Coordination of Expectations and the Informational Role of Policy

Transcription

1 Coordination of Expectations and the Informational Role of Policy Yang K. Lu y and Ernesto Pasten z First version: September, 26 This version: May 3, 28 Abstract This paper studies a novel, dynamic "informational role" of policy in economies with coordination failures, imperfect information and shocks. Imperfect information delivers episodes when a given equilibrium is persistently selected; and shocks deliver transitions when they occasionally reveal information about fundamentals. Policy a ects the rate at which information is revealed, so a government without information advantage can still improve coordination, manipulating current and future equilibrium selections. An application to optimal taxation shows that the informational role is irrelevant immediately after transitions, but suggests permanent small tax reductions to extend e cient episodes and severe transitory tax cuts to shorten ine cient ones. JEL codes: D8, E6, H2, H3 Keywords: optimal policy, taxation, coordination failures, learning, endogenous uctuations This paper has previously circulated under the title "Coordinating Expectations through the Informational Role of Taxation". We thank Bob King for his constant support and guidance, as well as insightful comments by Elias Albagli, Marios Angeletos, Ricardo Caballero, Christophe Chamley, Eduardo M.R.A. Engel, Simon Gilchrist, Francois Gourio, Peter Ireland, Larry Kotliko, Denny Lie, Bart Lipman, Franck Portier, Hyun Song Shin, Zheng Song, Jean Tirole, Adrien Verdelhan, and seminar participants at BU, BU-BC Green Line Meeting (Dec. 27), IDEI-TSE (Toulouse), the 27 LACEA Meeting (Bogota) and Banco Central de Chile. Both authors also gratefully acknowledge the 26 and 27 BU-Econ Research Summer Awards. Errors and omissions are exclusively ours. y yanglu@bu.edu, Boston University. 27 Bay State Road, Boston, MA z epasten@bu.edu, IDEI-TSE and Banco Central de Chile. Manufacture des Tabacs, Aile Jean-Jacques La ont, 21 allée de Brienne, 31 Toulouse, France.

2 1 Introduction Many deviations from the plain-vanilla Neoclassical model generate strategic complementarity the dependence of individuals payo s on the aggregation of decisions simultaneously made by others. In these situations, multiple rational expectations equilibria may exist, some of them ine - cient, which have been called as "coordination failures." 1 In dynamic economies, such coordination failures take the form of ine cient dynamic patterns when information about underlying fundamentals is imperfect and dispersed among agents, and shocks to those fundamentals occur. The aggregate dynamics follow a process marked by stable regimes and switches when sudden and substantial aggregate adjustments take place. Regimes arise when expectations sustain the selection of the same equilibrium for some time; and switches are triggered when the accumulation of shocks occasionally reveals public information about fundamentals, generating synchronized revisions of expectations. 2 We show in this paper that, in these environments, a government that possesses no control or information superiority about fundamentals can still improve coordination by making use of the "informational role" of its policy. We focus speci cally on taxation policy, but we argue that our results can be extrapolated to other policy instruments which, also unconnected to fundamentals, asymmetrically change agents payo s structure across their potential actions. The informational role" of policy refers to its ability to a ect the probability that any particular equilibrium is selected, and through the information conveyed by the history of past outcomes, its ability to a ect future equilibrium selections. Speci cally, this role takes action through three channels. First, policy a ects the magnitude of shocks necessary to trigger a transition between regimes. Hence, policy can either postpone ine cient transitions or facilitate desirable ones. Second, it extracts information about aggregate conditions, which may be used to manipulate agents learning, to improve the e ciency of future policy, or to conduct experiments. Thus, policy is also a tool for social learning. Finally, a policy design contingent upon which equilibrium takes place may be e ective in particular situations to rule out undesirable outcomes. To the best of our knowledge, this role for policy has remained unexplored. However, it deserves close attention from both empirical and theoretical points of view. Empirically, the dynamic pattern of these economies is an appealing representation of many environments, for instance, for the cyclicality of output in the U.S. post-wwii data, Hamilton [34] suggests. 3 In addition, Cochrane [2] reported di culties in attributing the bulk of aggregate uctuations to any single type of shock. His conclusions present a straightforward interpretation of this class of economies, where it 1 Some examples are: Diamond [25] in economies with trading frictions, Bryant [14] when production is specialized and there is imperfect information, Weitzman [55] when production has increasing returns, Cooper and John [21] in a generalization of neo-keynesian motivations, Kiyotaki [39] when investment has xed costs and there is imperfect competition, and for a summary, Cooper [22]. 2 Papers in this line are Azariadis [7], Woodford [56], Howitt and McAfee [38], Chamley [18], Frankel and Pauzner [29] and Caballero, Farhi and Hammour [15]. 3 Other examples span real investment and pricing rigidity along the business cycle, cycles of technology innovation, cross-sectional co-movements, nancial investment and asset prices, international capital ows, current account crises, and speculative attacks against an exchange rate peg. 2

3 is not one single shock but the accumulation of shocks (eventually of di erent natures) what drives uctuations. Thus, as a corollary, policy shocks have non-linear responses, sometimes negligible and sometimes large, which motivate a non-trivial and strategic optimal policy design. From a theoretical perspective, this optimal design di ers from standard recommendations. It has a corrective motive, which stands in contrast to the neoclassical idea that policy should minimize its intertemporal e ects. It also stresses that policy should be passive after transitions but aggressive in anticipation to some of those transitions. These features depart from the emphasis of the neo- Keynesian theory to the importance of smoothing uctuations with also smooth policies. As a vehicle to convey these ideas, we focus on optimal taxation in a labor economy in an approach inspired in Chamley [18]. Although this model has no quantitative ambition, it presents a transparent mechanism with which to generate unique equilibrium dynamics with regimes of high and low labor participation driven by expectations. It also abstracts from state variables like capital to isolate our policy study from other already studied ways of in uencing expectations. In this economy, agents live for only one period, but the history of aggregate activity and taxes is observable for them. Each agent is endowed with one indivisible unit of labor, which, if put to work, generates a payo that depends on the aggregate mass of participants. This complementarity is motivated as an exogenous production externality. If they decide to work, they also need to pay a xed cost, which is assumed to be heterogeneous among agents. However, there is a relevant mass of agents with cost of similar magnitude, called "the cluster." The relative position of the cluster in the distribution of costs is the key fundamental in this economy, which su ers small random shocks. If the fundamental is observable, the economy is essentially static, and while some regions on the cluster s location deliver one equilibrium, in general it has two self-ful lling equilibria: one in which the cluster participates and one in which they do not. In contrast, if agents cannot observe the fundamental, an intertemporal link is established wherein the information conveyed by past equilibrium realizations is heavily weighted when agents form expectations about the current equilibrium realization. Thus, in general, any given equilibrium is persistently selected, unless shocks have accumulated such that the cluster enters into the region in which the alternative equilibrium is dominant. This event conveys precise information about the state of the fundamental, triggering a synchronized revision of expectations and generating a switch to a new phase where the alternative equilibrium prevails. We introduce a benevolent government to this economy which nances the provision of a public good with a proportional tax levied from labor income (no debt allowed). Like individuals, the government cannot observe the cluster s costs but it can observe the history of aggregate activity. When the tax changes the reward of participation, it also controls the size of regions on the fundamental that generate transitions. Thus, as anticipated above, taxes have an informational role by a ecting both the transition probabilities and the information revealed either if there is a transition or if there is not. The dynamics of this economy are ine cient: there exist phases with low participation ("pessimism") despite the fact that equilibrium with high participation ("optimism") 3

4 can in general be supported with the same fundamentals. Thus, this informational role should be optimally used to eliminate or shorten episodes with low participation. To illustrate the solution of the optimal taxation problem, we rst assume that policy has no e ect on the current realization of the equilibrium, only on which equilibrium takes place in the next period. This arti cial assumption allows us to represent the problem as the maximization of the expected present value of welfare, where the aggregate dynamics are governed by two Markov regimes, and transition probability is partially controlled by policy. The authority must balance its short run objective, such as the provision of public good and participation in each regime, with a variety of dynamic e ects of taxation. For instance, a tax cut extends the expected duration of the optimism, but it also releases information that extends the expected duration of the future regime with low participation. Similarly, policy shortening bad times also shortens expected future good times, with the additional e ect that if such policy fails it depresses expectations in the current regime even further. These features generate a rich policy problem depending on beliefs about the fundamental that has no closed form solution. However, the simplicity of the economy allows us to analytically derive its general properties and to numerically simulate its optimal time path. From this restricted analysis we obtain a variety of conclusions. First, due to the complementarity, policy has an enhanced e ect on the aggregate participation in both regimes. Hence, the optimal tax rate should be lower than in an identical economy lacking this feature. In addition, the informational role implies a counter-cyclical design of policy with non-trivial timing and asymmetric design across regimes. At the onset of the "pessimistic" regime, policy has no power on expectations to pull the economy back to the "optimistic" phase, but when there is higher probability that fundamentals have improved, large and transitory positive policy shocks are optimal to break the aggregate pessimism. Conversely, at the onset of the "optimistic" phase there is no need to ensure the good standing of expectations via policy, but later on a sequence of small and permanent positive policy shocks maximizes ex-ante welfare. When we allow policy to a ect the equilibrium realization in the current period, in general it makes no di erence, since, despite the fact that policy has the potential to rule out the pessimistic equilibrium, such policy implies an unfeasible subsidy. However, further improvements may be introduced using a contingent scheme in particular situations. The optimal policy arising from its informational role dictates that the tax rate at the onset of the optimistic regime should be higher than the rate levied at the end of the previous pessimistic regime. Thus, the increase of taxes may generate equilibrium multiplicity, and thereby hastening the start of a new optimistic regime. One way to x such problem is to consider a lower rate if the low activity equilibrium takes place. This contingent design rules out the multiplicity, and therefore the higher tax rate derived in our previous analysis can be safely levied in the unique equilibrium with high activity. This paper makes contact with several strands of research. The attention to policy in early models of coordination failures focuses on the elimination of the Pareto inferior outcomes, which seems implausible in practice. Beginning with the research of Morris and Shin [43], global games and 4

5 equilibrium selection has enriched the understanding of speculative attacks, as it has also spread out to other topics. 4 However, these models remain essentially static. As shown by Chamley [18] and Frankel and Pauzner [29], a similar logic to global games may be applied to dynamic economies, in which if, for some reason, the history of individuals information is irrelevant, then there is unique equilibrium dynamics with the features presented above. Angeletos, Hellwig and Pavan [3] relax this condition, recovering multiplicity but still with the same general properties. In an independent branch, Woodford [57] has also used heterogeneity and strategic complementarity to justify pricing frictions, where the attention has been devoted to the implications of transparency of monetary policy on welfare. 5 Policy has in our study several positive and negative e ects on expected welfare, but in contrast to that literature, we focus on environments where the government has no information superiority about fundamentals. Thus, our focus is directed toward the impact of policy on the precision of other public signals, not upon if policy is a public signal itself. A natural benchmark for our approach is the literature of taxation in the Neoclassical economy. In regard to the study of the interplay of taxes and strategic complementarity, our only direct predecessor is Ennis and Keister [27], who show the ampli ed e ect of taxes on ex-ante welfare in a static economy with trading frictions. We study the dynamic e ects of policy in a simple and exible framework applied to production externalities, but this approach can be extrapolated to other applications. Finally, our paper is also related to the study of policy under model uncertainty. Many studies, such as those of Svensson and Williams [53], see this problem as an exogenous regimeswitching process, where the government uses its policy as a learning tool. The dynamics in our economy are also characterized by a regime-switching process, and experimentation is likewise presented as a potential use for policy. However, the endogeneity of transitions to policy makes its design with this sole motive suboptimal. This feature simpli es the informational role that taxes have, where learning is in some sense a by-product of a much more important goal: the coordination of expectations to shorten episodes with ine cient outcomes. The rest of the paper is organized as follows. Section 2 displays the economy inspired by Chamley [18], while Section 3 introduces the government, including a brief discussion about policy instruments. Section 4 presents the complete problem and proposes general properties of its solution. Section 5 uses a numerical example to examine the time series of an equilibrium policy and its e ects on welfare. Section 6 includes our conclusions, and the appendices contain proofs for the main results, the algorithm for the numerical solution, and gures used in the paper. 2 The Economy without a government In this section we describe the economy of labor participation inspired in Chamley [18] that serves as a laboratory for our policy analysis. The key elements of this economy are: i) strategic comple- 4 The equilibrium selection mechanism has been proposed by Carlson and van Damme [16]. Other applications are: Bannier [1], Goldstain and Pauzner ([31], [32]), Morris and Guimaraes [48], Shea [51] and Morris and Shin [46]. 5 See Hellwig [37], Morris and Shin [45], Svensson [52], Angeletos and Pavan [4] and Amador and Weill [1]. 5

6 mentarity in individuals decisions; ii) incomplete and dispersed information about "fundamentals", which is a set of variables that summarizes aggregate conditions; and iii) small shocks to fundamentals. The result is an economy with multiplicity of equilibria when information is complete, but with equilibrium uniqueness when information is incomplete and dispersed. When shocks are also introduced, the economy exhibits unique dynamics, characterized by a regime-switching process with stable episodes of low and high labor participation (called "pessimistic" and "optimistic" "regimes" or "phases", or simply regimes L and H), and sudden transitions between them with stochastic timing (called "transitions" or "switches.") The study of this economy is introduced sequentially. After describing the basic setting, we show the existence of two Pareto-ranked stable rational expectations equilibria (low and high participation) if fundamentals are perfectly observable. Later we introduce uncertainty on fundamentals to show the equilibrium selection mechanism, and the ways in which its interaction with shocks deliver regimes and transitions. 2.1 Setting Players. The economy is populated at every period t by a continuum of risk neutral agents with measure one, indexed by i 2 [; 1]. Each of them lives one period, owns one indivisible unit of labor, and simultaneously takes a binary decision: participating in a productive activity or not. Payo s. reward and a xed cost. If a given agent decides to work, her payo is the di erence between her participation If she decides to not work, she receives utility from leisure or home production, which is normalized to zero. Hence, the utility function takes the form U (a; A t ; c) = ( m(a t ) c if a = 1 if a = ; (1) where a denotes the binary decision of working or not. The participation reward is denoted by m(a t ), which depends on the mass of agents A t who also decide to work (also called along the paper as "aggregate participation" or just "participation".) The dependence of m(a t ) on A t is aimed to introducing strategic complementarity in agents decisions, so m >. Since our goal is the study of policy, this complementarity is motivated using a simple exogenous production externality, but this speci cation may be interpreted as the reduced form of more involved mechanisms. 6 We assume that this externality enters linearly in m(a t ), which, as we show below, allows for a simple closed form solution for the equilibrium set, m(a t ) = " + (1 & ") A t ; 6 It can be endogenously obtained, for example, as the result of trading frictions (Diamond [25]), imperfect competition and xed costs (Kiyotaki [39]) or nancially constrained rms and xed costs (Caballero, Fahri and Hammour [15]). It can also have an exogenous motivation, for instance, a productive sector with increasing returns to scale (Weitzman [55]) or a multisector economy with scope externalities (Beaudry and Portier [12]). 6

7 where "; & are arbitrary small positive numbers ensuring that m(a t ) 2 (; 1). The xed cost of participation c has a direct interpretation as disutility of working, including the opportunity cost from an alterative activity. More generally, it may also include a "menu cost", or production costs associated with an exogenous technology separable of the complementarity. Heterogeneity. [; 1], represented by, Agents are heterogeneous in their participation costs c, spanning in the range ( + for c 2 [ t ; t + ] f (c j t ) = ; (2) otherwise which is composed by two uniform distributions. There is a density of agents with cost c 2 [; 1], but there exists a relevant mass of agents with cost inside a smaller range, c 2 [ t ; t + ], with an additional density. This concentration of agents is called "the cluster", and its relative position in the support of costs depends on the parameter t. This parameter is interpreted as the key "fundamental" in this economy since it summarizes aggregate conditions. We assume that t 2 [; 1 ] to ensure that the cluster is fully contained in the support for costs [; 1], and restrict parameters and to satisfy F (1 j t ) = 1 8 t, i.e. = 1. The costs heterogeneity plays two key roles in the model: it is part of the argument to motivate equilibrium multiplicity when information is complete ( t observable), and when information is incomplete ( t unobservable), it introduces the dispersion of information needed to obtain equilibrium uniqueness. Shocks to fundamentals. The next modelling ingredient are shocks that shape an stochastic process for t. To specify this process, we rst assume that its support [; 1 ] is broken into an arbitrarily ne discrete and evenly distributed grid fw k g K k=1, such that w k = (k 1), with k = 1; :::; K, w 1 =, w K = 1, and denoting the length of each "step" in the grid, = 1 K 1. Thus, t follows a similar process to a random walk, but with two twists: it only takes value on the grid fw k g K k=1 and it can move only "one step a time" with probability p < 1 3. The rst assumption ensures that t is restricted in its support. The second assumption imposes that Pr [ t+1 = w k 1 j t = w k ] = p if k > 1; Pr [ t+1 = w k+1 j t = w k ] = p if k < K: When t is not located in its boundaries, there is a probability p that t+1 is located either in the immediate upper or in the immediate lower position of t in the grid. Thus, the probability that t+1 = t is 1 2p. But when t reaches the upper (lower) boundary, there is only probability p that t+1 is located in the immediate lower (upper) position, and a probability 1 p that t+1 = t. This process can be alternatively represented as the vector % t = [% 1t ; :::; % Kt ], such that each of 7

8 its elements is de ned as % kt = Pr ( t = w k ) with k = 1; :::; K, following a motion described by % t+1 = Q% t (3) where Q = 1 p p p 1 2p p p 1 2p p p 1 p 1 C A (4) There are two features that this speci cation for shocks is aim to capture. First, aggregate shocks hit by de nition to most agents in the economy (the cluster), but speci c individual characteristics generate heterogeneous e ect on their payo s. For example, an oil price shock has an heterogeneous pass through to cost across rms because of heterogeneity in productive technologies. And second, shocks are intrinsically small (the grid in t is arbitrary ne and it has a "one step a time" random process.) These two features re ect the main idea behind the study of strategic complementarity in macroeconomics: because there its a relevant mass of agents with a similar optimal reaction function, but its position in the distribution of reaction functions is unknown, small shocks accumulate their e ects on expectations (delivering regimes) to generate occasional large real adjustments (delivering transitions.) Information structure. Unless otherwise speci ed, all agents can observe their own realization of cost, and because their participation decisions are simultaneous, they cannot observe others costs and decisions. However, they have access to public information = t, which contains the history of past participation rates, so = t = A t 1 ; = t 1. Then, the information set of an agent living at t with participation cost c is = e t = fc; = t g. 2.2 Equilibrium when the fundamental is observable In this section, we obtain the equilibrium set assuming that the fundamental t is perfectly observable, so the relevant public information is = t = f t g. Under this assumption, this economy is reduced to an in nite sequence of static problems, so this section abstracts from time subindex. Linking our results to Cooper and John [21], we show that the participation reward function (1) and the heterogeneity (2) are the key features in the model to motivate multiplicity of rational expectations equilibria. We rst start de ning equilibrium under complete information. De nition 1 When the fundamental is observable, the equilibrium is a set () of cuto strategies c (), such that an agent optimally decides to participate if its cost c c (), so a equilibrium 8

9 cuto satis es the indi erence condition for the marginal participant: U (1; A () ; c ) = U (; A () ; c ) ; for a given equilibrium participation rate A (), which is de ned by: A () = c Z() df (c j ) : This equilibrium implies a xed point problem between the cuto strategy c () and the participation rate A (). To compute the equilibrium set, we rst use the heterogeneity in cost speci ed in (2) to de ne the mass of participants A (c; ) for an arbitrary cuto c 2 [; 1] and for a given, A (c; ) = Z c 8 >< c for c 2 [; ] df (c j ) = c + (c ) for c 2 (; + ) >: + c for c 2 [ + ; 1] (5) The next step is to obtain the participation reward for an agent with cost ec as a function of A (c; ), which is U (1; A (c; ) ; ec) = m(a (c; )) ec. Since the utility of not working has been normalized to zero, the equilibrium cuto must satisfy U (1; A (c ; ) ; c ) =, m(a (c ; )) = " + (1 & ") A (c ; ) = c (6) Combining (5) and (6), we obtain c () and the equilibrium mass of participants A () = A (c () ; ). Parameters " and & in the participation reward m (A t ) rule out sunspot equilibria with full or zero participation. 7 If the concentration of agents in the cluster is such that + > 1, there is multiplicity of equilibria for some realizations of, with c L = 8 >< fc L g if > c H () = fc >: L ; c H g if 2 [c L ; c H ] ; (7) fc H g if < c L " 1 (1 & ") and " + (1 & ") c H = 1 (1 & ") : Figure 1 shows the function m(a (c; )) for c 2 [; 1] and a 45 o line representing the right hand side of (6), such that an equilibrium c is obtained in their intersection. Given the complementarity (m > ) and the heterogeneity in costs (2), the function m(a (c; )) takes a discrete version of the 7 For any value of A, there are some agents with c < " for whom it is always worthwhile to work, so A >. Conversely, there is also a strictly positive mass of agents with c > 1 & such that not working is the dominant strategy, so A < 1. 9

10 S-shaped function required for equilibrium multiplicity (Cooper and John [21]). The equilibrium set (7) and Figure 1A show that two equilibria exists, c L and c H, when 2 [c L ; c H ], i.e. when costs for the cluster are in the center of the population distribution. The far left crossing in Figure 1A re ects one rational expectations equilibrium, in which agents expect that the cluster will not participate in the current period, and because of the externality, this expectation translates into low expected participation reward that makes optimal for the cluster to not participate. Given these expectations, even some agents with cost lower than the cluster do not participate, so participation is optimal only for those with c c L. Conversely, there is another rational expectations equilibrium in the far right crossing in Figure 1A, which re ects, for the same state of, that if agents expect that the cluster will participate, the high expected participation reward con rm that original prior. In this case, participation is optimal for some agents with cost higher than those in the cluster, in particular for those with c c H. The xed point (6) admits another solution, the middle crossing in the Figure 1A, but that solution is ignored in the analysis since it is locally unstable. 8 However, when the cluster in on the extremes of the population distribution, the equilibrium is unique. When the fundamental is "bad", > c H, as in Figure 1B, not participation is dominant for the cluster even assuming that they participate, so those expectations are not con rmed ex-post, and hence the unique equilibrium has low activity. With a similar logic, shown in Figure 1C, the unique equilibrium has high activity when fundamental is "good", meaning < c L. 2.3 Equilibrium when the fundamental is unobservable Now we study the equilibrium of our economy when the fundamental is unobservable. Agents now use their knowledge about past aggregate activity to form expectations about the fundamental, building a link between periods. Thus, we restore the use of a time index. The key distinction of this case respect to the one above is that now there is only one equilibrium at a time, in such a way that the economic dynamics are characterized by a regime-switching process alternating between "pessimism" (regime L) and "optimism" (regime H). This result is presented sequentially. First, we show that regimes arise when an initial state and conditions for transitions to happen are assumed common knowledge. Later, when we study transitions, we con rm these two assumptions: only under the assumed conditions, the accumulation of shocks fully reveals t in the state, which in turn triggers a synchronized expectations adjustment that generates a transition. Thus, a new regime starts with common knowledge. 9 The de nition of equilibrium in this case shares the same logic with De nition 1. Nevertheless, we state this notion as a new de nition to make explicit the modi cation introduced by uncertainty. 8 The slope of m(a (c; )) around that point is + > 1, so any disturbance around that solution implies that the economy converges to either of the other two solutions. In contrast, the far left and the far right solutions are stable since m(a (c; )) around them has slope < 1. 9 This argument is scketched below, but the reader is referred to Chamley [18] for formal proofs. 1

11 De nition 2 When the fundamental t is unobservable, but the set of public information = t contains the history of the past aggregate participation, = t = A t 1 ; = t 1, the equilibrium is a set ( t ; = t ) of cuto strategies c ( t ; = t ), such that an agent optimally decides to participate if its cost c c ( t ; = t ), so an equilibrium cuto satis es the indi erence condition for the marginal participant: E t nu (1; A ( t ; = t ); c t ) j e = t o n = E t U (; A ( t ; = t ); c t ) j = e o t for the information set of the marginal agent, e = t = fc t ; = t g, and for a given equilibrium participation rate A ( t,= t ), which is de ned by: A ( t ; = t ) = c (Z t;= t) df (c j t ) : To form expectations about t, agents use their information available at the beginning of each period, which has two components: the history of past equilibria = t (of public dominion), and their own participation cost c (private information.) In comparison with De nition 1, the computation of the equilibrium participation A t remains unchanged, but the indi erence condition de ning c ( t ; = t ) now holds in expected value. As the counterpart the xed point problem when t is observable in (6), E [m(a (c; t )) j = t ] is de ned as the expected participation reward for an arbitrary cuto strategy c and information set e= t = fc; = t g, so the equilibrium cuto c t solves the xed point problem, E t hm(a (c t ; t )) j = e i t = c t ; (8) with e = t = fc t ; = t g. The rest of the section is devoted to show that there is a unique solution for this problem, so ( t ; = t ) has a unique, time-varying element depending on t and the history of past participation = t. This time-variability is what mechanically drives regimes and transitions The rising of regimes We show now that a regime L ("pessimistic", with low participation) arises if the fundamental t is initially observed in a "bad" state and that this regime prevails if t c L 8t, which is common knowledge. Similarly, a regime H ("optimistic", with high participation) arises if t is initially observed in a "good" state and that this regime stands if t c H 8t, which is also common knowledge, with c L and c H de ned by (7). To precise what are "bad" or "good" initial states, we de ne two critical positions: w L, which is the highest position of t in its grid fw k g K k=1 such that c H is the only equilibrium cuto in (7); 11

12 and w H, which is the lowest position of t such that the unique equilibrium is c L. Thus, w L sup fw k : w k < c Lg ; w H inf fw k : w k > c H g : (9) Using w L and w H, we state Proposition 1, which shows uniqueness of the equilibrium cuto on the onset of a regime (t = 1) if the initial position is public information = 1, but the current position of the fundamental 1 is not observable. Proposition 1 For = 1 = f = w H g, the set of equilibrium cuto s ( 1 ; = 1 ) = fc L g. Conversely, for = 1 = f = w L g, the set of equilibrium cuto s ( 1 ; = 1 ) = fc H g. Proof. Because the argument is symmetrical, we show this proposition only for regime L, which is illustrated in Figure 2. The strategy to sketch the proof is: i) to form the left hand side of (8) using only public information; ii) to show that (8) has only one solution in that case; and iii) to show that this result is not modi ed when private information is taken into account. i) Given the process (3) for t, when agents observe = w H, they know that 1 could be located one step to the right (position w H+1, crossing once the 45 o line in Figure 2, with probability p), one step to the left (position w H 1, crossing twice, also with probability p), or in the same position (position w H, crossing once, with probability 1 2p). Thus, at t = 1, the payo function m(a (c; 1 )) for these possible location of 1 are respectively denoted in Figure 2 as m, m, and m, ( is the length of the steps.) Hence, only using public information, = 1 = f = w H g, E 1 [m(a (c; 1 )) j = 1 ] = pm(a (c; w H 1 )) + (1 2p) m(a (c; w H+1 )) + pm(a (c; w H+1 )) = pm + (1 2p)m + pm m e ii) This expected term is denoted as m e in Figure 2, which coincides with m when the candidate cuto c 2 [; w H 1 ) [ (w H+1 ; w H 1 + ) [ (w H+1 + ; 1]. Given = w H, m crosses only once the 45 o line, so the only candidate solution is c L in this range. To check if there is another solution, we concentrate on the complementary subspaces, [w H 1 ; w H+1 ] and [w H 1 + ; w H+1 + ]. As shown in Figure 2, m(a (c; 1 )) is convex around c = 1 and concave around c = 1 +. Thus, by Jensen s inequality, the expected payo m e is above m in the range [w H 1 ; w H+1 ] and below m in the range [w H 1 + ; w H+1 + ], ensuring that there is not another solution in these ranges. Therefore, using only public information = 1, the only solution is c L. iii) If agents take into account their private information (their own cost), uniqueness remains. Because the density inside the cluster is higher than outside, agents rationally assign higher probability to be part of the cluster rather than outside. Then, to compute the left hand side of (8), agents with cost laying in [w H 1 + ; w H+1 + ] assign higher probability than p to m v and lower probability than p to m v, attening even further the expected payo function in this critical range, and thus not a ecting the result of part ii). 12

13 This proposition relies on the fact that, respect to the case when t is observable, uncertainty on t decreases the e ect of aggregate participation A t on individuals expected reward m (A (c; t )). Thus, if the publicly observable shows equilibrium uniqueness at t =, the smoothness introduced by this force is strong enough to maintain uniqueness at t = 1. The next proposition extends this argument for t > 1. Proposition 2 For = 1 = f = w H g, but f j g t j=1 are unobservable, the set of equilibrium cuto s ( t ; = t ) = fc L g 8t 1 if the condition to trigger a transition t < c L is common knowledge and if it has not been violated 8j t. The number of periods where these conditions are satis ed is called a "regime L." Conversely, for = 1 = f = w L g, but f j g t j=1 are unobservable, the set of equilibrium cuto s ( t ; = t ) = fc H g 8t 1 if the condition to trigger a transition t > c H is common knowledge and if it has not been violated 8j t. The number of periods where these conditions are satis ed is called a "regime H." Proof. The sketch for this proof also focuses on regime L, uses Figure 3 and Figure 4 for illustration, and follows the same steps than the proof for Proposition 1. i) The expected return of participation for an arbitrary cuto c is P E t [m(a (c; t )) j = t ] = K k;t m(a (c; w k )); where t = [ 1t ; : : : ; Kt ] is a vector representing beliefs formed using only public information, so kt = Pr [ t = w k j = t ] for k = 1; : : : ; K. k=1 These beliefs evolve according to Bayes rule after participation is observed at the end of each period (or equivalently, at the beginning of the next), e t = 8 < : t = Qe t ; kt 1 1 P L i=1 it 1 for w k > w L ; (1) for w k w L where e t = [e 1t ; : : : ; e Kt ] are ex-post beliefs about t 1 after observing A t 1, e kt = Pr [ t 1 = w k j = t ] for k = 1; : : : ; K. Because the cluster do not participate if the equilibrium cuto is c L, and the distribution of costs outside the cluster is uniform, agents cannot infer the exact position of t 1 after observing A t 1. However, because agents know the condition that triggers a transition, if they observe the continuation of the regime, they infer that t 1 c L, so using the de nition of w L in (9), they update their beliefs in (1) knowing that Pr [ t 1 w L j = t ] =. In addition, beliefs t also consider the process for t in (3), with initial beliefs e 1H = 1 since = 1 = f = w H g. ii) Figure 3A shows the evolution of the likelihood % t implied by the process (3) for t. Figure 3B shows the evolution of beliefs t, taking into account the updating rule (1). The likelihood % t 13

14 converges to uniform [; 1], but beliefs t increasingly skew to higher positions of t, converging to lim i;t = t!1 ( 2 tan 1 2 r sin (r (i L)) for w i > w L for w i w L ; where r = 2(K L)+1. This expression is derived in the Appendix. This skewness ensures that, for any period t with t j c L 8j t, the xed point problem (8) has only one solution, ( t ; = t ) = fc Lg. Figure 4 shows a parametrized example for the evolution of the expected participation reward, with only one crossing with the 45 o line. iii) When private information is considered, Chamley [18] shows that c L remains the unique equilibrium cuto if < 2 (1 ) (Proposition 4). The intuition is similar to part iii) in Proposition 1: agents assign higher probability to be part of the cluster, attening the left hand side of (8) respect to the case with only public information, and then not altering the result of ii). These two propositions show that if the fundamental is observed at a given period t = in a state that almost generate multiplicity, the uncertainty on future states t and the evolution of beliefs implied by Bayes rule ensure the realization of the same equilibrium for all future periods as long as t remains outside of some critical regions, respectively t > w L and t < w H for regime L and regime H (with low and high participation.) The triggering of transitions Following the road map of this section, we now study transitions to con rm the conditions assumed above to motivate regimes. Proposition 3 During a regime L (c L is the unique equilibrium), a transition is triggered if t < c L, in which case the state of fundamental t is fully revealed at position = w L. Conversely, during regime H (c H is the unique equilibrium), a transition is triggered if t > c H, in which case the state of t is fully revealed at position = w H. Proof. Once again, we show this result for regime L. By de nition of the equilibrium trigger c L, agents with cost c c L decide to work during this regime. If t c L, the expectation of low participation is con rmed ex-post after the observation of the participation rate A (c L ; t), which is constant for any t c L since the complete cluster does not work. However, if t < c L, some strictly positive portion of the cluster, speci cally those agents with c 2 [ t ; c L ], decide to work even if they expect that c = c L. This unusual participation is observed at the end of period t, which together with the knowledge about the process for t (moving "one step a time"), provoke a synchronized revision on beliefs e t+1, assigning e Lt+1 = Pr [ t = w L j = t+1 ] = 1. Using propositions 1 and 2, this information generate the rising of a regime H with high participation. 14

15 Thus, the equilibrium cuto strategy is unique for any t and = t, 8 fc L >< g if t > c H, for any = t fc L ( t ; = t ) = g if t 2 [c L ; c H ] and = t = A t 1 (c L ) ; = t 1, for any = t 1 fc H >: g if t 2 [c L ; c H ] and = t = A t 1 (c H ) ; = t 1, for any = t 1 fc H g if t < c L, for any = t (11) Comparing this equilibrium set ( t ; = t ) in (11) to the equilibrium set () when t is observable in (7), dispersed information about t provides an equilibrium selection mechanism in the region [c L ; c H ]. Whether the equilibrium cuto strategy is c L or c H in each period critically depends on the observation of A t 1. In this sense, the equilibrium is Markov, because the last realization of participation summarizes all the relevant information contained in the history of participation. When shocks have accumulated such that t < c L or t > c H (respectively, for regimes L and H), precise information about t is revealed in a state where only the alternative equilibrium is dominant. Taking into account the stochastic process of t, these two features of ( t ; = t ) imply that the dynamics of this economy is unique, following a regime-switching process. Aggregate participation takes values A (c L ) or A (c H ) for some time, and when t reaches its switching threshold, there is a transition to the alternative regime. We close this section with two comments. Notice that transitions are unexpected, since skewness introduced by rational learning implies that beliefs become more "pessimistic" when time elapses in regime L, and more "optimistic" when time elapses in regime H. Also notice that the equilibrium in this economy has the potential to generate ine cient outcomes when t 2 [c L ; c H ]. This is because, in this region, a regime L is possible despite a Pareto superior regime H can be supported for the same fundamental. 3 Introduction of government The ine cient equilibrium dynamics shown above, when the fundamental is unobservable, serves as motivation for our study of policy. To start, Section 3.1 modi es the baseline economy displayed in Section 2.1 to introduce a policymaker who maximizes social welfare, levies taxes and provides a public good. Subsequently, Section 3.2 modi es the equilibrium of Section 2.2 to include taxes, showing the rst key result of the paper: policy partially controls the cuto s c L and c H. Also assuming t observable, Section 3.3 obtains a reduced form for the government s objective that is later used by Section 4 to study the informational role of policy when t is unobservable. To close this chapter, Section 3.4 informally discusses other interesting environments with similar features than our prototype economy, such that our results of Section 4 could be extrapolated to those cases. 15

16 3.1 Modi cation to the setting Players. The setting for private agents remains identical to Section 2. The government has an in nite horizon, and controls proportional taxes f t g 1 t= 2 [; 1]1 levied from the reward obtained by each participant. 1 For that, it disposes of a one-period commitment technology and use its tax revenues to nance the provision of a public good g (no debt is allowed.) Timing. Each time period is broken into three stages: 1. The government speci es taxes to be collected at the end of the current period; 2. Private agents (also called "the public") simultaneously take their decision of participation; 3. Aggregate participation is observed, taxes are collected, the public good is provided, and information sets of all players are updated. good, Payo s. Individuals utility function (1) is modi ed to take into account taxes and the public U (a; c; A t ; t ; g t ) = ( (1 t ) m(a t ) c + (g t ) if a = 1 (g t ) if a = ; (12) where () is the utility provided by the public good, such that >, <. The government maximizes the present discounted value of welfare across agents, 1X t= t 1R U (a; c; A t ; t ; g t ) df (c j t ) ; (13) where is the time discount factor. Because private agents live only one period, the government s objective is represented as the discounted sum of a sequence of "one-period welfare". Each of these terms is computed by the cross-sectional aggregation of individuals utility with measure F (c j t ). Heterogeneity and shocks. Both, the heterogeneity in cost F (c j t ) and the stochastic di usion process of t remain identical to Section 2, respectively de ned by (2) and (3). Information structure. The information set for private agents = e t also remains identical to Section 2, which is composed, unless otherwise speci ed, by their participation cost c (private information), and the history of participation = t = A t 1 ; = t 1 (public information). The government has no control on the fundamental t, and it has only access to public information = t. 1 A discussion about other policy instruments is included in the Appendix. 16

17 3.2 Equilibrium with taxes when the fundamental is observable Assuming that t is observable, the relevant public information set for private agents and the government is = t = f t g. Thus, as in Section 2.2, we abstract from the time index because the study of the equilibrium in this case can be conducted as in a static economy. Following De nition 1, the equilibrium is obtained by two equations and two unknowns: the equilibrium participation A is a function of the equilibrium cuto strategy c and the fundamental ; and the indi erence condition for the marginal participant de ning c is a function of A. The modi cation to the agents utility from (1) to (12) has no e ect on the equation for A, but it enters on the condition for c. The xed point problem re ects this modi cation, (1 ) m(a (c ; )) = c (14) which is the counterpart of (6) in the economy without government. Respect to that case, taxes decrease participation reward but does not a ect the value of not working (normalized to zero), so c is now a function of A and, but not on public spending g. This is because all agents, participants and no participants, equally enjoy the public good. In general, policy instruments asymmetrically a ecting the payo of individuals among their decisions (in this case, to work or not to work) are the type of policy that has the potential to exert an informational role. As before, the problem in (14) has also closed form solution by using the functional form of m (), the heterogeneity F (c j ) in (2), and the participation rate delivered by a given and an arbitrary cuto c, represented by A (c; ) in (5). Hence, in the relevant case when + > 1, the equilibrium set (; ) with taxes and when is observable is 8 >< fc L ()g if > c H () (; ) = fc >: L () ; c H ()g if 2 [c L () ; c H () ] ; (15) fc H ()g if < c L () with c L () = " 1 1 (1 & ") and " + (1 & ") c H () = 1 1 (1 & ") ; which keeps the same structure as () in (7), the equilibrium set when is also observable, but without taxes. However, the equilibrium (; ) in (15) now has an important property: taxes controls the equilibrium cuto s c L () and c H (). This control implies that taxes a ect aggregate participation in each equilibrium, as well as the extension of the regions in where only one equilibrium prevails. This control is also key when we study optimal policy when is unobservable in Section 4. The following proposition states some useful properties for that analysis. Proposition 4 For s = L; H, the equilibrium cuto strategies c s depend on taxes, such that: <, where the complementarity ampli es the negative e ect of taxes on c s; 17

18 <, taxes have a stronger e ect for c H than for c L ; c 2 >, taxes have an increasing e ect on cuto s; iv. When the equilibrium is unique on c H, a positive shift of taxes may generate multiplicity fc L ; c H g or uniqueness on c L ; v. When the equilibrium is unique on c L, a negative shift of taxes may generate multiplicity fc L ; c H g or uniqueness on c H. The properties stated in this proposition need no proof since they are direct results from c L () and c H () in (15). Property i: emphasizes that the e ect of taxes on cuto s is negative and ampli ed by the complementarity. This ampli cation e ect is explicitly obtained by di erentiating the xed point condition s ( t ) = m [A (c ; t )] + (1 m [A (c ; t )] = 1 (1 t @c (16) If taxes are higher, participation reward is lower, incentives to work are also lower and then the equilibrium participation cuto is lower too. This e ect is captured by the rst term m [] on the right of (16), which is negative. But, in addition, because the complementarity introduced by the productive externality in m [], lower participation feeds back on even lower participation reward, amplifying the e ect of taxes on cuto s. This feedback is captured by the second term on the right of (16), which is also @c <. If there would be =, so there would be no ampli cation e ect. Property ii: states that, because of the complementarity, the higher participation delivered by c H than by c L implies than the participation reward is also higher for c H. Hence, because taxes are proportional, they have a bolder e ect for c H than for c L. Property iii: remarks that, because of the complementarity, taxes have an increasingly negative e ect on equilibrium cuto s. Property iv: stresses that higher taxes increase the range of where the equilibrium with low participation is strictly dominant, and move the range with multiplicity to higher positions in the grid of. Finally, Property v: shows the converse e ect: lower taxes increase the range of where high participation is dominant and moves down the range with multiplicity. 3.3 A reduced form for the policy objective Because the focus of this paper is the study of the informational role of policy under uncertainty, we do not study optimal policy for the economy of Section 3.2, when t is observable. However, as a rst approximation to understand the e ect of taxes on welfare, in the following we obtain a reduced form for the policy objective (13) for this case. 18