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?
Hand In —Problem Set 2 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 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
Hand In —Problem Set 3 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 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
Hand In —Problem Set 4 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 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?
Hand In —Problem Set 5 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 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?
Microeconom
icPrinciplesIIEC202
Intr
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Lec
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0
Fra
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LondonSch
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Jan
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FrancescoNava
(LondonSch
oolofEco
nomics)
Microeconomic
PrinciplesIIEC202
January2013
1/10
Introdu
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FrancescoNava
(LondonSch
oolofEco
nomics)
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Not
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FrancescoNava
(LondonSch
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PrinciplesIIEC202
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(LondonSch
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FrancescoNava
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Not
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WeeklyCourseProgram
(Weeks
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.6.1
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up
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tary
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2.8.
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9.5
FrancescoNava
(LondonSch
oolofEco
nomics)
Microeconomic
PrinciplesIIEC202
January2013
7/10
CourseRequirements:Classwork
Cla
sses
follo
wle
ctu
res
wit
ha
one/
two
wee
ksla
g
Th
ew
ork
for
each
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FrancescoNava
(LondonSch
oolofEco
nomics)
Microeconomic
PrinciplesIIEC202
January2013
8/10
Not
es
Not
es
CourseRequirements:Examination
Th
ere
will
be
asi
ngl
eth
ree-
hou
r,fo
ur-
qu
esti
onp
aper
Th
e20
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er5
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oth
erq
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tion
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ew
orth
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FrancescoNava
(LondonSch
oolofEco
nomics)
Microeconomic
PrinciplesIIEC202
January2013
9/10
Materials
Ma
inT
extb
oo
k
Mic
roec
onom
ics,
Cow
ell,
Oxf
ord
Un
iver
sity
Pre
ss,
2005
[C]
Slid
es
Incl
ud
em
ater
ials
req
uir
edfo
rth
eev
alu
atio
ns
un
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erw
ise
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illb
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gges
ted
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ein
tere
sted
read
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pp
len
tary
an
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na
tive
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tbo
ok
An
Intr
od
uct
ion
toG
ame
Th
eory
,O
sbor
ne,
Oxf
ord
Pre
ss,
2003
[O]
Eco
nom
ics
ofC
ontr
acts
,S
alan
ie,
MIT
Pre
ss,
2005
[S]
FrancescoNava
(LondonSch
oolofEco
nomics)
Microeconomic
PrinciplesIIEC202
January2013
10/10
Not
es
Not
es
StaticCompleteInformationGames
EC202—LecturesI&II
FrancescoNava
LSE
January2013
FrancescoNava(LSE)
StaticCompleteInformationGames
January2013
1/27
Summary
GamesofCompleteInformation:
Definitions:
Game:Players,Actions,Payoffs
Strategy
BestResponse
SolutionConcepts[inpurestrategies]:
DominantStrategyEquilibrium
NashEquilibrium
PropertiesofNashEquilibria:
Non-ExistenceandMultiplicity
Inefficiency
Examples
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January2013
2/27
Notes
Notes
IntroductiontoGames
Anyenvironmentinwhichthechoicesofanindividualaffectthewellbeing
ofotherscanbemodeledasagame.
Whatpinsdownaspecificgame:
Whoparticipatesinagame
[Players]
Thechoicesthatparticipantshave
[Choices]
Thewellbeingofindividuals
[Payoffs]
Theinformationthatindividualshave
[RulesoftheGame]
Thetimingofeventsanddecisions
[RulesoftheGame]
FrancescoNava(LSE)
StaticCompleteInformationGames
January2013
3/27
IntroductiontoGames
InmostmodelsdiscussedinMT:
individualdecisionsdidnotaffectthewellbeingofothers
anydependencewouldjusthingefrom
equilibrium
prices
LecturesI,II&IIIdiscusscompleteinformationstrategicform
games.
Insuchenvironments:
Individualsknoweverything,butforthedecisionsmadebyothers
Alldecisionstakeplaceatonce
FrancescoNava(LSE)
StaticCompleteInformationGames
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)
StaticCompleteInformationGames
January2013
5/27
RepresentingSimoultaneousMoveCompleteInfoGames
StrategicForm
1\2
sm
s5,2
1,2
m0,0
3,5
ExtensiveForm
FrancescoNava(LSE)
StaticCompleteInformationGames
January2013
6/27
Notes
Notes
RepresentingSimoultaneousMoveCompleteInfoGames
StrategicForm
1\2
sm
s5,2
1,2
m0,0
3,5
ExtensiveForm
FrancescoNava(LSE)
StaticCompleteInformationGames
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)
StaticCompleteInformationGames
January2013
8/27
Notes
Notes
InformationandPureStrategies
Astrategyinagame:
isamapfrom
informationintoactions
itdefinesaplanofactionforaplayer
Inacompleteinformationstrategicformgame:
playershavenoprivateinformation
playersactsimultaneously
Inthiscontextastrategyisanyelementofthesetofactions
Forinstancea(pure)strategyforplayeriissimplya i∈Ai
FrancescoNava(LSE)
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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)
StaticCompleteInformationGames
January2013
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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,-
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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,-
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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
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DominantStrategyEquilibrium
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)
StaticCompleteInformationGames
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]
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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)
StaticCompleteInformationGames
January2013
16/27
Notes
Notes
Dominance:FinalConsiderations
Dominanceisoftenconsideredabenchmarkofrationality:
Rationalplayersneverchoosedominatedstrategies
Commonknowledgeofrationalitymeans:
playersonlyemploystrategiesthatsurviveiterativeelimination
Dominanceisasimpleconceptbuywithimportantlimitations:
Oftenthereisnodominantstrategyevenafteriteration
Itoftenleadstoinefficientoutcomes
Thusaweakernotionofequilibrium
needstobeintroducedtomodel
behaviorespeciallyforrichersetups
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StaticCompleteInformationGames
January2013
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NashEquilibrium:Introduction
Dominancewastheappropriatesolutionconceptifplayershadno
informationorbeliefsaboutchoicesmadebyothers
Theweakernotionofequilibrium
thatwillbeintroducedpresumesthat:
playershavecorrectbeliefsaboutchoicesmadebyothers
playerschoicesareoptimalgivensuchbeliefs
theenvironmentiscommonknowledgeamongplayers
Suchmodelallowsfortighterpredictionswhendominancehasnobite
FrancescoNava(LSE)
StaticCompleteInformationGames
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)
StaticCompleteInformationGames
January2013
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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)
StaticCompleteInformationGames
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)
StaticCompleteInformationGames
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
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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)
StaticCompleteInformationGames
January2013
23/27
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)
StaticCompleteInformationGames
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)
StaticCompleteInformationGames
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]
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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)
StaticCompleteInformationGames
January2013
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DSEimpliesNE
Fact
Anydominantstrategyequilibrium
isaNashequilibrium
Proof.
IfaisaDSEthena i∈b i(a′ −i)foranya′ −
iandi∈N.
WhichimpliesaisNEsincea i∈b i(a−i)foranyi∈N.
FrancescoNava(LSE)
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January2013
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Notes
Notes
StaticCompleteInformationGames
EC202—LectureIII:MixedStrategies
FrancescoNava
LSE
January2013
FrancescoNava(LSE)
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January2013
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Summary
GamesofCompleteInformation:
Definitions:
MixedStrategy
DominantStrategy
SolutionConcepts:
NashEquilibrium
Examples
NEExistence
FrancescoNava(LSE)
StaticCompleteInformationGames
January2013
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Notes
Notes
IntroductiontoMixedStrategies
Aproblematicaspectofthesolutionconceptsdiscussedinthefirsttwo
lectureswasthatequilibriadidnotalwaysexist.
Reasonsforthelackofexistencewere:
Non-convexitiesinthechoicesets
Discontinuitiesofthebestresponsecorrespondences
Todayweintroducemixedstrategieswhichsolvebothproblemsand
guaranteeexistenceofatleastaNashequilibrium.
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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)
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Notes
Notes
Simplex
ThesetofpossibleprobabilitydistributionsonasetBis:
calledsimplex
anddenotedby
∆(B)
Suchsetis:
Closed
Bounded
Convex
FrancescoNava(LSE)
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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)
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January2013
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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)
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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)
StaticCompleteInformationGames
January2013
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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
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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)
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January2013
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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)
StaticCompleteInformationGames
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)
StaticCompleteInformationGames
January2013
12/15
Notes
Notes
ThreePlayerExample
Agamewithmorethan2players:
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)
StaticCompleteInformationGames
January2013
13/15
NashEquilibriumExistence
Theorem
(NEExistence)
AnygamewithafinitenumberofactionspossessesaNashequilibrium.
Theorem
(PNEExistence)
Anygamewithconvexandcompactactionsetsandwithcontinuousand
quasi-concavepayofffunctionspossessesapurestrategyNashequilibrium.
Assumptionsinboththeoremsguaranteethatbestresponsemapsare
continuous(upper-hemicontinuous,closedandconvexvalued)onconvex
compactsetsandthusexistence...
FrancescoNava(LSE)
StaticCompleteInformationGames
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)
StaticCompleteInformationGames
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
Considerthefollowingeconomy:
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)
︸︷︷
︸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)
GamesofIncompleteInformation
January2013
1/22
Summary
GamesofIncompleteInformation:
Definitions:
IncompleteInformationGame
InformationStructureandBeliefs
Strategies
BestReplyMap
SolutionConceptsinPureStrategies:
DominantStrategyEquilibrium
BayesNashEquilibrium
Examples
EXTRA:MixedStrategies&BayesNashEquilibria
FrancescoNava(LSE)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
January2013
9/22
DominantStrategyEquilibrium
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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$isweaklydominantforthebuyer
FrancescoNava(LSE)
GamesofIncompleteInformation
January2013
15/22
BNEExampleII:EntryGame
Considerthefollowingmarketgame:
FirmI(theincumbent)isamonopolistinamarket
FirmE(theentrant)isconsideringwhethertoenterinthemarket
IfEstaysoutofthemarket,Erunsaprofitof1$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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
GamesofIncompleteInformation
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)
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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)
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January2013
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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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Summary
DynamicGames:
Definitions:
ExtensiveFormGame
InformationSetsandBeliefs
BehavioralStrategy
Subgame
SolutionConcepts:
NashEquilibrium
SubgamePerfectEquilibrium
PerfectBayesianEquilibrium
Examples:ImperfectCompetition
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January2013
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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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January2013
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Notes
Notes
WithoutPerfectRecall
Withoutperfectrecallassumptionthisispossible:
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January2013
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PerfectInformation
Anextensiveformgamehasperfectinformationifnotwonodesbelongto
thesameinformationstate
WithPerfectInformation
WithoutPerfectInformation
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January2013
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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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January2013
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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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January2013
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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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January2013
21/30
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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January2013
22/30
Notes
Notes
ComputingSPE—BackwardInduction
DefinitionofSPEisdemandingbecauseitimposesdisciplineonbehavior
eveninsubgamesthatoneexpectsnottobereached
SPEhoweveriseasytocomputeinperfectinformationgames
Backward-inductionalgorithmprovidesasimpleway:
Ateverynodeleadingonlytoterminalnodesplayerspickactionsthat
areoptimalforthem
ifthatnodeisreached
Atallprecedingnodesplayerspickanactionsthatoptimalforthem
ifthatnodeisreachedknowinghowalltheirsuccessorsbehave
Andsoonuntiltherootofthetreeisreached
ApurestrategySPEexistsinanyperfectinformationgame
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January2013
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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
Considerthefollowingeconomy:
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
FrancescoNava(LSE)
RepeatedGames
January2013
15/16
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.
FrancescoNava(LSE)
RepeatedGames
January2013
16/16
Notes
Notes
AdverseSelection
EC202LecturesXI&XII
FrancescoNava
LSE
January2013
FrancescoNava(LSE)
AdverseSelection
January2013
1/27
Summary
AdverseSelection:
1HiddenCharacteristics
2Uninformedpartymovesfirst
Monopoly:
Onetypeofconsumer
Multipletypesofconsumer
Competition
Definitions:
PoolingEquilibrium
SeparatingEquilibrium
InsuranceMarkets
FrancescoNava(LSE)
AdverseSelection
January2013
2/27
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
Firms’sDecisionProblem
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
Firms’sDecisionProblem
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
Considerthefollowingeconomy:
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
Example:CompetitioninInsuranceMarkets
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
Example:CompetitioninInsuranceMarkets
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
InsuranceMarkets:SeparatingEquilibriamaynotExist
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
InsuranceMarkets:SeparatingEquilibriamaynotExist
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
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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
CompleteInformationBenchmark
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
Beginbylookingatthecompleteinformationbenchmark:
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)
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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)
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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)]
Agentisriskaverseandhispreferencesare:
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)
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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)
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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
U(xc 1,xc 2)subjectto
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)
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February2013
7/24
ParetoOptimum
II
TheParetoOptimaofthiseconomyaresolutionsof:
max
xc 1,xc 2,xb 1,xa 2
U(xc 1,xc 2)subjectto
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
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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)
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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
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February2013
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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
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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
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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
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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
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February2013
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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)
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February2013
18/24
Notes
Notes
PigovianTaxes
Adifferentwaytosolvetheexternalityproblemisthroughtaxes:
Onecouldtaxproductionofgood1atrateT
Onecouldtaxconsumptionofgood1atratet
WhereT=qandt=r(fromthepreviousremedy)
Suchtaxrateswould:
solvetheexternalityproblemsincePO=CE
requirealotofinformationtobecomputedexactly
ThesetaxlevelsareoftencalledPigoviantaxesinhonorofPigouwhofirst
wroteaboutthem
in1928.
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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
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February2013
20/24
Notes
Notes
ACompensationMechanism
CompensatingMechanismshavebeendesignedtointernalizeexternalities
whenever:
thefirmsandconsumersknowallfundamentalsoftheeconomy
whilethegovernmentdoesnot
Suchmechanism
guaranteethatthegovernmentcan:
elicitthePigoviantaxratesfrom
producersandconsumers
setsucheffectivetaxratessothatPO=CE
Thelimitationstothisapproach(suggestedinVarian1994)arethat:
itrequiresalotofknowledgeonthepartofconsumersandproducers
itdoesnobetterthanmarketsforpollutionrights
FrancescoNava(LSE)
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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)
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February2013
22/24
Notes
Notes
CoaseTheorem
II
Thepreviousargumentcanberepeatedsolongasb′<c′
Moreoverasimilarargumentworksforthecaseinwhichb′>c′
ThusCoaseconcludedthatthefollowingresulthadtohold:
Theorem
Ifpropertyrightsareclearlydefinedandtransactioncostsarenegligible,
thepartiesaffectedbyanexternalitysucceedineliminatingany
inefficiencythroughthesimplerecourseofnegotiation.
Thetwoessentialingredientsforhisclaimare:
1Negligibletransactioncosts
2Welldefinedpropertyrights
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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)
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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)
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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)
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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
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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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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
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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)
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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)
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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)
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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,
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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)
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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
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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
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PublicGoods
February2013
16/17
Notes
Notes
LocalPublicGoods
Definition(LocalPublicGood)
Localpublicgoodsarepublicgoodsthatapplyonlytotheinhabitantsof
aparticulargeographicalarea(garbagecollection,publictransport,parks)
Tieboutwasthefirsttostudytheirtheoryin1956
Themainfeatureofthesemarketsisthatindividualsarefreetodecidein
whichcommunitytolive
Thusindividualswillmoveawayfrom
communitieswithtoofeworpoorly
financedpublicgoods
Tieboutshowedthatthisprocesshadanefficientequilibrium
ifthereare
perfectmobilityandperfectinformation
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February2013
17/17
Notes
Notes
ChoiceUnderUncertainty
Review
FrancescoNava
LSE
January2013
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ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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)
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ChoiceUnderUncertainty
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
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ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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
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ChoiceUnderUncertainty
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
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ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
January2013
15/17
FullInsuranceatFairPrices
Graphicallyfullinsuranceatactuariallyfairpricesoccurssince:
y
x
NI
FI
Y
YL
YpL
YpL
(1
p)/p
FrancescoNava(LSE)
ChoiceUnderUncertainty
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)
ChoiceUnderUncertainty
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:
CompleteInformationBenchmark
HiddenEffort
AgencyCost
GeneralPrincipalAgentModel
CompleteInformationBenchmark
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
Agentisriskaverseandhispreferencesare:
U(w,e)=E[u(w,e)]
Principalisriskneutralandhispreferencesare:
V(w)=E[q−w]
FrancescoNava(LSE)
MoralHazardEZ
January2013
4/21
Notes
Notes
SimplePrincipal-AgentModel:CompleteInfoI
Beginbylookingatthecompleteinformationbenchmark:
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
Principalisriskneutralandhispreferencesare:
V(w)=E[q−w]
Agentisriskaverseandhispreferencesare:
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