Rational Inattention and Multiple Equilibria

Transcription

1 Rational Inattention and Multiple Equilibria Martin Ellison and Alistair Macaulay March 31, 2017 Abstract We show that introducing rational inattention leads to multiple equilibria in a class of models which otherwise have a unique equilibrium. We apply our results to a heterogeneous agent New Keynesian (HANK) DSGE model and show that, whilst this version of the model has a single steady state under rational expectations, it has multiple steady states when households are rationally inattentive. A low unemployment steady state is associated with moderate inflation and a high job hiring rate, but there is also a trap-type steady-state that has high unemployment, low inflation and a low job hiring rate, albeit with steady-state inflation remaining positive. Both steady-states are shown to be locally stable. 1 Introduction Rational Inattention (RI) has rapidly established itself as a popular way of modelling how agents make decisions when they find information difficult to process. Following the seminal contributions of Sims (1998, 2003), agents in rational inattention models are assumed to be constrained in how much information they can process each period, which means they are restricted in the precision with which they can observe the economic variables relevant to their decisions. Models in this literature typically assume that agents completely understand the workings of the model in which they are operating, and so the only unknown variables relevant for their decisions that require information processing are exogenous shock processes (e.g. Mackowiak and Wiederholt, 2015). Those papers have rightly received a lot of attention for the valuable insights they provide, but there is no inherent reason to restrict RI models to these cases. We contribute to the literature by studying models where agents are ignorant of how endogenous variables are determined, and so they must allocate their limited processing capacity to endogenous variables. We show that this opens up new avenues of economic dynamics. In particular, in this paper we demonstrate that applying rational inattention to endogenous variables can create multiple equilibria in environments that have a unique equilibrium in the absence of information processing We thank Ricardo Reis, Paul Beaudry and Albert Marcet for their comments. 1

2 constraints. This is very different from most existing models of multiple equilibria, in that it does not rely on common knowledge of fundamentals and the ability of agents to co-ordinate their beliefs identified by Morris and Shin (2000). Multiplicity arises in our framework because agents endogenously restrict themselves to a small number of choices, even though the optimal choice without rational inattention is continuous. This result, known from the work of Matejka and Sims (2011) and Matejka (2015), introduces a non-linearity in the way agents respond to a variable over which they are inattentive. That non-linearity is already important when the variable over which agents are rationally inattentive is exogenous, but once agents are rationally inattentive over endogenous variables then the non-linearity is sufficient to generate multiple equilibria (or multiple steady states in dynamic models). All that is needed is that agents are rationally inattentive to one or more endogenous variables, which is justified if agents have some degree of ignorance about how individual decisions map into aggregate outcomes (see section 5.2 for a detailed discussion). This assumption, that prior beliefs are not necessarily tied to fundamentals, is conceptually similar what Veldkamp (2011) refers to as the diffuse prior. The only difference is that we assume agents know the plausible range in which the relevant endogenous variable could fall before they process any information, which means the models we study feature bounded priors, as required for the discreteness result in Matejka and Sims (2011). We demonstrate our approach in a simple example and then apply it to a version of the heterogeneous agent New Keynesian (HANK) model presented in Ravn and Sterk (2016). Their model has multiple steady states because they assume that firms can post a small number of job vacancies without cost, which introduces a non-linearity in the Phillips curve. If this assumption is removed then there is a single steady state. We do this and instead have rationally inattentive households processing information about the future job hiring rate, an endogenous variable relevant for deciding household precautionary saving. In keeping with agents being somewhat ignorant about how individual precautionary savings decisions affect the labour market, we assume that agents have bounded uniform priors on the job hiring rate. The combination of rational inattention and a bounded uniform prior is sufficient to generate an additional trap-like steady state with high unemployment, low but positive inflation and a low job hiring rate. The new steady state is similar to that obtained by Ravn and Sterk (2016) through their vacancy cost non-linearity, although the different mechanisms supporting these high unemployment steady states call for different policy actions. Generating multiple equilibria in economic models is not particularly difficult, and many authors have had varying degrees of success through imposing various non-linearities on their models. The advantage of our approach is that the rational inattention in our framework endogenously generates just the right sort of non-linearity needed to create multiple equilibria or steady-states, without the 2

3 need for imperfectly-motivated stories for exogenous non-linearities. The multiplicity we obtain does not rely on imposed non-linearities, only those generated within the model because of the rational inattention constraint. The rest of the paper continues in Section 2 by placing our contribution in the context of the literature. Sections 3, 4 and 5 progressively build a simple framework that acts as a proof of concept showing how rational inattention to endogenous variables can create multiple steady states. Section 6 applies our results to a version of the fully specified heterogeneous agent New Keynesian (HANK) model of Ravn and Sterk (2016). A final section concludes. 2 Related Literature This paper fits principally within the literature on rational inattention. For tractability, most existing models with RI have agents inattentive about a normally distributed shock variable, with Gaussian prior beliefs about that shock variable and a quadratic objective function 1. Matejka and Sims (2011) and Matejka (2015) show, however, that assuming a bounded prior belief leads to very different results to the quadratic-gaussian formulation. Specifically, they show that the optimal decision rule of an agent facing a rational inattention problem with a bounded prior entails the agent choosing to limit themselves to a discrete number of options, even when the optimal choice under perfect information is continuous. Our work uses this result as we restrict our analysis to cases in which the unknown variable about which agents process information has some natural bounds. In particular, we assume a uniform prior belief over this range to reflect ignorance about the variable throughout, which brings our models closely into line with the pricing model of Matejka (2015). Our expression of RI mirrors that in Matejka (2015) except for this interpretation of the prior. Rational inattention does not simply exist as a theoretical curiosity. The motivation behind studying such information frictions is that they are more realistic than models with perfect information, and so it is important that various studies have begun to find evidence in support of the theory. Bartos, Bauer, Chytilova, and Matejka (2016) find some evidence for RI in data on statistical discrimination, and Khaw, Stevens and Woodford (2016) present some experimental evidence. In fact, their experimental evidence specifically supports models of RI in which choices are discrete, despite a continuum of choices being available (and optimal under perfect information). This supports the form of RI in Matejka (2015) and in this paper over the quadratic-gaussian case studied by, among others, Mackowiak and Wiederholt (2009) 2. 1 This is convenient as the optimal posterior belief about the shock, after processing information, is also Gaussian. See Mackowiak and Wiederholt (2009) for an example. 2 Khaw, Stevens and Woodford also find that choice updating does not occur at every opportunity in their experiment, so they claim that the Woodford (2009) model of RI is the best fit to the data. Our model, and that of Matejka (2015), abstracts away from this problem 3

4 We are the first paper to incorporate RI with bounded prior beliefs into a general equilibrium setting. Mackowiak and Wiederholt (2015) solve a DSGE model with RI, but restrict themselves to studying agents who are inattentive about exogenous Gaussian shocks. They find that RI can by itself explain a large amount of the slow adjustment of macroeconomic variables to shocks seen in the data. Our approach differs from this in that we do not assume a Gaussian shock distribution and Gaussian prior household beliefs, and the agents in our model are inattentive over variables endogenous to the model, rather than over exogenous shocks. This is why the DSGE model we present as an example of our approach has two steady states, whereas the model in Mackowiak and Wiederholt (2015) has a unique steady state. Similarly, Luo, Nie, Wang and Young (2016) study the effects of rational inattention over uninsurable income shocks in a general equilibrium model. They too restrict themselves to studying inattention over exogenous Gaussian shocks, and Gaussian prior beliefs. Both of these papers have households who cannot process enough information to precisely predict their future environment, but who do perfectly understand the mechanisms at play within the model they inhabit - there is no uncertainty over endogenous variables except for through the effect of the shocks on those variables. In contrast, we assume that agents who cannot process large amounts of information about their environment are also unable to precisely pin down how endogenous model variables are determined, which drives our assumption of the ignorance prior (see section 5.2) and so drives our results. There are to our knowledge only two other papers that generate multiple equilibria through RI, and both rely on different mechanisms to the one presented in this paper. Matejka and Tabellini (2016) show multiplicity in the context of voting patterns, but their result relies on policy decisions affecting voters information processing capacity, which then feeds into the voting patterns of those people, and so the result is specific to that and a few other related contexts. Yang (2015) derives multiple equilibria in a simple co-ordination game, and so the result relies on strategic interactions that are unlikely to be present when considering large populations of agents. Moreover, the agents in Yang s model are restricted to a binary choice ( invest or don t invest ) and the result is for partial equilibria, whereas in our model agents choose from a continuum of options and the results are incorporated into a general equilibrium setting. After setting out our model in an abstract setting, we demonstrate its use in a heterogeneous-agent New Keynesian (HANK) DSGE model. This section of the paper therefore relates to the literature on DSGE models with search and matching frictions in the labour market (e.g. Gertler and Trigari (2009)) and incomplete markets (e.g. Huggett, 1993). Specifically, we study the model presented in Ravn and Sterk (2016) (hereafter RS), which has both of these features. The RS model has multiple steady states without rational inattention, for simplicity, assuming that updating takes place every period. 4

5 because the authors assume that firms can post some small number of vacancies without cost, which imposes a kink in the Phillips curve. We argue in section 6.2 that this assumption is unreasonable. We therefore remove this assumption, at which point the RS model has a unique steady state. We then introduce RI, and show that the multiplicity result is restored. The key mechanism in the RS model with perfect information or rational inattention is that future labour market conditions matter for current consumption choices because unemployment causes an income fall, so a higher unemployment probability in the future encourages more precautionary saving in the present to insulate consumption against such a shock. This idea has strong empirical support: Carroll and Dunn (1997), for example, find evidence that expenditure is significantly negatively correlated with unemployment expectations. This mechanism implies the possibility of self-fulfilling fluctuations: high unemployment expectations lead to a fall in consumption, and so output, and so to high unemployment. For this reason, a variety of models with this mechanism (such as Rendahl (2016) and Heathcote and Perri (2015)) also have multiple steady states; our version is interesting because the multiplicity is generated solely by rational inattention. We show in section 6.7 that the source of multiplicity is important for policy in our model. 3 Rational Inattention Problem Suppose we have an agent choosing x. There is a variable y which is relevant for the optimal choice of x, but the agent does not know it exactly. They can collect information about y from a variety of noisy signals, but there is a limit to how much information they can process. This limit takes the form of an information processing budget of κ, formalised in equation 4 below. The payoff function being maximised is U(y, x). The agent views y as exogenous and random 3. In maximising the expected payoff the agent must decide on an optimal decision rule to map the signals they are able to process to choices of x. A convenient way to express this problem is that the agent s decision rule is the joint probability density function over x and y. That is, given a particular y, the agent chooses how often they will choose each different possible level of x. They are aware that signals contain noise, so they are deciding how often, and by how much, they are willing to choose the wrong x for each level of y. The agent can spend their information processing budget on ensuring they make no mistakes at all when y is at the top of its possible range, but in doing so they must accept that they will make larger mistakes with higher probability when y is low. This follows the form of the rational inattention problem faced by firms 3 The variable y will in fact be endogenous to aggregate agent choices, but the agent does not take this into account. This will be explored in more detail in section 5.2 5

6 in Matejka (2015). The agent problem is therefore: f = arg max E[U(y, x)] = arg max U(y, x) ˆf(y, x)dydx (1) ˆf ˆf y x subject to x ˆf(y, x)dx = g(y) y (2) ˆf(y, x) 0 y, x (3) H[g(y)] E x H[ ˆf(y x)] κ (4) The function H[.] is the entropy of the distribution over which it operates. That is: H[g(y)] = g(y) log g(y)dy (5) The first constraint (equation 2) ensures that the marginal distribution of y obtained from the optimal joint pdf is consistent with the agent s prior belief about the distribution of y, g(y) 4. The second constraint (equation 3) is that the solution must be positive everywhere, as required for it to be a joint pdf. The final constraint (equation 4) is the information processing constraint. Entropy H[.] is a measure of the dispersion of a distribution. The first term of constraint 4 is the entropy of the prior. The prior reflects the information held by the agent about the distribution of y before receiving any signals. We will assume throughout that this prior is uniformly distributed (see section 5.2), so the prior is rather dispersed and entropy is high. The second term is the expected entropy of f(y x), the updated distribution over y believed by the agent after taking in the available signals 5. A precise knowledge of y would give a very low entropy, so the entropy difference from the prior would be large. It is this difference, how much the agent can learn from the signals, that is constrained in this model. As noted above, this means the agent faces a trade-off: they can distinguish between a few values of y which are close together, but that reduces the entropy of the posterior a great deal, so outside of that small range of y their posterior f(y x) must remain dispersed. When y is in that small range, the agent making that decision will be very accurate in choosing optimal x, but when y is outside of that range they will make large mistakes with a high probability. Alternatively, they can choose to allocate their information processing 4 In Matejka (2015), this marginal distribution of the variable subject to rational inattention is the true distribution of that variable. This will not be the case here, as y will be determined endogenously in the model. Instead, the marginal distribution obtained by integrating the joint pdf over x should be interpreted here as the distribution of y the agent is expecting to see from their ignorance prior (see section 5.2). 5 The information content of the signals is incorporated into the choice of x, so the conditional distribution of y given the choice of x tells us what the agent believes about y. 6

7 capacity to distinguishing between a small number of cases which are far apart. They are then never precise in estimating y, but they make large mistakes less often. This is what drives the result in Matejka (2015) that the agent optimally restricts themselves to a small number of discrete levels of x, even though a continuous range of x is available, when κ is sufficiently small that the information constraint binds. 4 Rational Inattention Solution This problem follows the setup of the firm problem in Matejka (2015), in which firms optimally choose from a small number of discrete prices even though there is a continuum of prices available to them. Matejka and Sims (2011) show for a general class of similar problems that there are three conditions required for this result: 1) e U (y, ŷ) is analytic and positive on the real line. 2) U(y, ŷ) is weakly decreasing in y ŷ and is non-constant. 3) The prior distribution g(y) has bounded support. To match these conditions to the agent problem above, note that we will assume a one-to-one mapping from y to optimal x, so the posterior estimate ŷ pins down x. In solutions we will not show the estimate ŷ, working instead with the choice of x that it implies. 4.1 Example This simple example shows the conditions above resulting in a discrete choice of x. Suppose: The first order condition under perfect information is: U(y, x) = xy x 2 (6) x = y 2 (7) Suppose y can take any value between 0 and 1, so optimal x is in the range [0, 0.5]. Assume that prior beliefs are uniform: g(y) [0, 1]. The optimal decision rule for κ = 0.5 is plotted in figure 1. As in Matejka (2015), the agent optimally chooses to restrict themselves to two levels of x, even though under perfect information the optimal choice of x is continuous in y. The logic behind this is discussed in section 3 above, and in detail in Matejka and Sims (2011) and Matejka (2015). As y increases (and so the optimal choice of x under perfect information increases), the probability an agent chooses the higher level of x in their menu increases. 7

8 x 0 0 y Figure 1: Optimal decision rule for κ = General aggregate response There is a large population of agents. They all face the same y, but we assume that they receive different noisy signals and/or they interpret those signals differently. Therefore for each y some agents choose each of the levels of x in the optimal menu which arises from the RI problem with uniform priors (equations 1-4). The proportions on each level of x are determined by the probabilities in the optimal joint pdf obtained as the decision rule from that agent problem. Therefore for each level of y we obtain mean x using: x(y) = xf(x y)dx (8) With the population of agents normalised to 1, this gives aggregate x as a function of y. With no processing capacity (κ = 0), agents must just maximise their expected payoff from their prior belief, as they cannot update their beliefs from this. They therefore choose the same level of x for all y. As the information processing capacity κ increases, some information about y is processed, so agents begin to set different levels of x for different y. For low values of κ, agents optimally restrict themselves to two values of x. As shown in Matejka (2015), as κ rises further, more choices of x are introduced into the optimal menu. As this occurs the aggregate response of x to y approaches the perfect information first order condition. Importantly, this aggregate response function has a wavelike shape around its perfect information equivalent, as shown in the example diagram below. 8

9 aggregate x y Figure 2: Aggregate x in the example model in section 4.1 with κ = 0 (magenta), κ = 0.5 (blue), κ = 1 (green) and in the unconstrained case (red) Consider the cases above where information processing is constrained but non-zero (the blue and green curves). The flat sections of the aggregate response function occur where changes in y do not lead to much change in the proportions of agents choosing each level of x in their menus. In figure 1 above, it can be seen that this is the case for extreme high and low values of y, and x(y) is flat in these regions accordingly. In contrast, as y moves from 0.4 to 0.6, large numbers of agents switch from choosing the low level of x to the high level, and this corresponds to the steep section of the corresponding aggregate response function in figure 2. The shape of the aggregate response curve in the less constrained (κ = 1) case has the same form, but with this greater information processing capacity agents choose from four levels of x, so there are four flat regions in the aggregate response curve. The shape of the aggregate response curve is therefore driven by the shape of the curves in the optimal decision rule: if the probability of choosing the low level of x in figure 1 fell linearly as y increased the aggregate response curve would be linear. In fact, the distribution of x y for the values of x in the optimal menu is logistic in shape, which is what gives rise to the wave-like shape of the aggregate response curve 6. 6 Matejka and McKay (2015) study in detail how RI leads to the logit model. 9

10 5 Simple two equation systems We now examine the sort of models in which this formulation can lead to multiple equilibria. We focus on cases where perfect information leads to a unique equilibrium. We also (for simplicity) restrict the analysis to models with only two equations: the response of x to y discussed above and one relating y to the aggregated choices of x. This helps to show what forces are needed to generate multiple equilibria, and to understand how they arise 7. The important point here is that y is not in fact uniformly distributed like the prior beliefs of the agents. In Matejka (2015) and many other models in the rational inattention literature, agents prior beliefs about the distribution of y are correct, but we explore situations where agents have priors that depart from reality. This might be particularly pertinent, for example, after structural breaks or times of turmoil. In particular, we assume a uniform ignorance prior, which implies that the agents do not understand how their decisions (and those of other agents) affect y. This is discussed in section 5.2 below. This form of the prior means that in our models y can be determined endogenously by the interaction of the aggregate response function x(y) and a second equation relating y and x. The equation to complete the system sets out how y must respond or relate to the aggregate chosen variable x: A(y) = x (9) The perfect information equivalent of the problem in section 3 can be written as: max U(y, x) (10) x This has first order condition: U x (y, x) = 0 (11) The solution to this is optimal x as a function of y. Write this function as ˆx(y). For the perfect information model to have a unique equilibrium it must therefore be that there is a unique y for which: A(y) = ˆx(y) (12) In contrast, equilibrium under rational inattention is defined by: A(y) = x(y) (13) Where x(y) is from the optimal solution to equations 1-4 and is defined in equation 8. We will consider cases where the perfect information first order 7 The DSGE model presented in section 6 shows that the method is not restricted to two equation systems. 10

11 condition is monotonically increasing 8. As discussed above, under rational inattention there are parts of the range of y in which x responds sharply to small movements in y, and other parts where x is shallow relative to the perfect information first order condition ˆx. It is this that will generate multiple equilibria for some models. When the optimal x is increasing with y, it will be the case that x (y) 0. If x(y) were to decrease with y it must be that the increase in y caused some agents to switch from one level of x to a lower level, even though the signals they get will be more convincingly informing them that y is high. This never occurs. The response function can be horizontal however, if a change in y causes no agent to switch from one level of x to another. This will often happen when y is at the extreme high (low) end of its range, as then all agents will be choosing the highest (lowest) level of x for all values of y around that extreme 9. With an aggregate response function x(y) that is always upward sloping, there are two cases in which rational inattention delivers multiple equilibria but perfect information does not. These are shown graphically below aggregate x y Figure 3: Aggregate x in the example model in section 4.1 with κ = 0 (magenta), κ = 0.5 (blue) and in the unconstrained case (red), with two possible forms for equation 9 in black 8 The same logic applies when the first order condition is monotonically decreasing. 9 When the perfect information first order condition is downward sloping in y the aggregate response function under rational inattention is always (weakly) downward sloping. 11

12 Equilibria occur where the relevant black line meets the relevant coloured line (red for perfect information, blue for rational inattention, magenta for zero information processing). Consider first the case of some information processing capacity (the blue aggregate response curve). In the first case (the lower black line), the slope of equation 9 is steeper than x(y) when the two lines first meet, but shallower than the maximum slope of x(y), and so the two lines cross again. The second case is the reverse of this. The first case could also have a third equilibrium, and in such instances equation 9 could also cross the perfect information first order condition from below, rather than from above as in the diagram above. This is the situation shown in section 5.1. Note that in both cases there is a unique equilibrium when there is no information processing, as then the prior pins down the choice of x made by every agent for all y. As information processing capacity κ increases, we see multiple equilibria. As κ rises further, the rational inattention aggregate response curve approaches the perfect information first order condition (see figure 2), and so we return again to a unique equilibrium. The crucial element common to these cases is that at least for some of the range of y between the equilibria, equation 9 must have slope between the maximum and minimum slopes of x(y), that is it must be strictly upward sloping for at least some of the relevant range of y. This result rules out a large class of models in which an increase in y is associated with an increase in x from one side of the model, but with a decrease in x from the other side. A simple partial equilibrium model of supply and demand, for example, would not give multiple equilibria whether it was consumers or firms assumed to be rationally inattentive, as the supply and demand curves do not have first derivatives of the same sign (for most goods). 5.1 Example The intuition behind this result that rational inattention can drive multiple equilibria can best be explained by continuing the simple example from section 4.1 in the κ = 0.5 case, where the agents choose from only two values of x. Add to this example a linear response of y to aggregate x: Ay + C = x (14) This is shown below for A = 0.7 and C = 0.1. Equilibria occur when this line crosses the aggregate response of x to y. There is a unique equilibrium in the perfect information case and three equilibria in the rational inattention case. 12

13 aggregate x y Figure 4: Aggregate x in the example model in section 4.1 with κ = 0.5 (blue) and in the unconstrained case (red), with equation 9 in black Under perfect information, there is only one (x, y) pair such that equations 14 and 7 (the response of x to y in perfect information) are both satisfied, (y = 0.5, x = 0.25). A y above this would lead to a higher x from the agent first order condition 7, but x would not increase by enough relative to y for equation 14 to be satisfied, and so the equilibrium is unique. The pair (y = 0.5, x = 0.25) is also an equilibrium in the rational inattention case 10. The only difference from the perfect information case here is that instead of all agents choosing x = 0.25, half choose x = and half choose x = The major differences from the perfect information model come when y increases a little from 0.5. Under perfect information, all agents respond the same way, by increasing their choice of x a little. Under rational inattention, however, the rise in y leads to a large number of agents switching from x = to x = This means that aggregate x rises sufficiently that there is another equilibrium at around (y = 0.64, x = 0.35) in which almost all agents choose the high level of x. In effect, agents are using their limited information processing capacity to decide if they face a high or low value of y; as y moves a little above 0.5 the majority of agents decide on high, and consume accordingly, whereas if they knew y more precisely they would choose x based on only a somewhat high y. A similar process gives rise to the equilibrium below y = This does not have to be the case, it just happens to be so for this example. 13

14 5.2 Discussion of the uniform prior In the example above the prior belief is uniform over the whole possible range of the variable to be estimated (y). This is the assumption of an ignorance prior, that the agent has no information at all about where y might lie before they begin taking on and processing information from signals available to them. This is consistent with the premise of the problem, that agents are not equipped with huge amounts of information about their environment when they make decisions. The idea is not without precedent; it has been suggested as an explanation in experimental decision making settings that agents start with a uniform prior and then update from that (e.g. Fox and Clemen, 2005) 11.. This is precisely what is happening here, except that the updating is rational given a signal and a processing constraint, rather than due to some behavioural heuristic. This is an important assumption for the results presented in this paper. As noted by Matejka (2015), rational inattention problems with Gaussian priors (such as Mackowiak and Wiederholt, 2009) have very different solutions. The discreteness result, which is the key mechanism underpinning the models in this paper, does however hold for general bounded prior distributions. An example of this is shown in appendix A. The models we study model the unknown variable y as endogenous to the aggregate x chosen by the agents facing the problem. Ignorance here therefore refers to the assumption that while agents decide how y will affect their choice of x, they do not understand how others will do so, and/or they are uncertain about how aggregate choices will impact the variable y. If economists cannot be sure of how the economic environment they face works, it is sensible to assume households and firms face similar model uncertainty HANK model 6.1 Motivation The specific form of rational inattention used above has agents gathering information about a variable relevant to their choices, over which they (crucially) have a bounded prior belief. In DSGE models with incomplete markets for unemployment insurance and search and matching frictions, future labour market prospects matter a great deal for the household decision, and yet it is unlikely that households can process enough information to precisely estimate their own future prospects. It is also plausible that households do not understand how 11 This is an extreme assumption in many cases. It is implausible to assume households are not aware of some persistence in most macroeconomic variables, for example. Our results also hold for other prior beliefs (see appendix A). The important point is that this prior belief does not necessarily represent an underlying distribution of the variable being estimated. 12 See Sousa and Sousa (2013) for some evidence of model uncertainty in asset markets. 14

15 aggregate consumption will respond to signals about future employment, or exactly how the labour market responds to aggregate consumption, so the assumption of an ignorance prior is reasonable here. These models are therefore a prime candidate for the application of the method set out in sections 3 to 5 above. Indeed, data from the Survey of Economic Expectations (SEE) shows some preliminary support for the idea of rational inattention of this kind over future labour market outcomes. The graph below is a histogram of responses to the survey question asking how likely people thought they were to lose their job in the next twelve months. Expectations are far from continuously distributed; they clump together around certain values 13, as predicted by rational inattention with a bounded prior belief. Density Lose job Figure 5: Expectations from Survey of Economic Expectations Model The diagram below is taken from figure 1 panel 1 in Ravn and Sterk (2016) (RS). Their simple HANK model has a borrowing limit of zero, so no agent can supply bonds to the market, so in equilibrium all workers consume their wages if employed (and their unemployment benefits if unemployed). This serves to reduce the number of types to the employed, the unemployed, and the assetrich (who own firms but do not enter the labour force), and so their model is analytically tractable. 13 Manski and Straub (2000) note this as a key feature of the data as well. 15

16 Figure 6: Figure 1 panel 1 from RS The blue line PC is the Phillips Curve; it is derived from the firm price setting problem. The red line EE is the Euler equation of employed workers (the only type not necessarily at their borrowing constraint), plotted for interest rate above the zero lower bound (R > 1) and at the zero lower bound (R = 1). The intended steady state equilibrium is I, and II is therefore a standard liquidity trap. III what the authors call the unemployment trap. This diagram shows clearly that the unemployment trap only exists because the Phillips Curve is vertical at a hiring rate of η. This arises because RS assume that there is a (low) level of vacancies that can be posted at no cost to the firm; the job finding rate η is the rate if the only hiring is through this informal channel. They justify that assumption with evidence that a large number of jobs are filled without ever being advertised. This assumption is only justified by their evidence if advertising is believed to be the only cost of posting a vacancy. If we also consider that there are administrative costs to creating a new position within a firm that are separate to the costs of advertising a vacancy, the link between informal hiring and costless hiring becomes more tenuous. In the interpretation of the search and matching framework in Rogerson and Shimer (2010), vacancy costs come from employing recruiters; it is quite possible that some unadvertised hires occur because the firm employs a recruiter who already has someone suitable for the firm on their books. In this case a lack of advertising does not justify the strong assumption of a fixed endowment of free vacancies for each firm, and a jump in vacancy costs beyond that. This is a promising model for the application of rational inattention: the solid portions of the lines in their figure fit into the general form in section 5 that mean rational inattention can give multiple equilibria. They could easily be the red and (either of the) black lines in figure 3. We therefore use the RS 16

17 model almost as it is, but remove the informal vacancy assumption and add in rational inattention in deriving EE. This leads to multiple equilibria in similar locations to I and III in the diagram above, and thereby demonstrates this way of using rational inattention in a detailed economic model. 6.3 Firms The firm problem is to set prices and choose vacancies to maximise profits 14. There are quadratic adjustment costs as in Rotemberg (1982), firms are monopolistic and households have CES preferences. The labour market matching function is: M(u s, v s ) = u α s vs 1 α (15) Wages are determined in a match by Nash bargaining. We assume (as RS do in drawing their figure 1) that worker bargaining power is zero. The firm therefore captures all surplus from the match, so the wage just compensates the worker for the disutility of working and does not change with labour market tightness. This means that the wage is constant in the hiring rate η. Using these assumptions and functional forms, the Phillips curve becomes: [ ys+1 ] 1 γ + γmc s = φ(π s 1)Π s φe s Λ s,s+1 (Π s+1 1)Π s+1 (16) y s where In steady state this becomes: mc s = w s e As + k q s (1 ω)e s Λ s,s+1 k q s+1 (17) φ(1 β)(π 1)Π = 1 γ + γ(w + kη α 1 α (1 β(1 ω))) (18) φ measures the extent of price adjustment costs, γ is the elasticity of substitution between goods in the consumer s problem, k is the cost of posting a vacancy, ω is the (fixed) job separation rate, η is the hiring rate and q is the vacancy filling rate, equal to η α 1 α, where α is the elasticity of the matching function wrt job searchers. Λ s,s+1 is the discount factor of the owners of the firm, who are assumed to be risk neutral. This therefore does not vary over time. Equation 18 is identical to equation (PC) in RS, except that assuming no worker bargaining power means that real wage w is constant, not a function of η, and λ f is dropped. This is the Lagrange multiplier on the constraint that vacancies must be weakly greater than the informal (free) amount. As we have dropped the assumption that firms can post a positive number of vacancies for free, the constraint is simply that vacancies cannot be negative. We will only study the region of the PC where vacancies (and so hiring) are strictly positive, so λ f is always zero in this version of the model. 14 This part of the model is identical to that in RS. Their paper sets out the firm problem in detail. 17

18 6.4 Households RS express the employed household s problem as: max V e s = subject to ( c 1 µ es 1 1 µ ζ ) + βω(1 η s+1 )V u s+1 + β(1 ω(1 η s+1 ))V e s+1 (19) P s c es + b s+1 R s = W + b s (20) b s+1 0 (21) RS note that the no-borrowing constraint is never binding for these households, so we can drop constraint 21. We express nominal wage W as the (fixed) real wage multiplied by the price level wp s, and substitute constraint 20 into the objective function. The problem is then to choose savings b s+1 to maximise: 1 [ w + b s b s+1 1 µ P s ] 1 µ 1 R s P s 1 µ ζ + βω(1 η s+1) 1 µ + β(1 ω(1 η s+1)) 1 µ [ ϑ + b s+1 P s+1 b s+2 ] 1 µ R s+1 P s+1 [ w + b s+1 b ] 1 µ s+2 P s+1 R s+1 P s+1 (22) Terms in V i s+2 have been omitted as they are not relevant for the decision of b s+1. The first order condition under perfect information is as in RS, so we have changed nothing so far: c µ es = βe R ( ) s ω(1 η s+1 )c µ u,s+1 Π + (1 ω(1 η s+1))c µ e,s+1 (23) s+1 Unemployed households are always at their borrowing constraint, and so their problem never matters for equilibrium determination 15. To get the steady state Euler equation (EE) Ravn and Sterk note that c es = w and c us = ϑ 16 due to the no-borrowing constraint, and they substitute for an interest rate rule: R s = max{ R Π δπ Π δπ η δ θ 1 α, 1} (24) As they do in drawing their figure 1, we will assume that the interest rate responds only to inflation, that is that δ θ = 0. This gives: Here R = 1 = β max{r Π δπ, 1} [ ( ϑ ) µ ] ω(1 η) + 1 ω(1 η) Π w δπ R Π 15 See RS for a detailed explanation. 16 ϑ is the unemployment benefit, or payoff to home production. (25) 18

19 6.5 Perfect Information Steady State This is the same as in RS, except that we are only considering the case in which R > 1, and we have removed the possibility of firms hiring at zero cost up to a certain level. The Phillips Curve (PC) and Euler Equation for employed households (EE) are shown below 17. The y axis is steady state inflation and the x axis is the steady state hiring rate (these are example parameters, the model has not been calibrated). Figure 7: Steady State relations under perfect information Without the forced non-linearity in the Phillips Curve from informal hiring there is only one steady state. 6.6 Rational Inattention This is more complicated than the simple cases analysed in section 5. In those problems, the agent chooses x while uncertain about a variable y, which is in turn directly linked back to aggregate x. Here, the employed household decides consumption while uncertain about the future hiring rate, but this decision also depends on future inflation, which is the variable linked to the hiring rate through the Phillips Curve. 17 Appendix B has the parameter values under which this is drawn. 19

20 The employed household problem with RI is as follows: f = arg max E[Vs e (η s+1, b s+1 )] ˆf = arg max ˆf η s+1 b s+1 V e s (η s+1, b s+1 ) ˆf(η s+1, b s+1 )dη s+1 db s+1 (26) subject to x ˆf(η s+1, b s+1 )dx = g(η s+1 ) η s+1 (27) ˆf(η s+1, b s+1 ) 0 η s+1, b s+1 (28) H[g(η s+1 )] E bs+1 H[ ˆf(η s+1 b s+1 )] κ (29) The function H[.] is the entropy of the distribution over which it operates. It is defined in equation 5 above. These constraints are explained in section 3. The major problem in solving this household problem is that it must be done numerically, which means choosing values for every element in Vs e aside from b s+1 and η s+1 (Vs e is set out in equation 22). We therefore assume that agents know all relevant parameters in the problem. They can observe current interest rates and prices. Here we consider the problem in steady state, so we assume agents make their decisions expecting inflation to remain constant. This implies (through the interest rate rule in equation 24) a constant interest rate. Households either know this or simply expect interest rates to remain constant as they do with inflation. We normalise current prices P s to 1, so current inflation enters the problem through R s and assuming constant inflation pins down expectations of P s+1. We run this problem many times, for a variety of possible inflation rates. This leaves b s and b s+2 left to pin down. In RS, the borrowing limit is zero. We relax that here, as at any point in time some households will think the future hiring rate is high and some will think it is low, because they have received different signals about the future. This means that different households would like to make different saving and borrowing decisions. This cannot happen in the RS model. We therefore allow employed households to save and borrow, but maintain the assumption that unemployed households cannot borrow, and so the unemployed households remain irrelevant for determining equilibrium. A household that remains employed for multiple periods will not therefore necessarily have b s = 0, and the optimal savings decision changes dependent on this. The model therefore generates a distribution of wealth b s in any given period (and for equilibrium the net asset position of the population of employed households must be zero). Newly unemployed households may therefore have some savings in some cases, but their Euler equation is still unimportant for the determination of steady state because they are still at their borrowing constraint for all inflation rates and hiring rates consistent with steady state and equilibrium, as in the RS 20

21 model 18. If a household employed in period s becomes unemployed in period (s + 1), they will therefore have b s+2 = 0. That just leaves b s+2 in the case where the agent is employed in period (s + 1). We assume that agents assume b e s+2 = b s, that is that their wealth will remain constant over time. This can be thought of as another manifestation of the agents ignorance about the future. The solution to the household problem under these assumptions for Π = 1.02 and b s = 0 is plotted below b(s+1) 0 eta(s+1) Figure 8: Optimal decision rule for κ = 0.5 This shows how the savings choices of employed households vary with η s+1. This decision rule varies with the household wealth b s. The more savings a household enters a period with (higher b s ), the greater the marginal benefit to saving further (in b s+1 ) as there are diminishing marginal returns to consumption in each period. We need to add one additional condition for steady state not required in the RS model, that the wealth distribution must be stable. The figure above shows that a household with zero wealth in period s will almost certainly choose to borrow if η s+1 is high, and so at Π = 1.02 and high η s+1 a steady state cannot have households with zero wealth, as the wealth distribution would not therefore be in steady state. The perfect information graph in RS is still valid under 18 That is, no unemployed household will choose to save if they have any of the saving levels chosen by employed households at any point along the black dashed line in figure 10 below, and inflation and the hiring rate are at the levels associated with that point on the line. There may be other steady states where the newly unemployed choose to save a little in anticipation of remaining unemployed, but they are ignored here. 21

22 this condition, as there all households choose zero savings and have zero wealth everywhere along the EE curve, so the wealth distribution is trivially in steady state. For each (Π, η) pair we start with an even distribution of wealth 19 and update this based on the household savings decisions made under that Π and η. We iterate this process until we find the stable wealth distribution for that (Π, η) pair. The stable distributions are robust to different starting distributions. We constrain wealth to the range [ 0.006, 0.006]. This means that no agent starting with b s = 0 (for sensible inflation rates) hits the constraints for any η. For each level of inflation examined, the net asset position of employed households in the stable wealth distribution is monotonically decreasing in the hiring rate. A higher hiring rate implies a lower probability of being unemployed in the next period, so a lesser incentive to save for all employed households. For each steady state inflation rate, we therefore find a single steady state hiring rate consistent with asset market equilibrium. The results of this are shown in figure 9. This approach nests that of Ravn and Sterk, as under perfect information all employed households make the same savings decision if they have the same starting wealth, so the situation where all have wealth zero and (η, Π) are such that they continue to do so is an equilibrium and a steady state. Their original EE is shown in red, and the EE under rational inattention is in black. A greater hiring rate makes more of the households select the lower savings rate in their menu, and a higher inflation rate encourages more saving. This is why both the perfect information and rational inattention Euler equations are upward sloping: for net savings to be zero, a higher hiring rate must be offset by higher inflation. The wave shape of the EE curve under rational inattention arises for the same reasons discussed in section 4, because households do not smoothly adjust their savings in response to the hiring rate as is the case in perfect information: at extreme η a small change in η does not change household decisions very much, and so the change in inflation required to maintain equilibrium and steady state is small. The reverse is true for η in the middle of its range: savings respond more to η in that range under rational inattention than under perfect information, and so the EE curve is steep there. The logistic shape of the probabilities in the optimal decision rule (see figure 8) therefore drives the multiplicity result. 19 That is, there are equal numbers of households on each wealth level across 251 possible levels. 22

23 Inflation Hiring Rate Figure 9: Steady state relations under perfect information and rational inattention (κ = 0.5) Under perfect information there is a unique steady state, with η = and Π = With rational inattention there are two steady states, a high employment steady state with η = , Π = and a low employment steady state with η = , Π = A Policy Question Laurence Ball (2014), among others, has argued for a higher inflation target in the US and other developed economies. His principal argument is that a higher inflation target implies higher long run nominal interest rates, and so helps to keep interest rates above their zero lower bound, and that inflation of 4% is not really much more costly than of 2%. Ball claims that the main costs of a higher inflation target come from inefficiencies caused by higher long run inflation, such as from distortions in cash holdings and the tax system 20. He also considers the risk that inflation would be more volatile under a higher target. In the standard New Keynesian model with homogeneous agents, the steady state of the economy is independent of the inflation target, so Ball gives no thought to the question of whether raising the inflation target might alter the long run level of real activity. That is not true in our model or in the RS model, and analysing this demonstrates the importance of distinguishing between the two ways of generating multiple steady states in the model. 20 See Mishkin (2011) for a more detailed list of this kind of cost 23