Professor Dr. Holger Strulik Open Economy Macro 1 / 34

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1 Professor Dr. Holger Strulik Open Economy Macro 1 / 34

2 13. Sovereign debt (public debt) governments borrow from international lenders or from supranational organizations (IMF, ESFS,...) problem of contract enforcement: no international government collateral? compare to individual borrowers and lenders (in strong states) for governments willingness rather than ability to repay debt matters. Yet, why borrow internationally? international risk diversification (consumption smoothing) investment Why repay sovereign debt? (cost of default) direct sanctions (a war? trade restrictions?) exclusion from international capital markets Professor Dr. Holger Strulik Open Economy Macro 2 / 34

3 Model: Reputation As A Collateral Eaton, J. and Gersovitz, M. (1981). Debt with potential repudiation: Theoretical and empirical analysis. Review of Economic Studies, 48(2), Here we consider a largely simplified version. endowment economy: Y t in period t government not explicitly modeled infinite planning horizon β: discount factor of future utility, 0 < β < 1. utility of (representative) consumer: V = β t U(C t) (1) t=0 Professor Dr. Holger Strulik Open Economy Macro 3 / 34

4 Budget constraint: B t+1 = (1 + r)b t + Y t C t (2) Insert (2) in (1): V = β t U ((1 + r)b t + Y t B t+1) t=0 FOC (wrt B t) for utility maximum: β t U (C t)(1 + r) β t 1 U (C t 1) = 0 (3) That is the (familiar) Euler equation: β(1 + r)u (C t+1) = U (C t) Professor Dr. Holger Strulik Open Economy Macro 4 / 34

5 Intertemporal budget constraint (invoking the No-Ponzi-End-Condition): t=0 ( ) t 1 C t = 1 + r t=0 ( ) t 1 Y t (4) 1 + r Sequence of endowments Y t: 0,1,0,1,0,... Suppose β(1 + r) = 1 Thus ideally, perfect consumption smoothing, C t = C at all times Invoking the formula for geometric series: t=0 ( ) t 1 1 C = C 1 + r r = C 1 + r r t=0 ( ) t 1 Y t = ( ) r 1 + r = r 1 + r t=0 ( 1 ) 2t 1 + r Professor Dr. Holger Strulik Open Economy Macro 5 / 34

6 Once more invoking the formula for geometric series: t=0 ( ) 2t 1 = 1 + r (1+r) 2 = 2 (1 + r) r(2 + r) (5) After substituting everything in (4) we obtain C: C = r 1 + r 1 (1 + r)2 1 + r r(2 + r) = r (6) That is: B 0 = r B 1 = 0 B 2 = r... C 0 = r = r C 1 = r 2 + r = r C 0 = r = r Professor Dr. Holger Strulik Open Economy Macro 6 / 34

7 Next, suppose: the individual (country) does not repay the first loan as punishment he (it) is forever excluded from international capital markets The implied sequence of utility is ( ) 1 U + βu(1) + β 2 U(0) + β 3 U(1) + β 4 U(0), r Compare this to ( ) ( ) ( ) ( ) ( ) 1 1 U + βu + β 2 1 U + β 3 1 U + β 4 1 U, r 2 + r 2 + r 2 + r 2 + r if the individual is sufficiently patient and risk averse it will smooth consumption and repay this is obvious if U(0) =, as for example for log-utility. Professor Dr. Holger Strulik Open Economy Macro 7 / 34

8 However the argument is not totally convincing: the individual cannot be cut-off from saving then there exists a savings strategy with default that beats debt repayment: Bulow, J., and Rogoff, K. (1989). Sovereign Debt: Is to Forgive to Forget?. American Economic Review 79, Strategy: default in period 1 and save 1 2+r C 1 = r = 1 + r 2 + r in period 2 save nothing and consume the return (1 + r) in period 3 save again 1 2+r starting in period 1 consumption rises by 1 = 1+r 2+r 2+r r 1+r compared to repayment strategy. Professor Dr. Holger Strulik Open Economy Macro 8 / 34

9 Problems: model predicts default at good economic states empirically default happens more in bad states in about 60 percent default happened when output was below trend (Tomz and Wright, 2007) the model can be fixed by introducing autocorrelated shocks (Aguiar and Gopinath, 2006) Conclusion (Bulow and Rogoff, 1989): reputation argument needs full exclusion from world market suppose only exclusion from debt market defaulting country is allowed to save (in other than creditor countries) conditional on getting debt, best response: default and invest the money to insure for the future the new contract doesn t need reputation (deposit as collateral) anticipating this, there will be no debt contract. Professor Dr. Holger Strulik Open Economy Macro 9 / 34

10 Borrowing for Investment (Obstfeld/Rogoff, Chapter 6.2) so far: endowment economy, no production now: lending to increase investment and thus next period s GDP investment is good for the lender, it increases the chance that the loan gets repaid problem: once the loan arrived, why not use it for consumption? Simple model: no uncertainty (no borrowing to smooth consumption) 2 periods. investment in capital K 2 representative agent utility U 1 = u(c 1) + βu(c 2) period 1: output Y 1 given, end of period capital stock K 1 = 0 period 2: Y 2 = F (K 2) investment K 2 partly financed by borrowing, D 2 debt of period 2 R: repayment end of period 2 for simplicity: direct sanctions: share η of resources lost after default. Professor Dr. Holger Strulik Open Economy Macro 10 / 34

11 Budget constraints: K 2 = Y 1 + D 2 C 1 C 2 = F (K 2) + K 2 R Individuals eat the end-of-period capital stock. Sanctions: in case of default, creditor sanctions reduce the country s period 2 resources by fraction η Thus repayment R = min {(1 + r)d 2, η [F (K 2) + K 2]} Observe: high investment K 2 reduces the incentive to default. Professor Dr. Holger Strulik Open Economy Macro 11 / 34

12 For comparison, hypothetical benchmark: commitment to repay (1 + r)d 2 always: max U 1 = u(y 1 + D 2 K 2) + βu [F (K 2) + K 2 (1 + r)d 2] K 2,D 2 FOCs: u (C 1) + βu (C 2)(F (K 2) + 1) = 0 (7) u (C 1) βu (C 2)(1 + r) = 0 (8) From (7) and (8): u (C 1) βu (C 2) = (1 + r) = F (K 2) + 1 (9) This should look familiar: marginal rate of substitution between utility from present and future consumption equals the gross return on investment. equalization of returns: r = F (K 2) Professor Dr. Holger Strulik Open Economy Macro 12 / 34

13 Without commitment: repayment never exceed sanction cost: R η [F (K 2) + K 2] case 1: borrower uses credit as he wishes case 2: borrower commits to use credit for investment note: investment K 2 makes sanctions more costly, increases the range of incentive compatibility. Case 1: lenders worry: will the country invest enough such that η [F (K 2) + K 2] (1 + r)d 2? Let D denote maximum debt without default. Professor Dr. Holger Strulik Open Economy Macro 13 / 34

14 Further simplifying assumptions: linear production: Y 2 = αk 2, α > r log utility: u(x) = log(x) Next: Solve the problem for both cases, default and non-default Compare utilities as a function of debt D 2 Determine the critical D 2 below which utility of non-default is higher. (i) In case of non-default: C 1 = Y 1 + D 2 K 2, C 2 = (α + 1)K 2 (1 + r)d 2 (10) Together (intertemporal budget constraint): C 1 + C2 α + 1 = Y1 + α r D2 α + 1 (11) Professor Dr. Holger Strulik Open Economy Macro 14 / 34

15 Utility maximization: max U 1 = log (Y 1 + D 2 K 2) + β log [(α + 1)K 2 (1 + r)d 2] (12) K 2 FOC: 1 β(α + 1) + = 0 Y 1 + D 2 K 2 (α + 1)K 2 (1 + r)d 2 1 β(α + 1) = (13) C 1 C 2 Insert C 2 = β(α + 1)C 1 into (11) and solve for optimal consumption: C 1 = 1 ( Y 1 + α r 1 + β α + 1 D2 C 2 = β(α + 1) 1 + β ) ( Y 1 + α r α + 1 D2 ) Professor Dr. Holger Strulik Open Economy Macro 15 / 34

16 Implied utility: U N = (1 + β) log [ ( 1 Y 1 + α r )] 1 + β α + 1 D2 + β log [(1 + α)β] (ii) In case of default: C 1 = Y 1 + D 2 K 2, C 2 = (1 η)(α + 1)K 2 Together (intertemporal budget constraint): C 1 + C 2 = Y1 + D2 (14) (1 η)(1 + α) Utility maximization: max U 1 = log (Y 1 + D 2 K 2) + β log [(1 η)(α + 1)K 2] (15) K 2 Professor Dr. Holger Strulik Open Economy Macro 16 / 34

17 FOC: 1 β(1 η)(1 + α) + = 0 C 2 = β(1 η)(1 + α)c 1 (16) C 1 C 2 Combine this with (14) to solve for optimal consumption: C 1 = 1 (Y1 + D2) 1 + β C 2 = β (1 + α)(1 η) (Y1 + D2) 1 + β Implied utility: U D = (1 + β) log ( ) 1 (Y1 + D2) + β log [β(1 + α)(1 η)] 1 + β Professor Dr. Holger Strulik Open Economy Macro 17 / 34

18 Compute utility difference: ( ) 1 + D 2 U D U N Y = (1 + β) log α r D 2 + β log(1 η) (17) α+1 Y 1 Inspect (17) and observe: utility difference negative for small debt ratios (for D 2 Y 1 0) since η < 1 utility difference increase in debt ratio eventually it gets positive The critical debt value beyond which lenders will not extend credit is obtained by solving U D U N = 0: D 2 = D ( 1 1 η 1 α r α+1 ) β 1+β 1 ( ) β 1 1+β 1 η Y 1 (18) Professor Dr. Holger Strulik Open Economy Macro 18 / 34

19 Observe from inspecting (18): (1 + r) > η(1 + α) is mild sufficient condition for D > 0 more powerful sanctions (greater η) increase the borrowing limit D more patience (higher β) and higher productivity (higher α) increase D higher interest on debt r reduces D Next step: maximize utility given the incentive compatibility constraint D 2 D: max L = log(y 1 + D 2 K 2) + β log [F (K 2) + K 2 (1 + r)d 2] λ ( D 2 D ) K 2,D 2 (Recall: inequality constraints, like D 2 D, require the Kuhn-Tucker FOC, i.e. either D 2 < D and λ = 0, or D 2 = D and λ > 0.) Professor Dr. Holger Strulik Open Economy Macro 19 / 34

20 FOCs (in general notation): u (C 1) = (1 + r)βu (C 2) + λ u (C 1) = [ 1 + F (K 2) ] βu (C 2) λ [ D 2 D ] = 0 Observe: if country is not debt constrained: D 2 < D such that λ = 0: solution coincides with commitment case if debt constrained: D 2 = D and λ > 0: U (C 1) > (1 + r)βu (C 2) F (K 2) > r i.e. domestic interest rate is above r, investment is below commitment case and the MRS is above the commitment case (first period consumption is too low ) Professor Dr. Holger Strulik Open Economy Macro 20 / 34

21 Case 2: Pre-commitment to investment: country sets K 2 before it gets credit (how?) it may nevertheless default but lenders are always willing to lend up to D 2(1 + r) η [F (F 2) + K 2] difference to inflexible upper bound D: country can always get more credit by committing to invest more. but the debt ceiling still distorts allocation (from first best under commitment to repay). Professor Dr. Holger Strulik Open Economy Macro 21 / 34

22 Debt Overhang Idea: probability of default increases with accumulated debt it may actually be in the self interest of creditors to write down some debt debt restructuring, haircuts Simple model: period 1: income Y 1, inherited debt D due in period 2 period 2: income Y 2 = AF (K 2) A: productivity shock with E(A) = 1, A [A, Ā] and prob density π(a) 100 percent depreciation: K 2 = investment in period 1 risk neutral consumers (linear utility) time preference rate=interest rate = 0 direct sanction as enforcement mechanism. Professor Dr. Holger Strulik Open Economy Macro 22 / 34

23 Domestic consumers: U 1 = C 1 + E(C 2) C 1 = Y 1 K 2 C 2 = AF (K 2) min [ηaf (K 2), D] Note: default more likely in a bad state (low A). max Y 1 K 2 + E(AF (K 2)) E min [ηaf (K 2), D] }{{} V (D,K 2 ) Note: E(AF (K 2)) = F (K 2) since E(A) = 1 V (D, K 2) is the expected repayment from creditors perspective, V is the market value of debt, D is the face value of debt. Professor Dr. Holger Strulik Open Economy Macro 23 / 34

24 Suppose there is default in case of low A, then A < V (D 2, K 2) = A D ηf (K 2 ) D ηf (K 2 and we have: ) Ā ηaf (K 2)π(A)dA + Dπ(A)dA D ηf (K 2 ) Note: prob. of default is not exogenous. It depends on investment K 2. FOC for optimal investment: 1 + F (K 2) K 2 V (D, K 2) = 0 Using Leibniz rule we obtain the FOC: [ F (K 2) 1 η A D ηf (K 2 ) Aπ(A)dA ] = 1 (19) Professor Dr. Holger Strulik Open Economy Macro 24 / 34

25 Observe from (19) LHS: expected return of investment net of expected penalty payments RHS: marginal rate of substitution between consumption this period and next period (MRS) the MRS ( u (C 1 ) βu (C 2 ) is unity because linear utility and no time preference ) from implicitly differentiating: dk 2 dd < 0 (not so easy to show) more inherited debt means a higher prob. of default and thus a lower return on investment and less investment (the debt overhang effect). How does the value of debt depend on debt? ( dv Ā dd = Dπ(A)dA + ηf (K 2) D ηf (K 2 ) A D ηf (K 2 ) Aπ(A)dA ) dk2 dd Professor Dr. Holger Strulik Open Economy Macro 25 / 34

26 Observe: the first term is positive conditional on repayment in full the value of debt is increasing in D the second term is negative higher face value of debt depresses investment and makes default more likely The second term dominates for large D Debt Laffer Curve Conclude: creditors can make themselves better off by writing down face value of debt haircut problem: coordinate forgiveness, eliminate free rider behavior one solution (?): a very large buyer buys most of the debt and forgives some of it problem: why sell to the large buyer, except at the higher ex-post price? Professor Dr. Holger Strulik Open Economy Macro 26 / 34

27 The SUW textbook discusses reduced-form example of debt overhang: Suppose: repayment D (face value) in good state repayment of 25 in bad state probability of good state π increasing debt reduces π, π(d) with π (D) < 0 expected repayment: V = π(d)d + (1 π(d)) 25 And thus V D = π (D)D + π π (D) 25 = π + π (D)(D 25) Observe: interior maximum if π declines fast enough... Professor Dr. Holger Strulik Open Economy Macro 27 / 34

28 Figure 12.5: The debt Laffer curve 45 o Expected Repayment D D* D Debt Observe: Optimal debt-forgiveness D D Expected repayment increases up to the amount D D Secondary market price of debt = V /D: ratio of debt-laffer-curve / 45 o -line Professor Dr. Holger Strulik Open Economy Macro 28 / 34

29 Numerical example: Face Value D = 100 prob of good state π(100) = 1/3 Expected repayment: 100/ /3 = 50 Secondary market price: 50/100 = 0.5 Haircut: 1/5 Face Value D = 80 prob of good state π(80) = 1/2 Expected repayment: 80/2 + 25/2 = 52.5 Secondary market price: 52.5/80 = 0.66 Professor Dr. Holger Strulik Open Economy Macro 29 / 34

30 Free-Rider Problem: in case of many creditors: who forgives debt? those who don t forgive benefit more from increase in market price collective action problem Other instruments of debt restructuring: third party buy-backs debt swaps Third party buy-backs a third party (IMF, ECB) buys debt on secondary market and immediately forgives it (destroys the paper) a pretty expensive method of debt relief why? prices adjust. Professor Dr. Holger Strulik Open Economy Macro 30 / 34

31 Returning to the example, suppose third party announces to buy 75 units of face value of debt: Result: outstanding debt after buy-back: 25 this can be paid at any state expected payment: 25 (this is the face value!) expected price per unit of debt: 1 the price jumps up to 1 at announcement, before buy-back takes place. creditors get 75 from third party and 25 from debtor before they expected to get 50 (net gain 50) the debtor pays 25 before it expected to pay 50 (net gain 25) third party pays 75 for debt relief of 25 method mainly benefits the creditors. Professor Dr. Holger Strulik Open Economy Macro 31 / 34

32 Debt Swaps: issuance of new debt in exchange for old debt new debt is senior, i.e. it is served first. For the example: suppose government issues 25 units of new debt with seniority new debt is default-free! price of new debt: 1 (face value) stock of old debt after the swap: D o value of old debt: D o 1/3 + 0 (since in a bad state new debt is served first) price of old debt (1/3D o )/D o = 1/3 price of old debt falls from 1/2 to 1/3 receipts from new debt can be used to swap 25/(1/3) = 75 units of old debt. Professor Dr. Holger Strulik Open Economy Macro 32 / 34

33 Results: outstanding old debt falls to 25 debtor expects to pay /3 = 33.3 it expects to pay = 16.6 less creditors expect to get /3 = 33.3 they get = 16.6 less this seems to be a bad deal for risk-neutral investors but risk averse investors may actually like it. Professor Dr. Holger Strulik Open Economy Macro 33 / 34

34 Example the Greek Debt Swap of March 2012: face value of debt 350 billion Euros 206 billion held by private creditors 177 billion thereof issued under Greek law new Greek law: bond swap if more than 2/3 of creditors agree private creditors agreed (including some holders of bonds issued under foreign law) 197 billion old debt was exchanged for 92 billion new debt with face value 92 billion debt write down 105 billion. During the Greek debt crisis we ve also seen: haircuts: 50% in 2011 (with about 83 % of bond holders participating) third party buy backs: since 2012 ECB buys Greek bonds outright default: 2015 on IMF loan... December 5th, 2016: 86 billion euros debt relief by EU governments. Professor Dr. Holger Strulik Open Economy Macro 34 / 34