An Introduction to Rational Inattention

An Introduction to Rational Inattention Lecture notes for the course Bounded Rationality and Macroeconomics December 2, 2005 1 Introduction The objective of modelling economic agents as being rationally inattentive is to capture the fact that people are constrained in their ability to acquire and process information. To do so mathematically, we need a model which 1. defines clearly what the limits to information acquisition and processing are; and 2. which information people do take into account, given their limits, and which information they disregard. There are many ways one could think of to achieve this goal The probably simplest way is to assume that people update their information sets only infrequently, with the probability of updating in a certain period being set exogenously. This is what Mankiw & Reis (2002) do. Alternatively, one could equip the agents in the model with biases that one observes in experiments, e.g. over-confidence, overestimation of low probabilities and underestimation of high probabilities, favouring confirmatory over contradictory evidence. Another alternative and this is the one that rational inattention chooses is to model agents as having only a probability distribution over the true state of the world, instead of knowing the state of the world with certainty. For instance, imagine looking at a thermometer or your watch: While being in most cases, at least highly correlated with the actual temperature or time, respectively, they probably do not tell you exactly the true value. 1

Sims s (2003) motivation for choosing the latter alternative over the previous ones is the following (p. 666): That people have limited information-processing capacity should not be controversial. It accords with ordinary experience, as do the basic ideas of the behavioral, learning, and robust control literatures. The limited information-processing capacity idea is particularly appealing, though, for two reasons. It accounts for a wide range of observations with a relatively simple single mechanism. And, by exploiting ideas from the engineering theory of coding, it arrives at predictions that do not depend on the details of how information is processed. 2 Motivation The basic idea of agents having only a probability distribution on the true state of the world is still empirically empty. What matters is which probability distribution agents have over the state of the world. Let us denote the state of the world by X, which is a random variable. Assume that the agent receives a signal on this state of the world; in the thermometer example, this would be the temperature displayed. 1 We will denote the signal by S. The appropriate mathematical concept for describing the relation between the true state X and the signal S is their joint probability distribution. As usual we denote the joint probability distribution by f (x, s). Now, rational inattention posits the following (see Sims, 2005): 1. The agent can choose this joint distribution. 2. The agent is restricted in the accuracy of her choice: she cannot observe the true state over all possible states of the world. 3. Given this restriction, the agent chooses the joint distribution rationally, i.e. optimally according to an objective function. 1 I am using the thermometer example, because here it is still relatively clear what the true state of the world actually is: when water starts freezing, the temperature is defined to be 0 C, and when it starts boiling we call this 100 C. Therefore, we can really measure deviations of a displayed temperature from the actual temperature. With time, this is already less easy... 2

Let me illustrate this in the context of the thermometer example: Point 1. means that you can choose between different thermometers to measure the temperature. Point 2. says that none of the available thermometers can always tell you the true temperature. However, some of them may always tell you the truth for some temperatures. Point 3. means that which thermometer you choose depends on the purpose of your measurement. For example, you might like the temperature in your apartment to be around 20 C. Therefore, you will choose for your apartment a thermometer which is accurate in the environment around 20 C, while you do not care whether it is widely inaccurate for temperatures around 100 C. In contrast, you probably want the thermometer in your car to be accurate around 0 C, because that is where the streets get icy, whereas you hardly care whether it is accurate for temperatures around 20 C. This idea could also be valuable for macroeconomic theory because of its potential to explain the pattern of relatively frequent changes of individual variables (e.g., prices) and, at the same time, sluggish behaviour of the corresponding aggregates (e.g., the observation that the consumer price index trails behind the business cycle by around two quarters and responds to monetary policy innovations only with a one-year delay). As Sims (2005, p. 2) puts it: One appeal of the rational inattention idea (...) is that it can in principle explain the observed patterns of inertial and random behavior by a mechanism with many fewer [compared to other macroeconomic models] 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. 3 Some needed basic concepts In engineering, a concept to describe the uncertainty embodied in the probability density function (pdf) of a random variable, is its entropy. The entropy is a single number and, for our purposes, best defined as H(X) E[log 2 f (X)], where f (X) is the pdf of X, see Sims (1998, p. 345). If X is normally distributed, X N(µ, σ 2 ), its entropy turns out to be H(X) = 1 2 log 2 (2πσ2 ). (1) Furthermore, we need the so-called mutual information. Consider two random variables X and S with Cov[X, S] = 0, i.e., the two variables are correlated. In this case, 3

each of these random variables tells us something about the respective other: If, e.g., Cov[X, S] > 0, we expect X to be relatively high when observing a high S. Thus, when observing one of the two variables, we obtain information on which values the second variable is likely to have. That is, we can calculate the conditional mean, the conditional variance and so on. Therefore, we experience a reduction in uncertainty on the distribution of X through observing S. This reduction in uncertainty can be quantified and is called the mutual information between X and S. It is defined as the prior uncertainty about X minus the posterior uncertainty on X, given the observation of S: I(X; S) = H(X) H(X S), (2) see Maćkowiak & Wiederholt (2005, p. 30). This shows that we will need conditional expectations and conditional variances as wells as, once again, the law of iterated expectations. Consider a function D which is defined on two random variables: D(X, S). Thus, it is itself a random variable. Recall that the conditional pdf f (x s) is defined as f (x s) f (x,s) f (s) ; therefore, f (x, s) = f (x s) f (s). The law of iterated expectations states: E X,S [D(X, S)] = = = = D(x, s) f (x, s) dx ds D(x, s) f (x s) f (s) dx ds ( ) D(x, s) f (x s) dx f (s) ds E X [D(X, s) s] f (s) ds = E S [ EX [D(X, S) S] ]. (3) Here, I denote by E Y the expectation calculated by using the marginal pdf of the random variable Y; i.e., E Y yf (y)dy 4 Allocating attention Maćkowiak & Wiederholt (2005) set up a model with monopolistic competition, so that each producer has market power for her own good and, therefore, can set the price of that good. The profit-maximising price (in logs) p i of producer i turns out to be (see pp. 12 and 13): p f i = p + α 1 y + α 2 z, (4) 4

where p is the log of the aggregate price level, y is the log of aggregate demand, and z is an idiosyncratic (individual) demand shock. (The superscript f indicates full information.) All these variables (p, y and z) are normally distributed. Since the sum of two normally distributed variables is also normally distributed, one can summarise the aggregate conditions as p + α 1 y. Thus, p f i = + α 2 z, (5) This is where rational inattention enters the scene. The firm is assumed not to be able to observe and z perfectly. All it can observe is signals S 1 on the aggregate-state variable and S 2 on the idiosyncratic-state variable. Maćkowiak & Wiederholt show that, given the observed signals, the profit-maximising price is p i = E[ S 1 ] + α 2 E[z S 2 ], (6) Whenever p f i and p i differ, there is a loss in profits. Denoting firm i s profits by π i, it turns out that this loss is quadratic in the deviation of p i from p f i : π(p f i,, z) π(p i,, z) = 2 (pf i p i ) 2, (7) where is a positive constant. This difference is always positive, as we would expect, because it should always be advantageous to know the true state with certainty. Maximising the expected profit under uncertainty amounts, due to this formula, to minimising the expected squared deviation of p i from p f i : [ min E,z,S 1,S 2 f (,S 1 ), f (z,s 2 ) 2 (pf i p i ) 2] (8) The minimisation is, as argued in the introduction to this handout, over the joint distribution of the true state and the signals: The firm chooses which signals to obtain, given the true state of the world. One can show that for quadratic objective functions as is the case here and Gaussian state variables, it is optimal to choose the joint density of the signals and the states also Gaussian. Due to the quadratic form of the objective function and due to equations (5) and (6) this minimisation problem reduces to: [ min f (,S 1 ), f (z,s 2 ) 2 E,z,S 1,S 2 (p f i E,z[p f i S 1, S 2 ]) 2]. (9) Due to the law of iterated expectations, this is equal to [ [ ]] min f (,S 1 ), f (z,s 2 ) 2 E S 1,S 2 E,z (p f i E,z[p f i S 1, S 2 ]) 2 S1, S 2 [ ] = min f (,S 1 ), f (z,s 2 ) 2 E S 1,S 2 Var[p f i S 1, S 2 ] = min f (,S 1 ), f (z,s 2 ) 2 Var[pf i S 1, S 2 ]. 5

The latter part of the equation is due to the fact that for Gaussian random variables, the conditional variance is constant, i.e. it does not depend on the values upon which it is conditional. Now, the aggregate-state variable and the idiosyncratic-state variable z are assumed to be uncorrelated. Therefore Var[p f i S 1, S 2 ] = Var[ S 1, S 2 ] + Var[α 2 z S 1, S 2 ] = σ S 2 1 + α 2 2σz S 2 2. (10) This expression has to be minimised. However, we cannot make it arbitrarily small (i.e., set it to zero)! This is the core of rational inattention: the minimisation has to respect the information flow constraint I(, z; S 1, S 2 ) = H( ) H( S 1 ) + H(z) H(z S 2 ) κ. Due to normality, we have H( ) H( S 1 ) + H(z) H(z S 2 ) κ 1 2 log 2 (2πσ2 ) 1 2 log 2 (2πσ2 S 1 ) + 1 2 log 2 (2πσ2 z ) 1 2 log 2 (2πσ2 z S 2 ) κ 1 2 log 2 (σ2 ) 1 2 log 2 (σ2 S 1 ) + 1 2 log 2 (σ2 z ) 1 2 log 2 (σ2 z S 2 ) κ. The objective function (10) is strictly decreasing in both decision variables σ 2 S 1 and σ 2 z S 2, while the l.h.s. of the constraint is strictly increasing in those variables. Hence, in the optimum the constraint will be fulfilled by equality, and we can use Lagrange s method to solve the minimisation problem. The Lagrangian is L = σ S 2 1 + α 2 2σz S 2 2 [ 1 λ 2 log 2 (σ2 ) 1 2 log 2 (σ2 S 1 ) + 1 2 log 2 (σ2 z ) 1 ] 2 log 2 (σ2 z S 2 ) κ. The first-order conditions are, therefore (recalling that log b x = log x / log b), L σ 2 S 1 = 1 + 1 2 λ 1 log 2 1 σ 2 S 1! = 0 λ = 2 (log 2) σ 2 S 1 ; (11) L σz S 2 = α 2 2 + 1 2 2 λ 1 1 log 2 σz S 2 2 λ = 2 α 2 2 (log 2) σz S 2 2 ; (12) as well as the information flow constraint itself. 6

Equating (11) with (12) yields σ 2 S 1 = α 2 2 σ 2 z S 2. (13) That is, the reduction of the two sorts of uncertainty occurs according to their relative contribution to the objective function. Plugging (13) into the information flow constraint delivers 1 2 log 2 (σ2 ) 1 2 log 2 (α2 2 σz S 2 2 ) + 1 2 log 2 (σ2 z ) 1 2 log 2 (σ2 z S 2 ) = κ log 2 (σ ) 2 log 2 (α 2 2 σz S 2 2 ) + log 2 (σz 2 ) log 2 (σz S 2 2 ) = 2 κ. Solving for σ 2 z S 2 yields σ 2 z S 2 = 2 κ+ 1 2 [log 2 (σ2 )+log 2 (σ2 z ) log 2 (α 2 2 )] σ 2 S 1 = α 2 2 2 κ+ 1 2 [log 2 (σ2 )+log 2 (σ2 z ) log 2 (α 2 2 )]. Bibliography Maćkowiak, Bartosz & Mirko Wiederholt (2005). Optimal Sticky Prices under Rational Inattention. Unpublished manuscript, URL: http://www.wiwi.huberlin.de/wpol/html/staff/mirko/optimalstickyprices.pdf. Mankiw, N. Gregory & Ricardo Reis (2002). Sticky Information versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve. Quarterly Journal of Economics, 117(4): 1295 1328. Sims, Christopher A. (1998). Stickiness. Carnegie Rochester Conference Series on Public Policy, 49: 317 356. Sims, Christopher A. (2003). Implications of Rational Inattention. Journal of Monetary Economics, 50(3): 665 690. Sims, Christopher A. (2005). Rational Inattention: A Research Agenda. url: http://sims.princeton.edu/yftp/riplus/ratinattplus.pdf ; originally prepared for the 7 th Deutsche Bundesbank Spring Conference in Berlin. 7