Self-Fulfilling Debt Crises with Long Stagnations

Q1. What is the Calvo (1988) mechanism, and why does it require a bimodal endowment process?

The Calvo mechanism generates multiple equilibrium interest rates for a given level of debt: lenders’ expectation that the borrower will default in the low-output state forces them to charge a high interest rate to break even, but the high rate raises the debt service burden and makes default more likely, validating the pessimistic expectation. For this self-confirming loop to sustain multiple stable equilibria, the interest rate correspondence — mapping debt levels to possible interest rates — must have an upward-sloping region at both the low and high rate. A unimodal endowment distribution generates a correspondence with a downward-sloping high-rate segment (higher debt → lower high interest rate), which is inadmissible and eliminates multiplicity. A bimodal distribution with well-separated high and low growth states, as observed empirically in crisis-prone countries, generates an upward-sloping correspondence at both rates, creating a region of intermediate debt levels where either rate is an equilibrium.

The second key model feature is the timing of moves: the borrower chooses current debt (amount borrowed today) rather than debt at maturity (the repayment obligation). When the borrower chooses debt at maturity, it implicitly pins down the default probability and therefore the interest rate, eliminating multiplicity. When the borrower chooses current debt, the interest rate is determined by lenders and can take either the high or the low value consistent with break-even pricing, given the chosen debt level.

Q2. How does the paper calibrate the endowment process and what does the estimation reveal?

The paper estimates a two-state Markov regime-switching model for annual GDP per capita growth for Argentina, Brazil, Italy, Portugal, and Spain using data from 1980 to 2017, and finds clear evidence of a bimodal distribution with persistent high- and low-growth regimes across all five countries. Estimated using a Bayesian MCMC algorithm with the Kim (1994) filter, the posterior means for the benchmark cross-country calibration are: low-growth rate gL = −1.0% per year, high-growth rate gH = 3.0% per year, persistence of low-growth state pL = 0.60, persistence of high-growth state pH = 0.80, and standard deviation of transitory shocks σ = 0.015. The average gap between gL and gH across countries is approximately 6 percentage points, more than three times the standard deviation of the transitory shock — confirming the bimodal structure that is essential for multiplicity. Both growth regimes are persistent, with the low-growth state having 60–80% persistence across countries.

The quantitative model uses these estimates together with standard parameters: risk-free rate R* = 3.5%, recovery rate κ = 75%, discount factor β = 0.75, and risk aversion γ = 3. The sunspot process governing equilibrium selection is i.i.d. with a 5% probability of the bad (high-rate) sunspot in each period.

Q3. What does the calibrated model predict for interest rates and when do self-fulfilling crises occur?

In the calibrated quantitative model, the multiplicity region is present only in the low-growth state and for intermediate debt levels; in the high-growth state, the multiplicity region is either empty or negligibly small. In the low-growth state with intermediate debt, the interest rate schedule features two admissible equilibria: a low-rate equilibrium consistent with a low probability of default (1.7% in the benchmark simulation) and a high-rate equilibrium consistent with a high probability of default (60.1%). Both are sustained by self-confirming expectations. The scenario simulation illustrates this starkly: two economies starting from identical wealth and facing identical growth-shock sequences but different sunspot realizations in period t=2 (when they are in the low-growth state) face interest rates of 4.0% versus 46.4% and next-period default probabilities of 1.7% versus 60.1%, respectively, with no difference in fundamentals.

The model also generates endogenous austerity: borrowers optimally refrain from increasing debt to avoid discrete jumps in interest rates, both at the fundamental threshold (driven by the growth regime) and at the expectations-driven threshold (driven by the sunspot). In low-growth states facing the bad sunspot, the borrower either bunches at a low debt level below the multiplicity region or makes a discrete jump above it, echoing the binary fiscal-adjustment dynamics observed in crisis episodes.

Q4. How does the model interpret the European debt crisis and the role of the ECB?

The model provides a direct interpretation of the European sovereign debt crisis: the southern European economies (Italy, Spain, Portugal) entered a low-growth state around 2009–2010, which created conditions for the Calvo mechanism to operate; spreads jumped to high-rate equilibria driven by expectations rather than by fundamentals alone. The ECB’s announcement of the Outright Monetary Transactions (OMT) program in September 2012 — the commitment to purchase sovereign bonds in secondary markets — shifted lenders’ beliefs from the bad-sunspot equilibrium to the good-sunspot equilibrium, collapsing spreads substantially even without actual intervention. In the model’s language, a credible lender of last resort can eliminate the bad equilibrium by committing to lend at the low-rate schedule, rendering the high-rate self-fulfilling expectations non-viable. The Argentine crisis of 1998–2002 fits the model as an alternative trajectory: Argentina entered the low-growth state with a 7% spread on 35% debt-to-GDP and, without a lender of last resort intervention, eventually defaulted in 2002 — consistent with the bad-sunspot equilibrium path.

Q5. What role does persistence of the low-growth state play?

The persistence of the low-growth state (pL) is the key parameter governing the severity of self-fulfilling crises: higher pL generates higher equilibrium interest rates in the bad-sunspot equilibrium and a larger multiplicity region. Intuitively, if the economy is likely to remain in the low-growth state for a long time, the probability of default conditional on entry into the bad equilibrium is very high, requiring lenders to charge very high interest rates to break even. The higher the interest rate, the more debt service costs compress fiscal space, making default even more likely and potentially sustainable at even lower debt levels. The paper shows in robustness exercises that the multiplicity result is robust to reasonable perturbations in pL, κ (recovery rate), σ (transitory shock standard deviation), gL, and gH, with the key ingredient being the bimodal structure of the endowment process rather than any single parameter value.

Q6. What are the policy implications for lenders of last resort?

The central policy implication is that a lender of last resort — such as the ECB or the IMF — is justified precisely when fundamentals are weak, not because fundamentals alone cause the crisis but because weak fundamentals create conditions in which expectations can trigger a self-fulfilling crisis. Intervening in the bad-sunspot equilibrium by committing to supply funds at low-rate terms makes the high-rate equilibrium infeasible: lenders cannot expect default because the lender of last resort ensures the borrower can always roll over at low rates. The model thus rationalizes the design of the OMT: a credible commitment with no limit on size is sufficient to rule out the bad equilibrium without necessarily requiring actual asset purchases. The paper notes that such interventions will also have effects on the economy outside the period of crisis, since the availability of backstop financing may affect the equilibrium path more broadly.

Key Concepts

Calvo (1988) mechanism

the self-fulfilling loop in sovereign debt markets in which high lender expectations of default require high interest rates for break-even pricing, which raise the actual default probability and thereby confirm the initial pessimistic expectations; generates multiple equilibrium interest rates for a given debt level.

state-dependent multiplicity

the feature of the model in which multiple interest rate equilibria arise only in periods of low and persistent growth (stagnation), not in high-growth regimes; the central quantitative finding that self-fulfilling crises are empirically plausible only when growth fundamentals are weak.

endogenous austerity

the borrower’s optimal choice to hold debt below the multiplicity region to avoid discrete jumps in interest rates triggered by either fundamentals or expectations; reflected in the flat portions of the debt policy function and consistent with fiscal consolidation patterns observed in crisis episodes.

sunspot variable

the exogenous coordination device that selects among the multiple equilibrium interest rate schedules; takes a bad or good realization each period according to an i.i.d. process, with the bad sunspot selecting the high-rate schedule and the good sunspot selecting the low-rate schedule in the low-growth state.