Optimal Policy with Credibility Concerns

Optimal Policy with Credibility Concerns Abstract This paper considers a reputation model of optimal taxation in which the public is unsure about the government type. A long-lived government can be of the trustworthy type (that commits to its announced tax rate) or the opportunistic type (that retains the ability to change the tax rate after announcing it). This paper determines the optimal commitment strategy for the trustworthy government –most of prior studies treat that strategy as exogenous – and nds that allowing the committed strategy to be cho-sen optimally has signi cant consequences for the equilibrium dynamics. The optimal committed strategy is found to vary with the patience of the two government types, the initial reputation of the government, and the elasticity of household production. This formulation can thus explain di¤erences in policy responses across governments in the face of similar credibility problems. 1 1 Introduction What are the optimal policies for a trustworthy government that is able to commit but su¤ers from public skepticism concerning its ability to commit?1 This is an important question for policymakers, but game theory and macroeconomics have not yet provided a satisfactory answer. Most of the literature discussing reputation games and the macroeconomic literature on reputation focus on the optimal strategies of opportunistic types (types that cannot commit), who may or may not mimic the strategies of trustworthy types (types that commit to some particular strategies). However, very little has been discussed on how the strategies of trustworthy types are constructed because they are often given exogenously in the model.2,3 This paper provides an answer to this question by allowing a trustworthy government to optimally choose its committed policies and announce them. A simple taxation model is constructed in the tradition of past work on optimal policy with limited commitment by Barro and Gordon (1983), Fischer (1990), and Phelan (2006). Three important results follow. First, we show that the optimal committed strategy of the trustworthy type changes with the patience of the opportunistic type. Therefore, once we consider an optimizing trustworthy government, an exogenously speci…ed committed strategy cannot always be incentive-compatible at any patience level of the opportunistic government. Second, we characterize the conditions under which it is optimal for the trustworthy government to build up its reputation by announcing policies that would likely not be implemented if the opportunistic type were in place. This strategic reputation building by the trustworthy government is new to the literature and produces equilibrium dynamics of reputation that are sharply di¤erent from those in Barro (1986) or Cukierman and Liviatan (1991). Third, we establish the uniqueness of the equilibrium announcement by adopting a re…nement in the spirit of Mailath, Okuno-Fujiwara and Postlewaite (1993) as a technique to resolve the well-known problem of equilibrium indeterminacy in signaling games. This model is a …nitely repeated reputation game in which a government that seeks to maximize the present value of its tax revenues plays against a continuum of small private1 Whether or not a government can commit is an issue that arises from a well-known time inconsistency problem (Kydland and Prescott, 1977). That is, it is often in the short-run best interest of a government to raise capital taxes, default on its debt obligations, or generate in‡ation surprises. 2 Examples include Kreps and Wilson (1982), Milgrom and Roberts (1982), Barro (1986), Fudenberg and Levine (1992), Cripps, Mailath and Samuelson (2004), Phelan (2006) and Liu (2009), in which the committed strategy is to play the same action each period. Examples allowing for history-dependent committed strategy include Schmidt (1993), Aoyagi (1996), Celentani, Fudenberg, Levine and Pesendorfer (1996), Evans and Thomas (1997) and Ghosh (2010). 3 Although any strategy can be sustained in equilibrium in an abstract game by constructing payo¤s properly, an arbitrary objective for a government can be problematic when we consider optimal policies. 2 sector households. Within each period, the government …rst announces a tax rate. Households then decide whether or not to engage in production with a …xed cost. After the production, the government decides the actual tax rate to implement. Households entertain the possibility that the government is one of two types, di¤ering with regard to their ability to commit: a ‘trustworthy’government, which by de…nition is sure to enact its announced tax rate; or an ‘opportunistic’ government, which retains the option of deviating from its announced rate. In the model, the optimal deviation is to the highest tax rate possible, which represents con…scation by the government of all household production. The type of government is private information and is …xed throughout the whole game. Households update their beliefs about the government type in a Bayesian fashion after observing the government’s actual taxation at the end of each period. The government in the model uses the tax announcement to control both the households’ perception of the likelihood that the announced tax rate will be implemented and how the households’beliefs concerning the government type will evolve once the announced tax rate is actually implemented. In equilibrium, both types of government use the same announcing strategy that maximizes the trustworthy government’s objective because it is both desirable and costless for the opportunistic government to hide its identity before the actual taxation by imitating the announcement of the trustworthy type. An important trade-o¤ arises when the trustworthy government chooses its announced tax rate. On the one hand, the trustworthy government has an incentive to separate itself from the opportunistic type. In particular, it would like to announce a low tax rate so that enacting such a rate would be costly and thus unlikely if the opportunistic type were in place. This bene…ts the trustworthy type because by enacting the announced policy, the government conveys more information about its identity and improves its reputation. As a result, there will be more production and thus a greater ‡ow of tax revenue in the future. On the other hand, rather counter-intuitively, the trustworthy government also has an incentive to pool with the opportunistic type. In particular, it would like to announce a high tax rate, i.e., a rate that would likely be enacted even if the current government were the opportunistic type. This raises the credibility of the current tax announcement and, in turn, reduces households’risk of con…scation. As a result, more households engage in production and more tax revenue is secured for the current period. We analyze how this trade-o¤ between current and future credibility is a¤ected by key parameters in the model, i.e., the patience of the two types of government, the initial reputation of the government, the elasticity of household production, and the time horizon of the game. In particular, the patience of the trustworthy type measures its incentive to separate 3 and the patience of the opportunistic type measures its incentive to mimic, which in turn determines the trustworthy type’s incentive to pool. If the opportunistic type’s incentive to mimic is strong, i.e., op is high, it will be too costly for the trustworthy type to separate itself from the opportunistic type. As a result, the equilibrium involves little learning with high taxation throughout the entire tenure of the government. If the opportunistic type’s incentive to mimic is weak, however, it is possible that the equilibrium tax announcement, once implemented, helps the government to regain public trust. Such reputation building occurs in equilibrium when the trustworthy type is su¢ cient patient or when the initial reputation of the government is su¢ ciently high. In this case, the household production increases over the tenure of the government as long as the announced tax rate is consistently implemented. The elasticity of household production determines the cuto¤s in the patience of the two government types, and the time horizon of the game has a very limited e¤ect on the equilibrium dynamics. These results suggest that the optimal committed strategy, i.e., the optimal policy announcements in our current context, can be drastically di¤erent even with a similar initial government reputation, elasticity of household production and time horizon of the game. The recent government debt problems in many developed countries o¤er real-life examples of such cases. Facing similar situations with their outstanding government debt and public criticism about the government budget problem, Britain and the United States adopt very di¤erent policies. Britain’s Conservative - Liberal Democrat government, which won power in 2010, has since implemented huge public spending cuts and tax hikes to slash a record de…cit inherited from the previous Labour administration. The coalition sticks to the tough austerity measures even though the economy is sliding into a recession and unemployment has reached a 16-year high (Bloomberg.com, 2012). In contrast, the U.S. government takes a backloaded approach in managing the budget problem and continues its stimulus plans, which has led to increases in the federal debt ceiling twice in less than a year (July 31, 2011 and January 26, 2012) and a downgrade of government credit rating by Standard & Poor’s in August 2011.4 This paper shares the view of Mailath and Samuelson (2001) that reputation building is an exercise not only in pooling with good types but also in separating oneself from bad types.5 Focusing on reputation building as separation, Mailath and Samuelson (2001) ab4 Alesina, Ardagna and Trebbi (2006) provide a theory and evidence to explain the delay in the stabilization of large and increasing budget de…cits. They …nd that in a presidential system, when faced with …scal crisis, the government is able to cut additional percentage points of GDP when compared with a parliamentary system. However, this explanation does not apply here because the U.S. is under a presidential system, whereas the U.K is under a parliamentary system. 5 See also Horner (2002). 4 stract from the strategic behavior of the bad type, which is assumed to always engage in the same single action. In contrast, this paper allows both types to act strategically and thus explores the interaction between incentives to separate and incentives to pool. It is precisely the competition in reputation building between trustworthy and opportunistic types that determines the optimal policies. A few papers prior to this one also studied models in which both trustworthy and opportunistic types behave strategically. Cukierman and Liviatan (1991) analyzed a case in which the optimal announcement by the trustworthy government reacts to its imperfect credibility. The current paper builds on their arguments by emphasizing an important reverse relationship: policy announcements can also be a powerful tool to manipulate government credibility. A similar idea is explored in King, Lu and Pasten (2008), who provided a numerical example in which reputation building by a trustworthy central bank implies gradualism in disin‡ation resembling the “Volcker disin‡ation”. However, King, Lu and Pasten (2008) do not explore the conditions under which reputation building is optimal for the trustworthy policymaker, which is the main focus of the current paper. Kambe (1999) studies a bargaining game and allows players to strategically announce bargaining postures to which they may become committed. However, the announced postures are required to be constant in his model, whereas in this paper, the government can announce non-constant policies and bene…ts from the additional ‡exibility. Wolitzky (2011) extends Kambe’s model to allow players to announce nonconstant postures, but his paper studies maxmin payo¤s and postures. In the current paper, we focus on sequential equilibria following the existing literature on reputation. Similar to the current paper, in the literature on unsecured debt, Cole, Dow and English (1995), D’Erasmo (2007) and Chatterjee, Corbae and Rios-Rull (2008) show the di¤erence in time preference across types as a central modeling component.6 Rather than focusing on the default or debt repayment decision, however, the current paper shows, in the context of unsecured debt, how the optimal level of debt (as a form of announced policy) can be determined by the borrower, who balances the trade-o¤ between the current and future credibility of debt repayments. In the current model, the government has all of the bargaining power with respect to the households, but it also has private information about its type. This feature links the current analysis to the literature on the informed-principal problem pioneered by Myerson (1983) and Maskin and Tirole (1992). In agreement with this literature, the optimal contract is in‡uenced by the principal’s incentive to hide or reveal its private information, which thus leads to a pooling or separating equilibrium. The current paper, however, also includes mixed strategies by the opportunistic type as equilibria to generate dynamics of credibility 6 See also Diamond (1989). 5 and provide a theory of reputation building. The design of policy announcements is the central focus of this paper, which relates the paper to models of strategic information transmission initiated by Crawford and Sobel (1982) and Sobel (1985). However, the announcement in this model is only cheap talk for the opportunistic type and not for the trustworthy type because the latter is committed to act as it announces. This nonstandard feature of signaling di¤erentiates this paper from the rest of the literature. Consequently, popular re…nements such as the intuitive criterion lead to counterintuitive results.7 Fortunately, a unique and plausible equilibrium can be reached when a re…nement in the spirit of Mailath, Okuno-Fujiwara and Postlewaite (1993) is applied. The remainder of the paper is organized as follows. Section 2 presents the setup of the game and de…nes the equilibrium. Section 3 uses a two-period example to illustrate the main results of the paper. Section 4 extends those results to a game with an arbitrary number of periods and studies the equilibrium dynamics. Section 5 establishes the uniqueness of the equilibrium. Finally, Section 6 concludes. 2 A Reputation Game This section describes a …scal policy game played between a government, which can be trustworthy or opportunistic, and a competitive private sector composed of atomistic households. In contrast to previous analyses of similar …scal games (e.g., Phelan, 2006), in which the trustworthy government simply sets an exogenously …xed tax rate, we work with a case in which the trustworthy government is able to choose the optimal tax policy and announce it at the beginning of each period. The equilibrium notion in the game will be Markov perfect. 2.1 Setup There are two parties: a continuum of households with costs to engage in production, and a government that lives for T periods and collects tax on household production each period. The government announces a proportional tax rate each period before the households’ production decisions but levies the taxes after the household production. The government can be of the trustworthy type (that always honors its announced tax rate) or the opportunistic type (that retains the ability to change the tax rate after the announcement). The government type is not directly observable by households (i.e., it is 7 For example, the intuitive criterion selects the worst equilibrium in which the government announces a 100 percent tax rate and no household produces; the re…nement by Grossman and Perry (1986) rules out all sequential equilibria. 6 private information to the government) and remains …xed throughout the game. Households only live for one period but are able to observe the whole public history of the game.8 Each household can produce one unit of output with a cost c or not produce. The production cost c is heterogeneous across households and distributed according to a distribution with c.d.f. G (c) = c and support [0,1].9 The parameter is greater than or equal to 1 ( > 1) and can be interpreted as the elasticity of household production.10 The timing is as follows: At the beginning of the game, nature chooses a government to be the trustworthy type (with probability 1 ) or the opportunistic type with probability (1 1 ). Then, in each of the T periods, 1. The government announces a tax rate t 2 [0; 1]. 2. Households choose whether to produce 1 unit or not. Let t be the fraction that produce.11 3. If the government is the trustworthy type, it must tax at the announced rate. If the government is the opportunistic type, it chooses whether the actual tax rate is t (to mimic the trustworthy type) or 100% (to con…scate all household production). Let t denote the probability of con…scation. The momentary payo¤ to the government of either type is t t if it taxes at rate t , or t for the opportunist government if it con…scates all household production. The objective of the government of either type is thus to maximize the lifetime tax revenues. The two types of governments are allowed to di¤er in how they discount future tax revenues, with tr and op denoting the time discount factor of the trustworthy government and the opportunistic government, respectively. Households are risk neutral with zero payo¤ if they do not produce. If they produce, the payo¤ is (1 c if the announced tax rate is implemented or c if the government t) con…scates. Therefore, if the government is less likely to con…scate all household production, more households with low realizations of production cost will produce. In a special case of = 1, i.e., the production cost is uniformly distributed, and the fraction of households that produce is simply the expected payo¤ or one minus the expected tax rate. 8 The assumption of short-lived households is not essential. As long as the rewards of production cannot be stored over periods, the maximization problem of households will be the same with a longer horizon. 9 The heterogeneous production costs among households serve to generate a smooth response of households to taxation, which is a general practice in macroeconomics when studying the e¤ects of policies. 10 We focus on the case with > 1 to have a well de…ned density g (c) = c 1 over the entire support [0,1]. In addition, > 1 guarantees that household production is a decreasing function with respect to the announced tax rate. 11 is publicly observable, but the individual production decision is private information to each household. Under this assumption, no individual household’s action has an impact on another’s decision or a¤ects the government action. 7 2.2 Markov strategies and Markov perfect equilibria Strategies are mappings from histories to actions. This paper restricts its attention to Markov strategies that condition actions solely on payo¤-relevant variables. In this incomplete information game, the only payo¤-relevant variable is the households’homogenous belief that the current government is of the trustworthy type.12 De…ne this belief to be the state variable of the game: t . Under the Markov restriction, we can write the strategies of both the households and the government as functions of t . 2.2.1 Households and the opportunistic government Given the announced tax rate t , the opportunistic government and households are essentially playing a simultaneous game: each takes the other’s action as given and chooses its best reaction. The households act before the government levies taxes but still take the opportunistic type’s action as given because of their atomistic feature, namely, the households’ individual production decision or belief does not a¤ect the play of the government or the future values of . As households make production decisions before the government levies the taxes, they form expectations regarding their after-tax output. Given the action of the opportunistic type t , the expected after-tax output is (1 t ) t , which is a probability-weighted average between being taxed at the rate t and being con…scated. t is the households’perception of how likely the government will implement the announced tax rate: ( t) = t + (1 t ) (1 t) : (1) This is the sum of two elements: the likelihood of a trustworthy government being present so that t will be implemented for certain and the likelihood of an opportunistic government being present but mimicking with probability (1 t ). Because t is the credibility of the current tax announcement, we call it short-term credibility to distinguish it from t , which measures the credibility of tax announcements in the long-run and is thus called long-term credibility in this paper. Given the long-term credibility t , there is a one-to-one mapping 12 Households in one generation share a homogenous belief because they are identical to each other except for the production cost. However, the individual realization of production cost has no information content about government type and therefore all households in one generation share a common information set. In fact, when households choose whether to produce or not, the identity of the government and the announced tax rate are both payo¤-relevant variables. The identity of the government is not observable by the households so it must be replaced by the households’best estimate of it. This extension of MPE to an incomplete information game is in the spirit of Ball (1995): "... actions depend on agents’ best estimates of payo¤-relevant variables." The announced tax rate itself is an optimal choice by the government, whose payo¤ depends on how many households produce and their best estimate of the government type. 8 between t and t . If the opportunistic type mimics for certain, i.e., t = 0, then t = 1. If it con…scates for certain, i.e., t = 1, then t = t . Randomizing between mimicking and con…scating with t 2 (0; 1) corresponds to t 2 ( t ; 1). Using the expected after-tax output as a common cuto¤, households will produce if they receive a cost c lower than (1 t ) ( t ) and will not produce otherwise. This individual optimization results in an aggregate best reaction function on the households’side conditional on the action of the opportunistic type, t : ( t ) = G [(1 t) ( t )] : (2) With the distribution G(x) = x , household production is [(1 t ) ( t )] in which the parameter represents the elasticity of household production with respect to the expected reward to production. Now we will turn to the strategy of the opportunistic type, which takes the household production t as given and chooses whether to con…scate it at the end of each period. If the government con…scates all household production in period t, it is known to be of the opportunistic type. Given that the government type is permanent in the game, the long-term credibility in period t + 1 is zero and remains zero afterward, which implies zero production in all future periods as part of the unique Markov perfect equilibrium outcome for the rest of the game.13 The payo¤ to con…scation is thus t . If the opportunistic government does tax at the announced rate in period t when it was expected to deviate with probability t , Bayes’rule implies an updated posterior belief about government type: t t = ; (3) t+1 = + (1 ) (1 ) ( t t) t t where t is the prior belief in period t that the government is of the trustworthy type, and ( t ) is the marginal probability of being taxed at rate t . The payo¤ to the opportunistic type for mimicking the trustworthy type in terms of levying taxes is thus t t + op Vt+1 ( t = t ). Vt+1 denotes the value of future tax revenues implied by the positive level of long-term credibility. 13 The intuition underlying this result is that if the opportunistic government will con…scate all household production for certain in the last period and if the public knows for certain the current government is opportunistic, no household will produce. Given the lack of production in the last period, there is no bene…t for the opportunistic type to hide its identity in the second to last period. It will thus con…scate all household production for certain, which again implies zero production if the government is known to be opportunistic. This logic can be applied to all preceding periods so that as long as the long-term credibility drops to zero, there will be no production in all future periods. More precisely, consider any period with = 0; if = 1 or = 1, household optimization implies = 0. If < 1 and < 1, government optimization requires + op V (0) >= + op V (0), which only holds for = 0. 9 The best reaction function of the opportunistic government is to choose t and maximize (1 t) t t + op Vt+1 ( t = t ) + t t , given the household production t and the associated belief t = G 1 ( t ) = (1 t ) as suggested by equation (2). The optimal choice of t should then satisfy the following "incentive compatibility constraints": If t t + op Vt+1 ( t = t) > t, the opportunistic type mimics, If t t + op Vt+1 ( t = t) 6 t, the opportunistic type con…scates all production, If t t + op Vt+1 ( t = t) = t, the opportunistic type plays a mixed strategy, (4) ( t ) = 0; ( t ) = 1; ( t ) 2 (0; 1) : For future reference, we call the …rst inequality the "pooling" incentive compatibility constraint and the second inequality the "separating" incentive compatibility constraint. In equilibrium, the probability of con…scation t chosen by the opportunistic type is indeed the same as that expected by the households. De…ne the equilibrium as the following: De…nition 1 Given the state variable t and the announced tax rate t , f^ t ; ^ t g is the equilibrium of the simultaneous game between households and the opportunistic government if ^ t = (^ t ) and ^ t = (^ t ). Denote this equilibrium as f^ t ( t ; t ) ; ^ t ( t ; t )g. This is the equilibrium of the subgame after a given announced tax rate, which is very similar to the Markov perfect equilibrium studied in Phelan (2006). However, the question of what tax rate the government would announce in the …rst place remains. This is the topic we now turn to. 2.2.2 Announcement As there will be no production in all future periods once the government is identi…ed to be the opportunistic type, we can conclude that in any equilibrium, both government types announce the same tax rate at Step 1 in each period t. Thus, nothing about the current government’s identity can be learned from observing the announced tax rate t , and the household’s decision is based on the current-period prior belief t , as described in the previous subsection. We then impose a working assumption that the opportunistic government uses the same announcement strategy as the trustworthy type. Section 5 will revisit this assumption and argue that it is indeed the equilibrium announcement strategy of the opportunistic type in an explicitly modeled signaling game. In addition, Section 5 will show that the equilibrium outcomes obtained under this assumption 10 are the same as in the signaling game once a re…nement in the spirit of Mailath, OkunoFujiwara and Postlewaite (1993) is adopted. Therefore, imposing this working assumption has no e¤ect on the equilibrium strategies or the equilibrium outcomes. Imposing this assumption, however, signi…cantly simpli…es the analysis in two ways. First, this assumption makes any tax announcement uninformative concerning the identity of the government. In turn, the households’belief about government type is updated only once in each period t after they observe the actual tax rate implemented by the government, and the updating is completely summarized by Bayes’rule as speci…ed in (3). Second, this assumption makes the trustworthy type a leader in the game because the tax announcement, regardless of the true identity of the policymaker, is set to maximize the trustworthy government’s payo¤. The stage game played between the government and the households in each of the T periods can thus be analyzed as if there were three players in the game with a trustworthy government setting the announcement …rst, and the households producing accordingly, followed by the opportunistic type deciding whether to con…scate all household production.14 The following de…nition formally de…nes the Markov perfect equilibrium of the whole T -period game. De…nition 2 The sequence of Markov strategy triplets f t ( t ) ; the Markov perfect equilibrium (MPE) if, in each period t,15 i) t ( t ) solves the trustworthy government’s optimization: Wt ( t ) = max t n t ^t + tr Wt+1 ^ t= t o t ( t) ; t ( t )gTt=1 is called ; where f^ t ( t ; t ) ; ^ t ( t ; t )g is the equilibrium given in De…nition 1 and ^ t ( t ; t ) = t + (1 ^ t ( t ; t )) according to (1); t ) (1 ii) t ( t ) = ^ t ( t ( t ) ; t ), t ( t ) = ^ t ( t ( t ) ; t ) ; iii) long-term credibility evolves as t+1 = t = t ( t ) according to (3), where t ( t ) = ^ ( ( ) ; ). t t t t The MPE in period t refers to the equilibrium strategy triplet f t ( t ) ; t ( t ) ; t ( t )g. As the leader in this game, the trustworthy type is able to use its tax announcements f t gTt=1 to a¤ect the household production f t gTt=1 and the opportunistic type’s con…scation 14 Even when the government in place is indeed trustworthy, it still needs to account for the reaction of the opportunistic type in terms of the con…scation probability because the households, when making production decisions, are generally concerned about a possible con…scation by an opportunistic government. 15 The subindex t of each function indicates that its functional form may be time-varying because of the …nite horizon of the game. 11 probabilities f t gTt=1 to maximize its life-time tax revenues. Backward induction can be applied to obtain an MPE solution.16 In each period t, the trustworthy government takes the future value function as given. The current tax announcement t a¤ects the equilibrium actions of the opportunistic type and the households through their best reaction functions (4) and (2), respectively, which in turn determine the current tax revenue t t and the evolution of long-term credibility through Bayes’ rule (3). Thus, when choosing the tax announcement, the trustworthy government faces a trade-o¤ between the momentary payo¤ governed by short-term credibility and the future value of revenues determined by the growth of long-term credibility. 3 A Two-period Model This section solves a two-period version of the model to illustrate the basic trade-o¤s facing a trustworthy government with imperfect credibility. On the one hand, a higher announced tax rate is more credible and thus implies more tax revenue for the current period. On the other hand, a lower announced tax rate, once implemented, can be more e¤ective in signaling the government type, thus improving its future reputation. The relative strength of these o¤setting forces critically depends on the ratio of the time preference for the two government types tr = op , the elasticity of household production , and the long-term credibility t . 3.1 Optimal tax announcement with full credibility First consider a benchmark case in which a trustworthy government possesses full credibility, = = 1. The expected after-tax output is (1 ) and the fraction of households who produce is (1 ) . Household production thus decreases with the announced tax rate, as shown in Panel A of Figure 1. This feature replicates the celebrated La¤er curve, which traces an inverted-U relationship between the announced tax rate and revenue. Panel B in Figure 1 depicts one example of the La¤er curve. The revenue-maximizing tax rate is: UC = arg max (1 16 ) = ( + 1) 1 : Backward induction can be applied here because the government announces a tax rate at the beginning of each period rather than a path of tax rates for the entire future at time 0. The former case can be thought of as a limited commitment case in which the trustworthy type has no access to the intertemporal commitment technology even though it can commit to the current period’s tax announcement. Thus, when the trustworthy government sets t in period t, it recognizes that f s gTs>t will be chosen by future trustworthy governments with similar objectives. 12 This rate is labeled as "unconstrained optimal" because it is free of constraints imposed by credibility concerns. We now turn to the case with imperfect credibility. 3.2 MPE in the …nal period Let us start with the …nal period, t = 2. With some inherited credibility 2 , the opportunistic type will always con…scate all household production, irrespective of 2 , because there is no future revenue to reward mimicking. In turn, the current government’s short-term credibility 2 is …xed at the level 2 , implying 2 = U C . The optimal tax rate U C yields the value functions of both types: V2 ( 2 ) = (1 UC) 2; W2 ( 2 ) = U C V2 ( 2 ): (5) Actually, for any …xed short-term credibility, the homogeneity feature of G ( ) ensures that the optimal tax rate is always equal to the unconstrained optimal rate, U C . This property is desirable because any variation in optimal tax rates obtained later can be attributed to the endogeneity of short-term credibility. Endogenous short-term credibility captures the opportunistic type’s best reaction to the tax announcement (recall Equation (1)) and is the essence of "credibility concerns." 3.3 MPE in the …rst period In the …rst period, t = 1, with some initial long-term credibility 1 , the trustworthy government takes V2 ( 2 ) and W2 ( 2 ) as given, and chooses 1 by solving:17 max 1 1 ^1 + tr W2 1= ^ 1 : where ^ 1 = 1 + (1 ^ 1 ) and f^ 1 ; ^ 1 g is the equilibrium given in De…nition 1. 1 ) (1 Therefore, when the trustworthy government decides the optimal tax announcement, it takes into account the best reaction of households and the opportunistic type, which takes one of the following three forms: 1) a pooling outcome ^ 1 = 1 in which the announced tax rate 1 satis…es the pooling incentive compatibility constraint: 1 ^1 + op V2 ( 1 ) > ^ 1 with ^ 1 = (1 17 1) : (6) The initial long-term credibility is the prior belief of households that the government in power is the trustworthy type. 13 2) a separating outcome ^ 1 = 1 in which the announced tax rate rating incentive compatibility constraint: 1 ^1 + op V2 (1) 6 ^ 1 with ^ 1 = (1 1) (7) ( 1) : 3) a mixed outcome ^ 1 2 ( 1 ; 1) in which the announced tax rate tunistic type play a mixed strategy: 1 ^1 + op V2 1= ^ 1 = ^ 1 with ^ 1 = (1 satis…es the sepa- 1 1) ( 1) 1 makes the oppor- : (8) The trustworthy type …rst maximizes conditioning on each form of the outcomes and then picks the optimal 1 ( 1 ) that generates the highest payo¤. The optimal tax rate thus determines the equilibrium strategies of households and the opportunistic type, f 1 ( 1 ) ; 1 ( 1 )g, and the equilibrium short-term credibility 1 ( 1 ). If U C satis…es either the pooling or the separating incentive compatibility constraint, i.e., 1 > (1 U C ), U C is the optimal tax rate among those U C ) = op or 1 > op = (1 that generate pooling or separating outcomes. Lemma 1 When UC UC When UC satis…es the pooling incentive compatibility constraint, = arg max 1 1 ^1 + tr W2 ( 1 ) where ^ 1 = (1 1) ; satis…es the separating incentive compatibility constraint, UC = arg max 1 1 ^1 + tr W2 (1) where ^ 1 = (1 1) ( 1) : Proof. Notice that when 1 is …xed at 1 ( 1 = 1 in a pooling outcome and 1 = 1 in a separating outcome), the trustworthy government’s tax revenue, 1 (1 ; is 1) 1 maximized at U C and its continuation value, tr W2 1 = 1 ; is independent of the current tax rate. This solution to the unconstrained maximization does not change after the incentive compatibility constraint is imposed because the constraint is not binding at U C . Notice that it is impossible for U C to satisfy the pooling and the separating incentive compatibility constraints simultaneously except for a special non-interesting case with full credibility and U C = 1 op . Conditional on generating the form of outcome that cannot be supported by U C or a mixed outcome, the optimal tax rate must induce the opportunistic type to be indi¤erent between mimicking and con…scating. The following lemma formalizes this property. 14 Lemma 2 Conditional on generating a mixed outcome or the form of outcome that cannot be supported by U C , the problem of the trustworthy type can be simpli…ed to: max 1 2[0;1] s.t. 1 ^1 + op V2 1= 1 ^1 + tr W2 ^ 1 = ^ 1 with ^ 1 = (1 1= ^ (9) 1 ^1 1) and ^ 1 2 [ 1 ; 1] : (10) Proof. As shown above, when the short-term credibility 1 is …xed, the solution to the unconstrained maximization (9) is U C : In the case of a pooling or a separating outcome, imposing the incentive compatibility constraint will alter the solution if U C is not in the constrained set. Maximization then dictates that the trustworthy type minimizes the deviation from U C to satisfy the pooling or separating incentive compatibility constraint, which implies binding constraint in either case, i.e. the payo¤s to mimicking and con…scating are identical with ^ 1 equal to 1 or 1 . In the case of a mixed outcome, the opportunistic type is already indi¤erent between mimicking and con…scating. Thus, the opportunistic type is always indi¤erent in any case. Lemma 2 implies that if U C does not satisfy the pooling incentive compatibility constraint, ~1 such that ~1 ^ 1 (~1 ; 1) + op V2 ( 1 ) = ^ 1 (~1 ; 1) is the optimal tax rate among those that generate pooling outcomes: ~1 ( 1 ) = 1 op (1 UC) 1 1 +1 > UC: (11) We call this form of outcome "constrained pooling." Correspondingly, an "unconstrained pooling" outcome refers to a pooling outcome supported by U C . If U C does not satisfy the separating incentive compatibility constraint, ^1 such that ^1 ^ 1 (^1 ; 1 ) + op V2 (1) = ^ 1 (^1 ; 1 ) is the optimal tax rate among those that generate separating outcomes: ^1 ( 1 ) = 1 op (1 UC) 1 +1 1 6 UC: (12) Similar to the pooling case, this form of outcome is called "constrained separating" and an "unconstrained separating" outcome refers to a separating outcome supported by U C . In the case of a mixed outcome, the optimal tax rate is: 1 = 2 +1 tr = op tr = op if tr = op < 2; 1 = 0 otherwise. The MPE in period 1 is determined by the outcome and its associated optimal tax rate 15 that yield the highest payo¤ for the trustworthy type. For easier reference, we label the MPE after the particular form of outcome that is in equilibrium. Proposition 1 A unique MPE exists in period 1 with the following properties: i. If tr = op > ( + 1) = , 1 6 U C and 1 weakly increases with 1 . If tr = op 6 ( + 1) = , 1 > U C and 1 weakly decreases with 1 . ii. There exist min ; op ; tr and max ; op ; tr such that if tr = op > ( + 1) = , 1 the MPE is mixed for values of 1 2 (0; min ), constrained separating for values of 1 2 1 1 1 [ min ; max ], and unconstrained separating for values of 1 2 [ max ; 1]; if tr = op < ( + 1) = and op > 1 U C , the MPE is mixed for values of 1 2 (0; min ), constrained pooling for values of 1 2 [ min ; max ], and unconstrained pooling for values of 1 2 [ max ; 1]. ; op ; tr such that the iii. If tr = op < ( + 1) = and op < 1 U C , there exists MPE is mixed for values of 1 2 (0; min( min ; )), constrained pooling for, if any, values of ), and unconstrained separating for values of 1 2 [ ; 1]. 1 2 [ min ; 3.4 Understanding the MPE in the …rst period To understand the MPE in the …rst period, it is useful to …rst review the trade-o¤s facing the trustworthy government with imperfect credibility in deciding the optimal tax announcement. Then we study how the trade-o¤s are a¤ected by the time discount factors of the two government types, the elasticity of household production, and the initial long-term credibility. Finally, we compare the MPE in our model with the results in the existing literature to highlight our contribution. 3.4.1 The trade-o¤s in setting the optimal tax rate Conditional on generating an outcome that can be supported by the unconstrained optimal rate U C , the trade-o¤ is the same as that when the trustworthy government has full credibility, as proved in Lemma 1, and can be captured by the conventional La¤er curve. That is, tax revenue …rst increases with the tax rate and then decreases because of the negative e¤ect of tax rate on the production base. In the more interesting case covered in Lemma 2, the trustworthy type always chooses the tax announcement from a set of rates that makes the opportunistic type indi¤erent between mimicking and con…scating. This feature introduces an extra role played by the tax announcement –it now determines the opportunistic type’s probability of con…scation, which constitutes a trade-o¤ for the trustworthy government unique to its subjection to public skepticism about its identity. 16 Using the indi¤erence condition (10), we can specify the set of tax rates that makes the opportunistic type indi¤erent as the following: ( 1) f 1 : f ( 1; 1) 2 [ 1 ; 1]g = [^1 ( 1 ) ; ~1 ( 1 )] where f ( 1 ; 1 ) is the level of short-term credibility, given 1 and 1 ; that equalizes the value of mimicking with the value of con…scating for the opportunistic type: ^ 1( 1; 1) = f ( 1; op 1) (1 (1 1) UC) +1 1 2 1 2 1; (13) and the boundaries of the set, ^1 ( 1 ) and ~1 ( 1 ), are speci…ed in Equations (12) and (11), respectively. Therefore, the constrained set ( 1 ) requires the tax rate to imply a level of short-term credibility that is consistent with the indi¤erence condition of the opportunistic type and can also be achieved by a feasible action of the opportunistic type 1 2 [0; 1]. That is, ^ 1 = f ( 1 ; 1 ) 2 [ 1 ; 1], since ^ 1 = 1 + (1 ^ t ). 1 ) (1 For all 1 2 ( 1 ), the function f ( 1 ; 1 ) describes how the short-term credibility ^ 1 , conditional on a given level of long-term credibility 1 , changes endogenously with the announced tax rate 1 , which essentially captures the best reaction of the opportunistic type. When the announced tax rate is lower, holding 1 constant, the tax-revenue gap between implementing and deviating from the announcement is larger. Thus, the opportunistic type is more tempted to deviate and con…scate all household production. The probability of con…scation 1 consequently increases, which implies a lower level of short-term credibility 1 to keep the opportunistic type indi¤erent. The e¤ect is captured by the positive derivative of the function ^ 1 ( 1 ; 1 ) with respect to 1 : @ ^ 1( 1; @ 1 1) = @f ( 1 ; @ 1 1) > 0: Therefore, although a lower announced tax rate makes it more rewarding to produce, the announcement also appears less credible to households. In other words, households face a greater risk of con…scation after production when the announced tax rate is lower. This credibility e¤ect o¤sets the conventional tax e¤ect on household production as described by the La¤er curve. As a result, the marginal gain from raising the announced tax rate is larger when the short-term credibility responds to it endogenously than when the short-term 17 credibility is …xed, as it is in the full credibility case:18 @ 1 ^1( 1; @ 1 1) = h @ 1 G (1 ^ 1) 1( 1; @ 1 i ) 1 > @ 1 G [(1 @ 1 1 )] > 0: The announced tax rate maximizing current tax revenue is therefore higher than the unconstrained optimal rate U C , with the discrepancy measuring the extent to which a trustworthy government accommodates imperfect credibility. Announcing a higher tax rate may bene…t current tax revenue, but it impedes the transmission of information from the government to households. Because a higher announced tax rate is more likely to be implemented by the opportunistic type as well, implementing it does not convey much information about the government type. The Bayesian learning equation (3) re‡ects such a learning e¤ect, in which higher short-term credibility 1 reduces future long-term credibility 2 . Because the trustworthy type’s tax revenue in the …nal period increases with its long-term credibility, a higher current tax rate imposes a loss in terms of future revenues through a slower growth of its long-term credibility: @ tr W2 1= ^ 1( 1; @ 1 1) 6 0: The trade-o¤ between the credibility e¤ect and the learning e¤ect of a tax announcement is present as long as the trustworthy government contemplates the generation of a mixed outcome or an outcome that cannot be supported by the unconstrained optimal rate U C . 3.4.2 The key determinants in the trade-o¤s It is well-known that the conventional trade-o¤ in choosing the tax rate is determined by the elasticity of household production . A more elastic supply enhances the negative e¤ect of tax rate on household production and reduces the revenue-maximizing tax rate. This conventional trade-o¤ is the only relevant one for the trustworthy government if either of the following conditions is met: 1) it wants to generate a pooling outcome and U C > ~1 ( 1 ) (i.e., U C can support a pooling outcome); 2) it wants to generate a separating outcome and U C 6 ^1 ( 1 ) ( i.e., U C can support a separating outcome). In all other cases, the trustworthy government faces an additional trade-o¤ between the credibility e¤ect and the learning e¤ect of a tax rate, as discussed in the previous subsection. This trade-o¤ will be a¤ected not only by the elasticity of household production but also by the time preference 18 The derivatives are positive because a reasonable tax rate should always be on the increasing side of the La¤er curve. 18 of the governments and the initial long-term credibility. The e¤ect of tr = op and (Part 1): To understand how various parameters in our model a¤ect the trade-o¤, it is useful to start with the optimization problem (9) without imposing the constraint ^ 1 2 [ 1 ; 1]. That is, we allow 1 to range from 0 to 1, and the implied ^ 1 = f ( 1 ; 1 ) can be lower than 1 or higher than 1. After taking into account the aforementioned credibility and learning e¤ects, the objective of the trustworthy government in (9), i.e., its present discounted value of current and future tax revenues, still exhibits a hump-shaped relationship with the announced tax rate. However, compared with the conventional La¤er curve, this modi…ed La¤er curve peaks at 1 rather than at U C . Whether this peak occurs before the tax rate reaches U C or not depends on the parameters tr = op and in the following way: 1 > UC if and only if tr = op 6 ( + 1) = : To understand this result, it is useful to observe that the trustworthy type’s objective function in period 1 di¤ers from the opportunistic type’s value of mimicking only in its weight on the future continuation value: W1 ( ) = max 1 1 ^1 + tr UC op V2 1= ^1 : op Given the elasticity of household production and in turn U C , a larger ratio of time preference tr = op results in the trustworthy type valuing its future revenue more than the opportunistic type, which enhances the trustworthy type’s incentive to invest in the growth of long-term credibility by announcing a lower tax rate in period 1. In contrast, a higher elasticity of household production implies a lower U C , which raises the opportunistic type’s reward in the …nal period if its identity is not revealed until then but reduces the trustworthy type’s reward in the …nal period. Thus, given the ratio of time preference tr = op , a higher elasticity of household production makes it not only more di¢ cult but also less rewarding for the trustworthy type to separate itself from the opportunistic type. If tr = op U C = 1, the objective of the trustworthy type is identical to the opportunistic type’s value of mimicking, which in turn equals the household production in period 1 because of the indi¤erence condition (10). Given > 1, the household production reacts negatively to a tax increase, @ ^ 1 =@ 1 6 0, even after accounting for the credibility e¤ect. In this case, decreasing the announced tax rate increases the payo¤ to the trustworthy type, and the optimal tax rate 1 is zero. In another extreme case in which the trustworthy type is myopic, i.e., tr = op U C = 0, the optimal tax rate that maximizes the momentary tax revenue 19 has to be greater than U C because of the credibility e¤ect of the tax announcement discussed in the subsection above. Because the optimal tax rate 1 is continuously decreasing in tr = op U C , a value of tr = op U C must exist between 0 and 1 that makes the optimal tax rate equal to U C . 1 ^1 The e¤ect of tr = op and (Part 2): Now we impose the constraint ^ 1 2 [ 1 ; 1] in the optimization problem (9), which is equivalent to choosing 1 from the set ( 1 ) = [^1 ( 1 ) ; ~1 ( 1 )]. 1 is the optimal tax rate if it belongs to the constrained set ( 1 ), i.e., f ( 1 ; 1 ) 2 [ 1 ; 1]. Otherwise, the optimization of the trustworthy type dictates the optimal tax rate to be the minimal deviation from 1 to satisfy the boundary condition, ^ 1 = f ( 1 ; 1 ) 2 [ 1 ; 1]. Because of the monotonicity of f ( 1 ; 1 ) with respect to 1 , when f ( 1 ; 1 ) < 1 , the optimal tax rate must be higher than 1 , which makes the lower bound of ^ 1 binding as a result of being the minimal deviation from 1 . The tax rate is then equal to ^1 ( 1 ) and generates a constrained separating outcome. For a constrained separating outcome to be in equilibrium, it is necessary that ^1 ( 1 ) 6 U C because otherwise, U C would be a dominant choice for the trustworthy type to generate a separating outcome. Given ^1 ( 1 ) > 1 , having a constrained separating MPE is only possible when U C > 1 . Similarly, when f ( 1 ; 1 ) > 1, the optimal tax rate must be lower than 1 and the upper bound of ^ 1 must be binding, which implies a constrained pooling outcome. The constrained pooling outcome could be in equilibrium only when U C fails to support a pooling outcome, i.e., U C 6 ~1 ( 1 ). Given ~1 ( 1 ) < 1 , having a constrained pooling MPE is only possible when U C 6 1 . Therefore, the parameters tr = op and not only determine whether 1 is higher or lower than U C conditional on a mixed MPE but also determine whether the MPE can be constrained pooling or separating. When tr = op 6 ( + 1) = , the MPE cannot be constrained separating, whereas when tr = op > ( + 1) = , the MPE cannot be constrained pooling. An immediate implication is that a candidate equilibrium tax rate has to lie between U C and 1 . The e¤ect of 1 : The e¤ect of long-term credibility 1 works through a¤ecting the boundaries of the constrained set ( 1 ). In particular, ^1 ( 1 ) increases with 1 whereas ~1 ( 1 ) decreases with 1 so the lower bound and the upper bound of ^ 1 = f ( 1 ; 1 ) both become tighter as 1 increases. As a result, with low values of 1 , the boundaries of ^ 1 do not bind and the optimal tax rate is 1 , which implies a mixed outcome. With high values of 1 , either the lower bound or the upper bound of ^ 1 binds depending on whether tr = op is greater than ( + 1) = . 20 When tr = op > ( + 1) = , 1 6 U C so that if the equilibrium tax rate di¤ers from 1 and U C , it is necessary that 1 = ^ 1 ( 1 ) and the MPE is constrained separating. A higher 1 moves ^1 ( 1 ) closer to U C , which is the tax rate that would maximize current tax revenues if the opportunistic type did not react to the change of tax rate. The shrinking gap between ^1 ( 1 ) and U C indicates a smaller loss in current tax revenue that the trustworthy government must bear to achieve a separating outcome when the opportunistic type reacts optimally to the tax change. Thus, conditional on a separating outcome being optimal, a larger initial stock of reputation makes it easier for the trustworthy type to separate itself from the opportunistic type. The gain from the larger stock of reputation arises because, +1 given the tax rate, a higher 1 increases the temptation for con…scation (1 ( 1 ) but 1) leaves the continuation value op V2 (1) unchanged. Similarly, when tr = op 6 ( + 1) = , it is easier for the trustworthy government with a higher initial 1 to generate a pooling outcome conditional on a pooling outcome being optimal because, given the tax rate, the continuation +1 value op V2 ( 1 ) increases but the temptation for con…scation (1 remains the same. 1) 3.4.3 The equilibrium tax rate The optimal tax rate in the optimization problem (9) can only be the equilibrium tax rate if it yields a higher payo¤ to the trustworthy type than the payo¤ yielded by the unconstrained optimal rate U C . This comparison is only relevant when U C can generate either a pooling outcome or a separating outcome, which is a case that is not covered in Lemma (2). Whether it is possible for U C to generate a pooling outcome or a separating outcome is determined by the comparison between op and 1 UC : ^1 (1) = ~1 (1) 6 UC if and only if op >1 UC; where ^1 (1) = ~1 (1) is the lower bound for ~1 ( 1 ) and the upper bound for ^1 ( 1 ). If ~1 (1) 6 U C , there must exist 1 2 [ max ; 1] such that ~1 ( 1 ) 6 U C and, in turn, U C 1 can generate a pooling outcome. If ^1 (1) > U C , there must exist 1 2 [ max ; 1] such that ^1 ( 1 ) > U C and, in turn, U C can generate a separating outcome. Therefore, to determine the MPE, the optimal outcome from the optimization problem (9) must be compared with an unconstrained pooling outcome when op > 1 U C and 1 2 [ max ; 1] and with an unconstrained separating outcome when op 6 1 U C and 1 1 2 [ max ; 1]. In all other cases, the MPE is determined directly by the optimization problem (9). When 1 is high enough, U C is always the dominant choice for the trustworthy type.19 19 Notice that the optimal tax rate in the optimization problem (9) can only take three values conditional 21 The MPE is either unconstrained pooling or unconstrained separating depending on the comparison between op and 1 U C . When 1 decreases, the MPE becomes constrained pooling or constrained separating, determined by the comparison between tr = op and ( + 1) = . When the trustworthy type is relatively impatient as compared to the opportunistic type ( tr = op 6 ( + 1) = ), the trustworthy type values its current tax revenue more than the future gain derived from having higher long-term credibility. To this end, a pooling outcome is desirable because it implies full short-term credibility of the current tax announcement ( ^ 1 = 1), which would encourage household production and in turn boost the tax revenue. Conditional on having a pooling outcome, a lower 1 implies a lower continuation value for the opportunistic type if it mimics the trustworthy type and thus makes it more di¢ cult for the trustworthy type to induce a pooling outcome. Therefore, as 1 decreases, the trustworthy type has to raise the tax rate from U C to achieve the pooling outcome because a higher tax rate reduces the temptation for the opportunistic type to con…scate all household production. When the trustworthy type is su¢ ciently more patient than the opportunistic type ( tr = op > ( + 1) = ), however, it values the future tax revenue more and thus is willing to invest in its reputation by inducing a separating outcome. Conditional on having a separating outcome, a lower 1 implies a lower gain from con…scation, which thus makes it more di¢ cult for the trustworthy type to induce a separating outcome. Therefore, as 1 decreases, the trustworthy type has to reduce the tax rate from U C to increase the temptation for the opportunistic type to con…scate all household production. In either case, the deviation of the equilibrium tax rate from U C implies a loss in current tax revenue. The loss keeps increasing as 1 decreases to the extent that it is no longer worthwhile for the trustworthy type to force either a constrained pooling outcome or a constrained separating outcome. At this point, the MPE becomes mixed and the equilibrium tax rate becomes …xed at 1 . Figure 2 plots two representative policy functions 1 ( 1 ) : The left-hand panel shows the case of equally patient government types: tr = op > 1 U C . The right-hand panel shows the case in which the trustworthy type is su¢ ciently more patient than the opportunistic type: tr = op > ( + 1) = . The elasticity of household production is = 2 so that the unconstrained optimal tax rate U C is 1=3. These two functions have the opposite patterns of monotonicity. However, in both cases, the gap between the equilibrium tax rate and the unconstrained optimal rate U C shrinks as long-term credibility grows, which re‡ects on 1 ; ~ 1 ( 1 ) or ^ 1 ( 1 ). When 1 = 1, the optimal tax rate with full credibility is U C . When 1: 6 = 1 op U C , ^ 1 (1) = ~ 1 (1) 6= U C . Thus, the payo¤ to the trustworthy type yielded by U C is strictly higher than the payo¤ yielded by either 1 or ^1 (1) = ~1 (1). Given that the payo¤ is continuous in 1 for 1 2 (0; 1], there must exist a cuto¤ in 1 above which the unconstrained optimal rate U C is the dominant choice. 22 the diminishing credibility concerns of the trustworthy type in designing its optimal tax announcement when its credibility capital accumulates. 3.4.4 A comparison with the literature In the reputation literature, it is typical to assume that the action of a trustworthy type, i.e., the tax rate in the current context, is …xed, e.g., Barro (1986) and Phelan (2006). However, an important consistency question in this reputation literature remains: is the …xed action consistent with what the trustworthy type would like to do in the case that it is able to choose? The answer from our model is no because the equilibrium tax rate varies with the credibility of the government. For example, when both types of government are equally patient , which is a standard assumption in the literature, a trustworthy government with high credibility in the …rst period favors a tax rate equal to U C , whereas it favors a tax rate higher than U C if the credibility is lower. It has been shown that with moderate credibility, the payo¤ to the trustworthy type increases with the announced tax rate 1 as long as 1 implies a nonzero probability of con…scation by the opportunistic type. Therefore, the trustworthy type keeps increasing the announced tax rate until the probability of con…scation reaches zero. Optimization by the trustworthy type implies a social welfare loss of imperfect credibility beyond what is predicted in the literature because household production is a decreasing function in the tax rate: lower credibility of a government results in a higher equilibrium tax rate and, in turn, a lower level of household production in the economy. Several papers in the reputation literature do feature history-/state-dependent committed strategies, e.g., Evans and Thomas (1997) and Ghosh (2010). However, the committed strategy is usually taken as given when studying impact of the opportunistic type’s patience on its equilibrium payo¤. The analysis here shows that the equilibrium tax rate that a government commits to not only varies with the state variable, 1 , but also changes with di¤erent levels of tr = op and op . As op changes, a pre…xed committed strategy cannot be always incentive compatible if we allow it to be chosen optimally. In particular, the optimal tax rate changes from a decreasing function with respect to 1 to an increasing function when op is reduced. 4 The Dynamics This section reports results concerning equilibrium dynamics when governments have a T period rather than a two-period horizon. 23 When there are more future periods, the opportunistic type’s incentive to mimic is strengthened. That is, ^ t ( ; ) > ^ t0 ( ; ) at any given ( ; ) if t 6 t0 . The strengthened incentive to mimic makes it increasingly di¢ cult for the trustworthy type to separate itself from the opportunistic type. On the other hand, if the trustworthy type is more patient than the opportunistic type, the di¤erence between types in terms of their future continuation values becomes more prominent when there are more future periods. In other words, the future continuation value is increasingly more valuable for the trustworthy type than it is for the opportunistic type. Thus, the trustworthy type is more willing to separate. Which force is dominant again critically depends on the ratio of time preference tr = op and the elasticity of household production . 4.1 Impatient trustworthy government We …rst report results for the case tr = op < ( + 1) = . Proposition 2 If tr = op < ( + 1) = , t > U C for all t. i) When the opportunistic type is su¢ ciently patient, op > 1 U C , the equilibrium tax announcement starts with a constant rate U C if T is su¢ ciently large. The equilibrium tax rate then increases over time, which induces constrained pooling outcomes, and then decreases over time toward U C , which induces mixed outcomes. ii) When the opportunistic type is not su¢ ciently patient op < 1 U C , the equilibrium tax announcement starts with a tax rate higher than U C . With a lower 1 , the equilibrium tax rate …rst increases over time, which induces constrained pooling outcomes, and then decreases over time toward U C , which induces mixed outcomes. With a higher 1 , the equilibrium tax rate converges to U C over time, which induces (possibly) an initial stage of constrained pooling outcomes followed by a stage of mixed outcomes until a separation occurs. The common feature of Cases i) and ii) is that the trustworthy type is relatively impatient as compared to the opportunistic type. Current tax revenue is thus more important to the trustworthy government than its reputation building. As a result, the optimal tax announcement features tax rates that are higher than the unconstrained optimal level U C because the current revenue is non-decreasing in at = U C . The di¤erence between Cases i) and ii) lies in the incentive of mimicking by the opportunistic type. The comparison between op and 1 U C captures such an incentive in an in…nite horizon game with a …xed tax rate equal to U C . In each period, the payo¤ of mimicking forever is the present discounted value of all future tax revenues U C = 1 op , and the payo¤ of con…scation is the current period household production . The opportunistic type prefers mimicking forever if and only if op > 1 UC. 24 When the opportunistic type has a strong incentive to mimic, op > 1 U C , it is easier for the trustworthy type to induce a pooling outcome. If the government has many periods to look forward to in its tenure, a pooling outcome can be sustained by the unconstrained optimal rate U C , which maximizes the current tax revenue. An unconstrained pooling outcome is thus optimal for the trustworthy government as long as it is feasible. As time elapses, fewer future periods remain to reward the opportunistic type for maintaining its reputation so that it is increasingly prone to deviating from the announced tax rate U C . To keep its tax announcement fully credible in the short run (i.e., to sustain a pooling outcome), the trustworthy type has to raise the announced tax rate from U C to reduce the opportunistic type’s temptation to con…scate all household production. When it becomes too expensive for the trustworthy type to sustain a pooling outcome, i.e. the revenue loss created by raising the tax rate from U C becomes too large, the equilibrium becomes a stage of mixed outcomes, with the equilibrium tax rate declining over time toward U C . Figure 3 plots the equilibrium time series f t ; t ; t ; t ; t t gTt=1 from a numerical example with parameters = 3, tr = op > 1 U C , T = 20 and 1 = 0:1. Associated with an equilibrium path of tax rates as described above, the long-term credibility remains constant at its initial level until the very end of the game, when the stage of mixed outcomes starts. As the government’s reputation improves during that stage, the opportunistic government takes greater chances to con…scate household production. Household production is the highest during the stage of unconstrained pooling outcomes and subsequently decreases over time in concert with the dynamics of the tax revenues. A higher initial long-term credibility 1 delays the timing for the equilibrium tax rate to rise from U C and it also shortens the stage of constrained pooling outcomes and the stage of mixed outcomes. A higher decreases the overall level of the equilibrium tax rate because U C decreases in , but it makes the tax rate start to rise earlier in the government’s tenure. In addition, the equilibrium tax rates are lower when the trustworthy government is more patient (higher tr ). Finally, A longer tenure (larger T ) prolongs the stage of pooling outcomes. When the opportunistic type’s incentive to mimic is weak, op < 1 U C , a pooling outcome is impossible if the announced tax rate is U C . In other words, the trustworthy government must bear a cost in terms of current tax revenue if it wants to induce a pooling outcome. In contrast, the unconstrained optimal rate U C can support a separating outcome if the government’s long-term credibility is su¢ ciently high,20 which makes it attractive for the trustworthy government to improve its reputation. 20 A constrained separating outcome is always dominated when 25 tr = op < ( + 1) = . If the initial long-term credibility 1 is low, it costs too much in terms of tax revenue for the trustworthy government to build its long-term credibility up to a level that makes the unconstrained separating outcome feasible. Thus, over the entire tenure of the government, no separating outcome will occur and the equilibrium path only involves a stage of constrained pooling outcomes followed by a stage of mixed outcomes, with the dynamics resembling those in Case i). If 1 is su¢ ciently high, however, an unconstrained separating outcome will occur on the equilibrium path.21 This full separation of two government types occurs after a stage of reputation building, with the equilibrium tax rate converging to the unconstrained optimal rate U C from above. It is possible for the stage of reputation building to be preceded by a stage of constrained pooling outcomes. Figure 4 plots the equilibrium dynamics from two numerical examples, which di¤er only in the initial long-term credibility 1 . A higher initial long-term credibility results in a faster convergence of the equilibrium tax rate to U C . Longterm credibility increases over time, together with the household production. The general pattern for the probability of con…scation and the tax revenue is to increase over time, with some ‡uctuations along the path. Similar to Case i), higher and tr values decrease the overall level of the equilibrium tax rate. Finally, a longer tenure (larger T ) adds more periods for the trustworthy government to enjoy the full credibility after separation. However, this extra payo¤ does not vary with the policy choices before the separation and thus has no impact on the equilibrium dynamics. 4.2 Patient trustworthy government We now turn to the case tr = op > ( + 1) = . Proposition 3 Only if tr = op > ( + 1) = , t 6 U C for all t.22 When the trustworthy type is su¢ ciently more patient than the opportunistic type and the initial long-term credibility is not extremely low, the equilibrium tax rate starts at a rate lower than U C and increases over time, which induces mixed outcomes until a separation occurs. 21 It is possible that the tenure is so short that by the end of the game, long-term credibility has not yet grown su¢ ciently high to sustain an unconstrained separating outcome. However, the equilibrium dynamics are similar except for the lack of an unconstrained separating outcome at the end. 22 The su¢ cient conditions for reputation building in equilibrium are: 1) tr = op > B2 > ( + 1) = , where B2 is de…ned as a cuto¤ in tr = op in period t (T 11 UC , the equilibrium tax rate is equal to UC . Otherwise, the equilibrium tax rate is equal op : This result is robust in the extension of our model that allows the government type to follow a Markov process as in Phelan (2006). short run. When the trustworthy type is su¢ ciently more patient than the opportunistic type and the initial reputation is not extremely low, the equilibrium dynamics follow the same pattern as depicted in Figure 5, even with T increased to in…nity. The extremely low announced tax rates at the initial stage of the game, once implemented, generate a rapid growth of government reputation until full credibility is reached. This pattern of equilibrium reputation stands in sharp contrast to the existing literature on government reputation in which the equilibrium reputation typically stays constant at its initial level until close to the end of the game and then increases gradually (see Figure 3). The existing literature produces such equilibrium dynamics of reputation either because the trustworthy government is not strategic (e.g., Barro, 1986) so that it cannot avoid being mimicked by the opportunistic type or because the two government types share the same time preference (e.g., Cukierman and Liviatan, 1991) so that the trustworthy type …nds it too costly to separate itself from the opportunistic type. History does have cases in which policymakers made tough announcements. George H. W. Bush stated “Read my lips, no new taxes” during his presidential campaign but later raised taxes under the pressure of Congress. Paul Volcker, who became the Chairman of the Federal Reserve in early 1980s, maintained tough disin‡ation policies despite the most severe recession since WWII that were caused by the policies, and successfully reduced the in‡ation from 13.5% in 1981 to 3.2% by 1983. Regardless of whether or not the policymakers kept their promises in the end, the question is why did they make tough commitments in the …rst place? This model provides an explanation: a patient, trustworthy policymaker wants to use tough announcements to build up its reputation quickly, and the less patient opportunistic policymaker has to make similar announcements, although he or she may not necessarily follow through with them, to hide the identity before the public takes action. 5 The Equilibrium Re…nement In the last two sections, the optimal tax announcement was derived under a working assumption regarding the signaling strategy of the opportunistic government: it must always imitate the announcement strategy of the trustworthy type. This assumption and its implied optimal announcement will be revisited in this section, in the context of a within-period signaling game in which the signaling strategies and households’beliefs are obtained in equilibrium. In a nutshell, the results are: (1) imitation is indeed the equilibrium signaling strategy of the opportunistic government so that the assumption is not binding; and (2) the optimal announcement derived under the assumption coincides with the unique equilibrium in the signaling game when a re…nement in the spirit of Mailath, Okuno-Fujiwara and Postlewaite (1993) is applied. We thus complete the solution to the …nitely repeated game, determining both within-period signaling and the intertemporal evolution of credibility. 5.1 The signaling game The within-period signaling game is between the government in place (now as the sender) and households (now as the receivers). At the beginning of each period, the households have a prior belief regarding the government type , which is given by the updated belief arising from the last period’s government tax action. The current government, with its type unobserved by the households, then sends a public message in terms of an announced tax rate . Contemplating this message, the households update their belief regarding the current government type from to ( ) in a manner that this section studies in detail. The new longterm credibility then serves as the basis for the interaction between household production ^ ( ; ) and the government’s con…scation probability ^ ( ; ), if the opportunistic type is in place. As is standard in the game theory literature, a suitable equilibrium concept for such a sender-receiver game with incomplete information is the "sequential equilibrium" proposed by Kreps and Wilson (1982). Denoting m (T R) and m (OP ) as the signaling strategies for the trustworthy and opportunistic types, respectively, a pure sequential equilibrium in this game can be de…ned as follows: De…nition 3 The strategies and beliefs fm (T R) ; m (OP ) ; ^ ( ; ) ; ( )g form a pure "sequential equilibrium" within any time period if: 25 D.3.1) m (T R) maximizes the trustworthy type’s payo¤: m (T R) = arg max ^ ( ; ) + tr W = ^ , where ^ = + (1 ) [1 ^ ( ; )] : D.3.2) m (OP ) maximizes the opportunistic type’s payo¤: m (OP ) = arg max [1 ^ ( ; )] h ^( ; )+ op V =^ i + ^ ( ; )^( ; ): h D.3.3) ^ ( ; ) maximizes the households’expected payo¤: ^ ( ; ) = G (1 25 i ^ ) : We omitted the opportunistic type’s strategy for its con…scation probability ^ ( ; ) in this de…nition because: (1) it is not part of the signaling equilibrium as it is neither the sender’s signaling strategy nor the receiver’s best reaction; (2) it is fully determined by the MPE that we derived in previous sections as long as replaces ; (3) its information is incorporated in the construction of short-term credibility ^ . D.3.4) ( ) is de…ned from the signaling strategies by Bayes’rule whenever possible: Pr ( jT R) if the denominator is positive, Pr ( jT R) + (1 ) Pr ( jOP ) ( ( 1 if m (T R) = 1 if m (OP ) = where Pr ( jT R) = ; Pr ( jOP ) = 0 otherwise 0 otherwise. ( ) Pr (T Rj ) = In this game, the pure sequential equilibrium has the following property:26 Lemma 3 In any pure sequential equilibrium, m (T R) = m (OP ) = a and addition, a can take any value from [0; 1]. (a) = : In Proof. Suppose m (T R) 6= m (OP ). The opportunistic type’s identity is revealed if households observe the message m (OP ). In turn, the opportunistic type will con…scate all household production for sure as its future credibility will never grow again. Anticipating this con…scation, no household will produce, which implies a zero payo¤ to the opportunistic type. Thus, the opportunistic type can always improve its gains by deviating to the same message sent by the trustworthy government, which makes distinct messages from the two types of government impossible in equilibrium. When m (T R) = m (OP ), Bayes’rule implies the belief on the equilibrium path to be . Take any arbitrary message a and let it be the announced tax rate for both types. Notice that the de…nition of sequential equilibrium only speci…es the belief on the equilibrium path and leaves the beliefs o¤ the equilibrium path unrestricted. We can then endow households with the beliefs that if the announced tax rate is di¤erent from a, the government must be the opportunistic type, ( 6= a) = 0. Thus, is positive only if = a and zero otherwise, making it optimal for both types to choose = a. In Sections 3 and 4, the assumption that the opportunistic type always imitates the announcement of the trustworthy type renders the announced message uninformative. Lemma 3 shows that any equilibrium of the signaling game must be a pooled one, so that the lack of information about government type in the announced message is an equilibrium result. However, the equilibrium message still has to be determined because any message 2 [0; 1] can be sustained as a sequential equilibrium by properly specifying the out-of-equilibrium beliefs. We call those beliefs "the supporting beliefs" for the equilibrium. To this end, we adopt a re…nement based on the de…nition of undefeated pure sequential equilibrium developed by Mailath, Okuno-Fujiwara and Postlewaite (1993, M-OF-P hereafter). 26 Similar to Mailath, Okuno-Fujiwara and Postlewaite (1993), and Taylor (1999), we restrict attention to pure strategy in the signaling rule. It is possible to construct a semi-separating sequential equilibrium if we allow for a mixed signaling strategy. However, this type of sequential equilibrium will also be ruled out using the re…nement proposed in the next subsection. 5.2 Disciplining o¤-equilibrium beliefs In the second paragraph on page 251 of M-OF-P, the authors summarize their criticism of the intuitive criterion and their proposed way of proceeding: Suppose that all players are equally forward looking and that an out-ofequilibrium message is to be interpreted as a clear signal that the informed player is of some type (or types). The types not sending that message should be making their choice (in particular of not sending that message) understanding the inference that will be made in the absence of that message. But if all types are sending the out-of-equilibrium message, the original message or yet some other message, understanding the beliefs that each message choice will lead to, these beliefs must then be part of a second sequential equilibrium. The equilibrium is not particularly important in itself, but what is important is that a proposed rule for making inferences can be consistent only if the resulting beliefs are of a sequential equilibrium. This paragraph suggests that the idea of their re…nement is to interpret the out-ofequilibrium messages as signals of some types of senders and that the beliefs based on those messages should not con‡ict with the senders’ incentives. However, di¤ering from other popular re…nements, e.g., Grossman and Perry’s Perfect Sequential Equilibrium (1986) and Cho and Krep’s Intuitive Criterion (1987), the re…nement of M-OF-P requires the out-ofequilibrium message to be part of an alternative equilibrium, so the incentives in sending a “signal”of this kind should be evaluated using the beliefs in the alternative equilibrium. The re…nement in this paper adopts this idea of evaluating senders’incentives and requires the out-of-equilibrium beliefs that support a sequential equilibrium to be "strongly coherent" in the following sense: De…nition 4 In an economy with a set of sender types , a candidate equilibrium fmC ; C ; C g, and an alternative equilibrium fmA ; A ; A g, the message gives rise to a "strongly coherent out-of-equilibrium belief" ' about a subset of types if: D.4.1) ; is an out-of-equilibrium message: 8! 2 , mC (!) = 6 D.4.2) A nonempty set of types sends in the alternative equilibrium and prefers the alternative equilibrium over the candidate equilibrium: 9 K 6= ? and K f! 2 jmA (!) = g such that: 8! 2 K; R (fmA ; and 9! 2 K; R (fmA; > R(fmC ; A; Ag ; !) > R(fmC ; A; A g ; !) C; C; C g; !) C g; !) where R (fm; ; g ; !) is the payo¤ to type ! in an equilibrium fm; ; g; D.4.3) The belief upon receiving is consistent with the set K: (~ ! ) Pr ( j~ !) ; (!) Pr ( j!) !2 8~ ! 2 K; ' (~ !j ) = P where (!) is the prior that the sender is of type ! and Pr ( j!) is the probability that type ! would send : 8 > if ! 2 K and R (fmA ; A ; A g ; !) > R(fmC ; C ; C g; !) < 1 Pr ( j!) = (14) 0 if ! 2 =K > : [0; 1] otherwise This re…nement in the spirit of M-OF-P eliminates any candidate equilibrium if its supporting belief concerning an out-of-equilibrium message is not strongly coherent, when such a strongly coherent belief exists. De…nition 5 (Strongly Coherent Sequential Equilibrium) A candidate sequential equilibrium fmC ; C ; C g is not "strongly coherent" if there exists a message A = mA (~ ! ) sent by ! ~ in an alternative equilibrium fmA ; A ; A g such that any strongly coherent belief about ! ~ upon receiving A , if it exists, is incompatible with the supporting belief for the candidate equilibrium: ' (~ ! j A ) 6= C (~ ! j A ) for any Pr ( A j~ ! ) satisfying (14). If such a message A does not exist, in other words, if the supporting beliefs for the candidate sequential equilibrium are strongly coherent whenever possible, we then say that the sequential equilibrium is "strongly coherent". To illustrate how this re…nement works, we start with the within-period signaling game at the …nal period. Because there is no future period, the opportunistic government always con…scates all household production, and the payo¤s to both types reduce to the momentary tax revenues. There is a continuum of pooling sequential equilibria with equilibrium tax rates ranging from zero to one. Given the inherited belief regarding government type derived from the previous periods, in each sequential equilibrium, the household production is = (1 ) . The equilibrium payo¤ is (1 ) to the trustworthy type and (1 ) to the opportunistic type. In deriving the optimal tax announcement in Sections 3 and 4, we impose the assumption on the announcement strategy of the opportunistic type, which implies ( ) = for all . As a result, the announced tax rate that maximizes the payo¤ to the trustworthy type is the unconstrained optimal rate U C = arg max (1 ) . We now show that among all pooling sequential equilibria, only the one with an equilibrium tax announcement equal to U C survives the re…nement that requires the out-of-equilibrium beliefs to be strongly coherent. To understand this result, …rst notice that among all pooling equilibria, the trustworthy type obtains the highest equilibrium payo¤ when the sequential equilibrium is pooling at U C . In contrast, the opportunistic type’s equilibrium payo¤ decreases with the equilibrium tax rate. Now consider those pooling equilibria with the announced tax rate H higher than U C . To sustain such an equilibrium tax rate, the households’ belief upon receiving an out-ofequilibrium message U C must be ( U C ) < because otherwise, the trustworthy government would obtain higher tax revenue by announcing U C rather than the equilibrium tax rate H . However, this supporting belief is not strongly coherent because the strongly coherent belief upon receiving U C is ' (T Rj U C ) = according to D.4.3) as both types of government are strictly better o¤ in the alternative equilibrium with the tax rate U C than in the candidate equilibrium with the tax rate H . Next consider pooling equilibria with the announced tax rate L lower than U C . The supporting belief upon receiving U C is again ( U C ) < to ensure that announcing L is optimal for the trustworthy type. Comparing the equilibrium payo¤s, the trustworthy type is strictly better o¤ in the U C equilibrium than in the L equilibrium, whereas the opportunistic type is strictly worse o¤. Thus, upon receiving an out-of-equilibrium message U C , the strongly coherent belief is ' (T Rj U C ) = 1, which would make announcing U C a pro…table deviation for the trustworthy type, thereby destroying the candidate equilibrium in which both types announce L : In contrast, the pooling equilibrium with the tax rate U C is strongly coherent. This equilibrium can be supported by the beliefs such that ( ) = 0 if < U C and ( ) = otherwise. Take any out-of-equilibrium message . If > U C , both types are strictly worse o¤ in the alternative equilibrium with the tax rate than in the candidate equilibrium with the tax rate U C . The set K de…ned in D.4.2) is empty and, in turn, no restrictions are placed on the out-of-equilibrium belief concerning . If < U C , only the opportunistic type is better o¤ in the alternative equilibrium with the tax rate . The strongly coherent out-of-equilibrium belief upon receiving is thus ' (T Rj ) = 0, which is identical to the supporting belief for the U C equilibrium. Therefore, the within-period signaling game at the …nal period of our repeated game has a unique, strongly coherent sequential equilibrium, in which the equilibrium tax rate is the same as that in the MPE de…ned by De…nition 2. This is not a coincidence: Proposition 4 The within-period signaling game in each period t has a unique, strongly coherent sequential equilibrium with the equilibrium tax rate equal to that in the MPE of period t de…ned by De…nition 2.27 6 Conclusion and Remarks This paper has presented a simple reputation game in which a trustworthy government can use announcements for its future policies as an instrument to accommodate public doubts concerning its ability to pre-commit or to facilitate public learning of its ability to commit. The unique Markov perfect equilibrium of the game reveals that it is optimal for a trustworthy government to separate itself from an opportunistic government only when the time discount factor of the opportunistic type is su¢ ciently low. If this condition does not hold, it will be too expensive for a trustworthy type to induce a separating outcome through active policymaking and, thus, reputation building will never occur in equilibrium. In addition, if the trustworthy type is su¢ ciently more patient than the opportunistic type, the optimal policy announcement will be aggressive in the sense that it is costly to ful…ll. In contrast, if both types share a similar patience, the optimal policy announcement will be more realistic and will likely be implemented. If we interpret the probability of getting reelected as a proxy for the patience of the government, the model then predicts that policy announcements are more conservative and more likely to be honored in countries with more intense interparty competition. If the popularity greatly di¤ers across party lines, it is more likely to observe aggressive policy announcements, and the chances that the government reneges on its announcements are higher. In addition, if the current government’s ancestor is known to be opportunistic or short-sighted, it is more likely for the current government to conduct reforms that may be costly in the short run to build a good reputation and thus would be bene…cial in the long run. 27 The author would like to thank an anonymous referee for pointing out the di¤erence between this re…nement and the re…nement of M-OF-P. In particular, the re…nement of M-OF-P requires the out-ofequilibrium message be preferred by all types who send the message in the alternative equilibrium. However, the re…nement in this paper argues that as long as there is a nonempty set of types that send an outof-equilibrium message in the alternative equilibrium and that prefer the alternative equilibrium, beliefs concerning this message should be consistent with the set. As a result, the re…nement in this paper rules out more sequential equilibria compared with the re…nement of M-OF-P. A very similar re…nement can be found in Taylor (1999).