## wpiea2020227-print-pdf - 1. These developments have brought the need to plan for future deleveraging strategies to the ## Source details **Canonical URL:** [wpiea2020227-print-pdf - 1. These developments have brought the need to plan for future deleveraging strategies to the](https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020227-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2020/english/wpiea2020227-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2020/english/wpiea2020227-print-pdf.pdf.json) --- ### Overview and research question - Studies constrained efficient borrowing paths for governments facing default risk in a quantitative sovereign default framework à la Eaton and Gersovitz (1981) with long-term debt. - Provides a benchmark for designing deleveraging plans for highly indebted countries and for comparing simpler policies intended to enhance fiscal discipline. ### Model and equilibrium concepts - Framework and preferences: - Infinite-horizon small open economy with stochastic endowment stream {y_t} taking values in Y = {y_1,...,y_J} and following a Markov process. - Preferences: E[Σ_{t=0}^∞ β^t u(c_t)] with β∈(0,1), u′>0, u′′<0. - Debt and default: - Long-term bonds as in Hatchondo and Martinez (2009) with coupon parameter ∈[0,1] determining exogenous debt duration. - Lenders are competitive and risk-neutral discounting at rate r. - Default writes off entire debt; continuation value U_t drawn from distribution f_j conditional on y_t = y_j (support (−1,1)); expected continuation values finite. - Equilibrium concepts: - Markov Perfect Equilibrium (MPE): government chooses default and borrowing each period taking future strategies as given. - Ramsey equilibrium: government commits at period 0 (or after default) to an income-history-contingent borrowing path but takes ex-post default decisions as optimal; internalizes how current borrowing affects future bond prices and past welfare. ### Main technical contribution and solution method - Reduction to low-dimensional state: - Effect of borrowing decisions in period t on welfare prior to t can be encapsulated by two single-dimensional variables (under conditions verified numerically can collapse into one variable), enabling tractable algorithm for Ramsey borrowing plan. - Numerical verification of optimality: - Although U's concavity cannot be proved analytically, computation of first-order derivatives and exploration of U over b′ allow numerical verification that the found solution is a global maximum (details in Appendix C). ### Calibration and validation - Calibrated MPE to match Mexico using quarterly data from 1993 to 2018. - Targeted moments and parameters (selected): - r = 0.01 - = 0.97 -  = 0.94 - " = 1.5% -  = 0.5 - E(y) = 1 -  = 0.035 (implied debt duration = 4.8 years) - d_0 = 0.17 (Avg debt = 44.2%) - d_1 = 1.2 (Avg spread = 3.3%) - Std spread target = 2.4% - E(exclusion duration) = 3 years (implied) - Grid and numerical settings: - 50 grid points for b; 21 grid points for y; interpolation: cubic spline for V and q; iteration tolerance: max deviation < 10 6. - Validation: - MPE captures relationship between spread, income, and debt aligned with estimates for a sample of emerging economies. ### Main quantitative findings and key statistics - Welfare gains from commitment: - Welfare gain from permanently switching to a Ramsey government ranges from 0.3% to 0.7%. - Deleveraging scenario: Ramsey deleveraging welfare gain equivalent to permanent consumption increase of 0.44% relative to MPE. - Spreads and market value: - Average spread: Ramsey 0.5%; Markov 3.3%. - Mean debt (%): Data 44.2; Markov 44.0; Ramsey 39.5; One-period 26.6. - Mean debt market value (%): Data 37.7; Markov 38.2; Ramsey 26.4. - Std dev spread: Data 2.4; Markov 2.4; Ramsey 0.4; One-period 0.7. - Borrowing patterns: - Ramsey does not lower average borrowing but “fans out” borrowing: reduces borrowing in later periods and in states where default risk is most affected by borrowing, and expands borrowing in remaining states. - Ramsey implements more procyclical fiscal policy than Markov because default risk is more sensitive to borrowing in low income states. - Ramsey may buy back debt in some low-income states; Markov never buys back debt. - Deleveraging success: - Starting from debt-to-income ratio of 50% (average income, h_0 = 0): - Markov initial spread 6.6%; defaults during first six years occur in 33% of simulation samples. - Ramsey defaults during first six years in 22% of samples. - Ramsey smooths initial adjustment and achieves faster subsequent debt reduction due to higher bond prices. - Simulated moments (Table 2, long-run averages from 1,000 samples of 500 quarters; reported averages across selected 88-quarter samples): - Corr(spread, y): Data -0.4; Markov -0.8; Ramsey -0.8; One-period -0.5. - (c)=(y)1.1: Data 1.1; Markov 1.1; Ramsey 1.4; One-period 1.4. ### Optimality conditions — Markov versus Ramsey (mechanisms and intuition) - Markov first-order condition (equation (8)): - Marginal benefit today: u′(c)[ q(b′;y) + @q(b′;y)/@b′ ·  ]. - Marginal cost tomorrow: expectation over j of F_j(V′_j) u′(c′_j) [  + (1) q′_j ]. - Envelope: @V(b;y)/@b = u′(c)[  + (1) q( ^b(b;y); y) ]. - Ramsey first-order condition (equation (9) and condensed form (11), (15)): - Ramsey internalizes the effect of b_{t+1}(y_t) on utility flows prior to t; adds weights h_{q,t} and h_{V,t} (or consolidated h_t) multiplying derivatives @q_t/@b_{t+1} and @V_t/@b_{t+1}. - Recursive laws of motion for weights: - h_{q,t} = (1) F_i(V_t)/(1 + r) [ h_{q,t-1} + u′(c_{t-1}) _{t-1} h_{V,t-1} ]. - h_{V,t} = f_i(V_t) [  + (1) q_t ]/(1 + r) [ h_{q,t-1} + u′(c_{t-1}) _{t-1} h_{V,t-1} ] + F_i(V_t) h_{V,t-1}. - Initial: h_{q,0} = 0; h_{V,0} = 1. - Auxiliary variable: - h_t = h_{q,t}/h_{V,t} + u′(c_t) _t when h_{V,t} ≠ 0. - Law of motion: h_t = F_i(V_t) (1) h_{t-1}/[ f_i(V_t) [  + (1) q_t ] h_{t-1} + (1 + r) F_i(V_t) ] + u′(c_t) _t. - Key differences and implications: 1. Weight on bond-price derivative: Ramsey uses h_t, typically h_t > u′(c_t) _t, so marginal cost of issuing debt is higher under Ramsey. 2. @q/@b interpretation differs: Ramsey considers effect of changing debt stock in t+1 alone, Markov considers how b′ affects future debt stocks. 3. Next-period bond prices q_{t+1;j} differ across frameworks because Ramsey assumes path from t+2 onward invariant to marginal change in b_{t+1}. ### Role of long-term debt and maturity - Long-term bonds create hedging benefits and time inconsistency: with income uncertainty and long-term bonds, MPE is not constrained efficient and Ramsey can improve welfare by constraining future borrowing. - Optimal ex-ante duration: - If Markov government could choose ex-ante duration and commit, it would choose duration of 2.3 years (optimal  = 0.0963 vs. benchmark  = 0.035, implied 4.8 years). - Ramsey government’s optimal ex-ante duration is much higher (over 30 years in numerical exploration). - Quantitative results robust to setting debt duration to Markov’s ex-ante optimum ("2.3 years vs. 4.8 years in our benchmark"). ### Policy implications and recommendations - Commitment to constrain future borrowing materially reduces sovereign risk and improves deleveraging outcomes. - Simpler commitment devices: - Commitment to an optimal sequence of debt limits captures 60% of Ramsey welfare gains (optimal debt-limits deleveraging welfare gain = 0.26%, equivalent to 59% of Ramsey’s gain). - Debt-limits implementation details (numerical search): - Optimal sequence: transition of 8 years (T = 32 quarters), final debt limit 38.25%, slope a_1 = -0.000235, a_0 = 0.0223. - Debt limits bind during deleveraging; they impose harsher initial austerity and lower welfare relative to Ramsey. - Designing deleveraging plans should account for: - Sensitivity of bond prices to borrowing in low-income states. - Benefits of front-loading or fanning-out borrowing across states and time to minimize default risk and welfare losses. - Choice of debt instruments matters: - Commitment to longer durations can be welfare improving, especially under Ramsey-type commitment. ### Deleveraging exercises and comparisons - High-debt initial state exercise (b/y = 50%, h_0 = 0): - Markov: initial spread 6.6%; 33% probability of defaults in first six years. - Ramsey: 22% probability of defaults in first six years; Ramsey never defaults in income paths where Markov repays. - Ramsey chooses slower deleveraging in year 1 but faster thereafter due to higher bond prices and lower default risk. - Debt-limits vs Ramsey: - Debt-limits path imposes faster deleveraging and higher average net transfers to lenders during deleveraging. - Optimal debt limits achieve 59% of Ramsey welfare gain. ### Robustness exercises and sensitivity - Role of U-shock volatility (_U): - In baseline, shock to continuation utility of defaulting plays minor role; defaults mostly driven by low income realizations. - Higher _U increases role of non-income default determinants, makes bond prices sensitive to borrowing at high incomes, and moderates Ramsey’s additional procyclicality. - Ramsey achieves significant welfare gains and spread reductions for different _U values; welfare gains decline with higher _U. - No income uncertainty (" = 0): - No hedging motive for long-term debt; optimal to issue one-period bonds; Markov and Ramsey coincide. - Optimal ex-ante duration exploration: - Markov-optimal ex-ante duration implies 2.3 years and produces lower spreads but default risk remains inefficiently high; Ramsey still improves welfare. - Could not find interior Ramsey-optimal ex-ante duration up to 30 years; welfare increases with duration on that interval. ### Conclusions and avenues for future research - Method: solves Ramsey constrained-efficiency problem in sovereign default model with long-term debt by reducing history effects to low-dimensional variables and computing constrained efficient borrowing paths. - Comparative results: - Ramsey eliminates more than 80% of default risk (empirically plausible calibration) without lowering average borrowing levels; increases borrowing and debt volatility. - Welfare gain of Ramsey vs Markov ranges from 0.3% to 0.7%. - Ramsey tends to implement more procyclical fiscal policy and may buy back debt in some low-income states. - Extensions for future work: - Joint decisions on currency denomination, maturity composition, reserves accumulation, and interaction with monetary policy. - Quantify role of fiscal rules and other debt management instruments under time inconsistency. *Source — content from the provided PDF chapter "wpiea2020227-print-pdf - 1. These developments have brought the need to plan for future deleveraging strategies to the" (canonical URL provided).* ### 1. These developments have brought the need to plan for future deleveraging strategies to the ### wpiea2020227-print-pdf - 1. These developments have brought the need to plan for future deleveraging strategies to the ### Overview and research question - The paper studies constrained efficient borrowing paths for governments facing default risk in a quantitative sovereign default framework à la Eaton and Gersovitz (1981) with long-term debt. - It provides a benchmark for designing deleveraging plans for highly indebted countries and for comparing simpler policies intended to enhance fiscal discipline. ### Model and equilibrium concepts - Framework: - Infinite-horizon small open economy with stochastic endowment stream {y_t} taking values in Y = {y_1,...,y_J} and following a Markov process. - Preferences: E[Σ_{t=0}^∞ β^t u(c_t)] with β∈(0,1), u′>0, u′′<0. - Long-term bonds as in Hatchondo and Martinez (2009): a bond issued at t pays coupon stream with parameter ∈[0,1] determining exogenous debt duration; lenders are competitive and risk-neutral discounting at rate r. - Default: if government defaults it writes off entire debt and the economy receives a continuation value U_t drawn from a distribution f_j conditional on y_t = y_j (support (−1,1)); expected continuation values under default are finite. - Two equilibrium concepts: - Markov Perfect Equilibrium (MPE): government chooses default and borrowing each period taking future strategies as given; standard concept in quantitative sovereign default literature. - Ramsey equilibrium: at period 0 (or after default) government commits to an income-history-contingent borrowing path but takes ex-post default decisions as optimal; this Ramsey planner internalizes how current borrowing affects future bond prices and thus past welfare, implementing the constrained efficient borrowing path. ### Solution method and tractability result - Main technical contribution: - The effect of borrowing decisions in period t on welfare prior to t can be encapsulated by two single-dimensional variables (which under conditions verified numerically can collapse into one variable). This enables an algorithm to solve for the Ramsey government’s borrowing plan. - Although convexity of the Ramsey problem cannot be shown analytically, computation of first-order derivatives allows numerical verification that the found solution is a global maximum. ### Calibration and validation - Calibrated MPE to match data from Mexico, a representative economy with sovereign risk and a standard reference in the literature. - Verified model implications for the relationship between spread, income, and debt are aligned with estimates for a sample of emerging economies. - The MPE match to data validates the model as a laboratory for comparing Markov vs. Ramsey governments. ### Main quantitative findings and statistics - Welfare gains: - The welfare gain from permanently switching to a Ramsey government ranges from 0.3% to 0.7%, with larger gains for lower income. - Spreads: - Average spread is 0.5% in the Ramsey economy and 3.3% in the Markov economy. - Borrowing patterns under Ramsey vs. Markov: - The Ramsey government does not lower average borrowing but "fans out" borrowing: it reduces borrowing in later periods and in states where default risk is most affected by borrowing, and expands borrowing in remaining states. - Because default risk is more sensitive to borrowing in low income states, the Ramsey government implements a more procyclical fiscal policy than the Markov government. - More dispersed borrowing can lead the Ramsey government to buy back debt in some low-income states (it internalizes that raising bond prices in period t may allow better issuance prior to t). The Markov government never buys back debt. - Deleveraging success: - Starting from a state with high debt, the Ramsey government has a higher probability of completing successful deleveraging without defaults than the Markov government. - Commitment under Ramsey allows smoothing of the initial adjustment during deleveraging by effectively reducing future default risk. - Simpler commitment devices: - Commitment to an optimal sequence of debt limits (a simpler policy plan) imposes harsher initial austerity and delivers 60% of the welfare gains achieved by the Ramsey government. - Optimal ex-ante debt duration: - If the Markov government could choose the optimal ex-ante debt duration and commit to it, it would choose a duration of 2.3 years (vs. 4.8 years in the authors’ calibration). - When the Ramsey government is assumed to issue debt with the Markov government’s optimal ex-ante duration, qualitatively similar results obtain. - The optimal ex-ante duration for the Ramsey government is much higher (over 30 years). ### Mechanisms and intuition - Ramsey vs. Markov difference: - The Ramsey government internalizes how increasing borrowing in period t raises default risk after t and thus lowers bond prices at t, shrinking feasible borrowing sets prior to t. By internalizing this intertemporal price effect, the Ramsey planner implements constrained efficient borrowing. - The Ramsey government therefore distorts Markov borrowing more in states where: (i) bond prices are more sensitive to borrowing, and (ii) changes in bond prices have a larger effect on past welfare. - Role of long-term debt and maturity: - Long-term bonds create hedging benefits from active trading; in presence of income uncertainty and long-term bonds, MPE is not constrained efficient and there is room for constraining future borrowing. - Related results: optimal ex-ante maturity choices differ across equilibrium concepts; Ramsey planner prefers much longer duration. ### Policy implications and recommendations - Enhancing long-term fiscal discipline can materially reduce sovereign risk and improve the success of fiscal programs aiming to reduce debt levels. - Commitment devices that constrain future borrowing (e.g., fiscal rules or optimal debt limits) can capture a substantial fraction of the gains from full Ramsey commitment (the optimal debt limits deliver 60% of Ramsey welfare gains). - Designing deleveraging plans should account for: - The sensitivity of bond prices to borrowing in low-income states. - The benefits of front-loading or fanning-out borrowing across states and time to minimize default risk and welfare losses. - Choice of debt instruments (maturity structure) matters: commitment to longer durations can be welfare improving, especially under Ramsey-type commitment. ### Relation to existing literature (high-level) - Builds on Eaton and Gersovitz (1981) sovereign default framework and subsequent long-term debt literature (e.g., Hatchondo and Martinez, Arellano, Aguiar and Gopinath). - Contrasts with papers finding MPE is constrained efficient in some settings (Aguiar et al., 2019) by showing that with income uncertainty and long-term bonds, hedging benefits make MPE not constrained efficient. - Complements analyses of commitment to repayment and to limiting future borrowing; quantifies gains from commitment to borrowing paths and from simpler instruments like debt limits. *Italic: Source — content from the provided PDF chapter "wpiea2020227-print-pdf - 1. These developments have brought the need to plan for future deleveraging strategies to the" (canonical URL provided).* ### 3.1 Optimality condition for the Markov government ### 3.1 Optimality condition for the Markov government ### Markov government: first-order condition - Optimality condition (equation (8)): - u′(c)[ q(b′;y) + @q(b′;y)/@b′ ·  ] = J ∑ j=1  j (y) F_j(V′_j) u′(c′_j) [  + (1) q′_j ] - Definitions used: - V′_j = V(b′;y_j) - q′_j = q( ^b(b;y_j); y_j ) - c′_j = ^c(b;y_j) -  = b′ (1) b denotes number of bonds issued in the period - Envelope condition: - @V(b;y)/@b = u′(c)[  + (1) q( ^b(b;y); y) ] - Interpretation: - Left-hand side: current marginal benefit from issuing an extra bond - Government collects q(b′;y) + @q(b′;y)/@b′ ·  additional units of the consumption good when issuing an extra bond. - Second term captures cost of lowering current bond price; a lower bond price reduces proceeds. - Marginal change in current consumption weighted by u′(c). - Right-hand side: cost of transferring more debt to future periods - In states with repayment next period, government pays coupon  and carries stream (1) of coupon obligations to future periods valued by q( ^b(b;y_j); y_j ). ### Role of bond-price derivative - @q/@b′ term captures how issuing more bonds today changes current bond prices and thus proceeds. - Marginal proceeds include both immediate q(b′;y) and the price effect through @q. --- ### 3.2 Optimality condition for the Ramsey government ### Differentiability and first-order condition - Proposition 1: - U is continuously differentiable w.r.t. b_{t+1}(y_t) for all t = 0,1,... and y_t ∈ Y_t. - Optimality condition (equation (9)) for b_{t+1}(~y_t): - Effect of b_{t+1}(~y_t) on utility flows before t along the income path y_0; ~y_1; ...; ~y_t (first line) + Probability-of-arriving-at-t without defaults (second line) · [ u′(c_t) [ q_t + @q_t/@b_{t+1}(~y_t) · _t ] ∑_{J} _j(~y_t) F_j(V_{t+1;j}) u′(c_{t+1;j}) [  + (1) q_{t+1;j} ] ] = 0 - V_{t+1;j} = V_{t+1}(⃗b; (~y_t; y_j)) - q_{t+1;j} = q_{t+1}(⃗b; (~y_t; y_j)) - c_{t+1;j} = c_{t+1}(⃗b; (~y_t; y_j)) - _t(⃗b; ~y_t) = b_{t+1}(⃗b; ~y_t) (1) b_t(⃗b; ~y_{t-1}) - Interpretation: - Ramsey government internalizes effects of b_{t+1} on utility flows prior to t (first line) in addition to the t and t+1 trade-off a Markov government accounts for. - Time-inconsistency arises because Markov government only accounts for effects on t and t+1. ### Sufficiency and numerical verification - Cannot prove U is concave; condition (9) not shown sufficient analytically. - Numerical method: compute @U/@b_{t+1}(~y_t) over a range of b_{t+1}(~y_t) and approximate U shape to verify global maximum (details in Appendix C). --- ### 3.2.1 Recursive optimality condition for the Ramsey government ### Condensing cumulated effects into state variables - b_{t+1}(y_t) affects: - q_t(⃗b;y_t) via repayment probability in t+1 - V_t(⃗b;y_t) via repayment value in t - Effect on bond prices in t-1 can be written (equation (10)): - @q_{t-1}(⃗b;y_{t-1})/@b_{t+1}(y_t) = Pr(y_i | y_{t-1}) F_i(V_t(⃗b;y_t)) (1)/(1 + r) · @q_t(⃗b;y_t)/@b_{t+1}(y_t) + Pr(y_i | y_{t-1}) f_i(V_t(⃗b;y_t)) (  + (1) q_t(⃗b;y_t) )/(1 + r) · @V_t(⃗b;y_t)/@b_{t+1}(y_t) ### Proposition 2: first-order condition in condensed form - Optimal borrowing plan ⃗b̃ satisfies (equation (11)): - h_{q,t}(·) · @q_t(⃗b̃;y_t)/@b_{t+1}(y_t) + h_{V,t}(·) · @V_t(⃗b;y_t)/@b_{t+1}(y_t) = 0 - Laws of motion for weights h_{q,t}, h_{V,t}: - h_{q,t} = (1) F_i(V_t)/(1 + r) [ h_{q,t-1} + u′(c_{t-1}) _{t-1} h_{V,t-1} ] (equation (12)) - h_{V,t} = f_i(V_t) [  + (1) q_t ]/(1 + r) [ h_{q,t-1} + u′(c_{t-1}) _{t-1} h_{V,t-1} ] + F_i(V_t) h_{V,t-1} (equation (13)) - Initial values: - h_{q,0}(⃗b̃;y_0) = 0 - h_{V,0}(⃗b̃;y_0) = 1 - Interpretation: - h_{q,t} weights how a change in q_t affects expected utility flows from period 0 to t−1. - h_{V,t} weights how a change in V_t affects expected utility flows from period 0 to t−1 plus direct effect on U. - Recursive structure arises from geometric coupon assumption. ### Auxiliary variable h_t and Lemma 3 - Define (if h_{V,t} ≠ 0): - h_t(⃗b;y_t) = h_{q,t}(⃗b;y_t)/h_{V,t}(⃗b;y_t) + u′(c_t(⃗b;y_t)) _t(⃗b;y_t) (equation (14)) - Simplified optimality condition (equation (15)): - u′(c_t) q_t + @q_t/@b_{t+1}(y_t) · h_t ∑_{j=1}^J _j(y_i) F_j(V_{t+1;j}) u′(c_{t+1;j}) [  + (1) q_{t+1;j} ] = 0 - Law of motion for h_t (equation (16)): - h_t = F_i(V_t) (1) h_{t-1}/[ f_i(V_t) [  + (1) q_t ] h_{t-1} + (1 + r) F_i(V_t) ] + u′(c_t) _t - Interpretation: - h_t encapsulates relative effects of changing bond prices in period t on welfare up to t. - h_t increases with repayment probability F_i(V_t), marginal utility u′(c_t), and issuances _t; decreases with density f_i(V_t). --- ### Comparison: Markov versus Ramsey optimality conditions - Three main differences between Ramsey (equation (15)) and Markov (equation (8)): 1. Weight on bond price derivative: - Ramsey uses h_t; Markov uses u′(c_t) _t. - Typically h_t > u′(c_t) _t (h_t−1 > 0 in simulations), so marginal cost of issuing debt higher under Ramsey. - Ramsey internalizes welfare effects of changing proceeds in all periods up to t. 2. Derivative @q_t/@b_{t+1}(y_t): - In Ramsey condition it reflects effect of changing debt stock in t+1 alone. - In Markov condition @q/@b′ reflects effects of how next-period debt stock b′ affects future debt stocks. 3. Next-period bond prices: - q_{t+1;j} differ across frameworks: in Ramsey, debt path from t+2 onwards unaffected by change in b_{t+1}(y_t), so q_{t+1;j} invariant; in MPE next-period bond prices change with b′. - This produces a discrepancy in marginal cost term: - J ∑ j=1  j (y) F_j(V_{t+1;j}) u′(c_{t+1;j}) [  + (1) q_{t+1;j} ] - Special cases: - If h = 0 (e.g., period 0 or after default), trade-offs in optimality conditions coincide though @q/@b_t and q_{t+1;j} may still differ. - With one-period debt ( = 1), Ramsey and Markov incentives coincide: h_t = u′(c_t) b_{t+1} and h is not a relevant state variable. --- ### Recursive representation and equilibrium recursion (constrained efficient recursion) - Using Lemma 3, constrained efficient borrowing plan solves recursion with additional history variable h: 1) V solves (R): - V(b;y_i;h) = u(c) + ∑_j _j(y_i) [ F_j(V′_j) V′_j + ∫_{V′_j}^1 U f_j(dU) ] - Resource constraint (R_c): c = y_i  b + q(b′; y_i; h′)  ; with b′ = (1) b +  - First-order condition (R_{b′}): - u′(c) q(b′;y_i;h′) + ∆q(b′;y_i;h′) h′ ∑_j _j(y) F_j(V′_j) u′(c′_j) [  + (1) q′_j ] = 0 - ∆q function: - ∆q(b′;y_i;h′) = ∑_j _j(y_i) f_j(V′_j) u′(c′_j) [  + (1) q′_j ]^2 /(1 + r) - V′_j = V(b′; y_j; h′) - c′_j = ^c(b′; y_j; h′) - q′_j = q( ^b(b′; y_j; h′); y_j; ^h(b′; y_j; h′) ) - Law of motion for h (R_{h′}): - h′ = F_i(V(b;y_i;h)) (1) h /[ f_i(V(b;y_i;h)) [  + (1) q(b′; y_i; h′) ] + (1 + r) F_i(V(b;y_i;h)) ] + u′(c)  2) Policy functions { ^c; ^b; ^h } satisfy (R_c), (R_{b′}), (R_{h′}) for all (b;y_i;h) given { q; V }. 3) Bond price q satisfies: - q(b′; y_i; h′) = 1/(1 + r) ∑_j _j(y_i) F_j( V(b′; y_j; h′) ) [  + (1) q( ^b(b′; y_j; h′); y_j; ^h(b′; y_j; h′) ) ] - Remarks on solution: - ∆q based on closed-form derivative: - @q_t(⃗b;y_t)/@b_{t+1}(y_t) = ∑_{j=1}^J _j(y) f_j(V_{t+1;j}) u′(c_{t+1;j}) [  + (1) q_{t+1;j} ] /(1 + r) - Policy functions solved by system of non-linear equations in (b′;h′). - Unique solution found in baseline parameterization; sign change of left-hand side of (R_{b′}) around root indicates global maximum (numerical details in Appendix C). --- ### Calibration (summary of parameter values and targets) - Data: quarterly data from 1993 to 2018 for Mexico (business cycle properties analogous to previous literature). - Utility: constant relative risk aversion u(c) = c^{1γ}/(1γ), γ ̸= 1. - Income process: discretization of log(y_t) =  log(y_{t-1}) + (1)  + "t. - Continuation utility of defaulting U ∼ N(V_D(y), _U); V_D(y) given by equation (17). - Exclusion stochastic with exit probability denoted (probability of exiting this period). - Default income loss: proportion d_0 + d_1 y. - Calibration targets and parameter values (from Table 1 and text): - r = 0.01 (standard) - = 0.97 (standard) -  = 0.94 (Mexico GDP) - _" = 1.5% (Mexico GDP) [notation in table:  " = 1.5%] -  = 0.5 - E(y) = 1 -  = 0.035 - d_0 = 0.17 (Avg debt = 44.2%) - d_1 = 1.2 (Avg spread = 3.3%) - _U calibrated to match Std spread = 2.4% (table lists _U unspecified but Std spread target 2.4%) - E(exclusion duration) = 3 years (implied calibrations) - Debt duration (implied) = 4.8 years (from  = 0.035 and targeting Mexican sovereign spreads) - Additional notes: - Volatility of sovereign spread targeted: mean = 3.3% and standard deviation = 2.3% (J.P. Morgan EMBI spread Q1 1994–Q1 2018) - Domestic risk aversion γ included in calibration to match consumption-volatility target - Four parameters calibrated via simulations: d_0, d_1, _U, γ --- ### Quantitative results (high-level summary as presented) - Main findings (Section 5 summary): - Economy with a Ramsey government features: - Significantly lower spread - Higher average market value of debt claims - Welfare gain from permanently switching to Ramsey government ranges from 0.3% to 0.7% - Ramsey government borrowing behavior: - Reduces borrowing in low-income states where bond prices are more sensitive to borrowing - Expands borrowing in remaining states - Deleveraging: - Ramsey government has higher probability of completing a successful deleveraging (without default) compared with a Markov government, even when smoothing initial adjustment - Robustness exercises: - Main quantitative findings do not depend on assumed debt duration - Optimal ex-ante debt duration significantly higher for Ramsey government than Markov government - Explores role of shocks to the utility of defaulting *Source: wpiea2020227-print-pdf - 3.1 Optimality condition for the Markov government* ### 5.1 Default risk and welfare ### 5.1 Default risk and welfare ### Long-run moments (Table 2) - Data vs simulations: long-run moments reported for Data, Markov, Ramsey, One-period. - Table 2 key figures (as reported): - Mean debt (% ): Data 44.2; Markov 44.0; Ramsey 39.5; One-period 26.6 - Mean debt mkt. value (%): Data 37.7; Markov 38.2; Ramsey 26.4 - Mean spread (in %): Data 3.3; Markov 3.3; Ramsey 0.5; One-period 0.2 - Std dev spread: Data 2.4; Markov 2.4; Ramsey 0.4; One-period 0.7 - Corr(spread, y): Data -0.4; Markov -0.8; Ramsey -0.8; One-period -0.5 - (c)=(y)1.1: Data 1.1; Markov 1.1; Ramsey 1.4; One-period 1.4 - Simulation details: - 1,000 samples of 500 periods (quarters) each; select samples of 88 periods without defaults and with the last default occurring at least 30 periods before the beginning of the sample. - Reported values are averages across selected samples. - Debt level in simulations calculated as present value of coupon payments discounted at the risk-free rate, i.e. b = (r+)·(?) - Market value of debt corresponds to q·b. ### Spread, borrowing, and market valuation - Constrained efficient (Ramsey) allocation features significantly lower spreads than the Markov equilibrium: - Ramsey government eliminates more than 80% of both the average level and the volatility of the sovereign spread (relative to the Markov government). - While mean face debt is lower with the Ramsey government, the mean market value of debt is higher — implying the Ramsey government does not simply borrow less on average to lower default risk; market value of debt (present discounted value of net payments to lenders under risk neutral, competitive lenders) is a better gauge of expected debt payments than face value. ### Welfare gains from commitment and consumption timing - Welfare gains from permanently switching from the MPE (Markov perfect equilibrium) to a Ramsey government: - Left panel of Figure 2: welfare gains between 0.3% to 0.7% (measured as permanent proportional change in consumption that would leave consumers indifferent). - With parameterization such that (1+r)<1 (as in much quantitative sovereign default literature), improved allocation under Ramsey characterized by higher consumption in earlier periods. - Deleveraging scenario welfare: - Welfare gain when deleveraging plan implemented by Ramsey government is equivalent to a permanent consumption increase of 0.44% relative to the MPE. ### One-period bonds vs long-term debt - If government issues only one-period bonds (no time inconsistency), default risk is lower than the benchmark MPE with long-term debt, but welfare is lower. - Consumers are better off with a Markov government that issues debt with a duration of 4.8 years than with one that issues one-period debt. - Rationale: one-period debt eliminates time inconsistency but magnifies exposure to shifts in borrowing opportunities (rollover risk), prompting the government to issue less debt and reduce consumption frontloading. ### Cyclicality of fiscal policy and borrowing incentives (Figure 3) - Borrowing policies (Markov and Ramsey) are hump-shaped across income levels: less issuance at very high and very low income realizations. - Two forces determine borrowing across income: - Consumption-smoothing: borrowing lower when income is higher. - Default-price effect: borrowing increases default risk and lowers bond price; effect stronger at lower income levels. - At sufficiently low income, bond-price sensitivity dominates and equilibrium borrowing can increase with income. - Ramsey-specific features: - Ramsey government shows a jump in borrowing at the lowest income realizations: Ramsey is less concerned about lowering bond prices where repayment is unlikely because lowering the price there has small effect on previous-period welfare. - Ramsey curbs borrowing more than Markov at moderately low income levels because bond prices and prior welfare are more sensitive to borrowing in those states; thus Ramsey implements a more procyclical fiscal policy. - For y_t < 0.96, derivative @q_t/@b_{t+1} becomes significantly negative in Ramsey economy; h_q_t/h_V_t is significantly away from zero for y_t > 0.92, so for 0.92 < y_t < 0.96 Ramsey optimally curbs borrowing more than Markov. - For income realizations above the mean, @q_t/@b_{t+1} ≃ 0 in Ramsey economy (debt chosen commands almost no default risk marginally). - Ramsey may choose to buy back debt in states where Markov never repays, because Ramsey internalizes the benefit of higher bond prices on prior welfare. ### Borrowing dispersion and role of history variable (Figure 4) - Ramsey borrowing path entails significant dispersion even conditional on income, especially at low income realizations — highlighting the disciplining role of a history variable. - Higher borrowing dispersion at lower incomes under Ramsey is reflected in higher consumption dispersion (as in Table 2). - Ramsey repays in some low-income realizations where Markov never repays, enabling Ramsey to borrow to buffer consumption drops over that income range. ### Cross-country evidence on sensitivity of spreads to debt - In a panel of 33 emerging economies: spread increases more with debt in years with high spread (statistical significance less than 1%). - Spread also increases more with debt in years with low growth (weak significance, less than 17%). - Appendix D shows the MPE captures the relationship between debt, spread, and income in the data. ### Deleveraging exercises (high-debt initial state) - Setup: simulations start with average income, debt-to-income ratio of 50%, and initial history h_0 = 0 for Ramsey government. - For Markov government: - Initial spread 6.6% - Defaults during first six years of deleveraging occur in 33% of simulation samples. - For Ramsey government: - Probability of default during first six years reduced to 22%. - In none of the income paths for which Ramsey defaults does the Markov government find it optimal to repay. - Dynamics: - In first year, Ramsey chooses on average a slower deleveraging and lowers net transfers to creditors (y_t - c_t), reflecting ability to moderate early adjustment due to domestic consumers' relative impatience and (1+r)<1. - After first year, Ramsey achieves faster debt reduction with similar net transfers in repayment states because higher bond prices enable collecting same proceeds with fewer bonds issued. - Lower probability of default under Ramsey is immediately reflected in lower sovereign spreads. - Volatility and welfare tradeoffs: - Ramsey increases consumption and debt volatility compared to Markov. - Ramsey welfare gain equivalent to permanent consumption increase of 0.44% relative to MPE. - Model likely understates gains from commitment because it omits windfall gains to bondholders and does not reflect lower spreads in higher aggregate income. ### Deleveraging with debt limits vs Ramsey - Markov government constrained by a sequence of debt limits solved numerically; debt limit constraint: government not forced to buy back bonds when limit < (1-)b. - Parameterization and search over family of debt limit sequences with b(0)=b_0; b(t)-b(t+1)=a_0 + a_1 t; b(t)=b(1) for t ≥ T. Optimization over {T; b(1); a_1}. - Optimal sequence of debt limits found: - transition of 8 years (T= 32), - final debt limit of 38.25%, - slope a_1 = -0.000235, - a_0 = 0.0223. - Outcomes with optimal debt limits: - Average debt path almost coincides with debt limits (limits binding in simulations); debt volatility close to zero during deleveraging. - Compared with Ramsey deleveraging, debt-limits path imposes much faster deleveraging and forces higher average net transfers to lenders during deleveraging. - Ramsey is more successful in reducing spreads than Markov with debt limits. - Welfare: optimal debt-limits deleveraging achieves welfare gain of 0.26%, equivalent to 59% of the welfare gain achieved by a Ramsey government. *Source: 5.1 Default risk and welfare (from provided IMF content).* ### 5.4 Robustness exercises ### 5.4 Robustness exercises ### Debt maturity and time inconsistency - The inefficiency studied hinges on the government issuing long-term debt. The paper assumes an exogenous maturity structure. - Appendix E.1 robustness checks: - Main quantitative results do not change if the debt duration is set to the optimal ex-ante value preferred by the Markov government: "2.3 years vs. 4.8 years in our benchmark". - Once the Ramsey government corrects the time inconsistency in borrowing decisions, the optimal ex-ante debt duration is significantly higher than in the benchmark ("more than 30 years"). ### Role of the shock to the utility cost of defaulting - The shock to the utility cost of defaulting enables exploiting first-order conditions to solve the Ramsey government’s problem. - The importance of the shock is given by the volatility of Uconditional on income ( U ), which in the benchmark calibration is disciplined by the sovereign spread volatility. - Findings for parameter values in Table 1: - The shock to U plays a minor role in determining default decisions. - For most income realizations, the government either repays or defaults almost surely regardless of the realization of U (as shown in the bottom-right panel of Figure 3). - Simulations confirm that defaults are mostly driven by low income realizations. - Interpretation and sensitivity: - Shocks to U can be interpreted as default determinants not related to income. - A higher  U captures a higher role for those non-income default determinants, making bond prices sensitive to borrowing even at high income realizations. - Equation (11) implies that in such scenarios the Ramsey planner would want to restrict borrowing at high income realizations. - Appendix E.2 verifies that higher values of  U moderate the additional procyclicality of fiscal policy chosen by the Ramsey government. - Appendix E.2 also shows that the Ramsey government achieves significant welfare gains and significant reductions in sovereign default risk for different values of  U. --- ### 6 Conclusions ### Method and policy mechanism - The paper solves a quantitative Eaton-Gersovitz sovereign default model with long-term debt in which a Ramsey government decides the entire borrowing plan taking as given ex-post optimal default decisions. - The Ramsey government improves upon the Markov government by taking into account how borrowing decisions in period t affect borrowing opportunities, and thus welfare, prior to t. - The effect of borrowing decisions in t on utility flows prior to t can be encapsulated by two single dimensional variables, and by one variable under conditions verified in the quantitative exercise, enabling a tractable algorithm to find the constrained efficient borrowing policy. ### Comparative results: Ramsey vs. Markov government - The Ramsey government distorts borrowing decisions more when: - bond prices are more sensitive to borrowing, and - changes in bond prices have a more significant effect on past utility flows. - For empirically plausible parameter values: - "more than 80% of the default risk is eliminated by a Ramsey government, without lowering average borrowing levels." - An efficient reduction in default risk prescribes a higher volatility in borrowing and debt levels. - The welfare gain of having a Ramsey government instead of a Markov government ranges from "0.3% to 0.7%", depending on initial income. - Fiscal policy characteristics under the Ramsey government: - More procyclical fiscal policy that includes debt buybacks in some states with low income. - This higher procyclicality is mitigated in economies where default determinants are less correlated with income. - Deleveraging and initial conditions: - Starting from a state with high debt, the Ramsey government has a higher probability of completing a successful deleveraging (without defaults) than the Markov government. - By effectively reducing future default risk, the Ramsey government can afford to smooth out the initial adjustment during the deleveraging path. - The Markov government’s commitment to a simple deleveraging plan consisting of a sequence of debt limits imposes harsher initial austerity and delivers "60% of the welfare gains achieved by the Ramsey government." ### Extensions and avenues for future research - The methodology could be extended to study other aspects of debt management where time inconsistency matters: - Issuing debt in local and foreign currency. - Joint fiscal and monetary policy decisions. - Issuing debt with different maturities. - Accumulating debt and assets (foreign reserves, see Bianchi et al., 2018). - Extending the analysis to quantify these issues and inform the role of fiscal rules to regulate other debt management tools is highlighted as an interesting avenue for future research. *Source: wpiea2020227-print-pdf - 5.4 Robustness exercises* ### References ### wpiea2020227-print-pdf - References ### References overview - Bibliographic record of works cited, focusing on sovereign debt, default, debt maturity, fiscal policy, and computational methods. - Key recurring themes in the literature cited: - Sovereign default models and mechanisms: Arellano (2008); Aguiar and Gopinath (2006, 2007); Eaton and Gersovitz (1981); Bulow and Rogoff (1988, 1991); Chatterjee and Eyigungor (2012, 2015); Bocola and Dovis (2019). - Debt maturity, dilution, and restructuring: Aguiar et al. (2019, 2020); Hatchondo and Martinez (2009, 2013, 2017); Hatchondo, Martinez, and Sapriza (2010); Bolton and Jeanne (2009). - Welfare, fiscal rules, and sovereign risk: Aguiar, Amador, and Fourakis (2020); Faraglia et al. (2019); Gelpern et al. (2020); IMF (2017); Thornton and Vasilakis (2017). - Computational and theoretical methods: Marcet and Marimon (2019); Mateos-Planas and Ríos-Rull (2016); Rottger (2019); Lorenzoni and Werning (2019). - Additional empirical and applied contributions on credit ratings, spreads, reserves, and pandemic-related debt issues: Jaramillo and Tejada (2011); Bianchi, Hatchondo, and Martinez (2018); Reinhart and Rogoff (2020); Gelpern, Hagan, and Mazarei (2020). ### Appendix A — Definition of the Markov Perfect Equilibrium - Notation and key objects: - x′ denotes the next-period value of variable x. - ^b denotes the borrowing strategy followed by future governments (number of bonds outstanding at the end of the period). - ^d denotes the defaulting strategy followed by future governments. - V denotes the continuation value under repayment when future governments follow strategies ^b and ^d. - Continuation value under repayment: V(b;y) = u(c) + Σ_{j=1}^J π_j(y) { ∫ U′ [ d′_j U′ + (1 − d′_j) V(b′; y_j) ] f(U′) dU′ } (equation (19)), where c = y − τ b + q(b′; y′)[b′ − (1 − τ) b], d′_j = ^d(b′; y_j; U′), and b′ = ^b(b;y). - Bond price condition (lenders break even in expectation): q(b′; y) = [ Σ_{j=1}^J π_j(y) { ∫ U′ [1 − ^d(b′; y_j; U′)] [ τ + (1 − τ) q(b′′_j; y_j) ] f(U′) dU′ } ] / (1 + r) (equation (20)). - Government repayment choice and value: V(b;y) = Max_{b′} { u(c) + Σ_{j=1}^J π_j(y) { ∫ U′ [ d′_j U′ + (1 − d′_j) V(b′; y_j) ] f(U′) dU′ } } subject to c and d′_j as above (equation (21)). - Government default decision: d = Max_{d ∈ {0,1}} { d U + (1 − d) V(b;y) } (equation (22)). - Markov Perfect Equilibrium characterization: strategies ^d and ^b such that: - (a) Given ^d and ^b, V satisfies (19). - (b) Given ^d and ^b, q satisfies (20). - (c) ^d solves (22) for all b, y, U. - (d) ^b solves (21) for all b, y. ### Appendix B — Proofs (selected highlights) - B.1 Proof of Proposition 1 — differentiability and propagation of effects: - Step 1: Differentiability in t+1. The derivative @V_{t+1}/@b_{t+1}(~y_t) = −u′(c_{t+1}(...)) [ τ + (1 − τ) q_{t+1}(...) ] exists and is continuous. - Step 2: Differentiability of effects in t. Expression for @q_t/@b_{t+1}(~y_t) given as a weighted integral over future states; implies @V_t/@b_{t+1}(~y_t) is continuously differentiable. - Step 3 and 4: The same logic extends backward through previous periods; derivatives exist and are continuous for s = t − 1, t − 2, ..., 0. - B.2 Proof of Proposition 2 — recursive structure of Ramsey optimality conditions: - The derivative of the Ramsey government’s objective w.r.t. b_{t+1}(~y_t) can be written as a function of derivatives of repayment values and bond prices up to t. - Lemma 4: For any n, @q_{t−n}/@b_{t+1}(~y_t) = Pr(~y_{t−n+1}; ... ; ~y_t | ~y_{t−n}) [ A^q_{t;t−n} @q_t/@b_{t+1}(~y_t) + B^q_{t;t−n} @V_t/@b_{t+1}(~y_t) ] (equation (25)). - Lemma 5: Recursive expressions for coefficients A^q_{t;t−n} and B^q_{t;t−n} (equations (27)–(30)), including base case for n = 1: - A^q_{t;t−1} = (1 − τ) F_i(V_t) / (1 + r) (equation (29)). - B^q_{t;t−1} = f_i(V_t) [ τ + (1 − τ) q_t ] / (1 + r) (equation (30)). - The Ramsey necessary condition (for the optimal borrowing plan ⃗b*) is: for all t and y_t ∈ Y_t, h^q_t(⃗b*; y_t) @q_t(⃗b*; y_t)/@b_{t+1}(y_t) + h^V_t(⃗b*; y_t) @V_t(⃗b*; y_t)/@b_{t+1}(y_t) = 0 (equation (39)). - Lemma 6: Law of motion for welfare weights h^q_t and h^V_t: - h^q_t = (1 − τ) F_i(V_t) / (1 + r) [ h^q_{t−1} + u′_{t−1} σ_{t−1} h^V_{t−1} ] (equation (37)). - h^V_t = f_i(V_t) [ τ + (1 − τ) q_t ] / (1 + r) [ h^q_{t−1} + u′_{t−1} σ_{t−1} h^V_{t−1} ] + F_i(V_t) h^V_{t−1} (equation (38)). - B.3 Proof of Lemma 3 — connection with first-order conditions and value derivatives: - Uses expressions (14), optimality condition (11), and the derivative @V_t/@b_{t+1}(y_t) equal to u′(c_t) [ q_t + @q_t/@b_{t+1}(y_t) σ_t(...) ] minus an expectation term involving next-period default densities. ### Appendix C — Computation - Markov Perfect Equilibrium (MPE) computation: - Iteration on the value and bond price functions as in Hatchondo et al. (2010). - Initial iteration corresponds to the final period of the finite-horizon model; equilibrium is the limit of the finite-horizon game. - Numerical settings: - 50 grid points for b. - 21 grid points for y. - Interpolation: cubic spline for V and q over grid [0:4;4]. - No extrapolation over b. - Iteration tolerance: maximum deviation across iterations for V and q below 10 6. - Ramsey government computation: - Solve the fixed point described for h′ and b′, with initial iteration identical to last period of finite-horizon model. - Assumes h ∈ [h̲; h̄].16 - Same grids for b and y as for MPE; linear interpolation over h. No extrapolation over b or h. - Computational challenges: solving non-linear equations (R_b′)-(R_h′) for (h′; b′) and ensuring global maximization. - Solving for h′ given (b; y_i; h; b′): - Solve nonlinear equation h′ − g(h′) = 0 where g(h′) = F_i(V)(1 − τ) h̄ f_i(V)[ τ + (1 − τ) q ] + (1 + r) F_i(V) h + u′(c) [ b′ − (1 − τ) b ] (equation (40)), with V = V(b; y_i; h) and q = q(b′; y_i; h′). - Figure 6 (described): Finding fixed point h′ = g(h′); example assumes (b, h) take average simulation values, mean income y_i, and b′ = ^b(b; h; y_i). Blue dashed line depicts g(h′); left vertical axis is the 45 degrees line and right axis is g(h′). - Numerical bounds and verification: - Imposed bounds: h = 0 and ̄h = 6, and verified these bounds are never binding in simulations.16 *Source: wpiea2020227-print-pdf - References* ### 1. Ifhg(h)>0, we imputeh ### wpiea2020227-print-pdf - 1. Ifhg(h)>0, we imputeh ### Solving for h′ when h′ − g(h′) = 0 (root-finding procedure) - Procedure steps: - 1. If hg(h)>0, we impute h′ = h. We do this to avoid extrapolating in h. - 2. If hg(h)<0, we impute h′ = h. - 3. If 1. and 2. do not hold, we search over a grid fh1;:::;hng starting from h1 = h and find the lowest index i with hig(hi)<0 and hi+1g(hi+1)>0. We then search for a root within the interval (hi; hi+1) using a bisection method. - Representative case: - Figure 6 presents a representative case where the root h′ satisfies h′g(h′) = 0 for (h; h). The figure shows the slope of g is significantly below the 45 degrees line, validating the conjecture of an increasing h′g(h′). - Uniqueness checks: - Alternative procedure 1: - a) Impute h′ = h only when hig(hi)>0 for all i = 1;:::;n. - b) Impute h′ = h only when hig(hi)<0 for all i = 1;:::;n. - c) Otherwise, use a bisection method to find a root in [h; h]. - Result: the fixed point originally found also satisfies fixed point conditions using this alternative procedure. - Alternative procedure 2: - Same as original but start searching for h′ in step 3 from the highest h on the grid (instead of from the lowest). - Result: no difference — the fixed point features a unique solution for h′g(h′) = 0 for all (b; yi; h; b′). ### Finding the global maximum for the Ramsey government’s borrowing plan - First-order condition and constructed function: - The borrowing plan chosen by the Ramsey government (⃗b) satisfies - @U(⃗b; yt)/@b t+1(yt) = Pr(yt|y0) [ [ht − u′(ct) t] @qt/@b t+1(yt) + @Vt/@b t+1(yt) ] = 0 for all t and yt. (Equation (41), arguments omitted for simplicity.) - For each initial state (b; h; yi), define - O(b′) = ∫b′b [ ... ] db′ (Equation (42)), where: - V′j = V(b′; yj; h′) - q′j = q(b′; yj; h′) - W(b; h; yi; b′) = u(c) + ∑j j(yi) [ F j(V′j) V′j + ∫V′j U f j(dU) ] - The value of h′ used in equation (42) is the one that solves (R h′) given (b; yi; h; b′). - Role and limitations of O(b′): - O approximates the shape of the Ramsey government’s objective with respect to b t+1 = b′ after arriving at t with (b t; h t; y) = (b; h; yi). - The integrand in equation (42) is proportional to the derivative of the Ramsey government’s objective; the constant of proportionality is Pr(yt|y0) h V t−1(⃗B; yt), where ⃗B = ⃗b for all components except for b t+1(yt). - O is useful to identify local maxima but not necessarily the global maximum because it omits the dependence of h V t−1(⃗B; yt−1) on the choice of b t+1(yt). - Numerical strategy: - Approximate O over a grid for b′ and use its shape to verify whether the candidate optimum is at boundaries b, b or interior. - If interior, find b′ = Argmax b′∈{b1;:::;bn} O(b′) and use a non-linear equation solver with initial guess b′ to solve equation (R b′). - Empirical shapes and checks: - Figure 7 shows the approximated shape of O for average values of (b, h, y) in the simulations; a flat segment corresponds to b′ choices where the government buys back so many bonds that current consumption is too low to make repayment optimal for almost any realization of the continuation value of defaulting U. - Uniqueness of local maxima: - After finding functional fixed points {V; ^b; ^h; ^c; q}, verify fixed point conditions when computing b′ via two alternative procedures: - i) choose the local maximum with the lowest b′ - ii) choose the local maximum with the highest b′ - Result: both local maxima coincide for all (b; h; yi) on the grid, indicating the solution found is a global maximum. ### Robustness checks and model variants - Robustness without income uncertainty: - Infinite-horizon model with " = 0 (no income uncertainty), all other parameters benchmark. - Result: No hedging motive for long-term debt; optimal to issue one-period bonds, eliminating time inconsistency and making optimal solution for Markov and Ramsey governments coincide. - Finding: Optimal paths of consumption with one-period bonds and those implied by the algorithm (long-term bonds with commitment) are almost identical. - Optimal ex-ante debt duration (E.1): - Method: choose  in an initial period with zero debt and income equal to the mean, commit to bonds with that  thereafter; welfare relative to one-period bonds. - Optimal ex-ante  = 0:0963, implying average debt duration of 2.3 years. - Observation: 2.3 years is around half the value used in the parameterization. - Simulation results for duration 2.3 years (Table 5): - Markov vs Ramsey: - Mean debt-to-income ratio (in %) : 35.4 (Markov) ; 35.1 (Ramsey) - Mean debt market value (in %) : 34.3 (Markov) ; 34.8 (Ramsey) - Mean spread (in %) : 1.5 (Markov) ; 0.5 (Ramsey) - Std dev spread : 1.8 (Markov) ; 0.6 (Ramsey) - (c)=(y : 1.3 (Markov) ; 1.4 (Ramsey) - Findings: - Markov government issues less debt and at a lower spread for duration of 2.3 years relative to benchmark long-term structure. - Default risk remains inefficiently high; a Ramsey government would choose a borrowing path with lower default risk and higher average borrowing. - Welfare gain from permanently switching to a Ramsey government in a period with no debt and mean income is 0.16% (compared with 0.42% for benchmark parameterization). - Additional numerical exploration: - Could not find an interior solution for ex-ante optimal duration for the Ramsey problem up to 30 years; welfare always increases with duration over that interval. - Duration of 30 years corresponds to  = 0:0016. - Shocks to the utility cost of defaulting (E.2): - Table 6 reports simulations for different values of U while keeping other benchmark parameters. - General patterns: - For both Ramsey and Markov governments, higher U implies lower debt levels and higher mean and standard deviation of sovereign spreads. - Higher U increases the mass of states where default is optimal; lenders anticipate worse outcomes and offer worse bond price schedules; government borrows less. - Higher U increases sensitivity of bond prices to debt at high income states, motivating the Ramsey government to distort borrowing in those states to moderate procyclicality. - Despite differences, Ramsey government achieves significant welfare gains and reductions in sovereign default risk for different U values; welfare gains decrease with U. - Selected numerical rows from Table 6 (rows in bold correspond to baseline calibration): - Markov: - U = 0.08: Std. dev. V D = 4.3 ; Mean debt (%) = 3.0 ; Mean spread (%) = 2.2 ; Std dev spread = 1.1 - U = 0.10: Std. dev. V D = 4.3 ; Mean debt (%) = 3.3 ; Mean spread (%) = 2.4 ; Std dev spread = 1.1 - U = 0.25: Std. dev. V D = 0.7 ; Mean debt (%) = 3.9 ; Mean spread (%) = 2.8 ; Std dev spread = 1.1 - U = 0.75: Std. dev. V D = 6.4 ; Mean debt (%) = 8.9 ; Mean spread (%) = 4.7 ; Std dev spread = 1.0 - U = 1.00: Std. dev. V D = 2.5 ; Mean debt (%) = 12.1 ; Mean spread (%) = 5.2 ; Std dev spread = 1.0 - Ramsey: - U = 0.08: Std. dev. V D = 0.4 ; Mean debt (%) = 0.5 ; Mean spread (%) = 0.4 ; Std dev spread = 1.4 ; Welfare gain (%) = 0.43 - U = 0.10: Std. dev. V D = 0.3 ; Mean debt (%) = 9.5 ; Mean spread (%) = 0.5 ; Std dev spread = 0.4 ; Welfare gain (%) = 0.45 - U = 0.25: Std. dev. V D = 3.4 ; Mean debt (%) = 5.0 ; Mean spread (%) = 0.9 ; Std dev spread = 0.6 ; Welfare gain (%) = 0.41 - U = 0.75: Std. dev. V D = 1.7 ; Mean debt (%) = 8.4 ; Mean spread (%) = 4.2 ; Std dev spread = 2.1 ; Welfare gain (%) = 0.17 - U = 1.00: Std. dev. V D = 1.3 ; Mean debt (%) = 8.4 ; Mean spread (%) = 3.5 ; Std dev spread = 1.0 ; Welfare gain (%) = 0.13 ### Cross-country empirical evidence (D) - Data and setup: - Use annual data from a sample of 33 emerging market countries, 1994 to 2015. - Subtract to GDP growth rates the average growth rate for each country (GDP assumed stationary in the model). - Remove country-year observations in which the country was in default according to Standard & Poor’s definition. - Empirical strategy mirrors Edwards (1984, 1986), Akitoby and Stratmann (2008), Jaramillo and Tejada (2011): regress log(spread) on explanatory variables. - Simulated regressions (Table 3) using Markov government simulations: - Regression (1): - Constant = -1.713 - Public debt to GDP = 0.167 - Real GDP growth = -0.045 - R2 = 0.637 - Regression (2): - Constant = 0.677 - Public debt to GDP = 0.106 - Real GDP growth = -0.0291 - Debt × I(GDPMean spread) = 0.014 - R2 = 0.816 - Notes: - Annual variables created from quarterly model simulations. - Dependent variable: log(spread). - Dummy I(...) = 1 when condition satisfied, 0 otherwise. - Panel regressions on actual data (Table 4): - Estimation: fixed effects panel regression with robust standard errors; model: log(Spread)it = + Xit + i + it. - Summary of coefficients (standard errors in parentheses): - Public debt to GDP: 0.020 (0.004) in Regression (1); 0.019 (0.004) in (2); 0.004 (0.002) in (3); 0.020 (0.004) in (4); 0.017 (0.003) in (5) - Real GDP growth: -0.042 (0.006); -0.034 (0.009); -0.031 (0.004); -0.036 (0.007); -0.036 (0.006) - Reserves to GDP: -0.033 (0.017); -0.033 (0.017); -0.022 (0.010); -0.033 (0.017); -0.027 (0.014) - Net gov borrowing to GDP: 0.020 (0.014); 0.021 (0.015); 0.017 (0.009); 0.012 (0.015); -0.023 (0.017) - VIX: 0.034 (0.003); 0.034 (0.003); 0.0194 (0.004); 0.035 (0.003); 0.028 (0.004) - Interaction terms: - Debt_it × I(growth_it < mean_i(growth)) = 0.002 (0.001) in Regression (3) - Debt_it × I(spread_it > mean_i(spread)) = 0.013 (0.001) in Regression (3) - NGB_it × I(growth_it < mean_i(growth)) = 0.020 (0.016) in Regression (4) - NGB_it × I(spread_it > mean_i(spread)) = 0.102 (0.016) in Regression (5) - Sample: - Observations = 523 for all regressions. - R-squared: 0.77 (Regression 1), 0.77 (2), 0.87 (3), 0.77 (4), 0.82 (5). - Number of countries = 33. - Findings and comparison with model: - All regressions show spread increases with debt and decreases with GDP growth. - Model is consistent and implies a remarkably similar coefficient for the growth rate; the model’s coefficient for debt is higher than the one in the data. - Regression (2) suggests spread may be more sensitive to debt when aggregate income is below long-run mean, but statistical significance weak (17% in simulations). - Regression (3) suggests spread increases more with debt when spread is above its mean. - Regression (4) suggests spread increases more with net government borrowing but at low significance (19%). - Regression (5) suggests spread increases more with net government borrowing in years with high spread. *Italic: Source — wpiea2020227-print-pdf (excerpts provided).* --- _Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020227-print-pdf.pdf_