Gambling for Redemption and Self-Fulfilling Debt Crises*

Transcription

1 Federal Reserve Bank of Minneapolis Research Department Staff Report May 211 Gambling for Redemption and Self-Fulfilling Debt Crises* Juan Carlos Conesa Universitat Autònoma de Barcelona imothy J Kehoe University of Minnesota, Federal Reserve Bank of Minneapolis, and National Bureau of Economic Research ABSRAC We develop a model of the sorts of sovereign debt crisis that have occurred recently in such European countries as Greece, Ireland, and Portugal he government sets its expenditure-debt policy optimally given a fixed probability of a recovery in fiscal revenues In doing so, the government can optimally choose to gamble for redemption, and the economy can be optimally driven to a level of debt with increased vulnerability to self-fulfilling debt crisis he model explains why, in contrast to the Mexican crisis of , where a loan package put together by US President Bill Clinton put an immediate end to the crisis, rescue packages put together by the European Union do not seem to have helped Greece, Ireland, or Portugal * We thank Isabel Correia and Pedro eles for helpful discussions he views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System

2 1 Introduction his paper develops a model of the sorts of sovereign debt crisis that we have witnessed in such European countries as Greece, Ireland, and Portugal he government sets its expenditure-debt policy optimally given a fixed probability of a recovery in fiscal revenues In doing so, the government can optimally choose to gamble for redemption, and the economy can be optimally driven into a situation of increasing vulnerability to speculative attacks in the form of self-fulfilling debt crisis Our paper provides a benchmark for analyzing government behavior in the face of fiscal pressures and a tool for testing the implications of alternative policy responses In our analysis, as in Cole and Kehoe (1996, 2), we characterize in a simple Markov structure the time consistent policy of a strategic government that is faced with non-strategic bond holders Figure 1: Debt to GDP ratios in Europe and the United States 12 1 United States 8 Euro zone Percent Year Source: OECD Economic Outlook 21 for the Euro one data and Economic Report of the President 211 for the US data he recent recession which continues in some countries and the policies intended to overcome it have generated very large government budget deficits and increases in government debt over all of the developed world In the Unites States, debt has gone up since 1

3 21, reaching 932 percent of GDP in 21 and is expected to reach 153 percent by 212 In the Euro area the level of government debt was going down systematically since the mid-nineties reaching a minimum in 27 of 659 percent of GDP Since then, it has grown up to 843 percent in 21 and is expected to reach 889 percent in 212 he pattern observed for the entire Euro area is common across most European countries, but it has been more extreme in countries like Greece, whose debt is up to 1259 percent of GDP in 21, or to 974 percent in Ireland Countries like Belgium or Italy, however, have debt to GDP ratios of 984 and 1195 percent respectively Portugal (with 827 percent) is about to enter a rescue package with the same level of debt as France, only slightly below that of Germany Figure 2: Debt to GDP ratios in selected European countries Italy 1 Greece 8 Percent 6 Germany Portugal 4 Spain 2 Ireland Year Source: OECD-Stat Looking at Figure 2, it is hard to argue that Portugal is subject to more pressures than Germany or that Greece is subject to more pressures than Italy In the context of our model, the answer is that countries like Greece, Ireland, Portugal and Spain have had budget deficits around 1 percent of GDP, or even more, in 29 and 21, while Germany and Italy experienced budget deficits of 3 and 5 percent, respectively he United States has also experienced these sorts of budget deficits around 1 percent and, until recently, did not appear to be subject to the 2

4 same sort of pressures he recent warning by Standard and Poors that it would downgrade US government debt serves as evidence for the mechanism in our model In our model, the crucial element putting countries at risk of suffering a self-fulfilling debt crisis is the drop in government revenues that occurs as the result of a recession in the private sector of the economy In terms of the fall in government revenues (deflated by the GDP deflator), we observe in Figure 3 that Greece, Ireland, and Portugal have all experienced much larger drops that has Germany, but so have Spain and Italy Our analysis indicates that Spain and Italy are also venerable to such crises, and the reason that such crises have not occurred yet in these countries is somewhat arbitrary Figure 3: Real government revenues in selected European countries 15 1 Germany Index 27 = Italy Portugal Greece Spain Ireland Year Source: OECD-Stat Our analysis shows that under certain conditions it is optimal for the government to gamble for redemption By this, we mean that the government does not undertake painful adjustments to reduce spending, hoping for a recovery of government revenues, and debt continues to increase Indeed, the government strategy follows a martingale gambling strategy that will send the economy into the crisis zone if the recovery does not happen soon enough Under other conditions, however, the government gradually reduces the level of debt to exit the crisis zone and avert the possibility of a liquidity crisis, as in Cole and Kehoe (2) 3

5 Notice that in our environment not running down debt (or running it up until default is unavoidable) is part of the optimal strategy under some circumstances In contrast, in their excellent overview of past default experiences Reinhart and Rogoff (29) argue that the reason some countries fail to adjust and are vulnerable to a potential crisis is because somehow both the governments and their lenders are fooling themselves into thinking that this time is different (this is indeed the title of their book) As such, countries vulnerability would be the result of selfdelusion and lack of rationality In contrast to their view, we provide a model in which such apparently irrational behavior can be an optimal response to either fundamentals or to expectations of a bailout by both borrowing governments and lenders that perfectly understand the risks of a crisis his paper is most related to those of Cole and Kehoe (1996, 2) In those papers, the authors provided a general equilibrium infinite horizon model of countries subject to the possibility of a self-fulfilling debt crisis every period he substantial difference is that in their framework debt crisis were liquidity crisis due to the inability to roll over debt that could arise under certain conditions As such, a decisive action of a third party providing a loan (or a bailout) would be enough to avert the problem Indeed, the model was intended to shed light on the financial troubles of Mexico in , and, in that occasion, the decisive action of the Clinton administration was enough to avert the crisis European Union packages for countries like Greece, Ireland, and Portugal have not had the same healing properties In fact, quite the opposite seems to be the case, with the spreads on bonds, relative to Germany s, continuing to rise despite the announcements and first implementations of the rescue packages his indicates more fundamental solvency problems than those present in a standard liquidity crisis of the type studied in Cole and Kehoe, as clearly discussed recently by Chamley and Pinto (211) for the Greek case Our model accommodates this issue Arellano (28) focuses on default incentives on international borrowing over the cycle, but does not determine the optimal behavior of government finances Similar frameworks have been used to analyze currency crisis following Calvo (1988) Our model establishes conditions under which a debt crisis can occur, and how that possibility shapes optimal government s behavior, but is silent about why at a particular point in time a crisis might or might not occur Indeed, once the government is in the crisis zone (and we show under what conditions a government will find it optimal to enter it) a potential crisis is 4

6 exclusively triggered by a non-fundamental random variable: a sunspot A very interesting related strand of the literature, starting with the work of Morris and Shin (1998), would relate the probability of such an event happening to lack of perfect knowledge within a global game framework While this strand of the literature focuses on why a crisis might happen at a particular point in time, we focus on the conditions under which a benevolent government might find it optimal to subject itself to such a vulnerable position Bailouts in the form of issuing credit at below market price to troubled economies (as those used in the presence of liquidity crisis) might or might not work In fact, the probability of a bailout might make it more attractive for a government to enter the crisis zone, becoming more vulnerable to a crisis Our model provides sharp predictions of when a third party bailout helps avert a liquidity crisis (as in the case of the Mexican crisis of ) and when such a strategy would only make things worse Since the response of the government is always optimal, however, the only policy prescription relate to the optimal form of the bailout A different scenario arises if one believes that a given government might be over-optimistic or trying to trick investors or the third parties capable of implementing the bailout with respect to the probability of a recovery It might be the case that an over-optimistic government is gambling for redemption, while it should be making sacrifices and decreasing its debt position A complete analysis of a game between two strategic players the government of the country and the third party, in the case of the European countries that we have discussed, the European Union, or more specifically, Angela Merkel is beyond the scope of this paper 2 General model he state of the economy in every period s= ( B, a, z-1, z) is the level of government debt B, whether or not the private sector is in normal conditions a = 1 or in a recession a =, whether or not default has occurred in the past z - 1, and the value of the sunspot variable z he country s GDP is 1-a 1-z y( az, ) = A Z y where 1 > AZ, > Before period, a = 1, z = 1 In period, a unexpectedly becomes a = and GDP drops from y = y to y Ay y = < In every period t, t = 1,2,, a t becomes 1 5

7 with probability p, 1> p > Once a = 1, it stays equal to 1 forever he drop in t productivity by the factor Z is the country s default penalty Once z t =, it stay equal to forever Here the default penalty occurs in the same period as the crisis We could have it occur in the next period, in which case 1-a 1- z-1 y( az, 1) A Z y - = While the results in our numerical experiments would change, this alternative assumption has no qualitative effect on our results Government tax revenue is q yaz (, ) where we assume, to keep things simple, that the tax rate is fixed he government s problem is to choose cgb,, ', z to solve V(; s p, p) = max u(, c g) + bev( s'; p, p) st c= (1- q) y( a, z) g+ zb= qy( a, z) + q( B', s; p, p) B' z = if z - 1 = Here z 1 is the decision not to default, and z is the decision to default We later specify the precise timing of the interaction of the government and international bankers, which determines the price of bonds qb ( ', spp ;, ) and the default decision z Some possibilities for ucg (, ) are ucg (, ) = (1- g)logc+ glogg ucg (, ) = (1- g)log c+ glog( g- g) ucg (, ) = log( c+ g-c- g), or we could impose even more curvature than that of the natural logarithm he last specification incorporates the idea that during a recession, when private consumption is low, the government has an incentive to increase public spending t In every period a random variable t is drawn from the uniform distribution on [,1] If 1, international bankers expect there to be a crisis and do not lend to the government if such a crisis would be self-fulfilling his allows us to set the probability of a self-fulfilling 6

8 crisis at an arbitrary level, 1, if the level of debt is high enough for such a crisis to be possible he timing within each period is like that in Cole and Kehoe (1996, 2): 1 t is realized, the aggregate state is st = ( Bt, at, zt- 1, zt), and the government chooses how much debt Bt 1 to sell 2 Each of a continuum of measure 1 of international bankers chooses how much debt bt 1 to purchase (In equilibrium, of course, bt 1 Bt 1) 3 he government makes it default decision z t, which determines y t, c t, and g t t 1 he crucial elements of this timing are that, first the government chooses how much debt B it wants to sell and this determines the price of the debt if there is no crisis and only later does the government choose where to default or not hat is the price of the debt depends on the amount of new debt being offered Bt 1, but where or not a crisis takes place depends on the amount of debt that has to be repaid B t 21 Bond prices International bankers are risk neutral with discount factor b so that the bond prices qspp (;, ) are determined only by the probability of default in the next period here is a continuum of measure one of bankers Each solves the dynamic programming problem he constraint b WbB (, ', sp ;, ) max x EW(( b', B'', s'; p, ) x qb ( ', sp ;, ) b' w zb ( ', sqb, ( ', sp ;, ); p, ) b x, b A A eliminates Ponzi schemes, but A is large enough so that the constraint does not otherwise bind We assume that the banker s endowment of consumption good w is large enough to rule out corner solutions in equilibrium here are four cutoff levels of debt: bapp ( ;, ), Bapp ( ;, ), a =,1: 7

9 1 If B b(; p, p), the government does not default when the private sector is in a recession even if international bankers do not lend, and, if B > b(; p, p), the government defaults when the private sector is in a recession if intentional bankers do not lend 2 If B b(1; p, p), the government does not default when the private sector is in normal conditions even if international bankers do not lend, and, if B > b(1; p, p), the government defaults when the private sector is in normal conditions if intentional bankers do not lend 3 If B B(; p, p), the government does not default when the private sector is in a recession if international bankers lend, and, if B > B(; p, p), the government defaults when the private sector is in a recession even if intentional bankers lend 4 If B B(1; p, p), the government does not default when the private sector is in a normal conditions if international bankers lend, and, if B > B(1; p, p), the government defaults when the private sector is in normal conditions even if intentional bankers lend Bond prices qb ( ', spp ;, ) are assumed to satisfy the property that, once default has occurred, international bankers do not lend: qb ( ',( Ba,,, z); p, p ) = Furthermore, during a crisis, international bankers do not lend: qb ( ',( Ba,,1, z); p, p ) = if B > b( a; p, p) and z> 1- p In addition, international bankers never lend an amount so large that the country would never repay: qb ( ',( Ba,,1, z); p, p ) = if B' > B(1; p, p), that is, even if the private sector recovers his last condition, however, we think of as being incorporated into the equilibrium bond price rather than as being a selffulfilling crisis 8

10 We will show that b(; p, p) < b(1; p, p), b(; p, p) < B(; p, p), b(1; p, p) < B(1; p, p), and B(; p, p) < B(1; p, p) Here, we first consider the case where b(1; p, p) < B(; p, p), that is, where b(; p, p) < b(1; p, p) < B(; p, p) < B(1; p, p) he first order condition for to the international bankers utility maximization problem implies that qb ( ', sp ;, ) EzB ( ', sqb, ( ', sp ;, ); p, ), which implies that ìï b if B' b(; p, p) b( p+ (1-p)(1- p) ) if b(; p, p) < B' b(1; p, p) qb ( ',( B,,1, z); p, p) = ï íb(1 - p) if b(1; p, p) < B' B(; p, p) bp(1 - p) if B(; p, p) < B' B(1; p, p) ï if B(1; p, p) < B' ïî ìï b if B' b(1; p, p) qb ( ',( B,1,1, z); p, p) = ï íb(1 - p) if b(1; p, p) < B' B(1; p, p) ï if B(1; p, p) < B' ïî Figure 4: Bond prices as a function of debt and conditions in the private sector 9

11 qb ( ', a ) qb ( ',1) qb ( ',) b () b (1) B () B (1) B ' Now consider the case where B(; p, p) < b(1; p, p), that is, where b(; p, p) < B(; p, p) < b(1; p, p) < B(1; p, p) Here the solution to the intentional bankers utility maximization problem implies that ìï b if B' b(; p, p) b( p+ (1-p)(1- p) ) if b(; p, p) < B' B(; p, p) qb ( ',( B,,1, z); p, p) = ï íbp if B(; p, p) < B' b(1; p, p) bp(1 - p) if b(1; p, p) < B' B(1; p, p) ï if B(1; p, p) < B' ïî here is yet a third case, which we will show is not degenerate, where b(; p, p) < b(1; p, p) = B(; p, p) < B(1; p, p) Here ìï b if B' b(; p, p) b( p+ (1-p)(1- p) ) if b(; p, p) < B' b(1; p, p) qb ( ',( B,,1, z); p, p) = ï í bp(1 - p) if b(1; p, p) < B' B(1; p, p) ï ïî if B(1; p, p) < B' 1

12 22 Definition of equilibrium An equilibrium is a value function for government V( s ) and policy functions B '( s ) and zb ( ', sqp, ;, ) and gb ( ', sqp, ;, ), a value function for bankers WbB (, ', sp ;, ) and policy correspondence b'( b, B', s; p, ), and a bond price function qb ( ', sp ;, ) such that 1 V() s and B '( s ) solve the government s problem at the beginning of the period: V( B, a, z, ; p, ) max u( c, g) EV( B', a', z, '; p, ) 1 st c (1 ) y( a, z( B', s, q( B', s; p, ); p, )) gb ( ', sqb, ( ', sp ;, ); p, ) zb ( ', sqb, ( ', sp ;, ); p, ) B yaz (, ) qb ( ', sp ;, ) B' 2 b'( b, B', s; p, ) solves the banker s problem and qb ( ', sp ;, ) is consistent with market clearing and rational expectations: B'( s) b'( b, B', s; p, ) qb ( ', sp ;, ) EzB ( ', sqb, ( ', sp ;, ); p, ) 3 zb ( ', sqp, ;, ) and gb ( ', sqp, ;, ) solve the government s problem at the end of the period: max ucg (, ) EV( B', a', z, '; p, ) st c (1 ) y( a, z) g zb y( a, z) qb' z or z 1, but z if z 1 3 Self-fulfilling liquidity crises Although we are able to prove some propositions about the equilibria of the general model, we need to resort to numerical examples to illustrate the possibilities and to do comparative statics analysis Before turning to the results for the general model, we study two special cases, where we can derive analytical characterizations of the equilibria in which we are interested he first is the case where a = 1, that is, where the private sector has recovered and where there is no incentive for the government to gamble for redemption his is the model of 11

13 Cole-Kehoe (1996, 2) without private capital Notice that we can easily modify the analysis of this case to study the limiting case where a and p, that is, where there is a recession but no possibility for recovery, simply by replacing y with Ay in what follows In this case, where self-fulfilling crises are possible, but where there is no incentive for the government to gamble for redemption, the optimal strategies of the government all involve either leaving debt constant or running it down to eliminate the possibility of a crisis In the next section, we consider the other extreme case, where recovery is possible but not self-fulfilling crises We start by assuming that p = Notice that, since a recovery has already occurred in the private sector, p is irrelevant o derive the optimal government policy, we solve he first order conditions are å s max b uc (, g) t' = t t ' t' st c = (1- q) y t' g + B = qy+ b B + t' t' t' 1 Bt ' = B Bt ' B(1; p, ) t ' b ug ((1- q) y, gt' ) = lt' l = t' 1 bl + t' he transversality condition is limt' lsb ³ t' + 1 he first order conditions imply that ug((1 - q) y, gt' ) = ug((1 - q) y, gt' -1) Consequently, g = + g = g ˆ t' 1 t' 1 B g ˆ B y b ( q ) = + - t' + 1 t' 12

14 Suppose that hen for Bs gˆ = qy-(1 - b) B = B Otherwise B t ' is explosive It is easy to show that too low of a ĝ results in a path B t ' that violates the transversality condition oo high of a ĝ results in a path for hits B(1; p, ) We will argue that this cannot be optimal We can calculate the value of being in state s= ( B, a, z 1, z) = ( B,1,1, z) as u((1 -q) y, qy-(1 -b) B) V( B,1,1, z; p, ) = - B t ' that he calculation of utility when default has occurred, when z =, is mechanical In this case B = and c= (1- q) Zy g = qzy Notice that, once a default has occurred, z and p are irrelevant Consequently, when s= ( B, a, z, z) = ( B,1,, z) -1 u((1 -q) Zy, qzy) V( B,1,, z; p,) =, Let us calculate b(1; p, ) Let V ( B, a, q; p, p ) be the value of not defaulting he utility of n repaying B even if the international bankers do not lend is bu((1 -q) y, qy) Vn ( B,1,; p,) = u((1 -q) y, qy- B) +, while the utility of defaulting V ( B, a, q; p, p ) is d u((1 -q) Zy, qzy) Vd ( B,1,; p,) = Consequently, b(1; p, ) is determined by the equation V ( b(1; p,),1,; p,) = V ( b(1; p,),1,; p,) n d 13

15 bu((1 -q) y, qy) u((1 -q) Zy, qzy) u((1 -q) y, qy- b(1; p,)) + = u((1 -q) Zy, qzy) -u((1-q) y, qy- b(1; p,)) = b ( u((1 -q) y, qy) -u((1-q) Zy, qzy) ) Let us now characterize B(1; p, ) We first find the stationary upper limit on debt s s B (1; p, ) and then show that B(1; p, ) = B (1; p, ) Suppose that international lenders set q = b when a = 1 hen, if the government decides to repay its debt while the value of defaulting is u((1 -q) y, qy-(1 -b) B) Vn ( B,1, b; p,) =, bu( qzy,(1 -q) Zy) Vd ( B,1, b; p,) = u( qzy,(1 - q) Zy + bb) + s Consequently, B (1; p, ) is determined by the equation s s V ( B (1; p,),1, b; p,) = V ( B (1; p,),1, b; p,) n s u((1 -q) y, qy -() B (1; p, )) s bu((1 -q) Zy, qzy) = u((1 - q) Zy, qzy+ bb (1; p,)) + d s s u((1 - q) Zy, qzy+ bb (1; p,)) -u((1-q) y, qy-(1 -b) B (1; p,)) b = s ( u((1 q) y, qy (1 b) B (1; p, )) u((1 q) Zy, qzy) ) s Suppose that, at t =, the government finds itself with B > B (1; p, ) hen it cannot be s optimal to run down its debt to B1 B (1; p, ) in one period, setting g = qy- B + bb 1 in t = and g = qy-(1 - b) B t 1 thereafter o demonstrate this, suppose, to the contrary, that 14

16 bu((1 -q) y, qy-(1 -b) B1 ) u((1 -q) y, qy- B+ bb1) + bu((1 -q) Zy, qzy) ³ u((1 - q) Zy, qzy+ bb ) + If we choose B 1 to maximize the left hand side of this inequality, the utility of not defaulting, we set B1= B his implies that bu((1 -q) y, qy-(1 -b) B ) u((1 -q) y, qy- B+ bb) + bu((1 -q) y, qy-(1 -b) B1 ) ³ u((1 -q) y, qy- B+ bb1) + s for any B1¹ B his, however, contradicts or assumption that B > B (1; p, ) Let us now consider the case where p > First, observe that b(1; p, p ) = b(1; p, ) Suppose that B > b(1; p, p) and the government decides to reduce B to b(1; p, p ) in periods, = 1,2,, he first order conditions for the government s problem imply that he government s budget constraints are g = g ( B ; p) t g ( B ; p) + B = qy+ b(1 - p) B 1 g ( B ; p) + B = qy+ b(1 - p) B 1 2 g ( B ; p) + B = qy+ b(1 - p) B -2-1 g ( B ; p) + B = qy+ bb(1; p, p) -1 Multiply each equation by ( b(1 - p)) t and adding, we obtain å å - 1 t 1 t 1 - t = = t= ( b(1 p)) g ( B ; p) B ( b(1 p)) qy ( b(1 p)) bb(1; p, p) 1 -( b(1 -p)) 1 -( b(1 -p)) g B B y b p 1 -b(1-p) 1 -b(1-p) -1 ( ; p) + = q + ( b(1 -p)) b (1;, p) 15

17 1 -b(1-p) -1 g ( B; p) = qy- ( B-( b(1 -p)) bb(1; p, p) ) 1 -( b(1 -p)) Notice that g ( B ; p) = lim g ( B ; p) = qy-(1 -b(1- p)) B We can compute the value V ( B; p ) of each of the policies of running down the debt in periods, = 1,2,, Letting V ( B; p ) be the value of the policy where there are still t t periods to go in running debt, we can write bpu((1 -q) Zy, qzy) V ( B; p) = u((1 - q) y, g ( B; p)) + b(1 - p) V- 1( B; p) + bpu((1 -q) Zy, qzy) V- 1( B; p) = u((1 - q) y, g ( B; p)) + b(1 - p) V-2( B; p) + bp ((1 ), ) 2 ( ; ) ((1 ), ( ; )) (1 ) 1 ( ; ) u -q V B u y g B V B Zy q p = - q p + b - p p + Zy b ((1 ), ) 1 ( ; ) ((1 ), ( ; )) u -q V B u y g B y q p = - q p + y Notice that g increases from g ( B; p ) to q y in period o calculate V ( B; p ), we use backwards induction V ( B ; p) = (1 + b(1 -p)) u((1 -q) y, g ( B ; p)) 2 bpu((1 -q) Zy, qzy) + bu((1 -q) y, qy) + b(1 -p) V ( B ; p) = (1 + b(1 - p) + ( b(1 -p)) ) u((1 -q) y, g ( B ; p)) 2 3 bpu((1 -q) Zy, qzy) + (1 + b(1 -p)) + ( b(1 -p)) 2 bu((1 -q) y, qy) 16

18 V ( B ; p) = (1 + b(1 - p) + ( b(1 - p)) + + ( b(1 -p)) ) u((1 -q) y, g ( B ; p)) b - p + b - p + + b -p 2-2 (1 (1 ) ( (1 )) ( (1 )) ) + ( b(1 -p)) -2 bu((1 -q) y, qy) bpu((1 -q) Zy, qzy) and, of course, V ( B; p) = V ( B; p) : ( b(1 -p)) 1 -( b(1 -p)) bpu((1 -q) Zy, qzy) p = - q p + V ( B ; ) u((1 ) y, g ( B ; )) 1 + b(1 - p) 1 + b(1 -p) + ( b(1 -p)) -2 bu((1 -q) y, qy) Notice that u((1 q) y, qy (1 b(1 p)) B ) bpu((1 q) Zy, qzy) V ( B; p) = b(1 -p) (1- b)(1 + b(1 -p)) o find B(1; p, p ), we solve év B p p V B p p V B p p p ù êë úû bu((1 -q) Zy, qzy) = u((1 - q) Zy, qzy+ b(1 - p) B(1; p, p))) max ( (1;, )), ( (1;, )),, ( (1;, ); ) Our arguments have produced the following analytical characterization of V( B,1,1, z; p, p ): ìï u((1 -q) y, Zy) if B b(1; p, p) 1 2 max év ( B; p), V ( B; p),, V ( B; p) ù êë úû if b(1; p, p) < B B(1; p, p), z 1-p V( B,1,1, z; p, p) = ï í u((1 -q) Zy, qzy) if b ( 1; p, p) < B B(1; p, p), 1- p< z u((1 -q) Zy, qzy) if B(1; p, p) < B ïî Some of the different possibilities for optimal government strategies which vary with the initial debt are illustrated in figure 5 17

19 Figure 5: Optimal debt policy with self-fulfilling crises B t B(1; p, p) always default crisis zone b(1; p, p ) t 4 Gambling for redemption without self-fulfilling crises Suppose now that a = and p = hat is, no self-fulfilling crises are possible, but the private sector is in a recession and faces the possibility p, 1 p, of recovering in every period We can also interpret this as the limiting case in which crises can occur, but the government and the international bankers assign probability to them We focus on the case where ucg (, ) is separable in c and g and later discuss other possibilities using a numerical experiment in which ucg (, ) = log( c+ g-c- g) For the case where ucg (, ) is separable, we argue that the optimal government policies is to increase its debt as long as a = In fact, if the country is unlucky in the sense that a = long enough, its government eventually defaults Consequently, the upper limits on the debt, B(; p,) and B(1; p, ) are crucial for our analysis Notice that, since B(1; p, ) = B(1;, ), because the probability of recovery does not matter once it has occurred, we have already derived a condition that determines B(1; p, ) 18

20 u((1 - q) Zy, qzy+ bb(1; p,)) -u((1-q) y, qy-(1 -b) B(1; p,)) b = ( u((1 -q) y, qy -() B(1; p, )) -u((1-q) Zy, qzy) ) Figure 7: Uncertainty tree with recession path highlighted a 1 a 1 a 1 a a 1 a a 1 a 1 a o determine B(; p,), we suppose that the government runs up its debt to some level B B(; p,), borrows B(; p,) < B' B(1; p,) at price b p, then repays the next period if the private sector recovers and on defaults otherwise a = he value of borrowing B(1; p, ) at price b p planning, repaying the current debt, and then repaying in the next period if the private sector recovers and on defaulting otherwise is Vn ( B,,1; p, ) = u((1 - q) Ay, qay + bpb(1; p, ) -B) æu((1 -q) AZy, qazy) bpu((1 - q) Zy, qzy) ö + b(1 - p) + è ç 1 -b(1-p) ()((1-p )) ø b p + u ((1-q) y, q y -() B (1; p,)) he value of borrowing B(1; p, ) at price b p and then defaulting is 19

21 Vd ( B,,1; p, ) = u((1 - q) AZy, qazy+ bpb(1; p, )) æu((1 -q) AZy, qazy) bpu((1 - q) Zy, qzy) ö + b(1 - p) + 1 b(1 p) (1 b)(1 b(1 p)) çè ø b p + u ((1-q) Zy, q Zy ) he equation that determines B(; p,) is, therefore, V ( B(; p,),,1; p,) = V ( B(; p,),,1; p,) n b p u((1 - q) Ay, qay+ bpb(1; p,) - B(; p,)) + u((1 -q) y, qy-(1 -b) B(1; p,)) b p = u((1 - q) AZy, qazy+ bpb(1; p,)) + u((1 -q) Zy, qzy) he government may, in fact, choose a lower level of the debt than B(1; p, ), but B(; p,) is the highest level of the debt B ' at which the government can borrow at price qb ( ', sp ;, ) optimal d Indeed, if the constraint B ' B(1; p,) does not bind, we can calculate the B ' by solving he first order condition is max u((1 - q) Ay, qay+ bpb' -B) æu((1 -q) AZy, qazy) bpu((1 - q) Zy, qzy) ö + b(1 - p) + 1 b(1 p) (1 b)(1 b(1 p)) çè ø b p + u ((1-q) y, q y -() B ') u ((1- q) Ay, qay+ bpb' - B) = u ((1-q) y, qy-(1 - b) B') g Letting B ˆ '( B ) be the solution to this problem, here are two cases: B'( B) min[ Bˆ '( B), B(1; p, )] g 2

22 1 he government chooses to never violate the constraint B B(; p,), and the optimal debt converges to B(; p,) if a sufficiently long 2 he government chooses in periods if a sufficiently long 41 Equilibrium with no default Let us first consider the case where government chooses to never violate the constraint B B(; p,) For this to be an equilibrium, two things must be true: 1 he expected discounted value of steady state utility at B = B(; p,) must be higher than that of defaulting after bankers have purchased B' = B(; p,) at price, u((1 -q) Ay, qay-(1 -b) B(; p,)) æu((1 -q) Ay, qay-(1 -b) B(; p, )) bpu((1 -q) y, qy-(1 - b) B(; p, )) ö + b(1 - p) + çè 1 -b(1-p) ()((1-p)) ø b p + u ((1-q) y, q y -() B (; p,)) ³ u((1 - q) AZy, qazy+ bb(; p,)) æu((1 -q) AZy, qazy) bpu((1 - q) Zy, qzy) ö + b(1 - p) + çè 1 -b(1-p) ()((1-p)) ø b p + u ((1-q) Zy, q Zy ) 2 he expected discounted value of steady state utility at B = B(; p,) must be higher than that of running up the debt one more time at price and defaulting otherwise, b p, repaying if the private sector recovers, 21

23 u((1 -q) Ay, qay-(1 -b) B(; p,)) æu((1 -q) Ay, qay-(1 -b) B(; p, )) bpu((1 -q) y, qy-(1 - b) B(; p, )) ö + b(1 - p) + çè 1 -b(1-p) ()((1-p)) ø b p + u ((1-q) y, q y -() B (; p,)), ³ u((1 - q) Ay, qay+ bpb'( B(; p,)) -B( ; p, )) æu((1 -q) AZy, qazy) bpu((1 - q) Zy, qzy) ö + b(1 - p) + çè 1 -b(1-p) ()((1-p)) ø b p + u ((1-q) y, q y -() B '( B (; p,))) where B'( B) min[ Bˆ '( B), B(1; p, )] If these two conditions are satisfied, the optimal government policy is the solution to the dynamic programming problem V B a u A y A y B B EV B a 1 a 1 a (, ) = max ((1- q) -, q - + b '- ) + b ( ', ') We write Bellman s equation explicitly as st B B(; p,) V( B, ) = max u((1 - q) Ay, qay+ bb' - B) + b(1 - p) V( B', ) + bp) V( B',1) he first order condition is V( B,1) = max u((1 - q) y, qy+ bb' - B) + bv( B',1) bu - q A y qa y+ bb- B = bev B a, 1-a 1-a g((1 ), ' ) B( ', ') while the envelope condition is V B a =-u - A y A y+ B- B 1 a 1 a B(, ) g((1 q) -, q - b ' ) he envelope condition implies V( B, a ) is decreasing in B A standard argument that the operator on the space of functions defined by Bellman s equations maps concave into concave implies that V( B, a ) is concave in B Now, using the first order condition, we can use standard arguments to show that the policy function for debt B '( Ba, ) is increasing in B while the policy function for government 22

24 spending gba (, ) is decreasing We can show that B '() and it is impossible for B '( B) unless the constraint B B(; p,) binds, which implies that B '( B, ) > B We have already argued that B '( B,1) the initial debt B = B Figure 6 illustrates some optimal government strategies as functions of Figure 7: Optimal debt policy gambling for redemption when B B(; p,) binds B t B(1; p, p) B(; p, p ) always default default unless a t 42 Equilibrium with eventual default Let us now consider the case where government chooses to violate the constraint B B(; p,) with its sale of debt in period, defaulting in period 1 unless the private sector recovers he optimal government policy along the bank of the uncertainty tree in which at is the solution to the finite horizon dynamic programming problem u((1 - q) y, qy+ () Bt + 1)) Vt( Bt,) = max u((1 - q) Ay, qay+ bbt+ 1- Bt) + b(1 - p) Vt+ 1( Bt+ 1,) + bp st B(1; p,) ³ B ³ B(; p,) + 1 B B(; p,) We solve this by backwards induction with the terminal value function 23

25 V ( B,) = max u((1 - q) Ay, qay+ bpb -B ) + 1 u((1 -q) Zy, qzy)) u((1 - q) y, qy+ () B + 1)) + b(1 - p) + bp st B B(1; p,) 1 We then choose the value of for which V( B,) is maximal As long the constraint B ³ + 1 B(; p,) binds, we can increase the value of V( B,) by increasing Figure 7: Optimal debt policy gambling for redemption when B B(; p,) does not bind B t B(1; p, p) B(; p, p ) always default default unless a t In figure 7, we illustrate two possibilities, which depend on B In one 1, and in the other 2 he algorithm for calculating the optimal policy on a grid of bond levels is a straightforward application of policy function iteration Since we work backwards from the period in which the government borrows at price p and will default in the next period unless a recovery of the private sector occurs, let us reverse our labeling of the value functions and define 24

26 V ( B,) = max u((1 - q) Ay, qay+ bpb' -B) u((1 -q) Zy, qzy)) u((1 - q) y, qy+ () B')) + b(1 - p) + bp st B(; p,) B B(1; p,) he steps of the algorithm are 1 Solve for the value function V ( B,) and the policy function B '( ) B on a grid of bonds B on the interval [ BB, (; p,)] We set the lower limit of B equal to any value, including a negative value In an application with a given initial stock of debt, we could set B B We have already solved this problem analytically he solution is B'( B) min[ Bˆ '( B), B(1; p, )] unless B '( B) B(; p, ), in which case B '( B ) B (; p, ) Consequently, B ˆ '( B) max B(; p, ), min[ B'( B), B(1; p, )] It turns out that the values of B for which B '( B) B(; p, ) are those for which it is not optimal to set Let t, and set B B(; p,) and 2 Solve for the value function V (,) t+ 1 B and the policy function B '( ) t 1 B in Bellman s equation u((1 - q) y, qy+ () B)) Vt + 1( B, ) = max u((1 - q) Ay, qay+ bb' - B) + b(1 - p) Vt ( B', ) + bp Let B t be the largest value of B for which Vt 1 ( B,) Vt( B,) 3 Repeat step 2 until B t B Let be such that B B We can prove that B B (;,) B 1 B B p Our 2 1 algorithm divides the interval [ BB, (; p,)] into subintervals [ BB, 1), [ B, B ),,

27 1 p If the initial capital stock [ B, B(;,)] B is in the subinterval [ B t, B t 1], then the optimal government policy is to increase B, selling debt B, B(; p,) < B B(1; p,), in period t 1, and defaulting in period t unless the private sector recovers he optimal sequence of debt is B, Bt 1'( B), Bt 2 Bt 1 B '( '( )), B( B1( ( Bt 1'( B)))) 26

28 5 Results of the general model For the general model in which a, 1 p, and 1, we can show that there are four qualitatively distinct ranges of the parameter space, as illustrated in figure 7 Figure 8: Four possible phase diagrams in general model case 1 case 2 case 3 case 4 b () b (1) B () B (1) B When debt is low enough, B b(; p, ), the government necessarily runs up its debt, gambling for redemption As debt becomes larger, however, the probability of a crisis increases and the price of bonds falls, as in figure 4 For high enough levels of debt, the benefits of running down debt to avoid a self-fulfilling crisis may dominate the benefits of gambling for redemption For small enough, this never happens, and the government eventually defaults unless the private sector recovers to a 1 Otherwise, debt will stop at one of the fixed points b(; p, p ), b(1; p, p ), or B(; p, p ) here are, of course, also degenerate cases where optimal debt is constant in an interval We will present numerical experiments for each of the four different cases and illustrate how they depend on the parameters A, Z, p, and 27

29 51 Calibration of the model and quantitative results In order to solve for the equilibrium we need to choose a functional form for the utility function We choose a log specification, ie for our benchmark scenario are the following: u c, g log( c) 1 log( g) he parameters we choose able 1: Parameter values in the Benchmark scenario Parameter Value arget A 9 Average gov t revenue loss (see Figure 3) Z 95 As in Cole and Kehoe (1996) p 2 Average recovery in 5 years 98 Real interest rate of safe bonds 2% 3 Real interest rate in crisis zone 5% 75 Consumers value c 3 times more than g 3 Government revenues as a share of output In Figure 9 we plot the policy functions, together with the debt thresholds (as described in Figure 8) Figure 9: Policy functions in good (left panel) and bad (right) times B'(B) 6 4 B'(B) First, notice the value of the thresholds he lower thresholds (in bad and good times, respectively) are for debt levels of 26 and 29 (relative to output in good times, which is 28

30 normalized to 1) Any debt above these levels makes the economy vulnerable to a panic, in which case a crisis would occur and default would be optimal Second, the upper thresholds (in bad and good times, respectively) are 138 and 141 Notice there are several countries that have levels of debt quite close to those Greece was at the end of 21 at 126, Italy was at 12, while Belgium or Ireland were around 1 If the economy is in good times then the optimal policy is to keep debt constant when the economy is in the safe region and decrease debt step by step when the economy is in the crisis zone his is exactly the policy prescription of Cole and Kehoe (2) In bad times the optimal policy is to gamble for redemption up to the point where the economy enters the crisis zone If the economy started in the crisis zone, then it would be optimal to decrease debt step by step to exit the crisis zone Notice that in our benchmark scenario we would be in Case 1, as described in Figure 8 Suppose now that the probability of a panic is very small, or alternatively that there exists an outside institution willing to bail out this economy by lending at the safe real interest rate (in our benchmark scenario 2%) hen, the upper thresholds go up to around 225, however the policy function shifts up, so that there is gambling for redemption all the way until the upper limit his situation is labeled as Case 3 in Figure 8 Figure 1: Policy function in bad times for small B'(B)

31 Notice that the existence of a potential bailout implies that it is optimal to gamble for redemption in situations where it wouldn t be optimal otherwise he specification of the economy as shown in able 1 is done having in mind one period as one year, and in our model economy we only allow for one-period debt Such a situation would mostly reflect developing economies that usually borrow very short term he crisis zone is between 3% and 14% GDP, which would imply that substantial amounts of debt would have to be refinanced every period, and as a consequence panics might happen at relatively low debt to GDP ratios However, most European countries borrow at longer maturities Including in the analysis debt of longer maturities would greatly complicate the analysis here is a simple way to think of it, though Imagine debt has a maturity of 6 years (which corresponds to the average maturity of debt for countries like Greece, Portugal or Spain) and is uniformly distributed over all the maturities hen, every period 17% (1/6) of the total debt must be refinanced According to that interpretation, we simply choose to rescale the model so that a period is 6 years We report the results in the following figure Figure 11: Policy functions in bad times with 6-year periods Normal Low B'(B) 1 B'(B) If the probability of a panic is equivalent to 3% in annual terms (left panel), then the optimal strategy is to gamble for redemption in the safe zone until the economy reaches a steady state at the lower threshold that separates the safe zone from the crisis zone If the economy starts in the crisis zone, then the optimal strategy would be to go back to the safe zone in one period he safe zone ends at 13% debt to GDP (measured in yearly terms) 3

32 Again here we observe that in response to a lower probability of a panic (or the access to financing at safe interest rates) the government finds optimal to gamble for redemption until the upper threshold of debt (which at the same time shifts to the left) his case is reported in the right panel hus, if there is a low probability of a panic (or some lender is willing to lend the government at the safe interest rate) then the optimal strategy is to gamble for redemption until debt reaches 18% of GDP in yearly terms 52 Sensitivity analysis In order to illustrate that many other scenarios are possible we will provide numerical results for various different scenarios in Figure 12 Figure 12: Policy function if probability of a recovery is high A = 7 Z = B'(B) 3 B'(B) P = 95 P = 99 31

33 B'(B) 1 B'(B) In the left panel of Figure 12 we show the changes when the crisis is much more severe than in the benchmark he output loss in bad times is 3% relative to good times Notice that the lower and upper bound in bad times shift to the left relative to the benchmark scenario he optimal policy generates three steady states In the safe zone the optimal policy prescribes running up the debt In the crisis zone there is a steady state between the two thresholds below which it is optimal to run down and above which it is optimal to run up until the safe zone if there is recovery Above that there is a small crisis zone where it is optimal to run down In the right upper panel of Figure 12 we show the changes when the penalty upon default is very large (5% of output) As usual if the economy is in the safe zone then the optimal policy is to run up until it hits the lower threshold If the economy is in the crisis zone then there are three areas First (for relatively low debt) run debt down, next keep it constant, and if debt is relatively high run it up until the steady state where you are indifferent between defaulting or not Finally in the bottom two panels of Figure 12 we examine the impact of increasing the probability of a recovery If the probability is 95% (left) then both lower debt thresholds are steady states, and eventually as the probability gets sufficiently close to 1% (right) the steady state is the lower debt threshold in good times 6 Interpretations and extensions 32

34 7 References Arellano, C (28), Default Risk and Income Fluctuations in Emerging Economies, American Economic Review, 98, Calvo, G (1988), Servicing the Public Debt: he Role of Expectations, American Economic Review, 78, Chamley, C and Pinto, B (211), Why Bailouts end Not o Work: An Example Motivated by Greece 21, he Economists Voice, 8, art 3 Cole, H and Kehoe, (1996), A Self-Fulfilling Model of Mexico s Debt Crisis, Journal of International Economics, 41, Cole, H and Kehoe, (2), Self-Fulfilling Debt Crises, Review of Economic Studies, 67, Morris, S and Shin, H (1998), Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks, American Economic Review, 88,