1. The structure of the economy

The economy we shall consider comprises three sectors: (i) consumers; (ii) firms; and (iii) 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. ^1/ International borrowing by the government is allowed, while consumers may borrow from or lend to the government, which is largely consistent with practice in borrowing countries. ^2/

Unlike most macro-dynamic models of the type previously considered, our model incorporates sovereign risk via the introduction of an upward-sloping debt supply function. Eaton and Gersovitz (1981) have shown that because of the moral hazard associated with sovereign risk, lenders will charge a risk premium on lending to countries, and could, on occasion, also apply lending limits. ^3/ This risk premium may vary directly with the stock of debt that the country holds since the probability of default is likely to increase with increased costs of servicing the debt; however, this could be offset if the costs associated with default also increase with debt accumulation. Sachs (1984), Sachs and Cohen (1982), and Cooper and Sachs (1985) have shown that a country, by adopting apparent growth-oriented policies, as well as policies that enhance its foreign exchange earning capacity, can shift the upward sloping debt supply curve outward, so that at each level of debt a lower risk premium is charged. Thus, for example, larger investment efforts or the adoption of efficiency-enhancing structural adjustment packages will result in a lower evaluation of country risk by lenders. The main conclusion of this literature is that the level of indebtedness of the country is likely to determine both the extent of additional credit that is likely to be made available, as well as the terms on which it will be made available. This proposition is best formalized by an upward sloping supply curve.

To incorporate this feature of international borrowing, we assume that the interest rate, i(z), charged on foreign debt, z, is:

where i_o is the interest rate prevailing internationally and i₁ω(z) is the country specific risk premium which varies directly with the stock of debt, z, held by the country. The possibility of a cutoff in debt is captured by the assumption that the function 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 it becomes prohibitive, so that an effective lending limit is reached. ^1/ With this specification, an exogenous increase in foreign interest rates is captured by an increase in i_o, while an exogenous shift in country-specific risk can be represented by a shift in the parameter i₁. ^2/

Equation (1) results from the combination of an interest parity relationship, i = i_o + $\stackrel{•}{\text{e}}$ and an equation relating to the determination of the relevant foreign interest rate, i.e., i_o = i_w + ρ, when i_w is the world interest rate and ρ is the risk premium attached to the home country. Recalling that $\stackrel{•}{\text{e}}$ = 0 (since this is a fixed exchange rate model) and assuming a specific form for the risk premium, i.e., ρ = i₁ω(z), yields equation (1). ^3/ Interest parity (given fixed exchange rates) requires that the domestic interest rate be equalized with the foreign interest rate. Hence expression (1) above will also represent the interest rate prevailing at home, at which domestic consumers can borrow or lend to the government.

2. Macroeconomic equilibrium

Summing the consumers budget constraint (3), the firm’s profit relation (8), and the government budget constraint (17), and noting that $\stackrel{•}{\text{z}}=\stackrel{•}{\text{a}}-\stackrel{•}{\text{b}}$, the rate of decumulation of foreign debt, – $\stackrel{•}{\text{z}}$, is equal to the current account balance. ^2/

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 consumption, λ,

and investment (I = $\stackrel{•}{\text{k}}$) 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 consumption must remain constant, at say, $\overline{\lambda }$. 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 (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. ^1/

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 consumption, λ, is not 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, which is analytically tractable, we shall focus first on the balanced-budget policy

where lump sum taxes T are continually adjusted to finance expenditures. ^2/ Using this model we analyze the long-run and short-run effects of a number of disturbances, focusing on the tradeoffs involved between the rate of capital accumulation, on the one hand, and the accumulation of foreign debt, on the other.

1. The long run

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, $\overline{\text{a}}$.

The steady state of the economy is reached when all variables cease to change, i.e., $\stackrel{•}{\lambda }=\stackrel{•}{\text{q}}=\stackrel{•}{\text{k}}=\stackrel{•}{\text{z}}=0$. With no depreciation of capital, steady-state investment is zero, which in turn implies that the shadow price of investment, (q) is equal to 1. Denoting the steady-state values by tildes, the long run can be described by:

These equations jointly determine the steady-state values of the marginal utility $\tilde{\lambda }$, consumption $\tilde{x}$, capital stock $\tilde{k}$, national debt $\tilde{z}$, and lump sum taxes $\tilde{T}$.

The equilibrium defined by (22a) - (22e) has a particularly simple recursive structure. The equilibrium level of the foreign debt $\tilde{z}$ 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 $\tilde{z}$, the domestic rate of interest is determined 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 $\tilde{x}$. Finally, given $\tilde{x}$, the equilibrium marginal utility of consumption $\tilde{\lambda }$ is determined by (22a).

Table 1 summarizes the long-run effects of shifts in the exogenous variables namely: the foreign interest rate (i_o), 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. ^1/

Table 1.

Long-Run Effects

D ≡ 2i' – i"$\tilde{\text{b}}$ > 0

Table 1.

Long-Run Effects

		Increase in
		io	i1	θ	g
Foreign debt	$\tilde{\text{z}}$	$-\frac{1}{\text{D}}$	$\frac{{\omega }^{\prime }\tilde{\text{b}}-\omega }{\text{D}}$	0	0
Capital stock	$\tilde{\text{k}}$	$-\frac{({\text{i}}^{\prime }-\text{i}"\tilde{\text{b}})}{{\text{f}}_{\text{kk}}\text{D}}$	$\frac{{\text{i}}^{\prime }({\omega }^{\prime }\tilde{\text{b}}+\omega )-\text{i}"\omega \tilde{\text{b}}}{{\text{f}}_{\text{kk}}\text{D}}$	$\frac{{\text{f}}_{\text{k}}\theta }{{\text{f}}_{\text{kk}}}$	0
Interest rate	$\tilde{\text{i}}$	$\frac{\text{i}\text{'}-\text{i}"\tilde{\text{b}}}{\text{D}}$	$\frac{\text{i}\text{'}(\omega \text{'}\tilde{\text{b}}+\omega )-\text{i}"\omega \tilde{\text{b}}}{\text{D}}$	0	0
Trade balance		$-\frac{[\text{i}-\tilde{\text{z}}(\text{i}\text{'}-\text{i}"\tilde{\text{b}})]}{\text{D}}$	$\frac{\text{i}\text{d}\tilde{\text{z}}}{{\text{di}}_1}+\frac{\text{zd}\tilde{\text{i}}}{{\text{di}}_1}$	0	0
Consumption	$\tilde{\text{x}}$	$\frac{{\text{f}}_{\text{k}}(\text{i}\text{'}-\text{i}"\tilde{\text{b}})}{{\text{f}}_{\text{kk}}\text{D}}$	$[\frac{\text{i}}{{\text{f}}_{\text{kk}}}-\tilde{\text{z}}]\frac{\text{d}\tilde{\text{i}}}{{\text{di}}_1}$	${\text{f}}_{\theta }-\frac{{\text{f}}_{\text{k}}{\text{f}}_{\text{k}\theta }}{{\text{f}}_{\text{kk}}}$	-1
		$+\frac{\text{i}-\tilde{\text{z}}(\text{i}\text{'}-\text{i}"\tilde{\text{b}})}{\text{D}}$	$-\frac{\text{i}\text{d}\tilde{\text{z}}}{\text{d}{\text{i}}_1}$
Lump sum taxes	$\tilde{\text{T}}$	$\frac{\overline{\text{a}}(\text{i}\text{'}-\text{i}"\tilde{\text{b}})}{\text{D}}$	$\overline{\text{a}}\frac{\text{d}\tilde{\text{i}}}{{\text{di}}_1}$	0	1

D ≡ 2i' – i"$\tilde{\text{b}}$ > 0

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Table 1.

Long-Run Effects

		Increase in
		io	i1	θ	g
Foreign debt	$\tilde{\text{z}}$	$-\frac{1}{\text{D}}$	$\frac{{\omega }^{\prime }\tilde{\text{b}}-\omega }{\text{D}}$	0	0
Capital stock	$\tilde{\text{k}}$	$-\frac{({\text{i}}^{\prime }-\text{i}"\tilde{\text{b}})}{{\text{f}}_{\text{kk}}\text{D}}$	$\frac{{\text{i}}^{\prime }({\omega }^{\prime }\tilde{\text{b}}+\omega )-\text{i}"\omega \tilde{\text{b}}}{{\text{f}}_{\text{kk}}\text{D}}$	$\frac{{\text{f}}_{\text{k}}\theta }{{\text{f}}_{\text{kk}}}$	0
Interest rate	$\tilde{\text{i}}$	$\frac{\text{i}\text{'}-\text{i}"\tilde{\text{b}}}{\text{D}}$	$\frac{\text{i}\text{'}(\omega \text{'}\tilde{\text{b}}+\omega )-\text{i}"\omega \tilde{\text{b}}}{\text{D}}$	0	0
Trade balance		$-\frac{[\text{i}-\tilde{\text{z}}(\text{i}\text{'}-\text{i}"\tilde{\text{b}})]}{\text{D}}$	$\frac{\text{i}\text{d}\tilde{\text{z}}}{{\text{di}}_1}+\frac{\text{zd}\tilde{\text{i}}}{{\text{di}}_1}$	0	0
Consumption	$\tilde{\text{x}}$	$\frac{{\text{f}}_{\text{k}}(\text{i}\text{'}-\text{i}"\tilde{\text{b}})}{{\text{f}}_{\text{kk}}\text{D}}$	$[\frac{\text{i}}{{\text{f}}_{\text{kk}}}-\tilde{\text{z}}]\frac{\text{d}\tilde{\text{i}}}{{\text{di}}_1}$	${\text{f}}_{\theta }-\frac{{\text{f}}_{\text{k}}{\text{f}}_{\text{k}\theta }}{{\text{f}}_{\text{kk}}}$	-1
		$+\frac{\text{i}-\tilde{\text{z}}(\text{i}\text{'}-\text{i}"\tilde{\text{b}})}{\text{D}}$	$-\frac{\text{i}\text{d}\tilde{\text{z}}}{\text{d}{\text{i}}_1}$
Lump sum taxes	$\tilde{\text{T}}$	$\frac{\overline{\text{a}}(\text{i}\text{'}-\text{i}"\tilde{\text{b}})}{\text{D}}$	$\overline{\text{a}}\frac{\text{d}\tilde{\text{i}}}{{\text{di}}_1}$	0	1

D ≡ 2i' – i"$\tilde{\text{b}}$ > 0

An increase in the level of international interest rates, represented by an upward shift in i_o 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 i_o, and maintain equality with the latter. This is brought about by a reduction in the level of external debt, $\tilde{\text{z}}$. 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 ($\tilde{\text{b}}$ = 0), then a rise in the more costly foreign debt does indeed result in a rise in the domestic interest rate, though not by the full amount. ^1/ If i" > 0 and domestic private residents are net debtors ($\tilde{\text{b}}$ < 0), the upward pressure on the domestic interest rate is increased. On the other hand, if i" > 0 and domestic private residents are large creditors ($\tilde{\text{b}}$ > 0), the response of the domestic interest rate is reduced, and indeed it may now fall. ^2/

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, while the financing requirement of the government rises. As noted, in the long run, the trade surplus must finance the interest costs i($\tilde{\text{z}}$)$\tilde{\text{z}}$ of the foreign debt. While 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 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, while 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. ^3/ 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.

Turning to the impact of an increase in the risk premium i₁, we see that this will raise or lower the marginal return or cost to consumers from holding additional debt, according to whether ω—ω′${\tilde{\text{b}}}_{<}^{>}0$. In the case where domestic consumers are net debtors ($\tilde{\text{b}}$ < 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 (i.e., increase $\tilde{\text{b}}$) 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 $\tilde{\text{b}}$. In the case where $\tilde{\text{b}}$ is sufficiently small so that ω—ω′$\tilde{\text{b}}$ > 0, external debt will still fall. However, if $\tilde{\text{b}}$ is sufficiently large to reverse this inequality, a higher risk premium will lead to a higher equilibrium stock of external debt. ^1/

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 i_o. 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. Focusing on 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.

Column 3 of Table 1 summarizes the effect of a productivity shock, θ, which is introduced as a shift operator in the production function. ^2/ 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 f_kθ, 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 i.e., ${{\text{f}}_{\theta }}_{<}^{>}0,$ and whether or not capital stock increases.

The last column of Table 1 summarizes the impact of a change in government expenditures. Since, in this model all government expenditures are merely for consumption purposes, and the government budget is balanced, a change in government expenditure affects only consumption by the private sector. Any increase (decrease) in government expenditures has to be matched fully by an increase (decrease) in taxation and hence a commensurate decrease (increase) in private consumption.

2. Transitional dynamics

The transitional dynamic behavior of the economy is determined by the system of differential equations, (14a) - (14g). 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, impact 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 that we will consider are all unanticipated permanent shocks. Table 2 summarizes the qualitative short run effects of the various shocks.

Table 2.

Qualitative Short-Run Effects

Table 2.

Qualitative Short-Run Effects

		Increase in
		io	i1	θ	g
Debt accumulation	$\stackrel{•}{\text{z}}\left(0\right)$	–	$<0\text{if}\frac{\text{d}\tilde{\text{z}}}{{\text{di}}_1}<0$	+	0
			$?\text{if}\frac{\text{d}\tilde{\text{z}}}{{\text{di}}_1}>0$
Investment	$\stackrel{•}{\text{k}}\left(0\right)$	–	–	+	0
Interest rate	i(0)	+(–1)	$+\omega \left(\tilde{\text{z}}\right)$	0	0
	$\begin{array}{l}\text{di}\left(0\right)\\ \overline{\text{d}}\text{t}\end{array}$	–	?	+	0
Trade balance		+	$<0\text{if}\frac{\text{d}\tilde{\text{z}}}{{\text{di}}_1}<0$	–	0
			$?\text{i}\text{f}\frac{\text{d}\tilde{\text{z}}}{{\text{di}}_1}>0$
Consumption	x(0)	?	?	?	–(= –1)
Lump sum taxes	T(0)	+(=ā)	+	0	–(= –1)

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Table 2.

Qualitative Short-Run Effects

		Increase in
		io	i1	θ	g
Debt accumulation	$\stackrel{•}{\text{z}}\left(0\right)$	–	$<0\text{if}\frac{\text{d}\tilde{\text{z}}}{{\text{di}}_1}<0$	+	0
			$?\text{if}\frac{\text{d}\tilde{\text{z}}}{{\text{di}}_1}>0$
Investment	$\stackrel{•}{\text{k}}\left(0\right)$	–	–	+	0
Interest rate	i(0)	+(–1)	$+\omega \left(\tilde{\text{z}}\right)$	0	0
	$\begin{array}{l}\text{di}\left(0\right)\\ \overline{\text{d}}\text{t}\end{array}$	–	?	+	0
Trade balance		+	$<0\text{if}\frac{\text{d}\tilde{\text{z}}}{{\text{di}}_1}<0$	–	0
			$?\text{i}\text{f}\frac{\text{d}\tilde{\text{z}}}{{\text{di}}_1}>0$
Consumption	x(0)	?	?	?	–(= –1)
Lump sum taxes	T(0)	+(=ā)	+	0	–(= –1)

a. Increase in the foreign interest rate

The previous section showed that an increase in i_o 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. ^1/ The rate of capital accumulation, therefore slows down as a result of this shock. With the stock of foreign debt fixed instantaneously, a one percentage point increase in the foreign interest rate leads to a corresponding immediate one 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 i_o on initial consumption is unclear. Equation (5) shows that the consumption varies inversely with the marginal utility of consumption, 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 depends upon the long-run marginal utility, $\tilde{\lambda }$, which may either increase or decrease, as previously discussed. Secondly, since we know that the long-run capital stock declines, capital will decumulate over the transition path. This would tend to increase consumption over the transitional path. 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 in 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 and this may occur at the expense of consumption.

The impact effect of the increase in external interest rates on the trade balance tends to increase 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).

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 i_o 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 (A5) and (A6) in the Appendix, for k and z in the form

$\begin{array}{c}\text{k}=\tilde{\text{k}}+{\text{A}}_1{\text{e}}^{{\mu }_1\text{t}}+{\text{A}}_2{\text{e}}^{{\mu }_2\text{t}}\hfill \\ \text{z}=\tilde{\text{z}}+\varphi \left({\mu }_1\right){\text{A}}_1{\text{e}}^{{\mu }_1\text{t}}+\varphi \left({\mu }_2\right){\text{A}}_2{\text{e}}^{{\mu }_2\text{t}}\hfill \end{array}$

we have

$\frac{\text{z}-\tilde{\text{z}}}{\text{k}-\tilde{\text{k}}}=\frac{\varphi \left({\mu }_1\right){\text{A}}_1{\text{e}}^{({\mu }_1-{\mu }_2)\text{t}}+\varphi \left({\mu }_2\right){\text{A}}_2}{{\text{A}}_1{\text{e}}^{({\mu }_1-{\mu }_2)\text{t}}+\varphi \left({\mu }_2\right)}$

where ϕ(μ₁) > 0, ϕ(μ₂) < 0 are defined in the Appendix. Since μ₂ is the dominant stable root (i.e., 0 > μ₂ > μ₁), it follows that as t→∞ (z - $\tilde{\text{z}}$)/(k - $\tilde{\text{k}}$) → 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 the upward shift in the cost of debt, i_o, 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 being at A, having lower stocks of both physical capital and foreign debt. In the limit, this is approached along the locus XX having the negative slope ϕ(μ₂). Initially, the declining capital stock and external debt causes the system to move in a southwesterly direction from 0. 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 the 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. ^1/

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Transitional Path in Response to Upward Shift in Cost of Debt

Citation: IMF Working Papers 1989, 054; 10.5089/9781451969474.001.A001

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c. 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 f_kθ > 0, i.e., the productivity shock impacts positively 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 consumption, $\tilde{\lambda }$, is now lower. Instantaneously λ(0) will likely fall, though by a smaller amount than in the long-run. Consequently, though 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, leads 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 pre-shock levels.

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Transitional Path in Response to Productivity Shock

Citation: IMF Working Papers 1989, 054; 10.5089/9781451969474.001.A001

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