JOURNAL OF Economic Dy namics

ELSEVIER Journal of Economic Dynamics and Control

19 (1995) 1141-l 163 & Contml

Large external debt and (slow) domestic growth A theoretical analysis

Daniel Cohen

Ecole Normale Supkrieure, Pans, France

CEPREMAP, Paris, France

CEPR, London, United Kingdom

(Received March 1992; final version received April 1994)

Abstract

This paper analyzes the optimal rescheduling strategy of lenders whose claims on a country show a discount on the secondary markets. The paper shows that lenders should proceed as follows. They should split the debt into two components: a performing and a nonperforming part. They should act ‘as if’ the debt amounted to the performing part and scale down how much money the borrower should service on that part only. As a result it is shown that the (efficient) servicing of the debt crowds in investment. The paper relates this result to the ‘debt overhang’ argument according to which too large a nominal claim may reduce investment and the market value of the debt.

Key words: External debt; Economic growth; Default JEL ch$cation: F3; 04

1. Introduction

Investment in the group of large debtor countries was depressed in the 1980s and many analysts argued that the debt crisis was a cause of this poor perfor- mance (rather than, say, the effect or the joint product of another cause) (see, e.g., World Economic Outlook, 1991, p. 63). The theoretical argument behind this analysis, known as the ‘debt overhang’ (D.O.) argument, runs as follows. If there

This paper is a revised and augmented version of a paper initially entitled How to Reschedule an

Heavily Discounted Debt’. 1 thank the two referees and an editor of the Journal for their helpful

comments. Errors are mine.

0165-1889/95/$09.50 6 1995 Elsevier Science B.V. All rights reserved

SSDl 016518899400822 Y

1142 D. Cohen / Journal of Economic Dynamics and Control 19 (1995) 1141-1163

is a discount on the secondary market of the debt, this points to the fact that, at least marginally, the lenders do not expect to be repaid in full. Repayments are then likely to be scaled down to the debtor’s resources. Debt is therefore operating like a (marginal) tax on the country’s resources, which explains its adverse effect on domestic investment. (See Sachs, 1989; Krugman, 1988; see also Claasens and Diwan, 1989, and Eaton, 1991, for useful surveys.)

The D.O. argument has been, to some extent, echoed by the Brady deals that have led to a few (sometimes significant) debt reductions (see, e.g., the World Debt Tables, 1990). Yet, even after the Brady deals were completed, the debt was still discounted. Does that mean that the write-offs were not thorough enough? This question takes us back to the initial question: By what benchmark should we gauge that the ‘debt overhang’ argument is valid? By inspecting the second- ary market alone? By comparing the investment rate to an appropriate bench- mark? These are the questions that this paper will address, and although I will only present a theoretical analysis, I will also indicate how it leads to an empirical analysis of these questions.

The punchline of the analysis is that observing a discount on the debt is not in itself sufficient for arguing that the debt will act as a tax on the economy. As we shall see, ‘efficient’ lenders will raise investment rather than reduce it, whatever the discount on the debt. The intuition behind this result is simply the following. If lenders are more patient than the debtors (as we shall assume), they will value growth more than the debtor itself. If they can, they will consequently try to tilt upward the pattern of investment of the debtor. In order to do that the lenders will implement a strategy in which they tell the lenders: ‘If you invest more, we shall reschedule your debt more generously.’ In practice, we shall see that this amounts to act ‘as if’ the debt only amounted to its market value and to let the debtor trust that banks are willing to let it grow as fast as the economy.

On the other hand, we shall also see that a lack of commitment capability of the lenders will work in the direction of the D.O. argument. In that case, our analysis will have a direct empirical implication: one should gauge the invest- ment ‘slowdown’ with respect to a financial autarky benchmark (rather than, say, with respect to the level that prevailed in seventies as is usually done), and the order of magnitude of the crowding out coefficient is approximately equal to the intertemporal elasticity of substitution of the representative agent in the economy. I will argue that empirical results do point towards such a result.

Section 2 spells out the model. It is a stochastic version of the model examined in Cohen and Sachs (1986). It also shares many similarities with Marcet and Marimon (1992) and Chari and Kehoe (1993). Section 3 calculates the socially efficient and the post-default growth rates of the economy. Section 4 analyzes the constrained first best of the lenders. Section 5 shows how an optimal rescheduling (based upon the distinction between performing and nonperform- ing asset) can achieve the equilibrium described in Section 4. Section 6 shows the

D. Cohen /Journal of Economic Dynamics and Control 19 (1995) 1141-1163 1143

link of this analysis with the ‘debt overhang’ literature and draw its empirical implications.

2. The set up

The model is a stochastic version of the model in Cohen and Sachs (1986) and Cohen (1991). To the extent that the model postulates a technology of produc- tion which exhibits constant returns to scale, it generates an endogenous growth equilibrium of the variety examined in Romer (1986). By analyzing an economy which is disturbed by stochastic shocks we are able to analyze explicitly how the lenders should deal with the discount on the debt. (In particular, we shall prove that the lenders will want - in general - to keep a positive discount on the debt. while this feature would play no role in a deterministic model.) Also, while Cohen and Sachs (1986) only analyze the first-best equilibrium, we also analyze in this paper the outcome of a game in which the lenders lack the capability to commit themselves.

2. I. Production

I will consider a one-good economy, in which the same good can be used indifferently for export, consumption, or investment. In each period, the avail- able stock of capital is a pre-determined variable. The production Qt is a linear function of existing capital:

Qt = K,. (1)

Capital can be increased through investment, and investment itself is a costly process. Let us assume (following Abel, 1978; Hayashi, 1982) that an increase I, of capital costs a larger amount J, which is a convex function of I, written as

in which 4 is a strictly positive number which measures the cost of adjustment. The investment decision I,, while taken at time t, increases the capital stock at

time t + 1, according to a stochastic law of motion:

K <+I = CK,(l - d) + ItIt + &+,I, (3)

in which d is the rate of depreciation of installed capital. One can think of 0, as of a stochastic shock which exogeneously increases (or decreases) the productivity

1144 D. Cohen J Journal of Economic Dynamics and Control I9 (I 995) 1141-I I63

of installed capital stock. Here, the investment decision I, must be taken before its productivity (0,+ i) is known. 0, is an iid stochastic variable which is worth:

0, = u with probability p,

= u with probability 1 - p. (4)

I shall refer to the event of probability p (where the productivity shock is U) as the ‘good state’ and to the event of probability (1 - p) (when productivity experiences a lower, possibly negative, shock) as the ‘bad state’ (see Genotte, Kharas, and Sadeq, 1987, for a model with a similar structure). I will call lJ the expected value of 8,:

(1 + 0) = p(l + u) + (1 - p)(l + 0). (4’)

Three technical conditions are needed in order to proceed with the analysis. They are stated in Eqs. (AlS), (A1.6) and (A1.9) in Appendix 1.

2.2. Preferences

I will assume that the country is inhabited by a representative agent who chooses simultaneously his debt, investment, and consumption decision. I also leave aside here the important issue of domestic taxation (see Helpman, 1988, for such an analysis and Cohen, 1991, Ch. 5, for the important issue of domestic monetization). The representative agent’s preferences are represented by an intertemporal expected utility function:

Uo = Eo&WC,L 0

(5)

in which C, is the aggregate consumption of the country at time t, and u(C) = (l/y) Cy when y < 1 and y # 0, or u(C) = 1ogC when y = 0. E0 is the conditional expectation operator as of time 0. Insofar as we are dealing with a Markovian stochastic process, the relevant information is the initial stock of capital K0 and the transition law (4).

2.3. External debt

In order to focus on the question raised in the title, I will simply assume that the country inherits an initial debt D,, (assumed to be short-term) which is large enough to be quoted below par on the markets, and I will investigate what is the

D. Cohen 1 Journal ofEconomic L$namics and Control I9 IlW5) 1141-1163 1145

optimal rescheduling strategy for the lenders (see, e.g., Cohen, 1988, for an analysis, in a three-period model, of the difficulties at hand when analyzing the transitional path of a stochastic model).

Following the Eaton and Gersovitz (1981) approach and my earlier work with Sachs, I shall assume that the country always has the ability to repudiate its stock of outstanding debt while the lenders can retaliate and impose on the borrower the following two sanctions:

A. A defaulting country is forced into financial autarky forever after it has defaulted (a lighter penalty would not change the results much).

B. The productivity of the capita1 of the defaulting country is reduced by a factor i, so that the post-default technology of production is

Q,=(l -i)K,.

In all that follows, I will assume that the lenders are risk-neutral, act competi- tively, and have access to a riskless rate of interest, r, which stays constant all along. These assumptions are warranted if one deals with a small developing country whose risk (driven, say, by idiosyncratic fluctuations) can be diversified away by the creditors. I will assume that /I is below l/(1 + r) to insure that the country will be constrained on the borrowing side. Furthermore, I will leave aside all bargaining issues and assume that the lenders can credibly make (at any point in time, but not necessarily for the entire future) a take-it-or-leave-it offer to the debtor (see Cohen and Verdier, 1990, for a bargaining approach to a ~ simplified - version of this model).

3. The optimal growth rates: The wealth-maximizing solution and the post-default equilibrium

In this section, I would like to calculate the optima1 investment strategy in the two extreme cases when, on the one hand, the country simply maximizes its wealth and when, on the other, it is forced down a post-default path.

3.1. The wealth-maximizing solution

Let us simply assume here that the representative agent can simply ‘sell’ the country to the world financial markets which, in turn, simply maximize the present discounted value of the country’s net output. Maximizing the country’s wealth (when - following the assumptions made in Section 3 ~ the world riskless rate of interest is r and when assuming that the country’s risk can be diversified

1146 D. Cohen /Journal of Economic Dynamics and Control 19 (1995) 1141-I 163

away) amounts to

(7)

The solution to this program is given in Appendix 1. Given the linearities in the model, IV, is shown to be a linear function of initial output:

Wo = GQo, (8)

and is obtained by picking up a fixed investment rate:

2 = LIQ,, (9)

associated to a fixed rate of gross investment:

j = Jt/Qt = X( 1 + + 4%). (10)

(All the technical conditions for the equilibrium to exist are spelled out in Appendix 1.)

The equilibrium growth rate of the economy oscillates. It is high in the good state of nature [ = (1 + u)(l + X - d)] and low in the bad state of nature [ = (1 + v)(l +x -d)].

In the sequel I will refer to this equilibrium as the socially eficient equilibrium.

It is the equilibrium which would be attained in a free-trade world without nations.

3.2. The post-default case

Let us assume now, as another extreme case, that the country has defaulted on its external debt. In this case, the representative agent must make his investment decision so as to allocate consumption optimally over time. Mathematically, the agent must solve the following program:

U,(Q) = mzx{u[Q[l - 2 - x(1 + &x)]l + BpU,[Q(l +u)(l +x -41

+ 8(1 -PP,CQU +4(1 + x -dll, (11)

in which U,, is the utility level that the agent can reach when the available output is Q. at initial time.

D. Cohen /Journal of Economic Dynamics and Control 19 (1995) 1141-I163 1147

The solution is spelled out in Appendix 1 where it is shown that the solution Ud(QO) can be written as

I; = Cj.Q'b if ;‘#O,

1 =---logQ, + Cd if ;l=O.

1-B (12)

The solution is also shown to involve a fixed investment rate:

x, = It/Q,, (13)

which is smaller than the socially efficient investment rate (obtained in the open economy case).

When one sets 1, = 0, the equilibrium attained is simply that which would prevail with financial autarky (no debt and no access to the world financial markets).

4. The ‘maximum repayment’ which can be extracted from an indebted country

In this section, I will consider the following simple problem. Let us assume that the face value of the debt is infinite and that the lenders can monitor both the investment and the repayment strategy of the debtor in such a way as to maximize the value of the transfers made abroad by the country. While it is assumed that the borrower will give up his sovereignty over his consumption and investment decision, he will nevertheless keep its sovereignty over the matter of defaulting: at any point in time, the borrower will remain free to break the lenders’ rule and to follow afterwards the post-default path defined by Eq. (12). In other words, the rules of the game in this section are as follows: the lenders montior the debtor’s economy so as to maximize the value of the transfers channelled abroad by the debtor, subject to the constraint that the program is never expected (neither today nor later on) to be dominated by a post-default path. Clearly, under this set of hypotheses, the value of the transfers channelled abroad by the debtor will provide an upper limit to the market value of any debt accumulated by the country.

Formally, the problem can be written as follows. Call Pl the amount of transfers abroad which is made by the debtor, yL the gross investment rate (inclusive of the cost of installation) achieved by the country, and C, the consumption left to the country. One has

C, = Qdl - y,) - Pt.

1148 D. Cohen /Journal of Economic Dynamics and Control 19 (1995) 1141-1163

Call

(14)

the level of utility which the lenders’ program is expected to deliver to the country. With these notations, the program that the lenders must solve is to maximize

(15)

subject to

U, 2 Ud(Q1), for all t.

in which U,(Q,) is the post-default level of utility - as defined in Eq. (12). This problem is solved in Appendix 2. Given the many linearities built into

this model, the problem boils down to finding a fixed (gross) investment rate y* and a fixed debt service ratio b* = P,/Q, which solves the problem (15). The solution is shown to involve an investment rate which lies between the socially efficient rate and the post-default rate. One can state:

Proposition 1. The ‘maximum repayment’ program which the lenders would like to monitor involves a fixed investment rate which is smaller than the socially optimum one but larger than the financial autarky rate. It involves a transfer of resources from the debtor which is a fixed fraction of GDP, a fraction which is smaller than the cost of default.

Even when it is the banks themselves which design the investment and consumption policy of the borrower, they will choose a lower investment rate than that which is socially desirable. The reason is that the banks must take care to avoid that the country may one day choose to default. A too rapid path of capital accumulation, even while socially dtsirable, will raise the post-default utility of the country and, if not carefully balanced, can be counterproductive to the banks. On the other hand, one sees however that the banks will choose an investment rate that is larger than the rate that would be chosen by a country which is financially autarkic. The intuition is quite simply that the lenders are more patient than the debtor [we assumed /I < l/(1 + r)]; they consequently value growth more than the country itself.

In the sequel, I will call P’: the ‘maximum repayment’ that the lenders can expect to receive from the debtor. Due to the linearities involved, V, can be

D. Cohen / Journal of‘ Economic Dynamics and Control 19 (I 995) Il4/-1163 1149

written as a linear function of current output:

V,* = z*Qt,

in which

b*

‘* =((l + r) - (1 + O)(l + x* - d))’

(16)

(17)

and where

1 + H = p(1 + v) + (1 - p)(l + tl),

In the simple case when u = u = 0 (the economy follows a deterministic path which is driven by capital accumulation alone), the simple result is that the

market value of the debt (as a fraction of GDP) is the usual term:

z* = h*/(r - n*),

in which n* = .x* - d is the equilibrium growth rate of the economy. As we show in Appendix 2, the key to the selection of the (constrained)

first-best is simply to consider the dual problem of selecting for any value of z (i.e., the value of the present discounted values of the transfers generated by the country) the choice of b and n which maximizes the country’s welfare. The constrained first-best is then simply characterized as the value of z* for which the corresponding welfare is just equal to the post-default level of welfare. The

higher Z* (due to a larger cost of debt repudiation) and the larger the socially efficient investment rate will be. In the extreme case when the country cannot repudiate the debt (2 = l), banks would simply choose the socially efficient rate

and get paid the entire wealth of the country. As we shall see in the next section, this dual problem gives the key to how the constrained first best may be

achieved in practice. Technically, this dual problem also helps understanding in another way why

the optimal investment rate is above the financial autarky rate. For any value of z*, the country pays

P, = z*C(l + r) - (1 + @(l + x - d)] Qt. (17’)

Payments are now negatively correlated to the growth rate or, equivalently, negatively correlated to the investment rate. This induces the country to invest more than when it is financially autarkic.

1150 D. Cohen /Journal of Economic Dynamics and Control 19 (I995) 1141-1163

5. How to implement the ‘maximum repayment’ scheme

I will now indicate how lenders can indeed capture the ‘maximum repayment’ scheme even when they cannot monitor indefinitely the choices of the borrower, and when the face value of the debt is large but finite. Consider the following decomposition of the debt:

D, = Vf + R,, (18)

in which D, is the face value of the debt, V: is the maximum value calculated above, and R, is the residual. Assume that the lenders fictitiously regard R, as a nonperforming asset and only insist on V: being serviced (while R, is automatically capitalized). Furthermore, assume that, each period, they ask (and are trusted to ask in the future) the borrower to transfer an amount P, which is what is necessary to keep VF growing at the expected rate of growth of the economy (which requires them to be able to observe at any time t the investment rate x, chosen by the debtor).

Under these assumptions P, must solve

(1 + r)VF - P, = (1 + O)(l + x - d)V,*,

in which (1 + O)(l +x -d) = p(1 + u)(l +x-d) + (1 - p)(l + u)(l +x -d) is the expected growth rate of the economy when the investment rate x has been selected by the debtor. P, is then given by

P, = [(l + r) - (1 + O)(l +x - d)] V:

= z*[(l + r) - (1 + @(l + x - d)] Qt, (19)

and the optimum investment decision chosen by the country will coincide with the ‘maximum repayment’ strategy designed in Eq. (17) (i.e., the borrower will exactly choose xt = x*, see Appendix 2).

Compared to the calculation in the previous section, one sees this strategy simply amounts to implement the dual problem in which lenders set how much they want to be paid (VF in present value terms) and - subject to this constraint - let the debtor freely maximize their welfare.

It is crucial to note that this fictitious decomposition of the debt into a performing and a nonperforming part is updated in each period. Indeed, along Eq. (19) I’* is only left to grow at a rate of (1 + O)(l + x - d) which is the average growth rate of the economy. If things go well, the actual growth rate will be larger and V, + 1 must be scaled up; conversely, V,, 1 will be scaled down if the

D. Cohen 1 Journal of Economic Dynamics and Control 19 (1995) 1141-I 163 1151

bad state occurs. Lenders must then obviously have a large enough nominal debt so as to be able to keep implementing the strategy (19) even when the country is indefinitely hit by a sequence of good shocks. This requires the face value of the debt to exceed a level h*Q (in which h* > z*; see Appendix 3 for its calculation).

The optimal rescheduling strategy amounts to act ‘as if’ the debt was only equal to VT (rather than to V: + R,): the creditors scale how much the country should pay on VT and acts ‘as if’ they were only interested in controlling V:‘s growth rate. In the deterministic model of Cohen and Sachs (1986) the creditors could safely write down the stock of the debt to VT. In this stochastic model, it is important instead that they keep an unserviced part of the debt in reserve (‘in case’ the country goes through a sequence of good shocks). When such an optimal rescheduling strategy is implemented, I’: is nevertheless the marker value of the debt: it is indeed simply equal to the maximum present value of all expected future payments. However, even though I’: is, ex post, the market value of the debt, it is crucial that the lenders do not explicitly base their rescheduling strategy upon the observed market of the debt. Indeed, if they were to do so, they would ask to be repaid:

P, = z(x)[(l + u) - (1 + 0)(1 + x - d)] Ql,

in which z(x)Qt is the observed market value of the debt (itself a function of the country’s investment strategy x), and now the country would be induced to bring down the market value of the debt.

These results can be summarized as follows:

Proposition 2. When the debt-to-GDP ratio is above ajoor value h*, the lenders can capture the ‘maximum repayment’ value V * by proceeding as follows. They shouldjctitiously split the debt into a performing and a nonperforming component, the performing component being equal to V * and act ‘as if’ the debt amounted to V* only. When this rescheduling strategy is undertaken, the equilibrium murket value of the debt is equal to V *.

Now, obviously, as time passes the size of the nonperforming asset grows relatively to the performing one, and some write-off of the debt may become possible without impairing the lenders’ ability to capture V *. One can actually show:

Corollary. When the debt-to-GDP rutio is above the threshold h*, the debt cun be written down to h* GDP without impairing the lenders’ return. If the write-ofli‘is repeated each time the economy goes into the bad state (and if the rescheduling is

1152 D. Cohen J Journal of Economic Dynamics and Control 19 (1995) 1141-l 163

undertaken according to Proposition 2’s technique), the lenders capture the ‘maximum repayment’ scheme while the market price of the debt is stabilized at a constant equilibrium price below par.

One important implication of the corollary to Proposition 2 is that it is not enough to observe a discount on the debt to warrant a write-off. The intuition is that the discount on the debt takes into account the possibility that the economy may go into a bad state. Lenders have no reason to write off the debt before that prediction materializes. It is only in the deterministic case when u = u that the optimal strategy is indeed to write off the debt ‘once and for all’ (in order to erase whatever backward shocks may have lifted the debt-to-GDP ratio above h*) and let the debt be quoted at par. In the stochastic case, instead, the lenders want to keep a strictly positive nonperforming asset R, = (h* - z*)QI so as to be able to absorb an infinite sequence of good shocks without getting short of nominal claims on the country.

6. The ‘debt overhang’ problem revisited

In view of Proposition 2, it appears that the face value of the debt is of little importance in assessing the optimal rescheduling strategy of the debt. This should come as no surprise: when they behave optimally, lenders get as much as the country can transfer and more nominal claims cannot imply less actual payments (see also Bulow and Rogoff, 1989a, b). This result, however, contra- dicts the ‘debt overhang’ argument according to which too large a nominal claim may discourage investment and reduce the market value of the debt (as in Sachs, 1989; Krugman, 1988). I would now like to indicate - within our framework of analysis - where this result comes from.

A key feature of the optimal rescheduling strategy described in Proposition 2 is that lenders should let the performing asset grow along the expected growth rate of this economy. As apparent from Eq. (19) this implies that the service of the debt is negatively correlated with the investment decision of the borrower. Even though such behaviour is in the lenders’ self-interest, I would now want to show that this is not a ‘time-consistent’ decision, that is: it is a decision which is an optimal one to take only if the lenders can commit themselves (in whatever way: sophisticated contracting or a built-in reputation) to actually implement it (in an interesting model which is solved by a simulation study, Marcet and Marimon, 1992, make a similar argument). Indeed, assume instead that lenders cannot commit themselves to implement the first-best strategy (19) and are simply expected to choose a memoriless strategy. Which equilibrium arises? In other words, what is the (Markovian) Nash subgame-perfect equilibrium of the noncooperative game between the lenders and the debtor? In such a game, the lenders are expected (by the borrower) to choose a feedback strategy, which here

D. Cohen /Journal of Economic &namics and Control 19 (1995) 1141-1163 II53

can be restricted to take the form:

P, = hQ,, (21)

while the debtor is expected (by the lender) to choose an investment strategy which, here, can be restricted to take the form:

I, = xQt. (22)

It is straightforward to see that the equilibrium of this game is nothing else

but the post-default path. Lenders ask P, = LQ, (the loss which the debtor would forego by defaulting) and the debtor chooses I, = .xIQ,.

In fact, one can change the rules of the game slightly and assume that the lenders announce first, each period, their repayment requirement and let the debtor choose his investment without changing the result (see Cohen and Michel, 1988, for a further elaboration on this point). Indeed, assume that both the lenders and the debtor expect, at some time t, that the post-default path will

be the equilibrium outcome of the game in future periods. Let P, = vQt, the payment chosen by the lenders. The country chooses its investment rate so as to solve

C(v) = + x(1 +:0.x) - v]’ + fl(l + p)Cd(l + x - 4;’ , (23)

in which

1 + /1 = p(1 + 24)’ + (1 - p)(l + 0);‘.

Because of the envelope theorem it is clear that C(v) is a decreasing function of v. On the other hand, by definition of Cd, C(n) = Cd. One consequently sees why

the lenders will always find it optimal to ask v = i. Asking v > i would induce the country to default, asking v < 1 would be a waste of money (from the lenders’ viewpoint). This shows that a period-by-period commitment over the repayment strategy is not enough to reach the constrained first best.

As apparent from Eq. (21), a memoriless rescheduling strategy acts as a tax on output: the borrower expects that the lenders will ask for as much as he can pay, and this is an amount which will be proportional to how much output he can generate. As a result, investment is inefficiently depressed even though (in this model, as in Sachs’ or Krugman’s) the collection of resources needed to service the debt is not in itself a source of distorsions.

One can actually go beyond the analysis of the sign of the crowding out effect and investigate the order ofmagnitude of the crowding out coefficient. Specialize our model to the simple case in which 4 = 0 (there is no adjustment cost). In that case, one can show (see Appendix 1) that the investment rate to prevail in the

1154 D. Cohen / Journal of Economic Dynamics and Control 19 (1995) 1141-l 163

D.O. case can be written as

where xi. is the investment rate to prevail when the country has to pay a fraction I of its resources to its creditors (in a time-consistent way) and x, is the financial autarky rate. l/( 1 - y) is nothing else but the intertemporal elasticity of substitu- tion of the representative agent in the economy.

Empirical application

The key empirical implication of the preceding analysis is that the D.O. argument cannot be gauged by simply observing a discount of the secondary market nor by comparing the investment rate of the rescheduling countries to, say, the investment rate that prevailed in the seventies (as is usually done in most empirical studies). One must compare the investment rate in the eighties to a financial autarky benchmark.

How to calculate such a benchmark? The financial autarky rate is the rate for which domestic investment is equal to domestic saving. One may then want to calculate the dependency of each of these two terms on the domestic interest rate and find the rate for which they are equal. In practice, I show (Cohen, 1993) that one can avoid this (empirically) difficult route and simply estimate the depen- dency of saving and investment on the trade balance in order to calculate the financial autarky rate (I do that for the developing countries in the sixties). Once the financial autarky rate is calculated, we are in a position to test whether investment was crowded out or in by the net transfers abroad paid in the eighties. For the group of rescheduling countries, I have shown that the answer is unambiguously in favor of the crowding out hypothesis.

Furthermore, I obtain a crowding out coefficient which is approximately equal to l/3. This is a reasonable value of the intertemporal elasticity of substitution. Both the sign and the magnitude of the effect consequently work, empirically, in the way of a confirmation of the D.O. argument, such as we interpreted it above.

7. Summary and conclusion

The key to an efficient rescheduling of the debt is a clear commitment that the lenders will always scale down appropriately, in the future as well as today, the flow of resources transferred by the country to its creditors in a manner which explicitly acknowledges the (ex post) market value of the debt. Such a strategy,

D. Cohen / Journal of Economic Dynamics and Control 19 (1995) 1141-I 163 1155

however, does not imply that the debt should be priced at par. It does imply, however, that the creditors should act ‘as if’ it only amounted to its market value. The empirical test of a debt overhang is a test of whether the investment rate is below or above the financial autarky rate. Empirical tests do suggest that this is actually what happened in the rescheduling countries.

Appendix 1: Optimal growth in the (totally) open and in the post-default economy case

A. I. The wealth-maximizing equilibrium

Let us first define, if it exists,

where (a) R,, 1 = [I?,(1 - d) + I;] [ 1 + &+ J, with Bt a stochastic process as in Eq. (4) in the text, and (b) I?, = 1.

Because of the linearities that are set in the model, we can turn Eq. (7) in the text into

in which Z?, = K,/Ko and it = I,/Ko. One can therefore write

W, = tiKo.

W. is the ‘wealth’ of the country. In a Fisherian world (with no friction on the world financial markets), this gives the amount that a country can consume (in present value terms), when it is originally without debt and is imposed, in the long run, a transversality condition:

lim Eo[e-“D,] = 0. L+7

By Bellman’s principle, 0 must be a solution to

0 = rn:x 1 - x(1 + $#Jx) + e [p(l + u) + (1 - p)(l +0)X1 +.x -4

(Al.l)

1156 D. Cohen /Journal of Economic Dynamics and Control 19 (1995) 1141-1163

The equilibrium value of X is

i_g+~~-l] with 1 + B = ~(1 + U) + (1 - p)(l + u).

Eq. (Al.l) yields that X is a solution to

The solution that is socially efficient is

r-e X’m+d-,/ii, (A1.4)

where

which exists if

,>2(1-s-d)/(s+d)i.

In addition, we impose X 2 d, which requires that

(A1.3)

(A1.5)

(A1.6)

A.2. The post-default case

Let us here simply ‘guess’ that the solution to Eq. (11) can indeed be written as

UAQo) = C,Q;.

Then the ‘guess’ will prove to be correct if

1 - A- x(1 + 44x) 1 [ ’ + fiC2p Y

(1 + u)(l + x - d) 1 Y + PC,(l - p) (1 + u)(l + x - d) 11 . (A1.7)

D. Cohen 1 Journal of Economic Dynamics and Control 19 (1995) 1141-1163 1157

One can simply write that

c = max L [l - L - x( 1 + +$.X,1;’ j. X y 1 -B(l +p)(l +x--d);”

where

1 + p = p(l + u)’ + (1 - p)(l + u)i’.

We assume

1 + ,u < 1 + e = p(l + u) + (1 - p)(l + 0).

The first-order condition for .x is

1 + +x /3(1 + p)(l + x - d);‘_’

1 - r. - x(1 + ffjx) = 1 - fi(l + p)(l + x - d)“’

which can also be written as

[

1 +x_d+ 1-2-x(1 +f+x) =(l +x--d)+

1 + 4x 1 B(1+/4 ’

(A1.8)

(A1.9)

(A1.lO)

the left-hand side is a decreasing function of x (its derivative is - +[l - i - x( 1 + +4x)1/(1 + 4~)’ and of 1, while the right-hand side is an

increasing function of x. With positive growth (x 2 d) the right-hand side decreases with 2 and p. One consequently sees why y < 1 and p < l/l + r will imply that the investment rate is smaller after default (2 > 0) or under financial autarky (i = 0) than under free access to the world financial markets (which is obtained with y = 1 and /I = l/(1 + r)).

A.3. Empirical implications

In order to get an order of magnitude of the effect of repudiation on investment, consider the simple case when 4 = 0. In that case, (A1.lO) can be written as

fl(1 + p)[r -d - 21 = (1 + x -A)‘-;‘,

which can be approximated (when x - d is small enough) as

B(1 + P) A

x=xR- 1-Y ’

1158 D. Cohen / Journal of Economic Dynamics and Control I9 (I 995) 1141-l I63

in which

x a

= d + B(l + P)k - 4 - 1 1-Y

is the financial autarky rate of investment. Assuming that the time period is short enough that /I(1 + cl) is sufficiently close to 1, one then gets that

In the case of repudiation, one then finds that investment is crowded out below financial autarky by a coefficient l/(1 - y) which is simply the intertem- poral elasticity of substitution of the representative consumer.

In the ‘time-consistent’ equilibrium which is described in Section 6, this coefficient can be interpreted as the effects of repaying a fraction 1 of output on the country’s investment rate.

Appendix 2

Because of the linear structure of the model, the lenders who want to extract the maximum repayment from the borrowers must find a payment strategy P = b*Q and a constant investment rate x* so as to solve the following problem:

subject to

EtQ t+l = (1 + x - 4U + ‘$Q,

and to

Eo&W(l - b - AQtl 2 UdQo). 0

(A2.1)

Because of the linearity of the model this inequality implies that similar ones will be held in the future.

D. Cohen J Journal of Economic Dynamics and Control 19 11995) 114/L1163 1159

Define o(x) as the solution to

dx)Qo = Eo f ~ Qt

f=O (1 + Y)”

when the investment rate is x. One has

o(x) =

[ 1 _ (1 + ml + x - 4 l

l+r 1 . (A2.2)

The problem faced by the lenders can then be written in the following more compact form: Find z*, the solution to

z* = n$a~ ho(x),

subject to

(A2.3)

1 [l -b-x(1 +&x)1’ > c_

i,l -B(l +p)(l +x-C@- /.

By duality this problem is simply that of finding the largest z* such that

1 [l - z*w(x)-r - x(1 + &5x)]; max -

X ?’ 1 - B(l + /J)(l + x - d);

= Cj

1 [l-i-x(1 +i$x)]’

S max y 1 - B(l + /L)(l + x - d)“’ (A2.4)

The geometry of the problem can readily be drawn as in Fig. 1 in the space (x, b). The indifference curves of the debtor are

; [l - tJ - x(1 +f$x)]:’ = C[l - b(l + /L)(l + .Y -u’);‘], (A2.5)

and reach their maxima on a line whose equation is

1 +x_-d+ 1 -b-x(1 +3&d =- 1 + lpx B(1 : PP + x - q-r,

1160 D. Cohen / Journal of Economic Dynamics and Control 19 (I 995) 1141-I I63

b

A

A-- -

X

Fig. 1

and which is negatively sloped; see Eq. (A1.lO). The utility CL is obtained when the maximum is at b = A. When the lenders design the optimal repayment scheme, they choose a point (b*(A), c*(A)) on the indifference curve CL so as to get the largest z* such that

b = z*[(l + r) - (1 + @(l + x -d)]. (A2.6)

The tangency point is on the right to (1, x,,): it involves a larger investment rate (x* > xd) and a lower repayment (b* < I) than the post-default point.

As z* is increased (corresponding to a larger 1) the tangency point shifts towards the south-east: the country invests more and pays more. In order to show this point, let us write the first-order condition which specifies the solution to (A2.4):

1 + 4x - z*(l + 0)

1 - z*[(l + r) - (1 + O)(l + x -d)] - x(1 + f+x)

/?(l + P)(l + x - 4-l = 1 - /?(l + p)(l + x -d)’

(A2.7)

D. Cohen J Journal ?fEconomic Dynamics and Control 19 (1995) 114/L1163 1161

which shows that

1 + 4x 2 z*(l + 0).

Eq. (A2.7) implies

[I1 + 4x’ i

1 -[l +x-d]‘-;-(l +x-d) /?(l + /_L) 1

+.y(l +$4x)

= 1 + z* (1 + 0) i F -J-Cl +x-d,i-;‘-(l +r)

B(l + p)

11 )

or equivalently,

+ @l + x -d) [

1

/I(1 + p)[l + x - d]Y 11 dX

1 =-

P dz* (’ + ‘) p(l

(1 +d +x -d)‘-;‘ - /I(1 + r) .

(A2.8)

(A2.10)

Because of inequality (A2.8) one sees that the term in brackets on the left-hand

side is necessarily positive. Since /I( 1 + Y) < 1 (the country is a ‘debtor’) and since we assumed x 2 d and 1 + 8 2 1 + p, one sees that the term in brackets in the right-hand side is also positive. This shows dx/dz* 2 0. As the country must

repay a larger debt, investment is crowded in (in the constrained first-best). Investment is therefore necessarily between the financial autarky rate (z* = 0) and the socially efficient rate (when z* is set to maximum value associated with

;Z = 1). This proves Proposition 1. Eq. (A2.4) also shows that the maximum repayment is obtained when the

debtor ,fveely chooses his investment decision x, when subject to the repayment strategy

P, = Z*[(l + r) - (1 + d)(l + x, - d)]Qt.

This shows that the lenders, in order to implement the maximum repayment strategy, need to commit themselves to such a repayment strategy but also need only to observe, on a period-by-period basis, the investment rate x, which is chosen by the debtor. This proves Proposition 2.

1162 D. Cohen /Journal of Economic Dynamics and Control I9 (1995) 1141-II63

Appendix 3

In order to implement the maximum repayment scheme along the line of Proposition 2, an infinite nominal debt is sufficient, but not necessary. Indeed, consider the threshold

h* = z* (1 + 4 - (1 + w + x* - 4( >z*),

(1 + Y) - (1 + U)(l + x* -d)

and assume that the face value of the debt is initially at D, = h*Q,. Requiring the debtor to pay P, = z*[(l + I) - (1 + O)(l + x* - d)]Q implies that the face value of the debt at time t + 1 is

D t+l = (1 + r)D, - P, = h*[l + x* - d][l + u]Qt.

Now two things can happen: If the good state materializes, one has

Q f+ 1 = (1 + u)(l + x* - d)Q, so that, in that case, D,, I = h*Q,+ i. If the bad state materializes, the debt-to-GDP ratio increases and the debtor can write down (u - u)(l + x* - d)Q, and bring the debt-to-GDP ratio back to h*. In all instances, one sees that the lenders can keep the debt-to-GDP ratio at the level h* indefinitely while implementing the maximum repayment scheme. This proves the corollary to Proposition 2.

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