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---

### Index rebalancings as passive demand shocks — conceptual framework
- Investors heterogeneous in activism; passive demand T_i,t(w_i,t) captures holdings of semi- and fully passive investors, is perfectly inelastic, and shifts with index weights w_t = (w_1t,...,w_Nt).
- Market-clearing for bond i with fixed supply B_i:
  - B_i = A_i,t + T_i,t(w_i,t), where A_i,t is active demand.
  - An exogenous increase in T_i,t reduces the supply available to active investors (residual supply).
- Reduced-form inverse demand elasticity estimated from price reactions to passive-demand shifts:
  - ˆη_i = (−) Δq_i,t / ΔT_i,t  (B_i − T_i,t) / q_i,t. (Equation (1) as presented)
- Identification challenges:
  - Changes in w_i,t may be endogenous to asset prices or coincide with issuances/redemptions.
  - Price reactions may reflect shifts in expected payoffs (intrinsic value), not only demand slope.
  - Illustrated cases:
    - Panel (a): fixed expected payoffs — higher T_i,t with inelastic active demand raises price.
    - Panel (b): higher expected payoffs following ΔT_i,t shift active demand upward, confounding elasticity estimates.

### Empirical strategy — using EMBIGD rebalancings to identify exogenous supply shifts
- Index focus: J.P. Morgan EMBIGD (emerging market sovereign and quasi-sovereign U.S. dollar-denominated bonds issued in international markets).
  - EMBIGD tracked by funds with combined assets under management (AUM) of around US$300 billion in 2018.
  - EMBIGD uses a cap rule (diversified face amount) rather than pure market-cap weights.
- Flows implied by rebalancings (FIR) for country c at rebalancing date t:
  - FIR_c,t ≡ Δ~T_c,t / q_c,t−1 (B_c,t−1 − w_c,t−1 W_t−1). (Equation (2))
  - Δ~T_c,t ≡ (w_c,t − w_BH,c,t) W_t. Definitions:
    - w_c,t ≡ q_c,t B_c,t f_c,t / (q_t I_t) — benchmark weight where f_c,t ≤ 1 is diversification coefficient and DFA = f_c,t B_c,t.
    - w_BH,c,t ≡ w_c,t−1 q_c,t / q_c,t−1 q_t / q_t−1 = q_c,t f_c,t−1 B_c,t−1 / (q_t I_t−1) — buy-and-hold weight.
    - W_t denotes AUM passively tracking EMBIGD.
  - Interpretation: a 1 p.p. FIR implies a 1% reduction in the supply of the country’s bonds available to active investors at the time of rebalancing.
- Endogeneity concerns and empirical remedies:
  - Restrict analysis to rebalancing events where B_c,t = B_c,t−1 (no new issuances, repurchases, or removals due to maturity).
  - Construct instrument Z_c,t based on a synthetic index using diversified face amounts only:
    - ~w_c,t ≡ f_c,t B_c,t / Σ_c f_c,t B_c,t. (Equation (3) general form)
    - Under B_c,t = B_c,t−1, instrument simplifies to:
      - Z_c,t ≡ ( f_c,t Σ_c f_c,t B_c,t − f_c,t−1 Σ_c f_c,t−1 B_c,t−1 ) / ( f_c,t−1 Σ_c f_c,t−1 B_c,t−1 ). (Equation (4))
  - Instrument variation sources:
    - Cross-event fluctuations due to changes in others' diversified face amounts (Σ_c f_c,t B_c,t).
    - Cross-country heterogeneity within an event induced by the EMBIGD cap rule, since f_c,t changes differently across countries when others are added/removed.
- Key insight: by restricting to constant face amounts and leveraging the cap rule, Z_c,t variation is driven by index composition mechanics and is uncorrelated with contemporaneous price shocks.

### Cap rule mechanics — example and implications
- Diversified face amount (DFA) computation (Equation (5) as presented):
  - DFA_c,t =
    - ICA_t × 2 if FA_c,t = FA_max,t
    - ICA_t + (ICA_t / (FA_max,t − ICA_t)) (FA_c,t − ICA_t) if FA_c,t > ICA_t
    - FA_c,t if FA_c,t ≤ ICA_t
  - ICA_t = Index Country Average; FA_max is the largest face amount.
- Illustrative 5-country example with entrant F:
  - Capping and DFA adjustments can occur even when face amounts remain constant, producing cross-country heterogeneity exploitable for identification.
- Implication: cap rule induces exogenous-like variation in ~w_c,t and Z_c,t when face amounts of others change.

### Estimation strategy and econometric specifications
- Estimation window and design:
  - Focus on 5-day symmetric windows around each rebalancing date (authors use [−5,5]; primary window for average reactions is 5-day symmetric).
  - Use an instrumented difference-in-differences design with first-stage instrument Z_c,t for FIR_c,t.
- Main 2SLS specification (Equation (6) as presented):
  - log(q_i,t,h) = θ_c(i),t + θ_b(i),t + γ 1_{h∈Post} + β (FIR_c(i),t × 1_{h∈Post}) + X_i,t + ε_i,t,h.
    - FIR_c(i),t instrumented by Z_c,t in first stage.
    - q_i,t,h is price of bond i at rebalancing event t, h trading days before or after release; h = 1 is first trading day after J.P. Morgan releases index composition.
    - θ_c(i),t country-month fixed effects; θ_b(i),t bond-characteristics-month fixed effects.
    - X_i,t includes monthly bond controls: face amount and beginning-of-month spread.
    - Coefficient β measures how a 1 p.p. increase in FIR influences average change in log price around rebalancing.
- Preferred specification:
  - Use bond-month fixed effects θ_i,t to exploit within- and across-event variation.
  - Leverage cap rule with month-1_{h∈Post} fixed effects to focus on cross-country variation within an event.
- Leads-and-lags dynamic specification (Equation (7) as presented):
  - log(q_i,t,h) = θ_c(i),t + θ_b(i),t + Σ_{h∉−2} γ_h 1_h + Σ_{h∉−2} β_h (FIR_c(i),t × 1_h) + X_i,t + ε_i,t,h.
    - 1_h dummy variables for trading day h in [−5,5].
    - Allows testing for parallel trends and exploring dynamic FIR effects.

### Empirical headline result (from section 5.3 summary)
- A 1 p.p. reduction in effective bond supply leads to a 30 basis point increase in bond prices (empirical estimate).

### Data, sample, and summary statistics (Section 3.4)
- Data sources and sample period:
  - Majority of variables sourced from J.P. Morgan.
  - Sample period: 2016 to 2018.
  - Individual bond prices from Datastream.
  - Bond characteristics from J.P. Morgan Markets.
  - Morningstar used for asset holdings of funds benchmarked against EMBIGD and EMBI Global Core for 2016–2017.
  - Final dataset: 131,820 bond-time observations for 751 bonds across 68 countries.
- Construction of FIR and passive-share adjustment:
  - Passive Share = 100 − Active Share, where Active Share estimated at country level using Cremers and Petajisto (2009) method.
  - Average Passive Share weighted by each fund’s AUM yields estimated passive fund share of 50%.
  - W_t computed by adjusting AUM benchmarked against EMBIGD using rescaling factor of 50%.
  - For comparison: bond-level Active Share yields a value-weighted average of 72%; Cremers and Petajisto (2009) show an average value-weighted Active Share fluctuating between 55% and 80%.
  - Appendix Table D2 reports results using alternative shares of passive funds; qualitative implications remain the same.
- Data cleaning and exclusion criteria:
  - Drop stripped spreads below 0 or above 5,000 basis points.
  - Drop observations of Zc,t below the 5th percentile or above the 95th percentile.
  - Exclude bond-month observations with daily returns below the 1st percentile or above the 99th percentile.
- Key summary statistics (Table 2):
  - log(Price): Mean 4.64; Std. Dev. 0.13; 25th Pctl 4.59; 75th Pctl 4.68; Min 3.07; Max 5.19.
  - Instrumented FIR (%): Mean −0.15; Std. Dev. 0.20; 25th Pctl −0.32; 75th Pctl 0.00; Min −0.66; Max 0.23.
  - Stripped spread (bps): Mean 278; Std. Dev. 288; 25th Pctl 128; 75th Pctl 3560; Min 4904.
  - EIR duration (%): Mean 6.36; Std. Dev. 3.92; 25th Pctl 3.48; 75th Pctl 7.71; Min −0.03; Max 19.08.
  - Average life (years): Mean 9.6; Std. Dev. 8.9; 25th Pctl 4.0; 75th Pctl 9.9; Min 1.0; Max 99.8.
  - Face amount (billion U.S. dollars): Mean 1.3; Std. Dev. 0.8; 25th Pctl 0.7; 75th Pctl 1.6; Min 0.5; Max 7.0.
  - Notes:
    - Stripped Spread definition as in source.
    - EIR Duration definition as in source.
    - Average Life definition as in source.
- FIR instrument distribution and first-stage evidence:
  - FIR and Zc,t residualized using rebalancing-month and country fixed effects.
  - First-stage scatter shows R-squared = 86% between FIR and Z instrument.
  - Instrumented FIR distribution:
    - Mean = −0.15
    - Median = −0.14
    - Std. Dev. = 0.21
    - Values range from −0.7% to 0.25%

### Recursive government problem and bond pricing (Section 4.3)
- Recursive equilibrium and state space:
  - Focus on a Recursive Markov Equilibrium (RME); state vector (h,B,s) with h current default status, B beginning-of-period stock of debt, s = (y,τ) exogenous states.
  - Passive demand specified as T′ = T(τ,B′) with τ exogenous following fτ(τ′|τ).
  - Market-clearing: B′ = A′(.) + T(τ,B′), where A′(.) is end-of-period active demand.
- Resource constraint and government value functions:
  - c(h = 0,B,y,τ;B′) = y + q(y,τ,B′)B′ − (1−λ)B − (λ + (1−λ)ν)B.
  - c(h = 1,y) = y − φj(y).
  - V(y,τ,B) = Max { V^r(y,τ,B), V^d(y) } with standard Bellman equations for repayment and default presented.
- Bond pricing and inverse supply elasticity:
  - q(y,τ,B′) = β⋆ E_{s′|s} [ R(y′,τ′,B′) ] Ψ(y,τ,B′), where β⋆ ≡ 1/r_f.
  - R(y′,τ′,B′) = (1 − d(y′,τ′,B′)) [ λ + (1 − λ) ( ν + q(y′,τ′,B′′) ) ].
  - Inverse supply elasticity ε ≡ ∂log q(.) / ∂log B′ decomposes into:
    - ∂log E_{s′|s} R′(.) / ∂log B′ (elasticity of expected repayment; weakly negative)
    - ∂log Ψ(.) / ∂log B′ (price decline due to downward-sloping active demand)
  - Ψ(.) acts as disciplining device limiting government debt in quantitative analysis.
- Secondary markets and link with empirical analysis:
  - Two instances of secondary market trading per period to match high-frequency elasticity.
  - High-frequency reduced-form elasticity:
    - ˆη = (−) ( ∆q / ∆T′ ) ( B′ − T(τ,B′) ) / q_{SM,0}(y,τ,B′ ).
  - Decomposition: ˆη = η + α, where
    - η ≡ (−) ( ∆Ψ / ∆T′ ) ( B′ − T(τ,B′) ) / Ψ_{SM,0}(y,τ,B′ )
    - α ≡ (−) ( ∆ER′ / ∆T′ ) ( B′ − T(τ,B′) ) / E_{y′,τ′|y,τ} R(y′,τ′,B′ )
  - Two channels emphasized:
    - Persistent τ affects future Ψ(.) and expected payoffs.
    - τ changes affect government’s policies B′(.) and d(.), feeding back into expected payoffs and Ψ(.).

### Calibration and targeted moments (Argentina benchmark, quarterly)
- Preferences and processes:
  - CRRA u(c) = c^{1−γ} / (1−γ) with γ = 2.00.
  - Output AR(1): log(y′) = ρ_y log(y) + ε′_y, ε′_y ∼ N(0,σ_y).
  - Default output cost φ(y) = max { d̄_0 y + d̄_1 y^2 , 0 } with d̄_0 < 0, d̄_1 > 0.
  - Passive demand proportional: T′ = τ × B′.
  - τ AR(1): log(τ′) = (1−ρ_τ) log(τ⋆) + ρ_τ log(τ) + ε′_τ, ε′_τ ∼ N(0,σ_τ).
- Downward-sloping Ψ(.) functional form:
  - Ψ(y,τ,B′) = exp [ −κ ( V_{s′|s}(R′(.)) / E_{s′|s}(R′(.)) ) × ( B′ − T′ − Ā ) ], with κ ≥ 0 and Ā average holdings of active investors.
- Calibration tables (Table 5):
  - Panel a: Fixed Parameters
    - γ Risk aversion 2.00
    - r Risk-free interest rate 0.01
    - λ Debt maturity 0.05
    - ν Debt services 0.03
    - θ Reentry probability 0.0385
    - ρ_y Output, autocorrelation 0.93
    - σ_y Output, shock volatility 0.02
    - τ⋆ Share of passive demand 0.123
    - ρ_τ FIR, autocorrelation 0.66
    - σ_τ FIR, shock volatility 0.02
    - d̄_0 Default cost—level −0.264
    - d̄_1 Default cost—curvature 0.31
  - Panel b: Calibrated Parameters
    - β Discount rate 0.951
    - κ Slope parameter 72.0
    - Ā Active investors demand 0.455
- Calibration targets and fit (Table 6 targeted moments):
  - E[SP] Bond spreads: Data 472bp, Model 476bp
  - σ(SP) Volatility of spreads: Data 200bp, Model 135bp
  - E[B/Y] Debt to output: Data 55%, Model 54%
  - E[Ψ] Inconvenience yield: Data 1.0, Model 1.002
  - ˆη Reduced-form elasticity: Data −0.3, Model −0.29

### Decomposition of reduced-form elasticity and role of persistence
- Decomposition: ˆη = η + α where η structural (Ψ channel) and α captures expected repayment channel.
- Quantitative findings (Table 7 and text):
  - Baseline: ˆη −0.29; η −0.17; Bias, 1 − η/ˆη 40%
  - Lower persistence (ρ_τ = 0.25): ˆη −0.24; η −0.18; Bias 25%
  - Low persistence (ρ_τ = 0.50): ˆη −0.26; η −0.17; Bias 34%
  - Higher persistence (ρ_τ = 0.80): ˆη −0.32; η −0.17; Bias 48%
- Interpretation:
  - The bias (share of reduced-form elasticity attributable to endogenous changes in expected repayment) increases with persistence of τ.
  - Structural elasticity η typically accounts for less than two-thirds of ˆη in baseline calibration.
- Conclusion: Issuers’ endogenous responses to supply-shifting shocks and changes in expected repayment are critical; neglecting them can bias identification of structural demand elasticities by more than one-third.

### Implications of a downward-sloping demand (Section 5.3)
- Comparison with perfectly elastic demand (κ = 0) — unconditional moments (Table 8):
  - E(SP) Bond spreads: Baseline 476bp; Perfectly elastic 910bp
  - σ(SP) Volatility of spreads: Baseline 135bp; Perfectly elastic 466bp
  - E(B/y) Debt to output: Baseline 54%; Perfectly elastic 52%
  - E(d) Default frequency: Baseline 3.84%; Perfectly elastic 4.74%
  - σ(B)/σ(y) Standard deviation of debt, relative to output: Baseline 1.06; Perfectly elastic 1.84
  - ρ(∆B,y) Correlation between issuances and output: Baseline 0.37; Perfectly elastic 0.51
  - ρ(SP,y) Correlation between spreads and output: Baseline −0.68; Perfectly elastic −0.51
- Mechanisms:
  - Government internalizes both default-probability channel and investors’ inelastic demand when choosing B′.
  - Two factors produce lower default rate and spreads under inelastic demand:
    1. Optimal debt policy B′(y,τ,B) constrained by the price impact of issuance (incentive to refrain from very large issuance).
    2. Pricing consequences: for low B′, q(.) higher in inelastic case due to lower default risk (not convenience yield).
  - Net effect: inelastic demand acts as a disciplining device lowering default risk and borrowing costs.
- Response to output shocks and pro-cyclicality:
  - Pro-cyclicality of debt issuance is dampened under inelastic demand.
  - Positive output shock: issuance response muted under inelastic demand; consumption expansion smaller.
  - Negative output shock: under inelastic demand, contraction in debt is less pronounced because spreads fall as active holdings decline.
- Distributional and business-cycle statistics:
  - Debt-to-output distribution less dispersed under inelastic demand; standard deviation of debt ~30% smaller.
  - Debt-to-output correlation with output smaller; spreads-output correlation larger in magnitude under inelastic demand.
- Welfare implications (certainty equivalent consumption, CEC):
  - CEC positive: households prefer world with inelastic investors because disciplining effect reduces default risk and borrowing costs.
  - CEC decreases as stock of debt increases due to higher inconvenience yield at higher B.
- Overall quantitative and empirical summary:
  - Empirical estimate: 1 p.p. reduction in effective bond supply → 30 basis point increase in bond prices.
  - Structural decomposition: over one-third of this response attributable to endogenous changes in expected repayment.
  - Inelastic demand significantly influences optimal debt issuance and default policy, dampens pro-cyclicality, reduces default risk and spreads, and biases reduced-form elasticity estimates if issuer responses ignored.

### Microfoundations and quantitative solution (Appendices B–C and Appendix figures)
- Active-investor demand first-order approximation yields closed-form price with Ψ(.) term:
  - qi,t = (Et(Ri,t+1) − rf) × [1 − κit Vt(Ri,t+1)/Et(Ri,t+1) × (Bi,t − Ti,t − ̄Ait)].
- Two investor interpretations produce analogous pricing kernels:
  - Risk-averse mean–variance investors aggregate into κRA,t.
  - Risk-neutral investors with VaR constraint aggregate into κVaR,t.
- Numerical solution details:
  - Discretization via Tauchen: 31 gridpoints for y; 15 for τ; B grid with 250 points from B = 0 to B = 1.2.
  - Algorithm uses Brent’s method, cubic spline interpolation, and a convexification device with εV ∼ N(1, σv2) where σv2 = 2.25×10−6.
- Within-period and impulse-response quantitative results:
  - For a 5% increase in passive demand:
    - Bond prices increase by 1%; about half of that explained by increase in expected repayment.
    - One-period-ahead default risk decreases more than 10%.
    - Government raises (lowers) its debt by less than 1% for a 5% increase (decrease) in passive demand.
  - Counterfactual with fixed debt shows larger immediate price impact (~1%) relative to optimal debt adjustment (~0.50%).
- Appendix figures illustrate diversified vs. non-diversified country face amounts and EMBI Global country-level weights for December 2018.

_Italic: Source: wpiea2024227-print-pdf — IMF Working Paper No. WP/2024/227._

### Section 4 formulates a sovereign debt model with endogenous default and inelastic investors,

### Section 4 formulates a sovereign debt model with endogenous default and inelastic investors,

### Index rebalancings as passive demand shocks — conceptual framework
- Investors are heterogeneous in activism; passive demand, T_i,t(w_i,t), captures holdings of semi- and fully passive investors, is perfectly inelastic, and shifts with index weights w_t = (w_1t,...,w_Nt).
- Market-clearing for bond i with fixed supply B_i: B_i = A_i,t + T_i,t(w_i,t), where A_i,t is active demand. An exogenous increase in T_i,t reduces the supply available to active investors (residual supply).
- Reduced-form inverse demand elasticity estimated from price reactions to passive-demand shifts:
  - ˆη_i = (−) Δq_i,t / ΔT_i,t  (B_i − T_i,t) / q_i,t . (Equation (1) as presented)
- Identification challenges:
  - Changes in w_i,t may be endogenous to asset prices or coincide with issuances/redemptions.
  - Price reactions may reflect shifts in expected payoffs (intrinsic value), not only demand slope — illustrated in Figure 1:
    - Panel (a): fixed expected payoffs — higher T_i,t with inelastic active demand raises price.
    - Panel (b): higher expected payoffs following ΔT_i,t can shift active demand upward, confounding elasticity estimates.

### Empirical strategy — using EMBIGD rebalancings to identify exogenous supply shifts
- Index focus: J.P. Morgan EMBIGD (emerging market sovereign and quasi-sovereign U.S. dollar-denominated bonds issued in international markets).
  - EMBIGD tracked by funds with combined assets under management (AUM) of around US$300 billion in 2018.
  - EMBIGD uses a cap rule (diversified face amount) rather than pure market-cap weights.
- Flows implied by rebalancings (FIR) for country c at rebalancing date t:
  - FIR_c,t ≡ Δ~T_c,t / q_c,t−1 (B_c,t−1 − w_c,t−1 W_t−1). (Equation (2))
  - Δ~T_c,t ≡ (w_c,t − w_BH,c,t) W_t. Definitions:
    - w_c,t ≡ q_c,t B_c,t f_c,t / (q_t I_t) — benchmark weight where f_c,t ≤ 1 is diversification coefficient and DFA = f_c,t B_c,t.
    - w_BH,c,t ≡ w_c,t−1 q_c,t / q_c,t−1 q_t / q_t−1 = q_c,t f_c,t−1 B_c,t−1 / (q_t I_t−1) — buy-and-hold weight.
    - W_t denotes AUM passively tracking EMBIGD.
  - Interpretation: a 1 p.p. FIR implies a 1% reduction in the supply of the country’s bonds available to active investors at the time of rebalancing.
- Endogeneity concerns and empirical remedies:
  - FIR can be affected by sovereign issuances or mechanically correlated with past price changes.
  - Restrict analysis to rebalancing events where B_c,t = B_c,t−1 (no new issuances, repurchases, or removals due to maturity).
  - Construct instrument Z_c,t based on a synthetic index using diversified face amounts only:
    - ~w_c,t ≡ f_c,t B_c,t / Σ_c f_c,t B_c,t. (Equation (3) general form)
    - Under B_c,t = B_c,t−1, instrument simplifies to:
      - Z_c,t ≡ ( f_c,t Σ_c f_c,t B_c,t − f_c,t−1 Σ_c f_c,t−1 B_c,t−1 ) / ( f_c,t−1 Σ_c f_c,t−1 B_c,t−1 ). (Equation (4) as presented)
  - Instrument sources of variation:
    - Cross-event fluctuations due to changes in others' diversified face amounts (Σ_c f_c,t B_c,t).
    - Cross-country heterogeneity within an event induced by the EMBIGD cap rule, since f_c,t changes differently across countries when others are added/removed.

### Cap rule mechanics — example and implications
- Diversified face amount (DFA) computation (Equation (5) as presented):
  - DFA_c,t =
    - ICA_t × 2 if FA_c,t = FA_max,t
    - ICA_t + (ICA_t / (FA_max,t − ICA_t)) (FA_c,t − ICA_t) if FA_c,t > ICA_t
    - FA_c,t if FA_c,t ≤ ICA_t
  - ICA_t = Index Country Average (average country-level face amount used to compute DFA); FA_max is the largest face amount.
- Illustrative 5-country example (A,B,C,D,E) with a new entrant F:
  - Panel a of Table 1: face amounts and DFA before and after rebalancing show capping for some countries and DFA adjustments even when face amounts remain constant.
  - Panel b: synthetic weights ~w_c,t and percentage changes Z_t illustrate heterogeneity: capped countries exhibit smaller decreases in weights when a new country is added, producing cross-country variation exploitable for identification.
- Key insight: by restricting to constant face amounts and leveraging the cap rule, Z_c,t variation is driven by index composition mechanics and is uncorrelated with contemporaneous price shocks.

### Estimation strategy and econometric specifications
- Estimation window and design:
  - Focus on 5-day symmetric windows around each rebalancing date (the authors use [−5,5] in leads-and-lags; primary window for average reactions is 5-day symmetric).
  - Use an instrumented difference-in-differences design with first-stage instrument Z_c,t for FIR_c,t.
- Main 2SLS specification (Equation (6) as presented):
  - log(q_i,t,h) = θ_c(i),t + θ_b(i),t + γ 1_{h∈Post} + β (FIR_c(i),t × 1_{h∈Post}) + X_i,t + ε_i,t,h.
    - FIR_c(i),t instrumented by Z_c,t in first stage.
    - q_i,t,h is price of bond i at rebalancing event t, h trading days before or after release; h = 1 is the first trading day after J.P. Morgan releases index composition (release occurs during trading hours on the last business day of each month, so h = 1 corresponds to that same day).
    - 1_{h∈Post} equals 1 in the h days after rebalancing and 0 in the h days before.
    - θ_c(i),t are country-month fixed effects; θ_b(i),t are bond-characteristics-month fixed effects (maturity, rating, bond type: sovereign or quasi-sovereign).
    - X_i,t includes monthly bond controls: face amount and beginning-of-month spread.
    - Coefficient of interest β measures how a 1 p.p. increase in FIR influences the average change in log price of bonds around rebalancing.
- Preferred specification:
  - Replace country-month and bond-characteristics-month fixed effects and bond controls with bond-month fixed effects θ_i,t to exploit within- and across-event variation.
  - Leverage cap rule with month-1_{h∈Post} fixed effects to focus on cross-country variation within an event.
- Leads-and-lags dynamic specification (Equation (7) as presented):
  - log(q_i,t,h) = θ_c(i),t + θ_b(i),t + Σ_{h∉−2} γ_h 1_h + Σ_{h∉−2} β_h (FIR_c(i),t × 1_h) + X_i,t + ε_i,t,h.
    - 1_h are dummy variables equal to 1 for trading day h in [−5,5].
    - Allows testing for parallel trends and exploring dynamic FIR effects.

*Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024227-print-pdf.pdf*

### 3.4    Data and Summary Statistics

### 3.4    Data and Summary Statistics

### Data sources and sample period
- Majority of variables sourced from J.P. Morgan.
- Sample period: 2016 to 2018.
- Individual bond prices from Datastream.
- Bond characteristics (maturity, duration, etc.) from J.P. Morgan Markets.
- Morningstar used to obtain asset holdings of funds benchmarked against the EMBIGD and EMBI Global Core for 2016–2017.
- Final dataset comprises 131,820 bond-time observations for 751 bonds across 68 countries.

### Construction of the FIR measure and passive-share adjustment
- A key input is the assets under management (AUM) of funds that track the EMBIGD,Wt.
- J.P. Morgan reports assets benchmarked to their indexes but does not distinguish passive vs. active funds.
- For each fund, compute Passive Share = 100 − Active Share, where Active Share is the measure developed by Cremers and Petajisto (2009).
  - Active Share is estimated at the country level (using country weights in index and funds’ portfolios).
  - For portfolios, bonds are assigned to a country only if included in the EMBIGD.
- Average Passive Share weighted by each fund’s AUM yields an estimated passive fund share of 50%.
- Wt is computed by adjusting AUM benchmarked against the EMBIGD using a rescaling factor of 50%.
- For comparison: bond-level Active Share yields a value-weighted average of 72%; Cremers and Petajisto (2009) show an average value-weighted Active Share fluctuating between 55% and 80%.
- Appendix Table D2 reports results using alternative shares of passive funds; qualitative implications remain the same.

### Data cleaning and exclusion criteria
- Drop stripped spreads below 0 or above 5,000 basis points.
- Drop observations of Zc,t below the 5th percentile or above the 95th percentile (to avoid extreme Zc,t driven by large, pre-announced changes in the EMBIGD).
- Exclude bond-month observations with daily returns below the 1st percentile or above the 99th percentile.
- Rationale: identification strategy relies on most information being known on the last business day of the month; extreme values or pre-announced index changes would violate this.

### Key summary statistics (Table 2)
- log(Price): Mean 4.64; Std. Dev. 0.13; 25th Pctl 4.59; 75th Pctl 4.68; Min 3.07; Max 5.19.
- Instrumented FIR (%): Mean −0.15; Std. Dev. 0.20; 25th Pctl −0.32; 75th Pctl 0.00; Min −0.66; Max 0.23.
- Stripped spread (bps): Mean 278; Std. Dev. 288; 25th Pctl 128; 75th Pctl 3560; Min 4904. [Note: preserve numeric entries exactly as in source table.]
- EIR duration (%): Mean 6.36; Std. Dev. 3.92; 25th Pctl 3.48; 75th Pctl 7.71; Min −0.03; Max 19.08.
- Average life (years): Mean 9.6; Std. Dev. 8.9; 25th Pctl 4.0; 75th Pctl 9.9; Min 1.0; Max 99.8.
- Face amount (billion U.S. dollars): Mean 1.3; Std. Dev. 0.8; 25th Pctl 0.7; 75th Pctl 1.6; Min 0.5; Max 7.0.
- Notes:
  - Stripped Spread is the difference between a bond yield-to-maturity and the corresponding point on the U.S. Treasury spot curve, where the value of collateralized flows are “stripped” from the bond.
  - EIR Duration measures the sensitivity of dirty prices to parallel shifts of the U.S. interest rates, expressed as the percentage change of dirty price if all U.S. interest rates change by 100 basis points.
  - Average Life is the weighted average period until principal repayment.
  - Sources: Bloomberg, Datastream, J.P. Morgan Markets, Morningstar Direct, and authors’ calculations.

### FIR instrument, distribution, and first-stage evidence (Figure 2 and narrative)
- FIR and instrument Zc,t are residualized using rebalancing-month and country fixed effects.
- First-stage scatter (Panel (a)) shows a clear positive relationship between FIR and the Z instrument with an R-squared value of 86%.
- Instrumented FIR distribution (Panel (b)):
  - Mean = −0.15
  - Median = −0.14
  - Std. Dev. = 0.21
  - Values range from −0.7% to 0.25%, with more negative than positive observations.
- Interpretation: more negative values are consistent with an increase over time in the number of bonds included in the EMBIGD; for countries with constant face amounts, inclusion of bonds from other countries typically reduces sample countries’ weights.

### Empirical implications for identification
- Restricting analysis to countries with constant face amounts mitigates confounding changes in weights due to additions in the EMBIGD.
- Excluding extreme Zc,t values reduces contamination from large, pre-announced index changes that would violate the assumption that most information is known on the last business day of the month.

_Italic: Source: wpiea2024227-print-pdf — 3.4 Data and Summary Statistics (IMF)._

### 4.3    Government Problem:  Recursive Formulation

### 4.3    Government Problem:  Recursive Formulation

### Recursive equilibrium and state space
- The analysis focuses on a Recursive Markov Equilibrium (RME) and represents the government’s infinite-horizon decision problem as a recursive dynamic programming problem.
- Passive demand is specified as T′ = T(τ,B′) with τ exogenous following a continuous Markov process fτ(τ′|τ).
- Market-clearing: B′ = A′(.) + T(τ,B′), where A′(.) is end-of-period active demand.
- State vector: (h,B,s) where h is current default status, B is beginning-of-period stock of debt, and s = (y,τ) are exogenous states.

### Resource constraint and government value functions
- Resource constraints:
  - c(h = 0,B,y,τ;B′) = y + q(y,τ,B′)B′ − (1−λ)B − (λ + (1−λ)ν)B.
  - c(h = 1,y) = y − φj(y).
  - q(y,τ,B′) is the price of a unit of debt; B′ − (1−λ)B are new issuances; (λ + (1−λ)ν)B are current debt services.
- Government’s choice over default d ∈ {0,1}:
  - V(y,τ,B) = Max { V^r(y,τ,B), V^d(y) }.
- Repayment Bellman equation (if not in default):
  - V^r(y,τ,B) = Max_{B′} u(c) + β E_{s′|s} V(y′,τ′,B′)
  - Subject to c = y + q(y,τ,B′)B′ − (1−λ)B − (λ + (1−λ)ν)B.
- Default value function (excluded from debt markets, passive demand zero during default; exit with probability θ):
  - V^d(y) = u(y − φ(y)) + β E_{s′|s} [ θ V(y′,τ′,0) + (1−θ) V^d(y′) ].

### Bond pricing and inverse supply elasticity
- Bond price faced by government (for exogenous state {y,τ}):
  - q(y,τ,B′) = β⋆ E_{s′|s} [ R(y′,τ′,B′) ] Ψ(y,τ,B′),
    - where β⋆ ≡ 1/r_f is the lenders’ discount factor,
    - R(y′,τ′,B′) ≡ R′(.) denotes next-period repayment function,
    - Ψ(y,τ,B′) captures the downward-sloping component of active demand.
- Next-period repayment function:
  - R(y′,τ′,B′) = (1 − d(y′,τ′,B′)) [ λ + (1 − λ) ( ν + q(y′,τ′,B′′) ) ],
    - with q(y′,τ′,B′′) the next-period bond price and B′′ ≡ B′(y′,τ′,B′).
- Bond price q(y,τ,B′) weakly decreases in B′ through:
  - higher expected default probability lowering expected repayment (R),
  - the downward-sloping active investor demand component Ψ(y,τ,B′).
- Inverse supply elasticity ε ≡ ∂log q(.) / ∂log B′ can be written as:
  - ε = ∂log E_{s′|s} R′(.) / ∂log B′ + ∂log Ψ(.) / ∂log B′.
  - First term: elasticity of expected repayment w.r.t. bond supply (weakly negative).
  - Second term: price decline due to downward-sloping active demand.
- The Ψ(.) mechanism acts as a disciplining device limiting government debt in quantitative analysis.

### Secondary markets and link with empirical analysis
- Introduce two instances of secondary market trading within a period to match high-frequency empirical elasticity measured on daily data.
- Timing within a period:
  1. Endowment y realized; initial states {y,τ,B}.
  2. Government chooses d(y,τ,B) and B′(y,τ,B).
  3. Primary and secondary market open; q_{SM,0}(y,τ,B′) denotes opening price.
  4. Next-period index weights τ′ are realized; bond prices updated.
  5. Secondary market closes; q_{SM,1}(y,τ′,B′) denotes closing price.
- High-frequency price reaction to index weight change ∆T′ ≡ T(τ′,B′) − T(τ,B′):
  - Reduced-form elasticity ˆη:
    - ˆη = (−) ( ∆q / ∆T′ ) ( B′ − T(τ,B′) ) / q_{SM,0}(y,τ,B′ ),
    - where ∆q ≡ q_{SM,1}(y,τ′,B′) − q_{SM,0}(y,τ,B′).
- Decomposition of ˆη into structural demand elasticity η and expected repayment change α:
  - ˆη = (−) ( ∆Ψ / ∆T′ ) ( B′ − T(τ,B′) ) / Ψ_{SM,0}(y,τ,B′ ) ≡ η
         + (−) ( ∆ER′ / ∆T′ ) ( B′ − T(τ,B′) ) / E_{y′,τ′|y,τ} R(y′,τ′,B′ ) ≡ α,
    - ∆Ψ ≡ Ψ_{SM,1}(y,τ′,B′) − Ψ_{SM,0}(y,τ,B′),
    - ∆ER′ ≡ E_{y′|y} R(y′,τ′,B′) − E_{y′,τ′|y,τ} R(y′,τ′,B′).
- Two mechanisms emphasized:
  - Persistent τ affects future Ψ(.) terms and future expected payoffs through Equations (17) and (18), so τ changes influence bond prices beyond the inelastic Ψ(.) channel.
  - Changes in τ affect government’s value function V^r(.), altering debt and default policies B′(.) and d(.), which feed back into expected payoffs and Ψ(.).

### Calibration (quarterly, Argentina benchmark)
- Preferences and processes:
  - CRRA u(c) = c^{1−γ} / (1−γ) with γ = 2.00.
  - Output AR(1): log(y′) = ρ_y log(y) + ε′_y, ε′_y ∼ N(0,σ_y).
  - Default output cost φ(y) = max { d̄_0 y + d̄_1 y^2 , 0 } with d̄_0 < 0, d̄_1 > 0 (Chatterjee and Eyigungor (2012) functional form).
  - Passive demand proportional: T′ = T(τ,B′) = τ × B′.
  - τ AR(1): log(τ′) = (1−ρ_τ) log(τ⋆) + ρ_τ log(τ) + ε′_τ, ε′_τ ∼ N(0,σ_τ).
- Downward-sloping Ψ(.) functional form:
  - Ψ(y,τ,B′) = exp [ −κ ( V_{s′|s}(R′(.)) / E_{s′|s}(R′(.)) ) × ( B′ − T′ − Ā ) ],
    - κ ≥ 0 characterizes elasticity of demand function,
    - Ā denotes average holdings of active investors.
  - κ and Ā assumed time-invariant for tractability.
- Table 5 (calibration) — fixed and calibrated parameters:
  - Panel a: Fixed Parameters
    - γ Risk aversion 2.00
    - r Risk-free interest rate 0.01
    - λ Debt maturity 0.05
    - ν Debt services 0.03
    - θ Reentry probability 0.0385
    - ρ_y Output, autocorrelation 0.93
    - σ_y Output, shock volatility 0.02
    - τ⋆ Share of passive demand 0.123
    - ρ_τ FIR, autocorrelation 0.66
    - σ_τ FIR, shock volatility 0.02
    - d̄_0 Default cost—level −0.264
    - d̄_1 Default cost—curvature 0.31
  - Panel b: Calibrated Parameters
    - β Discount rate 0.951
    - κ Slope parameter 72.0
    - Ā Active investors demand 0.455
- Calibration targets and model fit (Table 6, Targeted moments):
  - E[SP] Bond spreads: Data 472bp, Model 476bp
  - σ(SP) Volatility of spreads: Data 200bp, Model 135bp
  - E[B/Y] Debt to output: Data 55%, Model 54%
  - E[Ψ] Inconvenience yield: Data 1.0, Model 1.002
  - ˆη Reduced-form elasticity: Data −0.3, Model −0.29
- Model implications illustrated (Figure 4 summary):
  - Default occurs in states with high debt and low output.
  - Bond price q(.) is decreasing in B′ and increasing in y.
  - Counterfactual with perfectly elastic demand (q(.)/Ψ(.)) shows little effect of downward-sloping demand at low B′, but as B′ increases, repayment volatility reduces Ψ(.) and lowers q(.).

### Decomposing the reduced-form demand elasticity and persistence effects
- Decomposition (Equation 21): ˆη = η + α, where η is structural inverse demand elasticity from Ψ(.) changes and α captures changes in expected repayment.
- Quantitative findings (Figure 5 and Table 7):
  - The reduced-form inverse demand elasticity ˆη (orange) is always larger in magnitude than the structural elasticity η (blue); the vertical difference captures α.
  - On average, structural elasticity η accounts for less than two-thirds of reduced-form elasticity ˆη.
  - Table 7 (Persistence of shocks and demand elasticity):
    - Baseline: Reduced-form ˆη −0.29; Structural η −0.17; Bias, 1 − η/ˆη 40%
    - Lower persistence (ρ_τ = 0.25): ˆη −0.24; η −0.18; Bias 25%
    - Low persistence (ρ_τ = 0.50): ˆη −0.26; η −0.17; Bias 34%
    - Higher persistence (ρ_τ = 0.80): ˆη −0.32; η −0.17; Bias 48%
  - Interpretation:
    - The bias (portion of reduced-form elasticity attributable to endogenous changes in expected repayment) increases with the persistence of the τ process.
    - The more persistent the τ shock, the smaller the share of total price response attributable to the inelastic component Ψ(.).
- Conclusion from decomposition:
  - Accounting for issuers’ endogenous responses to supply-shifting shocks and the resulting changes in expected repayment is critical when identifying structural demand elasticities.
  - Neglecting these channels can introduce significant biases; even with temporary FIR shocks, the bias can exceed one-third of the reduced-form elasticity.

*Source: wpiea2024227-print-pdf - 4.3    Government Problem:  Recursive Formulation*

### 5.3    Implications of a Downward-sloping Demand

### 5.3    Implications of a Downward-sloping Demand

### Comparison with perfectly elastic demand: unconditional moments
- The paper compares the baseline model with inelastic investors to a counterfactual with perfectly elastic demand (κ= 0). All other model parameters remain the same.
- Table 8: Unconditional moments (Baseline vs Perfectly elastic)
  - E(SP) Bond spreads: Baseline 476bp; Perfectly elastic 910bp
  - σ(SP) Volatility of spreads: Baseline 135bp; Perfectly elastic 466bp
  - E(B/y) Debt to output: Baseline 54%; Perfectly elastic 52%
  - E(d) Default frequency: Baseline 3.84%; Perfectly elastic 4.74%
  - σ(B)/σ(y) Standard deviation of debt, relative to output: Baseline 1.06; Perfectly elastic 1.84
  - ρ(∆B,y) Correlation between issuances and output: Baseline 0.37; Perfectly elastic 0.51
  - ρ(SP,y) Correlation between spreads and output: Baseline -0.68; Perfectly elastic -0.51
- Key comparative finding: Despite similar levels of debt, default frequency and average spreads are lower in the inelastic demand (baseline) case than in the perfectly elastic case.

### Mechanisms: how downward-sloping demand alters policy and pricing
- Government internalizes both:
  - Effects of higher B′ on q(.) through changes in default probability.
  - Effects on q(.) through investors’ inelastic demand.
- Two factors explain lower default rate and spreads under inelastic demand:
  1. Optimal debt policy B′(y,τ,B) changes:
     - For large values of B (states where V(R′(.)) is high), an additional unit of B′ reduces bond price q(.) due to both higher default risk and investors’ inelastic behavior.
     - Government therefore refrains from issuing very large amounts of debt; inelastic demand imposes a limit on maximum issuance.
  2. Pricing consequences:
     - For small values of B′ (low default risk), q(.) is higher in the inelastic case than in the perfectly elastic case.
     - This higher bond price is not driven by a convenience yield: Ψ(.) is typically smaller than one given the calibration.
     - Instead, higher q(.) results from lower default risk, itself a consequence of reduced incentives to issue large amounts of B′.
- Net effect: Inelastic demand acts as a disciplining device that reduces default risk and increases bond prices by diminishing government incentives to issue additional debt.

### Response to output shocks and pro-cyclicality
- Standard limited-commitment/endogenous-default models: optimal bond policy is pro-cyclical (Arellano, 2008).
  - Government would like to issue more debt in "bad" times to smooth consumption, but higher spreads from higher default risk lead it to reduce debt; in "good" times it issues more due to cheaper credit.
- With inelastic investors:
  - Pro-cyclicality is dampened.
  - Impulse responses to endowment shocks (Figure 7):
    - Positive output shock:
      - Under perfectly elastic demand, government increases debt issuance more.
      - Under inelastic demand, response is muted because an extra unit of debt reduces bond prices through downward-sloping demand despite cheaper financing.
      - Cost: inelasticity prevents larger issuance when financing is cheap, lowering expansion in consumption.
    - Negative output shock:
      - Lower output raises borrowing costs and prompts debt reduction.
      - Under inelastic demand, reducing the stock of debt decreases the inconvenience yield demanded by investors, lowering spreads; contraction in debt is therefore less pronounced.

### Distributional and business-cycle statistics
- Debt-to-output distribution (Figure 8):
  - Under inelastic demand, distribution of B/y is significantly less dispersed.
  - Standard deviation of debt is about 30% smaller under inelastic investors (consistent with σ(B)/σ(y) values in Table 8: Baseline 1.06 vs Perfectly elastic 1.84).
  - Debt-to-output exhibits a smaller unconditional correlation with output when investors are inelastic (ρ(∆B,y): Baseline 0.37 vs Perfectly elastic 0.51).
  - This leads to a larger (in magnitude) correlation between spreads and output when investors are inelastic (ρ(SP,y): Baseline -0.68 vs Perfectly elastic -0.51).

### Welfare implications: certainty equivalent consumption (CEC)
- Definition: CEC is the proportional increase in consumption under the perfectly elastic case such that the household is indifferent between the perfectly elastic case and the inelastic case.
- CEC formula (from power utility and model value functions):
  - ̃x = [V(y,τ,B) / ̃V(y,τ,B)]^{1/(1−γ)} − 1, where ̃V(.) is the government’s value function under the perfectly elastic case.
- Findings (Figure 9):
  - CEC is positive: households prefer the world with inelastic investors because the disciplining device reduces default risk and borrowing costs sufficiently.
  - CEC decreases as the stock of debt increases, due to the higher inconvenience yield demanded by investors at higher B (as shown in Figure 6).

### Summary of quantitative empirical and model-based results (from section and conclusion)
- Empirical estimate: a 1 p.p. reduction in effective bond supply leads to a 30 basis point increase in bond prices.
- Structural model decomposition: over one-third of this response is attributable to endogenous changes in expected repayment of bonds.
- Overall implications:
  - Inelastic demand significantly influences optimal debt issuance and default policy.
  - Inelastic demand reduces government incentives to issue additional debt, lowers default risk, and reduces borrowing costs.
  - Inelastic investors dampen pro-cyclicality of debt policy and lead to less volatile debt-to-output dynamics.
  - Ignoring issuers’ endogenous responses biases demand elasticity estimates for risky assets.

*Source: IMF working paper section 5.3, "Implications of a Downward-sloping Demand."*

### Appendix Figure A1

### Appendix Figure A1

### Figure description
- Illustrates differences between the country-level face amount and their diversified versions, which the EMBIGD uses to generate the diversified bond weights.
- Data used are from December 2018.
- Axes/labels (as shown): "Diversified country face amount (Billion U.S. dollars)", "Country face amount (Billion U.S. dollars)", "Country face amount", "45 degree line".
- Sources listed in the figure: J.P. Morgan Markets, and authors’ calculations.

### Key quantitative findings from figure and surrounding note
- The figure compares country-level face amounts with diversified country face amounts used to compute diversified bond weights for EMBIGD.
- Data reference date: December 2018.

### Analytical framework: model of inelastic investors (Appendix B — core results)
- Market-value demand of active investors (first-order approximation around ̄πi) decomposes into:
  - An average, exogenous-mandate-driven component capturing fixed purchases (̄Ait).
  - An elastic deviation component proportional to ˆπi,t and a weighted-average investor parameter κit(Λ) ≡ 1 / Pj Λj Wj,t θj ξij (Equation (B1) → (B2)).
- Definition and interpretation:
  - ri,t+1 ≡ Rit+1 / qi,t − rf, with rf denoting the risk-free rate.
  - ̄Ait interpreted as active investors’ holdings aimed at satisfying fixed part of mandates (qi,t ̄Ait+1 = Pj Wj,t θj ξij).
- Closed-form bond price (Equation (B4)):
  - qi,t = (Et(Ri,t+1) − rf) × [1 − κit Vt(Ri,t+1)/Et(Ri,t+1) × (Bi,t − Ti,t − ̄Ait)], where κit(Λ) is the parameter summarizing downward-sloping demand.

### Microfoundations (B.2–B.3): two interpretations that yield analogous pricing kernels
- Risk-averse investors (mean–variance preferences, heterogeneous):
  - Investor j maximizes Et(TCj,t) − σj/2 Vt(TCj,t) with TCj,t = [xj,t − (1−θj) wt]′·rt+1.
  - Optimal portfolio: xj,t = 1/σj Σt−1 μt + (1−θj) wt (Equation (B5)).
  - Market-clearing yields active demand ̃At ≡ Pj Wj,t 1/σj Σt−1 μt and passive demand ̃Tt ≡ wt Pj Wj,t (1−θj).
  - Two-asset simplification yields price kernel (Equation (B7)) qi,t = (Et(Ri,t+1) − rf) × Ψi,t with
    - Ψi,t ≡ 1 − κRA,t Vt(Ri,t+1)/Et(Ri,t+1) × (Bi,t − Ti,t) and 1/κRA,t ≡ Pj Wj,t σj (Equations (B8)).
  - Key message: price elasticity captured by investors’ risk aversion aggregated via 1/κRA,t.

- Risk-neutral investors with VaR constraint:
  - Problem imposes Φ2 Vt([xj,t+1 − (1−αj) st+1]′·rt+1) − 1 ≤ 0; Φ2 captures intensity of regulatory risk constraint.
  - Optimal portfolio: xj,t = 1/(ρj Φ2) Σt−1 μt + (1−θj) wt (Equation (B9)), where ρj is Lagrange multiplier.
  - Aggregation yields price kernel (Equation (B10)):
    - qi,t = (Et(Ri,t+1) − rf) × [1 − κVaR,t Vt(Ri,t+1)/Et(Ri,t+1) × (Bi,t − Ti,t)], where 1/κVaR,t ≡ Pj Wj,t λj Φ2.
  - Key message: VaR constraint intensity and how binding it is in aggregate determine elasticity analogous to risk-aversion channel.

### Quantitative model (Appendix C — structure and calibration)
- Secondary markets introduced to reconcile model’s quarterly calibration with high-frequency empirical elasticity estimates (two secondary-market trades per period: before and after realization of index weights τ′).
- Timing of each period:
  1. Endowment y realized. Initial states: {y, τ, B}.
  2. Government chooses d(y, τ, B) and B′(y, τ, B).
  3. Primary and secondary market open; opening price qSM,0(y, τ, B′).
  4. Next-period index weights τ′ realized.
  5. Secondary market closes; closing price qSM,1(B′, y, τ′).
- Secondary-market pricing formulas:
  - qSM,0(y, τ, B′) = β⋆ Ey′,τ′|y,τ [R(y′, τ′, B′) ΨSM,0(y, τ, B′)] with ΨSM,0 defined in Equation (C2).
  - qSM,1(B′, y, τ′) = β⋆ Ey′|y [R(y′, τ′, B′) ΨSM,1(y, τ′, B′)] with ΨSM,1 defined in Equation (C4).
  - Difference between qSM,0 and qSM,1 arises only from update of τ′ while y and B′ remain fixed while market open.

### Equilibrium definition and solution method (C.2–C.3)
- Recursive Markov Equilibrium: collection of value functions V(·), Vr(·), Vd(·); policy functions d(·), B′(·); and bond prices q(·) satisfying three consistency conditions (government optimization, repayment function, and pricing consistency).
- Numerical solution:
  - Discretization via Tauchen:
    - 31 gridpoints for y.
    - 15 gridpoints for τ.
  - B grid: 250 equally spaced points between B = 0 and B = 1.2.
  - Algorithm steps include guesses for value functions and q, solving for B′(y, τ, B) using Brent’s method, cubic spline interpolation, computing Vd(y), convexifying default decision via i.i.d. shock εV ∼ N(1, σv2) with σv2 = 2.25×10−6, updating q via Equation (17), and iterating to convergence.

### Quantitative results and mechanisms (C.4; Figures C1–C2)
- Within-period effects of a passive-demand (τ) shock (Figure C1, panels a–c):
  - For a 5% increase in the passive demand (as a share of the total stock of debt):
    - Bond prices increase by 1%.
    - About half of that increase is explained by an increase in expected repayment.
    - One-period-ahead default risk decreases more than 10%.
  - Government adjusts stock of debt in response to τ′ shock; adjustment magnitude:
    - For a 5% increase (decrease) in the passive demand, the government raises (lowers) its debt by less than 1% (panel d).
  - Persistence of the τ process matters:
    - As persistence decreases (ρτ set to 0.50 and 0.25 in alternative parameterizations), implied changes on bond prices, expected repayment, and default risk decrease.

- Impulse-response dynamics (Figure C2):
  - Baseline impulse response to an increase in passive demand:
    - Government increases debt; effects are persistent.
    - Bond prices increase approximately 0.50% when government optimally adjusts debt.
  - Counterfactual with fixed debt:
    - If bond policy is held fixed, bond price increases about 1%.
    - Decomposition (counterfactual): initial bond-price increase driven by convenience price Ψ(.), which increases initially because the passive demand shock is persistent; Ψ(.) effects fade over time while higher debt raises default risk and influences future expected repayments.
  - Timing interplay:
    - Lenders anticipate government’s response (debt issuance), which attenuates the immediate price impact relative to the counterfactual with fixed debt.

### Implications for interpretation of empirical elasticity
- Structural elasticity differs from reduced-form estimates because government debt responses to passive-demand shocks offset part of immediate price effects.
- Persistence of the passive-demand (τ) process and the degree to which debt policy responds determine the magnitude and duration of price and default-risk responses.

*Source: Appendix Figure A1 and Appendices B–C of the provided PDF content.*

### Appendix Figure D1

### Appendix Figure D1

### Figure description
- Title: EMBI Global country-level weights in December 2018
- Weight (%) axis: 0 2 4 6 8 10 12
- The figure illustrates the EMBI Global country-level diversified and non-diversified weights for December 2018.
- Country-level weights are computed as the sum of the weights of all bonds from each country included in the index.

### Countries shown (in chart order)
- MEXICO
- CHINA
- INDONESIA
- TURKEY
- RUSSIA
- ARGENTINA
- BRAZIL
- PHILIPPINES
- COLOMBIA
- SOUTH AFRICA
- CHILE
- OMAN
- KAZAKHSTAN
- UKRAINE
- ECUADOR
- PERU
- PANAMA
- DOMINICAN REPUBLIC
- LEBANON
- MALAYSIA
- EGYPT
- URUGUAY
- VENEZUELA
- HUNGARY
- POLAND
- NIGERIA
- SRI LANKA
- ROMANIA
- CROATIA
- AZERBAIJAN
- JAMAICA
- COSTA RICA
- LITHUANIA
- EL SALVADOR
- ANGOLA
- PAKISTAN
- INDIA
- COTE D'IVOIRE
- MONGOLIA
- GHANA
- SERBIA
- KENYA
- PARAGUAY
- IRAQ
- JORDAN
- GUATEMALA
- SENEGAL
- ZAMBIA
- MOROCCO
- TRINIDAD AND TOBAGO
- BELARUS
- GABON
- BOLIVIA
- HONDURAS
- SLOVAK REPUBLIC
- NAMIBIA
- GEORGIA
- ARMENIA
- VIETNAM
- ETHIOPIA
- TUNISIA
- CAMEROON
- MOZAMBIQUE
- SURINAME
- PAPUA NEW GUINEA
- TAJIKISTAN
- BELIZE
- Non-diversified
- Diversified

### Notes and data provenance
- Note: The figure illustrates the EMBI Global country-level diversified and non-diversified weights for December 2018.
- Country-level weights are computed as the sum of the weights of all bonds from each country included in the index.
- Sources: J.P. Morgan Markets, and authors’ calculations.

*Inelastic Demand Meets Optimal Supply of Risky Sovereign Bonds — Working Paper No. WP/2024/227*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024227-print-pdf.pdf_