## _wp15286

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### Introduction: context, motivation, and main qualitative findings
- Government spending is an important policy tool in low-income countries (LICs) with distinctive features that shape fiscal transmission.
- LIC-specific features highlighted:
  - Lower public investment efficiency: "one dollar of investment expenditures yields much less than one dollar of effective public capital."
  - Low degree of home bias in government spending; substantial shares of public investment are spent on imports of material, machinery, and skilled labor.
  - Limited international capital mobility and limited asset market participation.
- External versus domestic financing matters qualitatively and quantitatively:
  - Under restricted capital mobility, external financing increases the resource envelope and mitigates private-sector crowding-out.
  - Capital inflows from external commercial borrowing or aid can appreciate the real exchange rate and reduce traded-good competitiveness.
- Quantitative baseline calibration examples:
  - With relatively low investment efficiency in the baseline, the impact (10-year cumulative) output multipliers for public investment:
    - under domestic borrowing are about 0.3 (−0.5),
    - under external commercial borrowing correspond to 0.4 (0.6).
  - Aid financing produces a slightly higher long-run output multiplier than commercial debt financing, because commercial debt requires larger posterior fiscal adjustments.
  - With external financing, a 10-year output multiplier increases from 0.6 to 1.1 when efficiency rises from 0.4 to 0.8.
- Roles emphasized:
  - Capital mobility: high international capital mobility diminishes the advantage of external financing because private households can adjust portfolios and de-leverage.
  - Home bias: higher home bias reduces crowding out of private consumption but causes real exchange rate appreciation and declines in traded output.
  - Public investment efficiency: improvements in marginal efficiency significantly raise the effect of public investment on output.

### Analytical model: objectives, setup, and mechanisms
- Purpose:
  - Simple two-sector framework to isolate key roles of external financing (aid) and home bias in determining private-sector responses.
- Key assumptions and structure (parameter notation preserved):
  - Government spending: ∆g_t = ρ ∆g_{t−1} + ε_t, with 0 < ρ < 1.
  - Fraction ω of government spending increases financed externally: ω ∆g_t = s ∆a^*_t, 0 ≤ ω ≤ 1.
  - Representative household utility: E_0 ∑_{t=0}^∞ β^t [ (c_t)^{1−σ}/(1−σ) − a_N (l^N_t)^{1+ψ}/(1+ψ) − a_T (l^T_t)^{1+ψ}/(1+ψ) + (m_t)^{1−ξ}/(1−ξ) ] with 0 < β < 1.
  - Consumption aggregator (CES): c_t = [ φ^{1/χ} (c^N_t)^{(χ−1)/χ} + (1−φ)^{1/χ} (c^T_t)^{(χ−1)/χ} ]^{χ/(χ−1)}.
  - Government spending aggregator (CES): g_t = [ (φ_G)^{1/χ} (g^N_t)^{(χ−1)/χ} + (1−φ_G)^{1/χ} (g^T_t)^{(χ−1)/χ} ]^{χ/(χ−1)}, 0 ≤ φ_G ≤ 1 (home bias).
  - Monetary policy: R_t = R E_t ( (π^N_{t+1}/π^N)^{φ_π} ) with φ_π > 1.
  - Analytical simplifications for tractability: σ = χ = 1 in closed-form derivations; steady-state normalizations set π = π^N = p^N = s = p^G = w^N = w^T = y = c = mc = 1.
- Key analytical results (reduced-form and comparative statics):
  - Reduced IS and NK Phillips specifications produce parameters:
    - Θ_π = φ (φ_π − 1),
    - Θ_g = (1−φ_G − ω) (1−ρ) ψ / (1+ψ),
    - Γ_c = (θ−1)(1+ψ) / ζφ,
    - Γ_g = (θ−1)(1−ω)ψ / ζφ.
  - Proposition 1: Under φ_π > 1 and φ_π < 1 + 2ζ(1+β) / (θ−1)(1+ψ), there is a unique rational expectations equilibrium and
    - ˆc_t = − ψ / (1+ψ) [ 1 − ω − (d / (d + n)) φ_G ] ∆g_t, with d ≡ (1−ρ)(1−βρ) and n ≡ (φ_π − 1)(1+ψ)(θ−1)ρ / ζ.
  - Real exchange rate in equilibrium:
    - ˆs_t = φ ψ / (1−φ) (1+ψ) [ 1 − ω − ( (d + n/φ) / (d + n) ) φ_G ] ∆g_t.
  - Proposition 2 (consumption multiplier):
    - M_c ≡ ∆c_t / ∆g_t = − ψ / (1+ψ) [ 1 − ω − (d / (d + n)) φ_G ].
    - Comparative statics:
      - ∂M_c / ∂ω > 0 — more external financing increases the consumption multiplier.
      - ∂M_c / ∂φ_G > 0 — more government spending home bias increases the consumption multiplier.
    - Extremes:
      - ω = 0 (all domestic financing): M_c < 0 (crowding out).
      - ω = 1 (all external financing): M_c > 0 (crowding in).
  - Proposition 3 (sectoral multipliers):
    - Traded output multiplier: M_y^T ≡ ∆y^T_t / ∆g_t = 1 / (1+ψ) (1 − ω − φ_G).
    - Non-traded output multiplier: M_y^N ≡ ∆y^N_t / ∆g_t = [ (d + n/(1+ψ)) / (d + n) ] φ_G.
    - Implications:
      - ∂M_y^T / ∂ω < 0 and ∂M_y^T / ∂φ_G < 0 — more external financing or home bias decrease traded output multiplier.
      - M_y^N ≥ 0 and ∂M_y^N / ∂φ_G > 0 — non-traded output increases with higher home bias.

### Quantitative model features and baseline calibration
- Model: small open New Keynesian model with LIC features added:
  - Two household types (savers fraction f, hand-to-mouth fraction 1−f), imperfect labor mobility, government consumption and inefficient public investment, limited household access to international financial markets, government external debt access, money targeting.
- Key calibrated parameter values and steady-state targets (exactly as in source):
  - Intertemporal elasticity of substitution set to 0.34, implying σ = 2.94.
  - Frisch labor elasticity for savers: ψ = 1; average Frisch labor elasticity: 0.25.
  - Inverse elasticity of real money balances: ξ = 3.1.
  - Discount factor: β = 0.98.
  - Share of savers: f = 0.25.
  - Degree of home bias in private consumption and investment: φ = 0.6.
  - Degree of home bias in government purchases: φ_G = 0.7.
  - Nontraded value added in steady state: 65 percent of GDP.
  - Elasticity of substitution between traded and nontraded goods: χ = 0.44.
  - Labor income shares: αN = 0.45, αT = 0.6.
  - Share of labor in nontraded sector implied: φl = 0.58.
  - Private capital depreciation: δ = 0.025 (annual depreciation rate of 10 percent).
  - Elasticity of substitution between varieties: θ = 6 (steady-state markup 20 percent); subsidy rate to nontraded firms u = 0.2.
  - Intra-temporal elasticity between sectoral labor: χl = 0.6.
  - Capital adjustment cost parameter: κ = 1.4.
  - Price rigidity for nontraded goods: ζ = 56.6 (price rigid for one year on average).
  - Public capital depreciation: δG = 0.0125 (annual rate of 5 percent).
  - Public investment efficiency baseline: ϵ = 0.4; alternative tested: ϵ = 0.8.
  - Public investment-to-output ratio: 0.045 (yields annual public capital-output ratio of 0.36).
  - Output elasticity w.r.t. public capital: αG = 0.11 (steady-state annual net rate of return to public capital 25.6 percent).
  - Fiscal adjustment magnitude: γ = 0.03.
  - Steady-state public debt-to-annual output ratio: 0.33, with 70 percent of public debt assumed external.
  - Degree of private-sector capital account openness baseline: v = 10 (limited mobility); sensitivity analysis explores smaller v’s.
- Solution method: log-linearize equilibrium system and solve with Sims’s (2001) algorithm; model frequency quarterly; calibration dataset based on 45 SSA countries, 2005–2012.

### Government spending experiments: consumption shocks, public investment shocks, and financing regimes
- Experiment design:
  - Government consumption shock: 1 percent of steady-state output at time zero, follows AR(1) (equation (41)).
  - Public investment shock: 1 percent of steady-state output at time zero; public investment accumulates into public capital with inefficiency kG t = (1 − δG) kG t−1 + ϵ gI t, 0 ≤ ϵ ≤ 1.
  - Financing modalities: domestic debt, external debt, and aid (when one source used others set to steady-state; domestic/external debt financing trigger gradual fiscal adjustments via tax rule log ( τt / τ ) = ρτ log ( τt−1 / τ ) + γ log ( sb t−1 / sb )).
  - Present-value cumulative multiplier at k quarters defined by Mx(k) (equation (48)).
- Main findings for government consumption shocks (baseline v = 10):
  - Domestic financing:
    - Impact output multiplier: 0.4; five-year multiplier: −0.4.
    - Domestic borrowing induces negative wealth effect on savers (anticipation of higher future taxes): savers increase labor supply and reduce consumption; hand-to-mouth consumption increases for first two quarters.
    - Private investment is crowded out; real exchange rate appreciates modestly; current account moves little.
    - If government purchases are almost all nontraded (φ_G = 0.999), impact multiplier rises from 0.4 to 0.7.
  - External debt financing:
    - Larger foreign exchange inflows appreciate real exchange rate much more than domestic financing.
    - Traded output short-run declines more strongly: traded output impact multiplier −0.3 with external financing versus 0.1 with domestic financing.
    - Trade deficits rise substantially; trade balance impact multiplier −0.5 with external financing versus almost zero with domestic financing.
    - Short-run consumption and investment are less negative / more positive than with domestic financing:
      - Impact output multiplier: 0.43 (external) versus 0.39 (domestic).
      - Five-year output multiplier: 0.2 (external) versus −0.4 (domestic).
      - Five-year cumulative investment multiplier: almost zero (external) versus −0.8 (domestic).
      - Consumption multiplier on impact rises from 0.1 (domestic) to 0.5 (external).
    - Over time, fiscal adjustments (higher income tax) dampen investment and nontraded output; debt service leads to small real depreciation later.
  - Caveat: analysis assumes constant risk premium; higher initial debt could raise premium non-linearly and necessitate larger fiscal adjustments.
- Public investment results:
  - Within each financing method:
    - Short-run multipliers slightly smaller than for consumption shocks but substantially larger in later years as public capital accumulates.
    - Government investment weakens negative wealth effects because households expect higher future production/income; consumption more positive; private investment still crowded out relative to consumption shock but may not fall below steady state due to higher marginal product of private capital.
  - Financing comparisons for public investment (reported 10-year output multipliers):
    - External debt financing: 0.6.
    - Aid financing: 0.7.
    - Domestic financing: −0.5.
  - Mechanisms: external financing increases resources available and mitigates crowding out, but creates stronger real exchange rate appreciation and traded-sector losses (Dutch disease).

### External financing: aid versus external debt and sensitivity to capital mobility
- Aid versus external debt:
  - Short-run fiscal effects similar; key difference: aid does not require repayment, so tax rate remains at steady state and savers do not anticipate higher future taxes.
  - Investment with aid financing outperforms external debt in later years because income tax rate is lower.
  - Aid yields better longer-horizon output performance among the three financing methods.
  - Both external debt and aid produce substantial real exchange rate appreciation, lowering traded output (Dutch disease).
  - Empirical reference: Rajan and Subramanian (2011) find manufacturing is negatively affected by aid inflows, consistent with model implications.
- Sensitivity to international capital mobility:
  - Importance of financing source diminishes when international capital mobility becomes high.
  - Comparative scenarios with v = 10 (baseline), v = 0.001, v = 0.0001 (nearly open) show that with higher capital mobility households counteract government external borrowing; the economy as whole borrows less externally and initial real appreciation discourages further household external borrowing.
  - With near-open capital account, government consumption under external financing yields similar crowding-out as domestic financing.

### Public investment efficiency, home bias in investment, and multiplier magnitudes
- Public investment efficiency:
  - Baseline overall efficiency ϵ = 0.4; marginal efficiency experiment sets ̄ϵ = 0.8 for new investment (rule (49)).
  - Steady-state annual rate of return to public capital: 25.6 percent.
  - With external financing, 10-year multiplier increases from 0.6 to 1.1 when marginal efficiency rises from 0.4 to 0.8.
  - Long-run (20-year) cumulative output multiplier under higher marginal efficiency reaches 1.6.
  - Policy implication: raising marginal efficiency (project selection, execution, governance) materially raises multipliers; sustaining growth effects requires a permanent increase in public investment.
- Home bias in public investment:
  - Baseline φ_G = 0.7; alternative for public investment φ_GI = 0.3 (while φ_GC = 0.7).
  - Lower home bias (φ_GI = 0.3) implies higher import content of investment and reduced effectiveness in promoting domestic production.
  - Example numerical effects (external debt financing):
    - Impact output multiplier with φ_GI = 0.3: 0.15.
    - Impact output multiplier with φ_GI = 0.7: 0.40.
  - With low home bias, traded output can outperform nontraded output; with domestic debt financing, higher demand for traded goods can generate real depreciation and positive traded output responses.
- Key numerical parameters and multipliers reiterated:
  - Public investment shock size: 1 percent of steady-state output.
  - Baseline ϵ = 0.4; marginal efficiency experiment ̄ϵ = 0.8.
  - Home-bias baseline: φ_G = 0.7; alternative φ_GI = 0.3.
  - Reported multipliers:
    - 10-year output multipliers: external debt 0.6, aid 0.7, domestic −0.5.
    - 10-year multiplier with higher marginal efficiency: 0.6 → 1.1.
    - 20-year cumulative with higher marginal efficiency: 1.6.
    - Impact output multiplier with external debt and φ_GI = 0.3: 0.15; with φ_GI = 0.7: 0.40.

### Policy implications, caveats, and research directions
- With limited international capital mobility, external financing (debt or aid) generally yields larger output multipliers than domestic financing because external financing raises available resources and mitigates crowding-out.
- Large foreign inflows associated with external financing appreciate the real exchange rate and reduce traded output; persistent traded losses can shrink multipliers (Dutch disease).
- High steady-state returns to public capital do not guarantee large long-run multipliers if marginal efficiency of new public investment is low.
- Public investment scaling-up delivers weaker short-run stimulative effects when home bias in investment spending is low (large import content).
- Further research suggested:
  - Estimate the model with sufficient quarterly data to evaluate empirical importance of LIC features.
  - Estimate country-specific parameters (marginal efficiency, home bias).
  - Disentangle government spending by financing source for better country-specific assessment.
  - Incorporate state-dependent government spending effects (zero-lower bound, deep recessions, government indebtedness) in future work.

*Italic: Source: _wp15286 - References.......................................................................................43*

### References.......................................................................................43

### _wp15286 - References.......................................................................................43

### Introduction: context, motivation, and main qualitative findings
- Government spending is an important policy tool in low-income countries (LICs) with distinctive features that shape fiscal transmission.
- Key LIC-specific features highlighted:
  - Lower public investment efficiency: "one dollar of investment expenditures yields much less than one dollar of effective public capital."
  - Low degree of home bias in government spending; substantial shares of public investment are spent on imports of material, machinery, and skilled labor.
  - Limited international capital mobility and limited asset market participation.
- External financing versus domestic financing matters qualitatively and quantitatively:
  - Under restricted capital mobility, external financing increases the resource envelope and mitigates private-sector crowding-out.
  - Capital inflows from external commercial borrowing or aid can appreciate the real exchange rate and reduce traded-good competitiveness.
- Quantitative examples from the baseline calibration:
  - With relatively low investment efficiency in the baseline, the impact (10-year cumulative) output multipliers for public investment:
    - under domestic borrowing are about 0.3 (−0.5),
    - under external commercial borrowing correspond to 0.4 (0.6).
  - Aid financing produces a slightly higher long-run output multiplier than commercial debt financing, because commercial debt requires larger posterior fiscal adjustments.
  - With external financing, a 10-year output multiplier increases from 0.6 to 1.1 when efficiency rises from 0.4 to 0.8.
- Role of capital mobility:
  - When international capital mobility is high, the advantage of external financing diminishes because private households can adjust portfolios and de-leverage foreign borrowing; the total external resources may not increase much compared with domestically financed spending increases.
- Role of home bias:
  - A higher degree of home bias generally makes government spending more expansionary by reducing crowding out of private consumption, but it also causes real exchange rate appreciation and decline in traded output.
  - Low home bias in LIC public investment reduces its expansionary impact relative to developed-economy contexts.
- Public investment efficiency:
  - Improvements in marginal efficiency significantly raise the effect of public investment on output; addressing planning, execution, and management problems matters for multipliers.

### Analytical model: objectives, setup, and mechanisms
- Purpose:
  - Use a simple analytical two-sector framework to isolate and demonstrate the key roles of external financing (aid) and home bias in determining private-sector responses to government spending increases.
  - Focus on two opposing effects:
    - External financing reduces private-sector crowding-out by increasing resources.
    - External financing aggravates traded-output losses via larger real exchange rate appreciation.
- Simplifying assumptions for the analytical model:
  - No labor mobility across sectors.
  - Abstract from limited asset market participation, investment, and external and domestic debt.
  - Monetary policy follows a forward-looking interest rate rule implemented via a money target rule.
  - External financing is modeled as foreign aid.
  - Households cannot access international capital markets (fully restricted international capital mobility).
  - Government spending follows an exogenous AR(1): ∆g_t = ρ ∆g_{t−1} + ε_t with 0 < ρ < 1 and i.i.d. ε_t.
  - Fraction ω of government spending increases is financed with external funds: ω ∆g_t = s ∆a^*_t, where 0 ≤ ω ≤ 1.
- Key model structure and definitions (preserve parameter notation and exact relations):
  - Representative household maximizes expected utility:
    E_0 ∑_{t=0}^∞ β^t [ (c_t)^{1−σ}/(1−σ) − a_N (l^N_t)^{1+ψ}/(1+ψ) − a_T (l^T_t)^{1+ψ}/(1+ψ) + (m_t)^{1−ξ}/(1−ξ) ].
    - Parameters: 0 < β < 1; σ inverse of intertemporal elasticity of substitution; a_j for j = N, T labor disutility weights; ψ inverse of Frisch elasticity; ξ inverse of elasticity for real money balances.
  - Budget constraint: c_t + b_t + m_t = w^N_t l^N_t + w^T_t l^T_t + R_{t−1} b_{t−1}/π_t + m_{t−1}/π_t + z_t + Ω_t.
  - Consumption aggregator (CES): c_t = [ φ^{1/χ} (c^N_t)^{(χ−1)/χ} + (1−φ)^{1/χ} (c^T_t)^{(χ−1)/χ} ]^{χ/(χ−1)} with χ > 0 and 0 < φ < 1.
    - Demands: c^N_t = φ (p^N_t)^{−χ} c_t and c^T_t = (1−φ) (s_t)^{−χ} c_t.
  - Government budget constraint: b_t + s_t a^*_t + m_t = g_t + z_t + R_{t−1} b_{t−1}/π_t + m_{t−1}/π_t.
  - Government spending aggregator (CES): g_t = [ (φ_G)^{1/χ} (g^N_t)^{(χ−1)/χ} + (1−φ_G)^{1/χ} (g^T_t)^{(χ−1)/χ} ]^{χ/(χ−1)} with 0 ≤ φ_G ≤ 1 (φ_G interpreted as home bias of government spending).
    - Government demand shares: g^N_t = φ_G (p^N_t / p^G_t)^{−χ} g_t and g^T_t = (1−φ_G) (s_t / p^G_t)^{−χ} g_t.
  - Market clearing for nontraded goods: y^N_t = ∫_0^1 y^N_t(i) di = c^N_t + g^N_t + (ζ/2) (π^N_t − 1)^2 p^N_t y^N_t.
  - Market clearing for traded goods: y^T_t = c^T_t + g^T_t − a^*_t.
  - Monetary policy specification for analytical tractability: R_t = R E_t ( (π^N_{t+1}/π^N)^{φ_π} ) with φ_π > 1 and R steady-state interest rate.
- Price and production structure:
  - Nontraded firms monopolistically competitive with Rotemberg quadratic price adjustment costs (parameter ζ > 0 controls price stickiness), elasticity of substitution θ > 1 across varieties (θ typically much bigger than one in NK literature).
  - Traded-good firms perfectly competitive with y^T_t = l^T_t.
- Analytical simplifications employed to derive transparent results:
  - Assume steady state g = a^* = 0.
  - Labor disutility parameters set to a_N = φ^{−ψ} and a_T = (1−φ)^{−ψ}.
  - Normalize steady-state values so that π = π^N = p^N = s = p^G = w^N = w^T = y = c = mc = 1, R = 1/β, y^N = l^N = c^N = φ, y^T = l^T = c^T = 1−φ.
  - For analytical clarity set σ = χ = 1 to show qualitative roles of ω (external financing) and φ_G (home bias); note main insights persist when σχ ≥ 1.

### Key mechanisms and implications derived from the analytical model
- External financing (ω closer to 1):
  - Mitigates private consumption and investment crowding-out under restricted capital mobility by expanding available resources.
  - Simultaneously tends to appreciate the real exchange rate, reducing traded output; net effect on aggregate output depends on magnitudes.
- Degree of home bias (φ_G):
  - Higher φ_G reduces crowding-out of private consumption from government consumption increases but also leads to real exchange rate appreciation and declines in traded output.
  - Low φ_G in LICs (large import content of public investment) weakens the expansionary impact of public investment.
- Capital mobility:
  - With high international capital mobility (low portfolio adjustment costs), the private sector can reallocate and de-leverage, diminishing the boost from external public financing; the mitigation of crowding out by foreign inflows is small in open economies.
- Public investment efficiency:
  - Low public investment efficiency in LICs reduces the immediate output impact of public investment scaling-up.
  - Increasing marginal efficiency can raise multipliers substantially (example: 10-year multiplier rises from 0.6 to 1.1 when efficiency increases from 0.4 to 0.8 under external financing).
- Interaction with debt and fiscal adjustment:
  - Commercial debt financing implies larger future fiscal adjustments (taxes) to service debt, which reduces long-run multipliers compared with aid financing.

*Italic: Source: _wp15286 - References.......................................................................................43*

### Appendix I details the equilibrium conditions, the steady state,

### _wp15286 - Appendix I details the equilibrium conditions, the steady state,

### Linearized equilibrium system and definitions
- Linearized equilibrium around steady state yields:
  - ˆc_t = E_t ˆc_{t+1} − (ˆR_t − E_t ˆπ_{t+1}), (17)
  - ˆπ^N_t = β E_t ˆπ^N_{t+1} + [ (θ−1)(1+ψ) / ζφ ] ˆc_t + [ (θ−1)(1−ω)ψ / ζφ ] ∆g_t, (18)
  - ˆR_t = φ_π E_t ˆπ^N_{t+1}, (19)
  - ˆs_t = ˆc_t + ( 1−φ_G − ω 1−φ ) (ψ / (1+ψ)) ∆g_t, (20)
  - ˆπ_t = ˆπ^N_t + (1−φ / φ) (ˆs_t − ˆs_{t−1}), (21)
  - ∆g_t = ρ ∆g_{t−1} + ε_t, (22)
  - Notation: ∆x_t = x_t − x and ˆx_t = ∆x_t / x.

- Interpretation:
  - Equations (17), (18), (19) correspond to the IS curve, the NK Phillips curve for non-traded goods inflation, and the interest rate rule, respectively.
  - Equation (20) pins down dynamics of the relative price of traded goods and involves labor and goods market equilibrium for traded goods and optimal labor decisions for traded-good production.
  - Equations (21) and (22) derive from the definition of inflation and the stochastic process for government spending.

- Reduced system after manipulation:
  - ˆc_t = E_t ˆc_{t+1} − Θ_π E_t ˆπ^N_{t+1} − Θ_g ∆g_t, (23)
  - ˆπ^N_t = β E_t ˆπ^N_{t+1} + Γ_c ˆc_t + Γ_g ∆g_t, (24)
  - and (22), where
    - Θ_π = φ (φ_π − 1),
    - Θ_g = (1−φ_G − ω) (1−ρ) ψ / (1+ψ),
    - Γ_c = (θ−1)(1+ψ) / ζφ,
    - Γ_g = (θ−1)(1−ω)ψ / ζφ.

- Equilibrium definition:
  - Definition 1: An equilibrium is a sequence of stochastic processes {ˆc_t, ˆπ^N_t}_{t=0}^∞ that satisfy the reduced IS curve (17) and NK Phillips curve (18), given {∆g_t}_{t=0}^∞.

### Existence, uniqueness, and equilibrium characterization
- Proposition 1 (conditions and characterization):
  - Assume monetary policy satisfies φ_π > 1 (the Taylor principle) and φ_π < 1 + 2ζ(1+β) / (θ−1)(1+ψ).
  - Then:
    - a) there is a unique rational expectations equilibrium {ˆc_t, ˆπ^N_t}_{t=0}^∞;
    - b) equilibrium consumption is characterized by
      - ˆc_t = − ψ / (1+ψ) [ 1 − ω − (d / (d + n)) φ_G ] ∆g_t, (25)
      - with d ≡ (1−ρ)(1−βρ) and n ≡ (φ_π − 1)(1+ψ)(θ−1)ρ / ζ.
  - Proof: See Appendix.

- Real exchange rate in equilibrium:
  - ˆs_t = φ ψ / (1−φ) (1+ψ) [ 1 − ω − ( (d + n/φ) / (d + n) ) φ_G ] ∆g_t. (26)
  - Comparative statics:
    - ∂ˆs_t / ∂ω < 0 — more external financing tends to appreciate the real exchange rate.
    - ∂ˆs_t / ∂φ_G < 0 — more government spending home bias (more spending on non-traded goods) generates non-traded inflationary pressures and real appreciation.

### Consumption multiplier and mechanisms
- Proposition 2: Consumption multiplier
  - M_c ≡ ∆c_t / ∆g_t = − ψ / (1+ψ) [ 1 − ω − (d / (d + n)) φ_G ]. (27)
  - Implications:
    - a) crowding-out or crowding-in: M_c ≶ 0 if 1 − ω ≶ (d / (d + n)) φ_G;
    - b) ∂M_c / ∂ω > 0 — more external financing increases the consumption multiplier;
    - c) ∂M_c / ∂φ_G > 0 — more government spending home bias increases the consumption multiplier.
  - Intuition and extreme cases:
    - ω = 0 (all financing domestic): M_c < 0 (crowding out). Mechanisms:
      - Income mechanism: government reduces transfers to finance spending, lowering private consumption.
      - Real interest rate mechanism: non-traded inflation rises, nominal interest rates increase (Taylor rule), expected future overall inflation rises less than expected non-traded inflation and nominal rate, real interest rate rises, reducing private consumption.
    - ω = 1 (all financing external): M_c > 0 (crowding in). Mechanisms:
      - External financing relaxes national budget constraint, enabling increases in both private and public consumption.
      - Real exchange rate appreciates, real interest rate decreases, stimulating private consumption.
    - Consumption multiplier increases in ω and in φ_G; sufficiently high φ_G can switch effects from crowding out to crowding in.

### Sectoral (traded and non-traded) output multipliers
- Proposition 3: Sectoral multipliers
  - Traded output multiplier:
    - M_y^T ≡ ∆y^T_t / ∆g_t = 1 / (1+ψ) (1 − ω − φ_G).
  - Non-traded output multiplier:
    - M_y^N ≡ ∆y^N_t / ∆g_t = [ (d + n/(1+ψ)) / (d + n) ] φ_G. (28)
  - Implications:
    - a) Traded output can decrease or increase: M_y^T ≶ 0 if ω + φ_G ≶ 1.
    - b) ∂M_y^T / ∂ω < 0 and ∂M_y^T / ∂φ_G < 0 — more external financing or home bias decrease the traded output multiplier.
    - c) M_y^N ≥ 0 and ∂M_y^N / ∂φ_G > 0 — non-traded output increases with higher home bias; M_y^N increases with φ_G.
  - Intuition and extreme cases:
    - ω = 1 and φ_G > 0: M_y^T < 0 — external financing appreciates real exchange rate, harming traded-sector competitiveness; traded sector has flexible prices so no demand-driven stimulus.
    - ω = 0: M_y^T can be positive as real exchange rate may appreciate less or depreciate, stimulating traded output.
    - φ_G = 0 (no public spending on non-traded goods): M_y^T > 0 if ω > 0.
    - φ_G = 1 (all public spending on non-traded goods): significant real appreciation can make M_y^T < 0.
    - Non-traded output multiplier unaffected by ω in this simple model due to assumed imperfect labor mobility; with perfect labor mobility, M_y^N would depend on ω.

### Summary of model insights from the analytical framework
- Degrees of external financing (ω) and government spending home bias (φ_G) are key determinants:
  - Both can reduce crowding-out of private consumption or generate crowding-in (increase M_c).
  - Both tend to appreciate the real exchange rate, discouraging traded output (reducing M_y^T).
- The analytical model abstracts from many LIC features; the paper proceeds to a quantitative model to incorporate additional features for quantitative assessment of consumption and investment multipliers.

### Quantitative model features (start of Section III)
- The quantitative framework is a small open New Keynesian model adapted from Berg et al. (2010a) that introduces LIC-relevant features:
  - 1) two types of households — savers and hand-to-mouth;
  - 2) some labor mobility across sectors (imperfect mobility);
  - 3) distinction between government consumption and public investment, which can be inefficient;
  - 4) limited access to international financial markets by households;
  - 5) government’s access to external government debt;
  - 6) money targeting.
- Households:
  - Fraction f are savers (a) and fraction 1−f are hand-to-mouth (h).
  - Savers maximize E_0 ∑_{t=0}^∞ β^t [ (c_{a,t})^{1−σ}/(1−σ) − (l_{a,t})^{1+ψ}/(1+ψ) + (m_{a,t})^{1−ξ}/(1−ξ) ], (29)
  - Savers budget constraint (30) and portfolio adjustment costs ac^b_{a,t} ≡ s_t v ( b^{h*}_{a,t} / b^{h*}_{a} − 1 )^2 b^{h*2}_a capture private-sector capital account openness.
  - Capital is sector specific with adjustment costs ac^i_{a,t} and law of motion k_{j,a,t} = (1−δ) k_{j,a,t−1} + i_{j,a,t}, j ∈ {N,T}. (31)
  - Consumption and investment are CES aggregates with elasticity χ and home bias φ; nontraded goods produced by monopolistically competitive firms with variety elasticity θ.
  - Savers supply labor to both sectors with imperfect mobility:
    - l_{a,t} = [ (φ_l)^{-1/χ_l} (l_{a,N,t})^{1+χ_l} / χ_l + (1−φ_l)^{-1/χ_l} (l_{a,T,t})^{1+χ_l} / χ_l ]^{χ_l/(1+χ_l)}. (32)
  - Aggregate real wage index: w_t = [ φ_l (w_{N,t})^{1+χ_l} + (1−φ_l) (w_{T,t})^{1+χ_l} ]^{1/(1+χ_l)}. (33)
- Hand-to-mouth households:
  - Inelastic labor supply (l_h,t = l_h ∀t) and consumption equals disposable income:
    - c_{h,t} = (1−τ_t) w_t l_h + s_t rm^* + z. (34)
- Firms:
  - Nontraded good firms are monopolistically competitive; traded good firms are perfectly competitive.
  - Nontraded sector technology:
    - y_{N,t}(i) = z_N [ k_{N,t−1}(i) ]^{1−α_N} [ l_{N,t}(i) ]^{α_N} (k_{G,t−1})^{α_G}. (35)
  - Demand for variety i: y_{N,t}(i) = [ P_{N,t}(i) / P_{N,t} ]^{−θ} y_{N,t}. (36)
  - Firms maximize expected discounted dividends with price-adjustment costs ac^p_{t}(i) ≡ ζ/2 [ π^N_t(i) / π^N_{t−1} − 1 ]^2 p_{N,t} y_{N,t}.
  - Implicit production cost (tax) ι and subsidy u are introduced so steady-state markup is zero; implicit cost revenue is rebated lump-sum to firms/savers.

*Source: Appendix I and start of Section III from the supplied content unit.*

### 2.    Traded Good Sector

### 2.    Traded Good Sector

### Traded good firm and aggregate output
- Representative traded good firm periodic profit function (as in equation (38)):
  - ΩTt ≡ (1−ι)st [ zT (kTt−1)1−αT (lTt)αT (kGt−1)αG ] − wTt lTt − rTt kTt−1 + ιst yTt.
- Total dividends from firms: Ωt = ΩNt + ΩTt.
- Total real output in period t: yt = pNt yNt + st yTt.

### Government: budget constraint, financing, and policies
- Flow budget constraint (equation (39)):
  - taxt + bd t + st bg∗ t + st a∗ t + mt = pG t ( gC t + gI t ) + z + Rt−1 bd t−1 / πt + st R∗ bg∗ t−1 / π∗ + mt−1 / πt.
- Tax revenue specification:
  - taxt = τt ( wt lt + rN t kN t−1 + rT t kT t−1 ).
- Borrowing interest rates: government faces same nominal interest rate R∗ as savers when borrowing externally.
- Government consumption and public investment:
  - CES baskets of traded and nontraded goods (as in (9)).
  - Public capital accumulation with inefficiency (equation (40)):
    - kG t = (1 − δG) kG t−1 + ϵ gI t, 0 ≤ ϵ ≤ 1.
    - ϵ measures investment inefficiency.
  - Exogenous processes for spending (equation (41)):
    - log ( gjt / gj ) = ρG log ( gjt−1 / gj ) + εjt,  j ∈ {C, I}.
- Financing modalities considered: domestic debt, external debt, and aid.
  - When one financing source is used, the other two are set to steady-state levels (example explained).
  - With domestic or external debt financing, total government debt rises and triggers gradual fiscal adjustments; tax rate adjusts to maintain debt sustainability:
    - log ( τt / τ ) = ρτ log ( τt−1 / τ ) + γ log ( sb t−1 / sb ),  γ ≥ 0 (equation (42)),
    - where sb t−1 = bd t−1 + st−1 bg∗ t−1 / yt−1.
- Money growth rule (government targets money): real money balance follows (equation (43)):
  - mt = φ mt−1 / πt, with nominal reserve money growing at constant rate φ.

### Aggregation and market clearing
- Aggregation of consumption and labor across household types (equation (44)):
  - xt = f xa t + (1 − f) xh t,  x ∈ { c, cN, cT, l, lN, lT }.
- Aggregation of saver-only variables (equation (45)):
  - xt = f xa t,  x ∈ { m, iN, iT, i, kN, kT, bd, bh∗, Ω, acb, aci }.
- Nontraded goods market clearing (equation (46)):
  - yN t = ( pN t )−χ [ φ ( c t + i t + aci t + acb t + acp t ) + φG ( pG t )χ g t ].
- Balance of payments / current account definition (equation (47)):
  - current account deficits = cad t = ct + it + pG t ( gC t + gI t ) + aci t + acb t + acp t + st (R∗ − 1) bh∗ t−1 + bg∗ t−1 / π∗ − yt − st rm∗ = st ( a∗ t + bh∗ t + bg∗ t − bh∗ t−1 + bg∗ t−1 / π∗ ).

### Solution approach and baseline calibration (quantitative model)
- Solution:
  - Log-linearize equilibrium system and solve with Sims’s (2001) algorithm for linear rational expectations models.
  - Model frequency: quarterly.
  - Calibration dataset: average LIC based on data of 45 SSA countries from 2005 to 2012.
- Key calibrated parameter values and steady-state targets (exact values as in the text):
  - Intertemporal elasticity of substitution set to 0.34, implying σ = 2.94.
  - Frisch labor elasticity for savers: ψ = 1; average Frisch labor elasticity: 0.25.
  - Inverse of intertemporal elasticity of real money balances: ξ = 3.1.
  - Discount factor: β = 0.98.
  - Share of savers: f = 0.25.
  - Degree of home bias in private consumption and investment: φ = 0.6.
  - Degree of home bias in government purchases: φG = 0.7.
  - Nontraded value added in steady state: 65 percent of GDP.
  - Elasticity of substitution between traded and nontraded goods: χ = 0.44.
  - Labor income shares: αN = 0.45, αT = 0.6.
  - Share of labor in nontraded sector (φl) implied: φl = 0.58.
  - Private capital depreciation: δ = 0.025 (implying annual depreciation rate of 10 percent).
  - Elasticity of substitution between varieties: θ = 6 (steady-state markup 20 percent); subsidy rate to nontraded firms u = 0.2.
  - Intra-temporal elasticity of substitution between sectoral labor: χl = 0.6.
  - Capital adjustment cost parameter: κ = 1.4.
  - Price rigidity for nontraded goods: ζ = 56.6 (price rigid for one year on average).
  - Public capital depreciation: δG = 0.0125 (annual rate of 5 percent).
  - Public investment efficiency baseline: ϵ = 0.4; alternative tested: ϵ = 0.8.
  - Public investment-to-output ratio calibrated to 0.045 (yields annual public capital-output ratio of 0.36 in steady state).
  - Output elasticity with respect to public capital: αG = 0.11 (steady-state annual net rate of return to public capital 25.6 percent).
  - Fiscal adjustment magnitude: γ = 0.03.
  - Steady-state public debt-to-annual output ratio: 0.33, with 70 percent of public debt assumed external.
  - Degree of capital account openness in private sector baseline: v = 10 (limited mobility); sensitivity analysis explores smaller v’s.

### Government consumption effects: experiments, multipliers, and financing regimes
- Method:
  - Analyze government consumption shock and public investment shock separately under three financing sources: domestic debt, external debt, and aid.
  - Government consumption shock size: 1 percent of steady-state output at time zero, decreases over time per AR(1) in (41).
  - Present-value cumulative multiplier at k quarters (equation (48)):
    - Mx(k) = [ ∑ s=0..k ( ∏ j=0..s r−1 j ) 4xs ] / [ ∑ s=0..k ( ∏ j=0..s r−1 j ) 4pG s g m s ],  x ∈ { y, yN, yT, c, i, tb }, m ∈ { C, I }.
- Main comparative findings:
  - Domestic financing (v = 10):
    - Negative wealth effect on forward-looking savers anticipating higher future taxes: savers increase labor supply; aggregate labor rises (hand-to-mouth have inelastic labor).
    - Savers’ consumption decreases; hand-to-mouth consumption increases for first two quarters due to higher income from government consumption.
    - Standard crowding-out of private investment: higher domestic borrowing increases interest rate demanded by savers, crowding out private investment.
    - Real exchange rate appreciates modestly; current account moves little.
    - Output multiplier on impact: 0.4; five-year multiplier: −0.4 (Table 2 referenced).
    - If government purchases are almost all nontraded goods (high home bias), impact multiplier rises from 0.4 to 0.7 (exercise with φG = 0.999).
  - External debt financing:
    - Large inflows of foreign exchange appreciate real exchange rate much more than with domestic financing.
    - Traded output declines more strongly short run: traded output multiplier on impact −0.3 with external financing versus 0.1 with domestic financing.
    - Trade deficits rise substantially; trade balance multiplier on impact −0.5 with external financing versus almost zero with domestic financing.
    - Short-run consumption and investment are less negative / more positive than with domestic financing:
      - Impact output multiplier slightly larger with external financing: 0.43 versus 0.39 with domestic financing.
      - Five-year output multiplier becomes positive: 0.2 with external financing versus −0.4 with domestic financing.
      - Cumulative investment multiplier five years after shock: almost zero with external financing versus −0.8 with domestic financing.
      - Consumption multiplier on impact rises from 0.1 (domestic) to 0.5 (external) due to stronger hand-to-mouth responses.
    - Mechanism: external financing expands domestic resource envelope short run, relieving crowding-out; but stronger real appreciation undermines traded output and worsens current account.
    - Over time, fiscal adjustments (higher income tax rate) dampen investment and nontraded output; debt service obligations lead to small real depreciation later, improving traded sector competitiveness.
  - General caveats:
    - Analysis assumes a constant risk premium; in practice, a higher-debt initial state could raise the premium non-linearly and necessitate larger fiscal adjustments, potentially offsetting expansionary effects of debt-financed spending.
- Noted contrasts and mechanisms:
  - Domestic financing resembles closed-economy neoclassical/NK behavior: negative wealth effect, crowding-out, small impact multipliers due to import leakages.
  - External financing generates stronger short-run expansion in aggregate demand and consumption but at the cost of larger real appreciation and deterioration in the trade balance; medium-term effects hinge on fiscal adjustments and debt sustainability.

*Source: IMF working paper chapter "2.    Traded Good Sector" (embedded equations, calibration, and fiscal experiments as presented in the source).*

### 3.    External Financing: Aid

### 3.    External Financing: Aid

### Aid versus External Debt: short-run and long-run fiscal effects
- Short-run fiscal effects with aid financing (dashed lines) are similar to those with external debt financing (dotted-dashed lines).
- Main distinction: aid financing does not require subsequent repayments, so the tax rate remains at the steady-state level and savers do not anticipate higher future taxes; therefore their consumption does not change much.
- In later years, investment with aid financing outperforms investment with external debt financing because the income tax rate is lower.
- Among the three financing methods, aid financing is the best in terms of output performance in the longer horizon, since persistent higher tax rates with debt financing keep output below the steady state.
- Both external debt and aid financing lead to substantial real exchange rate appreciation, lowering traded output—creating a Dutch disease mechanism:
  - Real exchange rate appreciation forces reallocation of production factors from the traded to the nontraded good sector to meet higher nontraded good demand.
  - If the loss of traded output is persistent, Dutch disease effects can prevail and output multipliers will be much smaller.
- Empirical note cited in text: Rajan and Subramanian (2011) find traded production (manufacturing) is negatively affected by aid inflows, consistent with the model implication that aid-financed spending increases reduce traded output.

### Sensitivity to International Capital Mobility
- The degree of external financing matters for government spending effects when international capital mobility is limited.
- The importance of accounting for financing sources largely vanishes when international capital mobility becomes high.
- Comparative scenario in text: impulse responses for a government consumption shock with external debt financing across degrees of capital mobility v = 10 (baseline), v = 0.001, and v = 0.0001 (nearly open capital account).
  - With higher capital mobility, households counteract government external borrowing, and the economy as a whole borrows less externally relative to limited mobility.
  - An initial real appreciation from rising external public debt discourages households from borrowing externally because the marginal benefit of additional borrowing in units of local goods decreases.
  - With fewer resources available for the domestic economy, government consumption with external financing has similar crowding-out effects as with domestic debt financing.

### Public Investment Effects: overview and multipliers
- Public investment increase scenario: an increase equal to 1 percent of steady-state output at time zero.
- Public investment accumulates to public capital used in production, increasing marginal productivity of private inputs.
- Within each financing method:
  - Multipliers for public investment are slightly smaller in the short run but substantially bigger in later years as productive public capital builds up.
  - Negative wealth effect from government investment is weakened because households expect more public capital and higher future production/income; consumption is more positive, crowding out private investment further relative to a government consumption shock.
  - Productivity increases imply output of both sectors and hence consumption are higher in later years compared to a government consumption shock.
  - Despite a higher income tax rate, private investment does not fall below the steady-state level because higher marginal product of private capital outweighs the disincentive from higher capital income tax.
- Financing-method comparisons for public investment:
  - Between domestic and external debt financing, private investment is crowded out more with domestic financing.
  - Public investment multipliers for output are bigger with the two external financing methods than with domestic financing.
  - Reported 10-year output multipliers:
    - External debt financing: 0.6
    - Aid financing: 0.7
    - Domestic financing: −0.5

### Public Investment Efficiency (marginal efficiency)
- Steady-state annual rate of return to public capital: 25.6 percent.
- Despite high steady-state return, Table 3 shows long-run output multipliers for public investment are much lower than 1 when marginal efficiency is low (baseline overall efficiency ϵ = 0.4).
- Thought experiment raises marginal efficiency (not overall efficiency) using rule (49): ϵ_t = ̄ϵ( g_I_t − g_I / g_I_t ) + ϵ( g_I / g_I_t ), if g_I_t > g_I, with ̄ϵ = 0.8 as marginal efficiency.
  - Purpose: capture higher marginal efficiency on new investment spending (example: joint ventures between Chinese companies and governments in Africa) and to avoid offsetting effects from higher steady-state public capital under higher overall efficiency.
- With higher marginal efficiency:
  - With external financing, the 10-year multiplier increases from 0.6 to 1.1.
  - The long-run (20-year) cumulative output multiplier increases to 1.6.
  - The increase in multipliers highlights the powerful role of marginal efficiency in raising public investment returns.
- Policy implications to raise marginal efficiency:
  - Improve institutional capacity and governance quality to reduce mistakes in selecting and supervising projects, corruption, and supply bottlenecks.
  - Sustain growth effects requires a permanent increase in public investment to replenish depreciated capital and move the economy to a new steady state with more public capital and output.

### Home Bias in Public Investment
- Baseline assumption: both types of spending consist of 70 percent of nontraded goods (φ_G = 0.7).
- In LICs, many public investment projects rely on imports of material, equipment, and skilled labor, so home bias in investment spending can be much lower than 0.7.
- Alternative explored: set home bias in public investment φ_GI = 0.3 while government consumption φ_GC = 0.7.
  - Definitions in text:
    - φ_GC = 0.7 and φ_GI = 0.3 for degrees of home bias in g_C_t and g_I_t.
    - Relative prices p_GC_t and p_GI_t defined by equations (50) and (51).
    - Term p_G_t(g_C_t + g_I_t) replaced by p_GC_t g_C_t + p_GI_t g_I_t in equilibrium.
- Effects of lower home bias (φ_GI = 0.3) versus baseline (φ_GI = 0.7):
  - When home bias is small, traded output outperforms nontraded output — opposite to pattern under higher home bias.
  - With domestic debt financing, higher government demand for traded goods leads to a real depreciation, generating positive responses of traded output and reversing earlier real appreciation.
  - Lower home bias implies a higher share of demand met by imports; thus public spending is less effective in promoting domestic production and associated with a smaller output multiplier.
  - Example numerical comparison with external debt financing:
    - Impact output multiplier under φ_GI = 0.3 is 0.15.
    - Impact output multiplier under φ_GI = 0.7 is 0.40.
- Short-run dynamics with aid financing are similar to dynamics under external debt financing.

### Key quantitative calibration values and parameters highlighted
- Public investment shock size: 1 percent of steady-state output at time zero.
- Baseline overall marginal efficiency in examples: ϵ = 0.4.
- Marginal efficiency experiment value: ̄ϵ = 0.8.
- Steady-state annual rate of return to public capital: 25.6 percent.
- Home-bias baseline: φ_G = 0.7; alternative φ_GI = 0.3.
- Reported multipliers:
  - 10-year output multipliers: external debt 0.6, aid 0.7, domestic −0.5.
  - 10-year multiplier raising under higher marginal efficiency: 0.6 to 1.1.
  - Long-run (20-year) cumulative output multiplier under higher marginal efficiency: 1.6.
  - Impact output multiplier with external debt and φ_GI = 0.3: 0.15; with φ_GI = 0.7: 0.40.

### Conclusions and research implications
- With limited international capital mobility, output multipliers are generally bigger with external (debt or aid) than with domestic financing because external financing increases resources available and mitigates crowding-out.
- Large inflows of foreign exchange with external financing appreciate the real exchange rate and reduce traded output.
- High returns to public capital do not guarantee large long-run output multipliers if marginal efficiency of public investment is low.
- Public investment is unlikely to deliver strong short-run stimulative effects when home bias in investment spending is low (large import content).
- Suggestions for further work:
  - Estimate the model with sufficient quarterly data to evaluate empirical importance of LIC features.
  - Estimate country-specific parameters (marginal efficiency, public spending home bias).
  - Disentangle government spending by financing source for better country-specific assessment.
- The paper focuses on LIC-relevant features but leaves out state-dependent government spending effects (e.g., zero-lower bound, deep recessions, government indebtedness) that are important for future research.

*Source: 3. External Financing: Aid (excerpt from the supplied PDF chapter).*

### 2.    Proof of Proposition 2

### 2.    Proof of Proposition 2

### Proof summary
- The expression (27) for the multiplier M_c follows from using the rational expectations solution (25) from Proposition 1 and the fact that c = 1.
- The crowding-out and crowding-in results, M_c S0, can be verified by applying the conditions(1−ω)T dφ_G d+n to (27).

### Parameter sign and domain conditions used
- 0≤ω≤1
- 0≤φ_G≤1
- 0<d<1
- n>0
- ψ>0
- These follow from the assumptions about the structural parameters.

### Effects of external financing on the multiplier (part b)
- The implications of external financing on the multiplier are determined by taking the derivative of (27) with respect to the parameter ω to obtain:
  - ∂M_c/∂ω = ψ/(1+ψ) >0, given that ψ >0.

### Sensitivity to home bias in government consumption (part c)
- Taking the derivative of (27) with respect to the parameter φ_G yields:
  - ∂M_c/∂φ_G = (ψ/(1+ψ)) (d/(d+n)) >0, given that ψ >0, 0<d<1, and n>0.

### Conclusion
- Q.E.D.

*2.    Proof of Proposition 2, _wp15286 - 2.    Proof of Proposition 2*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15286.pdf_