Proof of Lemma 2. We first solve the ND case. VND (defined in (14)) is a parabola with a minimum at
which rewrites as (20), using the t = 0 budget constraint. Whenever
which rewrites as (21), using the t = 0 budget constraint.
We now solve the PD case. VPD is a parabola with a minimum at
which rewrites as (23), using the t = 0 budget constraint. If
Claim 12 (Full Default) Assume Assumption 1. At t = 1, the Sovereign never chooses ST1 leading to default with probability 1 at t = 2 (i.e., default when either
Proof of Claim 12. If the Sovereign chooses ST1 leading to default with probability 1 at t = 2, then p1 = 0 (see (7)), the budget constraint at t = 1 is ST0 = e1 and the cost at t = 1 is:
Since the cost in the ND case is either
Assumption 1 implies
Claim 13 Let P− [θ1] and P+ [θ1] be two polynomials of degree 2 in θ1:
Proof. The coefficient of degree 2 of P− [θ1] is
Computations for Definition 3. Claim 13 shows that
and we find that
which holds true (Assumption 1).
For any value
Claim 13 implies that this rewrites P+ [θ1] ≥ 0 for
A straightforward consequence of Lemma 2 is:
Claim 14 For a given E (πD) and debt portfolio (ST0, LT) satisfying the t = 0 budget constraint, three intervals of θ1 are defined:
- Interval 1:
The PD solution exists and the ND solution is constrained.
- Interval 2:
The PD solution exists and the ND solution is interior.
- Interval 3:
The PD solution does not exist and the ND solution is interior.
Claims 15 to 18 are used in the proof of Proposition 4 below.
Claim 15 For a given E (πD) and debt portfolio (ST0, LT) satisfying the t = 0 budget constraint, if θ1 belongs to I2 and I3 defined in Claim 14, then the Sovereign chooses the interior ND solution at t = 1.
Proof. For θ1 ∈ I3, the interior ND solution is the only solution available. For θ1 ∈ I2, the choice of no default (i.e.,
The condition defining the lower bound of I2 rewrites:
so that an upper bound for the LHS in (46) is:
The condition defining the upper bound of I2 rewrites:
so that a lower bound for the RHS in (46) is:
Hence a sufficient condition for (46) is:
which holds true since
Claim 16 For a given E (πD), if θ1 belongs to I1, then there exists a threshold LT* (function of E (πD) and θ1) such that for any debt portfolio (ST0, LT) satisfying the t = 0 budget constraint:
If LT ≤ LT*, then the Sovereign chooses at t = 1 a value ST1 leading to no default,
If LT > LT*, then the Sovereign chooses at t = 1 a value ST1 leading to potential default.
Proof. The Sovereign chooses not to default iff:
This shows existence of the threshold LT*. Assumption 1 implies
Differentiating this condition implies:
We compute this derivative to show that it is non positive. First, we have:
We now turn attention to:
End of the proof.
The Claim below shows that, for
Claim 17 For a given E (πD) and debt portfolio (ST0, LT) satisfying the t = 0 budget constraint, if θ1 belongs to I1, then we have: the Sovereign chooses the PD solution (i.e., LT* = 0)10 iff
Proof. Given Claim 16, it is enough to show that, for LT = 0, the Sovereign chooses not to default iff
This is equivalent to
The Claim below shows that, for
Claim 18 For a given E (πD) and debt portfolio (ST0, LT) satisfying the t = 0 budget constraint, if θ1 ∈ I1, then the Sovereign chooses the constrained ND solution (i.e.,
Proof. The threshold LT* is
This latter inequality rewrites P+ [θ1] ≥ 0, which is equivalent to
This holds true (with
Proof of Proposition 4. The computations for Definition 3 show that
Proof of Corollary 5. Claim 14 shows that θ1 ∈ I2 ⋃ I3 iff:
A sufficient condition is then:
Claim 15 shows that the Sovereign chooses the interior ND solution when this condition holds. End of the proof.
Proof of Proposition 6. If the Sovereign issues a portfolio leading to ND, then the expected cost is either
If the Sovereign issues a portfolio leading to PD, then the expected cost is
Proof of Proposition 7. We first compute the optimal LT and the value of the expected cost at the optimum in the PD case. The FOC of the minimization of (29) is:
LT is positive (the denominator is positive, the numerator as well since
This value of LT satisfies:
where the second inequality follows from
The Sovereign chooses a portfolio (ST0, LT) implying no default iff the value (30) of the expected cost is smaller than the value (31) of the expected cost. This rewrites as:
One of the roots of
and Q  ≤ 0 since it rewrites exactly as
that is the ratio between the coefficient Q  of degree 0 and the coefficient C of degree 2 where:
We distinguish 2 subcases:
If C > 0, then
“between the roots” (in particular, for every ).
If C < 0, then
“outside the roots”. Either (and for every ), or (and ).
Taken together, these cases summarize as folkows: If C > Q , then
We now show that Condition (58) holds true for
which is implied by:
which always holds true for x > 1. As P+ [θ1] < 0 implies:
we have that the derivative of the RHS of Condition (59) is negative for
Using Condition (26) defining
which rewrites as:
and x > 1 implies that
Given x > 1, a lower bound of the LHS is
which is positive for x > 6 (this is the only use in the proof of the Assumption x > 6). Hence (63) holds true (and so does (62)).
Lastly, note that when
which holds true since it follows from
which holds true. End of the proof.
Proof of Proposition 8. We compute the expected costs in the two cases where the Sovereign chooses a portfolio leading to PD under
If the Sovereign chooses a portfolio leading to PD under
If the Sovereign chooses a portfolio leading to ND under
and the optimal LT solves the FOC:
The minimum expected cost follows, replacing LT by its value (67):
The Sovereign prefers the ND solution iff (68) is smaller than (66). Using
A sufficient condition for (68) smaller than (66) is then:
This rewrites as the following polynomial of degree 2 in
The coefficient of degree 2 is positive (convexity of 1/X). The discriminant is zero since it writes:
which simplifies to:
and holds true because:
It follows that the polynomial in a is always positive: the Sovereign always prefers the ND solution. End of the proof.
Proof of Proposition 9. The optimal portfolio (35) and (36) is computed in the proof of Proposition 7 above. The variations of LT and ST0 in
In the FOC (54), the E (πD) correspond to the price p0 (and then to the investors’ belief), the
Angeletos, George-Marios, 2002, “Fiscal Policy with Noncontingent Debt and the Optimal Maturity Structure”, Quarterly Journal of Economics 117, pp.1105–1131.
Arellano, C. and A. Ramanarayanan, 2010, “Default and the maturity structure in sovereign bonds,” Journal of Political Economy, 120(2), pp.187–232.
Bolton, P, and O. Jeanne, 2009, “Structuring and Restructuring Sovereign Debt: The Role of Seniority,” Review of Economic Studies, 76(3), pp.879–902.
Bolton, P. and O. Jeanne, 2008, “Structuring and Restructuring Sovereign Debt: The Role of a Bankruptcy Regime,” Journal of Political Economy.
Broner, F., G. Lorenzoni and S. Schmukler, 2013, “Why Do Emerging Economies Borrow Short Term?” Journal of the European Economic Association, 11, pp. 67–100.
Buera, F. and J. P. Nicolini, 2004, “Optimal Maturity of Government Debt without State Contingent Bonds,” Journal of Monetary Economics 51, pp.531–554.
Challe, E, F. Le Grand and X. Ragot, 2010, “Incomplete Markets, Liquidation Risk, and the Term Structure of Interest Rates,” Banque de France WP 301.
Cole, H. and T. Kehoe, 1996, “A Self-Fulfilling Model of Mexico’s 1994–95 Debt Crisis,” Journal of International Economics, 41, pp.309–30.
Conesa, J. and T. Kehoe, 2012, “Gambling for Redemption and Self-Fulfilling Debt Crises,” Federal Reserve Bank of Minneapolis Research Department Staff Report 465.
Conesa, J. and T. Kehoe, 2014, “Is It Too Late to Bail Out the Troubled Countries in the Eurozone?” American Economic Review, 104(5), pp.88–93.
Ghosal, S. and M. Miller, 2003, “Coordination Failure, Moral Hazard and Soreveign Bankruptcy Procedures,” Economic Journal 113, pp.276–304.
Grossman, H. and J. Van Huyck, 1988, “Sovereign Debt as a Contingent Claim: Excusable Default, Repudiation, and Reputation,” The American Economic Review 78(5), pp.1088–1097.
Haldane, A., A Penalver, V. Saporta and H S Shin, 2005, “Analytics of sovereign debt restructuring,” Journal of International Economics, 65(2), pp.315–333.
Hatchondo, J.C. and L. Martinez, 2009, “Long-duration bonds and sovereign default,” Journal of International Economics, 79, pp.117–125.
Hatchondo, J.C., Martinez, L. and C. Sosa Padilla, 2010, “Debt dilution and sovereign default risk,” Federal Reserve Bank of Richmond WP 10-08R.
Jeanne, O., 2009, “Debt Maturity and the International Financial Architecture,“ The American Economic Review, 99(5), pp.2135–2148.
Jeanne, O., J. Ostry and J. Zettelmeyer, 2008, “A Theory of International Crisis Lending and IMF Conditionality,” IMF Working paper 08/236 and CEPR Discussion Paper 7022.
Laubach, T., 2009, “New Evidence on the Interest Rate Effects of Budget Deficits and Debt,” Journal of the European Economic Association, 7(4), pp. 858–885.
Morris, S., and H.S. Shin, 1998, “Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks,” The American Economic Review, 88(3), pp.587–597.
Reinhart, V., and B. Sack, 2000, “The Economic Consequences of Disappearing Government Debt,” Brookings Papers on Economic Activity, 31(2000-2), pp.163–220.
The first author acknowledges support from the center of excellence MME-DII (ANR-11-LBX-0023-01). The authors thank Roger Guesnerie and IMF staff from various departments for helpful comments on an earlier version.
We may relax this assumption by considering default at t = 1 (the conditions of default being analogously defined as in t = 2). We may then consider an equilibrium where parameters do not lead to default at t = 1. We choose the simpler model presented here.
Under the assumptions of Rational Expectations and symmetric information, all the agents have the same expectation πD.
Under the assumption that investors are risk averse, in the case of a positive expected default probability (and only in this case), a risk premium would appear and p0 would decrease. The price differential between this case and the no default case would increase. This should not affect the intuition of the results.
The analysis of the coordination problems raised by issuance of LT debt is part of a companion paper.
In particular, the interval
As written above, the probability of default is correctly expected, conditionnal on (ST0, LT).
For s = M, H,
This is a slight abuse of notation since Claim 16 states that the PD solution is chosen for LT < LT* only.