Journal Issue
Share
Article

A Debt Overhang Model for Low-Income Countries

Author(s):
International Monetary Fund. Research Dept.
Published Date:
April 2008
Share
  • ShareShare
Show Summary Details

IMF Staff Papers55, 654–678. doi:10.1057/imfsp.2008.13;

published online 10 June 2008

Debt overhang—the relationship between heavy debt and low growth—is a fundamental concept in the literature that argues in favor of debt relief. Unfortunately, the majority of existing theoretical models are designed for middle-income countries suffering from heavy nonconcessional debt burdens. These models do not apply to low-income countries (LICs) where external loans are highly concessional and comprise a large share of debt. In addition, existing debt overhang models typically provide no explanation as to why the debtor country has excess debt in the first place.1

To fill these gaps, this paper formulates Cohen and Sachs’s (1986) sovereign debt model as a concessional lending problem and numerically demonstrates how a link between large debt and low growth may be generated in LICs. The model focuses on the effect of a cutoff, which is an income level above which the country loses its eligibility for aid assistance. Such cutoffs exist with multilateral concessional lending from the World Bank’s International Development Association (IDA) and by the IMF with loans under the Poverty Reduction and Growth Facility (PRGF).2

This paper shows that an LIC—when it has no effective tools to raise the country’s total factor productivity (TFP)—may have an incentive to accumulate a significant amount of concessional debt and allocate resources to consumption rather than investment. Such a country would manage its large debt at a very low cost by allowing the economy to stagnate around the cutoff, and thus would become permanently aid dependent. This is more than just a theoretical possibility; this paper provides empirical evidence of growth stagnation around the cutoff.

There are two types of agents in the model: an official creditor and an LIC debtor. The creditor lends at a fixed subsidized interest rate if the debtor country lies at or below the cutoff. Above the cutoff, the creditor lends at the world interest rate. The creditor can commit to the contracts but the debtor country cannot. The creditor thus imposes a participation constraint to prevent the debtor country from defaulting. Imposing a participation constraint is equivalent to imposing an endogenous debt ceiling constraint. The LIC debtor maximizes the representative agent’s welfare subject to this lending rule. Some researchers argue that the focus should be on “bad” governments that care about their own welfare rather than that of households. This paper shows, however, that a debt overhang problem may occur even with a benevolent government.

Last, this paper proposes policy implications for aid design and debt relief. For aid design, it finds that the existing eligibility criteria and graduation policies may be improved to provide stronger growth incentives and to reduce the cost of aid assistance. For debt relief, a one-time stock treatment can promote growth, given certain initial conditions and TFP.

I. Theoretical Literature

This paper is related to the sovereign debt and debt overhang literature. First, the model endogenizes debt sustainability 3 by incorporating enforcement mechanisms—an important topic in the sovereign debt literature. There are two main types of models that explain enforcement mechanisms in the literature: reputation and sanction models. In reputation models, debtors find it painful to be excluded from future credit markets. One classic reputation model is that of Eaton and Gersovitz (1981). They assume a concave utility function so that the country has an incentive to smooth consumption over time. The output path takes two values in turn, high and low. In this environment, the country does not want to be excluded from the international capital markets, because in financial autarky it cannot smooth consumption. Bulow and Rogoff (1989), on the other hand, show conditions under which reputation does not provide sufficient repayment incentives. Other aspects of reputation have been studied by, for example, Atkeson (1991) and Cole and Kehoe (1998).

In sanction models, debtors are penalized on default. A common way of introducing the default penalty is to assume a loss of a fraction of output on default. This can be, for example, the loss of access to short-term trade credits. Some researchers argue that sanction models fail to consider possible renegotiation processes and analyze the processes in the context of dynamic bargaining games.4 Yet debt renegotiation itself can be costly—Rose (2005) finds that debt renegotiation is associated with an economically and statistically significant decline in bilateral trade between a debtor and its creditors.

Cohen and Sachs (1986) incorporate components of both reputation and sanction models into their enforcement mechanisms. Their model is a neoclassical growth model in which the initial capital stock lies below the steady state. The country can borrow from abroad at the given world interest rate. As long as the country’s capital remains scarce, the world interest rate is lower than the initial marginal product of capital, so the country finds it painful to be excluded from external borrowing. The marginal product of capital decreases as the country accumulates capital, eventually converging to the world interest rate. In the steady state, the country’s default cost is merely the one that comes from sanctions. An important contribution of Cohen and Sachs is that they analyze sovereign debt dynamics in the context of growth. Thus, their model may be useful when thinking about development problems of an LIC with good growth prospects.

The theoretical literature on debt overhang that explains the relationship between large debt and low growth in LICs lags behind the empirical literature. The existing debt overhang models typically consider a case in which initial debt is so large that the country would be insolvent unless it received some form of debt relief (Krugman, 1988; and Sachs, 1989). In these models, excess debt reduces the supply of new loans by scaring off creditors; it also reduces the demand for new investment and discourages policy efforts to reform by acting like a distortionary tax where a fraction of future output is assumed to be used for repayments of the initial debt. This discourages domestic investment, resulting in low growth.

However, because some key features of LICs are not incorporated into these models, their applicability to this context is questionable. In particular, the majority of loans to LICs are highly concessional and are provided by official creditors who are neither profit maximizers nor risk neutral. This may generate a unique lending pattern—for example, contrary to the existing models, large debt may not discourage new official lending, as argued by Easterly (2002).

The model presented below formulates some specific LIC characteristics by considering the case in which an LIC debtor has no access to foreign private loans but has access to subsidized loans provided by a benevolent creditor.

II. Empirical Motivation

This work begins with empirical documentation showing that there is some economic stagnation around the cutoff. I run growth regressions using an unbalanced panel of 94 countries, of which 33 are LICs.5 The data set is taken from the Penn World Table Version 6.1 (Heston, Summers, and Aten, 2002), the World Bank’s World Development Indicators, and the Barro-Lee (1993) data set.

As for data on the cutoff, I use the operational cutoff, which was formally recognized by IDA donors in IDA8 in 1987. Prior to this date, a higher cutoff, known as the historical cutoff—initially set at $250 in 1964—was used for the IDA cutoff. The operational cutoff was introduced in the early 1980s because of the limited availability of IDA resources and the attention to poor performance in LICs. Both cutoffs are updated annually according to the world inflation rate using the SDR deflator.6

The dependent variable is the percentage annual growth rate of real GDP per capita. The explanatory variables are those typically included in a standard growth regression: the percentage of population growth (GPO), the percentage investment share of real GDP per capita (INV), the initial secondary schooling attained as a percentage of the total population in 19857(INIT_EDU), and the initial level of real GDP per capita in 1988 (INIT_RGDP). In addition to these variables, I include the variable of interest, a measure of proximity to the cutoff in the form of a Bartlett kernel:

where yit is country i’s GNI 8 per capita in year t, y¯t is the cutoff in year t, and b is a scaling factor that controls the width of the kernel band. Note that a negative coefficient for PROX implies that there is a negative relationship between the country’s growth rate and its proximity to the cutoff. The scaling factor, b, is set equal to ½, but I obtained similar results in the cases where b = ⅓ and b = ¼. Appendix IV reports the distribution of PROX and partial regression of the growth rate on PROX.

Table 1 shows the estimation results based on a pooled sample for 1988-2000. The ordinary least square (OLS) coefficient for PROX is negative and is statistically significant at the 5 percent level. To take into account endogeneity of PROX, INV, and INIT_RGDP. I also run two-stage least square (2SLS) regressions. Here, one-year lags of these variables are used as the instruments for PROX and INV, and the log of real GDP per capita in 1985 is used as the instrument for INIT_RGDP. The corresponding first-stage regressions are reported in Appendix V. The coefficient of PROX remains negative and significant at 5 percent.9 When the sample is restricted to those countries that lie below the cutoff, the significance level improves. This empirical evidence is consistent with the paper’s theoretical result—the existence of a cutoff could result in economic stagnation at or around it.10

Table 1.Growth Regressions
OLS2SLS
VariablesAll countriesAll countriesCountries below the cutoff
PROX-1.73**-1.38*-3.97**
(0.70)(0.76)(1.96)
GPO-0 90***-0.91***-1.13***
(0.14)(0.14)(0.29)
INV0.11***0.14***0.17***
(0.02)(0.02)(0.06)
INIT_EDU-5.32e-03-1.54e-030.05
(0.01)(0.01)(0.06)
INITRGDP-0.64***-0.88***-1.23
(0.24)(0.25)(1.04)
Note: Standard errors are in parentheses. OLS = ordinary least squares; 2SLS = two-stage least squares. *significantly different from 0 at the 10 percent level. “significantly different from 0 at the 5 percent level. ***significantly different from 0 at the 1 percent level.
Note: Standard errors are in parentheses. OLS = ordinary least squares; 2SLS = two-stage least squares. *significantly different from 0 at the 10 percent level. “significantly different from 0 at the 5 percent level. ***significantly different from 0 at the 1 percent level.

III. The Model

Official creditors typically fix their concessional interest rates; for example, the rates of the World Bank’s IDA and the IMF’s PRGF are 0.75 11 and 0.5 percent, respectively. I thus consider the following concessional lending rule: the lender who has full access to the world financial markets loans out funds at a fixed subsidized interest rate (r¯) as long as the country’s output per capita (y) is below the cutoff (y¯).12 The interest rates for concessional lending are thus set according to the following rule:

where r is the world interest rate. I assume that the borrower country has no access to foreign private financing, given that in practice, the majority of loans to LICs are offered by official lenders.

In addition, I impose a participation constraint to motivate the borrower to adhere to the contract.13 With this constraint, the borrower’s value function under repayment is required to be greater than or equal to its value function under default. The borrower country solves the following problem:

subject to

where c, x, k, and X denote consumption, investment, capital, and repayment obligation, respectively. The repayment obligation in period t, Xt, is defined by Xt=(1+r˜t)Dt, where Dt is concessional debt due in period t.14 β is the discount factor where r ≡ 1/β-1 is assumed in order for consumption in the steady state to be flat. The participation constraint is given by Equation (3). The LIC’s flow budget constraint is given by Equation (4). The production function is given by f(kt). The transition equation for capital is given by Equation (5), where δ is the rate of capital depreciation. The value function under default, vD(k), is the value function in autarky with penalties for violating the participation constraint:

where λ is the fraction of output lost. I assume that such a violation incurs two types of costs: the exclusion of the violator from future concessional lending and the loss of a fraction of the violator’s output. I also assume that when the participation constraint is binding, the LIC adheres to the borrowing contract.

In each period, the borrower country compares the value function under repayment, vR(k, X), with that under default, vD(k). When vR(k, X)vD(k) the country repays; otherwise it defaults. The value function under repayment, vR(.,.), is given by:

where vR(.,.) is increasing in k and is decreasing in X. Appendix III interprets the first-order conditions (FOCs) of the Bellman equation (equations (9) and (10)).

To make a connection with Cohen and Sachs’s model (1986), the participation constraint can be replaced with a debt capacity function h(k), which is defined implicitly by vR(k, h) = vD(k), where ∂vR(k,X)/∂X is strictly negative. In other words, given k, h(k) is uniquely determined and thus the case where h(k) is backward bending in k can be excluded. Thus the debt capacity function is well defined. The original value function under repayment can be rewritten as

This formulation is the same as that of Cohen and Sachs (1986),15 except that in this paper I numerically derive the value functions and the implied debt capacity function using the value function iteration method.16 I also extend their model to analyze the dynamics of concessional loans to LICs.

IV. The Numerical Results

Because one cannot solve this problem analytically unless the participation constraint is absent, I solve it numerically using the value function iteration method. I specify the functional forms of the utility and production functions as u(c) =c1–1/σ/(1–1/σ) and f(k) = Akη.

Calibration

Table 2 lists calibrated parameter values. Depreciation of capital is set at 0.1—a reasonable number in the real business cycle literature (for example, see Kydland and Prescott, 1982). The concessional interest rate is set at 0.75 percent, in line with existing official concessional lending practice. The value of the capital elasticity of the production function (η=0.33) is based on the findings in Gollin (2002) and is consistent with the existing real business cycle literature. The paper’s main results are robust to alternative values of η=0.3 and 0.4; a poverty trap arises with η= 0.2 if there is a slight change in other parameter values.

Table 2.Calibrated Parameter Values
Depreciation rate of capital (δ)0.1000Elasticity of intertemporal substitution (σ)0.45
Concessional interest rate (r¯)0.0075Fraction of output lost on default (λ)0.05
Capital elasticity of output (η)0.3300Cutoff level (fraction of U.S. steady-state output)0.15
Time preference (β)0.9500Level of TFP for the U.S.30.00
Note: TFP = total factor productivity.
Note: TFP = total factor productivity.

The discount factor is set at 0.95 and the world interest rate at 1/β–1=0.0526 in order to obtain flat consumption in the steady state. Debt overhang emerges more easily with a smaller value of β (or a higher r) because the borrower country will have a stronger incentive to consume in earlier periods (or because the benefits from concessional loans are larger). With a higher value of β, a poverty trap arises if there is a slight change in other parameter values.

The value for the elasticity of intertemporal substitution (σ = 0.45) is in line with the calibration results of Ogaki, Ostry, and Reinhart (1996). Using

Cooley and Ogaki’s (1996) two-step procedure,17 they estimate the lower and upper bounds of the intertemporal elasticity of substitution assuming that the elasticity is an increasing function of the level of wealth.18 The paper’s main conclusions are robust to alternative values of σ = ⅓ and ⅔ if there is a slight change in other parameter values.

The fraction of output lost on default (λ) is set at 0.05. This value needs to be sufficiently positive to maintain the paper’s main conclusions. If it is zero, then there will be no default cost in the steady state and thus the borrower country will have an incentive to default; as a result, no loans will be made.

The cutoff level (y¯) is set as a fraction of steady-state output in the United States (yUS). I use 0.15, because the purchasing-power-parity-adjusted real outputs per capita in most lower-middle-income countries are above this level. I calculate yUS by omitting the participation constraint and by assuming that U.S. TFP is 30 (note: this number is just a scaling factor) and the LIC-U.S. TFP ratio is ⅓.19 The paper’s main conclusions are sensitive to the LIC’s steady-state output level relative to the cutoff. More details are discussed at the end of this section.

Benchmark Economy

Consider the benchmark economy with initial income (y1) equal to 90 percent of the cutoff (or about 70 percent of steady-state output (yss)), and initial debt repayment obligation (X1) equal to 90 percent of the cutoff (or 100 percent of y1).

Figures 1 and 2 summarize the results from the numerical solution. To better understand the dynamics of concessional lending, these results are displayed along with those for nonconcessional lending. The only difference between these two loans is the interest rate level, where r˜=r for all t under nonconcessional loans. Thus, the nonconcessional debt capacity function hN(k) is defined implicitly by vR(k, hN) = vD(k), where ∂vR(k,X)/∂X is strictly negative and r˜=r for all t.

Figure 1.The Benchmark Economy: Numerical Results

Note: Panels (a)-(d) show how the benchmark economy will respond under the concessional lending (solid line) and nonconcessional lending (dotted line) schemes. The paths of output, consumption, investment, and repayment obligation are shown as a fraction of the steady-state output level (yss).

Figure 2.Endogenous Debt Capacity Ceilings: The Benchmark Case

Note: This figure shows the endogenous debt capacity function under concessional loans (h (k), solid line) and under nonconcessional loans (hN(k), dashed line). They are reported as a fraction of yss. The y-axis is the repayment obligation (X) as a fraction of the steady-state output level (yss). The x-axis is the level of capital per capita (K) as a fraction of the steady-state capital level (Kss). The vertical dotted line is the cutoff.

This discontinuity of the concessional debt capacity function (h(k))20 implies that the recipient country must drastically reduce its external debt precisely when it surpasses the cutoff. Such debt reduction is possible only through a steep decline in consumption, which the country may find too painful. For the benchmark case, this one-time cost of consumption reduction outweighs the long-run benefit of achieving the steady state where output is much higher than the cutoff (call this “high” steady state). Thus the country decides not to cross the line and converges to the cutoff (call this “low” steady state, that is, point a in Figure 2.) On the other hand, under nonconcessional lending, this perverse incentive is absent, and the country steadily grows to reach the high steady state, yet consumption in the short run is lower than in the case of concessional lending (Figure 1).

This result captures the paper’s debt overhang mechanism. Because the cost of servicing debt is kept artificially low, the country is motivated to carry a large amount of debt by consuming excessively and thus does not grow. Figure 3 shows that there exists a debt overhang threshold above which the country is trapped with large debt and no growth.

Figure 3.The Debt Overhang Threshold and Initial Conditions

Note: This figure shows a debt overhang threshold (solid) above which a country is trapped with large debt and no growth. It shows that if the country lies in region A or B, it converges to the low steady state, whereas if it lies in C or D, it achieves the high steady state. Arrears countries that lie above these ceilings (E and F) are outside the scope of this paper. The y-axis is the initial repayment obligation as a fraction of the cutoff. The x-axis is the initial income as a fraction of the cutoff. The dash-dotted line is the endogenous debt capacity ceilings. The vertical dotted line is the cutoff.

Whether or not the country is trapped in a debt overhang depends on the country’s initial conditions: initial debt and capital (Figure 3). The intuition is as follows: First, the higher the initial debt level, the more likely the country is to converge to the low steady state, because this allows the country to manage heavier debt at a low interest rate. Second, the lower the country’s initial capital, the larger the impact of short-run growth. As a result, the country tries to borrow a larger quantity of concessional loans to raise both investment and consumption in the short run, and thus it is more likely to be trapped in the low steady state. In short, the country converges to the high steady state only if initial debt is low enough, initial income is high enough, or both conditions hold.

Whether or not the country is trapped in a debt overhang is also conditional on the country’s TFP level (Figure 4). The higher the TFP, the higher the steady-state output level relative to the cutoff, and therefore the greater the long-run benefit of achieving the high steady state. With a higher TFP, the country thus finds it more costly to be trapped in the low steady state. Figure 4 shows that the debt overhang threshold shifts up with a higher TFP level. With a higher TFP level, the country is more likely to lie below the debt overhang threshold. Note that here the initial income level is kept the same across different TFP levels; initial capital levels are adjusted accordingly.

Figure 4.Debt Overhang Thresholds with Different TFPs

Note: The solid lines show the endogenous debt capacity ceilings and debt overhang threshold using the benchmark parameter values. The dotted lines show the corresponding figures when the total factor productivity (TFP) level is 1 percent higher than the benchmark economy. The y-axis is the size of the initial repayment obligation as a fraction of the cutoff. The x-axis is the initial income as a fraction of the cutoff. The vertical dotted line is the cutoff.

V. Policy Implications and Conclusions Implications for Aid Design

Even though the paper does not solve for the most efficient form of aid, 21 it does have implications for aid design. First, the model implies that the perverse incentive arising from per capita income eligibility exists under other forms of aid as well. For example, if a country repeatedly receives sufficiently large grants with a similar income per capita eligibility criterion, it may have an incentive to stagnate around the cutoff in order to maintain future grant eligibility.

Second, the model suggests that the existing eligibility and allocation rules could take into account other aspects, in addition to income per capita, in order to avoid the perverse incentive. In practice, some measures of TFP are already taken into account in the aid allocation formula of the multilateral development banks. For example, the IDA Country Policy and Institutional Assessment takes into account policy and structural indicators.

Third, the model implies that certain forms of graduation policies may provide stronger growth incentives as well as reduce the cost of aid assistance. For example, the model’s debt overhang may disappear if the concessional interest rates are allowed to increase with income levels. This implies that a more gradual move from concessional to nonconcessional lending provides the right incentive for growth. Indeed, the existence of “blend countries” suggests that something is at work in IDA and other multilateral development banks’ operational rules.

Implications for Debt Relief

The model implies that a one-time-debt-relief stock treatment may be effective in helping a country get out of the poverty trap and achieve growth. For example, suppose that the benchmark economy receives one-time relief that enables the country to move below the debt overhang threshold (solid line in Figure 3). The country now converges to the high steady state.

One-time debt relief may also be effective even if the country initially lies above the cutoff. Consider a country that has relatively high initial debt and lies in region B in Figure 3. Note that this country has an incentive to go back to the cutoff because the benefit of raising the debt ceiling by reducing capital is greater than the cost of lowering output. The country is thus better off reducing output until it eventually falls to the cutoff. Here upfront debt relief that moves the country from B to D is effective in achieving growth.

Note that if such a stock treatment is accompanied by factors that can raise TFP, such as productivity growth and an improvement in institutional quality, then the debt overhang threshold itself will shift upward (Figure 4) resulting in a larger number of countries that lie below the threshold. This means that if debt relief resources are used for development purposes that also directly raise TFP, more countries will be able to achieve growth given the same amount of debt relief. The above arguments provide some justification for the recent one-time-debt-relief stock treatment, known as the Multilateral Debt Relief Initiative (MDRI). The MDRI is a 100 percent debt stock cancellation by the IMF, the IDA, and the African Development Bank for a group of LICs; its goal is to free up resources to help countries achieve the United Nations’ Millennium Development Goals.

However, there are some caveats to this argument. First, for this type of stock treatment to work, it is important that recipient countries view it as a one-time event. If they do not, the poverty trap may reemerge if countries receive repeated debt relief with similar income per capita eligibility criteria. Second, the theoretical environment may be too efficient; that is, the model assumes that the country can efficiently reallocate freed resources from debt relief to productive activities. In reality, however, it may be quite difficult to handle a sudden increase in resources in the presence of weak institutions.

Conclusions

Whether an LIC is trapped in a debt overhang depends on its initial conditions and its TFP. The larger the initial debt, the stronger the incentives an LIC has to manage its debt at a low interest rate by becoming permanently aid dependent. The lower an LIC’s initial income, the more it tries to borrow a larger quantity of concessional loans to raise both investment and consumption in the short run and thus becomes more likely to be trapped in the low steady state. Last, the lower the level of TFP, the more likely it becomes that the benefit of remaining at the cutoff exceeds the long-run benefit of achieving the high steady state.

APPENDIX I Per Capita Income Cutoff: GDP vs. GNI

The model defines the cutoff in terms of GDP per capita. However, in practice, it is defined in terms of GNI per capita, which excludes the interest payments on external debt from GDP. Because a high level of external debt alters the level of capital stock above which the country loses eligibility, this new feature introduces an additional incentive for debt accumulation. However, as shown below, the paper’s key conclusion—a poverty trap occurs depending on the country’s TFP and initial conditions—still holds under the GNI case. This appendix solves a borrower’s problem in which the country’s output per capita is expressed in terms of GNI per capita—more specifically, it solves the problem of Equations (2)–(7) with y now defined as y=f(k)r˜X/(1+r˜).

The Bellman equation now has three state variables because r˜t+1 becomes a function of r˜t as well as kt and Xt:

subject to vR(k,X,r˜)vD(k),, where r˜ follows the rule given by Equation (1) and is equal to r¯ if f(k)r˜X/(1+r˜)y¯, and r otherwise. The debt capacity function h(k,r˜) is defined implicitly by vR(k,h,r˜)=vD(k),, where vR(k,X,r˜)/X is strictly negative.

I numerically solve for two value functions, vR(K,X,r¯) and vR(k, X, r), because r˜t can take only two values, r¯ and r. I solve these functions in the same manner as in Section IV using the same parameter values and functional forms. The numerical results for the benchmark economy (see Appendix Figures A1 and A2) are very similar to those presented in Section IV.

Figure A1.The Benchmark Economy with a GNI per Capita Cutoff: Numerical Results

Note: Panels (a)–(d) show how the benchmark economy will respond under the concessional lending (solid line) and nonconcessional lending (dotted line) schemes. The paths of output, consumption, investment, and repayment obligation are shown as a fraction of the steady-state output level (yss)

Figure A2.Endogenous Debt Ceilings with a GNI per Capita Cutoff: The Benchmark Case

Note: The solid lines show the endogenous debt capacity ceilings under concessional loans (h(k,r˜)),, and the dashed line shows those ceilings under nonconcessional loans (h(k)). They are shown as a fraction of the steady-state output level (yss). The y-axis is the repayment obligation (X) as a fraction of yss. The x-axis is capital per capita (k) as a fraction of the steady-state capital level (kss). There are three vertical lines: the left line is the cutoff and the middle and right (kr and kr¯) show the levels of capital at the points of discontinuity for h(k, r) and h(k,r¯),, respectively. The dotted lines (Fr and Fr¯) are the graphs of X=((1+r˜)/r˜)(f(k)y¯) where r˜ is equal to r and r¯, respectively.

In the GNI case, however, the corresponding endogenous debt capacity function is more complicated than Figure 2 in the text. Lines Fr and Fr¯are the graphs of X=((1+r˜)/r˜)(f(k)y¯),, where r˜ is equal to r and r¯, respectively. These lines determine the interest rate in the next period; for example, if r˜t=r¯, then r˜t+1=r if the country lies on the right-hand side of the line Fr¯, and r˜t+1=r¯ otherwise. kr and kr¯ are the levels of capital at the points of discontinuity for h(k, r) and h(k,r¯), respectively. k is the level of capital that satisfies y¯=Akη.

Could a poverty trap still occur in the GNI case? Yes. Suppose the country initially lies at point (a) in Appendix Figure A2 with r˜1=r¯. This implies that the interest rate in period 2 is also r¯, because the point (k1, X1) lies on the left-hand side of the line Fr¯. If the country chooses to cross kr¯ in period 2, the point (k2, X2) will lie on the right-hand side of the line Fr¯. Here, consumption in period 2 must be very low, because the country needs to reduce borrowing. This is the same story discussed in Section IV.

The new feature of the GNI case can be demonstrated as follows. Suppose the country initially lies on the right-hand side of the line Fr, say at point (b), with r˜1=r. This implies that the interest rate in period 2 is also r. Here, the country could significantly increase its borrowing and consumption in period 2—for example, by moving to point (c)—as long as the point (k2, X2) remains at or below h(k, r). However, this implies that the country would need to move back below h(k,r¯) in period 3, which requires consumption in that period to be very low. As a result, the country would not choose to converge to point (kr, h(kr, r)), that is, the point of discontinuity of h(k, r).

Oscillating solutions, in which the country chooses to move back and forth between the right- and left-hand sides of the line Fr, could also occur, depending on the initial conditions. This is because under the GNI cutoff rule, the problem has multiple discontinuities, as shown in the figure.

APPENDIX II No Participation Constraints

This appendix considers the environment in which the LIC fully precommits to honoring the conditions of the concessional lending scheme that is imposed by the creditor so that there is no need to impose a participation constraint. In practice, though, this is an unrealistic assumption because it allows the LIC unlimited access to the donor’s funds. Analyzing this nonparticipation constraint environment, however, is nonetheless useful to understand the role of a debt ceiling constraint.

In the absence of participation constraints, capital overshoots in period 1 as a result of a subsidized interest rate. The problem is given by

Subject to the intertemporal budget constraint

where K1, D1, and r˜1 are given. FOCs are given by

where μ is the shadow price. Initially, the country can borrow at the concessional interest rate (that is, r˜2=r¯) because I assume that initial output lies below the cutoff. At any level of capital above the cutoff, the country can borrow only at the world interest rate. The capital levels in period 2 and in the steady state, k2 and kss, are pinned down by r¯=f(k2)δ and r=f(kss)-δ (by equation (A.4)). These equations imply that k2 is greater than kss because the concessional interest rate is lower than the world interest rate (r¯<r.

Thus capital overshoots the steady state in period 1. However, from period 3 onward, capital is at its steady-state level (that is, kj = kss for j≥3), because as of period 2, the country no longer has access to concessional loans. Its capital level exceeds the cutoff, and the capital level is kss (from equation (A.4)). Consumption, too, overshoots in period 1(c1 > c2 = css). This is implied by the following Euler equations: u(c1)=β(1+r¯)u(c2)andu(c2)=β(1+r)u(c3),becauseβ(1+r¯)<1,β(1+r)=1,, and u(c) is decreasing in c. Once {k2, kss} and {c1, css} are pinned down, the path of debt, {D2, Dss} can be derived via the budget constraint. The dynamics of concessional loans without a participation constraint are thus characterized by the overshooting of capital and consumption in period 1 as a result of the low concessional interest rate. The donor’s budget, α, is determined by αD2(rr¯)/(1+r)..

APPENDIX III The FOCs of the Bellman Equation

The Bellman equation (equations (9) and (10)) can be rewritten as Equations (A.5) and (A.6)

because vR(k,X)vD(k)) and X ≤ h(k) are equivalent by construction, given the debt capacity function. In the following, I consider two cases: X h(k)) and X = h(k)), and interpret the corresponding FOCs.

CASE 1: Xh(k)

The FOC with respect to k is given byuc(c)=βvkR(k,X),, and by the envelope theorem, vkR(k,X)=uc(c)(fk(k)+1δ).

The Euler equation is given by

The FOC with respect to X is given by uc(c)/(1+r˜(k,X))=βvxR(k,X),, and by the envelope theorem, vXR(k,X)=uc(c). Combining the above two equations we get

Equation (A.8) can be rewritten as uc(c)=β(1+r¯)uc(c) if the country lies strictly below the cutoff, and uc(c)=β(1+r)uc(c) if the country lies strictly above the cutoff. Because β(1+r¯)<β(1+r)=1, the country has an incentive to overconsume in earlier periods under the concessional lending scheme.

CASE 2: X = h(k)

The Bellman equation (equations (A.5) and (A.6)) can be rewritten as

The FOC with respect to k is given by

and by the envelope theorem, vkR(k,X)=uc(c)(fk(k)+1δ),andvXR(k,X)=uc(c). Thus the Euler equation is given by

The numerical simulations suggest that h(.,.) is decreasing in k unless the country lies sufficiently above the cutoff. Thus there is upward pressure for c, whereas there is downward pressure for c—the country has an incentive to overconsume around the cutoff in earlier periods. To state this intuition more formally, rewrite Equation (A.9) as

One can interpret the bracketed term as the return from savings. The term β(1+r˜(k))/(1+r˜(k)hk(k)) can be interpreted as the effective discounting factor. If h(.,.) were constant so that hk(.,.) would be zero, then Equation (A.10) would reduce to the familiar expression with the return from savings given by fk(k) + 1- δ and the discounting factor given by β. The implication of the downward-sloping h (that is, hk(.,.) < 0) is twofold. First, the return from capital, the bracketed term in Equation (A.10), is suppressed. Second, the ratio in Equation (A.10) is less than 1, which effectively lowers the discount rate. Both effects act to depress savings.

APPENDIX IV A Description of PROX

There are more than 1,300 observations on PROX between 1987 and 2000, of which about 17 take positive values (that is, only 17 of the observations lie close to the cutoffs). Appendix Figure A3 shows the histogram of PROX excluding observations with PROX = 0. When I carry out a simple OLS regression of the growth rate of real GDP per capita on PROX, the coefficient for PROX is negative (–2.22) and is statistically significant at 1 percent.

Figure A3.Frequency of PROX

Note: This figure reports the histogram of PROX excluding observations with PROX = 0.

Appendix Figure A4 relates the growth rate with the level of GNI after removing regressors other than PROX. Recall that the empirical model is y = c + αPROX + γZ + ε, where y is the percentage per capita annual growth rate of real GDP, PROX is a measure of proximity to the cutoff, and Z represents the other explanatory variables. Denote c^,α^,andγ^ as the OLS coefficients. Appendix Figure A4 plots the OLS residuals of the growth rate excluding α^PROX(thatis,yc^γ^Z)andα^PROX against the percentage deviation of GNI from the threshold for IDA eligibility.

Figure A4.Ordinary Least Square Residuals Excluding alpha hat*PROX

Note: Remember that the paper’s empirical model is y= c + αPROX + γZ + ε, where y is the percent annual growth rate of real GDP per capita, PROX is a measure of proximity to the cutoff, and Z represents the other explanatory variables. Denote c^,α^,andγ^ as the OLS coefficients. The figure plots the ordinary least squares (OLS) residuals of the growth excluding α^PROX(thatis,yc^γ^Z)andα^PROX against the deviation of GNI from the threshold for IDA eligibility.

APPENDIX V First-Stage Regressions

See Table A1.

Table A1.First-Stage Regressions
Dependent Variables
Variables
PROXINVINIT_RGDP
First-Stage Regression: All Countries
GPO1.70e-030.03—0.01***
(2.45e-03)(0.06)(2.22e-03)
INIT_EDU1.08e-044 12e-035 22e-04**
(2.31e-04)(5 73e-03)(2 10e-04)
PROX(-1)0.92***0.59*-0 01
(0.01)(0.31)(0.01)
INV(-1)—4 54e-040.94***3.18e-03***
(3.89e-04)(0.01)(3.54e-04)
RGDP85-2.61e-030.34***1.00***
(4.13e-03)(0.10)(3.76e-03)
R-squared0.800.930.99
First-Stage Regressions: Countries That Lie Below the Cutoff
GPO-3 98e-03-5 30e-03-8 64e-03**
(3.98e-03)(0.09)(3.42e-03)
INIT_EDU1.28e-03**0.011.97e-03***
(6.21e-04)(0.01)(5.34e-04)
PROX(-1)0.90***1.06**1.10e-03
(0.02)(0.47)(0.02)
INV(-1)4.72e-040 91***1.59e-03**
(7.60e-04)(0.02)(6.53e-04)
RGDP85-0.010.130.10***
(0.01)(0.21)(0.01)
R-squared0.810.880.99
Note: Standard errors are in parentheses. *denotes significance at the 10 percent level, **denotes significance at the 5 percent level, and ***denotes significance at the 1 percent level.
Note: Standard errors are in parentheses. *denotes significance at the 10 percent level, **denotes significance at the 5 percent level, and ***denotes significance at the 1 percent level.
APPENDIX VI Examples

To provide some examples that support this paper’s key conclusions, this appendix reports a set of countries that have converged to (group A) and diverged from (group B) the cutoff. The group A countries have much higher concessional debt than the group B countries. The bottom two countries in group B were affected by the Asian financial crisis in 1997 (Figure A5).

APPENDIX VII Data

See Tables A2 and A3.

Table A2.Data
VariablesDefinitionSource
Growth rateThe growth rate of real GDP per capita (in percent)Constructed from real GDP per capita, Constant prices: Laspeyres (RGDPL) in Summers-Heston data set, version 6.1
GNIGNI per capita in current U.S. dollars, Atlas methodologyWorld Development Indicators
y¯The operational IDA cutoff in terms of GNI per capita in U.S. dollars, Atlas methodologyWorld Bank GNI/capita operational guidelines
GPOPercent population growth (a year)Constructed from population (POP) in Heston, Summers, and Aten, 2002
INVInvestment share of real GDP per capita (in percent a year)Heston, Summers, and Aten, 2002
INIT_RGDPThe log of real GDP per capita in 1985, constant prices: LaspeyresHeston, Summers, and Aten, 2002
INIT_EDUPercent of secondary schooling attained in the total population in 1985Barro-Lee data set
Note: IDA = World Bank International Development Association.
Note: IDA = World Bank International Development Association.
Table A3.Country or Regional Coverage of the Data Set
Low incomeLower-Middle incomeUpper-Middle incomeHigh income
BangladeshAlgeriaArgentinaAustralia
BeninBoliviaBarbadosAustria
CameroonBrazilBotswanaBelgium
Central African Rep.Dominican Rep.ChileCanada
Congo, Dem. Rep. ofEcuadorCosta RicaCyprus
Congo, Rep. ofEl SalvadorHungaryDenmark
Gambia, TheFijiMalaysiaFinland
GhanaGuatemalaMauritiusFrance
Guinea-BissauGuyanaMexicoGermany
HaitiHondurasPolandGreece
IndiaIndonesiaPanamaHong Kong SAR
KenyaIran, I.R. ofSouth AfricaIceland
LesothoJamaicaTrinidad and TobagoIreland
MalawiJordanTurkeyIsrael
MaliParaguayUruguay
Low incomeLower-Middle incomeUpper-Middle incomeHigh income
MozambiquePeruVenezuelaJapan
NepalPhilippinesKorea, Rep. of
NicaraguaSri LankaNetherlands
NigerSyrian Arab Rep.New Zealand
PakistanThailandNorway
Papua New GuineaTunisiaPortugal
RwandaSingapore
SenegalSpain
Sierra LeoneSwitzerland
TanzaniaUnited Kingdom
TogoUnited States
Uganda
Zambia
Zimbabwe

Figure A5.Examples

Source: World Bank, Global Development Finance.

Note: The dashed lines report concessional loans as a percent of GNI between 1987 and 2000 for countries that have converged to the cutoff (Group A) and that have diverged from the cutoff (Group B). The solid lines show the percent deviations from the per capita income cutoffs.

REFERENCES

    AtkesonAndrew1991“International Lending with Moral Hazard and Risk of Repudiation,”EconometricaVol. 59 (July) pp. 106989.

    AtkesonAndrew and MasaoOgaki1996“Wealth-Varying Intertemporal Elasticities of Substitution: Evidence from Panel and Aggregate Data,”Journal of Monetary EconomicsVol. 38 (December) pp. 50734.

    • Crossref
    • Search Google Scholar
    • Export Citation

    BarroRobert J. andJong-WhaLee1993“International Comparisons of Educational Attainment,”Journal of Monetary EconomicsVol. 32No. 3 pp. 36394.

    • Crossref
    • Search Google Scholar
    • Export Citation

    BarroRobert J. andJong-WhaLee and XavierSala-i-Martin2004Economic Growth (CambridgeMassachusetts, MIT Press, 2nd ed.).

    BulowJeremy and KennethRogoff1989“A Constant Recontracting Model of Sovereign Debt,”Journal of Political EconomyVol. 97 (February) pp. 15578.

    • Crossref
    • Search Google Scholar
    • Export Citation

    BorenszteinEduardo and AtishGhosh1989“Foreign Borrowing and Export Promotion Policies,”Staff Papers International Monetary FundVol. 36 (December).

    • Crossref
    • Search Google Scholar
    • Export Citation

    ChenXiao Philip B.Ender MichaelMitchell andChristineWells2003Regressions with Stata (Los Angeles: UCLA, Academic Technology Services, Statistical Consulting Group). Available via the Internet: www.ats.ucla.edu/stat/stata/webbooks/reg/default.htm.

    • Search Google Scholar
    • Export Citation

    CohenDaniel and JefferySachs1986“Growth and External Debt Under Risk of Debt Repudiation,”European Economic ReviewVol. 30 (June) pp. 52960.

    • Crossref
    • Search Google Scholar
    • Export Citation

    ColeHarold L. and Patrick J.Kehoe1998“Models of Sovereign Debt: Partial Versus General Reputations,”International Economic ReviewVol. 39 (February) pp. 5570.

    • Crossref
    • Search Google Scholar
    • Export Citation

    CooleyThomas F. and MasaoOgaki1996“A Time Series Analysis of Real Wages, Consumption, and Asset Returns,”Journal of Applied EconometricsVol. 11No. 2 pp. 11934.

    • Crossref
    • Search Google Scholar
    • Export Citation

    EasterlyWilliam1999“How Did Highly Indebted Poor Countries Become Highly Indebted? Reviewing Two Decades of Debt Relief,”Policy Research Working Paper No. 2225 (WashingtonWorld Bank).

    • Search Google Scholar
    • Export Citation

    EasterlyWilliam2002The Elusive Quest for Growth: Economists’ Adventures and Misadventures in the Tropics (CambridgeMassachusetts, MIT Press).

    • Search Google Scholar
    • Export Citation

    EatonJonathan and RaquelFernandez1995“Sovereign Debt,”NBER Working Paper No. 5131 (Cambridge, MassachusettsNational Bureau of Economic Research).

    • Search Google Scholar
    • Export Citation

    EatonJonathan and RaquelFernandez and MarkGersovitz1981“Debt with Potential Repudiation: Theoretical and Empirical Analysis,”Review of Economic StudiesVol. 48 pp. 289309.

    • Crossref
    • Search Google Scholar
    • Export Citation

    GollinDouglas2002“Getting Income Shares Right,”Journal of Political EconomyVol. 110 (April) pp. 45874.

    HestonAlan RobertSummers andBettinaAten2002“Penn World Table Version 6.1,”Center for International Comparisons at the University of Pennsylvania (PhiladelphiaUniversity of Pennsylvania). Available via the Internet: pwt.econ.upenn.edu/php_site/pwt_index.php.

    • Search Google Scholar
    • Export Citation

    International Monetary Fund2003“Debt Sustainability in Low-Income Countries—Towards a Forward-Looking Strategy” (Washington). Available via the Internet: www.imf.org/external/np/pdr/sustain/2003/052303.pdf.

    • Search Google Scholar
    • Export Citation

    International Monetary Fund and World Bank2004“Debt Sustainability in Low-Income Countries: Further Considerations on an Operational Framework and Policy Implications,” (Washington). Available via the Internet: www.imf.org/external/np/pdr/sustain/2004/091004.htm.

    • Search Google Scholar
    • Export Citation

    International Monetary Fund“Operational Framework for Debt Sustainability Assessments in Low-Income Countries—Further Considerations,” (Washington). Available via the Internet: www.imf.org/External/np/pp/eng/2005/032805.pdf.

    • Search Google Scholar
    • Export Citation

    International Monetary Fund2006“Review of Low-Income Country Debt Sustainability Framework and Implications of the MDRI,” (Washington). Available via the Internet: www.imf.org/external/np/pp/eng/2006/032406.pdf.

    • Search Google Scholar
    • Export Citation

    KrugmanPaul1988“Financing vs. Forgiving a Debt Overhang,”Journal of Development EconomicsVol. 29 (November) pp. 25368.

    KydlandFinn E. and Edward C.Prescott1982“Time to Build and Aggregate Fluctuations,”EconometricaVol. 50 (November) pp. 134570.

    OgakiMasao Jonathan D.Ostry and Carmen M.Reinhart1996“Saving Behavior in Low- and Middle-Income Developing Countries: A Comparison,”IMF Staff PapersVol. 43 (March).

    • Search Google Scholar
    • Export Citation

    RajanRaghuram2005“Straight Talk: Debt Relief and Growth,”Finance & DevelopmentVol 42 (June).

    RoseAndrew K2005“One Reason Countries Pay Their Debts: Renegotiation and International Trade,”Journal of Development EconomicsVol. 77 (June) pp. 189206.

    • Crossref
    • Search Google Scholar
    • Export Citation

    SachsJeffrey1989“The Debt Overhang of Developing Countries,”in Debt Stabilization and Development: Essays in Memory of Carlos Diaz Alejandroed. byGuillermoCalvo and others (OxfordBasil Blackwell).

    • Search Google Scholar
    • Export Citation

    World Bank2001IDA Eligibility Terms and Graduation Policies (WashingtonWorld Bank).

    YueVivian Z.2006“Sovereign Default and Debt Renegotiation” (New York: New York University, unpublished). Available via the Internet: www.econ.nyu.edu/user/yue/Yue_2006.pdf.

    • Search Google Scholar
    • Export Citation
*Junko Koeda is an economist with the IMF’s Middle East and Central Asia Department. The author would like to express thanks to Atish Ghosh, Carlos Vegh, anonymous referees, Yasuyuki Sawada, Yossi Yakhin, Leah Brooks, Chikako Yamauchi, Jianhai Lin, and Andrew Tweedie for helpful suggestions and comments.
1For theoretical models that explain debt accumulation in low-income countries, see for example, Easterly (1999); he shows that impatience could lead to overborrowing.
2Note, however, that in practice there are other criteria, such as the country’s creditworthiness and performance, that affect the determination of IDA loan eligibility in addition to per capita income eligibility criteria. This means that there may be some countries below the cutoff that are disqualified for IDA loans and some that are above the cutoff but are qualified for IDA loans, or both.
3For more details on the debt sustainability framework for low-income countries, see IMF (2003) and IMF and World Bank (2004, 2005, and 2006).
4For a summary of debt renegotiation literature, see for example, Eaton and Fernandez (1995) and Yue (2006).
5I exclude observations on Rwanda for 1994 from the sample where Cook’s distance, leverage, and studentized residuals exceed the conventional cutoffs. For conventional cutoffs and methodology, see Chen and others (2003, Chapter 2, 2.1).
6See World Bank (2001) for a detailed description of IDA eligibility criteria.
7The corresponding 1988 data are not available, so the 1985 data are used.
8GNI is commonly denoted as GNP. GNI is the new terminology under the 1993 System of National Accounts (SNA), replacing the old terminology—GNP—under the 1968 SNA.
9This significance level corresponds to the one-sided test.
10I also carried out regressions using specifications similar to the basic regression discussed in Chapter 12 of Barro and Sala-i-Martin (2004). The data I used here contain 79 countries, 13 of which are low-income countries. The dependent variable is the average real GDP per capita growth for 1990–2000. The explanatory variables are similar to those of Barro and Sala-i-Martin plus the average of PROX for 1990–2000. The 2SLS coefficient for the average of PROX becomes negative and insignificant but the t-values of all other regressors are considerably lower than in Barro and Sala-i-Martin (Table 12.3). This is probably because the sample size is only 79—one-third fewer than that of Barro and Sala-i-Martin—because the period covered (1990–2000) is much shorter.
11More precisely, this is the service charge that the World Bank currently imposes on the loans.
12The model assumes that the cutoff is expressed in terms of GDP per capita. In practice, however, it is defined in terms of GNI per capita, which excludes the interest payments on external debt from GDP. Appendix I presents the borrower’s problem where the cutoff is expressed in terms of GNI per capita. The main conclusions hold true for the GNI case.
13The cases with no participation constraints are discussed in Appendix II.
14The model is expressed with X rather than D, because otherwise the interest rates would be a function of the previous period’s capital (call this k-1), and thus k-1 would need to be treated as an additional state variable in the model’s recursive equation.
15An extension of Cohen and Sachs (1986) can be seen in Borensztein and Ghosh’s (1989) mode.
16Here, I use a two-period utility function (that is, u(c)+βu(c)) and an arbitrary debt capacity function (that is, h(k,r˜) = constant) as the starting functions; if the participation constraint is violated, the utility function is penalized. I obtain the same fixed point in the functional space.
17Following this two-step procedure, they first estimate the intratemporal parameters in a two-good model of tradable and non-tradable goods. Given these intratemporal parameters, they then estimate the intertemporal elasticity of substitution by applying the generalized method of moments to the Euler equation.
18Atkeson and Ogaki (1996) show that the intertemporal elasticity of substitution rises with the level of wealth.
19A cursory glance at the TFP ratio of 40 LICs relative to the United States between 1960 and 2000 shows that about one-fourth of LICs have TFP levels that are less than one-third the U.S. level and are stable.
20This discontinuity emerges in the amount of resources that can be borrowed (h(k)/(1 + rt + 1)) as a result of the jump in the interest rate that takes place as soon as the country’s income exceeds the cutoff.
21For more discussion towards an optimal debt relief proposal see, for example, Rajan (2005).

Other Resources Citing This Publication