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### Aid volatility: empirical facts
- Using data from 63 recipient countries from 1969 to 1995, Pallage and Robe (2001) report that aid volatility is about two to three times that of recipient countries’ output.
- Bulíř and Hamann (2003, 2008) find that aid is generally more volatile than tax revenues, especially in highly aid-dependent countries.
- Using more recent data, the authors find that 78 percent of low-income countries (LICs) and 93 percent of lower middle-income countries (LMICs) experience higher volatility in aid than in tax revenues.
- Appendix A: proportion of countries where aid volatility > tax revenue volatility by detrending method:
  - First difference series: LICs 78.3%   LMICs 92.5%
  - HP filter: LICs 82.6%   LMICs 94.3%
  - Linear trend: LICs 78.3%   LMICs 92.5%
- Calibration for the aid process across 23 countries: ρ_a = 0.72 and σ_a = 0.098.

### Main modeling features and setup
- Two-period analytical model:
  - Incorporates Dutch disease (DD) externality and hand-to-mouth households.
  - Government instruments: reserve accumulation and transfers (transfers may be productive with parameter αG).
  - Stochastic aid process (equation (2.1)): a*1 known; a*2 = a*2L with probability p and a*2 = a*2H with probability 1−p; E(a*2) = a*1.
  - Government budget constraints (period-1 and period-2) and household preferences specified (β < 1, σ relative risk aversion).
  - Output externality: y2 = y [ 1 + ξ (y − ch1)/y ] + αG (z1 / y), with ξ > 0 and αG > 0.
- Infinite-horizon DSGE quantitative model:
  - Two production sectors (nontraded N and traded T), heterogeneous households (fraction η savers, 1−η hand-to-mouth), public investment accumulating public capital k_G, reserves with adjustment cost υ_res.
  - Policy instruments: government transfers z_t, public investment g^I_t, reserves res^*_t.
  - Key calibration values used in simulations: β = 0.98; σ = 2; r* = 0.04; y = 1; R*0 = 0.13; a*1 = 0.05; p = 0.5; baseline ξ = αG = 0.
  - Reserves adjustment cost: υ_res = 0.0006. Portfolio adjustment cost υ = 500 (benchmark).

### Core analytical findings (two-period model)
- Ramsey-optimal fiscal spending and reserve accumulation rules are forward-looking, responding to both current and expected future aid (a*1 and a*2), not only contemporaneous aid.
- Optimal policy typically combines partial fiscal spending and partial reserve accumulation:
  - Under positive net return to reserves (1 + r* + ξ − αG > 0): γ ∈ (0,1) and ω ∈ (0,1) where γ and ω are spending and reserve response coefficients.
  - Comparative statics:
    - ∂γ/∂ξ < 0; ∂ω/∂ξ > 0 (stronger DD externality → more reserves, less current transfers).
    - ∂γ/∂αG > 0; ∂ω/∂αG < 0 (more productive spending → more current transfers, less reserve accumulation).
- Precautionary motive: stochastic aid increases reserve accumulation and reduces current transfers; larger consumption variance raises expected consumption growth via precautionary saving, lowering period-1 consumption.

### Quantitative model results: baseline shock and mechanisms
- Baseline shock: exogenous 1 percent increase in aid inflows above steady-state.
- Optimal interior allocation: reserves, government transfers, and public investment all rise following the aid shock.
- Mechanisms and roles:
  - Public investment raises production capacity, crowds in private investment, raises future tax revenues and wages, and acts as a form of saving (public capital as precautionary saving).
  - Reserve accumulation mitigates DD effects by dampening real exchange rate appreciation and provides interest income to government.
  - Transfers smooth consumption for hand-to-mouth households given large fraction of liquidity-constrained consumers.
- Aggregate dynamics:
  - Nontraded goods demand and output rise; traded goods output initially contracts due to DD but recovers as capital accumulates.
  - Aggregate labor falls due to a wealth effect from higher saver income.

### Alternative scenarios isolating instruments
- Alternative Scenario I — Excluding public investment (public investment held at deterministic steady state):
  - Ramsey planner uses only transfers and reserves.
  - Compared with baseline: larger increases in transfers and reserves; hand-to-mouth consumption increases more short-to-medium run.
  - Without public capital increase: smaller long-run expansion; output, consumption, and private investment remain significantly below baseline in later periods.
- Alternative Scenario II — Excluding reserve accumulation (υ_res = 1000, reserves virtually constant):
  - Aid channeled into transfers and public investment; both rise significantly more in the short run.
  - Larger goods demand → stronger nontraded production, real wage surge, private investment boom.
  - More procyclical fiscal policy: short-lived expansion but larger exposure to negative aid shocks.
  - Stronger DD effects: larger currency appreciation and sharper short-run fall in traded-goods output (though quicker rebound than baseline).
- Conclusion: public investment and reserve accumulation both save but are not perfect substitutes; each mitigates different channels of volatility and DD.

### Welfare implications and numerical magnitudes
- Welfare components:
  - Variance effects: higher σ_a increases consumption volatility and reduces welfare.
  - Mean effects: precautionary saving (reserves and public investment) raises mean consumption, increasing welfare.
  - In the calibrated model mean effects dominate.
- Numerical example (50-percent increase in aid volatility; government responds by accumulating more reserves and increasing public investment):
  - Savers’ welfare rises to 0.79 percent of steady-state consumption from 0.35 percent.
  - Hand-to-mouth households’ welfare rises to 1.11 percent of steady-state consumption from 0.49 percent.
- Welfare ranking across scenarios:
  - Baseline (all three instruments) yields highest aggregate welfare.
  - Savers are roughly indifferent between baseline and excluding reserves because higher capital income offsets fiscal instability in the no-reserves case.
  - Excluding reserves but retaining public investment yields higher welfare than excluding public investment.

### Sensitivity and robustness findings
- Share of hand-to-mouth households:
  - Benchmark 1−η = 0.75 (η = 0.25); alternative η = 0.45.
  - With smaller share of hand-to-mouth (η = 0.45): less reserve accumulation, more aid to public investment, stronger private investment response, larger output expansion, higher wages.
- Stronger DD externalities:
  - Comparison for ρ_z_T: benchmark ρ_z_T = 0.1 versus stronger ρ_z_T = 0.3.
  - Stronger DD (ρ_z_T = 0.3) → larger short-to-medium TFP decline in traded sector, slightly more reserve accumulation, less public investment, smaller private investment increase, smaller output expansion, and a slightly higher short-run transfer increase.
  - Real exchange rate appreciates slightly less under stronger DD, consistent with slightly more reserve accumulation.
- Lower public capital productivity:
  - Benchmark α_G = 0.39 versus lower α_G = 0.187.
  - Lower α_G = 0.187 → stronger reserve accumulation and reduced public investment; lower aggregate demand and weaker private investment and wage growth; public investment less effective as saving instrument; transfers increase more relative to benchmark beyond initial periods; private consumption increases less.
  - Decreasing public investment efficiency ǫ yields analogous results (less public investment, more reserve saving, weaker output and consumption response).

### Policy implications and recommendations
- Optimal policy mix under volatile aid:
  - Combine transfers, public investment, and reserve accumulation rather than relying on a single instrument.
  - Public investment serves dual roles: development-enhancing and precautionary saving (complements reserve accumulation).
  - Transfers remain essential for consumption smoothing of hand-to-mouth households.
- Guiding principles for LICs:
  - Higher share of financially constrained consumers → tilt policy toward transfers and reserve accumulation, less toward public investment.
  - Stronger DD externalities or lower public capital productivity → favor more reserve accumulation and relatively less public investment.
  - Higher aid volatility implies increased precautionary saving via both public investment and reserves to raise mean consumption despite higher variance effects.
- Broader implications:
  - Lessons extend to management of other volatile external incomes (e.g., natural resource revenues) and the design of combined reserve-fiscal tools to respond to external shocks.

*Source: wpiea2025154-source-pdf (IMF Working Paper)*

### 1.1   Related Literature ...............................................................................................

### 1.1   Related Literature

### Aid volatility: empirical facts
- Using data from 63 recipient countries from 1969 to 1995, Pallage and Robe (2001) report that aid volatility is about two to three times that of recipient countries’ output.
- Bulíř and Hamann (2003, 2008) find that aid is generally more volatile than tax revenues, especially in highly aid-dependent countries.
- Using more recent data, the authors find that 78 percent of low-income countries (LICs) and 93 percent of lower middle-income countries (LMICs) experience higher volatility in aid than in tax revenues.
- Appendix A provides data and calculation details and compares aid vs. tax revenue volatility.

### Identified channels and prior findings in the literature
- Aid volatility reduces the effectiveness of aid in promoting economic growth, health, and social development (Lensink and Morrissey (2000); Hudson (2015)).
- Unpredictable aid disbursements are linked to reduced private investment and government expenditure (Hudson and Mosley (2008)).
- Donor behavior and coordination in disbursement are important to improve predictability (Bulíř and Hamann, 2003; Arellano et al., 2009).
- Financial constraints in LIC households (limited access to capital and international markets) amplify consumption fluctuations from volatile aid (Pallage and Robe, 2003; Pallage et al., 2006; Arellano et al., 2009).

### Key gaps motivating this paper
- Existing literature has extensively studied macroeconomic implications of aid scaling-ups but paid comparatively less attention to the optimal configuration of policies to manage volatile aid.
- The paper fills this gap by jointly considering fiscal spending and foreign exchange reserve accumulation in a fully Ramsey-optimal policy framework.

### Main modeling features used to address the gap
- Two prevalent LIC features are incorporated:
  - Dutch disease (DD) externality: a growth externality where aid spending (analogous to a decrease in net exports) negatively affects future output.
  - Financially constrained (hand-to-mouth) households: unable to smooth consumption on their own.
- Government instruments considered: reserve accumulation, transfers to households, and public investment (which builds public capital).
- Models:
  - A simple two-period model with DD externality and hand-to-mouth households to derive analytical insights and characterize Ramsey-optimal policy rules.
  - A richer quantitative dynamic stochastic general equilibrium (DSGE) two-sector small-open-economy model calibrated to an average LIC, with learning-by-doing externalities à la van Wijnbergen (1984) to capture DD effects.

### Analytical findings from the simple model
- Ramsey-optimal fiscal spending and reserve accumulation rules include a forward-looking component (depend on expected future aid), departing from contemporaneous-only rules.
- Optimal policy typically combines partial fiscal spending and partial reserve accumulation of aid.
  - Baseline (no DD, transfers not productive): partial reserve accumulation arises from precautionary saving motives; transfers smooth consumption for hand-to-mouth households.
  - With higher aid volatility: government accumulates more reserves and allocates less aid to households in the current period.
- Effects of structural features:
  - DD externalities → greater optimal reserve accumulation and reduced current fiscal spending (to avoid harming future output).
  - Productive transfers → less reserve accumulation and increased current fiscal spending (because spending raises future output).

### Quantitative model results and policy implications
- Allowing public investment (public capital accumulation) is important: public investment increases welfare relative to a policy that only transfers aid and accumulates reserves.
  - Public investment smooths consumption indirectly by sustaining production and income over time; it functions as an alternative form of saving alongside international reserves.
- Under increased aid volatility, optimal responses include:
  - Higher public investment.
  - Higher reserve accumulation.
  - These support higher future transfers and consumption through two channels:
    - Mitigating real exchange rate appreciation and limiting the contraction in traded goods output, which supports future tax revenues.
    - Reserve savings generate interest income, increasing government revenues.
  - Public investment also crowds in private investment, raising private capital and output, which further increases government tax revenues.

### Welfare implications and numerical magnitudes (calibrated model)
- Aid volatility affects welfare through both variance and mean effects on consumption:
  - Variance effect: higher volatility increases consumption variance and reduces welfare.
  - Mean effect: precautionary savings via reserve accumulation and public investment raise the mean of consumption and increase welfare.
  - In the calibrated model the positive mean effect dominates.
- Exact calibrated welfare magnitudes under a 50-percent increase in aid volatility (assuming government responds by accumulating more reserves and increasing public investment):
  - Savers’ welfare rises to 0.79 percent of steady-state consumption from 0.35 percent.
  - Hand-to-mouth households’ welfare rises to 1.11 percent of steady-state consumption from 0.49 percent.

### Sensitivity and robustness findings
- A smaller share of hand-to-mouth households:
  - Government should shift spending toward public investment and reduce foreign reserve accumulation (less need to smooth consumption via fiscal transfers).
- More persistent or stronger DD externalities:
  - Greater contraction in traded goods production and less government resources available.
  - Slightly larger reserve accumulation than baseline to reduce currency appreciation and mitigate DD effects.
- Lower public capital productivity:
  - Government reallocates from public investment toward transfers and increases reserve accumulation as a savings device to smooth consumption.

*Source: wpiea2025154-source-pdf (IMF Working Paper)*

### 1.1 Related Literature

### wpiea2025154-source-pdf - 1.1 Related Literature

### Contribution to literature and positioning
- Extends literature on policy responses to manage foreign aid by explicitly incorporating aid volatility, reserve accumulation, and a detailed specification of fiscal spending including public investment.
- Connects and contrasts with prior approaches:
  - Builds on contemporaneous partial fiscal spending and reserve accumulation rules studied by Berg et al.(2010a) and Berg et al.(2015).
  - Relates to sterilized reserve accumulation literature (Buffie et al.(2008); Adam et al.(2009)) and to literature on sterilized foreign exchange interventions (Adler et al.,2019; Adrian et al.,2022; Alla et al.,2020; Basu et al.,2020; Benes et al.,2015; Cavallino,2019; Fanelli and Straub,2021; Faltermeier et al.,2022; Itskhoki and Mukhin,2023).
  - Complements work on aid-financed public investment and Big Push narratives (Sachs,2005; Adam and Bevan,2006; Agénor et al.,2008; Cerra et al.,2009; Chatterjee and Turnovsky,2007; Zanna et al.,2019).
  - Relates to optimal natural resource allocation literature that studies smoothing versus front-loaded public investment (Davis et al.,2001; Barnett and Ossowski,2003; Bems and de Carvalho Filho,2011; Takizawa et al.,2004; van der Ploeg and Venables,2011; International Monetary Fund,2012a,b; Berg et al.,2013; Araujo et al.,2016).

- Key distinctions:
  - Highlights public capital accumulation as a precautionary saving instrument under volatile aid, in addition to traditional development/growth roles.
  - Emphasizes the role of country characteristics (DD externalities, financially constrained households) in shaping optimal combinations of fiscal spending (including public investment) and reserve accumulation (FXI).

### Simple two-period model setup and assumptions
- Agents and frictions:
  - Representative hand-to-mouth households unable to smooth consumption across periods.
  - Government receives foreign aid, can accumulate international reserves, and makes transfers which may be productive.
  - Externality from net exports that induces Dutch disease (DD) effects.
  - Aid in period 2 is uncertain and volatile.
- Stochastic aid process (equation (2.1)):
  - a*1 known with certainty.
  - a*2 = a*2L with probability p; a*2 = a*2H with probability 1−p; a*2H > a*2L; E(a*2) = a*1.
- Government budget constraints (equations (2.2)–(2.4)):
  - Period-1: a*1 = z1 + res*1 − (1 + r*) res*0.
  - Period-2: res*2 = 0 and a*2L = z2L − (1 + r*) res*1 (if aid low), a*2H = z2H − (1 + r*) res*1 (if aid high).
- Households:
  - Preferences (equation (2.5)): U = (ch1)^(1−σ)/(1−σ) + β [ p (ch2L)^(1−σ)/(1−σ) + (1−p) (ch2H)^(1−σ)/(1−σ) ] with β < 1 and σ relative risk aversion.
  - Budget constraints: ch1 = y + z1 (equation (2.6)); ch2L = y2 + z2L (2.7); ch2H = y2 + z2H (2.8).
- Output externalities (equation (2.9)):
  - y2 = y [ 1 + ξ (y − ch1)/y ] + αG (z1 / y), with ξ > 0 and αG > 0 capturing DD externality and spending productivity.

### Aggregate constraints and intuition
- Balance of payments / aggregate constraints (equations (2.10)–(2.12)):
  - res*1 = (1 + r*) res*0 + a*1 + y − ch1.
  - 0 = (1 + r*) res*1 + a*2L + y2 − ch2L.
  - 0 = (1 + r*) res*1 + a*2H + y2 − ch2H.
- Trade-offs:
  - Accumulating reserves increases exports-like net supply improving period-2 output/consumption.
  - Accumulating reserves reduces transfers in period 1, which may lower period-2 output if transfers are productive.

### Ramsey solution — core analytical results
- Euler equation (equation (2.13)):
  - (ch1)^−σ = β (1 + r* + ξ − αG) [ p (ch2L)^−σ + (1−p) (ch2H)^−σ ].
- Optimal policies without aid volatility (Proposition 1; assume a*2L = a*2H = a*2 and res*0 = 0):
  - Optimal consumption in period 1 (equation (2.14)):
    - ch1 = [ (1 + r*) / m ] [ (a*1 + a*2) / (1 + r*) + y / p ]  (as given in source; variables n, m, yp defined below).
  - Optimal fiscal spending and reserve accumulation rules (equations (2.15) and (2.16)):
    - z1 = [ (1 + r*) / m ] ≡ γ a*1 + [ 1 / m ] a*2 + [ (1 + r*) / m ] y/p − y  (rule form provided in source).
    - res*1 = [ (n + ξ − αG) / m ] ≡ ω a*1 − [ 1 / m ] a*2 − [ (1 + r*) / m ] y/p + y.
  - Definitions:
    - n ≡ [ β (1 + r* + ξ − αG) ]^{1/σ}
    - m ≡ 1 + r* + n + ξ − αG
    - y_p ≡ y + [ (1 + ξ − αG)/(1 + r*) ] y
- Interpretation:
  - Optimal rules are forward-looking and respond to both current and future aid (a*1 and a*2), not only contemporaneous aid.
  - Under special assumptions (quadratic utility, ξ = αG = y = 0, β (1 + r*) = 1), transfers fully smooth consumption per permanent income hypothesis.

### Comparative statics on response coefficients (Corollary 1)
- Under positive net return to reserve accumulation (1 + r* + ξ − αG > 0):
  - a) γ ∈ (0,1); ∂γ/∂ξ < 0; ∂γ/∂αG > 0.
  - b) ω ∈ (0,1); ∂ω/∂ξ > 0; ∂ω/∂αG < 0.
- Implications:
  - It is optimal to both spend and save portions of current aid (γ and ω strictly between 0 and 1).
  - Higher DD externalities (higher ξ) increase optimal reserve accumulation (∂ω/∂ξ > 0) and reduce transfers today (∂γ/∂ξ < 0).
  - Higher spending productivity (higher αG) increases transfers today (∂γ/∂αG > 0) and lowers reserve accumulation (∂ω/∂αG < 0).

### Aid volatility and precautionary motives
- Reintroducing stochastic aid (equation (2.1)) prevents closed-form analytic solutions; use simulations calibrated to a quarterly parameterization consistent with later quantitative calibration:
  - β = 0.98
  - σ = 2
  - r* = 0.04
  - y = 1 (normalized)
  - R*0 = 0.13
  - a*1 = 0.05
  - p = 0.5
  - Baseline: ξ = αG = 0
  - Alternative cases: ξ = 0.05 (DD externality); αG = 0.02 (productive fiscal spending)
  - Aid mean preserved: E(a*2) = a*1 with a*2H > a*2L and a*2L, a*2H ≥ 0
- Key simulation findings (as summarized in text and figures referenced in the source):
  - Higher aid volatility (measured by a*2H − a*2L) leads to more reserve accumulation (res*1) and less government transfers (z1) in period 1 — consistent across baseline, DD, and productive-spending cases.
  - Decomposition (equation (2.18)) of expected consumption growth:
    - E[(ch2 − ch1)/ch1] = β(1 + r* + ξ − αG)^{−1} / (σ β (1 + r* + ξ − αG)) [Intertemporal Motive] + (1/2)(1 + σ) E[ ((ch2 − ch1)/ch1)^2 ] [Precautionary Motive].
    - Increase in consumption variance raises expected consumption growth, which corresponds to lower period-1 consumption (precautionary saving).
  - Structural parameter effects under volatility:
    - DD externality (ξ = 0.05) → higher reserve accumulation and lower transfers in period 1 compared with baseline.
    - Productive spending (αG = 0.02) → lower reserve accumulation and higher fiscal transfers in period 1 compared with baseline.
  - Mechanisms:
    - DD externality raises the benefit of reserve accumulation for future output and raises future consumption volatility, both increasing precautionary saving.
    - Higher transfer productivity raises future output and lowers the effective return on reserves (in the simple model), reducing precautionary saving and raising present spending.

### Transition to the quantitative model
- The paper proceeds to an infinite-horizon quantitative model with richer features:
  - Heterogeneous households, two production sectors (nontraded and traded), explicit public investment that accumulates public capital as an asset.
  - Model structure follows the real side of DSGE frameworks used for LICs (Berg et al.(2010a); Shen et al.(2018)).
  - Superscripts N and T denote nontraded and traded sectors; superscript * denotes foreign goods; variables without time subscript indicate steady-state values.

*Source: wpiea2025154-source-pdf - 1.1 Related Literature*

### 3.1 Households

### 3.1 Households

### Household types and setup
- Households are of two types: a fraction η are savers (superscripta) and the remaining 1−η are hand-to-mouth households (superscripth). Only savers have access to asset and capital markets; hand-to-mouth households are liquidity-constrained.

### Savers
- Savers hold a portfolio of assets, including private capital, domestic government bonds, and foreign assets, and adjust savings to smooth consumption.
- Representative saver maximizes expected utility:
  - E0 Σt=0 βt [ (ca_t)^{1−σ} / (1−σ) − (la_t)^{1+ψ} / (1+ψ) ] ≡ Ua_t, (3.1)
  - where Et is the conditional expectation operator, β is the discount factor, σ and ψ are inverses of the intertemporal elasticity of substitution for consumption and the labor Frisch elasticity, respectively.
- Budget constraint:
  - ca_t + iN,a_t + iT,a_t + ba_t + s_t b∗,a_t + aci,a_t + s_t ac∗,a_t
    = (1−τ)[w_t la_t + rN_t kN,a_{t−1} + rT_t kT,a_{t−1}] + r_{t−1} b a_{t−1} + s_t r∗ b∗,a_{t−1} + s_t rm∗ + Πa_t + z_t, (3.2)
- Foreign assets pay constant gross interest r∗ and are subject to portfolio adjustment costs:
  - ac∗,a_t ≡ υ/2 (b∗,a_t − b∗,a)^2, where parameter υ governs capital account openness.
- Private investment adjustment costs:
  - aci,a_t ≡ κ/2 [ ((iN,a_t / kN,a_{t−1}) − δ)^2 kN,a_{t−1} + ((iT,a_t / kT,a_{t−1}) − δ)^2 kT,a_{t−1} ].
- Law of motion for private capital:
  - k_{j,a,t} = (1−δ) k_{j,a,t−1} + i_{j,a,t}, j ∈ {N,T}, (3.3)
  - where δ is the capital depreciation rate.
- Consumption and investment CES aggregation of non-traded and traded goods:
  - c_t = [ φ^{1/χ} (cN_t)^{(χ−1)/χ} + (1−φ)^{1/χ} (cT_t)^{(χ−1)/χ} ]^{χ/(χ−1)}, (3.4)
  - Price index of consumption basket:
    - 1 = [ φ (pN_t)^{1−χ} + (1−φ) (s_t)^{1−χ} ]^{1/(1−χ)}. (3.5)
- Nontraded consumption bundle is a CES aggregate of varieties:
  - cN_t = [ ∫_0^1 cN_t(i)^{(θ−1)/θ} di ]^{θ/(θ−1)}, (3.6)
  - θ is the elasticity of substitution between varieties.
- Savers supply labor to both sectors; total labor supply:
  - la_t = [ (φ_l)^{−1/χ_l} (la,N_t)^{(1+χ_l)/χ_l} + (1−φ_l)^{−1/χ_l} (la,T_t)^{(1+χ_l)/χ_l} ]^{χ_l/(1+χ_l)}, (3.7)
  - Aggregate real wage index:
    - w_t = [ φ_l (wN_t)^{1+χ_l} + (1−φ_l) (wT_t)^{1+χ_l} ]^{1/(1+χ_l)}. (3.8)
- All savers’ optimal choices and equilibrium conditions are provided in Appendix B (as referenced).

### Hand-to-Mouth Households
- Utility for hand-to-mouth households:
  - Uh_t = [ (ch_t)^{1−σ} / (1−σ) − (lh)^{1+ψ} / (1+ψ) ]. (3.9)
- Labor is supplied inelastically (lh fixed).
- Hand-to-mouth budget constraint (consumption determined each period by current income):
  - ch_t = (1−τ) w_t lh + s_t rm∗ + z_t. (3.10)
- Hand-to-mouth households do not have access to capital markets and do not receive asset income.

### Key structural and functional roles of households (model implications)
- Only savers determine aggregate investment, capital accumulation, debt, dividends, and portfolio adjustment costs:
  - For x ∈ {iN,iT,kN,kT,b,b∗,Π,aci,ac∗}: x_t = η x_{a,t}. (3.22)
- Aggregation with two household types for per-capita variables x ∈ {c,cN,cT,l,lN,lT}:
  - x_t = η x_{a,t} + (1−η) x_{h,t}. (3.21)
- Remittances and transfers are identical across households:
  - rm∗,a_t = rm∗,h_t = rm∗_t; z_t is common.

*Source: IMF Working Paper content (3.1 Households).*

### Appendix Afor the sample description).  The persistence and standard

### wpiea2025154-source-pdf - Appendix Afor the sample description).  The persistence and standard

### Calibration and key parameter values
- Aid process persistence and volatility calibrated to means across 23 countries: ρ_a = 0.72 and σ_a = 0.098.
- Reserves adjustment cost parameter: υ_res = 0.0006.
- Households’ portfolio adjustment cost parameter chosen for benchmark: υ = 500 (mid-point of the range examined in Shen et al. (2018)), implying fairly limited capital account mobility.
- Reserves adjustment cost calibrated to match volatility of reserves-to-GDP ratio.
- Benchmark calibration and alternative parameter values are used in sensitivity analysis to assess robustness.

### Optimal policy framework
- Policy maker chooses instruments to maximize households’ welfare subject to model equilibrium conditions (equations (B.1)–(B.32) in Appendix B).
- Lagrangian formulation (equation (5.1)): L0 = max_y_t E0 Σ_{t=0}^∞ β^t [U(y_{t+1},y_t,y_{t-1},u_t) − λ_t F(y_{t+1},y_t,y_{t-1},u_t)].
- Aggregate utility: U(y_{t+1},y_t,y_{t-1},u_t) = η U^a_t + (1−η) U^h_t, where U^a_t and U^h_t defined in equations (3.1) and (3.9).
- Policy instruments: government transfers z_t, public investment spending g^I_t, and reserves res^*_t.
- First-order conditions summarized by equation (5.2); second-order accurate solution obtained by solving these FOCs with the non-linear equilibrium conditions using perturbation methods.

### Baseline scenario: response to a 1 percent aid inflow shock
- Shock: exogenous 1 percent increase in aid inflows above steady-state levels.
- Optimal allocation is interior: reserves, government transfers, and public investment spending all rise following the aid shock to smooth consumption and mitigate Dutch disease (DD) effects.
- Rationale and mechanisms:
  - Large share of hand-to-mouth consumers → government emphasizes consumption smoothing via transfers.
  - Public investment raises production capacity, boosts future output and consumption, increases marginal product of private capital and labor, and stimulates private investment.
  - Ramsey planner saves part of aid as foreign reserves to mitigate DD effects and to provide longer-run funds via interest earnings and reserve de-cumulation.
  - Despite reserve accumulation, the real exchange rate appreciates (decline in s_t) because some aid is spent domestically; appreciation reduces domestic-currency value of foreign remittances to hand-to-mouth consumers and weakens traded goods competitiveness.
  - Aggregate effects: higher private consumption and public investment raise demand, especially for domestic nontraded goods, increasing total output; traded goods output contracts initially due to DD and persistent reduction in sectoral productivity z^T_t (equation (3.15)), but capital accumulation eventually offsets this.
  - Aggregate labor input falls due to wealth effect from savers’ higher income.
- Public investment functions as a saving instrument, building public capital that generates long-lasting positive macroeconomic effects consistent with development and resource revenue management literature.

### Alternative scenarios: isolating roles of instruments
- Two alternative scenarios analyze removing one of the saving channels.

- Alternative Scenario I — Excluding public investment (public investment held at deterministic steady state):
  - Ramsey planner uses only transfers and reserve accumulation.
  - Compared to baseline: larger increase in transfers and reserves; hand-to-mouth consumption increases more in short-to-medium run.
  - Without public capital increase: smaller overall economic expansion; output, consumption, and private investment remain significantly below baseline beyond initial years.
  - Reserve accumulation increases more than baseline but cannot substitute for public investment’s saving role; public investment is crucial to support longer-term growth and consumption.

- Alternative Scenario II — Excluding reserve accumulation (very high reserves adjustment cost υ_res = 1000 so reserves stock virtually constant):
  - Aid inflow channeled into transfers and public investment, which rise significantly more in the short run relative to baseline.
  - Consequences:
    - Much larger goods demand → nontraded goods production and real wage surge; private investment boom.
    - Higher household income and consumption for both household types.
    - The same aid inflow becomes more expansionary, but spending increase is short-lived and follows aid persistence.
    - Greater fiscal procyclicality; in negative aid shocks this implies sudden decline in government spending and severe contraction.
    - Stronger Dutch disease effects: larger currency appreciation and sharper short-run fall in traded-goods output (though rebound is quicker than in baseline).
  - Conclusion: public investment and reserve accumulation both save but cannot fully substitute; reserves mitigate demand spikes and dampen Dutch disease.

### Welfare implications and aid volatility
- Welfare measures:
  - Unconditional welfare for household types: W_a = E Σ_{t=0}^∞ β^t U^a_t and W_h = E Σ_{t=0}^∞ β^t U^h_t; aggregate W = η W_a + (1−η) W_h.
  - Welfare ranked across scenarios: highest aggregate welfare achieved under baseline using all three instruments.
  - Savers are indifferent between baseline and alternative II (excluding reserves) because higher capital income offsets instability from procyclical fiscal policy.
  - Scenario II yields higher welfare than scenario I for all households, underscoring importance of public investment.

- Welfare decomposition under aid volatility (baseline with all three instruments):
  - Vary σ_a (standard deviation of aid) from 0.08 to 0.16 and evaluate macroeconomic volatility (Figure 7) and mean deviations (Figure 8).
  - Higher σ_a → higher volatility in consumption, output, labor; strengthens precautionary saving motives → more public capital, reserve accumulation, and private capital accumulation → higher average levels of output and consumption.
  - Two opposing welfare effects:
    - Variance effects: higher consumption volatility → welfare reducing.
    - Mean effects: precautionary saving and asset accumulation → higher long-run consumption → welfare improving.
  - Decomposition methodology (equations (5.3)–(5.7)): compute equivalent steady-state consumption compensations ζ_j, with components ζ_{j,M} (mean effects) and ζ_{j,V} (variance effects).
  - Empirical findings:
    - Variance effects (ζ_{a,V}, ζ_{h,V}) are negative and become more negative as σ_a rises.
    - Mean effects (ζ_{a,M}, ζ_{h,M}) are positive and increase as σ_a rises.
    - Overall mean effects dominate; example magnitudes: increasing σ_a from 0.098 to 0.147 (50 percent higher aid volatility) is associated with savers’ welfare rising to 0.79 percent of steady-state consumption (from 0.35 percent) and hand-to-mouth households’ welfare rising to 1.11 percent (from 0.49 percent).
  - Interpretation: despite higher volatility costs, increased uncertainty can improve welfare when precautionary motives lead to strong capital accumulation and higher long-run consumption.

### Sensitivity analysis
- Robustness checks consider alternative parameter values reflecting LIC characteristics:
  - (i) smaller share of hand-to-mouth households,
  - (ii) stronger DD externality of traded goods production,
  - (iii) less productive public capital (lower α_G).
- Benchmark policy still assumes Ramsey planner optimally uses all three instruments.
- Note: alternative aid process AR(1) in levels (a^*_t = (1−ρ_a) a^* + ρ_a a^*_{t−1} + ε_t) examined qualitatively yields similar welfare implications of higher aid volatility.

*Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025154-source-pdf.pdf*

### 6.1 Share of Hand-to-Mouth Households

### 6.1 Share of Hand-to-Mouth Households

### Sensitivity test: share of hand-to-mouth households
- Benchmark assumption: a relatively large share of hand-to-mouth households, with 1−η = 0.75 (η = 0.25).
- Alternative calibration: η = 0.45 (consistent with evidence from richer SSA economies, Demirguc‑Kunt and Klapper (2012)).

### Key quantitative findings
- When the economy has a smaller share of hand-to-mouth households (η = 0.25 versus η = 0.45):
  - The Ramsey planner accumulates less of the aid inflow as reserves.
  - More aid is allocated to public investment and less to transfers.
  - Stronger initial public investment attracts more private investment, yielding a relatively larger output expansion and higher wages.
  - Higher wages support consumption of both savers and hand-to-mouth households.
  - The smaller increase in reserve accumulation reflects reduced need for precautionary savings because more households can self-insure via financial markets.

### 6.2 Dutch Disease (DD) Effects

### Model setup
- DD effects captured through a growth externality on traded goods production (equation (3.15)).
- Two values compared for ρ_z_T (AR(1) parameter in traded sector for TFP):
  - Benchmark: ρ_z_T = 0.1 (solid lines)
  - Stronger externality: ρ_z_T = 0.3 (dashed lines)

### Key quantitative findings
- With a stronger DD externality (ρ_z_T = 0.3):
  - Productivity of the traded goods sector declines by a larger magnitude in the short-to-medium run but recovers faster later.
  - The higher ρ_z_T amplifies both initial decline and subsequent recovery in traded output.
  - The model shows slightly more reserve accumulation and less spending, particularly less public investment spending.
  - The larger productivity decline reduces the magnitude of the private investment increase.
  - Weaker public and private demand lowers aggregate goods demand, resulting in a smaller increase in output under ρ_z_T = 0.3.
  - Smaller buildup in public and private capital leads to a smaller increase in the marginal product of labor; the real wage rate does not rise as much.
  - To compensate for weaker income and consumption growth, the optimal spending strategy indicates a slightly higher increase in transfers in the shorter run, relative to ρ_z_T = 0.1.
  - In equilibrium, the real exchange rate appreciates slightly less under the more persistent DD externality, aligning with slightly more reserve accumulation under ρ_z_T = 0.3.

### 6.3 Public Capital Productivity

### Model setup
- Public capital productivity captured by α_G in equations (3.11) and (3.14).
- Two calibrations compared:
  - Benchmark: α_G = 0.39 (solid lines)
  - Lower productivity (LIC estimate from Gupta et al. (2014)): α_G = 0.187 (dashed lines)
- Sensitivity also performed for public investment efficiency ǫ in equation (3.19).

### Key quantitative findings
- With lower public capital productivity (α_G = 0.187):
  - Stronger accumulation of reserves and a reduction in public investment spending in response to the aid shock.
  - Relative reduction in public investment leads to lower aggregate demand, less growth in private investment and labor demand, and a smaller increase in the real wage compared to the benchmark.
  - Less productive public capital diminishes the role of investment spending as a saving instrument over the long run.
  - Reduced accumulation of public and private capital leads to markedly lower output when public capital is less productive.
  - Increased savings in the form of larger reserves help sustain transfers to households over time—transfers increase more relative to the benchmark beyond the initial periods.
  - Despite increased transfer payments and a less negative labor response under α_G = 0.187, private consumption increases less than under the benchmark, primarily due to the much smaller increase in real wages.
- Decreasing public investment efficiency ǫ yields analogous results:
  - Reduced allocation of aid toward public investment and more savings as reserves.
  - Less capital accumulation, weaker output expansion following the aid shock, and a smaller increase in consumption.
  - The exercise corresponds to decreasing the efficiency of public investment over time, which can mimic public investment adjustment costs due to absorptive capacity constraints.

### Policy implications and overarching conclusions

### Optimal policy mix under volatile aid
- Directing some aid spending to public investment, alongside transfers and reserve accumulation, enhances welfare compared with allocating aid solely to transfers and reserves.
- Transfers directly support hand-to-mouth consumption; public investment builds public capital that stabilizes production and income over the long term.
- Public investment can function as a precautionary saving instrument, akin to reserve accumulation, in the presence of volatile aid flows.

### Sensitivity and guiding principles for LICs
- Key LIC features that affect optimal policy allocation:
  - Higher proportion of financially constrained consumers → reduced allocation to public investment and more reserve accumulation.
  - Intensified DD externalities → reduced allocation to public investment and more reserve accumulation.
  - Lower productivity of public capital → reduced allocation to public investment and more reserve accumulation.
- Higher aid volatility may require increased public investment, along with transfers and reserve accumulation:
  - Higher aid volatility increases negative variance effects on welfare (e.g., consumption volatility).
  - Precautionary saving motives raise savings in both public investment and reserve accumulation, producing more positive mean effects and supporting higher consumption over the medium term.

### Broader implications
- Insights on spending and saving through public investment can inform management of other volatile external incomes, such as natural resource revenues.
- Findings contribute to policy discussions on combining reserve accumulation with fiscal policy (public investment and transfers) and other policy tools in response to external shocks.

*Source: 6.1 Share of Hand-to-Mouth Households; 6.2 Dutch Disease (DD) Effects; 6.3 Public Capital Productivity; Conclusion — wpiea2025154-source-pdf*

### Appendix A Aid vs. Tax Revenue Volatility in the Data

### Appendix A Aid vs. Tax Revenue Volatility in the Data

### Data and measurement
- Sample composition:
  - 23 low income countries (LICs) and 53 lower middle income countries (LMICs) based on World Bank’s classification of 2021.
- Definitions:
  - Aid is measured by the net official development assistance (ODA) received as percent of GDP, where ODA is in current US dollars (Organisation for Economic Co-Operation and Development,2022), and GDP is in current US dollars (World Bank,2022).
  - Tax revenue is general government tax revenues as percent of GDP from the World Development Indicators database (World Bank,2022).
- Sample period:
  - Overall sample period set from 1980 to 2019, but the sample period for each country is the period for which both aid and tax revenues have no missing values.
- Note:
  - Sample periods of each country are available upon request.

### Volatility comparison: aid vs. tax revenues (key statistics)
- Volatility measured by the standard deviation of detrended series as a share of GDP.
- Three de-trending methods used:
  - First difference series
  - HP filter (with a smoothing parameter of 6.25 for annual data Ravn and Uhlig,2002)
  - Linear trend
- Proportion of countries in which aid volatility is higher than tax revenues’ volatility:
  - First difference series: LICs 78.3%   LMICs 92.5%
  - HP filter: LICs 82.6%   LMICs 94.3%
  - Linear trend: LICs 78.3%   LMICs 92.5%

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### Appendix B Equilibrium Conditions

### Key definitions and conventions
- Lagrangian multipliers and Tobin’s Q:
  - Let λ_a_t be the Lagrangian multiplier for the budget constraint and Q_N_t and Q_T_t for the laws of motion of capital in the two production sectors.
  - Tobin’s Q for k_N_t and k_T_t are q_N_t ≡ Q_N_t λ_a_t and q_T_t ≡ Q_T_t λ_a_t.

### First-order conditions and household conditions (exact expressions)
- First order condition (FOC) for private consumption savers, c_a_t:
  - λ_a_t = (c_a_t)^{−σ} (B.1)
- Labor supply, savers, l_a_t:
  - (l_a_t)^{ψ} = λ_a_t w_t (1−τ) (B.2)
- Euler equation, domestic bonds:
  - λ_a_t = β E_t [ λ_a_{t+1} r_t ] (B.3)
- Holdings of foreign assets:
  - λ_a_t s_t [ 1 + υ ( b^{∗,a}_t − b^{∗,a} ) ] = β E_t [ λ_a_{t+1} s_{t+1} r^{∗} ] (B.4)
- Hand-to-mouth households’ budget constraint:
  - c_h_t = (1−τ) w_t l_h + s_t rm^{∗} + z_t (B.5)

### Capital and investment FOCs (exact expressions)
- FOC for capital nontraded sector, k_N_t:
  - q_N_t = β E_t [ (λ_a_{t+1}/λ_a_t) [ (1−τ) r_N_{t+1} − (κ/2) ( (i_N_{t+1}/k_N_t − δ)^2 ) + κ (i_N_{t+1}/k_N_t − δ) (i_N_{t+1}/k_N_t) + q_N_{t+1} (1−δ) ] ] (B.6)
- FOC for capital traded sector, k_T_t:
  - q_T_t = β E_t [ (λ_a_{t+1}/λ_a_t) [ (1−τ) r_T_{t+1} − (κ/2) ( (i_T_{t+1}/k_T_t − δ)^2 ) + κ (i_T_{t+1}/k_T_t − δ) (i_T_{t+1}/k_T_t) + q_T_{t+1} (1−δ) ] ] (B.7)
- FOC for investment nontraded sector, i_N_t:
  - q_N_t = 1 + κ ( i_N_t / k_N_{t-1} − δ ) (B.8)
- FOC for investment traded sector, i_T_t:
  - q_T_t = 1 + κ ( i_T_t / k_T_{t-1} − δ ) (B.9)

### Capital accumulation and allocation (exact expressions)
- Capital accumulation nontraded sector, k_N_t:
  - k_N_t = (1−δ) k_N_{t-1} + i_N_t (B.10)
- Capital accumulation traded sector, k_T_t:
  - k_T_t = (1−δ) k_T_{t-1} + i_T_t (B.11)
- Labor supplied to the nontraded goods sector:
  - l_N_t = φ_l [ (w_N_t / w_t) ]^{χ_l} l_t (B.12)
- Labor supplied to the traded goods sector:
  - l_T_t = [1−φ_l] [ (w_T_t / w_t) ]^{χ_l} l_t (B.13)

### Production, pricing, and factor shares (exact expressions)
- Production of nontraded goods:
  - y_N_t = z_N_t ( l_N_t )^{α_N} ( k_N_{t−1} )^{1−α_N} ( k_G_{t−1} )^{α_G} (B.14)
- Pricing condition:
  - (1−ι) p_N_t = θ/(θ−1) mc_N_t (B.15)
  - where
    - mc_N_t = [ (1−α_N)^{−(1−α_N)} (α_N)^{−α_N} ] [ ( r_N_t )^{1−α_N} ( w_N_t )^{α_N} ( z_N_t )^{−1} ( k_G_{t−1} )^{−α_G} ]
- Factors’ share in nontraded goods production:
  - (1−α_N) w_N_t l_N_t = α_N r_N_t k_N_{t−1} (B.16)
- Production of traded goods:
  - y_T_t = z_T_t ( l_T_t )^{α_T} ( k_T_{t−1} )^{1−α_T} ( k_G_{t−1} )^{α_G} (B.17)
- Labor demand in the traded goods sector, l_T_t:
  - w_T_t l_T_t = (1−ι) s_t α_T y_T_t (B.18)
- Capital demand in the traded goods sector, k_T_t:
  - r_T_t k_T_{t−1} = (1−ι) s_t (1−α_T) y_T_t (B.19)

### Stochastic processes and public sector (exact expressions)
- TFP process:
  - log( z_T_t / z_T ) = ρ_{zT} log( z_T_{t−1} / z_T ) + κ log( y_T_{t−1} / y_T ) (B.20)
- Government budget constraint:
  - p_G_t ( g_C + g_I_t ) + z_t + s_t res^{∗}_t + s_t ac_res_t = tax_t + (1−r_{t−1}) b + s_t r^{∗} res^{∗}_{t−1} + s_t a^{∗}_t (B.21)
  - where ac_res_t ≡ υ_res/2 ( res^{∗}_t − res^{∗} )^2 and tax_t = τ ( w_t l_t + r_N_t k_N_{t−1} + r_T_t k_T_{t−1} ).
- Foreign aid process:
  - log( a^{∗}_t / a^{∗} ) = ρ_a log( a^{∗}_{t−1} / a^{∗} ) + ε_t (B.22)
- Law of motion for public capital:
  - k_G_t = (1−δ_G) k_G_{t−1} + ǫ g_I_t (B.23)

### Aggregation, market clearing, and external sector (exact expressions)
- Aggregate consumption:
  - c_t = η c_a_t + (1−η) c_h_t (B.24)
- Aggregate labor:
  - l_t = η l_a_t + (1−η) l_h (B.25)
- Real wage index:
  - w_t l_t = w_N_t l_N_t + w_T_t l_T_t (B.26)
- Total private investment:
  - i_t = i_N_t + i_T_t (B.27)
- Aggregate output:
  - y_t = p_N_t y_N_t + s_t y_T_t (B.28)
- Market clearing condition for nontraded goods:
  - y_N_t = (p_N_t)^{−χ_dN_t}, where d_N_t = φ ( c_t + i_t + ac_i_t ) + φ_G ( p_G_t )^{χ_g} t, (B.29)
  - with ac_i_t = κ/2 [ ( (i_N_t / k_N_{t−1} − δ)^2 / k_N_{t−1} ) + ( (i_T_t / k_T_{t−1} − δ)^2 / k_T_{t−1} ) ].
- Balance of payments:
  - c_t + i_t + p_G_t ( g_C + g_I_t ) + ac_i_t − y_t + s_t [ ac^{∗}_t + ac_res_t − rm^{∗} − (r^{∗} − 1) ( res^{∗}_{t−1} + b^{∗}_{t−1} ) ] = s_t [ a^{∗}_t − ( b^{∗}_t − b^{∗}_{t−1} ) − ( res^{∗}_t − res^{∗}_{t−1} ) ] . (B.30)
  - where ac^{∗}_t ≡ υ/2 ( b^{∗}_t − b^{∗} )^2 / η and ac_res_t ≡ υ_res/2 ( res^{∗}_t − res^{∗} )^2.
- The relative price of consumption goods:
  - 1 = [ φ ( p_N_t )^{1−χ} + (1−φ) ( s_t )^{1−χ} ]^{1/(1−χ)} (B.31)
- The relative price of government purchases:
  - p_G_t = [ φ_G ( p_N_t )^{(1−χ)} + (1−φ_G) ( s_t )^{(1−χ)} ]^{1/(1−χ)} (B.32)

*Ramsey-Optimal Fiscal Spending and Reserve Accumulation Policies under Volatile Aid  
Working Paper No. WP/2025/154*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025154-source-pdf.pdf_