advanced microeconomics - forsiden · advanced microeconomics introduction i what you have...

Advanced Microeconomics


ECON5200 - Fall 2012


Introduction

I What you have done:

- consumers maximize their utility subject to budget constraintsand firms maximize their profits given technology and marketprices;

- no strategic behavior.

I What we will do:

- in many interesting situations, agents’optimal behaviordepends on the other agents’behavior;

- strategic behavior.

I Game theory provides a language to analyze such strategicsituations;

I Countless number of examples! Auctions, Bargaining, Pricecompetition, Civil Conflicts. . .


Introduction

Road map

I Static Game:

1. With Complete Information (I);

2. With Incomplete Information (II).

I Dynamic Game:

1. With Complete Information (II-III);

2. With Incomplete Information (III).


Strategic Games with Complete Information

Strategic Game with Pure Strategies

I N players with i ∈ I ≡ 1, ...,N;

I s ∈ S ≡ ∏i=1,..,N

Si pure strategy profile, si ∈ Si finite set;

I ui (s) payoff or utility;

I G ≡ 〈I , Sii , ui (s)i 〉 strategic form of finite game withpure strategy.



Strategic Game with Mixed Strategies

I σ ∈ ∆ (S) ≡ ∏i=1,..,N

∆ (Si ) mixed strategy profile, σi ∈ ∆ (Si );

I ui (σ) = ∑s∈S

∏j=1,..,N

σj (sj ) ui (sj ) expected utility;

I G ≡ 〈I , ∆ (Si )i , ui (σ)i 〉 strategic form of finite gamewith mixed strategy;

I Interpreting mixed strategies:

- as object of choice;

- as pure strategies of a perturbed game (see later in BayesianGames);

- as beliefs.



Equilibrium Concepts

I Nash Equilibrium ⇒ it is assumed that each player holds thecorrect expectation about the other players’behavior and actrationally (steady state equilibrium notion);

I Rationalizability ⇒ players’beliefs about each other’s actionsare not assumed to be correct, but are constrained byconsideration of rationality;

I Every Nash equilibrium is rationalizable.



Rationalizability

DefinitionIn G , si is rationalizable if there exists Zj ⊂ Sj for each j ∈ I suchthat:

1. si ∈ Zi ;

2. every sj ∈ Zj is a best response to some belief µj ∈ ∆ (Z−j ).

I Common knowledge of rationality;

I An action is rationalizable if and only if it can be rationalizedby an infinite sequence of actions and beliefs.



Example (1 - Rationalizability - See notes!)...



Strictly Dominance

Definitionsi is not strictly dominated if it does not exist a strategy σi :

ui (σi , s−i ) > ui (si , s−i ) , ∀s−i ∈ S−i



Strictly Dominance

I A unique strictly dominant strategy equilibrium (D,D):

I It is Pareto dominated by (C ,C ). Does it really occur??



Iterative Elimination of Strictly Dominated Strategies

DefinitionSet S0 = S , then for any m > 0 si ∈ Smi iff there does not existany σi such that:

ui (σi , s−i ) > ui (si , s−i ) , ∀s−i ∈ Sm−1−i

DefinitionFor any player i , a strategy is said to be rationalizable if and only ifsi ∈ S∞

i ≡⋂m≥0

Smi .



Example (2 - Beauty Contest - See notes!)...



Iterated Weak Dominance

I There can be more that one answer for iterated weakdominance;

I Not for iterated strong dominance.



Example (3 - Cournot vs Bertrand Competition - Proposedas exercise)

Example

I n profit-maximizer-firms produce qi quantity of consumptiongood at a marginal cost equal to c > 0;

I demand function is P = max 1−Q, 0 with Q ∈ ∑i=1...n

qi ;

I Find:

1. The rationalizable equilibria when n = 2;

2. The rationalizable equilibria when n > 2;

3. Compare your results with the Bertrand competition outcome.



Nash Equilibrium

Definitionσi ∈ ∆ (Si ) is a best response to σ−i ∈ ∆ (S−i ) if:

ui (σi , σ−i ) ≥ ui(σ′i , σ−i

)for all σ′i ∈ ∆ (Si )

Let Bi (σ−i ) ⊂ ∆ (Si ) be the set of player i best response.

Definitionσ is a Nash equilibrium profile if for each i ∈ I .

σi ∈ Bi (σ−i )



Nash Theorem

Theorem (Nash (1950))A Nash equilibrium exists in a finite game.

Theorem (Kakutani Fixed Point Theorem)Let X be a compact, convex and non-empty subset of Rn, acorrespondence f : X → X has a fixed point if:

1. f is non-empty for all x ∈ X;

2. f is convex for all x ∈ X;

3. f is upper hemi-continuous (closed graph).



Sketch Proof Nash Theorem

See notes!



Best Response Correspondence Example



The Kitty Genovese Problem/Bystander EffectI n identical people;

I x > 1 benefits if someone calls the police;

I 1 cost of calling the police;

What is the symmetric mixed strategy equilibrium with pequal to the probability of calling the policy?

I In equilibrium each player must be indifferent between callingor not the police;

I If i calls the police, gets x − 1 for sure;I If i doesn’t, gets:

0 with Pr (1− p)n−1

x with Pr 1− (1− p)n−1



The Kitty Genovese Problem/Bystander EffectI Indifference when:

x − 1 = x(1− (1− p)n−1

)I Equilibrium symmetric mixed strategy is p = 1− (1/x)1/(n−1)

I http://en.wikipedia.org/wiki/Murder_of_Kitty_Genovese



Zero-Sum Game

DefinitionA N-player game G is a zero-sum game (a strictly competitivegame) if ∑

i=1,..,N

ui (s) = K for every s ∈ S .



Zero-Sum Game

Definitionσi ∈ ∆ (Si ) is maxminimizer for player i if:

minσ−i∈∆(S−i )

ui (σi , σ−i ) ≥ minσ−i∈∆(S−i )

ui(σ′i , σ−i

)for each σi ∈ ∆ (Si )

A maxminimizer maximizes the payoff in the worst case scenario(saddle-point equilibrium)

TheoremLet G be a zero-sum game. Then σ ∈ ∆ (S) is a Nash Equilibriumiff, for each i , σ is a maxminimizer.



Example (4 - All-Pay Auction - Proposed as exercise)

I Two players submit a bid for an object of worth k;

I bi ∈ [0, k ] individual strategy space where bi is the bid;

I The winner is the player with the highest bid;

I If tie each player gets half the object, k/2;

I Each player pays her bid regardless of whether she wins;

I Find that:

1. No pure Nash equilibria exist;

2. The mixed strategy equilibrium is equal to the one representedhere below.



Example (4 - All-Pay Auction - Proposed as exercise)


Extensive Form Games

Representation of a Game

The games can be represented in two forms:

I Normal or strategic form (we have done);

I Extensive form.

The Extensive form contains all the information about a game:

I who moves when;

I what each player knows when he moves;

I what moves are available to him;

I where each move leads.

whereas a normal form is a ‘summary’representation.



Extensive Form

DefinitionA tree is a set of nodes and directed edges connecting these nodessuch that:

1. for each node, there is at most one incoming edge;

2. for any two nodes, there is a unique path that connect thesetwo nodes.

DefinitionAn extensive form game consists of i) a set of players (includingpossibly Nature), ii) a tree, iii) an information set for each player,iv) an informational partition, and v) payoffs for each player ateach end node (except Nature).



Extensive Form

DefinitionAn information set is a collection of points (nodes) such that:

1. the same player i is to move at each of these nodes;

2. the same moves are available at each of these nodes.

DefinitionAn information partition is an allocation of each node of the tree(except the starting and end-nodes) to an information set.

DefinitionA (behavioral) strategy of a player is a complete contingent-plandetermining which action he will take at each information set he isto move.



Extensive Form vs Normal Form


Strategic Games with Incomplete Information

Static Games with Incomplete Information

There are many realistic circumstances in which agents haveprivate information. Some examples are:

I A bidder does not know the other bidders’value in auction;

I Parties do not know the voters’preferences;

I An employer does not know the skills of the employee;

I Incumbent firm does not know whether the entrant isaggressive or not;

I ....



Bayesian Games


I ω ∈ Ω finite set of "states of nature";

I τi : Ω→ Ti types (signal) profile with ti ∈ Ti ;

I pi : Ω→ [0, 1] prior belief with pi (ω|ti ) ≥ 0

I σ ∈ ∆ (S) ≡ ∏i=1,..,N

∆ (Si ) strategy profile with

σi : Ti → ∆ (Si );

I υti ≡∑ω∈Ω pi (ω|ti ) ui (σ,ω) the expected payoff of type ti ;

I G ≡⟨I ,Ω, Sii , Tii , τii , pii , υti ti

⟩



Bayesian Games: Interpretation

I Ω is a set of possible states of nature that determine thephysical setup of the game (payoffs);

I Ti is the set of i ’s private types that encode player i ’sinformation/knowledge;

I pi is player i ’s interim belief about the state and the otherplayers’types.



Battle of the Sexes Revisited

I ω ∈ Ω ≡ ω1,ω2 with ω1 = meet and ω2 = avoid ;

I τ1 (ω1) = τ1 (ω2) = z ;

I m = τ2 (ω1) 6= τ2 (ω2) = x ;

I p1 (ω1|z) = p1 (ω2|z) = 1/2, p2 (ω1|m) = p2 (ω2|x) = 1;

I (1/2)Eu1 ((B, σ2) ,ω1) + (1/2)Eu1 ((B, σ2) ,ω2) player 1’sex-ante utility if she plays B.



Bayesian Nash Equilibrium

Definition (Harsanyi (1967/1968))A Nash equilibrium of a Bayesian Game is a Nash equilibrium of astrategic game characterized by:

- Set of players (i , ti ) with i ∈ I and ti ∈ Ti ;

- Set of strategies for each (i , ti );

- Payoff function for each (i , ti ) is given by υti .

Following Harsanyi (1967/1968) we transform a game ofincomplete information in a game with imperfect informationwhere Nature moves first.



Bayesian Nash Equilibrium

Definitionσ∗ ∈ ∆ (S) is a Bayesian Nash Equilibrium if:

E [υti (σ∗i (ti ) , σ

∗−i (t−i ) , ω)] ≥ E [υti (σi , σ∗−i (t−i ) , ω)]

for each σi ∈ ∆ (Si ), ti ∈ Ti and i ∈ I .



Example (5 - Building New Capacity - See notes!)



Example (6 - Public Good Provision - Proposed as exercise!)

I There are two players, i = 1, 2, who may either cooperate ordefeat in the provision of a public good;

I si ∈ Si ≡ 0, 1 is the players’strategy space, where 0 standsfor "defeat" and 1 for "cooperate";

I If agents decide to cooperate, then they sustain a cost ci ,which is private information;

I Common-Knowledge: ci ∼ P (·) over [c, c ] with c < 1 < c ;

I The individual payoff is ui (si , sj , ci ) = max (s1, s2)− ci si ;

I Find the BNE of the public good game.



Example (7 - Second-Price vs First-Price Auction - Proposedas exercise)

I n bidders whose private evaluation is v≤ vi ≤ v make a bidbi ≥ 0;

I Each bidder observes only his own evaluation but believes thatthe others’evaluations are iid and distributed according toF ∼ [v , v ];

I The player with the highest bid wins the auction by paying thesecond highest bid;

I Find:

1. that bi = vi is a weakly dominant strategy;

2. the BNE of a first-price auction (i.e. the player with thehighest bid wins the auction by paying his own bid).


Dynamic Games with Perfect Information


I We study dynamic games where players make a choicesequentially;

I We assume perfect information: Each player can perfectlyobserve the past actions;

I Best representation by using extensive form games.




Chain-Store Game

Stackelberg-Cournot Competition



Dynamic Games in Extensive Form


I H set of histories with ak equal to an action taken by a player:

- ∅ ∈ H;

- if(a1, ...ak

)∈ H then

(a1, ...al

)∈ H for each l < k;

- if(a1, ...ak , ...

)is an infinite sequence such that(

a1, ...ak)∈ H for each k ∈N then

(a1, ...ak , ...

)∈ H.

I Z set of terminal histories:

-(a1, ...ak

)∈ Z if it is an infinite sequence or @ ak+1 such that(

a1, ...ak+1)∈ H.



Dynamic Games in Extensive Form

I P : H\Z → I assignment function;

I A (h) = a| (h, a) ∈ H set of actions available to P (h);

I υi : Z → R;

I Γ ≡ 〈I ,H,P, υii 〉.



Strategies

DefinitionA strategy of player i ∈ I in Γ, σi , is a mapping from H to adistribution on the set of available action, σi (h) ∈ ∆ (Ai (h)) foreach non terminal history h ∈ H\Z for which P (h) = i (completecontingent plan).

For each strategy profile in Γ, let O (σ) the outcome of σ.



Nash Equilibrium

DefinitionA Nash equilibrium of a dynamic game with perfect information Γis a strategy profile σ∗ such that for each i ∈ I and for each σi ,O (σ∗) ≥i O

(σi , σ

∗−i).

Theorem (Zermelo 1913, Kuhn 1953)A finite dynamic game of perfect information has a pure-strategyNash equilibrium.



Backward Induction

Backward induction is the following procedure:

I Let L < ∞ be the maximum length of all histories;

I Find all nonterminal histories of L− 1 length and assign anoptimal action there. Eliminate unreached L-length terminalhistories and regard other L−length terminal histories asL− 1-length terminal histories;

I Find all nonterminal histories of L− 2 length and assign anoptimal action there. Eliminate unreached L− 1-lengthterminal histories and regard other L− 1-length terminalhistories as L− 2-length terminal histories;

I ....



Example (9 - Stackelberg-Cournot Game - See notes!)....



Example (10 - Hotelling Game and Product Differentiation -Proposed as exercise!)

I Consumers are distributed uniformly along the interval [0, 1];

I Two firms are located at the extremes and compete on prices;

I c is the cost of 1 unit of good and t is the transportation costby unit of distance squared;

I Consumers’payoff is U = s − p − td2 where s is the maxwillingness to pay, p is the market price and d is the distance;

I Find;

1. The NE of the game when firms’location is exogenously given;

2. The SPE of the game when firms decide first their locationand then compete on prices.



Subgame

DefinitionThe subgame of Γ following h ∈ H is the extensive-form gameΓ (h) ≡

⟨I ,H |h,P|h,

υi |hi

⟩where:

I h′ ∈ H |h ⇔ (h, h′) ∈ H;

I P|h (h′) = P (h, h′) for each h′ ∈ H |h;

I υi |h (h′) = υi (h, h′) for each h′ ∈ Z |h ⊂ H |h.

Let σi |h a strategy for player i of Γ (h) and O|h(σ|h)the outcome

of σ|h.



Subgame Perfect Equilibrium

DefinitionA subgame perfect equilibrium of an extensive form game withperfect information Γ is a strategy σ∗ such that for any i ∈ I andnon terminal history h ∈ H\Z for which P (h) = i , one has:

Oh(

σ∗|h

)≥i |h Oh

(σi , σ

∗−i |h

)for all strategy σi in the subgame Γ (h).



One-Shot-Deviation Principle

I To find a SPE we need to check a very large number ofincentive constraints;

I We can apply a principle of dynamic programming: OSDP;

Definitionσ′i in Γ (h) at h ∈ H\Z for i ∈ P (h) is called one-shot deviationfrom σi if σi |h and σ′i prescribe a different action only at the initialhistory (i.e. σ′i (∅) 6= σi (h) and σ′i (h

′) = σi (h, h′) for anyh′ 6= ∅ with (h, h′) ∈ H\Z ).



One-Shot-Deviation Principle

TheoremIn an extensive form game with perfect information Γ a strategy σ∗

is a SPE iff:Oh(

σ∗|h

)≥i |h Oh

(σi , σ

∗−i |h

)for any one-shot deviation σi from σ∗i |h at any h ∈ H\Z fori ∈ P (h).

Proof.(See notes!).



Example (11 - Bargaining Game - Proposed as exercise!)

I Two players use the following procedure to split 1kr :

- Players 1 offers player 2 an amount x ∈ [0, 1];- If player 2 accepts, then 1 gets 1− x , if 2 refuses neitherreceives any money;

I Find:

1. The SPE of the bargaining game;

2. Introduce the possibility of player 2 to make a counter-offer.Let δi be the individual discount factor. Find the SPE;

3. Find the SPE of the infinitely repeated version.


Dynamic Games with Complete Information

Infinite Repeated Game

Through the infinite repeated version of the dynamic game withperfect information we can answer to the following questions:

I When can people cooperate in a long-term relationship?

I What is the most effi cient outcome that arises as anequilibrium?

I What is the set of all outcomes that can be supported inequilibrium?



Infinite Repeated Game


I at ∈ A ≡ ∏i=1,..,N

Ai , ati ∈ Ai finite set;

I ui (at ) payoff or utility;

I G t ≡ 〈I , Aii , ui (at )i 〉 stage-game;

I F t ≡ co u (at ) |∀at ∈ A set of feasible payoffs;

I An infinite repeated game, G∞ (δ), is equal to the infiniterepetition of G t , where δ ∈ (0, 1) is the individual discountfactor.



Strategy

I See def. of strategy and SPE for dynamic games withcomplete information;

I An equilibrium strategy profile σ generates an infinitesequence of action profiles

(a1, a2, ...

)∈ A∞;

I The discounted average payoff is given by:

Vi (σ) = (1− δ)∞

∑t=0

δtui(at)



Min-Max Payoff

DefinitionThe min-max payoff is equal to:

v i = mina−imaxaiui (a)

I In the prisoner dilemma v i = 0 for i = 1, 2;

I The min-max payoff serves as a lower bound on equilibriumpayoffs in a repeated game.

LemmaPlayer i’s payoff in any NE for G∞ (δ) is at least as large as v i .



Example (12 - Infinite Repeated Prisoner Dilemma - Seenotes!)

g>0, l>0

When can (C ,C ) be played in every period in equilibrium?



Folk Theorem

We know that player i ’s (pure strategy) SPE payoff is neverstrictly below v i . The Folk Theorem shows that every feasible vistrictly above v i can be supported by SPE.

Definitionv ∈ F is strictly individually rational if vi is strictly larger than v ifor all i ∈ I . Let F ∗ ⊂ F be the set of feasible and strictlyindividually rational payoff profiles.



Folk Theorem

Theorem (Fudenberg and Maskin (1986))Suppose that F ∗ is full-dimensional. For any v ∈ F ∗, there existsa strategy profile σ and δ∈ (0, 1) such that σ is a SPE andachieves v for any δ ∈ (δ, 1).



Example (13 - Optimal Collusion - Infinite CournotCompetition - Proposed as exercise!)

I Dynamic Cournot duopoly model;

I Stage game equal to the static Cournot game and δ ∈ (0, 1);

I Using a "stick and carrot" strategy find the stronglysymmetric SPE.


Dynamic Games with Incomplete Information


I We consider dynamic games where past actions (by players ornature) are imperfectly observed;

I We treat them as an extension of dynamic game withcomplete information.




I There are no subgame out of the game itself;

I The pure NE are (O,O) and (V ,F ), but is the latter credible?




I N players with i ∈ I ≡ 1, ...,N and c denotes Nature;

I ht =(a1, a2, ..., ak

)∈ H set of histories with ak equal to an

action taken by a player;

I P : H\Z → I ∪ c, A (h) = a| (h, a) ∈ H set of actionsavailable to P (h);

I fc (a|h) is the probability that a occurs after h for whichP (h) = c ;

I υi : Z → R.




I Ii a partition of h ∈ H |P (h) = i with the propertyA (h) = A (h′) if h, h′ ∈ Ii ;

I Each Ii ∈ Ii is player i’s information set: the set of historiesthat player i cannot distinguish;

I A (Ii ) the set of action available at Ii ;

I Γ ≡ 〈I ,H,P, fc , Iii , υii 〉.



Extensive Games with Imperfect Information

I P (∅) = P (L,A) = P (L,B) = 1 and P (L) = 2;

I I1 = ∅ , (L,A) , (L,B) and I1 = L.



Mixed and Behavioral Strategies

DefinitionA mixed strategy of player i in an extensive game〈I ,H,P, fc , Iii , υii 〉 is a probability measure over the set ofplayer i’s pure strategies. A behavioral strategy of player i is acollection βi (Ii )Ii∈I1 of independent probability measures, whereβi (Ii ) is a probability measure over A(Ii ).

TheoremFor any mixed strategy of a player in a finite extensive game withperfect recall there is an outcome-equivalent behavioral strategy.

The Nash equilibrium of the game can be found in the usual way.We need a reasonable refinement of NE.



Perfect Bayesian Nash Equilibrium

I Recall that in games with complete information some NE maybe based on the assumption that some players will actsequentially irrationally at certain information sets off thepath of equilibrium;

I In those games we ignored these equilibria by focusing on SPE;

I We extend this notion to the games with incompleteinformation by requiring sequential rationality at eachinformation set: PBNE as equilibrium refinement of BNE;

I For each information set, we must specify the beliefs of theagent who moves at that information set.



Sequential Rationality

DefinitionA player is said to be sequentially rational iff, at each informationset he is to move, he maximizes his expected utility given his beliefsat the information set (and given that he is at the information set)- even if this information set is precluded by his own strategy.



Consistency

DefinitionGiven any strategy profile σ, an information set is said to be on thepath of play iff the information set is reached with positiveprobability according to σ.

DefinitionGiven any strategy profile σ and any information set Ii on the pathof play of σ, a player’s beliefs at Ii is said to be consistent with σiff the beliefs are derived using the Bayes’rule and σ.

This definition does not apply off the equilibrium path becauseotherwise we cannot apply the Bayes’rule.



Consistency

I Can the strategy (X ,T , L) be considered not consistent byusing our definition of consistency?

I We need to check consistency also off the equilibrium path by"trembling handing".



Perfect Bayesian Nash EquilibriumDefinitionA strategy profile is said to be sequentially rational iff, at eachinformation set, the player who is to move maximizes his expectedutility given:

1. his beliefs at the information set;

2. given that the other players play according to the strategyprofile in the continuation game.

DefinitionA Perfect Bayesian Nash Equilibrium is a pair (σ, µ) of strategyprofile and a set of beliefs such that:

1. σ is sequentially rational given beliefs µ;

2. µ is consistent (also off the equilibrium path) with σ.



Perfect Bayesian Nash Equilibrium

Example (13 - PBNE - See notes!)


Adverse Selection, Signaling and Screening

Economics of Information

I Fundamental welfare theorems rely on perfect observability ofall commodities to all market participants;

I Often in a transaction one party knows something that otherparties don’t know;

I We study three types of equilibria:

i. Adverse selection: An informed individual’s trading decisionsadversely affects uninformed market participants;

ii. Signaling : Informed individuals signal information about theirunobservable knowledge through observable signal;

iii. Screening : Uninformed parties develop mechanisms to screeninformed individuals with different information.



Adverse Selection (Akerlof, 1970)

I Consider a labor market with many identical potential firms;

I Firms want to hire workers to produce final good by adoptinga constant return to scale technology with labor as the soleinput;

I Firms are risk-neutral, price-taker and profit maximizer;

I The price of the firm’s output is equal to one;

I Workers differ in productivity, θ ∈[θ, θ]distributed according

to F (θ);

I r (θ) is the reservation wage of workers, i.e. the gain theymight obtain by working at home.



Adverse SelectionPerfect Observability

I Due to the assumptions of perfect competition and CRS thefirms would set a wage equal to:

w ∗ (θ) = θ

I Only the workers with θ|r (θ) ≤ θ would accept the offer;

I The equilibrium is Pareto optimal.



Adverse SelectionImperfect Observability

DefinitionIn a competitive labor market with unobservable productivitylevels, a competitive equilibrium is the pair w ∗,Θ∗ such that:

Θ∗ = θ|r (θ) < w ∗w ∗ = E (θ|θ ∈ Θ∗)

I The equilibrium is characterized by a fixed point;

I Typically this equilibrium will not be Pareto optimal.



Adverse Selection

I Suppose r (θ) = r and F (r) ∈ (0, 1);

I The Pareto optimal allocation is: if θ ≥ r accept employmentand if θ < r not accept;

I If w > r then Θ∗ =[θ, θ], if w < r then Θ∗ = ∅;

I In both cases w = E (θ): Firms are unable to distinguishamong workers’productivity. Thus, the equilibrium outcomeis ineffi cient:

i. If the share of good workers is large enough, then E (θ) > rand too many workers are hired;

ii. If the share of bad workers is large enough, then E (θ) < r andtoo few workers are hired.



Adverse Selection

I Adverse selection occurs when an informed individual’sdecision depends on her unobservable characteristics andadversely affects the uninformed agents;

I In the labor market context, adverse selection arises when onlyrelatively less capable workers accept a firm’s employmentoffer at any given wage;

I If r(θ) is no longer constant adverse selection arises(specifically if it is increasing!).



Adverse SelectionSuppose r(θ) ≤ θ for all θ and r ′(θ):



Adverse SelectionPossibility of multiple equilibria:



Example (14 - Credit Market and Adverse Selection -Proposed as exercise!)

I A borrower need a loan to finance a project of value I = 1;

I (r , x) is a loan contract, r ≥ 0 is the interest rate andx ∈ [0, 1] is a collateral;

I q is the probability of borrower’s default, in that casecollateral value is bx with b ∈ (0, 1);

I Two types of borrower, 0 < q1 < q2 < 1/2, whose mass is50% of pop;

I Credit market is perfect competitive and agents are riskneutral;

I Find the Pareto optimal allocation and the equilibrium withasymmetric information.



I Both the firms and the high-ability workers have incentives totransmit information;

I Market responses to the problem of adverse selection:

i. Signaling : Informed individuals (workers) choose their level ofeducation to signal information about their ability touninformed parties (the firms);

ii. Screening : Uninformed parties (firms) take steps to screen thevarious types of individuals on the other side of the market(workers).



Signaling (Spence 1973, 1974)

I Two types of workers: high ability, θH , and low ability, θL;

I The probability of high type workers ( exogenous share in thepopulation) is λ ∈ (0, 1);

I Perfect competitive markets (zero profits), whose profit’sfirms is π = θ − w .



SignalingFull Information

I Bertrand competition outcome;

I (wL,wH ) such that wH = θH and wL = θL;

I If abilities are not observable, then w = E (θ|θ ∈ Θ∗) whereΘ∗ is equal to the set of types accepting the contract.



SignalingImperfect Observability

I Suppose that before entering the job market a worker can getsome level of education, e, and the firms observe it;

I Assume that:

i. Education does not increase workers’productivity;

ii. Education is more costly for the low ability worker than for thehigh ability worker, also at the margin (single crossingproperty);

I Workers’payoff u = w − c (e, θ) with c (0, θ) = 0,ce (e, θ) > 0, cθ (e, θ) < 0 and ceθ (e, θ) < 0.



SignalingWelfare

Welfare effect is generally ambiguous:

I Signaling can lead to a more effi cient allocation of workers’labor and to a Pareto improvement;

I Signaling is a costly activity and workers’welfare may bereduced.



SignalingTiming

I Nature determines worker’s type;

I Worker chooses an education level contingent on his type;

I Conditional on the education level, firms make wage offerssimultaneously

I Worker decides which offer to accept, if any.



SignalingPerfect Bayesian Equilibrium

I We use the concept of Perfect Bayesian Equilibrium;

I The worker’s strategy is optimal given the firm’s strategy;

I µ (e) are the up-dated firms’beliefs that the worker ishigh-type;

I Each firm’s wage offer, following the choice e, is optimal giventhe belief µ (e), the worker’s strategy and the other firms’strategy.



SignalingPerfect Bayesian Equilibrium

The firms’(pure strategy) Nash equilibrium wage offers equal theworker’s expected productivity:

w (e) = µ (e) θH + (1− µ (e)) θL



SignalingSeparating Equilibrium

I Let e∗(θ) the educational equilibrium choice of type θ andw ∗(e) the equilibrium wage of a worker who displays aneducational level of e;

I In any separating equilibrium e∗(θH ) 6= e∗(θL) andw ∗ (e∗(θH )) = θH and w ∗ (e∗(θL)) = θL;

I e∗(θL) = 0, which implies that the low ability worker receivesutility equal to θL at the separating equilibrium.



Signaling

Example (15 - Education and Signaling - See notes!)

I θH = 2, θL = 1, c (e, θ) = eθ ;

I Find the separating PBNE.



SignalingPooling Equilibrium

I The two types of workers choose the same level of educatione∗(θL) = e∗(θH ) = e∗;

I This implies that w ∗ (e∗) = λθH + (1− λ) θL;

I Any education level between 0 and e ′ can be sustained as apooling equilibrium. e ′ corresponds to the level of educationsuch that low type receives zero utility with w = E [θ], e ′:w − c(e ′(θL), θL) = 0.



Signaling

Example (16 - Reputation Game - Proposed as exercise!)



Screening

I Workers’outside option is zero r(θi ) = 0;

I Jobs may differ in the “task level” t > 0 required. (Ex.different number of ours per week);

I The output of a type-θi worker is θi regardless of the worker’stask level. Higher task levels only affect workers’utility. Theyare costly for the workers;

I Workers’payoff u = w − c (t, θ) with c (0, θ) = 0,ct (t, θ) > 0, cθ (t, θ) < 0 and ctθ (t, θ) < 0.



ScreeningTiming

I Stage 1 : Firms simultaneously announce a menu of contracts(w , t). Each firm may announce any finite numbers ofcontracts;

I Stage 2: Workers decide whether they want to sign a contractand which one to sign:

- If indifferent between signing and not signing a contract, theworker will sign;

- If indifferent between two types of contract, the worker willchoose the contract with the lower task level.



ScreeningFull Observability

If abilities are observable equilibrium entails firms offering adifferent contract to each type:

I (w ∗H , t∗H ) = (θH , 0) for high ability workers;

I (w ∗L , t∗L ) = (θL, 0) for low ability workers;

I Workers accept contracts and firms earn zero profits.



ScreeningBreak-Even Line



ScreeningPooling Equilibria

No pooling equilibria exists:



ScreeningSeparating Equilibria

I If (wL, tL) and (wH , tH ) are the contracts signed respectivelyby the low- and the high-ability workers in a separatingequilibrium, then both contracts yield zero profits: wL = θLand wH = θH ;

I This implies that separating equilibria does not allow forcross-subsidies. Separating equilibria are on the break-evenlines;

I In any separating equilibrium, the low-ability workers acceptcontract (wL, tL) = (θL, 0). They receive the same contractas under full information.




I In any separating equilibrium the high ability workers acceptcontract (wH , tH ) = (θH , tH ) where tH is such that:θH − c(tH , θL) = θL − c(0, θL) = θL;

I This means that low type is indifferent between contract(θH , tH ) and contract (0, θL);

I If tH > tH , firms can offer contracts which attract high abilityworkers and make positive profits.




I A separating equilibrium exists if the pooled break-even line issuffi ciently far from θH ;

I This means that λ must be suffi ciently low;

I As in the signaling model, asymmetric information leads toPareto ineffi cient outcomes.


Principal-Agent Model


I A principal wants to delegate a task to an agent;

I Delegation benefits: Increasing returns associated with tasks’division, or by the principal’s lack of time or ability to performthe task himself;

I The agent and the principal have different objectives;

I If the agent has no private information, then the principalcould propose a contract that perfectly controls the agent’sbehavior ⇒ No incentives problems;

I When the agent has private information, then incentivesproblems arise.




I Why a theory of contract?;

I A principal delegates an action to a single agent through thetake-it-or-leave-it offer of a contract;

I One-shot relationship: No repetition is available to achieveeffi ciency;

I The principal proposes the contract, no bargaining issues;

I A benevolent court of law must be available. It enforces thecontract and imposes penalties if one of the contractualpartners adopts a behavior that deviates from the onespecified in the contract.




DefinitionA contract is a legally binding exchange of promises or agreementbetween parties.

I Different types of contract exist;

I Implicit contract: A contract that is self-enforcing. When thetwo parties play a game where the unique Subgame PerfectNash equilibrium of the game corresponds to the desiredoutcome;

I Explicit contract: Whenever the desired outcome is notSubgame Perfect we need an explicit contract. Internalizingcourt’s punishment agents do not have interest in deviatingfrom the agreement.




I Problem of delegating a task to an agent with differentobjectives and private information;

I Which private information?

I Moral hazard or hidden action: Endogenous uncertainty forthe principal;

I Adverse selection or ex-post hidden information: Exogenousuncertainty for the principal;

I Non verifiability: The principal and the agent share ex-postthe same information;

I No court of law can observe this information ⇒ agency costs.



Principal-Agent ModelHidden action

I An agent chooses actions that affect the value of trade or theagent’s performance;

I The principal cannot control those actions and they are notobservable either by the principal or by the court of law ⇒Actions are not contractible;

I Examples: Worker’s effort in performing a task, timingdevoted to a task, how safely a driver drives, green-investmentby regulated firms...




I With moral hazard the expected volume of trade dependsexplicitly on the agent’s effort;

I The realized production level is a noisy signal of the agent’saction;

I The principal wants to design a contract that induces thehighest effort from the agent despite the impossibility ofdirectly conditioning the agent’s reward on his action.




I To make the agent responsible for the consequences of hisactions the principal lets the agent bear some risk;

I Risk—sharing/effi ciency and rent/effi ciency trade-off.



Principal-Agent ModelHidden information

I An agent gets access to information that is not availableneither to the principal nor to the court of law;

I Examples: A tenant observes local weather conditions, expertsknow the diffi culty of the case, regulated firms have privateinformation on their costs,...;

I To achieve effi ciency, the contract must elicit the agent’sprivate information;

I The principal must give up some information rent to theprivately informed agent;

I Rent-effi ciency trade-off.


Moral Hazard

Moral Hazard

I The principal delegates the agent to perform a task;

I The worker chooses the intensity of effort, e ∈ 0,E, toperform the task. His effort positively affects the outputq ∈ 0,Q;

I The principal only cares about the output and don’t observeeffort;

I Since the effort is costly, the principal has to compensate theagent for incurring this cost;

I The agent’s compensation has to be contingent on theoutcome q that is a noisy signal of effort e.


Moral Hazard

Moral HazardRisk-sharing/effi ciency trade-off

I Pr q = Q |E = pE and Pr q = Q |0 = p0 with p0 < pE ;

I The risk-neutral principal’s utility q − w ;

I The agent’s utility u (w)− e with uw > 0, uww ≤ 0;

I The agent’s reservation u ≡ u (w);

I pEQ − E ≥ p0Q and pEQ − E ≥ u then e = E is effi cient.


Moral Hazard

Moral HazardTiming and risk-sharing/effi ciency

i. The principal offers a contract to the agent;

ii. The agent then accepts or refuses the contract;

iii. If the agent refuses the contract he gets a reservation utility u.If the contract is accepted, the agent then chooses the level ofeffort e ∈ 0,E, which is unobservable by the principal;

iv. Finally, as a result of the agent’s choice, a quantity q isproduced.


Moral Hazard

Moral HazardFull Information and risk-sharing/effi ciency

I If e is verifiable then the contract can specify the desiredeffort, e = E , and the contingent transfers, w ,w with w ifq = 0 and w if q = Q;

I The principal’s problem is:

maxw ,w

pEQ − (pEw + (1− pE )w)

s.t. : pE u (w) + (1− pE ) u (w)− E ≥ u (IR)

I Since the principal is risk-neutral and the agent is risk adverse,then perfect insurance, w = w s.t. u (w) = E + u.


Moral Hazard

Moral HazardIncomplete Information and risk-sharing/effi ciency

I If e is not verifiable, then the principal’s problem is:

maxw ,w

pEQ − (pEw + (1− pE )w)

s.t.:pE u (w) + (1− pE ) u (w)− E ≥ u (IR)

b ≡ u (w)− u (w) ≥ EpE − p0

(IC )

I Since pE > p0 then w ≥ w and no longer agent’sfull-insurance.


Moral Hazard

Moral HazardIncomplete Information and risk-sharing/effi ciency

Using the binding constraints:- u (w) = u + E − pE E

pE−p0 < u + E ;

- u (w) = u + E + (1−pE )EpE−p0 > u + E ;

- r ≡ E(wSB

)− wFB , risk-premium.


Moral Hazard

Moral HazardFull Information and rent/effi ciency

I Assume that also the agent is risk-neutral, u (w) = w , andhas limited liability, w ≥ w ;

I The principal’s problem is:

maxw ,w

pEQ − (pEw + (1− pE )w)

s.t.:

pEw + (1− pE )w − E ≥ u (IR)

w ,w ≥ w (LL)

I First best solution is not affected by LL.


Moral Hazard

Moral HazardIncomplete Information and rent/effi ciency

I Let b ≡ w − w and w ≡ w , then the principal’s problembecomes:

maxb,w

pEQ − (w + pE b)

s.t.:

w + pE b ≥ w + E (IR)

b ≥ EpE − p0

(IC )

w ≥ w (LL)

I IR is not an issue in the presence of LL, w = w andw = w + E

pE−p0 and R ≡p0EpE−p0 is the agent’s expected rent.


Moral Hazard

Moral HazardInference problem

I The principal’s goal is to detect what the agent has done byobserving related variables;

I Should the wage increase with the observed output level? Theanswer is, “Not necessarily”.


Moral Hazard

Moral HazardFull Inference and full information (Mirrlees, 1975)

I The output is q (e) = e + ε, with ε ∼ F (·) over R,limε→−∞

F (ε)f (ε) = 0;

I P’s max problem with full information:

maxe ,w (q)

E [q − w (q) |e]

s.t. : E [u (w (q))− e|e] ≥ u

I It is optimal for the P to full insure the A andeFB : he

(eFB

)= 1 with w (q) = h (e) ≡ u−1 (u + e).


Moral Hazard

Moral HazardFull Inference and incomplete information (Mirrlees, 1975)

I Consider the second-best setting and the following schedule(in terms of promised utility):

u =

U if q ≥ QU − P if q < Q

I The contract is defined by U,P,Q;I q = e + ε ⇒ q < Q if ε < Q − e, i.e. with probabilityF (Q − e);

I The agent’s expected utility is U − F (Q − e)P − e;

I To implement FB P = 1f (Q−eFB ) with U = u + e

FB + F (Q−e)f (Q−e) ;

I No cost to implement FB allocation but we need no LL.


Moral Hazard

Moral HazardLimited Inference and incomplete information (Mirrlees, 1975)

I q ∈ [0,Q ], e ∈ 0,E and MLRP: l (q) ≡ fE (q)−f0(q)fE (q)

withlq (q) > 0;

I P’s max problem:

maxw (q)

∫ Q

0(q − w (q)) fE (q) dq

∫ Q

0u (w (q)) fE (q) dq − E ≥ u (IR,λ)∫ Q

0u (w (q)) fE (q) dq − E ≥

∫ Q

0u (w (q)) f0 (q) dq (IC , µ)

I The FOC is (λ+ µl (q)) uw (w (q)) = 1, which implies thatwq (q) > 0.


Moral Hazard

Moral HazardFirst-Order Approach

I q ∈ [0,Q ], e ∈ [e−, e+] with F (q|e) and MLRP:l (q) ≡ fe (q|e)

f (q|e) with lq (q) > 0;

I P’s max problem:

maxw (q),e

∫ Q

0V (q − w (q)) f (q|e) dq

∫ Q

0u (w (q)) f (q|e) dq − ψ (e) ≥ u (IR,λ)

e = argmaxe

∫ Q

0u (w (q)) f (q|e) dq − ψ (e) (IC , µ)

I By using FOA, if the argmax of IC is unique and SOC aresatified, then we can replace IC by FOC .


Adverse Selection

Adverse Selection

This type of agency problem arises in many settings:

I Interaction between the shareholders of a firm and itsmanagers, or the firm and its workers: Private informationabout the productivity of the managers or the workers;

I Interaction between an investor and a firm, or a bank and itsmanagers: private information about the projects undertaken;

I Relationship between an insurance company and itscustomers: private information about the risks that thecustomer is facing;

I Price discrimination: private information about the customers’willingness to pay.


Adverse Selection

Adverse SelectionPrice discrimination: Full Information

I P (seller) produces q at cost C (q) with Cq ,Cqq > 0 and sellsat t;

I A (buyer) gets benefit θq with θ ∈

θ, θwith probabilities

µ, µ;

I Complete info P’s problem:

maxt ,q

t − C (q)

s.t. : θq − t ≥ 0

I FB allocation is Cq(qFB (θ)

)= θ and tFB (θ) = θqFB (θ).


Adverse Selection

Adverse SelectionPrice discrimination: Incomplete Information

I Incomplete info P’s problem:

max(t ,q),(t ,q)

µ(t − C

(q))+ µ (t − C (q))

s.t.:

θq − t ≥ 0(IR)

θq − t ≥ 0 (IR)

θq − t ≥ θq − t(IC)

θq − t ≥ θq − t (IC )

I Let r ≡ θq − t and r≡ θq − t the buyers’rent.


Adverse Selection


I The P’s problem is equal to:

max(t ,q),(t ,q)

µ(θq − C

(q)− r)+ µ

(θq − C (q)− r

)s.t.:

r ≥ 0(IR)

r ≥ 0 (IR)

r ≥ r +(θ − θ

)q(IC)

r ≥ r −(θ − θ

)q (IC )

I From(IC)and (IC ) q ≥ q, and

(IR)and (IC ) are not

binding.


Adverse Selection


I Rent/effi ciency trade-off: SB allocation in terms of q:

qSB : Cq(qFB

)= θ

qSB : Cq(qSB

)= θ − µ

µ

(θ − θ

)I In terms of transfers:

tSB = θqSB −(θ − θ

)qSB

tSB = θqSB

I qSB < qSB .


Adverse Selection

Adverse SelectionGeneral Framework

I q ∈ Q and θ ∈ Θ ≡[θ, θ]distributed according to F (·);

I Agent’s preference U (q, t; θ) = V (q, θ)− t;

I Principal’s preference t − C (q, θ);

I P’s problem under full info:

max(t ,q)

t − C (q, θ)

s.t.:V (q, θ)− t ≥ 0

I FB allocation is Cq(qFB (θ) , θ

)= Vq

(qFB (θ) , θ

)and

tFB (θ) = V(qFB (θ) , θ

).


Adverse Selection

Adverse SelectionGeneral Framework: Implementability

I Let r (θ) ≡ V (q (θ) , θ)− t (θ) the agent’s rent;I Then r (θ) ≥ 0 and r (θ) = maxθ V

(q(θ), θ)− t

(θ);

I Spence-Mirrlees condition (i.e. single-crossing property)Vqθ (q, θ) ≥ 0 for each q, θ;

TheoremIf single-crossing property holds, then (q (·) , r (·)) is incentivecompatible iff qθ (θ) ≥ 0

r (θ) = r (θ) +∫ θ

θVθ (q (s) , s) ds

Proof.(See notes!).


Adverse Selection

Adverse SelectionGeneral Framework: Optimality

P’s problem under incomplete info:

max(t(·),q(·))

∫ θ

θ[t (θ)− C (q (θ) , θ)] f (θ) dθ

s.t.:

V (q (θ) , θ)− t (θ) ≥ 0, ∀θ

V (q (θ) , θ)− t (θ) ≥ V(q(θ), θ)− t

(θ), ∀θ, θ


Adverse Selection

Adverse SelectionGeneral Framework: Optimality

By using the Theorem, the P’s problem is:

max(t(·),q(·))

∫ θ

θ[V (q (θ) , θ)− C (q (θ) , θ)− r (θ)] f (θ) dθ

s.t.:

r (θ) ≥ 0, ∀θ

r (θ) = r (θ) +∫ θ

θVθ (q (s) , s) ds, ∀θ, θ

qθ (θ) > 0

Additional assumption Vθ (q (θ) , θ) ≥ 0. (See notes!).

advanced microeconomics - forsiden · advanced microeconomics introduction i what you have...

Documents