Domestic and External Debt

Transcription

1 Domestic and External Debt Paola Di Casola Spyridon Sichlimiris May 14, 2014 PRELIMINARY DRAFT. Abstract We provide a model that links domestic and external government debt and default. The domestic economy features a scarcity of saving instruments because the government can restrict capital flows from the private sector. By issuing debt, the government can exploit its market power in the domestic asset market. Default counteracts adverse shocks (benefit) but hinders the access to cheap domestic funds (cost). The optimal behaviour prescribes more domestic borrowing in good times and more external borrowing in bad times. Default is more likely in bad times. Thus, we obtain an endogenous preference for domestic bondholders. The model can generate realistic external and domestic debt levels due to the endogenous costs associated with the loss of market power in the domestic market. Varying the magnitude of the shortage of assets, the average external to domestic debt ratio is high when the friction is severe or negligible and low when it is moderate. The predictions of the model are in line with various cross-country regularities on domestic and external debt. Key words: sovereign debt, domestic, external, default, shortage of assets JEL Codes: E21, E44, F30, F34, G15, O16 We are grateful to Tore Ellingsen, Lars Ljungqvist and seminar participants at Stockholm School of Economics, Toulouse School of Economics, Mannheim University and SUDSWEC conference 2013 for useful discussions and comments. Financial support from the Jan Wallander and Tom Hedelius Foundation is gratefully acknowledged. All remaining errors are our own. Stockholm School of Economics. Paola.DiCasola@hhs.se Stockholm School of Economics. Spyridon.Sichlimiris@hhs.se 1

2 1 Introduction There has been great progress in the understanding of sovereign debt and default in recent years. However, the link between domestic and external debt has remained largely unexplored. What determines the composition of domestic and external debt? How is the composition of debt linked to defaults? We address these and other related questions within an extension of the dynamic general equilibrium model of the kind developed by Arellano (2008). A key feature is that the domestic private sector cannot produce enough saving instruments for domestic investors. The need for saving instruments could be satiated if domestic investors had perfect access to an international frictionless financial market. The government may want to restrict capital flows 1 to exercise market power in the domestic economy. Given our assumptions, the government is the only agent that can exploit the arbitrage across markets by targeting bondholders using two debt instruments to insure against adverse shocks. Hence, domestic government debt embodies a dual role in our model: insurance and liquidity instrument. The insurance role comes from the ability to smooth out income shocks and the liquidity role from the additional saving instruments provided in the domestic economy. Importantly, the insurance role goes hand in hand with the liquidity role. In the presence of a government that cannot commit to repay the debt, the liquidity role of domestic debt can give rise to a commitment technology for the government to issue debt in the external market. In fact, default entails benefits and costs. Default counteracts adverse shocks but hinders the access to future cheap domestic funds. The former effect is prevalent in models á la Eaton and Gersovitz (1981). The latter effect constitutes an endogenous cost that originates from the liquidity role of domestic debt. Our contribution is not only to provide a model that encompasses both types of debt with endogenous default, but also to highlight the forces that link them. The optimal behaviour prescribes more domestic borrowing in good times and more external borrowing in bad times. Default is more likely in bad times. Thus, we obtain an endogenous preference for domestic bondholders. Domestic and external debt are linked through the shortage of assets. The ability of the private sector to provide stores of value determines the market power of the government as an additional liquidity provider. The domestic and external shares of average debt vary with the magnitude of this friction. When the friction is severe or negligible, countries can sustain high levels of debt, mostly external debt, with low default rates. When the friction is moderate, countries can sustain 1 This feature limits arbitrage across countries. Similar assumptions are used in Caballero and Krishnamurthy (2006) and Ventura (2012) to generate a shortage of assets. We can think of this limitation in trading as barriers induced by the government in the form of transaction costs, credit ceilings, entry barriers in banking sector, or explicit restriction on capital flows. See Kraay et al. (2005) and Abiad et al. (2008) for empirical evidence. Our model is agnostic on the factors that generate these restrictions. 2

3 lower levels of debt with a balanced composition and higher default rates. We know that governments default on either type of debt, the only difference being that domestic defaults are more difficult to detect than external ones. 2 The papers by Reinhart and Rogoff (2011) and Kohlscheen (2010) highlight that domestic and external defaults are positively correlated and often happen together. In the model selective default is never optimal because we assume the same costs for defaulting on any type of bondholder. However, the composition of debt at default can measure the endogenous preference for one type or the other of bondholders. Our simplifying assumption allows us to better identify the links between domestic and external debt and default. The decision of issuing domestic or external debt hinges on different market conditions in the two asset markets, but the risk of default links the two debt instruments. Favourable domestic market conditions affect the borrowing capacity of the government also in the external market. Our attempt is to interpret within one framework the following empirical regularities that characterize the sovereign debt and default. First, Figure 1 3 shows that the ratio of domestic debt over total debt is pro-cyclical for most of the countries. Second, Reinhart and Rogoff (2011) provide evidence to the fact that domestic defaults are less frequent than external ones and happen in more distressed periods. Third, looking at Figure 2, we can see that countries with low levels of total debt feature significant shares of domestic and external debt. On the other hand, countries with high levels of total debt seem to have mostly external debt. The relationship between total debt and its two components does not seem linear. The key ingredients of our model are: a) lack of commitment from the side of the government, b) incomplete markets, c) shortage of private assets in the domestic economy. The first two assumptions are needed in order to have a model where default arises in equilibrium. The third one is the financial friction we use to link domestic and external debt. The domestic economy is a small open endowment economy and comprises consumers and investors. Consumers face uninsured aggregate endowment shocks. A benevolent government maximizes their utility by providing insurance through borrowing, saving and defaulting. The domestic economy is characterized by a shortage of assets. This is modelled with domestic investors subject to asynchronicity between the need and availability of consumption goods. Outside this economy there are external investors who have access to a frictionless external asset market. The government can issue domestic debt to the domestic investors and external debt to the external investors. However, the government can 2 Reinhart and Rogoff (2011) mention that their data on domestic defaults are to be considered the lower bound of the real data. 3 We thank Ugo Panizza for providing us with the updated version of his dataset. See Appendix E for a description of the dataset. 3

4 Figure 1: Correlation between business cycle and ratio of domestic to total government debt. Source: own calculations on Panizza s dataset. decide not to repay the debt. In our model the government is the only strategic agent. The lack of commitment is reflected in the endogenous default probabilities. The government acts as a price maker in the domestic market and as a price taker in the external market. Therefore, the government has market power in the domestic market and prices of domestic bonds and external bonds differ. 4 However, the default risk contributes to reduce the gap between the two prices. As in standard sovereign debt models, prices for external bonds are constant for no default risk and decrease in the amount issued for positive default risk. Instead, prices for domestic bonds always decrease with the amount issued, but they decrease more slowly than prices of external bonds with positive default risk. In our model the default probability has a leverage effect in the domestic asset markets, apart from entailing a risk premium. The 4 The gap between prices of domestic and external bonds has been highlighted in the data. Du and Schreger (2013) report evidence of the failure of long-run interest rate parity between price of government bonds in emerging markets and the US in the period In particular, credit spreads in local-currency debt are lower than credit spreads in foreign-currency debt for emerging markets. The main reason is that local-currency debt features lower liquidity risk than foreign-currency debt. Although our model is about real variables, we can classify local-currency debt as domestic debt and foreign-currency debt as external debt. Corradin and Rodriguez-Moreno (2014) document a similar pricing anomaly for Euro area comparable bonds denominated in EUR and USD between 2008 and USD-denominated were cheaper than EURdenominated bonds, even after adjusting for exchange rate risk. The results of the empirical analysis suggest that the anomaly can be explained with the different conditions faced by the banks when using the bonds as collateral for liquidity operations with the ECB. 4

5 Figure 2: Average ratio of domestic, external and total government debt over GDP. Source: own calculations based on Ugo Panizza s dataset described in Panizza (2008). leverage effect comes from the fact that the default risk increases the prices of all domestic assets (government bonds and private assets) to reflect the decrease of the relative amount of safe assets in the economy. When the government debt is risky, prices of domestic and external bonds are closer. Our theoretical contribution is to show that the government chooses the ratio of domestic to total debt at the level where the unit price of domestic bonds is equal or higher than the unit price of external bonds. This implies that the government may or may not find it optimal to close the price difference between the two markets. This highlights that our model is able to capture the empirical finding that similar debt instruments can have different prices across markets. The major quantitative results are the following. First, we find that the government borrows more domestically in good times and more externally in bad times. As in the standard sovereign debt models in the literature, the incentives to default are stronger in bad times. Our model predicts more propensity to default for higher external-to-total debt 5

6 ratios. Given that selective default is never optimal in our model, this behaviour can be interpreted as stemming from the endogenous preference of the government for domestic bondholders. These results are in line with the first two empirical regularities mentioned above. Second, the model can generate realistic external and domestic debt levels due to the endogenous costs associated with the loss of market power in the domestic market. Domestic and external debt instruments are used by the government to insure against endowment shocks. Moreover, the domestic debt is a saving instrument for domestic investors. Since the government can restrict access to international markets to the domestic investors, it can exercise market power by providing an additional liquidity instrument. That gives the government the ability to issue domestic debt at low interest rates. Favourable domestic market conditions give rise to high external debt levels. Third, we find a hump-shaped relationship between domestic debt and the magnitude of the friction. Non-monotonic is also the relationship between the level of external debt and the magnitude of the friction and takes a U shape. The market power of the government is higher, the scarcer the available private assets in the domestic economy. This implies that the price elasticity of domestic government bonds is very high. The government can issue more external than domestic debt because of the price difference that arises between the two markets. As the shortage of assets becomes less severe, the market power decreases and the elasticity of domestic prices as well. This leads to an increase in the domestic-to-total debt ratio. External debt decreases because the price difference arising between the two markets is smaller than before. When domestic private assets are even more abundant, the domestic debt decreases and the external debt increases. The price of domestic bonds is close to the price of external bonds. These non-monotonic relationships resemble patterns in the data, when the empirical analysis is conducted with a proxy for the level of private assets in the economy. Our model predicts that very high levels of external government debt are associated with low levels of domestic government debt. The relationship between total debt and its two components resembles the one in the data. Fourth, the paper shows that default rates are low when domestic private assets are scarce. The endogenous default costs arising from the market power are high. On the contrary, the government finds it optimal to default more often when the shortage of domestic assets is moderate. In this case the government s market power is reduced, implying a lower utility from honouring the debt contract. Default incentives are low when domestic private assets are abundant because the government can exploit the price differential between the two markets. The fact that countries default at low levels of debt is considered a puzzle in the literature on international debt, as mentioned in Reinhart and Rogoff (2011). 6

7 We show that default rates do not depend only on the business cycle and the level of debt to repay, but also on the availability of stores of value in the domestic economy. Fifth, the welfare of the government is higher when the shortage of assets is more severe. This coincides with the case where the government has more sizeable market power and can issue high levels of external debt with low default risk. The welfare is higher also when the shortage of assets is negligible and the government can exploit the price differential between domestic and external market. Sixth, this paper shows that both types of government debt can be sustainable in equilibrium when domestic private assets are scarce, even without default punishment, i.e. when the country is only allowed to save during the default period. In this case domestic market prices adjust to a level where borrowing and repaying is more appealing than defaulting and saving. This feature is due to the endogenous default costs. Our novel mechanism attributes an indistinguishable insurance and liquidity role to the domestic debt. This dual role gives rise to high prices of domestic bonds which make saving unattractive. This leads to sustainable external debt levels even without exogenous default punishment. Importantly, our requirement for the existence of both types of debt is the shortage of assets in the domestic economy. The paper continues as follows. Section 2 reviews the literature and points out the contributions of the paper. Section 3 describes the model and section 4 the equilibrium concept used to solve it. Section 5 present the quantitative results. Section 6 discusses the results and section 7 the empirical evidence. Section 8 concludes. 2 Literature review Many papers have addressed the issue of shortage of assets and the role of government debt. The seminal paper by Woodford (1990) shows that government debt can benefit the economy when agents are liquidity constrained, in the sense that they receive investment opportunities in alternating periods and lack instruments to store endowments. Therefore, government bonds carry a liquidity premium. Holmström and Tirole (1998) use a similar framework but encompass the former paper as a special case. In fact, they allow the private market to produce savings instruments based on their expected returns, but with a limit due to ex-post agency problems. In case of aggregate uncertainty these private assets are not enough to solve the liquidity problem and government debt can be sold at a liquidity premium. Holmström and Tirole (2011) extend the previous result to an open economy. 5 In a similar framework of liquidity shortage, Caballero and Krishnamurthy 5 Holmström and Tirole (2011) explain that even if a country can trade in a global financial market, there can still exist a shortage of assets in that country because of the limited amount of international collateral, i.e. 7

8 (2006) and Farhi and Tirole (2012) obtain conditions for the existence of a bubble and argue that government debt could replace the bubble. Gourinchas and Jeanne (2012) study the general equilibrium implications of default risk in a model of shortage of safe assets. They highlight the leverage effect due to the presence of default risk, but the default is not modelled in their paper. Our model goes one step further, by including the feedback effect of the leverage possibility on the default decision. The papers in the literature that are closer to ours are Brutti (2011), Mengus (2013) and Gennaioli et al. (2014). These papers assign a liquidity role to government debt and introduce domestic and external debt with a risk of default. However, the focus of these papers is the distinction between domestic and external default, assuming that governments prefer external defaults but cannot discriminate across creditors. The choice of issuing domestic debt or external debt is not modelled in Brutti (2011) and Gennaioli et al. (2014). Mengus (2013) studies the issuance of the two types of debt, but the two-period horizon does not permit to analyse the implication for the continuation values and the feedback effects between new debt and default incentives. With respect to this literature, we use the shortage of assets inside the domestic economy to model explicitly how market forces drive the government s choices between domestic and external debt and the decision to default or not. The infinite-horizon setting allows us to take into account the implications of government s choices for the future. To the best of our knowledge, this is the first paper to model endogenously the choice and the interaction between domestic and external debt with the possibility of default in a general equilibrium framework. There is a large literature on sovereign debt, started with the seminal paper by Eaton and Gersovitz (1981), but it is focused on external debt. The first papers to asses the quantitative findings of sovereign debt models are Aguiar and Gopinath (2006) and Arellano (2008). Extensions of the basic models have been considered, 6 but it is difficult to reproduce levels of external debt-to-gdp comparable with the data. Our model builds on the basic framework in the sovereign debt literature but includes a domestic economy. Thanks to the assumption of the shortage of private assets in the domestic economy, we uncover the interaction between domestic and external debt and we claims backed up by tradable goods. Even in this environment government bonds may be needed. 6 Other papers have expanded the set of possibilities for the government, allowing for trading with international financial institutions (Boz, 2011), the issuance of long-term bonds (Hatchondo and Martinez (2009) and Chatterjee and Eyigungor (2012)) or the renegotiation of debt repayment and re-entrance into the financial market after default (Benjamin and Wright (2009), Yue (2010) and D Erasmo (2011)). Moreover, the political instability of the government (Cuadra and Sapriza (2008) and Hatchondo et al. (2009)), the risk-aversion of the external investors and their fear for model uncertainty have been taken into account (Lizarazo (2013) and Pouzo and Presno (2012)). Mendoza and Yue (2012) merge the sovereign debt literature with the RBC literature while Durdu et al. (2013) introduce news shocks. Only Benjamin and Wright (2009) have managed to produce levels of external debt-to-gdp comparable with the data when the model is not calibrated to match the average debt-to-gdp ratio. For a review of this literature, see Wright (2013), Aguiar and Amador (2013) and Stähler (2013). 8

9 are able to reproduce "realistic" levels of debt. Finally, our paper is related to the literature on the existence of debt. Bulow and Rogoff (1989) show that debt contracts of small open economies cannot be sustained by reputation for repayment. As long as the economy has access to "cash-in-advance" insurance contracts, default is always optimal. Therefore, borrowing is possible only if creditors can punish the defaulter. Many papers have challenged this result, 7 but the closest ones to our framework are Hellwig and Lorenzoni (2009) and Mengus (2014). The first paper shows that Bulow and Rogoff (1989) s result does not hold when the interest rate is not fixed. In a general equilibrium framework with multilateral lack of commitment, the interest rate adjusts to ensure repayment. When the interest rate is low enough that a bubble in international markets can emerge, external government debt can exist without punishment for default. Mengus (2014) shows that the existence of domestic unbacked government debt allows the government to sustain external debt, due to the endogenous internal costs of default. Our paper shows that external and domestic debts can be sustained without punishment if the government has access to cheap domestic funding that makes defaulting too costly. 3 Model The economy consists of a small open endowment economy and the rest of the world. The time of the model is discrete and there is no population growth. The small open economy is populated by domestic consumers, domestic investors and a government. The rest of the world is populated by external investors. We consider an incomplete markets model where the government is benevolent and represents only the domestic consumers. The government provides insurance to the domestic consumers against stochastic endowment shocks by borrowing/lending to investors through one-period bonds, but cannot commit to repay. Therefore, default arises in equilibrium in our model. The domestic asset market is characterized by a limited supply of private assets. The external asset market is frictionless. The government restrict capital outflows between domestic and external investors and provides the only intermediation, by issuing domestic bonds to domestic investors and external bonds to external investors. This assumption reflects the fact that the government can target bondholders when issuing bonds. 8 In the domestic market the government acts as price-maker by influencing the total supply of assets, whereas in the external market the government is a price-taker. We proceed with the description of the agents. 7 See Panizza et al. (2009) for a literature review on this topic. 8 For example, some countries issue bonds that are non-transferable or difficult to transfer. 9

10 3.1 Domestic Consumers There is a mass λ of identical, infinitely lived consumers. They are risk-averse and maximize their utility from consumption. Every period they receive an endowment that is stochastic and not storable. We consider the case where the endowment follows a Markov process. 9 The domestic consumers do not have access to any financial market. 10 Therefore, the only instrument the consumers can use to insure against the endowment risk is a transfer from the government. The representative consumer solves the problem max E 0 {c t } β t u(c t ) t=0 t=0 (1a) subject to c t = y t + τ t, (1b) where 0 < β < 1 is the discount factor, c t is consumption, y t is the endowment, τ t are the government transfers and u( ) is increasing and strictly concave. 3.2 Government The government is assumed to be benevolent and therefore maximizes the utility of the domestic consumers. The government does not take into account the utility of the domestic investors. As the government acts as an insurance provider in our model, it cares only about the agents who are risk averse, face endowment shocks and have no instrument available to insure against them. It has access to the domestic and the external financial market to issue one-period domestic and external bonds. The government can ex-ante identify the bondholders, but cannot commit to repay the debt back to any of them. We model the lack of commitment through the strategic default choice. At any time t the government observes and taxes all the endowment of the consumers Y t and decides whether to repay the debt or not. In case of repayment it can issue one-period bonds; in case of no repayment (default) it may suffer from an output loss and it is in financial autarky for a temporary period, 11 independently of the identity of the bondholders. Due to this assumption, partial 9 The Markov process for the endowment will allow us to rewrite the problem in a recursive form later on. 10 A rationale for this assumption is that their transaction costs are very high, so borrowing or lending is prohibitive for them. 11 We write the problem with the temporary exclusion from the market to make it more comparable with the standard sovereign debt models. In section 6.1 the role of the punishment structure is discussed more extensively. We will show that the possibility we give to the government to influence the total supply of assets in the domestic economy reduces the sensitivity of the model to the punishment structure. This is in stark contrast with the vast majority of the papers in the literature on sovereign debt and default that rely heavily 10

11 or selective default is never an optimal choice. As mentioned at the beginning, we want to identify the market conditions driving the choice of domestic and external issuance and therefore abstract from exogenous preferences for one or the other type of investors. 12 The problem of the government is max E 0 {C t, B β t u(c t ) t+1 D, BE t+1, d t} t=0 t=0 (2a) subject to C t q D t B D t+1 qe t B E t+1 = Y t B D t B E t if d t = 0, (2b) and C t = Y de f t if d t = 1, (2c) where Bt E and Bt D are external and domestic borrowing to repay in the current period and qt EBE t+1, qd t BD t+1 are the values of the current external and domestic borrowing. Positive values of Bt E and Bt D indicate borrowing, while negative values indicate saving. The variable d t {0, 1} represents the choice of default. The possibility of default gives rise to an endogenous borrowing constraint. The government saves by buying domestic private assets through the domestic investors or external risk-free assets through the external investors. Y t and C 13 t represent, respectively, the aggregate endowment and consumption of the domestic consumers. The difference T t = qt DBD t+1 + qe t BE t+1 BD t Bt E represents the total transfer from the government to the consumers. During default the government is assumed to lose a fraction of the endowment, hence Y de f t =(1 κ)y t. 14 on the punishment to produce quantitative results that replicate the data. 12 This simplifying assumption allows us to identify the main mechanism driving the issuance of domestic and external debt. Even if the exogenous costs from defaulting were different, our qualitative results would still hold. In fact, the model features an endogenous default cost because the government can affect the total supply of assets in the domestic economy and is the only intermediary between domestic and external markets. The two types of debt are linked and the two defaults would be linked as well, even if only partially. 13 Y t = λ 0 y tdi = λy t and C t = λ 0 c tdi = λc t. 14 Following Arellano (2008), the sovereign debt literature assumes an asymmetric cost during default. The relative loss in endowment declines with the endowment and is zero for low enough levels. This asymmetry in default cost is necessary to generate high default probability and high volatility of spreads. Without the asymmetric default cost, realistic default rates can be generated only by assuming a very impatient borrower. Mendoza and Yue (2012) are able to generate an endogenous asymmetric loss of output in their model because they merge the sovereign debt literature with the RBC literature. For the moment we do not assume this ad-hoc exogenous asymmetry in default cost in order to better analyse the mechanism of our model. In section 6.1 we discuss the effect of the punishment structure on our results. 11

12 3.3 Domestic Investors We intend to create a demand for stores of value in the domestic economy. We do it in the most convenient way. There is a mass 1 λ of risk-neutral domestic investors who live for two periods: in the first period they are young, in the second they are old. They receive a non-storable endowment when young and consume only when old. The endowment is stochastic and perfectly correlated with the endowment of the domestic consumers. Domestic investors have access to two types of assets: domestic private assets and one-period government bonds (in case the government issues them). Domestic private assets pay one unit of good in the following period. In case the government has not defaulted at time t, the problem of the representative domestic investor is subject to max E t c D,t c D,t t+1, l t+1, bt+1 D t+1 (3a) q l tl t+1 + q D t b D t+1 = yd t, (3b) c D,t t+1 = bd t+1 + l t+1 if d t+1 = 0, (3c) c D,t t+1 = l t+1 if d t+1 = 1, (3d) where the q l t l t+1 and qt DbD t+1 represent the investment of the endowment in the two assets. If the government had defaulted at time t, the value of bt+1 D is set to zero. 3.4 External Investors We want the external investors to act in a frictionless environment. Therefore, we model them as identical, infinitely lived, risk-neutral agents. We maintain the risk neutrality assumption of the original paper by Eaton and Gersovitz (1981) that external investors can diversify all their individual risk. They have access to a competitive international financial market where they can trade as much as needed of external risk-free assets. These assets pay one unit of good in the following period. The external investors can also trade external 12

13 government bonds. They can observe the endowment of the small open economy when it is realized. The representative external investor solves the problem 15 ( max E 0 {ct E, z t+1, b β E) t c E t t+1 E } t=0 t=0 (4a) subject to c E t + q z t z t+1 + q E t b E t+1 = z t + b E t if d t = 0, (4b) c E t + q z t z t+1 = z t if d t = 1, (4c) where the q z t z t+1 and qt EbE t+1 represent the investment in the two assets. 3.5 Asset Markets The domestic and the external asset markets are competitive. The main difference is that domestic investors have a demand for stores of value and the supply of private assets in the domestic economy is limited to a constant amount L. 16 These features give rise to an endogenous interest rate in the domestic market. In the external market the supply of risk-free assets is as much as needed ( unlimited ). The interest rate in the external market is exogenous to the small open economy. 15 We do not need any assumption about the impatience of the domestic consumers relatively to the impatience of the external investors to get government s borrowing in equilibrium. However, in the quantitative analysis we will assume that β β E. 16 The parameter L can be interpreted as a measure of financial development of the economy. A potential micro-foundation for L is described in Appendix D. 13

14 3.6 Timing t t+1 endowment Government s default decision Domestic Investors trade in domestic market, External Investors trade in external market, Government issues bonds if not default consumption The government and the young domestic investors receive their endowments. The external investors observe them. Afterwards, the government decides whether to default or not on the amount borrowed in the previous period. If the government defaults on its debt, it is in autarky for a temporary period and loses a fraction of its endowment. Asset markets open. Domestic young investors can trade domestic private assets and external investors can trade external risk-free assets. On the other hand, if the government does not default, it has access to the assets market and chooses the amount of bonds to issue in the domestic and the external market. When asset markets open, domestic investors can trade domestic government bonds and domestic private assets. External investors can trade external government bonds and external riskfree assets. Finally, agents consume. 4 Recursive equilibrium We consider the recursive competitive equilibrium. We first define the problem in a recursive form, then we define the equilibrium. Finally, we discuss the solution. 4.1 Problem The government s problem contains three state variables: Y, B E, B D. If the government does not default, it can trade with domestic and external investors. If the government defaults, the punishment is independent of the amount not repaid and the identity of the bondholders. Therefore, in equilibrium the government will default on the total debt or 14

15 not default at all. This implies that at the beginning of the period the only variable that matters, apart from the endowment of the domestic consumers, is the total amount of debt to repay. Hence, we notice that we can simplify the recursive formulation of the problem and use only one state variable representing the total debt maturing in the current period, no matter to whom it has to be paid. 17 The two state variables B E, B D can be replaced by one, B = B E + B D. The problem is greatly simplified because the value functions for the government contain only two state variables: Y and B. In this way we have managed to extend the standard sovereign debt model to account for the presence of domestic and external debt, without increasing the state space of the recursive problem. The value of the option V o can be written V o (Y, B) =max{v de f (Y), V c (Y, B)}. (5) The government decides whether to default or repay and trade in the asset markets with domestic and external investors. The value of defaulting V de f is [ ] V de f (Y) =u((1 κ)y)+βe θv o (Y,0)+(1 θ)v de f (Y ). (6) In case of default the government is excluded from the market for the current period and loses a portion of its endowment. From the next period it can re-enter the market with probability θ without repaying any of the past debt. 18 The assumption that the punishment for default is independent from the type of debt is the reason why we have only one value function of default. The continuation value of trading with the domestic and external investors V c is subject to V c (Y, B) = max B D,B E {u(y B + q D (B, B D, Y)B D + q E (B, Y)B E )+βev o (Y, B )} B = B D + B E. The government repays the total debt, issue new domestic and external bonds and gets the continuation value of the option from the next period. We can rewrite the value V c 17 This simplification implies that if the government at any period of time saves in one market and borrows in the other market, what matters for the default decision is the net debt. There can be default events where the government does not repay the debt owed in one market and in this way loses the right to its savings in the other market. Our assumption is that by damaging the credit relationship in one market, the government manages its credit relationship also in the other market, as in Cole and Kehoe (1998). 18 The use of a random variable for the re-entrance in the market allows us to write the value of default in a recursive way. This formulation is usual in standard sovereign debt models. 15

16 in a more compact form, by substituting the constraint B E = B B D into the objective function. V c (Y, B) =max B,B D {u(y B + q D (B, B D, Y)B D + q E (B, Y)(B B D )) + βev o (Y, B )}. (7) When the government has not defaulted, the representative domestic investor solves the problem subject to V D (Y, B, B D )=max{δ(b, Y)l +(1 δ(b, Y))(l + b D )} (8a) b D, l y D = q D (B, B D, Y)b D + q l (B, B D, Y)l, (8b) where δ(b D, Y) is the probability that the government chooses default in the following period. The young domestic investors choose the optimal allocation of their endowment in domestic government bonds, b D, and domestic private assets, l. The old domestic investors consume. If the government had defaulted in the previous period, the value of b D is set to zero. Now we turn to the problem of the representative external investor. The continuation value from trading with a non-defaulting government is { V End (Y, b E, z, B )=max (z + b E q E (B, Y)b E q z z ) b E, z + β E (1 δ(b, Y))EV End (Y, b E, z, B (Y, B )) } + β E δ(b, Y)EV Ed (Y, z ), (9) where δ(b D, Y) is the probability that the government chooses to default in the following period. The external investors decide the amount of external government bonds, b E, and external risk-free assets, z, to purchase. If the government chooses not to default in the next period, they get the continuation value from trading with a non-defaulting government; otherwise, they get the continuation value from not trading with the government because of the default event. If the government defaults, the continuation value for the representative external investor is { [ ]} V Ed (Y, z) =max (z q z z )+β E E θv End (Y,0,z, B (Y,0)) + (1 θ)v Ed (Y, z ). z (10) 16

17 The investors only trade external risk-free assets in the current period. In the following period they can trade again with the government with probability θ. Following Chatterjee et al. (2007), we define two sets which characterize the government s optimal decision to repay or to default. The repayment set is the set of values of current endowment such that, given a value of total debt B, the government prefers repayment to default: Rep(B) ={Y Ψ : V c (Y, B) V de f (Y)}. (11) Its complement, the default set, is the set of values of current endowment such that, given a value of total debt B, the government prefers default to repayment: D(B) ={Y Ψ : V de f (Y) > V c (Y, B)}. (12) These sets depend only on the total amount of debt to repay at the beginning of the period because we have simplified the problem and reduced the number of state variables. The identity of the bondholders does not affect the choice of default and therefore the default and repayment sets do not depend on B D and B E. Now we can define the endogenous default probability as the probability that, given the current endowment and amount of bonds issued, the next period s endowment belongs to the default set δ(b, Y) = D(B ) f (Y, Y)dY. (13) Even if the government can discriminate ex-ante between the two types of debt, selective default is not optimal because the cost of defaulting is the same for any type and amount of debt. Ex-post discrimination is not optimal. This implies a unique default probability that depends only on the aggregate endowment of the consumers and the total debt to be repaid. In this way we can better isolate the interaction between external and domestic debt as a result of different market conditions and not different default incentives. 4.2 Definition of Equilibrium A recursive competitive equilibrium is an initial condition B 0, pricing functions q l, q D,q E, q z, sets of value functions V c, V de f, V o for the government, V D, V End, V Ed for the domestic and external investors respectively; policy functions B D, B for the government, b D, b E, l, z for the investors; the repayment set Rep(B) and default set D(B) such that: 17

18 (a) Given the price functions q D and q E, B D, B, the repayment set Rep(B) and the default set D(B) satisfy the government s maximization problem (5) -(7). (b) Given the price functions q D and q l and the government s policy functions, b D l solve the maximization problem of the representative domestic investor (8a) -(8b). and (c) Given the price functions q E and q z and the government s policy functions, b E z solve the maximization problem of the representative external investor (9) -(10). and (d) Bond markets clear, that is 1 λ 2 0 b D di = B D in case of domestic borrowing and 1 0 be di = (B B D ) in case of external borrowing. Domestic asset markets clear, that is 1 λ 2 0 l di = L. 4.3 Solution In this model the government acts first. In case it does not default, it has to decide the amount of bonds to issue for the next period. The price of bonds depends on the amount borrowed and the government internalizes the effect of its choice. The price is taken as given by the investors. We start solving the problem of the external investors, expressed in (9) -(10). After using the Benveniste-Scheinkman Theorem, the first order condition for b E reads: q E (B, Y) =β E (1 δ(b, Y)), while the first order condition for z reads: q z = β E. (14) Here we can see the role of the assumption of risk-neutrality: the price of the risky asset - external government bond - is such that the expected profits from its purchase are equal to the profits from the purchase of the external risk-free asset. We can rewrite the price of risk-free assets z in terms of the interest rate they bear, r z, so that the first order conditions for z is r z = βe. (15) 18

19 Using the expression above, we can derive the price of the external government bonds only as function of the external risk-free interest rate and the default probability: q E (B, Y) = 1 δ(b, Y) 1 + r z. (16) The probability of default δ(b, Y) defines the risk premium that the government pays to the investors. The higher is the default probability, the lower is the price at which the government is able to sell government bonds to the investors. At equilibrium the quantity demanded and the quantity offered of government bonds are equalized, so that 1 0 be di =(B B D ). Now we move to the problem of the domestic investors, expressed in (8a) -(8b). We can use the no-arbitrage condition to derive the price of domestic government bonds in term of the price of the domestic private asset l: q D (B, B D, Y) =(1 δ(b, Y)) q l (B, B D, Y). As noted before, the price of domestic private assets is not constant, but depends on the total amount of bonds issued by the government, the amount of domestic bonds and the aggregate endowment of the consumers. Also in this case the expected return from the risky asset is equal to the return on the private asset. At equilibrium markets clear, so 1 λ 2 0 b D di = B D and 1 λ 2 0 l di = L. We can use the aggregate version of the budget constraint of the young domestic investors (8b) to get an expression for the price of domestic private assets: 19 Y D =(1 δ(b, Y)) q l (B, B D, Y)B D + q l (B, B D, Y)L, that becomes q l (B, B D, Y) = Y D (1 δ(b, Y))B D + L. (17) We can introduce a counter-factual price for the domestic economy. It is the price of domestic private assets if the government was not to issue any type of bonds and therefore depends only on the aggregate endowment shock to the domestic economy: q l (Y) = YD L. (18) 19 Notice that Y D represents the aggregate endowment of the domestic investors, i.e. 1 λ 2 0 y D di = 1 λ It is perfectly correlated with consumers endowment Y. 2 yd. 19

20 The amount of domestic private assets defines their price in case the government does not issue bonds. The larger the availability of private assets, the lower is their the price. When L < Y D, the shortage of assets is significant, when L Y D the shortage of assets is negligible. In particular, we will define the friction as "severe" when L is below Y D and close to 0 and "moderate" when it is below Y D but close to it. It is important to compare 18 and 14 because q l with q z are both independent from government s decisions. The price differential between the two markets is larger when the friction is severe or negligible. We can rewrite q l as a function of q l : q l (B, B D, Y) = q l (Y). (19) (1 δ(b,y))b D L + 1 The new formulation of q l shows how the government affects the price of domestic private assets. Domestic government bonds increase the supply of assets in the asset market. More precisely, the part of bonds that yield a return with certainty decreases the price of domestic private assets below the counter-factual q l. This effect is stronger, the larger is the fraction of bonds relatively to the total assets in the domestic economy. We can rewrite the price of domestic bonds q D (B, B D, Y) = (1 δ(b, Y))q l (Y). (20) (1 δ(b,y))b D L + 1 The relationship between q D and q l is similar to the relationship between q E and q z. The prices of domestic government bonds (17) and domestic private assets (20) coincide as long as the default probability is not strictly positive. When the default probability becomes strictly positive, the price of government bonds is lower than the price of domestic private assets because of the risk premium. However, comparing (16) and (20) we notice that the default probability affects the price of domestic government bonds through another channel, an additional general equilibrium effect. The probability of default reduces the supply of government bonds (hence the total supply of assets the economy) yielding the return with certainty. This implies higher prices for domestic bonds (the price of private assets increases as well in the presence of positive default probability). It is worth mentioning the role of the available private assets in the domestic economy. Formulas 19 and 20 show that the parameter L affects the price of domestic private assets and domestic bonds. First, for given values of B and B D both prices are decreasing in the value of L. This happens through the decrease of q l and the decrease of share of bonds relatively to the total assets in the domestic economy. The rate of decrease in prices is 20

21 higher, the higher the available private assets. Therefore, given Y, a unit domestic bond has a higher price when the shortage of assets is severe than when the shortage of assets is not severe. A second important effect is on the elasticity of both prices to the amount of debt issued by the government. The more the availability of domestic private assets, the lower is the elasticity of domestic asset prices to domestic and total debt issued. When the shortage of assets is severe, the price of domestic bonds decreases rapidly with the amount of domestic and total debt issued. This represents the government s market power, that is higher for lower amount of private assets. The market power gives rise to an endogenous default cost that is new to the standard sovereign debt models. This cost links domestic and external debt. Even if the exogenous costs from domestic and external defaults were different, the probabilities of defaulting on either type of debt would still be linked through this endogenous channel. Therefore, our model can be interpreted as the limiting case in which the exogenous costs of the two types of default are the same, but its qualitative results would still hold if they were different. 20. We expect that the government will choose to issue more domestic to total debt when the first effect dominates and less domestic to total debt when the second effect dominates. 21 In section 5 we investigate the role of L quantitatively. Having described the role of the default probability on the price of government bonds, we can define the highest level of borrowing at which the default premium does not play a role. The no-default-risk borrowing limit is B MIN = sup{b : D(B) = }. (21) This is the highest level of debt the government can attain before the default probability is strictly positive. Below this level the government bonds behave as the other assets. There exists a level of borrowing where the default event is an almost-sure event and therefore the price of government bonds is reduced to zero. The no-default borrowing limit for the government is 22 B MAX = inf{b : D(B) =Ψ}. (22) 20 For an intuition on how the quantitative results would change in case that selective default could be optimal, refer to Appendix C We solve the model for the case where only one type of debt can be issued at any time and therefore the default probability is referred only to one type of debt. 21 In Appendix B we solve the model only for the case of domestic debt, i.e. as if the domestic economy was a closed economy. In that framework we can better analyse the effect of L on B D because the domestic debt coincides with the total debt. 22 We can define the no-default-borrowing limit as in Zhang (1997) because our model satisfies the conditions of Proposition 1 in his paper. 21

22 This is the highest level of debt the government can attain before the probability of default is one. These limits define the set of values of total borrowing that the government can issue only by paying a risk premium to the investors. These two limits are particularly useful when we discuss the debt Laffer curve later in our analysis. Finally, we can define the lower bound for the government s domestic issuance. This limit arises from the constraint on the consumption of domestic investors. We define this limit: B D = L. (23) We can turn to the main theoretical results. The next proposition is an extension to our model of the correspective proposition proved in the literature. 23 Proposition 1. For any given Y, the default set for total past debt B 1 is a subset of the default set for total past debt B 2, where B 2 B 1. PROOF: See Appendix A. Regardless of the composition of domestic and external debt, the value function of repaying the debt is decreasing in the amount to repay. On the other hand, the value of default is independent from the amount not repaid. If the government prefers default to repaying some specific amount, it will prefer default to repaying any amount greater or equal than that. This ensures the monotonicity of the value of the option and the value of remaining in the contract with respect to the amount of the debt to repay. The significance of the proposition is that the government always faces a trade-off when issuing domestic and external debt. The additional utility from issuing one more unit of bonds today is compared to the additional expected discounted loss in the continuation utility. 24 Compared to the great majority of the models in the literature, our model allows the government to trade also in the domestic market. The choice of domestic and external debt for the government reflects the market conditions faced in each market. We obtain a monotonic relationship between domestic to total debt ratio and prices of government bonds. Proposition 2. Assume that B is the optimal total amount of debt given (B, Y). If the government s issuance of domestic debt is equal to B 2 D, it implies that qd (B, B 1 D, Y) > qe (B, Y), where B 1 D [0, B2 D). 23 For similar propositions refer to Eaton and Gersovitz (1981), Arellano (2008) and Chatterjee et al. (2007). 24 This effect cannot be seen with the use of derivatives, because our maximization problem is over a nonconvex constraint set. 22

23 PROOF: See Appendix A. The proposition indicates that the government would never optimally choose the domestic to total debt ratio such that the unit price of domestic bonds is lower than the unit price of external bonds. Depending on the marginal effect of the issuance on domestic bond prices, the government may find it optimal to keep a positive wedge between domestic and external prices or it may find it optimal to drive them to be equal. Given our assumption that the penalty for default is the same, no matter the type of debt not repaid, the government chooses the ratio of domestic to total debt solely on the basis of the different level and elasticity of prices in domestic and external bond markets. This shows that the government has optimally decided to discriminate between domestic and external bondholders The case of i.i.d. shocks Our aim is to show that the shortage of assets is the key to link domestic and external government debt. In order to better describe the mechanism driving the equilibrium levels of debt and the default choice, we now make some simplifying assumptions. We assume i.i.d. shocks to the endowment of the consumers, permanent autarky in case of default (θ = 0), no default output loss (κ = 0) and constant endowment for domestic investors (Y D = Y D ). Due to the assumption of i.i.d. shocks, the distribution of the endowment shock for the next period does not depend on the current endowment shock. Therefore the default probability depends only on B and not on Y. The same holds for q E : q E (B )= 1 δ(b ) 1 + r z. (24) However, prices in the domestic market would still depend on Y through Y D. Once we assume a constant endowment for domestic investors, the counter-factual price of the domestic economy is fixed: q l = YD L. (25) Hence, prices in the domestic economy depend only on B and B D : q l (B, B D )= q l, (26) (1 δ(b ))B D L

24 q D (B, B D )= (1 δ(b ))q l. (27) (1 δ(b ))B D L + 1 The default probability decreases the price of domestic and external government bonds, but the effect on the former ones is reduced. In fact, the default probability plays a dual role in the domestic market. On one hand, it represents the risk premium that the government has to pay to domestic investors with respect to the price of domestic private assets. This additional return reflects the fact that government bonds carry a default risk and decreases the price. On the other hand, the default probability decreases the amount of assets yielding a return with certainty and therefore it increases the price of domestic private assets and domestic bonds. This second effect is the endogenous leverage arising from the risky nature of government bonds and the shortage of stores of value. Government s decisions feed back into the price of all assets. The leverage effect is not present in the price of external government bonds, because the external asset market is not characterized by a shortage of stores of value. The next two propositions are extensions of propositions proved in the aforementioned literature. Proposition 3. If default is optimal for some debt level B, i.e. D(B) =, there is no contract of external or domestic debt available such that the government can roll over the existing debt. PROOF: See Appendix A. The intuition of the proposition follows from the fact that the model yields default in equilibrium. The default set is not empty when default is an optimal choice for some levels of endowment Y. Given our assumptions, the default probability depends only on the amount borrowed and not on the endowment level. If the government could roll over the debt for some B where the default set is not empty, it means that the government would optimally choose to roll over and not to default. This would be possible for any endowment, thereby contradicting the assumption that default is optimal for some B and some Y. Here we can see how the model differs from Cole and Kehoe (2000) and the related models that have self-fulfilling types of equilibria, where the government issues new debt to external investors before deciding whether to repay the past debt or to default. The lag between the issuance of new debt and the default decision gives rise to multiple equilibria. Some of these equilibria feature roll over of debt. 24

25 By using Proposition 3 we can show the following: Proposition 4. Given total debt B, if the government finds it optimal to default with Y 2, it finds it optimal with Y 1, where Y 1 Y 2. PROOF: See Appendix A. The proposition illustrates the monotonic relationship between endowment shocks and default incentives. The lower the endowment, the higher the marginal utility of consumption. The government has two ways of smoothing consumption: defaulting or issuing new debt (after repaying the past debt). Defaulting is more appealing when roll over of debt is not possible. The Proposition shows that for low levels of endowment the benefits of risk sharing using government debt are outweighed by the benefits of risk sharing using the default option. Proposition 1, 3 and 4 show that default incentives in our model work in a similar way as in the standard sovereign debt model. The insurance benefits of defaulting outweigh the insurance benefits of borrowing for low endowments and high debt to repay. However, our model adds a new channel through which default may be unattractive. When defaulting, the government loses the benefits of the market power. Proposition 2 shows how the government exploits the different market conditions. The market power is more sizeable when the shortage of assets is severe. However, the insurance motive is always stronger. Therefore, our model can produce high levels of debt even though default is highly valuable for insurance purposes. We can now introduce the endogenous "Laffer curves" of the model. We start with the case of the external market and the curve is the same as in the benchmark model in Arellano (2008). 25 Figure 3 represents the increase in current consumption due to external borrowing on the y-axis and the amount of external borrowing on the x-axis. The value B is the value of external borrowing that maximizes the current utility. The vertical red lines define the risky borrowing region, that is the region where there is borrowing with positive default probabilities. We know that in equilibrium the government will never choose a level of borrowing above B because it can attain the same level of consumption for a lower level of borrowing. In fact, here we are not representing the continuation utility from choosing B E, but we know that it decreases in B E. Until B MIN the increase in current consumption is linear in B E because the probability of default is zero. The marginal gain from borrowing 25 There is a sign difference between our paper and Arellano (2008). Positive values of B indicate borrowing instead of saving. 25

26 q E (B )B E B MIN B* B MAX B E Figure 3: The Laffer curve for the external market 1 is positive and constant: each additional unit of borrowing adds 1+r z unit of consumption. After that point the curve becomes concave, due to the risk premium to be paid to external investors. In the risky borrowing region the marginal gain from additional borrowing is positive but decreasing because the default risk increases. After B the marginal gain from borrowing becomes negative: the risk premium more than offsets the additional utility of borrowing one more unit. Figure 4 represents the Laffer curve for the domestic market. Differently from before, the curve is never linear. Until B MIN the curve is concave even though the probability of default is zero. In fact, domestic market prices increase in B D because the government increases the supply of assets in the economy. This determines the positive but decreasing marginal gain from borrowing. In the risky borrowing region a positive probability of default exists. As we mentioned above, the probability of default has a dual role in affecting the price of government bonds: a positive one and a negative one. The net effect on domestic bond prices is negative but to a lesser extent than on external bond prices. The marginal gain from borrowing decreases more quickly because of the additional effect of the default risk. After B the marginal gain from borrowing becomes negative: the risk premium more than offsets the additional utility of borrowing one more unit. The Laffer curve for this model coincides with Figure 3 for B = B E and with Figure 4 for B = B D. For any other combination of B D and B E, the Laffer curve is a linear combination of the two. The higher is the ratio of domestic to total debt, the more concave is the 26

27 q D (B,B D )B D B MIN B* B MAX B D Figure 4: The Laffer curve for the domestic market curve and therefore the faster the marginal gain from borrowing decreases. Combining that with the role of the continuation utility, we expect that the total debt arising in equilibrium is low when the domestic to total debt ratio is high. Proposition 2 shows that domestic to total debt ratio increases as long as prices in the domestic market are higher than in the external market. This implies that the domestic to total debt ratio is high when the price for each unit of domestic bond is lower than the unit of external bond and the elasticity of domestic prices to government debt is low. As we mentioned before, the severity of the shortage of assets determines these market conditions. The next section investigates quantitatively how the severity of the shortage of assets affects the equilibrium debt levels and default. 5 Quantitative results Having described how default incentives and pricing functions behave, we can turn the discussion into the quantitative importance of our model. The purpose of the following quantitative exercise is threefold. First, we want to show how the friction shapes the policy functions. Second, we will argue that the model s mechanism is able to produce levels of domestic and external debt that most of the quantitative models of sovereign debt and default cannot obtain. Finally, we conduct a welfare analysis. We investigate the compensation in consumption that has to be given to domestic consumers to move from a given 27

28 value of private assets to a higher amount of private assets. The model is solved through value function iteration on a discrete state space in Fortran. 26 The algorithm converges because it closely replicates the proof of the existence of an equilibrium Policy functions We first present the policy functions of the model to understand how the mechanism works and how the shortage of assets affect government s choices. We calibrate the model with standard values for yearly frequency. The endowment process of the consumers is assumed to be a log-normal AR(1) process with zero mean and persistence ρ equal to The standard deviation of the error, σ y, is equal to This process is discretized into a 31-state Markov chain with Tauchen-Hussey method (Tauchen and Hussey, 1991). We use 300 points for the assets grid. The parameter values are summarized in Table 1. The discount factor and the risk aversion of domestic consumers are, respectively, 0.9 and 2. The the risk-free interest rate in the external market is 0.04, thus implying a value for β E equal to In other words, the ratio β/β E of discounting rates is equal to We assume that the probability of re-entering the market in the period following default, θ, is equal to 0.9 and the fraction of output lost during default, κ is equal to 0. We choose these values because we want to show that our model can produce high values of debt even in the absence of harsh punishment for defaulting. The standard sovereign debt models assume an asymmetric default cost, following Arellano (2008). In section 6.1 we discuss in more detail how the punishment structure of our model affects the equilibrium outcomes. It is apparent that the qualitative features of our model remain the same, irrespectively of the punishment. The mass of domestic consumers, λ, is 0.33, thus implying that their average endowment is half the one of the whole domestic economy. 28 We choose this value in order to show that high debt levels are sustainable in equilibrium even when the government s size is larger than the domestic investors size. Later we will show that our results are qualitatively the same with different values of λ. Given the parameter values we solve the model and obtain the policy functions. Figure 26 We choose to work with a discrete state space because there is no clear evidence that smoothness-based interpolation techniques are the best way to solve these problems, as highlighted by Chatterjee and Eyigungor (2012). In their paper Hatchondo et al. (2010) find that interpolation techniques applied to the models described in Arellano (2008) and Aguiar and Gopinath (2006) work well. In fact, they are as precise as DSS methods with very fine grids and take much less computation time. On the other hand, our model differs in many dimensions from the aforementioned papers. Therefore, we proceed with a DSS and we conduct a robustness check by increasing the number of grid points for both state variables. 27 See the proof in the appendix of Aguiar and Gopinath (2004). 28 Notice that we can define the total endowment in the economy as Y tot = Y + Y D. Hence, the endowment of the young investors represents 1 λ 1+λ of the total endowment in the domestic economy. 28

29 discount factor β 0.95 risk aversion σ 2 relative mass of consumers λ 0.33 risk-free external interest rate r z 0.04 probability of reentry θ 0.9 default output loss κ 0 persistence of AR(1) ρ 0.95 error st.dev. of AR(1) σ y Table 1: Parameter values 5 and Figure 6 show the pricing functions of government bonds in two extreme cases. In Figure 5 we plot the price of domestic government bonds across various levels of bonds chosen and across the endowment shocks, in case that only domestic debt is issued. In this way we can better analyse how the function varies with the level of domestic debt chosen. As mentioned in the previous section, the pricing function is a convex function and the slope becomes steeper when the risk premium becomes positive. The price fall to zero when the probability of default is one. The price is higher and stays above zero for more values, the higher is the endowment level. This happens for two reasons: 1) the endowment of the domestic investors is also higher, thus increasing the price of domestic private assets at the same time; 2) the probability of default is lower because the endowment will remain in the upper region with high probability. In Figure 6 we plot the price of external government bonds across various levels of bonds chosen and across the endowment shocks, in case that only external debt is issued. The pricing function is a straight line because the interest rate is taken as given by the small open economy. The function becomes convex when the risk premium becomes positive. The price fall to zero when the probability of default is one. As in the previous case, the price is higher, the higher the endowment. This happens only for one reason: the probability of default is lower because the endowment will remain in the upper region with high probability. Without the extreme assumption that only one type of bond is issued, the pricing function would depend on the other type of bond issued through the default probability. The higher the amount of bonds issued in the other market, the steeper the slope of the pricing functions. Figure 7 shows how the pricing function of domestic bonds varies when the availability of domestic private assets vary. The price of domestic bonds decreases with L but it also becomes less steep. Therefore, the function becomes less sensitive to the amount of bonds issued with an increase in L. Under the assumption that only one type of debt is issued, we 29

30 can compare the pricing functions for domestic and external bonds in order to understand how market conditions vary in the two asset markets. The more private assets are available in the domestic economy, the larger is the set of B for which external bond prices are higher than domestic bond prices. On the other hand, we can notice the double role of the default probability in the domestic market. When the default probability becomes positive, the price of external bonds decreases steadily because the government has to pay a risk premium. The price of domestic bonds decreases more slowly because the default probability implies also a positive leverage effects on all the assets in the domestic economy. Therefore, in our model the positive effect of the default probability on the price of domestic bonds coming from the leverage does not offset the negative effects coming from the risk premium. Figure 8 shows the government s policy function for defaulting. If default is optimal for a given value of endowment and debt to repay, it is optimal for a higher value as well. Moreover, if default is optimal for a given value of debt to repay and endowment, it is optimal for a lower endowment as well. The policy function features a monotonic relationship with respect to the state variables. Figure 9 reports the optimal borrowing decision for the government in the domestic market. The amount of debt issued does not vary significantly with the amount of debt to repay, except at the point where the government defaults and the debt goes to zero. On the other hand, the debt issued in the domestic market is higher, the higher is the endowment level. Figure 10 shows a very different behaviour for the policy function in the external market. The amount of debt issued increases with the debt to repay and is higher, the lower the endowment level. At the point of default, the external debt drops to zero. The different behaviour in the two markets is motivated by the feature of the pricing functions described before. When the endowment is high, the domestic price is higher and therefore the set of values of new debt for which domestic prices are above external prices is larger. On the other hand, the government affects the price of domestic bonds even if the probability of default is null. The marginal gain from borrowing in the domestic market is always negative. Hence, the government finds it optimal to vary the levels of total new debt with respect to the old debt to repay through the external market. The behaviour of the total new debt issued with respect to the debt to repay is reported in Figure 11. As in standard sovereign debt models, the behaviour of total new debt is not monotonic: the debt is higher for lower endowments up to some level and then the relationship reverses. However, in our model this feature comes from the combination of the optimal choices in two markets. The optimal borrowing in each market features monotonicity with respect to past debt to repay. Figure 12 reports the value of the option for the government for different values of 30

31 q D Y high Y low Figure 5: Price function for domestic bonds, conditional on no external bonds issued. B D q Y high Y low Figure 6: Price function for external bonds, conditional on no domestic bonds issued. B E the aggregate endowment of the consumers. The value function becomes a straight line when default is the optimal choice. The value of the option is higher for higher levels 31

32 2.5 2 q D for L =2.1 q E for L =2.1 q D for L =3.7 q E for L = q B Figure 7: Price function conditional on no other type of bond issued, for two values of L. Y high Y low B Figure 8: Default decision. 32

33 B Y high Y low B Figure 9: Optimal domestic debt B E Y high Y low B 6.58 Figure 10: Optimal external debt. of endowment because the value of staying in the contract is higher for higher values of endowment. 33

34 4 3 Y low Y high 2 1 B B Figure 11: Optimal total debt V o Y high 5 Y low Figure 12: Value of the option. B 34

35 5.2 Simulation results After solving the model, we simulate 29 it for different values of the parameter L, in order to see how the results are affected by the severity of the shortage of assets. Figure 13 shows one of the main results of our analysis. The average equilibrium debt levels exhibit a nonmonotonic behaviour with respect to the magnitude of the friction. The main reason for this behaviour is the way that the government smooths the consumers consumption profile. As we have highlighted in the introduction, the main elements are: lack of commitment in the presence of incomplete markets and shortage of assets in the domestic economy. The lack of commitment together with incomplete markets assumption ensures that default can be an equilibrium outcome. The government can provide insurance to the consumers through default. Default is not unpleasant in this regard. However, the introduction of the shortage of assets carries endogenous costs from default. When the government defaults, it loses the benefits from its market power. When the government supplies risky assets in the economy, it affects the price of domestic private assets and the price of domestic bonds. The additional supply of risky bonds increases domestic asset prices. Due to the no-arbitrage argument that holds in equilibrium, this implies an increase in the price of bonds. Since the government would like to issue government debt at high price, a conflict between the insurance benefits and the loss of market power due to default arises. Our model captures these two effects of default in a parsimonious way. Figure 14 shows the price differential between the two markets. The price differential is higher for higher endowment levels, because the demand for stores of value from the domestic investors is higher. Comparing Figure 14 with Figure 15 we can see how the price differential in the two markets is reduced by the government s choices of borrowing. The difference between the two prices represents the gain the government gets from its market power. Two factors determine the optimal level of domestic debt: 1) the scarcer the supply of domestic private assets, the higher the market prices in the domestic market for a given amount of debt; 2) the scarcer the supply of domestic private assets, the higher the elasticity of market prices to domestic and total government debt issued. The amount of domestic debt issued increases with the amount of domestic private assets when private assets are very scarce. Domestic government bonds and private assets behave as complements as long as the second effect is dominant. After reaching a maximum, the level of domestic debt decreases when the shortage of assets is less severe. Domestic government bonds and private assets behave as substitutes because the first effect becomes dominant. In fact, when domestic private assets are abundant, the government finds it optimal to save in the domestic market. The level of external debt increases in the size of market power. When 29 We simulate the model for periods and eliminate the first 1000 periods. 35

36 the shortage of assets in the domestic market is severe or very negligible, the endogenous default costs for the government are high. Therefore, the default incentives are lower and the government can sustain high levels of debt in the external market. Our model is able to produce high levels of debt even in the absence of harsh punishment for defaults. Most of the papers in the literature on sovereign debt can reproduce external debt-to-output levels between 6% and 23%, by having harsher and asymmetric default costs. Only Benjamin and Wright (2009) obtain a value of 65% without asymmetric default costs, by explicitly modelling the debt renegotiation between government and lenders after the default decision. In our model external debt levels reach values between 20% and 81%, even when the punishment for default is not harsh. Therefore, the levels obtained are to be considered a lower bound of what our model can produce. In section 6 we will show how our debt levels are affected by introducing asymmetric default costs B D B E E(B /Y tot ) L/E(Y tot ) Figure 13: Average domestic and external government debt over GDP for different values of L, percentage values. 36

37 15 q *l for Y low q *l for Y high q z 10 q L/E(Y tot ) Figure 14: Prices of domestic private assets and external risk-free assets without government for different values of L. 37

38 3.5 3 E(q D ) E(q E ) 2.5 E(q) L/E(Y tot ) Figure 15: Average prices of domestic and external government debt over GDP for different values of L. 38

39 Figure 16 reports the relationship between domestic and external debt over GDP with respect to total debt over GDP as predicted by the model when the severity of the shortage of assets vary. The plot obtained with the simulated data closely resembles Figure 2. Our model is able to reproduce the pattern observed in the data for different countries across the years in terms of domestic and external debt ratios. High levels of external debt are associated with low levels of domestic debt. When total debt is low, the domestic and external shares are both significant. In terms of our mechanism, this happens because the government can issue high levels of external debt when the shortage of assets is severe. The sensitivity of domestic prices to the levels of domestic debt is so high that the government does not find it optimal to issue much domestic debt. On the other hand, the price differential between the two markets is so large that the default choice implies a large cost. Therefore, the external government debt can reach high levels without being very risky B D B E 70 E(B D /Y tot ), E(B E /Y tot ) E(B /Y tot ) Figure 16: Average domestic and external government debt over GDP with respect to average total debt over GDP. As shown in Neumeyer and Perri (2005), business cycle statistics for small open emerging economies and small open developed economies feature some differences. In emerging 39

40 economies consumption is more volatile than output, interest rate spreads on external debt are very volatile and counter-cyclical, the trade balance is counter-cyclical. In developed economies consumption is less volatile than output, interest rate spreads are less volatile then in emerging economies and acyclical, the trade balance is less counter-cyclical than in emerging economies. Table 2 shows how the model performs in terms of business cycle statistics when the severity of the friction varies. If we interpret L as a measure of financial development of the economy, we can compare business cycle statistics between emerging and developed economies. Consumption is much more volatile than GDP for low values of L but less and less for higher values of L, as in the data. In fact, for low levels of L the levels of total debt issued are high. For intermediate values of L the total debt issued is lower but the default probability is higher. For high values of L, the total debt issued is lower and the default risk is lower. The probability of default, the spread in the domestic and the external markets and their variability are very low. Hence, spreads are not highly correlated with the GDP. Our model shares this limitation with the standard sovereign debt models in the literature. It is difficult to obtain high default rates and volatile spreads without the use of asymmetric default costs. However, we can notice the non-monotonic relationship between default risk and availability of domestic private assets. The default risk is higher when the shortage of stores of value is moderate and is lower when the friction is severe or negligible. The default incentives for the government are low for low values of L. This feature comes from the endogenous default cost we have introduced in the sovereign debt model: for sizeable market power default costs are high and default events are more rare. The trade balance is negatively correlated with output, no matter the availability of domestic private assets. This means that capital outflows are more likely during bad times. The model predicts that more external debt is issued in bad times and default incentives are stronger in bad times. Hence, the economy cannot experience capital inflows in bad times. The correlation between domestic to total debt ratio and the GDP is positive, in line with the empirical evidence in Figure 1. As shown before, domestic debt is higher in good times. 40

41 L/Y tot E(B D /Y tot ) E(B E /Y tot ) σ(y) σ(c) P(de f ) E(Rs D ) σ(rs D, Y) ρ(rs D, Y) E(Rs E ) σ(rs E, Y) ρ(rs E, Y) ρ(b D /B, Y) NaN ρ(nx/y, Y) Table 2: Statistics for different values of L. All values are in percentage. 41

42 L/Y tot Cert. equiv. cons. E(B D /Y tot ) E(B E /Y tot ) P(de f ) Table 3: Welfare of the government for different values of L. All values are in percentage except the certainty equivalent consumption. 5.3 Welfare analysis for the government We conduct a welfare analysis for the government. We investigate the compensation in consumption that has to be given to domestic consumers to move from a world where the government has strong market power to a world where the government has less market power. Following Chatterjee and Eyigungor (2012), we assume that the initial debt is zero and compute the average value of the option with the invariant distribution of the Markov chain for the endowment process. We repeat the calculation for different values of L. In Table 3 we report the value of the certainty equivalent consumption. First, we can notice that the certainty equivalent consumption is higher when the shortage of assets is severe and the market power for the government is more sizeable. The highest welfare is achieved with high level of external and total debt and low default risk. The compensation in consumption that should be given to domestic consumers to move from a world with L equal to 5% of average GDP to a world with L equal to 37% of average GDP is 20% of their final consumption equivalent. Second, the welfare decreases while the severity becomes moderate. This coincides with a high level of domestic to total debt ratio. Third, the certainty equivalent consumption starts increasing again when the availability of private domestic assets is roughly one third of the average GDP. This happens because the friction starts becoming negligible above such value, but at the same time the government can benefit from the price differential between the two markets. For these levels of L external debt is high again and the default probability is high as well. 42

43 6 Discussion 6.1 Asymmetric default cost We assume an asymmetric output cost for the government in the periods of default and autarky. The form of this cost is the same as in Arellano (2008) and is used in most of the sovereign debt literature to align the statistics on default probability and interest rate spreads with the data: { γe(y) if Y > γe(y), Y de f = Y if Y γe(y). (28) The cost function penalizes the good states more than the bad states. Given that default incentives are stronger in bad states, the asymmetric default cost favours default events. More default events mean higher probability of default and interest rate spreads in the simulated data. Moreover, Chatterjee and Eyigungor (2012) mention that the asymmetric default cost generates more volatile spreads because it increases the sensitivity of the default output cost to the endowment shock. We solve the model using the same parameter values as in Table 1, except for the parameter κ that is not used and replace with the parameter γ. We fix γ to be one, that means that only endowment states above the average are penalized during default and autarky periods. Looking at the policy functions, we can understand how the government s optimal choices vary between the model with proportional default loss and asymmetric default loss. The policy functions do not feature monotonicity with respect to the endowment levels. Figure 17 shows that default incentives vary with the endowment depending on whether the endowment is above or below the average. Below the average default incentives are slightly stronger for higher endowment levels. Above the average default incentives are stronger for lower endowment levels. This implies that the default risk varies substantially when the endowment shock makes the economy move from above to below the average. This significant change in default risk will help generate high and volatile spreads in the simulated data. A similar discontinuity characterizes the optimal level of domestic and external debt. Figure?? resembles Figure 9 in the behaviour of B D with respect to B. The optimal level of domestic debt is still invariant to the amount of debt to repay, except for the point where the government defaults. On the other hand, the levels of domestic debt increase in the endowment levels if they are above the average. The domestic debt does not vary much with the endowment if the endowment is below the average. The optimal levels 43

44 of external debt are reported in Figure??. The external debt issued is still an increasing function of debt to repay. The relationship between optimal external debt and endowment level is changed because default is optimal for more values of B when the endowment is above the average. Apart from this discontinuity, the external debt is still decreasing in the endowment levels. Y high Y low B Figure 17: Default decision with asymmetric default costs. [figure to be added] [figure to be added] We report the business cycle statistics of the simulated data in Table 4 and compare with the results in Table 2. The levels of domestic debt are higher than before and the levels of external debt are lower than before. The default punishment for the high endowments is harsher than for the low endowments. Therefore, anything else equal, default incentives in good states are lower than in the previous case. Given that domestic debt is mostly issued in good states, the average level of domestic debt has increased for any value of L. Default incentives are now even stronger in bad states and external debt is mostly issued in bad states. This explains the lower average levels of external debt. The volatility of consumption is still decreasing in L. As expected, the probability of default has increased, in some cases up to ten times. It is still featuring an hump-shaped relationship with respect to the availability of private assets in the domestic economy. The same hump-shaped relationship characterizes the average interest rate spreads on domestic and external debt. In contrast 44

45 L/Y tot E(B D /Y tot ) E(B E /Y tot ) σ(y) σ(c) P(de f ) E(R D s ) σ(r D s, Y) ρ(r D s, Y) E(R E s ) σ(r E s, Y) ρ(r E s, Y) ρ(b D /B, Y) NaN ρ(nx/y, Y) Table 4: Statistics for different values of L with asymmetric default costs. All values are in percentage. with previous results, the variability of the spreads increases with L. Interest rate spreads are more counter-cyclical than before. The domestic-to-total debt ratio is still pro-cyclical and the trade balance is still counter-cyclical. 45

46 L/Y tot Cert. equiv. cons. E(B D /Y tot ) E(B E /Y tot ) P(de f ) Table 5: Welfare of the government for different values of L with asymmetric default costs. All values are in percentage except the certainty equivalent consumption. 46

47 6.2 No punishment after default to be added... 7 Empirics We conduct a cross-country empirical analysis to compare the predictions of our model with the data. We use the dataset constructed by Ugo Panizza 30 because it is the largest dataset on this topic. It contains data on domestic and external government debt over GDP for 126 countries for the period Debt is distinguished between domestic and external according to the place where it is issued and the legislation under which it is issued. 31 As a proxy for the severity of the shortage of assets in the domestic economy we use data on domestic credit to private sector over GDP compiled by the World Bank. We start the empirical analysis by fitting the best polynomial 32 for the relationship between domestic and external debt and total debt. Our unit of observation is the country. Figure 18 looks very similar to the predictions of our model, reported in Figure 16. We fit the best polynomial for the relationship between domestic and external and credit. The results are reported in Figure 19. The picture closely resembles the results obtained with our model by varying the availability of private assets in the domestic economy, as shown in Figure 13. Given the apparent similarities we conduct a more in-depth analysis in the next sections. 30 We use the updated version of his dataset, described in Panizza (2008). 31 Reinhart and Rogoff (2009) found that in most countries, for most of their history, domestic debt has been issued in local currency and held by domestic bondholders, while external debt has been issued in foreign currency and held by foreign bondholders. 32 The best polynomial is selected on the basis of the Information Criteria. 47

48 Figure 18: Predicted values of average domestic and external government debt over GDP with respect to average total debt over GDP. Results of fractional polynomial, numbers are percentage values. 48

49 Figure 19: Predicted values of average domestic and external government debt over GDP with respect to average domestic credit to private sector over GDP. Results of fractional polynomial, numbers are percentage values. 49