Efficient Sovereign Default

Efficient Sovereign Default Abstract In this article, I show that the key aspects of sovereign debt crises can be rationalized as part of the efficient risk-sharing arrangement between a sovereign borrower and foreign lenders in a production economy with informational and commitment frictions. The constrained efficient allocation involves ex post inefficient outcomes that resemble sovereign default episodes in the data and can be implemented with non-contingent defaultable bonds and active maturity management. Defaults and periods of temporary exclusion from international credit markets happen along the equilibrium path and are essential to supporting the efficient allocation. Furthermore, during debt crises, the maturity composition of debt shifts towards short-term debt and the term premium inverts as in the data. 1. Introduction The conventional framework to study sovereign debt crises, defined as periods of high interest rate spreads, is the incomplete-market approach that follows the seminal contribution of Eaton and Gersovitz (1981). This framework can account for the behaviour of interest rates, debt, and macroeconomic aggregates around sovereign debt crises. However, this framework is not well suited for policy analysis, since the frictions that lead to a failure of risk-sharing are not explicitly identified. In contrast, another strand of the literature studies the efficient risk-sharing arrangement between the sovereign borrower and foreign lenders by explicitly stating the underlying frictions. These models are suited for policy analysis, but they do not identify the securities that will be traded, so the relationship between data and model is less transparent. This article takes a first step towards bridging the gap between the literature on quantitative incomplete markets and that on constrained efficient risk-sharing arrangement. I show that the key aspects of sovereign debt crises can be rationalized as part of the efficient risk-sharing arrangement between a sovereign borrower and foreign lenders in a production economy with informational and commitment frictions. The efficient allocation can be implemented as the equilibrium outcome of a sovereign debt game with non-contingent defaultable debt of multiple maturities. Defaults and ex post inefficient outcomes along the equilibrium path are not a pathology; rather, they support the ex ante efficient outcome. Moreover, the equilibrium outcome path displays defaults when output is low, an inversion of the term structure of interest rates spreads when the spread level is high, and a negative association between the duration of outstanding debt and the probability of future defaults. I provide a different view on policies than the literature on incomplete markets. First, the negative correlation between the maturity of outstanding debt and the probability of a crisis emerges as a way to support the efficient allocation when only non-contingent defaultable debt of multiple maturities is available. Hence, the high reliance on short-term debt is just a symptom, and not a cause, of an imminent debt crisis, in contrast to the view in Cole and Kehoe (2000). Second, dilution of long-term debt is essential to replicate the insurance prescribed by the efficient allocation given the available assets. Hence, introducing and enforcing seniority clauses to avoid dilution of outstanding long-term debt will lower welfare. This contrasts with the predictions in Chatterjee and Eyigungor (2012) and Hatchondo et al. (2016). Finally, because ex post inefficient outcomes are part of the efficient allocation, interventions by a supranational authority aimed at reducing the inefficiencies in a sovereign default episode are not beneficial from an ex ante perspective. I consider a simple production economy in which imported intermediate inputs are used in production, similarly to Aguiar et al. (2009) and Mendoza and Yue (2012). The government cannot commit to repaying its debt and has private information about the state of the domestic economy.1 In the baseline economy, the source of private information is the relative productivity of the domestic non-tradable sector. One interpretation of this assumption is that the government has more information about the domestic economy than do foreign lenders, and it controls the released statistics.2 I first characterize the optimal risk-sharing arrangement between the government and the foreign lenders subject to the restrictions imposed by the lack of commitment and private information. Absent contracting frictions, the risk-neutral lenders would completely insure the risk-averse government against fluctuations in productivities, and the realization of the shock would have no effect on the continuation of the allocation. Both the private information and the government’s lack of commitment limit such insurance. In particular, because of the presence of private information, the provision of a dynamic incentive is needed to have insurance. In order for insurance payments to borrowers with currently low-productivity-shocks to be incentive compatible, there must be a cost associated with claiming to have low productivity. Lenders can impose such a cost by reducing the continuation value of the government through lowering its future consumption levels. The lack of commitment interacts with the incentive problem. In particular, when the government’s continuation value is low, the government is tempted to deviate from the efficient allocation by increasing current consumption by not repaying lenders and then living in autarky thereafter. To prevent such an outcome, the lenders must provide a sufficiently low amount of intermediate goods so that this kind of deviation is unprofitable. Enforcing a continuation value for the government close to autarky is ex post inefficient. That is, if the government and the lenders could renegotiate the terms of their agreement, committing not to do so again in the future, then both could be made better off. By increasing the government’s value when it is close to autarky, it is possible to avoid the drop in imported intermediate inputs, which depresses production and reduces the government’s ability to repay the lenders. The necessity of providing incentives ex ante requires that these ex post inefficient outcomes happen along the equilibrium path with strictly positive probability. I show that under appropriate sufficient conditions, any efficient allocation settles down to a stationary distribution with ex post inefficiencies. The theme that ex post inefficiencies along the equilibrium path are necessary to support the ex ante optimal arrangement in economies with incentive problems has been explored in various contexts (e.g.Green and Porter, 1984; Phelan and Townsend, 1991; Yared, 2010).3 A novel feature of my article is that there is no termination of the risk-sharing relationship. The optimal outcome has periods of temporary autarky (which are ex post inefficient), but cooperation eventually restarts after the domestic economy recovers and the government makes a payment to the lenders. I then turn to implementing an efficient allocation as an equilibrium outcome of a sovereign debt game, similar to what is considered by the literature on quantitative incomplete markets. The set of assets that the government can issue is restricted to non-contingent bonds of multiple maturities. The government has the option to default, which I define as suspending the principal and coupon payments specified by the bond contracts. The government is excluded from credit markets until a given partial repayment to the bondholders is made. The government can also impose a tax on the payments received by foreign exporters from the domestic firms for the imports of intermediate goods, capturing the idea that the government cannot commit to repay the foreign lenders. Along the equilibrium outcome path, there are defaults only when the government’s continuation value is equal to the autarkic value. In this sense, defaults in the model are infrequent. Moreover, defaults are associated with high indebtedness (relative to the maximal level of debt sustainable), low output, and ex-post inefficiencies. The crucial step in proving that the efficient allocation can be an outcome of the sovereign debt game is to show that it is possible to replicate the wealth transfers implied by the efficient allocation with non-contingent defaultable debt. Defaults and partial repayments introduce de facto implicit state contingencies in the bond contract. When there is full repayment, the state contingent returns implied by the efficient allocation are replicated by exploiting the variation in the price of long-term debt after a shock. After the realization of a low productivity shock, the continuation value for the sovereign borrower decreases and the probability of a default in the near future increases, reducing the value of the outstanding long-term debt. This reduction results in a capital loss for the lenders and provides some debt relief to the borrower after an adverse shock. The opposite happens after the realization of a high-productivity shock: the price of outstanding long-term debt goes up and the lenders realize a capital gain. The model can generate the features of output, consumption, imports, and exports that occur during and after debt crises. The proximate cause of a debt crisis is a sufficiently long string of low-productivity shocks, which lead the borrower’s continuation value to decrease until it reaches the value of autarky, when there is a default. The lack of commitment implies that the imports of intermediates must drop to prevent a deviation by the borrower. This drop in imports reduces output, consumption, and the payments made to the lenders. Once the economy receives a high-productivity shock in the non-tradable sector, output increases and the borrower runs a trade surplus to partially repay the defaulted debt. These repayments result in the gradual increase of the borrower’s continuation value; hence, consumption, production, and imported intermediate inputs used in production will also increase. The pattern for output and consumption is consistent with the evidence in Mendoza and Yue (2012) for a sample of twenty-three recent defaults.4 In these episodes, on average, output is 4.5% and 5.2% below trend, and consumption is 3.1% and 3.6% below trend in the year of the default and the year after, respectively.5 This association between default and output being below trend is not universal. In a larger sample, Tomz and Wright (2007) find that about a third of default episodes happen with output above trend. The drop in trade during default episodes predicted by the model is also in line with the data. For instance, see Rose (2005) and the references in the survey by Panizza et al. (2009).6 Along the path approaching default, the maturity composition of the sovereign debt shifts towards short-term debt. This shift occurs because, when the probability of future default is high, the price of the long-term debt is more sensitive to shocks. Therefore, a lower long-term debt holding is needed to replicate the debt relief that is implicit in the efficient allocation after a low realization of productivity in the non-tradable sector. Since the overall indebtedness of the sovereign borrower is increasing along the path approaching a default, it must be that the amount of short-term debt issued increases along with the probability of default. Thus, the maturity composition shortens as indebtedness increases. This is broadly consistent with the evidence in Broner et al. (2013) and Arellano and Ramanarayanan (2012). Both papers document that when spreads are high, the maturity of newly issued debt shortens. More directly, Buera and Nicolini (2010) document that the stock of outstanding debt in Argentina was more tilted towards shorter maturity in 2000 (at the onset of the default in January of 2002) relative to 1997 (see Figure 5 in their paper). Bocola and Dovis (2016) provide similar evidence for Italy in the current crisis. Furthermore, along the equilibrium path, the term spread curve inverts during debt crises as documented by Broner et al. (2013) and Arellano and Ramanarayanan (2012). Related Literature This article is related to several strands of literature. First, it is related to the literature on quantitative incomplete markets on sovereign default. Following Eaton and Gersovitz (1981), recent contributions include Aguiar and Gopinath (2006), Arellano (2008), Hatchondo and Martinez (2009), Benjamin and Wright (2009), Yue (2010), Arellano and Ramanarayanan (2012), Chatterjee and Eyigungor (2012), and Mendoza and Yue (2012). I depart from this literature by studying the best equilibrium outcome of the sovereign debt game when the maturity structure is sufficiently rich. As previously argued, the policy implications that arise from assuming that markets are incomplete differ substantially from those that arise from allowing for the possibility of complete markets. My work is also related to the literature on optimal contracting approach to sovereign borrowing (e.g.Atkeson, 1991; Thomas and Worrall, 1994, Kehoe and Perri, 2002; Kehoe and Perri, 2004; Aguiar et al., 2009; Tsyrennikov, 2013). More broadly, this paper is also related to the literature on dynamic contracting with informational and commitment frictions (e.g.Atkeson and Lucas, 1995; Ales et al., 2014). Green (1987), Thomas and Worrall (1990), and Atkeson and Lucas (1992) consider environments with only private information, while Kehoe and Levine (1993), Kocherlakota (1996), Alvarez and Jermann (2000), and Albuquerque and Hopenhayn (2004) consider only lack of commitment. The main contribution of my article relative to this strand of literature is to propose an implementation that relates the efficient outcome to the data series that are the focus of the empirical research: interest rates, default decisions, and maturity composition of debt. In this sense, this article takes a first step towards bridging the gap between the literature on quantitative incomplete markets and that on constrained efficient arrangements. The idea that managing the maturity composition of debt serves to replicate state contingent returns has been explored, among others, by Kreps (1982), Angeletos (2002), and Buera and Nicolini (2004). In these models, movements in the term structure of interest rates are generated by fluctuations in the equilibrium stochastic discount factor. I consider a small open economy environment with risk-neutral lenders where the international interest rate is orthogonal to the shocks in the domestic economy. The variation in price of long-term debt is generated through variation in the endogenous probability of future default. In this aspect, my article is closely related to the work of Arellano and Ramanarayanan (2012), who endogenize the time-varying maturity composition of debt in an Eaton and Gersovitz (1981) type of model. Consistent with my findings, these authors find that the maturity composition of debt shortens when the probability of default is high. The difference between their paper and mine is that they cannot assess the efficiency of such an equilibrium outcome. Moreover, they determine the maturity composition by trading off a hedging motive and a commitment-not-to-dilute motive. The hedging motive is that long-term debt is attractive because it allows for a state contingent return. The commitment-not-to-dilute motive is that short-term debt is a commitment device not to dilute outstanding long-term debt. In my article, the hedging motive alone is sufficient to shorten the duration of debt when the probability of default is high. The rest of the article is organized as follows. In Section 2, I describe the environment, while in Section 3, I characterize the efficient allocation. In Section 4, I define the sovereign debt game. In Section 5, I construct and characterize the default rule, bond prices, and holdings that support the efficient allocation as an equilibrium outcome of the sovereign debt game. Finally, after discussing the implementation in Section 6, I conclude the article in Section 7. 2. Environment In this section, I lay out the environment in which the source of private information is the productivity of the non-tradable sector. In the Online Appendix, I reinterpret this economy as a taste shock economy, as in Atkeson and Lucas (1992).7 Time is discrete and indexed by $$t=0,1,...$$. There are three types of agents in the economy: a large number of homogeneous domestic households, a benevolent domestic government, and a large number of foreign lenders. In addition, there are three types of goods: a domestic consumption good (non-traded), an export good, and an intermediate good. The source of uncertainty is a shock to the relative productivity of the domestic consumption (non-tradable) sector. 2.1. Preferences All agents are infinitely lived. The stand-in domestic household values a stochastic sequence of consumption of the domestic good, $$\{c_{t}\}_{t=0}^{\infty}$$, according to   \begin{equation} \mathbb{E}_{0}\sum_{t=0}^{\infty}\beta^{t}U\left(c_{t}\right), \end{equation} (1) where $$\beta\in(0,1)$$ is the discount factor, and the period utility function has constant relative risk aversion,   \begin{equation} U(c)=\frac{c^{1-\gamma}}{1-\gamma},\label{U_CRRA} \end{equation} (2) with $$\gamma>1$$. I assume that $$\gamma>1$$ to ensure that the government indebtedness increases after the negative productivity shock.8 The government is benevolent, and it maximizes the utility of the stand-in domestic household. Foreign lenders are risk neutral, and they value consumption of the export good. They discount the future with a discount factor $$q\in(0,1)$$, which should be thought of as the inverse of the risk-free interest rate in international credit markets. I allow the discount factor $$\beta$$ and $$q$$ to differ, but I will restrict my analysis to the case where $$q\geq\beta$$; that is, the domestic households discount the future at a weakly higher rate than the international interest rate. 2.2. Endowments and technology Foreign lenders have a large endowment of the intermediate good. They have access to a technology that transforms one unit of the intermediate good into one unit of the export good so the relative price between the export and the intermediate good is fixed at one. Each domestic household is endowed with one unit of labour in each period. There is a domestic production technology that transforms the intermediate good and labour into the domestic consumption good, $$c$$, and foreign consumption good, $$x$$, as follows:   \begin{gather} c\leq zF\left(m_{1},\ell_{1}\right)\,\,\text{ and }\,\,x\leq F\left(m_{2},\ell_{2}\right),\label{RCz1}\\ \end{gather} (3)  \begin{gather} m_{1}+m_{2}\leq m,\,\,\,\ell_{1}+\ell_{2}\leq1,\label{RCz2} \end{gather} (4) where $$m_{1}$$ and $$m_{2}$$ are the units of the intermediate good allocated to the production of the domestic and export good, respectively; $$m$$ is the total amount of intermediates used domestically; and $$\ell_{1}$$ and $$\ell_{2}$$ are the units of domestic labour allocated to domestic and export production, respectively. The production function $$F$$ has constant returns to scale; it is increasing and continuously differentiable, it satisfies the Inada condition $$\lim_{m\rightarrow0}F_{m}(m,\ell)=+\infty$$$$\forall\ell>0$$, and it is such that $$F(0,1)>0$$, so strictly positive output can be produced in autarky. For notational convenience, let $$f(m)=F(m,1)$$. The relative productivity of the domestic sector, $$z$$, is distributed according to a probability distribution $$\mu$$, and it is independent and identically distributed over time. For simplicity, let $$z$$ take on only two values, $$z\in\left\{ z_{L},z_{H}\right\}$$ with $$z_{L}<z_{H}$$.9 Due to the properties of constant-returns-to-scale technology, the technological restrictions imposed by (3)–(4) can be summarized by the following aggregate resource constraint   \begin{equation} \frac{c}{z}+x\leq f(m),\label{RCz} \end{equation} (RC) as well as the non-negativity conditions on $$c$$ and $$x$$. 2.3. Timing The timing of events within the period is as follows: Foreign lenders supply intermediate goods $$m_{t}\geq0$$; the productivity shock $$z_{t}$$ is realized according to $$\mu;$$ and real activity occurs: production, consumption, and exporting take place. Let $$z^{t}=(z_{0},z_{1},...,z_{t})$$. An allocation is a stochastic process $$\boldsymbol{x}\equiv\{m(z^{t-1}),c(z^{t}),x(z^{t})\}_{t=0}^{\infty}$$. An allocation $$\boldsymbol{x}$$ is feasible if it satisfies the resource constraint (RC) for all $$t,z^{t}$$. 2.4. Information Foreign lenders observe the amount of intermediate goods that the country imports, $$m$$, and the amount of exports, $$x$$. Moreover, they can observe the amount of resources, $$m_{1}$$ and $$\ell_{1}$$, employed in the domestic consumption (non-tradable) sector. However, they cannot see the amount of output produced with the inputs, because the realization of $$z$$ is privately observed by the domestic government. From (RC), foreign lenders can use their information about $$m$$ and $$x$$ to infer $$c/z$$ but not $$c$$ and $$z$$ separately. I collect assumptions made so far here: Assumption 1. The utility function is $$U\left(c\right)=c^{1-\gamma}/(1-\gamma)$$ with $$\gamma>1$$; the discount factor $$\beta$$ satisfies $$qz_{H}^{1-\gamma}/\mathbb{E}\left(z^{1-\gamma}\right)\leq$$$$\beta\leq q;$$ and the production function $$F\left(m,\ell\right)$$ is increasing, continuously differentiable, displays constant returns to scale, $$F(0,1)>0$$, and satisfies the condition $$\lim_{m\rightarrow0}F_{m}(m,\ell)=+\infty$$$$\forall\ell>0.$$ The productivity shock can take on two values, $$z\in\left\{ z_{L},z_{H}\right\}$$, and it is independent and identically distributed over time. The condition $$\beta>qz_{H}^{1-\gamma}/\mathbb{E}\left(z^{1-\gamma}\right)$$ ensures that the government is sufficiently patient so that in the high-productivity state it prefers to reduce its consumption below the autarky level, being rewarded with an increase in future consumption. 3. Efficient Allocation In this section, I define and characterize a (constrained) efficient allocation when lenders cannot separately observe $$c$$ and $$z$$, and the sovereign borrower cannot commit to repay. I establish that, under certain sufficient conditions, an efficient allocation has cyclical periods with ex post inefficient outcomes that resemble a sovereign default episode in the data. 3.1. Definition Private information and lack of commitment by the sovereign borrower impose constraints in addition to the RC, which an allocation must satisfy to be implementable. First, consider the restriction imposed by the fact that $$z$$ is privately observed by the borrower. By the revelation principle, it is without loss of generality to focus on the direct revelation mechanism in which the sovereign borrower reports his type. Define the continuation utility for the sovereign borrower associated with the allocation $$\boldsymbol{x}$$ after history $$z^{t}$$ (according to truth-telling) as   \begin{equation} v(z^{t})\equiv\sum_{j=1}^{\infty}\sum_{z^{t+j}}\beta^{j-1}\Pr(z^{t+j}|z^{t})U(c(z^{t+j})). \end{equation} (5) An allocation $$\boldsymbol{x}$$ is incentive compatible if and only if it satisfies the following (temporary) incentive compatibility constraint for all $$t,z^{t},z$$:   \begin{equation} U(c(z^{t-1},z_{t}))+\beta v(z^{t-1},z_{t})\geq U\left(z_{t}\left[f(m(z^{t-1}))-x(z^{t-1},z)\right]\right)+\beta v(z^{t-1},z)\label{IC}, \end{equation} (IC) where $$z_{t}\left[f(m(z^{t-1}))-x(z^{t-1},z)\right]$$ is the consumption of non-traded good if the borrower has productivity $$z_{t}$$ but exports the amount $$x\left(z^{t-1},z\right)$$ instead of $$x\left(z^{t-1},z_{t}\right)$$. The incentive compatibility constraint, (IC), captures the informational frictions in the economy. It ensures that the borrower has no incentive to engage in undetectable deviations. That is, after any history $$z^{t}$$, the borrower does not want to choose the action prescribed for type $$z\neq z_{t}$$. Second, consider the restrictions imposed by lack of commitment. To be implementable, an allocation $$\boldsymbol{x}$$ must satisfy the following sustainability constraint for all $$t,z^{t}$$:   \begin{equation} U(c(z^{t}))+\beta v(z^{t})\geq U(z_{t}f(m(z^{t-1})))+\beta v_{a},\label{SUST} \end{equation} (SUST) where $$v_{a}$$ is the value of autarky given by   \begin{equation} v_{a}\equiv\frac{\sum_{z}\mu(z)U(zf(0))}{1-\beta}. \end{equation} (6) The sustainability constraint, (SUST), requires that the borrower have no strict incentive to engage in detectable deviations. That is, after any history, the borrower cannot gain from increasing his consumption by failing to export $$x(z^{t})$$. As it is standard in the literature, I assume that after this detectable deviation, the borrower is punished with autarky. This entails two forms of punishment. First, the sovereign borrower cannot access credit markets to obtain insurance. Second, the borrower suffers a loss in production because he cannot use imported intermediate goods. Later, I will show this autarkic value is the worst equilibrium value of the sovereign debt game. A feasible allocation $$\boldsymbol{x}$$ is said to be efficient if it maximizes the present value of net transfers to the foreign lenders, $$x-m$$, subject to (RC), (IC), the sustainability constraint (SUST), and a participation constraint for the borrower,   \begin{equation} \sum_{t=0}^{\infty}\sum_{z^{t}}\beta^{t}\Pr(z^{t})U(c(z^{t}))\geq v_{0}\label{PC} \end{equation} (PC) for some feasible initial level of promised utility $$v_{0}\in\lbrack v_{a},\bar{v}]$$, with $$\bar{v}\equiv\lim_{c\rightarrow\infty}U(c)/(1-\beta)$$. An efficient allocation solves   \begin{equation} J(v_{0})=\max_{\boldsymbol{x}}\sum_{t=0}^{\infty}\sum_{z^{t}}q^{t}\Pr(z^{t})\left[x(z^{t})-m(z^{t-1})\right]\label{J} \end{equation} (J) subject to (RC), (IC), (SUST), and (PC). I will refer to the value $$J:[v_{a},\bar{v}]\rightarrow$$ as the Pareto frontier. The constraint set in (J) is not necessarily convex, because of the presence of $$U\circ f(m)$$, a concave function, on the right hand side of (SUST). Thus, randomization may be optimal. It is possible to rule out randomization as part of the efficient allocation by making an additional assumption following Aguiar et al. (2009). Assumption 2. Let $$m^*$$ be the statically efficient level of intermediates, i.e. $$m^*$$ such that $$f'(m^*)=1$$. Define $$H:[U(f(0)),U(f(m^{\ast}))]\rightarrow$$ as $$H(\underline{u})\equiv C(\underline{u})-f^{-1}\circ C(\underline{u})$$ with $$C=U^{-1}$$. $$H$$ is concave. If Assumption 2 is satisfied,10 then randomization is not optimal. 3.2. Near-recursive formulation The problem in (J) admits a near-recursive formulation using the borrower’s promised utility, $$v$$, as a state variable. The problem is not fully recursive, because the problem in period 0 is slightly different, as I explain next. From $$t\geq1$$, an efficient allocation solves the following recursive problem for $$v\in\lbrack v_{a},\bar{v}]$$:   \begin{equation} B(v)=\max_{m,c(z),v^{\prime}(z)}\sum_{z}\mu\left(z\right)\left[f(m)-m-\frac{c(z)}{z}+qB(v^{\prime}(z))\right]\label{P} \end{equation} (P) subject to   \begin{eqnarray} U\left(c(z)\right)+\beta v^{\prime}(z) & \geq & U\left(z\left[f(m)-y^{\ast}(z^{\prime})\right]\right)+\beta v^{\prime}(z^{\prime}) \forall z,z^{\prime}\label{ic}\\ \end{eqnarray} (7)  \begin{eqnarray} U\left(c(z)\right)+\beta v^{\prime}(z) & \geq & U\left(zf(m)\right)+\beta v_{a} \forall z\label{sust}\\ \end{eqnarray} (8)  \begin{eqnarray} v^{\prime}(z) & \geq & v_{a} \forall z\label{sustp}\\ \end{eqnarray} (9)  \begin{eqnarray} \sum_{z}\mu\left(z\right)\left[U\left(c(z)\right)+\beta v^{\prime}(z)\right] & = & v,\label{pkc} \end{eqnarray} (10) where $$B(v)$$ is the maximal present discounted value of net transfers, $$x-m=f(m)-c/z-m$$, that the foreign lenders can attain subject to a recursive version of the incentive compatibility constraint, (7), a recursive version of the sustainability constraint, (8), the fact that continuation utility must be greater than the value of autarky, (9), and the constraint that the recursive allocation delivers a value of $$v$$ to the sovereign borrower (the promise-keeping constraint), (10). The function $$B$$ traces out the utility possibility frontier. At $$t=0$$, for all $$v_{0}\in\lbrack v_{a},\bar{v}]$$, the problem in (J) can be expressed as   \begin{equation} J(v_{0})=\max_{m,c(z),v^{\prime}(z)}\sum_{z}\mu\left(z\right)\left[f(m)-m-\frac{c(z)}{z}+qB(v^{\prime}(z))\right] \end{equation} (11) subject to (7), (8), (9), and the participation constraint   \begin{equation} \sum_{z}\mu\left(z\right)\left[U\left(c(z)\right)+\beta v^{\prime}(z)\right]\geq v_{0}.\label{pc} \end{equation} (12) The difference between the Pareto frontier $$J$$ and the utility possibility frontier $$B$$ is that, in the utility possibility frontier, the promise-keeping constraint (10) requires that the allocation deliver exactly the promised utility $$v\in\lbrack v_{a},\bar{v}]$$ to the sovereign borrower. This is because for $$t\geq1$$, the promise-keeping constraint serves to maintain incentives from previous periods. In contrast, in the definition of the Pareto frontier $$J$$, the participation constraint (12) requires that the sovereign borrower receive at least$$v$$. This is because in period $$t=0$$ there are no incentives from previous periods to keep. In many applications, this asymmetry is irrelevant, because the participation constraint in (J) is binding. This is not the case here, because the utility possibility frontier $$B(v)$$ has an increasing portion, as I will later show. 3.3. Properties The next proposition establishes three properties of the efficient allocation that I will later use to characterize the equilibrium outcome which supports the efficient allocation. Proposition 1. Under Assumptions 1 and 2, the efficient allocation is such that (i)There are distortions in production. There exists $$v^{\ast}\in(v_{a},\bar{v})$$ such that $$m(v)=m^{\ast}$$ for all $$v\geq v^{\ast}$$. For all $$v\in\lbrack v_{a},v^{\ast})$$, $$m(v)$$ is strictly less than $$m^{\ast}$$ and is strictly increasing in $$v$$, and, in particular, $$m(v_{a})=0$$. (ii)The efficient allocation is dynamic: $$\forall v\in\lbrack v_{a},\bar{v}]$$, $$c(v,z_{L})>c(v,z_{H})$$ and $$v^{\prime}(v,z_{L})<v^{\prime}(v,z_{H})$$. (iii)There is insurance. Let $$b(v,z)\equiv x(v,z)-m(v)+qB(v^{\prime}(v,z))$$ be the lenders’ value after the realization of $$z$$. Then $$\forall v\in\lbrack v_{a},\bar{v}]$$, $$b(v,z_{H})>b(v,z_{L})$$. (iv)The value for the lenders when $$v=v_{a}$$ is positive, $$b(v_{a},z)>0$$ for all $$z.$$ The proof of this proposition can be found in the Online Appendix. Part (i) states that low levels of promised utility for the borrower are associated with imported intermediates that are below the statically efficient level, $$m^{\ast}$$, which is such that $$f^{\prime}(m^{\ast})=1$$. When the continuation value for the borrower is low, imports must be low to satisfy the sustainability constraint. Whenever the sustainability constraint is binding, $$m<m^{\ast}$$. In particular, at autarky it must be that $$m(v_{a})=0$$. In fact, if the foreign lenders supplied any $$m>0$$, the sovereign government could unilaterally achieve a lifetime utility of $$U(zf(m))+\beta v_{a}>U(zf(0))+\beta v_{a}=v_{a}$$. Thus, only $$m=0$$ is consistent with the promise-keeping and sustainability constraints at autarky. On the other hand, for continuation values high enough, $$v\geq v^{\ast}$$, the threat of autarky after an observable deviation is sufficiently harsh that the statically efficient amount of intermediate imports can be supported; that is, $$m(v)=m^{\ast}$$ for all $$v\geq v^{\ast}$$. It can be shown that $$m$$ is actually strictly increasing in the borrower’s promised value for $$v\in\lbrack v_{a},v^{\ast}]$$. This result is illustrated in Figure 1. Figure 1 View largeDownload slide Policy function for intermediate imports Figure 1 View largeDownload slide Policy function for intermediate imports Part (ii) states that the efficient allocation is dynamic, in the sense that it uses variation in the borrower’s continuation utility to provide incentives, thus allowing for higher transfers after the realization of a low-productivity shock. This feature of the efficient allocation is critical for it to be implementable as an outcome of the sovereign debt game. Part (iii) shows that the market value of debt is state-contingent; there is debt relief when the borrower has a high marginal utility of consumption (low $$z$$). Thus, the efficient allocation provides some, albeit imperfect, insurance. Part (iv) states that the market value of debt when the continuation value for the borrower equals autarky is strictly positive. In the implementation, there will be defaults when the value for the borrower equals autarky. This property implies that the value of debt must be positive during a default. This will in turn require that I allow for partial repayment of debt. 3.4. Optimality of ex post inefficiencies I now turn to the main result of this section: an efficient allocation calls for ex post inefficient outcomes with strictly positive probability, provided that a sufficient condition is satisfied. 3.4.1. Region with ex post inefficiencies The next proposition establishes that the utility possibility frontier is upward sloping for borrower values close to autarky. Lemma 1. There exists a $$\tilde{v}\in(v_{a},v^{\ast})$$ such that the utility possibility frontier $$B(v)$$ is strictly increasing over $$[v_{a},\tilde{v})$$ and decreasing over $$[\tilde{v},\bar{v}]$$. I refer to the interval $$[v_{a},\tilde{v})$$ as the region with ex post inefficiencies because for all $$v\in\lbrack v_{a},\tilde{v})$$, the market value of debt (and hence the value for the lenders) can be increased by providing higher utility to the borrower, thus making both existing lenders and the borrower better off. This is because supporting a continuation value for the borrower close to the autarkic level requires that a very low level of intermediate goods be employed in production so that the sustainability constraint (8) is satisfied. This depresses production and, consequently, the repayments that the lenders can receive in the period. In particular, when the borrower’s value is close to autarky, intermediates are close to zero (see Proposition 1 part (i)). Thus, because of the Inada condition on $$f$$, the marginal return from additional intermediates is large enough that the benefit from the extra production which can be obtained by increasing the borrower’s continuation value is larger than the cost to the lenders of providing the additional value to the borrower. Therefore, both agents can be made better off relative to autarky, and $$B$$ is upward sloping in a neighbourhood of $$v_{a}$$. In contrast, for sufficiently high promised values, $$v\geq v^{\ast}$$, the statically efficient level of intermediates can be supported. For such promised values, increasing the borrower’s value is costly and has no benefit for the lenders, and so $$B$$ is strictly decreasing for $$v\geq v^{\ast}$$. Therefore, because of the concavity of $$B$$, the utility possibility frontier must peak at some $$\tilde{v}\in(v_{a},v^{\ast})$$. Over the interval $$[\tilde{v},\bar{v}]$$, which I will refer to as the efficient region, $$B$$ is decreasing. These results are illustrated in Figure 2. Figure 2 View largeDownload slide Pareto and utility possibility frontiers Figure 2 View largeDownload slide Pareto and utility possibility frontiers 3.4.2. The efficient allocation transits to the region with ex post inefficiencies Any efficient allocation starts in the efficient region, because the participation constraint, (PC), in the programming problem (J) can hold as an inequality. For any borrower value, $$v$$, in the region with ex post inefficiencies, (PC) does not bind and $$J(v)=J(\tilde{v})=B(\tilde{v})>B(v)$$. It is optimal for the lenders to promise at least $$\tilde{v}$$ to the borrower. Instead, for $$v$$ in the efficient region (PC) in (J) binds and $$J(v)=B(v)$$. The question now is, Does an efficient allocation transit to the region with ex post inefficiencies after some history, or is the efficient region an ergodic set? Provided that a sufficient condition is satisfied, the continuation of any efficient allocation transits to the region with ex post inefficiencies after a sufficiently long (but finite) string of realizations of $$z_{L}$$. The essential piece of the argument is to show that following a realization of $$z_{L}$$, the continuation utility is strictly lower than the current one: $$v^{\prime}(v,z_{L})<v$$. To this end, I assume that the following sufficient condition holds: Assumption 3. The parameters $$z_{L},z_{H},\mu_{H},$$ and $$\gamma$$ satisfy the following conditions:11  \begin{align} &\mu_{H}z_{L}^{1-\gamma} \geq\mathbb{E}\left(z^{1-\gamma}\right)\label{Aeconomics}\\ \end{align} (13)  \begin{align} &\left(1-\mu_{H}\right)\left(\frac{z_{L}^{1-\gamma}}{\mathbb{E}\left(z^{1-\gamma}\right)}-1\right)+\mu_{H}\left(\frac{z_{H}}{z_{L}}\right)^{1-\gamma} \geq1\label{Atech}. \end{align} (14) Under this additional assumption, I can prove the following lemma: Lemma 2. Under Assumptions 1–3, for every $$v$$ in the efficient region, $$v'\left(v,z_{L}\right)<v$$. Lemma 2 is not obvious, because there is a tension between two countervailing forces. First, there is an incentive effect that calls for lowering $$v^{\prime}(v,z_{L})$$ below $$v$$. This is because lowering the continuation utility after a low-productivity shock helps to separate types and to provide more current consumption when the marginal utility of consumption is high. Second, there is a countervailing commitment effect: Lowering the continuation utility tightens future sustainability constraints. As is standard in economies with only lack of commitment, there is a motive to backload payments to the sovereign borrower in order to relax future sustainability constraints and allow for lower-production distortions in the future. Under Assumption 3, the incentive effect outweighs the commitment effect. To understand why, consider a necessary first-order condition from the problem   \begin{equation} B^{\prime}(v)=\frac{q}{\beta}\left[\mu_{L}B^{\prime}(v^{\prime}(v,z_{L}))+\mu_{H}B^{\prime}(v^{\prime}(v,z_{H}))\right]+\frac{f^{\prime}(m(v))-1}{U^{\prime}(z_{L}f(m(v)))f^{\prime}(m(v))},\label{DB2} \end{equation} (15) which, using the fact that $$B^{\prime}(v)\leq0$$ for $$v\geq\tilde{v}$$ and $$\beta\leq q$$, can be rewritten as   \begin{equation} \left[B^{\prime}(v^{\prime}(v,z_{L}))-B^{\prime}(v)\right]\geq\mu_{H}\left[B^{\prime}(v^{\prime}(v,z_{L}))-B^{\prime}(v^{\prime}(v,z_{H}))\right]-\frac{\beta}{q}\frac{f^{\prime}(m(v))-1}{U^{\prime}(z_{L}f(m(v)))f^{\prime}(m(v))}.\label{DB3} \end{equation} (16) Equation (16) illustrates the two forces operating in the model. The first term in square brackets on the right-hand side of (16) stands in for the incentive effect, while the second term stands in for the commitment effect. First notice that by the concavity of $$B$$, if the right-hand side of (16) is positive, then it must be that $$v^{\prime}(v,z_{L})<v$$. By Proposition 1, Part (ii), $$v^{\prime}(v,z_{H})>v^{\prime}(v,z_{L})$$ and, thus, the first term on the right-hand side of (16) is strictly positive. Absent any commitment problem (i.e. $$f^{\prime}(m)=1$$), the second term on the right-hand side of (16) is equal to zero. Therefore, the right-hand side is positive and, consequently, it will be true that $$v^{\prime}(v,z_{L})<v$$. When the sustainability constraint binds (i.e., $$f^{\prime}(m)>1$$), the second term on the right-hand side of (16), $$-\left[f^{\prime}(m)-1\right]/\left[U^{\prime}(z_{L}f(m))f^{\prime}(m)\right]$$, is negative; it is then not obvious that the right-hand side of (16) is positive. Thus, in this case, it is not guaranteed that $$v^{\prime}(v,z_{L})<v$$. Assumption 3 guarantees that this is indeed the case. Note that such assumption is met either if, for a given $$\mu_{H}$$, $$z_{L}^{1-\gamma}-z_{H}^{1-\gamma}$$ is sufficiently large or, for given $$z_{L}^{1-\gamma}-z_{H}^{1-\gamma}$$, $$\mu_{L}$$ is sufficiently small. Intuitively, if $$z_{L}^{1-\gamma}-z_{H}^{1-\gamma}$$ is sufficiently large, the benefit of separating the two types is large. It is very cheap to satisfy the promise-keeping constraint by providing consumption when the productivity shock is low, $$z=z_{L}$$. To provide a large spread in current consumption across types in an incentive-compatible way, it is necessary to have a large spread in continuation values, $$v^{\prime}(v,z_{H})-v^{\prime}(v,z_{L})$$. Thus, the first term of the right-hand side of (16) is large. Moreover, if $$\mu_{L}$$ is small, the cost of tightening future sustainability constraints by reducing the continuation value after $$z_{L}$$ is small, from an ex ante perspective. Inspecting (16), if $$\mu_{H}$$ is low, then the first term on the right-hand side is again large. Thus, if either $$z_{L}^{1-\gamma}-z_{H}^{1-\gamma}$$ is sufficiently large or $$\mu_{L}$$ is sufficiently small, the benefits from lowering $$v^{\prime}(v,z_{L})$$ below $$v$$ by relaxing the incentive compatibility constraint and the current sustainability constraint (incentive effect) are larger than the costs arising from higher-production distortions in the future after a low productivity shock (commitment effect). Under the assumptions in Lemma 2, for all $$v$$ in the efficient region, $$v^{\prime}(v,z_{L})$$ lies strictly below the 45-degree line, as illustrated in Figure 3. Let $$\Delta\equiv\min_{v\in\lbrack\tilde{v},\bar{v}]}\left\{ v-v^{\prime}(v,z_{L})\right\}$$. Because $$v^{\prime}(\cdot,z_{L})$$ is continuous as shown in Lemma B2 in the Online Appendix, it follows that $$\Delta>0$$. Thus, starting from any $$v_{0}\in\lbrack\tilde{v},\bar{v})$$, after a sequence of $$t$$ consecutive realizations of $$z_{L}$$, the borrower’s continuation value is less than $$v_{0}-\Delta t$$. Thus, after a sufficiently long string $$z^{T}=\left(z_{L},z_{L},...,z_{L}\right)$$ with $$T\leq(v_{0}-\tilde{v})/\Delta$$, the continuation utility transits to the region with ex post inefficiencies, $$[v_{a},\tilde{v})$$. The next proposition summarizes the argument above. Figure 3 View largeDownload slide Law of motion for borrower’s value Figure 3 View largeDownload slide Law of motion for borrower’s value Proposition 2. Under Assumptions 1–3, an ex ante efficient allocation transits to the region with ex post inefficiencies with strictly positive probability. 3.4.3. Role of the main ingredients The interaction between lack of commitment and private information is key to having ex post inefficient outcomes happening along the path in this production economy. Both lack of commitment and the fact that intermediates are used in production are crucial to generating an upward sloping portion of the utility possibility frontier. However, these two features alone cannot generate ex post inefficient outcomes associated with an ex ante efficient allocation. Without an incentive problem, any continuation of an efficient allocation is itself efficient. Thus, an efficient allocation never transits to the region with ex post inefficiencies of the utility possibility frontier. See Aguiar et al. (2009) for this result in a related environment. Private information alone generates a downward drift of the continuation utility (see Thomas and Worrall (1990) and Atkeson and Lucas (1992)) but does not generate ex post inefficiencies, because with commitment, there is no connection between low continuation values and production in the economy. The statically efficient amount of production can always be sustained. Low continuation values for the borrower only have distributional effects in that the lenders can appropriate larger shares of total undistorted production. Also in this case, continuations of efficient allocations are always on the Pareto frontier. Both contracting frictions are needed to obtain ex post inefficient outcomes as part of the ex ante optimal arrangement (Proposition 2). Lack of commitment and production are crucial for having an upward sloping portion of the utility possibility frontier (Lemma 1); private information is crucial for having the efficient allocation to transit to the region with ex post inefficiencies (Lemma 2). These features are also present in previous works that also establish the optimality of ex post inefficiencies in related environments, such as Phelan and Townsend (1991), Quadrini (2004), Clementi and Hopenhayn (2006), DeMarzo and Sannikov (2006), and DeMarzo and Fishman (2007). 3.5. Long-run properties The next proposition establishes that the efficient allocation features perpetual cycles that transit in and out of the region with ex post inefficiencies. Proposition 3. Under Assumptions 1–3, any efficient allocation converges to a unique non-degenerate stationary distribution. Moreover, the inefficient region, $$[v_{a},\tilde{v}]$$, is in the support of such distribution. The proof is relegated to the Online Appendix. The key to understand Proposition 3 is to understand the law of motion for continuation utility illustrated in Figure 3. These laws of motion define a unique ergodic set for promised utility. By Lemma 2, after a sufficiently long—and finite—string of draws of $$z_{L}$$, continuation utility transits to the region with ex post inefficiencies. This region, and the value of autarky in particular, is not an absorbing state. In the Online Appendix, I show that whenever $$z_{H}$$ is drawn, then the continuation is back in the efficient region. To show that autarky is not an absorbing state, a sufficient condition is that $$\beta>qz_{H}^{1-\gamma}/\mathbb{E}\left(z^{1-\gamma}\right).$$ This condition ensures that the government is sufficiently patient so that in the high-productivity state it prefers to reduce its consumption below the autarky level, being rewarded with an increase in future consumption. Thus, under these conditions, there is sufficient “mixing” that the existence of a unique limiting distribution is guaranteed. The limiting distribution has perpetual cycles that transit in and out of the region with ex post inefficiencies. This feature differentiates my environment from related dynamic contracting problems such as those in Quadrini (2004), Clementi and Hopenhayn (2006), DeMarzo and Sannikov (2006), DeMarzo and Fishman (2007), and Hopenhayn and Werning (2008), which also have ex post inefficiencies along the path. In all of these papers, in the long run, either the incentive problem disappears or there is an inefficient termination of the venture between the principal and the agent. In contrast, here the incentive problem does not disappear in the long run and the risk-sharing relationship does not terminate. The optimal allocation has periods of temporary autarky, but cooperation eventually restarts after the domestic economy recovers.12 The efficient allocation only pins down the transfers between the sovereign borrower and the foreign lenders. It is silent about bond prices, defaults, and so forth. In the next section, I show that the efficient allocation can be implemented given a set of assets typically considered in the quantitative sovereign default literature. 4. Sovereign Debt Game In this section, I describe the game that implements the efficient allocation. The set of securities that the sovereign borrower can issue is restricted to the ones considered in the literature on quantitative incomplete markets. I impose a set of rules that describe the treatment of the government in default. These rules stand in for the outcome of a renegotiation process and determine the payoff of debt in default. I take these rules as a feature of the environment, and I derive implications for the path of debt and bond prices that implement the efficient allocation given these rules. The fact that the constrained efficient allocation can be implemented implies that this set of rules is not restrictive. In Section 6, I discuss how the assumption about this process can be relaxed without affecting the main characteristics of equilibrium outcomes. Consider a game between competitive foreign lenders (bondholders and exporters of the intermediate good), competitive domestic firms, and a benevolent domestic government. The government can issue two types of non-contingent defaultable bonds: a one-period bond, $$b_{S}$$ (or foreign reserves if $$b_{S}<0$$) and a consol, $$b_{L}\geq0$$, that cannot be used for savings. One unit of the one-period bond promises to pay one unit of the export good tomorrow in exchange for $$q_{S}$$ units of the export good today. The consol is a perpetuity that promises to pay a coupon of one unit of the export good in every period starting tomorrow in exchange for $$q_{L}$$ units of the export good today. The government cannot commit to satisfying the terms of the bond contracts. The borrower can choose among three options regarding the repayment of inherited debt, $$\delta_{t}\in\left\{{\texttt{full}},{\texttt{partial}},{\texttt{suspend}}\right\} .$$When $$\delta_{t}=$$full, there is full repayment: the government pays in full the outstanding one-period bond and the coupon on the consol. When $$\delta_{t}=$$partial, there is partial repayment: the government pays a fraction $$r_{S}\in\lbrack0,1)$$ of outstanding obligations on one-period bonds and $$r_{L}/(1-q)$$ for each unit of consols outstanding with $$r_{L}\in\lbrack0,1)$$. After making this partial payment, the government does not make any more future payments to the holder of legacy long-term debt. Note that holders of the consol are treated equally irrespective of the time the consol was issued.13 Finally, $$\delta_{t}=$$suspend denotes suspension of payments: the government makes no payments in the current period. It is then excluded from international credit markets, and the defaulted debt is rolled over to the next period (missed interest payments are forgiven) when the government can choose a repayment policy $$\delta_{t+1}$$ for the notional amount of today’s debt obligations. I would say that the government is in default whenever it repays less than what is contractually specified, that is, $$\delta_{t}$$$$\neq$$full. I introduce partial repayment as a possible choice for the government to decentralize the efficient allocation. This is because the total value of debt in the efficient allocation (i.e., the present discounted value of net transfers to foreign lenders) is positive when the value for the borrower equals autarky. I will discuss this issue further in the proof of Proposition 4. In addition to issuing debt, the government can also tax the payments made by domestic firms to foreign exporters for the intermediate goods at a rate $$\tau_{t}\in\lbrack0,1]$$. Thus, foreign exporters receive an after-tax payment of $$p_{t}(1-\tau_{t})$$ per unit of intermediate good sold, where $$p_{t}$$ is the price of the intermediates in terms of the final good. We should think of this as the government interfering with private transactions.14 The sequence of events within the period is as follows: Foreign lenders set a price for intermediate inputs $$p_{t}$$ (how many units of export good for one unit of intermediate good). Domestic competitive firms choose the quantity of intermediate inputs they want to use, $$m_{t}$$. The productivity shock $$z_{t}$$ is realized and privately observed by the domestic government. The government picks a policy $$\pi_{t}=(\delta_{t},\boldsymbol{b}_{t+1},\tau_{t})$$ that consists of a repayment decision $$\delta_{t}$$, new bond holdings, $$\boldsymbol{b}_{t+1}=(b_{S,t+1},b_{L,t+1}),$$ and a tariff on imported intermediates, $$\tau_{t}$$. Bond holdings are bounded by a large positive constant $$\bar{B}$$ to rule out a Ponzi scheme. Bond prices $$\boldsymbol{q}_{t}=(q_{S,t},q_{L,t})$$ are consistent with foreign lenders’ optimality. Following Chari and Kehoe (1990), to formally define a sustainable equilibrium, let $$h^{t}=(h^{t-1},p_{t},m_{t},\pi_{t})$$ be a public history up to period $$t\geq0,$$ and let $$h^{-1}=\boldsymbol{b}_{0}=(b_{S,0},b_{L,0})$$ be the initial outstanding debt. It is convenient to define the following public histories when agents take action: $$h_{p}^{t}=h^{t-1}$$, $$h_{m}^{t}=(h^{t-1},p_{t})$$, and $$h_{\sigma}^{t}=(h^{t-1},p_{t},m_{t})$$. The price of the intermediate good, $$p=\left\{ p_{t}\right\} _{t=0}^{\infty}$$, the allocation rule for $$m=\left\{ m_{t}\right\} _{t=0}^{\infty}$$, the strategy for the government, $$\sigma=\left\{ \sigma_{t}\right\} _{t=0}^{\infty}$$, and the price of bonds, $$\boldsymbol{q} =\left\{ q_{S,t},q_{L,t}\right\} _{t=0}^{\infty}$$, are all functions of the relevant histories. 4.1. Problem of the government To set up the problem for the government, let $$Y(\tau,z)=zF(m_{t},1)-p_{t}m_{t}+\tau p_{t}m_{t}$$ be the amount of resources available to the government after production, repayments of intermediates, and collection of the tariff revenue. Taking as given $$p$$, $$m$$, and the price schedule for bonds, $$\boldsymbol{q}$$, after any history $$\left(h_{\sigma}^{t},z\right)$$, the strategy for the government, $$\sigma,$$ solves the problem   \begin{equation} W(h_{\sigma}^{t},z)=\max_{c,\pi=\left(\delta,b_{S}^{\prime},b_{L}^{\prime},\tau\right)}U(c)+\beta\mathbb{E}\left[W(h_{\sigma}^{t+1},z_{t+1})|h_{\sigma}^{t},\pi\right]\label{W} \end{equation} (W) subject to $$b_{S}^{\prime},b_{L}^{\prime}\leq\bar{B}$$ and the consolidated budget constraints of the government and the stand-in household.15 If there is no default (i.e. if $$\delta=$$full), the consolidated budget constraint is given by   \begin{equation} c+(b_{S,t}+b_{L,t})\leq Y(\tau,z)+q_{S,t}(h_{\sigma}^{t},\pi)b_{S}^{\prime}+q_{L,t}(h_{\sigma}^{t},\pi)(b_{L}^{\prime}-b_{L,t}),\label{bc0} \end{equation} (17) or, if there is partial repayment (i.e. if $$\delta=$$partial) by   \begin{equation} c+\left(r_{S}b_{S,t}+r_{L}\frac{b_{L,t}}{1-q}\right)r\leq Y(\tau,z)+q_{S,t}(h_{\sigma}^{t},\pi)b_{S}^{\prime}+q_{L,t}(h_{\sigma}^{t},\pi)b_{L}^{\prime},\label{bcr} \end{equation} (18) or, if there is default without any partial repayment (i.e. if $$\delta=$$suspend), then I impose the restriction that after $$\delta_{t}=$$suspend, there is a temporary exclusion from international credit markets; that is,   \begin{equation} c\leq Y(\tau,z)\,\,\,\text{and}\,\,\,\left(b_{S}^{\prime},b_{L}^{\prime}\right)=\left(b_{S,t},b_{L,t}\right).\label{bc1} \end{equation} (19) 4.2. Bond prices and other equilibrium objects The price of the imported intermediate, $$p_{t}$$, must be consistent with optimization by competitive foreign lenders who take the tariff level as given after all histories:   \begin{equation} 1=\mathbb{E}\left[p_{t}(h_{p}^{t})\left(1-\tau_{t}(h_{\sigma}^{t},z_{t})\right)|h_{p}^{t}\right].\label{p} \end{equation} (20) The allocation rule for the quantity of foreign intermediate goods, $$m_{t}$$, satisfies the optimality condition for the representative domestic competitive firm after all histories:   \begin{equation} \mathbb{E}\left[Q(h_{\sigma}^{t},z_{t})F_{m}(m_{t}(h_{m}^{t}),1)|h_{m}^{t}\right]=p_{t}(h_{p}^{t}),\label{m} \end{equation} (21) where $$Q(h_{\sigma}^{t},z_{t})=U^{\prime}\left(c(h_{\sigma}^{t},z_{t})\right)/\mathbb{E}\left[U^{\prime}\left(c(h_{\sigma}^{t},z_{t})\right)\right]$$ is the price that the representative domestic firm uses to evaluate profits. Given the government repayment policy, bond prices $$q_{S,t},q_{L,t}$$ are consistent with the maximization problem of the risk-neutral foreign lenders who discount the future at a rate $$q$$. For the one-period bond, if $$b_{S,t+1}\geq0$$, it must be that   \begin{equation} q_{S,t}(h^{t})=q\mathbb{E}\left[\chi_{S,t+1}(h^{t+1})|h^{t}\right],\label{q1} \end{equation} (qS) where $$\chi_{S,t+1}$$ is the ex post value of short-term debt:   \begin{equation} \chi_{S,t+1}(h^{t+1})=\left\{ \begin{array}{c@{\kern6pt}c} 1 & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{full}}\\[5pt] r_{S} & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{partial}}\\[5pt] q\mathbb{E}\left[\chi_{S,t+2}(h^{t+2})|h^{t+1}\right] & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{suspend}} \end{array}\right.\label{chi}\!\!. \end{equation} (22) Here the expectation in (qS) is taken at the end of period $$t$$ over next period’s value of the productivity shock, $$z_{t+1}.$$ The lenders understand the repayment rule of the government, and the price is simply the actuarially fair value of repayments. The only subtle part is that if the government does not repay the short bond at $$t+1$$ by setting $$\delta_{t+1}(h_{\sigma}^{t+1})=$$suspend, this bond still has value because it gives the holder the right to any partial repayment $$r_{S}$$ that the government may make on this defaulted debt either in period $$t+2$$ or some later period to regain access to the bond market. In this sense, when $$\delta_{t+1}=$$suspend, $$q\mathbb{E}\left[\chi_{S,t+2}(h^{t+2})|h^{t+1}\right]$$ represents the secondary market value of defaulted debt. Next, consider situations in which the government saves, in that $$b_{S,t+1}<0$$. If the history is such that the government is not excluded from saving, the rate on foreign assets will equal the world rate, that is, $$q_{S,t}(h^{t})=q$$. If instead the history is such that the government is being temporarily excluded from the international credit market, I adopt the convention that $$q_{S,t}(h^{t})=\infty$$. Allowing for exclusion from savings is necessary to decentralize the efficient risk-sharing arrangement. In such arrangement, detectable deviations are punished with permanent autarky. If there is no mechanism preventing the government to save after a detectable deviation, then the government could default and self-insure via saving. This would increase the incentive for the government to deviate and prevent to decentralize the efficient allocation with this market structure. Finally, the price for the consol must be such that   \begin{equation} q_{L,t}(h^{t})=q\mathbb{E}\left[\chi_{L,t+1}(h^{t+1})|h^{t}\right],\label{q8} \end{equation} (qL) where $$\chi_{L,t+1}$$ is the ex post value of the consol given by   \begin{equation} \chi_{L,t+1}(h^{t+1})=\left\{ \begin{array}{c@{\quad}c} 1+q_{L,t+1}(h^{t+1}) & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{full}}\\[5pt] \frac{r_{L}}{1-q} & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{partial}}\\[5pt] q\mathbb{E}\left[\chi_{L,t+2}(h^{t+2})|h^{t+1}\right] & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{suspend}} \end{array}\right.\!\!\!.\label{chiL} \end{equation} (23) If the government does not repay its consol at $$t+1,$$ this consol still has value because it gives the holder the right to any partial repayment $$r_{L}/(1-q)$$ that the government may make in future periods in order to regain access to the bond markets. 4.3. Equilibrium definition Given initial outstanding debt $$\boldsymbol{b}_{0}$$, a sustainable equilibrium is a strategy for the government, $$\sigma$$, a price rule for the foreign intermediate good, $$p$$, price rules for the government bonds, $$q_{S}$$ and $$q_{L}$$, and an allocation rule for the intermediate good, $$m$$, such that (i) given $$p$$, $$m$$, $$q_{S}$$, and $$q_{L}$$, the government’s strategy, $$\sigma$$, is the optimal policy associated with (W) for all $$(h_{\sigma}^{t},z)$$; (ii) given $$\sigma$$, the price and allocation rules $$p$$ and $$m$$ satisfy (20) and (21); (iii) given $$\sigma,$$ the price of the short-term bond $$q_{S}$$ satisfies (qS) whenever $$b_{S}\geq0$$, and it equals $$q$$ or $$\infty$$ when $$b_{S}<0,$$ where $$q_{S}=\infty$$ stands in for exclusion from saving abroad; (iv) given $$\sigma,$$ the price of the long-term bond $$q_{L}$$ satisfies (qL). The associated equilibrium outcome is denoted by $$\boldsymbol{y}=\left(\boldsymbol{x},\boldsymbol{g},\boldsymbol{p}\right)$$, where $$\boldsymbol{x}=\left\{ m(z^{t-1}),c(z^{t})\right\} _{t=0}^{\infty}$$, $$\boldsymbol{g}=\left\{ \delta(z^{t}),b_{L}(z^{t}),b_{S}(z^{t}),\tau(z^{t})\right\} _{t=0}^{\infty}$$, and $$\boldsymbol{p}=\left\{ p(z^{t-1}),\boldsymbol{q}(z^{t})\right\} _{t=0}^{\infty}$$. It is useful to characterize the set of outcomes that can be implemented as a sustainable equilibrium of the sovereign debt game. Such characterization gives us a set of necessary and sufficient conditions for an equilibrium outcome, which I will later show that are met by the efficient allocation. The logic of the characterization follows Abreu (1988) and Chari and Kehoe (1990) in using reversion to the worst equilibrium to characterize the set of equilibrium outcomes. The worst equilibrium outcome from the borrower’s perspective is the autarkic allocation. The autarkic allocation can be supported as an equilibrium outcome as follows: Zero intermediates and a price equal to zero for both short-term and long-term debt can be supported as part of a sustainable equilibrium, because if foreign lenders expect a tariff equal to 100% (full expropriation) and full default in any subsequent periods irrespective of the action chosen today by the government ($$\delta_{t}=$$suspend for all subsequent $$t$$), then the government has no incentive to choose something different than $$\tau_{t}=1$$ and $$\delta_{t}=$$suspend, confirming the lenders’ beliefs. The fact that the sovereign borrower cannot save after a deviation follows from an assumption commonly used in the literature (e.g.Atkeson, 1991; Aguiar et al., 2009) to rule out the classic Bulow and Rogoff (1989) result that no foreign debt can be sustained in equilibrium. The next Lemma characterizes the set of sustainable equilibrium outcomes: Lemma 3. Given initial outstanding debt $$\boldsymbol{b}_{0}$$, $$\boldsymbol{y}$$ is a sustainable equilibrium outcome if and only if it satisfies the constraints (RC), (IC), and (SUST) in addition to the budget constraints (17)–(19) and (20), (21), (qS) and (qL) along the equilibrium outcome path. Proof. It is evident that the conditions in Lemma 3 are necessary for an equilibrium outcome. Conditions (20), (21), (qS), and (qL) are part of the definition of a sustainable equilibrium. Budget feasibility from the government’s perspective requires the outcome to satisfy (17)–(19). Constraints (IC) and (SUST) ensure that the sovereign borrower has no strict incentive to engage in a detectable and undetectable deviation, respectively. In (SUST), I use the fact that the borrower’s value after a detectable deviation cannot be lower than the value of the worst equilibrium, $$v_{a}$$. To see that such conditions are also sufficient, consider an outcome that satisfies the conditions in the statement. Such outcome can be implemented as sustainable equilibrium by relying on trigger strategies that revert to the worst equilibrium after a detectable deviation from the government. Clearly, such an outcome satisfies the optimality conditions for the competitive agents, since it satisfies (20), (21), (qS), and (qL), and it is budget feasible for the government, since it satisfies the budget constraints (17)–(19). Then I am left to show that the government has no incentive to deviate from the prescribed path of plays. To this end, note the detectable deviations are not profitable, because the outcome satisfies (SUST). Moreover, undetectable deviations are not profitable, because the outcome satisfies (IC). Then, the government does not have a strict incentive to deviate from the proposed path of plays, concluding the proof. ∥ 5. Defaults, Bond Prices, and Maturity Composition I now show that any efficient allocation can be implemented as an equilibrium outcome of the sovereign debt game. This amounts to say that the best equilibrium outcome of the sovereign debt game is (constrained) efficient. Along the equilibrium outcome path, there are defaults only when the borrower’s continuation value is equal to the autarkic value. Defaults along the path do not trigger permanent exclusion from international credit markets. In the model, the borrower suspends payments to foreign lenders until a high-productivity shock is drawn and a given partial repayment on the defaulted debt is made. After such partial repayment, the borrower regains access to foreign borrowing and lending. It is important to note the distinction between a detectable deviation from the path of play, which may include a default, and what happens along the equilibrium path when there is a default. On-path defaults are excusable in the sense introduced by Grossman and Van Huyck (1988): They happen in well-understood circumstances, and the country regains access to international credit markets after making a partial repayment; therefore, on-path defaults do not trigger autarky forever. 5.1. Implementation In the rest of the article, I will assume that the efficient allocation satisfies the following properties: Assumption 4. The efficient allocation is such that (i)for all $$v>v_{a}$$, $$v^{\prime}(v,z_{L})<v$$ and there exists a $$\underline{v}\in(v_{a},\bar{v})\,$$ such that $$v^{\prime}(v,z_{L})=v_{a}$$ for all $$v\leq\underline{v}$$; (ii)$$v^{\prime}(\cdot,z)$$ is strictly increasing for all $$v\geq\underline{v}$$ and for all $$z.$$ Part (i) implies that starting from any $$v$$, there is a strictly positive probability of reaching autarky after a sufficiently long but finite string of low-productivity shocks. This property does not follow from Lemma, 2 since the Lemma does not immediately extend to the region of ex-post inefficiencies if lenders are more impatient than the borrower. This is because for $$v$$ such that $$B^{\prime}\left(v\right)>0$$, it is more costly to front-load utility for the relatively more patient lenders (the opposite of what happens for promised utility $$v$$ such that $$B^{\prime}\left(v\right)\leq0$$). In the Online Appendix, I provide two sets of sufficient conditions on primitives that ensure the efficient allocation satisfies property (i)). In particular, I show that if $$f\left(m^{\ast}\right)-f\left(0\right)$$ is sufficiently small or if $$\beta$$ is sufficiently close to $$q,$$ then the condition is satisfied. Moreover, if the efficient allocation displays partial insurance, in that $$U^{\prime}\left(c\left(v,z_{L}\right)\right)\geq U^{\prime}\left(c\left(v,z_{H}\right)\right),$$ then a stronger version of condition (13) ensures that part (i) holds.16 Part (ii) requires that the continuation values, $$v^{\prime}(\cdot,z_{L})$$ and $$v^{\prime}(\cdot,z_{H}),$$ be increasing in current promised utility. Both (i) and (ii) are satisfied in my simulations (e.g. Figure 3).17 For later reference, define the short-term spread as the difference between the interest rate implied by $$q_{S}$$ and the risk-free international interest rate: $$s_{S}\equiv1/q_{S}-1/q$$. The long-term spread is defined as the difference between the consol’s yield to maturity18 and the risk-free interest rate: $$s_{L}\equiv(1+q_{L})/q_{L}-1/q$$. The term premium is the difference between the long- and the short-term spreads: $$s_{T}\equiv s_{L}-s_{S}$$. The main result of this section is that there exists a sustainable equilibrium that decentralizes the efficient allocation such that default happens along the equilibrium path in the region of ex post inefficiencies when the government’s debt is high (relative to the maximal amount that can be sustained) and output is low. Furthermore, if the recovery rate on long-term bonds is sufficiently high relative to the recovery rate on short-term bonds,19 the equilibrium outcome is such that when spreads are low, the term premium is positive, but it delivers an inversion of the yield curve when spreads are high. Moreover, numerical simulations show that the maturity composition of debt shortens when spreads are high. Proposition 4. Under Assumptions 1–3 and $$\beta<q$$, given an efficient allocation that satisfies Assumption 4 and a set of recovery rates $$\left(r_{S},r_{L}\right)$$ with at least one rate greater than zero, a sustainable equilibrium exists that decentralizes it, with default happening along the equilibrium path when the continuation value for the borrower equals $$v_{a}$$. Moreover, if $$r_{L}\geq r_{S}$$, the equilibrium is such that the term premium is positive when borrower’s value is above $$v_{a}$$ and negative at $$v_{a}$$. The proof of the proposition is by construction. I construct the on-path default rule, bond holdings, tariffs, and prices that support the efficient allocation and are consistent with the sufficient condition for a sustainable equilibrium outcome in Lemma 3. Since the efficient allocation can be represented by a time-invariant function of borrower’s continuation utility and exogenous shocks, the on-path repayment rule, bond holdings, tariffs, and prices can also be expressed as a function of on-path continuation utility for the borrower and the current realization of the productivity shock $$z$$:   \begin{align} \bar{\tau},\bar{p}:\left[v_{a},\bar{v}\right] & \rightarrow\mathbb{R},\quad\bar{\delta}:\left[v_{a},\bar{v}\right]\times\left\{ z_{L},z_{H}\right\} \rightarrow\left\{{\texttt{full}},{\texttt{partial}},{\texttt{suspend}}\right\} \label{bar1}\\ \end{align} (24)  \begin{align} \bar{q}_{S},\bar{q}_{L},\bar{b}_{S},\bar{b}_{L} & :\left[v_{a},\bar{v}\right]\times\left\{ z_{L},z_{H}\right\} \rightarrow\mathbb{R}\label{bar2}. \end{align} (25) Note that I consider a decentralization in which $$\bar{\tau}$$ does not depend on $$z$$ to emphasize how active maturity management can replicate state contingent return implicit in the efficient allocation. A state contingent tariff can also provide insurance to the government; for instance, see the role of state contingent taxes on capital income in Aguiar et al. (2009).20 An outcome path $$\boldsymbol{y}$$ can be recovered in the natural way from (24), (25), and the law of motion for $$v$$ from the efficient allocation. Moreover, bond holdings and prices depend only on the continuation value after $$z$$ is realized, $$v^{\prime}(v,z)$$. With some abuse of notation, I can then write   \begin{eqnarray} \bar{q}_{S}(v,z) & = & \bar{q}_{S}(v^{\prime}(v,z)),\text{ }\bar{q}_{L}(v,z)=\bar{q}_{L}(v^{\prime}(v,z)),\text{ } \\ \end{eqnarray} (26)  \begin{eqnarray} \bar{b}_{S}(v,z) & = & \bar{b}_{S}(v^{\prime}(v,z))\text{, } \bar{b}_{L}(v,z)=\bar{b}_{L}(v^{\prime}(v,z)). \end{eqnarray} (27) The steps to construct the candidate equilibrium outcome path $$\boldsymbol{y}$$ from an efficient allocation $$\boldsymbol{x}$$ are as follows: (1) define the repayment policy; (2) use the repayment policy in the optimality conditions for the foreign lenders to calculate equilibrium bond prices; (3) choose short- and long-term debt to match the total value of debt (lenders’ value) after a realization of $$z$$ implied by the efficient allocation   \begin{equation} b(v,z)=f(m(v))-c(v,z)-m(v)+qB(v^{\prime}(v,z)),\label{bx} \end{equation} (28) and (4) use the optimality conditions for the domestic firms and the lenders to get tariffs and prices for the intermediate good. Consider first the repayment policy. The borrower defaults only when his continuation value is equal to the autarky value, $$v_{a}$$. For all other borrower values, $$v>v_{a}$$, there is full repayment. In particular,   \begin{equation} \bar{\delta}(v,z)=\left\{ \begin{array}{c@{\quad}c} {\texttt{suspend}} & \text{if }v=v_{a}\text{ and }z=z_{L}\\ {\texttt{partial}} & \text{if }v=v_{a}\text{ and }z=z_{H}\\ {\texttt{full}} & \text{if }v>v_{a}\text{ for all }z \end{array}\right.\!\!\!,\label{deltabar2} \end{equation} (29) where $$\bar{\delta}(v_{a},z_{L})=$$suspend because, as showed in Lemma B6 in the Online Appendix, when the borrower’s value is autarky, there are no capital flows in that $$m(v_{a})=0$$ and $$c(v_{a},z_{L})=f(0)$$. When $$z=z_{H}$$, $$c(v_{a},z_{H})<f(0)$$ and so there is an outflow of resources. This is matched by a partial repayment on defaulted debt. Given the repayment policy, bond prices are uniquely pinned down by the lenders’ optimality conditions. The price for short-term debt is given by   \begin{equation} \bar{q}_{S}(v)=\left\{ \begin{array}{@{}cc} q & \text{if }v>v_{a}\\[3pt] q\bar{R}_{S} & \text{if }v=v_{a} \end{array}\right.\!\!\!,\label{qbar1} \end{equation} (30) where $$\bar{R}_{S}$$ is the expected repayment in case of default for short-term debt, $$\bar{R}_{S}=r_{S}\bar{R}$$ where $$\bar{R}=\mu(z_{H})/(1-q\mu(z_{L}))$$. The price for long-term debt can be written recursively as   \begin{equation} \bar{q}_{L}(v)=\left\{ \begin{array}{@{}cc} q\sum_{i=L,H}\mu(z_{i})\left[1+\bar{q}_{L}(v_{i}^{\prime}(v))\right] & \ \text{if }v>v_{a}\\[3pt] \frac{q}{1-q}\bar{R}_{L} &\ \text{if }v=v_{a} \end{array}\right.\!\!\!,\label{qbar8} \end{equation} (31) where $$\bar{R}_{L}q/\left(1-q\right)$$ is the expected repayment in case of default for long-term debt, $$\bar{R}_{L}=r_{L}\bar{R}$$. For all $$v>v_{a}$$, the price of short-term debt is equal to that of a risk-free bond. The price of long-term debt is lower than the price of a risk-free consol, because there is always a positive probability that there will be a default over the relevant time horizon of the bond. The next Lemma shows that $$\bar{q}_{L}$$ is increasing in the continuation value for the borrower, and it establishes that the price of long-term debt increases after drawing $$z_{H},$$ in that $$\bar{q}_{L}\left(v_{H}^{\prime}(v)\right)>\bar{q}_{L}\left(v_{L}^{\prime}(v)\right)$$. Lemma 4. Under the assumptions in Proposition 4, $$\bar{q}_{L}:[v_{a},\bar{v}]\rightarrow$$ is the unique fixed point of the contraction mapping defined by the right-hand side of (31), it is increasing, and for all $$v$$ we have  \[ \frac{q}{1-q}>\bar{q}_{L}\left(v_{H}^{\prime}(v)\right)>\bar{q}_{L}\left(v_{L}^{\prime}(v)\right). \] The proof for this Lemma is provided in the Online Appendix. In the proof, I use the assumption that $$v^{\prime}\left(v,z\right)$$ is monotone in $$v$$ for all $$z,$$ part (ii) of Assumption 4. Given the functions for bond prices $$\bar{q}_{S}$$ and $$\bar{q}_{L}$$, in the no-default region, $$\bar{b}_{S}(v)$$ and $$\bar{b}_{L}(v)$$ are chosen to match the total value of debt (lenders’ value) implied by the efficient allocation after $$z_{H}$$ and $$z_{L}$$ defined in (28):   \begin{eqnarray} b(v,z_{H}) & = & \bar{b}_{S}(v)+\bar{b}_{L}(v)\left[1+\bar{q}_{L}(v_{H}^{\prime}(v))\right]\label{bL}\\ \end{eqnarray} (32)  \begin{eqnarray} b(v,z_{L}) & = & \bar{b}_{S}(v)+\bar{b}_{L}(v)\left[1+\bar{q}_{L}(v_{L}^{\prime}(v))\right].\label{bH} \end{eqnarray} (33) A unique solution to (32)–(33) is guaranteed by Lemma 4, which establishes that $$\bar{q}_{L}\left(v_{H}^{\prime}(v)\right)>\bar{q}_{L}\left(v_{L}^{\prime}(v)\right)$$, meaning that the price of the long-term bond falls after $$z_{L}$$ is realized relative to $$z_{H}$$. Thus, outside the default region, the maturity composition of debt is uniquely pinned down. Simple algebra shows that   \begin{eqnarray} \bar{b}_{L}(v) & = & \frac{b(v,z_{H})-b(v,z_{L})}{\bar{q}_{L}(v_{H}^{\prime}(v))-\bar{q}_{L}(v_{L}^{\prime}(v))}\label{bbar8}\\ \end{eqnarray} (34)  \begin{eqnarray} \bar{b}_{S}(v) & = & b(v,z_{H})-\bar{b}_{L}(v)\left[1+\bar{q}_{L}(v_{H}^{\prime}(v))\right].\label{bbar1} \end{eqnarray} (35) Notice that it is guaranteed that $$\bar{b}_{L}(v)>0$$, because, by Proposition 1 part (iii), $$b(v,z_{H})-b(v,z_{L})\,{>}\,0$$ and $$\bar{q}_{L}(v_{H}^{\prime}(v))-\bar{q}_{L}(v_{L}^{\prime}(v))>0$$. When the continuation value for the next period onward is equal to autarky, the split between long- and short-term debt is indeterminate; $$\bar{b}_{S}(v_{a})$$ and $$\bar{b}_{L}(v_{a})$$ must satisfy   \begin{equation} b(v_{a},z_{H})=r_{S}\bar{b}_{S}(v_{a})+r_{L}\frac{\bar{b}_{L}(v_{a})}{1-q}.\label{bval} \end{equation} (36) The other possible outcome follows from (36) and $$\delta(v_{a},z_{L})=$$suspend because   \begin{equation} b(v_{a},z_{L})=q\left[\mu(z_{H})b(v_{a},z_{H})+\mu(z_{L})b(v_{a},z_{L})\right]=\frac{q\mu(z_{H})}{1-q\mu(z_{L})}b(v_{a},z_{H}).\label{bvah} \end{equation} (37) Thus, for any given recovery rates $$\left(r_{S},r_{L}\right),$$ I can choose any $$\left(\bar{b}_{S}(v_{a}),\bar{b}_{L}(v_{a})\right)$$ that satisfies (36). Consequently, (37) will also be satisfied. Note that to be able to satisfy (36), at least one between $$r_{S}$$ and $$r_{L}$$ must be strictly positive. If both $$r_{S}$$ and $$r_{L}$$ are strictly positive, the maturity composition is undetermined. I resolve the indeterminacy by assuming that $$\bar{b}_{S}(v_{a})/\bar{b}_{L}(v_{a})=\lim_{v\rightarrow v_{a}}\bar{b}_{S}(v)/\bar{b}_{L}(v)$$. The recovery rates $$\left(r_{S},r_{L}\right)$$ are free parameters. They can be chosen to be sufficiently low so that $$\bar{b}_{S}$$ is strictly positive in the default region and for $$v$$ close to $$v_{a}$$ so that a non-full repayment has a natural interpretation. Finally, on-path tariff rates and prices for the intermediate good, $$\bar{\tau},\bar{p}$$, are given by21  \begin{equation} \mathbb{E}\left[\frac{U^{\prime}\left(c(v,z)\right)}{\mathbb{E}\left[U^{\prime}\left(c(v,z)\right)\right]}f^{\prime}(m(v))\right]=\frac{1}{1-\bar{\tau}(v)}=\bar{p}(v).\label{ptaubar} \end{equation} (38) Then the outcome path $$\boldsymbol{y}$$ constructed from the efficient allocation $$\boldsymbol{x}$$ using (29), (30), (31), (34), (35), (36), and (38) satisfies the optimality conditions (20) and (21), the equilibrium bond pricing equations (qS) and (qL), and the budget constraints (17)–(19) along the equilibrium path. Since the efficient allocation satisfies (IC) and (SUST), this concludes the proof of the first part of the Proposition, as it verifies that the constructed outcome satisfies the sufficient conditions for a sustainable equilibrium outcome in Lemma 3. I am left to characterize the behaviour of the term structure of interest rates. To this end, first notice that outside the default region, the short-term debt is risk-free; see (30). Thus, $$q_{S,t}=q$$ and $$s_{S,t}=0$$. From Lemma 4 we know that $$q_{L,t}<q/(1-q)$$. Therefore, the short-term spread is zero, while the long-term spread is positive. Consequently, the term spread $$s_{T}$$ is positive. When the borrower’s continuation value is in the default region, the term spread is given by   \begin{equation} s_{T}(v_{a})=\left(1+\frac{1}{\bar{q}_{L}(v_{a})}\right)-\frac{1}{\bar{q}_{S}(v_{a})}=-\left(\frac{1-\bar{R}r_{L}}{\bar{R}r_{L}}\right)-\frac{1}{q\bar{R}}\left(\frac{r_{L}-r_{S}}{r_{S}r_{L}}\right), \end{equation} (39) which is negative provided that the recovery rate on short-term debt $$r_{S}$$ is not too small relative to the recovery rate for long-term debt $$r_{L}.$$ A sufficient condition is $$r_{L}\geq r_{S}$$. ∥ Proposition 4 can be generalized to the case in which $$z$$ can take on more than two values, allowing for a richer maturity structure. For instance, if $$z$$ can take on $$N$$ values, then I can use $$N$$ types of the perpetuity considered in Hatchondo and Martinez (2009) that pay a coupon which decays exponentially at rate $$\alpha_{n}\in\lbrack0,1]$$. The one-period bond and the consol are special cases of this class of securities for $$\alpha$$ equal to $$1$$ and $$0$$, respectively. Provided that the return matrix satisfies a full-rank condition22 (which is satisfied when $$N=2$$ because of Lemma 4), the statement in Proposition 4 generalizes to the case with $$N>2$$. 5.2. Mechanism that replicates state-contingent returns The crucial step proving that the efficient allocation can be an outcome of the sovereign debt game was to show that it is possible to replicate the insurance provided by the efficient allocation, that is, the total value of debt falls after an adverse shock relative to a positive shock, $$b(v,z_{H})>b(v,z_{L})$$. How is insurance provided in the sovereign debt game? When there is default, partial repayments make the non-contingent debt de facto state-contingent. When there is full repayment, the fall in the value of debt after the realization of a low-productivity shock is obtained by diluting outstanding long-term debt, that is, by imposing a capital loss on the holders of outstanding long-term debt. After a low-productivity shock, the continuation value for the borrower decreases, the overall level of indebtedness increases, and the probability that there will be a default in the near future increases. This increase in the likelihood of a future default reduces the value of the outstanding long-term debt, resulting in a capital loss for the debt holders and a capital gain for the borrower. This capital loss on the debt holders after an adverse shock mimics the debt relief for the borrower associated with the efficient allocation. This capital loss in a low-productivity shock is compensated by a capital gain after a high-productivity shock, so on average the lenders break even. The maturity composition of debt is driven solely by the requirement of matching this differential value of government debt ex post. Note that in the region with ex post inefficiencies, a reduction in the value of long-term debt is not necessarily good from the government’s perspective. In fact, in such region there are sustainable equilibria that attain a higher value for the government as well as the legacy debt holders. However, such continuation strategies are not optimal from an ex ante perspective as explained in Proposition 2. Hence the equilibrium strategies in the best sustainable equilibrium prevent the government to pursue such welfare-enhancing policies. 5.3. Maturity shortens as interest rate spread increases I now turn to the implications for the optimal maturity composition of debt. The main finding is that the maturity of outstanding debt issued by the sovereign borrower shortens as the long-term spread increases. In particular, the amount of long-term debt decreases, while the amount of short-term debt increases for all $$v$$ in the efficient region. This result is illustrated in Figure 4.23 I cannot state a proposition for this result, but the findings are consistent with all of my numerical simulations. Figure 4 View largeDownload slide Bond prices and holdings, insurance and ex post variation of the long-term debt price Figure 4 View largeDownload slide Bond prices and holdings, insurance and ex post variation of the long-term debt price To understand this result, notice that outside the default region, the amount of long-term debt issued by the borrower is determined by (34). Given the ex post variation in the price of the consol, $$\bar{q}_{L}(v_{H}^{\prime}(v))-\bar{q}_{L}(v_{L}^{\prime}(v))$$, the long-term debt holdings are constructed to match the debt relief implied by the optimal contract after the realization of $$z_{L}$$, $$b(v,z_{H})-b(v,z_{L})$$. As is shown in Figure 4, the level of debt relief is approximately constant for all $$v$$ over the efficient region. The ex post variation in the price of the consol is larger the closer the borrower is to the default region. This is because as the borrower’s continuation value approaches the default threshold from above, it is more likely that a realization of a low-productivity shock will push the economy into default in the near future. Hence, the long-term debt price is more sensitive to the realization of the shock. Therefore, a lower holding of long-term debt is needed to replicate the same amount of insurance, that is, the same debt relief after a high taste shock. Since the overall level of indebtedness is increasing, it must be that $$\bar{b}_{S}$$ is increasing as the borrower’s continuation value approaches $$\tilde{v}$$, because $$\bar{b}_{L}$$ is falling at the same time. Therefore, in the efficient region, as the level of indebtedness and the spread on long-term debt $$s_{L}$$ increase, the maturity composition of debt shortens. In the region with ex post inefficiencies, $$[v_{a},\tilde{v}]$$, the ratio of short-term debt to long-term debt is not always decreasing in the borrower’s value under all parameterizations. This is because the ex post variation in the price of long-term debt is high, but also the amount of insurance, $$b(v,z_{H})-b(v,z_{L})$$, increases a lot in this region (Figure 4). Despite not necessarily being monotonically decreasing in this region, the maturity composition of debt is more tilted towards short-term debt than it is for continuation values associated with lower default probabilities. The decision rules for debt holdings obtained in my simulations can be used to characterize the pattern of debt issuances if the economy is hit by a sequence of low-productivity shocks that leads to a declining pattern of continuation values eventually triggering a default. Because in my simulations $$\bar{b}_{S}\left(v\right)$$ is decreasing in $$v$$ and $$\bar{b}_{L}\left(v\right)$$ is increasing in $$v$$ (Figure 4), along this path, the stock of short-term debt is increasing while the stock of long-term debt (consol) is decreasing. To achieve this outcome, the borrower issues more short-term debt than the one coming due, $$\bar{b}_{S}\left(v'\left(v,z_{L}\right)\right)>\bar{b}_{S}\left(v\right)$$, and buys back $$\bar{b}{}_{L}\left(v\right)-\bar{b}_{L}\left(v'\left(v,z\right)\right)>0$$ units the outstanding long-term debt.24 In this sense, the issuance pattern is broadly consistent with the evidence in Broner et al. (2013) and Arellano and Ramanarayanan (2012). In fact, along the path leading to a default, all new issuances are short-term debt. Once the economy recovers and the borrower’s promised value increases, the borrower issues more long-term debt $$\bar{b}_{L}\left(v'\left(v,z\right)\right)-\bar{b}{}_{L}\left(v\right)>0$$ and a lower amount of one period debt than the one coming due, $$\bar{b}_{S}\left(v'\left(v,z_{L}\right)\right)<\bar{b}_{S}\left(v\right)$$. 5.4. Assumptions on rules in default I now turn to discuss the conventions I choose for the government in default and whether they can be relaxed without affecting the main characteristics of equilibrium outcomes. To this end, it is important to understand the dynamics of payments prescribed by the efficient allocation. In the Online Appendix (Lemma B5), I show that when the borrower’s value equals the value of autarky, there are no capital flows when $$z$$ equals $$z_{L}$$, $$x\left(v_{a},z_{L}\right)=0$$, and there are outflows when $$z$$ equals $$z_{L}$$, $$x\left(v_{a},z_{H}\right)>0$$ and $$b\left(v_{a},z_{H}\right)>0$$ (see also Proposition 1 part iv). The efficient allocation only pins down total payments when the value of the borrower is autarky and it draws a positive productivity shock, as illustrated in equation (36). This implies a degree of indeterminacy at $$v_{a}$$. In fact, if the recovery rates double and the face value of debt is halved, the borrower makes the same payment and raises the same resources in the previous period as the price of debt doubles from equations (30) and (31). The fact that $$b\left(v_{a},z_{H}\right)>0$$ requires that at least one between $$r_{S}$$ and $$r_{L}$$ be strictly positive. Other than this, there are no other requirements on recovery rates. Total payments prescribed by the efficient allocation after the economy recovers do not depend on the length the country spent in temporary autarky, say $$n\geq0$$. This implies that if the recovery rates $$r_{S}$$ and $$r_{L}$$ do not depend on $$n$$, then the interest rates arrears must be forgiven so that the equilibrium payout received by lenders does not depend on $$n$$. An alternative way to implement the efficient allocation is to have interest rate arrears not being forgiven and recovery rates that depend on $$n$$. This clearly would not change the behavior of the equilibrium outcome leading to a default. In setting up the sovereign debt game, I assumed that the government settles with the holders of legacy debt by making a current payment. Nothing will change if the government could use a mix of current payments and newly issued debt to pay existing debt holders as part of the settlement agreement, as in Benjamin and Wright (2009). It is worth noticing that some payment must be done in the current period, as the total net export of the country is positive, $$x\left(v_{a},z_{H}\right)>0$$. Further note that I am ruling out the possibility for holders of legacy long-term debt to holdout by assuming that holders of legacy long-term debt receive no payments once a partial repayment is made. Finally, several authors (e.g. Tomz and Wright, 2013 and references therein) have argued that market access and interest rates differ depending on the “haircut” applied on defaulted debt and the history of previous default. This is inconsistent with the dynamics of the efficient allocation considered here. Adding an extra dimension of asymmetric information may help in this regard. In particular, one can account for this fact by introducing private information about the type of borrower, which is only revealed by a default (and not by other actions). Although interesting, this is outside the scope of this article. To summarize, in this section, I showed that an efficient allocation can be implemented with only non-contingent defaultable debt of multiple maturities. Along the equilibrium outcome path, defaults are associated with an ex post inefficient drop in output and trade, and inversion of the yield curve, and happen only when the borrower’s value is equal to autarky and the level of debt is high relative to the maximal amount of debt the country can support. When there is no default, capital gains or losses on outstanding long-term debt replicate the state contingent returns implied by the efficient allocation. Moreover, the maturity of outstanding debt shortens as interest rate spreads increase. 6. Discussion of Implementation 6.1. History dependence and debt dilution The strategies that support the efficient allocation are history dependent. In particular, the pricing function $$\boldsymbol{q}$$ does not only depend on the stock of outstanding debt and an indicator variable that records whether the government has access to international credit markets, as it is typically considered in the literature on quantitative sovereign default. The reason why history dependence is needed is connected to the debt-dilution problem. With long-term debt, any borrower’s action that increases the likelihood of future outright default is tantamount to a (partial) default, because it imposes a capital loss to the holders of the outstanding debt. The equilibrium that implements the efficient allocation treats outright default and this more subtle partial default in a parallel fashion: The borrower is punished if he deviates from the path of plays by diluting existing debt too much, or if after a positive shock, he does not reduce his level of indebtedness. This stands in contrast with standard sovereign debt models in which only outright default is punished with a trigger to autarky (with potential re-entry). This difference has important implications for how we think about the role of long-term debt, seniority, and pari passu clauses. Chatterjee and Eyigungor (2015) and Hatchondo et al. (2016) argue that within an equilibrium of the sovereign debt game in which only outright default is punished with trigger strategies, debt dilution is a problem and seniority clauses may be desirable. Moreover, absent rollover risk, short-term debt is very desirable and governments will opt to choose a maturity composition tilted towards short-term debt. This article makes clear that such results are generated by the asymmetric treatment of outright default and dilution. In the best equilibrium of the sovereign debt game analysed here, debt dilution is necessary for the best sustainable equilibrium to be equivalent to the efficient allocation. Hence, policies that introduce seniority to reduce the ability to dilute outstanding debt may not be warranted. 6.2. Comparison with Alvarez and Jermann (2000) The implementation I propose is also applicable to other environments than the one considered here. For instance, consider an economy with lack of commitment and public information,25 where the domestic agents discount more heavily than the international interest rate, $$\beta<q$$. Such condition arises in general equilibrium with a large number of countries, as shown in Alvarez and Jermann (2000). Under such condition, the efficient allocation is dynamic, in that $$v^{\prime}(v,z_{H})\neq v^{\prime}(v,z_{L})$$ for all $$v$$ in the ergodic set26 and it can be implemented as the best sustainable outcome of the sovereign debt game presented in Section 5. Moreover, the pattern of maturity composition of debt and the spreads leading to a default are similar. Alvarez and Jermann (2000) show that it is possible to implement the efficient allocation under lack of commitment with state contingent debt and endogenous debt limits. The main advantage of my implementation is that it neatly maps into the objects considered in applied works. In particular, my implementation can be used to derive prices for defaultable bonds, and it has implications for the maturity composition of debt. Finally, it is worth noticing that while the implementation I propose works in the environment considered by Alvarez and Jermann (2000), the converse is not true. Debt limits are not enough to implement the efficient allocation with private information. This follows from the fact that the efficient allocation with private information does not satisfy $$qU^{\prime}\left(c\left(z^{t}\right)\right)\geq\beta U^{\prime}\left(c\left(z^{t},z_{t+1}\right)\right)$$ for all $$z^{t},z_{t+1}$$. That is, the borrower is not always “borrowing constrained.” This observation follows from a version of the inverse Euler equation that holds in the economy considered here. Optimality requires that when the sustainability constraint does not bind, we have $$qU^{\prime}\left(c\left(z^{t-1},z_{H}\right)\right)<\beta\mathbb{E}U^{\prime}\left(c\left(z^{t-1},z_{H},z_{t+1}\right)\right)$$. So, there is at least one state $$z_{t+1}$$ for which $$qU^{\prime}\left(c\left(z^{t-1},z_{H}\right)\right)<\beta U^{\prime}\left(c\left(z^{t-1},z_{H},z_{t+1}\right)\right);$$ that is, when current productivity is high, $$z_{H},$$ the borrower is “saving constrained.” Thus, debt limits and Arrow securities are not enough to decentralize the efficient allocation. After certain histories, a minimal asset holding requirements would be needed. 7. Final Remarks In this article, I show that key aspects of sovereign debt crises can be rationalized as part of the efficient risk-sharing arrangement between a sovereign borrower and foreign lenders in an economy with informational and commitment frictions. Along the outcome path that supports an efficient allocation, sovereign default episodes happen because of the need to provide incentives, despite being ex post inefficient. This article takes a first step towards bridging the gap between the literature on quantitative incomplete markets and the literature on optimal contracts. This article is qualitative in nature; however, it shows that one can interpret the outcome of an optimal contracting problem through the lens of a standard sovereign debt game, and derive implications for interest rates and bond holdings, which are the focus of the applied literature. A quantitative evaluation of the model is a fruitful area for future research. Such extension would also be suited to study how ex ante efficiency would be reduced by policies designed to mitigate the ex post inefficiency, such as bailout. The implementation I propose—and its implications for the optimal maturity composition of debt—is also applicable to other environments than the one considered here.27 As I mentioned, exact implementation may require very large positions, so it may be interesting to think about approximate implementation by imposing a cap on the debt positions. Moreover, it may be interesting to study the maturity composition in the best outcome of the sovereign debt game when the number of maturities available is smaller than the cardinality of the state space. Finally, while the efficient allocation can be implemented as an equilibrium outcome of the sovereign debt game, the converse is not true. There is a continuum of equilibria and generically they are not efficient. Thus, despite the fact that agents are able to achieve the efficient outcome in a market setting, regulation by a supranational authority may indeed be helpful in avoiding inefficient equilibria. Acknowledgements This is a revised version of the first chapter of my dissertation at the University of Minnesota. I am indebted to V.V. Chari, Patrick Kehoe, and Larry Jones for valuable advice. I would like to thank Michele Tertilt and three anonymous referees for very helpful suggestions. I also want to thank my discussants, Manuel Amador, Andy Atkeson, Ryan Chahrour, Pablo Kurlat, Vivian Yue, as well as Mark Aguiar, Cristina Arellano, Philip Bond, Wyatt Brooks, Erzo Luttmer, Ellen McGrattan, Chris Phelan, Ali Shourideh, Andrea Waddle, Ivan Werning, Pierre Yared, and Ariel Zetlin-Jones for their useful comments. I acknowledge the financial support of the Hutcheson Fellowship from the Economics Department of the University of Minnesota and the hospitality of the International Economics Section at Princeton University. The usual disclaimers apply. Supplementary Data Supplementary data are available at Review of Economic Studies online. Footnotes 1. The main results of the article remain valid if I consider an environment in which the incentive problem arises because of moral hazard, as in Atkeson (1991) and Tsyrennikov (2013). That is, the efficient risk-sharing arrangement can be implemented as the equilibrium outcome of the sovereign debt game with non-contingent defaultable debt of multiple maturities, and defaults and ex post inefficient outcomes happen along the equilibrium path. 2. See Rogoff (2011) for such an argument. 3. Quadrini (2004), Clementi and Hopenhayn (2006), DeMarzo and Sannikov (2006), DeMarzo and Fishman (2007), and Hopenhayn and Werning (2008) study this in the context of firm dynamics with credit frictions. 4. See also the reference in the survey by Panizza et al. (2009) and Tomz and Wright (2013). 5. Author’s calculations. See section C in the Online Appendix for details. 6. This drop can have a non-trivial impact on output. Gopinath and Neiman (2014) present a model calibrated to replicate the crisis in Argentina in 2002 and show that the decline in imports of intermediate goods can account for up to a 5-percentage-point decline in the welfare-relevant measure of productivity. 7. Under this interpretation, the link between output and default is weakened and so it may help to account for the weak relationship found in the data by Tomz and Wright (2007). 8. I further discuss this issue in the Online Appendix after Lemma B1. 9. The characterization of the efficient allocation does not depend on the two-shock assumption, and it can be extended (see a previous version of the article). This assumption will play a role in the implementation. I will discuss such role after the proof of the main proposition in Section 5. 10. Assumption 2 is satisfied if the curvature in $$U$$ and $$f$$ is low. 11. To understand the plausibility of condition (13), consider the following back-of-the-envelope calculation. If $$z_{L}$$ is a big recession in which productivity falls by say 10% (relative to productivity in good time $$z_{H}=1$$), and it is fairly unlikely in that the probability of such recession is 5% (so $$\mu_{H}=0.95$$), then we have that (13) is satisfied if $$\gamma$$ is greater than 1.6. 12. This is because the sovereign borrower is the owner of the domestic production technology that can be also operated in autarky. 13. This is consistent with the pari passu clause present in the vast majority of sovereign debt contracts. 14. See Arellano et al. (2015) for evidence of this interference. 15. In the background, as in Aguiar et al. (2009), the stand-in domestic household supplies labour inelastically and receives lump sum transfers (or taxes if negative), $$LS_{t}$$, from the government. The stand-in household’s budget constraint is $$c_{t}=w_{t}+LS_{t}$$, where $$w_{t}=F_{\ell}(m_{t},1)$$ is the competitive wage rate. (17)–(19) represent the combined budget constraints of the benevolent government and the stand-in household. 16. If the property in part (i) of Assumption 4 is not satisfied by the efficient allocation, it is still possible to implement the efficient allocation but it requires defaults for $$v\neq v_{a}$$. 17. To generate Figure 3 and 4, I consider the following functional form and parameterization: $$F\left(m,\ell\right)=\left(\alpha m^{1-1/\rho}+\left(1-\alpha\right)\ell^{1-1/\rho}\right)^{\frac{\rho}{\rho-1}}$$ with $$\rho=3$$ and $$\alpha=0.45$$; the other parameters in the model are $$\gamma=2$$, $$\beta=0.95$$, $$q=0.96$$, $$z_{H}=1,$$$$z_{L}=0.65$$, and $$\mu\left(z_{H}\right)=0.8$$. This example is representative of several simulations I perform. 18. That is the implicit constant interest rate at which the discounted value of the bond’s coupons equals its price. Define $$q_{YM,L}$$ as $$q_{L}=\frac{q_{YM,L}}{1-q_{YM,L}}$$. The consol’s yield to maturity is $$1/q_{YM,L}=\frac{q_{L}}{1+q_{L}}$$. 19. Recent works have documented that this is actually true in the data. See, for example, Zettelmeyer et al. (2013) and Asonuma et al. (2015). 20. An attractive feature of the decentralization I propose is that it extends to pure exchange economies, while the decentralization based on state contingent tariff or taxes does not. 21. The necessity of tariff is related to the necessity of capital income taxes in the implementation for the efficient allocation in an economy with lack of commitment in Kehoe and Perri (2004) and Aguiar et al. (2009). 22. For $$\alpha_{i}\in\left\{ \alpha_{1}=0,\alpha_{2},...,\alpha_{N}=1\right\}$$, define $$\bar{q}_{_{i}}$$ in a similar way as in (31):   \[ \bar{q_{i}}(v^{\prime})=\left\{ \begin{array}{cc} q\sum_{\theta}\mu(\theta)\left[1+(1-\alpha_{i})\bar{q}_{i}(v^{\prime}(v,\theta))\right] & \text{if }v^{\prime}>v_{a}\\ q\frac{\bar{R}_{i}}{1-(1-\alpha_{i})q} & \text{if }v^{\prime}=v_{a} \end{array}.\right. \] Then, if the return matrix   \[ \underset{N\times N}{\bar{Q}(v)}\equiv\left[\begin{array}{ccc} 1+q_{1}(v^{\prime}(v,z_{1})) & ... & 1+q_{N}(v^{\prime}(v,z_{1}))\\ \begin{array}{c} 1+q_{1}(v^{\prime}(v,z_{2}))\\ ... \end{array} & \begin{array}{c} ...\\ ... \end{array} & \begin{array}{c} 1+q_{N}(v^{\prime}(v,z_{2}))\\ ... \end{array}\\ 1+q_{1}(v^{\prime}(v,z_{N})) & ... & 1+q_{N}(v^{\prime}(v,z_{N})) \end{array}\right] \] is invertible, then there exists a $$\bar{b}(v)=[\bar{b}_{_{1}},...,\bar{b}_{_{N}}]^{T}$$ that solves the analogue of (32)–(33) given $$\bar{Q}$$. 23. The parameters used to generate Figure 4 are the ones listed in footnote 17. The recovery rates are $$r_{S}=r_{L}=0.6$$. 24. The borrower must buy back his outstanding long-term debt because the long-term debt I consider is a consol. If I would have instead considered a perpetuity with coupon decaying at a rate $$\alpha$$, some of the long-term debt is coming due in every period and so the net issuance of long-term debt would be $$\bar{b}_{L}\left(v'\left(v,z\right)\right)-\left(1-\alpha\right)\bar{b}_{L}\left(v\right)$$. In such a case, it is possible that even if the stock of outstanding long-term debt is declining, net issuance can be positive. 25. In this case, defaults will not be associated with ex post inefficiencies. 26. This is not a property of the efficient allocation for an economy with no private information, lack of commitment (one sided), and $$\beta=q$$. 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Efficient Sovereign Default

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Abstract

Abstract In this article, I show that the key aspects of sovereign debt crises can be rationalized as part of the efficient risk-sharing arrangement between a sovereign borrower and foreign lenders in a production economy with informational and commitment frictions. The constrained efficient allocation involves ex post inefficient outcomes that resemble sovereign default episodes in the data and can be implemented with non-contingent defaultable bonds and active maturity management. Defaults and periods of temporary exclusion from international credit markets happen along the equilibrium path and are essential to supporting the efficient allocation. Furthermore, during debt crises, the maturity composition of debt shifts towards short-term debt and the term premium inverts as in the data. 1. Introduction The conventional framework to study sovereign debt crises, defined as periods of high interest rate spreads, is the incomplete-market approach that follows the seminal contribution of Eaton and Gersovitz (1981). This framework can account for the behaviour of interest rates, debt, and macroeconomic aggregates around sovereign debt crises. However, this framework is not well suited for policy analysis, since the frictions that lead to a failure of risk-sharing are not explicitly identified. In contrast, another strand of the literature studies the efficient risk-sharing arrangement between the sovereign borrower and foreign lenders by explicitly stating the underlying frictions. These models are suited for policy analysis, but they do not identify the securities that will be traded, so the relationship between data and model is less transparent. This article takes a first step towards bridging the gap between the literature on quantitative incomplete markets and that on constrained efficient risk-sharing arrangement. I show that the key aspects of sovereign debt crises can be rationalized as part of the efficient risk-sharing arrangement between a sovereign borrower and foreign lenders in a production economy with informational and commitment frictions. The efficient allocation can be implemented as the equilibrium outcome of a sovereign debt game with non-contingent defaultable debt of multiple maturities. Defaults and ex post inefficient outcomes along the equilibrium path are not a pathology; rather, they support the ex ante efficient outcome. Moreover, the equilibrium outcome path displays defaults when output is low, an inversion of the term structure of interest rates spreads when the spread level is high, and a negative association between the duration of outstanding debt and the probability of future defaults. I provide a different view on policies than the literature on incomplete markets. First, the negative correlation between the maturity of outstanding debt and the probability of a crisis emerges as a way to support the efficient allocation when only non-contingent defaultable debt of multiple maturities is available. Hence, the high reliance on short-term debt is just a symptom, and not a cause, of an imminent debt crisis, in contrast to the view in Cole and Kehoe (2000). Second, dilution of long-term debt is essential to replicate the insurance prescribed by the efficient allocation given the available assets. Hence, introducing and enforcing seniority clauses to avoid dilution of outstanding long-term debt will lower welfare. This contrasts with the predictions in Chatterjee and Eyigungor (2012) and Hatchondo et al. (2016). Finally, because ex post inefficient outcomes are part of the efficient allocation, interventions by a supranational authority aimed at reducing the inefficiencies in a sovereign default episode are not beneficial from an ex ante perspective. I consider a simple production economy in which imported intermediate inputs are used in production, similarly to Aguiar et al. (2009) and Mendoza and Yue (2012). The government cannot commit to repaying its debt and has private information about the state of the domestic economy.1 In the baseline economy, the source of private information is the relative productivity of the domestic non-tradable sector. One interpretation of this assumption is that the government has more information about the domestic economy than do foreign lenders, and it controls the released statistics.2 I first characterize the optimal risk-sharing arrangement between the government and the foreign lenders subject to the restrictions imposed by the lack of commitment and private information. Absent contracting frictions, the risk-neutral lenders would completely insure the risk-averse government against fluctuations in productivities, and the realization of the shock would have no effect on the continuation of the allocation. Both the private information and the government’s lack of commitment limit such insurance. In particular, because of the presence of private information, the provision of a dynamic incentive is needed to have insurance. In order for insurance payments to borrowers with currently low-productivity-shocks to be incentive compatible, there must be a cost associated with claiming to have low productivity. Lenders can impose such a cost by reducing the continuation value of the government through lowering its future consumption levels. The lack of commitment interacts with the incentive problem. In particular, when the government’s continuation value is low, the government is tempted to deviate from the efficient allocation by increasing current consumption by not repaying lenders and then living in autarky thereafter. To prevent such an outcome, the lenders must provide a sufficiently low amount of intermediate goods so that this kind of deviation is unprofitable. Enforcing a continuation value for the government close to autarky is ex post inefficient. That is, if the government and the lenders could renegotiate the terms of their agreement, committing not to do so again in the future, then both could be made better off. By increasing the government’s value when it is close to autarky, it is possible to avoid the drop in imported intermediate inputs, which depresses production and reduces the government’s ability to repay the lenders. The necessity of providing incentives ex ante requires that these ex post inefficient outcomes happen along the equilibrium path with strictly positive probability. I show that under appropriate sufficient conditions, any efficient allocation settles down to a stationary distribution with ex post inefficiencies. The theme that ex post inefficiencies along the equilibrium path are necessary to support the ex ante optimal arrangement in economies with incentive problems has been explored in various contexts (e.g.Green and Porter, 1984; Phelan and Townsend, 1991; Yared, 2010).3 A novel feature of my article is that there is no termination of the risk-sharing relationship. The optimal outcome has periods of temporary autarky (which are ex post inefficient), but cooperation eventually restarts after the domestic economy recovers and the government makes a payment to the lenders. I then turn to implementing an efficient allocation as an equilibrium outcome of a sovereign debt game, similar to what is considered by the literature on quantitative incomplete markets. The set of assets that the government can issue is restricted to non-contingent bonds of multiple maturities. The government has the option to default, which I define as suspending the principal and coupon payments specified by the bond contracts. The government is excluded from credit markets until a given partial repayment to the bondholders is made. The government can also impose a tax on the payments received by foreign exporters from the domestic firms for the imports of intermediate goods, capturing the idea that the government cannot commit to repay the foreign lenders. Along the equilibrium outcome path, there are defaults only when the government’s continuation value is equal to the autarkic value. In this sense, defaults in the model are infrequent. Moreover, defaults are associated with high indebtedness (relative to the maximal level of debt sustainable), low output, and ex-post inefficiencies. The crucial step in proving that the efficient allocation can be an outcome of the sovereign debt game is to show that it is possible to replicate the wealth transfers implied by the efficient allocation with non-contingent defaultable debt. Defaults and partial repayments introduce de facto implicit state contingencies in the bond contract. When there is full repayment, the state contingent returns implied by the efficient allocation are replicated by exploiting the variation in the price of long-term debt after a shock. After the realization of a low productivity shock, the continuation value for the sovereign borrower decreases and the probability of a default in the near future increases, reducing the value of the outstanding long-term debt. This reduction results in a capital loss for the lenders and provides some debt relief to the borrower after an adverse shock. The opposite happens after the realization of a high-productivity shock: the price of outstanding long-term debt goes up and the lenders realize a capital gain. The model can generate the features of output, consumption, imports, and exports that occur during and after debt crises. The proximate cause of a debt crisis is a sufficiently long string of low-productivity shocks, which lead the borrower’s continuation value to decrease until it reaches the value of autarky, when there is a default. The lack of commitment implies that the imports of intermediates must drop to prevent a deviation by the borrower. This drop in imports reduces output, consumption, and the payments made to the lenders. Once the economy receives a high-productivity shock in the non-tradable sector, output increases and the borrower runs a trade surplus to partially repay the defaulted debt. These repayments result in the gradual increase of the borrower’s continuation value; hence, consumption, production, and imported intermediate inputs used in production will also increase. The pattern for output and consumption is consistent with the evidence in Mendoza and Yue (2012) for a sample of twenty-three recent defaults.4 In these episodes, on average, output is 4.5% and 5.2% below trend, and consumption is 3.1% and 3.6% below trend in the year of the default and the year after, respectively.5 This association between default and output being below trend is not universal. In a larger sample, Tomz and Wright (2007) find that about a third of default episodes happen with output above trend. The drop in trade during default episodes predicted by the model is also in line with the data. For instance, see Rose (2005) and the references in the survey by Panizza et al. (2009).6 Along the path approaching default, the maturity composition of the sovereign debt shifts towards short-term debt. This shift occurs because, when the probability of future default is high, the price of the long-term debt is more sensitive to shocks. Therefore, a lower long-term debt holding is needed to replicate the debt relief that is implicit in the efficient allocation after a low realization of productivity in the non-tradable sector. Since the overall indebtedness of the sovereign borrower is increasing along the path approaching a default, it must be that the amount of short-term debt issued increases along with the probability of default. Thus, the maturity composition shortens as indebtedness increases. This is broadly consistent with the evidence in Broner et al. (2013) and Arellano and Ramanarayanan (2012). Both papers document that when spreads are high, the maturity of newly issued debt shortens. More directly, Buera and Nicolini (2010) document that the stock of outstanding debt in Argentina was more tilted towards shorter maturity in 2000 (at the onset of the default in January of 2002) relative to 1997 (see Figure 5 in their paper). Bocola and Dovis (2016) provide similar evidence for Italy in the current crisis. Furthermore, along the equilibrium path, the term spread curve inverts during debt crises as documented by Broner et al. (2013) and Arellano and Ramanarayanan (2012). Related Literature This article is related to several strands of literature. First, it is related to the literature on quantitative incomplete markets on sovereign default. Following Eaton and Gersovitz (1981), recent contributions include Aguiar and Gopinath (2006), Arellano (2008), Hatchondo and Martinez (2009), Benjamin and Wright (2009), Yue (2010), Arellano and Ramanarayanan (2012), Chatterjee and Eyigungor (2012), and Mendoza and Yue (2012). I depart from this literature by studying the best equilibrium outcome of the sovereign debt game when the maturity structure is sufficiently rich. As previously argued, the policy implications that arise from assuming that markets are incomplete differ substantially from those that arise from allowing for the possibility of complete markets. My work is also related to the literature on optimal contracting approach to sovereign borrowing (e.g.Atkeson, 1991; Thomas and Worrall, 1994, Kehoe and Perri, 2002; Kehoe and Perri, 2004; Aguiar et al., 2009; Tsyrennikov, 2013). More broadly, this paper is also related to the literature on dynamic contracting with informational and commitment frictions (e.g.Atkeson and Lucas, 1995; Ales et al., 2014). Green (1987), Thomas and Worrall (1990), and Atkeson and Lucas (1992) consider environments with only private information, while Kehoe and Levine (1993), Kocherlakota (1996), Alvarez and Jermann (2000), and Albuquerque and Hopenhayn (2004) consider only lack of commitment. The main contribution of my article relative to this strand of literature is to propose an implementation that relates the efficient outcome to the data series that are the focus of the empirical research: interest rates, default decisions, and maturity composition of debt. In this sense, this article takes a first step towards bridging the gap between the literature on quantitative incomplete markets and that on constrained efficient arrangements. The idea that managing the maturity composition of debt serves to replicate state contingent returns has been explored, among others, by Kreps (1982), Angeletos (2002), and Buera and Nicolini (2004). In these models, movements in the term structure of interest rates are generated by fluctuations in the equilibrium stochastic discount factor. I consider a small open economy environment with risk-neutral lenders where the international interest rate is orthogonal to the shocks in the domestic economy. The variation in price of long-term debt is generated through variation in the endogenous probability of future default. In this aspect, my article is closely related to the work of Arellano and Ramanarayanan (2012), who endogenize the time-varying maturity composition of debt in an Eaton and Gersovitz (1981) type of model. Consistent with my findings, these authors find that the maturity composition of debt shortens when the probability of default is high. The difference between their paper and mine is that they cannot assess the efficiency of such an equilibrium outcome. Moreover, they determine the maturity composition by trading off a hedging motive and a commitment-not-to-dilute motive. The hedging motive is that long-term debt is attractive because it allows for a state contingent return. The commitment-not-to-dilute motive is that short-term debt is a commitment device not to dilute outstanding long-term debt. In my article, the hedging motive alone is sufficient to shorten the duration of debt when the probability of default is high. The rest of the article is organized as follows. In Section 2, I describe the environment, while in Section 3, I characterize the efficient allocation. In Section 4, I define the sovereign debt game. In Section 5, I construct and characterize the default rule, bond prices, and holdings that support the efficient allocation as an equilibrium outcome of the sovereign debt game. Finally, after discussing the implementation in Section 6, I conclude the article in Section 7. 2. Environment In this section, I lay out the environment in which the source of private information is the productivity of the non-tradable sector. In the Online Appendix, I reinterpret this economy as a taste shock economy, as in Atkeson and Lucas (1992).7 Time is discrete and indexed by $$t=0,1,...$$. There are three types of agents in the economy: a large number of homogeneous domestic households, a benevolent domestic government, and a large number of foreign lenders. In addition, there are three types of goods: a domestic consumption good (non-traded), an export good, and an intermediate good. The source of uncertainty is a shock to the relative productivity of the domestic consumption (non-tradable) sector. 2.1. Preferences All agents are infinitely lived. The stand-in domestic household values a stochastic sequence of consumption of the domestic good, $$\{c_{t}\}_{t=0}^{\infty}$$, according to   \begin{equation} \mathbb{E}_{0}\sum_{t=0}^{\infty}\beta^{t}U\left(c_{t}\right), \end{equation} (1) where $$\beta\in(0,1)$$ is the discount factor, and the period utility function has constant relative risk aversion,   \begin{equation} U(c)=\frac{c^{1-\gamma}}{1-\gamma},\label{U_CRRA} \end{equation} (2) with $$\gamma>1$$. I assume that $$\gamma>1$$ to ensure that the government indebtedness increases after the negative productivity shock.8 The government is benevolent, and it maximizes the utility of the stand-in domestic household. Foreign lenders are risk neutral, and they value consumption of the export good. They discount the future with a discount factor $$q\in(0,1)$$, which should be thought of as the inverse of the risk-free interest rate in international credit markets. I allow the discount factor $$\beta$$ and $$q$$ to differ, but I will restrict my analysis to the case where $$q\geq\beta$$; that is, the domestic households discount the future at a weakly higher rate than the international interest rate. 2.2. Endowments and technology Foreign lenders have a large endowment of the intermediate good. They have access to a technology that transforms one unit of the intermediate good into one unit of the export good so the relative price between the export and the intermediate good is fixed at one. Each domestic household is endowed with one unit of labour in each period. There is a domestic production technology that transforms the intermediate good and labour into the domestic consumption good, $$c$$, and foreign consumption good, $$x$$, as follows:   \begin{gather} c\leq zF\left(m_{1},\ell_{1}\right)\,\,\text{ and }\,\,x\leq F\left(m_{2},\ell_{2}\right),\label{RCz1}\\ \end{gather} (3)  \begin{gather} m_{1}+m_{2}\leq m,\,\,\,\ell_{1}+\ell_{2}\leq1,\label{RCz2} \end{gather} (4) where $$m_{1}$$ and $$m_{2}$$ are the units of the intermediate good allocated to the production of the domestic and export good, respectively; $$m$$ is the total amount of intermediates used domestically; and $$\ell_{1}$$ and $$\ell_{2}$$ are the units of domestic labour allocated to domestic and export production, respectively. The production function $$F$$ has constant returns to scale; it is increasing and continuously differentiable, it satisfies the Inada condition $$\lim_{m\rightarrow0}F_{m}(m,\ell)=+\infty$$$$\forall\ell>0$$, and it is such that $$F(0,1)>0$$, so strictly positive output can be produced in autarky. For notational convenience, let $$f(m)=F(m,1)$$. The relative productivity of the domestic sector, $$z$$, is distributed according to a probability distribution $$\mu$$, and it is independent and identically distributed over time. For simplicity, let $$z$$ take on only two values, $$z\in\left\{ z_{L},z_{H}\right\}$$ with $$z_{L}<z_{H}$$.9 Due to the properties of constant-returns-to-scale technology, the technological restrictions imposed by (3)–(4) can be summarized by the following aggregate resource constraint   \begin{equation} \frac{c}{z}+x\leq f(m),\label{RCz} \end{equation} (RC) as well as the non-negativity conditions on $$c$$ and $$x$$. 2.3. Timing The timing of events within the period is as follows: Foreign lenders supply intermediate goods $$m_{t}\geq0$$; the productivity shock $$z_{t}$$ is realized according to $$\mu;$$ and real activity occurs: production, consumption, and exporting take place. Let $$z^{t}=(z_{0},z_{1},...,z_{t})$$. An allocation is a stochastic process $$\boldsymbol{x}\equiv\{m(z^{t-1}),c(z^{t}),x(z^{t})\}_{t=0}^{\infty}$$. An allocation $$\boldsymbol{x}$$ is feasible if it satisfies the resource constraint (RC) for all $$t,z^{t}$$. 2.4. Information Foreign lenders observe the amount of intermediate goods that the country imports, $$m$$, and the amount of exports, $$x$$. Moreover, they can observe the amount of resources, $$m_{1}$$ and $$\ell_{1}$$, employed in the domestic consumption (non-tradable) sector. However, they cannot see the amount of output produced with the inputs, because the realization of $$z$$ is privately observed by the domestic government. From (RC), foreign lenders can use their information about $$m$$ and $$x$$ to infer $$c/z$$ but not $$c$$ and $$z$$ separately. I collect assumptions made so far here: Assumption 1. The utility function is $$U\left(c\right)=c^{1-\gamma}/(1-\gamma)$$ with $$\gamma>1$$; the discount factor $$\beta$$ satisfies $$qz_{H}^{1-\gamma}/\mathbb{E}\left(z^{1-\gamma}\right)\leq$$$$\beta\leq q;$$ and the production function $$F\left(m,\ell\right)$$ is increasing, continuously differentiable, displays constant returns to scale, $$F(0,1)>0$$, and satisfies the condition $$\lim_{m\rightarrow0}F_{m}(m,\ell)=+\infty$$$$\forall\ell>0.$$ The productivity shock can take on two values, $$z\in\left\{ z_{L},z_{H}\right\}$$, and it is independent and identically distributed over time. The condition $$\beta>qz_{H}^{1-\gamma}/\mathbb{E}\left(z^{1-\gamma}\right)$$ ensures that the government is sufficiently patient so that in the high-productivity state it prefers to reduce its consumption below the autarky level, being rewarded with an increase in future consumption. 3. Efficient Allocation In this section, I define and characterize a (constrained) efficient allocation when lenders cannot separately observe $$c$$ and $$z$$, and the sovereign borrower cannot commit to repay. I establish that, under certain sufficient conditions, an efficient allocation has cyclical periods with ex post inefficient outcomes that resemble a sovereign default episode in the data. 3.1. Definition Private information and lack of commitment by the sovereign borrower impose constraints in addition to the RC, which an allocation must satisfy to be implementable. First, consider the restriction imposed by the fact that $$z$$ is privately observed by the borrower. By the revelation principle, it is without loss of generality to focus on the direct revelation mechanism in which the sovereign borrower reports his type. Define the continuation utility for the sovereign borrower associated with the allocation $$\boldsymbol{x}$$ after history $$z^{t}$$ (according to truth-telling) as   \begin{equation} v(z^{t})\equiv\sum_{j=1}^{\infty}\sum_{z^{t+j}}\beta^{j-1}\Pr(z^{t+j}|z^{t})U(c(z^{t+j})). \end{equation} (5) An allocation $$\boldsymbol{x}$$ is incentive compatible if and only if it satisfies the following (temporary) incentive compatibility constraint for all $$t,z^{t},z$$:   \begin{equation} U(c(z^{t-1},z_{t}))+\beta v(z^{t-1},z_{t})\geq U\left(z_{t}\left[f(m(z^{t-1}))-x(z^{t-1},z)\right]\right)+\beta v(z^{t-1},z)\label{IC}, \end{equation} (IC) where $$z_{t}\left[f(m(z^{t-1}))-x(z^{t-1},z)\right]$$ is the consumption of non-traded good if the borrower has productivity $$z_{t}$$ but exports the amount $$x\left(z^{t-1},z\right)$$ instead of $$x\left(z^{t-1},z_{t}\right)$$. The incentive compatibility constraint, (IC), captures the informational frictions in the economy. It ensures that the borrower has no incentive to engage in undetectable deviations. That is, after any history $$z^{t}$$, the borrower does not want to choose the action prescribed for type $$z\neq z_{t}$$. Second, consider the restrictions imposed by lack of commitment. To be implementable, an allocation $$\boldsymbol{x}$$ must satisfy the following sustainability constraint for all $$t,z^{t}$$:   \begin{equation} U(c(z^{t}))+\beta v(z^{t})\geq U(z_{t}f(m(z^{t-1})))+\beta v_{a},\label{SUST} \end{equation} (SUST) where $$v_{a}$$ is the value of autarky given by   \begin{equation} v_{a}\equiv\frac{\sum_{z}\mu(z)U(zf(0))}{1-\beta}. \end{equation} (6) The sustainability constraint, (SUST), requires that the borrower have no strict incentive to engage in detectable deviations. That is, after any history, the borrower cannot gain from increasing his consumption by failing to export $$x(z^{t})$$. As it is standard in the literature, I assume that after this detectable deviation, the borrower is punished with autarky. This entails two forms of punishment. First, the sovereign borrower cannot access credit markets to obtain insurance. Second, the borrower suffers a loss in production because he cannot use imported intermediate goods. Later, I will show this autarkic value is the worst equilibrium value of the sovereign debt game. A feasible allocation $$\boldsymbol{x}$$ is said to be efficient if it maximizes the present value of net transfers to the foreign lenders, $$x-m$$, subject to (RC), (IC), the sustainability constraint (SUST), and a participation constraint for the borrower,   \begin{equation} \sum_{t=0}^{\infty}\sum_{z^{t}}\beta^{t}\Pr(z^{t})U(c(z^{t}))\geq v_{0}\label{PC} \end{equation} (PC) for some feasible initial level of promised utility $$v_{0}\in\lbrack v_{a},\bar{v}]$$, with $$\bar{v}\equiv\lim_{c\rightarrow\infty}U(c)/(1-\beta)$$. An efficient allocation solves   \begin{equation} J(v_{0})=\max_{\boldsymbol{x}}\sum_{t=0}^{\infty}\sum_{z^{t}}q^{t}\Pr(z^{t})\left[x(z^{t})-m(z^{t-1})\right]\label{J} \end{equation} (J) subject to (RC), (IC), (SUST), and (PC). I will refer to the value $$J:[v_{a},\bar{v}]\rightarrow$$ as the Pareto frontier. The constraint set in (J) is not necessarily convex, because of the presence of $$U\circ f(m)$$, a concave function, on the right hand side of (SUST). Thus, randomization may be optimal. It is possible to rule out randomization as part of the efficient allocation by making an additional assumption following Aguiar et al. (2009). Assumption 2. Let $$m^*$$ be the statically efficient level of intermediates, i.e. $$m^*$$ such that $$f'(m^*)=1$$. Define $$H:[U(f(0)),U(f(m^{\ast}))]\rightarrow$$ as $$H(\underline{u})\equiv C(\underline{u})-f^{-1}\circ C(\underline{u})$$ with $$C=U^{-1}$$. $$H$$ is concave. If Assumption 2 is satisfied,10 then randomization is not optimal. 3.2. Near-recursive formulation The problem in (J) admits a near-recursive formulation using the borrower’s promised utility, $$v$$, as a state variable. The problem is not fully recursive, because the problem in period 0 is slightly different, as I explain next. From $$t\geq1$$, an efficient allocation solves the following recursive problem for $$v\in\lbrack v_{a},\bar{v}]$$:   \begin{equation} B(v)=\max_{m,c(z),v^{\prime}(z)}\sum_{z}\mu\left(z\right)\left[f(m)-m-\frac{c(z)}{z}+qB(v^{\prime}(z))\right]\label{P} \end{equation} (P) subject to   \begin{eqnarray} U\left(c(z)\right)+\beta v^{\prime}(z) & \geq & U\left(z\left[f(m)-y^{\ast}(z^{\prime})\right]\right)+\beta v^{\prime}(z^{\prime}) \forall z,z^{\prime}\label{ic}\\ \end{eqnarray} (7)  \begin{eqnarray} U\left(c(z)\right)+\beta v^{\prime}(z) & \geq & U\left(zf(m)\right)+\beta v_{a} \forall z\label{sust}\\ \end{eqnarray} (8)  \begin{eqnarray} v^{\prime}(z) & \geq & v_{a} \forall z\label{sustp}\\ \end{eqnarray} (9)  \begin{eqnarray} \sum_{z}\mu\left(z\right)\left[U\left(c(z)\right)+\beta v^{\prime}(z)\right] & = & v,\label{pkc} \end{eqnarray} (10) where $$B(v)$$ is the maximal present discounted value of net transfers, $$x-m=f(m)-c/z-m$$, that the foreign lenders can attain subject to a recursive version of the incentive compatibility constraint, (7), a recursive version of the sustainability constraint, (8), the fact that continuation utility must be greater than the value of autarky, (9), and the constraint that the recursive allocation delivers a value of $$v$$ to the sovereign borrower (the promise-keeping constraint), (10). The function $$B$$ traces out the utility possibility frontier. At $$t=0$$, for all $$v_{0}\in\lbrack v_{a},\bar{v}]$$, the problem in (J) can be expressed as   \begin{equation} J(v_{0})=\max_{m,c(z),v^{\prime}(z)}\sum_{z}\mu\left(z\right)\left[f(m)-m-\frac{c(z)}{z}+qB(v^{\prime}(z))\right] \end{equation} (11) subject to (7), (8), (9), and the participation constraint   \begin{equation} \sum_{z}\mu\left(z\right)\left[U\left(c(z)\right)+\beta v^{\prime}(z)\right]\geq v_{0}.\label{pc} \end{equation} (12) The difference between the Pareto frontier $$J$$ and the utility possibility frontier $$B$$ is that, in the utility possibility frontier, the promise-keeping constraint (10) requires that the allocation deliver exactly the promised utility $$v\in\lbrack v_{a},\bar{v}]$$ to the sovereign borrower. This is because for $$t\geq1$$, the promise-keeping constraint serves to maintain incentives from previous periods. In contrast, in the definition of the Pareto frontier $$J$$, the participation constraint (12) requires that the sovereign borrower receive at least$$v$$. This is because in period $$t=0$$ there are no incentives from previous periods to keep. In many applications, this asymmetry is irrelevant, because the participation constraint in (J) is binding. This is not the case here, because the utility possibility frontier $$B(v)$$ has an increasing portion, as I will later show. 3.3. Properties The next proposition establishes three properties of the efficient allocation that I will later use to characterize the equilibrium outcome which supports the efficient allocation. Proposition 1. Under Assumptions 1 and 2, the efficient allocation is such that (i)There are distortions in production. There exists $$v^{\ast}\in(v_{a},\bar{v})$$ such that $$m(v)=m^{\ast}$$ for all $$v\geq v^{\ast}$$. For all $$v\in\lbrack v_{a},v^{\ast})$$, $$m(v)$$ is strictly less than $$m^{\ast}$$ and is strictly increasing in $$v$$, and, in particular, $$m(v_{a})=0$$. (ii)The efficient allocation is dynamic: $$\forall v\in\lbrack v_{a},\bar{v}]$$, $$c(v,z_{L})>c(v,z_{H})$$ and $$v^{\prime}(v,z_{L})<v^{\prime}(v,z_{H})$$. (iii)There is insurance. Let $$b(v,z)\equiv x(v,z)-m(v)+qB(v^{\prime}(v,z))$$ be the lenders’ value after the realization of $$z$$. Then $$\forall v\in\lbrack v_{a},\bar{v}]$$, $$b(v,z_{H})>b(v,z_{L})$$. (iv)The value for the lenders when $$v=v_{a}$$ is positive, $$b(v_{a},z)>0$$ for all $$z.$$ The proof of this proposition can be found in the Online Appendix. Part (i) states that low levels of promised utility for the borrower are associated with imported intermediates that are below the statically efficient level, $$m^{\ast}$$, which is such that $$f^{\prime}(m^{\ast})=1$$. When the continuation value for the borrower is low, imports must be low to satisfy the sustainability constraint. Whenever the sustainability constraint is binding, $$m<m^{\ast}$$. In particular, at autarky it must be that $$m(v_{a})=0$$. In fact, if the foreign lenders supplied any $$m>0$$, the sovereign government could unilaterally achieve a lifetime utility of $$U(zf(m))+\beta v_{a}>U(zf(0))+\beta v_{a}=v_{a}$$. Thus, only $$m=0$$ is consistent with the promise-keeping and sustainability constraints at autarky. On the other hand, for continuation values high enough, $$v\geq v^{\ast}$$, the threat of autarky after an observable deviation is sufficiently harsh that the statically efficient amount of intermediate imports can be supported; that is, $$m(v)=m^{\ast}$$ for all $$v\geq v^{\ast}$$. It can be shown that $$m$$ is actually strictly increasing in the borrower’s promised value for $$v\in\lbrack v_{a},v^{\ast}]$$. This result is illustrated in Figure 1. Figure 1 View largeDownload slide Policy function for intermediate imports Figure 1 View largeDownload slide Policy function for intermediate imports Part (ii) states that the efficient allocation is dynamic, in the sense that it uses variation in the borrower’s continuation utility to provide incentives, thus allowing for higher transfers after the realization of a low-productivity shock. This feature of the efficient allocation is critical for it to be implementable as an outcome of the sovereign debt game. Part (iii) shows that the market value of debt is state-contingent; there is debt relief when the borrower has a high marginal utility of consumption (low $$z$$). Thus, the efficient allocation provides some, albeit imperfect, insurance. Part (iv) states that the market value of debt when the continuation value for the borrower equals autarky is strictly positive. In the implementation, there will be defaults when the value for the borrower equals autarky. This property implies that the value of debt must be positive during a default. This will in turn require that I allow for partial repayment of debt. 3.4. Optimality of ex post inefficiencies I now turn to the main result of this section: an efficient allocation calls for ex post inefficient outcomes with strictly positive probability, provided that a sufficient condition is satisfied. 3.4.1. Region with ex post inefficiencies The next proposition establishes that the utility possibility frontier is upward sloping for borrower values close to autarky. Lemma 1. There exists a $$\tilde{v}\in(v_{a},v^{\ast})$$ such that the utility possibility frontier $$B(v)$$ is strictly increasing over $$[v_{a},\tilde{v})$$ and decreasing over $$[\tilde{v},\bar{v}]$$. I refer to the interval $$[v_{a},\tilde{v})$$ as the region with ex post inefficiencies because for all $$v\in\lbrack v_{a},\tilde{v})$$, the market value of debt (and hence the value for the lenders) can be increased by providing higher utility to the borrower, thus making both existing lenders and the borrower better off. This is because supporting a continuation value for the borrower close to the autarkic level requires that a very low level of intermediate goods be employed in production so that the sustainability constraint (8) is satisfied. This depresses production and, consequently, the repayments that the lenders can receive in the period. In particular, when the borrower’s value is close to autarky, intermediates are close to zero (see Proposition 1 part (i)). Thus, because of the Inada condition on $$f$$, the marginal return from additional intermediates is large enough that the benefit from the extra production which can be obtained by increasing the borrower’s continuation value is larger than the cost to the lenders of providing the additional value to the borrower. Therefore, both agents can be made better off relative to autarky, and $$B$$ is upward sloping in a neighbourhood of $$v_{a}$$. In contrast, for sufficiently high promised values, $$v\geq v^{\ast}$$, the statically efficient level of intermediates can be supported. For such promised values, increasing the borrower’s value is costly and has no benefit for the lenders, and so $$B$$ is strictly decreasing for $$v\geq v^{\ast}$$. Therefore, because of the concavity of $$B$$, the utility possibility frontier must peak at some $$\tilde{v}\in(v_{a},v^{\ast})$$. Over the interval $$[\tilde{v},\bar{v}]$$, which I will refer to as the efficient region, $$B$$ is decreasing. These results are illustrated in Figure 2. Figure 2 View largeDownload slide Pareto and utility possibility frontiers Figure 2 View largeDownload slide Pareto and utility possibility frontiers 3.4.2. The efficient allocation transits to the region with ex post inefficiencies Any efficient allocation starts in the efficient region, because the participation constraint, (PC), in the programming problem (J) can hold as an inequality. For any borrower value, $$v$$, in the region with ex post inefficiencies, (PC) does not bind and $$J(v)=J(\tilde{v})=B(\tilde{v})>B(v)$$. It is optimal for the lenders to promise at least $$\tilde{v}$$ to the borrower. Instead, for $$v$$ in the efficient region (PC) in (J) binds and $$J(v)=B(v)$$. The question now is, Does an efficient allocation transit to the region with ex post inefficiencies after some history, or is the efficient region an ergodic set? Provided that a sufficient condition is satisfied, the continuation of any efficient allocation transits to the region with ex post inefficiencies after a sufficiently long (but finite) string of realizations of $$z_{L}$$. The essential piece of the argument is to show that following a realization of $$z_{L}$$, the continuation utility is strictly lower than the current one: $$v^{\prime}(v,z_{L})<v$$. To this end, I assume that the following sufficient condition holds: Assumption 3. The parameters $$z_{L},z_{H},\mu_{H},$$ and $$\gamma$$ satisfy the following conditions:11  \begin{align} &\mu_{H}z_{L}^{1-\gamma} \geq\mathbb{E}\left(z^{1-\gamma}\right)\label{Aeconomics}\\ \end{align} (13)  \begin{align} &\left(1-\mu_{H}\right)\left(\frac{z_{L}^{1-\gamma}}{\mathbb{E}\left(z^{1-\gamma}\right)}-1\right)+\mu_{H}\left(\frac{z_{H}}{z_{L}}\right)^{1-\gamma} \geq1\label{Atech}. \end{align} (14) Under this additional assumption, I can prove the following lemma: Lemma 2. Under Assumptions 1–3, for every $$v$$ in the efficient region, $$v'\left(v,z_{L}\right)<v$$. Lemma 2 is not obvious, because there is a tension between two countervailing forces. First, there is an incentive effect that calls for lowering $$v^{\prime}(v,z_{L})$$ below $$v$$. This is because lowering the continuation utility after a low-productivity shock helps to separate types and to provide more current consumption when the marginal utility of consumption is high. Second, there is a countervailing commitment effect: Lowering the continuation utility tightens future sustainability constraints. As is standard in economies with only lack of commitment, there is a motive to backload payments to the sovereign borrower in order to relax future sustainability constraints and allow for lower-production distortions in the future. Under Assumption 3, the incentive effect outweighs the commitment effect. To understand why, consider a necessary first-order condition from the problem   \begin{equation} B^{\prime}(v)=\frac{q}{\beta}\left[\mu_{L}B^{\prime}(v^{\prime}(v,z_{L}))+\mu_{H}B^{\prime}(v^{\prime}(v,z_{H}))\right]+\frac{f^{\prime}(m(v))-1}{U^{\prime}(z_{L}f(m(v)))f^{\prime}(m(v))},\label{DB2} \end{equation} (15) which, using the fact that $$B^{\prime}(v)\leq0$$ for $$v\geq\tilde{v}$$ and $$\beta\leq q$$, can be rewritten as   \begin{equation} \left[B^{\prime}(v^{\prime}(v,z_{L}))-B^{\prime}(v)\right]\geq\mu_{H}\left[B^{\prime}(v^{\prime}(v,z_{L}))-B^{\prime}(v^{\prime}(v,z_{H}))\right]-\frac{\beta}{q}\frac{f^{\prime}(m(v))-1}{U^{\prime}(z_{L}f(m(v)))f^{\prime}(m(v))}.\label{DB3} \end{equation} (16) Equation (16) illustrates the two forces operating in the model. The first term in square brackets on the right-hand side of (16) stands in for the incentive effect, while the second term stands in for the commitment effect. First notice that by the concavity of $$B$$, if the right-hand side of (16) is positive, then it must be that $$v^{\prime}(v,z_{L})<v$$. By Proposition 1, Part (ii), $$v^{\prime}(v,z_{H})>v^{\prime}(v,z_{L})$$ and, thus, the first term on the right-hand side of (16) is strictly positive. Absent any commitment problem (i.e. $$f^{\prime}(m)=1$$), the second term on the right-hand side of (16) is equal to zero. Therefore, the right-hand side is positive and, consequently, it will be true that $$v^{\prime}(v,z_{L})<v$$. When the sustainability constraint binds (i.e., $$f^{\prime}(m)>1$$), the second term on the right-hand side of (16), $$-\left[f^{\prime}(m)-1\right]/\left[U^{\prime}(z_{L}f(m))f^{\prime}(m)\right]$$, is negative; it is then not obvious that the right-hand side of (16) is positive. Thus, in this case, it is not guaranteed that $$v^{\prime}(v,z_{L})<v$$. Assumption 3 guarantees that this is indeed the case. Note that such assumption is met either if, for a given $$\mu_{H}$$, $$z_{L}^{1-\gamma}-z_{H}^{1-\gamma}$$ is sufficiently large or, for given $$z_{L}^{1-\gamma}-z_{H}^{1-\gamma}$$, $$\mu_{L}$$ is sufficiently small. Intuitively, if $$z_{L}^{1-\gamma}-z_{H}^{1-\gamma}$$ is sufficiently large, the benefit of separating the two types is large. It is very cheap to satisfy the promise-keeping constraint by providing consumption when the productivity shock is low, $$z=z_{L}$$. To provide a large spread in current consumption across types in an incentive-compatible way, it is necessary to have a large spread in continuation values, $$v^{\prime}(v,z_{H})-v^{\prime}(v,z_{L})$$. Thus, the first term of the right-hand side of (16) is large. Moreover, if $$\mu_{L}$$ is small, the cost of tightening future sustainability constraints by reducing the continuation value after $$z_{L}$$ is small, from an ex ante perspective. Inspecting (16), if $$\mu_{H}$$ is low, then the first term on the right-hand side is again large. Thus, if either $$z_{L}^{1-\gamma}-z_{H}^{1-\gamma}$$ is sufficiently large or $$\mu_{L}$$ is sufficiently small, the benefits from lowering $$v^{\prime}(v,z_{L})$$ below $$v$$ by relaxing the incentive compatibility constraint and the current sustainability constraint (incentive effect) are larger than the costs arising from higher-production distortions in the future after a low productivity shock (commitment effect). Under the assumptions in Lemma 2, for all $$v$$ in the efficient region, $$v^{\prime}(v,z_{L})$$ lies strictly below the 45-degree line, as illustrated in Figure 3. Let $$\Delta\equiv\min_{v\in\lbrack\tilde{v},\bar{v}]}\left\{ v-v^{\prime}(v,z_{L})\right\}$$. Because $$v^{\prime}(\cdot,z_{L})$$ is continuous as shown in Lemma B2 in the Online Appendix, it follows that $$\Delta>0$$. Thus, starting from any $$v_{0}\in\lbrack\tilde{v},\bar{v})$$, after a sequence of $$t$$ consecutive realizations of $$z_{L}$$, the borrower’s continuation value is less than $$v_{0}-\Delta t$$. Thus, after a sufficiently long string $$z^{T}=\left(z_{L},z_{L},...,z_{L}\right)$$ with $$T\leq(v_{0}-\tilde{v})/\Delta$$, the continuation utility transits to the region with ex post inefficiencies, $$[v_{a},\tilde{v})$$. The next proposition summarizes the argument above. Figure 3 View largeDownload slide Law of motion for borrower’s value Figure 3 View largeDownload slide Law of motion for borrower’s value Proposition 2. Under Assumptions 1–3, an ex ante efficient allocation transits to the region with ex post inefficiencies with strictly positive probability. 3.4.3. Role of the main ingredients The interaction between lack of commitment and private information is key to having ex post inefficient outcomes happening along the path in this production economy. Both lack of commitment and the fact that intermediates are used in production are crucial to generating an upward sloping portion of the utility possibility frontier. However, these two features alone cannot generate ex post inefficient outcomes associated with an ex ante efficient allocation. Without an incentive problem, any continuation of an efficient allocation is itself efficient. Thus, an efficient allocation never transits to the region with ex post inefficiencies of the utility possibility frontier. See Aguiar et al. (2009) for this result in a related environment. Private information alone generates a downward drift of the continuation utility (see Thomas and Worrall (1990) and Atkeson and Lucas (1992)) but does not generate ex post inefficiencies, because with commitment, there is no connection between low continuation values and production in the economy. The statically efficient amount of production can always be sustained. Low continuation values for the borrower only have distributional effects in that the lenders can appropriate larger shares of total undistorted production. Also in this case, continuations of efficient allocations are always on the Pareto frontier. Both contracting frictions are needed to obtain ex post inefficient outcomes as part of the ex ante optimal arrangement (Proposition 2). Lack of commitment and production are crucial for having an upward sloping portion of the utility possibility frontier (Lemma 1); private information is crucial for having the efficient allocation to transit to the region with ex post inefficiencies (Lemma 2). These features are also present in previous works that also establish the optimality of ex post inefficiencies in related environments, such as Phelan and Townsend (1991), Quadrini (2004), Clementi and Hopenhayn (2006), DeMarzo and Sannikov (2006), and DeMarzo and Fishman (2007). 3.5. Long-run properties The next proposition establishes that the efficient allocation features perpetual cycles that transit in and out of the region with ex post inefficiencies. Proposition 3. Under Assumptions 1–3, any efficient allocation converges to a unique non-degenerate stationary distribution. Moreover, the inefficient region, $$[v_{a},\tilde{v}]$$, is in the support of such distribution. The proof is relegated to the Online Appendix. The key to understand Proposition 3 is to understand the law of motion for continuation utility illustrated in Figure 3. These laws of motion define a unique ergodic set for promised utility. By Lemma 2, after a sufficiently long—and finite—string of draws of $$z_{L}$$, continuation utility transits to the region with ex post inefficiencies. This region, and the value of autarky in particular, is not an absorbing state. In the Online Appendix, I show that whenever $$z_{H}$$ is drawn, then the continuation is back in the efficient region. To show that autarky is not an absorbing state, a sufficient condition is that $$\beta>qz_{H}^{1-\gamma}/\mathbb{E}\left(z^{1-\gamma}\right).$$ This condition ensures that the government is sufficiently patient so that in the high-productivity state it prefers to reduce its consumption below the autarky level, being rewarded with an increase in future consumption. Thus, under these conditions, there is sufficient “mixing” that the existence of a unique limiting distribution is guaranteed. The limiting distribution has perpetual cycles that transit in and out of the region with ex post inefficiencies. This feature differentiates my environment from related dynamic contracting problems such as those in Quadrini (2004), Clementi and Hopenhayn (2006), DeMarzo and Sannikov (2006), DeMarzo and Fishman (2007), and Hopenhayn and Werning (2008), which also have ex post inefficiencies along the path. In all of these papers, in the long run, either the incentive problem disappears or there is an inefficient termination of the venture between the principal and the agent. In contrast, here the incentive problem does not disappear in the long run and the risk-sharing relationship does not terminate. The optimal allocation has periods of temporary autarky, but cooperation eventually restarts after the domestic economy recovers.12 The efficient allocation only pins down the transfers between the sovereign borrower and the foreign lenders. It is silent about bond prices, defaults, and so forth. In the next section, I show that the efficient allocation can be implemented given a set of assets typically considered in the quantitative sovereign default literature. 4. Sovereign Debt Game In this section, I describe the game that implements the efficient allocation. The set of securities that the sovereign borrower can issue is restricted to the ones considered in the literature on quantitative incomplete markets. I impose a set of rules that describe the treatment of the government in default. These rules stand in for the outcome of a renegotiation process and determine the payoff of debt in default. I take these rules as a feature of the environment, and I derive implications for the path of debt and bond prices that implement the efficient allocation given these rules. The fact that the constrained efficient allocation can be implemented implies that this set of rules is not restrictive. In Section 6, I discuss how the assumption about this process can be relaxed without affecting the main characteristics of equilibrium outcomes. Consider a game between competitive foreign lenders (bondholders and exporters of the intermediate good), competitive domestic firms, and a benevolent domestic government. The government can issue two types of non-contingent defaultable bonds: a one-period bond, $$b_{S}$$ (or foreign reserves if $$b_{S}<0$$) and a consol, $$b_{L}\geq0$$, that cannot be used for savings. One unit of the one-period bond promises to pay one unit of the export good tomorrow in exchange for $$q_{S}$$ units of the export good today. The consol is a perpetuity that promises to pay a coupon of one unit of the export good in every period starting tomorrow in exchange for $$q_{L}$$ units of the export good today. The government cannot commit to satisfying the terms of the bond contracts. The borrower can choose among three options regarding the repayment of inherited debt, $$\delta_{t}\in\left\{{\texttt{full}},{\texttt{partial}},{\texttt{suspend}}\right\} .$$When $$\delta_{t}=$$full, there is full repayment: the government pays in full the outstanding one-period bond and the coupon on the consol. When $$\delta_{t}=$$partial, there is partial repayment: the government pays a fraction $$r_{S}\in\lbrack0,1)$$ of outstanding obligations on one-period bonds and $$r_{L}/(1-q)$$ for each unit of consols outstanding with $$r_{L}\in\lbrack0,1)$$. After making this partial payment, the government does not make any more future payments to the holder of legacy long-term debt. Note that holders of the consol are treated equally irrespective of the time the consol was issued.13 Finally, $$\delta_{t}=$$suspend denotes suspension of payments: the government makes no payments in the current period. It is then excluded from international credit markets, and the defaulted debt is rolled over to the next period (missed interest payments are forgiven) when the government can choose a repayment policy $$\delta_{t+1}$$ for the notional amount of today’s debt obligations. I would say that the government is in default whenever it repays less than what is contractually specified, that is, $$\delta_{t}$$$$\neq$$full. I introduce partial repayment as a possible choice for the government to decentralize the efficient allocation. This is because the total value of debt in the efficient allocation (i.e., the present discounted value of net transfers to foreign lenders) is positive when the value for the borrower equals autarky. I will discuss this issue further in the proof of Proposition 4. In addition to issuing debt, the government can also tax the payments made by domestic firms to foreign exporters for the intermediate goods at a rate $$\tau_{t}\in\lbrack0,1]$$. Thus, foreign exporters receive an after-tax payment of $$p_{t}(1-\tau_{t})$$ per unit of intermediate good sold, where $$p_{t}$$ is the price of the intermediates in terms of the final good. We should think of this as the government interfering with private transactions.14 The sequence of events within the period is as follows: Foreign lenders set a price for intermediate inputs $$p_{t}$$ (how many units of export good for one unit of intermediate good). Domestic competitive firms choose the quantity of intermediate inputs they want to use, $$m_{t}$$. The productivity shock $$z_{t}$$ is realized and privately observed by the domestic government. The government picks a policy $$\pi_{t}=(\delta_{t},\boldsymbol{b}_{t+1},\tau_{t})$$ that consists of a repayment decision $$\delta_{t}$$, new bond holdings, $$\boldsymbol{b}_{t+1}=(b_{S,t+1},b_{L,t+1}),$$ and a tariff on imported intermediates, $$\tau_{t}$$. Bond holdings are bounded by a large positive constant $$\bar{B}$$ to rule out a Ponzi scheme. Bond prices $$\boldsymbol{q}_{t}=(q_{S,t},q_{L,t})$$ are consistent with foreign lenders’ optimality. Following Chari and Kehoe (1990), to formally define a sustainable equilibrium, let $$h^{t}=(h^{t-1},p_{t},m_{t},\pi_{t})$$ be a public history up to period $$t\geq0,$$ and let $$h^{-1}=\boldsymbol{b}_{0}=(b_{S,0},b_{L,0})$$ be the initial outstanding debt. It is convenient to define the following public histories when agents take action: $$h_{p}^{t}=h^{t-1}$$, $$h_{m}^{t}=(h^{t-1},p_{t})$$, and $$h_{\sigma}^{t}=(h^{t-1},p_{t},m_{t})$$. The price of the intermediate good, $$p=\left\{ p_{t}\right\} _{t=0}^{\infty}$$, the allocation rule for $$m=\left\{ m_{t}\right\} _{t=0}^{\infty}$$, the strategy for the government, $$\sigma=\left\{ \sigma_{t}\right\} _{t=0}^{\infty}$$, and the price of bonds, $$\boldsymbol{q} =\left\{ q_{S,t},q_{L,t}\right\} _{t=0}^{\infty}$$, are all functions of the relevant histories. 4.1. Problem of the government To set up the problem for the government, let $$Y(\tau,z)=zF(m_{t},1)-p_{t}m_{t}+\tau p_{t}m_{t}$$ be the amount of resources available to the government after production, repayments of intermediates, and collection of the tariff revenue. Taking as given $$p$$, $$m$$, and the price schedule for bonds, $$\boldsymbol{q}$$, after any history $$\left(h_{\sigma}^{t},z\right)$$, the strategy for the government, $$\sigma,$$ solves the problem   \begin{equation} W(h_{\sigma}^{t},z)=\max_{c,\pi=\left(\delta,b_{S}^{\prime},b_{L}^{\prime},\tau\right)}U(c)+\beta\mathbb{E}\left[W(h_{\sigma}^{t+1},z_{t+1})|h_{\sigma}^{t},\pi\right]\label{W} \end{equation} (W) subject to $$b_{S}^{\prime},b_{L}^{\prime}\leq\bar{B}$$ and the consolidated budget constraints of the government and the stand-in household.15 If there is no default (i.e. if $$\delta=$$full), the consolidated budget constraint is given by   \begin{equation} c+(b_{S,t}+b_{L,t})\leq Y(\tau,z)+q_{S,t}(h_{\sigma}^{t},\pi)b_{S}^{\prime}+q_{L,t}(h_{\sigma}^{t},\pi)(b_{L}^{\prime}-b_{L,t}),\label{bc0} \end{equation} (17) or, if there is partial repayment (i.e. if $$\delta=$$partial) by   \begin{equation} c+\left(r_{S}b_{S,t}+r_{L}\frac{b_{L,t}}{1-q}\right)r\leq Y(\tau,z)+q_{S,t}(h_{\sigma}^{t},\pi)b_{S}^{\prime}+q_{L,t}(h_{\sigma}^{t},\pi)b_{L}^{\prime},\label{bcr} \end{equation} (18) or, if there is default without any partial repayment (i.e. if $$\delta=$$suspend), then I impose the restriction that after $$\delta_{t}=$$suspend, there is a temporary exclusion from international credit markets; that is,   \begin{equation} c\leq Y(\tau,z)\,\,\,\text{and}\,\,\,\left(b_{S}^{\prime},b_{L}^{\prime}\right)=\left(b_{S,t},b_{L,t}\right).\label{bc1} \end{equation} (19) 4.2. Bond prices and other equilibrium objects The price of the imported intermediate, $$p_{t}$$, must be consistent with optimization by competitive foreign lenders who take the tariff level as given after all histories:   \begin{equation} 1=\mathbb{E}\left[p_{t}(h_{p}^{t})\left(1-\tau_{t}(h_{\sigma}^{t},z_{t})\right)|h_{p}^{t}\right].\label{p} \end{equation} (20) The allocation rule for the quantity of foreign intermediate goods, $$m_{t}$$, satisfies the optimality condition for the representative domestic competitive firm after all histories:   \begin{equation} \mathbb{E}\left[Q(h_{\sigma}^{t},z_{t})F_{m}(m_{t}(h_{m}^{t}),1)|h_{m}^{t}\right]=p_{t}(h_{p}^{t}),\label{m} \end{equation} (21) where $$Q(h_{\sigma}^{t},z_{t})=U^{\prime}\left(c(h_{\sigma}^{t},z_{t})\right)/\mathbb{E}\left[U^{\prime}\left(c(h_{\sigma}^{t},z_{t})\right)\right]$$ is the price that the representative domestic firm uses to evaluate profits. Given the government repayment policy, bond prices $$q_{S,t},q_{L,t}$$ are consistent with the maximization problem of the risk-neutral foreign lenders who discount the future at a rate $$q$$. For the one-period bond, if $$b_{S,t+1}\geq0$$, it must be that   \begin{equation} q_{S,t}(h^{t})=q\mathbb{E}\left[\chi_{S,t+1}(h^{t+1})|h^{t}\right],\label{q1} \end{equation} (qS) where $$\chi_{S,t+1}$$ is the ex post value of short-term debt:   \begin{equation} \chi_{S,t+1}(h^{t+1})=\left\{ \begin{array}{c@{\kern6pt}c} 1 & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{full}}\\[5pt] r_{S} & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{partial}}\\[5pt] q\mathbb{E}\left[\chi_{S,t+2}(h^{t+2})|h^{t+1}\right] & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{suspend}} \end{array}\right.\label{chi}\!\!. \end{equation} (22) Here the expectation in (qS) is taken at the end of period $$t$$ over next period’s value of the productivity shock, $$z_{t+1}.$$ The lenders understand the repayment rule of the government, and the price is simply the actuarially fair value of repayments. The only subtle part is that if the government does not repay the short bond at $$t+1$$ by setting $$\delta_{t+1}(h_{\sigma}^{t+1})=$$suspend, this bond still has value because it gives the holder the right to any partial repayment $$r_{S}$$ that the government may make on this defaulted debt either in period $$t+2$$ or some later period to regain access to the bond market. In this sense, when $$\delta_{t+1}=$$suspend, $$q\mathbb{E}\left[\chi_{S,t+2}(h^{t+2})|h^{t+1}\right]$$ represents the secondary market value of defaulted debt. Next, consider situations in which the government saves, in that $$b_{S,t+1}<0$$. If the history is such that the government is not excluded from saving, the rate on foreign assets will equal the world rate, that is, $$q_{S,t}(h^{t})=q$$. If instead the history is such that the government is being temporarily excluded from the international credit market, I adopt the convention that $$q_{S,t}(h^{t})=\infty$$. Allowing for exclusion from savings is necessary to decentralize the efficient risk-sharing arrangement. In such arrangement, detectable deviations are punished with permanent autarky. If there is no mechanism preventing the government to save after a detectable deviation, then the government could default and self-insure via saving. This would increase the incentive for the government to deviate and prevent to decentralize the efficient allocation with this market structure. Finally, the price for the consol must be such that   \begin{equation} q_{L,t}(h^{t})=q\mathbb{E}\left[\chi_{L,t+1}(h^{t+1})|h^{t}\right],\label{q8} \end{equation} (qL) where $$\chi_{L,t+1}$$ is the ex post value of the consol given by   \begin{equation} \chi_{L,t+1}(h^{t+1})=\left\{ \begin{array}{c@{\quad}c} 1+q_{L,t+1}(h^{t+1}) & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{full}}\\[5pt] \frac{r_{L}}{1-q} & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{partial}}\\[5pt] q\mathbb{E}\left[\chi_{L,t+2}(h^{t+2})|h^{t+1}\right] & \text{if }\delta_{t+1}(h_{\sigma}^{t+1})={\texttt{suspend}} \end{array}\right.\!\!\!.\label{chiL} \end{equation} (23) If the government does not repay its consol at $$t+1,$$ this consol still has value because it gives the holder the right to any partial repayment $$r_{L}/(1-q)$$ that the government may make in future periods in order to regain access to the bond markets. 4.3. Equilibrium definition Given initial outstanding debt $$\boldsymbol{b}_{0}$$, a sustainable equilibrium is a strategy for the government, $$\sigma$$, a price rule for the foreign intermediate good, $$p$$, price rules for the government bonds, $$q_{S}$$ and $$q_{L}$$, and an allocation rule for the intermediate good, $$m$$, such that (i) given $$p$$, $$m$$, $$q_{S}$$, and $$q_{L}$$, the government’s strategy, $$\sigma$$, is the optimal policy associated with (W) for all $$(h_{\sigma}^{t},z)$$; (ii) given $$\sigma$$, the price and allocation rules $$p$$ and $$m$$ satisfy (20) and (21); (iii) given $$\sigma,$$ the price of the short-term bond $$q_{S}$$ satisfies (qS) whenever $$b_{S}\geq0$$, and it equals $$q$$ or $$\infty$$ when $$b_{S}<0,$$ where $$q_{S}=\infty$$ stands in for exclusion from saving abroad; (iv) given $$\sigma,$$ the price of the long-term bond $$q_{L}$$ satisfies (qL). The associated equilibrium outcome is denoted by $$\boldsymbol{y}=\left(\boldsymbol{x},\boldsymbol{g},\boldsymbol{p}\right)$$, where $$\boldsymbol{x}=\left\{ m(z^{t-1}),c(z^{t})\right\} _{t=0}^{\infty}$$, $$\boldsymbol{g}=\left\{ \delta(z^{t}),b_{L}(z^{t}),b_{S}(z^{t}),\tau(z^{t})\right\} _{t=0}^{\infty}$$, and $$\boldsymbol{p}=\left\{ p(z^{t-1}),\boldsymbol{q}(z^{t})\right\} _{t=0}^{\infty}$$. It is useful to characterize the set of outcomes that can be implemented as a sustainable equilibrium of the sovereign debt game. Such characterization gives us a set of necessary and sufficient conditions for an equilibrium outcome, which I will later show that are met by the efficient allocation. The logic of the characterization follows Abreu (1988) and Chari and Kehoe (1990) in using reversion to the worst equilibrium to characterize the set of equilibrium outcomes. The worst equilibrium outcome from the borrower’s perspective is the autarkic allocation. The autarkic allocation can be supported as an equilibrium outcome as follows: Zero intermediates and a price equal to zero for both short-term and long-term debt can be supported as part of a sustainable equilibrium, because if foreign lenders expect a tariff equal to 100% (full expropriation) and full default in any subsequent periods irrespective of the action chosen today by the government ($$\delta_{t}=$$suspend for all subsequent $$t$$), then the government has no incentive to choose something different than $$\tau_{t}=1$$ and $$\delta_{t}=$$suspend, confirming the lenders’ beliefs. The fact that the sovereign borrower cannot save after a deviation follows from an assumption commonly used in the literature (e.g.Atkeson, 1991; Aguiar et al., 2009) to rule out the classic Bulow and Rogoff (1989) result that no foreign debt can be sustained in equilibrium. The next Lemma characterizes the set of sustainable equilibrium outcomes: Lemma 3. Given initial outstanding debt $$\boldsymbol{b}_{0}$$, $$\boldsymbol{y}$$ is a sustainable equilibrium outcome if and only if it satisfies the constraints (RC), (IC), and (SUST) in addition to the budget constraints (17)–(19) and (20), (21), (qS) and (qL) along the equilibrium outcome path. Proof. It is evident that the conditions in Lemma 3 are necessary for an equilibrium outcome. Conditions (20), (21), (qS), and (qL) are part of the definition of a sustainable equilibrium. Budget feasibility from the government’s perspective requires the outcome to satisfy (17)–(19). Constraints (IC) and (SUST) ensure that the sovereign borrower has no strict incentive to engage in a detectable and undetectable deviation, respectively. In (SUST), I use the fact that the borrower’s value after a detectable deviation cannot be lower than the value of the worst equilibrium, $$v_{a}$$. To see that such conditions are also sufficient, consider an outcome that satisfies the conditions in the statement. Such outcome can be implemented as sustainable equilibrium by relying on trigger strategies that revert to the worst equilibrium after a detectable deviation from the government. Clearly, such an outcome satisfies the optimality conditions for the competitive agents, since it satisfies (20), (21), (qS), and (qL), and it is budget feasible for the government, since it satisfies the budget constraints (17)–(19). Then I am left to show that the government has no incentive to deviate from the prescribed path of plays. To this end, note the detectable deviations are not profitable, because the outcome satisfies (SUST). Moreover, undetectable deviations are not profitable, because the outcome satisfies (IC). Then, the government does not have a strict incentive to deviate from the proposed path of plays, concluding the proof. ∥ 5. Defaults, Bond Prices, and Maturity Composition I now show that any efficient allocation can be implemented as an equilibrium outcome of the sovereign debt game. This amounts to say that the best equilibrium outcome of the sovereign debt game is (constrained) efficient. Along the equilibrium outcome path, there are defaults only when the borrower’s continuation value is equal to the autarkic value. Defaults along the path do not trigger permanent exclusion from international credit markets. In the model, the borrower suspends payments to foreign lenders until a high-productivity shock is drawn and a given partial repayment on the defaulted debt is made. After such partial repayment, the borrower regains access to foreign borrowing and lending. It is important to note the distinction between a detectable deviation from the path of play, which may include a default, and what happens along the equilibrium path when there is a default. On-path defaults are excusable in the sense introduced by Grossman and Van Huyck (1988): They happen in well-understood circumstances, and the country regains access to international credit markets after making a partial repayment; therefore, on-path defaults do not trigger autarky forever. 5.1. Implementation In the rest of the article, I will assume that the efficient allocation satisfies the following properties: Assumption 4. The efficient allocation is such that (i)for all $$v>v_{a}$$, $$v^{\prime}(v,z_{L})<v$$ and there exists a $$\underline{v}\in(v_{a},\bar{v})\,$$ such that $$v^{\prime}(v,z_{L})=v_{a}$$ for all $$v\leq\underline{v}$$; (ii)$$v^{\prime}(\cdot,z)$$ is strictly increasing for all $$v\geq\underline{v}$$ and for all $$z.$$ Part (i) implies that starting from any $$v$$, there is a strictly positive probability of reaching autarky after a sufficiently long but finite string of low-productivity shocks. This property does not follow from Lemma, 2 since the Lemma does not immediately extend to the region of ex-post inefficiencies if lenders are more impatient than the borrower. This is because for $$v$$ such that $$B^{\prime}\left(v\right)>0$$, it is more costly to front-load utility for the relatively more patient lenders (the opposite of what happens for promised utility $$v$$ such that $$B^{\prime}\left(v\right)\leq0$$). In the Online Appendix, I provide two sets of sufficient conditions on primitives that ensure the efficient allocation satisfies property (i)). In particular, I show that if $$f\left(m^{\ast}\right)-f\left(0\right)$$ is sufficiently small or if $$\beta$$ is sufficiently close to $$q,$$ then the condition is satisfied. Moreover, if the efficient allocation displays partial insurance, in that $$U^{\prime}\left(c\left(v,z_{L}\right)\right)\geq U^{\prime}\left(c\left(v,z_{H}\right)\right),$$ then a stronger version of condition (13) ensures that part (i) holds.16 Part (ii) requires that the continuation values, $$v^{\prime}(\cdot,z_{L})$$ and $$v^{\prime}(\cdot,z_{H}),$$ be increasing in current promised utility. Both (i) and (ii) are satisfied in my simulations (e.g. Figure 3).17 For later reference, define the short-term spread as the difference between the interest rate implied by $$q_{S}$$ and the risk-free international interest rate: $$s_{S}\equiv1/q_{S}-1/q$$. The long-term spread is defined as the difference between the consol’s yield to maturity18 and the risk-free interest rate: $$s_{L}\equiv(1+q_{L})/q_{L}-1/q$$. The term premium is the difference between the long- and the short-term spreads: $$s_{T}\equiv s_{L}-s_{S}$$. The main result of this section is that there exists a sustainable equilibrium that decentralizes the efficient allocation such that default happens along the equilibrium path in the region of ex post inefficiencies when the government’s debt is high (relative to the maximal amount that can be sustained) and output is low. Furthermore, if the recovery rate on long-term bonds is sufficiently high relative to the recovery rate on short-term bonds,19 the equilibrium outcome is such that when spreads are low, the term premium is positive, but it delivers an inversion of the yield curve when spreads are high. Moreover, numerical simulations show that the maturity composition of debt shortens when spreads are high. Proposition 4. Under Assumptions 1–3 and $$\beta<q$$, given an efficient allocation that satisfies Assumption 4 and a set of recovery rates $$\left(r_{S},r_{L}\right)$$ with at least one rate greater than zero, a sustainable equilibrium exists that decentralizes it, with default happening along the equilibrium path when the continuation value for the borrower equals $$v_{a}$$. Moreover, if $$r_{L}\geq r_{S}$$, the equilibrium is such that the term premium is positive when borrower’s value is above $$v_{a}$$ and negative at $$v_{a}$$. The proof of the proposition is by construction. I construct the on-path default rule, bond holdings, tariffs, and prices that support the efficient allocation and are consistent with the sufficient condition for a sustainable equilibrium outcome in Lemma 3. Since the efficient allocation can be represented by a time-invariant function of borrower’s continuation utility and exogenous shocks, the on-path repayment rule, bond holdings, tariffs, and prices can also be expressed as a function of on-path continuation utility for the borrower and the current realization of the productivity shock $$z$$:   \begin{align} \bar{\tau},\bar{p}:\left[v_{a},\bar{v}\right] & \rightarrow\mathbb{R},\quad\bar{\delta}:\left[v_{a},\bar{v}\right]\times\left\{ z_{L},z_{H}\right\} \rightarrow\left\{{\texttt{full}},{\texttt{partial}},{\texttt{suspend}}\right\} \label{bar1}\\ \end{align} (24)  \begin{align} \bar{q}_{S},\bar{q}_{L},\bar{b}_{S},\bar{b}_{L} & :\left[v_{a},\bar{v}\right]\times\left\{ z_{L},z_{H}\right\} \rightarrow\mathbb{R}\label{bar2}. \end{align} (25) Note that I consider a decentralization in which $$\bar{\tau}$$ does not depend on $$z$$ to emphasize how active maturity management can replicate state contingent return implicit in the efficient allocation. A state contingent tariff can also provide insurance to the government; for instance, see the role of state contingent taxes on capital income in Aguiar et al. (2009).20 An outcome path $$\boldsymbol{y}$$ can be recovered in the natural way from (24), (25), and the law of motion for $$v$$ from the efficient allocation. Moreover, bond holdings and prices depend only on the continuation value after $$z$$ is realized, $$v^{\prime}(v,z)$$. With some abuse of notation, I can then write   \begin{eqnarray} \bar{q}_{S}(v,z) & = & \bar{q}_{S}(v^{\prime}(v,z)),\text{ }\bar{q}_{L}(v,z)=\bar{q}_{L}(v^{\prime}(v,z)),\text{ } \\ \end{eqnarray} (26)  \begin{eqnarray} \bar{b}_{S}(v,z) & = & \bar{b}_{S}(v^{\prime}(v,z))\text{, } \bar{b}_{L}(v,z)=\bar{b}_{L}(v^{\prime}(v,z)). \end{eqnarray} (27) The steps to construct the candidate equilibrium outcome path $$\boldsymbol{y}$$ from an efficient allocation $$\boldsymbol{x}$$ are as follows: (1) define the repayment policy; (2) use the repayment policy in the optimality conditions for the foreign lenders to calculate equilibrium bond prices; (3) choose short- and long-term debt to match the total value of debt (lenders’ value) after a realization of $$z$$ implied by the efficient allocation   \begin{equation} b(v,z)=f(m(v))-c(v,z)-m(v)+qB(v^{\prime}(v,z)),\label{bx} \end{equation} (28) and (4) use the optimality conditions for the domestic firms and the lenders to get tariffs and prices for the intermediate good. Consider first the repayment policy. The borrower defaults only when his continuation value is equal to the autarky value, $$v_{a}$$. For all other borrower values, $$v>v_{a}$$, there is full repayment. In particular,   \begin{equation} \bar{\delta}(v,z)=\left\{ \begin{array}{c@{\quad}c} {\texttt{suspend}} & \text{if }v=v_{a}\text{ and }z=z_{L}\\ {\texttt{partial}} & \text{if }v=v_{a}\text{ and }z=z_{H}\\ {\texttt{full}} & \text{if }v>v_{a}\text{ for all }z \end{array}\right.\!\!\!,\label{deltabar2} \end{equation} (29) where $$\bar{\delta}(v_{a},z_{L})=$$suspend because, as showed in Lemma B6 in the Online Appendix, when the borrower’s value is autarky, there are no capital flows in that $$m(v_{a})=0$$ and $$c(v_{a},z_{L})=f(0)$$. When $$z=z_{H}$$, $$c(v_{a},z_{H})<f(0)$$ and so there is an outflow of resources. This is matched by a partial repayment on defaulted debt. Given the repayment policy, bond prices are uniquely pinned down by the lenders’ optimality conditions. The price for short-term debt is given by   \begin{equation} \bar{q}_{S}(v)=\left\{ \begin{array}{@{}cc} q & \text{if }v>v_{a}\\[3pt] q\bar{R}_{S} & \text{if }v=v_{a} \end{array}\right.\!\!\!,\label{qbar1} \end{equation} (30) where $$\bar{R}_{S}$$ is the expected repayment in case of default for short-term debt, $$\bar{R}_{S}=r_{S}\bar{R}$$ where $$\bar{R}=\mu(z_{H})/(1-q\mu(z_{L}))$$. The price for long-term debt can be written recursively as   \begin{equation} \bar{q}_{L}(v)=\left\{ \begin{array}{@{}cc} q\sum_{i=L,H}\mu(z_{i})\left[1+\bar{q}_{L}(v_{i}^{\prime}(v))\right] & \ \text{if }v>v_{a}\\[3pt] \frac{q}{1-q}\bar{R}_{L} &\ \text{if }v=v_{a} \end{array}\right.\!\!\!,\label{qbar8} \end{equation} (31) where $$\bar{R}_{L}q/\left(1-q\right)$$ is the expected repayment in case of default for long-term debt, $$\bar{R}_{L}=r_{L}\bar{R}$$. For all $$v>v_{a}$$, the price of short-term debt is equal to that of a risk-free bond. The price of long-term debt is lower than the price of a risk-free consol, because there is always a positive probability that there will be a default over the relevant time horizon of the bond. The next Lemma shows that $$\bar{q}_{L}$$ is increasing in the continuation value for the borrower, and it establishes that the price of long-term debt increases after drawing $$z_{H},$$ in that $$\bar{q}_{L}\left(v_{H}^{\prime}(v)\right)>\bar{q}_{L}\left(v_{L}^{\prime}(v)\right)$$. Lemma 4. Under the assumptions in Proposition 4, $$\bar{q}_{L}:[v_{a},\bar{v}]\rightarrow$$ is the unique fixed point of the contraction mapping defined by the right-hand side of (31), it is increasing, and for all $$v$$ we have  \[ \frac{q}{1-q}>\bar{q}_{L}\left(v_{H}^{\prime}(v)\right)>\bar{q}_{L}\left(v_{L}^{\prime}(v)\right). \] The proof for this Lemma is provided in the Online Appendix. In the proof, I use the assumption that $$v^{\prime}\left(v,z\right)$$ is monotone in $$v$$ for all $$z,$$ part (ii) of Assumption 4. Given the functions for bond prices $$\bar{q}_{S}$$ and $$\bar{q}_{L}$$, in the no-default region, $$\bar{b}_{S}(v)$$ and $$\bar{b}_{L}(v)$$ are chosen to match the total value of debt (lenders’ value) implied by the efficient allocation after $$z_{H}$$ and $$z_{L}$$ defined in (28):   \begin{eqnarray} b(v,z_{H}) & = & \bar{b}_{S}(v)+\bar{b}_{L}(v)\left[1+\bar{q}_{L}(v_{H}^{\prime}(v))\right]\label{bL}\\ \end{eqnarray} (32)  \begin{eqnarray} b(v,z_{L}) & = & \bar{b}_{S}(v)+\bar{b}_{L}(v)\left[1+\bar{q}_{L}(v_{L}^{\prime}(v))\right].\label{bH} \end{eqnarray} (33) A unique solution to (32)–(33) is guaranteed by Lemma 4, which establishes that $$\bar{q}_{L}\left(v_{H}^{\prime}(v)\right)>\bar{q}_{L}\left(v_{L}^{\prime}(v)\right)$$, meaning that the price of the long-term bond falls after $$z_{L}$$ is realized relative to $$z_{H}$$. Thus, outside the default region, the maturity composition of debt is uniquely pinned down. Simple algebra shows that   \begin{eqnarray} \bar{b}_{L}(v) & = & \frac{b(v,z_{H})-b(v,z_{L})}{\bar{q}_{L}(v_{H}^{\prime}(v))-\bar{q}_{L}(v_{L}^{\prime}(v))}\label{bbar8}\\ \end{eqnarray} (34)  \begin{eqnarray} \bar{b}_{S}(v) & = & b(v,z_{H})-\bar{b}_{L}(v)\left[1+\bar{q}_{L}(v_{H}^{\prime}(v))\right].\label{bbar1} \end{eqnarray} (35) Notice that it is guaranteed that $$\bar{b}_{L}(v)>0$$, because, by Proposition 1 part (iii), $$b(v,z_{H})-b(v,z_{L})\,{>}\,0$$ and $$\bar{q}_{L}(v_{H}^{\prime}(v))-\bar{q}_{L}(v_{L}^{\prime}(v))>0$$. When the continuation value for the next period onward is equal to autarky, the split between long- and short-term debt is indeterminate; $$\bar{b}_{S}(v_{a})$$ and $$\bar{b}_{L}(v_{a})$$ must satisfy   \begin{equation} b(v_{a},z_{H})=r_{S}\bar{b}_{S}(v_{a})+r_{L}\frac{\bar{b}_{L}(v_{a})}{1-q}.\label{bval} \end{equation} (36) The other possible outcome follows from (36) and $$\delta(v_{a},z_{L})=$$suspend because   \begin{equation} b(v_{a},z_{L})=q\left[\mu(z_{H})b(v_{a},z_{H})+\mu(z_{L})b(v_{a},z_{L})\right]=\frac{q\mu(z_{H})}{1-q\mu(z_{L})}b(v_{a},z_{H}).\label{bvah} \end{equation} (37) Thus, for any given recovery rates $$\left(r_{S},r_{L}\right),$$ I can choose any $$\left(\bar{b}_{S}(v_{a}),\bar{b}_{L}(v_{a})\right)$$ that satisfies (36). Consequently, (37) will also be satisfied. Note that to be able to satisfy (36), at least one between $$r_{S}$$ and $$r_{L}$$ must be strictly positive. If both $$r_{S}$$ and $$r_{L}$$ are strictly positive, the maturity composition is undetermined. I resolve the indeterminacy by assuming that $$\bar{b}_{S}(v_{a})/\bar{b}_{L}(v_{a})=\lim_{v\rightarrow v_{a}}\bar{b}_{S}(v)/\bar{b}_{L}(v)$$. The recovery rates $$\left(r_{S},r_{L}\right)$$ are free parameters. They can be chosen to be sufficiently low so that $$\bar{b}_{S}$$ is strictly positive in the default region and for $$v$$ close to $$v_{a}$$ so that a non-full repayment has a natural interpretation. Finally, on-path tariff rates and prices for the intermediate good, $$\bar{\tau},\bar{p}$$, are given by21  \begin{equation} \mathbb{E}\left[\frac{U^{\prime}\left(c(v,z)\right)}{\mathbb{E}\left[U^{\prime}\left(c(v,z)\right)\right]}f^{\prime}(m(v))\right]=\frac{1}{1-\bar{\tau}(v)}=\bar{p}(v).\label{ptaubar} \end{equation} (38) Then the outcome path $$\boldsymbol{y}$$ constructed from the efficient allocation $$\boldsymbol{x}$$ using (29), (30), (31), (34), (35), (36), and (38) satisfies the optimality conditions (20) and (21), the equilibrium bond pricing equations (qS) and (qL), and the budget constraints (17)–(19) along the equilibrium path. Since the efficient allocation satisfies (IC) and (SUST), this concludes the proof of the first part of the Proposition, as it verifies that the constructed outcome satisfies the sufficient conditions for a sustainable equilibrium outcome in Lemma 3. I am left to characterize the behaviour of the term structure of interest rates. To this end, first notice that outside the default region, the short-term debt is risk-free; see (30). Thus, $$q_{S,t}=q$$ and $$s_{S,t}=0$$. From Lemma 4 we know that $$q_{L,t}<q/(1-q)$$. Therefore, the short-term spread is zero, while the long-term spread is positive. Consequently, the term spread $$s_{T}$$ is positive. When the borrower’s continuation value is in the default region, the term spread is given by   \begin{equation} s_{T}(v_{a})=\left(1+\frac{1}{\bar{q}_{L}(v_{a})}\right)-\frac{1}{\bar{q}_{S}(v_{a})}=-\left(\frac{1-\bar{R}r_{L}}{\bar{R}r_{L}}\right)-\frac{1}{q\bar{R}}\left(\frac{r_{L}-r_{S}}{r_{S}r_{L}}\right), \end{equation} (39) which is negative provided that the recovery rate on short-term debt $$r_{S}$$ is not too small relative to the recovery rate for long-term debt $$r_{L}.$$ A sufficient condition is $$r_{L}\geq r_{S}$$. ∥ Proposition 4 can be generalized to the case in which $$z$$ can take on more than two values, allowing for a richer maturity structure. For instance, if $$z$$ can take on $$N$$ values, then I can use $$N$$ types of the perpetuity considered in Hatchondo and Martinez (2009) that pay a coupon which decays exponentially at rate $$\alpha_{n}\in\lbrack0,1]$$. The one-period bond and the consol are special cases of this class of securities for $$\alpha$$ equal to $$1$$ and $$0$$, respectively. Provided that the return matrix satisfies a full-rank condition22 (which is satisfied when $$N=2$$ because of Lemma 4), the statement in Proposition 4 generalizes to the case with $$N>2$$. 5.2. Mechanism that replicates state-contingent returns The crucial step proving that the efficient allocation can be an outcome of the sovereign debt game was to show that it is possible to replicate the insurance provided by the efficient allocation, that is, the total value of debt falls after an adverse shock relative to a positive shock, $$b(v,z_{H})>b(v,z_{L})$$. How is insurance provided in the sovereign debt game? When there is default, partial repayments make the non-contingent debt de facto state-contingent. When there is full repayment, the fall in the value of debt after the realization of a low-productivity shock is obtained by diluting outstanding long-term debt, that is, by imposing a capital loss on the holders of outstanding long-term debt. After a low-productivity shock, the continuation value for the borrower decreases, the overall level of indebtedness increases, and the probability that there will be a default in the near future increases. This increase in the likelihood of a future default reduces the value of the outstanding long-term debt, resulting in a capital loss for the debt holders and a capital gain for the borrower. This capital loss on the debt holders after an adverse shock mimics the debt relief for the borrower associated with the efficient allocation. This capital loss in a low-productivity shock is compensated by a capital gain after a high-productivity shock, so on average the lenders break even. The maturity composition of debt is driven solely by the requirement of matching this differential value of government debt ex post. Note that in the region with ex post inefficiencies, a reduction in the value of long-term debt is not necessarily good from the government’s perspective. In fact, in such region there are sustainable equilibria that attain a higher value for the government as well as the legacy debt holders. However, such continuation strategies are not optimal from an ex ante perspective as explained in Proposition 2. Hence the equilibrium strategies in the best sustainable equilibrium prevent the government to pursue such welfare-enhancing policies. 5.3. Maturity shortens as interest rate spread increases I now turn to the implications for the optimal maturity composition of debt. The main finding is that the maturity of outstanding debt issued by the sovereign borrower shortens as the long-term spread increases. In particular, the amount of long-term debt decreases, while the amount of short-term debt increases for all $$v$$ in the efficient region. This result is illustrated in Figure 4.23 I cannot state a proposition for this result, but the findings are consistent with all of my numerical simulations. Figure 4 View largeDownload slide Bond prices and holdings, insurance and ex post variation of the long-term debt price Figure 4 View largeDownload slide Bond prices and holdings, insurance and ex post variation of the long-term debt price To understand this result, notice that outside the default region, the amount of long-term debt issued by the borrower is determined by (34). Given the ex post variation in the price of the consol, $$\bar{q}_{L}(v_{H}^{\prime}(v))-\bar{q}_{L}(v_{L}^{\prime}(v))$$, the long-term debt holdings are constructed to match the debt relief implied by the optimal contract after the realization of $$z_{L}$$, $$b(v,z_{H})-b(v,z_{L})$$. As is shown in Figure 4, the level of debt relief is approximately constant for all $$v$$ over the efficient region. The ex post variation in the price of the consol is larger the closer the borrower is to the default region. This is because as the borrower’s continuation value approaches the default threshold from above, it is more likely that a realization of a low-productivity shock will push the economy into default in the near future. Hence, the long-term debt price is more sensitive to the realization of the shock. Therefore, a lower holding of long-term debt is needed to replicate the same amount of insurance, that is, the same debt relief after a high taste shock. Since the overall level of indebtedness is increasing, it must be that $$\bar{b}_{S}$$ is increasing as the borrower’s continuation value approaches $$\tilde{v}$$, because $$\bar{b}_{L}$$ is falling at the same time. Therefore, in the efficient region, as the level of indebtedness and the spread on long-term debt $$s_{L}$$ increase, the maturity composition of debt shortens. In the region with ex post inefficiencies, $$[v_{a},\tilde{v}]$$, the ratio of short-term debt to long-term debt is not always decreasing in the borrower’s value under all parameterizations. This is because the ex post variation in the price of long-term debt is high, but also the amount of insurance, $$b(v,z_{H})-b(v,z_{L})$$, increases a lot in this region (Figure 4). Despite not necessarily being monotonically decreasing in this region, the maturity composition of debt is more tilted towards short-term debt than it is for continuation values associated with lower default probabilities. The decision rules for debt holdings obtained in my simulations can be used to characterize the pattern of debt issuances if the economy is hit by a sequence of low-productivity shocks that leads to a declining pattern of continuation values eventually triggering a default. Because in my simulations $$\bar{b}_{S}\left(v\right)$$ is decreasing in $$v$$ and $$\bar{b}_{L}\left(v\right)$$ is increasing in $$v$$ (Figure 4), along this path, the stock of short-term debt is increasing while the stock of long-term debt (consol) is decreasing. To achieve this outcome, the borrower issues more short-term debt than the one coming due, $$\bar{b}_{S}\left(v'\left(v,z_{L}\right)\right)>\bar{b}_{S}\left(v\right)$$, and buys back $$\bar{b}{}_{L}\left(v\right)-\bar{b}_{L}\left(v'\left(v,z\right)\right)>0$$ units the outstanding long-term debt.24 In this sense, the issuance pattern is broadly consistent with the evidence in Broner et al. (2013) and Arellano and Ramanarayanan (2012). In fact, along the path leading to a default, all new issuances are short-term debt. Once the economy recovers and the borrower’s promised value increases, the borrower issues more long-term debt $$\bar{b}_{L}\left(v'\left(v,z\right)\right)-\bar{b}{}_{L}\left(v\right)>0$$ and a lower amount of one period debt than the one coming due, $$\bar{b}_{S}\left(v'\left(v,z_{L}\right)\right)<\bar{b}_{S}\left(v\right)$$. 5.4. Assumptions on rules in default I now turn to discuss the conventions I choose for the government in default and whether they can be relaxed without affecting the main characteristics of equilibrium outcomes. To this end, it is important to understand the dynamics of payments prescribed by the efficient allocation. In the Online Appendix (Lemma B5), I show that when the borrower’s value equals the value of autarky, there are no capital flows when $$z$$ equals $$z_{L}$$, $$x\left(v_{a},z_{L}\right)=0$$, and there are outflows when $$z$$ equals $$z_{L}$$, $$x\left(v_{a},z_{H}\right)>0$$ and $$b\left(v_{a},z_{H}\right)>0$$ (see also Proposition 1 part iv). The efficient allocation only pins down total payments when the value of the borrower is autarky and it draws a positive productivity shock, as illustrated in equation (36). This implies a degree of indeterminacy at $$v_{a}$$. In fact, if the recovery rates double and the face value of debt is halved, the borrower makes the same payment and raises the same resources in the previous period as the price of debt doubles from equations (30) and (31). The fact that $$b\left(v_{a},z_{H}\right)>0$$ requires that at least one between $$r_{S}$$ and $$r_{L}$$ be strictly positive. Other than this, there are no other requirements on recovery rates. Total payments prescribed by the efficient allocation after the economy recovers do not depend on the length the country spent in temporary autarky, say $$n\geq0$$. This implies that if the recovery rates $$r_{S}$$ and $$r_{L}$$ do not depend on $$n$$, then the interest rates arrears must be forgiven so that the equilibrium payout received by lenders does not depend on $$n$$. An alternative way to implement the efficient allocation is to have interest rate arrears not being forgiven and recovery rates that depend on $$n$$. This clearly would not change the behavior of the equilibrium outcome leading to a default. In setting up the sovereign debt game, I assumed that the government settles with the holders of legacy debt by making a current payment. Nothing will change if the government could use a mix of current payments and newly issued debt to pay existing debt holders as part of the settlement agreement, as in Benjamin and Wright (2009). It is worth noticing that some payment must be done in the current period, as the total net export of the country is positive, $$x\left(v_{a},z_{H}\right)>0$$. Further note that I am ruling out the possibility for holders of legacy long-term debt to holdout by assuming that holders of legacy long-term debt receive no payments once a partial repayment is made. Finally, several authors (e.g. Tomz and Wright, 2013 and references therein) have argued that market access and interest rates differ depending on the “haircut” applied on defaulted debt and the history of previous default. This is inconsistent with the dynamics of the efficient allocation considered here. Adding an extra dimension of asymmetric information may help in this regard. In particular, one can account for this fact by introducing private information about the type of borrower, which is only revealed by a default (and not by other actions). Although interesting, this is outside the scope of this article. To summarize, in this section, I showed that an efficient allocation can be implemented with only non-contingent defaultable debt of multiple maturities. Along the equilibrium outcome path, defaults are associated with an ex post inefficient drop in output and trade, and inversion of the yield curve, and happen only when the borrower’s value is equal to autarky and the level of debt is high relative to the maximal amount of debt the country can support. When there is no default, capital gains or losses on outstanding long-term debt replicate the state contingent returns implied by the efficient allocation. Moreover, the maturity of outstanding debt shortens as interest rate spreads increase. 6. Discussion of Implementation 6.1. History dependence and debt dilution The strategies that support the efficient allocation are history dependent. In particular, the pricing function $$\boldsymbol{q}$$ does not only depend on the stock of outstanding debt and an indicator variable that records whether the government has access to international credit markets, as it is typically considered in the literature on quantitative sovereign default. The reason why history dependence is needed is connected to the debt-dilution problem. With long-term debt, any borrower’s action that increases the likelihood of future outright default is tantamount to a (partial) default, because it imposes a capital loss to the holders of the outstanding debt. The equilibrium that implements the efficient allocation treats outright default and this more subtle partial default in a parallel fashion: The borrower is punished if he deviates from the path of plays by diluting existing debt too much, or if after a positive shock, he does not reduce his level of indebtedness. This stands in contrast with standard sovereign debt models in which only outright default is punished with a trigger to autarky (with potential re-entry). This difference has important implications for how we think about the role of long-term debt, seniority, and pari passu clauses. Chatterjee and Eyigungor (2015) and Hatchondo et al. (2016) argue that within an equilibrium of the sovereign debt game in which only outright default is punished with trigger strategies, debt dilution is a problem and seniority clauses may be desirable. Moreover, absent rollover risk, short-term debt is very desirable and governments will opt to choose a maturity composition tilted towards short-term debt. This article makes clear that such results are generated by the asymmetric treatment of outright default and dilution. In the best equilibrium of the sovereign debt game analysed here, debt dilution is necessary for the best sustainable equilibrium to be equivalent to the efficient allocation. Hence, policies that introduce seniority to reduce the ability to dilute outstanding debt may not be warranted. 6.2. Comparison with Alvarez and Jermann (2000) The implementation I propose is also applicable to other environments than the one considered here. For instance, consider an economy with lack of commitment and public information,25 where the domestic agents discount more heavily than the international interest rate, $$\beta<q$$. Such condition arises in general equilibrium with a large number of countries, as shown in Alvarez and Jermann (2000). Under such condition, the efficient allocation is dynamic, in that $$v^{\prime}(v,z_{H})\neq v^{\prime}(v,z_{L})$$ for all $$v$$ in the ergodic set26 and it can be implemented as the best sustainable outcome of the sovereign debt game presented in Section 5. Moreover, the pattern of maturity composition of debt and the spreads leading to a default are similar. Alvarez and Jermann (2000) show that it is possible to implement the efficient allocation under lack of commitment with state contingent debt and endogenous debt limits. The main advantage of my implementation is that it neatly maps into the objects considered in applied works. In particular, my implementation can be used to derive prices for defaultable bonds, and it has implications for the maturity composition of debt. Finally, it is worth noticing that while the implementation I propose works in the environment considered by Alvarez and Jermann (2000), the converse is not true. Debt limits are not enough to implement the efficient allocation with private information. This follows from the fact that the efficient allocation with private information does not satisfy $$qU^{\prime}\left(c\left(z^{t}\right)\right)\geq\beta U^{\prime}\left(c\left(z^{t},z_{t+1}\right)\right)$$ for all $$z^{t},z_{t+1}$$. That is, the borrower is not always “borrowing constrained.” This observation follows from a version of the inverse Euler equation that holds in the economy considered here. Optimality requires that when the sustainability constraint does not bind, we have $$qU^{\prime}\left(c\left(z^{t-1},z_{H}\right)\right)<\beta\mathbb{E}U^{\prime}\left(c\left(z^{t-1},z_{H},z_{t+1}\right)\right)$$. So, there is at least one state $$z_{t+1}$$ for which $$qU^{\prime}\left(c\left(z^{t-1},z_{H}\right)\right)<\beta U^{\prime}\left(c\left(z^{t-1},z_{H},z_{t+1}\right)\right);$$ that is, when current productivity is high, $$z_{H},$$ the borrower is “saving constrained.” Thus, debt limits and Arrow securities are not enough to decentralize the efficient allocation. After certain histories, a minimal asset holding requirements would be needed. 7. Final Remarks In this article, I show that key aspects of sovereign debt crises can be rationalized as part of the efficient risk-sharing arrangement between a sovereign borrower and foreign lenders in an economy with informational and commitment frictions. Along the outcome path that supports an efficient allocation, sovereign default episodes happen because of the need to provide incentives, despite being ex post inefficient. This article takes a first step towards bridging the gap between the literature on quantitative incomplete markets and the literature on optimal contracts. This article is qualitative in nature; however, it shows that one can interpret the outcome of an optimal contracting problem through the lens of a standard sovereign debt game, and derive implications for interest rates and bond holdings, which are the focus of the applied literature. A quantitative evaluation of the model is a fruitful area for future research. Such extension would also be suited to study how ex ante efficiency would be reduced by policies designed to mitigate the ex post inefficiency, such as bailout. The implementation I propose—and its implications for the optimal maturity composition of debt—is also applicable to other environments than the one considered here.27 As I mentioned, exact implementation may require very large positions, so it may be interesting to think about approximate implementation by imposing a cap on the debt positions. Moreover, it may be interesting to study the maturity composition in the best outcome of the sovereign debt game when the number of maturities available is smaller than the cardinality of the state space. Finally, while the efficient allocation can be implemented as an equilibrium outcome of the sovereign debt game, the converse is not true. There is a continuum of equilibria and generically they are not efficient. Thus, despite the fact that agents are able to achieve the efficient outcome in a market setting, regulation by a supranational authority may indeed be helpful in avoiding inefficient equilibria. Acknowledgements This is a revised version of the first chapter of my dissertation at the University of Minnesota. I am indebted to V.V. Chari, Patrick Kehoe, and Larry Jones for valuable advice. I would like to thank Michele Tertilt and three anonymous referees for very helpful suggestions. I also want to thank my discussants, Manuel Amador, Andy Atkeson, Ryan Chahrour, Pablo Kurlat, Vivian Yue, as well as Mark Aguiar, Cristina Arellano, Philip Bond, Wyatt Brooks, Erzo Luttmer, Ellen McGrattan, Chris Phelan, Ali Shourideh, Andrea Waddle, Ivan Werning, Pierre Yared, and Ariel Zetlin-Jones for their useful comments. I acknowledge the financial support of the Hutcheson Fellowship from the Economics Department of the University of Minnesota and the hospitality of the International Economics Section at Princeton University. The usual disclaimers apply. Supplementary Data Supplementary data are available at Review of Economic Studies online. Footnotes 1. The main results of the article remain valid if I consider an environment in which the incentive problem arises because of moral hazard, as in Atkeson (1991) and Tsyrennikov (2013). That is, the efficient risk-sharing arrangement can be implemented as the equilibrium outcome of the sovereign debt game with non-contingent defaultable debt of multiple maturities, and defaults and ex post inefficient outcomes happen along the equilibrium path. 2. See Rogoff (2011) for such an argument. 3. Quadrini (2004), Clementi and Hopenhayn (2006), DeMarzo and Sannikov (2006), DeMarzo and Fishman (2007), and Hopenhayn and Werning (2008) study this in the context of firm dynamics with credit frictions. 4. See also the reference in the survey by Panizza et al. (2009) and Tomz and Wright (2013). 5. Author’s calculations. See section C in the Online Appendix for details. 6. This drop can have a non-trivial impact on output. Gopinath and Neiman (2014) present a model calibrated to replicate the crisis in Argentina in 2002 and show that the decline in imports of intermediate goods can account for up to a 5-percentage-point decline in the welfare-relevant measure of productivity. 7. Under this interpretation, the link between output and default is weakened and so it may help to account for the weak relationship found in the data by Tomz and Wright (2007). 8. I further discuss this issue in the Online Appendix after Lemma B1. 9. The characterization of the efficient allocation does not depend on the two-shock assumption, and it can be extended (see a previous version of the article). This assumption will play a role in the implementation. I will discuss such role after the proof of the main proposition in Section 5. 10. Assumption 2 is satisfied if the curvature in $$U$$ and $$f$$ is low. 11. To understand the plausibility of condition (13), consider the following back-of-the-envelope calculation. If $$z_{L}$$ is a big recession in which productivity falls by say 10% (relative to productivity in good time $$z_{H}=1$$), and it is fairly unlikely in that the probability of such recession is 5% (so $$\mu_{H}=0.95$$), then we have that (13) is satisfied if $$\gamma$$ is greater than 1.6. 12. This is because the sovereign borrower is the owner of the domestic production technology that can be also operated in autarky. 13. This is consistent with the pari passu clause present in the vast majority of sovereign debt contracts. 14. See Arellano et al. (2015) for evidence of this interference. 15. In the background, as in Aguiar et al. (2009), the stand-in domestic household supplies labour inelastically and receives lump sum transfers (or taxes if negative), $$LS_{t}$$, from the government. The stand-in household’s budget constraint is $$c_{t}=w_{t}+LS_{t}$$, where $$w_{t}=F_{\ell}(m_{t},1)$$ is the competitive wage rate. (17)–(19) represent the combined budget constraints of the benevolent government and the stand-in household. 16. If the property in part (i) of Assumption 4 is not satisfied by the efficient allocation, it is still possible to implement the efficient allocation but it requires defaults for $$v\neq v_{a}$$. 17. To generate Figure 3 and 4, I consider the following functional form and parameterization: $$F\left(m,\ell\right)=\left(\alpha m^{1-1/\rho}+\left(1-\alpha\right)\ell^{1-1/\rho}\right)^{\frac{\rho}{\rho-1}}$$ with $$\rho=3$$ and $$\alpha=0.45$$; the other parameters in the model are $$\gamma=2$$, $$\beta=0.95$$, $$q=0.96$$, $$z_{H}=1,$$$$z_{L}=0.65$$, and $$\mu\left(z_{H}\right)=0.8$$. This example is representative of several simulations I perform. 18. That is the implicit constant interest rate at which the discounted value of the bond’s coupons equals its price. Define $$q_{YM,L}$$ as $$q_{L}=\frac{q_{YM,L}}{1-q_{YM,L}}$$. The consol’s yield to maturity is $$1/q_{YM,L}=\frac{q_{L}}{1+q_{L}}$$. 19. Recent works have documented that this is actually true in the data. See, for example, Zettelmeyer et al. (2013) and Asonuma et al. (2015). 20. An attractive feature of the decentralization I propose is that it extends to pure exchange economies, while the decentralization based on state contingent tariff or taxes does not. 21. The necessity of tariff is related to the necessity of capital income taxes in the implementation for the efficient allocation in an economy with lack of commitment in Kehoe and Perri (2004) and Aguiar et al. (2009). 22. For $$\alpha_{i}\in\left\{ \alpha_{1}=0,\alpha_{2},...,\alpha_{N}=1\right\}$$, define $$\bar{q}_{_{i}}$$ in a similar way as in (31):   \[ \bar{q_{i}}(v^{\prime})=\left\{ \begin{array}{cc} q\sum_{\theta}\mu(\theta)\left[1+(1-\alpha_{i})\bar{q}_{i}(v^{\prime}(v,\theta))\right] & \text{if }v^{\prime}>v_{a}\\ q\frac{\bar{R}_{i}}{1-(1-\alpha_{i})q} & \text{if }v^{\prime}=v_{a} \end{array}.\right. \] Then, if the return matrix   \[ \underset{N\times N}{\bar{Q}(v)}\equiv\left[\begin{array}{ccc} 1+q_{1}(v^{\prime}(v,z_{1})) & ... & 1+q_{N}(v^{\prime}(v,z_{1}))\\ \begin{array}{c} 1+q_{1}(v^{\prime}(v,z_{2}))\\ ... \end{array} & \begin{array}{c} ...\\ ... \end{array} & \begin{array}{c} 1+q_{N}(v^{\prime}(v,z_{2}))\\ ... \end{array}\\ 1+q_{1}(v^{\prime}(v,z_{N})) & ... & 1+q_{N}(v^{\prime}(v,z_{N})) \end{array}\right] \] is invertible, then there exists a $$\bar{b}(v)=[\bar{b}_{_{1}},...,\bar{b}_{_{N}}]^{T}$$ that solves the analogue of (32)–(33) given $$\bar{Q}$$. 23. The parameters used to generate Figure 4 are the ones listed in footnote 17. The recovery rates are $$r_{S}=r_{L}=0.6$$. 24. The borrower must buy back his outstanding long-term debt because the long-term debt I consider is a consol. If I would have instead considered a perpetuity with coupon decaying at a rate $$\alpha$$, some of the long-term debt is coming due in every period and so the net issuance of long-term debt would be $$\bar{b}_{L}\left(v'\left(v,z\right)\right)-\left(1-\alpha\right)\bar{b}_{L}\left(v\right)$$. In such a case, it is possible that even if the stock of outstanding long-term debt is declining, net issuance can be positive. 25. In this case, defaults will not be associated with ex post inefficiencies. 26. This is not a property of the efficient allocation for an economy with no private information, lack of commitment (one sided), and $$\beta=q$$. 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Published: Feb 1, 2018

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