## 1. Transition Paths

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---

### I. Bolivia's Current Trajectory
- Fiscal deficit:
  - Increased from 3.4 percent of GDP in 2014 to 8.1 percent of GDP in 2018, fell to 7.2 percent of GDP in 2019, then rose with Covid-19-related fiscal measures.
- Real effective exchange rate:
  - Appreciated by 34 percent since 2014, producing a moderate overvaluation and contributing to an external current account deficit averaging 4 percent of GDP in 2015-2019.
- International reserves:
  - Fell from $US 15.1 billion (45.5 % of GDP) in 2014 to $US 4.7 billion (12.2% of GDP) in mid-August 2021.
- Deposit withdrawals and pressures on the peg: moderate owing to administrative guidance and macroprudential measures.
- Policy trade-offs:
  - Fiscal adjustment reduces drain on national savings, sustains reserves, and helps avoid disorderly adjustment.
  - Continuing to fund fiscal deficits with foreign or private domestic savings reduces net national savings and crowds out private investment.
  - Moving to a floating exchange rate can lessen reserve drains, typically leads to real depreciation improving the current account, smooths consumption under nominal rigidities, and permits independent monetary policy.
  - A peg anchors inflationary expectations, can discipline fiscal policy, and may improve financial stability when unhedged financial liabilities are in dollars — but credibility risks can reverse this.

### II. Theoretical Approaches to Setting Exchange Rate Policy
- Mundell-Fleming implications:
  - With sticky prices, uncovered interest parity, and perfect asset substitutability: fiscal stimulus effective under a fixed exchange rate; monetary stimulus ruled out when maintaining the peg.
  - Under a float, fiscal stimulus partially dissipates via currency appreciation; monetary policy can be used to raise output accompanied by depreciation.
- Speculative-attack models (Krugman 1979; Flood and Garber 1984):
  - Recurring fiscal deficits under a fixed regime can lead to speculative attack and uncontrolled float; stochastic shocks increase the likelihood of attack or disorderly adjustment.
  - Floating regimes are not susceptible to these speculative-attack dynamics.
- Neo-Keynesian (Schmitt-Grohé and Uribe 2017):
  - Fixed exchange rate plus nominal wage rigidity leads to involuntary unemployment after negative shocks; flexible exchange rates allow real wage adjustment and labor market clearing.
- Reserves and fiscal implications (Bianchi and Sosa-Padilla 2020):
  - Reserves average 16 percent of GDP in pegged countries vs. 7 percent for floaters.
  - Fixed-rate countries need higher primary surpluses to build reserves for self-insurance.

### III. A Calibrated Model of the Bolivian Economy
- Model features:
  - Dynamic stochastic general equilibrium model for a small open commodity-exporting economy.
  - Upward-sloping Phillips curves and Calvo price adjustment for intermediate goods.
  - Ex-ante wage bargaining with wage stickiness and potential underemployment.
  - Two goods: home good and foreign good with detailed demand sources.
  - Households: Ricardian and non-Ricardian "hand-to-mouth".
  - Adjustment costs of investment limit crowding out; fiscal multiplier averages 0.4 for expenditure increases.
- Central bank balance sheet and quasi-fiscal mechanics:
  - Assets: government bonds and foreign reserves. Liabilities: monetary aggregates.
  - Quasi-fiscal (seignorage) revenues to central government accrue from differences between asset and liability returns.
- External sector and financial wedge:
  - Divergence from uncovered interest parity via a financial-accelerator-style wedge inversely related to Bolivia’s net foreign asset position.
  - Foreign and domestic bonds are imperfect substitutes; sterilized intervention is effective.
- Monetary strategies modeled:
  - Under a peg: central bank intervenes in currency market, adjusting money supply as it buys/sells foreign reserves.
  - Under inflation targeting: foreign reserves remain unchanged; central bank supplies money to keep nominal interest rate equal to policy rate.
- Model structure: detailed in Appendix I.

### IV. Two Steady States
- Calibration targets (model initial state reflecting end-2019):
  - External deficit: 5 percent of GDP
  - Fiscal deficit: 7 percent of GDP
  - Output growth: 3 percent
  - Potential output growth: 3.8 percent
  - Inflation rate: 1.8 percent
  - Foreign reserves: 25 percent of GDP
  - Central bank peg with target devaluation rate: 0.0 percent
- Two steady states obtained: fixed exchange rate (peg) and floating exchange rate with inflation target (IT).
- Key steady-state table values (preserved exactly):
  - Inflation Target: 2.0% (Fix) / 4.0% (IT) / 4.0% (Alternative)
  - Nominal devaluation: 0.0% / 0.0% / 0.0%
  - Nominal Interest Rate: 5.1% / 7.1% / 7.1%
  - Foreign Inflation: 2.0% / 2.0% / 2.0%
  - Central Bank Money / GDP: 25.0% / 25.0% / 21.4%
  - Foreign Reserves / GDP: 21.0% / 21.0% / 17.4%
  - Long run Government Debt / GDP: 40.0% / 40.0% / 40.0%
  - Quasifiscal gain / GDP: 1.2% / 1.7% / 1.4%
  - Government Expenditure / GDP: 14.0% / 14.4% / 14.2%
  - Government Investment / GDP: 10.0% / 10.0% / 10.0%
  - Long run sustainable primary deficit / GDP: -1.5% / -1.9% / -1.7%
  - NFA / GDP: 15.1% / 15.1% / 15.1%
  - Private external debt / GDP: 5.9% / 5.9% / 2.3%
  - NX / GDP: 0.1% / 0.1% / 0.1%
  - CA / GDP: 0.8% / 0.8% / 0.8%
  - Private Consumption / GDP: 66.9% / 66.9% / 66.9%
  - Private Investment / GDP: 8.0% / 8.0% / 8.0%
- Main steady-state insight:
  - Floating steady state allows a sustainable primary fiscal deficit 0.5 percent of GDP larger than with the fix and slightly higher government expenditure (0.5 percent of GDP).
  - Steady-state comparison abstracts from stochastic shocks and thus does not capture greater reserve self-insurance needs under a peg.

### V. Transition Paths to the Steady State
- Policymaker objective function components:
  - Aggregated household utility (Ricardian and non-Ricardian), adjustment cost of altering tax policy, and penalty term for divergences from steady-state public debt stock.
- Commitment assumptions:
  - Policymakers credibly commit to trajectory for exchange rate regime and control variables; private-sector expectations correspond to this commitment.
  - Linearized model yields certainty-equivalent solution paths despite stochastic shocks.
- Policy implementation rules modeled:
  - Transition to float: central bank immediately shifts to inflation target corresponding to floating steady state (4 percent) and stops intervening in FX market.
  - Transition to peg: nominal exchange rate remains fixed or depreciates at a steady 2 percent; domestic interest rates follow a standard Taylor rule; reserves meet financing gaps.
- Optimal fiscal outcomes:
  - Optimal fiscal deficit under inflation targeting is wider than under the peg, reflecting larger quasi-fiscal balance under higher inflation.
  - Under IT consumers enjoy a higher average level of utility due to better shock absorption (less underemployment and smaller declines in consumption).
- Quantitative transition dynamics under IT:
  - Currency: 11 percent nominal depreciation and 6 percent real depreciation on impact.
  - Cumulative nominal depreciation of 21 percent and real depreciation of 10 percent after five years.
  - Inflation rises to 7 percent in the first year of inflation targeting, then converges to the 4 percent target as interest rates are increased.
- Under the peg:
  - No change in nominal exchange rate; real exchange rate depreciates modestly via domestic disinflation below its 2 percent steady-state level, with adverse impacts on GDP and welfare due to nominal rigidities.
- Role of debt-weight parameter (ω1):
  - High ω1: fiscal balance reaches steady state after three years under IT and after four years under the peg.
  - Low ω1: foreign reserves decline rapidly under the peg, reflecting weak fiscal commitment and potential credibility problems.
  - Higher ω1 increases likelihood that optimal policy is self-reinforcing and credible.
- Assumption: commitment to chosen exchange rate regime is fully credible, ruling out ex ante switching under pressure.

### VI. Implications of the Exchange Rate Regime for Fiscal Policy
- Deterministic steady-state distinction: limited to seignorage/quasi-fiscal gains from higher IT inflation.
- With stochastic shocks on transition paths, floating rate advantages increase:
  - Fixed exchange rate plus labor market rigidity implies lower employment after negative shocks, increasing need for countercyclical fiscal support.
  - Floating rate accommodates monetary adjustments for the domestic business cycle, reducing need for preemptive conservative fiscal policy.
  - As shock variance increases, relative advantage of float over peg rises.

### VII. A Consistent Model with Regime Switching
- Model extension allows expected or unexpected changes in exchange rate regime after initial period.
- Expected transition scenario:
  - Authorities announce adoption of inflation targeting and the transition date; all agents know the transition date and adjust behavior accordingly.
  - Authorities adopt an optimal fiscal path with the exchange rate floating at the announced crossover point (three years after the initial period) and monetary authority switching to a 4 percent inflation target.
- Unexpected transition scenario:
  - Private agents do not anticipate a switch to inflation targeting; regime change occurs without prior private-sector expectation.

### Appendix 2: Transition scenarios, risk-off shock, and solution method
- Policy-optimal path assumptions:
  - Social planner maximizes intertemporal objective and implements indicated policies.
  - Expected transition (full commitment): transition takes place smoothly with steady fiscal deficit reduction and gradual depreciation once IT is adopted; intertemporal welfare comparable to immediate adoption.
  - Time consistency caveat: model restricts fiscal policies to be optimal given the regime and includes a penalty for debt deviations.
- Risk-off shock (modeled implementation and outcomes):
  - Modeled as a one-off increase in Ωu raising interest rates charged to Bolivians borrowing in dollars.
  - Quantitative responses:
    - Current account: switches from a 5 percent of GDP deficit to a surplus of 5 percent of GDP in just two periods.
    - Pegged regime (even allowing a two percent nominal devaluation): sudden large loss of reserves, increase in real interest rate, downward price adjustments with rising unemployment, larger loss in output and employment relative to IT.
    - Inflation targeting (IT): nominal exchange rate depreciates 19 percent on impact and another 5 percent in the following period; domestic prices rise in local currency terms but decline in dollar terms; loss in output and employment is significantly less than with the peg.
  - Policy implication: exchange rate flexibility offers advantages in responding to investor-preference shocks and can prevent untenable reserve and output outcomes.
- Model-implied required adjustment:
  - Cumulative real depreciation of 10 percent to reach sustainable equilibrium (same under IT or fixed regime); speed of convergence differs.
  - Bolivia experienced a 34 percent real appreciation over the past 7 years (larger than the model’s 10 percent depreciation).
  - An omega shock scenario may produce a 20 percent real depreciation on impact.
- Empirical lessons from other countries:
  - Successful managed transitions: Chile (1984-99), Israel (1995-2005), Poland (1990-2000), Russia (2005-14) — features: tranquil market conditions, adequate reserve coverage, careful communication, complementary institutions, gradual implementation.
  - Stressful transitions: Brazil (1999), Czech Republic (1997), Turkey (2001), Uruguay (2002) — some avoided large output losses and ultimately were effective.
- Bolivia-specific assessment:
  - Enabling factors: relatively benign external environment, institutional features such as a foreign currency auction mechanism that could be developed.
  - Constraints: reserve coverage is low, fiscal deficit is large.
  - Mitigants: up-front fiscal policy shift together with increased financing from international financial institutions may help close gaps.
- Model limitations and suggested extensions:
  - First-order linear approximation rules out some disorderly adjustment scenarios; risk-off approximates a sufficiently strong shock.
  - Credibility factors are hard to model; framework assumes rational expectations and optimal fiscal policy given the regime.
  - Suggested extensions: introduce explicit time consistency between regimes; perform second-order linearization; reassess nominal money demand elasticity with respect to inflation; consider informal limits on capital flows and moral suasion in capital account.

### Appendix I: Model structure (high-level inventory)
- Comprehensive DSGE with:
  - Heterogeneous households (non-Ricardian and Ricardian), habit persistence, money demand specification.
  - Labor market with union wage setting and Calvo wage/price stickiness and indexation.
  - Firms: final, intermediate, and domestic goods producers with monopolistic competition and investment adjustment costs.
  - Monetary policy regimes: inflation targeting rule and peg; central bank balance sheet and quasi-fiscal balance (qfa_t) explicitly modeled.
  - Fiscal policy: taxes, transfers, government debt dynamics, public capital accumulation with implementation lag L.
  - External sector: country risk premium function of NFA-to-GDP with parameter Ω_u; detailed balance of payments and NFA dynamics.
- Equations and first-order conditions for consumption, investment, bond holdings, money demand, wage and price-setting, firms' marginal costs, and government budget constraint provided in text.

### Appendix II: Linear Time Iteration Algorithm (solution method)
- Uses Rendahl (2017) LTI method to permit simulation with known future regime switches.
- Perturbation and linearization around a non-stochastic steady state x*; first-order Taylor expansion yields linear system with Jacobians A, B, C.
- Assumes solution form u_t = F u_{t−1}; iterative procedure updates F via F_{n+1} = −(B + C F_n)^{-1} A until convergence.
- Generalization around arbitrary point x̄ with constant G* term yields solution form u_t = E + F u_{t−1} with iterative updates for E and F.
- Application to regime-switching with known switch time T:
  - Solve backward from absorbing stable regime M1 at time T using period-specific F_T to compute F_{T−1}, E_{T−1}, etc.
  - Solutions are sequences of E's and F's that vary by period before T; agents internalize deterministic convergence toward the stable regime.
- Interpretation: algorithm enables analysis of dynamics and transitions from potentially unstable regimes toward a stable regime.

*Source: IMF Working Paper — "Fix vs. Float: Evaluating the Transition to a Sustainable Equilibrium in Bolivia" (Chapter/Section: 1. Transition Paths; Appendices I–II).*

### 1. Transition Paths _______________________________________________________________ 7

### 1. Transition Paths

### I. Bolivia's Current Trajectory
- Fiscal deficit increased from 3.4 percent of GDP in 2014 to 8.1 percent of GDP in 2018, fell to 7.2 percent of GDP in 2019, then rose with Covid-19-related fiscal measures.
- Real effective exchange rate appreciated by 34 percent since 2014, producing a moderate overvaluation and contributing to an external current account deficit averaging 4 percent of GDP in 2015-2019.
- International reserves fell from $US 15.1 billion (45.5 % of GDP) in 2014 to $US 4.7 billion (12.2% of GDP) in mid-August 2021.
- Deposit withdrawals and occasional pressures on the peg have been moderate due to administrative guidance and macroprudential measures.
- Policy trade-offs:
  - Fiscal adjustment reduces drain on national savings, sustains reserves, and helps avoid disorderly adjustment.
  - Continuing to fund fiscal deficits with foreign or private domestic savings reduces net national savings and crowds out private investment.
  - Moving to a floating exchange rate can lessen reserve drains, typically leads to real depreciation improving the current account, smooths consumption under nominal rigidities, and permits independent monetary policy.
  - A peg anchors inflationary expectations, can discipline fiscal policy, and may improve financial stability when unhedged financial liabilities are in dollars — but credibility risks can reverse this.

### II. Theoretical Approaches to Setting Exchange Rate Policy
- Mundell-Fleming model implications:
  - With sticky prices, uncovered interest parity, and perfect asset substitutability: fiscal stimulus effective under a fixed exchange rate; monetary stimulus ruled out when maintaining the peg.
  - Under a float, fiscal stimulus partially dissipates via currency appreciation; monetary policy can be used to raise output accompanied by depreciation.
- Speculative attack models (Krugman 1979; Flood and Garber 1984):
  - Recurring fiscal deficits under a fixed regime can lead to speculative attack and uncontrolled float; stochastic shocks increase the likelihood of attack or disorderly adjustment.
  - Floating regimes are not susceptible to these speculative-attack dynamics.
- Neo-Keynesian (Schmitt-Grohé and Uribe 2017):
  - Fixed exchange rate plus nominal wage rigidity leads to involuntary unemployment after negative shocks; flexible exchange rates allow real wage adjustment and labor market clearing.
- Reserves and fiscal implications (Bianchi and Sosa-Padilla 2020):
  - Reserves average 16 percent of GDP in pegged countries vs. 7 percent for floaters.
  - Fixed-rate countries need higher primary surpluses to build reserves for self-insurance.

### III. A Calibrated Model of the Bolivian Economy
- Model features:
  - Dynamic stochastic general equilibrium model for a small open commodity-exporting economy.
  - Upward-sloping Phillips curves and Calvo price adjustment for intermediate goods.
  - Ex-ante wage bargaining with wage stickiness and potential underemployment.
  - Two goods: home good (demand from government consumption, private consumption, public and private investment, exports) and foreign good (demand from domestic private investment and consumption).
  - Households: Ricardian and non-Ricardian "hand-to-mouth".
  - Adjustment costs of investment limit crowding out; fiscal multiplier averages 0.4 for expenditure increases.
- Central bank balance sheet:
  - Assets: government bonds and foreign reserves.
  - Liabilities: monetary aggregates.
  - Transfers quasi-fiscal (seignorage) revenues to central government.
  - Quasi-fiscal revenues accrue from differences between asset and liability returns.
- External sector:
  - Divergence from uncovered interest parity via a financial-accelerator-style wedge inversely related to Bolivia’s net foreign asset position.
  - Foreign and domestic bonds are imperfect substitutes; sterilized intervention is effective.
- Monetary strategies:
  - Under a peg: central bank intervenes in currency market, adjusting money supply as it buys/sells foreign reserves.
  - Under inflation targeting: foreign reserves remain unchanged; central bank supplies money to keep nominal interest rate equal to policy rate.
- Model structure detailed in Appendix I.

### IV. Two Steady States
- Calibration targets (model initial state reflecting end-2019):
  - External deficit: 5 percent of GDP
  - Fiscal deficit: 7 percent of GDP
  - Output growth: 3 percent
  - Potential output growth: 3.8 percent
  - Inflation rate: 1.8 percent
  - Foreign reserves: 25 percent of GDP
  - Central bank peg with target devaluation rate: 0.0 percent
- Two steady states obtained: fixed exchange rate (peg) and floating exchange rate with inflation target (IT).
- Key steady-state comparisons (as presented in table; values preserved exactly):
  - Inflation Target: 2.0% (Fix) / 4.0% (IT) / 4.0% (Alternative)
  - Nominal devaluation: 0.0% / 0.0% / 0.0%
  - Nominal Interest Rate: 5.1% / 7.1% / 7.1%
  - Foreign Inflation: 2.0% / 2.0% / 2.0%
  - Central Bank Money / GDP: 25.0% / 25.0% / 21.4%
  - Foreign Reserves / GDP: 21.0% / 21.0% / 17.4%
  - Long run Government Debt / GDP: 40.0% / 40.0% / 40.0%
  - Quasifiscal gain / GDP: 1.2% / 1.7% / 1.4%
  - Government Expenditure / GDP: 14.0% / 14.4% / 14.2%
  - Government Investment / GDP: 10.0% / 10.0% / 10.0%
  - Long run sustainable primary deficit / GDP: -1.5% / -1.9% / -1.7%
  - NFA / GDP: 15.1% / 15.1% / 15.1%
  - Private external debt / GDP: 5.9% / 5.9% / 2.3%
  - NX / GDP: 0.1% / 0.1% / 0.1%
  - CA / GDP: 0.8% / 0.8% / 0.8%
  - Private Consumption / GDP: 66.9% / 66.9% / 66.9%
  - Private Investment / GDP: 8.0% / 8.0% / 8.0%
- Main steady-state insight:
  - Direct comparison shows limited differences; floating steady state allows a sustainable primary fiscal deficit 0.5 percent of GDP larger than with the fix and slightly higher government expenditure (0.5 percent of GDP).
  - Model steady-state comparison abstracts from stochastic shocks and thus does not capture greater reserve self-insurance needs under a peg.

### V. Transition Paths to the Steady State
- Policymaker objective function includes:
  - Aggregated household utility (Ricardian and non-Ricardian), adjustment cost of altering tax policy, and penalty term for divergences from steady-state public debt stock.
- Policymaker commitment assumptions:
  - Policymakers credibly commit to trajectory for exchange rate regime and control variables; private-sector expectations correspond to this commitment.
  - Linearized model yields certainty-equivalent solution paths despite stochastic shocks.
- Policy implementation rules:
  - Transition to float: central bank immediately shifts to inflation target corresponding to floating steady state (4 percent) and stops intervening in FX market.
  - Transition to peg: nominal exchange rate remains fixed or depreciates at a steady 2 percent; domestic interest rates follow a standard Taylor rule; reserves meet financing gaps.
- Optimal fiscal outcomes:
  - Optimal fiscal deficit under inflation targeting is wider than under the peg, reflecting larger quasi-fiscal balance under higher inflation.
  - Under IT consumers enjoy a higher average level of utility due to better shock absorption (less underemployment and smaller declines in consumption).
- Quantitative transition dynamics under IT:
  - Currency undergoes an 11 percent nominal depreciation and 6 percent real depreciation on impact.
  - Cumulative nominal depreciation of 21 percent and real depreciation of 10 percent after five years.
  - Inflation rises to 7 percent in the first year of inflation targeting, then converges to the 4 percent target as interest rates are increased.
- Under the peg:
  - No change in nominal exchange rate; real exchange rate depreciates modestly via domestic disinflation below its 2 percent steady-state level, with adverse impacts on GDP and welfare due to nominal rigidities.
- Role of debt-weight parameter (ω1):
  - High ω1: fiscal balance reaches steady state after three years under IT and after four years under the peg.
  - Low ω1: foreign reserves decline rapidly under the peg, reflecting weak fiscal commitment and potential credibility problems.
  - Higher ω1 increases likelihood that optimal policy is self-reinforcing and credible.
- Assumption: commitment to chosen exchange rate regime is fully credible, ruling out ex ante switching under pressure.

### VI. Implications of the Exchange Rate Regime for Fiscal Policy
- Key distinctions:
  - In deterministic steady states, differences limited to seignorage/quasi-fiscal gains from higher IT inflation.
  - With stochastic shocks on transition paths, float has additional advantages:
    - Fixed exchange rate plus labor market rigidity implies lower employment after negative shocks, increasing need for countercyclical fiscal support.
    - Floating rate accommodates monetary adjustments for the domestic business cycle, reducing need for preemptive conservative fiscal policy.
    - As shock variance increases, relative advantage of float over peg rises.

### VII. A Consistent Model with Regime Switching
- Model extension allows expected or unexpected changes in exchange rate regime after initial period.
- Expected transition scenario:
  - Authorities announce adoption of inflation targeting and the transition date; all agents know the transition date and adjust behavior accordingly.
  - Authorities adopt an optimal fiscal path with the exchange rate floating at the announced crossover point (three years after the initial period) and monetary authority switching to a 4 percent inflation target.
- Unexpected transition scenario:
  - Private agents do not anticipate a switch to inflation targeting; regime change occurs without prior private-sector expectation.

*Source: IMF Working Paper — "Fix vs. Float: Evaluating the Transition to a Sustainable Equilibrium in Bolivia" (Chapter/Section: 1. Transition Paths).*

### Appendix 2 describes the solution method.

### wpiea2022043-print-pdf - Appendix 2 describes the solution method.

### Transition scenarios and policy-optimal path
- Policy path is optimal: the social planner (or the authorities) maximizes the intertemporal objective function and implements the indicated policies.
- Expected transition (full commitment by authorities; private agents know and expect the trajectory):
  - Transition takes place smoothly with a steady reduction in the fiscal deficit, and a gradual depreciation of the currency once the transition to the IT regime takes place.
  - This trajectory yields a level of intertemporal welfare comparable to, though slightly below, that resulting from immediate adoption.
- Time consistency caveat:
  - Even if the planner has an incentive to diverge at some point, full commitment assumption leads private agents to expect implementation.
  - The model restricts fiscal policies to be optimal given the exchange rate regime and includes a penalty for divergence from the steady-state debt level in the policymaker’s objective function (which tends to force the policymaker towards a time consistent path with respect to private expectations).

### Risk-off shock: mechanism and modeled implementation
- Risk-off shock description:
  - Adverse shift in asset preferences prompts investors to demand a higher risk premium to hold domestic assets.
  - Within the model, this results in a higher interest rate charged to Bolivians borrowing in dollars and, via imperfect substitution between domestic and foreign assets, a higher domestic interest rate.
  - If large, the effect is similar to a sudden stop.
- Modeled implementation:
  - The preference shift is implemented as a one-off increase in Ωu, the coefficient on debt in the interest rate specification, which effectively increases the interest rate that investors charge to hold domestic assets. (See: Equation 4 in Appendix 1.)
- Quantitative and qualitative model responses to a large unexpected risk-off shock:
  - Under either regime, the current account switches from a 5 percent of GDP deficit to a surplus of 5 percent of GDP in just two periods.
  - Under the pegged regime (even allowing for a two percent nominal devaluation):
    - There is a sudden large loss of reserves and increase in the real interest rate as authorities struggle to maintain the designated exchange rate.
    - Prices adjust downward but, due to price stickiness, unemployment increases.
    - Outcome: larger loss in output and employment relative to IT.
  - Under inflation targeting (IT):
    - The nominal exchange rate depreciates 19 percent on impact, and another 5 percent in the following period.
    - Domestic prices rise in local currency terms but decline in dollar terms.
    - Outcome: loss in output and employment is significantly less than with the peg.
- Policy implication:
  - The differential response highlights the advantages of exchange rate flexibility in response to shocks to investor preferences.
  - The scenario illustrates pressures on prices, employment, reserves, and output that may make continued adherence to a fixed exchange rate untenable after a large shock and may force a switch to a floating exchange rate.

### Comparison with other exchange rate assessments
- Model-implied required adjustment:
  - The model points to a cumulative real depreciation of 10 percent to bring Bolivia into a sustainable equilibrium (change between initial conjuncture and steady state).
  - The required 10 percent adjustment is the same under IT or a fixed regime; the difference lies in speed of convergence.
- Comparison with observed and alternative-model adjustments:
  - Bolivia experienced a real exchange rate appreciation of 34 percent over the past 7 years (larger than the model’s 10 percent depreciation).
  - Some alternative models imply larger depreciations; part of the discrepancy is attributable to the large fiscal adjustment implemented in this model to reach the steady state.
  - If alternative models assume an immediate shutdown of foreign financing, results may be more comparable to the omega shock scenario, where a sudden rise in the risk premium produced a 20 percent real exchange rate depreciation realized on impact.

### Experience with transitions and areas for further development
- Empirical lessons from other countries:
  - Successful managed transitions: Chile (1984-99), Israel (1995-2005), Poland (1990-2000), Russia (2005-14) — typically tranquil market conditions, adequate reserve coverage, careful communication, complementary institutions (e.g., derivatives markets), gradual implementation often involving one-time step devaluation followed by sliding peg and widening trading band.
  - Stressful transitions: Brazil (1999), Czech Republic (1997), Turkey (2001), Uruguay (2002) — although Brazil and Czech Republic avoided large output losses and transitions were ultimately effective.
- Bolivia-specific assessment:
  - Some enabling factors are present: relatively benign external environment, institutional features such as a foreign currency auction mechanism that could be further developed.
  - Constraints: reserve coverage is low, and the fiscal deficit is large.
  - Mitigants: an up-front shift in fiscal policy together with increased financing from international financial institutions may help close gaps.
- Model limitations and future improvements:
  - The model’s first-order linear approximation rules out some disorderly adjustment scenarios from repeated negative shocks; the risk-off scenario approximates a shock strong enough to prompt disorderly adjustment.
  - Credibility factors are difficult to model fully; current framework attributes rational expectations to private agents and imposes optimal fiscal policy given the regime.
  - Suggested extensions:
    - Introduce a requirement of time consistency between exchange rate regimes.
    - Perform second-order linearization of the model.
    - Reassess nominal money demand elasticity with respect to inflation (stronger negative correlation would yield a lower quasi-fiscal gain from IT).
    - Consider informal limits on capital flows and aspects of moral suasion in Bolivia’s capital account (the model currently assumes free capital flows).

### Conclusion: tradeoffs, welfare, and policy implications
- The small-country open-economy DSGE model elucidates tradeoffs between fixed and floating/IT regimes for Bolivia and generates stable transition paths under either regime when fiscal policy is time consistent and optimized for the chosen exchange rate path.
- Welfare and policy takeaways:
  - There are welfare benefits from adopting a floating regime, but these gains are relatively modest.
  - Continued adherence to a fixed exchange rate remains a plausible policy alternative, though it requires a larger fiscal adjustment and a higher path for the fiscal balance.
  - In risk-off scenarios (replicating conditions like the 2013 emerging market “taper tantrum”), the relative welfare advantage of exchange rate flexibility increases significantly.

*Source: wpiea2022043-print-pdf - Appendix 2 describes the solution method.*

### Appendix I. Model Structure

### Appendix I. Model Structure

### Model overview
- The model incorporates features from the canonical Pillar IV DSGE macroeconomic model of the financial programming initiative of the 2.0 of ICD.
- Agents: continuum of households indexed by i ∈ [0,1]. Households in [0,ω) are non-Ricardian (no financial market access, consume disposable income each period). Households in (ω,1] are Ricardian (access financial markets, own physical capital).
- Utility for all agents: 
  - u_t(C_t,N_t) = z_{u,t} ln(C_t − hab C_{t−1}) − γ_N (N_t^{1+σ_L})/(1+σ_L) + γ_m + (1/(1−σ_M)) (M_t/P_t)^{1−σ_M}
  - Parameters: σ_L (inverse of Frisch elasticity), σ_M (elasticity of money demand), γ_N, γ_m (scale parameters), hab (habit persistence), z_{u,t} follows an ARMA process.
- Non-competitive labor market introduces a wedge between the marginal rate of substitution and the real wage (Schmitt-Grohé and Uribe (2005) framework).

### Households — Non-Ricardian
- Budget constraint (nominal terms):
  - (1+τ_{c,t}) P_{c,t} C_{v,t} + M_{v,t} = (1−τ_{w,t}) ∫ W_{j,t} N_{v,t}^j dj + M_{v,t−1} + P_{c,t} T_{v,t}
- First-order conditions (consumption and money demand), with Lagrange multiplier Λ_{v,t}:
  - (1+τ_{c,t}) Λ_{v,t} P_{c,t} = z_{u,t} (C_{v,t} − hab C_{v,t−1})^{−1} − β hab E_t[ z_{u,t+1} (C_{v,t+1} − hab C_{v,t})^{−1} ] − γ_m (M_{v,t})^{σ_M} + β E_t[ Λ_{v,t+1} ] − Λ_{v,t} = 0
  - (expression presented in the source as the first-order condition combining money demand and multipliers)

### Households — Ricardian
- Resource constraint (nominal terms):
  - (1+τ_{c,t}) P_{c,t} C_{p,t} + P_{x,t} X_{p,t} + M_{p,t} + B_{p,t} + S_t B_{p,t}^* = R_{t−1} B_{p,t−1} + S_t R_{t−1}^* B_{p,t−1}^* + (1−τ_{w,t}) ∫ W_{j,t} N_{p,t}^j dj + [ (1−τ_{k,t}) R_{t}^k + τ_{k,t} δ Q_{t−1} ] K_{p,t−1} + γ_{ccpm} (1−ω) S_t C̄ P_{cpm,t}^* + T_{p,t} + ξ_{p,t} + M_{p,t−1}
- Notation and components retained exactly:
  - C_{p,t} per-capita Ricardian consumption; X_{p,t} investment; P_{x,t} nominal price of investment good; B_{p,t} nominal government bond paying nominal rate R_t; S_t nominal exchange rate (domestic per unit foreign); B_{p,t}^* foreign-currency bond.
  - After-tax capital income: (1−τ_{k,t}) R_{t}^k + τ_{k,t} δ Q_{t−1}
  - Commodity export revenue per-capita: γ_{ccpm} (1−ω) S_t C̄ P_{cpm,t}^*
  - Q_t nominal price of installed capital; δ depreciation rate.
- Investment adjustment cost:
  - f(X_{p,t}/X_{p,t−1}) = (a/2) (X_{p,t}/X_{p,t−1} − 1)^2, with a controlling investment adjustment speed.
- Capital accumulation:
  - K_{p,t} = (1−δ) K_{p,t−1} + z_{x,t} X_{p,t} (1 − f(X_{p,t}/X_{p,t−1})), where z_{x,t} is investment-specific shock (ARMA).
- First-order conditions (consumption, domestic and foreign bonds, investment, capital, money) are provided in the text, including:
  - Consumption: (1+τ_{c,t}) Λ_{p,t} P_{c,t} = z_{u,t} (C_{p,t} − hab C_{p,t−1})^{−1} − β hab E_t[ z_{u,t+1} (C_{p,t+1} − hab C_{p,t})^{−1} ] − Λ_{p,t} + β E_t[ Λ_{p,t+1} R_t ] = 0
  - Foreign bond Euler: −Λ_{p,t} S_t + β E_t[ Λ_{p,t+1} S_{t+1} R_t^* ] = 0
  - Investment FOC involves μ_t, z_{x,t}, f and f′ as in the text.
  - Money demand: γ_m (M_{p,t})^{σ_M} + β E_t[ Λ_{p,t+1} ] − Λ_{p,t} = 0
- Derivatives of f given explicitly:
  - f′(X_{c,t}/X_{c,t−1}) = a (X_{c,t}/X_{c,t−1} − 1) (1/X_{c,t−1})
  - f′(X_{c,t+1}/X_{c,t}) = a (X_{c,t+1}/X_{c,t} − 1) (1/X_{c,t})^2

### Labor markets and wage setting
- Labor unions set wages on behalf of households; continuum of unions, one per labor type d.
- Restrictions for union d:
  - Resource constraint: N_{j,t} = ∫_0^1 N_{t}(d,i) di (Eq1)
  - Labor demand for type d: N_{j,t} = (W_{j,t}/W_t)^{−ε_w} N_{t}^d (Eq2), where ε_w elasticity of substitution across labor types, N_t^d aggregate labor demand, W_t aggregate nominal wage index, W_{j,t} union wage.
- Wage adjustment: each period probability of not being able to adjust wages is θ_w; probability of setting optimally is 1−θ_w. When unable, indexation rule applies:
  - W_t = W_{t−1} g_z π_{t−1}^{χ_w} π̄^{(1−χ_w)}, with χ_w wage indexation parameter and g_z long-run productivity growth.
- Value of nominal wage s periods after last re-optimization:
  - W_{j,t+v} = W_t^* ∏_{k=1}^v ( g_z π̄^{(1−χ_w)} π_{t+k−1}^{χ_w} )
  - Real wage analog: w_{j,t+v} = w_t^* X_{t,v}^w, with X_{t,v}^w defined in text.
- Union chooses labor N_t(d) maximizing weighted utility; optimality condition:
  - U_N(N_{j,t}) = (1−τ_{w,t}) w_t [ ω λ_{v,t}^v + (1−ω) λ_{p,t}^p ] mmt_{t}^w (co-state variable notation preserved)
- If wages flexible, first-order condition simplifies to mark-up relation:
  - [ (1−ω) 1/MRS_{t+v}^p + ω 1/MRS_{t+v}^v ] w_t^* = ε_w (ε_w − 1)
- Equilibrium relation with wage dispersion:
  - N_t = ν_{w,t} N_t^d, with ν_{w,t} > 1 measuring inefficiency from wage dispersion. Recursive expression for ν_{w,t} given in text. If wages fully flexible, ν_{w,t} = 1.
- Aggregate real wage index:
  - w_t = [ θ_w ( w_{t−1}/w_t g_z π_{t−1}^{χ_w} π̄^{1−χ_w} )^{1−ε_w} + (1−θ_w) (w_t^*)^{1−ε_w} ]^{1/(1−ε_w)}

### Firms — Final goods producers
- Three types of goods producers: final goods, intermediate goods, domestic goods (homogenous, exported).
- Final consumption good aggregator (home and foreign inputs):
  - C_t = [ (1−α_c)^{1/η_c} (C_{H,t})^{(η_c−1)/η_c} + (α_c)^{1/η_c} (C_{F,t})^{(η_c−1)/η_c} ]^{η_c/(η_c−1)}, with η_c elasticity of substitution, α_c foreign share.
- First-order demands:
  - C_{H,t} = (1−α_c) (p_{H,t})^{−η_c} C_t
  - C_{F,t} = α_c (p_{F,t})^{−η_c} C_t
  - where p_{H,t} = P_{H,t}/P_{c,t}, p_{F,t} = P_{F,t}/P_{c,t}.
- Investment good aggregator similarly:
  - X_t = [ (1−α_x)^{1/η_x} (X_{H,t})^{(η_x−1)/η_x} + (α_x)^{1/η_x} (X_{F,t})^{(η_x−1)/η_x} ]^{η_x/(η_x−1)}
  - First-order conditions:
    - X_{H,t} = (1−α_x) (p_{H,t}/p_{x,t})^{−η_x} X_t
    - X_{F,t} = α_x (p_{F,t}/p_{x,t})^{−η_x} X_t
  - p_{x,t} = P_{x,t}/P_{c,t} relative price of investment.

### Firms — Domestic and intermediate goods
- Domestic good produced by perfectly competitive firm combining intermediate goods:
  - Y_{H,t} = [ ∫_0^1 Y_{j,t}^{1−1/ε_H} dj ]^{ε_H/(ε_H−1)}, with ε_H elasticity of substitution across varieties.
- Demand for intermediate variety j and price index:
  - Y_{H,j,t} = (P_{H,j,t}/P_{H,t})^{−ε_H} Y_{H,t,d}
  - P_{H,t} = [ ∫_0^1 P_{H,j,t}^{1−ε_H} dj ]^{1/(1−ε_H)}
- Demand for final domestic goods:
  - Y_{H,t}^d = C_{H,t}^* + C_{g,t} + X_{g,t} + C_{H,t} + X_{H,t}, where C_{H,t}^* foreign demand (exports), X_{g,t} public investment.
- Intermediate goods producers (monopolistic) production:
  - Y_{H,p,t} = z_{y,t} K_{p,t−1}^α Z_t^{1−α−α_g} (N_{p,t})^{1−α} (K_{t−1}^g)^{α_g}, with α ∈ (0,1), Z_t permanent productivity such that Z_{t+1}/Z_t = g_{z,t}, g_{z,t} follows AR(1) with parameters in text, z_{y,t} transitory productivity shock, K_{t−1}^g public capital.
- Labor composite used by firm l:
  - N_{p,t} = [ ∫_0^1 N_{j,t}^{(ε_w−1)/ε_w} dj ]^{ε_w/(ε_w−1)} (Eq3)
- Firms minimize cost subject to composite labor; demand for labor variety:
  - N_{p,j,t} = (W_{j,t}/W_t)^{−ε_W} N_{p,t}
  - Aggregate demand: N_{j,t} = (W_{j,t}/W_t)^{−ε_W} N_t^d
- First-order conditions (marginal cost relations):
  - W_t = (1−α) MC_{H,t} Y_{p,t} / N_{p,t}
  - R_t^k = α MC_{H,t} Y_{p,t} / K_{p,t−1}
- Marginal cost:
  - MC_{H,t} = 1/Z_t^{1−α−α_g} G_K_{g,t−1}^{α_g} (R_k,t)^{α} (W_t)^{1−α} (expression as in text)

### Price setting and Calvo pricing
- Calvo pricing for goods: fraction 1−θ_H reset prices optimally each period; θ_H follow indexation rule when not adjusting:
  - When not adjusted: P_{H,l,t} = P_{H,l,t−1} π_{t−1}^{ι_H} π̄^{1−ι_H}, with ι_H price indexation parameter.
- Firms maximize expected discounted profits with probability structure (β θ_H)^{v} and first-order condition provided. Recursive representation of optimality via f_t^H and f_{t+1}^H given in text.
- Price dispersion and aggregation:
  - Aggregate supply: Y_{H,t} = ν_{H,t} Y_{H,t}^d, with ν_{H,t} = ∫ (P_{H,l,t}/P_{H,t})^{−ε_H} dl capturing price distortion. Recursive expression for ν_{H,t} given.

### Monetary policy
- Two regimes modeled: inflation targeting (flexible exchange rate) and peg.
- Inflation targeting rule (nominal interest rate rule):
  - R_t / R̄ = (R_{t−1}/R̄)^{ρ_R} ( (π_t/π̄)^{φ_π} )^{1−ρ_R} exp(z_{m,t}), where ρ_R smoothing, φ_π sensitivity to inflation deviations, z_{m,t} monetary shock (ARMA).
- Peg regime: nominal devaluation rate constant:
  - d_t = S_t / S_{t−1} = d̄
- Central bank balance sheet:
  - M_t = B_{cb,t} + S_t B_{cb,t}^*
- Central bank flow of funds:
  - M_t − M_{t−1} + R_{t−1} B_{cb,t−1} + R_{t−1}^* S_t B_{cb,t−1}^* = B_{cb,t} + S_t B_{cb,t}^* + P_{c,t} qfa_t
  - qfa_t (quasi-fiscal balance) is function of returns on external/domestic assets, domestic inflation, and the real exchange rate.
- Regime-specific balance sheet adjustments:
  - Inflation targeting: central bank adjusts money supply so short-term interest rate aligns with policy rate; holdings of external assets constant; net domestic assets adjust.
  - Peg: central bank adjusts external asset holdings to maintain exchange rate target; money supply accommodates changes in external asset holdings; changes in government assets at central bank lead to changes in foreign asset holdings.

### Fiscal policy
- Government collects taxes on consumption, capital, labor; receives quasi-fiscal balance and commodity revenues; issues public debt.
- Central bank holds fraction α_g of government debt; households hold remainder.
- Government budget constraint (real terms):
  - p_{H,t} (1−γ_{ccpm}) C̄ p_{cpm,t}^* + T a x_{t}^c + T a x_{t}^p + T a x_{t}^k + a_t + qfa_t = p_{H,t} (C_{g,t} + X_{g,t}) + (R_t−1/π_t) a_{t−1} + Tr_t
  - Definitions: T a x components in text; T a x_{t}^k = τ_{k,t} ( r_t^k − δ q_{t−1}/π_t ) K_{t−1}; T a x_{t}^p = τ_{p,t}^w N_t; T a x_{t}^c = τ_{c,t} C_t.
- Marginal tax rates, government consumption and investment are constant in the baseline; fiscal instruments could instead follow fiscal rules.
- Government sets transfers optimally, maximizing objective:
  - U_{p,t} = U(C_t,N_t,m_t) + ω_1 ( (a_t^g − ā^g)^2 ) + ω_2 (Tr_t − Tr̄)^2
  - U(C_t,N_t,m_t) is weighted average of Ricardian and non-Ricardian utilities; ω_2 term penalizes abrupt fiscal instrument changes; ω_1 penalizes public debt deviations from steady-state.
- Public capital accumulation:
  - K_{g,t} = (1−δ) K_{g,t−1} + z_{xg,t} A_{g,t−L}
  - Authorized budget a_{g,t−L} and implementation lag L modeled; X_{g,t} = ∑_{n=0}^{L−1} a_n A_{g,t−n}, with ∑ a_n = 1. z_{xg,t} productivity shock in public investment.

### External sector and current account
- External interest rate: R_t^* = R̄_t^* − Ω_u [ exp( vvv_t / nnn v_t GGGPP_t − vvvvv̄_nnn v̄ GGGPP̄ ) − 1 ]^{−1} (Eq 4 in source)
  - Country risk premium is negative function of NFA-to-GDP ratio; Ω_u elasticity of country risk to NFA-to-GDP.
- GDP definition:
  - GGDP_t = p_{H,t} Y_{H,t} + rrrr_t C̄ P_{cpm,t}^*
- Non-commodity exports:
  - C_{t}^{h*} = ( p_{t}^H / rrrr_t )^{−ε_e} C_{t}^*
- Balance of payments / net foreign assets dynamics:
  - rrrr_t ( nfa_t − nfa_{t−1} ) = [ p_{t}^H C_{t}^{H*} + C̄ rrrr_t P_{cpm,t}^* ] − rrrr_t ( C_{t}^m + X_t^m ) + [ (R_{t−1}^*/π_t^*) − 1 ] rrrr_t nfa_{t−1}^*
  - Net foreign asset position in domestic currency: nfa_t = a_t^* + a_{cb,t}^*

*Source: Appendix I. Model Structure — wpiea2022043-print-pdf*

### Appendix II. Linear Time Iteration Algorithm

### Appendix II. Linear Time Iteration Algorithm

### Overview
- Describes the linear time iteration (LTI) solution method proposed by Rendahl (2017).
- LTI generalizes traditional perturbation to allow simulation of models with regime changes at a known time in the future.
- Rendahl (2017) contains a general presentation.

### Perturbation setup and linearization
- Model written as:
  - 퐸퐸
    푡푡
    [
    퐹퐹
    (
    푥푥
    푡푡
    ,푥푥
    푡푡+1
    ,푥푥
    푡푡+2
    )
    ]
    =0  (A0)
- Solution is sought around the non-stochastic steady state 푥푥
  ∗ such that:
  - 퐹퐹
    (
    푥푥
    ∗
    ,푥푥
    ∗
    ,푥푥
    ∗
    )
    =0
- First-order Taylor expansion around 푥푥
  ∗ yields (A1) with Jacobian matrices 퐽퐽
  푝푝 evaluated at 푥푥
  ∗.
- Simplified linear system:
  - 0 =퐽퐽
    푥푥
    푡푡
    (
    푥푥
    푡푡
    −푥푥
    ∗
    )
    +퐽퐽
    푥푥
    푡푡+1
    (
    푥푥
    푡푡+1
    −푥푥
    ∗
    )
    +퐽퐽
    푥푥
    푡푡+2
    (
    푥푥
    푡푡+2
    −푥푥
    ∗
    )
- Defining 푢푢
  푡푡 = (푥푥
  푡푡+1 −푥푥
  ∗) and Jacobians A, B, C gives:
  - 퐴퐴푢푢
    푡푡−1
    +퐵퐵푢푢
    푡푡
    +퐶퐶푢푢
    푡푡+1
    =0  (A2)

### Solution form and basic iteration
- Assume solution form:
  - 푢푢
    푡푡
    =퐹퐹푢푢
    푡푡−1  (A3)
  - F is unknown matrix to be found.
- Iterative procedure (single-equation representation):
  - Substitute (A3) into (A2) yielding:
    - 0 =퐴퐴푢푢
      푡푡−1
      +퐵퐵푢푢
      푡푡
      +퐶퐶퐹퐹
      푝푝
      푢푢
      푡푡
      (A4)
    - Solve for 푢푢
      푡푡 to get new approximation:
      - 퐹퐹
        1
        =−
        (
        퐵퐵+퐶퐶퐹퐹
        푝푝
        )
        −1
        퐴퐴
- Iteration summary:
  1. Assume initial guess 퐹퐹
     푝푝 and compute 퐹퐹
     1 =−(퐵퐵+퐶퐶퐹퐹
     푝푝)−1퐴퐴
  2. Compute 퐹퐹
     2 =−(퐵퐵+퐶퐶퐹퐹
     1)−1퐴퐴
  3. Continue until convergence: 퐹퐹
     푛푛 →퐹퐹

### Generalization around an arbitrary point
- Can solve (A1) around any point 푥푥 ̅ (not necessarily steady state) where:
  - 퐹퐹
    (
    푥푥 ̅,푥푥 ̅,푥푥 ̅
    )
    =퐺퐺
    ∗
- Linear approximation becomes:
  - 퐴퐴
    ∗
    푢푢
    푡푡−1
    +퐵퐵
    ∗
    푢푢
    푡푡
    +퐶퐶
    ∗
    푢푢
    푡푡+1
    +퐺퐺
    ∗
    =0   (A5)
- Solution form:
  - 푢푢
    푡푡
    =퐸퐸+퐹퐹푢푢
    푡푡−1
- Iterative updates:
  - 퐹퐹
    푛푛+1
    =−
    (
    퐵퐵
    ∗
    +퐶퐶
    ∗
    퐹퐹
    푛푛
    )
    −1
    퐴퐴
    ∗
  - 퐸퐸
    푛푛+1
    =
    (
    퐵퐵
    ∗
    +퐶퐶
    ∗
    퐹퐹
    푛푛
    )
    −1
    (
    퐶퐶
    ∗
    퐸퐸
    푛푛
    +퐺퐺
    ∗
    )
- Convergence criterion:
  - 퐸퐸
    푛푛
    →퐸퐸   and 퐹퐹
    푛푛
    →퐹퐹

### Application to regime-switching models (known switch time)
- Consider two regimes M1 and M2 with:
  - M1: 퐴퐴푢푢
    푡푡−1
    +퐵퐵푢푢
    푡푡
    +퐶퐶푢푢
    푡푡+1
    =0
  - M2: 퐴퐴
    ∗
    푢푢
    푡푡−1
    +퐵퐵
    ∗
    푢푢
    푡푡
    +퐶퐶
    ∗
    푢푢
    푡푡+1
    +퐺퐺
    ∗
    =0
- Assumptions:
  - Regime M1 satisfies conditions similar to Blanchard and Kahn (1980) and is absorbing: the system returns to M1 and will not return to M2 after time T.
- Known time T where system will be in M1. At time T:
  - 푢푢
    푡푡
    =퐹퐹
    푇푇
    푢푢
    푡푡−1
- For period T−1 (in M2) use known F_T to solve for F and E at T−1:
  - From:
    - 퐴퐴
      ∗
      푢푢
      푡푡−1
      +퐵퐵
      ∗
      푢푢
      푡푡
      +퐶퐶
      ∗
      퐹퐹
      푇푇
      푢푢
      푡푡
      +퐺퐺
      ∗
      =0
    - Solve to get:
      - 퐹퐹
        푇푇−1
        =−
        (
        퐵퐵
        ∗
        +퐶퐶
        ∗
        퐹퐹
        푇푇
        )
        −1
        퐴퐴
        ∗
      - 퐸퐸
        푇푇−1
        =−
        (
        퐵퐵
        ∗
        +퐶퐶
        ∗
        퐹퐹
        푇푇
        )
        −1
        퐺퐺
        ∗
- For T−2:
  - 퐹퐹
    푇푇−2
    =−
    (
    퐵퐵
    ∗
    +퐶퐶
    ∗
    퐹퐹
    푇푇−1
    )
    −1
    퐴퐴
    ∗
  - 퐸퐸
    푇푇−2
    =−
    (
    퐵퐵
    ∗
    +퐶퐶
    ∗
    퐹퐹
    푇푇−1
    )
    −1
    퐺퐺
    ∗

### Interpretation and implications
- Solutions for the regime-change model are sequences of E’s and F’s that vary by period before T.
- Agents adjust decision rules for every period before T, internalizing deterministic convergence toward the stable, absorbing regime M1.
- The algorithm enables analysis of dynamics and transitions from potentially unstable regimes toward a stable regime.

*Appendix II. Linear Time Iteration Algorithm — Fix vs. Float: Evaluating the Transition to a Sustainable Equilibrium in Bolivia (Working Paper No. WP/22/43)*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022043-print-pdf.pdf_