## wpiea2026151-source-pdf ## Source details **Canonical URL:** [wpiea2026151-source-pdf](https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026151-source-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2026/english/wpiea2026151-source-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2026/english/wpiea2026151-source-pdf.pdf.json) --- ### Private sector structure and production - Production across three sectors follows Cobb-Douglas technologies: - q_{j,t} = a_{j} z_{t}^{\nu_{j}} k_{j,t+1}^{\phi_{j}} S_{j,t}^{\epsilon_{j}} (e_{b,t} L_{j,t})^{(1-\phi_{j}-\epsilon_{j})}. - q_{x,t} = a_{x} z_{t}^{\nu_{x}} k_{x,t+1}^{\phi_{x}} S_{x,t}^{\epsilon_{x}} H^{\rho} (e_{b,t} L_{x,t})^{1-\phi_{x}-\epsilon_{x}-\rho}. - q_{n,t} = a_{n} z_{t}^{\nu_{n}} k_{n,t+1}^{\phi_{n}} S_{n,t}^{\epsilon_{n}} (e_{n,t} e_{b,t} L_{n,t})^{(1-\phi_{n}-\epsilon_{n})}. - Inputs used: - All sectors: capital (k), low-skill labor (L), high-skill labor (S), public infrastructure (z). - Sector x additionally uses sector-specific factor H. - Low-skill human capital augmented by primary education quality/quantity e_{b}; formal sector low-skill productivity further influenced by efficiency wage effort e_{n}. - Physical capital (k), infrastructure (z), and educational capital (S_{b}, S_{u}) combine imported machinery (a_{im}) with formal and informal sector inputs a_{in} and a_{ij}. ### Factor pricing and firm optimization - Competitive firms set factor prices equal to marginal value products: - For sector x: (1-φ_{x}-ε_{x}-ρ) q_{x,t} / L_{x,t} = w_{x,t}, ε_{x} q_{x,t} / S_{x,t} = w_{s,t}, φ_{x} q_{x,t} / k_{x,t+1} = r_{x,t}, ρ q_{x,t} / H = r_{h,t}. - For nontraded formal sector n: P_{n,t}(1-φ_{n}-ε_{n}) q_{n,t} / L_{n,t} = w_{n,t}, P_{n,t} ε_{n} q_{n,t} / S_{n,t} = w_{s,t}, P_{n,t} φ_{n} q_{n,t} / k_{n,t+1} = r_{n,t}. - For sector j: P_{j,t}(1-φ_{j}-ε_{j}) q_{j,t} / L_{j,t} = w_{j,t}, P_{j,t} ε_{j} q_{j,t} / S_{j,t} = w_{s,t}, P_{j,t} φ_{j} q_{j,t} / k_{j,t+1} = r_{j,t}. ### Households, preferences, and constraints - Two household types: - Saving households: capital owners, skilled and low-skill formal labor; utility V = Σ_{t=0}^∞ β^t (c_{2,t}^{1-1/σ}-1)/(1-1/σ); budget constraint as in the source (schematic [8]). - Liquidity constrained households ("ex ante poor"): unemployed, low-skill in agriculture and informal sectors; consume entire income each period; utility V = Σ_{t=0}^∞ β^t (c_{1,t}^{1-1/σ}-1)/(1-1/σ). - Representative schematic identities retained exactly from the source for the saving household budget constraint and liquidity-constrained household income (see source for full algebraic expressions). ### Labor market, effort, and informality - Formal-sector low-skill effort: - e_{n,t} = g_{0} + g_{1} ln(w_{n,t}/P_{t}) + g_{2} ln(w_{j,t}/P_{t}) + g_{3} u_{t}. - Solow condition: g_{1} = 1. - Informal sector and agriculture: self-employment and family-run operations; flexible wages; no efficiency-wage channel. - Total labor supply fixed at \bar L_{t}; transitions across sectors driven by public investment in upper-level education i_{u,t} and formal-sector job openings. - Aggregate low-skill in x and j: - \bar L_{xj,t} = L_{x,t} + L_{j,t} = L_{x,0} + L_{j,0} + #_{xj}(S_{t} - S_{0}) + (L_{n,t} - L_{n,0}). ### Human capital and public capital dynamics - Educational capital: - Basic S_{b} and upper-level S_{u}. - (1+g) \tilde z_{t} = i_{z,t} + (1+%_{z,t}) \tilde z_{t+1}. - S_{b,t} = i_{b,t+6} + (1+%_{b}) S_{b,t+1}; S_{u,t} = i_{u,t+8} + (1+%_{u}) S_{u,t+1}. - Aggregate skill: S_{t} = S_{0} + ι_{1} (S_{u,t+1} - S_{u,0}); e_{b,t} = 1 + ι_{2} (S_{b,t+1} - S_{b,0}). ### Modeling role summary - Private sector block captures tradable, informal non-tradable, and formal non-tradable production with Cobb-Douglas technologies, competitive factor pricing, and endogenous formal-sector effort. - Household heterogeneity enables distributional analysis of fiscal adjustments and financing choices via factor incomes, asset returns, and transfers. - Public investment in S_{b}, S_{u}, and z are central channels through which public policy affects private-sector productivity and labor allocation. --- ### Market clearing, closure, and horizons - Markets with flexible wages/prices clear for skilled labor, low-skill labor in sector x∧j, and two non-traded goods. - Aggregate identities (as presented): - S_t = S_x,t + S_n,t + S_j,t - ̄L_xj,t = L_x,t + L_n,t - q_n,t = c_n,t + a_kn " q=j,n,x (i_q,t + AC_q,t) + a_zn (I_z,t + m_t) + a_sn i_s,t - q_j,t = c_j,t + a_kj " q=j,n,x (i_q,t + AC_q,t) + a_zj (I_z,t + m_t) + a_sj i_s,t - Model closure: growth in net foreign debt equals the current account deficit; compact accounting identity preserved as in the source. - Time horizons: - Medium term: 10-year period emphasized for transitional analysis. - Very long run: 50-year horizon used to represent steady-state / balanced growth path. - Cautions: - Mapping macro aggregates to decile outcomes calibrated from limited distributional moments; long-run decile differentials may not persist unchanged over 50 years. - Greater weight placed on 10-year results; 50-year comparisons treated cautiously. Sensitivity analysis provided in appendix. ### Distributional mapping methodology - Goal: map small set of macro outputs to plausible income growth by decile via growth incidence curve (GIC). - Baseline GIC specification: quadratic g(p) = a + b p + c p^2 with decile midpoints p_i = 0.05, 0.15, ..., 0.95. - Two aggregate constraints used to solve parameters: total growth G_y and bottom-̄y percent growth G_̄y (weighted-average and bottom-x conditions), leaving one free parameter resolved using economic-shape constraints. - Shape constraints employed: - Pareto tail: enforce g_10 ≃ g_9 (or adjust if unconstrained solution violates). - Lorenz convexity: ensure cumulative shares s_i^(1) remain ascending. - Discrete implementation: - y_i^(1) = y_i^(0) (1 + g_i); s_i^(1) = s_i^(0) (1 + g_i) / sum_j s_j^(0) (1 + g_j). - Gini approximated with trapezoidal rule over decile points. --- ### Scenario summaries and key quantitative outcomes Baseline scenario (no aid interruption) - Assumptions: - External grants continue at roughly end-2024 level into the medium term. - Concessional financing sustains government development programs. - 2025 national budget public investment plan: modest scale-up equivalent to about +0.5 percent of GDP each in education and infrastructure spending (relative to 2024), fully implemented with grant support. - Key macro outcomes: - Modest long-run increase in real GDP of about 4.2 percent (relative to a no-investment counterfactual). - Unemployment stabilizes around 4 percent. - Private investment gradually rises. - Distributional outcomes: - Starting Gini (2016 HIES benchmark): about 0.35. - Gini declines to 0.31 after 10 years and to around 0.30 in the 50-year steady state. - Annualized income growth for bottom two deciles in the baseline: about 8.1 percent and 4.7 percent, respectively. - Mechanisms: sustained public investment raises productivity, stimulates employment, and narrows wage gaps; aid complement keeps domestic fiscal burden low. Non-concessional financing scenario (borrowing-only) - Assumptions: - All grant aid permanently cut starting in 2025. - Lost financing fully replaced by external commercial loans and domestic debt issuance. - No significant domestic revenue mobilization or expenditure reform assumed. - Public investment plans continued via non-concessional financing. - Macroeconomic outcomes: - Short run: public investment sustained by new debt. - By year 5: real GDP falls about 0.8 percent below baseline. - Unemployment rises modestly; agricultural wages rise relatively faster but overall incomes diminish versus baseline. - Public debt stock rapidly increases; debt service consumes fiscal space. - Distributional outcomes: - Gini rises to about 0.45 (versus 0.35 baseline) in the medium term. - Poorest decile annual income growth drops to roughly 4.4 percent. - Top decile annual income growth around 1.9 percent. - Poverty and inequality worsen; borrowing-only response is substantially more regressive. - Policy caveat: model omits banking-sector, currency substitution, and dollarization dynamics; crowding-out channel may be weaker in practice for Liberia. Uncompensated aid-cut scenario - Assumptions: - Aid cut met with no compensatory financing; government cuts public investment and spending drastically. - Macroe outcomes: - By year 5: real GDP about 1.4 percent lower than with continued aid. - By year 5: consumption growth 2.7 percentage points lower; formal sector output declines by more than 1 percent. - Private investment drops sharply and remains subdued. - Labor and long-run effects: - By year 10: income growth for agriculture and informal activities around 2.3 percentage points lower than baseline. - At 20 years: GDP level significantly below baseline; productive capacity reduced. - Distributional outcomes (medium term and long run): - Medium-term Gini rises to about 0.41 (relative to 0.31 baseline reported elsewhere). - Medium-term poverty headcount climbs to 44.4 percent (baseline 31.6 percent; initial 33.9 percent). - Bottom-decile income growth may contract (example: poorest decile instead of growing at 3.9 percent per year experiences a contraction of about 6.7 percent). - Long-run: top decile income grows by about 7.7 percentage points; bottom eight deciles lose. - Net finding: uncompensated cut yields worst macro and distributional outcomes among scenarios. Domestic Revenue Mobilization (DRM) scenario - Strategy: "borrow-and-reform" — interim borrowing to smooth shock while implementing ambitious DRM to replace aid over time. - Liberia-specific targets: - Eventual increase in total tax revenue by about 13 pp of GDP (relative to 2025 GDP) once fully phased in. - Recent tax-to-GDP ratio assumed about 16 percent of GDP (IMF,2024). - Informal sector roughly 39 percent of GDP and 60 percent of employment (The World Bank,2023a). - Major DRM components (phased ~a decade): - Broadening GST base in agriculture and informal sector to raise C-efficiency by about 6.5 percentage points over ten years. - Introducing infrastructure user fees phased to eventually generate an additional 6.5 pp of GDP in revenue over ten years. - Macroe outcomes: - By year 5: consumption about 3.1 percent lower and real GDP 0.7 percent lower than baseline. - Informal labor incomes about 0.9 percent lower by year 10 versus baseline. - Public debt increases early but stabilizes and declines as revenues rise. - Long run: consumption and investment rebound; real GDP growth improves; public investment becomes more self-financed. - Distributional outcomes: - Gini rises to about 0.40 under DRM (baseline 0.31). - Bottom two deciles face income growth reductions about 15.7 percentage points and 8.9 percentage points lower than baseline over the medium term. - Top decile income growth in DRM about 2 percent (roughly 1.5 percentage points above baseline for the decile). - In long-run unconstrained projection: Gini falls to approximately 0.28 and bottom-decile income growth improves by 2–3 percentage points. - Feasibility note: achieving a 13 pp of GDP revenue increase is extremely ambitious; scenario assumes GST rate increases and transfer cuts in early years for fiscal credibility. --- ### Distributional sensitivity and robustness (GIC and damping) - Five GIC specifications explored: quadratic (baseline), piecewise-linear (three segments), quadratic without Pareto tail, log-linear, square-root-linear. - Damping parameter - in {0.85, 0.90, 0.95, 1.00} applied to cross-decile growth persistence. - -=0.95 implies half-life ≈ 14 years; -=0.90 ≈ 7 years; -=0.85 ≈ 4 years. - Empirical long-run caps applied in some exercises: projected Gini ceiling ∈ {0.60, 0.65}; top-decile share cap ∈ {0.45, 0.50}. - Bootstrap settings: - Medium term: 5,000 draws. - Long run: 1,000 draws. - Initial income shares perturbed via multiplicative log-normal noise with σ = 0.06. - Medium-term benchmark (quadratic with -=0.90, perturbed shares) results: - Scenario ranking invariant: Baseline → DRM → Non-concessional financing → Uncompensated cuts. - Gini outcomes (selected): Baseline declines from 0.35 to 0.33; Uncompensated cuts rises to about 0.384. - Poverty headcount and mean income (10-year horizon, selected values): - Baseline: HC: 33.9% → 30.1%; Mean: $1773 → $1952. - Non-concessional financing: HC: 33.9% → 41.1%; Mean: $1773 → $1762. - DRM: HC: 33.9% → 41.4%; Mean: $1773 → $1762. - Uncompensated cuts: HC: 33.9% → 41.9%; Mean: $1773 → $1749. - Long-run summary (quadratic, -=0.90, selected reported values): - Baseline: Mean income $1,773 → $3,619; Poverty headcount 33.9% → 4.9%. - DRM: Mean income $1,822; Poverty headcount 31.1% (long run). - Non-concessional financing: Mean income equivalent to DRM; Poverty headcount 50.7%. - Uncompensated cuts: Mean income virtually unchanged long run; Poverty headcount 57.0%. - Robustness conclusion: scenario ranking (best to worst distributional outcome) invariant across GIC specifications and damping values. ### Policy-relevant implications (model-consistent) - Preferred strategy within the model: phased DRM paired with interim borrowing ("borrow-and-reform") dominates pure non-concessional borrowing and uncompensated cuts on macro and distributional grounds. - Recommended reform directions: - Improve tax collection and broaden tax base. - Rationalize expenditure. - Preserve or retarget government transfers (social protection) to protect the poor during adjustment. - Seek additional concessional support and improve expenditure efficiency. - Risks emphasized: - Heavy reliance on non-concessional borrowing creates unsustainable debt and regressive adjustment later. - Uncompensated aid loss leads to prolonged stagnation, high poverty, and extreme inequality. - Model limitations noted: analysis is real-sector focused and excludes banking-sector dynamics, monetary transmission, and dollarization effects relevant for Liberia. *Source: Excerpts from the supplied chapter content of wpiea2026151-source-pdf.* ### 2.1 The private sector........................................4 ### 2.1 The private sector ### Production functions and inputs - Production across all three sectors follows Cobb-Douglas production functions: - q_{j,t} = a_{j} z_{t}^{\nu_{j}} k_{j,t+1}^{\phi_{j}} S_{j,t}^{\epsilon_{j}} (e_{b,t} L_{j,t})^{(1-\phi_{j}-\epsilon_{j})}. - q_{x,t} = a_{x} z_{t}^{\nu_{x}} k_{x,t+1}^{\phi_{x}} S_{x,t}^{\epsilon_{x}} H^{\rho} (e_{b,t} L_{x,t})^{1-\phi_{x}-\epsilon_{x}-\rho}. - q_{n,t} = a_{n} z_{t}^{\nu_{n}} k_{n,t+1}^{\phi_{n}} S_{n,t}^{\epsilon_{n}} (e_{n,t} e_{b,t} L_{n,t})^{(1-\phi_{n}-\epsilon_{n})}. - All sectors use capital (k), low-skill labor (L), high-skill labor (S), and public infrastructure (z). - Sector x additionally uses a sector-specific factor H (e.g., land or natural resource). - Human capital of low-skill labor is augmented by primary education quality and quantity e_{b}; in the formal sector, low-skill productivity is also influenced by efficiency wage effort e_{n}. - Physical capital (k), infrastructure (z), and educational capital (S_{b}, S_{u}) combine imported machinery (a_{im}) with formal and informal sector inputs a_{in} and a_{ij}, i↔{k,z,S_{b},S_{u}}. ### Factor pricing and firm optimization - Competitive firms maximize profits by equating marginal value products to factor prices: - (1-φ_{x}-ε_{x}-ρ) q_{x,t} / L_{x,t} = w_{x,t}, ε_{x} q_{x,t} / S_{x,t} = w_{s,t}, φ_{x} q_{x,t} / k_{x,t+1} = r_{x,t}, ρ q_{x,t} / H = r_{h,t}. - P_{n,t}(1-φ_{n}-ε_{n}) q_{n,t} / L_{n,t} = w_{n,t}, P_{n,t} ε_{n} q_{n,t} / S_{n,t} = w_{s,t}, P_{n,t} φ_{n} q_{n,t} / k_{n,t+1} = r_{n,t}. - P_{j,t}(1-φ_{j}-ε_{j}) q_{j,t} / L_{j,t} = w_{j,t}, P_{j,t} ε_{j} q_{j,t} / S_{j,t} = w_{s,t}, P_{j,t} φ_{j} q_{j,t} / k_{j,t+1} = r_{j,t}. ### Household types and demand side - Two household types: - Saving households: capital owners, skilled and low-skill formal labor. Income includes net-of-tax factor earnings, returns from foreign and domestic bonds, shares of remittances, land rents, and government transfers. They maximize an iso-elastic utility V = Σ_{t=0}^∞ β^t (c_{2,t}^{1-1/σ}-1)/(1-1/σ) subject to budget constraint [8]. - Liquidity constrained households (the "ex ante poor"): unemployed, low-skill workers in agriculture and informal sectors. They consume their entire income each period; utility V = Σ_{t=0}^∞ β^t (c_{1,t}^{1-1/σ}-1)/(1-1/σ). - Saving household budget constraint (schematic, as in [8]): - P_{t} b_{t} - b_{f,t} = (1+f_{w})(w_{n,t} L_{n,t} + w_{s,t} S_{t+1}) + Σ_{q=j,n,x} [r_{q,t} + f_{q}(r_{q,t} + %P_{q,t})] k_{q,t+1} + (1+f_{h})(1+() r_{h,t} H + (1+a_{t}) T_{t}) + (1+r_{f})/(1+g) b_{f,t+1} + (1+r_{t+1})/(1+g) P_{t} b_{t+1}/P_{k,t} - Σ_{q=j,n,x} (i_{q,t} + AC_{q,t}) - 2(b_{f,t} - \bar b_{f})^{2} / P_{c,t} c_{2,t} + μ_{t} z_{t+1}. - Liquidity constrained household income identity (schematic, as in [10]): - P_{c,t} c_{1,t} = (1+f_{wx})(w_{x,t} L_{x,t} + r_{h} H + (1+f_{wj}) w_{j,t} L_{j,t} + a_{t} T_{t}). ### Labor market, effort, and informality - Formal-sector low-skill work effort e_{n,t} depends on own wage, informal wage, and unemployment rate u: - e_{n,t} = g_{0} + g_{1} ln(w_{n,t}/P_{t}) + g_{2} ln(w_{j,t}/P_{t}) + g_{3} u_{t}. - g_{1} = (∂e_{n,t}/∂(w_{n,t}/P_{t})) (w_{n,t}/P_{t} / e_{n,t}) = 1 (Solow condition). - Informal sector and agriculture: dominated by self-employment and family-run operations; no efficiency wage considerations and flexible wages. - Total labor supply fixed at \bar L_{t}; transitions between sectors driven by: - public investment in upper-level education i_{u,t} (upgrades low-skill to high-skill), - new formal-sector job openings. - Low-skill labor in x and j sectors evolves as: - \bar L_{xj,t} = L_{x,t} + L_{j,t} = L_{x,0} + L_{j,0} + #_{xj}(S_{t} - S_{0}) + (L_{n,t} - L_{n,0}). ### Human capital and sectoral dynamics - Two types of educational capital: basic (S_{b}) and upper-level (S_{u}). - Dynamics of public and private capital stocks: - (1+g) \tilde z_{t} = i_{z,t} + (1+%_{z,t}) \tilde z_{t+1}. - S_{b,t} = i_{b,t+6} + (1+%_{b}) S_{b,t+1}; S_{u,t} = i_{u,t+8} + (1+%_{u}) S_{u,t+1}. - Aggregate skill S_{t} = S_{0} + ι_{1} (S_{u,t+1} - S_{u,0}); e_{b,t} = 1 + ι_{2} (S_{b,t+1} - S_{b,0}). ### Summary of modeling role for private sector - The private sector block captures production across tradable, informal non-tradable, and formal non-tradable sectors with Cobb-Douglas technologies, factor pricing under competition, and endogenous labor effort in the formal sector. - Household heterogeneity (saving vs. liquidity constrained) allows analysis of distributional impacts of fiscal adjustments and financing choices by tracing factor income, asset returns, and transfers across groups. - Human capital accumulation (S_{b}, S_{u}) and public infrastructure (z) are central channels by which public investment affects private-sector productivity and sectoral labor allocation. *Source: 2.1 The private sector — extracted from the provided chapter content.* ### 2.3 Market Clearing Conditions ### 2.3 Market Clearing Conditions ### Market clearing and accounting closure - Flexible wages and prices equate demand to supply in: - the market for skilled labour, - the market for low-skill labour in sectorx↑j, - the markets for the two non-traded goods. - Aggregate market-clearing identities (as presented): - S_t = S_x,t + S_n,t + S_j,t - ̄L_xj,t = L_x,t + L_n,t - q_n,t = c_n,t + a_kn " q=j,n,x (i_q,t + AC_q,t) + a_zn (I_z,t + m_t) + a_sn i_s,t - q_j,t = c_j,t + a_kj " q=j,n,x (i_q,t + AC_q,t) + a_zj (I_z,t + m_t) + a_sj i_s,t - Model closure: growth in net foreign debt equals the current account deficit. Adding public and private budget constraints yields the accounting identity (compactly represented in the source): - #d_t + #dc_t + #b_f,t = r_d ↑g 1+g d_t↑1 + r_dc ↑g 1+g dc_t↑1 + r_f ↑g 1+g b_f,t↑1 + ) 2 (b_f,t ↑ ̄b_f)^2 + P_z,t (I_z,t + m_t) + P_s,t i_s,t + P_t (c_1,t + c_2,t) + P_k,t " q=j,n,x (i_q,t + AC_q,t) ↑P_n,t q_n,t ↑P_j,t q_j,t ↑q_x,t ### Long-run concept and horizons - Long run defined by convergence properties rather than calendar time. - Two interpretation horizons emphasized: - 10-year period: most short- and medium-term adjustments in capital accumulation, labor reallocation, and income distribution occur. - 50-year very long run: the new steady state or balanced growth path once reforms and shocks have fully manifested. - The framework does not impose formal constraints on the long run; 10 years used for transitional analysis, 50 years for fully realized steady-state comparisons. - Caution: mapping from macro aggregates to decile-level outcomes is calibrated from a limited set of distributional moments; growth differentials across deciles may not persist unchanged over five decades. - Greater weight placed on medium-term (10-year) results; very-long-run distributional comparisons treated more cautiously. - Appendix presents sensitivity analysis under alternative specifications, varying growth damping assumptions, and empirical bounding of long-run Gini values. ### Distributional assessment approach (overview) - Goal: generate a plausible profile of income growth by decile from a small set of macro outputs. - Three main steps: 1. Specify a functional form for the growth incidence curve g(p). 2. Determine parameters of g(p) using aggregate constraints (total growth G_y and bottom- ̄y percent growth G_̄y) and initial distribution data. 3. Impose additional economic-theory constraints (properties of Lorenz curves and heavy-tailed income distributions). - Given g(p), compute the implied new income distribution and Gini coefficient. ### Quadratic growth incidence curve specification - Choice: quadratic functional form - g(p) = " + & p + . p^2, 0 ⇐ p ⇐ 1 - p represents population percentile (p = 0 poorest, p = 1 richest). - Discrete implementation by decile: - decile i assigned g_i = " + & p_i + . p_i^2 with p_i = 0.05, 0.15, ..., 0.95 for deciles 1..10. - Rationale: - Quadratic is a transparent, parsimonious choice balancing flexibility and risk of overfitting. - AppendixA.2 considers piecewise-linear, log-linear and square-root linear GICs and robustness checks. ### Using aggregate constraints to solve for GIC parameters - Unknown parameters: (", &, .). Two linear equations provided by: - Weighted-average condition: sum_{i=1}^{10} s_i^(0) (1 + g_i) = 1 + G_y, which implies sum_{i} s_i^(0) g_i = G_y. - Bottom-̄y condition for bottom x deciles: sum_{i=1}^{x} s_i^(0) (1 + g_i) = (1 + G_̄y) sum_{i=1}^{x} s_i^(0), which implies sum_{i=1}^{x} s_i^(0) g_i = S^(0)_̄y G_̄y. - These yield two linear constraints in ", &, .. A third condition from economic theory is needed to determine uniqueness. ### Pareto tail and Lorenz convexity constraints - Pareto tail constraint: - Empirical incomes have heavy upper tails approximated by Pareto beyond high percentiles. - Impose non-decreasing GIC at high percentiles: top decile's growth >= ninth decile's growth (g_10 ≃ g_9). - For quadratic g(p), derivative g'(p) = & + 2. p; at p = 1 the condition is & + 2. ≃ 0. - Implementation: choose the free parameter so that g_10 = g_9 if the unconstrained solution has g_10 < g_9. - Lorenz convexity constraint: - Lorenz curve must remain convex; cumulative income shares must stay ordered (no lower decile overtakes a higher decile). - Ensure g(p) does not produce pathological jumps for low deciles; adjust . so that cumulative shares s_i^(1) remain ascending (s_1^(1) < s_2^(1) < … < s_10^(1)). - Practical rule: pick . to satisfy Pareto tail condition with equality unless unconstrained solution already meets it; fine-tune . marginally if needed for Lorenz convexity. - Result: unique g(p) satisfying aggregate constraints and g_10 = g_9 that typically preserves Lorenz convexity. ### Projecting decile incomes and inequality measures - Given g(p) and g_1..g_10: - Projected average income for decile i: y_i^(1) = y_i^(0) (1 + g_i). - Total income of decile i: Y_i^(1) = Y_total^(0) · s_i^(0) · (1 + g_i). - New share: s_i^(1) = s_i^(0) (1 + g_i) / sum_{j=1}^{10} s_j^(0) (1 + g_j), with sum_i s_i^(1) = 1. - Lorenz curve: - Assume within-decile uniform income; Lorenz curve between decile points is piecewise linear. - Cumulative share for bottom k deciles: L_k^(1) = sum_{i=1}^{k} s_i^(1). - Gini coefficient approximation: - Gini = 1 - 2 * (sum_{k=0}^{10} (L_k^(1) + L_{k+1}^(1)) · 0.1 / 2) (trapezoidal approximation over decile points). - Coarse decile data yield an approximate Gini. - By construction, s_i^(1) remain ascending, enabling comparison with initial Gini and calculation of poverty measures via a specified poverty line. ### Baseline scenario (no aid interruption) - Assumptions: - Liberia continues to receive external grants at roughly the end-2024 level into the medium term. - Concessional financing remains available to sustain government development programs. - The 2025 national budget’s public investment plan entails a modest scale-up equivalent to about +0.5 percent of GDP each in education and infrastructure spending (relative to 2024), and can be fully implemented thanks to ongoing grant support. - Key macro outcomes: - Modest long-run increase in real GDP of about 4.2 percent (relative to a no-investment counterfactual). - Unemployment stabilizes around 4 percent. - Private investment gradually rises. - Distributional outcomes: - Starting Gini (2016 HIES benchmark): about 0.35. - Gini declines to 0.31 after 10 years. - Gini falls further to around 0.30 in the very long run (50-year steady state). - Annualized income growth for bottom two deciles in the baseline: about 8.1 percent and 4.7 percent, respectively. - Mechanisms: - Sustained public investment in education and infrastructure raises productivity economy-wide, stimulates employment, and allows real wages to grow. - Productivity gains concentrated in sectors employing lower-skilled and informal workers, narrowing wage gaps. - With aid complementing development spending, fiscal burden on domestic resources remains low; only modest policy adjustments needed to maintain debt sustainability. - Interpretation notes: - The 2016 HIES benchmark is a calibration anchor, not a precise 2025 distributional estimate. - Liberia has experienced major shocks since 2016 (pandemic, food and inflation episodes); level and shape of current inequality are uncertain. - Distributional results are scenario comparisons around a common starting distribution; medium-term results are weighted more heavily than very-long-run projections. ### Non-concessional financing scenario (first aid-cut scenario) - Assumptions: - All grant aid is permanently cut starting in 2025. - Authorities fully replace lost financing with a combination of external commercial loans and domestic debt issuance. - No significant domestic revenue mobilization or expenditure reform is assumed to offset the aid shortfall. - Public investment plans are attempted to be continued by relying on non-concessional financing (bond markets or domestic banks). - Purpose: - Illustrate short-term benefits of maintaining spending versus medium- and long-term risks to debt sustainability and equity when concessional aid is replaced by non-concessional financing. *Source: wpiea2026151-source-pdf - 2.3 Market Clearing Conditions* ### 5.1 Macroeconomic Outcomes ### 5.1 Macroeconomic Outcomes ### Macroeconomic outcomes under non-concessional borrowing (borrowing-only scenario) - Short-run: government sustains planned public investment by replacing grants dollar-for-dollar with new debt, avoiding an abrupt collapse in public spending. - Medium-term fiscal dynamics: - Public debt stock rapidly increases as external commercial borrowing and domestic bond issuance accumulate. - Additional interest payments and amortization create mounting fiscal pressure, forcing sharper fiscal adjustments a few years into the scenario. - Government measures required to stabilize debt include raising indirect taxes and implementing deep cuts in government transfers; these measures are sharper than in the baseline and stem from reliance on non-concessional financing. - Debt servicing quickly consumes fiscal space, crowding out benefits of investment. - Real economy: - Real GDP falls about 0.8 percent below baseline by year 5. - Unemployment rises modestly as formal and informal sectors contract; the traditional (agriculture) sector sees a slight uptick as some peri-urban workers return to agriculture and farm-to-market roads benefit farming communities. - Real wages in agriculture rise faster than in other sectors, but overall employment and income still diminish compared to the no-aid-cut baseline. - Private sector effects: - Higher taxes depress consumption; transfer cuts reduce household income. - Heavy domestic borrowing drives up interest rates in the model, crowding out private investment. - Caveat: model is a real-sector framework and does not include banks, financial markets, or currency substitution. Liberia is highly dollarized and has a shallow domestic debt market, so the standard crowding-out channel may be weaker in practice than the model implies. - Fiscal sustainability risk: - Debt-to-GDP ratio climbs more rapidly; debt service as a share of revenue spikes. - Government does not undertake major tax base expansion in this scenario, so debt service must be met largely by expenditure cuts and a limited increase in the GST. - Liberia’s limited space to absorb shocks means a permanent aid loss met with non-concessional borrowing could push the country into debt distress. - Even after stringent consolidation (GST hike and transfer cuts), debt remains higher and more vulnerable to shocks, increasing risk of a debt crisis. - Net assessment: any short-run continuity in public investment is offset by debt accumulation and subsequent harsh fiscal measures needed to restore fiscal sustainability. ### Distributional impacts of non-concessional borrowing - Regressivity channels: - Indirect tax (GST) increase disproportionately burdens poor households who spend a larger share of income on consumption. - Cuts to government transfers and subsidies reduce income or increase costs for lower-income households. - Contraction in employment disproportionately affects informal and unskilled workers. - Upper-income households are relatively advantaged by higher interest yields on government bonds and savings, offsetting their share of GST increases. - Distributional outcomes: - Gini coefficient rises to about 0.45 (versus 0.35 baseline) in the medium term. - Income growth for the poorest decile drops to roughly 4.4 percent annually (much lower than the 8 percent+ under baseline). - Top decile’s income grows around 1.9 percent per year. - Poverty and inequality worsen significantly compared to baseline; borrowing-only response is substantially more regressive. - Policy implication: higher consumption taxes and transfer cuts amplify disparities; absent targeted mitigation, fiscal adjustment tends to be regressive. ### Overview of the uncompensated aid cut scenario - Definition: aid cut met with no compensatory financing, drastic cuts to public investment and spending, minimal gains from domestic revenue mobilization. - Outcome: sustained reduction in development spending and a smaller budget consistent with the permanent loss of external grants; serves as a worst-case benchmark for full impact of aid withdrawal when no offsets are available. --- ### 6 Domestic Revenue Mobilization (DRM) scenario ### Strategy and design - Approach: "borrow-and-reform" — use initial borrowing to smooth the shock, while implementing an ambitious DRM strategy to replace aid over time. - Rationale: tax reforms and administrative capacity buildup require time; interim deficit financing is used to maintain public investment in early years. - Liberia-specific assumptions: - Scenario targets an eventual increase in total tax revenue by about 13 pp of GDP (relative to 2025 GDP) once fully phased in. - Assumes recent tax-to-GDP ratio was about 16 percent of GDP (IMF,2024). - Recognizes the need to broaden the tax net into the informal sector (informal sector roughly 39 percent of GDP and 60 percent of employment (The World Bank,2023a)) and improve compliance. - Major DRM components (phased over roughly a decade): - Broadening the GST base in agriculture and the informal sector to raise C-efficiency by about 6.5 percentage points over ten years. - Introducing infrastructure user fees phased to eventually generate an additional 6.5 pp of GDP in revenue over ten years. - Distributional caveat: GST base broadening and user fees can be regressive if poorly designed; government transfers partly offset the burden in the model. ### Macroeconomic outcomes under DRM - Early years: - Government resorts to external commercial loans and domestic borrowing to keep public investment funded, preventing abrupt fiscal contraction. - Simultaneously implements tax measures and new fees that gradually raise revenue. - Medium term: - By year 5, consumption is about 3.1 percent lower and real GDP 0.7 percent lower than baseline. - Both formal and informal sector output fall slightly below baseline in the medium term; growth of informal labor incomes lags (about 0.9 percent lower by year 10). - Public debt has increased early on but begins to stabilize as domestic revenues rise. - Long term: - After about a decade, additional revenue substantially reduces financing gaps; public debt stabilizes and eventually declines as a share of GDP, containing debt-servicing costs. - Consumption and investment rebound; real GDP growth improves with a more sustainable fiscal outlook. - Increased tax revenue helps self-finance public investment programs; sectoral outputs and real wages rise across the economy in the long run. - Long-run outcome still worse than baseline but less scarring than the borrowing-only scenario. - Feasibility and constraints: - Achieving a 13 percentage point increase in the revenue-to-GDP ratio is extremely ambitious and requires deep tax administration reforms and political will. - Scenario includes raising the GST rate and cutting government transfers in early years to ensure fiscal credibility. ### Distributional impacts of DRM - Comparative outcomes: - DRM produces more moderate distributional effects than the borrowing-only scenario because adjustment is spread across borrowing and taxation and over time. - All aid-cut responses remain inferior to the no-aid-cut baseline. - Key numbers: - Gini coefficient rises to about 0.40 under DRM (baseline 0.31). - Bottom two deciles see income growth rates about 15.7 percentage points and 8.9 percentage points lower than in the baseline path over the medium term. - Top decile income growth in DRM is about 2 percent or roughly 1.5 percentage points above baseline for the decile. - In long-run unconstrained projection, Gini falls to approximately 0.28 under DRM. - Bottom-decile income growth improves by 2–3 percentage points in the long run under DRM. - By contrast, borrowing-only case shows bottom decile income growth significantly lower by 8.6 percent relative to baseline. - Mechanisms for more equitable outcomes: - Diversifying the fiscal response avoids placing the entire adjustment on consumption taxes and transfer cuts; burden can be shifted to broader taxation including potentially more progressive taxes. - Properly designed user fees and restored transfers over time can help reverse some inequality increases once revenues and debt are under control. - Policy message: pairing moderate borrowing with aggressive domestic revenue mobilization yields a more equitable and fiscally sustainable path than relying solely on non-concessional borrowing. *Source: Excerpt from IMF working paper chapter section "5.1 Macroeconomic Outcomes" and subsequent sections in the supplied PDF content.* ### 7.1 Macroeconomic Outcomes ### 7.1 Macroeconomic Outcomes ### Macroeconomic consequences of an uncompensated aid cut - With grant revenues gone and no alternative financing, the government must slash expenditures, especially capital investments, to avoid unsustainable deficits. - Key public investment projects in infrastructure, education, health, etc., are scaled back or halted to align spending with available financing. - Over the first five years, the model projects a cumulative contraction relative to the baseline: - By year 5, real GDP is about 1.4 percent lower than it would have been with continued aid. - Consumption growth is 2.7 percentage points lower by year 5. - Formal sector output declines by more than 1 percent relative to baseline. - Private investment drops sharply and remains subdued throughout the projection period. - Reduced public investment, which normally crowds in private investment by providing complementary infrastructure, can trigger a vicious cycle of declining private-sector activity and deteriorating aggregate demand. ### Labor market, productivity, and long-run growth potential - Labor market conditions deteriorate: - Job creation slows markedly as public investment is scaled back and the private sector contracts. - Real wage growth stagnates across all sectors; workers see little to no income growth over time. - Labor mobility declines, particularly in the informal sector, making it harder for workers to move from low-productivity rural work to higher-productivity urban or formal jobs. - By year 10, income growth for workers in agriculture and other informal activities is around 2.3 percentage points lower than in the baseline. - The cumulative effect is lower future growth potential and a substantially reduced productive capacity: - By the end of the 20-year horizon, the level of GDP is significantly below the baseline path and the economy’s productive capacity has suffered. ### Debt service, maintenance, and fiscal space - Legacy obligations constrain adjustment: - Existing debt obligations and essential maintenance must still be paid from a now-smaller budget. - In the absence of new aid or concessional loans, a larger share of revenues must go to servicing existing debt. - Basic recurrent expenditures are assumed maintained to prevent degradation of the capital stock. - These legacy costs consume fiscal space, leaving very little room for new growth-enhancing investments and producing a low-level equilibrium: low investment, low productivity growth, and persistent poverty. - The model highlights a self-reinforcing cycle: low public spending → weak private response → stagnant or declining incomes → limited domestic revenue → continued low public spending. Breaking this cycle without external support would be exceedingly difficult. ### Distributional impacts (section 7.2, summarized here) - The uncompensated-cut scenario produces the worst distributional outcomes among the scenarios examined: - Gini coefficient in the medium term remains at 0.41 relative to 0.31 at baseline. - Income growth in the bottom decile is 17.2 percentage points below the baseline. - Income growth in the upper decile is about 1.5 percentage points above the baseline. - Poverty headcount climbs to 44.4 percent, relative to 31.6 percent in the baseline and 33.9 percent in initial headcount. - Long-run distributional outcomes: - Median income growth becomes negative by about 1.6 percentage point below baseline. - Income growth for the two lowest deciles is reduced by 10.6 and 8.1 percentage points, respectively, compared to the baseline. - Example implication: instead of the poorest decile’s income growing at 3.9 percent per year, it experiences a contraction of about 6.7 percent. - At steady state, top decile income grows by about 7.7 percentage points, reflecting a widening premium to capital and skilled labor. - Dynamics underlying distributional shifts: - Cuts to education reduce the supply of skilled labor over time. - Reduced infrastructure investment stalls job creation in modern sectors, limiting pathways for productivity improvements for low-skill labor. - Fiscal contraction becomes neutral or regressive as resources are diverted to debt service and essential costs, disproportionately harming the poor. - Overall finding: a permanent, uncompensated aid cut would push the economy into prolonged stagnation and extreme inequality, underscoring the importance of continued development aid or credible alternative financing/reforms. ### Policy-relevant implications from the modeling framework - The choice of policy response determines whether an aid cut leads to a temporary setback or deep, lasting regression. - Within the model, a phased domestic revenue mobilization (DRM) strategy dominates both pure non-concessional borrowing and uncompensated cuts on macroeconomic and distributional grounds. - Recommended reform directions (model-consistent): - Improve tax collection and broaden the tax base. - Rationalize expenditure. - Preserve or retarget government transfers (social protection) to protect the poor during adjustment. - Seek additional concessional support and improve expenditure efficiency where possible. - Risks and cautions: - Heavy reliance on non-concessional borrowing is high-risk: it may avert an immediate collapse in investment but creates unsustainable debt and shifts adjustment burdens into the future. - Limited borrowing space in low-income countries makes expensive debt particularly dangerous, worsening inequality and fiscal strain. - Some beneficial options (cutting wasteful spending, improving public financial management) are not fully modeled but could generate credibility gains and make fiscal resources more productive. - Model scope limitations: - The analysis is real-sector focused and excludes detailed banking-sector dynamics, monetary transmission, and dollarization effects, which are relevant for Liberia but lie outside the model. *Source: Excerpt from the chapter “7.1 Macroeconomic Outcomes” and related sections in the supplied IMF chapter content.* ### 1. Baseline ### wpiea2026151-source-pdf - 1. Baseline ### Overview - The annex conducts a systematic distributional sensitivity analysis to assess how much distributional results depend on the growth incidence curve (GIC) functional form, constraining assumptions, persistence of cross-decile wedges, empirical caps on inequality, and initial income-share uncertainty. - The 10-year horizon is treated as the medium term; the 50-year horizon represents outcomes across steady states after full convergence. - Source:Authors’ simulations. ### Alternative GIC specifications (five specifications considered) - Quadratic (baseline) - Functional form: g(p) = a + b p + c p^2 with p evaluated at decile midpoints (p_i = 0.05, 0.15, ..., 0.95). - Over-identified: three parameters, two moment conditions (G_y and G_80), one free degree used to satisfy three nonlinear inequality constraints: - Pareto tail: g_10 ≃ g_9 + % with % drawn from {0, 0.5, 1.0} percentage points. - Lorenz convexity. - Gini and top-decile-share ceilings (binding only in long-run exercise). - Piecewise-linear (three segments) - Knots at p = 0.50 and p = 0.80: - g(p) = a1 + b1 p if p ≤ 0.50 - g(p) = a2 + b2 p if 0.50 < p ≤ 0.80 - g(p) = a3 + b3 p if p > 0.80 - Four free parameters (a1, b1, b2, b3). Same Pareto, Lorenz, and cap constraints apply. - Quadratic without Pareto tail - Same as quadratic but without the Pareto tail condition; Lorenz convexity and empirical Gini cap constraints remain (highest historically observed Gini in recent years is 0.65). - Log-linear (exactly identified) - Functional form: g(p) = a + b ln(p). - Exactly identified by the two aggregate moments; no shape constraints. - Log steepness: ln(0.05) ⇑↑ 3.0 and ln(0.95) ⇑↑ 0.05. - To prevent extreme long-run spreads, the spread between highest and lowest decile growth rates is capped at 10 percentage points in long horizons; when raw spread exceeds bound, the profile is compressed toward its mean while preserving log shape: g_i ↖ ḡ + 0.10 max(g) ↑ min(g) ⇒ (g_i ↑ ḡ). - Square-root-linear (exactly identified) - Functional form: g(p) = a + b √p. - Concave in p but milder than log: √0.05 ⇑ 0.22 versus √0.95 ⇑ 0.97 (ratio roughly 4:1 vs ~60:1 for log). - Same 10-percentage-point spread cap applied. ### Handling cross-decile growth persistence and empirical Gini/top-decile caps - Damping of cross-decile growth differentials implemented via exponential decay parameter - (delta) in (0, 1]: - Effective growth in year t: g_{i,t} = ḡ + -^{t-1} · ˜g_i, where ˜g_i = g_i − ḡ. - Grid of - used: {0.85, 0.90, 0.95, 1.00}. - Interpretations: - -=0.95 implies a half-life of approximately 14 years. - -=0.90 implies roughly 7 years. - -=0.85 implies about 4 years. - Empirical bounds on inequality applied in long-run exercises only: - Projected Gini constrained below a ceiling drawn from {0.60, 0.65}. - Projected top-decile income share constrained below {0.45, 0.50}. - Bootstrap and perturbation settings: - Medium term: 5,000 bootstrap draws. - Long run: 1,000 bootstrap draws. - Initial income shares perturbed independently in each draw using multiplicative log-normal noise (σ = 0.06) to reflect uncertainty in the 2016 HIES calibration. - Medium-term benchmark: quadratic GIC with moderate damping (- = 0.90) and non-binding caps. ### Medium-term results (adjusted Q-GIC; quadratic with -=0.90; perturbed initial shares) - Scenario ranking (by distributional outcome): Baseline (most progressive) → DRM → Non-concessional financing → Uncompensated cuts. - Gini outcomes: - Baseline: Gini declines from 0.35 to 0.33. - Uncompensated cuts: Gini rises to about 0.384. - Poverty headcount and mean income (10-year horizon): - Baseline: HC: 33.9% → 30.1%; Mean: $1773 → $1952. - Non-concessional financing: HC: 33.9% → 41.1%; Mean: $1773 → $1762. - DRM: HC: 33.9% → 41.4%; Mean: $1773 → $1762. - Uncompensated cuts: HC: 33.9% → 41.9%; Mean: $1773 → $1749. - Interpretation: - Baseline produces modest rightward shift of income distribution and decline in poverty. - The three aid-cut scenarios compress the bottom of the distribution and raise poverty headcounts. - DRM and non-concessional financing are nearly indistinguishable in the medium term, reflecting DRM’s short-run regressivity via GST broadening and user fees. ### Long-run results (adjusted Q-GIC; example reported with -=0.90 quadratic) - Baseline: - Mean income: $1,773 → $3,619. - Poverty headcount: 33.9% → 4.9%. - Gini density concentrated at low values (sustained progressive growth). - DRM: - Mean income: $1,822 (long run). - Poverty headcount: 31.1% (long run). - Mechanism: long-run returns to revenue-financed human capital investment raise productivity of low-skill labor, offsetting initial regressive burden. - Non-concessional financing: - Mean income: equivalent to the DRM scenario (reported as equivalent to DRM mean). - Poverty headcount: 50.7%. - Interpretation: distributional incidence of debt service falls disproportionately on liquidity-constrained households, yielding high long-run poverty despite similar mean income. - Uncompensated cuts: - Mean income: virtually unchanged in the long run. - Poverty headcount: 57.0%. - Worst long-run outcome: top decile gains over 10 percentage points of total income share while bottom eight deciles all lose. - Comparative patterns: - Baseline Lorenz curve shifts inward (toward equality) in both medium and long run. - Aid-cut scenarios shift Lorenz curves outward; DRM’s long-run Lorenz curve is closer to the initial distribution than non-concessional financing or uncompensated cuts. - Under DRM long run, bottom deciles experience near-zero share changes and top decile gains only modestly; uncompensated cuts produce large top-end gains and broad losses across bottom deciles. ### Robustness and sensitivity conclusions - The five GIC specifications span tightly constrained to fully unconstrained functional forms and broadly capture plausible shapes (symmetric, segmented, globally concave). - If scenario ranking remains consistent across these specifications, distributional conclusions are robust to choice of functional form and constraining assumptions. - Medium-term ranking is robust to perturbation of initial shares and to the adjusted quadratic GIC; baseline remains most progressive, other scenarios produce higher expected inequality and greater uncertainty. *Source:Authors’ simulations.* ### 1. Baseline ### 1. Baseline ### Kernel density and income distribution (Figure A.13 notes) - Kernel density estimates of the bootstrapped 50-year Gini (quadratic specification,-=0.90) are reported for each scenario. - Notes: Authors’ simulations. ### Poverty headcount and mean income — Baseline (long run, 50 years) - HC: 33.9% ! 4.9% - Mean: $1773 ! $3619 - Plots show initial (gray) and final (red) income density. - Shaded areas represent populations below the $3/day poverty line. ### Comparative scenario outcomes shown in Figure A.13 (long run, 50 years) - Non-concessional Financing - HC: 33.9% ! 50.7% - Mean: $1773 ! $1818 - DRM - HC: 33.9% ! 31.1% - Mean: $1773 ! $1822 - Uncompensated Cuts - HC: 33.9% ! 57.0% - Mean: $1773 ! $1742 - Each panel shows initial (gray) and final (red) income density for one scenario at 50 years (quadratic specification,-=0.90). ### Long-run Gini sensitivity and specification robustness (Figure A.14 summary) - The long run Gini is reported for five GIC specifications across four damping values (- = 0.85, 0.90, 0.95, 1.00). - Grouped bars show the median 50-year Gini for each GIC specification at each damping parameter-. Gini and top-share caps are imposed. - The dashed line marks the empirical ceiling (0.65); the dotted line marks the initial Gini (0.35). ### Key quantitative and qualitative findings on specification sensitivity - The scenario ranking (from best to worst distributional outcome) is invariant across all five specifications and all four damping values-: 1. Baseline (continued aid) — lowest Gini 2. DRM — second lowest Gini 3. Non-concessional financing — more regressive than DRM 4. Uncompensated cuts — worst distributional outcome - There is more variation across specifications in the long-run steady state results than in the medium term, particularly at higher values of- (less damping). - Concave specifications (log-linear and sqrt-linear), which weight bottom-decile growth more heavily, produce different absolute Gini levels compared with quadratic and piecewise specifications; however, they do not change the scenario ranking. - The three constrained quadratic variants (strict, piecewise-3, no Pareto) produce very similar results to each other, indicating neither the Pareto tail assumption nor the choice between polynomial and piecewise functional forms is a quantitatively important driver of the distributional results. - The sensitivity analysis spans: - Five GIC specifications (polynomial, piecewise-linear, concave forms) - With and without Pareto tail and Lorenz convexity constraints - Four growth damping assumptions - Three tail-premium values - Two Gini caps - Two top-share caps - Perturbed initial distributions - Conclusion: The ranking of policy responses is driven by aggregate growth moments from the structural model rather than by distributional-mapping assumptions. *Source: Authors’ simulations. Closing the Gap: A LIC’s Policy Options Under an Aid Shock — Working Paper No. WP/2026/151* --- _Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026151-source-pdf.pdf_