Advanced Microeconomics

Transcription

1 Advanced Microeconomics ECON Fall 2012

2 Introduction What you have done: - consumers maximize their utility subject to budget constraints and firms maximize their profits given technology and market prices; - no strategic behavior. What we will do: - in many interesting situations, agents optimal behavior depends on the other agents behavior; - strategic behavior. Game theory provides a language to analyze such strategic situations; Countless number of examples! Auctions, Bargaining, Price competition, Civil Conflicts...

3 Introduction Road map Static Game: 1. With Complete Information (I); 2. With Incomplete Information (II). Dynamic Game: 1. With Complete Information (II-III); 2. With Incomplete Information (III).

4 Strategic Games with Complete Information Strategic Game with Pure Strategies N players with i I {1,..., N}; s S S i pure strategy profile, s i S i finite set; i=1,..,n u i (s) payoff or utility; G I, {S i } i, {u i (s)} i strategic form of finite game with pure strategy.

5 Strategic Games with Complete Information Strategic Game with Mixed Strategies σ (S) (S i ) mixed strategy profile, σ i (S i ); i=1,..,n u i (σ) = σ j (s j ) u i (s j ) expected utility; s S j=1,..,n G I, { (S i )} i, {u i (σ)} i strategic form of finite game with mixed strategy; Interpreting mixed strategies: - as object of choice; - as pure strategies of a perturbed game (see later in Bayesian Games); - as beliefs.

6 Strategic Games with Complete Information Equilibrium Concepts Nash Equilibrium it is assumed that each player holds the correct expectation about the other players behavior and act rationally (steady state equilibrium notion); Rationalizability players beliefs about each other s actions are not assumed to be correct, but are constrained by consideration of rationality; Every Nash equilibrium is rationalizable.

7 Strategic Games with Complete Information Rationalizability Definition In G, s i is rationalizable if there exists Z j S j for each j I such that: 1. s i Z i ; 2. every s j Z j is a best response to some belief µ j (Z j ). Common knowledge of rationality; An action is rationalizable if and only if it can be rationalized by an infinite sequence of actions and beliefs.

8 Strategic Games with Complete Information Example (1 - Rationalizability - See notes!)...

9 Strategic Games with Complete Information Strictly Dominance Definition s i is not strictly dominated if it does not exist a strategy σ i : u i (σ i, s i ) > u i (s i, s i ), s i S i

10 Strategic Games with Complete Information Strictly Dominance A unique strictly dominant strategy equilibrium (D, D): It is Pareto dominated by (C, C ). Does it really occur??

11 Strategic Games with Complete Information Iterative Elimination of Strictly Dominated Strategies Definition Set S 0 = S, then for any m > 0 s i Si m any σ i such that: iff there does not exist u i (σ i, s i ) > u i (s i, s i ), s i S m 1 i Definition For any player i, a strategy is said to be rationalizable if and only if s i Si Si m. m 0

12 Strategic Games with Complete Information Example (2 - Beauty Contest - See notes!)...

13 Strategic Games with Complete Information Iterated Weak Dominance There can be more that one answer for iterated weak dominance; Not for iterated strong dominance.

14 Strategic Games with Complete Information Example (3 - Cournot vs Bertrand Competition - Proposed as exercise) Example n profit-maximizer-firms produce q i quantity of consumption good at a marginal cost equal to c > 0; demand function is P = max {1 Q, 0} with Q Find: 1. The rationalizable equilibria when n = 2; 2. The rationalizable equilibria when n > 2; q i ; i=1...n 3. Compare your results with the Bertrand competition outcome.

15 Strategic Games with Complete Information Nash Equilibrium Definition σ i (S i ) is a best response to σ i (S i ) if: u i (σ i, σ i ) u i ( σ i, σ i ) for all σ i (S i ) Let B i (σ i ) (S i ) be the set of player i best response. Definition σ is a Nash equilibrium profile if for each i I. σ i B i (σ i )

16 Strategic Games with Complete Information 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 R n, a correspondence 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).

17 Strategic Games with Complete Information Sketch Proof Nash Theorem See notes!

18 Strategic Games with Complete Information Best Response Correspondence Example

19 Strategic Games with Complete Information The Kitty Genovese Problem/Bystander Effect n identical people; x > 1 benefits if someone calls the police; 1 cost of calling the police; What is the symmetric mixed strategy equilibrium with p equal to the probability of calling the policy? In equilibrium each player must be indifferent between calling or not the police; If i calls the police, gets x 1 for sure; If i doesn t, gets: 0 with Pr (1 p) n 1 x with Pr 1 (1 p) n 1

20 Strategic Games with Complete Information The Kitty Genovese Problem/Bystander Effect Indifference when: x 1 = x (1 (1 p) n 1) Equilibrium symmetric mixed strategy is p = 1 (1/x) 1/(n 1)

21 Strategic Games with Complete Information Zero-Sum Game Definition A N-player game G is a zero-sum game (a strictly competitive game) if u i (s) = K for every s S. i=1,..,n

22 Strategic Games with Complete Information Zero-Sum Game Definition σ i (S i ) is maxminimizer for player i if: min u i (σ i, σ i ) min u ( ) i σ i, σ i for each σi (S i ) σ i (S i ) σ i (S i ) A maxminimizer maximizes the payoff in the worst case scenario (saddle-point equilibrium) Theorem Let G be a zero-sum game. Then σ (S) is a Nash Equilibrium iff, for each i, σ is a maxminimizer.

23 Strategic Games with Complete Information Example (4 - All-Pay Auction - Proposed as exercise) Two players submit a bid for an object of worth k; b i [0, k] individual strategy space where b i is the bid; The winner is the player with the highest bid; If tie each player gets half the object, k/2; Each player pays her bid regardless of whether she wins; Find that: 1. No pure Nash equilibria exist; 2. The mixed strategy equilibrium is equal to the one represented here below.

24 Strategic Games with Complete Information Example (4 - All-Pay Auction - Proposed as exercise)

25 Extensive Form Games Representation of a Game The games can be represented in two forms: Normal or strategic form (we have done); Extensive form. The Extensive form contains all the information about a game: who moves when; what each player knows when he moves; what moves are available to him; where each move leads. whereas a normal form is a summary representation.

26 Extensive Form Games Extensive Form Definition A tree is a set of nodes and directed edges connecting these nodes such that: 1. for each node, there is at most one incoming edge; 2. for any two nodes, there is a unique path that connect these two nodes. Definition An extensive form game consists of i) a set of players (including possibly Nature), ii) a tree, iii) an information set for each player, iv) an informational partition, and v) payoffs for each player at each end node (except Nature).

27 Extensive Form Games Extensive Form Definition An 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. Definition An information partition is an allocation of each node of the tree (except the starting and end-nodes) to an information set. Definition A (behavioral) strategy of a player is a complete contingent-plan determining which action he will take at each information set he is to move.

28 Extensive Form Games Extensive Form vs Normal Form

29 Strategic Games with Incomplete Information Static Games with Incomplete Information There are many realistic circumstances in which agents have private information. Some examples are: A bidder does not know the other bidders value in auction; Parties do not know the voters preferences; An employer does not know the skills of the employee; Incumbent firm does not know whether the entrant is aggressive or not;...

30 Static Games with Incomplete Information Bayesian Games N players with i I {1,..., N}; ω Ω finite set of "states of nature"; τ i : Ω T i types (signal) profile with t i T i ; p i : Ω [0, 1] prior belief with p i (ω t i ) 0 σ (S) i=1,..,n (S i ) strategy profile with σ i : T i (S i ); υ ti ω Ω p i (ω t i ) u i (σ, ω) the expected payoff of type t i ; G I, Ω, {S i } i, {T i } i, {τ i } i, {p i } i, {υ ti } ti

31 Static Games with Incomplete Information Bayesian Games: Interpretation Ω is a set of possible states of nature that determine the physical setup of the game (payoffs); T i is the set of i s private types that encode player i s information/knowledge; p i is player i s interim belief about the state and the other players types.

32 Static Games with Incomplete Information Battle of the Sexes Revisited ω Ω {ω 1, ω 2 } with ω 1 = meet and ω 2 = avoid; τ 1 (ω 1 ) = τ 1 (ω 2 ) = z; m = τ 2 (ω 1 ) = τ 2 (ω 2 ) = x; p 1 (ω 1 z) = p 1 (ω 2 z) = 1/2, p 2 (ω 1 m) = p 2 (ω 2 x) = 1; (1/2) Eu 1 ((B, σ 2 ), ω 1 ) + (1/2) Eu 1 ((B, σ 2 ), ω 2 ) player 1 s ex-ante utility if she plays B.

33 Static Games with Incomplete Information Bayesian Nash Equilibrium Definition (Harsanyi (1967/1968)) A Nash equilibrium of a Bayesian Game is a Nash equilibrium of a strategic game characterized by: - Set of players (i, t i ) with i I and t i T i ; - Set of strategies for each (i, t i ); - Payoff function for each (i, t i ) is given by υ ti. Following Harsanyi (1967/1968) we transform a game of incomplete information in a game with imperfect information where Nature moves first.

34 Static Games with Incomplete Information Bayesian Nash Equilibrium Definition σ (S) is a Bayesian Nash Equilibrium if: E [υ ti (σ i (t i ), σ i ( t i ), ω)] E [υ ti (σ i, σ i ( t i ), ω)] for each σ i (S i ), t i T i and i I.

35 Static Games with Incomplete Information Example (5 - Building New Capacity - See notes!)

36 Static Games with Incomplete Information Example (6 - Public Good Provision - Proposed as exercise!) There are two players, i = 1, 2, who may either cooperate or defeat in the provision of a public good; s i S i {0, 1} is the players strategy space, where 0 stands for "defeat" and 1 for "cooperate"; If agents decide to cooperate, then they sustain a cost c i, which is private information; Common-Knowledge: c i P ( ) over [c, c] with c < 1 < c; The individual payoff is u i (s i, s j, c i ) = max (s 1, s 2 ) c i s i ; Find the BNE of the public good game.

37 Static Games with Incomplete Information Example (7 - Second-Price vs First-Price Auction - Proposed as exercise) n bidders whose private evaluation is v v i v make a bid b i 0; Each bidder observes only his own evaluation but believes that the others evaluations are iid and distributed according to F [v, v]; The player with the highest bid wins the auction by paying the second highest bid; Find: 1. that b i = v i is a weakly dominant strategy; 2. the BNE of a first-price auction (i.e. the player with the highest bid wins the auction by paying his own bid).

38 Dynamic Games with Perfect Information Dynamic Games with Perfect Information We study dynamic games where players make a choice sequentially; We assume perfect information: Each player can perfectly observe the past actions; Best representation by using extensive form games.

39 Dynamic Games with Perfect Information Dynamic Games with Perfect Information Chain-Store Game Stackelberg-Cournot Competition

40 Dynamic Games with Perfect Information Dynamic Games in Extensive Form N players with i I {1,..., N}; H set of histories with a k equal to an action taken by a player: - H; ( ) ( ) - if a 1,...a k H then a 1,...a l H for each l < k; ( ) - if a 1,...a k,... is an infinite sequence such that ( ) ( ) a 1,...a k H for each k N then a 1,...a k,... H. Z set of terminal histories: ( ) - a 1,...a k Z if it is an infinite sequence or a k +1 such that ( a 1,...a k +1) H.

41 Dynamic Games with Perfect Information Dynamic Games in Extensive Form P : H\Z I assignment function; A (h) = {a (h, a) H} set of actions available to P (h); υ i : Z R; Γ I, H, P, {υ i } i.

42 Dynamic Games with Perfect Information Strategies Definition A strategy of player i I in Γ, σ i, is a mapping from H to a distribution on the set of available action, σ i (h) (A i (h)) for each non terminal history h H\Z for which P (h) = i (complete contingent plan). For each strategy profile in Γ, let O (σ) the outcome of σ.

43 Dynamic Games with Perfect Information Nash Equilibrium Definition A Nash equilibrium of a dynamic game with perfect information Γ is a strategy profile σ such that for each i I and for each σ i, O (σ ) i O ( σ i, σ i ). Theorem (Zermelo 1913, Kuhn 1953) A finite dynamic game of perfect information has a pure-strategy Nash equilibrium.

44 Dynamic Games with Perfect Information Backward Induction Backward induction is the following procedure: Let L < be the maximum length of all histories; Find all nonterminal histories of L 1 length and assign an optimal action there. Eliminate unreached L-length terminal histories and regard other L length terminal histories as L 1-length terminal histories; Find all nonterminal histories of L 2 length and assign an optimal action there. Eliminate unreached L 1-length terminal histories and regard other L 1-length terminal histories as L 2-length terminal histories;...

45 Dynamic Games with Perfect Information Example (9 - Stackelberg-Cournot Game - See notes!)...

46 Dynamic Games with Perfect Information Example (10 - Hotelling Game and Product Differentiation - Proposed as exercise!) Consumers are distributed uniformly along the interval [0, 1]; Two firms are located at the extremes and compete on prices; c is the cost of 1 unit of good and t is the transportation cost by unit of distance squared; Consumers payoff is U = s p td 2 where s is the max willingness to pay, p is the market price and d is the distance; Find; 1. The NE of the game when firms location is exogenously given; 2. The SPE of the game when firms decide first their location and then compete on prices.

47 Dynamic Games with Perfect Information Subgame Definition The subgame of Γ following h H is the extensive-form game Γ (h) I, H h, P h, { } υ i h where: h H h (h, h ) H; P h (h ) = P (h, h ) for each h H h ; υ i h (h ) = υ i (h, h ) for each h Z h H h. i Let σ i h a strategy for player i of Γ (h) and O h ( σ h ) the outcome of σ h.

48 Dynamic Games with Perfect Information Subgame Perfect Equilibrium Definition A subgame perfect equilibrium of an extensive form game with perfect information Γ is a strategy σ such that for any i I and non terminal history h H\Z for which P (h) = i, one has: ) ( ) O h (σ h i h O h σ i, σ i h for all strategy σ i in the subgame Γ (h).

49 Dynamic Games with Perfect Information One-Shot-Deviation Principle To find a SPE we need to check a very large number of incentive constraints; We can apply a principle of dynamic programming: OSDP; Definition σ i in Γ (h) at h H\Z for i P (h) is called one-shot deviation from σ i if σ i h and σ i prescribe a different action only at the initial history (i.e. σ i ( ) = σ i (h) and σ i (h ) = σ i (h, h ) for any h = with (h, h ) H\Z).

50 Dynamic Games with Perfect Information One-Shot-Deviation Principle Theorem In an extensive form game with perfect information Γ a strategy σ is a SPE iff: ) ( ) O h (σ h i h O h σ i, σ i h for any one-shot deviation σ i from σ i h i P (h). at any h H\Z for Proof. (See notes!).

51 Dynamic Games with Perfect Information Example (11 - Bargaining Game - Proposed as exercise!) Two players use the following procedure to split 1kr: Find: - Players 1 offers player 2 an amount x [0, 1]; - If player 2 accepts, then 1 gets 1 x, if 2 refuses neither receives any money; 1. The SPE of the bargaining game; 2. Introduce the possibility of player 2 to make a counter-offer. Let δ i be the individual discount factor. Find the SPE; 3. Find the SPE of the infinitely repeated version.

52 Dynamic Games with Complete Information Infinite Repeated Game Through the infinite repeated version of the dynamic game with perfect information we can answer to the following questions: When can people cooperate in a long-term relationship? What is the most effi cient outcome that arises as an equilibrium? What is the set of all outcomes that can be supported in equilibrium?

53 Dynamic Games with Complete Information Infinite Repeated Game N players with i I {1,..., N}; a t A A i, ai t A i finite set; i=1,..,n u i (a t ) payoff or utility; G t I, {A i } i, {u i (a t )} i stage-game; F t co {u (a t ) a t A} set of feasible payoffs; An infinite repeated game, G (δ), is equal to the infinite repetition of G t, where δ (0, 1) is the individual discount factor.

54 Dynamic Games with Complete Information Strategy See def. of strategy and SPE for dynamic games with complete information; An equilibrium strategy profile σ generates an infinite sequence of action profiles ( a 1, a 2,... ) A ; The discounted average payoff is given by: V i (σ) = (1 δ) t=0 δ t u i ( a t )

55 Dynamic Games with Complete Information Min-Max Payoff Definition The min-max payoff is equal to: v i = min a i max u i (a) a i In the prisoner dilemma v i = 0 for i = 1, 2; The min-max payoff serves as a lower bound on equilibrium payoffs in a repeated game. Lemma Player i s payoff in any NE for G (δ) is at least as large as v i.

56 Dynamic Games with Complete Information Example (12 - Infinite Repeated Prisoner Dilemma - See notes!) g >0, l>0 When can (C, C ) be played in every period in equilibrium?

57 Dynamic Games with Complete Information Folk Theorem We know that player i s (pure strategy) SPE payoff is never strictly below v i. The Folk Theorem shows that every feasible v i strictly above v i can be supported by SPE. Definition v F is strictly individually rational if v i is strictly larger than v i for all i I. Let F F be the set of feasible and strictly individually rational payoff profiles.

58 Dynamic Games with Complete Information Folk Theorem Theorem (Fudenberg and Maskin (1986)) Suppose that F is full-dimensional. For any v F, there exists a strategy profile σ and δ (0, 1) such that σ is a SPE and achieves v for any δ (δ, 1).

59 Dynamic Games with Complete Information Example (13 - Optimal Collusion - Infinite Cournot Competition - Proposed as exercise!) Dynamic Cournot duopoly model; Stage game equal to the static Cournot game and δ (0, 1); Using a "stick and carrot" strategy find the strongly symmetric SPE.

60 Dynamic Games with Incomplete Information Dynamic Games with Incomplete Information We consider dynamic games where past actions (by players or nature) are imperfectly observed; We treat them as an extension of dynamic game with complete information.

61 Dynamic Games with Incomplete Information Dynamic Games with Incomplete Information There are no subgame out of the game itself; The pure NE are (O, O) and (V, F ), but is the latter credible?

62 Dynamic Games with Incomplete Information Dynamic Games with Incomplete Information N players with i I {1,..., N} and c denotes Nature; h t = ( a 1, a 2,..., a k ) H set of histories with a k equal to an action taken by a player; P : H\Z I {c}, A (h) = {a (h, a) H} set of actions available to P (h); f c (a h) is the probability that a occurs after h for which P (h) = c; υ i : Z R.

63 Dynamic Games with Incomplete Information Dynamic Games with Incomplete Information I i a partition of {h H P (h) = i} with the property A (h) = A (h ) if h, h I i ; Each I i I i is player i s information set: the set of histories that player i cannot distinguish; A (I i ) the set of action available at I i ; Γ I, H, P, f c, {I i } i, {υ i } i.

64 Dynamic Games with Incomplete Information Extensive Games with Imperfect Information P ( ) = P (L, A) = P (L, B) = 1 and P (L) = 2; I 1 = {{ }, {(L, A), (L, B)}} and I 1 = {{L}}.

65 Dynamic Games with Incomplete Information Mixed and Behavioral Strategies Definition A mixed strategy of player i in an extensive game I, H, P, f c, {I i } i, {υ i } i is a probability measure over the set of player i s pure strategies. A behavioral strategy of player i is a collection {β i (I i )} Ii I 1 of independent probability measures, where β i (I i ) is a probability measure over A(I i ). Theorem For any mixed strategy of a player in a finite extensive game with perfect 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.

66 Dynamic Games with Incomplete Information Perfect Bayesian Nash Equilibrium Recall that in games with complete information some NE may be based on the assumption that some players will act sequentially irrationally at certain information sets off the path of equilibrium; In those games we ignored these equilibria by focusing on SPE; We extend this notion to the games with incomplete information by requiring sequential rationality at each information set: PBNE as equilibrium refinement of BNE; For each information set, we must specify the beliefs of the agent who moves at that information set.

67 Dynamic Games with Incomplete Information Sequential Rationality Definition A player is said to be sequentially rational iff, at each information set he is to move, he maximizes his expected utility given his beliefs at the information set (and given that he is at the information set) - even if this information set is precluded by his own strategy.

68 Dynamic Games with Incomplete Information Consistency Definition Given any strategy profile σ, an information set is said to be on the path of play iff the information set is reached with positive probability according to σ. Definition Given any strategy profile σ and any information set I i on the path of play of σ, a player s beliefs at I i is said to be consistent with σ iff the beliefs are derived using the Bayes rule and σ. This definition does not apply off the equilibrium path because otherwise we cannot apply the Bayes rule.

69 Dynamic Games with Incomplete Information Consistency Can the strategy (X, T, L) be considered not consistent by using our definition of consistency? We need to check consistency also off the equilibrium path by "trembling handing".

70 Dynamic Games with Incomplete Information Perfect Bayesian Nash Equilibrium Definition A strategy profile is said to be sequentially rational iff, at each information set, the player who is to move maximizes his expected utility given: 1. his beliefs at the information set; 2. given that the other players play according to the strategy profile in the continuation game. Definition A Perfect Bayesian Nash Equilibrium is a pair (σ, µ) of strategy profile 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.

71 Dynamic Games with Incomplete Information Perfect Bayesian Nash Equilibrium Example (13 - PBNE - See notes!)

72 Adverse Selection, Signaling and Screening Economics of Information Fundamental welfare theorems rely on perfect observability of all commodities to all market participants; Often in a transaction one party knows something that other parties don t know; We study three types of equilibria: i. Adverse selection: An informed individual s trading decisions adversely affects uninformed market participants; ii. Signaling: Informed individuals signal information about their unobservable knowledge through observable signal; iii. Screening: Uninformed parties develop mechanisms to screen informed individuals with different information.

73 Adverse Selection, Signaling and Screening Adverse Selection (Akerlof, 1970) Consider a labor market with many identical potential firms; Firms want to hire workers to produce final good by adopting a constant return to scale technology with labor as the sole input; Firms are risk-neutral, price-taker and profit maximizer; The price of the firm s output is equal to one; Workers differ in productivity, θ [ θ, θ ] distributed according to F (θ); r (θ) is the reservation wage of workers, i.e. the gain they might obtain by working at home.

74 Adverse Selection, Signaling and Screening Adverse Selection Perfect Observability Due to the assumptions of perfect competition and CRS the firms would set a wage equal to: w (θ) = θ Only the workers with {θ r (θ) θ} would accept the offer; The equilibrium is Pareto optimal.

75 Adverse Selection, Signaling and Screening Adverse Selection Imperfect Observability Definition In a competitive labor market with unobservable productivity levels, a competitive equilibrium is the pair {w, Θ } such that: Θ = {θ r (θ) < w } w = E (θ θ Θ ) The equilibrium is characterized by a fixed point; Typically this equilibrium will not be Pareto optimal.

76 Adverse Selection, Signaling and Screening Adverse Selection Suppose r (θ) = r and F (r) (0, 1); The Pareto optimal allocation is: if θ r accept employment and if θ < r not accept; If w > r then Θ = [ θ, θ ], if w < r then Θ = ; In both cases w = E (θ): Firms are unable to distinguish among workers productivity. Thus, the equilibrium outcome is ineffi cient: i. If the share of good workers is large enough, then E (θ) > r and too many workers are hired; ii. If the share of bad workers is large enough, then E (θ) < r and too few workers are hired.

77 Adverse Selection, Signaling and Screening Adverse Selection Adverse selection occurs when an informed individual s decision depends on her unobservable characteristics and adversely affects the uninformed agents; In the labor market context, adverse selection arises when only relatively less capable workers accept a firm s employment offer at any given wage; If r(θ) is no longer constant adverse selection arises (specifically if it is increasing!).

78 Adverse Selection, Signaling and Screening Adverse Selection Suppose r(θ) θ for all θ and r (θ):

79 Adverse Selection, Signaling and Screening Adverse Selection Possibility of multiple equilibria:

80 Adverse Selection, Signaling and Screening Example (14 - Credit Market and Adverse Selection - Proposed as exercise!) A borrower need a loan to finance a project of value I = 1; (r, x) is a loan contract, r 0 is the interest rate and x [0, 1] is a collateral; q is the probability of borrower s default, in that case collateral value is bx with b (0, 1); Two types of borrower, 0 < q 1 < q 2 < 1/2, whose mass is 50% of pop; Credit market is perfect competitive and agents are risk neutral; Find the Pareto optimal allocation and the equilibrium with asymmetric information.

81 Adverse Selection, Signaling and Screening Both the firms and the high-ability workers have incentives to transmit information; Market responses to the problem of adverse selection: i. Signaling: Informed individuals (workers) choose their level of education to signal information about their ability to uninformed parties (the firms); ii. Screening: Uninformed parties (firms) take steps to screen the various types of individuals on the other side of the market (workers).

82 Adverse Selection, Signaling and Screening Signaling (Spence 1973, 1974) Two types of workers: high ability, θ H, and low ability, θ L ; The probability of high type workers ( exogenous share in the population) is λ (0, 1); Perfect competitive markets (zero profits), whose profit s firms is π = θ w.

83 Adverse Selection, Signaling and Screening Signaling Full Information Bertrand competition outcome; (w L, w H ) such that w H = θ H and w L = θ L ; If abilities are not observable, then w = E (θ θ Θ ) where Θ is equal to the set of types accepting the contract.

84 Adverse Selection, Signaling and Screening Signaling Imperfect Observability Suppose that before entering the job market a worker can get some level of education, e, and the firms observe it; Assume that: i. Education does not increase workers productivity; ii. Education is more costly for the low ability worker than for the high ability worker, also at the margin (single crossing property); Workers payoff u = w c (e, θ) with c (0, θ) = 0, c e (e, θ) > 0, c θ (e, θ) < 0 and c eθ (e, θ) < 0.

85 Adverse Selection, Signaling and Screening Signaling Welfare Welfare effect is generally ambiguous: Signaling can lead to a more effi cient allocation of workers labor and to a Pareto improvement; Signaling is a costly activity and workers welfare may be reduced.

86 Adverse Selection, Signaling and Screening Signaling Timing Nature determines worker s type; Worker chooses an education level contingent on his type; Conditional on the education level, firms make wage offers simultaneously Worker decides which offer to accept, if any.

87 Adverse Selection, Signaling and Screening Signaling Perfect Bayesian Equilibrium We use the concept of Perfect Bayesian Equilibrium; The worker s strategy is optimal given the firm s strategy; µ (e) are the up-dated firms beliefs that the worker is high-type; Each firm s wage offer, following the choice e, is optimal given the belief µ (e), the worker s strategy and the other firms strategy.

88 Adverse Selection, Signaling and Screening Signaling Perfect Bayesian Equilibrium The firms (pure strategy) Nash equilibrium wage offers equal the worker s expected productivity: w (e) = µ (e) θ H + (1 µ (e)) θ L

89 Adverse Selection, Signaling and Screening Signaling Separating Equilibrium Let e (θ) the educational equilibrium choice of type θ and w (e) the equilibrium wage of a worker who displays an educational level of e; In any separating equilibrium e (θ H ) = e (θ L ) and w (e (θ H )) = θ H and w (e (θ L )) = θ L ; e (θ L ) = 0, which implies that the low ability worker receives utility equal to θ L at the separating equilibrium.

90 Adverse Selection, Signaling and Screening Signaling Example (15 - Education and Signaling - See notes!) θ H = 2, θ L = 1, c (e, θ) = e θ ; Find the separating PBNE.

91 Adverse Selection, Signaling and Screening Signaling Pooling Equilibrium The two types of workers choose the same level of education e (θ L ) = e (θ H ) = e ; This implies that w (e ) = λθ H + (1 λ) θ L ; Any education level between 0 and e can be sustained as a pooling equilibrium. e corresponds to the level of education such that low type receives zero utility with w = E [θ], e : w c(e (θ L ), θ L ) = 0.

92 Adverse Selection, Signaling and Screening Signaling Example (16 - Reputation Game - Proposed as exercise!)

93 Adverse Selection, Signaling and Screening Screening Workers outside option is zero r(θ i ) = 0; Jobs may differ in the task level t > 0 required. (Ex. different number of ours per week); The output of a type-θ i worker is θ i regardless of the worker s task level. Higher task levels only affect workers utility. They are costly for the workers; Workers payoff u = w c (t, θ) with c (0, θ) = 0, c t (t, θ) > 0, c θ (t, θ) < 0 and c tθ (t, θ) < 0.

94 Adverse Selection, Signaling and Screening Screening Timing Stage 1: Firms simultaneously announce a menu of contracts (w, t). Each firm may announce any finite numbers of contracts; Stage 2: Workers decide whether they want to sign a contract and which one to sign: - If indifferent between signing and not signing a contract, the worker will sign; - If indifferent between two types of contract, the worker will choose the contract with the lower task level.

95 Adverse Selection, Signaling and Screening Screening Full Observability If abilities are observable equilibrium entails firms offering a different contract to each type: (w H, t H ) = (θ H, 0) for high ability workers; (w L, t L ) = (θ L, 0) for low ability workers; Workers accept contracts and firms earn zero profits.

96 Adverse Selection, Signaling and Screening Screening Break-Even Line

97 Adverse Selection, Signaling and Screening Screening Pooling Equilibria No pooling equilibria exists:

98 Adverse Selection, Signaling and Screening Screening Separating Equilibria If (w L, t L ) and (w H, t H ) are the contracts signed respectively by the low- and the high-ability workers in a separating equilibrium, then both contracts yield zero profits: w L = θ L and w H = θ H ; This implies that separating equilibria does not allow for cross-subsidies. Separating equilibria are on the break-even lines; In any separating equilibrium, the low-ability workers accept contract (w L, t L ) = (θ L, 0). They receive the same contract as under full information.

99 Adverse Selection, Signaling and Screening Screening Separating Equilibria In any separating equilibrium the high ability workers accept contract (w H, t H ) = (θ H, t H ) where t H is such that: θ H c( t H, θ L ) = θ L c(0, θ L ) = θ L ; This means that low type is indifferent between contract (θ H, t H ) and contract (0, θ L ); If t H > t H, firms can offer contracts which attract high ability workers and make positive profits.

100 Adverse Selection, Signaling and Screening Screening Separating Equilibria A separating equilibrium exists if the pooled break-even line is suffi ciently far from θ H ; This means that λ must be suffi ciently low; As in the signaling model, asymmetric information leads to Pareto ineffi cient outcomes.

101 Principal-Agent Model Principal-Agent Model A principal wants to delegate a task to an agent; Delegation benefits: Increasing returns associated with tasks division, or by the principal s lack of time or ability to perform the task himself; The agent and the principal have different objectives; If the agent has no private information, then the principal could propose a contract that perfectly controls the agent s behavior No incentives problems; When the agent has private information, then incentives problems arise.

102 Principal-Agent Model Principal-Agent Model Why a theory of contract?; A principal delegates an action to a single agent through the take-it-or-leave-it offer of a contract; One-shot relationship: No repetition is available to achieve effi ciency; The principal proposes the contract, no bargaining issues; A benevolent court of law must be available. It enforces the contract and imposes penalties if one of the contractual partners adopts a behavior that deviates from the one specified in the contract.

103 Principal-Agent Model Principal-Agent Model Definition A contract is a legally binding exchange of promises or agreement between parties. Different types of contract exist; Implicit contract: A contract that is self-enforcing. When the two parties play a game where the unique Subgame Perfect Nash equilibrium of the game corresponds to the desired outcome; Explicit contract: Whenever the desired outcome is not Subgame Perfect we need an explicit contract. Internalizing court s punishment agents do not have interest in deviating from the agreement.

104 Principal-Agent Model Principal-Agent Model Problem of delegating a task to an agent with different objectives and private information; Which private information? Moral hazard or hidden action: Endogenous uncertainty for the principal; Adverse selection or ex-post hidden information: Exogenous uncertainty for the principal; Non verifiability: The principal and the agent share ex-post the same information; No court of law can observe this information agency costs.

105 Principal-Agent Model Principal-Agent Model Hidden action An agent chooses actions that affect the value of trade or the agent s performance; The principal cannot control those actions and they are not observable either by the principal or by the court of law Actions are not contractible; Examples: Worker s effort in performing a task, timing devoted to a task, how safely a driver drives, green-investment by regulated firms...

106 Principal-Agent Model Principal-Agent Model Hidden action With moral hazard the expected volume of trade depends explicitly on the agent s effort; The realized production level is a noisy signal of the agent s action; The principal wants to design a contract that induces the highest effort from the agent despite the impossibility of directly conditioning the agent s reward on his action.

107 Principal-Agent Model Principal-Agent Model Hidden action To make the agent responsible for the consequences of his actions the principal lets the agent bear some risk; Risk sharing/effi ciency and rent/effi ciency trade-off.

108 Principal-Agent Model Principal-Agent Model Hidden information An agent gets access to information that is not available neither to the principal nor to the court of law; Examples: A tenant observes local weather conditions, experts know the diffi culty of the case, regulated firms have private information on their costs,...; To achieve effi ciency, the contract must elicit the agent s private information; The principal must give up some information rent to the privately informed agent; Rent-effi ciency trade-off.

109 Moral Hazard Moral Hazard The principal delegates the agent to perform a task; The worker chooses the intensity of effort, e {0, E }, to perform the task. His effort positively affects the output q {0, Q}; The principal only cares about the output and don t observe effort; Since the effort is costly, the principal has to compensate the agent for incurring this cost; The agent s compensation has to be contingent on the outcome q that is a noisy signal of effort e.

110 Moral Hazard Moral Hazard Risk-sharing/effi ciency trade-off Pr {q = Q E } = p E and Pr {q = Q 0} = p 0 with p 0 < p E ; The risk-neutral principal s utility q w; The agent s utility u (w) e with u w > 0, u ww 0; The agent s reservation û u (ŵ); p E Q E p 0 Q and p E Q E û then e = E is effi cient.

111 Moral Hazard Moral Hazard Timing 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 û. If the contract is accepted, the agent then chooses the level of effort e {0, E }, which is unobservable by the principal; iv. Finally, as a result of the agent s choice, a quantity q is produced.

112 Moral Hazard Moral Hazard Full Information and risk-sharing/effi ciency If e is verifiable then the contract can specify the desired effort, e = E, and the contingent transfers, {w, w} with w if q = 0 and w if q = Q; The principal s problem is: max E Q (p E w + (1 p E ) w) w,w s.t. : p E u (w) + (1 p E ) u (w) E û (IR) Since the principal is risk-neutral and the agent is risk adverse, then perfect insurance, w = w s.t. u (w) = E + û.

113 Moral Hazard Moral Hazard Incomplete Information and risk-sharing/effi ciency If e is not verifiable, then the principal s problem is: s.t.: max p E Q (p E w + (1 p E ) w) w,w p E u (w) + (1 p E ) u (w) E û b u (w) u (w) (IR) E p E p 0 (IC ) Since p E > p 0 then w w and no longer agent s full-insurance.

114 Moral Hazard Moral Hazard Incomplete Information and risk-sharing/effi ciency Using the binding constraints: - u (w) = û + E p E E p E p 0 < û + E; - u (w) = û + E + (1 p E )E p E p 0 > û + E; - r E ( w SB ) w FB, risk-premium.

115 Moral Hazard Moral Hazard Full Information and rent/effi ciency Assume that also the agent is risk-neutral, u (w) = w, and has limited liability, w ŵ; The principal s problem is: s.t.: max p E Q (p E w + (1 p E ) w) w,w p E w + (1 p E ) w E û (IR) w, w ŵ (LL) First best solution is not affected by LL.

116 Moral Hazard Moral Hazard Incomplete Information and rent/effi ciency Let b w w and w w, then the principal s problem becomes: max b,w p E Q (w + p E b) s.t.: w + p E b ŵ + E (IR) b E p E p 0 (IC ) w ŵ (LL) IR is not an issue in the presence of LL, w = ŵ and w = ŵ + E p E p 0 and R p 0E p E p 0 is the agent s expected rent.

117 Moral Hazard Moral Hazard Inference problem The principal s goal is to detect what the agent has done by observing related variables; Should the wage increase with the observed output level? The answer is, Not necessarily.

118 Moral Hazard Moral Hazard Full Inference and full information (Mirrlees, 1975) The output is q (e) = e + ε, with ε F ( ) over R, lim ε F (ε) f (ε) = 0; P s max problem with full information: max E [q w (q) e] e,w (q) s.t. : E [u (w (q)) e e] û It is optimal for the P to full insure the A and e FB : h e ( e FB ) = 1 with w (q) = h (e) u 1 (û + e).

119 Moral Hazard Moral Hazard Full Inference and incomplete information (Mirrlees, 1975) Consider the second-best setting and the following schedule (in terms of promised utility): { U if q Q u = U P if q < Q The contract is defined by {U, P, Q}; q = e + ε q < Q if ε < Q e, i.e. with probability F (Q e); The agent s expected utility is U F (Q e) P e; 1 To implement FB P = f (Q e FB ) with U = û + efb F (Q e) + f (Q e) ; No cost to implement FB allocation but we need no LL.

120 Moral Hazard Moral Hazard Limited Inference and incomplete information (Mirrlees, 1975) q [0, Q], e {0, E } and MLRP: l (q) f E (q) f 0 (q) f E (q) l q (q) > 0; P s max problem: with Q max (q w (q)) f E (q) dq w (q) 0 Q 0 Q 0 u (w (q)) f E (q) dq E û (IR, λ) u (w (q)) f E (q) dq E Q 0 u (w (q)) f 0 (q) dq (IC, µ) The FOC is (λ + µl (q)) u w (w (q)) = 1, which implies that w q (q) > 0.

121 Moral Hazard Moral Hazard First-Order Approach q [0, Q], e [e, e + ] with F (q e) and MLRP: l (q) f e (q e) f (q e) with l q (q) > 0; P s max problem: Q 0 e = arg max ê Q max V (q w (q)) f (q e) dq w (q),e 0 u (w (q)) f (q e) dq ψ (e) û (IR, λ) Q 0 u (w (q)) f (q ê) dq ψ (ê) (IC, µ) By using FOA, if the argmax of IC is unique and SOC are satified, then we can replace IC by FOC.

122 Adverse Selection Adverse Selection This type of agency problem arises in many settings: Interaction between the shareholders of a firm and its managers, or the firm and its workers: Private information about the productivity of the managers or the workers; Interaction between an investor and a firm, or a bank and its managers: private information about the projects undertaken; Relationship between an insurance company and its customers: private information about the risks that the customer is facing; Price discrimination: private information about the customers willingness to pay.

123 Adverse Selection Adverse Selection Price discrimination: Full Information P (seller) produces q at cost C (q) with C q, C qq > 0 and sells at t; A (buyer) gets benefit θq with θ { θ, θ } { } with probabilities µ, µ ; Complete info P s problem: max t,q s.t. : θq t 0 FB allocation is C q ( q FB (θ) ) = θ and t FB (θ) = θq FB (θ).

124 Adverse Selection Adverse Selection Price discrimination: Incomplete Information Incomplete info P s problem: max µ ( t C ( q )) + µ (t C (q)) (t,q),(t,q) s.t.: θq t 0 ( IR ) θq t 0 (IR) ( ) θq t θq t IC θq t θq t (IC ) Let r θq t and r θq t the buyers rent.

125 Adverse Selection Adverse Selection Price discrimination: Incomplete Information The P s problem is equal to: max µ ( θq C ( q ) r ) + µ ( θq C (q) r ) (t,q),(t,q) s.t.: r 0 ( IR ) r 0 (IR) r r + ( θ θ ) q ( IC ) r r ( θ θ ) q (IC ) From ( IC ) and (IC ) q q, and ( IR ) and (IC ) are not binding.

126 Adverse Selection Adverse Selection Price discrimination: Incomplete Information Rent/effi ciency trade-off: SB allocation in terms of q: q SB : C q (q ) FB = θ ( ) q SB : C q q SB = θ µ ( ) θ θ µ In terms of transfers: t SB = θq SB ( θ θ ) q SB t SB = θq SB q SB < q SB.

127 Adverse Selection Adverse Selection General Framework q Q and θ Θ [ θ, θ ] distributed according to F ( ); Agent s preference U (q, t; θ) = V (q, θ) t; Principal s preference t C (q, θ); P s problem under full info: max t C (q, θ) (t,q) s.t.: V (q, θ) t 0 FB allocation is C q ( q FB (θ), θ ) = V q ( q FB (θ), θ ) and t FB (θ) = V ( q FB (θ), θ ).

128 Adverse Selection Adverse Selection General Framework: Implementability Let r (θ) V (q (θ), θ) t (θ) the agent s rent; Then r (θ) 0 and r (θ) = max θ V ( q ( θ ), θ ) t ( θ ) ; Spence-Mirrlees condition (i.e. single-crossing property) V qθ (q, θ) 0 for each q, θ; Theorem If single-crossing property holds, then (q ( ), r ( )) is incentive compatible iff q θ (θ) 0 Proof. (See notes!). θ r (θ) = r (θ) + V θ (q (s), s) ds θ

129 Adverse Selection Adverse Selection General Framework: Optimality P s problem under incomplete info: s.t.: θ max [t (θ) C (q (θ), θ)] f (θ) dθ (t( ),q( )) θ V (q (θ), θ) t (θ) 0, θ V (q (θ), θ) t (θ) V ( q ( θ ), θ ) t ( θ ), θ, θ

130 Adverse Selection Adverse Selection General Framework: Optimality By using the Theorem, the P s problem is: s.t.: θ max [V (q (θ), θ) C (q (θ), θ) r (θ)] f (θ) dθ (t( ),q( )) θ r (θ) 0, θ r (θ) = r (θ) + q θ (θ) > 0 θ θ V θ (q (s), s) ds, θ, θ Additional assumption V θ (q (θ), θ) 0. (See notes!).