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«Discussion Paper Series 1: Economic Studies No 34/2005 Discussion Papers represent the authors’ personal opinions and do not necessarily reflect ...»

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Rational inattention: a research agenda

Christopher A. Sims

(Princeton University)

Discussion Paper

Series 1: Economic Studies

No 34/2005

Discussion Papers represent the authors’ personal opinions and do not necessarily reflect the views of the

Deutsche Bundesbank or its staff.

Heinz Herrmann

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ISBN 3–86558–090–4 Non-technical summary Implications of rational inattention The majority of macro models currently in use assume that the market players have rational expectations and can process information immediately and without restrictions. These assumptions are justified, first, in that they make the models easier to solve. Second, the Lucas critique negates certain forms of limited rational expectations. Rational expectations and the unrestricted ability to process information also imply, however, that prices and market players’ behaviour adapt quickly and without error to new information. Yet this does not match what is generally observed in reality. At the same time, numerous empirical studies show that market players’ expectations are not rational in many cases.

Recently, theories have been developed that model imperfect information and deviations from rational expectations or cost-inducing information procurement and processing and study their implications. This includes, for example, learning literature (eg Sargent (1993), Evans und Honkapohja (2001)), robust control literature (eg Hansen and Sargent (2001), Giannoni (1999)) and behavioural economics literature (eg Laibson (1997), Benábou and Tirole (2001)). This paper is a contribution to this corpus of literature and adopts another way of modelling imperfect information of market players. It assumes that the market players have limited capacities to process information if shocks hit the economy and the market players receive and analyse signals. In this work the author makes use of the coding theory developed by engineering science. This is based on the idea that information initially flows through a “channel” before it reaches the market players.

Depending on the degree of capacity to process information, it may then be fraught with errors which, in turn, influence the decisions of the market players.

The advantage of this approach over the more psychological approaches in some of the recent literature is that it develops predictions about deviations from perfect optimizing behavior from a minimal set of assumptions, without reliance on possibly controversial psychological detail. Its advantage over approaches that simply postulate “measurement error” in people's observations is that the form of observation errors is predicted by the theory, from the structure of people's optimization problems.

Nichttechnische Zusammenfassung

Implikationen rationaler Unaufmerksamkeit

Die Mehrzahl der gegenwärtigen Makromodelle nimmt an, dass die Marktteilnehmer rationale Erwartungen haben und Informationen sofort und ohne Einschränkungen verarbeiten können. Diese Annahmen werden zum einen dadurch gerechtfertigt, dass so die Modelle einfacher zu lösen sind. Zum anderen verneint die Lucas Kritik bestimmte Formen begrenzter rationaler Erwartungen. Rationale Erwartungen und die unbegrenzte Fähigkeit, Informationen zu verarbeiten, implizieren aber auch, dass Marktteilnehmer und Preise schnell und ohne Fehler neue Informationen verarbeiten.

Das trifft in der Realität aber im allgemeinen nicht zu. Gleichzeitig zeigen viele empirische Untersuchungen, dass die Erwartungen der Marktteilnehmer oft nicht rational sind.

In neuerer Zeit sind Theorien entwickelt worden, die unvollständige Informationen und Abweichungen von rationalen Erwartungen modellieren oder auch die Beschaffung kostenträchtiger Informationen und die die sich daraus ergebenden Implikationen studieren. Dazu gehört z. B. die Literatur über das Lernen (siehe Sargent, 1993 und Evans und Honkapohja, 2001), die Literatur zur robusten Steuerung (siehe Hansen und Sargent, 2001, Giannoni, 1999) und die Literatur zum ökonomischen Verhalten (siehe Laibson, 1997, Benabou und Tirole, 2001). Dieses Papier ist ein Beitrag zu dieser Literatur; es beschreitet aber einen anderen Weg, um unvollständige Informationen von Marktteilnehmer abzubilden. Es nimmt an, dass die Marktteilnehmer begrenzte Fähigkeiten haben, Informationen zu verarbeiten, wenn Schocks die Volkswirtschaft treffen und die Marktteilnehmer Signale empfangen und verarbeiten. In dieser Arbeit verwendet der Verfasser die Kodierungstheorie der Ingenieurswissenschaften. Sie basiert auf der Idee, dass Informationen zunächst durch einen Kanal fließen, bevor sie beim Marktteilnehmer ankommen. Je nachdem welche Kapazität zur Verfügung steht, um Informationen zu verarbeiten, kann sie mit Fehlern belastet sein, die dann die Entscheidungen der Marktteilnehmer beeinflussen.

Der Vorteil dieses Ansatzes gegenüber einigen mehr psychologischen Ansätzen besteht darin, dass er Voraussagen über die Abweichungen vom perfekten optimierenden Verhalten macht und dabei mit einer geringen Anzahl von Annahmen auskommt und nicht auf psychologische Details angewiesen ist, die möglicherweise umstritten sind. Der Vorteil gegenüber der einfachen Annahme von Messfehlern in den Beobachtungen besteht darin, dass die Form des Beobachtungsfehlers von der Theorie erklärt werden und von der Struktur des Optimierungsproblems der Menschen abgeleitet werden kann.


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ABSTRACT. The literature applying information-theoretic ideas to economics has so far considered only Gaussian uncertainty. Ex post Gaussian uncertainty can be justified as optimal when the associated optimization problem is linear-quadratic, but the literature has often assumed Gaussian uncertainty even where it cannot be justified as optimal. This paper considers a simple two-period optimal saving problem with a Shannon capacity constraint and non-quadratic utility. It derives an optimal ex post probability density for wealth in two leading cases (log and linear utility) and lays out a general approach for handling other cases numerically. It displays and discusses numerical solutions for other utility functions, and considers the feasibility of extending this paper’s approaches to general non-LQ dynamic programming problems. The introduction of the paper discusses approaches that have been taken in the existing literature to applying Shannon capacity to economic modeling, making criticisms and suggesting promising directions for further progress.


In a pair of earlier papers1 I have argued for modeling the observed inertial reaction of economic agents to external information of all kinds as arising from an inability to attend to all the information available, and for treating that inability as arising from finite Shannon capacity. Shannon capacity is a measure of information flow rate that is inherently probabilistic. It uses the reduction in the entropy of a probability distribution as the measure of information flow. The entropy of a distribution is a global measure of the uncertainty implied by the distribution, relative to some base distribution. Because of this dependence on the base, the entropy of a distribution is not uniquely defined, but if we consider the joint distribution of two random vectors or variables, the expected reduction in entropy of one of the two achieved by observing the other of the two, the mutual information implied by the joint distribution, is uniquely defined, independent of any base. This measure Date: August 18, 2005.

This research was supported in part by NSF grant SES-0350686. The paper was originally prepared for and presented at the Deutsche Bundesbank Spring Conference, May 2005.

(Sims, 2003, 1998). The earlier paper contains an appendix arguing that Shannon capacity makes sense as a model of inattention. The later one gives explicit solutions for some simple economic models with a linear-quadratic structure.

c 2005 by Christopher A. Sims. This document may be reproduced for educational and research purposes, so long as the copies contain this notice and are retained for personal use or distributed free.


of mutual information can be derived from a few reasonable axioms, but it is pervasive less because of its axiomatic appeal than because has proved to be exactly the concept appropriate for studying information flows in physical communication channels. A Shannon “channel” is a set of possible inputs, a set of possible outputs, and a conditional distribution for outputs given inputs. From these elements, it is possible to calculate a tight upper bound for the mutual information between inputs and outputs, which is called the channel’s capacity. It is the measure of information flow we use in characterizing modems or internet connections in bits per second or bytes per second. Shannon showed that no matter what we might wish to send through the channel, whether music, text, or spreadsheets, and no matter what the physical nature of the channel — wires, optical cables, radio transmission, or a messenger service — it is possible to send information through the channel at a rate arbitrarily close to capacity.

Economists, particularly macroeconomists, have recognized the need to account for the inertia in observed economic behavior and have modeled it with a variety of devices — menu costs, adjustment costs, information delay, implementation delay, etc. As my two earlier papers argued, these mechanisms can match the observed pattern — slow, smooth cross-variable responses, combined with less smooth idiosyncratic randomness — only by postulating elaborate inertial schemes that are both difficult to connect to observation or intuition and critically important in making model behavior realistic. One appeal of the rational inattention idea (that is, of modeling agents as finite-capacity channels) is that it can in principle explain the observed patterns of inertial and random behavior by a mechanism with many fewer free parameters. Another is that it fits well with intuition; most people every day encounter, or could very easily encounter, much more information that is in principle relevant to their economic behavior than they actually respond to. The notion that this is because their are limits to “attention”, and that such limits might behave like finite Shannon capacity, is intuitively appealing.


A number of recent papers in macroeconomics and finance have used informationtheoretic ideas (Ma´ kowiak and Wiederholt, 2005; Luo, 2004; Mondria, 2005; Moscarini, c 2004; Van Nieuwerburgh and Veldkamp, 2004a,b; Peng and Xiong, 2005; Peng, 2005).

While these papers develop some valuable insights, it is worth noting that they have made assumptions, to allow tractable modeling, that are hard to defend and can lead to anomalous results. Some of these limitations are common to all or nearly all of the papers.

II.1. Not allowing fully endogenous choice of the form of uncertainty. It is central to the idea of modeling individuals as capacity-constrained that the nature,

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not just the quantity of their uncertainty about external signals (prices, income, wealth, asset yields, etc.) is subject to choice. The power of information theoretic ideas arises from the fact that the available joint stochastic processes for channel input and channel output are, to an arbitrarily good approximation, limited only by the capacity of the channel, not by its physical nature. In a model of an optimizing agent, the agents’ objective function will therefore determine the stochastic process for the joint behavior of actions and external signals. The articles cited in the previous paragraph, with the partial exception of Luo’s, postulate directly a simple parameteric form for this joint process, without deriving that form from the model’s objective function.

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