Macroeconomic Applications of Global Games Pau Roldan NYU April 28, 2014 1 / 50
Motivation Global Games in Macroeconomics Many macroeconomic phenomena can be understood as the outcome of self-fulfilling expectations, higher-order beliefs and information processing in an environment of strategic uncertainty. It is natural to model such scenarios as games in which players interact in coordination and their payoffs depend on own actions, actions of others and economic fundamentals. Two immediate problems arise: One might need to keep track of the infinite hierarchy of higher-order beliefs. This can 1 become intractable. When economic fundamentals are common knowledge, coordination can give rise to 2 multiplicity. Global games offer a tractable and stylized approach: Definition A global game is said to be a game of incomplete information within an strategic environment in which players receive private signals on unknown economic fundamentals. 2 / 50
Motivation Global Games in Macroeconomics First introduced by Carlsson and van Damme (1993). Two players i = 1 , 2 choose action a i ∈ { 0 , 1 } . Payoffs: a 2 = 1 a 2 = 0 a 1 = 1 θ , θ θ − 1,0 a 1 = 0 0, θ − 1 0,0 Common knowledge: ◮ If θ > 1, each player has a dominant strategy to invest. ◮ If θ ∈ [0 , 1], both invest and both not invest are two pure Nash equilibria. ◮ If θ < 0, each player has a dominant strategy not to invest. Incomplete information: ◮ Common prior is θ ∼ unif ( ±∞ ). ◮ Each player receives signal x i = θ + ε i , ε i ∼ N (0 , 1 /β ). ⋆ Player’s posterior: θ | x i ∼ N ( x i , 1 /β ). ⋆ Player’s belief about other player’s signal: x − i | x i ∼ N ( x i , 2 /β ). ◮ Solution: ⋆ Player i ’s switching strategy: a i ( x ) = 1 [ x > x ⋆ ] . ⋆ Player i assigns posterior probability: � √ β ( x ⋆ − x ) � P [ x − i > x ⋆ ] = 1 − Φ √ 2 ⋆ The unique strategy that survives infinite deletion of dominated strategies is x ⋆ = 1 / 2 Proof: See Morris and Shin (2003). 3 / 50
Motivation Global Games in Macroeconomics Main results: Multiplicity in coordination games arises from the implicit assumption of common 1 knowledge of the fundamental. Uniqueness obtains as a perturbation away from perfect information. 2 The entire hierarchy of beliefs can be conveniently captured in sufficient statistics by 3 means of iterative deletion of dominated strategies. Intuitively: ◮ The private signal may offer very precise information about the fundamental but it provides little or no information about the information embedded in others’ signals. ◮ Even if the idiosyncratic noise is tiny, players remain highly uncertain about others’ actions. 4 / 50
Literature on Coordination in Macroeconomics Coordination games with equilibrium multiplicity: ◮ Currency and debt crises: Krugman (1979), Flood and Garber (1984), Obstfeld (1986, 1996), Chari and Kehoe (2003), Cole and Kehoe (2000), Calvo (1988), Broner (2007). ◮ Bank runs: Diamond and Dybvig (1983). ◮ Sociopolitical change: Atkeson (2001), Edmond (2005). Global games of regime change: ◮ Morris and Shin (1998, 1999, 2004), Hellwig (2002), Hellwig, Mukherji and Tsyvinski (2006), Goldstein and Pauzner (2000), Angeletos and Werning (2006), Chamley (1999, 2003), Angeletos, Hellwig and Pavan (2006, 2007), Goldstein, Ozdenoren and Yuan (2011), Rochet and Vives (2004), Corsetti, Guimares and Roubini (2006). Applications: Speculative attacks, debt crises, bank runs, investment crashes, adoption of new technologies, liquidity crashes, socio-political change... Today: “Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks” (Morris and Shin; AER, ‘98). “Crises and Prices” (Angeletos and Werning; AER, ‘06). “Self-Fulfilling Currency Crises: Role of Interest Rates” (Hellwig, Mukherji, and Tsyvinski; AER, ‘06). “Signaling in a Global Game: Coordination and Policy Traps” (Angeletos, Hellwig and Pavan; JPE, ‘06). “A Theory of the Onset of Currency Attacks” (Morris and Shin; HB chapter, ‘99) “Dynamic Global Games of Regime Change” (Angeletos, Hellwig and Pavan; ECMA, ‘07) 5 / 50
Outline Motivation 1 Uniqueness in Static Global Games 2 Morris and Shin (1998) Multiplicity in Static Global Games 3 Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006) Uniqueness and Multiplicity in Dynamic Global Games 4 Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007) Discussion and Open Questions 5 6 / 50
PART I Static global games of regime change 7 / 50
Outline Motivation 1 Uniqueness in Static Global Games 2 Morris and Shin (1998) Multiplicity in Static Global Games 3 Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006) Uniqueness and Multiplicity in Dynamic Global Games 4 Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007) Discussion and Open Questions 5 8 / 50
Morris and Shin (1998) Uniqueness in a model of attacks to the status quo The game: ◮ Measure-one continuum of agents, i ∈ [0 , 1]. Each chooses an action a i ∈ { 0 , 1 } (attack status quo or not), with a cost c ∈ (0 , 1) of attacking. � 1 ◮ Status quo is abandoned (attack is successful) if A > θ , where A ≡ 0 a j dj is size of the attack and θ ∈ R is an exogenous fundamental (strength of the status quo). Key elements: ◮ Regime outcome: R ( θ ) ≡ 1 [ A >θ ] ◮ Individual payoff (ex-post): U ( a i , A , θ ) = a i ( R ( θ ) − c ) ◮ Marginal payoff: π ( A , θ ) ≡ U (1 , A , θ ) − U (0 , A , θ ) ◮ Coordination: π ( · ) increases with A . Information: ◮ If θ is common knowledge: ⋆ There is a set [ θ, θ ] ⊆ [0 , 1] such that all attack if θ ≤ θ , none attack if θ ≥ θ . ⋆ If θ ∈ ( θ, θ ), both attack and no attack are self-fulfilling equilibria. ◮ Suppose θ is observed with noise: ⋆ Nature draws: θ ∼ N ( z , 1 /α ) ⋆ Private signals: x i = θ + ξ i where ξ i ∼ N (0 , 1 /β ). 9 / 50
Morris and Shin (1998) Uniqueness in a model of attacks to the status quo Focus on monotone BNE . Why? ◮ The cdf of the agent’s posterior about θ is decreasing in x . ◮ For x < x , where x solves P [ θ ≤ 0 | x ] = c , attack is strictly dominant. ◮ For x > x , where x solves P [ θ ≥ 1 | x ] = 1 − c , not attack is strictly dominant. ◮ Intuitively, there should be a switching point x ⋆ ∈ [ x , x ]. Characterization : ◮ There is a threshold x ⋆ ∈ R such that attack iff x ≤ x ⋆ . � √ β ( x ⋆ − θ ) ◮ Aggregate size of the attack is A ( θ ) = P [ x ≤ x ⋆ | θ ] = Φ � . ◮ Status quo is abandoned iff θ ≤ θ ⋆ , where θ ⋆ solves θ ⋆ = A ( θ ⋆ ), that is θ ⋆ = Φ �� β ( x ⋆ − θ ⋆ ) � (1) ◮ x ⋆ solves the indifference condition P [ θ ≤ θ ⋆ | x ⋆ ] = c , that is � β x ⋆ + α z �� �� − θ ⋆ 1 − Φ β + α = c (2) β + α ◮ Equations (1) and (2) jointly determine solution for thresholds ( θ ⋆ , x ⋆ ). 10 / 50
Morris and Shin (1998) Uniqueness in a model of attacks to the status quo Proposition The equilibrium is unique iff β ≥ α 2 (3) 2 π and is in monotone strategies. Uniqueness holds as perturbation away from common knowledge iff (3) holds. There is uniqueness if the quality of public information is not too high. Else, public information would foster higher coordination. Proposition Let R ( θ ) ≡ 1 [ A ( θ ) >θ ] be the regime outcome. Then, as β → + ∞ , � 1 if θ < θ ∞ ≡ 1 − c R ( θ ) − → 0 otherwise When the noise in private information is small and θ ≈ θ ∞ , a small variation in θ can trigger a large variation in the size of the attack and in the regime outcome. 11 / 50
Morris and Shin (1998) Uniqueness in a model of attacks to the status quo Results are not robust to the existence of (endogenous) public information : The role of market prices as private-information aggregators: 1 ⋆ Angeletos and Werning (2006). ⋆ Hellwig, Mukherji and Tsyvinski (2006). The strategic feedback between speculators and policy-maker decisions: 2 ⋆ Angeletos, Hellwig and Pavan (2006). ⋆ Goldstein, Ozdenoren and Yuan (2011). Unlike private information, the use of public information permits agents to make inference about others’ actions, reinforces coordination and re-establishes multiplicity through non-fundamental volatility. 12 / 50
Outline Motivation 1 Uniqueness in Static Global Games 2 Morris and Shin (1998) Multiplicity in Static Global Games 3 Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006) Uniqueness and Multiplicity in Dynamic Global Games 4 Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007) Discussion and Open Questions 5 13 / 50
Angeletos and Werning (2006) Multiplicity through prices of trade Restore multiplicity through market-based information aggregation. First stage : ◮ Trade over risky asset with dividend f ( θ ) = θ at price p . ◮ Agent i invests in k units of risky asset: − e − γ w i v ( w i ) = w i = w 0 , i − pk i + fk i ◮ Asset supply is stochastic: √ K s ( ε ) = ε/ δ where ε ∼ N (0 , 1), 0 < δ < + ∞ (exogenous public noise). Second stage : ◮ Agents observe p and play ` a la Morris and Shin (1998). 14 / 50
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