Central bank reputation with noise
What this paper finds — and why it matters
Layer 1 — Overview
Research Question. How does noise in the mapping from central bank actions to realized inflation affect the existence and character of reputational equilibria in monetary policy? Specifically, can a central bank that faces uncertainty about whether it is perceived as “hawkish” or “dovish” sustain a pure strategy separating equilibrium, and how should each type behave as a function of its current reputation?
Model and Methodology. Amador and Phelan build on the monopolistic-competition, cash-in-advance framework of Chari, Christiano, and Eichenbaum (1998) and extend it to allow for (i) two central bank types — hawkish (type 1, high penalty γ₁ for inflationary actions) and dovish (type 2, lower penalty γ₂ < γ₁) — whose identity is private information; (ii) type switching governed by a Markov process, with probability δ that a hawkish bank is replaced by a dovish one and probability ε that a dovish bank is replaced by a hawkish one; and (iii) noise between the central bank’s chosen action μᵢ and realized money growth μₐ, which is drawn from a density f(μₐ|μᵢ) with full support. The equilibrium concept is pure symmetric Markov perfect equilibrium, in which all strategies are functions only of the public Bayesian posterior ρ that the current central bank is hawkish. The paper proceeds analytically to characterize no-pooling results and then computationally to demonstrate existence of separating equilibria.
Main Findings.
No pooling equilibria exist (analytical). Propositions 2 and 3 establish that no pure symmetric Markov equilibrium can have both types choosing the same positive action for any reputation ρ, as long as γ₁ ≠ γ₂ and Assumption 1 (pricing distortion sufficiently severe) holds. The intuition: if both types pool, realized inflation is uninformative, reputation does not change, and there are no dynamic incentives — but different static incentives (γ₁ ≠ γ₂) then imply different optimal actions, a contradiction.
Without sufficient noise, separating equilibria also fail to exist. In the no-noise limit, Bayesian updating forces the dovish bank’s reputation to jump to its maximum after one period of mimicking the hawkish action, making mimicry cheap when the discount factor β is high or the type-persistence probability ε is low. This makes the incentive-compatibility constraint for the dovish bank very difficult to satisfy, potentially precluding existence of a separating equilibrium.
With sufficient noise, pure strategy separating equilibria exist and have appealing properties (computational). The benchmark parameterization sets α = 1, σ = 5, β = 0.99, h(μ) = 0.5μ², ε = δ = 0.02, and the noise distribution such that the hawkish type’s unconstrained target would deliver mean inflation of 2% and the dovish type’s 3%. Under these parameters:
- In the full-information (known-type) world: price P = 1.313 for the hawkish type and P = 1.338 for the dovish type, with E[log(c) − αc] = −1.0297 and −1.0320 respectively, versus the efficient benchmark of −1.
- In the reputational equilibrium, both types choose lower inflationary actions than they would absent reputation considerations — because reputation is valuable (higher ρ lowers household prices and thus improves welfare for both types).
- Both types’ optimal actions are U-shaped in reputation ρ: they are most restrained — choosing the lowest inflationary actions — when ρ is middling (interior), because Bayesian updating is most sensitive (and thus the reputation cost of inflating is greatest) at interior beliefs, while it is difficult to move extreme beliefs.
- Average equilibrium inflation is 2.1%, which lies below the weighted average of unconstrained type targets (2.5% given equal switching probabilities), demonstrating that reputation concerns compress inflation outcomes.
Ergodic distribution of reputation remains interior. Starting from ρ = 0.5, expected reputation conditional on being hawkish stays below 0.63 and conditional on being dovish stays above 0.38, reflecting that noise and type switching prevent reputation from collapsing to its extremes.
Welfare implications. The hawkish type is made worse off by ongoing household uncertainty (relative to the reference game in which type is immediately revealed), while the dovish type is made better off. Households are better off under continuing uncertainty than under immediate revelation, unless reputation is near its maximum — because uncertainty suppresses inflationary temptations for both types.
Scope Conditions. Results apply within a monopolistic-competition, cash-in-advance economy with discrete time, infinite horizon, and Markov strategies. The no-pooling result requires Assumption 1 (the pricing distortion is sufficiently severe that the central bank has a positive incentive to inflate from μ = 0). The no-noise existence failure is an informal argument holding fixed discount and type-switching parameters. Computational results are specific to the benchmark parameterization but are verified to be robust to variation in β, σ, γ₁, γ₂, ε, and δ.
Layer 2 — Q&A
Q1: What is the fundamental time-inconsistency problem in the underlying Chari et al. (1998) economy, and how does the paper extend it?
A1: In the Chari et al. (1998) monopolistic-competition cash-in-advance economy, households exploit market power when setting prices, and the cash-in-advance constraint depresses consumption efficiency; this creates an ex-post temptation for the central bank to inflate and partially offset these distortions, even though in equilibrium such inflation is anticipated and only worsens inefficiencies. Equilibrium consumption equals (1/α) × ((σ−1)/σ) × (β/(1+μ)), compounding a monopoly distortion (σ−1)/σ < 1 and a cash-in-advance distortion β/(1+μ) < 1 below the efficient level 1/α. Amador and Phelan add household uncertainty about the central bank’s type — captured by the Bayesian posterior ρ that the bank is hawkish — allowing reputation to be endogenously determined and to feed back into equilibrium pricing.
Q2: Why does reputation matter only through differences in inflation costs γᵢ and not through differences in effective discount factors alone?
A2: Proposition 1 establishes that if γ₁ = γ₂ (equal inflation penalties), then even if the two types have different effective discount factors β₁ = β(1−δ) ≠ β₂ = β(1−ε), there exists a pooling Markov equilibrium in which both types choose the same action μ* and reputation plays no role. When both types have identical static incentives, they will always choose the same action given that reputation doesn’t affect payoffs in such an equilibrium. Hence the relevant dimension of heterogeneity for reputation to matter is the inflation cost parameter γᵢ, not patience.
Q3: What is the formal argument that no pooling equilibrium can exist when γ₁ ≠ γ₂?
A3: Propositions 2 and 3 provide the formal argument. If both types pool at any reputation ρ with a common positive action μ, Bayesian updating implies that ρ⁺ is independent of the money growth realization μₐ. The first-order condition for type i then reduces to the static condition (∂E[log(c) − αc|μ]/∂μ) = γᵢh’(μ), which cannot hold simultaneously for types 1 and 2 since γ₁ ≠ γ₂ and h’(μ) > 0 for μ > 0. This logic rules out pooling at the stationary reputation ρ* = ε/(δ+ε) in Proposition 2 and at any reputation where μ > 0 in Proposition 3.
Q4: Why does noise facilitate the existence of separating equilibria?
A4: Without noise, if types separate, observing the hawkish action reveals the bank is hawkish with certainty, pushing reputation to its maximum (1−δ) in a single period. This makes mimicry extremely cheap for the dovish type when β₂ is large or ε is small: the incentive compatibility condition requires that the dovish type’s static gain from choosing its own action exceeds the value gain from jumping to the best possible reputation, which is a very stringent requirement. With noise, mimicry generates only a probabilistic shift in beliefs rather than a discrete jump to the extreme, so the dovish type must maintain the hawkish action repeatedly to achieve a reputational gain — making mimicry costly enough that the incentive compatibility condition can be satisfied.
Q5: What is the “reference game” and what analytical purpose does it serve?
A5: The reference game is a variant in which the central bank’s type is fixed and is revealed to households immediately after they set prices at date t = 0. From t = 1 onward, the game reduces to the full-information, single-type game of Section 4. This allows the authors to isolate the “direct” effect of reputation — the fact that expected type affects equilibrium prices today — from the “indirect” or strategic effect of the central bank actively managing its reputation. In the numerical example, the reference-game prices form the upper dashed line in Figure 1, while the actual game’s prices form the lower solid line, with the gap between them attributable to the central bank’s incentive to restrain inflation in order to protect reputation.
Q6: What are the equilibrium price and welfare levels in the benchmark numerical example, and how do they compare to efficient and full-information benchmarks?
A6: The efficient benchmark delivers log(c) − αc = −1 with consumption c* = 1/α = 1. Under full information with only the hawkish type present, P = 1.313 and E[log(c) − αc] = −1.0297; under only the dovish type, P = 1.338 and E[log(c) − αc] = −1.0320. In the reputational equilibrium, prices lie below the full-information mixed benchmark for any given ρ (the solid line in Figure 1 lies below the dashed reference-game line), reflecting that the central banks’ desire to maintain reputation leads both types to restrain inflation beyond what the direct price effect alone would induce.
Q7: How does the U-shape of optimal central bank actions in reputation arise, and what does it imply for policy?
A7: The U-shape arises because Bayesian updating is most powerful at interior beliefs: for extreme reputations (near ε or 1−δ), any given realization of money growth moves the posterior relatively little, so the reputational cost of inflating is small. For interior (middling) reputations, the same action shifts the posterior substantially, making reputation more sensitive to inflation choices and thus increasing the marginal cost of inflating. Both types therefore choose their minimum inflationary actions at middling reputations. The policy implication is that a hawkish central bank with a very low reputation (following a run of high realized inflation outcomes) should not dramatically tighten, because further contraction does relatively little for its reputation until nature delivers enough favorable realizations to move it to a more interior range.
Q8: What happens to the ergodic distribution of reputation and inflation, and what does this imply about the persistence of reputational dynamics?
A8: Starting from ρ = 0.5, expected reputation remains in the interior: above 0.38 for the dovish type and below 0.63 for the hawkish type. The ergodic distribution of ρ (Figure 5) concentrates at interior values rather than the poles, showing that noise and type switching prevent reputation from stabilizing at extremes. The ergodic inflation distribution (Figure 6) has an average of 2.1%, compared to 2% under an all-hawkish world and 3% under an all-dovish world. Because ε = δ (types are equally likely in the long run), the unconstrained-type-weighted average would be 2.5%, so reputational incentives reduce equilibrium average inflation by approximately 0.4 percentage points.
Q9: Who gains and who loses from ongoing type uncertainty relative to immediate revelation?
A9: The hawkish type’s value function (Figure 3a) lies below the reference-game dashed line for intermediate reputations, indicating that the hawkish type is made worse off by uncertainty — it must bear the cost of restraining inflation beyond what is statically optimal in order to signal its type, but the households partially “blame” it for high realized inflation regardless. The dovish type (Figure 3b) is made better off under continuing uncertainty because its reputation benefits from households’ inability to perfectly distinguish types. Households (Figure 3c) are better off under uncertainty unless reputation is very high, because uncertainty suppresses inflation temptations for both types and keeps prices lower.
Q10: What happens to equilibrium behavior under robustness checks on key parameters?
A10: When the discount factor β or the elasticity of substitution σ decreases, both types inflate more and prices rise. When the hawkish type’s penalty γ₁ decreases (becomes less hawkish), both types inflate more and prices rise. When the dovish type’s penalty γ₂ decreases (becomes more dovish), the dovish type inflates more and, somewhat counterintuitively, the hawkish type inflates less, leaving prices roughly unchanged but slightly higher. When switching probabilities ε or δ increase, prices rise and both types inflate more, analogously to a decrease in β. Across all robustness exercises, the dovish type never inflates less than the hawkish type — consistent with Proposition 1’s implication that the inflation-cost difference γ₁ − γ₂ is the fundamental driver of separation.
Key Concepts
Hawkish type (type 1): A central bank that receives a relatively large negative payoff γ₁h(μᵢ) for taking inflationary actions, where γ₁ > γ₂. In the paper’s own sense, this type is not behavioral — it optimizes fully and can choose any action — but has a strong intrinsic cost to inflation, making it prefer lower money growth rates ceteris paribus.
Dovish type (type 2): A central bank with a lower penalty parameter γ₂ < γ₁ for inflationary actions. Like the hawkish type, it is fully strategic and optimizing, differing only in the magnitude of its intrinsic inflation cost.
Reputation (ρ): The Bayesian posterior probability that households assign to the current central bank being the hawkish type. It is the single payoff-relevant state variable in the Markov equilibrium, evolving through Bayes’ rule applied to realized money growth and type-switching probabilities.
Pure symmetric Markov perfect equilibrium: An equilibrium in which all households set the same price and consume the same amount (symmetry), and all strategies — prices P(ρ), central bank actions μ₁(ρ) and μ₂(ρ), and household consumption c(μₐ, ρ) — depend on history only through the current reputation ρ (Markov). The paper focuses exclusively on pure (non-mixed) strategy equilibria.
Pooling equilibrium: An equilibrium in which both types choose the same action μ₁(ρ) = μ₂(ρ) at some reputation ρ. The paper proves analytically that no pooling equilibrium can exist when γ₁ ≠ γ₂ and the pricing distortion is sufficiently severe (Assumption 1).
Separating equilibrium: An equilibrium in which μ₁(ρ) ≠ μ₂(ρ) for all ρ, so that realized money growth outcomes are informative about type and reputation evolves non-trivially. The paper argues that sufficient noise is necessary for such equilibria to exist.
Effective discount factor (βᵢ): The discount factor net of type-switching: β₁ = β(1−δ) for the hawkish type (which survives as hawkish with probability 1−δ) and β₂ = β(1−ε) for the dovish type. Central banks care only about payoffs while they are active, so effective discounting captures both time preference and expected tenure.
Noise (disconnection between actions and outcomes): The stochastic wedge between the central bank’s chosen action μᵢ and realized money growth μₐ, governed by a density f(μₐ|μᵢ) with full support. In the paper’s framework, noise is not merely a nuisance but a structural feature that makes reputational equilibria possible by preventing single-period complete revelation of type.