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This paper provides recursive methods for analyzing the credible equilibria of an incomplete information monetary policy game. The policy game is one in which a discretionary government has a short-run temptation to engage in a surprise monetary expansion. It also has the opportunity to imitate a government that always implements the ex ante optimal monetary policy actions. We derive an algorithm for computing the set of credible equilibrium payoffs. We use this algorithm to explore the idea that incomplete information monetary policy games of this kind have fewer equilibria and less indeterminacy than their complete information counterparts.

This paper considers the credibility of monetary policy in settings in which the government's attachment to a low monetary growth objective may initially be unknown. We assume two types of government. The first pursues a low monetary growth policy regardless of the behavior of private agents; the second seeks to maximize the utility of the representative household. This second or ‘discretionary’ type faces a short-run temptation to increase the monetary growth rate above the low ex ante optimal level. While it has no means of directly committing to a monetary policy, it can imitate a government with a preference for low monetary growth and in doing so, persuade private agents that low monetary growth is likely in the future. We investigate how this imitation option affects the structure of credible equilibria and the corresponding set of credible monetary policies. In pursuing this objective, we derive recursive methods for numerically characterizing credible equilibria in incomplete information policy games. These games are explicit in their treatment of private agents; they incorporate a simple, yet fully specified dynamic general equilibrium model. Our main qualitative result is that when the government is the discretionary type, but the public are initially uncertain of this, then as the representative household's discount factor converges to 1, the set of credible equilibrium payoffs collapses to a neighborhood of the ex ante optimal one. This result builds on, and extends to a monetary policy setting, the earlier game theoretic work of Fudenberg and Levine (1989). It implies that competitive allocations characterized by high monetary growth and inflation are not credible when the discretionary government is patient and can imitate. Imitation may or may not occur in equilibrium, but the option to imitate constrains the set of credible allocations and policies. Essentially, it provides a discretionary government with a means of managing the expectations of private agents. The contrast between incomplete and complete information games in which the government is known at the outset to be discretionary is striking. For all values of the discount factor, the latter admit low payoff equilibria. Such equilibria are characterized by severe ‘expectation traps’ in which firms anticipate high future monetary growth and raise prices at a correspondingly rapid rate. In the physical environment of this paper, the resulting inflation acts as a tax on the labor input, deterring effort and reducing consumption. A failure to accommodate creates a recession, but does not dampen these inflationary expectations. Consequently, it is optimal for the discretionary government to accommodate; it lacks the escape route that imitation provides in an incomplete information game. Moreover, in complete information settings, the degree of multiplicity is increasing in the government's discount factor and is large, not small, when the government is patient. A standard Folk theorem ensures that low monetary growth-low inflation allocations can be sustained as credible equilibria when the government is patient enough (see, for example, Ireland (1997) or Chari et al. (1998)). Such multiplicity and, in particular, the existence of extremely severe expectation trap equilibria is often seen as a problem for complete information models.1 While our main qualitative result is theoretically valuable, it is a limiting result. It does not help in determining whether a particular allocation and policy are credible given a specific configuration of parameters for agent preferences, technologies and initial beliefs. To address this sort of question, a quantitative evaluation of the set of credible equilibria in incomplete information games is needed. To this end, we derive an algorithm that exploits the recursive structure of such equilibria. We show that their history is completely summarized by three state variables: private agent beliefs (over government types), a policy statistic and the continuation payoff to the government. Additionally, these games admit an equilibrium value correspondence that gives combinations of beliefs, policy statistics and payoffs that are consistent with the initial period of a credible equilibrium. Adapting arguments in Abreu et al. (1990), we show that these equilibrium value correspondences solve a functional equation analogous to the Bellman equation in dynamic programming. We use this equation to obtain an iterative procedure for computing them. This allows us to numerically evaluate the set of equilibrium payoffs at any configuration of parameters and, hence, determine whether a particular allocation and policy is credible or not. The remainder of the paper proceeds as follows. Section 2 lays out the basic economic environment, while Section 3 describes the game. Section 4 verifies that a discretionary government can, through imitation, build a reputation for choosing a low monetary growth rate, and, hence, if it is sufficiently patient, guarantee itself a payoff close to the ex ante optimal one. In Section 5, we isolate the recursive structure of credible equilibria, and obtain the algorithm for computing their value correspondences. In Section 6, this algorithm is put to work and the set of credible equilibrium payoffs computed. We explore how this set depends on the parameters of the environment. In particular, we explore its dependence on the discount factor. 1.1. Related literature The literature on the credibility of monetary policy has two strands. The first makes very reduced form assumptions about the behavior of private agents, but considers a variety of different information structures. In some models, the preferences of the government are unknown; in others, the government's information or policy action is unobserved. The second strand embeds a large government into a fully specified general equilibrium model.2 This approach has the advantage of relating the costs and benefits of different policy actions to the underlying preference and technology parameters of the economy. Once these parameters are calibrated, a quantitative assessment of credible equilibria and the allocations they are capable of sustaining can be conducted. Papers pursuing this approach, however, concentrate on complete information games in which the government is known to be discretionary. Our paper combines an information structure considered in the first strand, with a fully specified model of the sort considered in the second. The information structure we assume is similar to that in the reduced form models of Backus and Driffill (1985) and Barro (1986). We depart from these earlier contributions both in our explicit treatment of private agents and in our focus on the entire set of credible equilibria in an infinite horizon game. Backus and Driffill (1985), and Barro (1986) focus on specific equilibria in finite horizon settings. Our basic physical environment is similar to that in Chari et al. (1998), who coin the term expectation traps, and Ireland (1997). Both of these papers are explicit in their modelling of private agents, but restrict attention to games in which the government is known to be discretionary. Other contributions to the second strand of the literature described above include Chang (1998) and Sleet (2001). Fernandez-Villeverde and Tsyvinski (2003), Phelan and Stacchetti (2001) and Sleet and Yeltekin (2006), build on Abreu et al. (1990), to obtain recursive methods for analyzing fully specified fiscal policy games in various settings.

In this paper we analyze a monetary policy game in which there is uncertainty over the government's type. The game incorporates an explicit treatment of the choice problems of private agents and links the costs and benefits of policy to the parameters of these agents’ preferences and technologies. We show that in contrast to complete information policy games, severe expectation traps are eliminated as equilibrium outcomes when the government is patient. In our game, a discretionary policymaker can manage the expectations of agents by imitating governments committed to particular actions. This effectively allows the policymaker to escape an expectation trap. Methodologically, we show that equilibria in these settings have a recursive structure which allows us to obtain an algorithm for numerically characterizing them. We have made two strong simplifying assumptions about the government. First, we have assumed that government types are permanent. The permanent types case has been considered in the reduced form literature and is a natural starting point; it simplifies the analysis by making revelation once and for all. It would be interesting, however, to consider models in which the government's type follows a richer process.11 Our second simplifying assumption is the existence of a government that always plays the Ramsey policy action. An important extension of the model would provide micro-foundations for this government. For example, suppose there are both fixed and flexible price firms and two groups of households, one who chooses its money holdings after the government has moved and the other before. This second group would find the real value of any cash it had previously set aside reduced in the aftermath of monetary injection as flexible price firms raise their prices. Consequently, its consumption would fall. A government type that cared sufficiently about the (current) well being of this group of households might then avoid monetary injections even if they stimulated output and the consumption of the first group of households. We conjecture that such a government might resemble the commitment type of the current paper.