Dynamic Rational Inattention David Dillenberger; R. Vijay Krishna; Philipp Sadowski Summer 2015 Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 1 / 17
Static Attention Constraints Rational Inattention starting with Sims (1998, 2003). Processing information is costly. Information is a (constrained) choice. Cost is understood as opportunity cost (unspeci…ed foregone options). Typical formulation: limit on how much information can be processed. Information often measured via Shannon entropy. Decision theoretic description: subjective information constraint (mental or physical). Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 2 / 17
Extant Dynamic Models Sims (2003, 2006), Ma´ ckowiak and Wiederholdt (2009) Static constraint applies every period. Achievements: Price stickiness when states are persistent, asymmetric reaction to shocks, sticky investment, ... Shortcomings: Ignores intertemporal opportunity costs. Precludes interpretations such as expertise mental fatigue. Not usually calibrated (necessary domain not obvious). Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 3 / 17
Recursive Rep. with Intertemporal Attention Constraint Let S be a set of payo¤ relevant states of the world. Recursive Anscombe-Aumann Choice Problem (observable): f 2 x speci…es state contingent lottery over consumption today; and 1 continuation problem for tomorrow. 2 Formally, X ' K ( F ( ∆ ( C � X ))) . In…nite Horizon Attention Constraint (subjective): ( P , ω 0 ) 2 ω determines partition of S today; and 1 attention constraint for tomorrow. 2 Timeline: Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 4 / 17
Recursive Representation: V ( x , ω , π ) = 0 2 3 1 Z @ max 4 5 π ( s ) A ( P , ω 0 ) 2 ω ∑ f 2 x ∑ ( u s ( c ) + δ V ( y , ω 0 max s , π s )) df ( s ) [ c , y ] I 2 P s 2 I C � X where V 0 ( x ) = V ( x , ω 0 , π 0 ) represents % on X . Subjective parameters: Π : Markov transition matrix with π s ( s 0 ) = Π ( s , s 0 ) and stationary distribution π 0 ( u s ) : State contingent utility functions δ : Discount factor ω 0 : In…nite Horizon Attention Constraint The vector (( u s ) , Π , δ , ω 0 ) is uniquely identi…ed from behavior. Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 5 / 17
Markov Decision Process for Information Choice Example Attention Stock θ 2 Θ . Periodic attention income κ � 0. Rate of deteriotation of stock γ 2 [ 0 , 1 ] . Learning the partition P of S costs attention c ( P ) . Stock evolves according to θ t + 1 = γ ( θ t � c ( P t )) + κ . Constraint is ( θ t � c ( P t )) � 0 for all t . Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 6 / 17
MDP for Information Choice II Example (Continued) Attention cost of choosing P after Q is c ( P j Q ) = ( 1 � b ) H µ ( P ) + bH µ ( P j Q ) H µ ( P ) is the entropy of P (given µ on S ) H µ ( P j Q ) is the relative entropy of P with respect to Q . Note that H µ ( P j P ) = 0. κ measures ability of decision maker (DM) to process new information. b measures degree to which DM can gain expertise. Attention stock θ t at any time captures attention capacity, depletion captures fatigue. Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 7 / 17
Application: Familiarity Bias Bias in favor of choosing among familiar options may stem from expertise in processing information F t G is RACP that o¤ers consumption acts f : S ! [ 0 , 1 ] in all periods until t and acts in G thereafter. Consumption preferences are state wise monotone. De…nition The DM displays familiarity bias between two menus of consumption acts F and G after t periods if F ∞ � FtG and G ∞ � GtF . Comparative statics link "more familiarity bias" to higher b . Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 8 / 17
MDP for Information Choice III Example DM cannot engage in acquiring information in two consecutive periods. Example The feasible set of partitions at any period solely depends on the realization of the state in the previous period. Example DM is endowed with an initial attention stock K , which he draws down every time he chooses to learn. Generates, for example, decreasing reservation wage in stationary environment. Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 9 / 17
In…nite Horizon Attention Constraints (IHACs) Let P be the set of all partitions of S . Let Ω be the space of IHACs, � P� Ω S � Ω ' K ... P 1 s 11 1 ... s ... P 2 2 s 11 ... 1 ... s P 1 3 1 P s 12 1 2 ... ... P 2 s 12 3 ... P 3 s 12 1 ... s 1 ... P 2 s 1 2 ... s ... 3 P 1 P s 2 2 ... P 1 3 ... s 3 P 2 3 ... 3 P 3 A minimal IHAC does not contain dominated subtrees in terms of the …neness of available partitions. Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 10 / 17
Main Result Theorem The binary relation % on X satis…es our axioms, if and only if there is a Dynamic Rational Inattention Representation (( u s ) , Π , δ , ω 0 ) where ω 0 is a minimal IHAC. The parameters (( u s ) , Π , δ , ω 0 ) are uniquely identi…ed from behavior. Standard assumptions in dynamic contexts: Strategic Rationality, Separability, and Stationarity. All are violated in our model. Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 11 / 17
Related Menu Choice Literature Uncertainty about Uncertainty about state of vNM taste the world Static with fixed information Dekel, Lipman, Rustichini Dillenberger, Lleras, Sadowski, (ECMA 2001) Takeoka (JET 2014) Static with information Ergin, Sarver De Oliveira, Denti, Mihm, choice (ECMA 2010) Ozbek (2015) Dynamic with fixed Krishna, Sadowski information process (ECMA 2014) Dynamic with information Dillenberger, Krishna, choice process Sadowski Not menu choice: Ellis (2014), Caplin and Dean (2015), Matejka and McKay (2015), ... Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 12 / 17
Known Axioms Continuous Monotone Preferences (DLR/DLST) Aversion to Randomization (ES/DDMO) Standard properties for consumption streams (no information required): Independence, History Independence, Stationarity, Worst Element (KS) Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 13 / 17
Relaxing Strategic Rationality De…nition A Contingent Plan from x is a function ξ x : S ! x . An Incentivized Contingent Commitment to plan ξ x is the following set. � f ( s ) � � if f = ξ x ( s ) ICC ( ξ x ) = g 2 F : 9 f 2 x with g ( s ) = l � ( s ) otherwise Axiom (Indi¤erence to Incentivized Contingent Commitment) For every x 2 X there is ξ x such that ICC ( ξ x ) � x. Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 14 / 17
Relaxing Separability Axiom (State Contingent Separability) If f 2 x and f 0 2 F ( ∆ ( C � X )) are such that f 0 1 ( s ) = f 1 ( s ) and f 0 2 ( s ) = f 2 ( s ) for all s 2 S, then ( x n f f g ) [ f f 0 g � x. Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 15 / 17
Relaxing Stationarity % which is represented by V ( � , ω 0 , π 0 ) is qualitatively of the same type as tomorrow’s preferences represented by V ( � , ω 0 s , π s ) , denoted by % ω 0 s . We can only impose this qualitative stationarity , if % ω 0 s is constructed based on RACPs that all give rise to the same optimal initial information choice. Observation: The optimal initial partition changes if and only if there is a violation of Independence. We state the following relaxation of Stationarity more formally in the paper: Axiom (Qualitative Stationarity) If % restricted to the RACPs used in the construction of % ω 0 s satis…es Independence, then % ω 0 s satis…es all the previous Axioms and Qualitative Stationarity. Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 16 / 17
Main Contributions We provide the …rst truly dynamic model of rational inattention and 1 demonstrate its usefulness in applications. We …rst uniquely identify a subjective decision process from choice 2 behavior. We show that IHACs provide a uni…ed view of a plausible class of 3 preferences that violate the central tenets of Separability and Stationarity. This signi…cantly increases the scope of well understood models of dynamic decision making. Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 17 / 17
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