## _wp0896

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---

### Introduction: purpose and scope
- Investigates interaction between monetary policy and accumulation of foreign reserves under floating exchange rates.
- Central concern: official acquisitions of foreign currency may be interpreted as interventions aimed at influencing the exchange rate, potentially undermining credibility of monetary policy goals such as price stability.
- Framework features:
  - Sterilized interventions that internalize the cost of intervention implied by monetary policy decisions.
  - Signalling role of sterilized interventions (as in Mussa (1981)): interventions can be perceived as signals about future monetary policy and influence equilibrium exchange and inflation rates through expectations.
  - Objective: establish a trade-off between faster reserve accumulation and monetary policy credibility.
- Applicability: relevant to central banks accumulating reserves whose stated inflation objectives lack credibility.

### Theoretical model: basic set-up
- Two-country world, two-stage game:
  - Stage 0: intervention amount x0 chosen (positive x0 = purchase of foreign currency).
  - Stage 1: monetary authority sets inflation rate π1.
- Monetary authority objective:
  - L_MA = θ(y1 − y)^2 + α(π1 − π)^2. (Equation (1))
- Reserve objective:
  - L_R = (R − x0)^2. (Equation (2)); initial reserves normalized to zero; R > 0.
- Intervention fully sterilized; cost (capital commitment) of intervention:
  - C = (s0 − s1) x0. (Equation (3))
- Intervention authority minimizes:
  - L_INT = γ L_R + β C, with γ and β weights.
- Price/output/inflation relationships:
  - Lucas supply: y1 = b(π1 − π^e_1). (Equation (4))
  - s1 − s0 = π1 (relative PPP, foreign inflation normalized to zero).
- Parameters: θ, α, β, γ ≥ 0; b sensitivity of output to inflation surprises.

### Complete information — centralization benchmark
- Centralized objective:
  - L_c = θ(y1 − y)^2 + α(π1 − π)^2 + γ(R − x0)^2 + β(s0 − s1) x0.
- Stage 1 (rational expectations) result:
  - π1 = π^e_1 = π + bθ/α y + β/(2α) x0. (Equation (6))
- Stage 0 first-order condition:
  - x0 = 4αγ/(4αγ − β 2) R + 2αβ/(4αγ − β 2) π. (Equation (7))
- Substitution gives:
  - π1 = π^e_1 = 4αγ/(4αγ − β 2) π + bθ/α y + 2βγ/(4αγ − β 2) R. (Equation (8))
- Second-order condition:
  - 4αγ − β 2 > 0. Under this condition coefficients in (7) and (8) are positive; x0 and π1 increase with π and R; π1 increases with y.
- Special case β = 0:
  - π1 = π^e_1 = π + bθ/α y and x0 = R (no internalized intervention cost).

### Institutional arrangements in practice — separation and centralization
- Institutional alternatives:
  - Centralization: central bank controls intervention and uses own funds.
  - Separation: intervention delegated to government agency (Treasury).
  - Shared responsibility and transitions also occur.
- Separation — equilibrium (rational expectations):
  - π_1 = π^e_1 = π + (b θ)/α y. (Equation (10))
  - x_0 = R + (β)/(2γ) π + (b θ β)/(2α γ) y. (Equation (11))
- Comparative complete-information result:
  - If R>0 and 4αγ − β^2 > 0, inflation under separation is lower than under centralization.
  - L_MA^c − L_MA^s > 0 for R>0 (separation yields better macro stabilization).
  - L_INT^c − L_INT^s > 0 (separation improves intervention-authority trade-off).
  - Reserve-targeting optimality ambiguous under complete information.

### Incomplete information: setup and intuition
- Private-sector uncertainty about nominal inflation target π:
  - Prior: π ~ N(π^e_0, Σ^π_0).
- Price setters observe noisy signal of interventions:
  - I0 = x_0 + μ_0, μ_0 ~ N(0, σ^2). σ^2 measures opaqueness.
- Timeline: prior → intervention set and noisy signal revealed → price setters update expectations (π^e_1, x^e_0) → monetary policy set.

### Stage 1 under incomplete information — channels
- Centralized first-order condition (Equation (12)) identifies:
  - Cost-of-intervention channel: terms involving β/(2α) x_0 and unexpected intervention component (x^e_0 − x_0).
  - Expectations channel: interventions affect π^e_1 via the intervention signal.
- Separation (Equation (13)): cost-of-intervention channel disappears; expectations channel remains.

### Stage 0 and learning — Proposition 1
- Intervention policy form under both arrangements:
  - x_0 = m_i π^e_0 + n_i R + p_i (π − π^e_0) + q_i y. (Equation (14))
- Price setters' updated beliefs:
  - π^e_1i = π^e_0 + λ_i (x_0 + μ_0 − m_i π^e_0 − n_i R − q_i y). (Equation (15))
  - x^e_0i = m_i π^e_0 + n_i R + q_i y + λ_i (x_0 + μ_0 − m_i π^e_0 − n_i R − q_i y). (Equation (16))
- Rewritten:
  - π^e_1i = π^e_0 + λ_i [p_i (π − π^e_0) + μ_0]. (Equation (17))
  - x^e_0i = m_i π^e_0 + n_i R + q_i y + λ_i [p_i (π − π^e_0) + μ_0].
- Interpretation:
  - p_i (π − π^e_0) captures credibility effects: if π < π^e_0, then interventions reduced (p_i(π − π^e_0) < 0), reducing reserve accumulation.
  - λ_i p_i measures intensity/speed of learning; if 0 < λ_i p_i < 1, interventions reduce variance of π and bring π^e_1 closer to π.

### Corollary 1 — volatility reduction after intervention
- Variance of target inflation in period 1:
  - Σ^π_1 = (1 − λ_i p_i) Σ^π_0.
- If 0 < λ_i p_i < 1, uncertainty about the inflation target is reduced after official interventions (Σ^π_1 < Σ^π_0).
- (1 − λ_i p_i) measures information content of interventions; larger λ_i p_i → faster learning and credibility gains.

### Simulation setup (section 5.1): parameter assumptions
- Weights and values:
  - θ = 0.3α and β = 0.1α.
  - Simulations performed for α = 10.
  - Additional parameter assumptions: b = 1, Σπ0 = 1 and σ2 = 10.
- γ denotes weight of reserve-targeting goal; coefficients λi and pi obtained numerically; used to compute mi, ni, qi.
- Various other parameter combinations tried; qualitative results robust.

### Simulation findings (figure 2 panels a–f) — intervention sensitivities and learning
- Panel (a) — coefficients of πe0 (mi):
  - mc and ms equal for all γ; mc = ms → 0 as γ rises.
- Panel (b) — coefficients of R (ni):
  - nc and ns → 1 as γ → ∞; persistent gap nc > ns (centralization more sensitive to reserve target).
- Panel (c) — coefficients of (π − πe0) (pi):
  - pc and ps → 0 as γ → ∞; for positive γ, pc significantly greater than ps.
  - Interpretation: interventions under centralization much more sensitive to gaps between true inflation target and public expectation.
- Panel (d) — coefficients of y (qi):
  - Separation: qs > 0; centralization: qc < 0 → separation more adequate for reserve accumulation.
- Panels (e) and (f) — learning rates:
  - λp measures speed of learning.
  - For positive γ, learning faster under centralization than separation (λc pc > λs ps).
  - λc pc − λs ps achieves maximum when γ ≈ 1.5 (which corresponds to 0.15α when α = 10).

### Intuition: why centralization can be more informative
- Centralization combines cost-of-intervention and expectations channels; separation leaves only expectations channel.
- Thus centralization can make interventions more informative about future monetary policy.
- Caveat: informational advantage depends on γ; for γ large (close to α) interventions driven by reserve target and learning is negligible.

### Trade-off between reserve accumulation and credibility building (section 5.2)
- Difference in interventions:
  - xs0 − xc0 = (ns − nc) R + (ps − pc) (π − πe0) + (qs − qc) y.
- Simulation implications: ns < nc, ps < pc, qs > qc.
  - Assuming π < πe0 and R < y, net effect: intervention is larger under separation than under centralization.
  - Larger gap between π and πe0 increases xs0 − xc0.
- Summary for positive finite γ:
  - Centralization → faster public learning about inflation target.
  - Separation → faster accumulation of international reserves.
- Interpretation:
  - Separation isolates monetary and intervention policies, permitting faster reserve accumulation without large adverse effects on inflation expectations.
  - Centralization improves credibility-building speed via more informative interventions.

### Macroeconomic stability and reserve targeting (section 5.3)
- Simulation targets: y = π = 1 and R = 1/2.
- Macroeconomic stability (E[L_MA|i]):
  - Separation provides more stable economy than centralization for low γ (roughly below 1.5).
  - Centralization superior when γ is large.
  - Performance difference significant only when government credibility is low (πe0 sufficiently large).
- Key summaries:
  - If πe0 close to π: negligible difference between arrangements.
  - If πe0 lacks credibility and γ sufficiently large: centralization leads to more stable economy.
  - If πe0 lacks credibility and γ small enough: separation leads to more stable economy.
- Reserve-targeting (E[L_R|i]):
  - As γ increases: mi, pi, qi → 0 and ni → 1, so x0 → R and E[L_R|c], E[L_R|s] → 0.
  - For small and intermediate γ, centralization can be superior for achieving targeted reserve level for certain πe0; for intermediate γ (≈ 1.5) and large πe0, separation becomes superior.
  - As γ → ∞ the two arrangements perform equally well.

### Policy-relevant conclusions and trade-offs (section 5.3.3 and concluding remarks)
- No uniform superiority:
  - Authorities face a trade-off between speed of reserve accumulation and speed/intensity of credibility-building (and associated macroeconomic stability).
- Broad implications under reasonable parametrizations:
  - Separation better for fast reserve accumulation.
  - Centralization better for macroeconomic stability and faster anti-inflation credibility formation.
- Conjectures about γ:
  - γ expected to be significantly larger than β.
  - Reserve targeting not expected to dominate output stabilization; plausible γ in simulations illustrated between γ = 2 to γ = 3 (with α = 10 and β = 1).
  - Within γ in [2, 3], graphs show separation yields faster reserve accumulation while centralization improves anti-inflationary credibility speed.
- Additional remarks:
  - Results differ from models where interventions are purely speculative (Vitale (2003)); here interventions also aim to achieve a reserve level.
  - Suggested future research: dynamic extensions with multiple stages of interaction and learning; uncertainty about reserve target.

*Source: _wp0896 - 1. Introduction; 3.2 Institutional Arrangements in Practice; 4 Incomplete Information; 5.1 Simulation of Stage 0 Parameters; 5.2–5.3 Results and Conclusions.*

### 1. Introduction.........................................................................................................

### _wp0896 - 1. Introduction

### Introduction: purpose and scope
- Investigates interaction between monetary policy and accumulation of foreign reserves under floating exchange rates.
- Key concern: official acquisitions of foreign currency may be interpreted by market participants as interventions aimed at influencing the exchange rate, potentially undermining credibility of monetary policy goals such as price stability.
- Develops a theoretical framework where official interventions (government transactions in the foreign exchange market) are implemented by policymakers with simultaneous goals of macroeconomic stabilization and reserve accumulation under floating exchange rates.
- Assumes interventions are always sterilized and internalizes the cost of intervention implied by monetary policy decisions.
- Emphasizes signalling: sterilized interventions can be perceived as signals about future monetary policy decisions (as in Mussa (1981)) and may influence equilibrium exchange and inflation rates through expectations.
- Objective: establish a trade-off between faster reserve accumulation and monetary policy credibility.
- Applicability: relevant to central banks accumulating reserves whose stated inflation objectives lack credibility.

### Theoretical Model: basic set-up
- Two-country world: domestic and foreign economies.
- Two-stage game:
  - Stage 0: intervention amount x0 chosen (positive x0 = purchase of foreign currency).
  - Stage 1: monetary authority sets inflation rate π1 directly.
- Macroeconomic objective function (monetary authority):
  - L_MA = θ(y1 − y)^2 + α(π1 − π)^2. (Equation (1))
- Reserve objective (intervention authority):
  - L_R = (R − x0)^2. (Equation (2))
  - Initial reserves normalized to zero; R > 0 (government attempting to accumulate foreign reserves).
- Intervention is fully sterilized: when intervention authority buys (sells) x0 of foreign currency, it sells (buys) domestic assets to keep liquidity constant in stage 0; monetary authority resets liquidity in stage 1 to achieve π1.
- Cost (capital commitment) of intervention:
  - C = (s0 − s1) x0. (Equation (3))
  - If buying foreign currency followed by nominal appreciation, intervention authority incurs capital loss (C > 0); reversed movements yield profits.
- Intervention authority minimizes:
  - L_INT = γ L_R + β C, with γ and β weights on reserve targeting and intervention cost.
- Price/output/inflation relationships:
  - Lucas supply curve: y1 = b(π1 − π^e_1). (Equation (4))
  - Relative purchasing power parity and foreign inflation normalized to zero imply: s1 − s0 = π1.
- Parameters: θ, α, β, γ ≥ 0; b sensitivity of output to inflation surprises.

### Complete information: benchmark analyses
- Focuses on case where all parameters and choices are common knowledge and observed by price setters.
- Compares two institutional arrangements:
  - Centralization: same policymaker sets monetary and intervention policies.
  - Separation: intervention decisions delegated to separate authority (analyzed later in the section beyond excerpt).

3.1 Centralization — key derivations and implications
- Centralized objective:
  - L_c = L_MA_c + L_INT_c = θ(y1 − y)^2 + α(π1 − π)^2 + γ(R − x0)^2 + β(s0 − s1) x0.
- Solve game by backward induction (stage 1 then stage 0).
- Stage 1 first-order condition yields:
  - π1 = π + b/(2θ/(α + b^2 θ)?) [text provides expression as:]
  - π1 = π + b/2 θ/(α + b/2 θ) (π^e_1 − π) + bθ/(α + b/2 θ) y + β/(2(α + b/2 θ)) x0. (Equation (5))
- With rational expectations simplifies to:
  - π1 = π^e_1 = π + bθ/α y + β/(2α) x0. (Equation (6))
- Implication: with β > 0 (cost of intervening internalized), intervention level x0 influences monetary authority’s chosen inflation π1. If β = 0, recover Barro and Gordon type result π1 = π^e_1 = π + bθ/α y and x0 = R.
- Stage 0 first-order condition yields:
  - x0 = 4αγ/(4αγ − β 2) R + 2αβ/(4αγ − β 2) π. (Equation (7))
- Substituting x0 into inflation expression gives:
  - π1 = π^e_1 = 4αγ/(4αγ − β 2) π + bθ/α y + 2βγ/(4αγ − β 2) R. (Equation (8))
- Second order condition requires 4αγ − β 2 > 0; under this condition all coefficients in (7) and (8) are positive, so x0 and π1 increase with π and R; π1 also increases with y.
- Economic interpretation and parameter discussion:
  - β must be strictly greater than zero to prevent infinite financial losses; empirical evidence suggests speculative profit is not a main determinant of interventions, so β may be small but positive.
  - Growing emphasis on inflation stabilization (α) and reserve management (γ) in recent years supports plausibility of 4αγ − β 2 > 0.
  - When β = 0, π1 = π^e_1 = π + bθ/α y and x0 = R (monetary authority fully pursues reserve target without trade-off).

*Source: _wp0896 - 1. Introduction.*

### 3.2  Institutional Arrangements in Practice

### _wp0896 - 3.2  Institutional Arrangements in Practice

### Institutional arrangements: overview
- Institutional alternatives in practice include:
  - Centralization: the central bank has full jurisdiction over foreign exchange intervention and uses its own funds to finance costs.
  - Separation: intervention decisions are delegated to a government agency other than the monetary authority (typically the Treasury).
  - Shared responsibility: costs and/or decision rights are split between the monetary authority and the Treasury.
  - Transitions between centralized and separated frameworks occur in practice.
- Empirical references cited: Vitale (2003) for industrial countries; Moser-Boehm (2005) for emerging markets; Schwartsman (2004) for Brazil.

### Separation: objectives and equilibrium (section 3.3)
- Monetary authority minimizes:
  - L_MA^s = θ(y_1 − y)^2 + α(π_1 − π)^2
- Intervention authority minimizes:
  - L_INT^s = γ(R − x_0)^2 + β(s_0 − s_1) x_0
- Monetary authority first-stage reaction (given y_1 = b(π_1 − π^e_1)):
  - π_1 = π + (b^2 θ)/(α + b^2 θ) (π^e_1 − π) + (b θ)/(α + b^2 θ) y   (equation (9))
- Under rational expectations this simplifies to:
  - π_1 = π^e_1 = π + (b θ)/α y   (equation (10))
- Intervention authority solution in stage 0:
  - x_0 = R + (β)/(2γ) π + (b θ β)/(2α γ) y   (equation (11))

### Centralization versus separation: main comparative results (section 3.4)
- Both the intervention level x_0 and inflation π_1 depend on institutional structure.
- If authorities desire to accumulate reserves (R>0) and second-order condition holds:
  - 4αγ − β^2 > 0
  - equations (7) and (11) imply x_0 will be positive with certainty, so inflation under separation is lower than under centralization.
- Macroeconomic stabilization comparison:
  - L_MA^c − L_MA^s = β[β π + 2γ R] { α β [β π + 2γ R] + 2 b θ y (4αγ − β^2) } / (4αγ − β^2)^2
  - Because 4αγ − β^2 > 0 and R>0, the expression above is positive for positive R, implying separation yields better macro stabilization.
- Intervention authority welfare:
  - L_INT^c − L_INT^s > 0, so under separation intervention authorities achieve a better trade-off between reserve targeting and cost minimization.
- Reserve-targeting outcome is ambiguous:
  - When evaluating intervention decisions solely in terms of reserve targeting (ignoring β(s_0 − s_1) x_0), equally plausible parameter values can favor either separation or centralization; no definitive ranking under complete information.

### Incomplete information: setup and intuition (section 4)
- Incomplete information modeled as private-sector uncertainty about the monetary authority’s nominal inflation target π.
- Price setters have prior: π is normally distributed with mean π^e_0 and variance Σ^π_0, so initial best prediction is π^e_0.
- Price setters observe a noisy signal of interventions: x_0 + μ_0, with μ_0 ~ N(0, σ^2). σ^2 measures opaqueness of interventions.
- Timeline (Figure 1): price setters form prior → intervention is set and noisy signal revealed → price setters update expectations (π^e_1 and x^e_0) → monetary policy set in stage 1.

### Stage 1 analysis under incomplete information (section 4.1)
- Centralized case: monetary authority minimizes
  - θ(y_1 − y)^2 + α(π_1 − π)^2 + γ(R − x_0)^2 + β(s_0 − s_1) x_0
  - Subject to π^e_1 = π^e_1 + (b θ)/α y + (β)/(2α) x^e_0, y_1 = b(π_1 − π^e_1), s_1 − s_0 = π_1, x_0 given.
- First-order condition yields:
  - π_1 = π + (b θ)/α y + (β)/(2α) x_0 + (b^2 θ)/(α + b^2 θ) (π^e_1 − π) + (β)/(2α) (b^2 θ)/(α + b^2 θ) (x^e_0 − x_0)   (equation (12))
  - Identified channels:
    - Cost-of-intervention channel: terms involving β/(2α) x_0 and the unexpected intervention component (x^e_0 − x_0).
    - Expectations channel: interventions affect π^e_1 via the intervention signal.
- Separation case: monetary authority faces π^e_1 = π^e_1 + (b θ)/α y and minimizes L_MA^s = θ(y_1 − y)^2 + α(π_1 − π)^2.
  - First-order condition yields:
    - π_1 = π + (b θ)/α y + (b^2 θ)/(α + b^2 θ) (π^e_1 − π)   (equation (13))
  - Cost-of-intervention channel disappears (intervention costs not internalized), but the expectations channel remains because price setters use the intervention signal to update π^e_1.

### Stage 0 analysis and learning: Proposition 1 (section 4.2)
- Proposition 1 (summary):
  - Under both centralization and separation, intervention policy has the form:
    - x_0 = m_i π^e_0 + n_i R + p_i (π − π^e_0) + q_i y   (equation (14))
  - Price setters' expectations:
    - π^e_1i = π^e_0 + λ_i (x_0 + μ_0 − m_i π^e_0 − n_i R − q_i y)   (equation (15))
    - x^e_0i = m_i π^e_0 + n_i R + q_i y + λ_i (x_0 + μ_0 − m_i π^e_0 − n_i R − q_i y)   (equation (16))
  - Coefficients m_i, n_i, p_i, q_i, λ_i are given in the appendix; i = c (centralization) or s (separation).
- Interpretations and implications:
  - The p_i (π − π^e_0) term shows that lack of credibility about π affects interventions: if π < π^e_0 (monetary authority perceived as more inflationary than reality), then p_i (π − π^e_0) < 0 → less intervention and less reserve accumulation.
  - Interventions are also a function of output target y via q_i y.
  - Credible announcements of x_0 would reveal π because x_0 is a simple function of π; but because announcements about π lack credibility, announcements about x_0 are also not credible in equilibrium.
  - Larger x_0 raises π^e_1i and x^e_0i via λ_i, so interventions influence monetary policy through the expectations channel even when intervention costs are not internalized.

- Rewriting with (14):
  - π^e_1i = π^e_0 + λ_i [p_i (π − π^e_0) + μ_0]   (equation (17))
  - x^e_0i = m_i π^e_0 + n_i R + q_i y + λ_i [p_i (π − π^e_0) + μ_0]
  - If π < π^e_0 (credibility building), x_0 will be reduced proportionately to π − π^e_0 and, if 0 < λ_i p_i < 1, this brings π^e_1 down closer to π.
  - The product λ_i p_i measures the intensity of learning from interventions; the closer λ_i p_i is to 1, the faster the learning.

### Corollary 1: volatility reduction after intervention
- Corollary 1:
  - Under both centralization and separation, the variance of the target level of inflation in period 1 is:
    - Σ^π_1 = (1 − λ_i p_i) Σ^π_0
  - If 0 < λ_i p_i < 1, uncertainty about the target level of inflation is reduced after official interventions (Σ^π_1 < Σ^π_0).
  - The factor (1 − λ_i p_i) measures the information content of official interventions about the inflation target. Larger λ_i p_i → faster learning and faster credibility gains.

### Comparative analysis under incomplete information (section 5)
- Proposition 1 and Corollary 1 do not by themselves rank centralization versus separation.
- The model is simulated in the subsequent section (section 5) to compare centralized and separated decisions in terms of:
  - Learning about the inflation target (speed and intensity of credibility gains).
  - Speed of accumulation of reserves.

_italics_ _wp0896 - 3.2  Institutional Arrangements in Practice_

### 5.1  Simulation of Stage 0 Parameters

### _wp0896 - 5.1  Simulation of Stage 0 Parameters

### Simulation setup and parameter assumptions
- The simulation examines sensitivity of interventions to four determinants: πe0, R, (π−πe0) and y, under two institutional settings (centralization versus separation).
- Weights and parameter values used (as stated):
  - θ = 0.3α and β = 0.1α (weights for output targeting and cost of intervention, given inflation-target weight α).
  - Simulations performed for α = 10.
  - Additional parameter assumptions: b = 1, Σπ0 = 1 and σ2 = 10.
  - Various other parameter combinations were tried and qualitative results proved robust.
- The coefficients λi and pi (for i = c, s) were obtained numerically with MATHEMATICA and used to compute mi, ni, qi (for i = c, s).
- γ denotes the weight of the reserve targeting goal; figures plot mi, ni, pi, qi for various γ.

### Panel results and interpretation (figure 2: panels a–f)
- Panel (a) — Coefficients of πe0 (mi, i = c, s):
  - mc and ms are equal for all values of γ.
  - mc and ms converge to 0 as γ rises.
  - Interpretation: interventions equally sensitive to direct effect of πe0 under both institutional settings.
- Panel (b) — Coefficients of R (ni, i = c, s):
  - Both nc and ns converge to 1 as γ → ∞.
  - Persistent gap: nc > ns, implying interventions are more sensitive to reserve target under centralization than under separation.
- Panel (c) — Coefficients of (π−πe0) (pi, i = c, s):
  - pc and ps both converge to 0 as γ → ∞.
  - For positive γ, pc is significantly greater than ps.
  - Interpretation: amount intervened is much more sensitive under centralization than under separation to gaps between true inflation target and public’s initial expectation.
  - If π < πe0 (monetary authority building anti-inflationary credibility), pi(π−πe0) is negative in both cases but significantly smaller under centralization than under separation, implying reserve accumulation may be undermined by credibility-building intentions—particularly under centralization.
- Panel (d) — Coefficients of y (qi, i = c, s):
  - Under separation: qs > 0 (amount intervened positively affected by y).
  - Under centralization: qc < 0 (amount intervened negatively affected by y).
  - Interpretation: separation seems more adequate for reserve accumulation.
- Panels (e) and (f) — Learning rates about the inflation target:
  - λp measures speed of learning about the inflation target; larger λp implies faster learning.
  - For positive γ, learning is faster under centralization than under separation (λc pc > λs ps).
  - The difference in learning speeds (λc pc − λs ps) achieves a maximum when γ is about 15% of the weight attributed to the goal of stabilizing inflation.
    - Note: λc pc − λs ps achieves its peak when γ = 1.5. Since α = 10, this corresponds to 0.15α.

### Intuition behind centralization advantage in information content
- Under centralization, interventions affect inflation through two channels: the cost-of-intervention channel and the expectations channel.
- Under separation, impact is reduced to the expectations channel only.
- Therefore, under centralization the link between future monetary policy and current intervention choices is stronger; interventions are more informative about future monetary policy and lead markets to make more accurate predictions about the inflation target.
- Caveat: the informational advantage of centralization depends on γ. For γ close to the weight on inflation targeting (α = 10 in simulations), λc pc and λs ps are both very small and interventions are driven almost entirely by the reserve target R; for large γ there is no significant learning about the inflation target under either arrangement.

### Trade-off between reserve accumulation and credibility building (section 5.2)
- Given x0 = mi πe0 + ni R + pi (π−πe0) + qi y and mc = ms, the difference between separation and centralization is:
  - xs0 − xc0 = (ns − nc) R + (ps − pc) (π − πe0) + (qs − qc) y
- Simulation implications:
  - ns < nc, ps < pc, qs > qc.
  - Assuming π < πe0 and R < y, the first term (ns−nc)R is negative while the other two terms are positive.
  - Since |ns − nc| < |qs − qc| (panels (b) and (d)), intervention is unquestionably larger under separation than under centralization.
  - The larger the difference between π and πe0, the larger the positive gap between xs0 and xc0.
- Summary for positive and finite γ:
  - When decisions are centralized, public learning about the inflation target occurs faster than when decisions are separated.
  - When decisions are separated, international reserves can be accumulated at a faster pace than when decisions are centralized.
- Interpretation:
  - Separation isolates monetary and intervention policies, permitting faster reserve accumulation without large adverse effects on inflationary expectations.
  - Centralization improves credibility-building speed via more informative interventions.

### Centralization versus separation — macroeconomic stability and reserve targeting (section 5.3)
- Targets used in simulations (without loss of generality): y = π = 1 and R = 1/2.
- Macroeconomic stability measure: expected loss LMA|i = θ(y1 − y)2 + α(π1 − π)2, for i = c, s.

Macroeconomic stability (5.3.1)
- Simulation findings (figure 3):
  - Behavior of E[(y1 − y)2], E[(π1 − π)2], and E[LMA|c] − E[LMA|s] similar in shape, differing in magnitude.
  - Separation provides a more stable economy than centralization for low values of γ (roughly below 1.5).
  - Centralization becomes a better option when γ is large.
  - Performance difference between arrangements is significant only when government credibility is low (πe0 sufficiently large).
- Key summary bullets:
  - When the inflation target is credible (πe0 close to π), there is no significant difference in macroeconomic stability between separation and centralization.
  - When the inflation target lacks credibility and γ is sufficiently large, centralization leads to a more stable economy in both inflation and output.
  - When the inflation target lacks credibility and γ is small enough, separation leads to a more stable economy in both inflation and output.
- Intuition:
  - If πe0 is close to π, the expectations channel is weak and interventions are not essential as signals; both channels (expectations and cost-of-intervention) have limited differential impact across arrangements, so choice of arrangement is negligible for stability.
  - If πe0 ≫ π, signaling via interventions matters; centralization can be superior for intermediate γ (around 1.5) because expectations channel is strong and centralization speeds learning, improving stabilization.
  - For small γ and large πe0, separation is better because cost-of-intervention channel makes inflation larger under centralization and expectations channel is weak.
  - For large γ, both channels fade and choice becomes irrelevant to macroeconomic stability.

Reserve targeting (5.3.2)
- Expected reserve-targeting loss: E[LR|i] = E[(R − x0)2 | i], i = c, s.
- As γ increases:
  - mi, pi, qi → 0 and ni → 1, so x0 → R and E[LR|c], E[LR|s] → 0 regardless of πe0.
- For small γ:
  - Direct effect of πe0 on x0 is same under both arrangements (mc = ms).
  - Sensitivity to lack of credibility (π − πe0) is larger under centralization (pc > ps).
  - For small γ, pc > mc; as γ rises pc eventually becomes smaller than mc.
  - Panel (a) of figure 4: for sufficiently small γ, E[LR|c] decreases in πe0; for intermediate γ, E[LR|c] increases in πe0 and converges to zero as γ → ∞.
  - Panel (b): For separation, E[LR|s] increases in πe0 for small and intermediate γ and converges to zero as γ → ∞.
- Panel (c) of figure 4:
  - For small and intermediate values of γ and πe0, centralization is superior for achieving targeted reserve level.
  - For intermediate γ (close to 1.5) and large πe0, separation becomes superior in reserve accumulation due to greater isolation between monetary and intervention decisions.
  - As γ → ∞ the two arrangements perform equally well regardless of πe0.

### Discussion and policy-relevant conclusions (5.3.3 and concluding remarks)
- Trade-offs:
  - Neither separation nor centralization is uniformly superior; authorities trade off reserve accumulation speed against macroeconomic stability and credibility-building speed.
  - For large enough γ, fast reserve accumulation can be achieved by separation, but at the cost of weakened signalling about inflation target and reduced macroeconomic stability.
  - For small γ, faster reserve accumulation can be achieved by centralization, but at the cost of larger inflationary effects via the cost-of-intervention channel.
- Reasonable conjectures about γ:
  - γ expected to be significantly larger than β, meaning reserve accumulation is typically more important than speculative motives to intervene.
  - Reserve targeting goal is not expected to be more important than stabilizing output.
  - These conjectures suggest γ lies within a narrower interval than 0 to 5; illustration: γ confined to γ = 2β = 2 to γ = θ = 3.
    - Under γ in [2, 3], graphs show faster reserve accumulation achieved by separation, while anti-inflationary credibility is improved faster by centralization.
- Relation to literature:
  - Results differ from Vitale (2003) because Vitale’s model examines interventions solely for speculative reasons; here interventions also aim to achieve a certain level of foreign reserves.
- Overall concluding remarks:
  - The model incorporates the cost of intervention and the interaction between sterilized foreign exchange interventions and monetary policy.
  - Under complete information, separation isolates monetary policy from intervention costs and generally improves macroeconomic stabilization; reserve-targeting superiority is ambiguous.
  - Under incomplete information, interventions influence public expectations and can raise expected inflation; a trade-off emerges between reserve accumulation and credibility building that translates into a trade-off between reserve accumulation and macroeconomic stability.
  - Under reasonable parametrizations, separation is better for fast reserve accumulation while centralization is better for macroeconomic stability.
  - Future research directions: dynamic version of the model to include multiple stages of interaction and learning; uncertainty about the reserve target.

*Source: _wp0896 - 5.1  Simulation of Stage 0 Parameters (sections 5.1–6 as provided)*

### References

### _wp0896 - References

### References list (selected items as listed)
- [1] Backus, David and John Driffill, 1985, “Inflation and Reputation,” American Economic Review, vol. 75(3), p. 530-38.
- [2] Barro, Robert J., and D. Gordon, 1983, “Rules, Discretion, and Reputation in a Model of Incomplete Information,” Journal of Monetary Economics, 12, 101-120.
- [3] Barro, Robert J., 1986, “Reputation in a Model of Monetary Policy with Incomplete Information,” Journal of Monetary Economics, 17, 1-20.
- [4] Bhattacharya, Utpal, and Paul Weller, 1997, ”The Advantage to Hiding One’s Hand: Speculation and Central Bank Intervention in the Foreign Exchange Market,” Journal of Monetary Economics, Vol. 39, p. 251-277.
- [5] Dominguez, Katherine and Jeffrey Frankel, 1993a, Does Foreign Exchange Intervention Work?, Institute of International Economics, Washington D.C.
- [6] Dominguez, Katherine and Jeffrey Frankel, 1993b, “Foreign Exchange Intervention Matter: An Empirical Assessment”, in On Exchange Rates, J. Frenkel (ed.), The MIT Press, Cambridge MA.
- [7] Edison, Hali, 1993, “The Effectiveness of Central Bank Intervention: A Survey of the Literature After 1982,” Princeton University, Department of Economics, Special Paper in International Economics, No. 18.
- [8] Eggertsson, Gauti B., “How tofight deflation in a liquidity trap: committing to being irresponsible,” IMF Working Paper, WP/03/64 (Washington: International Monetary Fund).
- [9] Granger, C. W. J., and Paul Newbold, Forecasting Economic Time Series, second edition, Academic Press, Orlando, FL.
- [10] Jeanne, Olivier, and Romain Ranciere, 2006, “The optimal level of international reserves in emerging market countries: formulas and applications,” IMF Working Paper 06/229 (Washington: International Monetary Fund).
- [11] Kaminsky, Graciela, and Karen Lewis, 1996, “Does foreign exchange intervention signal future monetary policy?,” Journal of Monetary Economics, Vol. 37, pages 285-312.
- [12] Lewis, Karen, 1995, “Are the Foreign Exchange Intervention and Monetary Policy Relates and Does it Really Matter?,” Journal of Business, Vol. 68, No. 2.
- [13] Lyons, Richard, 2001, The Microstructure Approach to Exchange Rates, The MIT Press (Cambridge, Massachusetts).
- [14] Moser-Boehm, Paul, 2005, ”Governance Aspects of Foreign Exchange Interventions,” BIS Papers, No. 24, Foreign Exchange Market Intervention in Emerging Markets: Motives, Techniques and Implications, Monetary and Economic Department, Bank of International Settlement.
- [15] Mussa, Michael, 1981, The Role of Official Intervention, Group of Thirty, New York.
- [16] Neely, Christopher, 2001, “The Practice of Central Bank Intervention: Looking Under the Hood,” Central Banking 11, p. 24-37, Federal Reserve Bank of Saint Louis.
- [17] Obstfeld , Maurice, and Kenneth Rogoff, 1998, “Foundations of International Macroeconomics,” The MIT Press: Cambridge, Massachusetts.
- [18] Sarno, Lucio, and Mark P. Taylor, 2001, “Official Intervention in the Foreign Exchange Market: Is it Effective, and, If so, How Does it Work?,” Journal of EconomicLiterature,Vol.39,No.3,p.839-868.
- [19] Schwartsman, Alexandre, 2004, “A ”Nova Politica Cambial” Nao e Nova,” Folha de Sao Paulo (in Portuguese).
- [20] Stein, J., 1989, “Cheap Talk and the Fed: A Theory of Imprecise Announcements,” American Economic Review, 79, p. 32-42.
- [21] Vitale, Paolo, 1999, ”Sterilised Central Bank Intervention in the Foreign Exchange Market,” Journal of InternationalEconomics, Vol. 49, p. 245-267.
- [22] Vitale, Paolo, 2003, “Foreign Exchange Intervention: How to Signal Policy Objectives and Stabilise the Economy,” Journal of Monetary Economics, Vol.50,p.841-870.

### Appendix — Proofs: key analytical results and equilibrium characterizations

- General setup and Bayesian updating
  - Observed signal I0 = x0 + μ0.
  - Projection theorem for jointly normal variables used to obtain:
    - πe1 ≡ E[π | I0] = πe0 + λ[I0 − E(I0)], (equation (18))
    - xe0 ≡ E[x | I0] = E(I0) + λ[I0 − E(I0)], (equation (19))
  - λ = Σπ,I0 Σ−1I0,I0, with Σa,b = Cov(a,b).

- Centralization case: intervention authority minimization problem (stage 0)
  - Objective: Min x0 E[ θ(y1 − y)2 + α(π1 − π)2 + γ(R − x0)2 + β(s0 − s1)x0 | ΩMA ] subject to model equations and information set ΩMA.
  - Conjectured functional form for x0 (equation (20)):
    - x0 = mc πe0 + nc R + pc (π − πe0) + qc y
    - mc, nc, pc, qc are constants to be solved for consistency with equilibrium.
  - Updated expectations given conjecture:
    - πe1 = πe0 + λc [I0 − mc πe0 − nc R − qc y]
    - xe0 = mc πe0 + nc R + qc y + λc [I0 − mc πe0 − nc R − qc y]
    - where λc = pc Σπ0 / (p2c Σπ0 + σ2)
  - First-order conditions and consistency yield closed-form coefficients:
    - mc = 2αβ / (4αγ − β2)
    - nc = 4αγ / (4αγ − β2)
    - pc = 2α[αβ + b2(2α+β)θλc] / { α [ 4γ(α + b2 θ) − β2 ] − 2b2 β(2α+β)θλc + b2(2α+β)2 θλ2c }
    - qc = − 2bθ [ β(λc − 1) + 2αλc ] / (4αγ − β2)
  - λc satisfies the second-order condition:
    - α [ 4γ(α + b2 θ) − β2 ] − 2b2 β(2α+β)θλc + b2(2α+β)2 θλ2c > 0
    - and solves λc = pc Σπ0 / (p2c Σπ0 + σ2).
  - Conclusion: with these coefficient values, the conjectured x0 is consistent with agents’ expectations and constitutes a Nash equilibrium.

- Separation case: intervention authority minimization problem (stage 0)
  - Objective: Min x0 E[ γ(R − x0)2 + β(s0 − s1)x0 | ΩINT ] subject to model equations and information set ΩINT.
  - Conjectured functional form for x0 (equation (21)):
    - x0 = ms πe0 + ns R + ps (π − πe0) + qs y
  - Updated expectations:
    - πe1 = πe0 + λs [I0 − ms πe0 − ns R − qs y]
    - xe0 = ms πe0 + ns R + qs y + λs [I0 − ms πe0 − ns R − qs y]
    - where λs = ps Σπ0 / (p2s Σπ0 + σ2)
  - First-order conditions and consistency yield closed-form coefficients:
    - ms = β(α + b2 θ) / [ 2αγ + b2 θ(2γ − βλs) ]  (multiplied by (α + b2 θ) in numerator as derived)
    - ns = 2γ(α + b2 θ) / [ 2αγ + b2 θ(2γ − βλs) ]
    - ps = αβ / { 2 [ αγ + b2 θ(γ − βλs) ] }
    - qs = bβθ(α + b2 θ) / { α [ 2αγ + b2 θ(2γ − βλs) ] }
  - λs satisfies the second-order condition:
    - λs < γ(α + b2 θ) / (b2 βθ)
    - and solves λs = ps Σπ0 / (p2s Σπ0 + σ2).
  - Conclusion: with these coefficient values, the conjectured x0 is consistent with agents’ expectations and constitutes a Nash equilibrium in the separation case.

- Proof of Corollary 1 (variance reduction from signal)
  - Using projection theorem, conditional variance of π given I0:
    - Σπ1 ≡ Var[π | I0] = Σπ0 − Σπ,I0 Σ−1I0,I0 ΣI0,π
  - Given Σπ,I0 = pi Σπ0 and ΣI0,I0 = p2i Σπ0 + σ2,
    - Σπ1 = Σπ0 − p2i Σπ02 / (p2i Σπ0 + σ2) = (1 − λi pi) Σπ0
    - where λi = pi Σπ0 / (p2i Σπ0 + σ2).

*Source: _wp0896 - References (Appendix and References section).*

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