Attention, Coordination, and Bounded Recall

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions

Alessandro Pavan

Northwestern University

Chicago FED, February 2016 Useful modelization for: - production or network externalities - incomplete markets - business cycles - large Cournot-Bertrand games - elections ...

Most of the literature: exogenous information structure

Many phenomena of interest: attention (info. acquisition) is central

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Motivation

Many socioeconomic environments - large group of agents - actions under dispersed information Most of the literature: exogenous information structure

Many phenomena of interest: attention (info. acquisition) is central

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Motivation

Many socioeconomic environments - large group of agents - actions under dispersed information

Useful modelization for: - production or network externalities - incomplete markets - business cycles - large Cournot-Bertrand games - elections ... Many phenomena of interest: attention (info. acquisition) is central

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Motivation

Many socioeconomic environments - large group of agents - actions under dispersed information

Useful modelization for: - production or network externalities - incomplete markets - business cycles - large Cournot-Bertrand games - elections ...

Most of the literature: exogenous information structure Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Motivation

Many socioeconomic environments - large group of agents - actions under dispersed information

Useful modelization for: - production or network externalities - incomplete markets - business cycles - large Cournot-Bertrand games - elections ...

Most of the literature: exogenous information structure

Many phenomena of interest: attention (info. acquisition) is central Equilibrium and eﬃ cient allocation of attention

- perfect recall

- bounded recall

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions This paper

Flexible (yet rich) framework

- complementarity or substitutability in actions

- rich set of payoﬀ interdependencies Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions This paper

Flexible (yet rich) framework

- complementarity or substitutability in actions

- rich set of payoﬀ interdependencies

Equilibrium and eﬃ cient allocation of attention

- perfect recall

- bounded recall Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Questions

What payoﬀ interdependencies create ineﬃ ciency in eq. allocation of attention?

How does ineﬃ ciency in attention relate to ineﬃ ciency in use of information?

What is the eﬀect of bounded recall?

What policies can alleviate such ineﬃ ciencies? (related work) Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Related literature (incomplete)

Eﬃ cient use of information and social value of information Radner (1977), Vives (JET 1984, 2013) Morris and Shin (AER 2002) Angeletos and Pavan (AER, 2004, Ecma 2007, Jeea, 2009) ... Information acquisition/(in)attention in coordination settings Vives and Van Zandt (2007) Hellwig and Veldkamp (Restud, 2009) Amir and Lazzati (2014) Ma´ckowiak and Wiederholt (AER, 2009, 2012) Myatt and Wallace (Restud 2012) → Szkup and Trevino (2013), Yang (2013) Colombo, Femminis and Pavan (Restud 2014) → Tirole (2014), Denti (2016) ... Memory Benabou Tirole (JPE 2004) Wilson (2004), Kocer (2010) ... Analogy-based equilibrium Jehiel (JET 2005) 2 Equilibrium allocation of attention

3 Eﬃ cient allocation of attention

4 Bounded recall

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Plan

1 Model (perfect recall) 3 Eﬃ cient allocation of attention

4 Bounded recall

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Plan

1 Model (perfect recall)

2 Equilibrium allocation of attention 4 Bounded recall

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Plan

1 Model (perfect recall)

2 Equilibrium allocation of attention

3 Eﬃ cient allocation of attention Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Plan

1 Model (perfect recall)

2 Equilibrium allocation of attention

3 Eﬃ cient allocation of attention

4 Bounded recall Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Model Actions and gross payoﬀs

ui ki , k j j=i, θ { } 6 Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Model Actions and gross payoﬀs

Continuum of agents with payoﬀs:

2 u k , K , θ, σ k where: k R —individual action ∈ K = k0dΨ(k0) —aggregate action σ 2 =R (k K)2dΨ(k ) — dispersion k 0 − 0 θ RR—underlying uncertainty ("fundamentals") ∈

Assumptions: u( ) quadratic in (k,K,θ), linear in σ 2 · k u( ) s.t. equilibrium and ﬁrst-best unique and bounded · Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Examples

Investment spillovers (Angeletos and Pavan AER 2004)

ui = Rki c(ki) − 1 2 R = (1 a)θ + aK and c(ki) = k − 2 i Beauty contest (Morris and Shin AER 2002)

2 ui = (1 r) (ki θ) r (L(ki) L¯) − − · − − · −

2 2 2 2 L(ki) k0 ki dΨ(k0) = (ki K) +σ and L¯ = L(k)dΨ(k) = 2σ ≡ − − k Z Z Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Examples

Monetary economies (Woodford 2005, Colombo, Femminis and Pavan, 2014, Llosa and Venkateswaran, 2015)

u(θ,Ci,Ni) V (Ci) Ni ≡ − v v 1 v 1 −v − Ci = chi dh Z[0,1] α Yi = θ Ni

phchidh piYi T [0,1] ≤ − Z

Cournot and Bertrand games (Vives JET 1984)

1 2 ui = (a θK) ki k − · − 2 i N = 1,234,576 sources of information:

1 yl = θ + εl with εl N(0,η− ) l = 1,...,N ∼ l

i i N Agent i’s "impressions" x = (xl)l=1 with

1 i i i i − x = yl + ξ with ξ N 0, z tl l = 1,...,N l l l ∼ l · where

ηl : accuracy tl : transparency/clarity i zl : attention

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Model Information and attention

Common prior: 1 θ N(0,π− ) ∼ θ i i N Agent i’s "impressions" x = (xl)l=1 with

1 i i i i − x = yl + ξ with ξ N 0, z tl l = 1,...,N l l l ∼ l · where

ηl : accuracy tl : transparency/clarity i zl : attention

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Model Information and attention

Common prior: 1 θ N(0,π− ) ∼ θ N = 1,234,576 sources of information:

1 yl = θ + εl with εl N(0,η− ) l = 1,...,N ∼ l Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Model Information and attention

Common prior: 1 θ N(0,π− ) ∼ θ N = 1,234,576 sources of information:

1 yl = θ + εl with εl N(0,η− ) l = 1,...,N ∼ l

i i N Agent i’s "impressions" x = (xl)l=1 with

1 i i i i − x = yl + ξ with ξ N 0, z tl l = 1,...,N l l l ∼ l · where

ηl : accuracy tl : transparency/clarity i zl : attention Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Model Attention cost and net payoﬀs

i i i N Attention cost: C(z ) where z = (zl)l=1

C0 zi > 0, all zi = 0 · n 6 i limzn ∞ Cn0 (z ) = ∞ · → convex (results extend to concave, e.g., entropy reduction) ·

i i E.g. C(z ) = c ∑l zl i i E.g. C(z ) = ∑l g(zl)

...but also C(zi) = µ(zi;y) (entropy reduction)

Net payoﬀ 2 i u ki,K,σ ,θ C(z ) k − Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Model Timing

agents allocate attention zi

update their beliefs based on xi

commit their actions ki

payoﬀs realized Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Plan

1 Model (perfect recall)

2 Equilibrium allocation of attention

3 Eﬃ cient allocation of attention

4 Bounded Recall Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Equilibrium use of information (Angeletos and Pavan, Ecma 2007)

Optimality: j j k j = E[ κ + α(K κ) x ; z ] − | where

κ = κ0 + κ1θ (complete-info. equilibrium action)

α ukK equilibrium degree of coordination ukk ≡ | | −→ Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Equilibrium allocation of attention

Theorem There exists a unique symmetric equilibrium. In this eq., the attention zˆ that each agent assigns to the various sources of information is s.t., for any source n = 1,...,N that receives strictly positive attention,

ukk zˆn = κ1γn | | s 2Cn0 (zˆ)tn where (1 α)πn − 1 αρn γn − is "inﬂuence" of the source ≡ N (1 α)πs πθ + ∑s=1 1−αρ − s and where

ηszˆsts πs πs = is endogenous precision and ρs = is endogenous "publicity" zˆsts + ηs ηs Given equilibrium allocation of attention zˆ, equilibrium actions are given by

i N i i N k = κ0 + κ1 ∑ γnxn all i [0,1], almost all x R . n=1 ∈ ∈ Agent’seq. continuation payoﬀ (ﬁxing k( ;z ˆ)): · i ukk i i Ui(z ;z ˆ) = E[u(K,K,σ k,θ)] + Var[ki K z ,zˆ,k( ;z ˆ)] C(z ) 2 − | · − Private value of attention

ukk ∂Var[k K z,k( ;z)] | | − | · − 2 · ∂zn private aversion to dispersion reduction in dispersion (ﬁxing eq. strategy· k( ;z)) · Result generalizes Colombo, Femminis, Pavan (Restud 2014)

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Private value of attention

Envelope reasoning: hold k( ;z ˆ) ﬁxed · Private value of attention

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Private value of attention

Envelope reasoning: hold k( ;z ˆ) ﬁxed ·

Agent’seq. continuation payoﬀ (ﬁxing k( ;z ˆ)): · i ukk i i Ui(z ;z ˆ) = E[u(K,K,σ k,θ)] + Var[ki K z ,zˆ,k( ;z ˆ)] C(z ) 2 − | · − Result generalizes Colombo, Femminis, Pavan (Restud 2014)

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Private value of attention

Envelope reasoning: hold k( ;z ˆ) ﬁxed ·

Agent’seq. continuation payoﬀ (ﬁxing k( ;z ˆ)): · i ukk i i Ui(z ;z ˆ) = E[u(K,K,σ k,θ)] + Var[ki K z ,zˆ,k( ;z ˆ)] C(z ) 2 − | · − Private value of attention

ukk ∂Var[k K z,k( ;z)] | | − | · − 2 · ∂zn private aversion to dispersion reduction in dispersion (ﬁxing eq. strategy· k( ;z)) · Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Private value of attention

Envelope reasoning: hold k( ;z ˆ) ﬁxed ·

Agent’seq. continuation payoﬀ (ﬁxing k( ;z ˆ)): · i ukk i i Ui(z ;z ˆ) = E[u(K,K,σ k,θ)] + Var[ki K z ,zˆ,k( ;z ˆ)] C(z ) 2 − | · − Private value of attention

1 Model (perfect recall)

2 Equilibrium allocation of attention

3 Eﬃ cient allocation of attention

4 Bounded Recall Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Eﬃ ciency

Welfare : ex-ante utility of representative agent

Deﬁnition

Eﬃ cient allocation consists of attention z∗ along with action rule k∗( ;z∗) that jointly maximize · 2 E[u(k,K,σ ,θ) z] C(z) k | − Team problem

Planner’s problem: control incentives but cannot transfer information Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Eﬃ cient use of information (Angeletos and Pavan, Ecma 2007)

Given attention z, eﬃ ciency in actions requires that k ( ;z) solves ∗ ·

k∗(x;z) = E[κ∗ + α∗(K κ∗) x ; z ] x, − | ∀ where κ∗ = κ∗ + κ∗θ FB 0 1 −→

uσσ 2ukK uKK aversion to volatility α∗ − − = 1 ≡ ukk + uσσ − aversion to dispersion

socially optimal degree of coordination Recall that eq.

ukk zˆn = κ1γn | | s 2Cn0 (zˆ)tn

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Eﬃ cient allocation of attention

Theorem

Eﬃ ciency in attention requires that, for any n for which zn∗ > 0,

ukk + uσσ zn∗ = κ1∗γn∗ | | s 2Cn0 (z∗)tn where

(1 α )πn − ∗ 1 αρn γn∗ − is eﬃ cient "inﬂuence" of the source ≡ N (1 α∗)πs πθ + ∑s=1 1−α ρ − ∗ s

ηszs∗ts πs∗ πs = is endogenous precision and ρs = is endogenous publicity z∗s ts + ηs ηs Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Eﬃ cient allocation of attention

Theorem

Eﬃ ciency in attention requires that, for any n for which zn∗ > 0,

ukk + uσσ zn∗ = κ1∗γn∗ | | s 2Cn0 (z∗)tn where

(1 α )πn − ∗ 1 αρn γn∗ − is eﬃ cient "inﬂuence" of the source ≡ N (1 α∗)πs πθ + ∑s=1 1−α ρ − ∗ s

ηszs∗ts πs∗ πs = is endogenous precision and ρs = is endogenous publicity z∗s ts + ηs ηs

Recall that eq.

ukk zˆn = κ1γn | | s 2Cn0 (zˆ)tn Welfare under eﬃ cient use of information (for given attention z)

w∗(z) E[u(κ∗,κ∗,0,θ)] L ∗(z) C(z), ≡ − − where u(κ∗,κ∗,0,θ) is welfare under FB allocation and

ukk + uσσ ∂Var[k K z,k ( ;z)] | | − | ∗ · − 2 · ∂zn social aversion to dispersion reduction in dispersion (ﬁxing eﬀ. strategy· k ( ;z) ) ∗ ·

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Eﬃ cient allocation of attention

Envelope reasoning Holding k ( ;z), Var[K κ k ( ;z),z] independent of z ∗ · − ∗ | ∗ · Social value of attention

ukk + uσσ ∂Var[k K z,k ( ;z)] | | − | ∗ · − 2 · ∂zn social aversion to dispersion reduction in dispersion (ﬁxing eﬀ. strategy· k ( ;z) ) ∗ ·

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Eﬃ cient allocation of attention

Envelope reasoning Welfare under eﬃ cient use of information (for given attention z)

w∗(z) E[u(κ∗,κ∗,0,θ)] L ∗(z) C(z), ≡ − − where u(κ∗,κ∗,0,θ) is welfare under FB allocation and

ukk + uσσ ∂Var[k K z,k ( ;z)] | | − | ∗ · − 2 · ∂zn social aversion to dispersion reduction in dispersion (ﬁxing eﬀ. strategy· k ( ;z) ) ∗ ·

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Eﬃ cient allocation of attention

Envelope reasoning Welfare under eﬃ cient use of information (for given attention z)

w∗(z) E[u(κ∗,κ∗,0,θ)] L ∗(z) C(z), ≡ − − where u(κ∗,κ∗,0,θ) is welfare under FB allocation and

ukk + 2ukK + uKK L ∗(πx,πz) | |Var[K κ∗ k∗( ;z),z] ≡ 2 − | · ukk + uσσ + | |Var[k K k∗( ;z),z] 2 − | · Holding k ( ;z), Var[K κ k ( ;z),z] independent of z ∗ · − ∗ | ∗ · Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Eﬃ cient allocation of attention

Envelope reasoning Welfare under eﬃ cient use of information (for given attention z)

w∗(z) E[u(κ∗,κ∗,0,θ)] L ∗(z) C(z), ≡ − − where u(κ∗,κ∗,0,θ) is welfare under FB allocation and

ukk + uσσ ∂Var[k K z,k ( ;z)] | | − | ∗ · − 2 · ∂zn social aversion to dispersion reduction in dispersion (ﬁxing eﬀ. strategy· k ( ;z) ) ∗ · Social value of attention

ukk + uσσ ∂Var[k K z,k ( ;z)] | | − | ∗ · − 2 · ∂zn social aversion to dispersion reduction in dispersion (ﬁxing eﬀ. strategy· k ( ;z) ) ∗ ·

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Equilibrium vs eﬃ cient allocation of attention

Private value of attention

ukk ∂Var[k K z,k( ;z)] | | − | · − 2 · ∂zn private aversion to dispersion reduction in dispersion (ﬁxing eq. strategy· k( ;z)) ·

Social value of attention

ukk + uσσ ∂Var[k K z,k ( ;z)] | | − | ∗ · − 2 · ∂zn social aversion to dispersion reduction in dispersion (fixing eff. strategy· k ( ;z) ) ∗ · Motivation Model Equilibrium Efficiency BoundedRecall Conclusions Effi cient allocation of attention

Eﬃ ciency in attention requires

- eﬃ ciency in use of information: k( ;z) = k ( ;z) · ∗ ·

- private = social aversion to dispersion uσσ = 0 ⇔ Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Plan

1 Model (perfect recall)

2 Equilibrium allocation of attention

3 Eﬃ cient allocation of attention

4 Bounded Recall Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Bounded Recall

Idea: posteriors correct, but agents cannot recall inﬂuence of individual sources Given attention z j, posterior beliefs about θ continues to be Normal with mean

j N i πn ηszsts x¯ = ∑n=1 δ nxn, with δ n N and πs ≡ πθ + ∑s=1 πs ≡ zsts + ηs N and precision πθ + ∑s=1 πs However, agent is unable to decompose x¯j into various impressions x j (x j ,...,x j ). ≡ 1 N Equivalently, unable to decompose his posteriors into

θ˜ xi | n Measurability constraint on k(x j;zz) Distinction relevant only in strategic setting Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Bounded Recall

For simplicity: πθ = 0 Theorem In unique symmetric equilibrium, given allocation z#, actions given by

i i k = κ0 + κ1x¯ For any source that receives strictly positive attention in eq.,

# # # # ukk ∂Var k K;z ,k( ;z ) ukk ∂Var K κ;z ,k( ;z ) C (z#) = | | − · | | (1 α) − · n0 2 z 2 z − ∂ n − − ∂ n Novel eﬀect:

# # ukk ∂Var K κ;z ,k( ;z ) | | (1 α) − · − 2 − ∂zn private aversion to volatility of own’saverage action reduction in volatility (ﬁxing eq. strategy k#( ;z)) · · Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Bounded vs Perfect Recall

Theorem Let zˆ be eq. allocation of attention with perfect recall. There exist publicity thresholds ρ0,ρ00 [0,1] s.t., starting from zˆ, any agent with bounded recall is better oﬀ by ∈

(a) locally increasing attention to sources for which ρ [ρ ,ρ ]; n ∈ 0 00 (b) locally decreasing attention to sources for which ρ / [ρ ,ρ ]. n ∈ 0 00 Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Bounded vs Perfect Recall

Reallocation of attention towards sources of average (endogenous) publicity zsts ρn = zsts + ηs

Sources of low publicity: useful to forecast θ

Sources of high publicity: useful to forecast K

Sources of intermediate transparency: good compromises Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Bounded vs Perfect Recall

Previous result about best responses extends to equilibrium N Suppose C(z) = c ∑s=1 zs Theorem Let zˆ be eq. attention with perfect recall and z# eq. attention with bounded # recall. There exist thresholds t0,t00 R++ s.t. zn > zˆn only if tn [t0,t00]. ∈ # # ∈ Furthermore for any n for which tn [t ,t ], z < zˆn only if z = 0. ∈ 0 00 n n Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Eﬃ ciency under Bounded Recall

...see paper! Eﬃ ciency in allocation of attention requires (a) absence of externalities from action-dispersion (b) eﬃ ciency in use of information

Bounded recall: reallocation of attention towards sources with intermediate transparency

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Conclusions

Attention in large economies with - complementarity / substitutability in actions - rich set of payoﬀ interdependencies - rich information structure Bounded recall: reallocation of attention towards sources with intermediate transparency

Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Conclusions

Attention in large economies with - complementarity / substitutability in actions - rich set of payoﬀ interdependencies - rich information structure

Effi ciency in allocation of attention requires (a) absence of externalities from action-dispersion (b) effi ciency in use of information Motivation Model Equilibrium Efficiency BoundedRecall Conclusions Conclusions

Attention in large economies with - complementarity / substitutability in actions - rich set of payoﬀ interdependencies - rich information structure

Eﬃ ciency in allocation of attention requires (a) absence of externalities from action-dispersion (b) eﬃ ciency in use of information

Bounded recall: reallocation of attention towards sources with intermediate transparency Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions Conclusions

Future work - endogenous sources / social learning (e.g., capital mkts information aggregation) →

- "optimal" recall strategy

- dynamics (optimal stopping)

- fully ﬂexible info. structures (attention-based correlated eq.) Motivation Model Equilibrium Eﬃciency BoundedRecall Conclusions

Thank You!