Temi di discussione(Working papers)Sovereign debt and reserves with liquidityand productivity crisesby Flavia Corneli and Emanuele TarantinoNumber 1012 - June 2015

The purpose of the Temi di discussione series is to promote the circulation of workingpapers prepared within the Bank of Italy or presented in Bank seminars by outsideeconomists with the aim of stimulating comments and suggestions.The views expressed in the articles are those of the authors and do not involve theresponsibility of the Bank.Editorial Board: Giuseppe Ferrero, Pietro Tommasino, Piergiorgio Alessandri,Margherita Bottero, Lorenzo Burlon, Giuseppe Cappelletti, Stefano Federico,Francesco Manaresi, Elisabetta Olivieri, Roberto Piazza, Martino Tasso.Editorial Assistants: Roberto Marano, Nicoletta Olivanti.ISSN 1594-7939 (print)ISSN 2281-3950 (online)Printed by the Printing and Publishing Division of the Bank of Italy

SOVEREIGN DEBT AND RESERVESWITH LIQUIDITY AND PRODUCTIVITY CRISESby Flavia Corneli* and Emanuele Tarantino**AbstractDuring the recent financial crisis developing countries have continued to accumulateboth sovereign reserves and debt. To account for this empirical fact, we model the optimalportfolio choice of a country that is subject to liquidity and productivity shocks. Wedetermine the equilibrium level of debt and its cost through a contracting game between acountry and international lenders. Although raising debt increases the sovereign exposure toliquidity and productivity crises, the simultaneous accumulation of reserves can mitigate thenegative effects of such crises. This mechanism rationalizes the complementarity betweendebt and reserves.JEL Classification: F34, F40, G15, H63.Keywords: sovereign debt, international reserves, liquidity shock, productivity shock,strategic default.Contents1. Introduction .......................................................................................................................... 52. The model ............................................................................................................................ 92.1 The lending game ......................................................................................................... 92.2 The investment game and the productivity shock ...................................................... 102.3 The liquidity shock ..................................................................................................... 112.4 The decision to default ............................................................................................... 123. Model solution ................................................................................................................... 123.1 Benchmark economy without liquidity shock ............................................................ 133.2 Economy with liquidity shock .................................................................................... 143.3 Equilibrium definition ................................................................................................ 163.4 Equilibrium analysis ................................................................................................... 164. Numerical simulations ....................................................................................................... 205. Foreign lenders' intervention ............................................................................................. 236. Conclusions ........................................................................................................................ 26References .............................................................................................................................. 27Appendix: Proofs .................................................................................................................... 29Figures .................................................................................................................................... 42_______________________________________* Bank of Italy, Directorate General for Economics, Statistics and Research.** University of Mannheim, Department of Economics.

1 Introduction 1During the recent financial crisis several developing countries have kept accumulating bothsovereign debt and reserves. These include Chile, Colombia, Malaysia, South Africa and Turkey,among others (Figure 1). The return on reserves is significantly lower than the cost of sovereigndebt (Rodrik, 2006), thus this piece of evidence has posed a puzzle (Alfaro and Kanczuk, 2009;Obstfeld et al., 2010): why don’t countries repay debt instead of raising reserves? We develop astatic model of optimal portfolio choice of a country that is subject to liquidity and productivityshocks. We find that, although raising debt increases the country’s exposure to liquidity andproductivity crises, the contemporaneous accumulation of reserves mitigates the consequencesof these crises on sovereign welfare. We therefore show that, in the sovereign’s optimal strategy,reserves can complement debt.Figure 1: External Debt and Reserve Accumulation — Average % Variation 2008-2011. Source:World Bank WDI.In principle, reserves and debt can be substitute means to address liquidity crises.decreasing its level of debt, a country reduces its exposure to liquidity shocks. By accumulatingreserves, the country holds resources that can be liquidated and injected if a liquidity shockoccurs. Yet, Rodrik (2006) and Mohd Daud and Podivinsky (2011) show that the accumulationof reserves is not accompanied by a reduction of debt. Moreover, Dominguez et al. (2012),Bussière et al. (2014), and Broner et al. (2013a), among others, document that during therecent crisis countries depleted their sovereign reserves and then rapidly replenished them. Thisevidence suggests that reserves and debt can be in a relationship of complementarity.We analyze the problem of sovereign reserves and debt accumulation in a model with liquidityand productivity shocks that draws on Bolton and Jeanne (2007). A risk-neutral country borrows1 This paper benefited from comments by Arpad Abraham, Mark Aguiar, Joshua Aizenman, Javier Bianchi,Patrick Bolton, Nicola Borri, Christopher Carroll, Fabio Castiglionesi, Giancarlo Corsetti, Bill Craighead, MarcoDella Seta, Andrea Finicelli and Nicola Gennaioli. Finally, we thank the participants at the American EconomicAssociation annual meeting 2012 (Chicago), and at the University of Bologna seminars. The views expressed arethose of the authors and do not necessarily reflect those of the Bank of Italy.By5

from international lenders and decides on the investment of these resources into a liquid (reserves)and an illiquid (production technology) asset. Debt and reserves are chosen by the country,instead international lenders set the interest rate on debt. The value of sovereign output dependson the productivity of the country’s technology and the share of borrowed resources that thesovereign invests in public expenditure; that is, those resources that are not invested in reserves.The interaction between the sovereign and international lenders is shaped by two frictions. First,the country cannot commit not to default on debt (Aguiar and Amador, 2015). Second, thecountry does not have access to additional borrowing in the event of a liquidity shock. 2 By thesetwo frictions, we introduce asset incompleteness in our model. 3A simple trade-off determines the equilibrium value of reserves.On the one hand, theydistract resources from the production technology. On the other hand, they can be injected inthe event of a liquidity crisis to avoid default. Moreover, consistently with the literature andprevalent anecdotal evidence, they cannot be seized by investors should the country decide todefault. The trade-off behind the accumulation of debt is as follows. By increasing its borrowingthe country holds more resources for investment. However, issuing debt raises the country’sexposure to liquidity and productivity crises. Indeed, debt increases both the likelihood that thecountry is hit by a liquidity shock and the probability that the due repayment to lenders fallsshort of realized output, thereby rendering more likely a default on the productivity shock.To establish the relationship between reserves and debt, we solve for two distinct set-ups.First, we consider a benchmark economy without liquidity shock and characterize the conditionsdetermining the equilibrium value of debt and reserves. In this environment, the motive forreserve accumulation is only related to the country’s genuine interest in maximizing the returnof its portfolio.Then, we introduce the possibility that a liquidity shock hits the country.Following Cole and Kehoe (2000), this shock occurs with a probability that increases in the levelof sovereign debt. Moreover, the country can only use its reserves to avoid default following theliquidity shock. Thus, we ask: will the country borrow less than in the benchmark economy, soto reduce the likelihood that the liquidity shock occurs? Alternatively, will the country borrowmore and at the same time accumulate reserves to use them should the liquidity shock occur?In our model of portfolio allocation, a relationship of complementarity arises at equilibriumbetween debt and reserves: an increase in the value of debt spurs the country to raise its reserveholdings. The country increases its borrowing so to have additional resources for investment.Although a larger stock of debt renders the occurrence of the liquidity shock and the default onthe productivity shock more likely, the accumulation of reserves allows the country to mitigatethe consequences of these shocks on country’s welfare. At equilibrium, lenders respond to an2 In an extension, we relax this assumption and show that our main results remain the same.3 Asset incompleteness is crucial to disentangle the relationship between the sovereign decision to defaultand the cost of sovereign debt (Arellano, 2008): with noncontingent assets, risk-neutral competitive lendersincorporate the probability of default in the premium set on the debt contract.6

increase in the probability of default caused by larger sovereign debt and reserves by raising thecost of debt.We conduct numerical simulations that illustrate our theoretical finding that debt and reserveare complements to the country. We show that, using levels of the parameters taken from theliterature and consistent with empirical evidence, when subject to the risk of a liquidity shock itcan be optimal for the country to increase, at the same time, its stocks of debt and reserves withrespect to a benchmark economy without liquidity shock. What are the implications of theseresults for country’s output? We find that, at equilibrium, a higher level of reserves is associatedwith higher expected output. This outcome is consistent with Benigno and Fornaro (2012) andDominguez et al. (2012), who find a positive relationship between reserve accumulation andGDP growth. In these numerical exercises, we obtain values of the variables of interest that areclose to those featured by emerging and developing economies. In particular, reserves amount toabout 30% of expected GDP, slightly above 26%, the IMF estimate for emerging and developingcountries for 2011. Moreover, the equilibrium level of debt is equal to about 55% of GDP,somewhat larger than the IMF estimate of about 36%. Finally, the equilibrium spread set byinternational lenders is slightly below the estimate of 5.44% for EMBI in Borri and Verdelhan(2011), and close to the 9-year excess return of 4.6% obtained by Broner et al. (2013b).We extend our theoretical model by relaxing the assumption that foreign lenders cannotsupply additional funds in the event of a liquidity shock. There, we allow the country to issuemore debt when hit by a liquidity shock. We find that an equilibrium exists in which the countryissues high-return debt and uses those resources to overcome the crisis; moreover, we show thatthe complementarity between debt and reserves is preserved at equilibrium.These results offer new insights to the literature that studies the relationship between sovereignreserves and debt. In our setup as in the extant papers, there may be circumstances where thereduction in debt and the accumulation of reserves are substitute means to deal with the prospectof a liquidity crisis. The country might reduce the amount of external debt and, thus, its exposureto a liquidity crisis. Or, it could increase the amount of reserves to inject the neededresources when the shock occurs. We show that, at the equilibrium of a model of sovereignportfolio allocation, reserves and debt can be in a relationship of complementarity: by raisingboth reserves and debt the country can, at the same time, increase its total resources for investmentin the production technology and use its liquidity in the event of a crisis. The novelelement of our analysis compared to previous studies is that reserves play a dual role. Theyact as a consumption smoothing asset when the sovereign decides to default, thereby offeringpost-default resources for consumption (as in, e.g., Alfaro and Kanczuk, 2009; Bianchi et al.2013; Jeanne and Rancière, 2011; among others). Moreover, they can be directly used in orderto avoid default, should the liquidity shock occur.Our model is related to Alfaro and Kanczuk (2009), who study the interaction between debt7

issuance and reserve holdings for a country that can be hit by sudden stops. They propose a smallopen endowment economy model to show that a standard quantitative setup cannot account forthe significant amount of reserves that countries hold. Building on Alfaro and Kanczuk (2009),Bianchi et al. (2013) allow the country to choose the maturity of its debt structure and find that,in equilibrium, the country decides to hold a positive amount of reserves since they representan insurance not only against default, but also against future increases in the borrowing cost.Our work complements the one of Bianchi et al. (2013) in that we abstract from the analysis ofoptimal debt maturity and sovereign risk aversion and focus instead on a risk-neutral country’sinvestment decision, in which reserves also represents an alternative and imperfect substitute foroutput.Moreover, the present work is related to Aizenman and Marion (2004). In a two-period model,they study the relationship between debt and reserves and find that the optimal borrowingchoice increases with reserves. They then introduce political uncertainty and find that politicalinstability and corruption reduce the optimal size of reserves and increase the one of externaldebt, altering the relationship between reserves and debt. They however abstract from the roleof reserves in the event of sudden stops and the analysis of the relationship between reservesand the cost of debt.We also contribute to two other strands of the literature. The first one studies the optimalcontractual arrangements in the presence of commitment problems and non-contingent contracts.In analogy to Aguiar and Gopinath (2006), Arellano (2008) and Yue (2010), we show thatdefault arises at equilibrium after an adverse shock occurs, consistently with prevalent empiricalevidence. However, while these papers study the relationship between default risk and output,consumption and foreign debt, we look at the interaction between default risk, sovereign debtand reserves, and the implications of this relationship for output and the cost of debt.The second related strand studies the role of sovereign reserves as a buffer against liquidityshocks. 4Jeanne and Rancière (2011) model reserves as an insurance contract that allowscountries to smooth the absorption of the output costs caused by a capital outflow. They find aclosed form expression for the optimal amount of sovereign reserves that is able to account forthe average level of reserves accumulated by emerging economies since 1980. Differently fromour setup, in their model debt level is exogenous and the country is risk-averse. We show thateven a risk neutral country can decide to hold positive amounts of reserves at equilibrium forreasons related to the management of its portfolio. 5In the next section, we present our main model. In Section 3, we discuss the equilibrium4 Obstfeld et al. (2010) and Calvo et al. (2012) provide evidence supporting this motive behind reserveaccumulation. See Bussière et al., 2014, for a survey on this subject.5 See Carroll and Jeanne (2009) and Guerrieri and Lorenzoni (2011) for models of precautionary savings.Extending our model to account for country’s risk-aversion would most likely lead to an even higher equilibriumlevel of reserves, all the rest being equal (Corneli and Tarantino, 2011).8

analysis and illustrate the qualitative features of the model’s equilibrium. In Section 4, we performnumerical simulations of our theoretical setting and, in Section 5, we analyze an extensionin which foreign lenders can intervene with fresh capital in the event of a liquidity crisis. Section6 concludes.2 The modelConsider a sovereign country with access to foreign borrowing. At stage zero, the country decidesthe the face valued of debt D it wants to borrow from foreign lenders and its allocation. Thelenders set the cost of D. At stage one, if the country is hit by a liquidity shock it decideswhether to default or inject the resources needed to redeem the shock and bring the productiontechnology to completion; otherwise the game proceeds to stage three. At stage three, theproductivity shock realizes and at stage four the sovereign can again choose whether to default.Figures 2 illustrates the timing of the game. 6 We solve the model by backward inductionand the equilibrium concept we employ is the Sub-game Perfect Nash Equilibrium (SPNE) inpure strategies. In what follows, we present in detail how we model each relevant node and themain ingredients of the game.t = 0t = 1t = 2t = 3t = 4LendingGameδInvestmentGameD, RLiquidityShockη(D), EDefaultDecisionFigure 2: TimelineProductivityShockz ∼ F (z)DefaultDecision2.1 The lending gameOur world is populated by a continuum of atomistic and risk-neutral foreign lenders (indexed byi ∈ I). The total mass of lenders is large, ensuring that perfect competition prevails and lendersdo not extract any rent; also, lenders have unlimited access to funds at the riskless interest rate(that, for simplicity, we assume to be zero).The sovereign borrows from a subset of mass 1 of lenders. As in Bolton and Jeanne (2007),lenders participate in a bidding game following the sovereign’s announcement of a fund-raisinggoal D, which represents the face value of country’s debt. Lenders move first by each simultaneouslysubmitting a bid. The sovereign then decides which bid(s) to accept.6 Figure 9 illustrates the full game-tree.9

The lenders’ payoff is equal to the value of the repayment D discounted by the probabilitythat the sovereign repays in full. At the bidding stage of the game each lender i makes an offeron the rate of return, r(i), that insures break even. That is, lender i solves a problem of thefollowing sort:D = D(1 + r(i))P rob{The Country is Solvent}1/(1 + r(i)) = P rob{The Country is Solvent}δ(i) = P rob{The Country is Solvent}.Thus, throughout the model we will denote by δ(i) = 1/(1 + r(i)) ∈ (0, 1) the inverse of the rateof return r(i) that lender i asks in exchange for a loan D. Accordingly, the Nash equilibriumof the lending game is defined by a set of bids (δ(i)) i∈I such that, for all i, bid δ(i) maximizesthe lender i’s payoff taking all the other bids δ(j), with j ≠ i, as given. At equilibrium thesovereign squeezes all the surplus from the lending relationship and (randomly) selects among aset of identical bids, δ(j) = δ(i) = δ, so we can focus on a representative sovereign-lender pair.2.2 The investment game and the productivity shockThe country decides the amount of debt to borrow from the lender (D) and how to invest thesefunds in public expenditure, g, and/or reserves, R. Specifically, the sovereign budget constraintis given byδD = g + R ⇐⇒ g = δD − R. (1)This implies that, once D and δ are determined, the country’s decision on the value of R pinsdown the resources invested in g. The country is risk-neutral. It allocates its resources bymaximizing the expected value its portfolio, E(W ), which corresponds to the sovereign expectedwelfare. E(W ) is given by the amount of liquid resources, R, and the expected value of theproduction technology, Y , after repaying debt, D.Reserves yield the risk-less interest rate and act as a safe and liquid technology that can becarried over from stage zero to the end of the game. The production technology is the risky andilliquid asset: it requires capital, that is public expenditure (g), as sole input and is subject to aproductivity shock, z. The receipts generated by the production technology materialize only atstage four, after the country has decided on the allocation of its resources and the uncertaintyover the productivity shock resolves.The production function that determines the value of the illiquid asset, Y , is linearly affected10

y the productivity shock. Thus, using (1):Y (z, g) = zY (g) = zY (δ, D, R),where z is a random variable drawn from a continuous density function f(z), and F (z) is thecdf induced by f(z). Throughout the paper we assume that z follows a continuous uniformdistribution with positive density over [1−c, 1+c]. The function Y (δ, D, R) is twice differentiable,with Y ′ ≥ 0 and Y ′′ < 0, is increasing in δ and D (as ∂Y/∂δ = DY ′ ≥ 0 and ∂Y/∂D = δY ′ ≥ 0),decreasing in R (∂Y/∂R = −Y ′ ≤ 0) and satisfies standard Inada Conditions (lim g→0 Y = 0,lim g→0 Y ′ = ∞ and lim g→∞ Y ′ = 0).2.3 The liquidity shockThe country is exposed to the risk that a liquidity crisis occurs at stage one, before the productiontechnology’s receipts materialize. Following Chang and Velasco (2000), the liquidityshock hits the country with probability η(D) ∈ (0, 1), and its consequences are that the productiontechnology needs a further injection of resources E ∈ [0, δD) at stage two in order to becompleted, with lim D→0 η(D) = 0, lim D→ ¯D η(D) = 1, and η ′ , η ′′ > 0 for all D ∈ [0, ¯D].This formalization captures, in a reduced form, the definition of sudden stop as a large andabrupt decline of capital inflows (as in, e.g., Caballero and Panageas, 2007). 7 The liquidity shockof our model can be interpreted as a withdrawal of private capital that causes disruptions inthe production technology unless additional resources are injected. Rodrik (2006) and Obstfeldet al. (2010) document that to overcome capital outflows countries need to sink resources forabout 10% of their GDP. Moreover, assuming that the probability of the shock increases withdebt is consistent with the evidence on the negative relationship between indebtedness and thelikelihood of a liquidity crisis (Cole and Kehoe, 2000).In the main model, if the liquidity shock occurs the country can only use its reserves toinject E, while it cannot dismantle the capital invested in the production technology or borrowadditional resources from lenders. This is consistent with Rodrik (2006), who finds that forcountries with larger holdings of reserves it is easier to face the consequences of capital outflows.Conversely, if it decides not to tackle the shock, the country defaults on the production technologyand retains reserves. In Section 5, we will relax this assumption and look at the case inwhich the country can borrow E from financial markets to intervene in case of a liquidity crisis.7 We would obtain analogous results by assuming that an exogenous fraction of debt D needs to be reimbursedat stage two.11

2.4 The decision to defaultThere are two stages at which the country may choose to default. The first is after the realizationof the liquidity shock (stage two). The second is after the realization of the productivity shock,when uncertainty over the value of output realizes (stage four). The main cost of default is thatthe country loses the entire value of the production technology (as in, e.g., Bolton and Jeanne,2007). 8 Instead, even after defaulting the country keeps its reserves (e.g., Aizenman and Marion,2004; Alfaro and Kanczuk 2009). In what follows, we discuss the determinants of the sovereigndecision to default.The country defaults on the productivity shock whenever the revenue generated by theproduction technology is lower than the face value of debt, or zY ≤ D. Hence, at equilibriumthis decision depends on the realized level of the productivity shock z, but also on the outcomeof the investment and lending games.To analyze the choice to default after the liquidity shock, throughout the paper we distinguishbetween two sub-games. If the country plays sub-game N, it chooses to default if hit by theliquidity shock. Instead, if it plays sub-game F , the country chooses to inject the neededresources (E) in the event of a shock. We find this a useful distinction because it allows us toeasily characterize all possible branches of the game-tree. 9 Country’s decision to play sub-gameF must be credible, that is, it must be feasible and time-consistent. Feasibility requires thatat the investment game the country accumulates a level of reserves larger than E (resourceconstraint). Time consistency requires that the decision not to default is sub-game perfect: ifthe country’s payoff from the continuation-game following the shock is less than E then it willprefer to default on the shock and keep its reserves (time-consistency constraint).3 Model solutionIn order to disentangle the role of reserves in our framework, we first solve the model in absenceof the liquidity shock. Then, we reintroduce the liquidity shock and study whether the countryinjects the needed resources should the shock happen. All formal proofs are relegated to theappendix.8 See Alfaro and Kanczuk (2005) for a discussion on output losses in the event of default and Gennaioli et al.(2014) for a model of costly sovereign defaults.9 Figure 10 illustrates the branches of the game-tree corresponding to the two sub-games F and N.12

3.1 Benchmark economy without liquidity shockThe country defaults at the final stage if:zY (δ, D, R) − D + R ≤ R ⇐⇒ z ≤ ¯z(δ, D, R) ≡DY (δ, D, R) . (2)The value of ¯z in (2) corresponds to the threshold of the productivity shock above which thecountry repays its creditors and keeps the residual.defaults on debt and loses the entire value of output.Instead, if z falls below ¯z, the countryAt stage zero, the country decides on the value of D and how much of δD it invests in reservesR. Specifically, it solves the following maximization problem:∫ ∞max E(W (δ, D, R)) = R + (zY (δ, D, R) − D)dF (z), (3)D,R∈[0,δD] ¯z(δ,D,R)under the condition that the value of R is feasible, that is, R ∈ [0, δD]. In turn the lender setsthe optimal value of δ to break-even in expectation:∫ ∞δD = D dF (z) ⇐⇒ δ = 1 − F (¯z(δ, D, R)). (4)¯z(δ,D,R)Proposition 1 characterizes the equilibrium results in the benchmark without liquidity shock.PROPOSITION 1. The sovereign country chooses D ∗ and R ∗ in the viable range to solve thefollowing first-order conditions∫ ∞¯z(δ,D,R)1 − Y ′ (δ, D, R)(zδY ′ (δ, D, R) − 1)dF (z) = 0 (5)∫ ∞¯z(δ,R,D)zdF (z) = 0. (6)Given the vector of equilibrium values of debt and reserves that solve (5) and (6), it exists aunique equilibrium discount factor δ ∗ ∈ (0, 1) that solves the lender zero-profit condition.The first-order conditions in Proposition 1 have a natural interpretation. Specifically, inexpression (5) the equilibrium value of debt solves the trade-off between the increase in theexpected marginal return of the production technology due to an additional unit of debt (equalto δY ′ ≥ 0) and the marginal cost in terms of a higher repayment due at maturity (equal to1−F (¯z)). Expression (6) determines the country’s investment in reserves: the equilibrium valueof R equates their marginal rate of return (equal to 1) to the marginal rate of return of theproduction technology. This second term is given by the marginal productivity of capital (Y ′ )13

discounted by the probability that the realization of the productivity shock is larger than ¯z,because otherwise the country defaults and loses the entire value of output.In Section 4, we carry out numerical simulations using a Cobb-Douglas production function.In that context, two parameters influence the equilibrium value of reserves resulting from (6):the income share of capital and the variability of the production technology. All else being equal,an increase in the income share of capital increases the marginal rate of return of the productiontechnology, thereby decreasing the amount of resources invested in reserves. Analogously, themarginal rate of return of the production technology increases if the support of the productiontechnology widens. The reason is that the country disregards the lower tail of the distribution ofz. This again implies that the reserves the country accumulates should decrease, as they becomerelatively less valuable.In the appendix, we give the conditions for the unicity of the equilibrium of the sovereigninvestment game as resulting from conditions (5) and (6), and we check that they are satisfiedin our numerical simulations. Finally, the deal signed at the initial stage between the lenderand the sovereign can be implemented by a debt contract in which the country receives δ ∗ D ∗ atstage zero against the promise to repay D ∗ at stage four. Hence, the lender earns (D ∗ − δ ∗ D ∗ )(or, equivalently, D ∗ r ∗ /(1 + r ∗ ), with r ∗ /(1 + r ∗ ) = 1 − δ ∗ ) provided the country repays in full,zero otherwise. In the following section, we will introduce the liquidity shock in our model’seconomy and study how it shapes sovereign borrowing and investment decisions.3.2 Economy with liquidity shockIn this section, a liquidity shock can hit the country at stage one. We study whether, relative tothe benchmark economy in Section 3.1, the country reduces the amount of accumulated debt, soas to lower the probability that the liquidity shock occurs, or it increases the amount of reserves,so as to alleviate the consequences of the shock on its welfare.If the liquidity crisis does not occur at stage one, at stage four the country defaults after therealization of the productivity shock if z ≤ ¯z, where ¯z is as defined in (2). Conversely, if theliquidity shock occurs at stage one, we distinguish between two cases, depending on whether thecountry injects (sub-game F ) or not (sub-game N) the resources needed to overcome the shock(E).Sub-game NAt stage two the sovereign defaults after the liquidity shock, thus, with probabilityη it loses output but keeps the accumulated liquidity (R). The lender anticipates country’sdecision and it sets δ to break-even in expectation:∫ ∞δD = D(1 − η(D)) dF (z) ⇐⇒ δ = (1 − η(D))(1 − F (¯z(δ, D, R))). (7)¯z(δ,D,R)14

The country’s choice on debt (D) and reserves (R) is obtained by solving the followingmaximization problem:∫ ∞max E(W N(δ, D, R)) = R + (1 − η(D)) (zY (δ, D, R) − D)dF (z). (8)D,R∈[0,δD] ¯z(δ,D,R)On the one hand, the decision to default after the liquidity shock implies that the sovereign willnot inject E to overcome the crisis. On the other hand, it will obtain the net expected payoffgenerated by the production technology only with probability 1 − η (that is, if the shock doesnot take place).Sub-game FAt stage four the country defaults after the realization of the productivity shockif z ≤ ¯z, as in (2). At stage two, the country liquidates a portion of its reserves to inject E. Anecessary condition for this to happen is that the country has accumulated enough liquidity, orR ≥ E (resource constraint).The decision to redeem the shock must also be sub-game perfect. Given the sovereign choiceof D and R and the discount factor (δ) offered by the lender, the country must have an incentiveto continue with the production technology instead of defaulting strategically at stage two,otherwise the decision to inject E would not be time-consistent. That is, the following timeconsistencyconstraint has to hold:R − E +∫ ∞¯z(δ,D,R)∫ ∞¯z(δ,D,R)(zY (δ, D, R) − D)dF (z) ≥ R(zY (δ, D, R) − D)dF (z) ≥ E.At stage zero, the lender solves the following zero-profit condition:∫ ∞δD = D dF (z) ⇐⇒ δ = 1 − F (¯z(δ, D, R)). (9)¯z(δ,D,R)This formulation of the lender’s problem takes into account that the sovereign does not defaultafter the liquidity shock occurs and always brings the production technology to completion. Thecountry decides on the accumulation of D and the value of R by solving∫ ∞max E(W F (δ, D, R)) = R + (zY (δ, D, R) − D)dF (z) − η(D)E, (10)D,R∈[E,δD] ¯z(δ,D,R)under the time-consistency constraint. The first term in (10) reflects the fact that the countryis certain about R, because reserves are not lost in the event of default. The second term in(10) is the expected value of the production technology net of the face value of debt D: this15

second term is positive by construction, because the expected value of the receipts is truncateddownwards by ¯z. The third term in (10) corresponds to the expected cost borne by the countryin the event of a liquidity shock.3.3 Equilibrium definitionThe equilibrium of the game with liquidity shock is defined by the vector {δ, D, R} such that thecountry maximizes its expected welfare and the lender breaks even in expectation. If the countrydoes not default on the liquidity shock (that is, when sub-game F is played), the country’s actionsmust also be feasible and sub-game perfect, insofar as both the time-consistency and resourceconstraints must be satisfied.DEFINITION 1. Denote by {δF ∗ , D∗ F , R∗ F } the vector that characterizes the equilibrium of thegame in which the country injects E should the liquidity shock occur (sub-game F ), and by{δN ∗ , D∗ N , R∗ N } the vector that characterizes the equilibrium of the game in which the countrydefaults should the liquidity shock occur (sub-game N).At {δ ∗ F , D∗ F , R∗ F }, the resource constraint R ∗ F ≥ E,and the time-consistency constraintmust be satisfied.∫ ∞¯z(δ ∗ F ,D∗ F ,R∗ F ) (zY (δ ∗ F , D ∗ F , R ∗ F ) − D)dF (z) ≥ EAt the SPNE of the game, the country chooses to play sub-game F if, and only if, conditionE(W (δF ∗ , D∗ F , R∗ F )) ≥ E(W (δ∗ N , D∗ N , R∗ N )) is satisfied and the relevant constraints hold true.Otherwise, the SPNE of the game features the choice of sub-game N.We first determine the conditions for the existence and unicity of {δ ∗ N , D∗ N , R∗ N } and {δ∗ F , D∗ F , R∗ F }.Then, we assess the strategic relationship between the country’s and lender’s choice variables.The country’s final decision depends on the economy’s parameters. Therefore, we analyze bymeans of numerical simulations the country’s choice between sub-game F and sub-game N andits consequences on the level of sovereign reserves, debt, output and welfare.3.4 Equilibrium analysisSub-game N To begin with, we study the country’s decision to accumulate debt and reserves,and the lender choice of δ, the discount factor, in the scenario featuring country’s default in theevent of a liquidity shock (sub-game N).16

PROPOSITION 2. In the sub-game in which the country defaults following the liquidity shock,the sovereign country chooses D ∗ N and R∗ Nconditions(1 − η(D))∫ ∞¯z(δ,D,R)in the viable range to solve the following first order∫ ∞(zδY ′ (δ, D, R) − 1)) dF (z) − η ′ (D) (zY (δ, D, R) − D)dF (z) = 0 (11)¯z(δ,D,R)∫ ∞1 − (1 − η(D))Y ′ (δ, D, R) zdF (z) = 0. (12)¯z(δ,D,R)Given the pair of equilibrium values of debt and reserves that solve (11) and (12), it exists aunique equilibrium discount factor δ ∗ N∈ (0, 1) that solves the lender’s zero-profit condition (7).The basic trade-offs that shape the solution to the investment and lending problems in thebenchmark economy also influence the equilibrium results in Proposition 2. The crucial differenceis that the country also needs to take into account that the accumulation of a higher amount ofdebt increases the likelihood that a liquidity shock occurs. Consequently, the value of the LHS ofthe first-order condition in (11) is lower than the one of the corresponding first-order conditionin the benchmark, (5), dictating a lower level of debt, ceteris paribus. By the same token, thepossibility that a liquidity shock occurs implies that the LHS of the first order condition in (12)is larger than the one in the benchmark (6). Hence, the country has an incentive to accumulatea higher level of reserves than in the benchmark, ceteris paribus. 10The main takeaway of Proposition 2 is that the accumulation of sovereign reserves mightarise at equilibrium independently of the decision to inject resources following a liquidity shock.The reason is that reserves are a liquid asset that the country retains in case of default.In the following corollary, we derive the relationship between agents’ choice variables atequilibrium in sub-game N.COROLLARY 1. If Y ′ is sufficiently small at {δN ∗ , D∗ N , R∗ N }, we find that:(i) dδ ∗ N/dD < 0, dδ ∗ N/dR ≤ 0, (ii) sign{dR ∗ N/dD} = sign{dD ∗ N/dR} > 0.Moreover,(iii) dR ∗ N/dδ > 0, dD ∗ N/dδ < 0.The equilibrium values of debt and the discount factor are in an inverse relationship; thatis, as D increases the equilibrium value of δ decreases. Intuitively, as the country gets moreindebted, the probability of default following the productivity crisis rises. Repayment concerns10 The conditions for the unicity of the equilibrium of the sovereign investment game are in the proof. We willverify that they are satisfied when we carry out numerical simulations with specific functional forms.17

induce the lender to reduce the value of the discount factor. We also find that an increase inthe amount of reserves triggers a decrease in the value of the discount factor: as the countryaccumulates more liquid assets, the expected value of output decreases and so does what thecountry can pledge to the lender.We also find that there is a direct relationship between debt and reserves, showing thatthese variables are in a relationship of complementarity. Reserves cannot be taken over by thelender, thus they mitigate the adverse consequences of a default for country’s welfare. Thus, thesovereign raises the level of debt to have more resources for investment. Moreover, since largerdebt renders default more likely, it hoards more reserves to retain them in default states.Finally, we show that if the discount factor increases, also the value of reserves increasesat equilibrium, because the country has more resources to invest in this asset. Conversely, anincrease in the discount factor reduces the amount of debt at equilibrium. Intuitively, an increasein the discount factor raises the size of the country’s total resources, so the sovereign can decreasethe amount of debt and reduce the probability of the liquidity shock without sacrificing the scaleof the investment in the production technology.The sufficient condition under which we derive the results in Corollary 1 is that Y ′ is sufficientlysmall at equilibrium. If Y ′ is small, the opportunity cost of accumulating reserves issmall and this, of course, stimulates the country to raise reserves, ceteris paribus. However, inthe proof of Corollary 1 we show that most of our results (and in particular the ones on therelationship between R and D) goes through under the alternative sufficient condition that theequilibrium level of reserves is small enough.Sub-game F We now turn to the equilibrium analysisis when the country does not defaultfollowing the liquidity shock (sub-game F ).PROPOSITION 3. If the country does not default following the liquidity shock, then we findthat:1. In the unconstrained equilibrium, the sovereign country chooses DF ∗ and R∗ Frange to solve the following first-order conditions:in the viable∫ ∞¯z(δ,D,R)∫ ∞1 −(zδY ′ (δ, D, R) − 1)dF (z) − η ′ (D)E = 0, (13)¯z(δ,D,R)zY ′ (δ, D, R)dF (z) = 0. (14)2. If the resource constraint is binding, at the equilibrium the optimal level of reserves isR ∗ F = E. 18

3. If the time-consistency constraint is binding, at the equilibrium the optimal level of reservesR ∗ Fis determined by the following condition∫ ∞¯z(δ,D,R)(zY (δ, D, R) − D)dF (z) = E.The value of DF∗ in the two classes of constrained equilibrium results from the correspondingfirst order condition. Finally, given DF∗ and R∗ F , it exists a unique equilibrium discount factor∈ (0, 1) that solves the lender zero-profit condition.δ ∗ FIn the unconstrained case, the country sets the levels of D and R as prescribed by (13) and(14), respectively. The optimal value of D is determined by two conflicting forces: on the onehand, a higher level of D raises the expected revenue of the production technology; on the otherhand, if D increases both the repayment due to the lender and the probability that the countryis hit by the liquidity shock increase.When deciding on the value of reserves, the countrysolves the same trade-off as in the benchmark economy: it is optimal to increase the amountof reserves as long as the marginal gain from one additional unit of resources invested in liquidassets (equal to 1) outweighs the expected marginal gain from investing the same amount in theproduction technology. Note that the LHS of the first-order condition that determines R in theunconstrained scenario, (14), is smaller than the corresponding one in sub-game N, (12). Thereason is that, in sub-game N, the country can bring the production technology to completiononly if it is exempted by the liquidity shock. This reduces the marginal return of the productiontechnology and boosts investment in reserves, ceteris paribus. 11If the country’s choice is constrained by either the resource or the time-consistency constraint,then the equilibrium level of reserves is determined by the binding condition and the level ofdebt is set to solve the corresponding first-order condition. Moreover, we find that there is nopure-strategy Nash equilibrium at which both the resource and the time-consistency constraintsare binding. If the resource constraint is binding, the country would like to accumulate a lowerlevel of reserves than E, but the presence of the resource constraint fixes R exactly at E. In otherwords, the resource constraint sets a value of R that is larger than in an unconstrained optimum.The time-consistency constraint works in the opposite direction. Since the expected output afterrepaying the lender in the event of no default (that is, the LHS of the time-consistency constraint)is decreasing in R, 12 a binding time-consistency constraint implies that although the countrywould like to accumulate a larger level of reserves, it has to reduce R in order to satisfy theconstraint with an equality. Thus, the time-consistency constraint pins down a higher value of11 As for Propositions 1 and 2, the conditions for the unicity of the equilibrium of the sovereign investmentgame are in the appendix. We will verify that they are satisfied when we carry out numerical simulations withspecific functional forms.12 We show this in the proof of Proposition 3.19

R than in an unconstrained optimum. Overall, these considerations imply that an equilibriumin which both constraints are simultaneously binding cannot exist.In the following corollary, we analyze the relationship between model’s choice variables insub-game F at equilibrium.COROLLARY 2. If Y ′ is sufficiently small at {δF ∗ , D∗ F , R∗ F }, we find that:(i) dδ ∗ F /dD < 0, dδ ∗ F /dR ≤ 0, (ii) sign{dR ∗ F /dD} = sign{dD ∗ F /dR} > 0.Moreover,(iii) dR ∗ F /dδ > 0, dD ∗ F /dδ < 0.These results hinge on the same insights developed after Corollary 1, with the difference thatin sub-game F , the country’s incentive to accumulate reserves is augmented by the need to injectE in the event of a liquidity crisis. This implies that the relationship of complementarity betweenreserves and debt is even stronger here than in sub-game N. As discussed in the Introduction,these results offer new insights on the role of reserves and the relationship between reserves anddebt accumulation.4 Numerical simulationsIn this section, we present numerical simulations that illustrate the results of the theoreticalframework. 13 We first investigate how the choice of reserves, debt, and the cost of debt change asthe variability of the production technology increases; that is, as the investment in the productiontechnology becomes relatively riskier than reserves. In a second set of simulations, we study howportfolio allocation and the cost of debt change as the value of E, the amount of resourcesthat needs to be injected to overcome a liquidity shock, rises. We assume throughout that theproduction function is a Cobb-Douglas of the following type:zY (δ, D, R) = z(δD − R) α .We maintain the assumption that the productivity shock is distributed as a uniform randomvariable:z ∼ U(1, c 2 /3), z ∈ [1 − c; 1 + c].Moreover, the probability of the liquidity shock, η(D) ∈ (0, 1), is modeled as an exponentialfunction,η(D) = 2 D/ ¯D − 1,13 For the sake of the exposition, we focus on the cases in which an equilibrium in pure strategies exists and iswell-defined.20

and fulfills the properties laid out in Section 2.3 (namely, lim D→0 η(D) = 0, lim D→ ¯D η(D) = 1,η ′ , η ′′ > 0 for all D ∈ (0, ¯D)).Income share of capital α 0.3Resource injection (as % of GDP) E 10%Liquidity shock parameter ¯D 4Support of the productivity process c [1, 2.5]Table 1: Calibration ParametersOur calibration is parsimonious, we need to set only four parameters to deliver a numericalsimulation of our theoretical framework. Table 1 reports the values of the four parameters thatdetermine the behavior of the model variables in the first set of simulations, i.e., the one in whichwe let the variance of the productivity shock vary. The value of the income share of capital (α)is taken from the literature. The amount of resources that needs to be injected in the event of aliquidity shock (E) amounts to 10% of expected GDP, consistently with the estimates in Rodrik(2006) and Obstfeld et al. (2009). The liquidity shock parameter cannot be estimated from thedata. We therefore choose it as to obtain that, in equilibrium, the probability of a liquidity crisisis about 10%, which is the unconditional probability of a sudden stop in Jeanne and Rancière(2011). 14 Finally, the parameter determining the support of the productivity process, and itsvariance, ranges between 1 and 2.5. We make it vary to study how country’s and lender’s choicesare affected by an increase in the variability of the production technology.The simulations in Figure 3 plot our variables of interest as the variance of the underlyingproductivity process rises.In particular, the figure reports the rate of return r chosen bythe lender, together with the equilibrium level of debt and reserves in the benchmark withoutliquidity shock, and in the economy with liquidity crises. Finally, the lower-right panel plots thevalue of expected welfare when the country plays sub-games F and N. 15Although ours is not a fully quantitative model, the simple parametrization in Table 1 allowsus to match the value of our variables of interest to that featured by emerging and developingcountries.Reserves amount to about 30% of expected GDP, slightly above 26%, the IMFestimate for emerging and developing countries in 2011. The level of debt is about 55% of GDP,somewhat higher than the IMF estimate of 36%. Finally, the rate of return r = 1/δ−1 set by thelender oscillates around 1.6: 16this is consistent with the rate of return generated by a 10-year14 In our model, this probability is determined by the equilibrium level of debt chosen by the country. Notethat, in these simulations, the value of η(D) does not change by much as the value of ¯D varies.15 The economy without liquidity shock represents a benchmark that the sovereign cannot implement. For thisreason, we do not report the welfare of the country in that case. Note also that, at the value of the parametersin Table 1, the resource and time-consistency constraints never bind.16 Recall that the discount factor δ is equal to 1/(1 + r). That is, in our model δ is inversely correlated withthe cost of debt: an increase in the value of δ set by the lender reflects an increase in the probability that thelender expects the country to be solvent and stands for a decrease in the cost of debt.21

ond capitalized at an annual interest rate of about 5%. Since we set the risk free interest rateto zero, the rate of return coincides with the spread set on sovereign debt. Thus, the spread inour simulations is slightly below the estimate of 5.44% for EMBI in Borri and Verdelhan (2011)and close to the 9-year excess return of 4.6% obtained by Broner et al. (2013b).In this first set of simulations, the country does not default following a liquidity shock atequilibrium (that is, it plays sub-game F ). The lender anticipates the country’s decision andsets a lower rate of return r than in the case with default on the liquidity shock (sub-game N).Moreover, the country accumulates larger stocks of debt and reserves than in the benchmarkeconomy, thus showing the relationship of complementarity between debt and reserves. Notethat the equilibrium value of reserves is well above E, the amount of resources needed in theevent of a liquidity shock. The relationship of complementarity arises also when looking at howreserves and debt vary with the variance of the productivity process. Indeed, we find that as thevalue of the variance increases, the country raises its holdings of both debt and reserves. It alsoturns out that, as debt and reserves increase, the rate of return set by the lender rises, reflectingthe higher probability of default on the shocks. Taken together, these simulations confirm theresults in Corollary 2.Figure 4 shows that the value of expected output increases in the variance of the productiontechnology. Conditionally on the country not defaulting following the productivity shock, theexpected value of output E(Y |z ≥ ¯z) increases because the sovereign enjoys the upper tail of thedistribution of the productivity shock, whereas it disregards the lower tail. Moreover, E(Y ), theexpected value of output, increases in the variance of z because country’s total resources (δD)and the amount of resources invested in the production technology (δD − R) increase with thevariance of the productivity process. Higher levels of reserves are associated with higher expectedoutput, an outcome that is consistent with Benigno and Fornaro (2012) and Dominguez et al.(2012). Finally, the output of the country when it uses its reserves to redeem the liquidity shock(sub-game F ) is larger than if it defaults following that shock (sub-game N). This is because,by using reserves to inject the needed resources following a liquidity shock, the country does notlose its output.Figure 5 plots the variables of interest for different values of E, the resources needed toovercome the liquidity crisis. Specifically, in this figure we let E vary from 0.02 to 0.12, thatis, from around 4% to 24% of country’s expected GDP. The value of c, the parameter thatdetermines the variance of the productivity process, is set at 2.25, the other parameters are as inTable 1. Note that E influences only the country’s choices in sub-game F , the reason is that inthe benchmark economy and when the country chooses to default following the liquidity shock(sub-game N), the expressions determining the optimal levels of reserves, debt and the cost ofdebt are independent of E.For low values of E, the country decides not to default if the liquidity shock occurs (that is,22

it plays sub-game F at equilibrium). Instead, when E is particularly large, then the countrydecides to default on the liquidity shock (sub-game N). As E rises above a threshold, thetime-consistency constraint binds and the levels of debt and reserves are as in the constrainedequilibrium of Proposition 3. In particular, the constrained level of reserves is lower than whenthe time-consistency constraint does not bind. In these simulations as in those of Figure 3, debtand reserves are in relationship of complementarity. The country accumulates more debt andreserves than in the benchmark economy. At the same time, debt and reserves increase as theamount of liquid resources E needed in the event of a liquidity shock rises.5 Foreign lenders’ interventionIn the main model we assume that, if the liquidity shock occurs, the country can inject theresources E and complete the production technology only by using its reserves. In other words,the lender cannot intervene in the event of a liquidity crisis. In this extension, we relax thisfriction and assume that, if the liquidity crisis happens at stage one, the country can issue newdebt to cover E. We interpret this as a form of foreign lenders’ intervention to tackle the shock.We change the baseline model of Section 2 by introducing a new lending game in stageone. More specifically, before taking the decision to default on the liquidity shock, the countryapproaches the lender announcing a fund raising goal of ˆD to cover E.decides on the discount factor ˆδ. Figure 6 illustrates the new timing of the game.In turn, the lenderThe objective of this section is to assess the impact of stage-one lenders’ intervention onstage zero lending and investment decisions. Following Section 3.2, if the liquidity crisis doesnot occur at stage one, at stage four the country defaults after the realization of the productivityshock if z ≤ ¯z, where ¯z is as defined in (2). Conversely, if the liquidity shock occurs at stageone, we distinguish between three cases. In the first, the country uses its reserves to inject theresources needed to overcome the shock (sub-game F ). In the second, it issues additional debtˆD to cover E (sub-game F L). In the third, it chooses to default (sub-game N). The first andthe third cases have been analyzed in Propositions 2 and 3. In what follows, we describe howwe solve for the equilibrium of sub-game F L.Sub-game F L At stage one, the country sets ˆD by maximizing its continuation payoff postliquidityshock. The lender responds by setting the value of ˆδ that insures break-even in expectation.23

The new stage-zero maximization problem follows:( ∫ ∞max F L(δ, D, R))D,R∈[0,δD]= R + η(D)+(1 − η(D))ẑ(δ,D,R,∫ ˆD)∞¯z(δ,D,R)(zY (δ, D, R) − (D + ˆD))dF (z) + ˆδ ˆD)− E(zY (δ, D, R) − D)dF (z),where the value of ˆD and ˆδ are determined by the stage-one lending game.This formulation of the investment game takes into account that if the liquidity crisis doesnot occur at stage one, at stage four the country defaults if z ≤ ¯z as in (2). Conversely, ifthe liquidity shock takes place at stage one, the threshold value of z that triggers country’sdefault at stage four is ẑ = (D + ˆD)/Y (δ, D, R) (see the proof of Proposition 4 for details), withẑ > ¯z, ceteris paribus. Due to stage-one lenders’ intervention, the country uses ˆδ ˆD to inject theneeded resources in the event of a shock and these resources cannot be invested in the productiontechnology. This raises the likelihood of a default following the productivity shock.The country’s stage-zero investment problem is not constrained by the resource and timeconsistencyconstraints.The resource constraint is redundant because the country can usethe new credit line to inject the needed resources. The time-consistency constraint is redundantbecause when deciding on ˆD, the country maximizes its continuation payoff post-liquidity shock,implying that the time-consistency constraint always holds true.F L.Finally, at stage zero the lender sets δ to solve the zero profit condition:( ∫ ∞∫ ∞)δD = D (1 − η(D)) dF (z) + η(D) dF (z)¯z(δ,D,R)ẑ(δ,D,R, ˆD)δ = (1 − η(D))(1 − F (¯z(δ, D, R))) + η(D)(1 − F (ẑ(δ, D, R, ˆD))).Proposition 4 summarizes the equilibrium features of the stage-one lending game in sub-gamePROPOSITION 4. At stage one, the country borrows ˆD ∗ = E/ˆδ ∗ . Given ˆD ∗ , it exists aunique value of ˆδ ∈ (0, 1) that solves the lender’s zero-profit condition, ˆδ ∗ .The sovereign decisions on D and R differ from those obtained under sub-games F and N inSection 3.4, because in the continuation game the sovereign will obtain additional credit afterthe liquidity shock occurs. To illustrate the results of the investment and lending games undersub-game F L and the country’s equilibrium choice, we perform a numerical simulation. We usethe same value of the parameters and the functional forms as in Section 4. Our objective is toassess whether, at equilibrium, the country chooses to borrow from the foreign lenders’ shouldthe liquidity shock occur, and the consequences of this choice on the values of debt, reserves,discount factor (or, equivalently, the rate of return) and welfare.24

Figure 7 reports the same equilibrium outcomes obtained under sub-game F as in Figure 3(that is, without stage-one intervention), and adds the equilibrium levels of rate of return, debtand reserves when the country can issue new debt in the event of a liquidity crisis. 17 The lowerrightpanel shows that, with this parametrization, the country prefers to issue fresh debt if hit bya liquidity shock instead of depleting its reserves. The upper-left panel shows the rates of returncharged when the country can decide to borrow in stage one: the rate of return set at stagezero is slightly lower than the one charged without foreign lenders’ intervention. Instead, thestage-one rate of return is higher than the stage-zero rate, because stage-one lenders anticipatethat the probability of default after the realization of the production technology increases withforeign lenders’ intervention. Finally, the level of debt and reserves chosen by the sovereign atstage zero is lower than without stage-one lenders’ intervention.Corroborating the conclusions obtained without lenders’ intervention, Figure 7 shows thateven when the country can obtain additional credit at stage one debt and reserves appear tobe in a relationship of complementarity. Indeed, a comparison with Figure 3 shows that theirvalue increases with respect to the benchmark economy as well as when the variability of theproduction technology increases. The lower-right panel illustrates the welfare implications offoreign lenders’ intervention, and shows that the country will always choose to borrow fromforeign lenders in stage one instead of making use of its reserves to overcome a liquidity crisis.When it can resort to lenders at stage one, the country will keep reserves in the event of a liquidityshock, while it will pay back its larger debt only if the value of the production technology islarge enough.This exercise shows that the country accumulates reserves independently from their use inthe event of a liquidity crisis. Indeed, even though reserves are no longer necessary to overcome aliquidity crisis, they are still a valuable asset since the country can retain them in case of default.Consequently, the optimal portfolio allocation still prescribes a positive amount of investmentin reserves.Figure 8 shows the overall cost of debt with and without the possibility of foreign lenders’intervention in stage one. We show that the sum of rate of returns r = 1/δ − 1 and ˆr = 1/ˆδ − 1weighted by the borrowed amounts D and ˆD is larger than the rate of return in sub-game F .The combined effect of the accumulated level of reserves and the higher probability of defaultimplies that the total cost of debt set by lenders is higher when the country can borrow fromforeign lenders following a liquidity crisis, compared to a world in which the country has toliquidate its reserves to overcome a liquidity shock. This shows that when liquidity crises occur,the possibility to obtain additional credit does not necessarily ease the pressure on the sovereignspreads.17 The outcomes obtained under sub-game N are the same as in Figure 3, with and without foreign lenders’intervention, and are not reported because under the parameterization in Table 1 sub-game N is never chosen atequilibrium.25

6 ConclusionsWe develop a model of optimal portfolio choice of a risk neutral country that is subject toliquidity and productivity shocks, and study sovereign optimal choice of debt and reserves. Inour model, on the one hand reserves distract resources from the production technology, on theother hand, they have two important upsides for the country: first, they can be used in theevent of a liquidity crisis; second, they cannot be seized by investors should the country decideto default.Our main result is that, in the sovereign’s optimal strategy, reserves and debt are in arelationship of complementarity. Although raising debt increases the country’s exposure toliquidity and productivity crises, the contemporaneous accumulation of reserves mitigates theconsequences of these crises on sovereign welfare. The numerical simulations of our theoreticalmodel account for a number of stylized facts arising from data on emerging economies: theaccumulation of reserves is not necessarily accompanied by a reduction of debt. Moreover,sovereign debt cost increases with the level of debt. Finally, the value of output is positivelyrelated with the amount of sovereign reserves.The rapid increase in the emerging economies’ sovereign reserves has been often consideredone of the elements that has fueled global imbalances and that could destabilize internationalcapital markets by distorting asset prices. In our model, sovereign decision to hoard reserves isthe result of an optimal policy strategy. Therefore, the analysis in this paper prescribes that,if the level of sovereign reserves is judged to be excessively high, the international communityshould act on the underlying economic conditions that shape a country’s investment decisions.In our future research, we would like to develop a dynamic model along the lines of, e.g.,Arellano and Ramanarayanan (2012), Bianchi et al. (2013) and Aguiar and Amador (2013) tostudy the optimal maturity structure that maximizes the returns of country’s portfolio allocation.Also, we would like to build a growth model with debt and reserves to determine whether thenegative relationship between sovereign indebtedness and growth arising from the data (Reinhartand Rogoff, 2011) is affected by the accumulation of reserves.26

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[29] Reinhart, C. M., Rogoff, K. S., 2011. From Financial Crash to Debt Crisis. AmericanEconomic Review 101, 1676-1706.[30] Rodrik, D., 2006. The Social Cost of Foreign Exchange Reserves. International EconomicJournal 20, 253-266.[31] Yue, V., 2010. Sovereign Default and Debt Renegotiation. Journal of International Eocnomics80, 176-187.Appendix: ProofsProof of Proposition 1. First note that the country and the lender solve respective problemssimultaneously, so each takes the choice of the other as given.We begin by analyzing the problem of the sovereign country. The sovereign country solvesthe investment game in (3):∫ ∞max E(W (δ, D, R)) = R + (zY (δ, D, R) − D)dF (z). (1)D,R∈[0,δD] ¯z(δ,D,R)The optimal value of reserves and debt, D ∗ and R ∗ , results from the following two first-orderconditions with respect to D and R, respectively:E(W (δ, D, R)) D =∫ ∞¯z(δ,R,D)E(W (δ, D, R)) R = 1 − Y ′ (δ, D, R)(zδY ′ (δ, D, R) − 1)dF (z) = 0 (2)∫ ∞¯z(δ,R,D)zdF (z) = 0, (3)Note that R = δD is not an equilibrium because, for any given value of δ and D, Y ′ (0) =∞ implies that (3) never holds when all resources are spent in reserves. For (2) and (3),we denote the first derivative of country’s expected welfare with respect to a variable x asE(W (δ, D, R)) x . The notation for the second order derivatives is analogous. Specifically, thesecond order conditions for (3) are: 18∫ ∞E(W ) DD = δY ′′ zdF (z) + 1 ( ) Y − δDY′ 2(4)¯zY Y∫ ∞E(W ) RR = Y ′′ zdF (z) + 1 ( ) DY′ 2(5)¯zY Y∫ ∞E(W ) DR = E(W ) RD = −δY ′′ zdF (z) + DY ′Y (Y − δDY ′ ) , (6)318 For the ease of the exposition, in the following we drop functional notation.¯z29

and the sufficient conditions for unicity require that E(W ) DD < 0 and E(W ) DD E(W ) RR >(E(W ) DR ) 2 at the equilibrium.We now turn to the analysis of the lender problem. The equilibrium value of δ ∈ (0, 1),denoted δ ∗ , that solves lender’s zero-profit condition exists and is unique if, and only if, 1 − F (¯z)in (4) satisfies the following four conditions.1. The first condition insures that, as δ approaches zero, (1 − F (¯z)) takes a finite value.Formally,∃ limδ→0(1 − F (¯z)) < ∞.To show that this condition holds, first note that as δ → 0 the value of ¯z(δ, D, R) goes toinfinity. Indeed, a nil value of δ implies that the country has no resources to invest andlim δ→0 Y (δ, D, R) = 0. Consequently, F (¯z(δ, D, R)) goes to 1 andlim(1 − F (¯z)) = 0.δ→02. The second condition insures that, as δ approaches one, the value of (1 − F (¯z)) is smallerthan one, orlim(1 − F (¯z)) < 1.δ→1This condition is clearly satisfied: as δ → 1 the value of ¯z(δ, D, R) is equal tothe value of F (¯z) taken at ¯z =DY (D−R)is strictly below one.D, andY (D−R)3. The third condition concerns the curvature of (1 − F (¯z)). Specifically, the necessary (butnot sufficient) condition for uniqueness requires the concavity of (1 − F (¯z(δ, D, R))) withrespect to δ.We first compute the first derivative of (1 − F (¯z)) with respect of δ, and find it is strictlyincreasing:∂(1 − F (¯z))∂δ∂F (¯z) d¯z= −∂z dδ == f(¯z) D2 Y ′> 0.Y 2The second derivative with respect to δ follows:∂ 2 (1 − F (¯z))∂δ 2= df(¯z)(D 2 Y ′Y+ D 3 ′′f(¯z)dδ Y 2Y − 2(Y )′ ) 2< 0. (7)2 Y 4The first term is nil because z follows a uniform distribution, so ∂f(¯z)/∂δ = 0. To establishthe sign of (7) we note that the term in curly brackets is negative, thus rendering the allexpression negative.30

4. The final and necessary condition insuring that an interior solution different from thetrivial one δ = 0 exists and is unique requires thatTaking limits, we find that:lim ∂(1 − F (¯z))/∂δ > 1.δ→0∂(1 − F (¯z))limδ→0 ∂δD 2 Y ′= lim = ∞. (8)δ→0 Y 2Indeed, since a nile value of δ implies that the country has not resources to invest,and, by the Inada conditions,limδ→01Y 2 = ∞lim Y ′ = ∞.δ→0The result in (8) also uses the assumption that z follows a uniform distribution, implyingthat ∂f(¯z)/∂δ = 0. 19We conclude that, under our assumptions, it exists a unique value of δ, denoted δ ∗ , that satisfiesthe lender zero-profit condition in (4).Q.E.D.Proof of Proposition 2. As remarked in the proof of Proposition 1, since the country and thelender choices take place simultaneously, each takes the choice of the other as given.sovereign country solves the investment game in (8):∫ ∞max E(W N(δ, D, R)) = R + (1 − η(D)) (zY (δ, D, R) − D)dF (z). (9)D,R∈[0,δD] ¯z(δ,D,R)The optimal value of reserves and debt, DN ∗ and R∗ N , results from the the following two first-orderconditions with respect to D and R, respectively:∫ ∞E(W N (δ, D, R)) D = −η ′ (D)+(1 − η(D))¯z(δ,R,D)∫ ∞(zY (δ, D, R) − D)dF (z)¯z(δ,R,D)E(W N (δ, D, R)) R = 1 − (1 − η(D))Y ′ (δ, D, R)The(zδY ′ (δ, D, R) − 1)dF (z) = 0 (10)∫ ∞¯z(δ,R,D)zdF (z) = 0. (11)19 Of course, we are implicitly restricting our attention to the cases in which ¯z(δ, D, R) lies in the support forz, [1 − c, 1 + c], since otherwise the lender would never serve the country.31

As in the proof of Proposition 1, R = δD cannot be an equilibrium because, for any given valueof δ and D, Y ′ (0) = ∞ implies that (11) would never hold true when all resources are spent inreserves. The second order derivatives follow: 20∫ ∞∫ ∞E(W N ) DD = −η ′′ (zY − D)dF (z) − 2η ′ (zδY ′ − 1)dF (z)¯z¯z( ∫ ∞+(1 − η) δ 2 Y ′′ zdF (z) + 1 ( ) )Y − δDY′ 2, (12)¯zY Y( ∫ ∞E(W N ) RR = (1 − η) Y ′′ zdF (z) + 1 ( ) )DY′ 2, (13)Y Y¯z∫ ∞E(W N ) DR = E(W N ) RD = η ′ Y ′ zdF (z)¯z∫ ∞+(1 − η)(−δY ′′ zdF (z) + DY )′Y (Y − δDY ′ ) . (14)3The sufficient conditions for unicity dictate that E(W N ) DD < 0 and E(W N ) DD E(W N ) RR >(E(W N ) DR ) 2 . Throughout our analysis, we assume that these conditions hold true and verifythat they indeed are satisfied in our numerical simulations.As far as the lender problem in (7) is concerned, the analysis follows the same steps as inthe proof of Proposition 1, because η(D) does not depend on δ. Therefore, it exists also in thissetup a unique value of δ, denoted δN ∗ , that satisfies the lender zero-profit condition in (7) for agiven couple of D and R that solves (10) and (11).Q.E.D.Proof of Corollary 1. We begin with the results in (i). The lender zero-profit condition in thecase of sub-game N is as in (7). In the proof of Proposition 2 we argue that it exists a uniquevalue of δ ∗ Nthat solves (7), that is¯zZP N (δ ∗ N, D, R) = δ ∗ N − (1 − η(D))(1 − F (¯z(δ ∗ N, D, R))) = 0. (15)Applying the Implicit Function Theorem, we find thatdδ ∗ NdR = −ZP N(δN ∗ , D, R) RZP N (δN ∗ , D, R) δ≤ 0 (16)dδ ∗ NdD = −ZP N(δ ∗ N , D, R) DZP N (δ ∗ N , D, R) δ< 0. (17)Indeed, by definition of δN ∗ , ZP N(δN ∗ , D, R) δ > 0: at δN ∗ a marginal increase of δ raises thedifference between the RHS and the LHS in (7). In what follows, to ease the exposition we drop20 In the rest of this proof we drop functional notation.32

functional notation. The derivative of (15) with respect to R isZP N (δN, ∗ D, R) R = (1 − η)f(¯z) DY ′≥ 0, (18)Y 2implying that the sign of (16) is clearly positive. As for the derivative of (15) with respect toD, this is equal toZP N (δ ∗ N, D, R) D =f(¯z)(1 − η) Y − δN ∗ DY ′+ η ′ (1 − F (¯z)) > 0 (19)Y Yif Y ′ is small or R is small. Indeed, the second term in (19) is clearly positive, whereas the signof the first term depends on the sign ofY − δN ∗ DY ′,YandY − δDY ′Y≥ 0 ⇐⇒YδD ≥ Y ′ ≥ 0, (20)which holds true if Y ′ is sufficiently small. Alternatively, due to the concavity of the productionfunction Y , (20) is positive if reserves are small.We now turn to the results in (ii) and (iii). The equilibrium value of D in sub-game N, DN ∗ ,results from (11). By the Implicit Function Theorem, we obtain that:dD ∗ NdR = − E(W N(δ, DN ∗ , R)) DRE(W N (δ, DN ∗ , R)) DD> 0 (21)dD ∗ Ndδ= − E(W N(δ, D ∗ N , R)) DδE(W N (δ, D ∗ N , R)) DD< 0. (22)To establish the signs of these conditions, first recall that the condition for the unicity of the equilibriumof the investment game requires that E(W (δ, D, R)) DD < 0 at the equilibrium (see theproof of Proposition 2). From the proof of Proposition 2, we also know that E(W N (δ, D, R)) DRis equal toE(W N (δ, D ∗ N, R)) DR =∫ ∞¯z(η ′ Y ′ − (1 − η)δY ′′ )zdF (z) + (1 − η) D∗ N Y ′Y 3 (Y − δD ∗ NY ′ ) .(23)The first term in (23) is clearly positive (Y ′′ < 0 and η ′ > 0). The second term is positiveif Y − δDY ′ > 0, which, as discussed above (see (20)), is true if either Y ′ or R are small atequilibrium. This implies that, under the same conditions, (23), and (21), are positive.To determine the sign of (22) we need to discuss the sign of E(W (δ, DN ∗ , R)) Dδ, which is33

given by∫ ∞E(W N (δ, DN, ∗ R)) Dδ = DN∗ (δ(1 − η)Y ′′ − η ′ Y ′ )zdF (z) +¯z(∫ ∞(1 − η) zY ′ dF (z) − (¯zδY ′ − 1) ∂¯z ). (24)∂DThe first term is clearly negative (for Y ′′ < 0 and η ′ > 0, Y ′ > 0). As for the second term, thefirst part is small if Y ′ is sufficiently small. As for the second part, we show that the value of¯zδY ′ − 1 is negligible. Indeed, (10) implies that, at DN ∗ , the following holds true:η ′1 − η∫ ∞¯z(zY − D ∗ N)dF (z) =¯z∫ ∞¯z(zδY ′ − 1)dF (z)= (¯zδY ′ − 1) +with ζ > 0, small. Since, by construction, ¯zY − D = 0, (25) can be rewritten as(¯zδY ′ − 1) = 1 ∫ ∞(η ′ (zY − D1 − ηN)dF ∗ (z) − (1 − η)¯z+ζ∫ ∞¯z+ζ∫ ∞¯z+ζ(zδY ′ − 1)dF (z) (25))(zδY ′ − 1)dF (z) ≈ 0. (26)For ζ small, the term in squared brackets corresponds to (10) and thus it is (approximatively)equal to zero at equilibrium. All this implies that (24) and (22) are negative.Finally, using the first-order condition in (12) and applying the Implicit Function Theorem,we have that, at R ∗ N :dR ∗ NdD = −E(W N(δ, D, RN ∗ )) RDE(W N (δ, D, RN ∗ )) RR> 0 (27)dR ∗ Ndδ= − E(W N(δ, D, R ∗ N )) RδE(W N (δ, D, R ∗ N )) RR> 0. (28)By the condition for the unicity of the equilibrium of the investment game E(W N (δ, D, RN ∗ )) RR

each takes the choice of the other as given. The sovereign country solves the investment gamein (10):∫ ∞max E(W F (δ, D, R)) = R + (zY (δ, D, R) − D)dF (z) − η(D)E. (30)D,R∈[0,δD] ¯z(δ,D,R)Under the time-consistency and resource constraints∫ ∞¯z(δ,D,R)(zY (δ, D, R) − D)dF (z) ≥ E (31)R ≥ E, (32)and, of course, under R ≤ δD. We solve the problem by maximizing the following expression,where we denote the Lagrange multipliers to the time-consistency and resource constraints by λand µ, respectively (we will show that the third constraint is always satisfied at equilibrium):L(D, R, λ, µ) = R +∫ ∞¯z(δ,D,R)(zY (δ, D, R) − D)dF (z) − η(D)E( ∫ )∞+λ (zY (δ, D, R) − D)dF (z) − E − µ(R − E).¯z(δ,D,R)The relative Kuhn-Tucker conditions are given below:∫ ∞L(D, R, λ, µ) R = 1 − (1 + λ) zY ′ (δ, D, R)dF (z) − µ = 0¯z(δ,D,R)∫ ∞L(D, R, λ, µ) D = (1 + λ) (zδY ′ (δ, D, R) − 1)dF (z) − η ′ (D)E = 0L(D, R, λ, µ) λ =∫ ∞¯z(δ,D,R)¯z(δ,D,R)(zY (δ, D, R) − D)dF (z) − E = 0L(D, R, λ, µ) µ = R − E = 0( ∫ )∞λ (zY (δ, D, R) − D)dF (z) − E = 0¯z(δ,D,R)µ(R − E) = 0µ ≥ 0, λ ≥ 0There are four cases to be discussed.Case A: λ = µ = 0. In this case the two constraints are both slack. Then, the optimal levels of35

debt and reserves are determined by, respectively:∫ ∞¯z(δ,D,R)∫ ∞1 −(zδY ′ (δ, D, R) − 1)dF (z) − η ′ (D)E = 0 (33)¯z(δ,D,R)zY ′ (δ, D, R)dF (z) = 0, (34)which correspond to the conditions in Proposition 3. Note that, since Y ′ → ∞ as R → δD,R = δD cannot be an equilibrium.Case B: λ > 0, µ = 0.The time-consistency constraint is binding, whereas the resourceconstraint is slack (meaning that R > E at this candidate equilibrium). The optimal levels ofdebt and reserves and the Lagrange multiplier λ are determined by the following three conditions:∫ ∞(1 + λ)1 − (1 + λ)∫ ∞¯z(δ,D,R)¯z(δ,D,R)∫ ∞(zδY ′ (δ, D, R) − 1)dF (z) − η ′ (D)E = 0, (35)¯z(δ,D,R)zY ′ (δ, D, R)dF (z) = 0, (36)(zY (δ, D, R) − D)dF (z) − E = 0. (37)The optimal value of reserves is computed by solving the time-consistency constraint for R,however it cannot be such to deplete all resources (i.e., R = δD) because in this case Y (0) = 0and (37) would never hold. The value of the time-consistency constraint multiplier is obtainedby solving (36) for λ,λ = 1 − ∫ ∞zY ′ (δ, D, R)dF (z)¯z(δ,D,R)∫ ∞¯z(δ,D,R) zY , (38)′ (δ, D, R)dF (z)and D results from (35) after plugging λ from (38). Note that reaction function for R = R(δ, D)that solves (37) is lower than the one that solves (34), ceteris paribus. This follows from twoconsiderations: first, the time-consistency constraint is decreasing in R and, second, the timeconsistencyconstraint is slack in Case A. Moreover, since λ > 0, (35) is larger than (33), sothe reaction function for D = D(δ, R) that solves (35) is larger than the one that solves (33),ceteris paribus.Case C: λ = 0, µ > 0. The time-consistency constraint is slack, whereas the resource constraintis binding. Then, the optimal level of debt, reserves and the Lagrange multiplier µ are determined36

y the following conditions:∫ ∞¯z(δ,D,R)∫ ∞1 −(zδY ′ (δ, D, R) − 1)dF (z) − η ′ (D)E = 0, (39)¯z(δ,D,R)zY ′ (δ, D, R)dF (z) − µ = 0, (40)R = E (41)Thus the equilibrium value of reserves is R = E < δD, the optimal value of D is computed using(39) and µ results from (40). In this case, the expression for the reaction function D = D(δ, R)that solves (39) is equal to the one that solves (33), as (33)=(39). Instead, the expression ofR = R(δ, R) that solves (40) is lower than in Case A, ceteris paribus. The reason is that for agiven D, in Case A the value of R(δ, D) that solves (34) must be such that1 −∫ ∞¯zzY ′ (δ, D, R)dF (z) = 0 < µ.Case D: λ > 0, µ > 0. In this case, both the time-consistency constraint and the resourceconstraint are binding. The candidate equilibrium values of D, R, λ and µ are given by thefollowing four conditions:∫ ∞1 − (1 + λ) zY ′ (δ, D, R)dF (z) − µ = 0, (42)¯z(δ,D,R)∫ ∞(1 + λ) (zδY ′ (δ, D, R) − 1)dF (z) − η ′ (D)E = 0, (43)∫ ∞However, notice that¯z(δ,D,R)¯z(δ,D,R)(zY (δ, D, R) − D)dF (z) − E = 0, (44)R = E. (45)∫ ∞lim (zY (δ, D, R) − D)dF (z) > ER→E¯z(δ,D,R)∀E ∈ [0, δD),implying that the time-consistency constraint is always violated at R = E. To show this, wefirst note that∂∂R∫ ∞¯z(δ,D,R)∫ ∞(zY (δ, D, R) − D)dF (z) = −Y ′ (δ, D, R) zdF (z) < 0,¯z(δ,D,R)that is, the LHS of the time-consistency constraint is decreasing in R. Second, if E = 0 we have37

that, by construction, ∫ ∞(zY (δ, D, E) − D)dF (z) > 0. Instead, if E → δD,¯z(δ,D,E)∫ ∞¯z(δ,D,E)(zY (δ, D, E) − D)dF (z) − E → 0,because the country would be investing all available resources in reserves, implying that therevenue generated by the production technology would be nil, the discount factor set by thelender equal to zero and lim E→δD E → 0. All this implies that the time-consistency and resourceconstraints cannot both be binding, and the candidate equilibrium associated to Case D cannotbe an equilibrium of the investment game.In the following, we derive the second order derivatives of the maximand in (30): 21∫ ∞E(W F ) DD = δ 2 Y ′′ zdF (z) − η ′′ E + 1 ( ) Y − δDY′ 2, (46)¯zY Y∫ ∞E(W F ) RR = Y ′′ zdF (z) + 1 ( ) DY′ 2, (47)¯zY Y∫ ∞E(W F ) DR = E(W F ) RD = −δY ′′ zdF (z) + DY ′Y (Y − δDY ′ ) . (48)3¯zThe usual sufficient conditions for unicity prescribe that E(W F ) DD < 0 and E(W F ) DD E(W F ) RR >(E(W F ) DR ) 2 . Throughout our analysis, we assume that these conditions hold true and we verifythat they indeed are satisfied in our numerical simulations.Finally, the analysis of the lender problem in (9) follows the same steps as in the proof ofProposition 1. Therefore, it exists also in this framework a unique value of δ, denoted δ ∗ F , thatsatisfies the lender zero-profit condition in (7) for a given couple of equilibrium values of D andR. Q.E.D.Proof of Corollary 2. We begin with the conditions in (i). The lender zero-profit condition inthe case of sub-game F is as in (9). In the proof of Proposition 2 we show that it exists a uniquevalue of δ ∗ Fthat solves (9), that isZP F (δ ∗ F , D, R) = δ ∗ F − (1 − F (¯z(δ ∗ F , D, R))). (49)By the Implicit Function Theorem we find thatdδ ∗ FdR = −ZP F (δF ∗ , D, R) RZP F (δF ∗ , D, R) δ≤ 0 (50)dδ ∗ FdD = −ZP F (δ ∗ F , D, R) DZP F (δ ∗ F , D, R) δ21 Note that, for the ease of the exposition, we are dropping functional notation.< 0. (51)38

By definition of δF ∗ , ZP F (δF ∗ , D, R) δ > 0, because, at δF ∗ , a marginal increase of δ raises thedifference between the RHS and the LHS in (9). 22 Moreover,ZP F (δ ∗ F , D, R) R = f(¯z) DY ′thus the sign of (50) is clearly positive. AlsoZP F (δ ∗ F , D, R) D = f(¯z)Y Y − δDY ′Y 2 ≥ 0, (52)Y 2 > 0 (53)if Y ′ small. Indeed, the sign of (53) depends on the sign of (Y − δDY ′ )/Y , which, as shown inthe proof of Corollary 1 (see (20) and the ensuing discussion), is positive if Y ′ is sufficiently smallor, due to the concavity of the production function Y , when reserves are small at equilibrium.We now turn to the derivation of the results in (ii) and (iii). The equilibrium value of Din the unconstrained case of sub-game F results from (13)=(33). Using the Implicit FunctionTheorem, we obtain that:dD ∗ FdR = − E(W F (δ, DF ∗ , R)) DRE(W F (δ, DF ∗ , R)) DD> 0 (54)dD ∗ Fdδ= − E(W F (δ, D ∗ F , R)) DδE(W F (δ, D ∗ F , R)) DD< 0. (55)To begin with, the condition for the unicity of the equilibrium of the investment game impliesthat E(W F ) DD < 0 at the equilibrium (see the proof of Proposition 3). From the proof ofProposition 3 we also know that E(W F (δ, D, R)) DR is given by∫ ∞E(W F (δ, DF ∗ , R)) DR = −δY ′′ zdF (z) + D∗ F Y ′¯zY 3 (Y − δD ∗ F Y ′ ) . (56)The first term in (56) is clearly positive (Y ′′ < 0). The second term is positive if Y −δD ∗ F Y ′ > 0,which, as discussed above, is positive if Y ′ is small or R is small enough. This implies that (21)is positive under the same conditions.To determine the sign of (55) we need to discuss the sign of E(W F ) Dδ , which is given byE(W F (δ, D ∗ F , R)) Dδ = D ∗ FThe first term is clearly negative (Y ′′∫ ∞¯zδY ′′ zdF (z) +∫ ∞¯zzY ′ dF (z) − (¯zδY ′ − 1) ∂¯z∂D . (57)< 0). As for the second term, it tends to zero if Y ′ issufficiently small. Finally, we show below that the value of ¯zδY ′ − 1 is negligible. Recall that22 In what follows, to ease the exposition we drop functional notation.39

(33) implies that, at the equilibrium, the following holds true:(¯zδY ′ − 1) +∫ ∞¯z∫ ∞¯z+ζ(zδY ′ − 1)dF (z) = η ′ E(zδY ′ − 1)dF (z) = η ′ E(¯zδY ′ − 1) =(η ′ E −∫ ∞¯z+ζ)(zδY ′ − 1)dF (z) ≈ 0, (58)with ζ positive but small. The last equality in (58) follows from the fact that, by (33), the termin squared brackets goes to zero for ζ negligible.Finally, using (34) and applying the Implicit Function Theorem:dR ∗ FdD = −E(W F (δ, D, RF ∗ )) RDE(W F (δ, D, RF ∗ )) RR> 0 (59)dR ∗ Fdδ= − E(W F (δ, D, R ∗ F )) RδE(W F (δ, D, R ∗ F )) RR> 0. (60)The unicity of the equilibrium of the investment game requires that E(W F (δ, D, R ∗ F )) RR < 0.Then, since E(W F (δ, D, R ∗ F )) RD = E(W F (δ, D, R ∗ F )) DR the sign of (59) is the same as the signof (54). Finally, E(W F (δ, D, R ∗ F )) Rδ is equal toE(W F (δ, D, R ∗ F )) Rδ = −Y ′′ D∫ ∞¯z( ) YzdF (z) + D 2 ′ 2. (61)YThe first term is clearly positive, whereas the second term is positive and goes to zero if Y ′ (·) issmall. This condition insures that the all expression is positive and also (60) is positive.Q.E.D.Proof of Proposition 4. We first consider the impact of foreign lenders’ intervention when thecountry decides to inject resources after the shock occurs.If the liquidity shock occurs at stage one, at stage four the country defaults ifzY (δ, D, R) − (D + ˆD) ≤ 0 ⇐⇒ z ≤D + ˆDY (δ, D, R)≡ ẑ(δ, D, R, ˆD).Since the country uses ˆδ ˆD to inject the needed resources in the event of a shock and theseresources cannot be invested in the production technology, the threshold that prescribes defaultat stage four does not depend on ˆδ, but only on ˆD.At stage one, the country chooses ˆD to maximize its continuation payoff (post-liquidity40

shock): 23maxˆDR +∫ ∞ẑ(zY − (D + ˆD))dF (z) + ˆδ ˆD − E (62)subject toˆδ ˆD − E ≤ 0. (63)The constraint in (63) implies that foreign lenders’ intervention covers no more than the nominalvalue of the shock (E). In turn, the lender sets ˆδ to solveˆδ ˆD = ˆD∫ ∞ẑdF (z) ⇐⇒ ˆδ = 1 − F (ẑ).The solution of stage one lender’s problem follows the same steps as in the proof of Proposition1. This means that it exists a unique value of ˆδ ∈ (0, 1), denoted ˆδ ∗ , that solves the lenderzero-profit condition for a given value of ˆD.The first-order condition of the country’s borrowing problem in (62) is equal to:ˆδ − (1 − F (ẑ)) = 0 ⇐⇒ ˆδ = (1 − F (ẑ)). (64)Note that (64) coincides with the zero-profit condition that determines ˆδ ∗ . Thus, the onlyequilibrium of the game features the country borrowing ˆD ∗ = E/ˆδ ∗ , that is, as much as thelender is willing to provide at the discount factor ˆδ that solves lender’s zero-profit condition.We now turn to the analysis of the case in which the country defaults after the liquidityshock. Foreign lenders’ intervention is unfeasible in these circumstances. Indeed, at stage one,the country sets ˆD to maximize its continuation payoff, which is equal to R + ˆδ ˆD: given thatthe country defaults on the occurrence of the liquidity shock, the revenue of the productiontechnology is nil at stage four. International lenders anticipate this and thus refuse to provideliquidity.Q.E.D.23 In what follows, to ease the exposition we drop functional notation.41

FiguresFigure 3: Numerical simulations with respect to the variance of the productivity shock (c 2 /3).Figure 4: Numerical simulations — Value of output.42

Figure 5: Numerical simulations with respect to the resources needed in the event of a liquidityshock (E).t = 0t = 1t = 2t = 3t = 4LendingGameLiquidityShockDefaultDecisionProductivityShockDefaultDecisionInvestmentGameLendingGameˆδ, ˆDFigure 6: Timeline with foreign lenders’ intervention43

Figure 7: Numerical simulations with foreign lenders’ intervention.Figure 8: Cost of debt with foreign lenders’ intervention.44

t=0 - Lending game Lender❅ ❅❅ δ❅t=0 - Investment game Country❅ ❅❅ D, R❅t=1 - Liquidity shock Nature❅η ❅ 1 − η ❅❅Liquidity shockt=2 - Default decisionCountry❅No Default ❅ Default❅❅{R}No liquidity shock❄t=3 - Productivity shock Nature❅ ❅ z ❅❅Nature❅ ❅ z ❅❅t=4 - Default decision CountryNo Default ❅ Default ❅❅❅{R + zY − D − E} {R − E}CountryNo Default ❅ Default ❅❅❅{R + zY − D} {R}Figure 9: Game-tree and country’s payoffs45

Liquidity shock❄t=2 - Default decisionCountry❅ ❅No Default❅ Default❅{R}t=3 - Productivity shock Nature❅ ❅ z ❅❅Sub-game Nt=4 - Default decision Country❅No Default❅ Default ❅❅{R + zY − D − E} {R − E}Sub-game FFigure 10: Economy with liquidity shock, sub-games N and F46

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"TEMI" LATER PUBLISHED ELSEWHERE2012F. CINGANO and A. ROSOLIA, People I know: job search and social networks, Journal of Labor Economics, v.30, 2, pp. 291-332, TD No. 600 (September 2006).G. GOBBI and R. ZIZZA, Does the underground economy hold back financial deepening? Evidence from theitalian credit market, Economia Marche, Review of Regional Studies, v. 31, 1, pp. 1-29, TD No. 646(November 2006).S. MOCETTI, Educational choices and the selection process before and after compulsory school, EducationEconomics, v. 20, 2, pp. 189-209, TD No. 691 (September 2008).P. PINOTTI, M. BIANCHI and P. BUONANNO, Do immigrants cause crime?, Journal of the EuropeanEconomic Association , v. 10, 6, pp. 1318–1347, TD No. 698 (December 2008).M. PERICOLI and M. TABOGA, Bond risk premia, macroeconomic fundamentals and the exchange rate,International Review of Economics and Finance, v. 22, 1, pp. 42-65, TD No. 699 (January 2009).F. LIPPI and A. NOBILI, Oil and the macroeconomy: a quantitative structural analysis, Journal of EuropeanEconomic Association, v. 10, 5, pp. 1059-1083, TD No. 704 (March 2009).G. ASCARI and T. ROPELE, Disinflation in a DSGE perspective: sacrifice ratio or welfare gain ratio?,Journal of Economic Dynamics and Control, v. 36, 2, pp. 169-182, TD No. 736 (January 2010).S. FEDERICO, Headquarter intensity and the choice between outsourcing versus integration at home orabroad, Industrial and Corporate Chang, v. 21, 6, pp. 1337-1358, TD No. 742 (February 2010).I. BUONO and G. LALANNE, The effect of the Uruguay Round on the intensive and extensive margins oftrade, Journal of International Economics, v. 86, 2, pp. 269-283, TD No. 743 (February 2010).A. BRANDOLINI, S. MAGRI and T. M SMEEDING, Asset-based measurement of poverty, In D. J. Besharovand K. A. Couch (eds), Counting the Poor: New Thinking About European Poverty Measures andLessons for the United States, Oxford and New York: Oxford University Press, TD No. 755(March 2010).S. GOMES, P. JACQUINOT and M. PISANI, The EAGLE. A model for policy analysis of macroeconomicinterdependence in the euro area, Economic Modelling, v. 29, 5, pp. 1686-1714, TD No. 770(July 2010).A. ACCETTURO and G. DE BLASIO, Policies for local development: an evaluation of Italy’s “PattiTerritoriali”, Regional Science and Urban Economics, v. 42, 1-2, pp. 15-26, TD No. 789(January 2006).E. COCOZZA and P. PISELLI, Testing for east-west contagion in the European banking sector during thefinancial crisis, in R. Matoušek; D. Stavárek (eds.), Financial Integration in the European Union,Taylor & Francis, TD No. 790 (February 2011).F. BUSETTI and S. DI SANZO, Bootstrap LR tests of stationarity, common trends and cointegration, Journalof Statistical Computation and Simulation, v. 82, 9, pp. 1343-1355, TD No. 799 (March 2006).S. NERI and T. ROPELE, Imperfect information, real-time data and monetary policy in the Euro area, TheEconomic Journal, v. 122, 561, pp. 651-674, TD No. 802 (March 2011).A. ANZUINI and F. FORNARI, Macroeconomic determinants of carry trade activity, Review of InternationalEconomics, v. 20, 3, pp. 468-488, TD No. 817 (September 2011).M. AFFINITO, Do interbank customer relationships exist? And how did they function in the crisis? Learningfrom Italy, Journal of Banking and Finance, v. 36, 12, pp. 3163-3184, TD No. 826 (October 2011).P. GUERRIERI and F. VERGARA CAFFARELLI, Trade Openness and International Fragmentation ofProduction in the European Union: The New Divide?, Review of International Economics, v. 20, 3,pp. 535-551, TD No. 855 (February 2012).V. DI GIACINTO, G. MICUCCI and P. MONTANARO, Network effects of public transposrt infrastructure:evidence on Italian regions, Papers in Regional Science, v. 91, 3, pp. 515-541, TD No. 869 (July2012).A. FILIPPIN and M. PACCAGNELLA, Family background, self-confidence and economic outcomes,Economics of Education Review, v. 31, 5, pp. 824-834, TD No. 875 (July 2012).

2013A. MERCATANTI, A likelihood-based analysis for relaxing the exclusion restriction in randomizedexperiments with imperfect compliance, Australian and New Zealand Journal of Statistics, v. 55, 2,pp. 129-153, TD No. 683 (August 2008).F. CINGANO and P. PINOTTI, Politicians at work. The private returns and social costs of political connections,Journal of the European Economic Association, v. 11, 2, pp. 433-465, TD No. 709 (May 2009).F. BUSETTI and J. MARCUCCI, Comparing forecast accuracy: a Monte Carlo investigation, InternationalJournal of Forecasting, v. 29, 1, pp. 13-27, TD No. 723 (September 2009).D. DOTTORI, S. I-LING and F. ESTEVAN, Reshaping the schooling system: The role of immigration, Journalof Economic Theory, v. 148, 5, pp. 2124-2149, TD No. 726 (October 2009).A. FINICELLI, P. PAGANO and M. SBRACIA, Ricardian Selection, Journal of International Economics, v. 89,1, pp. 96-109, TD No. 728 (October 2009).L. MONTEFORTE and G. MORETTI, Real-time forecasts of inflation: the role of financial variables, Journalof Forecasting, v. 32, 1, pp. 51-61, TD No. 767 (July 2010).R. GIORDANO and P. TOMMASINO, Public-sector efficiency and political culture, FinanzArchiv, v. 69, 3, pp.289-316, TD No. 786 (January 2011).E. GAIOTTI, Credit availablility and investment: lessons from the "Great Recession", European EconomicReview, v. 59, pp. 212-227, TD No. 793 (February 2011).F. NUCCI and M. RIGGI, Performance pay and changes in U.S. labor market dynamics, Journal ofEconomic Dynamics and Control, v. 37, 12, pp. 2796-2813, TD No. 800 (March 2011).G. CAPPELLETTI, G. GUAZZAROTTI and P. TOMMASINO, What determines annuity demand at retirement?,The Geneva Papers on Risk and Insurance – Issues and Practice, pp. 1-26, TD No. 805 (April 2011).A. ACCETTURO e L. INFANTE, Skills or Culture? An analysis of the decision to work by immigrant womenin Italy, IZA Journal of Migration, v. 2, 2, pp. 1-21, TD No. 815 (July 2011).A. DE SOCIO, Squeezing liquidity in a “lemons market” or asking liquidity “on tap”, Journal of Banking andFinance, v. 27, 5, pp. 1340-1358, TD No. 819 (September 2011).S. GOMES, P. JACQUINOT, M. MOHR and M. PISANI, Structural reforms and macroeconomic performancein the euro area countries: a model-based assessment, International Finance, v. 16, 1, pp. 23-44,TD No. 830 (October 2011).G. BARONE and G. DE BLASIO, Electoral rules and voter turnout, International Review of Law andEconomics, v. 36, 1, pp. 25-35, TD No. 833 (November 2011).O. BLANCHARD and M. RIGGI, Why are the 2000s so different from the 1970s? A structural interpretationof changes in the macroeconomic effects of oil prices, Journal of the European EconomicAssociation, v. 11, 5, pp. 1032-1052, TD No. 835 (November 2011).R. CRISTADORO and D. MARCONI, Household savings in China, in G. Gomel, D. Marconi, I. Musu, B.Quintieri (eds), The Chinese Economy: Recent Trends and Policy Issues, Springer-Verlag, Berlin,TD No. 838 (November 2011).A. ANZUINI, M. J. LOMBARDI and P. PAGANO, The impact of monetary policy shocks on commodity prices,International Journal of Central Banking, v. 9, 3, pp. 119-144, TD No. 851 (February 2012).R. GAMBACORTA and M. IANNARIO, Measuring job satisfaction with CUB models, Labour, v. 27, 2, pp.198-224, TD No. 852 (February 2012).G. ASCARI and T. ROPELE, Disinflation effects in a medium-scale new keynesian model: money supply ruleversus interest rate rule, European Economic Review, v. 61, pp. 77-100, TD No. 867 (April2012).E. BERETTA and S. DEL PRETE, Banking consolidation and bank-firm credit relationships: the role ofgeographical features and relationship characteristics, Review of Economics and Institutions,v. 4, 3, pp. 1-46, TD No. 901 (February 2013).M. ANDINI, G. DE BLASIO, G. DURANTON and W. STRANGE, Marshallian labor market pooling: evidencefrom Italy, Regional Science and Urban Economics, v. 43, 6, pp.1008-1022, TD No. 922 (July2013).G. SBRANA and A. SILVESTRINI, Forecasting aggregate demand: analytical comparison of top-down andbottom-up approaches in a multivariate exponential smoothing framework, International Journal ofProduction Economics, v. 146, 1, pp. 185-98, TD No. 929 (September 2013).A. FILIPPIN, C. V, FIORIO and E. VIVIANO, The effect of tax enforcement on tax morale, European Journalof Political Economy, v. 32, pp. 320-331, TD No. 937 (October 2013).

2014G. M. TOMAT, Revisiting poverty and welfare dominance, Economia pubblica, v. 44, 2, 125-149, TD No. 651(December 2007).M. TABOGA, The riskiness of corporate bonds, Journal of Money, Credit and Banking, v.46, 4, pp. 693-713,TD No. 730 (October 2009).G. MICUCCI and P. ROSSI, Il ruolo delle tecnologie di prestito nella ristrutturazione dei debiti delle imprese incrisi, in A. Zazzaro (a cura di), Le banche e il credito alle imprese durante la crisi, Bologna, Il Mulino,TD No. 763 (June 2010).F. D’AMURI, Gli effetti della legge 133/2008 sulle assenze per malattia nel settore pubblico, Rivista dipolitica economica, v. 105, 1, pp. 301-321, TD No. 787 (January 2011).R. BRONZINI and E. IACHINI, Are incentives for R&D effective? Evidence from a regression discontinuityapproach, American Economic Journal : Economic Policy, v. 6, 4, pp. 100-134, TD No. 791(February 2011).P. ANGELINI, S. NERI and F. PANETTA, The interaction between capital requirements and monetary policy,Journal of Money, Credit and Banking, v. 46, 6, pp. 1073-1112, TD No. 801 (March 2011).M. BRAGA, M. PACCAGNELLA and M. PELLIZZARI, Evaluating students’ evaluations of professors,Economics of Education Review, v. 41, pp. 71-88, TD No. 825 (October 2011).M. FRANCESE and R. MARZIA, Is there Room for containing healthcare costs? An analysis of regionalspending differentials in Italy, The European Journal of Health Economics, v. 15, 2, pp. 117-132,TD No. 828 (October 2011).L. GAMBACORTA and P. E. MISTRULLI, Bank heterogeneity and interest rate setting: what lessons have welearned since Lehman Brothers?, Journal of Money, Credit and Banking, v. 46, 4, pp. 753-778,TD No. 829 (October 2011).M. PERICOLI, Real term structure and inflation compensation in the euro area, International Journal ofCentral Banking, v. 10, 1, pp. 1-42, TD No. 841 (January 2012).E. GENNARI and G. MESSINA, How sticky are local expenditures in Italy? Assessing the relevance of theflypaper effect through municipal data, International Tax and Public Finance, v. 21, 2, pp. 324-344, TD No. 844 (January 2012).V. DI GACINTO, M. GOMELLINI, G. MICUCCI and M. PAGNINI, Mapping local productivity advantages in Italy:industrial districts, cities or both?, Journal of Economic Geography, v. 14, pp. 365–394, TD No. 850(January 2012).A. ACCETTURO, F. MANARESI, S. MOCETTI and E. OLIVIERI, Don't Stand so close to me: the urban impactof immigration, Regional Science and Urban Economics, v. 45, pp. 45-56, TD No. 866 (April2012).M. PORQUEDDU and F. VENDITTI, Do food commodity prices have asymmetric effects on euro areainflation, Studies in Nonlinear Dynamics and Econometrics, v. 18, 4, pp. 419-443, TD No. 878(September 2012).S. FEDERICO, Industry dynamics and competition from low-wage countries: evidence on Italy, OxfordBulletin of Economics and Statistics, v. 76, 3, pp. 389-410, TD No. 879 (September 2012).F. D’AMURI and G. PERI, Immigration, jobs and employment protection: evidence from Europe before andduring the Great Recession, Journal of the European Economic Association, v. 12, 2, pp. 432-464,TD No. 886 (October 2012).M. TABOGA, What is a prime bank? A euribor-OIS spread perspective, International Finance, v. 17, 1, pp.51-75, TD No. 895 (January 2013).L. GAMBACORTA and F. M. SIGNORETTI, Should monetary policy lean against the wind? An analysis basedon a DSGE model with banking, Journal of Economic Dynamics and Control, v. 43, pp. 146-74,TD No. 921 (July 2013).M. BARIGOZZI, CONTI A.M. and M. LUCIANI, Do euro area countries respond asymmetrically to thecommon monetary policy?, Oxford Bulletin of Economics and Statistics, v. 76, 5, pp. 693-714,TD No. 923 (July 2013).U. ALBERTAZZI and M. BOTTERO, Foreign bank lending: evidence from the global financial crisis, Journalof International Economics, v. 92, 1, pp. 22-35, TD No. 926 (July 2013).

R. DE BONIS and A. SILVESTRINI, The Italian financial cycle: 1861-2011, Cliometrica, v.8, 3, pp. 301-334,TD No. 936 (October 2013).D. PIANESELLI and A. ZAGHINI, The cost of firms’ debt financing and the global financial crisis, FinanceResearch Letters, v. 11, 2, pp. 74-83, TD No. 950 (February 2014).A. ZAGHINI, Bank bonds: size, systemic relevance and the sovereign, International Finance, v. 17, 2, pp. 161-183, TD No. 966 (July 2014).M. SILVIA, Does issuing equity help R&D activity? Evidence from unlisted Italian high-tech manufacturingfirms, Economics of Innovation and New Technology, v. 23, 8, pp. 825-854, TD No. 978 (October2014).G. BARONE and S. MOCETTI, Natural disasters, growth and institutions: a tale of two earthquakes, Journalof Urban Economics, v. 84, pp. 52-66, TD No. 949 (January 2014).2015G. BULLIGAN, M. MARCELLINO and F. VENDITTI, Forecasting economic activity with targeted predictors,International Journal of Forecasting, v. 31, 1, pp. 188-206, TD No. 847 (February 2012).A. CIARLONE, House price cycles in emerging economies, Studies in Economics and Finance, v. 32, 1,TD No. 863 (May 2012).G. BARONE and G. NARCISO, Organized crime and business subsidies: Where does the money go?, Journalof Urban Economics, v. 86, pp. 98-110, TD No. 916 (June 2013).P. ALESSANDRI and B. NELSON, Simple banking: profitability and the yield curve, Journal of Money, Creditand Banking, v. 47, 1, pp. 143-175, TD No. 945 (January 2014).R. AABERGE and A. BRANDOLINI, Multidimensional poverty and inequality, in A. B. Atkinson and F.Bourguignon (eds.), Handbook of Income Distribution, Volume 2A, Amsterdam, Elsevier,TD No. 976 (October 2014).M. FRATZSCHER, D. RIMEC, L. SARNOB and G. ZINNA, The scapegoat theory of exchange rates: the firsttests, Journal of Monetary Economics, v. 70, 1, pp. 1-21, TD No. 991 (November 2014).FORTHCOMINGM. BUGAMELLI, S. FABIANI and E. SETTE, The age of the dragon: the effect of imports from China on firmlevelprices, Journal of Money, Credit and Banking, TD No. 737 (January 2010).G. DE BLASIO, D. FANTINO and G. PELLEGRINI, Evaluating the impact of innovation incentives: evidencefrom an unexpected shortage of funds, Industrial and Corporate Change, TD No. 792 (February2011).A. DI CESARE, A. P. STORK and C. DE VRIES, Risk measures for autocorrelated hedge fund returns, Journalof Financial Econometrics, TD No. 831 (October 2011).D. FANTINO, A. MORI and D. SCALISE, Collaboration between firms and universities in Italy: the role of afirm's proximity to top-rated departments, Rivista Italiana degli economisti, TD No. 884 (October2012).M. MARCELLINO, M. PORQUEDDU and F. VENDITTI, Short-Term GDP Forecasting with a mixed frequencydynamic factor model with stochastic volatility, Journal of Business & Economic Statistics,TD No. 896 (January 2013).M. ANDINI and G. DE BLASIO, Local development that money cannot buy: Italy’s Contratti di Programma,Journal of Economic Geography, TD No. 915 (June 2013).J. LI and G. ZINNA, On bank credit risk: sytemic or bank-specific? Evidence from the US and UK, Journalof Financial and Quantitative Analysis, TD No. 951 (February 2015).

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