PRELIMINARY AND INCOMPLETE Capital Flows, Default, and Renegotiation in a Small Open Economy: Bargaining in a Greek Tragedy 1

Transcription

1 PRELIMINARY AND INCOMPLETE Capital Flows, Default, and Renegotiation in a Small Open Economy: Bargaining in a Greek Tragedy 1 M. Udara Peiris 2 Anna Sokolova 3 Dimitrios P. Tsomocos 4 October 6, We are grateful for helpful discussions with Kieran Walsh. Peiris work on this study has been supported by the Russian Academic Excellence Project First Version: May 23, ICEF, NRU Higher School of Economics, Russian Federation. upeiris@hse.ru 3 Department of Economics, NRU Higher School of Economics, Russian Federation asokolova@hse.ru 4 Saïd Business School and St. Edmund Hall, University of Oxford. Dimitrios.Tsomocos@sbs.ox.ac.uk

2 Abstract A distinguishing feature of the 2008 economic crisis in Europe were large numbers of non-performing loans to households and firms that in turn affected bank and ultimately sovereign solvency. Cross-border debt, originated by the private sector, was affected by national debt renegotiation outcomes. To study this, we develop a model where default and its associated costs originate from a decentralised economy but where the bargaining for national debt is made at a national level. We consider both the decision to default and the (convex) rate of default. The decision to default affects consumption smoothing possibilities while the convex rate of default determines feasible allocations. As a consequence pecuniary externalities pervade, levels of debt and default are high, and optimal macroprudential policies are countercyclical and higher rates of default are welfare improving. We perform a quantitative application to Greece. Keywords: open economy, capital flows, debt, default, renegotiation JEL Codes:

3 1 Introduction The growth of private sector capital flows over the last two decades has emphasised the need to study models of decentralised borrowing. On the other hand, the post-2008 period in Europe, Greece in particular, shows that private sector debt is often bailed out and negotiated by the sovereign. Greek banks were indebted to foreign lenders via the bond and interbank markets, and when spreads rose Greeks banks required state support to maintain solvency. The protracted negotiations on bailouts focused on repayment rates as much as fiscal and other institutional reforms. Our contribution to the study of cross border flows lies in the final aspect. In particular, that the pecuniary externality arising from decentralised borrowing affects convex default rates which in turn affect the level of debt undertaken. Consequently, optimal macroprudential policies need to consider both the incidence and rate of default, and we find that higher default rates are potentially welfare improving. This paper develops a finite horizon small open economy model with a risk neutral lender and a risk averse representative borrower that display three features: 1) private sector cross border flows (modelled as decentralized borrowing), 2) national debt negotiated by the government (modelled as Nash Bargaining), and 3) convex rates of default affecting allocations (market incompleteness provides the channel through which they affect equilibrium prices and allocations). These three features, combined, allow us to study the properties of an economy displaying constrained suboptimality, as in Geanakoplos and Polemarchakis (1986), strengthening the potential role for macroprudential policies. We develop the ideas of Jeske (2006), Wright (2006) and Kim and Zhang (2012) which highlight importance of and channels through which decentralised debt affects capital flows and default, Yue (2010) who introduces Nash bargaining as a means of determining default decisions in a centralised economy and Dubey et al. (2005a), Peiris and Tsomocos (2015) and Walsh (2015b) who describe the interaction between market incompleteness and default rates on equilibrium allocations. We first describe a decentralised economy where individual households wishing to default bargain collectively with a lender, taking the decisions of all other households and lenders as given. The outcome of this bargaining process is a repayment rate on outstanding debt that lenders and borrowers agree on. We then show that this economy is equivalent to a decentralised economy where households decide on their individual rates of default by evaluating the (non-pecuniary) costs and benefits of defaulting. As this equivalent economy fits into standard general equilibrium methodology, we then 1

4 use it to derive optimal macroprudential policies. Finally we compare our results with a centralised analogue of our economy and show that welfare and default rates are lower under a centralised economy. Recent years brought forth vast literature that addresses the issue of sovereign default risk within the context of small open economy models. The pioneering paper by Eaton and Gersovitz (1981) studies the risk of sovereign default in an endowment economy where a government chooses the amount of borrowing to smooth consumption of its residents. Every period the government may choose to default on its debt, causing permanent financial market exclusion. Later contributions modify this model to bring its business cycle characteristics closer to those observed in developing countries throughout recent years. Arellano (2008) calibrate their model to Argentina, assuming that default causes temporary market exclusion and output losses. The model is able to reproduce some business cycle characteristics, and account for the fact that defaults usually occur in recessions. In a similar context Aguiar and Gopinath (2006) compare quantitative performance of their model under permanent and transitory shocks to productivity and conclude that emerging markets volatility is mostly explained through permanent shocks. Most sovereign default models assume that the decisions on how much to borrow and how much to default are made by the government. This means that at the borrowing stage the effect of borrowing on the bond price is taken into account. Kim and Zhang (2012) introduces an alternative framework, in which the borrowing decisions are made by individual households, whereas the repayment decision is made by a benevolent government. This context yield an externality, as households do not internalize the effect of borrowing on default rates and the bond price. While most sovereign default models assume that default occurs by full amount, there are several papers that determine the default rates endogenously by modeling a renegotiation process. Bulow and Rogoff (1989) relate the decisions to default on sovereign debt to gains from international trade. The sovereign and the creditors take turns making renegotiation offers; the higher the country s gains from trade, the more it will repay the creditors. Yue (2010) studies the consequences of renegotiation by extending the framework of Eaton and Gersovitz (1981). In Yue (2010), after the endowment shock occurs, the borrower has a choice to either repay in full, initiate a renegotiation process or fully default, triggering financial autarky. If renegotiation occurs, the default rate will solve the Nash bargaining problem between the borrower and the lander. The higher the borrower s bargaining power and the value of autarky, the higher the default rate. Asonuma and Trebesch (2016) develop a framework in which the borrower may also choose to initiate a preemptive restructuring before the realization of the 2

5 fiscal shock. Compared to ex-post renegotiation, preemptive restructuring is less costly, but the losses are certain the borrower will only choose this option if the probability of default is high. Arellano and Bai (2014) introduce a model in which two borrowing countries simultaneously renegotiate with a common lender according to the Nash bargaining protocol. With three participants, the relative bargaining power of the lender decreases, which causes a reduction in repayment rates on both countries debts. In all papers mentioned above renegotiation happens without delay, as there is no asymmetry of information between the creditor and the borrower. Bai and Zhang (2012) construct a model that features renegotiation over debt repayment under asymmetric information: the government doesn t know the creditors reservation value. In this model delays may arise in equilibrium, as renegotiation process may take several periods. The authors show that the presence of secondary markets may reduce the delays, as prices will reveal part of the private information. Benjamin and Wright (2013) and Bi (2008) propose models in which delays occur because both creditor and the borrower optimally choose to wait for a future improvement in fundamentals. Literature that models sovereign defaults as a choice made by a benevolent government has to assume that defaults are associated with some kind of costs. This assumption is supported by empirical studies showing that defaults lead to capital market exclusion, a decrease in FDI flows (see Fuentes and Saravia 2010), a reduction in trade between borrowing and lending countries (see Rose 2005). Furthermore, Reinhart and Rogoff (2011) provide evidence linking sovereign defaults and banking crises, while Arteta and Hale (2008) show that defaults are associated with a reduction of foreign credit to the private sector. Most sovereign default models assume ad hoc default costs. A notable exception is Mendoza and Yue (2012). In their model default triggers capital market exclusion for both the sovereign and the private firms. Being unable to borrow, firms are forced to substitute imported inputs with inputs that do not require working capital financing. As inputs are not perfect substitutes, this triggers a productivity loss. In Gennaioli et al. (2014) default impairs liquidity of domestic banks that hold government bonds, which in turn reduces credit and investment. Our equivalent formulation of the Nash Bargaining process as a nonpecuniary default cost is related to the literature from Shubik and Wilson. (1977) and Dubey et al. (2005b) and applied in Tsomocos (2003), Goodhart et al. (2005) and Goodhart et al. (2006), De Walque et al. (2010) and Goodhart et al. (2016). Our model shares many features of Peiris and Tsomocos (2015) and Walsh (2015a) and Walsh (2015b). The former studies a two period large open international economy with incomplete markets and 3

6 default while the latter default in a small open dynamic incomplete markets economy. In these latter two papers, the marginal cost of default depends on the level of wealth, so the propensity to default depends on business cycle fluctuations. Section 2 outlines the underlying economy where default rates are the outcome of Nash Bargaining between the lender and domestic government then shows the equivalence with one where there is no negotiation, but where private agents internalise the cost of their private decisions to default. Section 3 develops this equivalent framework to study the welfare properties of our economy, calibrated to Greek data, and study optimal macroprudential policies. 2 Model 2.1 Nash Bargaining Framework We describe a decentralised economy along the lines of Yue (2010). There are three time periods t = {0, 1, 2}. Uncertainty is realised at the beginning of the second period and one of 2 states are realised, s = {H, L}. In the last period there is no further uncertainty. As a consequence, in total there are 5 date events, 1 in the first period, 2 in the second period and 2 in the last period. There are two countries, Home and Foreign. Home is inhabited a unit measure of identical households, h. In total there two goods each period: final output can be used to consume and invest as invested as capital for a competitive return next period R (the gross return), and labour: supplied inelastically in unit measure by households each period for a competitive wage w. Home countries is inhabited by a production technology (Firm) that transforms capital and labour into final output each period using a constant returns to scale technology and distributed any profits to Home households. Firms are endowed with a domestic productive technology A but can supplement this by (when available, the total factor productivity is A(1+ )) borrowing from abroad in the intraperiod capital market. Access to the intraperiod capital market is predicated on Home households having the ability to access foreign capital markets. Labour is supplied at the beginning of each period before production of final output occurs. Households enter date 0, having rented k 1 units of capital and supply a unit of labour l 0 to firms. In addition, to the total rental income from factors of production, households can borrow from the Foreign country an amount b 0 to be repaid in the second period at gross interest rate I 0. In the second period, households not wishing to honour the full debt due (b 0 I 0 ) 4

7 may send a request to the government to bargain with lenders on a reduced final repayment. The government proceeds with Nash Bargaining with the lenders taking the total requests of households as a given, and acting in their interests. The default rates, δ that are the outcome of the Nash Bargaining are then reported to households who decide whether to accept, in which case they repay the agreed rate, or do not, in which case they default fully but incur the costs of doing so. The cost of reneging is that they cannot access the foreign debt market in the second period. Bargaining occurs before any production decisions, so households that do not agree on the aggregate bargaining outcome can only rent their second period capital and last period labour investment to firms that are also denied the ability to access the capital markets. Thus the pecuniary costs of defaulting are two-fold: autarky and lower final period total factor productivity. Default does not occur in the last period as there is no uncertainty, so we ignore it. 1 The timing of events is in Figure 1 We restrict attention to rational expectations equilibria. 1 Formally speaking there are three possibilities: full default, in which case expecting this there would be nothing lent; partial repayment, in which case the default rate is priced into the interest rate and both borrowers and lenders are indifferent about the repayment rate; and full repayment. As only the latter two are possible and equivalent, we ignore default. 5

8 Figure 1: Timeline A 1 and A 2 become known Full repayment Renegotiation Full default Firms cannot borrow Firms cannot borrow t = 0 t = 1 t = 2 Households Firms Households Firms choose c 0, b 0, k 0 Households borrow Firms choose c 1, b 1, k 1 Firms produce Households choose δ 1 produce borrow choose c Firms Firm productivity depends on two components: a part A that is domestic and depends on the realization of the state of the world and a fixed part that has to be refinanced each period with external debt. Firm s profit in each period is: Π = A(1 + )K α L 1 α RK wl κ (1) Define output as Y = A(1 + )K α L 1 α Optimality implies: R = α Y K w = (1 α) Y L (2) (3) Foreign Household The economy is populated by identical households with lifetime utility: 2 2 c f 0 + (s) β τ c f τ (s) (4) s=1 τ=1 6

9 Where (s) > 0 is the probability of the corresponding state occurring and S s=1 (s) = 1. The budget constraints are: c f 0 + b f 0 + s f 0 = y f 0 (5) c f 1(s) + s f 1(s) = y f 1 (s) + I w 0 s 0 + (1 δ 1 (s))i 0 b f 0 (6) c f 2(s) = y f 2 (s) + I w 1 (s)s f 2(s) + I 1 (s)b f 2(s) (7) Foreign bonds, s f, are riskless and traded at an exogenous world interest rate I w. First order conditions necessary for a solution: 2 1 = βi 0 (s)(1 δ 1 (s)), (8) 1 = β s=1 2 (s)i1 w (s), (9) s=1 1 = βi w 1 (s). (10) No arbitrage implies I 1 (s) = I w 1 (s). (11) Home Households The economy is populated by identical households with lifetime utility: u(c 0 ) + S (s) s=1 2 β τ u(c τ (s)) (12) Where (s) > 0 is the probability of the corresponding state occurring and S s=1 (s) = 1. In the numerical section we use log utility. The budget constraints are: τ=1 c 0 + k 0 = w 0 l 0 + R 0 k 1 + b 0 (13) c 1 (s) + k 1 (s) + (1 δ 1 (s))b 0 I 0 = w 1 (s)l 1 (s) + R 1 (s)k 0 + b 1 (s) (14) c 2 (s) + b 1 (s)i 1 (s) = w 2 (s)l 2 (s) + R 2 (s)k 1 (s). (15) Labour is supplied inelastically l 0 = l 1 (s) = l 2 (s) = 1, s {H, L}. First 7

10 order conditions necessary for a solution: u(c 0 ) c 0 = βi 0 u(c 0 ) c 0 u(c 1 (s)) c 1 (s) No arbitrage implies Equilibrium = β 2 s=1 2 s=1 (s)(1 δ 1 (s)) u(c 1(s)) c 1 (s), (16) (s)r 1 (s) u(c 1(s)) c 1 (s), (17) = βi 1 (s) u(c 2(s)) c 2 (s). (18) R 2 (s) = I 1 (s). (19) As both economies are small open economies relative to the world, equilibrium requires that, given the solution to the Nash bargaining problem, the no arbitrage conditions hold I 1 (s) = I w 1 (s), (20) R 2 (s) = I 1 (s), (21) optimality (first order) conditions hold, asset markets clear b f 0 = b 0 (22) b f 1(s) = b 1 (s) (23) b f 2(s) = b 2 (s) (24) and expectations of agents are correct. The Nash Bargaining problem evaluates the difference between the value function of repayment and autarky for each agent, weighted by parameter. Formally, for each s {H, L}: { ] arg maxδ [0,1] [( 1 (H) 1 (s)) (( f δ 1 (s) = 1(s)) 1 if 1 (H) 1 (s) > 0 (25) 1, else. s.t. 1 (H) 1 (s) 0 (26) f 1(s) 0 (27) (28) 8

11 where, for the home household, 1 (H) 1 (s) = {V R} h 1(s) {V A} h 1(s), where and {V R} h 1(s) = u h (c h 1(s)) + βu h (c h 2(s)) = u h (w 1 (s)l 1 (s) + R 1 (s)k 0 + b 1 (s) k 1 (s) (1 δ 1 (s))b 0 I 0 ) + βu h (w 2 (s)l 2 (s) + R 2 (s)k 1 (s) b 1 (s)i 1 (s)) (29) {V A} h 1(s) = u h (w 1 (s)l 1 (s) + R 1 (s)k 0 k 1 (s)) + βu(w 2 (s)l 2 (s) + R 2 (s)k 1 (s)). (30) Similarly, for the foreign household f = {V R} f 1(s) {V A} f 1(s), where and {V R} f 1(s) = y f 1 (s) + I w 0 s 0 + (1 δ 1 (s))i 0 b f 0 s f 1(s) b f 1(s) + β{y f 2 (s) + I w 1 (s)s f 1(s) + I 1 (s)b f 1(s)}. (31) {V A} f 1(s) = y f 1 (s) + I w 0 s 0 s f A1 (s) + β{y f 2 (s) + I1 w (s)s f A1 (s)}. (32) Note that Autarky is the absence of financial trade between the foreign and domestic household. Under Autarky the foreign household can still trade with the rest of the world. The surplus for the foreign household is the higher initial wealth from repayment f = (1 δ 1 (s))i 0 b f 0 s f 1(s) b f 1(s) + s f A1 (s) + β{i1 w (s)s f 1(s) + I 1 (s)b f 1(s) I1 w (s)s f A1 (s)} (33) = (1 δ 1 (s))i 0 b f 0 (34) An interior solution to the Nash Bargaining problem requires the following expression to have a zero u h (c h 1(s)) c h 1(s) 1 A solution to this exists if s S, 1 (s), f 1(s) > 0. 1 (H) 1 (s). (35) f 1(s) We will proceed our analysis by considering an equivalent economy (in terms of allocation) where debtors incur a non-pecuniary punishment proportional to default rates. 9

12 2.2 Non-Pecuniary costs In this economy autarky is defined implicitly, but remains an out of equilibrium outcome. However as they are appropriately defined, at δ 1 (s) = 1 Define λ h 1(s) = B 1 (s) 1 1 (H) 1 (s) > 0 (36) f 1(s) 0 (37) (38) 1 (H) 1 (s). (39) f 1(s) Where all components in λ are the aggregate values of the economy and which households take as given. From definition, λ h 1(s) = + at δ 1 (s) = 1 and is an increasing function of 0 δ 1 (s) 1. Specify borrower s expected utility as: { S 2 } u(c 0 ) + (s) β τ u(c τ (s)) βλ 1 (H) 1 (s) max {δ 1 (H) 1 (s), 0} (40) s=1 τ=1 The budget and resource constraints of the economy are identical to before. There is one additional optimality condition per state for households, u h (c h 1(s)) b c h 1 (s) = λ 1 (s) 1(s) (41) = B 1 (s) 1 1 (H) 1 (s). f 1(s) (42) As b 1 (s) = B 1 (s), it is trivial that the condition in the non-pecuniary economy that determines the (interior) default rate, is identical to that of the Nash Bargaining model Existence of Equilibria (to be completed) We prove existence of equilibria in the equivalent non-pecuniary default costs (NPDC) economy. The proof proceeds in two steps. In the first step, we construct another economy with the same fundamentals but where default rates are exogenously given (we call it a δ-equilibria. We show that equilibrium generically exists for any vector of default rates. In the second step we introduce the additional first order condition in the equivalent non-pecuniary costs economy and show that there exists a vector of default rates that satisfies the first order condition for default. 10

13 Step One: Existence of δ-equilibrium Note that the underlying economy is an incomplete markets one with two assets. The vector of default rates affects the span of feasible allocations. Proof of existence is standard. Step Two: Existence of NPDC equilibrium default optimality condition Consider the household u h (c h 1(s)) b c h 1 (s) B 1 (s) 1 1(s) 1 (H) 1 (s). (43) f 1(s) Note that the first term reaches a maximum at δ 1 (s) = 1 while the second term reaches a minimum. Vice versa for δ 1 (s) = 0. There are three possibilities: (i) the expression is negative for all default rates, (ii) the expression is positive for all default rates, (iii) the expression fluctuates between negative and positive. In the third case, the intermediate value theorem tells us that a zero exists and an interior value of default rate occurs. If the expression is always negative then the bargaining outcome is full repayment while if the expression is always positive, the bargaining outcome is full default and autarky. However at full default, the surplus to foreign households (lenders) is zero so the second term is infinite. As the set of allocation is compact (see step 1), marginal utilities are bounded above by infinity and below by zero. Hence the expression cannot be positive at δ 1 (s) = 1. Therefore, the equilibrium default rates lie in the interval 0 δ 1 (s) < 1 and the NPDC equilibrium generically exists. 3 Macroprudential Policy and Taxing Capital Flows We first characterise the constrained efficient allocation in the δ-equilibrium. We perturb the vector of default rates to maximise the sum of utilities of foreign and home households. We then attempt to find tax rates on debt to support a vector of default rates that maximise the weighted sum of welfares. 3.1 Calibrated Model for Greece Lambda-equilibrium We start by investigating the equilibrium in the full model. We consider a model with risk-neutral lenders and assume that in period 1 there are two possible realizations of the productivity shock, A 1 (L) and A 1 (H). The 11

14 standard deviation of productivity shock is set to 1% matching 1% obtained by Bi,Traum (2012) after performing bayesian estimation of their model on Greek data. We set the risk-free interest rate to 2 % which compares to % observed for 10-year German bond yields (average for January 2010 to October 2014). We follow Angelopoulos et al. (2010) and set the share of capital income to We set the productivity loss from autarky to 40%, and we calibrate the bargain power to let the model produce equilibrium with partial default in one state ( = 0.24). We use logarithmic utility function. Given these model parameters, we find the lambda-equilibrium that produces partial default in low state with a default rate at 11.58%. The interest rate on Greek bonds in our model is 8.3% (compared to 12.81% observed from January 2010 to January 2015 for Greek 10-year bonds). 2 The debt-to-gdp ratio in the default state is 65% Borrowing limits and equilibrium default rates In this subsection we examine the relationship between model parameters and the properties of λ-equilibrium. In the model with pecuniary costs default rates are determined via the bargaining process: higher bargaining power of the borrower means more default. In the equilibrium this implies higher interest rate, and higher costs of debt repayment. Furthermore, if the bargaining power is too high, i.e. bargaining would yield full default in both states, then borrowing ex ante would not occur at all. These considerations suggest that small changes in may have large nonlinear effects on the equilibrium structure. Our model could potentially yield equilibria with either risk-free borrowing, partial defaults or no borrowing. The analysis below shows that changes in can affect the type of equilibrium the economy is facing. If in equilibrium a given state s is associated with partial default, then the marginal gain from defaulting in this state should equal zero. If, alternatively, state s is associated with full repayment, then the marginal gain from defaulting is non-positive. In other words, { u (c 1 (s))i 0 b 0 λ(s) = 0 for δ(s) > 0, (44) u (c 1 (s))i 0 b 0 λ(s) < 0 for δ(s) = 0. Consider an economy described above, with two possible realizations of productivity shock. Furthermore, consider δ-equilibrium in which default rates are exogenous. For this scenario, Figure 2 depicts levels of marginal gains in 2 Taken from St Louis FRED database Long-Term Government Bond Yields: 10-year: Main (Including Benchmark). 12

15 0 0 0 low and high states for different combinations of δ 1 (L) and δ 1 (H). Note that marginal gains from defaulting in a given state are higher, when default rate in the alternative state is substantial. For example, given δ 1 (L), the higher the δ 1 (H), the higher the gain from raising δ 1 (L). The intuition behind this result is as follows. Given default rate in state s, when expected default in states s 1 is high, equilibrium on the financial market implies higher compensation for the lenders, I 0 ; these higher costs of repayment raise the gain from defaulting in state s. Conversely, marginal gains from defaulting in a given state are lower, when default rate in this state is high. This result comes from the definition V R(s) V A(s) 1 δ 1 (s) of λ(s) associated with the bargaining process: λ(s) = 1. Intuitively, higher δ(s) raises the surplus of the borrower relative to the surplus of the lender, and increases the punishment λ(s). To determine λ-equilibrium, we compare marginal gains for high and low states. Figure 3a plots the zero level lines for marginal gains from defaulting in the two states, u (c 1 (s))i 0 b 0 λ(s) = 0, for = Note that the lines do not intersect: there does not exist a combination of positive δ 1 (L), δ 1 (H) < 1 that gives λ-equilibrium with partial defaults in both states. However, there is a pair δ 1 (L), δ 1 (H) consistent with partial default Figure 2: Marginal gains from defaulting (a) Low state (b) High state d H d H d L d L 1 Notes: Warm colors (lighter in grayscale) indicate positive marginal gain from defaulting, that is, u (c 1 (s))i 0 b 0 λ(s) > 0; cold colors (darker in grayscale) indicate negative marginal gain. The lines plot combinations of δ 1 (L), δ 1 (H) for which the marginal gain is zero. 13

16 in low state and full repayment in high state. To see this, consider the point δ 1 (L) = 0.07, δ 1 (H) = 0. In this δ-equilibrium the marginal gain from defaulting in low state is zero, while the marginal gain from defaulting in high state is negative thus, λ(l) and λ(h) support λ-equilibrium at δ 1 (L) = 0.07, δ 1 (H) = 0. Now let us compare this economy with the one in which the borrower has higher bargaining power, Figure 3b. As before, zero level lines of marginal gains do not intersect, indicating there is no equilibrium with partial defaults in both state. Unlike on Figure 3a, we now observe that the zero level line for default in high state is above that corresponding to the low state. Let us again consider the intersection between the horizontal axis, δ 1 (H) = 0, and the line u (c 1 (L))I 0 b 0 λ(l) = 0, given by δ 1 (L) = 0.19, δ 1 (H) = 0. For this combination of default rates the marginal gain from defaulting in high state is positive, meaning that δ 1 (H) = 0 cannot be supported as λ-equilibrium. In fact, for every combination of δ 1 (L), δ 1 (H) marginal gain from defaulting more in one of the states (or in both) is positive. This suggests that there is no λ-equilibrium with positive borrowing. This result suggests that an increase in the borrower s bargaining power may compromise her ability to borrow. On Figure 4 we further investigate the relationship between the borrower s bargaining power and the equilibrium outcomes. When the borrower s bargaining power is low, she is forced to fully repay the debt in both states, and Figure 3: Marginal gains and equilibrium (a) = 0.22 (b) = 0.28 d H u 0 (c 1 (H))I 0 b 0! 6(H) < u 0 (c 1 (L))I 0 b 0! 6(L) > 0 u 0 (c 1 (L))I 0 b 0! 6(L) = 0 u 0 (c 1 (H))I 0 b 0! 6(H) = 0 d H u 0 (c 1 (L))I 0 b 0! 6(L) > 0 u 0 (c 1 (L))I 0 b 0! 6(L) = 0 u 0 (c 1 (H))I 0 b 0! 6(H) = 0 u 0 (c 1 (H))I 0 b 0! 6(H) < u 0 (c 1 (H))I 0 b 0! 6(H) > u 0 (c 1 (H))I 0 b 0! 6(H) > u 0 (c 1 (L))I 0 b 0! 6(L) < u 0 (c 1 (L))I 0 b 0! 6(L) < d L d L 1 Notes: 14

17 borrowing is risk-free. When the bargaining power increases beyond 0.19, the default rate in the low state rises. At higher δ 1 (L) households wish to borrow more, as the repayment on the bonds now correlates negatively with the productivity shock, and issuing more debt promotes consumption smoothing across states L and H. As a result, consumers welfare increases. This effect remains in place as long as is below 0.24: once it increases beyond this thresholds, positive borrowing becomes unsustainable. Figure 4: Bargaining power, default and borrowing E0W h 0 /1(L) b Notes: One conclusion we can draw from this analysis is that, even though an increase in the bargaining power of the borrower may promote consumption smoothing and raise welfare, bargaining power that is too high will undermine the ability to borrow ex ante Pareto-improving delta-equilibria We now take model parameters from the previous exercise and compare welfare for different combinations of default rate in an equilibrium where default rates are set exogenously. We find that there exist combinations of default 15

18 Figure 5: Home welfare depending on δ 1 (L) and δ 1 (H) d H d L 1 1 Notes: The figure shows home welfare in the δ-equilibrium, depending on δ 1 (L) and δ 1 (H). The welfare is defined as E 0 W 0 = log(c 0 ) + βe 0 log(c 1 ) + β 2 E 0 log(c 2 ). rates that are pareto-improving. All these combinations are characterised by higher default rates in the low state; the combinations result in the same allocation. Compared to the lambda-equilibrium allocation, this allocation generates higher borrowing, and higher default rates. Figure 5 plots the welfare of the home country for all possible combinations of δ 1 (L), δ 1 (H). The maximum is obtained for a combination of default rates that are in linear relation to each other. Because for all the pareto-optimum combinations of default rates we have δ 1 (H) < δ 1 (L), we normalize δ 1 (H) = 0 and repeat the exercise. Figure 6 gives a 2-dimentional plot of welfare for different δ 1 (L), assuming δ 1 (H) = 0. Maximum welfare is obtained at δ1(l) = This level of default is not feasible in λ-equilibrium: at δ 1 (L) = 0.29 the marginal gain from defaulting in low state, u(uh (c h 1 (L)) δ 1 I 0 b 0 λ 1 (L), is negative: the utility loss λ 1 (L) 16

19 u 0 [c1(h)] b0i0! 6(H) u 0 [c1(l)] b0i0! 6(L) outweighs the marginal benefit associated with higher consumption. Thus, under δ1(l) = 0.29 the households would like to reduce the default rate. The λ-equilibrium is obtained at ˆδ 1 L = 0.12, where the marginal gain from defaulting in low state equals zero, and the marginal gain from defaulting in high state, where δ 1 (H) = 0, is negative. Table 1 compares equilibrium values of the endogenous variables in the λ-equilibrium with the δ-equilibrium that maximizes home welfare. In the δ-equilibrium that maximizes home welfare consumption smoothing is complete. Figure 6: Home welfare depending on δ 1 (L) E0W h / $ 1(L); max E 0 W h / 6 1 (L); 6! equilibrium / 1 (L) Notes: The figure shows home welfare in the δ-equilibrium, depending on δ 1 (L) and assuming δ 1 (H) = 0. The welfare is defined as E 0 W 0 = log(c 0 ) + βe 0 log(c 1 ) + β 2 E 0 log(c 2 ). 17

20 Table 1: Comparing λ-equilibrium and δ-equilibrium with maximum welfare Variable λ-equilibrium max E 0 W δ-equilibrium Welfare home Welfare foreign b δ 1 (L) I k c c 1 (L) c 1 (H) Notes: The table presents Macro-prudential policy We now try to find a combination of policy tools that can pareto-improve the lambda-equilibrium allocation. We introduce consumption subsidies that are financed through a lump-sum tax: c 0 (1 τ c 0) + τ 0 + k 0 = w 0 l 0 + R 0 k 1 + b 0 (45) c 0 τ c 0 = τ 0 (46) We perform a search for optimal τ0. c Figure 7 depicts home welfare depending on the subsidy. Home welfare is increasing over τ c.?? depicts the marginal gain from defaulting in high state depending on the subsidy. Note that as subsidy rises, so does the marginal gain from defaulting in the high state. At τ c = ɛ we have u (c 1 (H))b 0 = λ 1 (H) raising the subsidy further means that equilibrium with default in low state no longer exists. As we only search for pareto-improvements within the class of equilibria that generate default in low state, we do not consider equilibrium allocations for τ c > Thus, within the class of equilibria for which δ 1 (L) 0, δ 1 (H) = 0 the partooptimal subsidy is found at τ c = therefore, a pareto-improving policy subsidizes consumption in period 0. Table 2 compares the optimal allocation to the λ-equilibrium with no consumption subsidy. In the optimum allocation, consumption is redistributed toward present, and the agents borrowing is higher. Higher borrowing means that at each level of δ 1 (L) the difference between {V R} L 1 and {V A}L 1 becomes lower, as the amount the agents have to repay rises. In the risk-neutral λ-equilibrium 18

21 Table 2: Comparing λ-equilibria depending on consumption subsidy Variable τ c = 0 τ c = 0.01 Welfare home Welfare foreign b δ 1 (L) I k c c 1 (L) c 1 (H) Notes: The table presents... the cost of defaulting, λ 1 (L), is determined via: λ 1 (L) = 1 V RL V A 1 (L) 1 δ 1 (L) (47) Lower {V R} L 1 {V A}L 1 gives the home country an advantage in the bargaining process: the difference between renegotiation and autarky for home country becomes smaller, so if the renegotiation fails they would lose less. As a result, the cost of defaulting becomes smaller, which leads to higher equilibrium default rate. Therefore, taxing period 0 consumption gives the home country an advantage in the bargaining process. 19

22 Figure 7: Home welfare depending on τ c E0W h 0 u 0 [c1(h)] b0i0! 6(H) = c #10-3 Notes: The figure shows home welfare in the λ-equilibrium, depending on τ c and assuming δ 1 (H) = 0. The welfare is defined as E 0 W 0 = log(c 0 ) + βe 0 log(c 1 ) + β 2 E 0 log(c 2 ). 4 Centralized Economy 4.1 Solution We now consider a centralized version of our economy in which the government chooses the amount of borrowing and internalizes its effect on bond prices and default rates. We assume that the investors are risk-neutral. For brevity we will consider an economy with log utility and only two realizations of productivity shocks: A 1 (H) and A 1 (L), we assume that default occurs in the low state. In this case expected repayment rate is: E 0 (1 δ 1 ) = 1 π(l)δ 1 (48) where π(l) and π(h) are probabilities of low and high states. The risky interest rate is given by: I w I 0 = (49) 1 π(l)δ 1 The planner takes into account the way the arbitrageur sets δ 1 : u (c 1 (L))b 0 I 0 = 1 {V R} h 1 {V A}h 1. (50) 1 δ 1 (L) Combining (50) and (49), and applying the above definition of the risky interest rate, we obtain the repayment rate in the low state,: 1 δ 1 (L) = 1 (V DR V DA )(1 π(l)) u (c 1 (L))b 0 I w 1 (V DR V DA )π(l) 20 (51)

23 The risky interest rate is: [ u (c 1 (L))b 0 I w 1 I 0 = (V DR V DA )π(l) ] u (c 1 (L))b 0 (1 π(l)) (52) We write the maximization problem for the case when the equilibrium entails default in low state and repayment in high state: L = u(c 0 ) + βπ(l) [u(c 1 (L)) + βu(c 2 (L))] + β(1 π(l)) [u(c 1 (H)) + βu(c 2 (H))] + [ +µ 0 A0 (1 + )k 1 α + b 0 (c 0 + k 0 ) ] + [ +µ 1 (L) A 1 (L)(1 + )k0 α + b 1 (L) (c 1 (L) + k1 L + 1 {V R} L 1 {V A}L 1 u (c 1 (L)) +µ 1 (H)(A 1 (H)(1 + )k0 α + b 1 (H) [ ] u (c 1 (L))b 0 I w 1 ({V (c 1 (H) + k1 H R}L 1 {V A}L 1 )π(l) + )) + u (c 1 (L))(1 π(l)) +µ 2 (s) [A 2 (s)(1 + )k 1 (s) α (c 2 (s) + b 1 (s)i1 w )] + ] +γ [{V R R} L 1 (u(c 1(L)) + βu(c 2 (L))) + ] +γ [{V A A} L 1 (α(1 + αβ) ln k 0 + Σ), for s = H, L and Σ = (1 + αβ) ln A 1 (L) + β ln A 2 (L) + ln FOCs for consumption: (αβ) αβ (1+αβ) 1+αβ. L c 0 = u (c 0 ) µ 0 = 0 (53) L c 1 (L) = βπ(l)u (c 1 (L)) µ 1 (L) γ R u (c 1 (L)) + L c 1 (H) L c 2 (L) L c 2 (H) +µ 1 (L) 1 µ 1 (H) 1 {V R} L 1 {V A}L 1 [u (c 1 (L))] 2 u (c 1 (L)) π(l) {V R} L 1 {V A}L 1 1 π(l) [u (c 1 (L))] 2 u (c 1 (L)) (54) = β(1 π(l))u (c 1 (H)) µ 1 (H) (55) = β 2 π(l)u (c 2 (L)) µ 2 (L) γ R βu (c 2 (L)) (56) = β 2 (1 π(l))u (c 2 (H)) µ 2 (H) (57) ) ] + 21

24 FOCs for borrowing and capital inversment: L = µ 0 µ 1 (H)I w 1 b 0 1 π(l) L = µ 0 + µ 1 (L)αA 1 (L)(1 + )k0 α 1 + k 0 L b 1 (L) L k 1 (L) L b 1 (H) L k 1 (H) (58) +µ 1 (H)αA 1 (H)(1 + )k0 α 1 A α(1 + αβ) γ (59) k 0 = µ 1 (L) µ 2 (L)I w 1 (60) = µ 1 (L) + µ 2 (L)αA 2 (L)(1 + )k 1 (L) α 1 (61) = µ 1 (H) µ 2 (H)I w 1 (62) = µ 1 (H) + µ 2 (H)αA 2 (H)(1 + )k 1 (H) α 1 (63) FOCs for values of repayment and autarky: L {V R} L 1 L {V A} L 1 With some algebra we obtain: = µ 1 (L) 1 +µ 1 (H) 1 = µ 1 (L) 1 µ 1 (H) 1 1 u (c 1 (L)) + (64) 1 π(l) u (c 1 (L)) 1 π(l) + γr (65) 1 u (c 1 (L)) 1 π(l) u (c 1 (L)) 1 π(l) + γa (66) µ 0 = u (c 0 ) (67) µ 1 (L) = 1 βπ (c 1 (H)) + u (c 1 (L)) L (68) 1 + u (c 1 (L)) βi w u (c 2 (L)) The first-order condition for borrowing b 0 is: u (c 0 ) = βu (c 1 (H))I w (69) Compared to decentralized economy, in this version the planner understands that b 0 has only an indirect effect on consumption in the low state, as the 22

25 repayment (1 δ 1 )b 0 I 0 is pinned down by the planner thus, it does not enter into the planner s decision directly. The first order condition for capital then reads: u (c 0 ) = βπαa 1 (L)(1 + )k α 1 0 u (c 1 (L)) [ 1 u (c 1 (H)) u (c 1 (L)) u (c 1 (L)) βi w u (c 2 (L)) +β(1 π(l))αa 1 (H)(1 + )k α 1 0 u (c 1 (H)) + +βπ(l) α(1 + αβ) 1 k 0 1 u (c 1 (H)) βi w u (c 2 (L)) 1 + u (c 1 (L)) βi w u (c 2 (L)) ] + (70) Compare this with the decentralized case. In the decentralized economy capital influences consumption only through its return, A 1 (s)(1 + )αk0 α 1. In the centralized version the planner takes into account the effect of capital accumulation on the interest rate and amount of repayment. Raising capital means changing the gap between values of default and repayment, {V R} L 1 {V A} L 1. On the one hand, higher k 0 means higher value of renegotiation, {V R} L 1. On the other hand, it also implies higher value of autarky, {V A}d 1. By changing the gap the planner influences both the default rate and the interest. For the borrowing in period 1 in the low state we have that: u (c 1 (L)) = βi w u (c 2 (L)) (71) 1 u (c 1 (H)) + u (c 1 (L)) 1 u (c 1 (L)) {V R}L 1 {V A}L 1 u (c [u (c 1 (L))] 2 1 (L)) 1 u (c 1 (H)) + u (c 1 (L)) 1 u (c 1 (H)) {V R}L 1 {V A}L 1 u (c [u (c 1 (L))] 2 1 (L)) The value of borrowing in low state reflect its impact on current and future consumption, the value of repayment {V R} L 1 and the amount of repayment, (1 δ 1 )b 0 I 0. The remaining first-order conditions are similar to those observed in a decentralized economy setup: I w = αa 2 (L)(1 + )k 1 (L) α 1 (72) u (c 1 (H)) = βi w u (c 2 (H)) (73) I w = αa 2 (H)(1 + )k 1 (H) α 1 (74) 4.2 Centralized Economy: Numerical Results We calculate the centralised outcome of the economy presented earlier. Figure 8 depicts partial equilibrium in a centralized economy for different values 23

26 of b 0. The solution of this model is achieved at δ (L), where expected welfare reaches its maximum. Note, however, that at δ 1 = δ 1(L) the marginal gain from defaulting in high state is positive: that is, in high state the arbiter would allow the country to default by positive amount. Thus, δ 1 = δ 1(L) can only be an equilibrium outcome if the government can commit to repaying fully in high state. Figure 8: Centralized economy solution u 0 [c1(h)] b0i0! 6(H) u 0 (c0)! -I w 0 u0 (c1(h)) /1(L) E0W h / $ 1 (L); max E 0 W h b 0 Notes: Figure 9 compares partial equilibria (less the first-order condition for b 0 ) for centralized and decentralized economies. For each b 0 centralized model yields higher default rate and higher expected welfare. Table 3 compares the decentralized λ-equilibrium, the maximum achieved under δ-equilibrium with the centralized λ-equilibrium with commitment (over δ 1 (H) = 0). The centralized economy solution yields higher expected welfare: by borrowing more, the planner is able to influence the outcome of renegotiation towards more default: with high b 0 the difference between 24

27 /1(L) Figure 9: Centralized vs. Decentralized E0W h / 6 1 (L); 6! equilibrium / $ 1 (L); max E 0 W h 0 Centralized Decentralized b 0 Notes: values of renegotiation and autarky is low, which gives the borrower an advantage in the bargaining process. However, the planner is constrained by the fact that δ 1 (L) has to be determined via a bargaining gain, which prevents full consumption smoothing (achieved in optimal δ-equilibrium). The difference between centralized and decentralized solutions arises because choices that affect consumption also affect the solution of the Nash Bargaining problem. In the decentralized economy households do not internalize this link, and their choices over borrowing and capital investment are affected only by rates of return that they observe. But decisions concerning capital accumulation and borrowing affect the gain from renegotiation over debt repayment, which in turn affects the bargaining outcome. When home country has little to gain from renegotiating with the creditors, it repays less. By choosing lower capital and higher borrowing the planner can reduce gains from renegotiation and improve home country s strategic position in the bargaining process. Kim and Zhang (2012) also compare models with defaults under decen- 25

28 Table 3: Comparison with the centralized economy Variable Decentralized λ-eqm Optimal δ-eqm Centralized Economy Welfare home Welfare foreign b d L I k c c 1 (L) c 1 (H) Notes: The table presents... tralized and centralized borrowing. In their study the difference in the outcomes arises because under decentralized borrowing household cannot internalize the effect of borrowing on the bond price. Our model also features this effect. Unlike Kim and Zhang (2012), we assume that the default rate depends on the outcome of renegotiation, and as a result our model can generate partial defaults in equilibrium. In a centralized version of the model the planner takes into account the effect of borrowing on both the bond price and the repayment rate. 26

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