A. Sovereign Government’s Problem

At the beginning of period t, an endowment shock y_t is realized, and the country inherits an asset position b_t and a credit standing s_t from the last period. The credit standing can be either good (s_t = 0), indicating that the sovereign is current on its debt service, or bad (s_t = 1), meaning that the economy is in default.

If the current period begins with a good credit standing (s_t = 0), then the country either repays the maturing debt or defaults on it. Let V (y_t, s_t, b_t) be the country’s lifetime value function from period t on with the current endowment y_t, credit standing s_t and existing asset position b_t. The value function V (y_t, 0, b_t) is then given by

where V ^R is the value function for repaying debt, and V^D the one for default. If the sovereign fully repays the maturing debt, then it can borrow or save by trading in one-period bonds. Moreover, the government enters the next period with a good credit standing: s_t+1 = 0. Thus, the value of repaying debt is given by

Where b_t+1 is the one-period bond traded in this period, and q_t+1(y_t, b_t+1) is the corresponding price.

If default is chosen, a proportional output loss γy_t and exclusion from the capital markets ensue. The defaulting government then enters the subsequent period with a bad credit standing and a defaulted debt level, b_t+1 = (1 +r)b_t. To keep the model tractable, we use the risk free interest rate r, instead of a bond-specific interest rate, to compute the present value of the defaulted debt. The value function for default, V^D, is given by

If the sovereign starts a period with a bad credit standing s_t = 1, it stays in autarky, suffers a proportional output loss and may settle the defaulted debt through renegotiating with the debt holder on debt reduction. The renegotiation process is a dynamic game that can last more than one period. In equilibrium, this bargaining game determines an endogenous debt recovery rate α(y, b) ∈ [0, 1] and when the agreement on α(y, b) can be reached. Thus, the value of staying in default and continuing renegotiation from period t on, V (y_t, 1, b_t), is equal to the expected payoff that the borrower can get from the bargaining game from period t on. We denote this payoff as ∆_B(y_t, b_t):

The determination of this payoff will be discussed in detail in the debt renegotiation problem.

Default is preferable when V^D ≥V ^R. Thus, we can characterize the default policy of a sovereign government in good standing by a default set y^D(b) ⊆Y, following Arellano (2007). The default set y^D(b) is defined as follows

This default set specifies a set of endowments for which default is optimal given the debt position b. Consequently, we can compute the probability of default, θ(b), as the probability that the endowment shock falls in the default set, given the government’s debt position b.

B. Debt Renegotiation Problem

Based on Merlo and Wilson (1995), we model the debt renegotiation problem as a two-player stochastic bargaining game with complete information. This is a stochastic bargaining game in that both the endowment process and the identity of the proposer are stochastic. In each period, a state is realized, which determines the cake (i.e., the set of possible utility vectors to be agreed upon in that period), and the proposer is randomly selected. For simplicity, we assume that each player has a constant probability of being selected as proposer in each period. That is, the identity of the proposer is independent of the cake size. Let ϕ denote the probability that the borrower, B, can propose in a period, and (1 − ϕ) is the probability that the lender, L, proposes in each period. The frequency with which a player is selected as proposer is a parsimonious way to capture the bargaining power acquired through one’s ability to enjoy the first-mover advantage. The proposer may either propose a debt recovery rate or pass. If she proposes, then the remaining party chooses to accept or to reject the proposal. If the proposal is accepted, then the defaulting country immediately repays its reduced debt arrears, and then enters the next period with an upgraded credit standing of s_t+1 = 0 without any outstanding debt. If the proposal is rejected, a new output state is realized and the game repeats until an agreement is reached. In the case that the proposer chooses to pass, both players enter the next period and continue the game.

We now define some basic concepts of the game. A stochastic bargaining game may be indexed by (C, β, $\frac{1}{1+r}$), where for each endowment state y ∈ Y, C(y) ⊂ R² is a cake representing the set of feasible utility vectors that may be agreed upon in that state. β and $\frac{1}{1+r}$ are the discount factors for B and L, respectively⁴. A payoff function is an element ∆(y) ∈ C(y), where ∆_i(y) is the utility to player i, i = B, L.

As in Merlo and Wilson (1995), we focus on a game with stationary strategies, that is, the players’ actions depend only on the current state and the current offer. In equilibrium, the proposer’s strategy is to propose when the proposal would be accepted for sure and to pass otherwise. The other player acts passively: she accepts when a proposal is made and waits if the proposer passes. Therefore, we can denote the proposer i’s equilibrium strategy as a simple stopping function τ_i, with τ_i = 0 when i proposes and τ_i = 1 when i passes; the other player accepts or waits accordingly. A stationary subgame perfect (SSP) equilibrium is then defined as the players’ equilibrium stationary strategies τ_B and τ_L, and the payoff functions, ∆_B and ∆_L, associated with these strategies for player B and L.

We then proceed to characterize the SSP strategies and payoffs. The expected payoff for the borrower B in period t, E ∆_B(y_t, b_t), is given by:

Here the superscript denotes the identity of the proposer. So ${{\Delta }}_B^B$ represents the borrower’s payoff when the borrower himself is the proposer, and ${{\Delta }}_B^L$ refers to the borrower’s payoff when the lender is the proposer.

Correspondingly, the expected payoff for the lender L in period t, E ∆_L(y_t, b_t), is given by:

where ${{\Delta }}_L^B$ denotes the lender’s payoff when the borrower is the proposer, and ${{\Delta }}_L^L$ refers to the lender’s payoff when himself is the proposer.

First consider the case when the borrower B is the proposer. We refer the proposed debt recovery rate as α^B and the value of proposing as $V_B^{prop}$. When B proposes and the proposal is accepted, the sovereign repays the agreed amount of debt, α^Bb, immediately and enters the next period with a good credit standing and no outstanding debt. Thus, $V_B^{prop}$ is given by

And the lender’s payoff from acceptance, $V_L^{acpt}$, is

When B chooses to pass, the sovereign stays in autarky, suffers the proportional output loss in this period and then enters the next period continuing the bargaining game. We denote the value of passing as $V_B^{pass}$:

The lender’s value of rejecting an offer (or waiting) is denoted as $V_L^{rejt}$, which is given by

In equilibrium, the proposer B’s strategy is to propose an α^B* that solves the following problem

The two constraints specify the set of debt recovery rates acceptable for both players to settle the debt in the current period. If this set is nonempty, then an agreement can be reached in the current period. The stopping function τ_B(y_t, b_t) = 0, meaning that the borrower proposes in this period, and the equilibrium debt recovery rate α^B* is the smallest element in this set. In other words, the proposer B can extract extra surplus to the extent that the lender is indifferent between accepting and rejecting. However, if this set is empty, then the borrower would choose to pass (τ_B(y_t, b_t) = 1), and hence to delay the renegotiation by one period. This is often the case when the current endowment is low and the future ones are expected to be higher. In these circumstances, the borrower is only willing to repay a small proportion of the debt arrears in the current period, smaller than what the lender expects to get in the future with a better endowment realization (and when the lender himself may get to be the proposer). Therefore, there is no debt recovery rate that the borrower is willing to offer and the lender is willing to accept. In the end, the borrower just passes and one more period of delay arises.

We now characterize the players’ payoff functions. If α^B* exists, then the borrower’s payoff, ${{\Delta }}_B^B$, is the value of proposing,

And the lender’s payoff, ${{\Delta }}_L^B$, is the value of accepting,

Otherwise, the borrower’s payoff is the value of passing,

And the lender’s payoff is the value of waiting (equivalent to the value of rejecting):

Similarly, when the lender is the proposer, we denote the proposed debt recovery rate as ${\alpha }_t^L$. The value for the lender to propose an acceptable debt recovery rate is $V_L^{prop}$, given by

And the value for the borrower to accept this proposal is $V_B^{acpt}$:

When L chooses to pass or her proposal is rejected, the value of choosing pass is denoted as $V_L^{pass}$,

And the borrower’s value in this case, $V_B^{rejt}$, is given by

Again, in equilibrium, the proposer L solves the following problem

Similar to our argument above, if the minimum debt recovery rate acceptable for L is too large for B, then no α^L* exists, and L chooses to pass (τ_L(y_t, b_t) = 1); otherwise, L proposes and τ_L(y_t, b_t) = 0.

When L proposes, the payoffs for both players, ${{\Delta }}_B^L$ and ${{\Delta }}_L^L$, are given by

When L passes, the payoffs are:

An SSP equilibrium is a fixed point of the above recursive system(Equations (7) to (26))⁵.

C. International Investors’ Problem

The last part of the model specifies the international investors’ problem. There is an infinite number of risk neutral and competitive international investors, one of whom is randomly chosen to trade with the sovereign government in each period. Investors choose the face value of the bond to maximize expected profits, taking the bond prices as given

where π_t is the profit from trading one-period bonds. More explicitly, the expected profit E(π_t) can be expressed as

When b_t+1 ≥ 0, the government saves by purchasing bonds. Since international investors commit to repaying debts, the government’s saving is risk free. When b_t+1 < 0, the government borrows by issuing new bonds, and default and debt restructuring are possible future events. In this case, the expected present value of repayment consists of two terms. The first represents the repayment in the following period, while the second reflects the expected debt recovery in the case of default. As noted in the government’s problem, the probability of repayment is (1 −θ_t+1), and that of default is θ_t+1. In the case of default, debt renegotiation begins in period t + 2. E(∆_L(y_t+2, b_t+2)) is the expected present value of the payoff that the debt holder can get from the renegotiation game.

From the zero-profit condition of the investors, we obtain the price function as follows:

Finally, note that the term $-\frac{E\left({{\Delta }}_L\right)}{\left(1+r\right)b_{t+1}}$ is just the expected present value of the debt recovery rate, which lies in the interval [0, 1], and hence the bond price ranges from 0 to the price of a risk free bond. Consequently, the interest rates of sovereign bonds are always higher than or equal to the risk free rate. We define the bond spread as the difference between the country interest rate and the risk free interest rate.

A. Calibration

The model is calibrated to match features of the Argentine economy. We define one period as a quarter.

We use a constant relative risk aversion (CRRA) utility function:

where σ is the risk aversion coefficient. This parameter is set to 2, a standard value in real business cycle studies. We set the parameter of proportional output loss γ to be 2%, based on Sturzenegger’s (2002) estimate. The quarterly risk-free interest rate r is set to 1%, following Yue (2006).

As mentioned in the model set up, we assume that the exogenous endowment stream follows a log-normal AR(1) process:

where ӯ is the mean output and is normalized to 1. The autocorrelation coefficient ρ and the standard deviation of endowment innovations, σ_ε, are computed from the quarterly real GDP data of Argentina from 1980 Q1 to 2005 Q4. The GDP data are seasonally adjusted and are taken from the Ministry of Economy and Production (MECON). The data are detrended using a Hodrick-Prescott filter with a smoothing parameter of 1600. We then construct a discrete Markov representation of the endowment process as a 51 state Markov chain employing the quadrature procedure by Hussey and Tauchen (1991).

In the last part of the calibration, we set the sovereign’s time discount factor β and the sovereign’s probability of being selected as proposer in each period, ϕ, simultaneously to match the frequency of default and the average debt recovery rate using the simulated method of moments (SMM).

Argentina defaulted 4 times from 1824 to 1999 as documented in Reinhart, Carmen M. and Savastano (2003), and then defaulted a fifth time in late 2001. As a consequence, its long-term quarterly default frequency is about 0.70%. In the last 25 years (for which the quarterly GDP data are available), however, Argentina has defaulted twice, once in 1982 and once in late 2001, resulting in an average quarterly default frequency of 1.92%, much higher than the long-term default frequency. In this calibration, we set the parameter values to match the short-term default frequency. The reason is that in our model, the default decisions and the equilibrium delay lengths are interdependent: states in which defaults are chosen largely determine the renegotiation delay lengths, and the expected delay lengths, in turn, affect the default decisions ex ante. Moreover, both default decisions and equilibrium delay lengths are essentially determined by the output process. Given that the available short series of quarterly GDP data are consistent with a quarterly default frequency of 1.92%, if we artificially set parameter values to match the long-term default frequency (i.e., 0.7%), then we may get a biased result for the average delay length in the quantitative analysis. Therefore, we aim to match the short-term default frequency of 1.92%. The average debt recovery rate in the most recent Argentine debt restructuring was approximately 35%(Sturzenegger and Zettelmeyer (2005)).

The time discount factor is found to be 0.96, and the sovereign’s probability of being proposer, ϕ, is found to be 0.60. Our calibrated model generates a default frequency of 2.19% and an average debt recovery rate of 38.47%, which closely match the corresponding data moments. The calibration results are summarized in Table 2. Table 3 compares the targeted data statistics and the model-generated statistics.

Table 2.

Model Parameter Values

Table 2.

Model Parameter Values

Parameter	Value	Sources
Coefficient of Risk Aversion	σ=2	RBC literature
Output Loss in Default	γ=2%	Sturzenegger (2002)
Risk Free Interest Rate	r=1%	Yue (2006)
Std. Dev. of Endowment Innovations	σε=0.047	MECON
Autocorr. Coef. of Endowment	ρ=0.83	MECON
Discount Factor	β=0.96	Matched by Calibration
Borrower’s Probability of Being Proposer	ϕ=0.60	Matched by Calibration

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Table 2.

Model Parameter Values

Parameter	Value	Sources
Coefficient of Risk Aversion	σ=2	RBC literature
Output Loss in Default	γ=2%	Sturzenegger (2002)
Risk Free Interest Rate	r=1%	Yue (2006)
Std. Dev. of Endowment Innovations	σε=0.047	MECON
Autocorr. Coef. of Endowment	ρ=0.83	MECON
Discount Factor	β=0.96	Matched by Calibration
Borrower’s Probability of Being Proposer	ϕ=0.60	Matched by Calibration

Table 3.

Target Statistics and Model Statistics

Table 3.

Target Statistics and Model Statistics

Target Statistics	Data	Model
Default Frequency	1.92%	2.19%
Debt Recovery Rate	35.00%	38.47%

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Table 3.

Target Statistics and Model Statistics

Target Statistics	Data	Model
Default Frequency	1.92%	2.19%
Debt Recovery Rate	35.00%	38.47%

B. Numerical Results on Equilibrium Properties

In this part, we present some of the equilibrium properties of the model. Figure 1 plots the equilibrium default probabilities for each state of output and debt-to-mean output ratio. It is clear that default probabilities are higher with lower endowment realizations and higher levels of debt. In other words, for a given level of debt, the sovereign is more likely to default in bad endowment states. This result supports our argument that defaults are usually associated with bad output realizations, providing incentives to wait for a larger “cake” in the post-default debt renegotiations.

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Equilibrium Default Probabilities

Citation: IMF Working Papers 2008, 038; 10.5089/9781451869002.001.A001

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We then present in Figure 2 the equilibrium propose/pass choices made by the borrower and the lender for each endowment state and defaulted debt-to-mean output ratio⁶. On the vertical axes, “Propose” represents the case that a proposal is made and accepted, i.e., an agreement is reached. “Pass”, on the contrary, implies waiting and hence one period of delay.

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Equilibrium Propose/Pass Choices of the Borrower and the Lender

Citation: IMF Working Papers 2008, 038; 10.5089/9781451869002.001.A001

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As a confirmation of Proposition 1, the two graphs in Figure 2 are identical, meaning that for a given endowment state and a defaulted debt-to-mean output ratio, the borrower and the lender would make the same propose/pass choice no matter who is selected as proposer.

Furthermore, Figure 2 shows the states in which an agreement can be attained. We observe three interesting results from this figure: first, for any given debt-to-mean output ratio, there is a threshold output level, under which “pass” is chosen; second, for low levels of defaulted debt (e.g., debt-to-mean output ratios smaller than 0.3), the threshold output state increases in debt; third, when the defaulted debt levels are large enough, the output threshold does not change with the debt level. The first two results are very intuitive: debt settlements only occur in sufficiently good output states, and with a higher defaulted debt level, the sovereign needs a better endowment realization (and hence more available resources) to settle the debt. We will discuss the third result later, after we present the renegotiated debt repayment schedules.

We now plot the equilibrium debt recovery rates proposed by each player at each state (Figure 3). In this figure, a zero debt recovery rate implies that the proposer chooses to “pass” in the particular state. In general, the debt recovery rates proposed by the borrower are smaller than those proposed by the lender. This is because the proposer enjoys the “first-mover advantage”, which results in lower equilibrium debt recovery rates when the borrower is the proposer and higher ones when the lender is the proposer. In addition, the shapes of the two debt recovery schedules are similar: both of them are convex and decreasing in the defaulted debt level, meaning that a lower level of defaulted debt results in a much larger debt recovery rate. These debt recovery schedules are also similar to that obtained from a Nash Bargaining game, except that in the latter case there is no delay in equilibrium.

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Equilibrium Debt Recovery Rates

Citation: IMF Working Papers 2008, 038; 10.5089/9781451869002.001.A001

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Furthermore, we can take a look at the total debt repayments proposed by each player, as shown in Figure 4.

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Proposed Debt Repayments

Citation: IMF Working Papers 2008, 038; 10.5089/9781451869002.001.A001

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When the proposed debt repayment is zero, it means the proposer passes. Again, the debt repayments proposed by the lender are generally larger than those proposed by the borrower. More interestingly, for a given output state, except for the low defaulted debt region (e.g., the debt-to-mean output ratios smaller than 0.3), the proposed repayment does not change with the debt level. In other words, for any given endowment realization, there is a maximum amount that the sovereign is willing to repay, regardless of the defaulted debt level. This explains why the threshold output state is invariant of the defaulted debt level in the large debt region. And again, this result coincides with that from Nash bargaining, which is not surprising given that the bargaining structure in this paper is a variant of the Rubinstein game. Finally, this maximum amount of repayment increases in the output realization.

C. Simulation Results

We conduct 500 rounds of simulations, with 2000 periods per round, and then extract the last 200 periods in each round to study the limiting distribution of the model economy.

First, we show the ergodic distribution of the equilibrium delay length in Figure 5. Equilibrium delay length ranges from zero to about 160 periods (a period is a quarter), while falling between 0 and 20 quarters most frequently.

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Ergodic Distribution of the Equilibrium Delay Length

Citation: IMF Working Papers 2008, 038; 10.5089/9781451869002.001.A001

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The average delay length, together with other business cycle statistics generated by the model are reported in Table 4. For comparison, we also report the corresponding data statistics in that table. The debt service-to-GDP ratio is computed by dividing Argentina’s short-term debt and debt service for long-term debt by its GDP from 1980 to 2005. The debt data are taken from the Global Development Finance database of the World Bank. Seasonally adjusted consumption and current account data from 1980Q1 to 2005Q4 are taken from MECON. Bond spreads are annualized quarterly spreads on 3-year foreign currency denominated bonds issued by the Argentine government, taken from Broner, Fernando and Schmukler (2005).

Table 4.

Business Cycle Statistics from the Model and the Data

Table 4.

Business Cycle Statistics from the Model and the Data

Statistics	Data	Model
Average Renegotiation Delay Length (Quarters)	13.3	19.91
Std. Dev. of Bond Spreads	1.72%	1.37%
Average Bond Spreads	5.16%	3.46%
Correlation between Spreads and Output	−0.54	−0.51
Debt Service-to-GDP	14.88%	9.28%
Std. Dev. of CA/Output	5.40%	4.00%
Correlation between CA/Output and Output	−0.91	0.12
Consumption Std. Dev./Output Std. Dev.	1.03	0.99

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Table 4.

Business Cycle Statistics from the Model and the Data

Statistics	Data	Model
Average Renegotiation Delay Length (Quarters)	13.3	19.91
Std. Dev. of Bond Spreads	1.72%	1.37%
Average Bond Spreads	5.16%	3.46%
Correlation between Spreads and Output	−0.54	−0.51
Debt Service-to-GDP	14.88%	9.28%
Std. Dev. of CA/Output	5.40%	4.00%
Correlation between CA/Output and Output	−0.91	0.12
Consumption Std. Dev./Output Std. Dev.	1.03	0.99

The model generates an average delay length of about 20 quarters, slightly longer than the actual delay length experienced by the Argentine government. This result suggests that the incentive to wait for a larger “cake” can indeed result in long “beneficial” delays in debt renegotiation processes. The simulated average delay length is a bit too long because of our assumption that the sovereign fully repays the reduced debt arrears right after the agreement. In the real world, the debt arrears are not fully repaid in one step; instead, they are restructured or swapped to new ones, and will be repaid in the future. These debt swap and restructuring provide some state contingency to the debt repayment schedules, which can shorten the renegotiation process. In the extreme case with complete market and state-contingent debt repayment schedules, there would be no delays at all. Therefore, our result can be considered as the “upper bound” of the beneficial delay length.

The model simulation closely matches the volatility of bond spreads, which has been hard to explain in the literature. The existing quantitative models on sovereign default either assume exogenous return to the capital markets without repaying anything after default (for example, Model I in Aguiar and Gopinath (2006) and Arellano (2007)), or use Nash Bargaining to model the debt renegotiation process (for example, Yue (2006)). In these two cases, the expected length of exclusion from the capital markets is more or less fixed. In the present model, however, the expected length of exclusion is uncertain and depends on the defaulted debt level and the output process. Therefore, the existence of equilibrium delays adds more uncertainty to the debt settlement outcome and contributes to the larger volatility of bond spreads. The model-simulated average bond spread falls below the data statistic, but it is closer to reality than those obtained by other quantitative analysis of sovereign debt. This improvement stems from both the higher default frequency that we target and the additional “waiting cost” incurred during delays, which reflects the forgone earnings from re-investing the defaulted debt.

Furthermore, the model captures the countercyclical behavior of the bond spreads. Bond spreads are higher in bad endowment states due to the higher default risk and the larger potential “waiting cost” following default. The latter arises because the players have to wait longer for the economy to recover from a deeper recession before they can settle the defaulted debt.

The model generates an average debt service-to-GDP ratio of 9.28%, lower than the data statistic 14.88%. Note that we include both short-term debt and debt service of long-term debt in calculating the data statistic, while in the model we only have quarterly (short-term) debt. Therefore, it is not surprising that the the debt-to-GDP ratio generated by the model is a little lower than that observed in data.

The model simulation matches the current account volatility, but it fails to capture the countercyclical behavior of the current account. In this model, the sovereign faces tighter borrowing constraints in recessions than in booms, and therefore has to rely more on its own savings, rather than outside financing, to smooth consumption in bad times. Consequently, the sovereign has to save more in booms, resulting in a procyclical current account.

Finally, the simulation generates a consumption stream that is a little less volatile than output. It is typical of emerging economies that consumption is more volatile than output (Neumeyer and Perri (2004)). Aguiar and Gopinath (2006) and Yue (2006) match this fact by assuming a stochastic trend for the output process, implying that output shocks are permanent. The present model has only transitory output shocks and the sovereign aims to smooth consumption over time. Its ability to smooth consumption is quite limited due to the potential default risk, so that the volatility of consumption generated by the simulation is just a little less than that of output.

A. Output Process and Equilibrium Delay Length

In this part, we investigate the effects of output properties on the equilibrium delay length. Table 5 reports the average delay lengths under different values of the autocorrelation coefficient ρ and the standard deviation of endowment innovations, σ_ε, holding other parameters at their benchmark values.

Table 5.

Delay Lengths under Different Output Processes

Table 5.

Delay Lengths under Different Output Processes

Endowment Autocorr. Coef.	Delay Length (Quarters)
ρ = 0.99	44.01
ρ = 0.83	19.91
ρ = 0.50	10.39
ρ = 0.35	6.12
ρ ∈ [−0.3, 0.3]	∞
ρ < −0.3	∞
Std. Dev. of Endowment Innovations
σε = 0.01	12.32
σε = 0.047	19.91
σε = 0.2	35.72
σε = 0.25	59.44

View Table

Table 5.

Delay Lengths under Different Output Processes

Endowment Autocorr. Coef.	Delay Length (Quarters)
ρ = 0.99	44.01
ρ = 0.83	19.91
ρ = 0.50	10.39
ρ = 0.35	6.12
ρ ∈ [−0.3, 0.3]	∞
ρ < −0.3	∞
Std. Dev. of Endowment Innovations
σε = 0.01	12.32
σε = 0.047	19.91
σε = 0.2	35.72
σε = 0.25	59.44

The upper panel shows how the delay length varies with the endowment autocorrelation coefficient. A higher autocorrelation implies more persistent endowment shocks, and hence slower recovery after default. Therefore, the players have to wait longer for a sufficiently large “cake” to materialize. Consistent with this intuition, our results show that as the autocorrelation increases from 0.35 to 0.99, the renegotiation process lengthens from 6.12 to 44.01 quarters on average.

When the autocorrelation coefficient is too low in absolute value (i.e., when ρ ∈ [−0.3, 0.3]), the output process is close to i.i.d. with very small volatility⁷. In this case, it is always optimal for the sovereign to stay in autarky forever after default. Intuitively, when the output stream is not volatile, the benefits of smoothing consumption can be smaller than the cost of repaying the reduced debt arrears. As a result, the renegotiation delay length is infinity. When the output realizations are negatively correlated with large absolute values of ρ (i.e., ρ < −0.3), the output process is again volatile, but the sovereign still chooses to stay in autarky forever once it defaults. The reason is that on the one hand, it is least costly to settle the defaulted debt in good output states, suggesting that the players should reach an agreement in booms; on the other hand, given the negatively autocorrelated output stream, a good state today implies a bad one tomorrow. A bad output state tomorrow tightens the borrowing constraint and limits the sovereign’s ability to smooth consumption. Therefore, a costly debt repayment does not provide much benefit for the sovereign in terms of consumption smoothing. In the end, the economy chooses not to settle the debt even in good states.

The lower panel delivers the effects of endowment volatility on delay length. The equilibrium delay length increases as the volatility increases. This result is intuitive: to settle the defaulted debt, we need large positive shocks to materialize. With a more volatile output stream, it is less likely that a good enough shock will occur. Consequently, the players have to wait longer after default for a sufficiently good realization to settle the defaulted debt.

B. Other Parameters and Renegotiation Delay Length

Table 6 shows the equilibrium delay length under different values for the time discount factor β, the sovereign’s probability of being selected as proposer, ϕ, the proportional output loss parameter γ and the risk-free interest rate r.

Table 6.

Delay Lengths under Different Parameter Values

Table 6.

Delay Lengths under Different Parameter Values

Discount Factor	Delay Length (Quarters)	Interest Rate	Delay Length (Quarters)
β = 0.99	16.20	r = 0.01	19.91
β = 0.96	19.91	r = 0.05	8.59
β = 0.85	13.77	r = 0.10	6.51
B’s Prob. of Being Proposer		Output Loss
ϕ = 0.90	10.12	γ = 0.005	23.96
ϕ = 0.60	19.91	γ = 0.02	19.91
ϕ = 0.30	24.29	γ = 0.10	8.68

View Table

Table 6.

Delay Lengths under Different Parameter Values

Discount Factor	Delay Length (Quarters)	Interest Rate	Delay Length (Quarters)
β = 0.99	16.20	r = 0.01	19.91
β = 0.96	19.91	r = 0.05	8.59
β = 0.85	13.77	r = 0.10	6.51
B’s Prob. of Being Proposer		Output Loss
ϕ = 0.90	10.12	γ = 0.005	23.96
ϕ = 0.60	19.91	γ = 0.02	19.91
ϕ = 0.30	24.29	γ = 0.10	8.68

The upper left panel of Table 6 reports the effects of the time discount factor, holding all other parameters fixed at their benchmark values. Interestingly, the average delay length does not move monotonically as the time discount factor changes. The time discount factor affects the equilibrium delay length through two channels: from the perspective of payoff maximization in the bargaining, a more patient sovereign is willing to wait longer, since its patience enables it to be at a stronger position and thus to enjoy a higher payoff; from the perspective of consumption smoothing, however, a more patient government would like to settle the debt sooner so that it can reaccess the capital markets and smooth consumption in the future. With a large discount factor (β = 0.99), the second effect dominates the first one, so the equilibrium delay length is shorter than the benchmark case (β = 0.96). When the discount factor is small (e.g., β = 0.85), on the contrary, the first effect dominates, that is, the sovereign cares more about its bargaining payoff, again resulting in a shorter-than-benchmark delay length. In sum, the equilibrium delay length is short when the sovereign is very patient, then it is longer as the sovereign becomes less patient, and finally it is short again when the sovereign is very impatient.

The lower left panel presents the effect of changing the borrower’s probability of being selected as proposer. As argued before, this probability captures the borrower’s bargaining power. Stronger bargaining power of the government results in a lower debt recovery rate, making it possible for the sovereign to repay the debt arrears even in a not-so-good state after a shorter period of waiting. Also, settling the debt sooner brings two benefits: regaining access to the capital markets sooner and suffering from the output loss for a shorter period of time. Consequently, as the sovereign’s probability of being proposer becomes larger, the average delay length is shorter.

The upper right panel delivers the effects of the world interest rate. As the risk-free interest rate increases, the debt-holder in the bargaining becomes more impatient (her discount factor is $\frac{1}{1+r}$). Therefore, she is eager to reach an agreement and get repaid sooner at the cost of a smaller debt repayment. For the sovereign, repaying less and settling the debt sooner are also beneficial. Therefore, the equilibrium delay length is shorter with a higher risk-free interest rate.

Lastly, the lower right panel reports the delay lengths under different values of post-default proportional output loss. Intuitively, when the loss is larger, staying in default is more costly, forcing the sovereign to settle its debt sooner. As a result, the average delay length decreases in the proportional output loss.