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 February 5, 2018 Abstract Strategic default models were developed to explain sovereign default in emerging markets. They do not explain pre-crisis debt dynamics in Greece, the first rich country to default. These dynamics include much larger values for debt/gdp than implied by standard punishments; rapidly rising debt just prior to the crisis; and pro-cyclical fiscal policy in the period leading up to the crisis. We replace the assumption of absence of commitment to repay with an assumption that the sovereign commits to repay up to a limit based on ability-to-pay. We argue that limited commitment is more feasible for rich countries with stronger institutions. We calibrate to the 2010 Greek crisis, and demonstrate that in contrast to the strategic default model, the calibrated model matches Greek debt dynamics prior to the crisis. We use the calibrated model to explain incentives to take on risk, default probability, endogenous haircuts, and default duration. Keywords: Ability to Pay, Fiscal Limits, Sovereign Default, Strategic Default JEL Code: F34 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 Prior to the Great Recession of 2008, sovereign default was confined to emergingmarket and poor countries. With the recession, government debt/gdp in most industrial countries increased substantially, in many cases to unprecedented levels. These large debt ratios, accompanied by financing diffi culties in some European countries, raised the possibility of sovereign default by rich countries. confirmed this possibility. The Greek default subsequently Strategic default models were created to explain default in emerging economies (Eaton and Gersovitz 1982, Aguiar and Gopinath 2006, and Arellano 2008). They do not explain debt dynamics leading to the first debt crisis in a rich country, the Greek crisis. We focus on three particular features of the Greek debt crisis, which culminated in the first quarter of First, Greece reached debt/gdp ratios well above one hundred percent prior to the crisis, values much larger than any calibrated in strategic default models. 1 Second, prior to the crisis, output was low and Greek debt was rising rapidly, not falling as required by the strategic default model. 2 Third, Greece had countercyclical fiscal policy over the ten years prior to default, behavior more typical of rich countries than of emerging markets. 3 We modify the Arellano (2008) model of strategic default to enable it to match the behavior of debt leading up to the Greek crisis. In Arellano (2008), the sovereign uses debt to smooth consumption in the face of stochastic endowment shocks and to tilt it due to impatience, but she is not committed to repay debt. Failure to repay entails a 1 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). Chatterjee and Eyigungor (2012) use longer-maturity debt to achieve ratios as high as those in Argentina. 2 Falling debt whenever the probability of a crisis is positive is an analytical result of the strategic default model, not a calibrated result. The appendix in Arellano (2008), repeated in Schmitt-Grohe and Uribe (2017, p.515), presents a proof when income shocks are iid, but the proof seems to hold with Markov income shocks, whereby the price of debt would depend on the current income state. Sargent, Thomas J. and John Stachurski (2017) show falling debt prior to a crisis in their simulations of the Arellano (2008) model. Paluszynski (2017) and Bocola and Dovis (2016) introduce modificatins of 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) introduce self-fulfilling crises, together with a minimum government consumption and a fixed tax rate, similar to our limited ability-to-pay. 3 In strategic default models, consumption is more volatile than output. The social planner would implement this with lump-sum taxes and pro-cyclical fiscal policy, consistent with pro-cyclical fiscal policy in many emerging economies. 2

3 punishment. The sovereign optimally (strategically) defaults when the gains from nonrepayment of debt exceed the cost of punishment to default. The punishment is calibrated to match empirical values of output loss following defaults, together with estimated values of temporary financial market exclusion. For standard values of the punishment, the model cannot generate the large values for debt/gdp, recently observed in many rich countries, or the debt-dynamics preceeding the Greek crisis. We replace the assumption that the sovereign cannot commit to repay debt with the assumption the sovereign can make a limited commitment to repay. The commitment is limited because the sovereign faces an upper bound on her ability-to-pay, where this upper bound depends on current and expected future values of income. 4 Under the implicit contract with limited commitment, the sovereign commits to repay debt up to this upper bound. When this upper bound exceeds debt, she repays debt. However, when the upper bound is less than debt, repayment falls short of debt, constituting our definition of default. A sovereign could achieve credibility for this limited commitment by repaying debt in states, for which, in the absence of commitment, she would have strategically chosen default. Stronger institutions, which are more typical of rich countries, support the ability to commit. The commitment to pay up to ability allows large equilibrium values for debt/gdp when ability is large. Additionally, limited commitment yields different incentives for risk-taking, enabling us to explain debt dynamics in Greece leading up to the crisis. As debt increases, the number of future income states, in which the sovereign would replace contractual debt payments with the lower ability-to-pay amount, increases. For these income states, payments are given by ability-to-pay irrespective of the magnitude of debt, creating the incentive to increase debt. The interest rate does rise as debt increases, due to increased risk of default, but the rise is mitigated by the commitment to repay up to ability in default. The incentive to take on increased debt can dominate the response of the interest rate to debt, such that the sovereign optimally raises debt in low-income states. 5 Stiglitz (1981) first demonstrated that the possibility of default cuts off the lower portion of the risk distribution, thereby incentivizing risk-taking. The different incentives for risk-taking in the limited-commitment mode yield two predictions for pre-crisis debt dynamics which are opposite those of the strategic default 4 There is a large literature which tries to assess a sovereign s maximum ability to raise tax revenue and/or reduce government spending, including 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. 5 We show this dominance analytically under certain restrictions and our callibrated model contains it. 3

4 model. First, prior to a crisis, debt is rising, not falling. This is because incentives created by Stiglitz risk-taking dominate those created by the increase in the interest rate. Second, it yields counter-cyclical fiscal policy because the interest rate response to an increase in the possibility of default is not strong enough to reverse consumptionsmoothing incentives. We demonstrate quantatively that the limited commitment model can match the characteristics of debt leading up to the Greek crisis, by calibrating the model to match the Greek crisis. Our calibrated model allows large debt ratios, based on large ability to repay. It captures the timing of the crisis, together with rising debt prior to the crisis. With limited commitment, the sovereign retains incentives to smooth consumption, even when crisis probability is positive, yielding counter-cyclical fiscal policy. Additionally, we use the results of the calibrated model to describe other features of debt crises under the limited commitment contract, including incentives for risk-taking, the probability of default, and the magnitude and duration of endogenous haircuts. Haircuts and default duration are endogenously determined since the sovereign repays in default up to ability-to-pay. 6 The model s answer to the title question about why rich countries with high values for debt/gdp have not defaulted is two-fold. First, rich countries have made a limited commitment to repay up to ability-to-pay. Second, for all rich countries except Greece, ability-to-pay has exceeded values of debt. However, under our assumption that abilityto-pay is limited, these countries are not immune to a future sovereign debt crisis. The crisis in Greece occurred as the fall in output incentivized a large increase in debt, after which output fell further, reducing ability-to-pay below debt. 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. Our model differs from this literature due to our assumption that the sovereign can make a limited commitment to repay, a commitment which requires strong institutions, more often associated with rich 6 Other papers addressing duration and magnitude of default, include Benjamin and Wright (2008) and Arellano, Mateos-Planot and Rios-Rull (2013). 4

5 countries. The idea of the limited commitment borrows from Grossman and Van Huyck s (1988) concept of "excusable default," whereby lenders forgive some portion of debt in some states, and from Bulow and Rogoff (1989), in which renegotiation in some states is part of the implicit contract. We can view implementation of the limited commitment contract as supported by the threat of a massive punishment, should the sovereign renege and fail to repay what she is able. With the threat of the massive punishment, the sovereign adheres to her limited commitment and optimally chooses to repay debt when she is able. When debt exceeds ability to repay, she makes payments equal to ability-to-pay. We can only speculate on the nature of the massive punishment since the sovereign would never choose this alternative, but surely it would involve massive financial sector disruption, by assumption, greater than any we have ever witnessed. Poor countries with a negligible financial system would not be able to support this contract. The assumption of commitment is standard in models of monetary policy (Walsh, 2017), but not in the literature on optimal sovereign default. This paper brings together the literature above on strategic default and that on default due to limited ability-to-pay. The later 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 and yields default when ability-to-pay falls below outstanding debt. In this literature, debt/gdp ratios look more like those in the data, but there is no assumption that the sovereign behaves optimally. The limited commitment model presented here assumes optimal behavior by the sovereign and also yields default when ability-to-pay falls below outstanding debt. 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 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 5

6 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 The sovereign can trade in one-period debt 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 + A h (1 ι) y, (2) where where D is the face value of the debt, A h is the repayment the sovereign would make in a default in income state h, 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. Both the decision to default, ι, and the value of repayments in default, A h, are determined in equilibrium based on incentives created by limited commitment. 2.1 Limited Commitment The strategic default model assumes that the sovereign cannot commit to repay debt. We modify this by assuming limited commitment. In particular, the sovereign can commit to repay debt up to her limited ability-to-pay. When debt is less than ability-to-pay, full repayment occurs and there is no default. When debt exceeds ability-to-pay, the sovereign repays according to ability, but since this is less than contractual debt, the sovereign is in default. The model requires a specification for ability-to-pay. 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 6

7 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 (Collard, Habib, Rochet 2015). 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). Richer countries tend to have better institutions and therefore higher ability-to-pay relative to GDP. And since we are interested in countries with the ability to borrow, ability-to-pay must include expected future ability to raise taxes and reduce spending. This brief review of the literature demonstrates that there no consensus how to determine the ability-to-pay. We do not made a contribution to these microfoundations in this paper. We choose an ad hoc specification with simple parameterization, consistent with the literature and with the ability of the country to borrow. The first criteria is that ability-to-pay must be based on the entire expected present value of future income. A sovereign whose country has low current income and high expected future income should have higher ability-to-pay than one with the same current income and low expected future income. Second, a sovereign could not repay the entire expected present value of future income. Third, ability-to-pay relative to GDP should be increasing in GDP. We use two parameters, one linear (δ > 0) and one non-linear (0 < ψ < 1), to adjust the expected present value of income downward to ability-to-pay, and calibrate those parameters according to the crisis experience of Greece. Based on these considerations, we assume that the ability to repay in endowment state h is given by A h and is given by A h = t=0 ( ) t ψ [E (y (t) y (0) = y h ) δ], (3) 1 + r where y h represents current endowment income, r represents the risk-free interest rate, and E is notation for the rational expectation. 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. We assume that ability-to-pay, conditional on income, is known, although we realize 7

8 this is an important assumption to relax in future work. The ability to make the credible limited 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 and output 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. The basic strategic default model, exempified in Arellano (2008), has no payment in default. Punishment is a trigger strategy, invoked when the sovereign does not repay her full contractual obligations. Therefore, she has no incentive to make a partial repayment in default. Other authors have used partial repayment in default to raise debt/gdp ratios, where the fraction of debt repayed can be exogenous (Schmitt-Grohe and Uribe 2017) or endogenous through the financial contract (Salomao 2017) or through debt renegotiation (Yue 2010). Arellano, Mateos-Planot and Rios-Rull (2013) add a cost to carrying unrepaid debt. These modifications of the basic strategic default model do not base partial repayment on ability-to-pay and therefore create different incentives characterizing debt dynamics. The limited commitment assumption 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 since they would incur a massive punishment. The "massive punishment" serves to implement the limited commitment as an equilibrium behavior. 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 absence of ability-to-pay, is excusable and acts as insurance. 7 The massive punishment would likely involve severe destruction to the financial system because violation of the commitment is an event that was anticipated with zero probability. A poor sovereign which has been unable to establish commitment would not face the same kind of punishment 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. 8 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, as for a poor country. The commitment puts a zero probability on strategic default, implying much larger financial disruption. 8

9 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 recession causes the default. 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. 9 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 which implements the limited commitment contract. Therefore a sovereign never chooses strategic default. 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. 9 Arellano, Mateos-Planos, and Rios-Rull (2013) also claim that a sovereign continues to borrow in default. 9

10 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) since the value of repayments in the marginal state equals the value of debt. 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 repayment equal to 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. 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) 10

11 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 assumption that the sovereign pays the minimum ( of debt and ability-to-pay in each ) ˆ state implies that payments in states with default A j=1 jf (j h) dj are positive, raising the price of debt and reducing the interest rate, compared with a model in which default entails zero payments. 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) 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) 11

12 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. 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, 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. Therefore, next period s debt is subject to a state-dependent upper bound. We refer to this upper bound as a debt limit 12

13 and note that it is increasing in the state. This Laffer curve behavior of the proceeds from borrowing is also present in strategic default models. Using equation (11), compare the effect of an increase in debt (D ) on proceeds from borrowing (q D ) in this model with that in the model of strategic default in which either there are no debt repayments in default ( A j=0; 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. Under strategic default, the value in equation (11) for Aˆ D, where we interpret Aˆ as debt repayments, must be negative with D suffi ciently larger than Aˆ, for the net value to exceed the punishment, justifying default. Therefore, the second term in equation (11) is large and negative, instead of zero, implying that an increase in borrowing creates a smaller increase in the proceeds from borrowing than under limited commitment. This is because the price of debt falls relatively more as debt rises.. This implied 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 limited-commitment contract, the sovereign is more likely to choose to borrow to smooth consumption even in the neighborhood of default. 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 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) = max D [u (c) + βev (y, D )]. 13

14 Since the sovereign defaults in states j < ˆ, and repays in others, we can rewrite the value function as { [ ˆ 1 ]} V (y, D) = max u (c) + β V (y, A(y )) f (j) dj + V (y, D ) f (j) dj. D j=1 j=ˆ 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. 10 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 abilityto-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 11 u (c) (q D ) j β c D j=ˆ ( u (c ) c ) f (j h) dj = 0, 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 10 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. 11 The term multiplying ˆ D vanishes since at ˆ, A (y ) = D. (15) 14

15 marginal utility of current consumption equals the expected marginal utility of future consumption, conditional on repayment, multiplied by β (1 + r ). 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. 12 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 Probability of Default The probability of default is the probability of transiting from current income state h to income state ˆ or lower, where income state ˆ is the largest income state in which default occurs, given D. Since income is autoregressive, the probability of transiting to a lower income value is higher the smaller is h, equivalently, the lower is current income. The probability of transiting to income state ˆ or lower, from a given state above, is also higher the larger is ˆ. The value for ˆ is increasing in debt. Therefore, the probability of transiting from income state h to income state ˆ or lower is decreasing in income (the higher h, the lower the probability of transiting to a low enough value) and increasing in debt (which makes ˆ high). Therefore, the probability of default is increasing in debt and decreasing in income. 12 With no repayments in default, (q D ) D is smaller, implying that its inverse is larger, raising u(c) c, thereby reducing consumption through smaller D. This is the effect of the large increase in the interest rate in the neighborhood of default, offsetting the effect of the Stiglitz risk-taking. 15

16 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. In the model, external debt and government debt are identical, while they clearly differ in the data. Actual governments can choose government debt, but not external debt. We must choose which to match, given our model. We seek to match total government debt. In our 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 could be more relevant. However, Reinhart and Rogoff (2008) found no statistical difference in the incidence of default to domestic versus foreign creditors, implying a difference in political will to repay domestic versus foreign nationals is not central to the default decision. We use value function iteration with the choice variable being next period s debt, conditional on the current value of debt, the current output state, and the equilibrium price of new debt. Default occurs whenever the stochastic endowment shock reduces the ability-to-pay below outstanding 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 The upper bound on the debt grid is larger than any 16

17 sovereign ever chooses, but the lower bound is a constraint. 13 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 endogenous 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 Deadweight Cost of Default A deadweight cost to default is a financial market friction, which reduces the value of payments creditors receive in default, without reducing what debtors pay. Our model has no deadweight loss in default, implying no financial market frictions. Its addition would imply a larger response of the interest rate to an increase in debt, an adjustment which seems necessary in the calibration. Therefore, we add a small deadweight loss to default, allowing the increase in the interest rate 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. The deadweight loss reduces the price of debt and raises the interest rate, requiring 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 13 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. Moments like mean debt are sensitive to the lower bound, and we therefore omit these moments. 14 In the strategic default model, a large degree of impatience keeps the sovereign from accumulating a large quantity of assets, implying an endogenous lower bound. 17

18 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 > 0 ), reducing D 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 due to the larger increase in the interest rate as debt rises. 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. 18

19 3.3 Remaining Parameters There are four remaining parameter values, ψ, δ, 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), 15 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 to our data 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 15 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). 16 The measure of debt/gdp in the data measures GDP at an annual rate. The model value measures GDP at a quarterly rate. For purposes of calibrating and simulating the model, we make the two consistent by multiplying the data value of debt/gdp by four to obtain the ratio with GDP expressed at a quarterly rate. We report values of debt/gdp and debt/mean GDP by reconverting to the more commonly used measures, which value GDP at annual rates. 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. 18 In the model, there is no trend and the mean value of output is unity. 19

20 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, δ, 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 1. 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 0.93 crisis timing and three data targets δ 0.56 crisis timing and three data targets β 0.99 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 20

21 data on the eve of the crisis and in the period of the crisis requires that ψ and δ 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 δ 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 an impatient sovereign. The calibrated value of β is very close to the inverse of the gross risk-free interest rate (0.9908). Our model does not require much impatience due to the Stiglitz-type risk-taking created with the high value of ω, implying a deadweight loss in financial markets of only 0.20 percent of the value of repayments. Additional risk-taking, as a result of debt accumulation due to impatience, is unnecessary. 19 Table 2 compares model values with those targeted in the data. In presenting these results, we express GDP at an annual rate to conform with standard presentations. The model fits the data well, matching values for average and crisis debt almost perfectly and for precrisis debt within percentage points. 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. 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. 21