metadata: authors: 0: author: Cohen, Daniel
1: author: Verdier, Thierry
count: World
docna: 0: docna: Debt, debt relief, and growth : a bargaining approach
docty: Policy Research Working Paper
subtopic: Economic Theory & Research,Environmental Economics & Policies,Strategic Debt Management,Banks & Banking Reform,Financial Intermediation
teratopic: Environment,Macroeconomics and Economic Growth,Finance and Financial Sector Development
abstracts: cdata!: This report compares and identifies two
            ways that Governments can "up-front" the
            adjustment effort: accumulating reserves; and engaging in an
            equity swap. The authors compare these methods with a
            constant rescheduling agreement which assumes that no
            reserves can be accumulated and that tax collections go to
            the creditors. The paper analyzes the outcome of a
            "memoryless" rescheduling agreement. The model
            exhibits two potential Laffer curves effects. In one, the
            lenders would want to reduce the vulnerability of the debtor
            to their sanctions (that is, required taxation). In the
            other, the debtor would actually prefer less growth than
            more. The role of reserves is studied and it is determined
            that their use can improve a country's welfare, over
            the case of the rescheduling agreement. The country must,
            however, be able to commit itself to a tax rate before
            negotiations start. Otherwise, reserves are useless.
            Debt-equity swaps are also studied. The outcome - always
            Pareto - dominates the outcome of the rescheduling
            equilibrium. Banks always gain a fraction of the
            country's capital above the share of output that they
            gain in the rescheduling equilibrium. Thus, banks are
            relatively less "impatient" than the country to
            reach a debt-relief agreement.
repnme: repnme: Debt, debt relief, and growth : a
            bargaining approach
display_title: Debt, debt relief, and growth : a
            bargaining approach
listing_relative_url: /research/1991/09/699885/debt-debt-relief-growth-bargaining-approach
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Environment,Macroeconomics and Economic Growth,Finance and Financial Sector Development
origu: International Economics Department
no_of_pages: 32
model_extractions: **Policy, Research, and External Affairs|**

# **Debt and International Finance**

WPS 762 This paper -a product of the Debt and International Finance Division, Intemational Economics - Department is part of a larger effort in PRE to understand the impact of debt reduction on developing countries. Copies are available free from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Sheilah King-Watson, room S8-040, extcnsion 31047 (27 pages).

Debtor countries in the 1980s paid creditors with taxes the governments had to levy on their citizens. Because taxes must be collected in a distortionary way, however, governments are tempted to "up-front" the adjustment effort. Doing so helps relieve investment and growth of the burden of expected future taxation.

Governments have two ways of up-fronting: accumulating reserves and engaging in an equity swap. Cohen and Verdier compare these methods with a constant rescheduling agreement. In the rescheduling agreement, it is assumed that no reserves can be accumulated and that all tax collections go to the creditors. Their findings: _Rescheduling agreement._ A "memoryless" (past-independent) rescheduling game was studied. It is Pareto-inefficient. Two Laffer curves can take place. In one, the lenders would want to reduce the vulnerability of the debtor to their sanctions (that is, required taxation). In the other, the debtor would actually prefer less growth than more.

_The role of reserves._ Use of reserves can improve a country's welfare, over the case of the rescheduling agreement. The country must, however, be able to commit itself to a tax rate before negotiations start. Otherwise, reserves are useless.

_Debt-equity swaps._ The outcome always Pareto-dominates the outcome of the rescheduling equilibrium. Banks always gain a fraction of the country's capital above the share of output that they gain in the rescheduling equilibrium. Thus banks are relatively less "impatient" than the country to reach a debt-relief agreement.

The PRE Working Paper Series disseminates the findings of work under way in the Bank's Policy, Research, and Extcmal Affairs Complex. Anobjective ofthe series is to get these findings out quickly, even if presentations are less than fully polished. The findings, interpretations, and conclusions in these papers do not necessarily represent official Bank policy.

Produced by the PRE Dissemination Center
# Debt, Debt Relief, and Growth: A Bargaining Approach

by Daniel Cohen (CEPREMAP, Paris and CEPR, London) and Thierry Verdier (DELTA, Paris) Table of Contents

|I.|Introduction|1|
|---|---|---|
|I.|Framework of Analysis|2|
|-II.|Characterization of the Memoriless Subgame Perfect Equilibrium|5|
|IV.|The Role of Reserves|10|
|V .|Debt Relief and Debt Equity Swaps|15|
|Appe|ndix 1: Existence and Uniqueness of the Stationary<sup>Subgame</sup><br>Perfect Equilibrium|18|
|Appe|ndix 2: Proof of Proposition 2|21|
|Appe|ndix 3: Proof of Proposition 3|24|
|Refer|ences|27|

The paper was started while both authors were visiting the<sup>Debt and International Finance</sup> Division, International Economics Department of the World<sup>Bank. The authors thank</sup> Jonathan Eaton for his useful comments on a preliminary draft.
# I - INTRODUCTION

The net transfers that the government of the indebted LDCs were asked to make to their creditors in the 1980's had to be taxed (one way or another) on their residents. Taxes are bad, because they must be collectea in a distortionary way, and this explains why governments are usually tempted to up-front the adjustment effort that they must undertake so as to relieve (as much as possible) investment and growth from the burden of expected future taxation. (Surprise hyperinflation is often a favorite candidate for such manipulation of the time pattern of taxes).

Upfronting the burden of the adjustment, however, is only good if the debtor country can make sure that it can trade more payments to-day against less in the future. Assume however that the debt which the country owes is infinite (in face value terms). The trade-off between current and later payments can only be achieved through one of the two following channels: 1) the accumulation of reserves by the country which may help it separate the decision to tax the economy from the decision to service the debt of assuming, obviously, that the reserves cannot be seized). 2) a comprehensive debt relief agreement which upfronts the burden of the adjustment (In the paper we analyze the outcome of a massive debt equity swap). It is the comparison of each of these channels to a constant rescheduling agreement which forms the core of the paper.

The analysis in this paper draws on the sequential bargaining approach to negotiation pioneered by Rubinstein (1982) and already applied to the LDC debt problem by Bulow and Rogoff (1989), Fernandez and Rosenthal (1990) and O'Connell (1988) among others. This approach is merged with the (endogeneous) growth model of debt repudiation of Cohen and Sachs (1986) and with the "tax model" of debt of Helpman (1988a and 1988b). Contrarily to the model of debt rescheduling of Bulow and Rogoff, the rescheduling equilibrium that we obtain is typically Pareto-inefficient because of the bad timing of taxation which it induces. Our model will exhibit -in addition to this basic inefficiencytwo potential Laffer curves effects. One is the usual disincentive 1
effect of taxation that may be counterproductive to both the lenders and the debtor. Another one is a case of immiserizing growth that is detrimental to the debtor only. To the extent (as we shall prove) that the bargaining power of the debtor is reduced by a faster growth rate, we shall see that it may indeed happen that a good news for growth (say a better productivity of capital) turns out to be a bad news for the debtor's welfare (and a good news for the creditors only).

Assuming away these two Laffer curves effect, we shall proceed to investigate how the Pareto-inefficiency of the bargaining equilibrium could be removed.

We first investigate whether the debtor can raise its welfare by accumulating reserves (assumed now to be non seizable by the creditors) which may help It separate (and redesign optimally) the decision to tax their domestic economy and pay their creditors. We shall see thataccumulating reserves may help the country raise its bargaining stance against the creditors but only if it can commit Itself to a tax schedule before .he negotiation starts.

We then examine the welfare implications of a comprehensive debt-equity swap. While always Pareto-improving, we show that the deal always turns out to be relatively more to the advantage of the banks than to the advantage of the country. The intuition Is simply the following. As long as no deal is struck, the country must Invest alone (albeit inefficiently). The banks are therefore relatively less impatient than the country to negotiate a deal.

Section 11 sets up the model. Section III analyzes the outcome of a memoriless rescheduling agreement. Section IV investigates the role of reserves. Section V concludes with the analysis of a debt equity swap.

# 11 - FRAMEWORK OF ANALYSIS

1 - Let us first describe the country's economy. Following Cohen and Sachs (1986) and Cohen (1991), we assume that the country has access to a technology of production which can be characterized as follows. Production shows a constant return to scale technology: 2 (1) Qt a Kt
In which Kt is the stock of installed capital. Kt evolves through a law of motion:

# (2) K t+1 1 Kt(1-d) +It

in which d is the depreciation rate and It is the flow of new installments. It can be obtained through an installment technology where (3) Jt =I( t have to be spent in order to get It new capital. Because of the constant return to scale nature of the technology of production with respect to the storable asset, the model exhibits an endogeneous (but non exploding) growth feature qualitatively similar to that analyzed in Romer (1986).

Investment and production is undertaken by a representative private agent. The government's resources are obtalned through taxes on GDP. Call etthe taa -ate at time t, government's resources are given by (4) Tt =tQt The government's utility is that of the representative agent and Is assumed to be a linear function of present and future consumption (we do not analyze, here, the effect of finite intertemporal elasticity of substitution). We let (S) UO = E 1 Ct 0 such utllity, in which Ct=Qt-Jt-Tt is the net consumption of the representative private agent at time t.

2 - Let us now describe the International environment. Throughout all this paper we shall assune that the country owes an infinite debt to its creditors (the banks). The creditors and the debtor must negotiate 3
how much money the debtor will be willing to pay. Let us assume that the banks are only interested in maximizing the present discounted value of the net transfers (Pt) performad by the debtor. Call V this value. If ISO Is the discount factor of the banks

# (6) V 0 = Z3Pt

In section V we assume that the (net) payments by the debtor and the government resource may diverge by allowing the debtor to accumulate (non-seizable) reserves. Here, let us simply assume that all resources collected by the government are paid to the creditors and let the negotiation be directly carried over the tax rate t which the government is required by the debtor to impose on the private agents; let Pt=9t Qt the net transfers performed by the debtor.

3 - Let us now describe the bargaining structure out of which the payment will be extracted from the country. We shall slightly diverge from Rubinstein alternative offer structure and consider the t' w following games. Call G the games in which the banks make an offer to the country. If the country accepts to pay whichever transfer P was asked by the bank, we go to the next period and the banks make another offer. If the country refuses the offer from the banks, two things happen. On the one hand, the banks can impose (without a loss to themselves) sanctions to the debtor. On the other hand, the period after, it becomes the country's turn to make an offer. If accepted, we move to the next stage and the country makes another offer; if refused, sanctions are Imposed on the country and it becomes the banks' turn to make an offer. Presented In extensive form, the game evolves as follows: 4
# G = Banks make offers

If y / If no. Sanctions A Q are Imposed G0 _GI_ _GI_ = Country makes offers If yes If no. Sanctions A Qt are imposed GI Go0 III - CHARACTERISATION OF THE MEMORILESS SUBGAME PERFECT EQUILIBRIUM I - In this section, we shall restrict our attention to the case when only memoriless (past independent) strategies can be Implemented. Furthermore, we shall keep the assumption that no reserves can be accumulated (say that they could entirely be seized by the creditors) and that all tax co-sections by the government<sup>go to the creditors.</sup> For any equilibrium tax burden eGt imposed on them, the private agents choose to maximize E13 11-et-y,] Qt In which Yt is the share of 0 total investment, Jt/Qt, in GDP. When t is time invariant, one can show that the private agents' strategy is one in which a fixed Investment rate and a fixed growth rate is chosen.<sup>Call x=</sup> - Q the e Investment rate associated to this equilibrium, the growth rate of the economy is then simply: (7) n = a x (e) - d 5
In which **x(G)** Is simply: (8) x(e) - Arg max l-9-x(ll/2 * a x) x 1 -9 rI +a x-dj (In conformity with Romer's model, growt' is a positive function of investment).

Let us call e and 01 the equilibrium tax rates which are obtalned in each of the games Go and GV, (when the banks or the country have a first mover advantage). Let n0 and n1 the equilibrium growth rates which they are asociated to, and yo and y1 the corresponding gross Investment t/Q t). Those parameters form a system which must solve the following equations: <!-- Start of picture text --> 1- I_o-yo c -Y) 1(ln )- 1-i;F;<br>(a) 001 (I = I-- + I11Ii- +<br>(9)<br>(b) -g 0(1+n1I ) ( no) 1-% ( 0(I+n 0 )<br><!-- End of picture text --> Equation (9-a) states that the banks must make an offer e0 which Is acceptable to the country. (Note that the investment rate, if the country was to refuse the banks' offer, is equal to the investment rate It would choose in the game G1). Equation (9-b) states that the country must make an offer which Is acceptable to the banks. In order to obtain a palatable solution, let us simply assume here that the time between two offers becomes infinitely small. Let g 1+8 h and let h go towards zero (do 5 8I). In that case 9 and _e1_ converge one towards another. The equilibrium is then simply characterized as follows 6
<!-- Start of picture text --> (e -n<br>(a) e "x; g<br>(10) (b) n u a x - d<br>-<br>(c) x Arg Max 1-x(1+l/2 a x)<br>x<br><!-- End of picture text --> One can obtain a simple representation of the equilibrium through the following diagram In growth and payments.

<!-- Start of picture text --> n<br>Banks bargaining power<br>I ,Country's response<br>_~~~~~~~<br><!-- End of picture text --> FIGURE The downward sloping curve (equation (lOb) and (lOc)) Indicates the reaction of the growth rate of the economy to a tax burden e Imposed on the country by the banks. The upward sloping curve (equation 10-a) shows that the bargaining power of the banks Is an increasing function of the growth rate of the economy (since 8, t 80, equation (10-a) shows an Increasing relationship between 0 and n). Appendix 1 analyzes In detail 7
the conditions under which equations (10) have a unique equilibrium with positive growth. Equation (A1.3) in the appendix offers the relevant condition for existence. We assume it to hold in the sequl.

In the limiting case when a =a5p the bargaining power cf the banks is simply such as to impose e = 2 . The syster (10) then becomes a recursive one and growth can be directly calculated out of (10-b) and (10-c) when is substituted to 9. Otherwise, when a < 8 the share obtained by the banks is always superior to A/2.

There are twro potential "Laffer Curves" effects associated with the equilibrium depicted In (10). The first one is the usual Laffer-Curve effect. From the banks' view point, all that matter is to maximize . When 9 is too large, the disincentive to grow may overcome (from the banks and -a fortiorifrom the debtor's view point) the benefit arising from a larger tax rate. In a subgame perfect equilibrium, however, one can not rule out that this Pare.o-inefficient outcome will occur. Indeed, in the simple case when 80 = the tax rate is e = A. If, say, A = 100%, the country is totally vulnerable to the sanction imposed by the banks the Laffer Curve effect will be obtained if and only if the optimal tax rate is below 50%. To the extent that current taxation is harmless, the incentive to tax as much as possible on a period-by-period basis explains why a memoriless subgame perfect equilibrium cannot rule out this Laffer Curve effect.

The second Laffer Curve effect which can be obtained in this equilibrium is a case of "immiserlzing growth" for the country. As equation (10-a) shows, in effect, the fastest the growth rate, the weaker the bargaining position of the country becomes (whenever a 10* _>a05_ The Intuition behind this result is simply that the larger discount factor of the country makes it more eager to settle for a deal when the benefits of the d&-: are increased. Now assume that a positive shock, say a permanent reduction of the installation cost of capital (the parameter * In equation (3)) was to occur. The effect can easily be shown to increase the equilibrium growth rate of the economy and the share 6 which is obtained by the creditors. The impact on the welfare of the debtor is however ambiguous as these two effects have an opposite 8
effect on the country. The easiest way to see how this "immiserizing growth" effect can be obtained Is simply to consider t. e case when the growth rate of the country Is exugeneous (there are no Investment decision to be made, x=O). One can show that the formula (10-a) still apply so that the welfare of the debtor is simply: Take the extreme case when A=l, which corresponds to an extremely vulnerable country. In that case, thi utility of the country is simply When 51 is large enough, this function is a decreasia function of n, and the utility of the country is at its minimum when the growth rate is the fastest, 8O=n.

Whether these Laffer Curve effect are active or not, however, is not key to our analysis. In all instances, the equilibrium characterized by the system (10) is Pareto-inefficient. Indeed, a constant tax structure can always be improved upon. Because of the distortionary nature of the taxation of output, there is always room for a Paleto-improvement which would allow to trade more payments to the creditors to-day, against less so in the future. The next section will explore what are the consequences of attempting to overcome this inefficiency.

# We can summarize the results in this section as follows:

Proposition I :There exists a unique time-invariant memoriless subgame perfect equilibrium to the bargaining game between the creditors and the debtor. It is Pareto-inefficient. Two "Laffer-Curve" effects can take place, one where the lenders would want to reduce the vulnerability of 9
the debtor to their sanctions, the other one where the debtor would prefer less growth to more.

# III - THE ROLE OF RESERVES

Let us now investigate whether the accumulation of reserves can help the country alleviate the distortionary burden of taxation. In order to anaiyze this question, we shall make the following alteration to the rules that we played in the rescheduling game of the preceding section. Rather than assuming that the game Go or GI is indefinitely repeated we shall simply assume that, once an agreement Is reached (in the relevant game), the negociation stops and the country routinely transfers at each the payment Qt or 0 Q that were agreed upon. Call GI# 1-1,2, these new games. In the case when there are no Laffer curves, It is straightforward to see that the equilibrium that Is obtained (without reserves) In the games Gi is exactly the same as in the game GI' Indeed, when they make an offer %., the banks must make sure that the country will not prefer refusing their offer and move to the game G1 . fI there are no Laffer curve, they will consequently raise eO up to the maximum value of 9 for which (9-a) is satisfied. The same will hold in the game G1 so that (9-b) will also be satisfied.

Let us now assume, In addition, that t-.a country can hold reserves that cannot be seized by the creditors . There now exists a potential wedge between the country's current tax burden and the country's payments to its creditors. Call Ft the net flows of accumulated reserves at time t and let t the tax burden Imposed on the country. 0t keeps representing the net transfers to the creditors. We therefore have (17) ht Qt - et Qt + Ft We assume In the sequel that reserves pay a rate of interest which Is equal to 80, the creditors' rate of time preference. In order to simplify the analysis, we shall limit our study to the following two simple games.

1) CLM I The country and the creditor first negociate the value a0 or 01 that the country Is required to pay (stage 1 of Game A). Once 10
this Is done, the country can choose (and credibly commit itself to) a flat tax rate _T s_ e that may differ from the value 6 that Is agreed upon. We assume that the discrepancy can be financed by a one-shot accumulation of reserves which -perhapsinvolves a negative Initial negative consumption.

2) £ : The order of play Is reversed with respect to the Game A. First, we _ass_ ume that the country can commit Itself to a flat tax rate **_T._** Then, the negociation with the creditors start as to how much payments 0Qt the country should make.

# Anallsis g h Game A

Let us first analyze the second stage of the game. Assume that the country Is (already) required to pay a f. action f Qt Of Its output each period. Will it then find it profitable to accumulate reserve so as to alleviate the distortionary burden of taxes ?

Assume that the country builds up Initially a stock of reserve R which it uses to reduce (say uniformly) the tax burden on the private economy. The amount of tax reduction that reserves can buy Is simply given as (when writing the equations in continuous time) In which n.t Is the rate of growth which is associated with the tax rate _T._ The utility that the country can get out of this accumulation of reserves can be written as 11
> Since 0O 1 ao-n T t , a I-n* **so** that one can write

I - G - y(n ) (18) UR _5 - 1_<sup>_f._n</sup> Now, the utility that the country would obtain by not accumulating reserves Is simply Inspection of equations (18) and (19) shows that UOt UI . The country cannot raise its welfare by increasing reserves and reducing taxes. The reason is that the potential benefits of lower taxation, a higher growth rate, are partially captured by the lenders.

If one moves back to stage I of game A, it is consequently obvious that nothing is changed with respect to the rescheduling game that prevailed without reserves.

# Analysis of Game _

Let us now assume that the country is already committed to a pattern of taxation r when it enters into the rescheduling game G0 or I with the banks.

Call R(G,r) the amount of reserves that the country must accumulate when a deal 6 is struck with the creditors, while it is already committed to Imposing a tax rate r on the economy. Assume that there is a small time interval h between the offers that are<sup>made in the games Go</sup> and G I One has: 9 - T R(.) - n 0 (U 4 ao h) (in which nT Is the equilibrium growth rate that is achieved by the country when the tax rate r is implemented).

12
* In the game Go (resp. G1 ) the banks (resp.the country) can make an offer eO (resp. _e1)_ that Is acceptable if

where n (1=0.1) are the optimum growth rates that are selected by the country in the game Gi when no deal is struck in that game and when it expects to strike a deal at the next round.

Making use of the following equalities and letting h go towards zero, one finds that the equilibrium values (go, **n)** are a solution to 13
<!-- Start of picture text --> e 81-' [ 1 T r - y(nT) 1 -A-y(n-)<br>8 0 -n so8+8-2~ n- ~ n - Bi 1<br>a1 - n-<br>n +<br>86 i-2 n dn n-<br>I + . (Go + d) T y(n_ r) _____<br><!-- End of picture text --> This system yields a solution _9*(T)_ and a*(r). Moreover we should only consider donrestic tax rates that satisfies _T 5 @*(x)_ in order to have positive reserves.

We can show (In appendix 2) that V _T_ such that 0 :s **_Ts_** _5@(Y):_ where UR Is the utility of the country in the memoriless rescheduling equilibrium of section III and U(T) Is the level of welfare that the country can get If It expects to be asked to pay 04(6) by the bankers. We therefore see that in the game B, the country necessarily obtains a higher utility level than In the game of section III. Hence when the country has the ability to accumulate reserves and to commit itself to a pattern of tax, then It can Improve its welfare compared to the case of the rescheduling game. We can summarize our results as follows: ErMPoujign : When It can commit itself, ahead of the negotiation, to a given tax rate, the country can raise its welfare above the memoriless rescheduling equilibrium. Otherwise, reserves serve no purpose.

14
# IV - DEBT RELIEF AND DEBT EQUITY **SWAPS**

In thls section, we now assume that the banks and the country can bargain on a comprehensive debt relief agreement of the following form. We assume that the country can surrender Its sovereignty on a piece HO. If Its stock of capital and transfer its property to the creditors. In exchange, the creditors can write-off the debt that is due by the country.

The structure of the bargaining process is the same as the one that we assumed in section 11. In the game GoP the banks offer to write-off the country's debt against a fraction of the country's capital. If the country refuses, It has a pay a cost AQO0<sup>and may make an alternative</sup> offer in the game G1. If the creditors then refuse the country's offer, sanctions are again imposed on the country and the game switch back to G0 . Whenever a deal Is accepted, the game Is terminated: the debt is written-off; the creditors and the country each undertake the investment that maximize the welfare that they can obtain from the piece of capital that they eventually own.

Call Ho and H1 the offers that are respectively made by the creditors (In Game G0 ) and by the debtor (In Game G1). Assume that the time horizon between two offers is h. One may then write the condition of acceptability of the offers Hnand HI as follows in which fi and y are respectively the growth rate and the investment rate that the country chooses In order to maximize its utility when the debt Is cancelled. Because of the homogeneity of degree I of utility with respect to the initial stock of capital, this decision obviously

**15**
does not depend on the outcome H or H1 of the game. We can then write (l,Af) as a solution to Similarly, we define n and x as the corresponding choices of the creditors, when they maximize the utility that they can generate out of H1K (i=1,2): Finally ni and yb (i=O,1) are the optimum growth and investment rates that are selected by the country in the games Gi when no deal is struck in that game and when it expects to strike a deal at the next round.

Substituting H1 for its value as a function of Ho from equation (22-b) into equation (22-a) and letting the time intei val h shrink to zero, one finds that the equilibrium value H' is simply: <!-- Start of picture text --> (25) H 6 nb lA-yb<br>W;0; a 2n %+ SZ- n b I-<br><!-- End of picture text --> We can prove the following: 16
Propositioq 3 The outcome of the debt-equity swap always Pareto-dominates the outcome of the rescheduling equilibrium. The banks always gain a fraction of the country's capital that is above the share (of output) that they gain in the rescheduling equilibrium.

The fact the debt-equity swap Pareto-dominates the rescheduling equilibrium is not much of a surprise: by writing down the debt, the banks avoid imposing on the country an inefficient taxation of output. What is more of interest is the comparison of the debt-equity swap outcome with the rescheduling equilibrium. When<sup>6</sup> 0=<sup>6</sup> i. one can see directly from equation (25) that H > A the banks gain a larger fraction than in the rescheduling game. This hierarchy is shown (in appendix 2) to be always valid. This result may be interpreted as showing that the banks are relatively less "impatient" than the country to reach a debt relief agreement. The intuition behind this result is simply the following. If the country refuses the banks' offer in game Go (or if the banks refuse its offer in the game G1). then the country has to undertake an investment Yb that will be partially ripped-off by the banks in the next round, if an offer is accepted. (The country cannot discriminate between the investment that will raise its own post-deal capital from that which will go to the banks). This explains why the banks are "relatively" less impatient to reach a deal than is the country: as long as no deal is reached, the cost of investment is (albeit inefficiently) carried by the country alone.

17
# APPENDIX 1: EXISTENCE AND UNIQUENESS OF THE STATIONARY SUBGAME PERFECT EQUILIBRIUM

The two equations caracterizing the equilibrium in equations (10) are Equation (Al.l) defines a curve x(e) (or n(G)) and equation (Al.2) defines a curve 6(x) (or e(n)). It is easy to characterize the curve x(G). It Is given by: (1+0 ax)(8 1 + d - ax) + ax (1 + 0 ax ° a (1-0) that is Oed Iled The relevant solution must satisfy: 0 S x s _a d -< -- (in order to have a finite wealth for the country). Hence and the following conditions on the parameters must hold (A1.3) 2 la - (8+d)Ic < [(8 + d) 2_(61 r) ]1 1 ~~~18
(in order to have x i ao+d 21a(I-O) (a .d)i - and -2) a * (8.+d) It is easy also to see that there exists Aand 8such that x(QA) ( and x (68) Q One can write . Hence taor S c (OGA). the growth rate In the economy will be positive

* and for e (a AO 1) the growth rate in the economy will be negative and aways higher than -d. (For **_B_** c (0* 1) It Is equal to -d).

Moreover: n (0) a ax(O) - d < r and dn . a dx (8+d a a0 Wd - ax) * < 0 Now equation (41.2) defines a curve 6(n) which Is Increasing In n for ne [lo . Indeed, ai; a - 80 2; A > Vn oMoreover we have that Vn **£** 1-d.n(lO): G(n) c [O.A1 and V 0 c (O,A1. n(S) c [-d,n(O)1 hence: the function p : [-d,n(0) -> R n -- n(O(n)l Is a continuous function from l-d,n(0)j into 1-d.n(O)I.There Is consequently a fixed point n such that V(n) **a** n. Moreover since **_p(n)m_** dn dO ° and 4p is strictly decreasing over 1-d.n(O)I one sees that the 19
fixed point is unlique. We have therefore shown that and the subgame stattonayi equilibrium exists and is **uniaue** and a solutiOn to E , @ 5 o~~.9 **_(n)_** QED Nlotes: . i then n is negativeA with- d s n s 0 and and e +| ; 8i . if eA P. A then n is positive and 6e | + i

20
# APPENDIX 2 - PROOF OF PROPOSITION 2

The acceptability conditions can be written as follows From (A2.2), we conclude that for 'r s **_e*(T),_** one<sup>has that</sup> no **s** n T - _y(nT_ eT Moreover : _U(T) =_ **_T_ -** and **_IT_** 0 0~~~~~~~~~ 21
# Thus, **since 80s 8a** and **c t 0,** we have

<!-- Start of picture text --> ao-no I, yf yXn) ai r A1n y(ne)<br>U(t) 8-.. .. _...<br>a +a -2 ^ a- nY 80 +a 2 no a -n<br>r 1- ~ 0<br>1-A 1 _ - n)<br>8 -n' '1 y(nr) 1 *o+1 2 n<br>It . 8e8 . ; ,, .I 1 ynTs + +8 -2ne<br>a1 +0-2 na_ 1 a1 n<br>* -n I - n)<br>2<br>, o0+df n J 1 -ne<br>I U(s) t ° ~~~~~~~~~~~ I -y(nT) ~ ~~~~~~~~~~~~~~~~~r 1 -y( - 1+<br>61 i-2~ n ~ fl 41t81 - n<br>1-A 1- (n<br>ao a1- 2 n<br>8 - n<br><!-- End of picture text -->

**Thus. s§c. nT 1 S** na,we also have that:

<!-- Start of picture text --> 8 - n-<br>1 - I I - y(n)<br>80 *681 - 2 n<br>81 - ne<br><!-- End of picture text --> 1

**u s**
!~~~~~~~~~~~~~~~~~~~ Using tA2.2) and the fact that the equilibrium<sup>growth rate nR</sup> obtained In section III Is such that: <!-- Start of picture text --> 1- A - 7 (nR)<br>l +¢tn 4d) o R xIUn<br>0 aI61 2 n<br>I o(n +d) -U<br><!-- End of picture text --> we can conclude that n- a n R and also that U(tr1 - I OlGO + d) a I + ffnR + d)- UR Thus U(r) a UR . Q.E.D 23
# APPENDIX 3 - POOF OF PROPOSITION 3

Let us prove that the debt-equity swap Pareto-dominates the rescheduling agreement.

Let us first analyze the country's welfare. From the definition cf H' in equation ( ), one can write the country's welfare as in which In the particular case when a 0 81* one sees that e 5 1/2.

From (A2.3) one sees first that: xA Xb S x (xb = h is obtained when n b = 0 ; xb x is obtained when _itb_ 1). One can also write 24
To the extent that the f irst term in bracket in the right-hand side Is necessarily positives and since the solution to the equation is nothing else but the solution to the rescheduling equilibrium. one sees that xb > x (x the rescheduling solution) and that the level of welfare reached by the country (which is nothing else but the right-hand side in equation (A2.4)) is above the rescheduling level (which is the right-hand side of equation A2.5).

Let us now turn to the banks' pay-off.

One can write Since the f irst term In bracket is positive (and because A s 1) one can write So that H' a A b Since nb a no, this shows that H : A o : the banks gain a larger fraction of the country's capital stock than the share 6 that they obtained In the recheduling game. This proves the second part of proposition 3.

In order to show that this does improve their pay-off one has to see that 25
I 8:-t0 (since fS > nb. and 8, t **_ao)_ Si** **_ab_ Ob** so that This shows that the pay-off of the banks which Is is certainly above the rescheduling level <!-- Start of picture text --> A 00<br><!-- End of picture text --> Indeed this latter payment can be written and the first and the second term In equation (A2.6) respectively dominate the corresponding terms In equation (A2.7).

# Q.E.D

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