Macroeconomic Applications of Global Games Pau Roldan NYU April - - PowerPoint PPT Presentation

Macroeconomic Applications of Global Games

Pau Roldan

NYU

April 28, 2014

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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:

1

One might need to keep track of the infinite hierarchy of higher-order beliefs. This can become intractable.

2

When economic fundamentals are common knowledge, coordination can give rise to 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.

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Motivation

Global Games in Macroeconomics

First introduced by Carlsson and van Damme (1993). Two players i = 1, 2 choose action ai ∈ {0, 1}. Payoffs: a2 = 1 a2 = 0 a1 = 1 θ,θ θ − 1,0 a1 = 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 xi = θ + εi, εi ∼ N(0, 1/β). ⋆ Player’s posterior: θ|xi ∼ N (xi, 1/β). ⋆ Player’s belief about other player’s signal: x−i|xi ∼ N (xi, 2/β). ◮ Solution: ⋆ Player i’s switching strategy: ai(x) = 1[x>x⋆]. ⋆ Player i assigns posterior probability:

P[x−i > x⋆] = 1 − Φ √β(x⋆ − x) √ 2

• ⋆ The unique strategy that survives infinite deletion of dominated strategies is

x⋆ = 1/2 Proof: See Morris and Shin (2003).

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Motivation

Global Games in Macroeconomics

Main results:

1

Multiplicity in coordination games arises from the implicit assumption of common knowledge of the fundamental.

2

Uniqueness obtains as a perturbation away from perfect information.

3

The entire hierarchy of beliefs can be conveniently captured in sufficient statistics by 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.

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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)

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Outline

1

Motivation

2

Uniqueness in Static Global Games Morris and Shin (1998)

3

Multiplicity in Static Global Games Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006)

4

Uniqueness and Multiplicity in Dynamic Global Games Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007)

5

Discussion and Open Questions

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PART I Static global games of regime change

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Outline

1

Motivation

2

Uniqueness in Static Global Games Morris and Shin (1998)

3

Multiplicity in Static Global Games Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006)

4

Uniqueness and Multiplicity in Dynamic Global Games Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007)

5

Discussion and Open Questions

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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 ai ∈ {0, 1}

(attack status quo or not), with a cost c ∈ (0, 1) of attacking.

◮ Status quo is abandoned (attack is successful) if A > θ, where A ≡

1

0 ajdj 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(ai, A, θ) = ai(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:

xi = θ + ξi where ξi ∼ N (0, 1/β).

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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⋆. ◮ Aggregate size of the attack is A(θ) = P[x ≤ x⋆|θ] = Φ

√β(x⋆ − θ)

• .

◮ 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

1 − Φ

• β + α

βx⋆ + αz β + α − θ⋆

• = c

(2)

◮ 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 ≥ α2 2π (3) 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 → +∞, R(θ) − →

• 1

if θ < θ∞ ≡ 1 − c

• therwise

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.

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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:

1

The role of market prices as private-information aggregators:

⋆ Angeletos and Werning (2006). ⋆ Hellwig, Mukherji and Tsyvinski (2006). 2

The strategic feedback between speculators and policy-maker decisions:

⋆ 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.

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Outline

1

Motivation

2

Uniqueness in Static Global Games Morris and Shin (1998)

3

Multiplicity in Static Global Games Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006)

4

Uniqueness and Multiplicity in Dynamic Global Games Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007)

5

Discussion and Open Questions

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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:

v(wi) = −e−γwi wi = w0,i − pki + fki

◮ 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).

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Angeletos and Werning (2006)

Multiplicity through prices of trade

Definition

1

An equilibrium is a price function P(θ, ε), individual strategies for investment and attacking, k(x, p) and a(x, p), and their corresponding aggregates, K(θ, p) and A(θ, p), such that: k(x, p) ∈ arg max

k∈R

E[v(w0 + (f (θ) − p)k)|x, p] K(θ, p) = E[k(x, p)|θ, p] K(θ, P(θ, ε)) = K s(ε) a(x, p) ∈ arg max

a∈{0,1}

E[U(a, A(θ, p), θ)|x, p] A(θ, p) = E[a(x, p)|θ, p]

2

The equilibrium regime outcome is R(θ, ε) ≡ 1[A(θ,P(θ,ε))>θ].

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Angeletos and Werning (2006)

Multiplicity through prices of trade

When f (θ) = θ, equilibrium price is P(θ, ε) = θ − ε/

• δp, where

δp = β2δ γ2 Thus, unlike before, public information improves with private information (higher β means higher δp). In Morris and Shin (1998):

◮ Precision of public information was fixed, so that sufficiently precise private

information ensured uniqueness.

Now:

◮ Better private information improves public information and reduces strategic

uncertainty to ensure multiplicity.

◮ There is multiplicity even for a small deviation of 1/β or 1/δp from zero (the

perfect-information benchmark).

Figures 16 / 50

Angeletos and Werning (2006)

Multiplicity through prices of trade

Proposition (Nonfundamental volatility)

There are multiple equilibria if either source of noise is small: δβ√β > γ2√ 2π. Uniqueness is no longer a (small) perturbation away from perfect info.

Proposition (Regime outcome in perfect information)

As either source of noise vanishes (β → +∞ or δ → +∞),

1

There exists a passive equilibrium R(θ, ε) → 0.

2

There exists an aggressive equilibrium R(θ, ε) → 1. whenever θ ∈ (θ, θ). Even in perfect information (θ common knowledge), regime outcome is still fully sunspot-driven.

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Outline

1

Motivation

2

Uniqueness in Static Global Games Morris and Shin (1998)

3

Multiplicity in Static Global Games Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006)

4

Uniqueness and Multiplicity in Dynamic Global Games Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007)

5

Discussion and Open Questions

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Hellwig, Mukherji and Tsyvinski (2006)

Multiplicity through the role of interest rates

Restore multiplicity through information inherent in equilibrium interest rates, but not through the information structure itself.

• 1. Traders:

Each trader i ∈ [0, 1] is endowed with one unit of domestic currency. Two possible actions: buy domestic bond or exchange one-for-one for dollars. Net returns: Devaluation No devaluation Dollar 1 Domestic bond r r

• 2. Central Bank (CB):

Net value of maintaining the peg is θ − C(r, A), where

◮ θ ∈ R: Unobserved fundamental (strength of CB’s commitment). ◮ A ∈ [0, 1]: Loss of foreign reserves (total of dollars withdrawn by traders). ◮ C(r, A): Cost of defending the peg, in terms of losses of foreign reserves (A) and

interest rates (r).

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Hellwig, Mukherji and Tsyvinski (2006)

Multiplicity through the role of interest rates

First stage:

◮ Nature draws θ ∈ unif (±∞) (improper common prior). ◮ Traders receive unbiased private signal:

xi ∼ N(θ, 1/β)

Second stage:

◮ Domestic bond market and CB open. ◮ Traders submit contingent bids

ai(xi, r) ∈ [0, 1] di(xi, r) = 1 − ai(xi, r)

◮ Supply of bonds is exogenous: S(s, r), with S1(·) > 0, S2(·) ≥ 0 and

s ∼ N(0, 1/δ)

Third stage:

◮ CB decides whether to maintain peg after observing θ, r and A. ◮ Devaluation occurs iff

θ ≤ C(r, A)

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Hellwig, Mukherji and Tsyvinski (2006)

Multiplicity through the role of interest rates

Definition

A symmetric PBE consists of a bidding strategy a(xi, r), and interest rate function R(θ, s), a reserve loss function A(θ, s) and beliefs p(xi, r) on the posterior probability of a devaluation, such that:

1

a(xi, r), A(θ, r), and R(θ, s) satisfy a(xi, r)      = 1 if p(xi, r) > r ∈ [0, 1] if p(xi, r) = r = 0 if p(xi, r) < r (4) A(θ, r) =

• a(x, r)
• βφ(
• β(x − θ))dx

1 − A(θ, R(θ, s)) = S(s, R(θ, s))

2

For all r such that {(θ, s) : r = R(θ, s)} = ∅, p(xi, r) satisfies Bayes’ law. Key multiplicity channel will operate through r in (4):

◮ r determines direct payoff of holding domestic bond (LHS)... ◮ ...but also affects expectations of likelihood of devaluation (RHS). 21 / 50

Hellwig, Mukherji and Tsyvinski (2006)

Multiplicity through the role of interest rates

If equilibrium is in thresholds, then p(xi, r) = P[θ ≤ θ⋆(r)|xi, r]. For r ∈ (0, 1), there is a unique threshold x⋆(r) such that p(x⋆(r), r) = r (5) that is, indifference between domestic bonds and dollars if x = x⋆(r). Then, reserve losses are A(θ, r) = Φ(√β(x⋆(r) − θ)), so θ⋆(r) solves θ⋆(r) = C(r, A(θ⋆(r), r)) (6) and devaluation iff θ ≤ θ⋆(r). Market clearing:

◮ Assume S(s, r) = Φ(s − γΦ−1(r)); γ ≥ 0 price-elasticity of bond supply. ◮ Bond market clears: 1 − A(θ, r) = S(s, r), or

x⋆(r) − γ √β Φ−1(r) = θ − s √β for all (θ, s).

◮ Thus, z ≡ θ − s/√β is a sufficient statistic, and r is chosen, for all z, from:

R(z) ≡

• r ∈ [0, 1] : z = x⋆(r) −

γ √β Φ−1(r)

• An equilibrium is a joint solution to (5) and (6), conditional on r ∈ R(z), ∀z.

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Hellwig, Mukherji and Tsyvinski (2006)

Multiplicity through the role of interest rates

Multiplicity:

◮ There exists a unique solution for thresholds {x⋆(r), θ⋆(r)}, which is continuous in r. ◮ However, ∃z for which R(z) may be a multi-valued correspondence. Formal statement

Intuitively:

◮ Market clearing (1 − A = S) and marginal trader condition (r = p(x, r)) imply:

r = Φ

• β(1 + δ)
• θ⋆(r) − z − γ

Φ−1(r) √β(1 + δ)

• ◮ Three effects:

⋆ Payoff effect (+):

Higher r makes bonds pay more and be more attractive than dollars.

⋆ Devaluation effect (±):

Higher r affects θ⋆(r) and thus probability of devaluation (direction depends on C(·)).

⋆ Maket-clearing effect (+):

Higher r means higher x⋆(r), marginal agent’s expectation of θ is higher, makes him less

• ptimistic about devaluation and makes bond more attractive.

◮ Result: Multiplicity may arise if both ⋆ devaluation effect is negative (makes bond less attractive because it increases the probability

• f devaluation through θ⋆), and

⋆ it more than offsets payoff and market-clearing effects. 23 / 50

Hellwig, Mukherji and Tsyvinski (2006)

Multiplicity through the role of interest rates

Examples: Two examples when θ ∈ (0, 1] is common knowledge and γ > 0:

◮ C(A, r) = r, devaluation iff θ ≥ r. ◮ C(A, r) = A, devaluation iff θ ≥ A.

Figure:

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Outline

1

Motivation

2

Uniqueness in Static Global Games Morris and Shin (1998)

3

Multiplicity in Static Global Games Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006)

4

Uniqueness and Multiplicity in Dynamic Global Games Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007)

5

Discussion and Open Questions

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Angeletos, Hellwig and Pavan (2006)

Multiplicity through signaling of policy

Restore multiplicity through signaling of policy prior to Morris and Shin game. First stage:

◮ Nature draws θ ∼ unif (±∞) ◮ Policy-maker (PM) learns θ and chooses c ∈ [c, c] ⊂ (0, 1). ◮ PM’s payoff:

UPM(R, θ, c, A) ≡ (1 − R)(θ − A) − ψ(c) where ψ(·) is the cost of policy intervention, with ψ′(·) > 0 and ψ(c) = 0.

Second stage:

◮ Investors observe c and private signal xi = θ + ξi, ξi ∼ N(0, 1/β). ◮ Choose whether or not to attack using ex-post utility

Ui = ai(R − c)

Third stage:

◮ PM observes A and abandons status quo (R = 1) iff UPM(1, ·) ≥ UPM(0, ·). ◮ This means, again, that R = 1[A>θ], so that we can write

UPM(θ, c, A) = max{0, θ − A} − ψ(c)

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Angeletos, Hellwig and Pavan (2006)

Multiplicity through signaling of policy

Definition

A symmetric PBE consists of a strategy for the PM c : R → [c, c], a (symmetric) strategy for the agents a : R × [c, c] → {0, 1}, and a cdf µ : R × R × [c, c] → [0, 1], such that c(θ) ∈ arg max

c∈[c,c]

UPM(θ, c, A(θ, c)) a(x, c) ∈ arg max

a∈{0,1}

a +∞

−∞

1[A(θ,c)>θ]dµ(θ|x, c) − c

• where µ(θ|x, c) is obtained from c(·) using Bayes’ rule and

A(θ, c) ≡ +∞

−∞

a(x, c)

• βφ(
• β(x − r))dx

R(θ) ≡ 1[A(θ,c(θ))>θ]

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Angeletos, Hellwig and Pavan (2006)

Multiplicity through signaling of policy

If c = c = c, the game is identical to Morris and Shin (1998): a(x, c) = 1[x<x⋆] and R(θ) = 1[θ<θ⋆], where x⋆ ≡ θ⋆ + Φ−1(θ⋆)/

• β

and θ⋆ ≡ Φ

• β(x⋆ − θ⋆)
• = 1 − c

(7) If c < c, there are multiple equilibria.

Proposition

Suppose c < c. Then,

1

No-intervention equilibrium: There is an equilibrium in which c(θ) = c, ∀θ, a(x, c) = 1[x<x⋆] and R(θ) = 1[θ<θ⋆].

2

Intervention equilibrium: For any c⋆ ∈ (c, c], there is an equilibrium in which c(θ) = c⋆ for certain intermediate values θ ∈ [θ⋆, θ⋆⋆].

Formal statement of proposition 28 / 50

Angeletos, Hellwig and Pavan (2006)

Multiplicity through signaling of policy

• 1. PM’s point of view:

From (7), higher c means a smaller range of values θ for which SQ is abandoned. PM could use this fact to reduce size of attack. However, this would signal that

◮ θ is not too low: the cost of information for setting c > c may exceed the value of

maintaining status quo; if θ is too low, it is dominant to set c(θ) = c and let everyone attack.

◮ θ is not too high: the size of the attack would otherwise (i.e, if c(θ) = c) be too small

to justify the cost of intervention.

Thus, only for intermediate θ can this policy hope to succeed.

• 2. Investors’ point of view:

If investors agree on a policy a(x, c) that is insensitive to c, PM policy will remain inactive. If investors coordinate on a strategy a(x, c) that is decreasing in c and θ ∈ [θ⋆, θ⋆⋆], then PM can ex-ante decrease size of the attack by raising c.

Conclusion

Multiplicity if agents coordinating on different interpretations of, and different reactions to, the same policy choices induces different incentives for the PM, and vice versa.

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PART II Dynamic global games of regime change

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Dynamic Global Games

Literature

Classic references: Frankel and Pauzner (2000), Burdzy, Frankel and Pauzner (2001), Brudzy and Frankel (2005), Giannitsarou and Toxvaerd (2003). Applications and refinements: Chamley (1999, 2003), Levin (2000), Matsui (1999), Oyama (2004), Chassang (2010). Herding or social learning literature: Banerjee (1992) and Bikhchandani, Hirschleifer and Welch (1992). In this section:

“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)

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Outline

1

Motivation

2

Uniqueness in Static Global Games Morris and Shin (1998)

3

Multiplicity in Static Global Games Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006)

4

Uniqueness and Multiplicity in Dynamic Global Games Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007)

5

Discussion and Open Questions

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Morris and Shin (1999)

Towards a dynamic global game of regime change

Morris and Shin (1998) in continuous time. Elements of the theory:

◮ Time is t ∈ {∆, 2∆, 3∆, . . . }, for small ∆ > 0. ◮ The fundamental is a Brownian motion:

dθ = z √ ∆dt; z ∼ N(0, 1)

◮ Fundamental θ(t) is observed at the end of period t. ◮ Each speculator i receives a private signal:

xi(t) = θ(t) + ηi; ηi ∼ N(0, ǫ∆), with ǫ ≈ 0

◮ Information sets at instant t: ⋆ Central bank: ΩCB(t) ≡ {θ(t)}. ⋆ Speculators: Ωi(t) ≡ {θ(t − ∆), xi(t)}.

Strategies:

◮ A strategy for speculator i is at

i ≡ {ai(t)}, where ai(t) : Ωi(t) → {0, 1}.

◮ The CB observes θ(t) and abandons (R(t) = 1) iff

A(t) ≡ 1 aj(t)dj ≥ a(θ(t)) where a(·) is minimum attack size that triggers devaluation, and a′(θ) ≥ 0.

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Morris and Shin (1999)

Towards a dynamic global game of regime change

Characterization of the evolution of beliefs which trigger the change of sentiment and, in turn, precipitate the attack:

Proposition (Uniqueness)

For ǫ sufficiently small, there is a stochastic process {h(t)} with the property h(t) ≥ h(t − ∆) if θ(t − ∆) ≤ θ(t − 2∆) such that R(θ(t)) = 1[θ(t)≤h(t)]. When speculators are sufficiently well informed, the onset of an attack can be characterized by ‘tripping over’ a hurdle process h.

◮ The hurdle process moves in opposite direction to the fundamentals.

Intuitively:

◮ When ǫ is small, speculator puts less weight on θ−∆ and more on x. ◮ If fundamentals are thought to deteriorate, indifferent speculator believes: ⋆ other speculators to attack, and ⋆ an attack to be more likely to succeed. ◮ Indeed, hurdle process then increases and CB is more likely to abandon status quo. 34 / 50

Outline

1

Motivation

2

Uniqueness in Static Global Games Morris and Shin (1998)

3

Multiplicity in Static Global Games Angeletos and Werning (2006) Hellwig, Mukherji and Tsyvinski (2006) Angeletos, Hellwig and Pavan (2006)

4

Uniqueness and Multiplicity in Dynamic Global Games Morris and Shin (1999) Angeletos, Hellwig and Pavan (2007)

5

Discussion and Open Questions

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Angeletos, Hellwig and Pavan (2007)

Learning from past outcomes as a source of multiplicity

Endogenous information (here, past information and learning over time) fosters coordination and breaks down the uniqueness result. Morris and Shin game played repeatedly in discrete time.

◮ Action: ait ∈ {0, 1}; Regime outcome: Rt ≡ 1[At>θ], θ ∈ R. ◮ The game continues for as long as the status quo is in place. ◮ Individual payoff of player i ∈ [0, 1]:

Πi ≡

+∞

• t=1

ρt−1(1 − Rt)πit πit ≡ ait(Rt+1 − c)

Information:

◮ Nature draws θ ∼ N(z, 1/α) at t = 0. ◮ Private signals:

˜ xit = θ + ξit where ξit ∼ N(0, 1/ηt).

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Angeletos, Hellwig and Pavan (2007)

Learning from past outcomes as a source of multiplicity

Where βt ≡ t

τ=1 ητ (cumulative private precision), assume ∀t ∈ Z+,

+∞ > βt ≥ α2 2π and lim

t→+∞ βt = +∞

Focus on symmetric perfect monotone PBE:

Equilibrium definition ◮ Define recursively:

xt = βt−1xt−1 + ηt ˜ xt βt and βt = βt−1 + ηt with β1 = η1 and x1 = ˜ x1.

◮ Posterior beliefs:

θ|˜ xt ∼ N βtxt + αz βt + α , 1 βt + α

• ◮ Note:

⋆ xt is a sufficient statistic for ˜

xt (the whole history) with respect to θ and R.

⋆ Thus, all private information can be summarized into a single-dimensional object. 37 / 50

Angeletos, Hellwig and Pavan (2007)

Learning from past outcomes as a source of multiplicity

Lemma

Any monotone equilibrium is characterized by a sequence {x⋆

t , θ⋆ t }+∞ t=1 with

x⋆

t ∈ R ∪ {−∞}, θ⋆ t ∈ (0, 1) and θ⋆ t ≥ θ⋆ t−1, such that:

1

At any t, an agent attacks (refrains) iff xt < x⋆

t (xt > x⋆ t ),

2

The status quo is in place in period t iff θ > θ⋆

t−1,

Each period, θ⋆

t and x⋆ t ≡ X(θ⋆ t , βt) solve :

θ⋆

t

= At(θ⋆

t )

= P[xt ≤ x⋆

t |θ⋆ t ]

= Φ(

• βt(x⋆

t − θt))

c = P[θ ≤ θ⋆

t |x⋆ t , θ > θ⋆ t−1]

The sequence {θ⋆

t } is nondecreasing:

◮ The status quo cannot be in place in one period without also being in place in the

previous.

The sequence {x⋆

t } may be nonmonotonic:

◮ Periods where some agents attack (x⋆

t > −∞) may alternate with period where

nobody attacks (x⋆

t = −∞).

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Angeletos, Hellwig and Pavan (2007)

Learning from past outcomes as a source of multiplicity

Alternatively, let U(θ⋆

t , θ⋆ t−1, βt, α, z) be the net payoff from attacking for the

marginal agent, i.e. the agent with signal xt = X(θ⋆

t , βt).

Definition of U

Then, (θ⋆

t−1, θ⋆ t ) belongs to the equilibrium path iff

UMG

t

≡ U(θ⋆

t , θ⋆ t−1, βt, α, z) = 0

RESULT 1: Existence and recursive characterization

Proposition

The strategy a∞ is a monotone equilibrium iff there exists a sequence of thresholds {x⋆

t , θ⋆ t }+∞ t=1 such that

1

For all t, at(˜ xt) = 1 if xt < x⋆

t and at(˜

xt) = 0 if xt > x⋆

t .

2

For t = 1, θ⋆

1 solves U(θ⋆ 1 , −∞, β1, α, z) = 0 and x⋆ 1 = X(θ⋆ 1 , β1).

3

For any t ≥ 2, either θ⋆

t = θ⋆ t−1 > 0 and x⋆ t = −∞, or θ⋆ t > θ⋆ t−1 > 0 is a solution to

U(θ⋆

t , θ⋆ t−1, βt, α, z) = 0 and x⋆ t = X(θ⋆ t , βt).

A monotone equilibrium always exists. Learning takes the form of a truncation in the support of beliefs about θ. Knowledge that regime has survived past attacks translates into knowledge that θ is above a threshold θ⋆

t−1.

Proposition offers a recursive algorithm to compute equilibria.

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Angeletos, Hellwig and Pavan (2007)

Learning from past outcomes as a source of multiplicity

RESULT 2: Multiplicity

Proposition

1

U(θ⋆

t , θ⋆ t−1, βt, α, z) is continuous, nonmonotonic in θ⋆ t when θ⋆ t−1 ∈ (0, 1), and

strictly decreasing in θ⋆

t−1 and z for θ⋆ t−1 < θ⋆.

2

If θ⋆

t−1 > θ∞ ≡ 1 − c, UMG t

= 0 does not have a solution for βt sufficiently high.

3

If θ⋆

t−1 < θ∞ ≡ 1 − c, UMG t

= 0 has a solution for βt sufficiently high. Example: Suppose β2 low s.t. ∄ attacks at t = 2, and β3 high s.t. ∃ attacks at t ≥ 3. Three classes of equilibria: one with no attacks for all t, and two with no attacks at t = 2 but possible attacks at any t ≥ 3.

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Angeletos, Hellwig and Pavan (2007)

Learning from past outcomes as a source of multiplicity

RESULT 3: Timing and number of attacks cannot be predicted

◮ If θ and z are sufficiently high, the status quo survives in any monotone equilibrium. ◮ However, ∃t < +∞ such that, at any t ≥ t, an attack can occur, yet does not

necessarily take place.

◮ Furthermore, any number of attacks is possible.

RESULT 4: Coordination cycles

◮ After the most aggressive attack for a given period occurs, the game enters a phase of

tranquility, during which no attack is possible.

◮ This phase is longer the slower is the arrival of private information. 41 / 50

PART III Discussion and Open Questions

42 / 50

Discussion

Global games offer an appealing approach to episodes of regime change that are triggered by self-fulfilling prophecies. We have seen that a tiny departure from common knowledge entirely eliminates multiplicity of equilibria. This prediction is not robust in many dimensions, namely when there is acquisition

• f (endogenously-formed) public information.

This means that these models have:

◮ Little predictive power. ◮ Little to say about policy. 43 / 50

What’s missing?

Role of policy:

◮ Policy may itself be an endogenous source of information and restore multiplicity

(Angeletos, Hellwig and Pavan (2006)).

◮ Policy maker needs to be integrated as an extra player in the game before evaluating

effectiveness of its policies (Goldstein, Ozdenoren and Yuan (2011)).

Predictive power:

◮ Multiplicity in the theory opens the door to non-fundamental volatility. ◮ Stylized dynamic settings (AHP (2007)) fail to provide predictions. ◮ Linking theory more closely to business cycles might improve predictive power. ⋆ Changes in the volatility of productivity shocks may have a real impact on the economy

(Bloom (2009), Van Nieuwerburgh and Veldkamp (2006)).

⋆ The role of extrinsic shocks to expectations, called sentiments (Angeletos and La’O (2009)). ⋆ Higher uncertainty about fundamentals discourage investment and may lead the economy to

self-reinforcing uncertainty traps of high uncertainty and low activity (Fajgelbaum, Schaal and Taschereau-Dumouchel (2014)).

I would like to think of scenarios in which growth emerges from the trade-off between a safe capital investment opportunity and the risk of exchanging resources for foreign currency in a global game environment.

44 / 50

APPENDIX

45 / 50

Angeletos and Werning (2006): Uniqueness versus multiplicity

Figure: Morris and Shin (1998) uniqueness result Figure: Angeletos and Werning (2006) multiplicity result

Back 46 / 50

Hellwig, Mukherji and Tsyvinski (2006): Formal statement of multiplicity result

Lemma

Suppose the devaluation threshold θ⋆(r) is continuously differentiable. Then, there exist multiple market-clearing interest rate functions (that is, R(z) is a multi-valued correspondence), whenever for some r ∈ (0, 1), dθ⋆ dr > γ + √ 1 + δ √β(1 + δ) 1 φ(Φ−1(r))

Back 47 / 50

Angeletos, Hellwig and Pavan (2006): Formal statement multiplicity

Proposition

Suppose c < c. There are multiple equilibria:

1

There is an equilibrium in which c(θ) = c for all θ, a(x, c) = 1[x<x⋆(c)], and R(θ) = 1[θ<θ⋆(c)], where x⋆(c) ≡ 1 − c + Φ−1(1 − c)/√β and θ⋆(c) ≡ 1 − c.

2

For any c⋆ ∈ (c, c], there is an equilibrium in which a(x, c) = 1[x<x or (x,c)<(˜

x,c⋆)],

R(θ) = 1[θ<˜

θ] and

c(θ) =

• c⋆

if θ ∈ [˜ θ, ˜ ˜ θ] r

• therwise

where ˜ x, ˜ θ and ˜ ˜ θ are given by ˜ θ = ψ(c⋆) ˜ ˜ θ = ˜ θ + 1 √β

• Φ−1
• 1 −

c 1 − c ˜ θ

• − Φ−1(˜

θ)

• ˜

x = ˜ ˜ θ + Φ−1(˜ θ) √β

Back 48 / 50

Angeletos, Hellwig and Pavan (2007): Equilibrium definition

Conditional on the status quo being at place at the beginning of period t, let: at : Rt → [0, 1] be the strategy for period t. at ≡ {aτ}t

τ=1 be the strategy profile up to period t, with a∞ ≡ {aτ}+∞ τ=1.

pt(θ; at) be the probability that the status quo is abandoned in period t when all agents follow strategy at. Ψt(θ|˜ xt; at−1) be the cdf of posterior beliefs in period t, where ˜ xt ≡ {˜ xτ}t

τ=1.

Definition

A strategy a∞ is part of a symmetric monotone PBE if at(xt) (probability of attack) is non increasing in xt and independent of past actions, and is such that at(˜ xt) ∈ arg max

a∈[0,1]

• pt(θ; at)dΨt(θ|˜

xt; at−1) − c

• a
• for all t ≥ Z+ and all ˜

xt ∈ Rt.

Back 49 / 50

Angeletos, Hellwig and Pavan (2007): Recursive equilibrium

Define the functions u : R × [0, 1] × R × R2

+ × R → [−c, 1 − c]

X : [0, 1] × R+ → R U : [0, 1] × R × R2

+ × R → [−c, 1 − c]

by: u(x, θ⋆, θ⋆

−1, β, α, z)

≡    1 −

Φ √β+α βx+αz

β+α −θ⋆

Φ √β+α βx+αz

β+α −θ⋆ −1

• if θ⋆ > θ⋆

−1

−c if θ⋆ ≤ θ⋆

−1

X(θ⋆, β) ≡ θ⋆ + Φ−1(θ⋆) √β U(θ⋆, θ⋆

−1, β, α, z)

≡      limx→−∞ u(x, θ⋆, θ⋆

−1, β, α, z)

if θ⋆ = 0 u(X(θ⋆, β), θ⋆, θ⋆

−1, β, α, z)

if θ⋆ ∈ (0, 1) limx→+∞ u(x, θ⋆, θ⋆

−1, β, α, z)

if θ⋆ = 1 Clearly, since X(θ⋆, β) is the solution to θ⋆ = A(θ⋆) (that is, to θ⋆ = Φ(√β(x⋆ − θ⋆)) for x⋆), then U(θ⋆, θ⋆

−1, β, α, z) = 0

Back 50 / 50