthatleavesthe

individualindifferentbetweenthelotteryXandthecertainoutcomex CE

ForanygivenlotteryXwehavethatif:

anindividualisriskaversethenx CE<E(X)

anindividualisriskneutralthenx CE=E(X)

anindividualisrisklovingthenx CE>E(X)

FrancescoNava(LSE)


January2013

7/17

RiskAversion

Consideralottery

X=a,b,f(a)=p,f(b)=1−pandariskaverse

individual:

m

u(m

)

CEb

u(b)

a

u(a)

c=pa

+(1

p)b

pu(a

)+(1

p)u

(b)

c

u(c)

Risk

Ave

rsio

n u'

'<0

FrancescoNava(LSE)


January2013

8/17

Notes

Notes

RiskLoving

Consideralottery

X=a,b,f(a)=p,f(b)=1−pandariskloving

individual:

m

u(m

)

CEb

u(b)

au(

a)

c=pa

+(1

p)b

pu(a

)+(1

p)u

(b)

c

u(c)

Risk

Lov

ing

u''>

0

FrancescoNava(LSE)


January2013

9/17

RelativeRiskAversion(Pratt)

Thecoefficientofrelativeriskaversionisdefinedby:

R(x)=−xu′′ (x)

u′(x)

Itisameasureofriskaversionofindividuals

PreferencesudisplayconstantrelativeriskaversionCRRAifR′=0

AnyCRRApreferencetakestheform:

u(x)=

αxγ+

βfor

α>0,

γ∈(0,1)&∀β

IfpreferencesareCRRAthenforanyk>0:

E(u(X))=u(x CE)⇔

E(u(kX))=u(kxCE)

Riskaversiondoesnotchangewithproportionalchangesinthestakes

FrancescoNava(LSE)


January2013

10/17

Notes

Notes

AbsoluteRiskAversion

Thecoefficientofabsoluteriskaversionisdefinedby:

A(x)=−u′′ (x)

u′(x)

Itisameasureofriskaversionofindividuals

PreferencesudisplayconstantrelativeriskaversionCARAifA′=0

AnyCARApreferencetakestheform:

u(x)=−

αe−

γx+

βfor

α>0,

γ>0&∀β

IfpreferencesareCARAthenforanyk>0:

E(u(X))=u(x CE)⇔

E(u(k+X))=u(k+x CE)

Riskaversiondoesnotchangewithadditivechangesinthestakes

FrancescoNava(LSE)


January2013

11/17

ASimpleInsuranceModelI

Considerthefollowingdecisionproblemfacedbyariskaverseindividual:

TherearetwopossiblestatesoftheworldH,S

TheindividualcanbehealthyHorsickS

Theprobabilityofbeingsickisp

Theincomeofanindividualis:

Yifhealthy

Y−Lifsick

Letydenotetheconsumptionifhealthyandxifsick

Preferencesatisfyu′′∈(−

∞,0)and:

pu(x)+(1−p)u(y)

FrancescoNava(LSE)


January2013

12/17

Notes

Notes

ASimpleInsuranceModelII

Consumerscanbuyinsurancecoveragez∈[0,L]

Theunitpriceofinsuranceisq

Thereforethetotalpremiumisqz

Iftheydoso,theirconsumptioninthetwostatesbecomes:

y=Y−qz

x=Y−L−qz+z=Y−L+(1−q)z

Ifsotheproblemofaconsumerbecomes:

maxzpu(x)+(1−p)u(y)

FOCwithrespecttozrequires:

p(1−q)u′(x)=(1−p)qu′ (y)

FrancescoNava(LSE)


January2013

13/17

ASimpleInsuranceModelIII

FOCcanbewrittenintermsofMRSas:

u′(x)

u′(y)=1−p

pq

1−q

Thusaconsumeroftypetwants:

FullInsurance:

z=Lifq=p

UnderInsurance:

z<Lifq>p

OverInsurance:

z>Lifq<p

Thisisthecasebecauseu′′<0implies:

q

= > <

p⇔

u′(x)

u′(y)

= > <

1⇔

x

= < >

yFrancescoNava(LSE)


January2013

14/17

Notes

Notes

ASimpleInsuranceModelIV

Supposethataninsurancecompanyissellingthecontract

Itsprofitsonthecontractsoldaregivenby:

(1−p)qz−p(1−q)z=(q−p)z

Thecompany’profitsare:

positiveifq>p

negativeifq<p

zeroifq=p

Theinsurancepriceisactuariallyfairifq=p

FrancescoNava(LSE)


January2013

15/17

FullInsuranceatFairPrices

Graphicallyfullinsuranceatactuariallyfairpricesoccurssince:

y

x

NI

FI

Y

YL

YpL

YpL

(1

p)/p

FrancescoNava(LSE)


January2013

16/17

Notes

Notes

Under-InsuranceatUnfairPrices

Graphicallyunder-insuranceoccursifq>psince:

y

x

NI

FI

Y

YL

YpL

YpL

a >

(1

p)/p

a

PI

FrancescoNava(LSE)


January2013

17/17

Notes

Notes

MoralHazardEZ

LecturesXV&XVII(Easy)

FrancescoNava

LSE

January2013

FrancescoNava(LSE)

MoralHazardEZ

January2013

1/21

Summary

HiddenActionProblem

aka:

1MoralHazardProblem

2PrincipalAgentModel

Outline

SimplifiedModel:


HiddenEffort

AgencyCost

GeneralPrincipalAgentModel


HiddenEffort

AgencyCost

FrancescoNava(LSE)

MoralHazardEZ

January2013

2/21

Notes

Notes

Outline:MoralHazardProblem

Thebasicingredientsofamoralhazardmodelareasfollows:

Aprincipalandanagent,areinvolvedinbilateralrelationship

PrincipalwantsAgenttoperformsometask

Agentcanchoosehowmuchefforttodevotetothetask

Theoutcomeofthetaskispinneddownbyamixofeffortandluck

PrincipalcannotobserveeffortandcanonlymotivateAgentby

payinghimbasedontheoutcomeofthetask

Timing:

1Principalchoosesawageschedulewhichdependsonoutcome

2Agentchooseshowmuchefforttodevotethetask

3Agent’seffortandchancedeterminetheoutcome

4Paymentsaremadeaccordingtotheproposedwageschedule

FrancescoNava(LSE)

MoralHazardEZ

January2013

3/21

AsimplePrincipal-AgentModel

Considerthefollowingsimplifiedmodel:

Ataskhastwopossiblemonetaryoutcomes: q,q

withq<q

Agentcanchooseoneoftwoeffortlevels:e1,e2withe 1<e 2

Theprobabilityofthehighoutputgivenefforte iis:

p i=Pr(q=q|e i)

Assumethatp 1<p 2—iemoreeffort⇒

betteroutcomes

Principalchoosesawageschedulew


U(w,e)=E[u(w,e)]


V(w)=E[q−w]

FrancescoNava(LSE)

MoralHazardEZ

January2013

4/21

Notes

Notes

SimplePrincipal-AgentModel:CompleteInfoI


PrincipalcanobservetheeffortchosenbyAgent

PrincipalpicksawageschedulewithatdependonAgent’seffort

Agent’sreservationutilityisu—iewhathegetsifheresigns

ThustheparticipationconstraintofAgentis:

U(w

i,e i)=u(wi,e i)≥u

BypickingwagesappropriatelyPrincipaldefactochoosese iandwi

TheproblemofPrincipalthusisto:

maxe i,wiE[q|ei]−wi+

λ[u(w

i,e i)−u]

RecallthatE[q|ei]=p iq+(1−p i)q

FrancescoNava(LSE)

MoralHazardEZ

January2013

5/21

SimplePrincipal-AgentModel:CompleteInfoII

RecalltheproblemofPrincipal:

maxe i,wiE[q|ei]−wi+

λ[u(w

i,e i)−u]

Thelowestwagev ithatinducesefforte ifrom

Agentis:

u(v i,ei)=u

ThusPrincipalchoosestoinduceefforte ∗ifandonlyif:

e ∗∈argmaxe i∈ e 1,e2E[q|ei]−v i

Principaltheninducessucheffortchoicebyofferingwages:

wi∗=

v i

ife i=e ∗

v i−

εife i6=e ∗

CompleteinfoimpliesthatFOCforthewagerequiresMC=Price:

1/u w(w

i,e i)=

λ

FrancescoNava(LSE)

MoralHazardEZ

January2013

6/21

Notes

Notes

SimplePrincipal-AgentModel:IncompleteInfoI

NowconsiderthecaseinwhicheffortisunobservableforPrincipal:

SupposethatPrincipalprefersAgenttoexerthighefforte 2

Principalcanonlyconditionwagew(q)onoutcomeq:

w(q)=

wifq=q

wifq=q

Agent’sparticipationconstraintate irequires:

U(w(q),e i)=p iu(w,ei)+(1−p i)u(w,ei)≥u

(PC(e))

Agent’sincentiveconstraintguaranteesthathepickshigheffort:

U(w(q),e 2)≥U(w(q),e 1)

(IC)

FrancescoNava(LSE)

MoralHazardEZ

January2013

7/21

SimplePrincipal-AgentModel:IncompleteInfoII

Theproblemofaprincipalwhowantstheagenttoexerte 2isto:

Maximizehisprofitsbychoosingw(q)subjectto:

1Agent’sparticipationconstraintate 2

ietheagentpreferstoexerthigheffortthantoresign

2Agent’sincentiveconstraint

ietheagentpreferstoexerthigheffortthanloweffort

ThustheLagrangianofthisproblemis:

maxw,wE[q−w(q)|e

2]+

λ[U(w(q),e 2)−u]+

+µ[U(w(q),e 2)−U(w(q),e 1)]

Recallthat:

U(w(q),e i)=p iu(w,ei)+(1−p i)u(w,ei)

E[q−w(q)|e

i]=p i[q−w]+(1−p i)[q−w]

FrancescoNava(LSE)

MoralHazardEZ

January2013

8/21

Notes

Notes

SimplePrincipal-AgentModel:IncompleteInfoIII

WritingoutLagrangianexplicitlythePrincipal’sproblembecomes:

max

w,wp 2[q−w]+(1−p 2)[q−w]+

+λ[p2u(w,e2)+(1−p 2)u(w,e2)−u]+

+µ[p2u(w,e2)+(1−p 2)u(w,e2)−p 1u(w,e1)−(1−p 1)u(w,e1)]

Firstorderconditionsforthisproblemare:

p 2[−1+

λu w(w,e2)+

µu w(w,e2)]−p 1

µu w(w,e1)=0

(1−p 2)[−1+

λu w(w,e2)+

µu w(w,e2)]−(1−p 1)µu w(w,e1)=0

Byrearrangingitispossibletoshowthat:

1Both

µand

λarepositiveifuisincreasingandconcave

2IncentiveConstraintbindssince

µ>0

3Participationconstraintathigheffortbindssince

λ>0

4Wagesw,w

arefoundbysolvingthetwoconstraintsIC&PC

FrancescoNava(LSE)

MoralHazardEZ

January2013

9/21

SimplePrincipal-AgentModel:IncompleteInfoIV

Foranexplicitcharacterizationletubeadditivelyseparableinwande:

u(w,e)=

υ(w)+

η(e)

Firstorderconditionsinthisscenariobecome:

p 2[−1+

λυw(w)+

µυw(w)]−p 1

µυw(w)=0

(1−p 2)[−1+

λυw(w)+

µυw(w)]−(1−p 1)µ

υw(w)=0

Solvingwefindthat

λ,µ>0(condition(L)parallelsthecompleteinfo):

λ=

p 2υw(w)+1−p 2

υw(w)>0

(L)

µ

[ 1−p 1 p 2

] =(1−p 2)

[ 1 υw(w)−

1υw(w)

] >0

(M)

υ(w)=u+

η(e2)

p 1p 2−p 1−

η(e1)

p 2p 2−p 1

υ(w)=u−

η(e2)1−p 1

p 2−p 1+

η(e1)1−p 2

p 2−p 1

FrancescoNava(LSE)

MoralHazardEZ

January2013

10/21

Notes

Notes

SimplePrincipal-AgentModel:ExampleI

Example:e∈0,1,u(w,e)=2w

1/2−e,u=1,

q=4,q=0,p 1=3/4,p 0=1/4

CompleteInfo:whatarew1,w0,e∗?

Wagesw1andw0arefoundbyPC(e):

2w1/2

1−1=1⇒

w1=1

2w1/2

0−0=1⇒

w0=1/4

Optimalefforte ∗=1isfoundbycomparingprofits:

3 4q+1 4q−w1>1 4q+3 4q−w0

3 44−1=2>3 4=1 44−1 4

Thustheagentisfullyinsuredbytheprincipal

FrancescoNava(LSE)

MoralHazardEZ

January2013

11/21

SimplePrincipal-AgentModel:ExampleII

IncompleteInfo:whatarew,w,ifprincipalwantse ∗=1?

Wagesw=25

/16andw=1/16arefoundbysolvingPC(1)andIC:

3 4(2w1/2−1)+1 4(2w1/2−1)=1

3 4(2w1/2−1)+1 4(2w1/2−1)=1 4(2w1/2)+3 4(2w1/2)

Ifprincipalwantse ∗=0,awagew∗=1/4satisfyingPC(0)suffices:

2w1/2

∗−0=1

Theprincipal,however,preferse ∗=1since:

3 4(q−w)+1 4(q−w)>1 4q+3 4q−w∗

3 4(4−25 16)+1 4(−

1 16)=29 16>3 4=1 44−1 4

Theprincipalcannotfullyinsuretheagentwithincomplete

informationsinceitwouldunderminetheincentivestoexerteffort

FrancescoNava(LSE)

MoralHazardEZ

January2013

12/21

Notes

Notes

Principal-AgentModel

Considerasomewhatmoregeneralsetupinwhich:

Agentchoosesanyeffortlevele∈e1,...,en

Agent’sreservationutilityisstillu

Outputqcantakeoneofmvaluesq1,...,qm

Theprobabilityofoutputtakesvalueq jgivenefforte iisp ij>0


V(w)=E[q−w]


U(w,e)=E[u(w)−e]

PrincipalmovesfirstandtakesAgent’sresponseasgiven

FrancescoNava(LSE)

MoralHazardEZ

January2013

13/21

Principal-AgentModel:CompleteInfoI

Let’sbeginbyanalyzingthecompleteinfomodel:

PrincipalcanobserveAgent’sefforteandoutputq

Agent’sparticipationconstraintremains:

U(w,e)≥u

Principalcanchoosewageswijthatdependonbothe iandq j...

...thisisequivalenttoPrincipalpickingbothe iandwj...

...sincePrincipalcouldchooseawageschedulesuchthat:

wij=

wj

ife iisoptimalforPrincipal

wstU(w,ei)<uife iisnotoptimalforPrincipal

FrancescoNava(LSE)

MoralHazardEZ

January2013

14/21

Notes

Notes

Principal-AgentModel:CompleteInfoII

ThustheproblemofPrincipalbecomes:

maxi,w

∑n j=1p ij[q j−wij]+

λ∑n j=1p ij[u(wij)−e i−u]

Firstorderconditionswithrespecttowagesrequirethat:

1u′(w

ij)=

λ⇒

wijisindependentofiandj

PrincipalinsurestheriskaverseAgent

SincePCbindsattheoptimaleffortletwi=u−

1(ei+u)

Principalchoosestheeffortlevele∗tomaximizeprofits:

maxi∈1,...,n∑n j=1p ijqj−wi=maxi∈1,...,n∑n j=1p ijqj−u−

1(ei+u)

Infact,efforte ∗canbesustainedbythewageschedule:

wij=

u−

1(e∗+u)

ife i=e ∗

w<u−

1(ei+u)

ife i6=e ∗

ThisisefficientsinceAgentisfullyinsuredagainstrisk

FrancescoNava(LSE)

MoralHazardEZ

January2013

15/21

Principal-AgentModel:IncompleteInfoI

Principalobservesoutputq,butnotefforte:

Wagescanonlydependonoutputwj

Agentchooseshiseffortinprivate

GivenawageschedulewjAgent’sproblembecomes:

maxi∈1,...,n∑n j=1p ij[u(wj)−e i]

Thus,ifAgentchoosesefforte i,then(n−1 )incentiveconstraints:

∑n j=1p ij[u(wj)−e i]≥

∑n j=1p kj[u(wj)−e k]

(IC(k))

mustbindforanyk6=i.

Moreover,Agent’sPCmuststillbindattheoptimaleffort

FrancescoNava(LSE)

MoralHazardEZ

January2013

16/21

Notes

Notes

Principal-AgentModel:IncompleteInfoII

Principalchoosesthewageswjtomaximizehiswellbeingsubjectto:

1Agent’sparticipationconstraint(PC)

2Agent’sincentiveconstraints(IC)

InparticulartheproblemofPrincipalis:

maxw,i∑n j=1p ij[q j−wj]+

λ∑n j=1p ij[u(wj)−e i−u]+

+∑k6=iµk

[ ∑n j=1[piju(wj)−p kju(w

j)]−

e i+e k]

Firstorderconditionswithrespecttowagewjrequires:

1u′(w

j)=

λ+

∑k6=iµk

[ 1−p kj

p ij

]Thewagesarenolongerconstant⇒

Agentisnotfullyinsured

Principalpaysmorewhenoutputrevealsthatanactionmore

favorabletohimislikelytohavebeenchosenbyAgent

FrancescoNava(LSE)

MoralHazardEZ

January2013

17/21

Principal-AgentModel:IncompleteInfoIII

MainConclusionswithMoralHazard:

ComparedtocompeteinfoPrincipal:

1paysAgentmorewhenoutputishigh

2paysAgentlesswhenoutputisbad

3nolongerprovidesfullinsurancetoAgentonthevariableoutput

HedoessotoprovideincentivesforAgenttoexerteffort...

...sinceafullyinsuredAgentwouldhavenomotivestoexerteffort

ThisconclusionsrelyontheinformationproblemofPrincipaland

...wouldholdevenifPrincipalwereriskaverse

TheyalwaysholdsolongasAgentisriskaverse

FrancescoNava(LSE)

MoralHazardEZ

January2013

18/21

Notes

Notes

Principal-AgentModel:ExampleI

Example:e∈0,1

/2,1,u(w,e)=2w

1/2−e,u=1,

q=4,q=0,p 1=3/4,p 1

/2=1/2,p 0=1/4

CompleteInfo:whatarew0,w1/2,w1,e∗?

Wagesw0,w1/2andw1arefoundbyPC(e):

2w1/2

1−1=1⇒

w1=1

2w1/2

1/2−1/2=1⇒

w1/2=9/16

2w1/2

0−0=1⇒

w0=1/4

Optimalefforte∗=1isfoundbycomparingprofits:

3 4q+1 4q−w1>1 2q+1 2q−w1/2>1 4q+3 4q−w0

3 44−1=2>1 24−9 16=23 16>1 44−1 4=3 4

Thustheagentisfullyinsuredbytheprincipalatw1

FrancescoNava(LSE)

MoralHazardEZ

January2013

19/21

Principal-AgentModel:ExampleII

IncompleteInfo:whatarew,w,ifPwantse∗=1?

Wagesw=25

/16andw=1/16arefoundbysolvingPC(1)andIC:

3 4(2w1/2)+1 4(2w1/2)−1=1

3 4(2w1/2)+1 4(2w1/2)−1=1 2(2w1/2)+1 2(2w1/2)−1 2

≥1 4(2w1/2)+3 4(2w1/2)

IfPwantse∗=1/2,sameconstraintsbindinthisexample,thus:

w=25

/16

&w=1/16

IfPwantse∗=0,awagew=1/4satisfyingPC(0)suffices:

2w1/2−0=1

FrancescoNava(LSE)

MoralHazardEZ

January2013

20/21

Notes

Notes

Principal-AgentModel:ExampleIII

Theprincipalinthisexample,however,preferse∗=1since:

3 4(q−w)+1 4(q−w)>1 2(q−w)+1 2(q−w)>1 4q+3 4q−w

3 4(4−25 16)+1 4(−

1 16)>1 2(4−25 16)+1 2(−

1 16)>1 44−1 4

Ingeneralthewagesthatsupporte∗=1/2ande∗=1donotneed

tobethesame!!!

Thebindingconstraintscouldchangeleadingtochangesinwages

InthisexamplethisdoesnothappensincealltheICbindatonce

FrancescoNava(LSE)

MoralHazardEZ

January2013

21/21

Notes

Notes

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