## _wp06224 — References (Appendices and Main Findings)

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---

### Introduction and main insight
- Purpose: Formulate Cohen and Sachs’s (1986) sovereign debt model as a concessional lending problem tailored to low-income country (LIC) environments.
- Key mechanism: An incentive cutoff—an income level above which a country loses eligibility for concessional aid (examples: IDA and PRGF).
- Main insight:
  - LICs may have an incentive to accumulate significant concessional debt and favor consumption over investment, stagnating around the cutoff and becoming permanently aid-dependent.
  - This outcome can arise even with a benevolent government that maximizes household welfare.
- Empirical linkage: The paper provides empirical evidence of growth stagnation around the cutoff.
- Policy relevance: Implications for debt relief, including evaluation of one-time stock treatments such as the Multilateral Debt Relief Initiative (MDRI), described as a 100 percent debt stock cancellation by the IMF, the IDA, and the African Development Bank for a group of low-income countries.

### Theoretical literature and model framing
- Enforcement mechanisms summarized:
  - Reputation models (Eaton and Gersovitz (1981)): exclusion from credit markets induces repayment when countries value consumption smoothing.
  - Sanction models: default is penalized (loss of output fraction or access to short-term trade credits); renegotiation may be costly (Rose (2005)).
  - Cohen and Sachs (1986): combine reputation and sanction elements within a neoclassical growth model where access to borrowing affects marginal product of capital dynamics.
- Distinction for LICs:
  - Loans to LICs are highly concessional and provided largely by official creditors who are not profit-maximizers nor risk-neutral; large debt may not deter new official lending.
- Model structure highlights:
  - Two agents: a benevolent official creditor offering subsidized loans below/at the cutoff and lending at world interest rate above the cutoff; creditor can commit, debtor cannot.
  - A participation constraint by the creditor implies an endogenous debt ceiling.
  - Debtor (representative-agent LIC) maximizes welfare subject to the lending rule and participation constraint.

### Empirical motivation and results (1988–2000 growth regressions)
- Data and sample:
  - Unbalanced panel of 94 countries, of which 33 are low-income countries.
  - Data sources: Summers-Heston data set (version 6.1), World Bank’s World Development Indicators, and the Barro-Lee data set.
  - Cutoff: operational cutoff formally recognized by IDA donors in IDA8 in 1987; historical cutoff initially set at $250 in 1964; both cutoffs updated annually using the SDR deflator.
- Growth regression specification:
  - Dependent variable: percentage annual growth rate of real GDP per capita.
  - Explanatory variables: GPO; INV; INIT_EDU; INIT_RGDP.
  - Proximity-to-cutoff measure PROX defined using a Bartlett kernel with scaling factor b; b = 1/2 (similar results for b = 1/3 and b = 1/4).
  - A negative coefficient on PROX implies a negative relationship between growth and proximity to the cutoff.
- OLS and 2SLS findings:
  - Sample period for pooled estimations: 1988-2000.
  - OLS: PROX coefficient negative and statistically significant at the 5 percent level.
  - Endogeneity addressed via 2SLS with instruments: PROX(-1), INV(-1), and RGDP85 (log real GDP per capita in 1985).
  - Observations on Rwanda for 1994 excluded as outlier based on Cook’s distance, leverage, and studentized residuals.

### Appendix B-2: OLS coefficients and significance (1988–2000)
- OLS reported coefficients (standard errors in parentheses):
  - Column 1 (All Countries): PROX: -1.73** (0.70); GPO: -0.90*** (0.14); INV: 0.11*** (0.02); INIT_EDU: -5.32e-03 (0.01); INIT_RGDP: -0.64*** (0.24)
  - Column 2 (All Countries): PROX: -1.38* (0.76); GPO: -0.91*** (0.14); INV: 0.14*** (0.02); INIT_EDU: -1.54e-03 (0.01); INIT_RGDP: -0.88*** (0.25)
  - Column 3 (Countries Below the Cutoff): PROX: -3.97** (1.96); GPO: -1.13*** (0.29); INV: 0.17*** (0.06); INIT_EDU: 0.05 (0.06); INIT_RGDP: -1.23 (1.04)
- Significance notation:
  - *** at the 1 percent level; ** at the 5 percent level; * at the 10 percent level.
- Additional note:
  - Restricting the sample to countries below the cutoff improves the significance for PROX, consistent with theory that a cutoff could result in stagnation at or near it.
  - The stated significance for PROX corresponds to a one-sided test.

### Appendix B-1 and B-2 first-stage / PROX description
- PROX distribution and simple regression:
  - Over 1,300 observations on PROX between 1987 and 2000.
  - About 1/7 of observations take positive values (close to the cutoffs).
  - A simple OLS of growth on PROX yields PROX coefficient = -2.22, significant at 1 percent.
- First-stage instruments and results (coefficients and standard errors):
  - Instruments: PROX(-1), INV(-1), RGDP85.
  - First-stage (All countries) R-squared: 0.80; 0.93; 0.99 for the respective dependent variables.
  - Selected coefficients (All countries): PROX(-1): 0.92*** (0.01); INV(-1): 0.94*** (0.01) in relevant equations; RGDP85: 1.00*** (3.76e-03) for INIT_RGDP.
  - First-stage (Below cutoff) R-squared: 0.81; 0.88; 0.99.
  - Selected coefficients (Below cutoff): PROX(-1): 0.90*** (0.02); INV(-1): 0.91*** (0.02); RGDP85: 0.10*** (0.01) for INIT_RGDP.
- Significance notation as above; standard errors are in parentheses.

### Model setup, calibration, and numerical solution
- Concessional lending rule:
  - Lender loans at fixed subsidized interest rate r% as long as country GNI per capita y is below cutoff y(); otherwise loans at world interest rate r.
- Borrower problem:
  - Objective: maximize Σ β^(t+1) u(c_t, k_t, D_t) (sequential, infinite horizon).
  - Constraints summarized from the source: participation constraint (3); LIC flow budget constraint (4); transition equation for capital (5) with k_{t+1} = (1-δ)k_t + ...; interest rule (1).
- Default value function:
  - v^D(k): autarky value with penalties of exclusion from future concessional lending and loss of fraction λ of output; specified in equation (8) with λ.
- Functional forms:
  - Utility: u(c) = c^(1-σ)/(1-σ)
  - Production: f(k) = A k^η
- Calibration parameters used for numerical solution:
  - η = 1/3
  - σ = 0.451
  - δβ = 0.945
  - δ = 0.1
  - λ = 0.05
  - World interest: r where 1/(1+r) ≡ β and r = 0.0582 (given as 1/(rβ−1)=0.0582 in source text)
  - TFP A = 10
  - Concessional interest rate A (concessional rate) = 1 percent
  - Cutoff y()ss y fixed at 0.7796 of steady state output
- Donor implicit cost metric:
  - α_t ≡ Σ_{t=1}^∞ [(1/(1+r))^{t} (r_t - r)] (expression preserved from source)
- Initial conditions for benchmark economy:
  - Initial income equal to 90 percent of the cutoff, or 70 percent (precisely 70.47 percent) of steady state output ss y
  - Initial debt-output ratio D/y = 1
- Solution method:
  - Value function iteration used to derive value functions and implied debt capacity function.

### Numerical results and dynamics (benchmark and comparative cases)
- Figures described (from source):
  - Figure 1: Endogenous debt ceilings h(K)/Yss as function of K/Kss.
  - Figure 2: Time series for output, consumption, investment, debt as fractions of ss y, comparing concessional loans (solid) and non-concessional loans (dashed).
  - Figure 3: Regions A–F in initial income/cutoff vs. initial debt/output space; solid line = debt overhang threshold; dash-dotted line = endogenous debt ceilings; vertical dotted line = cutoff.
  - Figure 4: Debt overhang thresholds with TFP 1% and 3% higher than benchmark.
- Key model dynamics and quantitative insights:
  - The discontinuity in h(·) from the interest rule forces a recipient to drastically reduce external debt when surpassing the cutoff, achievable only via a steep decline in consumption.
  - In the benchmark, the one-time consumption cost exceeds long-run benefits of reaching the high steady state; the country converges to the cutoff ("low" steady state).
  - Under non-concessional lending (higher interest), the perverse incentive is absent and the country reaches the high steady state, although short-run consumption is lower than under concessional lending.
- Debt overhang mechanism:
  - Artificially low cost of servicing debt (concessional rates) motivates large debt accumulation and excessive consumption, resulting in low growth.
- Dependence on initial conditions and TFP:
  - Higher initial debt increases likelihood of converging to the low steady state.
  - Lower initial capital/income increases short-run borrowing incentives and likelihood of being trapped in the low steady state.
  - Higher TFP raises the steady state output relative to the cutoff and shifts the debt overhang threshold upward, easing escape from the trap.

### Policy implications and recommendations
- Effectiveness of one-time debt relief:
  - One-time stock treatment can be effective if it moves the country below the debt overhang threshold (e.g., benchmark economy moved below the solid line in Figure 3 converges to the high steady state).
  - A country in region B with relatively high initial debt can be moved to region D via upfront debt relief and achieve growth.
- Complementary policies:
  - Policies that raise TFP amplify debt relief effectiveness: productivity or institutional improvements shift the debt overhang threshold upward (Figure 4).
  - Using freed-up resources to directly raise TFP increases the impact of debt relief (insight relevant to MDRI).
- Caveats and risks:
  - Stock treatment must be perceived as one-time; repeated debt relief with similar income eligibility may create incentives to stagnate around the cutoff to remain eligible.
  - The model assumes efficient allocation of freed resources; weak institutions may prevent effective use of sudden resource increases.
  - The model does not establish whether debt relief is the most efficient form of aid; alternatives that directly raise TFP might be more welfare-maximizing.

### Appendix A: No participation constraints — alternative benchmark
- Assumptions:
  - LIC fully precommits to honoring concessional lending conditions; no participation constraint; unlimited access to donor funds.
- Planner’s problem:
  - Maximize ∑_{t=1}^{∞} β^{t-1} u(c_t, k_t) subject to the intertemporal budget constraint and transition dynamics.
- First-order conditions and steady states:
  - μ_t = u'_c(1,t) for certain indices; β μ_{t+1} = u'_c(2,t) Π_{s=1}^{t-1} (1 + r_s)/(1 + r_{s-1}) as in source.
  - k_2 solves f'(k_2) = δ + r_2; steady-state k_ss solves f'(k_ss) = δ + r.
  - Because concessional rate < world rate, k_2 > k_ss (capital overshoots).
- Dynamics:
  - Capital overshoots in period 1; k_j = k_ss for j ≥ 3.
  - Consumption overshoots in period 1: c_2 > c_ss implied by Euler equations u'(c_3) = β(1 + r_2) u'(c_2) and u'(c_2) = β(1 + r_1) u'(c_1).
  - Debt path D_t derived from budget constraint once capital and consumption are pinned down.
- Donor’s budget expression preserved as in source.

### Conclusions (concise)
- The model explains how debt overhang can be generated in LICs through an aid-eligibility cutoff creating perverse incentives.
- Key determinants of being trapped:
  - Larger initial debt → stronger incentive to remain aid-dependent at low interest.
  - Lower initial income/capital → greater short-run borrowing and higher likelihood of trap.
  - Lower TFP → greater relative benefit of remaining at the cutoff versus attaining the high steady state.
- Policy implication: a one-time stock debt relief can help escape the poverty trap if freed resources are used effectively and especially if complemented by policies that raise TFP.

*Source: Appendix B-2 of the provided IMF working paper content.*

### References...................................................................................................15

### _wp06224 - References...................................................................................................15

### Introduction
- Purpose: Formulate Cohen and Sachs’s (1986) sovereign debt model as a concessional lending problem tailored to low-income country (LIC) environments.
- Key mechanism: An incentive cutoff—an income level above which a country loses eligibility for concessional aid (examples: IDA and PRGF).
- Main insight:
  - LICs may have an incentive to accumulate significant concessional debt and favor consumption over investment, stagnating around the cutoff and becoming permanently aid-dependent.
  - This outcome can arise even with a benevolent government that maximizes household welfare.
- Empirical linkage: The paper provides empirical evidence of growth stagnation around the cutoff.
- Policy relevance: Implications for debt relief, including evaluation of one-time stock treatments such as the Multilateral Debt Relief Initiative (MDRI), described as a 100 percent debt stock cancellation by the IMF, the IDA, and the African Development Bank for a group of low-income countries.

### Theoretical literature and model framing
- Enforcement mechanisms in sovereign debt literature:
  - Reputation models (e.g., Eaton and Gersovitz (1981)): exclusion from credit markets induces repayment when countries value consumption smoothing.
  - Sanction models: default is penalized (e.g., loss of a fraction of output or access to short-term trade credits); renegotiation may be costly and reduce trade (Rose (2005)).
  - Cohen and Sachs (1986): combine reputation and sanction elements within a neoclassical growth model where access to borrowing affects the marginal product of capital dynamics.
- Gaps in existing debt-overhang models for LICs:
  - Most models focus on middle-income countries with non-concessional private debt; they typically assume initial insolvency requiring debt relief (Krugman (1988); Sachs (1989)).
  - LICs differ: loans are highly concessional and provided largely by official creditors who are not profit maximizers nor risk-neutral; thus large debt may not deter new official lending.
- Model specification highlights:
  - Two agents: a benevolent official creditor offering subsidized loans below/at the cutoff and lending at world interest rate above the cutoff; creditor can commit, debtor cannot.
  - Imposition of a participation constraint by the creditor implies an endogenous debt ceiling.
  - The debtor (representative-agent LIC) maximizes welfare subject to the lending rule and the participation constraint.

### Empirical motivation and results
- Data and sample:
  - Unbalanced panel of 94 countries, of which 33 are low-income countries.
  - Data sources: Summers-Heston data set (version 6.1), World Bank’s World Development Indicators, and the Barro-Lee data set.
  - Cutoff: operational cutoff formally recognized by IDA donors in IDA8 in 1987; historical cutoff initially set at $250 in 1964; both cutoffs updated annually using the SDR deflator.
- Growth regression setup:
  - Dependent variable: percentage annual growth rate of real GDP per capita.
  - Explanatory variables: percentage of population growth (GPO); percentage investment share of real GDP per capita (INV); initial secondary schooling as percentage of total population in 1985 (INIT_EDU); initial real GDP per capita in 1988 (INIT_RGDP).
  - Proximity to cutoff measure: PROX, defined using a Bartlett kernel with scaling factor b controlling bandwidth:
    - b is set equal to 1/2, with similar results for b = 1/3 and b = 1/4.
    - A negative coefficient on PROX implies a negative relationship between growth and proximity to the cutoff.
- Estimation and robustness:
  - Sample period for pooled estimations reported: 1988-2000.
  - OLS result: coefficient for PROX is negative and statistically significant at the 5 percent level.
  - To address endogeneity of PROX, INV, and INIT_RGDP, two-stage least squares (2SLS) regressions were run:
    - Instruments: one-year lags for PROX and INV; log of real GDP per capita in 1985 for INIT_RGDP.
    - First-stage regressions and distribution of PROX are reported in the appendices (Appendix B-1 and Appendix B-2).
- Data treatment note:
  - Observations on Rwanda for 1994 excluded due to outlier diagnostics (Cook’s distance, leverage, and studentized residuals exceeding conventional cutoffs).

### Policy implications and debt relief
- Debt relief effectiveness:
  - The model indicates that a one-time stock treatment can be effective depending on a country’s initial conditions and total factor productivity (TFP).
  - This theoretical result is used to interpret the potential impact of initiatives like the MDRI (100 percent debt stock cancellation for a group of LICs).
- Mechanism for persistent aid dependence:
  - Concessional lending tied to an income cutoff can create incentives for LICs to remain around the cutoff, maintaining access to subsidized loans and avoiding transitions to market-rate borrowing that would raise borrowing costs.

_Italic: Source: Content extracted from _wp06224 - References...................................................................................................15 (PDF) provided by the user._

### Appendix B-2. The coefficient of PROX remains negative and significant at 5 percent.

### Appendix B-2. The coefficient of PROX remains negative and significant at 5 percent.

### Empirical evidence: growth regressions 1988-2000
- OLS results reported for:
  - All Countries (column 1)
  - All Countries (column 2)
  - Countries Below the Cutoff (column 3)
- Reported coefficients and standard errors (parentheses):
  - PROX: -1.73** (0.70); -1.38* (0.76); -3.97** (1.96)
  - GPO: -0.90*** (0.14); -0.91*** (0.14); -1.13*** (0.29)
  - INV: 0.11*** (0.02); 0.14*** (0.02); 0.17*** (0.06)
  - INIT_EDU: -5.32e-03 (0.01); -1.54e-03 (0.01); 0.05 (0.06)
  - INIT_RGDP: -0.64*** (0.24); -0.88*** (0.25); -1.23 (1.04)
- Significance notation:
  - *** significantly different from 0 at the 1 percent level
  - ** significantly different from 0 at the 5 percent level
  - * significantly different from 0 at the 10 percent level
- Note: standard errors are in parentheses.
- Additional empirical note:
  - Restricting the sample to countries below the cutoff improves the significance level for PROX, consistent with the theoretical result that the existence of a cutoff could result in economic stagnation at or around it.
  - One-sided test: the stated significance level for PROX corresponds to the one-sided test.

### Model setup: concessional lending with a cutoff and participation constraint
- Concessional lending rule (interest rate rule):
  - Lender with full access loans at fixed subsidized interest rate r% as long as country's GNI per capita y is below cutoff y().
  - Otherwise loans at world interest rate r.
- Participation constraint:
  - Value function under repayment must be greater than or equal to value function under default.
- Borrower maximization problem (sequential, infinite horizon):
  - Objective: maximize sum of discounted utility: max Σ β^(t+1) u(c_t, k_t, D_t) (equation structure preserved from source).
  - Flow budget and transition equations summarized from constraints (3)–(7) in source:
    - Participation constraint (3)
    - LIC flow budget constraint (4)
    - Transition equation for capital (5): k_{t+1} = (1-δ)k_t + ...
    - Interest rate follows rule (1)
- Default value function:
  - v^D(k) is the autarky value function with penalties: exclusion from future concessional lending and loss of fraction λ of output.
  - Default value function specified in equation (8) with λ entering the penalties.

### Functional forms, calibration, and numerical solution
- Functional forms:
  - Utility: u(c) = c^(1-σ)/(1-σ)
  - Production: f(k) = A k^η
- Parameter values used for numerical solution:
  - η = 1/3
  - σ = 0.451
  - δβ = 0.945
  - Rate of capital depreciation δ = 0.1
  - Fraction of output lost upon default λ = 0.05
  - World interest rate r where 1/(1+r) ≡ β and r = 0.0582 (given as 1/(rβ−1)=0.0582 in source text)
  - TFP level normalized at A = 10
  - Concessional interest rate A (concessional rate) = 1 percent
  - Cutoff level y()ss y fixed at 0.7796 of steady state output
- Donor implicit cost of concessional lending:
  - α_t ≡ Σ_{t=1}^∞ [(1/(1+r))^{t} (r_t - r)] (expression preserved as in source)
- Initial conditions for benchmark economy:
  - Initial income equal to 90 percent of the cutoff, or 70 percent (precisely 70.47 percent) of steady state output ss y
  - Initial debt-output ratio D/y = 1
- Numerical method:
  - Value function iteration method used to numerically derive value functions and implied debt capacity function.

### Results and dynamics: benchmark case and mechanisms
- Figures summarized:
  - Figure 1: Endogenous debt ceilings (benchmark case) — shows h(K)/Yss as function of K/Kss (graphical)
  - Figure 2: The benchmark economy — time series for output, consumption, investment, debt as fractions of ss y, comparing concessional loans (solid) and non-concessional loans (dashed)
  - Figure 3: Initial conditions and the debt overhang threshold — shows regions A–F in initial income/cutoff vs. initial debt/output space; solid line is debt overhang threshold; dash-dotted line is endogenous debt ceilings; vertical dotted line is cutoff
  - Figure 4: Debt overhang thresholds with different TFPs — shows threshold shifts when TFP is 1% and 3% higher than benchmark
- Key qualitative quantitative insights:
  - The discontinuity in h(·) produced by the interest rule implies the recipient must drastically reduce external debt when surpassing the cutoff, achievable only via a steep decline in consumption.
  - For the benchmark case, the one-time consumption cost outweighs long-run benefit of reaching the high steady state; country converges to the cutoff ("low" steady state).
  - Under non-concessional lending (higher interest), the perverse incentive is absent and the country reaches the high steady state, though short-run consumption is lower than under concessional lending.
- Debt overhang mechanism summarized:
  - Artificially low cost of servicing debt (concessional rates) motivates large debt accumulation and excessive consumption, resulting in low growth.
- Dependence on initial conditions and TFP:
  - Higher initial debt increases likelihood of converging to the low steady state.
  - Lower initial capital/income increases short-run borrowing incentives and likelihood of being trapped in the low steady state.
  - Higher TFP raises the steady state output relative to the cutoff and shifts the debt overhang threshold upward, making escape from the trap more likely.

### Implications for debt relief and policy recommendations
- One-time debt relief (stock treatment) can be effective if it moves the country below the debt overhang threshold:
  - Example: benchmark economy receiving one-time relief moving it below the solid line in Figure 3 converges to the high steady state.
  - A country with relatively high initial debt in region B can be moved to region D via upfront debt relief and achieve growth.
- Complementary policies that raise TFP amplify the effectiveness of debt relief:
  - Productivity growth or institutional improvement shift the debt overhang threshold upward (Figure 4), increasing the number of countries that lie below the threshold for a given amount of debt relief.
  - Using freed-up resources for development purposes that directly raise TFP increases the impact of debt relief (insight relevant to MDRI).
- Caveats and risks:
  - Stock treatment must be perceived as one-time; repeated debt relief with similar income eligibility may create incentives to stagnate around the cutoff to remain eligible.
  - The model assumes efficient allocation of freed resources; weak institutions may prevent effective use of sudden increases in freed resources.
  - The model does not establish whether debt relief is the most efficient form of aid; alternatives that directly raise TFP might be more welfare-maximizing.

### Conclusions (concise)
- The paper presents a theoretical model explaining how debt overhang can be generated in LICs through an aid-eligibility cutoff creating perverse incentives.
- Key determinants of being trapped in debt overhang:
  - Larger initial debt → stronger incentive to remain aid-dependent at low interest.
  - Lower initial income/capital → greater short-run borrowing and higher likelihood of trap.
  - Lower TFP → greater relative benefit of remaining at the cutoff versus attaining the high steady state.
- Policy implication: a one-time stock debt relief can help escape the poverty trap if freed-up resources are used effectively, and particularly if complemented by policies that raise TFP.

*Source: Appendix B-2 of the provided IMF working paper content.*

### Appendix A: No participation constraints

### Appendix A: No participation constraints

### Model setup and assumptions
- The LIC fully precommits to honoring the conditions of the concessional lending scheme so that there is no participation constraint.
- This environment allows unlimited access to the donor’s funds (recognized as an unrealistic assumption), but is useful to understand the role of a debt ceiling constraint.
- The planner’s problem (maximization) is:
  - ∑_{t=1}^{∞} β^{t-1} u(c_t, k_t)
- The intertemporal budget constraint is as given in the source (complex expression with sums over t and s), where r^1, r^2, k_1, D_1,t are given.
- Initially, the country can borrow at the concessional interest rate (i.e. r^2 = r%), because initial output lies below the cutoff. At any level of capital above the cutoff, borrowing is only at the world interest rate.

### First-order conditions and steady states
- First-order conditions (as stated):
  - c_{1,t}: μ_t = u'_c(1,t) for 1,t = ?
  - For 2,t ≥ 1: β μ_{t+1} = u'_c(2,t) Π_{s=1}^{t-1} (1 + r_s) / (1 + r_{s-1}) (expression shown in source as (A-2))
  - For 1,t ≥ 1: '() tt rfk δ ++ = − % (expression shown in source as (A-3))
- μ is the shadow price.
- Capital levels:
  - Period-2 capital k_2 satisfies k_2: f'(k_2) = δ + r_2 (implied by (A-3) and source text).
  - Steady-state capital k_ss satisfies f'(k_ss) = δ + r (world rate) (by (A-3) in source).
- Because the concessional interest rate is lower than the world interest rate, k_2 is greater than k_ss (k_2 > k_ss), implying capital overshoots the steady state in period 1.

### Dynamics: overshooting, consumption, and debt path
- Capital:
  - Capital overshoots steady state in period 1.
  - From period 3 onwards, capital is at its steady state level (k_j = k_ss for j ≥ 3), because as of period 2 the country no longer has access to concessional loans and its capital level exceeds the cutoff (k_2 > k_ss and k_3 = k_ss with k_ss θ > 2 in source notation).
- Consumption:
  - Consumption overshoots in period 1 (c_2 > c_ss implied by Euler equations shown in source).
  - Euler equations presented:
    - u'(c_3) = β(1 + r_2) u'(c_2)
    - u'(c_2) = β(1 + r_1) u'(c_1)
  - Since 1)1(<+rβ and 1)1(=+rβ are given in the source and u' is decreasing in c, the overshooting result follows.
- Debt:
  - Once k_2 and k_ss (and associated consumption levels) are pinned down, the path of debt D_t can be derived via the budget constraint.
- Donor’s budget:
  - The donor’s budget is given by α ≡ (1/() ) (2 r r r D + −) as stated in the source with notation: )('cu},{2ss kk},{1ss cc},{2ss DD )1/()( 2 r r r D + − ≡ α α, determined by u' and the steady-state variables in the source.

### Appendix B-1: A Description of PROX
- Data coverage:
  - There are over 1300 observations on PROX between 1987 and 2000.
  - About 1/7 of observations take positive values (i.e., only 1/7 of the observations lie close to the cutoffs).
- Empirical finding:
  - A simple OLS regression of the growth rate of real GDP per capita on PROX yields a coefficient for PROX of -2.22, statistically significant at 1 percent.
- The histogram of PROX excludes observations with PROX = 0 (figure described in source).

### Appendix B-2: The First-Stage Regressions
- Instruments:
  - Instruments for PROX and INV are the one-year lags PROX(-1) and INV(-1).
  - Instrument for INIT_RGDP is RGDP85 (log of real GDP per capita in 1985).
- First-stage regression results (coefficients and standard errors as reported):
  - First-stage regression: All countries (Dependent variables: PROX, INV, INIT_RGDP)
    - GPO: 1.70e-03 (2.45e-03); 0.03 (0.06); -0.01*** (2.22e-03)
    - INIT_EDU: 1.08e-04 (2.31e-04); 4.12e-03 (5.73e-03); 5.22e-04** (2.10e-04)
    - PROX(-1): 0.92*** (0.01); 0.59* (0.31); -0.01 (0.01)
    - INV(-1): -4.54e-04 (3.89e-04); 0.94*** (0.01); 3.18e-03*** (3.54e-04)
    - RGDP85: -2.61e-03 (4.13e-03); 0.34*** (0.10); 1.00*** (3.76e-03)
    - R-squared: 0.80; 0.93; 0.99
  - First-stage regressions: Countries that lie below the cutoff (Dependent variables: PROX, INV, INIT_RGDP)
    - GPO: -3.98e-03 (3.98e-03); -5.30e-03 (0.09); -8.64e-03** (3.42e-03)
    - INIT_EDU: 1.28e-03** (6.21e-04); 0.01 (0.01); 1.97e-03*** (5.34e-04)
    - PROX(-1): 0.90*** (0.02); 1.06** (0.47); 1.10e-03 (0.02)
    - INV(-1): 4.72e-04 (7.60e-04); 0.91*** (0.02); 1.59e-03** (6.53e-04)
    - RGDP85: -0.01 (0.01); 0.13 (0.21); 0.10*** (0.01)
    - R-squared: 0.81; 0.88; 0.99
- Significance notation (as in source):
  - *** significantly different from 0 at the 1 percent level
  - ** significantly different from 0 at the 5 percent level
  - * significantly different from 0 at the 10 percent level
- Note: standard errors are in parentheses.

### Appendix B-3: Data definitions and country coverage
- Variables (definitions and sources as listed):
  - Growth Rate: The growth rate of real GDP per capita (in percentage). Constructed from Real GDP per capita, Constant Prices: Laspeyres (RGDPL) in Summers-Heston data set, version 6.1.
  - GNI per capita: GNI per capita in current US$, Atlas methodology. Source: World Development Indicators.
  - The operational IDA cutoff in terms of GNI per capita in US$ Atlas methodology. Source: World Bank GNI/capita operational guidelines.
  - GPO: Percentage of population growth (per annum). Constructed from population (POP) in Summers-Heston data, version 6.1.
  - INV: Investment share of Real GDP per capita (in percentage per annum). Source: Summers-Heston data set, version 6.1.
  - INIT_RGDP: Real GDP per capita, Constant Prices: Laspeyres (in percentage) in 1985. Source: Summers-Heston data set, version 6.1.
  - INIT_EDU: Percentage of secondary schooling attained in the total population in 1985. Source: Barro-Lee data set.
- Country or Regional Coverage (grouped by income classification as listed in source):
  - Low-Income: Bangladesh; Benin; Cameroon; Central African Rep.; Congo, Dem. Rep.; Congo, Rep.; Gambia, The; Ghana; Guinea-Bissau; Haiti; India; Kenya; Lesotho; Malawi; Mali; Mozambique; Nepal; Nicaragua; Niger; Pakistan; Papua New Guinea; Rwanda; Senegal; Sierra Leone; Tanzania; Togo; Uganda; Zambia; Zimbabwe.
  - Lower-Middle: Algeria; Bolivia; Brazil; Dominican Rep.; Ecuador; El Salvador; Fiji; Guatemala; Guyana; Honduras; Indonesia; Jamaica; Jordan; Paraguay; Peru; Philippines; Sri Lanka; Syrian Arab Rep.; Thailand; Tunisia; Trinidad & Tobago; Turkey; Uruguay; Venezuela, RB.
  - Upper-Middle: Argentina; Barbados; Botswana; Chile; Costa Rica; Hungary; Malaysia; Mauritius; Mexico; Panama; South Africa; South Africa (listed once in source); Venezuela, RB (also appears under Lower-Middle in source list); etc. (countries listed in source text under Upper-Middle).
  - High-Income: Australia; Austria; Belgium; Canada; Cyprus; Denmark; Finland; France; Germany; Greece; Hong Kong SAR; Iceland; Ireland; Israel; Italy; Japan; Korea, Rep.; Netherlands; New Zealand; Norway; Portugal; Singapore; Spain; Switzerland; United Kingdom; United States.
  - (The source lists country names under each income heading; entries above preserve the country names as given in the source.)

*Source: _wp06224 - Appendix A: No participation constraints*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06224.pdf_