## 1. The net impact is that the real interest rate drops and the real exchange rate depreciates.

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---

### Main result
- Net effect of an aid decline (under the case with no change in the monetary base): the real interest rate drops and the real exchange rate depreciates.
- Nominal variables move in the same direction when the monetary base is unchanged.

### Policy scenarios and immediate impacts
- Central bank / government options to finance an increased budget deficit after an aid cut:
  - Sell government debt to the private sector.
  - Monetize the increase in government debt (central bank purchases of government bonds, increasing the money supply).
  - Sell foreign exchange reserves (sterilized: match government bond purchases with reserve sales, keeping the monetary base unchanged).
- Comparative short-run impacts:
  - Adjusting government expenditures (cut spending to keep deficit unchanged):
    - Initial response: a real depreciation of the currency and a small decline in private consumption.
    - For a very persistent reduction in aid, the real interest rate rises slightly, supporting lower current consumption.
  - Sales of debt to the private sector (government maintains spending):
    - Larger fall in consumption than with expenditure cuts.
    - Currency depreciation is more muted relative to the expenditure-cut case.
    - Real interest rate increases (due to higher debt and future tax expectations).
  - Monetization of the deficit (central bank finances government by increasing money supply):
    - Persistent decline in aid leads to persistently lower consumption and a depreciated currency.
    - In the first period effects are similar to sales of debt, but beyond the first period consumption remains relatively higher under monetization than under sales of debt.
    - Inflation increases substantially as money supply rises in the flexible-price economy.
  - Sales of foreign exchange reserves:
    - If reserves can fully offset the decline in aid, the economy can be insulated and contemporaneous changes in consumption, real exchange rate, inflation, and the real interest rate can be zero.
    - If reserve use is constrained, reserves may be used up to a threshold and the remaining deficit must be financed by the other channels above.

### Key numerical / model notes (preserved exactly)
- Base case used in numerical simulations: uninterrupted inflow of the equivalent of two percent of GDP in aid is shown as the first row in Table 1.
- Aid process specification (AR(1)):
  - log(ãt / ã) = ρa log(ãt−1 / ã) + εt; εt ~ N(0, σa^2)
  - Persistence parameter: ρa
  - Standard deviation of the shock: σa
- Government budget consolidated (log-linearized and steady-state relationships are explicitly given in the model).
- Calibration assumptions for analytical tractability:
  - Households’ steady-state holdings of government debt: bc = 0.
  - Technology parameters set so that zN = $1−β and zT = (1−$)1−β.
  - Utility is logarithmic in consumption and real money balances: γ = ζ = 1.
  - Cost subsidy: ϕ = 1/(1−θ) (to offset monopolistic competition).
- Representative first-order log-linear conditions:
  - Euler: êbc_t = E_t êbc_{t+1} − (1/ϕ)[bR_t − E_t bπ_{t+1}]
  - Money-interest link: bR_t = (R − 1)ε bc_t − (R − 1)ζ bm_t
  - Internal balance: b_s b s_t = b_c b c_t + b_τ b τ_t + 'G ( (ḡ/ḡy_N) ) b g_t
  - External balance: #_s b s_t = #_c b c_t + b_τ b τ_t − (1−'G)( (ḡ/ḡy_T) ) b g_t − ( (ã̄/ḡy_T) ) b ã_t + ( (res̄/ḡy_T) )(c res̃_t − c res̃_{t−1})
  - Consolidated government budget constraint (log-linearized) appears as equation (24) in the source text.

### Analytical dynamics and graphical intuition
- Internal balance (IB) is downward-sloping in (c, s) space; external balance (EB) is upward-sloping. Aid reductions shift EB left; different financing choices shift IB or EB in characteristic ways.
- Under the expenditure-cut policy, equilibrium moves from point “a” to point “b”.
- Under sales of debt or monetization, the short-run equilibrium moves from point “a” to point “c”.
- Sales of debt introduce intertemporal considerations: increased bc_t raises expected future taxes (bτ_t), which further depresses consumption over time.
- Monetization raises inflation more than other options because of the immediate increase in money supply in a flexible-price environment.
- Selling reserves can fully offset the aid shock if reserves are sufficient and non-binding minimum thresholds are not violated; otherwise, partial use of reserves combined with other policies is likely.

---

### 5.1 Full-Scale Model Setup

### Households (model structure preserved)
- Two household types:
  - Savers (asset holders) denoted with (a).
  - Hand-to-mouth consumers denoted with (h).
- Savers:
  - Maximize E_0 Σ_{t=0}^∞ β^t [ (c_a_t)^{1-σ}/(1-σ) - {l(l_a_t)^{1+φ}/(1+φ)} + {m(m_a_t)^{1-ζ}/(1-ζ)} ] (equation (29)).
  - Budget constraint (30) (full expression preserved in source).
  - Foreign assets pay gross nominal rate R̃; portfolio adjustment costs ac̃_t = ϑ/2 (b̃_t - b̃̄)^2.
  - Investment adjustment costs ac_{i;a}_t = κ/2 [ (ξ_{iN;a}_t k_N;a_{t-1} - ι)^2/k_N;a_{t-1} + (ξ_{iT;a}_t k_T;a_{t-1} - ι)^2/k_T;a_{t-1} ].
  - Capital law of motion (31): k_{j;a}_t = (1-δ) k_{j;a}_{t-1} + i_{j;a}_t for j∈{N,T}.
  - Consumption and investment are CES aggregates with elasticity ω and home bias ϕ.
  - Total labor supply (32) and aggregate real wage index (33) as specified.
- Hand-to-mouth households:
  - c_h_t = (1-τ_t) w_t l_h_t + s_t rm̃.
  - Utility U_h_t = [ - (c_h_t)^{1-σ}/(1-σ) - {l̄ - l_h_t}^{1+φ}/(1+φ) ].
  - Large share of hand-to-mouth households in Haiti implies labor market outcomes significantly affected by their labor supply decisions; remittances constant.

### Firms
- Nontraded goods sector:
  - Monopolistically competitive, indexed i∈[0,1].
  - Production (34): y_N_t(i) = z_N [ k_N_{t-1}(i)^{1-β_N} l_N_t(i)^{β_N} ] k_G_{t-1}^{β_G}.
  - Aggregation: CES with elasticity θ.
  - Demand (35): y_N_t(i) = ( p_N_t(i) / p_N_t )^{-θ} y_N_t.
  - Rotemberg price adjustment costs ac_p_t(i) = ψ_p/2 [ (π_N_t(i)/π_N - 1)^2 p_N_t y_N_t ].
- Traded goods sector:
  - Perfect competition; production y_T_t = z_T [ k_T_{t-1}^{1-β_T} l_T_t^{β_T} ] k_G_{t-1}^{β_G}.
  - Firms maximize period profits (37).
- Total output: y_t = p_N_t y_N_t + s_t y_T_t.

### The public sector
- Government expenditures: g_t = g_C_t + g_I_t.
- Public capital accumulation: k_G_t = (1-δ_G) k_G_{t-1} + ϕ g_I_t with 0<ϕ<1.
- Government purchases are CES aggregates with elasticity ω and home bias ϕ_G.
- Tax revenues: tax_t = τ_t [ w_t l_t + r_N_t k_N_{t-1} + r_T_t k_T_{t-1} ].
- Consolidated government budget constraint (38) (text preserved).
- Monetary and fiscal policy scenarios as in the simple model; government expenditure cuts assumed reductions in public investment.

### Aggregation and market clearing
- Aggregate variables x_t = f x_a_t + (1-f) x_h_t for x ∈ {c, c_N, c_T, l, l_N, l_T} (39).
- Assets and capital held only by savers: x_t = f x_a_t for x ∈ {m, i_N, i_T, k_N, k_T, b_c, b̃_t, ac_i, ac̃} (40).
- Nontraded goods clearing (41) (text representation).
- Balance of payments/current account condition (42) (text representation).

### Model calibration and solution
- Solution method: algorithm of Schmitt-Grohe and Uribe (2004); welfare computed accordingly.
- Frequency: quarterly.
- Calibrated to Haitian economy averages for 2005-2015.
- Inflation set to 8.5% annually.
- Share of savers in the economy set to about 20%.

### Key parameter values (preserved exactly)
- β = 0.98
- σ = 2.32 (implying intertemporal elasticity of substitution of 0:43)
- Inverse Frisch labor supply elasticity = 1/0.8
- ω_l = 0:6
- ω = 0:44
- θ = 6
- ϕ = 0:5
- ϕ_G = 0:6
- β_N = 0:45
- β_T = 0:6
- δ = 0:025
- λ = 1:4
- β_G = 0.21
- ϕ (public investment efficiency) = 0:4

---

### 5.2 Policy Analysis Revisited

### Methodology and welfare measure
- Impulse responses computed for a persistent decline in foreign aid inflows (shock: decrease in aid of 2% of GDP).
- Welfare measured as present discounted value of life-time utility conditional on the aid decline and government policy:
  - Asset holders: V_a_0 = E_0 Σ_{t=0}^∞ β^t [ (c^a_t)^{1-θ}/(1-θ) + { l (l^a_t)^{1+φ}/(1+φ)} + { m (m^a_t)^{1-ζ}/(1-ζ)} ].
  - Hand-to-mouth: V_h_0 = E_0 Σ_{t=0}^∞ β^t [ -(c^h_t)^{1-θ}/(1-θ) + { - l l^h_t^{1+φ}/(1+φ) } ].
- Aggregate welfare is a weighted average with household shares.

### Impulse-response overview (shock: aid down 2% of GDP)
- Four policy responses considered:
  - Expenditure cuts (red dashed lines).
  - Sales of debt to the private sector (solid blue lines).
  - Monetization of the increased deficit (black circles lines).
  - Sales of foreign reserves (dash-dot green lines).
- Sales of foreign exchange reserves can fully offset the effects of the decline in aid inflows, leaving other variables unchanged provided reserve thresholds are not binding.
- Under other policies:
  - Real exchange rate depreciates.
  - Trade deficit declines.
  - Overall reduced GDP levels.
  - Time-paths differ across policies and across tradable vs non-tradable sectors and household types.

### Policy scenario findings (detailed)
- Adjusting Government Expenditure (expenditure cuts)
  - Government cuts reduce demand, non-tradable sector contracts due to nominal inertia.
  - Real depreciation raises tradable sector output and employment; trade deficit decreases.
  - GDP and private consumption fall; private investment initially crowded-in as asset holders smooth consumption.
  - If cuts are in public investment, lower public capital reduces production in both sectors over longer horizons.
  - Labor market: real wages decrease in the short run.
  - Household impacts:
    - Asset holders: persistent reduction in consumption, temporary reduction in labor supply, draw on foreign assets.
    - Hand-to-mouth: short-run decrease in consumption and utility despite higher remittances value.
  - Welfare: hand-to-mouth welfare decline of 0.11% relative to steady state, about 9 times larger than asset holders (source: Table 3).

- Sales of Debt to the Private Sector
  - Government issues debt; asset holders shift savings to government bonds and reduce private capital investment.
  - Investment decline reduces demand across sectors; non-tradable contraction dominates.
  - Asset holders: consumption falls by about 1%; small welfare decline due to smoothing.
  - Hand-to-mouth: large decline in consumption and much larger welfare loss.

- Monetization of the Deficit
  - Triggers much larger real depreciation; nominal exchange rate jumps.
  - Short-run: expansion in non-tradable sector and decline in traded goods production.
  - Aggregate consumption rises sharply in first period, then falls below baseline.
  - Hand-to-mouth: short-run increase in income, consumption and leisure; subsequent decline.
  - Asset holders: high inflation erodes asset values → large negative wealth effect, persistent labor supply increase, significant fall in consumption, private investment crowded out.
  - Welfare: asset holders suffer largest welfare losses under monetization; hand-to-mouth see relatively modest welfare decline.

- Sales of Reserves
  - Can fully offset the loss of aid inflows when reserve thresholds are not binding, leaving other variables effectively unchanged.

### Welfare outcomes (Table 3, percent changes relative to steady state)
- Hand-to-Mouth:
  - Expenditure Cut: -0.108
  - Sales of Debt: -0.313
  - Monetization: -0.024
  - Reserves Sales: 0 0 0
- Asset Holders:
  - Expenditure Cut: -0.012
  - Sales of Debt: -0.006
  - Monetization: -0.029
  - Reserves Sales: 0 0 0
- Aggregate:
  - Expenditure Cut: -0.090
  - Sales of Debt: -0.256
  - Monetization: -0.025
  - Reserves Sales: 0 0 0

### Aggregate conclusions and policy recommendation
- Consistent results across three models (static open-economy equilibrium, simplified dynamic model, full-scale DSGE):
  - Aid cut matched by equivalent reduction in government spending → real exchange rate depreciation; static model shows real interest rate decline to maintain external balance; DSGE shows similar dynamics.
  - Aid cuts matched by bond sales → real interest rate increases (crowding-out); real exchange rate indeterminate in static model, depreciation in dynamic models.
  - Monetization → real interest rate higher than pre-aid-reduction rate; real depreciation and sharply higher inflation; large nominal depreciation.
  - Reserves sales → substitute fully for lost aid inflow in all models.
- Welfare implications from full-scale DSGE:
  - Unless foreign reserves can fully compensate, welfare is reduced.
  - Expenditure cuts and sales of debt create larger welfare losses for liquidity-constrained (hand-to-mouth) households due to reductions in after-tax wages.
  - Monetization produces worst welfare outcomes for asset holders because very high inflation erodes asset values.
- Policy suggestion from study:
  - When sales of foreign reserves are constrained, a policy mix of expenditure cuts and depreciation through monetary expansion would be a good option.

### Model parameterization and key calibration values (from Full-Scale DSGE Model)
- β 0.98 Discount factor
- θ 2.32 Inverse of intertemporal elasticity of substitution for consumption
- φ 1/0.8 Inverse of Frisch labor supply elasticity
- ρ 0.5 Home bias in private consumption and investment
- G 0.6 Home bias in government consumption and investment
- σ 0.44 Elasticity of substitution between traded and nontraded goods
- λ_l 0.54 Steady-state labor share in nontraded goods sector
- σ_l 0.6 Elasticity of substitution between labor supply to the two sectors
- f 0.2 Fraction of asset holders
- κ 6 Elasticity of substitution between nontraded goods varieties
- θ_p 37.2 Price adjustment cost parameter
- δ 0.025 Depreciation rate of private capital
- γ 1.4 Investment adjustment cost parameter
- N̄ 0.45 Labor income share in nontraded output
- T̄ 0.60 Labor income share in traded output
- Ḡ 0.21 Output elasticity with respect to public capital
- δ_G 0.012 Depreciation rate of public capital
- η 0.4 Public investment efficiency in steady state.
- π 4 1.085 CPI inflation (annual rate)
- c/y 88% Private consumption to GDP ratio
- i/y 18% Private investment to GDP ratio
- g_C/y 7.6% Government consumption to GDP ratio
- g_I/y 10.9% Public investment to GDP ratio
- (s_res)/y 19.2% Foreign exchange reserves to GDP ratio (annual)
- (s_a)/y 5.7% Foreign aid to GDP ratio

---

### Appendix — Analytical structure and key takeaways (A & B summaries)

### Simple dynamic model structure and policy scenarios (preserved)
- Four monetary and fiscal policy scenarios:
  - Scenario (1): Adjustment in expenditures — g_t satisfies (51) with b_c t = b_c; ξ_t = ξ; res*_t = res*; m_t = ⟨m_t-1⟩_t
  - Scenario (2): Debt sales to the private sector — b_c t satisfies (51) with g_t = g; res*_t = res*; m_t = ⟨m_t-1⟩_t; ln(ξ_t/ξ) = % ln(b_c t-1 / b_c)
  - Scenario (3): Monetization of the deficit — m_t satisfies (51) with g_t = g; b_c t = b_c; ξ_t = ξ; res*_t = res*
  - Scenario (4): Sales of foreign exchange reserves — res*_t satisfies (51) with g_t = g; b_c t = b_c; ξ_t = ξ; m_t = ⟨m_t-1⟩_t
- Log-linearized system (A.2) — deviations bx_t = ln(x_t / x̄) with equations (59)–(74) and aggregated relations (75)–(83).

### Analytical solutions and common objects
- Minimum state variable method; linear conjecture for inflation:
  - bπ_t = {a} ba*_t + {b} db_c t-1 + {m} bm_t-1 + {res} cres*_t-1
- Nominal interest rate relation (85) feeds into Euler equation (59) to derive inflation dynamics of form:
  - bπ_t = A E_t bπ_t+1 + A_a ba*_t + A_b db_c t-1 + A_m bm_t-1 + A_res cres*_t-1
- Stationary root selection: {m} = 1 chosen (alternative root non-stationary).

### Scenario-specific analytical highlights (preserved expressions)
- Scenario (1) (expenditure adjustment):
  - {g}_m = 1
  - {g}_a = [1 + A (1 − ρ_a)]^{-1} A_a
  - {g}_b = −(1 + A)^{-1} A_b
  - {g}_res = −(1 + A)^{-1} A_res
  - Real interest rate br_t closed-form in terms of ba*_t, db_c t-1, cres*_t-1 (see B.1).
- Scenario (2) (debt sales):
  - {b}_m = 1; {b}_a and {b}_res solved numerically given calibration; db_c t evolves per (101).
- Scenario (3) (monetization):
  - Internal balance: b_s bs_t = b_c bc_t
  - bs_t and bc_t contemporaneously equal to debt-sale scenario coefficients (m)_{a,res}
  - br_t expression: br_t = [ (m)_a (ρ_a − 1) ψ ] ba*_t − [ (m)_res ψ ] cres*_t-1
  - Money-related coefficient {m} = 1.
- Scenario (4) (reserves sales):
  - Inflation predetermined: bπ_t = {res}_b db_c t-1 + {res}_m bm_t-1 with {res}_m = 1 and {res}_b = (1 + A)^{-1} A (res)_b.
  - Consumption, real exchange rate, and real interest rate are predetermined and independent of the foreign aid shock under this scenario.

### Key analytical takeaways
- Linear-conjecture inflation across scenarios: bπ_t = {·}_a ba*_t + {·}_b db_c t-1 + {·}_m bm_t-1 + {·}_res cres*_t-1.
- Stationary solution for money-related coefficient in multiple scenarios is {m} = 1.
- Debt sales and monetization produce identical initial contemporaneous responses for bs_t and bc_t, but diverge over time due to debt dynamics.
- Reserve-sales scenario makes consumption and real exchange rate responses predetermined and independent of the aid shock.
- Real interest rate dynamics derive from br_t = ψ (E_t bc_t+1 − bc_t) with closed-form expressions in each scenario.

*Source: wp18198 - 1. The net impact is that the real interest rate drops and the real exchange rate depreciates; 5.1 Full-Scale Model Setup; 5.2 Policy Analysis Revisited; Appendix A & B (model and policy analysis).*

### 1. The net impact is that the real interest rate drops and the real exchange rate depreciates.

### 1. The net impact is that the real interest rate drops and the real exchange rate depreciates.

### Main result
- Net effect of an aid decline (under the case with no change in the monetary base): the real interest rate drops and the real exchange rate depreciates.
- Nominal variables move in the same direction when the monetary base is unchanged.

### Policy scenarios and their immediate impacts
- Central bank / government options to finance an increased budget deficit after an aid cut:
  - Sell government debt to the private sector.
  - Monetize the increase in government debt (central bank purchases of government bonds, increasing the money supply).
  - Sell foreign exchange reserves (sterilized: match government bond purchases with reserve sales, keeping the monetary base unchanged).
- Comparative short-run impacts (from the analytical solutions and discussion):
  - Adjusting government expenditures (cut spending to keep deficit unchanged):
    - Initial response: a real depreciation of the currency and a small decline in private consumption.
    - For a very persistent reduction in aid, the real interest rate rises slightly, supporting lower current consumption.
  - Sales of debt to the private sector (government maintains spending):
    - Larger fall in consumption than with expenditure cuts.
    - Currency depreciation is more muted relative to the expenditure-cut case.
    - Real interest rate increases (due to higher debt and future tax expectations).
  - Monetization of the deficit (central bank finances government by increasing money supply):
    - Persistent decline in aid leads to persistently lower consumption and a depreciated currency.
    - In the first period effects are similar to sales of debt, but beyond the first period consumption remains relatively higher under monetization than under sales of debt.
    - Inflation increases substantially as money supply rises in the flexible-price economy.
  - Sales of foreign exchange reserves:
    - If reserves can fully offset the decline in aid, the economy can be insulated and contemporaneous changes in consumption, real exchange rate, inflation, and the real interest rate can be zero.
    - If reserve use is constrained, reserves may be used up to a threshold and the remaining deficit must be financed by the other channels above.

### Key numerical / model notes preserved exactly from the source
- Base case used in numerical simulations: uninterrupted inflow of the equivalent of two percent of GDP in aid is shown as the first row in Table 1.
- Aid process specification (AR(1)):
  - log(ãt / ã) = ρa log(ãt−1 / ã) + εt; εt ~ N(0, σa^2)
  - Persistence parameter: ρa
  - Standard deviation of the shock: σa
- Government budget consolidated (log-linearized and steady-state relationships are explicitly given in the model; see the model equations and the government budget constraint in the source).
- Calibration assumptions used for analytical tractability:
  - Households’ steady-state holdings of government debt: bc = 0.
  - Technology parameters set so that zN = $1−β and zT = (1−$)1−β.
  - Utility is logarithmic in consumption and real money balances: γ = ζ = 1.
  - Cost subsidy: ϕ = 1/(1−θ) (to offset monopolistic competition).
- First-order log-linear conditions (representative forms preserved in the source):
  - Euler: êbc_t = E_t êbc_{t+1} − (1/ϕ)[bR_t − E_t bπ_{t+1}]
  - Money-interest link: bR_t = (R − 1)ε bc_t − (R − 1)ζ bm_t
  - Internal balance: b_s b s_t = b_c b c_t + b_τ b τ_t + 'G ( (ḡ/ḡy_N) ) b g_t
  - External balance: #_s b s_t = #_c b c_t + b_τ b τ_t − (1−'G)( (ḡ/ḡy_T) ) b g_t − ( (ã̄/ḡy_T) ) b ã_t + ( (res̄/ḡy_T) )(c res̃_t − c res̃_{t−1})
  - Consolidated government budget constraint (log-linearized) appears as equation (24) in the source text.

### Analytical insights and dynamics
- Graphical depiction: internal balance (IB) is downward-sloping in (c, s) space; external balance (EB) is upward-sloping. Aid reductions shift EB left; different financing choices shift IB or EB in characteristic ways (see Figure 7 and Figure 8 descriptions).
- Under the expenditure-cut policy, equilibrium moves from point “a” to point “b” (initial to new equilibrium).
- Under sales of debt or monetization, the short-run equilibrium moves from point “a” to point “c”.
- Sales of debt introduce intertemporal considerations: increased bc_t raises expected future taxes (bτ_t), which further depresses consumption over time.
- Monetization raises inflation more than other options because of the immediate increase in money supply in a flexible-price environment.
- Selling reserves can fully offset the aid shock if reserves are sufficient and non-binding minimum thresholds are not violated; otherwise, partial use of reserves combined with other policies is likely.

### Simulations and model extensions
- The static model’s numerical simulations are summarized in Table 1 (base case and alternative financing cases); full dynamic behavior is explored in a DSGE framework calibrated to capture key features of Haiti.
- The full DSGE model for Haiti includes:
  - two types of households (savers and hand-to-mouth consumers),
  - nominal rigidities in the non-tradable sector,
  - investment in private capital and access to international financial markets,
  - public infrastructure investment with inefficiency elements.
- Computations for the DSGE and simple model were performed in Matlab using the fsolve nonlinear system solver.

*Source: wp18198 - 1. The net impact is that the real interest rate drops and the real exchange rate depreciates.*

### 5.1  Full-Scale Model Setup

### 5.1  Full-Scale Model Setup

### Households
- Two household types:
  - Savers (asset holders) denoted with (a).
  - Hand-to-mouth consumers denoted with (h).
- Savers:
  - Choose consumption c_a_t, labor l_a_t, real money balances m_a_t, investment i_N;a_t and i_T;a_t, capital k_N;a_t and k_T;a_t, domestic government bonds b_c;a_t and foreign assets b̃;a_t to maximize expected utility E_0 Σ_{t=0}^∞ β^t [ (c_a_t)^{1-σ}/(1-σ) - {l(l_a_t)^{1+φ}/(1+φ)} + {m(m_a_t)^{1-ζ}/(1-ζ)} ] (equation (29) in source).
  - Budget constraint (30): c_a_t + m_a_t + i_N;a_t + i_T;a_t + b_c;a_t + s_t b̃;a_t + ac_{i;a}_t + s_t ac̃_t = (1-τ_t)[ w_t l_a_t + r_N_t k_N;a_{t-1} + r_T_t k_T;a_{t-1} ] + m_a_{t-1}/π_t + R_{t-1} b_c;a_{t-1}/π_t + s_t R̃ b̃_{t-1}/π̃_t + s_t rm̃ + a_t.
  - Foreign assets pay a constant gross nominal rate R̃ and depreciate at foreign inflation π̃; portfolio adjustment costs ac̃_t = ϑ/2 (b̃_t - b̃̄)^2, where b̃̄ is steady-state real foreign assets and ϑ governs capital account openness.
  - Receive remittances rm̃ (assumed constant). Labor and capital rental income taxed at rate τ_t.
  - Investment adjustment costs: ac_{i;a}_t = κ/2 [ (ξ_{iN;a}_t k_N;a_{t-1} - ι)^2/k_N;a_{t-1} + (ξ_{iT;a}_t k_T;a_{t-1} - ι)^2/k_T;a_{t-1} ] (text representation).
  - Capital law of motion (31): k_{j;a}_t = (1-δ) k_{j;a}_{t-1} + i_{j;a}_t for j∈{N,T}, where δ is depreciation rate.
  - Total investment i_a_t = i_N;a_t + i_T;a_t.
  - Consumption and investment are CES aggregates of nontraded and traded goods with elasticity of substitution ω and home bias ϕ (defined in equation (4)). Nontraded goods are a CES aggregate of differentiated intermediate goods with elasticity of substitution θ.
  - Total labor supply (32): l_a_t = [ ϕ_l^{-1/ω_l} (l_{a;N}_t)^{(1+ω_l)/ω_l} + (1-ϕ_l)^{-1/ω_l} (l_{a;T}_t)^{(1+ω_l)/ω_l} ]^{ω_l/(1+ω_l)}, where ϕ_l is steady-state share of labor in nontraded sector and ω_l>0 elasticity of substitution between sectoral labor supplies.
  - Aggregate real wage index (33): w_t = [ ϕ_l (w_N_t)^{1+ω_l} + (1-ϕ_l) (w_T_t)^{1+ω_l} ]^{1/(1+ω_l)}.
- Hand-to-mouth households:
  - Consume all available income every period: c_h_t = (1-τ_t) w_t l_h_t + s_t rm̃.
  - Period utility U_h_t = [ - (c_h_t)^{1-σ}/(1-σ) - {l̄ - l_h_t}^{1+φ}/(1+φ) ] (text representation).
  - Decide total labor supply optimally and allocate labor between sectors as savers do.
  - Note: large share of hand-to-mouth households in Haiti implies labor market outcomes significantly affected by their labor supply decisions; remittances constant but exchange rate fluctuations matter for their income and consumption.

### Firms
- Nontraded goods sector:
  - Monopolistically competitive firms indexed i∈[0,1].
  - Production uses labor l_N_t(i), private capital k_N_{t-1}(i), and public capital k_G_{t-1} with technology (34): y_N_t(i) = z_N [ k_N_{t-1}(i)^{1-β_N} l_N_t(i)^{β_N} ] k_G_{t-1}^{β_G}, where z_N is sector-specific TFP and β_G is output elasticity of public capital.
  - Aggregation: y_N_t = [ ∫_0^1 y_N_t(i)^{(θ-1)/θ} di ]^{θ/(θ-1)} (CES).
  - Demand for each good (35): y_N_t(i) = ( p_N_t(i) / p_N_t )^{-θ} y_N_t.
  - Firms maximize net present value of profits discounted by stochastic discount factor Λ_t (Λ_a_t = Λ_a_0), with Rotemberg (1982) price adjustment costs ac_p_t(i) = ψ_p/2 [ (π_N_t(i)/π_N - 1)^2 p_N_t y_N_t ] where π_N_t = p_N_t / p_N_{t-1} and π_t is inflation.
- Traded goods sector:
  - Perfectly competitive.
  - Firms choose labor and capital to maximize period-t profits (37): s_t y_T_t - w_T_t l_T_t - r_T_t k_T_{t-1}, given technology y_T_t = z_T [ k_T_{t-1}^{1-β_T} l_T_t^{β_T} ] k_G_{t-1}^{β_G}.
- Total output: y_t = p_N_t y_N_t + s_t y_T_t.

### The Public Sector
- Government expenditures split between government consumption g_C_t and public investment g_I_t, with total g_t = g_C_t + g_I_t.
- Government investment is a constant share of total expenditures and accumulates into public capital:
  - k_G_t = (1-δ_G) k_G_{t-1} + ϕ g_I_t with 0<ϕ<1, δ_G depreciation of public capital, and ϕ capturing public investment efficiency.
- Total government purchases g_t are a CES aggregate of traded and nontraded goods with elasticity of substitution ω and home bias ϕ_G (as in equation (11)).
- Tax revenues based on total factor income: tax_t = τ_t [ w_t l_t + r_N_t k_N_{t-1} + r_T_t k_T_{t-1} ].
- Consolidated government budget constraint (38):
  - p_G_t (g_C_t + g_I_t) = tax_t + s_t ã_t + [ β (b_c_t - R_{t-1} b_c_{t-1}/π_t) ] + [ μ (m_t - m_{t-1})/π_t ] - s_t (res̃_t - res̃_{t-1}) (domestic credit itemization in text).
- Monetary and fiscal policy scenarios as in the simple model, with the further assumption that any government expenditure cuts are reductions in public investment spending.

### Aggregation and Market Clearing
- Aggregate variables x_t = f x_a_t + (1-f) x_h_t for x ∈ {c, c_N, c_T, l, l_N, l_T} (39).
- Since only savers have access to asset and capital markets, aggregate real money balances, investment, capital, debt, and dividends are x_t = f x_a_t for x ∈ {m, i_N, i_T, k_N, k_T, b_c, b̃_t, ac_i, ac̃} (40).
- Nontraded goods market clearing (41):
  - y_N_t = (p_N_t)^{-ω} [ ϕ (c_t + i_t + ac_i_t + ac_p_t) + ϕ_G (p_G_t)^{ω} g_t ] (text representation).
- Balance of payments/current account condition (42):
  - c_t + i_t + p_G_t g_t + ac_i_t + ac_p_t - y_t + s_t ac̃_t - s_t (R̃ - 1) b̃_{t-1}/π̃_t - s_t rm̃ = s_t [ ã_t - (b̃_t - b̃_{t-1}/π̃_t) ] - s_t (res̃_t - res̃_{t-1}) (text representation).

### Model Calibration
- Solution method: algorithm of Schmitt-Grohe and Uribe (2004); welfare computed accordingly.
- Frequency: quarterly.
- Calibration targets and data:
  - Calibrated to Haitian economy averages for 2005-2015.
  - Inflation set to 8.5% annually.
  - Share of savers in the economy set to about 20%.
- Preference and technology parameters:
  - Discount factor β = 0.98.
  - Risk aversion parameter σ = 2.32 (implying intertemporal elasticity of substitution of 0:43 as per Ogaki et al. (1996) for Haiti).
  - Inverse Frisch labor supply elasticity = 1/0.8 (reflecting relatively inelastic labor supply).
  - Intratemporal elasticity of substitution of labor between sectors ω_l = 0:6.
  - Elasticity of substitution between traded and nontraded goods ω = 0:44.
  - Elasticity of substitution between varieties of nontraded goods θ = 6 (matching steady-state markup of 20 percent).
  - Degree of home bias in private consumption and investment ϕ = 0:5.
  - Home bias for government purchases ϕ_G = 0:6.
  - Labor income shares: β_N = 0:45 and β_T = 0:6.
  - Depreciation rate of private capital δ = 0:025 (implying annual depreciation rate of 10 percent).
  - Investment adjustment cost parameter λ = 1:4.
  - Output elasticity of public capital β_G = 0.21 (corresponding to return on public capital ≈ 20%).
  - Public investment efficiency ϕ (efficiency parameter) = 0:4.

*Source: wp18198 - 5.1  Full-Scale Model Setup*

### 5.2  Policy Analysis Revisited

### 5.2  Policy Analysis Revisited

### Methodology and welfare measure
- Impulse responses are computed to convey dynamics of key variables in response to a persistent decline in foreign aid inflows (shock considered: decrease in aid of 2% of GDP).
- Welfare is measured as the present discounted value of life-time utility, conditional on the aid decline and on government policy.
  - For asset holders, welfare is V_a_0 = E_0 Σ_{t=0}^∞ β^t [ (c^a_t)^{1-θ}/(1-θ) + { l (l^a_t)^{1+φ}/(1+φ)} + { m (m^a_t)^{1-ζ}/(1-ζ)} ] (expression preserved from source).
  - For hand-to-mouth consumers, welfare is V_h_0 = E_0 Σ_{t=0}^∞ β^t [ -(c^h_t)^{1-θ}/(1-θ) + { - l l^h_t^{1+φ}/(1+φ) } ] (expression preserved from source).
- Aggregate welfare is a weighted average of V_a_0 and V_h_0 with weights from household shares.

### Impulse-response overview (shock: aid down 2% of GDP)
- Four policy responses considered (lines/colors as in figures):
  - Expenditure cuts (red dashed lines).
  - Sales of debt to the private sector (solid blue lines).
  - Monetization of the increased deficit (black circles lines).
  - Sales of foreign reserves (dash-dot green lines).
- Sales of foreign exchange reserves can fully offset the effects of the decline in aid inflows, leaving other variables unchanged provided reserve thresholds are not binding.
- Under the remaining policy choices:
  - Real exchange rate depreciates.
  - Trade deficit declines.
  - Overall reduced GDP levels.
  - Time-paths differ across policies and across tradable vs non-tradable sectors and household types.

### Policy scenario findings

- Adjusting Government Expenditure (expenditure cuts)
  - Government cuts reduce demand for goods, higher propensity for non-tradable goods.
  - Nominal inertia in non-tradable sector causes contraction of non-tradable output, employment and wages.
  - Real depreciation raises output and employment in tradable sector; trade deficit decreases.
  - Overall: GDP and private consumption fall; private investment is initially crowded-in as asset holders draw on assets to smooth consumption.
  - Assumption: entire reduction in expenditures is cuts in public investment → lower public capital → negative effects on production of traded and non-traded goods over longer horizon (muted if public investment inefficiencies exist).
  - Labor market: real wages decrease at least in the short run, dominated by reduced demand in non-tradable sector.
  - Household impacts:
    - Asset holders: persistent reduction in consumption, temporary reduction in labor supply, draw on foreign assets to support consumption and private investment.
    - Hand-to-mouth: consumption determined by wage income and foreign remittances; currency depreciation increases remittances but is outweighed by lower labor income → short-run decrease in consumption and utility.
  - Welfare implication: hand-to-mouth welfare decline of 0.11% relative to steady state, about 9 times larger than asset holders (source: Table 3).

- Sales of Debt to the Private Sector
  - Government issues new debt to finance increased deficit, drawing resources from private sector.
  - Asset holders hold more savings as government bonds and reduce investment in private capital.
  - Decline in investment demand affects both sectors; real depreciation expands tradable sector, non-tradable contracts.
  - Dominant non-tradable effects → decline in overall GDP and real wages.
  - Asset holders: negative wealth effect raises labor supply initially; higher future taxes reduce labor supply later; consumption falls by about 1%; relatively small welfare decline due to smoothing.
  - Hand-to-mouth: decline in after-tax wages produces strong negative income effect; despite increasing labor supply, they suffer large decline in consumption and much larger welfare loss. Policy option hurts hand-to-mouth households most.

- Monetization of the Deficit
  - Triggers much larger real depreciation; nominal exchange rate jumps significantly.
  - Real depreciation increases tradable sector competitiveness but makes traded goods more expensive for domestic consumers → substitution toward non-tradable goods.
  - Short-run: expansion in non-tradable sector and decline in traded goods production.
  - Aggregate consumption: rises sharply in first period, then falls below baseline and below expenditure-cut scenario thereafter.
  - Hand-to-mouth: higher foreign remittances value plus initial wage increase → short-run increase in income, consumption and leisure; as nominal rigidities dissipate and wages return, consumption falls.
  - Asset holders: high inflation erodes asset values → large negative wealth effect, higher and persistent labor supply, significant fall in consumption, private investment crowded out.
  - Welfare: asset holders suffer large welfare losses under monetization, larger than under alternative policies; hand-to-mouth see relatively modest welfare decline.

- Sales of Reserves
  - Can fully offset the loss of aid inflows when reserve thresholds are not binding, leaving other variables effectively unchanged.

### Welfare outcomes (Table 3, percent changes relative to steady state)
- Hand-to-Mouth:
  - Expenditure Cut: -0.108
  - Sales of Debt: -0.313
  - Monetization: -0.024
  - Reserves Sales: 0 0 0 (reported as 000 in source)
- Asset Holders:
  - Expenditure Cut: -0.012
  - Sales of Debt: -0.006
  - Monetization: -0.029
  - Reserves Sales: 0 0 0 (reported as 000 in source)
- Aggregate:
  - Expenditure Cut: -0.090
  - Sales of Debt: -0.256
  - Monetization: -0.025
  - Reserves Sales: 0 0 0 (reported as 000 in source)

### Aggregate conclusions and policy recommendation
- Consistent results across three models (static open-economy equilibrium, simplified dynamic model, full-scale DSGE):
  - Aid cut matched by equivalent reduction in government spending → real exchange rate depreciation; static model shows real interest rate decline to maintain external balance; DSGE shows similar dynamics.
  - Aid cuts matched by bond sales → real interest rate increases (crowding-out); real exchange rate indeterminate in static model, depreciation in dynamic models.
  - Monetization → real interest rate higher than pre-aid-reduction rate; real depreciation and sharply higher inflation; large nominal depreciation.
  - Reserves sales → substitute fully for lost aid inflow in all models.
- Welfare implications from full-scale DSGE:
  - Unless foreign reserves can fully compensate, welfare is reduced.
  - Expenditure cuts and sales of debt create larger welfare losses for liquidity-constrained (hand-to-mouth) households due to reductions in after-tax wages.
  - Monetization produces worst welfare outcomes for asset holders because very high inflation erodes asset values.
- Policy suggestion from study: when sales of foreign reserves are constrained, a policy mix of expenditure cuts and depreciation through monetary expansion would be a good option.

### Model parameterization and key calibration values (from Full-Scale DSGE Model)
- β 0.98 Discount factor
- θ 2.32 Inverse of intertemporal elasticity of substitution for consumption
- φ 1/0.8 Inverse of Frisch labor supply elasticity
- ρ 0.5 Home bias in private consumption and investment
- G 0.6 Home bias in government consumption and investment
- σ 0.44 Elasticity of substitution between traded and nontraded goods
- λ_l 0.54 Steady-state labor share in nontraded goods sector
- σ_l 0.6 Elasticity of substitution between labor supply to the two sectors
- f 0.2 Fraction of asset holders
- κ 6 Elasticity of substitution between nontraded goods varieties
- θ_p 37.2 Price adjustment cost parameter
- δ 0.025 Depreciation rate of private capital
- γ 1.4 Investment adjustment cost parameter
- N̄ 0.45 Labor income share in nontraded output
- T̄ 0.60 Labor income share in traded output
- Ḡ 0.21 Output elasticity with respect to public capital
- δ_G 0.012 Depreciation rate of public capital
- η 0.4 Public investment efficiency in steady state.
- π 4 1.085 CPI inflation (annual rate)
- c/y 88% Private consumption to GDP ratio
- i/y 18% Private investment to GDP ratio
- g_C/y 7.6% Government consumption to GDP ratio
- g_I/y 10.9% Public investment to GDP ratio
- (s_res)/y 19.2% Foreign exchange reserves to GDP ratio (annual) (reported as (s_res)=4y19.2% in source)
- (s_a)/y 5.7% Foreign aid to GDP ratio (reported as (s_a)=y5.7% in source)

*Source: wp18198 - 5.2  Policy Analysis Revisited (PDF chapter).*

### References

### wp18198 - References (Appendix A & B: Model and Policy Analysis)

### A. Appendix — A Simple Dynamic Model: structure and equilibrium conditions
- Model blocks and key equilibrium relationships (equation numbering preserved):
  - Households
    - Consumption Euler equation (43)
    - Money demand (44)
    - Labor supply (45)
  - Non-traded goods firms
    - Production of non-tradables (46)
    - Pricing condition (47)
    - Real marginal cost (48)
  - Traded goods firms
    - Production of tradables (49)
    - Labor demand (50)
  - Government consolidated budget constraint (51)
  - Monetary and fiscal policy rules: four scenarios
    - Scenario (1): Adjustment in expenditures — g_t satisfies (51) with b_c t = b_c; ξ_t = ξ; res*_t = res*; m_t = ⟨m_t-1⟩_t
    - Scenario (2): Debt sales to the private sector — b_c t satisfies (51) with g_t = g; res*_t = res*; m_t = ⟨m_t-1⟩_t; ln(ξ_t/ξ) = % ln(b_c t-1 / b_c)
    - Scenario (3): Monetization of the deficit — m_t satisfies (51) with g_t = g; b_c t = b_c; ξ_t = ξ; res*_t = res*
    - Scenario (4): Sales of foreign exchange reserves — res*_t satisfies (51) with g_t = g; b_c t = b_c; ξ_t = ξ; m_t = ⟨m_t-1⟩_t
  - Aggregate constraints and definitions
    - Aggregate labor (52)
    - GDP definition (53)
    - Internal balance condition (54)
    - External balance condition (55)
    - CPI definition (56)
    - Government price index (57)
    - Exogenous foreign aid process (58)

- Log-linearized system (A.2) — deviations denoted with bx_t = ln(x_t / x̄):
  - Log-linear Euler equation (59), money demand (60), labor supply (61)
  - Production and pricing relations (62)–(66)
  - Consolidated government budget (67)
  - Policy scenarios translated into log-linear restrictions (Scenario (1) – (4) as in A.1)
  - Aggregate labor (68), real GDP (69), internal and external balance conditions (70)–(71)
  - CPI and government price index log-definitions (72)–(73)
  - Aid AR(1) process in logs (74)

### A.3 Further manipulations — labor market and aggregate output
- Labor demand and aggregation:
  - Non-tradable sector labor demand: (75)
  - Tradable sector labor demand: (76)
  - Aggregate labor with calibration assumption β_N = β_T = β: (77)
- Equilibrium real wage expressed in terms of bs_t, bc_t, bξ_t (78) with definition of [3]
- Equilibrium total labor (79) and wage income measure (80)
- Sectoral outputs:
  - NT sector output (81)
  - T sector output (82)
  - Aggregate output (83) with definition of [4]
- Government budget constraint rewritten (84) (note: changes in government debt measured as simple deviations db_c t, not log-deviations)
- Money market relation (85): b_R t = (R-1)ψ b_c t − (R-1)χ b_m t
- Internal balance linear relation (86): b_s bs_t = b_c b_c t + b_ξ b_ξ t + 'G ( ḡ / ȳ_N ) b_g t (coefficients listed)
- External balance linear relation (87): #_s b_s t = #_c b_c t + b_ξ b_ξ t + (1-γ_G)(ḡ / ȳ_T) b_g t − (ā / ȳ_T) b_a* t − (res̄ / ȳ_T)(cres*_t − cres*_t-1) (coefficients listed)

### B. Policy Analysis in the Simple Dynamic Model — analytical details and solutions
- Overview: Four policy scenarios analyzed analytically in Appendix B with linear solutions for inflation and endogenous variables in terms of state variables (aid shock ba*_t, lagged debt db_c t-1, lagged money bm_t-1, and lagged reserves cres*_t-1).

- Common objects and methods:
  - Minimum state variable method; linear conjecture for inflation:
    - bπ_t = {a} ba*_t + {b} db_c t-1 + {m} bm_t-1 + {res} cres*_t-1 (used across scenarios)
  - Nominal interest rate relation from money market (85) feeds into Euler equation (59) to derive inflation dynamics of the form:
    - bπ_t = A E_t bπ_t+1 + A_a ba*_t + A_b db_c t-1 + A_m bm_t-1 + A_res cres*_t-1
  - Stationary root selection for {m} is critical (stationary solution generally {m} = 1 under calibrations stated)

- B.1 Adjusting Government Expenditures (Scenario (1))
  - Policy specification: db_c t = 0, b_ξ t = 0, cres*_t = 0, bm_t = bm_t-1 − bπ_t
  - Government expenditures expression from budget (84): ḡ b_g t = [ … ] (88) (depends on bs_t, bc_t, ba*_t, db_c t-1, bm_t-1 − bπ_t, cres*_t-1)
  - Internal balance (89) and external balance (90) expressed as:
    - b(g)_s bs_t = b(g)_c bc_t + 'G (ā / ȳ_N) ba*_t − 'G (1 / ȳ_N R') db_c t-1 + 'G (m̄ / ȳ_N)(bm_t-1 − bπ_t) + 'G (res̄ / ȳ_N) cres*_t-1
    - #-form for external balance with analogous terms (90)
  - Solutions for bs_t and bc_t in linear form (91)–(92):
    - bs_t = (g)_a ba*_t + (g)_b db_c t-1 + (g)_m (bm_t-1 − bπ_t) + (g)_res cres*_t-1
    - bc_t = (g)_a ba*_t + (g)_b db_c t-1 + (g)_m (bm_t-1 − bπ_t) + (g)_res cres*_t-1
  - Inflation dynamics: system (94) gives solution for {g}_a, {g}_b, {g}_m, {g}_res:
    - Stationary solution for {g}_m: {g}_m = 1
    - {g}_a = [1 + A (1 − ρ_a)]^{-1} A_a
    - {g}_b = −(1 + A)^{-1} A_b
    - {g}_res = −(1 + A)^{-1} A_res
  - Real exchange rate and consumption (95)–(96):
    - bs_t and bc_t expressed in terms of ba*_t, db_c t-1, bm_t-1, cres*_t-1 using (g) and {g} coefficients
  - Real interest rate br_t = ψ (E_t bc_t+1 − bc_t): closed-form in terms of ba*_t, db_c t-1, cres*_t-1

- B.2 Sales of Debt to the Private Sector (Scenario (2))
  - Policy specification: b_g t = 0, cres*_t = 0, bm_t = bm_t-1 − bπ_t, and tax rule b_ξ t = % db_c t-1 (with % >0 small)
  - Internal balance reduces to (97): b_s bs_t = b_c bc_t + b_ξ (−% db_c t-1)
  - External balance (98): #_s bs_t = #_c bc_t − (ā / ȳ_T) ba*_t − (res̄ / ȳ_T) cres*_t-1 + b_ξ (−% db_c t-1)
  - Solutions for bs_t and bc_t as (99)–(100): linear in ba*_t, db_c t-1, cres*_t-1 with coefficients (b)_{a,b,res}
  - Debt issuance dynamics: government budget yields db_c t expressed as (101):
    - (1 / ȳ) db_c t = 
(b)_a ba*_t + 
(b)_b db_c t-1 + 
(b)_m (bm_t-1 − bπ_t) + 
(b)_res cres*_t-1
  - Inflation dynamics produce a system (103) for {b}_a, {b}_b, {b}_m, {b}_res:
    - Stationary solution typically {b}_m = 1; {b} solved numerically under calibration
    - {b}_a and {b}_res derived from linear system using calibrated matrices in text
  - Real interest rate follows from Euler equation using solution for consumption

- B.3 Monetization (Scenario (3))
  - Policy specification: b_g t = 0, db_c t = 0, b_ξ t = 0, cres*_t = 0 — deficit monetized via money supply
  - Internal balance simplifies to (104): b_s bs_t = b_c bc_t
  - External balance (105): #_s bs_t = #_c bc_t − (ā / ȳ_T) ba*_t − (res̄ / ȳ_T) cres*_t-1
  - Solutions for bs_t and bc_t (106)–(107):
    - bs_t = (m)_a ba*_t + (m)_res cres*_t-1
    - bc_t = (m)_a ba*_t + (m)_res cres*_t-1
    - Coefficients (m)_{a,res} coincide with debt-sale scenario contemporaneously
  - Real interest rate (108): br_t = [ (m)_a (ρ_a − 1) ψ ] ba*_t − [ (m)_res ψ ] cres*_t-1
  - Money supply path implied by government budget (109): bm_t = 
(m)_a ba*_t + 
(m)_b db_c t-1 + 1/(·) (bm_t-1 − bπ_t) + 
(m)_res cres*_t-1
  - Inflation solution via minimum state variable method leads to system (110):
    - Stationary root {m} = 1
    - {m}_a, {m}_b, {m}_res solved from linear system; closed-form expressions given in text

- B.4 Sales of Foreign Exchange Reserves (Scenario (4))
  - Policy specification: b_g t = 0, db_c t = 0, b_ξ t = 0, bm_t = bm_t-1 − bπ_t; deficit financed by reserves sales
  - Internal balance same as monetization: (111) b_s bs_t = b_c bc_t
  - External balance (112) includes contemporaneous reserve change cres*_t − cres*_t-1
  - Reserve dynamics from government budget constraint yield (113): #^{(res)}_s bs_t = #^{(res)}_c bc_t + (m̄ / ȳ_T)(bm_t-1 − bπ_t) − (1 / ȳ_T R') db_c t-1
  - From (111) and (113) obtain bs_t and bc_t (114)–(115):
    - bs_t = ^{(res)}_m (bm_t-1 − bπ_t) + ^{(res)}_b db_c t-1
    - bc_t = (b_s / b_c) bs_t = (res)_m (bm_t-1 − bπ_t) + (res)_b db_c t-1
    - ^{(res)} coefficients defined in text
  - Inflation dynamics reduce to:
    - bπ_t = A E_t bπ_t+1 + bm_t-1 + A^{(res)}_b db_c t-1
    - Solution for inflation is predetermined: bπ_t = {res}_b db_c t-1 + {res}_m bm_t-1 with {res}_m = 1 and {res}_b = (1 + A)^{-1} A (res)_b
  - Consequence: consumption, real exchange rate, and real interest rate are predetermined and independent of the foreign aid shock under this scenario:
    - bs_t = [ ^{(res)}_b − ^{(res)}_m {res}_b ] db_c t-1
    - bc_t = [ (res)_b − (res)_m {res}_b ] db_c t-1
    - br_t = −ψ [ (res)_b − (res)_m {res}_b ] db_c t-1

### Key analytical takeaways (preserving textual expressions and numeric roots)
- The linear-conjecture solution for inflation across scenarios takes the general form:
  - bπ_t = {·}_a ba*_t + {·}_b db_c t-1 + {·}_m bm_t-1 + {·}_res cres*_t-1
- Stationary solution for the money-related coefficient in multiple scenarios is {m} = 1 (the alternative root {m} = −1/ϑ is non-stationary and discarded).
- Under:
  - Scenario (1) (expenditure adjustment): inflation coefficients {g}_m = 1; {g}_a = [1 +  (1 − ρ_a)]^{-1} _a; {g}_b = −(1 + )^{-1} _b; {g}_res = −(1 + )^{-1} _res.
  - Scenario (2) (debt sales): {b}_m = 1 (stationary), {b}_a and {b}_res solved numerically given calibration; {b} coefficients satisfy system (103).
  - Scenario (3) (monetization): {m}_m = 1; {m}_a and {m}_res given by closed-form expressions in (110).
  - Scenario (4) (reserves sales): inflation is predetermined: bπ_t = {res}_b db_c t-1 + {res}_m bm_t-1 with {res}_m = 1 and {res}_b = (1 + )^{-1}  (res)_b.
- Consumption and real exchange rate responses differ across scenarios:
  - Debt sales and monetization produce identical initial (contemporaneous) responses for bs_t and bc_t, but diverge over time due to debt dynamics.
  - Reserve-sales scenario makes consumption and real exchange rate responses predetermined and independent of the aid shock.
- Real interest rate dynamics derive from br_t = ψ (E_t bc_t+1 − bc_t) and are expressed in closed form in each scenario (representative expressions: br_t in B.1, B.2, B.3, B.4).

*Source: wp18198 - References (Appendix A & B of the supplied PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18198.pdf_