## _wp1692

## Source details

**Canonical URL:** [_wp1692](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1692.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1692.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1692.pdf.json)

---

### Model structure (economy, preferences, and consumption aggregator)
- Economy populated by a unitary mass of identical atomistic consumers and a government.
- Representative consumer maximizes:
  - U_t = E_t[sum_{s=0}^{+∞} β^s u(C_{t+s})],
  - u(C_t) = C_{t}^{1−γ} − 1  / (1−γ),
  - Consumption aggregator:
    - C_t = [ α^{1/η} M_t^{(η−1)/η} + (1−α)^{1/η} N_t^{(η−1)/η} ]^{η/(η−1)},
    - M_t is consumption of imported good; N_t is consumption of nontraded good.

### Financial structure, budget constraints, and market access
- Consumer cannot access foreign assets; can only hold liabilities of domestic government.
- Government issues domestic bonds B^g_t (domestic currency) and holds dollar reserves B_t (foreign assets); government is benevolent and manages reserves to maximize domestic welfare; reserves act as precautionary savings against export-value shocks.
- Consumer budget constraint (expressed in dollars):
  - B^g_t E_t + P_{Mt} M_t = P_{Xt} X_t + (1 + i^g_{t−1}) B^g_{t−1} E_t + T_t.
- Government budget constraint (including central bank operations):
  - (1 + i^g_{t−1}) B^g_{t−1} E_t + B_t + T_t = B^g_t E_t + (1 + i_{t−1}) B_{t−1}.
  - Government can perform sterilized “open market” operations changing B_t and B^g_t by the same amounts.
- Consolidated current account identity:
  - B_t − B_{t−1} = P_{Xt} X_t − P_{Mt} M_t + i_{t−1} B_{t−1}.
- Exogenous stochastic processes: P_{Xt} X_t, P_{Mt}, N_t, and i_{t−1}.
- Endogenous variables: M_t and B_t, with nonnegativity restriction b_t ≥ 0.

### Detrending, normalized variables, and first-order conditions
- Detrended (lower-case) variables (expressed in terms of imports except n_t and c_t):
  - m_t = G^{−t} M_t,
  - x_t = G^{−t} P_{Xt} X_t / P_{Mt},
  - n_t = G^{−t} N_t,
  - b_t = G^{−t} B_t / P_{Mt},
  - c_t = G^{−t} C_t,
  - G is the trend growth factor in income.
- Normalized aggregate budget constraint:
  - b_t + m_t = (1 + r_t)/G * b_{t−1} + x_t,
  - 1 + r_t = (1 + i_{t−1}) / (1 + π_t), π_t = P_{Mt} / P_{Mt−1} − 1.
- First-order condition for consumption (when b_t ≥ 0 not binding), after detrending:
  - c_t^{1/η − γ} m_t^{−1/η} = β G^{−γ} E_t[ (1 + r_{t+1}) c_{t+1}^{1/η − γ} m_{t+1}^{−1/η} ].
- State vector and policy functions:
  - State s_t = (x_t, n_t, r_t, b_{t−1}).
  - Endogenous policy functions c(s_t), m(s_t), b(s_t) satisfy first-order condition and consumption aggregator:
    - c(s_t) = [ α^{1/η} m(s_t)^{(η−1)/η} + (1−α)^{1/η} n_t^{(η−1)/η} ]^{η/(η−1)}.

### Carry cost and the target level of reserves
- Carry cost δ defined as:
  - δ = G/γ/β − (1 + r).
- Interpretation:
  - G/γ/β is the autarkic steady-growth path rate (consumer indifference rate); (1 + r) is return on reserves.
  - Condition for a finite target: G/γ/β > 1 + r (i.e., δ > 0).
  - If δ = 0, precautionary motives induce unbounded accumulation; if δ > 0, opportunity cost yields finite target b^*.
- First-order condition rewritten using δ:
  - c^{1/η−γ}_t m^{−1/η}_t = 1 / [1 + δ/(1 + r)] E_t[ (1 + r_{t+1})/(1 + r) c^{1/η−γ}_{t+1} m^{−1/η}_{t+1} ].
- Quasi-fiscal cost (model expression):
  - (1 + i^g_t) E_t E_{t+1} − (1 + i_t) (currency-consistent expression).
  - If no exchange rate uncertainty one period ahead, uncovered interest parity holds:
    - 1 + i_t = (1 + i^g_t) E_t E_{t+1}, and quasi-fiscal cost equals zero.
  - Quasi-fiscal cost is a transfer between government and domestic bondholders; not necessarily a national welfare cost.
- Observability and calibration role:
  - δ is not directly observable; it is calibrated from preference parameters.
  - Distinguish target b^* from unconditional average E(b): E(b) generally > b^* due to policy-function concavity and zero lower bound on reserves.

### Calibration sample and data construction (summary)
- Country sample: 21 developing countries with relatively little private capital flows (long-term PPG debt ≥ 75% of total external debt; not classified fragile; ≥ 15 consecutive years of data).
- Data: annual from 1960 to 2014.
- Construction:
  - Use WDI volume and value indexes for exports and imports to construct x_t and P^M_t.
  - Nontradable output N_t constructed as GDP^r_t − P_X X_t.
  - Normalize n_t so average = 1 per country.
  - Base year t^* = 2005 where P_N_{t^*} = P_M_{t^*}.
  - CES index implies (1 − α)/α = N_{t^*}/M_{t^*}; α determined from 2005 cross-country average.
- Estimated AR(1) processes for x_t, n_t, and imports own real rate of interest:
  - For x_t: y = 0.676, ρ = 0.778, σ = 0.161
  - For n_t: y = 13.56%, ρ = 0.877, σ = 0.107
  - For r_t: y = 0.186, ρ = 0.0, σ = 12.9%
- Discretization: Tauchen and Hussey (1991) with five gridpoints for x and three gridpoints for n and r.

### Benchmark parameters and carry cost
- Benchmark parameters:
  - γ = 2
  - α = 0.36
  - η = 1
  - G = 1.046
  - β = 0.99
- Average growth rates in sample: n and x are respectively 4.3% and 4.9%; model set common G = 4.6%.
- Under benchmark calibration carry cost equals δ = 6.9%.

### Quantitative results: target and average reserves (benchmarks and sensitivity)
- Reserves measured as reserves-to-imports ratio in months:
  - ρ_t = 12 * b_t / m_t.
- Averaging convention:
  - Average reserves E(ρ) computed over 5,000 simulations of 200 periods.
  - Target ρ^* = b^*/m^* (long-run levels without shocks).
- Benchmark outcomes:
  - Average level E(ρ) = 4.6 months of imports.
  - Target level ρ^* = 3.3 months of imports.
  - Average in data = 4.5 months of imports.
- Sensitivity highlights:
  - Both ρ^* and E(ρ) decrease with carry cost δ.
  - Target diverges to infinity as δ → 0; target exceeds 15 months of imports when δ = 2 percent.
  - With carry cost ≥ 5.8 percent (if β ≤ 1), average reserves = 6.1 months of imports, target = 4.6 months of imports.
  - Effects of risk aversion γ depend on whether δ or β is held constant:
    - Holding δ constant (adjust β): higher γ increases desired reserves.
    - Holding β constant: increasing γ reduces intertemporal elasticity and increases carry cost, which decreases the target.
    - Increasing γ above 2 has limited impact.
    - Reducing γ below 2 significantly increases optimal reserves: average reserves ≈ 15 months of imports if γ = 1.
  - Reserves target decreases with elasticity of substitution η.
  - Increasing import share α raises desired reserves.
- Robustness:
  - Benchmark results robust to increasing γ above 2 or changing β as long as β remains positive.
  - If β and γ both equal 1, optimal level of reserves ≈ 30 months of imports on average.

### Impulse response functions: key responses and variance decomposition
- Responses to shocks (Figure 2 summaries):
  - Negative export-income shock:
    - A 30 percent fall in x causes imports to fall by 20 percent.
    - Government runs down reserves by more than one and a half months of imports in response to that 30 percent fall in x.
  - Negative nontraded-output shock:
    - A 20 percent fall in n reduces reserves by about one half of a month of imports.
  - Unexpected increase in ex-post imports own real rate of interest (real import price shocks):
    - An unexpected increase in the price of imports decreases both imports and reserves.
- Probabilities under discretization:
  - Simulated export shock occurs with 22 percent probability.
  - Nontradable output shock occurs with 17 percent probability.
  - Real interest rate shock occurs with 17 percent probability.
- Variance decomposition of reserves (Table 3 contributions to V ar(b)):
  - Contributions: Shock x = 6.866; Shock n = 0.287; Shock r = 0.257; Total = 8.660.
  - Interpretation: Most variation in reserves explained by shocks to export income x.

### Unconditional distribution of reserves: model versus data
- Model features:
  - Reserves spend substantial time close to the zero lower bound.
  - Relatively fat tail of observations with very high levels of reserves.
  - Visual impression: model does not imply tight targeting of reserves.
- Empirical data features:
  - Reserves stay relatively close to their average level.
  - Conclusion: real-world governments appear more concerned about keeping reserves close to a target than the model implies; reluctance to run down reserves may reflect signaling concerns.

### Policy-rule analysis: linear and asymmetric rules, calibration, and welfare
- Proposed linear rule for reserves (equation (9)):
  - b_t = (1 + r_t)/(1 + r) b_{t−1} + λ(x_t − x) + μ( b̂ − b_{t−1} ).
  - λ = marginal propensity to save export income deviation; μ = speed of convergence toward target b̂.
- Optimized linear-rule calibration (benchmark):
  - b̂ = 0.22
  - λ = 0.35
  - μ = 0.2
  - Interpretation: λ = 0.35 implies country should accumulate 35 percent of export income in excess of the average in reserves each period.
- Comparison to certainty-equivalence:
  - λ_CE = (1 − ρ_x) G / (1 + r − ρ_x G).
  - Under benchmark calibration λ_CE = 0.907.
  - Optimized λ much lower than λ_CE because consumers are both impatient and risk-averse.
- Convergence:
  - Deviations from target under optimized linear rule have a half-life of about three years.
- Welfare metrics:
  - U_max: welfare with optimal (non-linear) reserves use.
  - U_min: average welfare with zero reserves.
  - Welfare gain from optimal reserves management (U_max − U_min) equivalent to permanent consumption increase by 0.57 percent.
  - Removing stochastic shocks entirely implies welfare-equivalent permanent consumption increase close to 5 percent.
  - U_lin: average welfare under linear rule.
  - Best linear rule provides 91.3 percent of welfare gains from best non-linear rule: (U_lin − U_min)/(U_max − U_min) = 91.3 percent.
- Sensitivity and asymmetry:
  - Welfare under linear rule insensitive to precise level of b̂ between zero and 2 b^*; setting λ and μ appropriately is more important.
  - Asymmetric linear rules (different λ^+, λ^−, μ^+, μ^−) yield small extra welfare:
    - Allowing asymmetry in μ raises welfare by 0.5 percent of the non-linear-rule welfare gains.
    - Allowing asymmetry in λ raises welfare by 0.9 percent of the non-linear-rule welfare gains.
- Policy-relevant takeaways:
  - Reserves primarily buffer export-income shocks.
  - Optimal policy uses reserves actively in response to shocks rather than strictly holding reserves at a fixed target.
  - Simple linear rules with calibrated λ and μ capture most (about 91.3 percent) of welfare gains from fully optimal non-linear reserve management.
  - Policy recommendation (model-based): prioritize the rule for accumulating/depleting reserves in response to shocks (choice of λ and μ) over precise calibration of the reserve target b̂.

*Source: 2.1 Model structure (content unit from provided IMF PDF).*

### 2.1    Model structure  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .6

### 2.1    Model structure

### Economy, preferences, and consumption aggregator
- Economy populated by a unitary mass of identical atomistic consumers and a government.
- Representative consumer maximizes:
  U_t = E_t[sum_{s=0}^{+∞} β^s u(C_{t+s})],
  with u(C_t) = C_{t}^{1−γ} − 1  / (1−γ),
  and consumption aggregator
  C_t = [ α^{1/η} M_t^{(η−1)/η} + (1−α)^{1/η} N_t^{(η−1)/η} ]^{η/(η−1)},
  where M_t is consumption of imported good and N_t is consumption of nontraded good.

### Financial structure, assets, and access
- Consumer cannot access foreign assets; can only hold liabilities of domestic government.
- Government issues domestic bonds B^g_t (expressed in domestic currency) and holds dollar reserves B_t (foreign assets).
- Government is benevolent and manages reserves to maximize domestic welfare; reserves act as precautionary savings against export-value shocks.

### Consumer budget constraint (expressed in dollars)
- B^g_t E_t + P_{Mt} M_t = P_{Xt} X_t + (1 + i^g_{t−1}) B^g_{t−1} E_t + T_t,
  where X_t and M_t are quantities of exports and imports; P_{Xt} and P_{Mt} are their dollar prices; E_t is domestic currency per dollar; i^g_{t−1} is domestic currency interest rate; T_t is lump-sum transfer from government.

### Government budget constraint (including central bank operations)
- (1 + i^g_{t−1}) B^g_{t−1} E_t + B_t + T_t = B^g_t E_t + (1 + i_{t−1}) B_{t−1},
  where i_{t−1} is the dollar nominal interest rate.
- Government can perform sterilized “open market” operations changing B_t and B^g_t by the same amounts (domestic bonds must be purchased by residents who lack access to foreign assets).

### Current account identity (consolidated)
- B_t − B_{t−1} = P_{Xt} X_t − P_{Mt} M_t + i_{t−1} B_{t−1}.

### Exogenous processes and endogenous variables
- Exogenous stochastic processes: value of exports P_{Xt} X_t, price of imports P_{Mt}, output of nontradable good N_t, and dollar nominal interest rate i_{t−1}.
- Endogenous: imports M_t and reserves B_t.
- Restriction: reserves must be nonnegative, b_t ≥ 0.

### Detrending and normalized (lower-case) variables (expressed in terms of imports except n_t and c_t)
- m_t = G^{−t} M_t,
  x_t = G^{−t} P_{Xt} X_t / P_{Mt},
  n_t = G^{−t} N_t,
  b_t = G^{−t} B_t / P_{Mt},
  c_t = G^{−t} C_t,
  where G is the trend growth factor in income.

### Normalized aggregate budget constraint
- b_t + m_t = (1 + r_t)/G * b_{t−1} + x_t,
  where r_t denotes the imported goods own real rate of interest between t−1 and t,
  and 1 + r_t = (1 + i_{t−1}) / (1 + π_t), with π_t = P_{Mt} / P_{Mt−1} − 1.

### First-order condition for consumption (when b_t ≥ 0 is not binding)
- In dollar terms:
  u′(C_t) ∂C_t/∂M_t = β E_t[ (1 + r_{t+1}) u′(C_{t+1}) ∂C_{t+1}/∂M_{t+1} ].
- After detrending:
  c_t^{1/η − γ} m_t^{−1/η} = β G^{−γ} E_t[ (1 + r_{t+1}) c_{t+1}^{1/η − γ} m_{t+1}^{−1/η} ].

### State, policy functions, and equilibrium characterization
- State at time t: s_t = (x_t, n_t, r_t, b_{t−1}).
- Endogenous policy functions: c(s_t), m(s_t), b(s_t).
- Policy functions satisfy the detrended first-order condition and the consumption aggregator identity:
  c(s_t) = [ α^{1/η} m(s_t)^{(η−1)/η} + (1−α)^{1/η} n_t^{(η−1)/η} ]^{η/(η−1)}.

*Source: 2.1 Model structure (content unit from provided IMF PDF).*

### 2.2    Carry cost and the target level of reserves

### 2.2    Carry cost and the target level of reserves

### Carry cost and existence of a target
- If G/γ/β > 1 + r, the level of reserves b_t tends to fluctuate around a “target” level, b^*.
- The target b^* is the long-run level that reserves converge to in the absence of shocks (x_t, n_t, and r_t equal to their averages x, n, and r).
- Shock realizations generate a stochastic dispersion of reserves, while the policy function attempts to bring reserves back to b^*.

### Definition and interpretation of the carry cost δ
- Carry cost is defined as: δ = G/γ/β − (1 + r).
- First term G/γ/β is the interest rate that would prevail under financial autarky in the deterministic steady-growth path and is the rate at which the representative consumer is indifferent between borrowing or saving in the deterministic model.
- Second term (1 + r) is the rate of return the consumer earns on reserves.
- Condition (4) (G/γ/β > 1 + r) requires δ to be positive.
  - If δ = 0, there is no opportunity cost to accumulating reserves and precautionary motives induce unbounded accumulation.
  - If δ > 0, accumulating reserves entails an opportunity cost that ensures a finite target level.
- δ summarizes combined effects of the discount factor, growth rate, and elasticity of intertemporal substitution on willingness to borrow against future income.
- First-order condition can be written:
  - c^{1/η−γ}_t m^{−1/η}_t = 1 / [1 + δ/(1 + r)] E_t[ (1 + r_{t+1})/(1 + r) c^{1/η−γ}_{t+1} m^{−1/η}_{t+1} ].
- In calibration the carry cost δ is treated as the exogenous parameter (G retained as separate because it appears in the budget constraint).

### Quasi-fiscal cost versus carry cost
- Quasi-fiscal cost frequently measured as spread between cost of issuing sterilization bonds and return on reserves; in model written as:
  - (1 + i^g_t) E_t E_{t+1} − (1 + i_t).
- For consistency, interest rates must be expressed in the same currency; equation (6) uses the dollar so local currency interest must be adjusted for depreciation.
- Consumer’s first-order condition for holding government debt:
  - c^{1/η−γ}_t m^{−1/η}_t = β G^{−γ} E_t[ (1 + i^g_t)/(1 + π_{t+1}) E_t E_{t+1} c^{1/η−γ}_{t+1} m^{−1/η}_{t+1} ].
- If no exchange rate uncertainty one period ahead (E_{t+1} known in period t), uncovered interest parity applies:
  - 1 + i_t = (1 + i^g_t) E_t E_{t+1}.
  - Quasi-fiscal cost in (6) equals zero.
- Under perfect financial integration the government should reproduce same allocation as in equilibrium with openness; hence no quasi-fiscal cost unless government accumulates more reserves than private sector would.
- With exchange rate risk, a wedge arises from exchange rate risk premium; domestic interest rate could be higher or lower than UIP-implied level and a positive quasi-fiscal cost may exist.
- Quasi-fiscal cost is a cost for the government but a gain for holders of domestic bonds (domestic residents); thus it is a transfer, not necessarily a welfare cost for the country as a whole.
- Difference between external debt interest rate and return on reserves is another commonly used measure but does not generally coincide with carry cost δ; external borrowing rates may understate true cost of holding reserves.

### Observability and calibration role of δ
- Carry cost δ is a theoretical construct not directly observable from market data because it involves a counterfactual interest rate (autarkic steady-growth path rate).
- In calibration δ results from assigned values to fundamental preference parameters (e.g., discount factor, risk aversion).
- Distinguish target level b^* from unconditional average E(b): E(b) is generally higher than b^* because
  - policy function concavity implies faster convergence toward target when below target than when above, and
  - zero-bound on reserves (reserves cannot be negative) lifts unconditional average above target (e.g., if b^* = 0 then E(b) > 0).

### Calibration sample and data construction (summary)
- Country sample: 21 developing countries with relatively little private capital flows (long-term PPG debt ≥ 75% of total external debt; not classified fragile; ≥ 15 consecutive years of data). Countries listed in text.
- Data are annual from 1960 to 2014.
- Use WDI volume and value indexes for exports and imports to construct x_t and P^M_t; nontradable output N_t constructed as GDP^r_t − P_X X_t.
- Normalize n_t so average = 1 per country; determine x_t path using ratio n_t/x_t in a base year t^* where P_N_{t^*} = P_M_{t^*} (base year t^* = 2005).
- CES index implies (1 − α)/α = N_{t^*}/M_{t^*}; α determined from 2005 cross-country average.
- Estimated AR(1) processes for x_t, n_t, and imports own real rate of interest (1 + r_t = (1 + i_{t−1})/(1 + π_t), where i_t is one-year eurobond interest rate in U.S. dollars).
- Estimated sample averages reported in Table 1:
  - For x_t: y = 0.676, ρ = 0.778, σ = 0.161
  - For n_t: y = 13.56%, ρ = 0.877, σ = 0.107
  - For r_t: y = 0.186, ρ = 0.0, σ = 12.9%
- Tauchen and Hussey (1991) discretization: five gridpoints for export income x and three gridpoints for nontraded income and real interest rate.
- Benchmark parameters (Table 2):
  - γ = 2
  - α = 0.36
  - η = 1
  - G = 1.046
  - β = 0.99
- Average growth rates in sample: n and x are respectively 4.3% and 4.9%; model set common G = 4.6%.
- Under benchmark calibration carry cost equals δ = 6.9%.

### Quantitative results: target and average reserves (benchmarks and sensitivity)
- Reserves reported as reserves-to-imports ratio in months: ρ_t = 12 * b_t / m_t.
- Average reserves E(ρ) is average over 5,000 simulations of 200 periods.
- Target reserves ρ^* is b^*/m^* where b^* and m^* are long-run levels in model without shocks.
- Under benchmark calibration:
  - Average level of reserves E(ρ) = 4.6 months of imports.
  - Target level of reserves ρ^* = 3.3 months of imports.
  - Average in data = 4.5 months of imports; target close to 3-months-of-imports rule of thumb.
- Sensitivity analysis (Figure 1 summary):
  - Both target ρ^* and average E(ρ) decrease with carry cost δ.
  - Target diverges to infinity as δ → 0; target already exceeds 15 months of imports when δ = 2 percent.
  - With carry cost ≥ 5.8 percent (if β ≤ 1), average reserves = 6.1 months of imports, target = 4.6 months of imports.
  - Effects of risk aversion γ depend on whether δ or β is held constant:
    - Holding δ constant (adjust β): higher γ increases desired reserves (greater willingness to smooth consumption).
    - Holding β constant: increasing γ reduces intertemporal elasticity and increases carry cost, which decreases the target.
    - Increasing γ above 2 has limited impact because higher risk aversion (raises reserves) is offset by lower intertemporal elasticity (raises carry cost).
    - Reducing γ below 2 significantly increases optimal reserves: average reserves ≈ 15 months of imports if γ = 1.
  - Reserves target decreases with elasticity of substitution η between tradable and nontradable goods (higher η → goods more substitutable → lower precautionary benefit → lower reserves).
  - Increasing share of imports in consumption α raises exposure to external shocks and increases desired reserves.
- Robustness conclusions:
  - Benchmark results are robust to increasing γ above 2 or changing β as long as β remains positive.
  - If β and γ both equal 1 (plausible set that maximizes optimal reserves), optimal level of reserves ≈ 30 months of imports on average.

*Source: _wp1692 - 2.2    Carry cost and the target level of reserves*

### 4.2    Impulse response functions

### _wp1692 - 4.2    Impulse response functions

### Impulse response functions: qualitative description and key responses
- Figure 2 reports impulse response functions of imports (j), nontraded output (n), and reserves (ρ, months of imports) to shocks in export income (x), nontraded output (n), and the real interest rate (r).
- Response to a negative export-income shock:
  - A 30 percent fall in export income x causes imports to fall by 20 percent.
  - The government runs down reserves by more than one and a half months of imports in response to that 30 percent fall in x.
- Response to a negative nontraded-output shock:
  - A 20 percent fall in nontraded output reduces reserves by about one half of a month of imports.
- Response to an unexpected increase in the ex-post imports own real rate of interest (real import price shocks, mostly P_Mt movements; nominal interest shocks i_t are much less volatile):
  - An unexpected increase in the price of imports decreases both imports and reserves.

### Probabilities under discretization
- Under the Tauchen-Hussey discretization used in simulations:
  - The simulated export shock occurs with 22 percent probability.
  - The nontradable output shock occurs with 17 percent probability.
  - The real interest rate shock occurs with 17 percent probability.

### Variance decomposition of reserves
- Most of the variation in reserves is explained by shocks to export income x rather than shocks to nontraded output or the real interest rate.
- Table 3: Contributions of the shocks to the variance of reserves (Shock x n r; V ar(b)):
  - V ar(b): 6.866 0.287 0.257 8.660
  - (Interpretation as presented: entries correspond to contributions of shocks x, n, r, and total variance when all three shocks are present.)

### Unconditional distribution of reserves: model versus data
- Figure 3 compares the unconditional distribution of the ratio of reserves to their average level (ρ_t / E[ρ_t]) in the model simulation and in empirical data.
- Model simulation features:
  - Reserves spend a substantial amount of time close to the zero lower bound.
  - There is a relatively fat tail of observations with very high levels of reserves.
  - Visual impression: the model does not give the impression that the government tries to keep reserves close to a target.
- Empirical data features:
  - Reserves stay relatively close to their average level.
  - Conclusion: real-world governments appear more concerned about keeping reserves close to a target than implied by the model — reluctance to run down reserves in bad shocks may reflect fear of sending a bad public signal about the state of the economy (literature note: Aizenman and Sun, 2012).

---

### Transition to policy-rule analysis (section 4.3) — summary of findings relevant to impulse responses and policy design
- The model implies reserves should be mostly used in response to export-income shocks x_t; this motivates consideration of simple linear policy rules that buffer export shocks.
- A proposed linear rule for reserves (equation (9)):
  - b_t = (1 + r_t)/(1 + r) b_{t−1} + λ(x_t − x) + μ( b̂ − b_{t−1} ).
  - λ is the marginal propensity to save export income (fraction of export-income deviation saved in reserves).
  - μ captures the speed of convergence of reserves toward the target b̂.
- Certainty-equivalence is a special case of the linear rule when μ = 0 and λ is determined by imports following a random walk; in that case reserves b_t are nonstationary.

### Optimized linear-rule calibration and comparisons
- Under the benchmark calibration, the optimized linear-rule parameters that maximize average welfare across simulations are:
  - b̂ = 0.22
  - λ = 0.35
  - μ = 0.2
- Comparison to non-linear optimal policy:
  - The optimized linear rule approximates the non-linear rule well when precautionary motives are weaker (high export income and high reserves).
  - Discrepancies appear when income and reserves are low because the optimal policy becomes notably nonlinear.
  - The optimal linear-rule target b̂ = 0.22 is slightly larger than the non-linear-policy implied target of 0.18.
  - Rationale: a linear rule provides less buffering to shocks, so a higher stock of reserves is preferable under the linear rule.

### Interpretation of λ and relation to certainty equivalence
- Under the optimized linear rule:
  - λ = 0.35 implies that in any given period the country should accumulate in reserves 35 percent of its export income in excess of the average.
- Under certainty equivalence (closed-form expression shown in the appendix):
  - λ_CE = (1 − ρ_x) G / (1 + r − ρ_x G), where ρ_x is the autocorrelation coefficient in export income and the growth factor G = [β(1 + r)]^{1/γ}.
  - Under the benchmark calibration, λ_CE = 0.907.
  - Conclusion: the optimized λ is significantly lower than λ_CE because consumers are both impatient (desire to borrow) and risk-averse (desire to insure), leading to more sparing use of reserves than under certainty equivalence.

### Speed of convergence and comparison to other models
- Deviations from the target under the optimized linear rule have a half-life of about three years — relatively quick convergence.
- This is much quicker than calibrated Schmitt-Grohé and Uribe (2003) models, where foreign-assets behavior is close to certainty equivalence with only a weak stationary force.

### Welfare implications: non-linear rule, linear approximation, and gains
- Welfare benchmark definitions:
  - U_max: welfare when reserves are used optimally (non-linear rule).
  - U_min: average welfare when reserves are set to zero (no reserves).
  - The welfare gain from optimal reserves management (U_max − U_min) is equivalent to a permanent increase in consumption by 0.57 percent.
  - Removing stochastic shocks entirely implies a welfare-equivalent permanent consumption increase close to 5 percent (larger than typical business-cycle welfare cost estimates because export income is volatile in the sample).
- Performance of the optimized linear rule:
  - Define U_lin as average welfare under the linear rule.
  - The welfare gain from the linear rule as a share of the gain from unconstrained optimal reserves is (U_lin − U_min)/(U_max − U_min).
  - The best linear rule provides 91.3 percent of the welfare gains from the best non-linear rule.
- Sensitivity analysis:
  - Welfare under the linear rule is remarkably insensitive to the level of the target b̂ (tested between zero and 2 b* in simulations).
  - It is far more important to set λ and μ at appropriate levels, especially ensuring they are not set too low — i.e., how reserves are accumulated/deccumulated in response to shocks matters more than the precise target level.
  - Optimal b̂ has only minor implications for optimal λ and μ (higher b̂ implies only small changes in optimal λ and μ).

### Asymmetric rules
- Considered asymmetric linear rules where λ and μ depend on whether x_t − x or b̂ − b_{t−1} are positive or negative:
  - b_t = (1 + r_t)/(1 + r) b_{t−1} + λ^+ (x_t − x)^+ + λ^− (x_t − x)^− + μ^+ (b̂ − b_{t−1})^+ + μ^− (b̂ − b_{t−1})^−.
- Welfare gains from allowing asymmetry:
  - Allowing asymmetry in μ raises welfare by 0.5 percent of the welfare gains from the best non-linear rule.
  - Allowing asymmetry in λ raises welfare by 0.9 percent of the welfare gains from the best non-linear rule.
  - Conclusion: asymmetric rules provide only small additional welfare improvements.

### Key policy-relevant takeaways
- Reserves are primarily used to buffer export-income shocks; export-income shocks explain most of reserves’ variance.
- Optimal policy uses reserves actively in response to shocks rather than maintaining reserves strictly at a target.
- Simple linear rules with calibrated λ and μ can capture most (about 91.3 percent) of the welfare gains from fully optimal non-linear reserve management.
- Policy focus recommendation (model-based): prioritize the rule for accumulating/depleting reserves in response to shocks (choice of λ and μ) over precise calibration of the reserve target b̂.

*Source: _wp1692 - 4.2    Impulse response functions (excerpts provided).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1692.pdf_