## wpiea2020085-print-pdf — Introduction, Section 3.1, and Section 6.4

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---

### Overview and purpose
- Macroeconomic models assign expectations about future policy a large role in current outcomes; policy is typically modeled under commitment or discretion.
- Governments attempt to influence beliefs via announcements (forward guidance, inflation targets, fiscal rules, timing of policies) that rarely bind future choices but shift expectations.
- This paper develops a rational-expectations theory of government credibility based on reputation in game theory (Kreps and Wilson,1982; Milgrom and Roberts,1982).
- Key insight: a rational government can mimic “behavioral” types (stubborn plan-followers) to earn reputation, because the private sector is uninformed about the government’s type and updates beliefs after announcements and actions.

### Model setup and objectives
- Government minimizes discounted expected loss:
  - L0 = E0[Σ_{t=0}^∞ β^t[(y⋆ − y_t)^2 + γ π_t^2]] where γ≥0 and β∈(0,1).
- Phillips curve:
  - π_t = κ y_t + β E_t[π_{t+1}] with κ≥0.
- Government imperfectly controls inflation:
  - π_t = g_t + σ ε_t, with ε_{iid} ∼ N(0,1).
- Behavioral types:
  - Set C of types c ∈ C, each committed to an inflation plan (a^c_t)_{t≥0}.
  - Government is rational with probability 1−z; behavioral types have total probability z with density ν over C.

### Timing, announcements, and inference
- At time 0 the government announces targets a = (a_t)_{t≥0} for all future periods.
- If government is behavioral type c, it announces (a^c_t)_t for sure. The rational type chooses an announcement r (possibly in C).
- Observing announcement c ∈ C, the private sector assigns probability 1 that the government is either rational or behavioral type c.
- Posterior probability (starting reputation) after announcement:
  - p_0(c; z, μ) = z ν(c) / [ z ν(c) + (1−z) μ(c) ]  (equation (4))
- Posterior update after observing realized π_t:
  - p_{t+1} = p_t · f_ε(π_t − a_t) / [ p_t · f_ε(π_t − a_t) + (1−p_t) · f_ε(π_t − g⋆_t) ]
  - Alternative form:
    - p_{t+1} = p_t + p_t (1−p_t) [ f_ε(π_t − a_t) − f_ε(π_t − g⋆_t) ] / [ p_t f_ε(π_t − a_t) + (1−p_t) f_ε(π_t − g⋆_t) ]  (equation (5))
- Reputation moves most when (i) p_t is away from 0 and 1 and (ii) realized inflation is closer to either announced target a_t or rational strategy g⋆_t.

### Parametrization of behavioral types and announced plans
- Behavioral type c parameterized by (a_0, ω, χ) with path:
  - a^c_t = (a_0 − χ) e^{−ω t} + χ
- Recursive form:
  - a^c_{t+1} = χ + e^{−ω}(a^c_t − χ) = φ_c(a_t).
- Plans bounded above by static Nash inflation π^N: a_0, χ ∈ A = [0, π^N].
- C contains constant, decreasing, and increasing paths depending on parameters.

### Government’s Bellman problem after announcement (continuation equilibrium)
- For announced plan c, rational government solves:
  - L_c(p, a) = min_g E[ (y⋆ − y)^2 + γ π^2 + β L_c(p', φ_c(a)) ]
  - Subject to:
    - π = g + ε
    - π = κ y + β[ p' φ_c(a) + (1−p') g⋆(p', φ_c(a)) ]
    - p' = p + p(1−p) [ f_ε(π − a) − f_ε(π − g⋆(p,a)) ] / [ p f_ε(π − a) + (1−p) f_ε(π − g⋆(p,a)) ]
- Expected inflation in Phillips curve is weighted average between φ_c(a) (behavioral action) and g⋆(p', φ_c(a)) (expected rational choice).
- By choosing current inflation levels, the rational type influences reputation evolution p' and thus expected inflation and continuation value.

### Analytical properties and lemmas
- Continuation equilibrium defined: for announcement c, a pair (L_c, g⋆_c) where L_c solves the Bellman given g⋆_c and g⋆_c corresponds to solution of that Bellman.
- Lemma 1:
  - In any continuation equilibrium the rational type’s reputation is a supermartingale:
    - E[p_{t+1} | rational, F_t] ≤ p_t
  - Implication: the planner cannot design a policy that generates expected reputational gains; expected reputation cannot be increased by strategy.

### Reputational equilibrium and K-equilibrium
- Reputational equilibrium (given initial z): distribution μ_z over C plus continuation equilibria {L_c, g⋆_c} and posterior p_0 satisfying:
  1. Posterior p_0 set by Bayes’ rule (equation (4)) given μ_z.
  2. μ_z minimizes starting reputation-adjusted loss:
     - L⋆_r(μ_z, z) = ∫_C L_c( p_0(c), a_0(c) ) dμ_z(c)
- In reputational equilibrium planner is indifferent among plans in supp(μ_z):
  - L_c(p_0(c), a_0(c)) = L_{c'}(p_0(c'), a_0(c')) for c, c' ∈ supp(μ_z)
  - Any plan not in support must satisfy L_c(p_0(c), a_0(c)) ≤ L_{c'}(1, a_0(c')) for c ∈ supp(μ_z), c' ∉ supp(μ_z).
- K-equilibrium (for given p_0 ∈ [0,1]):
  - c⋆_K(p_0) = argmin_c L_c(p_0, a_0(c))
  - Interest centers on limit as p_0 → 0 of c⋆_K(p_0).

### Construction of reputational equilibria (procedural)
- Given k ∈ R, partition plans by whether L(1,c) ≷ k. Plans with L(1,c) > k have μ(c) = 0.
- For remaining plans find p_0(c) solving L(p_0(c), c) = k.
- Bayes’ rule implies μ(c) such that p_0(c) = z ν(c) / [ z ν(c) + (1−z) μ(c) ].
- Normalize μ(c) so it integrates to 1 over C; this pins down k and μ.

### Main qualitative results (Introduction summary)
- Optimal announced policy typically features inflation starting high and diminishing gradually (except perhaps when initial reputation is very high).
- Gradual disinflation is more credible: a higher target today than tomorrow boosts gains from sticking to the plan and slows reputational losses enough to offset higher announced inflation’s negative effects on expected inflation.
- Contrast with patient-limit reputation literature where static zero-inflation plan can achieve commitment payoffs; with imperfect control gradualism emerges.
- Limit as initial reputation z → 0:
  - At z = 0 the only Markov equilibrium is repetition of static Nash (high inflation, natural output).
  - Small reputation z > 0 creates large departures from Nash.
  - Gradualist property persists along the zero-reputation limit.

### Numerical implementation: benchmark parametrization (Table 1)
- β = 0.995 — Discount factor (2% real interest rate)
- γ = 60 — Inflation weight (King, Lu, and Pastén (2016))
- σ = 1% — Std of control shock (King, Lu, and Pastén (2016))
- κ = 0.17 — Slope of Phillips curve (King, Lu, and Pastén (2016))
- y⋆ = 5% — Output target (King, Lu, and Pastén (2016))

### Continuation equilibrium: numerical observations
- L_c(p,a):
  - L is decreasing in p: higher reputation reduces expected inflation ⇒ higher current output.
  - L has convex–concave shape: near p = 0 or p = 1 the public is confident; reputation near bounds is harder to move.
  - At high p, a lower current target a is unambiguously better; at low p a lower a can increase expected reputation loss.
- Deviation behavior g⋆(p,a) − a:
  - As p → 0, governments tend to deviate more from the announced target.
  - Discontinuity at p = 0: government reverts to Nash inflation regardless of announcements because Bayes’ rule prevents p from moving.
  - Lower a ⇒ rational types set mean inflation further from a (costlier to keep low a).
- Expected reputation dynamics:
  - E[p′] is always below p (Lemma 1).
  - For announcements equal to Nash inflation, g = a for all p and reputation does not move.
  - Announcements lower than Nash lead to average reputation declines; lower announcements yield larger expected reputation loss.
  - Reputation is harder to move when p near 0 or 1; government ‘spends’ more reputation when p is larger.

### Announcements in the K-equilibrium and distributional properties
- K-equilibrium characterizations:
  - At p0 = 1: planner can promise zero inflation throughout; rational type intends to break promise but private sector fully believes it.
  - For p0 < 1: planner values incentivizing future governments and prefers plans with higher initial inflation a0 and gradual disinflation (a0 > χ).
  - Even as p0 → 0, planner prefers a0 > χ (gradual disinflation).
- Loss minimization over (a0, ω, χ):
  - K-equilibrium minimum achieved at ω, χ > 0: initial promise of gradual disinflation that does not converge to zero inflation.
  - Low χ ⇒ plans imply very low future inflation that is hard to sustain ⇒ planner prefers slower decay (small ω).
  - High χ ⇒ planner can use faster decay to provide short-run incentives.
- Credibility at vanishing p:
  - Plans with lower asymptote χ are less credible.
  - Plans with steeper promised descent (higher decay) are less credible.
- Limiting distribution μ⋆ = lim_{z→0} μ_z:
  - Planner tends to choose gradual plans with a0 > χ:
    - P(a0 > χ) = 71.3%.
    - P(a0 > 5 χ) = 18%.
  - Asymptote χ is more tightly clustered across announced plans than initial inflation a0.
  - Density over decay rate ω: planner does not play plans with very low ω often; μ⋆ falls sharply as ω becomes small.
  - Relationship to K-equilibrium: plans with lower K-equilibrium loss are announced more often; initial reputation in those plans is lower which lowers their equilibrium value and yields mixed strategy of announcements.

### Comparative statics (selected)
- Control shock variance σ:
  - As σ increases (more noise), deviations become less observable ⇒ adherence to plans decreases.
  - Result: average plan has higher asymptote χ, slightly higher a0, and slower decay ω as σ increases.
- Discount factor β and Phillips slope κ:
  - Varying impatience (1/β − 1; benchmark 2% annual):
    - More impatient planner ⇒ average plans start higher, converge to lower inflation, and have faster decay.
  - Varying Phillips curve slope κ:
    - Steeper Phillips curve ⇒ planner announces lower inflation throughout.

### Comparative perspectives with other solution concepts
- Ramsey plan (full commitment):
  - Linear-quadratic; shocks only affect via variance.
  - Ramsey plan achieves near-complete disinflation: Ramsey inflation ≈ 0 about 1.5 years after announcement.
  - Ramsey plan starts at a high initial inflation (≈ three quarters of the way to Nash inflation) then disinflates.
  - Contrast: reputational equilibrium announcements typically do not converge to zero (low probability χ = 0); K-equilibrium starts above Ramsey plan.
- Sustainable plans with reputation/punishments:
  - With perfect control and punishment expectation ξ, if ξ harsh enough the Ramsey plan is recoverable; otherwise planner best-responds to expectations when Ramsey not sustainable.
  - With imperfect control, private sector may trigger punishment on-path (false positives); deterrence must be weighed against false-positive costs.
  - Imperfect control prevents promises of zero inflation or Ramsey plan even when punishment expectations exist.

### Section 3.1: Reputation-building incentives (first-order and bounds)
- Output response to inflation (from solving the Phillips curve):
  - ∂y/∂π = 1/κ [1 − β ∂p′/∂π (φc(a) − g⋆(p′, φc(a)) + (1 − p′) ∂g⋆(p′, φc(a))/∂p′)]  (equation (7))
- Channels by which inflation affects current output:
  - Direct effect: 1/κ · 1.
  - Expectation-shifting effect: β 1/κ (−∂p′/∂π) (φc(a) − g⋆(p′, φc(a))).
  - Reputation effect: β 1/κ (−∂p′/∂π) (1 − p′) ∂g⋆(p′, φc(a))/∂p′.
- Lemma 2:
  - In any continuation equilibrium, the rational type’s choice of inflation is bounded above by Nash:
    - ∀c ∈ C: g⋆c(p,a) ≤ πN.
  - Nash-level inflation first-order condition:
    - πN = y⋆ κ 1 − β + κ 2 γ.
- Credibility of a plan (definition (8)):
  - C(p,a;c) = E[(1 − β) (πN − π)/(πN − a) + β C(p′c(p,a), φc(a))]
  - Alternate form:
    - C(p,a;c) = (1 − β) (πN − [pa + (1 − p) g⋆c(p,a)])/(πN − a) + β E[C(p′c(p,a), φc(a))]
- Credibility limits:
  - C_K(c) = lim_{p→0} C(p, a0(c); c)
  - C⋆(c) = lim_{z→0} ∫ C(p0(c), a0(c); c) dμ_z(c)

### Section 6.4: Recursive plans with reputation (setup and planner’s problem)
- Government composed of two agents each period: a “morning” planner and an “afternoon” policy maker.
- Timing:
  - Morning of period t: planner announces a policy recommendation for the afternoon of t+1 (given earlier announcement for the afternoon of t).
  - Afternoon of each period: policy maker chooses inflation π.
- Policy maker types:
  - Commitment type: stubbornly follows the recommendation.
  - Rational type: chooses whether to follow it.
- Planner and private sector do not know the policy maker’s type; they form statistical inference from past inflation realizations.
- Planner’s recursive formulation when facing target a and current reputation p:
  - v_R(p,a) = min_{g,a'} E[(y−y^⋆)^2 + γ π^2 + β v_R(p',a')]
  - Subject to:
    - π = g + ε
    - π = κ y + β ( p' a' + (1−p') g_R(p',a') )
    - p' = p + p(1−p) * [ f_ε(π−a) − f_ε(π−g_R(p,a)) ] / [ p f_ε(π−a) + (1−p) f_ε(π−g_R(p,a)) ]
- Planner’s initial value of interest:
  - J_R = lim_{p→0} min_a v_R(p,a)

### Numerical and qualitative findings for recursive plans
- Full reputation:
  - Planner announces zero inflation throughout and intends to break it (no effect on expectations).
- Moderate initial reputation:
  - Planner announces a positive level of initial inflation that converges to zero inflation.
- Low initial reputation:
  - The plan stops converging to zero and produces a pattern similar to the K-equilibrium.
- Recursive plans differ from reputational/K-equilibria in two ways:
  1. Reputational and K-equilibrium: inflation targets for all periods are chosen at the beginning (pre-announced).
  2. Recursive plan: targets chosen sequentially and allow feedback between reputation evolution and future targets.
- Two opposing effects:
  - Flexibility benefit: recursive plan can tailor future announcements to current credibility assessments.
  - Pre-announcement benefit: reputational and K-equilibria use partial future credibility to induce present expectations.
- In chosen parametrization, gains from pre-announcing appear to be of the same magnitude as gains from flexibility.
- Time-consistency result:
  - Sequential announcements do not capture the commitment benefit of pre-announcing, so recursive plan has a higher long-run asymptote for inflation than reputational and K-equilibrium plans — a second form of time-inconsistency in the Ramsey plan.

### Quantitative comparison (Table 2 — preserved exactly)
- Initial inflation:
  - Ramsey: 1.40%
  - K-equilibrium: 1.63%
  - ‘Average’ recursive plan: 1.58%
  - Recursive plan: 1.58%
- Long-run inflation:
  - Ramsey: 0%
  - K-equilibrium: 0.44%
  - ‘Average’ recursive plan: 0.65%
  - Recursive plan: 0.65%
- Value of loss function:
  - Ramsey: 0.3364
  - K-equilibrium: 0.7552
  - ‘Average’ recursive plan: 0.7589
  - Recursive plan: 0.7554

### Main policy implications and concluding insights
- Can reputation substitute for commitment?
  - Reputation combined with imperfect control creates incentives for staying close to announced targets.
  - Central bank’s optimal policy after announcement trades off surprise inflation benefits against the risk that a deviation is discovered, making reputation an important state variable under discretion.
- Gradualism:
  - Gradualism is a pervasive feature of optimal plans: planner backloads low inflation to make reputation maintenance both easy and valuable.
  - Gradualism arises from incentive considerations, not from inflation inertia per se.
  - Gradualist property holds at positive reputation levels and in the limit as initial reputation vanishes to zero; the limit is interpreted as a sensible refinement of the game.
- Interpretation:
  - Reputation effectively modifies the incentive constraint in the recursive planning problem; when reputation is low, large option values of sticking to the plan arise and are larger when the plan is backloaded.

*Source: wpiea2020085-print-pdf — Introduction, Section 3.1, and Section 6.4*

### Introduction6

### Introduction6

### Overview
- Macroeconomic models assign expectations about future policy a large role in current outcomes; policy is typically modeled under commitment or discretion.
- Governments attempt to influence beliefs via announcements (forward guidance, inflation targets, fiscal rules, timing of policies) that rarely bind future choices but shift expectations.
- This paper develops a rational-expectations theory of government credibility based on reputation in game theory (Kreps and Wilson,1982; Milgrom and Roberts,1982).
- Key insight: a rational government can mimic “behavioral” types (stubborn plan-followers) to earn reputation, because the private sector is uninformed about the government’s type and updates beliefs after announcements and actions.

### Model setup
- The government minimizes discounted expected loss:
  L0 = E0[Σ_{t=0}^∞ β^t[(y⋆ − y_t)^2 + γ π_t^2]]
  where γ≥0 and β∈(0,1).
- Phillips curve:
  π_t = κ y_t + β E_t[π_{t+1}] with κ≥0.
- Government imperfectly controls inflation:
  π_t = g_t + σ ε_t, with ε_{iid} ∼ N(0,1).
- Behavioral types: set C of types c ∈ C, each committed to an inflation plan (a^c_t)_{t≥0}.
- The government is rational with probability 1−z; behavioral types have total probability z with density ν over C.

### Timing and announcements
- At time 0 the government announces targets a = (a_t)_{t≥0} for all future periods.
- If the government is behavioral type c, it announces (a^c_t)_t for sure. The rational type chooses an announcement r (possibly in C).
- Observing announcement c ∈ C, the private sector assigns probability 1 that the government is either rational or behavioral type c.
- At time t the government sets inflation: behavioral type sets g_t = a^c_t; the rational type chooses g_t strategically. Realized inflation includes noise, so the private sector updates beliefs via Bayes’ rule.

### Beliefs and reputation dynamics
- Posterior probability the government is behavioral type c after announcement (starting reputation):
  p_0(c; z, μ) = z ν(c) / [ z ν(c) + (1−z) μ(c) ]  (equation (4))
- Update after observing realized π_t (with g⋆_t denoting rational strategy):
  p_{t+1} = p_t · f_ε(π_t − a_t) / [ p_t · f_ε(π_t − a_t) + (1−p_t) · f_ε(π_t − g⋆_t) ]
- Alternative form emphasizing movement:
  p_{t+1} = p_t + p_t (1−p_t) [ f_ε(π_t − a_t) − f_ε(π_t − g⋆_t) ] / [ p_t f_ε(π_t − a_t) + (1−p_t) f_ε(π_t − g⋆_t) ]  (equation (5))
- Reputation moves most when (i) p_t is away from 0 and 1 and (ii) realized inflation is closer to either announced target a_t or rational strategy g⋆_t.

### Parametrization of behavioral types
- Each behavioral type c is parametrized by (a_0, ω, χ) with path:
  a^c_t = (a_0 − χ) e^{−ω t} + χ
- Recursive form: a^c_{t+1} = χ + e^{−ω}(a^c_t − χ) = φ_c(a_t).
- Plans bounded above by static Nash inflation π^N: a_0, χ ∈ A = [0, π^N].
- C contains constant, decreasing, and increasing paths depending on parameters.

### Government’s Bellman problem after announcement
- For announced plan c, the rational government’s recursive problem (loss function L_c) is:
  L_c(p, a) = min_g E[ (y⋆ − y)^2 + γ π^2 + β L_c(p', φ_c(a)) ]
  subject to:
  π = g + ε
  π = κ y + β[ p' φ_c(a) + (1−p') g⋆(p', φ_c(a)) ]
  p' = p + p(1−p) [ f_ε(π − a) − f_ε(π − g⋆(p,a)) ] / [ p f_ε(π − a) + (1−p) f_ε(π − g⋆(p,a)) ]
- Expected inflation in the Phillips curve is a weighted average between φ_c(a) (behavioral action) and g⋆(p', φ_c(a)) (expected rational choice).
- By choosing current inflation levels, the rational type influences reputation evolution p' and thus affects expected inflation and continuation value.

### Key analytical properties and lemmas
- Continuation equilibrium defined: for announcement c, a pair (L_c, g⋆_c) where L_c solves (6) given g⋆_c and g⋆_c corresponds to solution of (6).
- Observation: given decay and asymptote parameters, starting a plan at different initial announcement a_0 is equivalent to arriving at current announcement a as continuation equilibrium unfolded.
- Lemma 1: In any continuation equilibrium the rational type’s reputation is a supermartingale:
  E[p_{t+1} | rational, F_t] ≤ p_t
  i.e., the planner cannot design a policy that generates expected reputational gains.
- Implication: the planner cannot expect to increase reputation over time by strategy; it can design plans that provide incentives to deliver on promises, but not raise expected reputation.

### Reputational equilibrium and K-equilibrium
- Reputational equilibrium (given initial z): a distribution μ_z over C plus continuation equilibria {L_c, g⋆_c} and posterior p_0 satisfying:
  1. Posterior p_0 set by Bayes’ rule (4) given μ_z.
  2. μ_z minimizes starting reputation-adjusted loss:
     L⋆_r(μ_z, z) = ∫_C L_c( p_0(c), a_0(c) ) dμ_z(c)
- In reputational equilibrium, planner is indifferent among plans in supp(μ_z):
  L_c(p_0(c), a_0(c)) = L_{c'}(p_0(c'), a_0(c')) for c, c' ∈ supp(μ_z)
  and any plan not in support must satisfy L_c(p_0(c), a_0(c)) ≤ L_{c'}(1, a_0(c')) for c ∈ supp(μ_z), c' ∉ supp(μ_z).
- K-equilibrium (for given p_0 ∈ [0,1]): announcement c and continuation equilibrium where
  c⋆_K(p_0) = argmin_c L_c(p_0, a_0(c))
- Interest centers on limit as p_0 → 0 of c⋆_K(p_0).

### Construction of reputational equilibria
- Procedure: given k ∈ R, partition plans by whether L(1,c) ≷ k. Plans with L(1,c) > k have μ(c) = 0.
- For remaining plans find p_0(c) solving L(p_0(c), c) = k.
- Bayes’ rule implies μ(c) such that p_0(c) = z ν(c) / [ z ν(c) + (1−z) μ(c) ].
- Normalize μ(c) so it integrates to 1 over C; this pins down k and μ.

### Main results (from introduction summary)
- Optimal announced policy typically features inflation starting high and diminishing gradually (except perhaps when initial reputation is very high).
- Gradual disinflation is more credible: a higher target today than tomorrow boosts gains from sticking to the plan and slows reputational losses enough to offset higher announced inflation’s negative effects on expected inflation.
- Contrast with reputation literature in the patient limit where a static zero-inflation plan can achieve commitment payoffs; with imperfect control gradualism emerges.
- Limit as initial reputation z → 0: at z = 0 the only Markov equilibrium is repetition of static Nash (high inflation, natural output), but as usual even small reputation z > 0 creates large departures from Nash. The gradualist property of optimal plans persists along the zero-reputation limit.

### Literature context and layout
- Builds on Barro (1986), Backus and Driffill (1985), Sleet and Yeltekin (2007), Dovis and Kirpalani (2019); key departure is assumption of imperfect control over inflation which smooths inference and enables tradeoffs that shape optimal plans.
- Related to literature on sustainable plans (Abreu, Pearce, and Stacchetti,1990; Chari and Kehoe,1990; Phelan and Stacchetti,2001) and models using imperfect control (Phelan,2006; Cukierman and Meltzer,1986; Faust and Svensson,2001).
- Distinguishes from cheap-talk models (Turdaliev,2010) because announcements cannot be sent by all behavioral types.
- Empirical/analytical focus in paper: Sections 2–7 develop model, equilibrium notions, main results, comparative statics, connections to other models, and concluding remarks.

*Source: wpiea2020085-print-pdf - Introduction6*

### 3.1  Reputation-building incentives

### wpiea2020085-print-pdf - 3.1  Reputation-building incentives

### Reputation-building incentives (first-order conditions)
- Output response to inflation (from solving the Phillips curve):
  - ∂y/∂π = 1/κ [1 − β ∂p′/∂π (φc(a) − g⋆(p′, φc(a)) + (1 − p′) ∂g⋆(p′, φc(a))/∂p′)] (equation (7))
- Three channels by which inflation affects current output (terms in equation (7)):
  - Direct effect: 1/κ · 1.
  - Expectation-shifting effect: β 1/κ (−∂p′/∂π) (φc(a) − g⋆(p′, φc(a))).
  - Reputation effect: β 1/κ (−∂p′/∂π) (1 − p′) ∂g⋆(p′, φc(a))/∂p′.

### Reputation and credibility (definitions and bounds)
- Nash-level inflation (stage-game or Markov equilibrium when p = 0): πN.
- First-order condition in Nash: πN = y⋆ κ 1 − β + κ 2 γ.
- Lemma 2: In any continuation equilibrium, the rational type’s choice of inflation is bounded above by Nash:
  - ∀c ∈ C: g⋆c(p,a) ≤ πN.
- Credibility of a plan (remaining credibility in state (p,a); definition (8)):
  - C(p,a;c) = E[(1 − β) (πN − π)/(πN − a) + β C(p′c(p,a), φc(a))]
  - Alternate form: C(p,a;c) = (1 − β) (πN − [pa + (1 − p) g⋆c(p,a)])/(πN − a) + β E[C(p′c(p,a), φc(a))]
- Credibility limits:
  - C_K(c) = lim_{p→0} C(p, a0(c); c)
  - C⋆(c) = lim_{z→0} ∫ C(p0(c), a0(c); c) dμ_z(c)

### Numerical implementation: parametrization (benchmarks)
- Parameter choices (Table 1: BenchmaRK calibRation):
  - β = 0.995 — Discount factor (2% real interest rate)
  - γ = 60 — Inflation weight (King, Lu, and Pastén (2016))
  - σ = 1% — Std of control shock (King, Lu, and Pastén (2016))
  - κ = 0.17 — Slope of Phillips curve (King, Lu, and Pastén (2016))
  - y⋆ = 5% — Output target (King, Lu, and Pastén (2016))

### Continuation equilibrium after announcement c (numerical observations)
- Loss function L_c(p,a):
  - L is decreasing in p: higher reputation reduces expected inflation ⇒ higher current output.
  - L has convex–concave shape: near p = 0 or p = 1 the public is confident; reputation near bounds is harder to move.
  - At high p, a lower current target a is unambiguously better (reduces expectations); at low p a lower a can increase expected reputation loss.
- Deviation behavior g⋆(p,a) − a:
  - As p → 0, governments tend to deviate more from the announced target.
  - Discontinuity at p = 0: government reverts to Nash inflation regardless of announcements because Bayes’ rule prevents p from moving.
  - Lower a ⇒ rational types set mean inflation further from a (costlier to keep low a).
- Expected reputation dynamics:
  - E[p′] is always below p (Lemma 1).
  - For announcements equal to Nash inflation, g = a for all p and reputation does not move.
  - Announcements lower than Nash lead to average reputation declines; lower announcements yield larger expected reputation loss.
  - Reputation is harder to move when p near 0 or 1; government ‘spends’ more reputation when p is larger.

### Announcements in the K-equilibrium (preferred plans as function of p0)
- K-equilibrium characterizations (Figure 5):
  - Shown variables: decay rate 1 − e^{−ω} (in percent), initial inflation a0, and asymptote χ.
  - At p0 = 1: planner can promise zero inflation throughout; rational type intends to break promise but private sector fully believes it.
  - For p0 < 1: planner values incentivizing future governments and prefers plans with higher initial inflation a0 and gradual disinflation (a0 > χ).
  - Even as p0 → 0, planner prefers a0 > χ (gradual disinflation).
- Loss minimization over (a0, ω, χ) (Figure 6):
  - K-equilibrium minimum achieved at ω, χ > 0: initial promise of gradual disinflation that does not converge to zero inflation.
  - Low χ ⇒ plans imply very low future inflation that is hard to sustain ⇒ planner prefers slower decay (small ω).
  - High χ ⇒ planner can use faster decay to provide short-run incentives.
  - χ cannot approach Nash inflation (too easy to keep; small gains).

### Credibility results (vanishing reputation)
- Credibility C at vanishing p (Figure 7; using loss-minimizing a0 at parameters):
  - Plans with lower asymptote χ are less credible.
  - Plans with steeper promised descent (higher decay) are less credible.
  - Interpretation caveat: fast-decay plans reach near-constant targets quickly, which influences measured credibility.

### Distribution of announcements in the reputational equilibrium
- Average announced plan as function of initial reputation z (Figure 8):
  - For intermediate z: average disinflation path starts at about half Nash inflation, converges toward about a tenth of Nash, halving distance each period.
  - As z → 0 planner puts more weight on plans converging to higher asymptote χ.
- Limiting distribution μ⋆ = lim_{z→0} μ_z (Figure 9):
  - Planner tends to choose gradual plans with a0 > χ:
    - P(a0 > χ) = 71.3%.
    - P(a0 > 5 χ) = 18%.
  - Asymptote χ is more tightly clustered across announced plans than initial inflation a0.
  - Density over decay rate ω: planner does not play plans with very low ω often; μ⋆ falls sharply as ω becomes small.
  - Relationship to K-equilibrium: plans with lower K-equilibrium loss are announced more often; initial reputation in those plans is lower which lowers their equilibrium value and yields mixed strategy of announcements.

### Comparative statics
- Control shock variance σ (Figure 10):
  - As σ increases (more noise), deviations become less observable ⇒ adherence to plans decreases.
  - Result: average plan has higher asymptote χ, slightly higher a0, and slower decay ω as σ increases.
- Discount factor β and Phillips slope κ (Figure 11):
  - Varying impatience (1/β − 1; benchmark 2% annual):
    - More impatient planner ⇒ average plans start higher, converge to lower inflation, and have faster decay.
    - Rationale: greater expected inflation bias when planner impatient ⇒ planner uses resilience (higher a0, steeper descent, lower χ).
  - Varying Phillips curve slope κ:
    - Steeper Phillips curve (same inflation gives smaller output boom) ⇒ planner announces lower inflation throughout.
    - Logic: weaker incentives to create surprise inflation when Phillips curve steeper.

### Other models (comparative perspectives)
- Ramsey plan (full commitment; Section 6.1):
  - Problem is linear-quadratic; shocks only affect via variance.
  - Ramsey plan achieves near-complete disinflation: Ramsey inflation ≈ 0 about 1.5 years after announcement.
  - Ramsey plan starts at a high initial inflation (≈ three quarters of the way to Nash inflation) then disinflates.
  - Contrast: reputational equilibrium announcements typically do not converge to zero (low probability χ = 0); K-equilibrium starts above Ramsey plan.
- Sustainable plans with expectations as threats (Section 6.2):
  - With perfect control and punishment expectation ξ, if ξ harsh enough the Ramsey plan is recoverable; otherwise planner best-responds to expectations when Ramsey not sustainable.
- Sustainable plans with reverting triggers (Green and Porter style; Section 6.3):
  - Private sector triggers punishment when |π − a|/a > D, punishment expectations ξ, and stochastic return θ to normal regime.
  - Imperfect control implies private sector may trigger punishment on-path (false positives); deterrence must be weighed against false-positive costs.
  - With imperfect control, planner cannot promise zero inflation nor the Ramsey plan even when able to target lower inflation via punishment expectations.

*Source: wpiea2020085-print-pdf - 3.1  Reputation-building incentives*

### 6.4  Recursive plans with reputation (inspired byDovis and Kirpalani,2019)

### 6.4 Recursive plans with reputation (inspired by Dovis and Kirpalani,2019)

### Setup and timing
- Government composed of two agents each period: a “morning” planner and an “afternoon” policy maker.
- Timing:
  - Morning of period t: planner announces a policy recommendation for the afternoon of t+1 (given an earlier announcement for the afternoon of t).
  - Afternoon of each period: policy maker chooses inflation π.
- Policy maker types:
  - Commitment type: stubbornly follows the recommendation.
  - Rational type: chooses whether to follow it.
- Planner and private sector do not know the policy maker’s type; they form statistical inference from past inflation realizations.
- Difference from Dovis and Kirpalani (2019): retain imperfect control and the new Keynesian Phillips curve used in the baseline.
- Equilibrium concept: recursive plans (with reputation).

### Planner’s problem (recursive formulation)
- When facing target a and current reputation p, the policy maker attains value
  v_R(p,a) = min_{g,a'} E[(y−y^⋆)^2 + γ π^2 + β v_R(p',a')]
  subject to
  π = g + ε
  π = κ y + β ( p' a' + (1−p') g_R(p',a') )
  p' = p + p(1−p) * [ f_ε(π−a) − f_ε(π−g_R(p,a)) ] / [ p f_ε(π−a) + (1−p) f_ε(π−g_R(p,a)) ]
- Interested in the low-reputation case: planner’s initial value
  J_R = lim_{p→0} min_a v_R(p,a)

### Numerical and graphical findings (Figure 15 and description)
- Full reputation: planner announces zero inflation throughout and intends to break it (no effect on expectations).
- Moderate initial reputation: planner announces a positive level of initial inflation that converges to zero inflation.
- Low initial reputation: the plan stops converging to zero and produces a pattern similar to the K-equilibrium.

### Relation to prior models and interpretation (section 6.5)
- Two key aspects differentiating this model from Ramsey/sustainable plans: reputation and imperfect control.
- Classification illustrated by Figure 16:
  - Reputation without noise: Barro (1986); Backus and Driffill (1985) → recover Ramsey plan (zero inflation throughout in their setup).
  - Imperfect control without reputation: Green and Porter (1984) → with high enough threat, planner chooses inflation below Nash; threat structure can induce constant-in-time inflation targets.
  - Reputation with imperfect control: reputational and K-equilibria → combine dynamics from targets and reputation.
- Main insights:
  - Introducing noise with reputation allows the planner to exploit the dynamic interaction between targets and reputation.
  - Recursive plans with reputation differ from Dovis and Kirpalani (2019) because that paper assumes perfect control and an old-style Phillips curve; both differences matter.
  - Intermediate combinations (no noise plus forward-looking Phillips curve, and noise plus old-style Phillips curve) yield a flat optimal plan.

### Gains from preannouncement and flexibility (section 6.6)
- Recursive plan differs from reputational and K-equilibria in two ways:
  1. Reputational and K-equilibrium: inflation targets for all periods are chosen at the beginning of time (pre-announced).
  2. Recursive plan: targets chosen sequentially and allow feedback between reputation evolution and future targets.
- Two opposing effects:
  - Flexibility benefit: recursive plan can tailor future announcements to current credibility assessments.
  - Pre-announcement benefit: reputational and K-equilibria use partial future credibility to induce present expectations.
- To disentangle effects, construct an “average” recursive plan:
  - Solve for recursive plan.
  - Take expected path of announcements accounting for reputation drift down over time.
  - Project this path onto (ω, χ, a_0) space of announcements and solve for continuation equilibrium after announcing this projected plan.
- In the chosen parametrization, gains from pre-announcing appear to be of the same magnitude as gains from flexibility.
- Time-consistency result: because sequential announcements do not capture the commitment benefit of pre-announcing, the recursive plan has a higher long-run asymptote for inflation than reputational and K-equilibrium plans — a second form of time-inconsistency in the Ramsey plan.

### Quantitative comparison (Table 2)
- Model outcomes (values preserved exactly as reported):
  - Initial inflation:
    - Ramsey: 1.40%
    - K-equilibrium: 1.63%
    - ‘Average’ recursive plan: 1.58%
    - Recursive plan: 1.58%
  - Long-run inflation:
    - Ramsey: 0%
    - K-equilibrium: 0.44%
    - ‘Average’ recursive plan: 0.65%
    - Recursive plan: 0.65%
  - Value of loss function:
    - Ramsey: 0.3364
    - K-equilibrium: 0.7552
    - ‘Average’ recursive plan: 0.7589
    - Recursive plan: 0.7554

### Concluding implications (section 7 and end of discussion)
- Main question: can reputation substitute for commitment?
- Findings:
  - Reputation combined with imperfect control creates incentives for staying close to announced targets.
  - Central bank’s optimal policy after announcement trades off surprise inflation benefits against the risk that a deviation is discovered, making reputation an important state variable under discretion.
  - Gradualism is a pervasive feature of optimal plans: planner backloads low inflation to make reputation maintenance both easy and valuable.
  - Gradualism arises from incentive considerations, not from inflation inertia per se.
  - Gradualist property holds at positive reputation levels and in the limit as initial reputation vanishes to zero; the limit is interpreted as a sensible refinement of the game.
  - Reputation effectively modifies the incentive constraint in the recursive planning problem; when reputation is low, large option values of sticking to the plan arise and are larger when the plan is backloaded.

*Source: wpiea2020085-print-pdf (section 6.4), https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020085-print-pdf.pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020085-print-pdf.pdf_