Why Don t Rich Countries Default? Explaining Debt/GDP and Sovereign Debt Crises

Transcription

1 Why Don t Rich Countries Default? Explaining Debt/GDP and Sovereign Debt Crises Betty C. Daniel Department of Economics University at Albany SUNY September 13, 2017 Abstract We build a rich country default model to better understand the Greek default and the European experience with rising debt/gdp. Our model differs from the literature in that we assume a rich sovereign is limited in her ability to repay and makes a credible commitment to repay when able. With this massive punishment to default, debt reaches the limit on ability to repay first, implying that default is determined by solvency. We calibrate to the 2010 Greek crisis, and explain why: countries with debt/gdp ratios higher than the value of standard default punishments avoid default; defaulting sovereigns always repay something; crises follow an increase in debt; debt ratios are heterogeneous across countries, while haircuts and default duration are heterogeneous across defaults. Keywords: Ability to Pay, Fiscal Limits, Sovereign Default, Sudden Stop, Strategic Default, Solvency The author thanks participants in seminars at University at Albany, University of California at Santa Cruz, and University of Southern California for helpful comments on the paper. 1

2 1 Introduction Following the Great Recession of 2008, government debt/gdp in most industrial countries has increased substantially, in many cases to unprecedented levels. These large debt ratios, accompanied by financing diffi culties in some European countries, raised the possibility that default might not be confined to emerging and poor countries. 1 The Greek default subsequently confirmed this possibility. Yet, many countries whose debt ratios reached new peaks, did not default. How can we explain default decisions in rich countries which have experienced a substantial rise in debt/gdp? Sovereign default models, including the seminal papers by Eaton and Gersovitz 1982, Aguiar and Gopinath 2006, and Arellano 2008, have focused on explaining default in emerging economies. In these models, the sovereign is assumed to use debt to smooth consumption in the face of stochastic endowment shocks, but she is not committed to repay debt. The sovereign optimally defaults when the gains from non-repayment of debt exceed the cost of punishment to default. Since most rich countries have avoided default as debt ratios have risen, sometimes to levels exceeding one hundred percent of GDP, application of this model seems to require very large default punishments. Even Greece sustained debt ratios well above one hundred percent before she defaulted. The value of punishments is calibrated using data surrounding actual default events. Around default events, output tends to be lower, and countries tend to be excluded from international financial markets for a limited period of time. Based on this evidence, the consensus in the strategic default literature is that punishments to default are 1) output loss and 2) temporary exclusion from credit markets. Empirically, output loss is more economically significant. Although recessions surrounding default can be large and protracted, the basic model has diffi culty generating debt ratios as high as those in the data for emerging markets. 2 It is diffi cult to imagine these costs approaching one hundred percent of GDP, as required for some rich countries to have avoided default recently. We take seriously the idea that the punishment to strategic default must be very large to explain recent experience in rich countries. We propose an alternative source of punishment, relevant for rich countries only. We assume that a rich country has strong enough institutions to have established a credible commitment to repay when able. This 1 We define default as failure to repay contractual debt obligations. 2 Many authors have worked to get debt/gdp ratios as high as those in emerging markets. This includes work on the damage default inflicts on the financial system and therefore on output (Brutti 2011, Gennaioli, Martin, and Rossi 2014, Padilla 2014, Perez 2015), and models which generate partial repayment, either exogenously (Uribe and Schmitt-Grohe 2017), or endogenously through an additional financial contract Salomao (2017), or through debt renegotiation (Yue 2010). 2

3 commitment implies that the probability of a default when the country can repay a strategic default is zero. If a country, which has credibly committed to forego strategic default, reneges and defaults, then the default violates expectations upon which the financial system is anchored. The output cost of the ensuing financial system chaos is likely to be immense. A large enough punishment to strategic default assures that a sovereign never fails to repay what she is able, always choosing to pay what she can to avoid the immense punishment. Default for a country facing this punishment is never strategic, but could occur when the sovereign faces limits on ability to repay. The contribution of our paper is to construct a model of default for rich countries. Since rich countries face different incentives from poor and emerging countries, debt dynamics in the neighborhood of a crisis will differ, enabling us to answer the title question as well as to establish other characteristics of rich-country crises. Our model builds on the seminal papers by Eaton and Gersovitz (1982), Aguiar and Gopinath (2006), and Arellano (2008), which assume that the sovereign acts as a central planner, maximizing expected utility of the representative agent when endowment income is subject to stochastic shocks. We make two primary departures from their assumptions. First, we modify the punishment to default to include an immense punishment for strategic default. Second, we assume that a sovereign has limited ability to repay. Together these modifications imply that a sovereign optimally chooses default based on solvency rather than strategy. We calibrate the model to match the Greek crisis, which culminated in the first quarter of Our model captures the timing of the crisis, together with rising debt prior to the crisis, as the optimal response of sovereign borrowing to stochastic changes in GDP. In contrast, the strategic default model would have predicted falling debt as the crisis approached. 3 Our alternative model of default addresses questions which have challenged 4 the simple strategic default model: 1) Why do rich countries with debt/gdp higher than the value of standard punishments not default? 2) Why does a sovereign in default always repay something? 3) Why do crises follow an increase in debt which sometimes ends in a sudden stop? 4) Why does debt become risky for different countries at different levels of 3 Falling debt whenever the probability of a crisis is positive is an analytical result of the strategic default model, not a calibrated result. Paluszynski (2017) and Bocola and Dovis (2016) modify the strategic default model to obtain rising debt in the neighborhood of default. Paluszynski uses learning about a rare disaster to explain that debt could increase in the early phase of a disaster while agents do not expect default and continue to borrow. However, debt must be falling as the crisis approaches. Bocola and Dovis (2016) use a model of self-fulfilling crises, together with the addition of a minimum government consumption and a fixed tax rate, similar to our limited ability-to-pay. 4 In many cases, authors have modified the original models to provide answers to these questions. We discuss their contributions at the end of the paper when we present our model s answers. 3

4 debt/gdp? 5) Why are haircuts and default duration highly heterogeneous across default events? Our model allows large debt ratios due to the commitment to repay when able. For rich countries, the ability to repay exceeds the value of standard punishments to default. Since agents repay according to ability, even in default, the interest rate response to an increase in the probability of default is much smaller than when default repayments are zero. Optimal fiscal policy is procyclical. The sovereign smooths consumption when output is low by increasing debt, even when the probability of default is positive. 5 Default occurs when ability-to-pay falls short of output. In default, the sovereign agrees to pay what she is able and borrows again based on expected future ability to pay. If output is high enough next period, she repays her debt and emerges from the default. If not, she again pays what she is able and borrows based on expected ability to pay. Most crises are resolved quickly, but some take a long time. Emergence from a crisis often comes when output rises, increasing ability to pay. Haircuts are highly variable across defaults, and the longest lasting defaults have the largest haircuts. Both characterizations are consistent with empirical evidence on emerging-market crises. 6 This paper is related to the large and growing literature on strategic default. Eaton and Gersovitz (1982), Aguiar and Gopinath (2006), and Arellano (2008) are the seminal papers. Aguiar and Amador (2013) and Aguiar, Chatterjee, Cole, and Strangeby (2016) provide extensions and survey extensions offered by others. Increasing equilibrium debt/gdp ratios to match those in emerging markets has been addressed by Uribe and Schmitt-Grohe (2017), and Salomao (2017), among others. Partial repayment is addressed by Salomao (2017) and Arellano, Mateos-Planos, and Rios-Rull (2013), as well as in models with renegotiation such as Yue (2011). Paluszynski (2017) and Bocola and Dovis (2016) address the time path for debt near a crisis. The paper also relates to another large literature on ability to pay and sustainable debt based on solvency concerns. This literature argues that there is a limit on ability to raise tax revenue and/or reduce government spending and includes Davig, Leeper, and Walker (2011), Bi (2012), Bi, Leeper and Leith (2013), Daniel and Shiamptanis (2013, 2017), D Erasmo, Mendoza, and Zhang (2015), and Collard, Habib, and Rochet (2015) among others. A limitation of this paper is that neither ability-to-pay nor the immense punishment 5 There is an endogenous limit to the increase in debt, as demonstrated later in the paper, but debt does not fall as in the strategic default model. 6 Sturzenegger and Zettelmeyer (2008), Benjamin and Wright (2008), and Cruces and Trebesch (2011) provide evidence. 4

5 are micro-founded internally in the model, even though both assumptions can be at least partially justified with outside literature. 7 The Arellano (2008) model of strategic default has made a fundamental contribution to understanding debt crises in emerging economies even though the punishment to default is not internally microfounded in that paper. We believe that our model of rich-country default can make a significant contribution to understanding default in rich countries, leaving the work on internally micro-founding the ability-to-pay and the immense punishment for future research. Our model is purposefully simple, making only two departures from the strategic default model and yet offers answers to the title question in addition to other questions about crises which have challenged the simple strategic default model. The paper is organized as follows. Section 2 presents the theoretical model, including the debt contract and optimizing behavior by the sovereign. Section 3 provides the calibration to the Greek crisis. Section 4 provides a quantitative description of financial crises using the calibrated model. Section 5 addresses the questions posed above, and Section 6 concludes. 2 Theoretical Model The domestic economy is small and open and subject to stochastic endowment shocks. We assume that endowment income on each date is drawn from a bounded distribution, indexed by j {1, j}. The bounded distribution approximates a distribution in which income is determined by ln y t = ln ȳ + ρ ln y t 1 + ɛ t 0 < ρ < 1 ɛ t N(0, σ 2 ɛ) such that high income today implies high expected future income. The value of j determines the value of income and therefore the income state of the economy. We assume that the sovereign is a benevolent dictator who maximizes the expected present value of utility of the representative agent, given by U = E β t u(c t ). (1) t=1 7 A large literature on fiscal limits, cited above, argues for fiscal limits on ability to repay. There is also a large literature on the effects of government default on the financial sector and the consequent effects on output, including Brutti 2011, Gennaioli, Martin, and Rossi 2014, Padilla 2014, Perez 2015, but to my knowledge no literature on the effects of a probability zero default. 5

6 The sovereign can trade in a limited set of financial contracts with risk-neutral international creditors, allowing consumption-smoothing and consumption-tilting based on the country s rate of time preference relative to the world s. The budget constraint is given by q D = c + ιd + (1 ι) A h y, (2) where where D is the face value of the debt, q is the price of debt, primes denote next period s values, and ι is the indicator function which takes on a value of one when the sovereign has chosen repayment and a value of zero in default. The financial contract we propose differs from that in the literature and is the foundation for our model. 2.1 Debt Contract We modify the standard debt contract with two assumptions: first, a limit on the ability to pay; and second, the threat of a massive punishment for a sovereign who fails to pay what she is able. The ability-to-pay is related to the government s ability to extract tax revenue from the population to use for debt repayment. Therefore, ability is related positively to national wealth and income and to the effi ciency of the tax collection process, and negatively to corruption and political aversion to tax payment. Quality of institutions matters, and quality tends to increase with income. Davig, Leeper and Walker (2011), and Bi (2012) define the maximum level of debt the country can repay as the "fiscal limit", and they motivate the limit by the top of the Laffer curve for distortionary taxes. However, the concept can be more general (Daniel and Shiamptanis 2013). The fiscal limit can be based on the maximum level of the primary surplus that a country could raise. It can include the inability to reduce government spending, perhaps due to the dependence of economic activity on the provision of public goods, together with the inability to raise taxes for other reasons, including political diffi culties (as in Bi, Leeper, and Leith 2013) and tax evasion (as in Daniel 2014). We assume that the ability to repay in endowment state h is given by A h and is determined by the expected present value of the country s endowment income net of a minimum level of consumption, c, according to A h = t=0 ( ) t ψ [E (y (t) y (0) = y h ) c], (3) 1 + r where y h represents current endowment income, r represents the risk-free interest rate, 6

7 and E is notation for the rational expectation. We assume ψ 1, such that the sovereign is not necessarily able to make repayments equal to the excess of expected income over minimum consumption forever. The minimum consumption should be interpreted as politically feasible, not as consumption at an Aiyagari (1994) debt limit. 8 The assumption of declining ability to extract a surplus for repayments over time is motivated by statements like that of Greece in the spring of 2016 that it cannot sustain a primary surplus of 3.5 percent of GDP indefinitely, as required by German demands. The autoregressive behavior of income implies that A h is increasing in y h, such that higher income today yields higher ability-to-pay. Additionally, the specification implies that ability-to-pay relative to GDP is increasing in GDP. 9 We assume that ability-to-pay, conditional on income, is known, although we realize this is an important assumption to relax in future work. We also assume a massive punishment to strategic default. This follows from our assumption assume that some countries, generally wealthy ones, have successfully established a credible commitment to avoid strategic default. This commitment allows accumulation of larger debt ratios, giving the sovereign more flexibility in smoothing consumption. Countries could have gained credibility for this commitment over time by foregoing strategic default when debt reached levels for which strategic default would have been optimal. Most rich countries have publicly pledged to repay debt the US backs debt with the full faith and credit of the US government and their behavior, over time as debt has accumulated, has been consistent with their pledge, allowing them to gain credibility. If a country, which has credibly committed to forego strategic default, reneges and defaults, then the default violates expectations upon which the financial system is anchored. The output cost of the ensuing financial system chaos could be immense. We cannot verify this with data since no country would rationally make this choice, implying that the outcome is unobservable. 10 The immense punishment to the violation of a credible promise implies that the sovereign s commitment is an equilibrium from which she does not deviate. 11 A poor sovereign which has been unable to establish commitment 8 Values for c and ψ replace Aiygari values of zero and unity. Even so, since the sovereign can borrow, actual consumption is always well above c, even in default. 9 Expected future output relative to current GDP falls less with an increase in current GDP than does minimum consumption relative to GDP. 10 The literature on the financial system disruption and consequent output loss from default, including Brutti 2011, Gennaioli, Martin, and Rossi 2014, Padilla 2014, Perez 2015, among others, presumes that default has a positive ex ante probability, equivalently that there was no credible commitment to avoid strategic default. Our commitment puts a zero probability on strategic default. 11 We could also motivate this with a Nash bargain where the surplus to be divided between the players includes the massive punishment. The borrower then agrees to the Nash bargain, the division of this very large surplus, subject to his ability to repay. With a large enough punishment, the ability-to-pay binds, and she repays what she is able. 7

8 would not face the same kind of punishment. The punishment does not necessarily rely on an explicit act by the creditor, but instead is the endogenous response of the economic system to sovereign default of an "inexcusable" magnitude. This follows the "excusable" default literature (Grossman and Van Huyck 1988), whereby excusable default provides insurance against bad outcomes for the borrower and inexcusable ones do not occur due to the massive punishment for them. In our framework, once a sovereign has credibly committed to repay when able, a strategic default is inexcusable and elicits a massive punishment, while a default due to solvency is excusable and acts as insurance. As long as the sovereign pays what she is able in default, our model contains no explicit default punishment. We could add costs of disrupting the financial system, as in the strategic default literature, but the empirical evidence on the costs of actual defaults is controversial. The data are consistent with the assumption that default and recession occur together, but Yeyati and Panizza (2011) argue that the default causes the recession. Additionally, Benjamin and Wright (2008) provide evidence that many countries emerge from default with debt higher than that for which they entered, suggesting ability to borrow while in default. 12 Given the inconclusive evidence on the punishments, we simplify by omitting them. 2.2 Equilibrium We assume that the domestic economy is small and open. Additionally, it has access to a risk-neutral international creditor whose opportunity cost is an exogenous fixed risk-free foreign interest rate. We define equilibrium below. Equilibrium is a set of policy functions for consumption c (D, y) and government debt holdings D (D, y), a cutoff value for states determining repayment ˆ (D), and a price function for debt q (D, y) such that the sovereign maximizes utility, equation (1), subject to its budget constraint, equation (2), and bond prices assure risk-neutral lenders an expected return equal to the exogenous risk-free rate of return Default A sovereign whose debt is less than her ability to repay, optimally chooses repayment to avoid the massive punishment. Therefore a sovereign never chooses strategic default. 12 Arellano, Mateos-Planos, and Rios-Rull (2013) also claim that a sovereign continues to borrow in default. 8

9 As long as the sovereign in default makes debt payments equal to ability to pay, there are no punishments to default. And there is a massive punishment when the sovereign fails to make repayments up to her ability. Therefore, the sovereign optimally chooses to repay the minimum of debt and ability, implying that the budget constraint for the country is effectively q D = c + min {A h, D} y, (4) since the government will choose default if D > A h, and otherwise will choose repayment. We follow Arellano (2008) and determine cutoff values for income as a function of the face value of debt, above which the sovereign makes contractual repayments and below which she defaults. We let ˆ be the cutoff value, given by the income state in which the face value of debt equals ability to pay. States with income below the cutoff (yˆ ) are default states, and states above are repayment states. We assume that repayment occurs with income equal to yˆ. For values of D < A 1, debt is safe and ˆ = 1, its lower support. For higher values of debt, the cutoff state is implicitly defined by D = Aˆ for D A 1. (5) As debt rises, ability-to-pay is equal to debt only if the income state rises, allowing the increase in Aˆ. Therefore, ˆ, defined as the lowest income state in which the sovereign repays, is increasing in debt Haircuts The size of the "haircut" in default depends on the ability to repay relative to outstanding debt. The sovereign, currently in state h with face value of debt D, optimally chooses to repay min {A h, D}. Therefore, the size of the "haircut" in state h (H h ) is given by Debt Price and Interest Rate H h = D A h D. When debt is large enough to be risky, the price of debt is decreasing in debt. The return on debt is determined such that the international creditor expects to receive the risk-free interest rate. Define ˆ as the lowest income state in which repayment occurs. 9

10 The arbitrage relationship, governing the interest rate set in the current period (r ) in state h for next period s debt (d ), is given by (1 + r )d = (1 + r ) d [1 F (ˆ h)] + ˆ j=1 A j f (j h) dj, (6) where r is the world risk-free interest rate, f (j h) is the density function for the distribution of income levels indexed j, conditional on beginning in state h, and F (ˆ h) = ˆ j=1 f (j h) dj is the cumulative distribution in state ˆ, conditional on beginning in state h. The probability of repayment is given by [1 F (ˆ h)]. The arbitrage relation in equation (6) requires that the value of debt (d ) multiplied by the gross risk-free interest rate (1 + r ) equal contractual repayments [(1 + r ) d ], multiplied by the probability of repayment [1 F (ˆ h)], plus repayments in each default state, (A j j < ˆ ), multiplied by their probabilities (f (j h) dj). Defining the price of debt (q) as and the face value of debt (D) as q r, D (1 + r) d, (7) equation (6) implies that the price of debt is q = D [1 F (ˆ h)] + ˆ j=1 A jf (j h) dj (1 + r ) D. (8) The derivative of the price of debt with respect to its face value is given by q D = [ D Aˆ D ] ˆ f (ˆ h) ˆ 1 + r D A j=1 jf (j h) dj (1 + r ) (D ) 2. (9) 10

11 Recognizing that Aˆ = D from equation (5) and simplifying yields ˆ q = A j=1 jf (j h) dj D (1 + r ) (D ) 2 0. (10) When the face value of debt is low enough that it is less than ability to pay in the worst state (D < A 1 ), all debt is safe (ˆ = 1; F (1 h) = 0), and q = 1 1+r. Since the integral has unity as the upper and lower limit, the derivative is zero. However, once the face value of debt rises above A 1, ˆ rises, and the price of debt falls as debt rises. Compare the effect of an increase in debt on the interest rate in this model with that in the model of strategic default in which either there are no debt repayments in default (Arellano 2008) or the repayments are some fixed fraction of debt. In the strategic default model, default occurs only if the gains to default, based on the difference between what the sovereign owes and what she repays, exceed the value of the punishment. With this alternative contract, the value in equation (9) for D Aˆ, where we interpret Aˆ as debt repayments, must be positive and large enough to exceed the punishment, justifying default. Therefore, the first term in equation (9) is large and positive, instead of zero, implying that an increase in borrowing creates a larger rise in the interest rate (and a larger fall in the price of debt) than the debt contract we specify. This large increase in the interest rate is responsible for the result that the sovereign saves when there is a positive probability of default, even though consumption-smoothing would require borrowing. Since the interest rate rises less with the ability-to-pay contract, the sovereign chooses to borrow to smooth consumption even in the neighborhood of default. When debt is risky, an increase in the face value of debt increases resources from borrowing (q D ) by less than the price of debt. taking the derivative with respect to D yields Multiplying equation (8) by D and ˆ D (q D ) = [1 F (ˆ h)] + f (ˆ h) [Aˆ D ]. (11) D 1 + r Noting that Aˆ = D from equation (5) and simplifying yields (q D ) D = [1 F (ˆ h)] 1 + r 0. (12) When D < A 1, all debt is safe and F (ˆ h) = 0. The effect of an increase in the face value of debt on the proceeds from borrowing is the inverse of the gross risk-free interest rate, equivalently the price of debt. However, once debt is large enough to be risky, 11

12 implying that the probability of default is positive (F (ˆ h) > 0), an increase in D requires a decrease in q such that the proceeds from borrowing rise by less than 1 1+r. The foregoing implies that there is an upper bound on borrowing (q D ). From equation (12), q D is increasing in D until D reaches the ability to pay in the highest state possible next period, conditional on the current state. Define this state as j h. Using equation (8) with j replacing ˆ, and F ( j h ) = 1, the upper bound on sovereign borrowing is determined by the expected present-value of repayments in default, conditional on income in the initial state. q D (q D ) ub = j A j=1 jf (j h) dj, (13) (1 + r ) where h is the initial state. Higher initial income implies a higher upper bound due to the autoregressive behavior of income. The upper bound on q D also implies an endogenous upper bound on D. Once the face value of debt rises so much that q D = (q D ) ub, the sovereign will not choose further increases in D. Larger D would be accompanied by a proportionate fall in q such the increase in future debt obligations would not be accompanied by an increase in borrowing proceeds and current consumption, a suboptimal move. When the probability of default is positive, the domestic interest rate carries a defaultrisk premium, given by r r = 1 q (1 + r ) = (1 + r ) ˆ j=1 (D A j ) f (j h) dj D ˆ j=1 (D A j ) f (j h) dj, (14) where the second equality uses equation (8). Note that an increase in debt, which causes ˆ to rise, creates a discrete jump in the interest premium by increasing the number of states with default repayments Probability of Default The probability of default is increasing in debt and decreasing in income. A sovereign optimally chooses default when ability to pay is less than debt. Ability to pay is increasing in income. Therefore, for a given level of debt, the probability of default is the probability that future income falls suffi ciently to reduce ability to pay below debt owed (D ), equivalently the probability of transiting from the current state h to a state below ˆ, where the value of ˆ is determined by the value of D. This probability is higher, the lower is income, due to the autoregressive nature of shocks, and the higher is debt, since 12

13 ˆ is increasing in debt Debt Rises Near a Crisis The dynamic behavior of debt, in response to shocks to income, is determined by the optimizing behavior of the sovereign. We represent the expected present value of utility for the sovereign with a value function, which depends on the exogenous state given by income (y), and on the face value of debt (D), according to V (y, D) = u (c) + βev (y, D ). Since the sovereign defaults in states j < ˆ, and repays in others, we can rewrite the value function as V (y, D) = u (c) + β [ ˆ V (y, A(y )) f (j) dj + j=1 1 j=ˆ ] V (y, D ) f (j) dj. The only distinction between repayment states and default states is initial debt, implying different arguments for the future value functions in repayment versus default states, but not different functions. 13 Maximization is subject to a budget constraint, given by equation (4), which depends on the current income state h, and which allows default with repayments equal to ability to repay, whenever ability is less than contractual debt repayments. The derivative of the value function with respect to debt differs in default and repayment states. In a repayment state V (y, D j ˆ ) D ( ) u (c) =. c However, for values of j putting the system into default states, the derivative of the value function with respect to debt is zero since, in default, the sovereign pays its ability irrespective of actual debt. The first order condition for the choice of next period s debt is given by 14 u (c) (q D ) j β c D j=ˆ ( u (c ) c ) f (j h) dj = 0 13 We could formally include the alternative value function with strategic default in which the sovereign suffers the massive punishment. Our assumption is that its value is so small that it would never be chosen in equilibrium, allowing us to omit it from the problem. 14 The term multiplying ˆ D vanishes since at ˆ, A (y ) = D. 13

14 where c should be understood as depending on j. Substituting from equation (12) yields u (c) c = β (1 + r ) ( j u(c ) j=ˆ ) f (j h) dj {( c u (c j = β (1 + r ) ) E f (j h) dj c j=ˆ ) } (j > ˆ ). The right hand side of equation (15) is the expected marginal utility of consumption next period, conditional on obtaining states in which repayment occurs. Since repayment in default states is not related to the amount borrowed, states below ˆ are not included in the integral for expected future marginal utility of consumption. At the optimum, the marginal utility of current consumption equals the expected marginal utility of future consumption, conditional on repayment, multiplied by β (1 + r ). (15) Since consumption is higher in states in which repayment occurs, the marginal utility of expected future consumption, conditional on repayment, is lower than unconditional marginal utility of expected future consumption. Therefore, when default is possible, the marginal utility of current consumption must be lower and current consumption higher. A positive probability of default next period (ˆ > 1) decreases the right hand side of the Euler equation (15) because expected future marginal utility is included only for repayment states, and consumption is higher in those states than in default states. The lower expected marginal utility of future consumption requires that the marginal utility of current consumption also fall, thereby increasing current consumption. Therefore, when the economy enters states low enough that default is possible, it raises consumption through an increase in borrowing. Therefore, consumption and the choice of debt next period are higher when the probability of default is positive. The possibility of paying only what the sovereign is able and not actual debt repayments cuts off the lower portion of the risk distribution encouraging the sovereign to increase consumption and debt, thereby taking on more risky behavior. This is Stiglitz s (1981) classic result that the availability of bankruptcy increases risk-taking behavior. The result is opposite that in the strategic default model, in which the sovereign saves in states for which the probability of default is positive. 3 Calibrated Model We calibrate the model to match the timing for the beginning of the Greek crisis and the behavior of Greek government debt prior to the Greek financial crisis. We seek to match total government debt, not just external debt as in strategic default models. In our 14

15 model, repayment is limited by ability to pay, and limits on ability can be due to distortionary taxation or on political will to redistribute away from the general populace toward wealthier bondholders. Both imply that the relevant debt is total government debt. An alternative way to justify calibration to government debt would be to replace consumption in the model with government consumption, as in Bocola and Dovis (2016), and reinterpret the budget constraints in terms of government income and spending. Alternatively, if there were limited political will to transfer resources toward foreign nationals, then external debt would be more relevant. However, Reinhart and Rogoff (2008) found no statistical difference in the incidence of default to domestic versus foreign creditors, implying no difference in political will to repay domestic versus foreign nationals. We use value function iteration with the choice variables being the decision to default or repay current debt and next period s debt, conditional on the current value of debt, the current output state, and the equilibrium price of new debt. 3.1 Standard and Estimated Parameter Values The external interest rate (r ) is set at the average value of the German ten year bond rate over the period yielding r = The coeffi cient of relative risk aversion (σ) takes on its standard value, yielding σ = 2. We estimate the autoregressive parameter for real Greek GDP and its standard error using quarterly OECD data from 1960Q1 to 2008Q2. We detrend and demean the log of the data and obtain values of ρ = and σ ɛ = 0.028, yielding a standard deviation of output of We approximate the behavior of the data using a discrete approximation with fifty-one output states based on Tauchen s (1986) method of approximating an autoregressive series with a Markov chain. We solve the model by creating a grid for the face value of debt with 1000 points, equally spaced from 0 to 1.75, the largest value for ability-to-pay. The upper bound on the debt grid is larger than any sovereign ever chooses, but the lower bound is a constraint. 15 Aiyagari, Marcet, Sargent, and Seppala (2002) show that precautionary savings in response to stochastic income creates a downward drift to the optimal behavior of government debt. They suggest imposing a lower bound to match behavior of actual sovereigns, who do not accumulate large assets. Battaglini and Coate (2008) and Barseghyan, Battaglini and Coate (2013), create political economy models with endoge- 15 We also calibrated with a debt grid which had a lower bound of , something no country has ever reached. This country swings through periods with very large assets. The calibrated parameters are slightly different, but the fundamental characteristics of the model near a debt crisis are the same. 15

16 nous lower bounds on debt. Our exogenous lower bound on debt prevents the sovereign from accumulating large amounts of assets in good times, consistent with empirical evidence, and with political economy models, although these models have endogenous lower bounds. 3.2 Deadweight Cost of Default Our model has no deadweight loss in default, a simplifying assumption that is not realistic. Therefore, we add a small deadweight loss to default to offset some of the Stiglitz-type risk-taking in the neighborhood of default. In the model, the sovereign already pays the maximum she is able in default, implying that we cannot add anything to these repayments. Therefore, we assume that, in default, the sovereign continues to pay her full ability, but the lender receives only a fraction ω of this repayment. This requires revision of equation (8) to yield q = D [1 F (ˆ h)] + ˆ j=1 ωa jf (j h) dj (1 + r ) D. (16) We view this deadweight loss as the administrative cost of the default and not as an explicit punishment to default. With the deadweight loss, the interest premium becomes r r = 1 q (1 + r ) = (1 + r ) ˆ j=1 (D ωa j ) f (j h) dj D ˆ j=1 (D ωa j ) f (j h) dj. This revision changes the derivative of the price of debt and current borrowing with respect to the face value of debt, equations (10) and (12), and the Euler equation (15), to yield ˆ q = ωa j=1 jf (j h) dj D (1 + r ) (D ) 2 [1 ω] f (ˆ h) ˆ < 0 (17) 1 + r D u (c) c (q D ) D = [1 F (ˆ h)] f(ˆ h)aˆ (1 ω) ˆ D = β (1 + r ) 0, (18) 1 + r ( ) u(c ) f (j h) dj c j j=ˆ [1 F (ˆ h)] f(ˆ h)aˆ (1 ω) ˆ D, (19) where we have used Aˆ = D. With ω < 1, repayments in default per unit of debt are lower, implying a lower price of debt and a smaller reduction in the price when debt rises. For increases in next period s debt which raise the value of ˆ ( ˆ D 16 > 0 ), reducing

17 the number of repayment states, deadweight loss (ω < 1) implies that the price of debt takes a downward jump, with the interest premium taking a corresponding upward jump. Additionally, the proceeds from additional borrowing (q D ) do not rise as much. In equation (19), the term multiplying ˆ D implies a discrete increase in the cost of raising debt beyond the next critical bound at which ˆ increases. At such a bound, a small increase in debt does not yield significant future debt relief in default because ability-topay in the next higher state almost matches the debt. However, since the creditor suffers a deadweight loss in default, he requires a discrete reduction in the price of debt with the increase in ˆ. Therefore, due to the fall in the price of debt, the sovereign receives little additional consumption from increasing debt just beyond the critical barrier, and she shoulders additional debt in repayment states. Together these incentives act to keep debt below the bounds at which ˆ changes, and they are larger, the larger the deadweight loss (the smaller is ω). When debt is risky, the deadweight loss allows a larger increase in the interest rate as debt rises, mitigating Stiglitz risk-taking. The deadweight cost of default adds an additional parameter, ω, for calibration. 3.3 Remaining Parameters There are four remaining parameter values, ψ, c, ω, and β which we calibrate to match four features of the data: (1) the timing of the crisis, (2) the value of average debt over the full business cycle preceding the crisis (2005Q1:2007Q4), 16 the values of debt/gdp on two dates: (3) pre-crisis (2009Q4) and (4) crisis (2010Q1). We obtain data on the values of debt using Eurostat data on quarterly values of debt relative to GDP, beginning in 2006Q1, and annual values for We convert these values to our measure of debt, which is detrended real debt relative to detrended real GDP in the median state, by multiplying the Eurostat data on debt/gdp by actual detrended real GDP relative to detrended mean GDP, computed using OECD data on real GDP. 17 This measure does not change with changes in real GDP. We refer our data 16 We measure the business cycle preceeding the crisis using the discretized states. Greek output falls to state 23 in 2005Q1, 0.4 standard deviations below the mean, after a period of being in higher states and returns to state 23 in 2008Q1. Therefore, we measure the business cycle as the first period of recession (2005Q1) through the last period of the subsequent boom (2007Q4). 17 This requires that we use quarterly interpolations of the annual debt data for 2005, the first year of our sample. To justify this calculation, assume that GDP has trends due to a nominal and a real component and that debt shares these trends. Therefore, we can view the ratio of debt/gdp as the ratio of real detrended debt to real detrended GDP. We obtain real detrended debt relative to real detrended GDP by multiplying debt/gdp by real detrended GDP relative to the mean of real detrended GDP. 17

18 measure simply as "debt,", consistent with the definition of debt in the model. 18 For output, we detrend and demean the OECD data on real GDP and place each observation into one of the 51 output states of the model by choosing the output state closest to the detrended and demeaned value. Our data on the Greek interest rate premium is the difference between the interest rate on ten year government bonds for Greece and Germany from the ECB Statistical Data Warehouse. Our model defines the crisis date as the first period in which Greece s ability to pay is less than debt. In 2010Q1, Greece suffered a reduction in output, reducing ability to pay. Greece did not have scheduled debt repayments in this period, implying that there were no observations on repayments, either missed or made. However, Greece began austerity programs and the ECB softened rules on collateral for ECB loans, implying that Greece expected financing diffi culties once maturity dates arrived. This evidence implies that the first period in which Greek debt exceeded ability to pay was 2010Q1, leading us to use this date as the first period of the crisis. To obtain model values, we generate a time series on the sovereign s choice of debt and default conditional on the initial value of debt given by the data and on output states from the data. We choose the start date as 2005Q1, the first date of the previous recession. This requires that the sovereign choose debt over the entire business cycle preceding the one which created the crisis, as well as over the beginning of the business cycle created by the financial crisis. Our first step in matching model values with the data is to narrow the choices of the parameter values to those which exactly match the timing of the crisis. Therefore, our calibration strategy requires that beginning on the start date (2005Q1), the sovereign chooses to repay in all periods leading up to the crisis and chooses not to repay in the crisis period. Candidate parameter values must imply that the sovereign choose next period s debt on each date beginning with 2005Q1 and ending with 2010Q1, such that debt is below realized ability to pay through 2009Q4 and above realized ability in 2010Q1. This requires that the sovereign choose debt consistent with the actual repayment and default decisions for a total of 21 periods. Given these restrictions, we finalize the choices for ψ, c, ω, and β by matching three additional features of the data: the average value of debt over the previous business cycle (2005Q1:2007Q4), the value of debt in the period prior to the crisis (2009Q4), and the value of debt in the crisis period (2010Q1). Parameter values and the sources for their calibration are given in Table In the model, there is no trend and the mean value of output is unity. 18

19 Table 1: Parameter Values Parameter Value Source σ 2 standard value r German ten year bond yield (quarterly value) σ ɛ regression estimate using real GDP data (1960Q1:2008Q2) ρ regression estimate using real GDP data (1960Q1:2008Q2) ψ 0.77 crisis timing and three data targets c 0.62 crisis timing and three data targets β crisis timing and three data targets ω crisis timing and three data targets Consider how the four different parameter values affect the model values of our data targets. All are important in matching crisis timing. The requirement that we match debt data on the eve of the crisis and in the period of the crisis requires that ψ and c be chosen such that ability-to-pay was above actual Greek debt on the eve of the crisis but was below debt on the date of the crisis. The value for β partially determines the sovereign s propensity to take on debt in alternative states and is important in determining the average value of debt. The values for ψ and c and β are jointly responsible for determining average debt over the previous business cycle and values for debt as the crisis unfolds. The value for ω is important in determining the sharp increase in borrowing leading up to the crisis. The closer ω is to unity, the smaller the deadweight loss and the steeper is the increase in debt leading up to the crisis due to greater Stiglitz-type risk-taking. Our calibration does not require a particularly impatient sovereign. The inverse of the gross risk-free interest rate in our calibration of β is 0.991, compared with our calibrated value for β of Our model does not require much impatience due to the Stiglitztype risk-taking created with the high value of ω, implying a deadweight loss in financial markets of only 0.70 percent of the value of repayments. result of debt accumulation due to impatience, is unnecessary. 19 Additional risk-taking, as a Table 2 compares model values with those targeted in the data. The model fits the data well, matching values for average and crisis debt almost perfectly and for precrisis debt within percentage points. 19 Bocola and Dovis (2017) modify the strategic default model with a minimum government consumption (analogous to a minimum consumption in our model) and calibrate to Italy, yielding a large β. 19

20 Table 2: Model Fit Timing Average Debt Pre-crisis Debt Crisis Debt Model 2010Q Data 2010Q Figure 1 plots the actual time path for debt relative together with the model-generated time path as a test of model fit. Model debt moves similarly to the data, rising during the initial four quarters of recession, becoming flat as income approaches the mean, eventually falling, and then rising sharply with the declining output which accompanies the worldwide financial crisis. Figure 1: General Fit of Model model debt data debt Q1 2006Q1 2007Q1 2008Q1 2009Q1 2010Q1 Model replicates data well. 3.4 Greek Crisis Consider how the Greek crisis unfolded, using the calibrated model. Over the business cycle from 2005Q1 to 2007Q4, our measure of Greek debt, real detrended debt relative to mean real detrended GDP, averaged about one hundred percent. The period began with four quarters of recession with output averaging 0.4 standard deviations below the median, during which debt was rising, and ended with eight quarters of boom with output 20

21 near trend (output averaged 0.2 standard deviations below trend) and debt falling. This relatively tranquil period ceased with the worldwide financial crisis which sent Greek GDP back to recession in 2008Q1, with output 0.3 standard deviations below trend. The recession worsened throughout the year with output at year end 0.7 standard deviations below trend. Greek debt increased, as it had in the previous recession. The difference is that, this time, after four quarters of moderate but worsening recession, output fell dramatically in 2009Q1, reaching 1.2 standard deviations below trend, instead of rising back toward trend, as it had following the previous four-quarter recession. Output remained in recession throughout 2009, averaging 1.1 standard deviations below trend. Over this period, debt was increasing, with the increase accelerating as the recession worsened. In 2010Q1, output fell further reaching 1.4 standard deviations below trend. Output this low, together with Greece s debt accumulated over eight quarters of worsening recession, meant that ability to pay was lower than debt, dating the crisis. 20 Therefore, our model implies a crisis with output 1.4 standard deviations below trend and with real detrended debt equal to 110 percent of mean real detrended GDP. We do not model Greece after 2010Q1 because the data is no longer market-determined, as required by our model. Greece receives its first offi cial bailout in May 2010, providing additional debt that the private sector might not have provided and affecting the timing of default. This scenario makes the Greek crisis sound entirely like bad luck instead of bad policy, and the bad luck of the financial crisis did contribute. However, Belgium and Italy were also high-debt countries whose output took sharp decreases with the financial crisis. We compute the probability of crisis for Greece over the next ten years from our start date of 2005Q1, and obtain a probability of 4 percent over the next ten years. This nonnegligible possibility of a crisis is due to the value of debt relative to ability to pay, enabling a period of strong recession to send the country into crisis. Either greater ability to pay or parameters delivering lower debt would have been required to reduce Greek risk in the wake of the financial crisis. A model of strategic default does not fit the Greek crisis. Greece was accumulating debt during the periods prior to the crisis when output was low and the probability of a crisis one quarter ahead was substantial, peaking at 34 percent in 2009Q4. The model 20 Greece did not actually have scheduled debt repayments in this period, implying that there were no observations on repayments, either missed or made. However, Greece began austerity programs and the ECB softened rules on collateral for ECB loans, implying that Greece expected financing diffi culties once maturity dates arrived. 21

22 of strategic default would have Greece reducing debt during each quarter of this year since the probability of a crisis one period ahead was positive. The result in the strategic default model, that debt rises when the probability of a crisis is positive, is an analytical result, implying that we cannot "recalibrate" the strategic default model to match the behavior of debt in Greece. 21 Attributing the Greek default to ability to pay instead of to a strategic decision carries a policy implication for creditors. With strategic default, Greece would be refusing to pay when she is able. Threats of additional punishments would be warranted to force Greece to pay something closer to her ability. However, if the Greek default is due to inability to pay, additional threats and/or additional punishments can extract no additional payments. 4 Quantitative Results on Debt Dynamics and Crises The sovereign s policy function together with simulations based on the policy function allow us to characterize properties of fiscal policy, debt dynamics near a crisis, and properties of crises themselves. 4.1 Fiscal Policy is Counter-Cyclical The sovereign s policy function implies that fiscal policy is counter-cyclical. When a sovereign is in an income state below the median and has debt lower than ability to pay in the same state, she generally chooses a modest increase in debt. There are exceptions to this behavior when debt is very close to ability-to-pay, which we discuss below. The sovereign chooses D to maximize her objective function, given by current utility plus the discounted continuation value. Figure 2 plots the sovereign s objective function on the vertical axis and next period s debt (D ) on the horizontal axis. The country is in state 20, 0.7 standard deviations below the median. Initial debt equal to 1.03 (implying that real detrended debt is 103 percent of mean real detrended GDP), the starting value on the horizontal axis, and is less than ability to pay, given by 1.32 percent of mean GDP (not shown). 21 Bocola and Dovis (2017) claim to have "recalibrated" the strategic default model to obtain countercyclical fiscal policy. However, they made a major modification to the model to obtain the result, by adding a minimum government consumption. Minimum consumption, together with a fixed tax rate is a limit on ability to pay, consistent with one of our two departures from the strategic default model. 22