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---

### 2.1 Lack of Commitment and Uncertainty
- Lack of commitment in issuance
  - The IGCP announces securities and provides a target for the amount it expects to issue the week prior to an auction, but there is no commitment to that target.
  - Data show several instances of ex-post deviation from the target; although there is a targeted amount, the amount issued in a given auction is uncertain from the bidder’s perspective.
  - Figure2 highlights ex-post deviations from the target in 8 Portuguese auctions. A value of 1 represents auctions where the target is met; filled squares represent deviations above or below target. Even before the debt crisis, the agency would regularly deviate from the ex-ante target.
  - Brenner et al.(2009) survey (48 countries): in response to “Does the treasury (or the central bank) have the right to change the quantity of the debt that is being sold after viewing the demand?”, 30 out of 48 countries reported having some discretion on how much to issue, regardless of a target being announced.

- Debt crisis and increased auction flexibility
  - During the European debt crisis, debt management offices increased flexibility of issuance mechanisms.
  - April 2010 IGCP changes:
    1. running multiple auctions simultaneously for treasury bonds;
    2. providing an interval, instead of an amount, as a target;
    3. setting the target range for the sum across the auctions being ran simultaneously.
  - April 2010 coincided with intensifying of the crisis in Greece, multiple downgrades of Greek debt, and a bailout in May.
  - February 2011: same type of changes introduced for auctions of treasury bills.
  - 2011 OECD Survey: “In response to uncertainty and volatility, auction calendars have become more flexible in most jurisdictions, auctions were held more frequently and multiple series per auction were introduced.”
  - By providing a range as target the government does not commit to a particular amount; this flexibility can be thought of as a way to ensure that the target (range) is met, i.e. there would be no failed auctions.

- Changes in demand during the crisis (Portuguese bid-level data, Figure3)
  - Prices normalized so the marginal price equals 1.
  - Left panel (normal times) vs right panel (crisis period): during the crisis the demand schedule is much steeper and inelastic, with more dispersion of bids.
  - Implication: government discretion on the amount borrowed matters when different borrowing decisions impact the value of debt (right panel). Discretion on quantity sold together with default risk are the key characteristics separating outcomes under the two protocols.
  - Data show no evidence of persistent investor heterogeneity; analysis assumes investors are symmetric.

- Government spending uncertainty as a driver of crisis
  - Unanticipated large government deficits were an important driver of the sovereign debt crisis in Europe (example: late 2009 Greek disclosure of much higher budget deficits triggered downgrades and spread increases).
  - For Portugal, higher public spending and lower resources to finance that spending led to elevated borrowing and, together with a prolonged recession, contributed to the sovereign debt crisis.
  - Figure4: difference between expected and actual spending as a percentage of expected spending (expected spending taken from the government’s spending proposal submitted for the year ahead):
    - Deviations between expected and actual spending are mostly positive and go up to 45% above the 1 year ahead expectation.
    - From 2010 through 2014 (the shaded area), deviations were not only positive but also higher than before and after the crisis.
  - Evidence motivates modeling uncertainty regarding financing needs via public spending surprises privately observed by the government; this informational asymmetry drives differences in bidding across auction protocols.
  - Supply of debt is random ex-ante — given realization of financing needs and after observing prices demanded by investors, the government chooses how much to borrow optimally (the government reserves discretion on quantities sold).

### 4.1 Environment (Model setup and timing)
- Time and agents
  - Time is discrete and infinite, t = {0, 1, 2, . . . }.
  - Small open economy; government borrows from a continuum of competitive, risk neutral, deep-pocketed foreign investors with discount factor R↑1.
  - Government maximizes expected discounted utility: E(∑t=0! t u(c t )), where u is a “nice function” and !↓(0, 1) is the discount factor.
- States and shocks
  - Public exogenous state s↓S is Markov and governs endowment y(s) and expected public spending g(s).
  - Private exogenous state T↓T determines a budget surprise "T and is i.i.d. over time.
- Debt instrument and issuance
  - Government issues defaultable long-term bonds promising a stream of exponentially declining coupon payments: at time t a unit of the bond promises to pay (1↑ρ) t+k↑1 (ρ+&) of the consumption good in period t+k.
  - Government choice variable B→; gross issuance implied is ω = B→ ↑(1↑ρ) B.
  - Issuance cost i(s,B,B→)↔0 is incurred at auctions.
- Default and reentry
  - Default consequences:
    - Transition to bad credit standing: exclusion from financial markets and flow utility cost h(s).
    - Regains good standing with probability ' through restructuring: face value of pre-default debt (B) reduced by fraction (.
- Timing within a period
  1. Exogenous state variables realized.
  2. If in good standing, government chooses whether to default.
  3. If in good standing and repays (d=0):
     - (a) Government runs an auction;
     - (b) Investors submit bid functions after observing public states;
     - (c) Government chooses B→ and P c given aggregate bid function.
  4. If defaults (d=1) or in bad standing, excluded from markets; next period regains access with probability ' or remains excluded with probability (1↑').

- Key conceptual points
  - Long-term bond structure and repeated auctions imply current bond values depend on entire future path of fiscal policy.
  - Dynamic dilution: issuing additional future long-term debt can reduce the value of claims held by current investors via higher default probability.
  - Auction protocol interacts with dilution and affects incentives to borrow, prices, default, and welfare.

- Numeric and notation highlights (preserved exactly)
  - Bond payment specification: (1↑ρ) t+k↑1 (ρ+&).
  - Gross issuance: ω = B→ ↑(1↑ρ) B.
  - Issuance cost: i(s,B,B→)↔0.
  - Reentry probability: ' ; reaccess probability complement (1↑').
  - Haircut fraction at restructuring: (.

### 4.2 Optimal Bidding
- Lenders
  - Lenders are competitive, risk neutral; each takes other lenders’ strategies and government strategy as given.
  - Lenders’ actions aggregate into a market demand curve observed by the government along with private state T.
- Equilibrium bid conditions (preserve equations as in source)
  - In equilibrium lenders bid only marginal prices P c (.) and bids for incremental issuance at n↔(1↑ρ)B must satisfy:
    - 0 = E({p(n)↔P c (T)} - Q(B(T)) ↑ ε(p(n),P c (T)|B(T))). (equation (2) in source)
  - Under the UP (uniform price) benchmark:
    - p(n) = Q(B(n)). (equation (3))
  - In the DP (discriminatory price) strategy:
    - p(n) = E[Q(B(T)) | p(n)↔P c (T)]. (equation (4))
- Interpretation
  - UP prices are pinned down by value of debt at each state; DP prices depend on investors’ beliefs about government's borrowing distribution.
  - Restriction to symmetric pure strategy equilibria for tractability; pins down individual quantities and abstracts from investor coordination problems.

### 4.3 Government’s Problem
- Value when in good standing
  - V(s,T,B) = max d↓{0,1} { (1↑d) V R (s,T,B) + d ! V D (s,T,B) }.
- Value under default
  - V D (s,T) = (1↑!) { u( y(s)↑g(s)↗"T ) - h(s) } + ! E{ ' V (s→,T→,(1↑()B) ) + (1↑') V D (s→,T→,B) }.
  - Government consumes endowment net of realized public spending while excluded from markets.
  - Default cost h(s) measured in utils.
  - With probability ' government restructures pre-default debt with haircut ( and reaccesses markets; with probability (1↑') remains excluded.
- Conditional on repaying, government solves
  - V R (s,T,B) = max { c↔0, P c >0, B→ } { (1↑!) u(c) + ! E[ V (s→,T→,B→) ] }
    subject to
    - c + (ρ+&) B + g(s) ↗ "T = y(s) + # (s,B,B→) (1↑i(s,B,B→))/(1↑i(s,B,B→))? [text shows complex fraction; preserved as in source]
  - Auction revenue (or reverse auction cost):
    - # (s,B,B→) = ∫0 B→ (1↑ρ) B ε( p(s,B,n), P c (s,B,B→)) dn.
  - In symmetric pure strategy equilibrium P c (s,B,B→) = p(s,B,B→).
- Government internalizes how choices affect current auction revenue # and continuation value E[V(·)|s].

- Value of debt Q j (s,B→) under protocol j (exact equation preserved)
  - Q j (s,B→) = R↑1 E{ (1↑d→ j ) [ (&+ρ) + (1↑ρ) Q j (s→, B j (s→,B→,T→)) ] + d→ j Q D j (s→,B→) }.
- Value of bond upon default
  - Q D j (s,B) = R↑1 { ' (1↑() Q j (s,(1↑() B ) ) + (1↑') E[ Q D j (s→,B) | s ] }.
- Q(s,B→) can differ across protocols because:
  1. Different protocols imply different budget sets -> different default decisions.
  2. Even with identical default decisions, continuation values differ because protocols induce different borrowing distributions B(·), and long-term debt makes future fiscal policy relevant to current Q.
- Dynamic dilution channel: future issuance reduces existing claim values; interaction with protocol affects borrowing incentives and welfare.

### 4.4 Equilibrium
- Definition 2 (Equilibrium) — recursive equilibrium consists of { V, V R , V D }, price equations { Q, Q D }, bid function p, and policy rules { d, B, P c } that satisfy:
  1. Price equations satisfy their functional equations given policy rules.
  2. Bid function satisfies zero-profit conditions for investors given policy rules and prices.
  3. Policy rules solve government’s problem given values and bids.
  4. Value functions satisfy their functional equations given bids.
  5. Bids and policy rules are consistent with auction clearing.
- (Full detailed list of conditions is in the appendix of the source.)

### 5 Calibration

- 5.1 Data (case study and sources)
  - Case study: Portuguese economy; detailed sovereign debt auction data and a protocol switch during the sovereign debt crisis motivate the choice.
  - Data sources:
    - Eurostat Annual National Accounts for real and nominal GDP, 1995-2022.
    - Monthly long-term government bond yields for Portugal and Germany from ECB Interest Rate Statistics, 1999-2022.
    - Government debt securities: BPStat general government statistics.
    - Annual realized government expenditures and revenues and one year ahead expectations, 2003-2022.
      - Realized from Portuguese Public Finance Council (CFP).
      - Year ahead estimates from government’s annual budget proposal reports (submitted every October).

- 5.2 Functional Forms and Parameters (preserved exactly)
  - Period = year.
  - Annual risk-free real interest rate r = 0.02.
  - Bond maturity rate ρ and coupon value & set to Paluszynski (2023) values:
    - ρ = 0.212
    - & = 0.050
  - Haircut fraction ( set to 0.535 to match 2012 Greek restructuring.
  - Utility: constant relative risk aversion u(c) = c 1↑% /(1↑%) with % = 2.
  - Default utility cost parameterization: h(y t ) = max{0,(1↑h 0 ) + h 1 log y t } following Bianchi and Mondragon (2022).
  - Time series estimations:
    - AR(1) for detrended log real per capita GDP (1995-2019): y t = μ y + ) y y t↑1 + * t .
    - AR(1) for detrended year-ahead expectation of log real per capita public spending (2003-2019): g t = μ g + ) g g t↑1 + + t .
    - Innovations correlated; estimate ) *,+ = corr( ˆ * t , ˆ + t ) using OLS residuals.
    - Spending shocks assumed lognormal: " t ⇒ log-normal(0, , " ); standard deviation , " estimated using log differences between real public spending and its year-ahead expectation.
  - Computational and technical features:
    - Preference shocks included (Generalized Type One Extreme Value distribution with scale parameter , m and correlation parameter ) m ) to ensure a pure strategy equilibrium computable.
    - Issuance cost function included to rule out “maximum dilution” behavior.
  - Parameters set from literature and exogenous choices:
    - R 1.02
    - % 2
    - ρ 0.212
    - & 0.050
    - ' 0.154
    - ( 0.535
  - Estimated parameters:
    - μ y 0.005
    - ) y 0.802
    - , * 0.019
    - μ g -0.388
    - ) g 0.773
    - , + 0.054
    - ) *,+ 0.397
    - , " 0.115
  - Calibrated parameters (by SMM):
    - ! 0.932
    - h 0 0.912
    - h 1 0.333
  - Role of calibrated parameters:
    - h 0 controls average penalty and is closely tied to average indebtedness.
    - h 1 controls how penalties change across states, influencing elasticity of bond price near average debt and average spreads.
    - ! (impatience) governs how quickly government approaches implied debt limits and affects mean and variation in distance between realized debt and implied limits; thus influences spread volatility and average level.

- 5.3 Targeted Moments (preserve numeric table entries)
  - Internal rate of return r(s,B→) defined as:
    - r(s,B→) = (ρ+&) / Q(s,B→) ↑ ρ.
  - Table 3 targeted moments (Data vs Model):
    - E[b→/y]: Data 48.91% ; Model 49.49%
    - E[r↑r f ]: Data 0.61% ; Model 0.63%
    - ,(r↑r f ): Data 1.02 p.p. ; Model 1.02 p.p.
  - Discussion:
    - Standard sovereign debt models typically underpredict spread volatility; Portugal is an extreme case with spreads much more volatile than their mean.
    - Accounting for the actual use of a DP (discriminatory price) in Portuguese sovereign debt auctions helps generate higher marginal spreads and makes the government more willing to borrow into higher spreads relative to a UP (uniform price) auction.
    - Incorporating the actual auction protocol improves match to the relative volatility of spreads that many models fail to attain.

### 5.4 Validation (Simulated and empirical moments; DP vs UP comparison)
- Simulation details and empirical computation
  - Model moments generated from simulations that extend to 10,000 years and are repeated 1,000 times.
  - Empirical moments involving spreads computed using annual data and average spreads from 1999 to 2010.
  - Empirical moments using average bid spreads computed for treasury bond auctions. Other empirical moments computed using annual data 1995–2019.
  - Two primary market spreads:
    - average bid spread, r_bid, computed as r_bid(s,B,B→) = (ρ+&) / p(s,B,B→) ↑ρ
    - average spread on the last bid accepted, r_marg, computed as r_marg(s,B,B→) = (ρ+&) / p(s,B,B→) ↑ρ
    - where p(s,B,B→) is the average price of bids executed in an auction and p(s,B,B→) is the price of the last bid accepted in an auction.

- Table 4: Moments of the Ergodic Distribution (Data, Discriminatory, Uniform) — preserved entries
  - E[r↑rε] 0.61% 0.63% 0.26%
  - E[r_bid↑rε] 0.79% 0.66% 0.26%
  - E[r_marg↑rε] 0.82% 1.01% 0.26%
  - ,(r↑rε) 1.02 p.p. 1.02 p.p. 0.14 p.p.
  - Default Rate - 0.99% 0.43%
  - E[b→/y] 48.91% 49.49% 53.98%
  - ,(tb/y) 4.35 p.p. 2.40 p.p. 2.01 p.p.
  - ,(c)/,(y) 1.49 1.52 1.53
  - corr(tb/y,y) -0.48 -0.12 -0.16
  - corr(tb/y,r↑rε) 0.18 -0.14 -0.11
  - corr(y,r↑rε) -0.54 -0.23 -0.35
  - corr(y,r_marg↑rε) -0.76 -0.34 -0.35
  - corr(y,r_bid↑rε) -0.76 -0.58 -0.35

- Key validation findings
  - Average spreads under the UP are much lower and less volatile than under the DP.
  - The average spread on the last bid accepted, E[r_marg↑rε], highlights the willingness to borrow more on the margin under the UP.
  - The difference between the average secondary market spread, E[r↑rε], and the average bid spread, E[r_bid↑rε], highlights the extent of static dilution in the DP auction.
  - Under the DP, investors require spreads higher than those measured in the secondary market (underpricing), while often overpaying for their first bids; average spread for accepted bids is lower than the spread on the marginal bid but still exceeds the secondary market spread.
  - Incorporating the observed auction protocol and data on measured forecast errors yields a significant difference between secondary market spreads and marginal spreads similar in magnitude to the data.

- Default rate
  - Model’s implied default rate (Table 4) is close to a 0.96% figure mentioned in-text: hypothetical Portugal default in 2011 absent a bailout yields 2 defaults in about the last 170 years → 1.2% default rate, “which is close to our 0.96%.”

- National accounts mapping and volatilities
  - Model variables: GDP y and government spending G = g↗" have empirical counterparts and were disciplined with data.
  - Model consumption c and trade balance tb = y↑c↑G correspond conceptually to data net exports and the implied consumption residual y↑G↑tb = c + i.
  - Data moments involving c use the implied residual (consumption plus private investment), producing a relative volatility of consumption well above 1, which the model matches quite well.
  - The model does not fully reproduce observed volatility of the trade balance.
  - Both data and model display relatively weak correlations between the trade balance and either output or spreads; the strongest in the data is a moderate ↑0.48 between the trade balance and output, which the model does not match.

- Correlations of output with spread measures
  - Strongest association in data is with the average spread for bids accepted in a given auction (closely followed by the spread on the marginal bid).
  - The model underestimates the absolute strength of these correlations but reproduces the ordering of magnitudes.
  - Calibrated model generates spread volatility observed in the data while inducing a correlation between spreads and output that is close, but smaller, than observed.

- Comparison of auction protocols (DP vs UP) — static and dynamic effects
  - Decomposition exercise:
    1. Compare using UP or DP this period, keeping all future auctions fixed under the DP (isolates static dilution).
    2. Compare using DP or UP for all auctions in every period (captures both static and dynamic effects).
  - Static effects:
    - Low debt example B = 0.3 at means of y and g: borrowing distributions and marginal revenue similar under UP and DP; static dilution minimal.
    - High debt example B = 0.55 at means of y and g: government borrows more under the DP because investors submit lower prices under DP due to static dilution and revenue is weakly increasing under DP as opposed to UP.
    - Forces significant when government starts highly indebted and faces negative surprise spending.
  - Dynamic effects:
    - Future decision rules differ across protocols; Q(·) weakly lower under DP.
    - Under DP, supercharged dilution motive over time lowers value of debt claims.
    - Bid schedules:
      - Under UP, bid schedule overlaps with value of debt.
      - Under DP, investors bid weakly below the value of debt for non-marginal bids (static dilution).
      - For buy-backs in reverse auction, supply denotes bids weakly above the value of debt for non-marginal buy-back bids.
    - Net result: DP implies more dilution over time, lower asset value, lower bids, government sustains less debt, borrows less overall, and defaults more often compared to UP.
    - Removing dynamic channel of DP shifts UP borrowing distributions to the right (UP gives government more commitment).

- Government and lender payoffs (equivalent variation calculations)
  - Government welfare and equivalent variation:
    - Government welfare: expected discounted utility net of preference shocks for initial state (s0,B0).
    - Equivalent variation -: percentage increase in the consumption path under DP that makes the government indifferent between DP allocation and UP allocation.
    - Utility is CRRA with relative risk aversion coefficient %⇑ = 1; - solves (1+-)^(1+%) E[V_DP] = E[V_UP].
    - Figure 10 findings:
      - Equivalent variation - is strictly positive for every initial state → the UP is preferred to the DP under the calibrated model.
      - Differences smallest at high endowment and zero debt (top right) and increase as initial endowment decreases and initial debt increases (toward bottom left).
      - Differences increase as default becomes more likely, up to point where default is certain.
      - Insurance benefits of DP are too costly relative to dilution; gains from switching are larger when government faces high debt and low endowment.
  - Lender welfare and equivalent variation:
    - Lender welfare: beginning-of-period value to the lender of holding a bond, Q_ante(s0,B).
    - Equivalent variation for lenders -_L(s0,B0) = Q_UP_ante(s0,B) ↑ Q_DP_ante(s0,B).
    - Figure 11 findings:
      - Differences in prices are very close to zero absent default risk.
      - Differences become meaningful as country nears default; static dilution and dynamic dilution lower prices under DP.
      - -_L is non-negative: lenders’ value is larger under the UP.

- Discussion and validation conclusions
  - Under calibrated model, the insurance component of the DP is more than offset by dilution effects (static and dynamic).
  - UP protects investors from being diluted within an auction and provides better incentives on government borrowing over time.
  - UP is preferred by government and lenders when default risk is a concern: switching to a UP is a Pareto improvement in the calibrated environment.
  - Mechanism consistent with observed switch in Portugal from DP to UP in aftermath of sovereign debt crisis; Portugal stopped issuing securities with maturity longer than one year from 2011 to 2014 and switched protocol upon return to markets for those maturities.
  - Timing of protocol change rationalized by switching costs: government (and lenders) wait until gains from switching exceed switching costs; gains larger in states with high debt and low endowment, consistent with crisis-period switching.
  - Overall validation statement:
    - Calibrated model under the DP matches standard moments in the Portuguese economy regarding debt, spreads and business cycle statistics.
    - DP enables model to generate spreads whose volatility significantly exceeds their mean, addressing a usual shortcoming of this class of models.
    - Comparing DP and UP when default risk matters: switching to UP can yield gains up to 0.6% of permanent consumption and is Pareto-improving for both the small open economy and foreign lenders.
    - Dynamics are key: although DP can perform better in a single-auction setting, when default risk is relevant the UP dominates by protecting against static dilution and improving intertemporal borrowing incentives.

### Appendices — selected technical and robustness highlights
- Appendix A: Full Definition of Equilibrium
  - Stationary symmetric recursive equilibrium consists of value functions V, V^R, V^D; price functions Q, Q^D; bid price function p; policy functions B→, ω, P^c, d.
  - Seven conditions listed including default decision optimality, borrowing decision optimality, asset pricing, bid optimality, and auction clearing.

- Appendix B: Omitted Proofs — Key results
  - Theorem 1 (Revenue Equivalence): If B→ is a random variable independent of the auction protocol, then ex-ante expected revenue in the auction is the same under both protocols.
  - Proposition 1 (Dominant strategy): Bidding marginal prices is a dominant strategy for investors.

- Appendix C: Investor heterogeneity — empirical findings
  - Across-dealer and within-dealer standard deviation definitions provided.
  - Empirical patterns (2005–2020):
    - For both securities, prior to crisis standard deviation very small; temporary increase during crisis; return to zero afterwards (more pronounced for treasury bills).
    - For treasury bonds, after the crisis variation does not fully return to pre-crisis zero; remains slightly higher; coincides with protocol switch from DP to UP.
    - Time-series pattern of average within-dealer SD: 1) increasing towards 2008; 2) drop in 2009; 3) higher 2010–2012; 4) decrease starting 2013; 5) almost flat bid functions from 2014 onward.
    - Standard deviation across and within investors roughly same magnitude; steeper aggregate bid functions reflect both wider individual bid ranges and heterogeneity across dealers.
  - Persistent heterogeneity analysis:
    - Ranking over time appears independent of dealer—no persistent pattern detected.
    - Dealer fixed effects fairly close to each other; individual and time fixed effects account for less than 5% of variation of rankings.

- Appendix D: Two Period Environment, Alternative Specification (closed-form insights)
  - Environment with multiplicative taste shock " privately observed by government; default value vd with exponential parametrization.
  - Commitment to borrowing rule b(")=b(")UP: numerical comparison (commitment to b(")UP) yields:
    - Ex-ante welfare: E[V(")UP]=1.754 and E[V(")DP]=1.761 (covariance term larger under DP; welfare higher under DP when financing needs large).
  - Closed-form DP solution (linear u(x)=x) and UP closed-form expressions preserved in text:
    - UP boxed expressions:
      - b(")= [&"↓&R"(1+μ)↓&R']1/μ y
      - p(")= R↓1 & μ" (1+μ)↓&R '
    - DP boxed expressions:
      - b(")= y [1↓ ˆ" exp(!("↓ˆ")↓!2&R("2 ↓ˆ"2))]1/μ
      - p(")= &ˆ" exp(!("↓ˆ")↓!2&R("2 ↓ˆ"2))

- Appendix E: Robustness (selected welfare comparisons preserved)
  - Utility variations (all other parameters same):
    - Linear Utility : E[V(")UP]=1.754 > E[V(")DP]=1.698
    - Log Utility : E[V(")UP]=↓0.0986 < E[V(")DP]=↓0.0973
    - CRRA, (=2: E[V(")UP]=↓2.0265 < E[V(")DP]=↓2.0259
    - CRRA, (=4: E[V(")UP]=↓0.8065 < E[V(")DP]=↓0.8059
    - CRRA, (=8: E[V(")UP]=↓0.5146 < E[V(")DP]=↓0.5137
  - Distribution and scenario robustness entries preserved in source (multiple parameter combinations reported).

- Appendix F: Computational Details (grids and convergence)
  - Private exogenous state includes vector m of i.i.d. preference shocks; m distributed Generalized Type One Extreme Value with scale parameter )m and correlation parameter *m.
  - Issuance costs i(s,B,B→)↗0 with sine-based specification to prevent “maximum dilution”.
  - Objects solved: continuation value functions W and WD; price functions Q and QD; expected default probability ,(s,B→).
  - Grids:
    - s: GDP y(s) grid — 23 points evenly spaced in logs across [E[log(y(s))]↓3)[log(y(s))], E[log(y(s))]+3)[log(y(s))]].
    - g(s) grid — 17 points evenly spaced in logs across analogous range.
    - B grid — 241 evenly spaced points on [0, 1.2].
    - T grid — 31 points evenly spaced spanning six of the logged variable’s long run standard deviations and centered at one.
  - Iteration and convergence:
    - Update WD and QD given baseline objects; solve government problem to update W, Q, ,(. Stop when sup-norm distance < 10↓5. Use relaxation updates fnext(·)= -j fold(·)+(1↓ -j) fnew(·). DP requires smoothing for bid schedules and auction revenue updates.

_Italic: Source: wpiea2025151-source-pdf_

### 2.1  Lack of Commitment and Uncertainty

### 2.1  Lack of Commitment and Uncertainty

### Lack of commitment in issuance
- The IGCP announces securities and provides a target for the amount it expects to issue the week prior to an auction, but there is no commitment to that target.
- The data show several instances of ex-post deviation from the target; although there is a targeted amount, the amount issued in a given auction is uncertain from the bidder’s perspective.
- Figure2 highlights this lack of commitment by presenting instances of ex-post deviations from the target in 8 Portuguese auctions. A value of 1 represents auctions where the target is met, while the filled squares represent deviations above or below target. Even before the debt crisis, the agency would regularly deviate from the ex-ante target.
- Brenner et al.(2009) surveyed treasury ministries and central banks around the world (answers from 48 countries). In response to the question “Does the treasury (or the central bank) have the right to change the quantity of the debt that is being sold after viewing the demand?”, 30 out of 48 countries reported having some discretion on how much to issue, regardless of a target being announced.

### Debt crisis and increased auction flexibility
- During the European debt crisis, debt management offices increased flexibility of issuance mechanisms.
- In April 2010 the IGCP increased flexibility by: 
  1) running multiple auctions simultaneously for treasury bonds; 
  2) providing an interval, instead of an amount, as a target; 
  3) setting the target range for the sum across the auctions being ran simultaneously.
- April 2010 coincided with intensifying of the crisis in Greece, multiple downgrades of Greek debt, and a bailout in May.
- In February 2011 the same type of changes were introduced for auctions of treasury bills.
- The 2011 Survey of the OECD Working Party on Public Debt Management noted: “In response to uncertainty and volatility, auction calendars have become more flexible in most jurisdictions, auctions were held more frequently and multiple series per auction were introduced.”
- By providing a range as target the government does not commit to a particular amount; this flexibility can be thought of as a way to ensure that the target (range) is met, i.e. there would be no failed auctions.

### Changes in demand during the crisis
- Bid-level data for Portuguese debt auctions (Figure3) show demand schedule changes during the crisis (as documented in Alves Monteiro (2022)):
  - Prices normalized so the marginal price equals 1.
  - Left panel (normal times) vs right panel (crisis period): during the crisis the demand schedule is much steeper and inelastic, with more dispersion of bids.
- Implication: government discretion on the amount borrowed matters when different borrowing decisions impact the value of debt (right panel). Discretion on quantity sold together with default risk are the key characteristics separating outcomes under the two protocols.
- The changes in Figure3 could be driven by differences across dealers or by dispersion within bid functions. The data show no evidence of persistent investor heterogeneity. As investors do not present persistent differences, the analysis assumes investors are symmetric.

### Government spending uncertainty as a driver of crisis
- Unanticipated large government deficits were an important driver of the sovereign debt crisis in Europe (example: late 2009 Greek disclosure of much higher budget deficits triggered downgrades and spread increases; Copelovitch et al.(2016) view this as the trigger of the Eurozone debt crisis).
- For Portugal, higher public spending and lower resources to finance that spending led to elevated borrowing and, together with a prolonged recession, contributed to the sovereign debt crisis.
- Figure4 plots the difference between expected and actual spending as a percentage of expected spending (expected spending taken from the government’s spending proposal submitted for the year ahead):
  - Deviations between expected and actual spending are mostly positive and go up to 45% above the 1 year ahead expectation.
  - From 2010 through 2014 (the shaded area), deviations were not only positive but also higher than before and after the crisis.
- The evidence motivates modeling uncertainty regarding financing needs via public spending surprises that are privately observed by the government; this informational asymmetry drives differences in bidding across auction protocols.
- The supply of debt is random ex-ante — given realization of financing needs and after observing prices demanded by investors, the government chooses how much to borrow optimally (the government reserves discretion on quantities sold).

*Italic: Source: wpiea2025151-source-pdf - 2.1  Lack of Commitment and Uncertainty*

### 4.1  Environment

### 4.1  Environment

### Model setup and timing
- Time is discrete and infinite, t = {0, 1, 2, . . . }.
- Small open economy; government borrows from a continuum of competitive, risk neutral, deep-pocketed foreign investors with discount factor R↑1.
- Government maximizes expected discounted utility: E(∑t=0! t u(c t )), where u is a “nice function” and !↓(0, 1) is the discount factor.
- Public exogenous state s↓S is Markov and governs endowment y(s) and expected public spending g(s).
- Private exogenous state T↓T determines a budget surprise "T and is i.i.d. over time.
- Government issues defaultable long-term bonds modeled as contracts promising a stream of exponentially declining coupon payments: at time t a unit of the bond promises to pay (1↑ρ) t+k↑1 (ρ+&) of the consumption good in period t+k.
- Government choice variable B→; gross issuance implied is ω = B→ ↑(1↑ρ) B.
- Issuance cost i(s,B,B→)↔0 is incurred at auctions.
- Default consequences:
  - Transition to bad credit standing: exclusion from financial markets and flow utility cost h(s).
  - Regains good standing with probability ' through restructuring: face value of pre-default debt (B) reduced by fraction (.
- Timing within a period:
  1. Exogenous state variables realized.
  2. If in good standing, government chooses whether to default.
  3. If in good standing and repays (d=0):
     - (a) Government runs an auction;
     - (b) Investors submit bid functions after observing public states;
     - (c) Government chooses B→ and P c given aggregate bid function.
  4. If defaults (d=1) or in bad standing, excluded from markets; next period regains access with probability ' or remains excluded with probability (1↑').

### Key conceptual points
- Long-term bond structure and repeated auctions imply current bond values depend on entire future path of fiscal policy.
- Dynamic dilution: issuing additional future long-term debt can reduce the value of claims held by current investors via higher default probability.
- Auction protocol interacts with dilution and affects incentives to borrow, prices, default, and welfare.

### Numeric and notation highlights
- Bond payment specification: (1↑ρ) t+k↑1 (ρ+&).
- Gross issuance: ω = B→ ↑(1↑ρ) B.
- Issuance cost: i(s,B,B→)↔0.
- Reentry probability: ' ; reaccess probability complement (1↑').
- Haircut fraction at restructuring: (.

---

### 4.2  Optimal Bidding

- Lenders are competitive, risk neutral; each takes other lenders’ strategies and government strategy as given.
- Lenders’ actions aggregate into a market demand curve observed by the government along with private state T.
- In equilibrium lenders bid only marginal prices P c (.) and bids for incremental issuance at n↔(1↑ρ)B must satisfy:
  - 0 = E({p(n)↔P c (T)} - Q(B(T)) ↑ ε(p(n),P c (T)|B(T))). (equation (2) in source)
- Under the UP (uniform price) benchmark:
  - p(n) = Q(B(n)). (equation (3))
- In the DP (discriminatory price) strategy:
  - p(n) = E[Q(B(T)) | p(n)↔P c (T)]. (equation (4))
- UP prices are pinned down by value of debt at each state; DP prices depend on investors’ beliefs about government's borrowing distribution.
- Restriction to symmetric pure strategy equilibria for tractability; this pins down individual quantities and abstracts from investor coordination problems.

---

### 4.3  Government’s Problem

- Value when in good standing:
  - V(s,T,B) = max d↓{0,1} { (1↑d) V R (s,T,B) + d ! V D (s,T,B) }.
- Value under default:
  - V D (s,T) = (1↑!) { u( y(s)↑g(s)↗"T ) - h(s) } + ! E{ ' V (s→,T→,(1↑()B) ) + (1↑') V D (s→,T→,B) }.
  - Government consumes endowment net of realized public spending while excluded from markets.
  - Default cost h(s) measured in utils.
  - With probability ' government restructures pre-default debt with haircut ( and reaccesses markets; with probability (1↑') remains excluded.
- Conditional on repaying, government solves:
  - V R (s,T,B) = max { c↔0, P c >0, B→ } { (1↑!) u(c) + ! E[ V (s→,T→,B→) ] }
    subject to
    - c + (ρ+&) B + g(s) ↗ "T = y(s) + # (s,B,B→) (1↑i(s,B,B→))/(1↑i(s,B,B→))? [text shows complex fraction; preserved as in source]
  - Auction revenue (or reverse auction cost):
    - # (s,B,B→) = ∫0 B→ (1↑ρ) B ε( p(s,B,n), P c (s,B,B→)) dn.
  - In symmetric pure strategy equilibrium P c (s,B,B→) = p(s,B,B→).
- Government internalizes how choices affect current auction revenue # and continuation value E[V(·)|s].

- Value of debt Q j (s,B→) under protocol j:
  - Q j (s,B→) = R↑1 E{ (1↑d→ j ) [ (&+ρ) + (1↑ρ) Q j (s→, B j (s→,B→,T→)) ] + d→ j Q D j (s→,B→) }.
- Value of bond upon default:
  - Q D j (s,B) = R↑1 { ' (1↑() Q j (s,(1↑() B ) ) + (1↑') E[ Q D j (s→,B) | s ] }.
- Q(s,B→) can differ across protocols because:
  1. Different protocols imply different budget sets -> different default decisions.
  2. Even with identical default decisions, continuation values differ because protocols induce different borrowing distributions B(·), and long-term debt makes future fiscal policy relevant to current Q.
- Dynamic dilution channel: future issuance reduces existing claim values; interaction with protocol affects borrowing incentives and welfare.

---

### 4.4  Equilibrium

- Definition 2 (Equilibrium): Given an auction protocol, a recursive equilibrium consists of value functions { V, V R , V D }, price equations { Q, Q D }, bid function p, and policy rules { d, B, P c } that satisfy:
  1. Price equations satisfy their functional equations given policy rules.
  2. Bid function satisfies zero-profit conditions for investors given policy rules and prices.
  3. Policy rules solve government’s problem given values and bids.
  4. Value functions satisfy their functional equations given bids.
  5. Bids and policy rules are consistent with auction clearing.
- (Full detailed list of conditions is in the appendix of the source.)

---

### 5  Calibration

### 5.1  Data
- Case study: Portuguese economy; detailed sovereign debt auction data and a protocol switch during the sovereign debt crisis motivate the choice.
- Data sources:
  - Eurostat Annual National Accounts for real and nominal GDP, 1995-2022.
  - Monthly long-term government bond yields for Portugal and Germany from ECB Interest Rate Statistics, 1999-2022.
  - Government debt securities: BPStat general government statistics.
  - Annual realized government expenditures and revenues and one year ahead expectations, 2003-2022.
    - Realized from Portuguese Public Finance Council (CFP).
    - Year ahead estimates from government’s annual budget proposal reports (submitted every October).

### 5.2  Functional Forms and Parameters
- Period = year.
- Annual risk-free real interest rate r = 0.02.
- Bond maturity rate ρ and coupon value & set to Paluszynski (2023) values:
  - ρ = 0.212
  - & = 0.050
- Haircut fraction ( set to 0.535 to match 2012 Greek restructuring.
- Utility: constant relative risk aversion u(c) = c 1↑% /(1↑%) with % = 2.
- Default utility cost parameterization: h(y t ) = max{0,(1↑h 0 ) + h 1 log y t } following Bianchi and Mondragon (2022).
- Time series estimations:
  - AR(1) for detrended log real per capita GDP (1995-2019): y t = μ y + ) y y t↑1 + * t .
  - AR(1) for detrended year-ahead expectation of log real per capita public spending (2003-2019): g t = μ g + ) g g t↑1 + + t .
  - Innovations correlated; estimate ) *,+ = corr( ˆ * t , ˆ + t ) using OLS residuals.
  - Spending shocks assumed lognormal: " t ⇒ log-normal(0, , " ); standard deviation , " estimated using log differences between real public spending and its year-ahead expectation.
- Computational and technical features:
  - Preference shocks included (Generalized Type One Extreme Value distribution with scale parameter , m and correlation parameter ) m ) to ensure a pure strategy equilibrium computable.
  - Issuance cost function included to rule out “maximum dilution” behavior.
- Parameters estimated and calibrated:
  - Parameters set from literature and exogenous choices:
    - R 1.02
    - % 2
    - ρ 0.212
    - & 0.050
    - ' 0.154
    - ( 0.535
  - Estimated parameters:
    - μ y 0.005
    - ) y 0.802
    - , * 0.019
    - μ g -0.388
    - ) g 0.773
    - , + 0.054
    - ) *,+ 0.397
    - , " 0.115
  - Calibrated parameters (by SMM):
    - ! 0.932
    - h 0 0.912
    - h 1 0.333
- Role of calibrated parameters:
  - h 0 controls average penalty and is closely tied to average indebtedness.
  - h 1 controls how penalties change across states, influencing elasticity of bond price near average debt and average spreads.
  - ! (impatience) governs how quickly government approaches implied debt limits and affects mean and variation in distance between realized debt and implied limits; thus influences spread volatility and average level.

### 5.3  Targeted Moments
- Model calibrated using SMM to match key Portuguese moments.
- Internal rate of return r(s,B→) defined as:
  - r(s,B→) = (ρ+&) / Q(s,B→) ↑ ρ.
- Table 3 targeted moments (Data vs Model):
  - E[b→/y]: Data 48.91% ; Model 49.49%
  - E[r↑r f ]: Data 0.61% ; Model 0.63%
  - ,(r↑r f ): Data 1.02 p.p. ; Model 1.02 p.p.
- Discussion:
  - Standard sovereign debt models typically underpredict spread volatility; Portugal is an extreme case with spreads much more volatile than their mean.
  - Accounting for the actual use of a DP (discriminatory price) in Portuguese sovereign debt auctions helps generate higher marginal spreads and makes the government more willing to borrow into higher spreads relative to a UP (uniform price) auction.
  - Incorporating the actual auction protocol improves match to the relative volatility of spreads that many models fail to attain.

*Italic: Source: wpiea2025151-source-pdf - 4.1  Environment (IMF working paper chapter).*

### 5.4  Validation

### 5.4  Validation

### Simulated and empirical business cycle moments
- Table 4 presents simulated business cycle moments under both protocols (Discriminatory, Uniform), along with their empirical counterparts (Data).
- Model moments are generated from simulations that extend to 10,000 years and are repeated 1,000 times.
- Empirical moments involving spreads are computed using annual data and average spreads from 1999 to 2010.
- Empirical moments using average bid spreads were computed for treasury bond auctions. Other empirical moments were computed using annual data starting from 1995 and up to 2019.
- In addition to the standard secondary market spread, two primary market spreads are calculated:
  - average bid spread, r_bid, computed as r_bid(s,B,B→) = (ρ+&) / p(s,B,B→) ↑ρ
  - average spread on the last bid accepted, r_marg, computed as r_marg(s,B,B→) = (ρ+&) / p(s,B,B→) ↑ρ
  - where p(s,B,B→) is the average price of bids executed in an auction and p(s,B,B→) is the price of the last bid accepted in an auction.

- Table 4: Moments of the Ergodic Distribution (Data, Discriminatory, Uniform)
  - E[r↑rε] 0.61% 0.63% 0.26%
  - E[r_bid↑rε] 0.79% 0.66% 0.26%
  - E[r_marg↑rε] 0.82% 1.01% 0.26%
  - ,(r↑rε) 1.02 p.p. 1.02 p.p. 0.14 p.p.
  - Default Rate - 0.99% 0.43%
  - E[b→/y] 48.91% 49.49% 53.98%
  - ,(tb/y) 4.35 p.p. 2.40 p.p. 2.01 p.p.
  - ,(c)/,(y) 1.49 1.52 1.53
  - corr(tb/y,y) -0.48 -0.12 -0.16
  - corr(tb/y,r↑rε) 0.18 -0.14 -0.11
  - corr(y,r↑rε) -0.54 -0.23 -0.35
  - corr(y,r_marg↑rε) -0.76 -0.34 -0.35
  - corr(y,r_bid↑rε) -0.76 -0.58 -0.35

- Key validation points from the moments and model behavior:
  - Average spreads under the UP are much lower and less volatile than under the DP.
  - The average spread on the last bid accepted, E[r_marg↑rε], highlights the willingness to borrow more on the margin under the UP.
  - The difference between the average secondary market spread, E[r↑rε], and the average bid spread, E[r_bid↑rε], highlights the extent of static dilution in the DP auction.
  - Under the DP, investors require spreads higher than those measured in the secondary market (underpricing), while often overpaying for their first bids; average spread for accepted bids is lower than the spread on the marginal bid but still exceeds the secondary market spread.
  - Incorporating the observed auction protocol and data on measured forecast errors yields a significant difference between secondary market spreads and marginal spreads similar in magnitude to the data.

- Default rate discussion:
  - The model’s implied default rate (reported in Table 4) is close to a 0.96% figure mentioned in-text: if Portugal would have defaulted in 2011 absent a bailout, that hypothetical yields 2 defaults in about the last 170 years → 1.2% default rate, “which is close to our 0.96%.”

- National accounts mapping and volatilities:
  - Model variables: GDP y and government spending G = g↗" have empirical counterparts and were disciplined with data.
  - Model consumption c and trade balance tb = y↑c↑G correspond conceptually to data net exports and the implied consumption residual y↑G↑tb = c + i.
  - Data moments involving c use the implied residual (consumption plus private investment), producing a relative volatility of consumption well above 1, which the model matches quite well.
  - The model does not fully reproduce the observed volatility of the trade balance.
  - Both data and model display relatively weak correlations between the trade balance and either output or spreads; the strongest in the data is a moderate ↑0.48 between the trade balance and output, which the model does not match.

- Correlations of output with spread measures:
  - The strongest association in the data is with the average spread for bids accepted in a given auction (closely followed by the spread on the marginal bid).
  - The model underestimates the absolute strength of these correlations but reproduces the ordering of magnitudes.
  - Unlike some prior literature (Aguiar et al. (2016)), the calibrated model generates spread volatility observed in the data while inducing a correlation between spreads and output that is close, but smaller, than observed.

### Comparison of auction protocols (DP vs UP) — static and dynamic effects
- Decomposition exercise:
  - First: compare using UP or DP this period, keeping all future auctions fixed under the DP (isolates static dilution).
  - Second: compare using DP or UP for all auctions in every period (captures both static and dynamic effects).

- Static effects (future auctions fixed under DP):
  - For low levels of debt (example B = 0.3 at means of y and g), borrowing distributions and marginal revenue are very similar under UP and DP; static dilution is minimal.
  - For higher levels of debt (example B = 0.55 at means of y and g), government borrows more under the DP because:
    - Investors submit lower prices under DP due to static dilution.
    - Revenue is weakly increasing under DP as opposed to UP, giving incentive to borrow more under DP.
  - These forces are particularly significant when the government starts highly indebted and faces negative surprise spending.

- Dynamic effects (protocols applied in all future auctions):
  - Future decision rules differ across protocols; value of debt Q(·) depends on those differences and is weakly lower under DP.
  - Under DP, supercharged dilution motive over time lowers the value of debt claims.
  - Bid schedules:
    - Under UP, the bid schedule overlaps with the value of debt.
    - Under DP, investors bid weakly below the value of debt for bids that might not be marginal (static dilution).
    - For buy-backs in a reverse auction, supply denotes bids weakly above the value of debt for buy-back bids that might not be marginal.
  - Net result: DP implies more dilution over time, lower asset value, lower bids, government sustains less debt, borrows less overall, and defaults more often compared to UP.
  - Removing dynamic channel of DP shifts UP borrowing distributions to the right (UP gives government more commitment).

### Government and lender payoffs (equivalent variation calculations)
- Government welfare and equivalent variation:
  - Government welfare: expected discounted utility net of preference shocks for initial state (s0,B0).
  - Equivalent variation -: percentage increase in the consumption path under DP that makes the government indifferent between DP allocation and UP allocation.
  - Utility is CRRA with relative risk aversion coefficient %⇑ = 1; - solves (1+-)^(1+%) E[V_DP] = E[V_UP].
  - Figure 10 (heat map) findings:
    - Equivalent variation - is strictly positive for every initial state → the UP is preferred to the DP under the calibrated model.
    - Differences are smallest at high endowment and zero debt (top right) and increase as initial endowment decreases and initial debt increases (toward bottom left).
    - Differences increase as default becomes more likely, up to point where default is certain.
    - Insurance benefits of DP are too costly relative to dilution; gains from switching are larger when government faces high debt and low endowment.

- Lender welfare and equivalent variation:
  - Lender welfare: beginning-of-period value to the lender of holding a bond, Q_ante(s0,B).
  - Equivalent variation for lenders -_L(s0,B0) = Q_UP_ante(s0,B) ↑ Q_DP_ante(s0,B).
  - Figure 11 (heat map) findings:
    - Differences in prices are very close to zero absent default risk.
    - Differences become meaningful as country nears default; static dilution and dynamic dilution lower prices under DP.
    - -_L is non-negative: lenders’ value is larger under the UP.

### Discussion and validation conclusions
- Under the calibrated model, the insurance component of the DP is more than offset by dilution effects (static and dynamic).
- UP protects investors from being diluted within an auction and provides better incentives on government borrowing over time.
- UP is preferred by government and lenders when default risk is a concern: switching to a UP is a Pareto improvement in the calibrated environment.
- This mechanism is consistent with the observed switch in Portugal from DP to UP in the aftermath of the sovereign debt crisis; Portugal stopped issuing securities with maturity longer than one year from 2011 to 2014 and switched protocol upon return to markets for those maturities.
- Timing of protocol change can be rationalized by switching costs: government (and lenders) wait until gains from switching exceed switching costs; gains are larger in states with high debt and low endowment, consistent with crisis-period switching.
- Historical context: prior to the 2010s, sovereign debt crises in Europe were not expected; in low default-risk environments DP and UP are closer in performance, which may explain earlier use of DP.

- Overall validation statement:
  - The calibrated model under the DP matches standard moments in the Portuguese economy regarding debt, spreads and business cycle statistics.
  - The DP enables the model to generate spreads whose volatility significantly exceeds their mean, addressing a usual shortcoming of this class of models.
  - Comparing DP and UP when default risk matters: switching to UP can yield gains up to 0.6% of permanent consumption and is Pareto-improving for both the small open economy and foreign lenders.
  - Dynamics are key: although DP can perform better in a single-auction setting, when default risk is relevant the UP dominates by protecting against static dilution and improving intertemporal borrowing incentives.

*Source: 5.4 Validation — wpiea2025151-source-pdf*

### References

### wpiea2025151-source-pdf - References

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### Appendix A: Full Definition of Equilibrium
- A stationary symmetric recursive equilibrium consists of:
  1. Value functions V, V^R, V^D;
  2. Price functions Q, Q^D;
  3. Bid price function p;
  4. Policy functions B→, ω, P^c, d.
- Conditions that must be satisfied:
  1. Default decision optimality: given V^R and V^D, d solves the government’s default or repayment decision and V is the resulting value function.
  2. Borrowing decision optimality: given V and p, {B→, P^c, ω} solve the government’s repayment problem and V^R is the resulting value function.
  3. Asset pricing in good standing: given d, B→ and Q^D, Q satisfies the functional equation defining the value of debt while in good standing.
  4. Value of default: given V, V^D is the value function for the government upon default.
  5. Asset pricing in default: given Q, Q^D satisfies the functional equation defining the value of a defaulted bond.
  6. Bid optimality: given Q, B→ and P^c, p satisfies the bid optimality condition of ex-ante zero profits for investors.
  7. Auction clearing: given p, P^c and B→, the sum of accepted bid quantities equals the debt issuance, ω ↑ B→ ↓ (1 ↓ !) B.

### Appendix B: Omitted Proofs — Key results
- Theorem 1 (Revenue Equivalence)
  - Statement: If B→ is a random variable independent of the auction protocol, then ex-ante expected revenue in the auction is the same under both protocols.
  - Proof sketch (as presented): Expected revenue under discriminatory protocol E[!^D(b→)] is expressed via integrals over bids and the cdf G. After changes in order of integration and substitution of equilibrium expressions, the expression simplifies to E[!^U(b→)], showing revenue equivalence.
- Proposition 1 (Dominant strategy)
  - Statement: Bidding marginal prices is a dominant strategy for investors.
  - Proof sketch (as presented): For a finite set of marginal prices P_c(" ; p) induced by shocks " with finite support, consider two realizations with P_c,1 > P_c,2. Showing by contradiction that deviating from bidding marginal prices cannot increase expected payoff because acceptance probabilities are identical and the value of debt depends only on marginal price. Under discriminatory protocol, bidding above marginal price strictly reduces unitary profit; under uniform price protocol, the deviation becomes weakly dominated in the limit ρ ↘ 0. Concludes bidding marginal prices is strictly dominant under discriminatory and weakly dominant under uniform protocol.

### Appendix C: Investor heterogeneity — empirical findings and measures
- Objective: Assess differences across dealers using variation in the price of the first bid (lowest yield/highest price) as the most informative bid.
- Definitions and statistics:
  - Across-dealer standard deviation for year t:
    - SD_t = (1 / M_t) sum_{j=1}^{M_t} sqrt( (1/N) sum_{i=1}^N (p_{i1j} − p_{1j})^2 )
  - Within-dealer standard deviation for dealer i and year t:
    - SD_{i,t} = (1 / M_t) sum_{j=1}^{M_t} sqrt( (1/K_j) sum_{k=1}^{K_j} (p_{ikj} − p_{ij})^2 )
- Empirical patterns (treasury bills and treasury bonds, 2005–2020 as shown in figures):
  - For both securities, prior to the crisis the standard deviation is very small.
  - There is a temporary increase during the crisis period followed by a return to zero afterwards (more pronounced for treasury bills).
  - For treasury bonds, after the crisis the variation does not fully return to pre-crisis zero; it remains at slightly higher levels. This change coincides with the protocol switch for treasury bond auctions from discriminatory to uniform price.
  - Time-series pattern of average within-dealer standard deviation across dealers:
    1. Increasing towards 2008;
    2. A drop in 2009 before the crisis;
    3. Higher from 2010 to 2012;
    4. A decrease starting in 2013, particularly accentuated in 2014;
    5. Almost flat bid functions from 2014 onward.
  - Some dealers exhibit more disperse bid functions than others during the crisis period.
  - Conclusion: The standard deviation across and within investors is roughly the same magnitude; steeper aggregate bid functions reflect both wider individual bid ranges and heterogeneity across dealers.
- Persistent heterogeneity analysis:
  - Method: Rank first bids (lowest yields) across dealers per auction; analyze ranking R_it with regression R_it = %_i + ρ_it.
  - Finding: Ranking over time appears independent of dealer—no persistent pattern detected.
  - Dealer fixed effects %_i are fairly close to each other, with few exceptions (exceptions more significant for dealers active over shorter periods).
  - Individual and time fixed effects account for less than 5% of the variation of rankings across investors and over time.

*Content derived solely from the supplied PDF content.*

### Appendix D: Two Period Environment, Alternative Specification

### Appendix D: Two Period Environment, Alternative Specification

### Environment and Parameterization
- Preferences over consumption streams: E["u(c0)+&u(c1)] with multiplicative taste shock,".
- " is privately observed by the government; drawn from a continuous distribution with support on ["L, "H] with "L < "H and cdf G. Assumption: g(")=G→(")>0 on ["L, "H].
- Default value parameterization: vd = y(1↓exp(↓z)) with z distributed exponentially with cdf F(z)=1↓exp(↓μz) and z = ↓ln(1↓vd/y).
- Implied distribution: F(vd)=1↓(1↓vd/y)μ and F→(vd)=(μ/y)(1↓vd/y)μ↓1. For μ=1 this collapses into a uniform distribution on [0,y].

### Commitment to a Borrowing Rule
- Government can commit to a borrowing rule b(")=b(")UP (the optimal borrowing rule under UP), where " is observed ex-post and commitment fixes distribution of b→ across protocols.
- With linear utility, welfare decomposes as: E["!(")] = E["]E[!(")]+C(",!(")) — difference across protocols determined by covariance (insurance component from multiplicative taste shock, ").
- Numerical comparison (commitment to b(")UP):
  - Ex-ante welfare: E[V(")UP]=1.754 and E[V(")DP]=1.761.
  - Conclusion: covariance term is larger under DP; welfare tends to be higher under DP particularly when financing needs in the first period are large.
- Figure 4 described (panels):
  - (a) Borrowing decisions as function of ".
  - (b) Bid schedules: static dilution present; bid schedule under DP lower than under UP.
  - (c) Revenue: insurance benefits—higher revenue in bad states at expense of lower revenue in good states.
  - (d) Value functions: welfare higher under DP in many cases.

### Cost of Not Committing and Static Dilution
- Allowing government to choose optimally for each realization of " given the price schedule:
  - UP welfare unchanged (government already optimal under UP).
  - Difference in welfare under DP measures static dilution from lack of commitment.
- This specification allows a closed-form solution under specified functional forms.

### Closed-Form Solution — Discriminatory Price Protocol (DP)
- Linear preferences u(x)=x. Equilibrium conditions for DP:
  - Actuarially fair price for investors and government first-order condition imply:
    - p(b)=1/(1↓G("(b))) ∫"H"(b) Q(b("))dG(")
    - "p(b("))=&F(y↓b("))
    - Combined optimality condition:
      "1↓G(")!
      "H
      "F(y↓b(x))dG(x)=&RF(y↓b("))
- Monotonicity Proposition:
  - Proposition 1: If u is strictly increasing and concave, &≃(0, 1), and f(vd)=F→(vd)>0 on [v,v], then b(" ; p) is non-decreasing in ".
- Key derivation steps:
  - Define n(")↑F(y↓b(")) as probability of repayment at ".
  - Define N(")= + "H n(x)dG(x) and N→(")=↓n(")dG(").
  - Obtain functional equation:
    N→(")/N(") = ↓" &R dG(")/(1↓G("))
  - For G(x)=1↓exp(↓!x) and dG(x)=!exp(↓!x), solving yields:
    log(N(")) = ↓"2 ! 2&R + C ⇒ N(")=K.exp(↓"2 ! 2&R)
  - This leads to:
    n(")=K " &R exp(!"↓"2 ! 2&R)
- Equilibrium selection assumptions:
  1) There exists ˆ" such that government first-order condition holds at b→=0 (n(ˆ")=1).
  2) At ˆ", n→(ˆ")=0 (selects the lowest such ").
- Solving n→(")=0 yields root condition:
  1+!"↓!"2 &R =0 ⇒ rewrite as "2 ↓&R"↓&R! =0
  - ˆ" is given by:
    ˆ" = (&R+ .(&R)2 +4 &R !)  (expression as derived in text)
- Constant K determined by 1 = n(ˆ") = K ˆ" &R exp(, !ˆ" ↓ !ˆ"2 2&R - !ˆ" -), yielding:
  K = &R ˆ" exp(, !ˆ"2 2&R - !ˆ" -) ↓!ˆ" -
- Final closed-form n("):
  n(")= ˆ" exp(!("↓ˆ")↓!2&R("2 ↓ˆ"2)) = ˆ" exp(↓!&R("↓ˆ"), +ˆ"2 ↓&R --)
- Mapping back to b→ via n(")=F(y↓b→(")).

### Closed-Form Solution — Specific F(·) and Mapping to b and p
- For vd = y(1↓exp(↓z)) with exponential z and F(vd)=1↓(1↓vd/y)μ:
  - Using 1↓vd/y = b(")/y when vd = y↓b(").
  - DP optimality condition yields closed-form:
    b(")= y [1↓ ˆ" exp(!("↓ˆ")↓!2&R("2 ↓ˆ"2))]1/μ
  - Corresponding price p(b(")) simplifies to:
    p(")= &ˆ" exp(!("↓ˆ")↓!2&R("2 ↓ˆ"2))
- Under Uniform Price Protocol (UP) with positive borrowing:
  - UP first-order condition yields:
    b(") = [&"↓&R"(1+μ)↓&R']1/μ y  (exact expression as derived)
  - UP price:
    p(") = R↓1 & μ" (1+μ)↓&R '  = p(") (expression as in text)
- Summary boxed expressions from text:
  - For a uniform price auction:
    - b(")= [&"↓&R"(1+μ)↓&R']1/μ y
    - p(")= R↓1 & μ" (1+μ)↓&R '
  - For a discriminatory price auction:
    - b(")= y [1↓ ˆ" exp(!("↓ˆ")↓!2&R("2 ↓ˆ"2))]1/μ
    - p(")= &ˆ" exp(!("↓ˆ")↓!2&R("2 ↓ˆ"2))

### Appendix E: Robustness
- Utility function variations (all other parameters same):
  - Linear Utility : E[V(")UP]=1.754 > E[V(")DP]=1.698
  - Log Utility : E[V(")UP]=↓0.0986 < E[V(")DP]=↓0.0973
  - CRRA, (=2: E[V(")UP]=↓2.0265 < E[V(")DP]=↓2.0259
  - CRRA, (=4: E[V(")UP]=↓0.8065 < E[V(")DP]=↓0.8059
  - CRRA, (=8: E[V(")UP]=↓0.5146 < E[V(")DP]=↓0.5137
- Distribution of " (CRRA with (=2, vd uniform on [v,v]):
  - "⇑Exp(1): E[V(")UP]=↓2.0265 < E[V(")DP]=↓2.0259
  - "⇑U(0, 5): E[V(")UP]=↓3.5773 < E[V(")DP]=↓3.5762
  - "⇑U(0, 10): E[V(")UP]=↓5.6877 < E[V(")DP]=↓5.6851
  - "⇑N(3, 2): E[V(")UP]=↓4.1615 < E[V(")DP]=↓4.1599
- Distribution of vd (CRRA with (=2, " exponentially distributed with !=1):
  - vd ⇑U(v,v): E[V(")UP]=↓2.0265 < E[V(")DP]=↓2.0259
  - vd ⇑N(u(0.2)=↓5, 1.5): E[V(")UP]=↓2.0254 < E[V(")DP]=↓2.0248
- Output growth scenarios (CRRA (=2, " exponential with !=1, vd uniform):
  - y1 = y0 : E[V(")UP]=↓2.0265 < E[V(")DP]=↓2.0259
  - y1 = 1.05⇓y0 : E[V(")UP]=↓1.9677 < E[V(")DP]=↓1.9671
  - y1 = 0.95⇓y0 : E[V(")UP]=↓2.0896 < E[V(")DP]=↓2.0891
- Budget deficits specification (additive deficit shock):
  - c = y+!(b("))↓b0↓" with CRRA (=2, vd uniform, " exponentially distributed with !=1 truncated to [0, 1]):
    - E[V(")UP]=↓2.8976 < E[V(")DP]=↓2.8952

### Appendix F: Computational Details
- Private exogenous state includes vector m of i.i.d. preference shocks for the government; enter additively into decision problems; unbounded to ensure positive probability for every feasible action (randomization ensures equilibrium existence).
- m distributed Generalized Type One Extreme Value with scale parameter )m and correlation parameter *m for computational tractability; choice probabilities and ex ante expected values can be written analytically in terms of choice values.
- Issuance costs i(s,B,B→)↗0 included to prevent counterfactual “maximum dilution” behavior; functional form as in Fourakis (2023) with a sine-based specification that imposes strict limit on one-period-ahead default probability threshold pd and continuity in issuance scale.
- Objects solved and used to assess convergence:
  1. Continuation value functions W(s,T,B,B→)=E[V(s→,T→,m,B,B→)|s] and WD(s,T,B)=E[VD(s→,T→,m,B)|s].
  2. Price functions Q(s,B→) and QD(s,B) and expected probability of default ,(s,B→).
- Grids used:
  1. s≃S: GDP y(s) grid — 23 points evenly spaced in logs across [E[log(y(s))]↓3)[log(y(s))], E[log(y(s))]+3)[log(y(s))]].
     - Expected public spending g(s) grid — 17 points evenly spaced in logs across [E[log(g(s))]↓3)[log(g(s))], E[log(g(s))]+3)[log(g(s))]].
  2. B≃B: grid for b uses 241 evenly spaced points on [0, 1.2].
  3. T≃T: grid for "(T) uses 31 points evenly spaced spanning six of the logged variable’s long run standard deviations and centered at one (average log is zero).
- Iteration and convergence:
  1. Given baseline objects, generate new guesses for WD(s,T,B) and QD(s,B) using restructuring structure upon regaining market access.
  2. Solve government problem in good standing to generate new guesses for W(s,T,B,B→), Q(s,B→), and ,(s,B→).
  3. Check sup-norm distance across all objects; stop if < 10↓5. Otherwise update guesses using fnext(·)= -j fold(·)+(1↓ -j) fnew(·) with j≃{V,Q}.
  - Notes: price functions typically require more smoothing than value functions; DP requires smoothing for bid schedules and auction revenue updates.

*Appendix D, E, and F content from the provided IMF working paper chapter.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025151-source-pdf.pdf_