Growth, External Debt, and Sovereign Risk in a Small Open Economy
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An optimizing growth model for a highly indebted small open economy is constructed and analyzed. An important innovation in the model is the incorporation of sovereign risk through the specification of an upward-sloping foreign debt supply function. The model is used to examine the interaction between external debt and growth in response to various policies and exogenous disturbances. It is shown that structural policies intended to reduce the fiscal deficit or increase productivity can lead to trade-offs in their effect on capital accumulation and the stock of debt.


An optimizing growth model for a highly indebted small open economy is constructed and analyzed. An important innovation in the model is the incorporation of sovereign risk through the specification of an upward-sloping foreign debt supply function. The model is used to examine the interaction between external debt and growth in response to various policies and exogenous disturbances. It is shown that structural policies intended to reduce the fiscal deficit or increase productivity can lead to trade-offs in their effect on capital accumulation and the stock of debt.

Recent research analyzing capital accumulation and growth in the international economy has been increasingly grounded in the underlying intertemporal optimizing behavior of private agents.1 Most of this literature, which can be viewed as deriving from the traditional optimal growth models pioneered by Cass (1965) and Koopmans (1965), focuses on small open economies. With few exceptions, it assumes that such economies face a perfect world capital market for debt and are free to borrow or lend as much as desired at the given world rate of interest.2 These models investigate the relationship between the rate of capital accumulation, the current account, and debt, in response to various types of disturbances. In particular, the responses to various types of fiscal disturbances and exogenous relative price disturbances have now been considered quite extensively.3 With a perfect world capital market, the dynamic adjustment has a simple recursive structure. On the one hand, the dynamic adjustment within the economy is driven by the accumulation of capital and does not depend directly on the stock of foreign debt. On the other hand, the current account and the stock of debt itself mirror the stable adjustment of the capital stock.4

This paper develops an intertemporal optimizing growth model for a small open developing economy, in order to study the dynamic interaction between growth and debt in response to various exogenous shocks and policies. There is little doubt that such economies require external capital, since typically they cannot generate adequate resources domestically to achieve the growth rates that may lead to an improvement in living standards.

But the assumption that such economies face a perfectly elastic supply of debt is clearly unrealistic. Experience with external borrowing in such economies has shown that debt repayments are not always made on time. Overborrowing, resulting from inadequate perceptions of domestic growth potential, has occurred on occasion. Long gestation lags in investment projects have led to difficulties in meeting repayment com-mitments in some cases (Kharas (1983) and Kharas and Shishido (1986)). International capital markets are likely to react to their perception of a country’s ability to repay, with lenders requiring a risk premium on the rate at which they are willing to lend to such economies, as well as, in some cases, imposing credit ceilings on borrowers.5

We incorporate this idea by assuming that the developing economy faces an upward-sloping supply schedule for debt, which embodies the risk premium associated with lending to a sovereign borrower. The analysis shows the effect of such a constraint on borrowing on the interaction between the dynamics of growth and debt accumulation in a fundamental way. In particular, the simple recursive dynamic structure associated with a perfectly elastic supply of debt breaks down. This is because the marginal cost of capital facing firms, and therefore determining their investment decisions, is now dependent upon the outstanding stock of national debt. Conditions in the international capital market therefore become important in determining the growth of capital in the domestic economy.

The paper analyzes the dynamic effects of various disturbances on key macroeconomic variables. Particular attention is devoted to considering shocks associated with the debt-supply schedule confronting the small developing economy. These shocks include: an increase in the level of the exogenously given world interest rate, and an increase in the risk premium associated with the country’s debt. The former reflects a general tightening in the world credit market, and the latter describes a deterioration in the country-specific borrowing opportunities. The effects of a productivity shock, which is taken to reflect some kind of structural efficiency-enhancing measure, are also considered. Finally, since fiscal policy is an important element in growth, debt accumulation, and adjustment strategies, the model considers the impact of changes in government expenditure as well. Overall, the model thus enables us to analyze the impact of both demand management and structural policies on variables such as household consumption-saving decisions, investment decisions of the firm, and the current account, and hence, debt accumulation. The paper stresses the interdependence between the rates of accumulation of capital and debt in the course of the dynamic adjustment and the possible trade-offs between them. However, the discussion focuses primarily on the growth aspects and does not address debt strategies or debt overhang issues.

The rest of the paper is organized as follows. Section I specifies the analytical framework. Section II discusses the long-run effects and dynamics of a balanced-budget policy; debt-financing deficit policy is discussed in Section III. Section IV briefly considers alternative specifications of the debt function. The main conclusions of the paper are summarized in the final section. For convenience, all technical matters relating to the solutions are relegated to the Appendix.

I. The Analytical Framework

In this section the economy’s structure is outlined, followed by a derivation of macroeconomic equilibrium.

The Structure of the Economy

The economy we shall consider comprises three sectors: (1) consumers; (2) firms; and (3) the government. For analytical tractability all consumers and firms are assumed identical, enabling us to focus on the representative unit in each group. The economy produces a single traded good. The model is real, with the only financial asset held by domestic residents being a traded bond.6 International borrowing by the government is allowed, and consumers may borrow from or lend to the government, which is largely consistent with practice in borrowing countries.7

Upward-Sloping Supply Schedule for Debt

A key element of the model is the upward-sloping supply schedule for debt, reflecting the degree of risk associated with lending to the economy. In its simplest form, this is expressed by assuming that the interest rate i(z) charged on foreign debt z is


where i0 is the interest rate prevailing internationally and i, ω(z) is the country-specific risk premium, which varies with the stock of foreign debt held by the country.

Various forms for the function i(z) may reasonably be postulated. For example, it is possible that the function may be kinked, being positively sloped up to a stock of debt, say, z = z*, when an absolute borrowing limit is reached, and vertical thereafter. The possibility of a cutoff in debt is also obtained, in perhaps a slightly more convenient way, by the assumption that ω(z) is convex (ω"(z) > 0). In this case, the fact that the interest rate rises at an increasing rate with the level of indebtedness means that at some point z* it becomes prohibitive, so that an effective borrowing limit is reached.8 In any event, we assume that we are on the upward-sloping portion of the curve, rather than at any absolute borrowing ceiling. With this formulation, an exogenous increase in the world interest rate is described by an increase in i0, while an exogenous shift in country-specific risk is represented by a change in i1.

The specification (1), which postulates the cost of debt to increase with the absolute level of foreign debt (possibly up to some ceiling) was first introduced by Bardhan (1967). More recently, it was incorporated by Obstfeld (1982) in his analysis of terms of trade shocks and by Eaton and Turnovsky (1983) in their study of exchange rate dynamics under covered interest parity. Other authors, such as Sachs (1984), Sachs and Cohen (1982), and Cooper and Sachs (1985), have shown how a country, by adopting growth-oriented policies, as well as policies that enhance foreign exchange earning capacity, can shift the upward-sloping supply function outward, so that at each level of debt a lower risk premium is charged. These effects can be incorporated by assuming that the risk premium depends upon the level of debt relative to some measure of earning capacity, such as capital or output.9 However, as will be shown in Section IV below, our essential qualitative results are not altered by these modifications.

The analysis also assumes a form of risk-adjusted interest parity. That is, the domestic interest rate prevailing in the small economy is equal to i(z), the rate at which it can borrow from abroad. Under this assumption, equation (1) also represents the interest rate prevailing at home, at which domestic consumers can borrow from or lend to the government.


The representative consumer chooses his or her consumption and bond holdings to maximize the intertemporal utility function:


subject to the budget constraint:


and the initial condition:


where x is consumption, l is labor supply assumed to be fixed, β is the consumer rate of time discount, w is the wage rate, π is profit distributed by firms to households, b is the stock of debt (or bonds) held by domestic residents, and T denotes lump-sum taxes.10 The utility function is assumed to be concave.11 The present-value Hamiltonian for the consumer problem may be written as

where a denotes the total stock of government debt (a = z + b), and λ, the Lagrange multiplier associated with the budget constraint, is the marginal utility of wealth measured in terms of units of foreign bonds. The required optimality condition for the individual’s consumption decision is12
The optimal dynamic path is determined by the budget constraint, along with13

as well as the transversality condition:


Equation (6) determines optimal bond accumulation by equating the marginal rate of return to consumers on consumption (the left-hand side) to the marginal cost of an additional unit of debt facing consumers.14


Firms produce output using capital and the fixed supply of labor through a production function y = f(k, l), which is assumed to possess standard neoclassical properties. Net profit of the representative firm at each point in time is therefore given by


where I is the rate of investment.

The function C(I) represents installation costs associated with the purchase of I units of new capital. It is assumed to be an increasing convex function of I:C′ > 0, C″ > 0. In addition, we assume C(0) = 0, C′(0) = 1, so that the total cost of zero investment is zero, and the marginal cost of the initial installation is unity. This formulation of the installation function follows the original specification of adjustment costs introduced by Lucas (1967) and Gould (1968).15 More recent work by Hayashi (1982) and others postulates an installation function that depends upon k as well as I. This modification makes little difference to our analysis, and for simplicity we retain the simpler specification.

The firm’s optimization problem is to


subject to


and the initial condition

k(0) =ko.

Two further points about this specification of the firm’s optimization problem should be noted. First, equation (10) abstracts from physical depreciation. Second, the firm is assumed to finance investment purely from retained earnings and therefore does not need to borrow.

Writing the Hamiltonian function for firms as


where q is the Lagrange multiplier associated with the accumulation equation (10), the optimality conditions are


together with the transversality condition


Equation (12) determines the level of investment in each period by equating the marginal cost of investment to the shadow price of capital. Equation (13) can be rewritten as


Here, the left-hand side, which equals the sum of the marginal physical product of capital deflated by q, and the percentage rate of change in the shadow price of capital—that is,

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—is the rate of return on investing in a unit of capital. Optimality requires that this return should be equated, at each point in time, to the interest rate.16 In view of the assumption about labor supply, equation (14) requires that the wage rate be set equal to the marginal physical product of the fixed supply of labor.


The remaining agent in the domestic economy is the government, which operates in accordance with its budget constraint:


Thus, government spending, g, plus the interest obligations on outstanding government debt i(z)a, must be financed either by imposing additional taxes, which we assume are of the lump-sum type, or by issuing additional debt. Of course, the government retains the choice of borrowing at home or abroad; that is, ȧ = ż + .

Macroeconomic Equilibrium

Summing the consumers’ budget constraint (3), the firm’s profit relation (8), and the government budget constraint (17), and noting that ż = ȧ, the rate of decumulation of foreign debt, –ż, is equal to the current account balance.17


The current account balance is simply the sum of the trade balance, f(k) – (x + C(I) + g), plus the service account, which in this case is only debt service, –i(z)z.

By combining the optimality conditions derived for the individual sectors above, together with the accumulation equations, macroeconomic equilibrium can be described by the following equations:


The pair of static equations (19a) and (19b) determine consumption as a function of the marginal utility of wealth, λ:


and investment

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as a function of its shadow price q:


The latter equation will be recognized as being the Tobin q theory of investment. Substituting for x and I into (20a)–(20e) describes the dynamics of the economy.

When capital is perfectly mobile internationally, so that the interest rate facing the small economy is fixed at the given world rate, the dynamic structure simplifies drastically. In order for a steady-state equilibrium to exist in such an economy, the discount rate, β, must equal the world interest rate, and this implies that the marginal utility of wealth must remain constant at, say, λ. The dynamics of the accumulation of capital k and its shadow price q, which constitute the core dynamics of the economy, are jointly determined by equations (20c) and (20b), respectively. The long-run equilibrium in such an economy has the property that it depends upon the initial stocks of real and financial assets. This, in turn, means that temporary shocks have permanent effects.18

The fact that the economy faces an upward-sloping supply function for debt changes the dynamics in a fundamental way. With the domestic interest rate depending upon the stock of national debt, the discount rate β is no longer tied to the world interest rate, and the marginal utility of wealth λ is no longer constant. Instead, it depends upon the accumulation of both foreign debt z and government debt a, and thereby the dynamics of the entire economy become highly interdependent.

In order to complete the specification of the dynamics, government budgetary policy needs to be specified. As a useful benchmark that is analytically tractable, we shall focus first on the balanced-budget policy

a˙ = 0,orT =g +i(z)a¯,

where lump sum taxes T are continually adjusted to finance expenditures.19 Using this model we analyze the long-run and short-run effects of a number of disturbances, focusing on the trade-offs involved between the rate of capital accumulation, on the one hand, and the accumulation of foreign debt, on the other.

II. Balanced-Budget Policy

The dynamics of the system involve forward-looking behavior. The short-run transition is therefore determined in part by the long-run steady state. Hence, it is convenient to begin with a consideration of the latter. Since the government is assumed to maintain a balanced budget, its stock of debt remains fixed at, say, a.

The Long Run

The steady state of the economy is reached when all variables cease to change; that is,

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. With no depreciation of capital, steady-state investment is zero, which in turn implies that the shadow price of investment, (q), is equal to unity. With the steady-state values denoted by tildes (¯), the long run can be described by


These equations jointly determine the steady-state values of the marginal utility

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, consumption x, capital stock
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, national debt z, and lump-sum taxes T.

The equilibrium defined by equations (22a)–(22e) has a particularly simple recursive structure. The equilibrium level of the foreign debt z is determined by (22b), which requires that consumers equate the costs (or returns) of buying or selling the marginal bond to their rate of time preference. Since the debt schedule facing the economy and, hence, the consumer is upward-sloping, the cost (or return) of holding a marginal bond equals the interest rate plus the impact of the additional bond on the interest rate. Having determined z, one can determine the domestic rate of interest by the debt-supply schedule. Equation (22c) then determines the long-run capital stock by equating the marginal physical product of capital to the domestic interest rate. At the same time, equation (22e) determines the required lump-sum taxes necessary to finance government expenditures and interest on outstanding government debt. Equation (22d) is the steady-state balance of payments equilibrium condition. In the long run, the country must run a balance of trade surplus of sufficient magnitude to finance the interest on the outstanding foreign debt. Given the capital stock and, hence, output and the level of interest payments, this determines the level of consumption x. Finally, given x, the equilibrium marginal utility of wealth

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is determined by (22a).

Table 1 summarizes the long-run effects of permanent shifts in the exogenous variables—namely, the foreign interest rate (i0), the risk premium (i1) a productivity shock (θ), and an increase in government expenditure (g)—on the stock of debt, the capital stock, the domestic interest rate, the trade balance, consumption, and taxes.20

Table 1.

Long-Run Effects

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Note: D ≡ 2i′ – ib > 0 (see Appendix); i0 denotes the foreign interest rate, i1 denotes the risk premium, θ denotes a productivity shock, and g denotes government expenditure.

Increase in the Foreign Interest Rate

An increase in the level of international interest rates, represented by an upward shift in i0, raises the marginal return or cost to consumers from holding an additional bond. Since in the long run this marginal return must equal the fixed rate of time discount β, this requires a reduction in the former to offset the effect of the higher i0 and to maintain equality with the latter. This equalization is brought about by a reduction in the level of external debt z. However, the effect upon the domestic interest rate is subject to countervailing considerations and is a priori unclear. On the one hand, the upward shift in the debt-supply schedule leads to a substitution away from foreign debt to domestic debt, putting upward pressure on the domestic interest rate. On the other hand, the reduction in the long-run stock of external debt lowers the risk premium, thereby tending to reduce the domestic rate. The overall effect depends upon which of these factors dominates, and this in turn depends critically upon the nature of the risk-premium function ω(z). Intuition would suggest that the domestic interest rate will, on balance, rise, and this in fact is likely to be so under plausible conditions. For example, if the debt-supply schedule is linear (i″ = 0), or if domestic private residents hold no bonds (b = 0), then a rise in the foreign interest rate does indeed result in a rise in the domestic interest rate, though not by the full amount. If i″ > 0 and domestic private residents are net debtors (b < 0), the upward pressure on the domestic interest rate is increased. However, if i″ > 0 and domestic private residents are large creditors (b > 0), the response of the domestic interest rate is reduced, and indeed it may now fall.21

The response of the domestic capital stock depends upon that of the domestic interest rate. Taking the more plausible case where the latter rises, the equilibrium capital stock falls, so that the reduction in long-run national debt is accompanied by a lower long-run capital stock. At the same time, the financing requirement of the government rises, requiring an increase in lump-sum taxes. As noted, in the long run the trade surplus must finance the interest costs i(z)z of the foreign debt. Although one effect of the higher foreign interest rate is to reduce the stock of foreign debt, this is likely to be offset by a higher domestic interest rate. The net effect on total interest payments is unclear and depends upon the function ω(z). In the case where this function is linear, the decline in foreign debt dominates the higher domestic interest rate, and overall interest payments decline. Assuming this case prevails, the long-run trade balance, which was, and remains, in surplus to meet the economy’s interest obligations, is reduced. What happens to domestic consumption is uncertain, even in the simplest case. For example, although the linear debt schedule leads to a reduction in the capital stock and, hence, output, less output is now devoted to interest payments on international debt.22 The net effect on consumption depends upon which of these influences dominates. Moreover, since utility is a function of consumption, an upward shift in the cost of debt may or may not lead to a corresponding reduction in steady-state welfare.

Increase in Risk Premium

An increase in the risk premium i1 will raise or lower the marginal return or cost to consumers from holding additional debt, according to whether ω – ω′b ≷ 0. In the case where domestic consumers are net debtors (b < 0), this quantity is certainly positive, raising the marginal cost to them of incurring additional debt. As a result, they reduce their level of debt (that is, increase b) so as to maintain the equality between the marginal cost of debt and the fixed rate of time preference β. However, if consumers are net creditors, the marginal return to holding more debt may either rise or fall with i1 even for the simplest debt function, depending upon the level of b. In the case where b is sufficiently small, so that ω – ω′b > 0, external debt will still fall. However, if b is sufficiently large to reverse this inequality, a higher risk premium will lead to a higher equilibrium stock of external debt.23

The response of the domestic interest rate depends upon whether the upward shift in the marginal cost of debt brought about by i1 more than offsets the decline resulting from the likely reduction in the stock of external debt. Again, this depends upon the form of the function ω(z). For plausible debt-supply functions, such as the linear or constant-elasticity type, the interest rate will certainly rise. However, a decline cannot be ruled out, although it is even less likely to occur than in response to an upward shift in io. The stock of capital moves counter to the interest rate and, therefore, almost certainly will fall. The equilibrium trade balance will rise or fall to cover external payments commitments. In terms of the linear debt-supply function, if domestic residents are net debtors, external debt as well as interest payments decrease and, therefore, the trade surplus declines; otherwise, it could move in either direction. Finally, the effects on long-run consumption and utility are unclear, although both will decline if net interest payments increase.

Productivity Shock

The third column of Table 1 summarizes the effect of a productivity shock, θ, which is introduced as a shift operator in the production function.24 Such a disturbance has no effect on the steady-state stock of external debt, and the equilibrium trade balance, too, remains unaffected. The effect on the capital stock depends on the sign of fkθ—that is, the impact of the productivity shock θ on the marginal physical product of capital. If the shock is productivity-enhancing, capital stock will increase. The effect on consumption depends on whether or not the shock increases output—that is, fθ ≷ 0—and whether or not capital stock increases.

Increase in Government Expenditure

The last column of Table 1 summarizes the impact of a change in government expenditure. Since in this model the long-run stocks of debt and capital are determined by equations (22b) and (22c) and are independent of g, the only effect of an increase in government expenditure is to cause an equal reduction in private consumption. With the government budget being balanced, the increased government expenditure must be matched by an equal increase in lump-sum taxation.

Transitional Dynamics

The transitional dynamic behavior of the economy is determined by the system of differential equations, (19a), (19b), and (20a)–(20e). The solution of this fourth-order dynamic system is described in the Appendix. In the short run the stock variables, k and z, may be regarded as predetermined. Any response to shocks will, therefore, have an effect on the two shadow prices, q(0) and λ(0). As shown in the Appendix, these prices are determined by the expected long-run responses of both the capital stock and the foreign debt. In this respect, the dynamics of the system are forward-looking.

In this section we will examine the transitional effects of exogenous shocks; namely, shifts in the debt schedule, productivity shocks, and changes in the government expenditure levels. The shocks to be considered are all unanticipated permanent changes. Table 2 summarizes the qualitative short-run effects of the various shocks.

Table 2.

Qualitative Short-Run Effects

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Note: i0 denotes foreign interest rate, i1 denotes risk premium, θ denotes a productivity shock, and g denotes government expenditure.

Increase in the Foreign Interest Rate

The previous section showed that an increase in i0 leads to a long-run decline in external debt and almost certainly in the stock of capital as well. Assuming this to be so, both of these long-run effects contribute to an immediate fall in the shadow price of investment q(0), and therefore in the level of investment.25 The rate of capital accumulation therefore slows down as a result of this shock. With the stock of foreign debt fixed instantaneously, a 1 percentage point increase in the foreign interest rate leads to a corresponding immediate 1 percentage point increase in the domestic interest rate, although over time the domestic rate will decline to the smaller steady-state response. The higher short-term interest with the fixed stock of debt means that the cost of servicing the debt also immediately increases, although it too declines over time as both the interest rate and the stock of debt decline to their steady-state values.

The effect of an increase in i0 on initial consumption is unclear. Equation (5) shows that consumption varies inversely with the marginal utility of wealth λ. However, we are unable to establish unambiguously the initial response of λ(0) and, hence, that of consumption. Various factors are at work. First, the short-run marginal utility λ(0) depends upon the long-run marginal utility

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, which may either increase or decrease, as previously discussed. Second, since we know that as the long-run capital stock declines, capital will decumulate over the transition path, consumption will tend to increase in the short run. An offsetting impact on consumption, however, results from the fact that long-run external debt is known to decline, and hence the debt stock must also be reduced during the transition. Since the long-run debt of the government is known to be fixed at ā, the private supply of debt has to increase, which may occur at the expense of consumption.

An increase in the foreign interest rate has the effect on the trade balance of increasing the surplus. Although the debt-servicing cost increases on impact and the effect on consumption is uncertain, the decline in investment alone is sufficient to increase (reduce) the surplus (deficit). In the short run, the declining capital stock will be accompanied by a declining stock of foreign debt.

The contrast between the short-run and long-run effects of an upward shift in the cost of debt can be usefully summarized at this point. An increase in i0 leads to long-run declines in both the capital stock and the level of external debt. Despite a somewhat higher interest rate, the long-run cost of debt servicing declines and this requires a smaller trade surplus. In the short run, prior to any adjustment in the stock of debt, the domestic interest rate responds by the full amount of the increase in the foreign rate. Consequently, debt-servicing requirements in the short run are increased. At the same time, since higher interest rates in the long run lead to a reduced capital stock, investment must decline. The cumulative effect of these changes is that in the short run the trade balance improves to meet the increased interest payments, as well as to allow the stock of foreign debt to decline.

Further characterization of the transitional dynamic paths followed by k and z can be obtained by using the dominant eigenvalue method suggested by Calvo (1987). Writing the solutions for k and z (equations (28) and (29) in the Appendix) in the form


we have


(see Appendix for a definition of Φ(μ1) > 0, Φ (μ2) < 0). Since μ2 is the dominant stable root (that is, 0 > μ2 > μ1), it follows that as t→∞(Z – Z)/(kk)→ Φ(μ2) < 0. That is, z and k asymptotically approach their respective steady-state values along a ray having a negative slope = Φ(μ2). The initial phase of the path can be determined by evaluating dz/dk, and d2z/dk2 at the initial instant t = 0. In the present case of an upward shift in the cost of debt, i0, we have already shown that dz(0)/dk(0) > 0, and imposing additional weak conditions, we can further establish that d2z(0)/dk(0)2 > 0.

The transitional path followed by k and z is illustrated in Figure 1. The starting point is at the origin, with the new steady-state equilibrium at A, having lower stocks of both physical capital and foreign debt. In the limit, this equilibrium is approached along the locus XX, which has a negative slope Φ(μ2). Initially, the declining capital stock and external debt causes the system to move in a southwesterly direction from O. The convexity of this locus, and the subsequent convergence along the negative ray, implies that the stock of foreign debt overadjusts during the transition. After declining to point B, z then begins to increase to its new equilibrium level. Intuitively, the convexity of the transitional locus OA implies an increasing decline in k relative to z over time. This in turn leads to an eventual decline in output, which exceeds the decline in other components of the current account. The initial current account surplus eventually becomes a deficit, and at that time additional external debt is incurred.26

Figure 1.
Figure 1.

Transitional Path in Response to Upward Shift in Cost of Debt

Citation: IMF Staff Papers 1990, 002; 10.5089/9781451947069.024.A007

Increase in Risk Premium

The qualitative short-run responses to an increase in the marginal cost of debt, i1, are given in the second column of Table 2. As before, these responses also depend on the long-run response of the level of external debt, which as shown above, depends upon (ω′b – ω). If ω′b < ω, so that z falls, the results are qualitatively the same as those for an increase in io and for essentially the same reasons. Once again, the short-run interest rate exceeds the long-run rate. If, however, ω′b < ω, so that long-run debt increases, the short-run responses are less clear. Even though one can establish that there will be an initial decline in the shadow cost of capital q(0), causing an immediate decumulation of capital, it is now possible for this to be accompanied by a current account deficit leading to an increase in foreign debt. In any event, even if the external debt does not decline on impact, it will have to rise at some point during the transitional path.

Productivity Shock

The productivity shock is simpler to analyze. The reason is that since it has no long-run effect on national debt, the only long-run response driving short-run behavior is the change in the capital shock. In order to be concrete, assume fkθ > 0; that is, the productivity shock has a positive effect on the marginal physical product of capital. In this case, the increase in the long-run level of the capital stock will lead to an instantaneous increase in the shadow price q(0), causing investment demand to increase and an increased accumulation of capital to take place. Moreover, since consumption increases in the long run, the steady-state value of the marginal utility of wealth

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is now lower. Instantaneously, λ(0) will likely fall, although by a smaller amount than in the long run. Consequently, although consumption increases immediately it will not increase by the full amount of the increase expected in the steady state. This is so because the expansion in the economy stimulates consumption as capital is accumulated over the transition.

There is no immediate effect on the interest rate or interest payments. The higher level of investment and the likely increase in consumption lead to a deterioration in the trade balance and, therefore, in the current account. Foreign debt begins to accumulate, resulting in a rising interest rate. However, this is only temporary. Over time, as capital is accumulated, output increases. Eventually, output exceeds the levels of domestic demand by an amount in excess of that of additional interest payments, leading to a surplus on the current account. At that time, foreign debt is reduced and eventually returns to its original level. The adjustment path followed by capital and external debt is illustrated in Figure 2. The dynamics of debt are mirrored in the behavior of interest rates. Increasing current account deficits and accumulating debt lead to rising domestic interest rates on account of the increasing risk premium. When the current account switches from a deficit to a surplus and the process of debt reduction begins, interest rates start to decline, also returning to their preshock levels.

Figure 2.
Figure 2.

Transitional Path in Response to Productivity Shock

Citation: IMF Staff Papers 1990, 002; 10.5089/9781451947069.024.A007

Increase in Government Expenditure

Since the long-run stocks of

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and z remain unaffected by changes in government expenditures, and since the long-run behavior of these state variables determines the dynamics, changes in government expenditures do not affect any variable other than consumption and revenues. Again, because of the balanced-budget assumption, as government expenditures increase (decrease) a corresponding increase (decrease) in revenues occurs, leading to a corresponding decrease (increase) in consumption.

III. Debt-Financed Deficit Policy

Up to this point it has been assumed that the government’s budget is continuously balanced. This assumption is a simplifying one, made in part to maintain analytical tractability.27 The present section briefly considers a modification wherein the domestic government finances its deficit with the issuance of debt instruments.28

At the outset it may be noted that such a debt-financing policy requires an accommodating one-for-all adjustment in the initial level of lump-sum taxes, in order to be sustainable in the long run. Formally, this requirement arises from the fact that with the stock of government debt constrained to adjust continuously, the number of unstable roots to the dynamic system (3) exceeds the number of “jump” variables (2). In order to obtain a viable solution therefore, an additional jump variable is required. This may be accomplished by appropriately choosing the initial level of lump-sum taxes. The steady state of the model is now defined by29


where equations (23a)–(23e) are identical to the equations describing the steady state for the balanced-budget case; (23f) is an additional relationship linking changes in government debt, total national debt, and capital stock, which (as shown in the Appendix) must additionally be satisfied. The coefficients, Ψi’s, are defined as


where, in turn, μ1 μ2 are the stable (negative) roots of the dynamic system (see Appendix for details), and Φ(μi) is defined as


The steady-state stock levels of

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, z, and a are jointly determined by equations (23b), (23c), and (23d) and depend (through (23f)) upon the initial conditions ko, ao, and zo. Because of this dependency, even temporary changes can give rise to permanent effects.30 Once
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and z are determined as above, the balance of payments condition (23d) determines steady-state consumption x, while
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may now be obtained from (23a). The required accommodation in lump-sum taxes is determined from (23e). Once T is set in this manner it is not thereafter adjusted, and the subsequent deficit is bond financed while the stock of government bonds follows an adjustment path. The short-run and long-run effects of the various disturbances can be analyzed in a manner similar to that done previously and may be briefly summarized.

An upward shift in the cost of debt, io, will lower both the steady-state stock of foreign debt and physical capital, as previously. The long-run interest rate will also increase, although by less than the increase in io. Long-run government debt may either rise or fall. This is because the higher interest rate, accompanied by the lower foreign debt, may or may not raise the marginal return (or cost) of purchasing an additional bond to consumers. If it does, a will have to rise, in order to maintain the equality of the marginal return with the fixed rate of consumer discount β. But if this marginal return falls, then a will have to fall for the same reason.

The short-run responses to an upward shift in the debt schedule can be analyzed as before. Again, we can establish that the long-run declines in

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and z generate short-run decumulations in these quantities. The interest rate rises more in the short run and the higher short-run interest costs to the government cause a short-run increase in the rate of accumulation of government debt.

Although the changes in the marginal cost of debt i1 and government expenditure give rise to virtually the same responses as before, more substantive differences arise with respect to a productivity shock. Not only does it lead to a long-run increase in capital stock as before, but now this increase is accompanied by higher long-run levels of both national and government debt. The higher marginal productivity of capital resulting from the increase in productivity raises the domestic interest rate, thereby encouraging more borrowing from abroad. The higher level of external debt raises the marginal rate of return of additional bonds to domestic consumers, and government debt will have to rise in order to maintain equality with β. The initial responses are also in the direction of the long-run effects.

IV. Alternative Specifications of the Debt Function

As noted earlier, it has been argued that country creditworthiness may not be a function only of the stock of debt. Factors that serve to increase productivity or growth, or foreign exchange earnings, may all indicate an increased capacity to service debt, and hence shift the debt-supply schedule outwards. To incorporate this idea, the debt function can be altered in at least two interesting ways. Note that the above argument basically requires debt to be scaled in some way in its effect on the rate of interest. The central point appears to be that creditors are not as concerned with the absolute amount of debt held by the country as with its ability to service the debt, which is better represented by debt relative to some measure of servicing capacity. Thus, alternative specifications might be to use either a debt-output ratio (z/y) or a debt-capital ratio (z/k) in equation (1) above.31

With employment fixed and output dependent on capital alone, these two modifications are almost identical. Moreover, neither of them alters the results in any fundamental way. For example, if one were to use the relationship

i(z/k)=io +i1ω(z/k)

and assume a balanced government budget, one finds that the two critical steady-state equilibrium relationships—equations (22b and 22c)—are modified to


The interdependence between z and k in the determination of the interest rate now destroys the recursivity in the long-run equilibrium noted earlier. These two equations now jointly determine the steady-state stocks of capital and external debt. The changes in these equilibrium levels then drive the short-run dynamics as before. Generally, the qualitative behavior of the system is changed little. The only point worth noting is that a productivity disturbance will now affect the steady-state stocks of both debt and capital.

V. Conclusion

Although it has long been recognized that small developing economies can borrow only at a premium and therefore face an upward-sloping supply of debt, most existing growth models of such economies treat the supply of debt as being infinitely elastic. This paper has departed from this assumption and analyzed the dynamic consequences of various kinds of disturbances in an economy facing an upward-sloping supply of debt. The presence of such a constraint changes the growth and capital accumulation dynamics in fundamental ways from those that occur under the more usual, but less realistic, assumption of perfect capital mobility.

In order to highlight the issues involved, we have based most of our analysis on the simplest such model and then modified it in alternative ways. The basic model is one in which the government maintains a balanced budget, the cost of debt rises with its absolute level, and employment remains fixed. We have emphasized the trade-offs that exist between the rate of capital accumulation, on the one hand, and the rate of accumulation of foreign debt, on the other, and analyzed how these respond to the various exogenous disturbances.

Rather than summarizing the specific results in detail, it is more useful to conclude by reviewing the source of the dynamic trade-off between the accumulation of capital and external debt. In the basic model, this stems directly from the two long-run equilibrium relationships—equations (22b) and (22c)—both of which reflect the upward-sloping supply of debt. These two equations sequentially determine the long-run equilibrium stocks of debt, z, and capital,

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, and it is the changes in these variables resulting from the various shocks that in turn drive the short-run dynamics. The nature of the trade-off is therefore determined by the relative adjustments in these two equilibrium quantities, and these are dependent upon the origin of the shock.

In the case of the foreign interest rate, we have seen that under plausible conditions, an increase in i0 will lower the long-run external debt, while raising the domestic interest rate, thereby reducing the equilibrium capital stock. Capital and debt will therefore both decline in the long run. Both also begin to decline in the short run, although at some point along the transitional path foreign debt will overshoot its long-run decline and will begin to increase. A higher risk premium, although almost certainly raising the long-run domestic interest rate and lowering the capital stock, may, or may not, lower long-run external debt as well. That depends upon the relative size of the country’s external debt to the net credit of its private sector. The stock of capital and debt, therefore, may, or may not, move together following this particular disturbance.

An efficiency-enhancing productivity shock, however, leads to no long-run trade-off between capital and debt in the basic model; the long-run capital stock rises, while the level of debt remains unchanged. There is, however, a short-run trade-off. The accumulation of capital along the transitional path generates an initial current account deficit and an initial increase in debt. But this is only temporary, as eventually the higher output resulting from the additional capital generates a current account surplus and the stock of debt returns to its original level. Finally, the minimal role for fiscal policy in the basic model also follows directly from these two equilibrium relationships and the fact that they are independent of government expenditure. This, however, would cease to be true if labor supply were endogenous. In that case, although the capital-labor ratio would remain unchanged in the long run, the constancy of this ratio would be achieved by proportionate increases in both capital and labor in response to an expansionary policy.

The simplicity of the model permits an explicit analysis of the dynamic interdependence between capital accumulation and debt, thereby helping to provide an intuitive understanding for the nature of the trade-offs involved between them. The analysis suggests that circumstances in which policymakers wish to enhance capital accumulation, while simultaneously reducing external debt, may require a carefully specified combination of policies. The present analysis provides a framework for undertaking this kind of analysis using more realistic models. Inevitably, such models will cease to be analytically tractable and will need to be studied using numerical simulation methods. But such an extension should provide important insights into the appropriate policies for improving the growth performance in developing economies.

APPENDIX Solution of Dynamic Systems

This Appendix provides the solutions for the dynamic equation systems presented in Sections II and III.

Dynamic Properties with Balanced Budget

Under the assumption of a balanced budget, the dynamic structure of the system (20a)–(20e) is a fourth-order system, which may be expressed in linearized form about the steady-state equilibrium

[λ˙q˙k˙z˙] =[0000i −fkk0xII0 −fk −λ(2i −ib)ii +iz][λ −λ˜q −q˜kz − −k˜z˜],(24)

where the elements appearing in the matrix are evaluated at steady state. The dynamic properties of the economy depend upon the eigenvalues of the characteristic equation of (24); namely

μ(μ −i)μ(μ −i) −[izμ +xa14] =μ(μ −i) +fkkIIiφ(μ),(25)

where the element a14 ≡ –λ(2i’ – i”b).

Assuming 2i’ – i”b > 0, the following properties can be established:

  • The product of the four roots is positive, implying that there are either zero, two, or four positive roots

  • The sum of the roots is positive, ruling out the case of zero positive roots

  • The coefficient of μ in equation (25) is negative, ruling out the possibility of all roots being positive.

We are therefore left with the case of two positive and two negative roots, which may be ordered as follows:


The dynamics are therefore a saddle point, with the stock variables k and z evolving gradually over time, and the shadow prices λ, q being allowed to undergo instantaneous jumps in response to new information.

The quantities φ(μ1), φ(μ2), as defined in equation (25), are both critical parts of the solution. By direct evaluation of the characteristic equation, one can establish that


where μ1, μ2, are ordered as above.

The analysis is focused on stable adjustment paths beginning from given initial capital stock k0 and stock of national debt z0. The solutions for k, z, λ, and q along such paths are

k =k˜ +[φ(μ2)dk˜ −dz˜]φ(μ1) −φ(μ2)eμ1t +[dz˜ −φ(μ1)dk˜]φ(μ1) −φ(μ2)eμ2t(28)
z =z˜ +φ(μ1)[φ(μ2)dk˜ −dz˜]φ(μ1) −φ(μ2)eμ1t +φ(μ2)[dz˜ −φ(μ1)dk˜]φ(μ1) −φ(μ2)eμ2t(29)
λ =λ˜ +a14φ(μ1)μ1[φ(μ2)dk˜ −dz˜φ(μ1) −φ(μ2)]eμ1t +a14φ(μ2)μ2[dz˜ −φ(μ1)dk˜φ(μ1) −φ(μ2)]eμ2t(30)
q =q˜ +μ1I[φ(μ2)dk˜ −dz˜φ(μ1) −φ(μ2)]eμ1t +μ2I[dz˜ −φ(μ1)dk˜φ(μ1) −φ(μ2)]eμ2t,(31)

where d

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k0, dz = z – z0 denote the long-run changes in k and z from their respective initial starting points.

One can eliminate eμ1t, eμ2t to define the two loci

λ −λ˜ = −a14φ(μ1) −φ(μ2)[φ(μ2)μ2 −φ(μ1)μ1](z −z˜)a14φ(μ1)φ(μ2)φ(μ1) −φ(μ2)[1μ1 −1μ2](k −k˜)(32)
q −q =μ1 −μ2I[φ(μ1) −φ(μ2)](z −z˜) +μ2φ(μ1) −μ1φ(μ2)I[φ(μ1) −φ(μ2)](k −k˜).(33)

These are both three-dimensional planes relating the respective instantaneous shadow prices to the slowly evolving dynamic variables k and z. They are the analogues to the usual two-dimensional stable adjustment paths associated with saddle points.

The solutions reported in equations (28)–(31) form the basis for the analysis of the short-run dynamics in response to the various disturbances presented in Section III. The different shocks identified differ simply in terms of how they affect the long-run equilibrium stock of capital and national debt.

Dynamic Properties with Debt-Financed Deficit

When the government’s budget is permitted to be out of balance through the use of debt financing, the dynamics of the economy are governed by a fifth-order system, which may be stated in linearized form as follows:

[λ˙q˙k˙za˙˙] =[000λ(2i −i"b)λiiifkki0xx0II00fk00(i +iz)a¯i00i][λ −λ˜q −q˜kza − − −k˜za˜˜](34)

This system may be shown to have two stable roots (μ1, μ2 < 0) and three unstable roots μ3, μ4, μ5 > 0). Analogous to equations (28)–(31), the stable solution, starting from initial stocks k0,z0, is

k =k˜ +A1eμ1t +A2eμ2t(35)
z =z˜ +φ(μ1)A1eμ1t +φ(μ2)A2eμ2t(36)
λ =λ˜ +iμ1[a14i +λia˜μ1 −i]φ(μ1)A1eμ1t +iμ2[a14i +λiaμ2 −i]φ(μ2)A2eμ2t(37)
q =q˜ +μ1IA1eμ1t +μ2IA2eμ2t(38)
a =a˜ +a˜iμ1 −iφ(μ1)A1eμ1t +a˜iμ2 −iφ(μ2)A1eμ2t,(39)

where φ (μi) and a14 are defined as previously, and, for notational convenience, A1 and A2 are given by

A1 =φ(μ2)dk˜ −dz˜φ(μ1) −φ(μ2),A2 =dz˜ −φ(μ1)dk˜φ(μ1) −φ(μ2).(40)

The assumption that the stock of government debt evolves continuously from the initial level ao can be shown to imply the relationships

Ψ1(z(t) −z0) +Ψ2(k(t) −k0) +Ψ3(a(t) −a0) =0(41)
Ψ1(z˜ −z0) +Ψ2(k˜ −k0) +Ψ3(a˜ −a0) =0(41)


Ψ1 ≡ a˜i[μ1μ2 −Ifkk −i(μ1 +μ2 −i)]>0(42)
Ψ2 ≡ a˜i2Iφ(μ1)φ(μ2)<0(43)
Ψ3 ≡  −(μ1 −i)(μ2 −i)(μ1 +μ2 −i)>0.(44)

The significance of equation (41) is that the dynamics of government debt a becomes tied to that of k and z. There are in fact only two linearly independent unstable roots. As a further consequence, equation (41’) imposes an additional constraint on the steady-state equilibrium, one which involves the initial points k0, z0, and a0.


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Jagdeep S. Bhandari is an economist in the European Department and is currently on leave of absence from West Virginia University where he is a Professor of Economics. He holds a Ph.D. from Southern Methodist University. He is also an attorney and counselor-at-law, having obtained a J.D. degree from Duquesne University and an LL.M. degree from Georgetown University.

Nadeem Ul Haque, an economist in the Developing Country Studies Division of the Research Department, holds degrees from the London School of Economics and Political Science and the University of Chicago.

Stephen J. Turnovsky is a Professor of Economics at the University of Washington and a Research Associate at the National Bureau of Economic Research. He holds a Ph.D. from Harvard University.

The authors are grateful to Mohsin Khan for useful comments on an earlier draft.


See, for example, Buiter (1987), Brock (1988), Frenkel and Razin (1987), Matsuyama (1987), Obstfeld (1982, 1989), and Sen and Turnovsky (1989a, 1989b).


An early exception is Bardhan (1967) who analyzes optimal borrowing by a small open economy that faces an upward sloping supply curve for debt of the form to be introduced in this paper. Reference should also be made to a recent paper by Otani and Villanueva (1988), which analyzes the accumulation of capital and external debt in a neoclassical economy and also assumes an imperfect market for loans. However, that paper adopts a very different approach and emphasizes different issues (that is, the role of human capital formation) from those addressed in the present paper.


For example, Buiter (1987), Brock (1988), and Obstfeld (1989) consider different fiscal shocks, including different types of disturbances in government expenditure and some forms of taxes; Matsuyama (1987) analyzes input price shocks; and Sen and Turnovsky (1989a, 1989b) discuss various types of disturbances in terms of trade and tariffs.


However, the stock of foreign debt does play an indirect role in determining the dynamics of the capital stock through the intertemporal national budget constraint. For a more detailed discussion of the dynamic structure with a perfect world capital market, see Sen and Turnovsky (1989a, 1989b).


See, for example, Eaton and Gersovitz (1980, 1981), Sachs (1984), and Cooper and Sachs (1985). See Edwards (1984) for an empirical investigation of the risk premium.


Note that since the model is real, no prices or nominal variables need be considered.


Even though in some countries the private sector has borrowed abroad, implicit or explicit government guarantees have essentially underwritten this debt, making private debt indistinguishable from government debt, insofar as the foreign creditor is concerned.


This enables us to incorporate the Eaton and Gersovitz (1980, 1981) argument.


Otani and Villanueva (1988) assume that the risk premium is a positive function of the country’s debt-to-export ratio.


For simplicity, labor is assumed to be fixed. Since in a developing country context the endogeneity of labor is not likely to be a critical issue, this assumption is not viewed as being particularly restrictive.


If b > 0, the consumers are creditors, whereas if b < 0, then they are debtors. Examination of the budget constraint shows that if consumers are creditors, then acquisition of increasingly costly debt by the government adds to disposable income, and vice versa.


Subscripts and primes (‘) denote derivatives.


This formulation assumes that the representative agent, in choosing his or her holding of debt, takes account of his or her decision on the aggregate debt of the economy and therefore on the prevailing domestic interest rate. This is possible even if the number of such agents—say, n—is large. Suppose aggregate debt holdings b = Σbi, where bi = b/n is the holding of each representative agent; the optimality condition for each such agent is

β −λ˙λ =i −ib/n.

As long as n < ∞, this condition is of the form represented in equation (6), with the number of agents n being simply absorbed in the coefficient ix. We are grateful to Ed Buffie for drawing this point to our attention.


Note that the cost of debt depends on whether the consumer is a net creditor or a net debtor. In the former case, the marginal cost exceeds the interest rate; in the latter case, the opposite is true.


Note that this specification implies that in the case where disinvestment may occur, C(I) < 0 for low rates of disinvestment. This may be interpreted as reflecting the revenue obtained as capital is sold off. The possibility that all changes in capital are costly can be incorporated by introducing sufficiently large fixed costs, so that C(0) > 0. This does not alter our analysis in any substantive way.


This is ensured by an appropriate adjustment in q at each point in time.


Hereafter, the labor variable will be suppressed for convenience.


See Sen and Turnovsky (1989a) for an example of such a model.


In Section III below, we shall also discuss a form of debt financing, which, in order to be sustainable in the long run, needs to be accompanied by a once-and-for-all change in lump-sum taxes.


The condition 2i’ – i”b > 0 given in Table 1 is essentially a stability condition; see Appendix.


Assuming a convex debt function of the form i = i0 + i1 zα, α > 1, this will be so if the ratio b/a > 1/α.


These results may be usefully compared with the long-run effects of an increase in the foreign interest rate under the limiting assumption where the debt-supply function is horizontal. In such a case, the domestic interest rate rises by the same amount as does the foreign interest rate, leading to a larger fall in the domestic capital stock than in the present case. The stock of external debt can be shown to decline by an amount that is proportional to the reduction in the capital stock, with the resulting effect on the long-run trade balance being ambiguous, depending upon the stock of external debt.


For example, for the constant-elasticity convex debt function i = i0 + i1zα, α ≥ 1, the quantity ω – ω’b = zα–1[z – αb]. The criterion determining whether external debt falls or rises with i1 depends upon whether the ratio of the country’s external debt to the net credit of its private sector is greater or less than a; that is, whether z/b ≶ α.


For this case, the production function is changed to f(k, θ).


For obvious reasons we restrict our discussion to the plausible case where the increase in the foreign interest rate leads to an increase in the long-run domestic interest rate and a corresponding decline in the long-run capital stock. The perverse case, where the long-run interest rate declines can be analyzed similarly but is of little practical interest.


This contrasts with the dynamics under the limiting case of uncovered interest parity when the paths followed by z and k can both be shown to be mono-tonic, following a permanent shock; see Sen and Turnovsky (1989a, 1989b).


As discussed in the Appendix, the dynamic structure of the balanced-budget variant involves four differential equations. With debt finance, the resulting dynamics is of the fifth order; the additional source of dynamics is the evolution of the stock of government debt ȧ.


The technical details are discussed in the Appendix.


For expositional convenience we restrict our discussion to the case where the debt schedule is linear.


The impact of changes in z0-that is, changes in the stock of debt held by the country—can be used to study the dynamics of the effects of debt forgiveness schemes. In this model, interest relief schemes are equivalent to negative foreign interest rate shocks.


These alternative specifications would transform equation (1) to

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IMF Staff papers, Volume 37 No. 2
Author: International Monetary Fund. Research Dept.