## ONLINE APPENDIX: Two Targets, Two Instruments — Monetary and Exchange Rate Policies in Emerging Market Economies

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**Canonical URL:** [ONLINE APPENDIX: Two Targets, Two Instruments — Monetary and Exchange Rate Policies in Emerging Market Economies](https://www.imf.org/-/media/websites/imf/imported-publications/external/pubs/ft/sdn/2012/_sdn1201appx.pdf)

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### Analytical framework and objectives
- Aggregate demand: y = −β(re − u) (real exchange rate e appreciates when e increases).
- Aggregate supply (Lucas surprise Phillips curve): π = e − y.
- Capital inflows respond to expected real interest differential: Δk = k( r* − r + Δe(·) ).
- Current account: ca = −α e − y.
- Balance of payments identity: ca + k = ΔR.
- Central bank chooses real interest rate r and sterilized FX intervention ΔR.
- Shocks:
  - Foreign interest rate shock r* with mean zero and variance *2 rσ .
  - Aggregate demand shock u with mean zero and variance 2 uσ .
- Timing: wage-setters form expectations eπ (expected inflation) before policy choices; shocks observed after expectations formation but before policy choice. All capital inflows occur in period 1 and remain in period 2; period‑2 variables are zero.
- Simplifying assumptions used in analytic solutions: α = 0, β = 0. Central bank objective function maximizes W = −{(y − 0)2 + (π − 0)2 + a( e − 0)2 + bR2 + c(ΔR)2} (weights a, b, c ≥ 0; output deviation weight normalized to unity).

### Discretionary monetary policy with sterilized FX intervention (Regime D)
- Optimal policies (first-order conditions) yield:
  - Optimal interest rate under discretion: rD = * (1) (1/tilde b) / ((1) + (1/tilde b) + b c + b)  (expression shown in source as equation (7)).
  - Optimal reserves accumulation under discretion: RD = * (1) (1/tilde b) ( … ) (expression shown in source as equation (8)).
  - Under a capital inflow shock (*0r < 0) the central bank lowers r and accumulates reserves (ΔR > 0); responses are less than one-for-one.
  - Under a negative aggregate demand shock (0u < 0) the central bank lowers r and decreases reserves.
- Macro outcomes under discretion (impact responses shown in source):
  - Output gap yD = * ( ) [(1)(1/tilde b) / (abcb + …)] (see equation (9)).
  - Inflation πD = * ( ) (/ ) [(1)(1/tilde b) / (abcb + …)] (see equation (10)).
  - Real exchange rate eD = (1)(1)(a u + r u … ) / (2(1)(1/tilde b) … ) (see equation (11)).
- Key qualitative implications:
  - Capital inflow shock (*0r < 0) or positive demand shock (0u > 0) → overheating: y > 0, π > 0, e > 0 (appreciation).
  - Even absent shocks, an inflationary bias exists: π = ( / ) 0 y a > 0 when the central bank has incentive to generate surprise inflation; credibility issues transmit this bias even if the central bank does not have the incentive.

### Inflation targeting with sterilized FX intervention (Regime IT)
- Binding IT constraint: πIT = 0, eIT = 0 (equation (12) implies r u = −ru = 0 so optimization reduces to FX intervention).
- Objective simplifies; optimal intervention under IT (equation (14)):
  - RIT = * ( ) ( − b u + r ) / (b c + …) (expression shown in source).
- Exchange rate under IT (equation following (14)):
  - eIT = * (1/2) ( c u − r ) / (b c + …) (expression shown in source).
- Comparison with discretion:
  - Central bank intervenes more under IT than under discretion: RIT > RD (inequality shown as (15)).
  - Despite larger intervention, exchange rate is more appreciated under IT than under discretion in response to an inflow shock: eIT > eD (text and inequality (16)).
  - Rationale: IT pins π and the output gap to targets, so deviations fall on e and R, making e more appreciated under IT even with more FX intervention.

### Welfare and regime ranking (qualitative)
- Regimes with FX intervention (two instruments) dominate non-intervention regimes (one instrument) in welfare terms because the intervention option can be set to zero if undesirable.
- Welfare comparison between discretion and IT depends on:
  - Magnitude of shocks (capital inflows and aggregate demand) relative to actual or perceived inflationary bias (0)y> or perceived e y>:
    - Large shocks → discretionary flexibility superior.
    - Low credibility/high perceived inflationary bias → IT discipline superior.

### Multilateral considerations and strategic interaction
- Extended capital flow equation introduces spillovers and deflection parameter θ ≥ 0 capturing capital reallocation across countries.
- Under uncoordinated discretionary (Nash) policies:
  - Optimal r̂Nash and R̂Nash expressions incorporate θ and show each country sets policy rates lower and intervenes less than under coordination.
  - Under coordination (θ = 0) the Nash solutions collapse to the coordinated solutions (equations (20) and (21), equivalent to the single-country discretion solutions (7) and (8)).
- Key multilateral results:
  - In Nash (uncoordinated) equilibrium, countries undertake too little sterilized intervention relative to coordination (inequality (22)).
  - Policy interest rates are maintained too low in Nash relative to coordination (inequality (23)).
  - Net effect on exchange rate ambiguous; typically, exchange rate appreciates more in Nash as countries attempt to deflect capital to others by appreciating their currency.

### Simulation results under discretionary policies (figures summarized)
- Figure A1: Policy responses to a 2.5 percentage-point increase in domestic demand under discretionary policies (with and without FX intervention). Variables shown over horizon t = 0 to 10: Policy interest rate, Reserves, Inflation, Output gap, Shock to domestic demand, Real exchange rate (appreciation when increases).
- Figure A2: Policy responses to a capital inflow shock based on a 5 percentage-point decline in the world interest rate under discretionary policies (with and without FX intervention). Variables shown over horizon t = 0 to 10: Policy interest rate, Reserves, Real exchange rate, Output gap, Cumulative capital inflow, Inflation.

### Initial impact responses to a temporary inflow shock — numerical comparison (Table A1)
- The table reports initial (on-impact) changes in percent for policy interest rate r, FX reserves R, inflation π, and real exchange rate e under four cases:

- IT, 2 instruments (R and r):
  - r = -1.3
  - R = 4.4
  - π = 0
  - e = 5.4

- IT, 1 instrument (r):
  - r = -2.0
  - R = 0
  - π = 0
  - e = 7.8

- Discretionary Policy (Non-IT), 2 instruments:
  - r = -1.5
  - R = 3.7
  - π = 1.0
  - e = 5.0

- Discretionary Policy (Non-IT), 1 instrument:
  - r = -2.2
  - R = 0
  - π = 1.3
  - e = 6.9

- Notes from the simulation summary:
  - Welfare is always higher under two-instrument policies than one-instrument policies within this model.
  - In the absence of an inflationary bias, discretionary policies dominate IT because IT constrains the central bank.
  - If there is an inflationary bias or low credibility, IT can improve welfare by anchoring expectations.

*Source: _sdn1201appx*

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_Source: https://www.imf.org/-/media/websites/imf/imported-publications/external/pubs/ft/sdn/2012/_sdn1201appx.pdf_