Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Attention, Coordination, and Bounded Recall Alessandro Pavan Northwestern University Chicago FED, February 2016
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Motivation Many socioeconomic environments - large group of agents - actions under dispersed information
Motivation Model Equilibrium Efficiency Bounded Recall 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 ...
Motivation Model Equilibrium Efficiency Bounded Recall 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 Bounded Recall 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
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions This paper Flexible (yet rich) framework - complementarity or substitutability in actions - rich set of payoff interdependencies
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions This paper Flexible (yet rich) framework - complementarity or substitutability in actions - rich set of payoff interdependencies Equilibrium and efficient allocation of attention - perfect recall - bounded recall
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Questions What payoff interdependencies create inefficiency in eq. allocation of attention? How does inefficiency in attention relate to inefficiency in use of information? What is the effect of bounded recall? What policies can alleviate such inefficiencies? (related work)
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Related literature (incomplete) Efficient 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)
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Plan Model (perfect recall) 1
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Plan Model (perfect recall) 1 Equilibrium allocation of attention 2
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Plan Model (perfect recall) 1 Equilibrium allocation of attention 2 Efficient allocation of attention 3
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Plan Model (perfect recall) 1 Equilibrium allocation of attention 2 Efficient allocation of attention 3 Bounded recall 4
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Model Actions and gross payoffs � � k i , { k j } j � = i , θ u i
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Model Actions and gross payoffs Continuum of agents with payoffs: � � k , K , θ , σ 2 u k where: k ∈ R — individual action � k � d Ψ ( k � ) — aggregate action K = � ( k � − K ) 2 d Ψ ( k � ) — dispersion σ 2 k = θ ∈ R — 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 Bounded Recall Conclusions Examples Investment spillovers (Angeletos and Pavan AER 2004) u i = Rk i − c ( k i ) c ( k i ) = 1 2 k 2 R = ( 1 − a ) θ + aK and i Beauty contest (Morris and Shin AER 2002) u i = − ( 1 − r ) · ( k i − θ ) 2 − r · ( L ( k i ) − ¯ L ) � � � � 2 d Ψ ( k � ) = ( k i − K ) 2 + σ 2 k � − k i ¯ L ( k ) d Ψ ( k ) = 2 σ 2 L ( k i ) ≡ and L = k
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Examples Monetary economies (Woodford 2005, Colombo, Femminis and Pavan, 2014, Llosa and Venkateswaran, 2015) u ( θ , C i , N i ) ≡ V ( C i ) − N i � � � v v − 1 v − 1 v C i = [ 0 , 1 ] c hi dh Y i = θ α N i � [ 0 , 1 ] p h c hi dh ≤ p i Y i − T Cournot and Bertrand games (Vives JET 1984) u i = ( a − θ K ) · k i − 1 2 k 2 i
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Model Information and attention Common prior: θ ∼ N ( 0 , π − 1 θ )
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Model Information and attention Common prior: θ ∼ N ( 0 , π − 1 θ ) N = 1 , 234 , 576 sources of information: ε l ∼ N ( 0 , η − 1 y l = θ + ε l with ) l = 1 ,..., N l
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Model Information and attention Common prior: θ ∼ N ( 0 , π − 1 θ ) N = 1 , 234 , 576 sources of information: ε l ∼ N ( 0 , η − 1 y l = θ + ε l with ) l = 1 ,..., N l Agent i ’s "impressions" x i = ( x i l ) N l = 1 with � � − 1 � � x i l = y l + ξ i ξ i z i with l ∼ N 0 , l · t l l = 1 ,..., N l where η l : accuracy t l : transparency/clarity z i attention : l
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Model Attention cost and net payoffs Attention cost: C ( z i ) where z i = ( z i l ) N l = 1 � z i � � > 0 , all z i � = 0 · C n · lim z n → ∞ C � n ( z i ) = ∞ · convex (results extend to concave, e.g., entropy reduction) � � E.g. C ( z i ) = c ∑ l z i l E.g. C ( z i ) = ∑ l g ( z i l ) ...but also C ( z i ) = µ ( z i ; y ) (entropy reduction) Net payoff � � k i , K , σ 2 − C ( z i ) k , θ u
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Model Timing agents allocate attention z i update their beliefs based on x i commit their actions k i payoffs realized
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Plan Model (perfect recall) 1 Equilibrium allocation of attention 2 Efficient allocation of attention 3 Bounded Recall 4
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Equilibrium use of information (Angeletos and Pavan, Ecma 2007) Optimality: k j = E [ κ + α ( K − κ ) | x j ; z j ] where κ = κ 0 + κ 1 θ (complete-info. equilibrium action) α ≡ u kK − → equilibrium degree of coordination | u kk |
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Equilibrium allocation of attention Theorem There exists a unique symmetric equilibrium. In this eq., the attention ˆ z that each agent assigns to the various sources of information is s.t., for any source n = 1 ,..., N that receives strictly positive attention, � | u kk | z n = κ 1 γ n ˆ 2 C � n ( ˆ z ) t n where ( 1 − α ) π n 1 − αρ n γ n ≡ is "influence" of the source ( 1 − α ) π s π θ + ∑ N s = 1 1 − αρ s and where η s ˆ z s t s is endogenous precision and ρ s = π s π s = is endogenous "publicity" z s t s + η s ˆ η s Given equilibrium allocation of attention ˆ z , equilibrium actions are given by � � k i = κ 0 + κ 1 all i ∈ [ 0 , 1 ] , almost all x i ∈ R N . ∑ N n = 1 γ n x i n
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Private value of attention Envelope reasoning : hold k ( · ; ˆ z ) fixed
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Private value of attention Envelope reasoning : hold k ( · ; ˆ z ) fixed Agent’s eq. continuation payoff (fixing k ( · ; ˆ z ) ): z ) = E [ u ( K , K , σ k , θ )]+ u kk U i ( z i ; ˆ 2 Var [ k i − K | z i , ˆ z )] − C ( z i ) z , k ( · ; ˆ
Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Private value of attention Envelope reasoning : hold k ( · ; ˆ z ) fixed Agent’s eq. continuation payoff (fixing k ( · ; ˆ z ) ): z ) = E [ u ( K , K , σ k , θ )]+ u kk U i ( z i ; ˆ 2 Var [ k i − K | z i , ˆ z )] − C ( z i ) z , k ( · ; ˆ Private value of attention −| u kk | · ∂ Var [ k − K | z , k ( · ; z )] ∂ z n 2 private aversion to dispersion · reduction in dispersion (fixing eq. strategy k ( · ; z ) )
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