Dynamic Rational Inattention David Dillenberger; R. Vijay Krishna; - - PowerPoint PPT Presentation

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

1

consumption today; and

2

continuation problem for tomorrow.

Formally, X ' K (F (∆ (C X))) . In…nite Horizon Attention Constraint (subjective): (P, ω0) 2 ω determines

1

partition of S today; and

2

attention constraint for tomorrow.

Timeline:

Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 4 / 17

Recursive Representation: V (x, ω, π) = max

(P,ω0)2ω ∑ I 2P

@max

f 2x ∑ s2I

2 4

Z

C X

(us (c) + δV (y, ω0

s, πs)) df (s) [c, y]

3 5 π (s) 1 A where V0 (x) = V (x, ω0, π0) represents % on X. Subjective parameters:

Π : Markov transition matrix with πs(s0) = Π(s, s0) and stationary distribution π0 (us) : State contingent utility functions δ : Discount factor ω0 : In…nite Horizon Attention Constraint

The vector ((us) , Π, δ, ω0) 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 θt+1 = γ (θt c (Pt)) + κ. Constraint is (θt c (Pt)) 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 jQ ) = (1 b) Hµ (P) + bHµ (P jQ ) Hµ (P) is the entropy of P (given µ on S) Hµ (P jQ ) is the relative entropy of P with respect to Q.

Note that Hµ (P jP ) = 0.

κ measures ability of decision maker (DM) to process new information. b measures degree to which DM can gain expertise. Attention stock θt 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 FtG 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, Ω ' K PΩS

s

1

s

2

P

s3 P

1 1

P

2 1

P

1 2

... ... ... ...

s

1

s

2

s

3

P

3 1

P

3 3

P

3 2

s

1

s

3

s

2

P

12 2

P

12 1

P

12 3

P

11 2

P

11 1

... ... ... ... ... ... ... ... ... ...

s

1

s

3

s

2

...

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 ((us) , Π, δ, ω0) where ω0 is a minimal IHAC. The parameters ((us) , Π, δ, ω0) 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 vNM taste Dekel, Lipman, Rustichini (ECMA 2001) Ergin, Sarver (ECMA 2010) Uncertainty about state of the world Dillenberger, Lleras, Sadowski, Takeoka (JET 2014) De Oliveira, Denti, Mihm, Ozbek (2015) Dillenberger, Krishna, Sadowski Krishna, Sadowski (ECMA 2014) Dynamic with information choice process Static with information choice Static with fixed information Dynamic with fixed information process

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 ξx : S ! x. An Incentivized Contingent Commitment to plan ξx is the following set. ICC(ξx) =

• g 2 F : 9 f 2 x with g(s) =

f (s) if f = ξx (s) l (s)

• therwise
• Axiom (Indi¤erence to Incentivized Contingent Commitment)

For every x 2 X there is ξx such that ICC(ξx) 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) = f1 (s) and

f 0

2 (s) = f2 (s) for all s 2 S, then (xn ff g) [ ff 0g x.

Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 15 / 17

Relaxing Stationarity

% which is represented by V (, ω0, π0) is qualitatively of the same type as tomorrow’s preferences represented by V (, ω0

s, πs), denoted

by %ω0

s.

We can only impose this qualitative stationarity, if %ω0

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 %ω0

s satis…es

Independence, then %ω0

s satis…es all the previous Axioms and Qualitative

Stationarity.

Dillenberger, Krishna, Sadowski () Dynamic Rational Inattention Summer 2015 16 / 17

Main Contributions

1

We provide the …rst truly dynamic model of rational inattention and demonstrate its usefulness in applications.

2

We …rst uniquely identify a subjective decision process from choice behavior.

3

We show that IHACs provide a uni…ed view of a plausible class of 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