1. Introduction

External public debt can have nonlinear impacts on economic growth. Thus, at low levels of indebtedness, an increase in the proportion of external public debt to GDP could promote economic growth; however, at high levels of indebtedness, an increase in this proportion could hurt economic growth. This article studies the non-monotonic relationship between external public debt and economic growth via a model of endogenous growth.

The theory of economic growth examines the relationship between external debt and growth using some contributions from international finance. Thus, ^{Krugman (1989)} shows the debt relief Laffer curve (with the shape of an inverted U), where the nominal value of debt of a country and its actual expected payment are related. On the upward segment of the curve, debt and expected payments increase because the risk of default is low; in the descending segment, the level of debt increases but expected payments begin to descend because the risk of default is very high. He concludes that when a country is on the descending segment of the curve, the country suffers from debtoverhang.¹ In this situation of debtoverhang, external debt obligations act as a tax on investment.

Using the above concepts, researchers have studied the effects of external over-indebtedness. Thus, ^{Cohen (1993)} extends the model of endogenous growth of ^{Cohen and Sachs (1986)} to formalize the relationship between external-indebtedness and investment.² Consequently, Cohen presents and compares three economic scenarios. In the first one, there is free access to the global financial market, and the rate of investment (and production) is greater than with financial autarky. In the second scenario, there is a credit restriction with soft repayment, and the investment rate is lower than the free access case but higher than the financial autarky case. In the third one, there is a credit restriction with forced repayment, and the investment rate is lower than with financial autarky. Therefore, the investment rate rises in the first scenario, before falling in the second and third scenarios. Thus, the relationship between external debt and investment (and growth) is non-linear. In addition, Cohen concludes that the third scenario would correspond to the concept of debt overhang, where external debt acts as a tax on investment, hurting economic growth.

Moreover, ^{Saint-Paul (1992)} shows an endogenous growth model with overlapping generations, where an increase in public debt reduces the growth rate of the economy. ^{Adam and Bevan (2005)} develop a model of endogenous growth with individuals who live two periods. They study various ways to finance public deficits. An increase in domestic public debt slows growth, while an increase in external public debt, financed in concessional terms, but rationed, helps growth. ^{Aizenman, Kletzer and Pinto (2007)} show a model of endogenous growth with restrictions in tax revenues and public debt. In general, they find that the higher the public debt the lower the growth. Finally, ^{Checherita-Westphal, Hallett and Rother (2014)} present a growth model with public capital and debt, where the public deficit is equal to public investment. In their model, the relationship between debt and growth is nonlinear. So, in the steady state, the optimal debt to GDP ratio can be determined where growth is maximized.

In order to study the relationship between external public debt and economic growth, this article presents an endogenous growth model for a small open economy.

The economy produces two goods, tradable (manufacturing) and non-tradable (non-manufacturing). The tradable sector produces domestic technological knowledge through learning by doing (^{Romer, 1989}). This knowledge is used in the non-tradable good sector. Therefore, in this model there are two learning externalities.³ The government taxes households with a lump-sum tax to finance spending on tradable goods and interest payments on its external debt, and the difference between these expenditures and tax revenue, if any, is covered by external public debt. The foreign lenders perceive a country risk that depends positively on the level of external public debt. More-over, the government collects taxes through another lump-sum tax on households to finance the purchase of non-tradable goods. Households consume a constant fraction of their disposable income, and own the two types of capital. They can borrow abroad, subject to a foreign credit constraint. In this article, country risk is fully transferred to the private sector.⁴ Thus, interest rate parity adjusted by country-risk is assumed, and as a result, the interest rate on the two types of capital, external private debt and public debt is equal to the world interest rate plus the country risk premium.

I study how the economy responds, in the steady state, to an increase in the proportion of external public debt to GDP and I obtain a nonlinear relationship between the ratio of external public debt to GDP and the growth rate. That is, my results show an inverted U-shaped curve connecting external public debt and economic growth. This nonlinearity is the result of two opposite effects on the growth rate of the economy when the proportion of external public debt to GDP increases. The positive effect is as follows: when the proportion of external public debt to GDP increases, the relative price of the non-tradable good decreases (the real exchange rate depreciates) and the tradable sector, leader in technological terms, attracts resources. Therefore, the proportion of labor employed in the manufacturing sector increases and the ratio of non-tradable to tradable capital diminishes, increasing the growth rate of the economy. The negative effect is as follows: when the proportion of external public debt to GDP increases, the country risk premium increases and interest payments on total external debt increases. Therefore, household disposable income falls, the proportion of savings to GDP declines and resources for capital accumulation decrease, thus the growth rate of the economy decreases.

At a high external public debt to GDP ratio, the economic growth that is stimulated by the depreciation of the real exchange rate and the attraction of resources towards the tradable sector, is offset by the exit of resources to the exterior (due to the burden of external debt), and the consequent decrease in the savings to GDP ratio.

The results of this paper are related to ^{Cohen (1993)} and ^{Checherita-Westphal, Hallett and Rother (2014)}, where nonlinear relationships between debt and growth are also presented, although in their models, the external debt affects economic growth through different channels than those presented here. The result that public spending on tradable goods leads to a depreciation of the real exchange rate, stimulating the tradable sector, is related, inversely, with ^{Korinek and Serven (2010)}. They affirm that the accumulation of international reserves (the extension of credit to foreigners for the purchase of domestic tradable goods) leads to a depreciation of the real exchange rate, stimulating the tradable sector and triggering the desired learning effects.

The results I obtained for an economy with endogenous growth, with two goods, two learning externalities, where the external public debt acts positively and negatively on economic growth, are not present in the literature and contribute to a better understanding of the relationship between external public debt and economic growth. Moreover, the existence of a maximum level of external public debt means that those responsible for public finances should be prudent in handling external public debt, to avoid high debt levels and prevent the kind of situations that occurred in Latin America in the 1980s and in the European periphery in recent years.

In the empirical literature, there is evidence showing the existence of this non-linearity between public debt and growth, for both developing and developed countries. Thus, ^{Pattillo, Poirson and Ricci (2002}, ²⁰¹¹⁾ study the contribution of the proportion of external debt to GDP to the growth of per capita GDP for 93 developing countries between the years 1969-1998. They find that the contribution of external debt (current net value) on growth is nonlinear, in the form of an inverted U. The critical point, where the contribution of external debt to growth becomes negative, is between 35 and 40% of GDP. The negative impact of the high level of external debt on growth operates through adverse effects on the formation of physical capital and total factor productivity (see ^{Pattillo, Poirson and Ricci 2004}). Analyzing 55 low-income countries between the years 1970 to 1999, ^{Clements, Bhattacharya and Nguyen (2003)} argue that the servicing of external debt negatively affects public investment and, indirectly, growth.

Recently, ^{Reinhart and Rogoff (2010)} argue that the relationship between public debt to GDP ratio and growth for advanced and emerging countries is weak at levels of public debt to GDP ratio lower than 90%, but the relationship is negative for ratios greater than 90%. Similarly, ^{Caner, Grennes and Koehler-Geib (2010)} determine the critical level, where an increase in average public debt ratio to GDP decreases the average annual growth for developed and developing countries between the years 1980-2008. They conclude that for the total sample of countries, the threshold stands at 77.1% of GDP. For developing economies, the critical level is at 64% of GDP. Also, ^{Checherita-Westphal and Rother (2012)} show that the relation between public debt to GDP ratio and growth in per capita income has an inverted U shape, for a sample of 12 countries in the euro area, with data since 1970. The threshold is between 90-100% of GDP, and could even begin at levels of 70-80% of GDP. The main channels through which public debt affects the rate of growth are private saving, public investment and total factor productivity. However, in a detailed review of the empirical literature, ^{Panizza and Presbitero (2013)} argue that the non-linear relationship between public debt and growth, with a threshold of 90% of GDP, is not robust across samples, specifications and estimation techniques.

It is important to mention that in the empirical literature there is also evidence of a relationship which is always negative between debt and growth; for example, ^{Kumar and Woo (2010)} show an inverse relationship between initial public debt and growth of per capita GDP for advanced and developing economies for the period 1970-2007.

The paper is organized as follows. In section 2, I develop an endogenous growth model of a small open economy. In section 3, I redefine the model in stationary variables. In section 4, I demonstrate the existence and stability of the steady state, as well as the nonlinear relationship between external public debt and growth. I present my conclusions in section 5.

2. The economy

In this model, the economy is small, so the world market determines the price of the tradable good and the world interest rate. Moreover, there is a country risk that depends positively on the external public debt. The tradable and non-tradable goods are produced using physical capital, labor and domestic technological knowledge. For simplicity, the tradable sector is the only one that generates domestic technological knowledge through learning by doing. Knowledge overflows to the non-tradable sector. Capital is specific in both sectors. The representative firm, in both the tradable and the non-tradable sector, maximizes profits taking the externality as given. The government imposes two lump sum taxes on households, one to finance spending on tradable goods and interest payments on the external public debt, and the other to finance the purchase of non-tradable goods. The government borrows from the rest of the world to purchase tradable goods. The representative household consumes a constant fraction of its disposable income. Households can borrow abroad, and have an external credit constraint. The total supply of labor is constant, and there is free mobility of labor between the two productive sectors.

3. The model in stationary variables

Given that the variables K _T and K _N show a constant, common rate of growth, it is necessary to define the model variables as stationary variables, that is, variables that are constant in the steady state. Thus, z = K _N /K _T is defined as a stationary variable. Furthermore, given that L is constant, it is normalized to one (L = 1). Thus, the equilibrium condition in the labor market is: n + (1 − n) = 1, where n is the fraction of labor employed in the tradable sector and (1 − n) is the fraction of the labor employed in the non-tradable sector. As n is constant in the steady state, the variable n is also stationary. Similarly, as the relative price of the non-tradable good must be constant in the steady state, p _N is another stationary variable.

Taking into consideration the externalities E ₁ and E ₂ , the production functions in stationary variables are:

Also, since D _G = θ _G Y _T , then d = D _G /K _T = θ _G A _T n ^1−α . Using the Equation (2), the rate of interest must be:

The marginal conditions for the tradable sector in stationary variables are:

The first order conditions for the non-tradable sector in stationary variables are:

I assume that α > β, so the tradable sector is more capital intensive than the non-tradable sector. ^{Turnovsky (1997)} shows that the dynamic in dependent economies changes when one sector is more capital intensive than the other.

The static condition of efficient allocation of labor between the two sectors is obtained by equating (24) and (26):

This condition says that the value of the marginal product of labor in both sectors must be equal at all times. With Equation (28), the level of p _N is:

Then, the growth rates of K _T and K _N are obtained in stationary variables. Substituting I _T =K˙T, I _N = K˙N, and Ḋ _H =θHK˙T into Equation (15), and dividing by K _T , one obtains:

Now, I determine K˙N/K _N in function of K˙T/KT. Taking logarithms and time derivatives of z = K _N /K _T , one obtains:

As will be apparent later, n is always in a steady state and is constant. Taking logarithms and time derivatives of Equation (29), I obtain:

Equating (31) and (32), I have:

Substituting Equation (33) into Equation (30), I have:

Substituting T _T and T _N , Equations (9) and (10), with >Y˙T=ATn1-αK˙T in the above equation, I obtain:

Finally, substituting D _H = θ _H K _T and the production functions, Equations (21) and (22), in Equation (35) and solving for K˙T/KT, the growth rate of K _T is obtained:

where r is defined by Equation (23). Similarly, I obtain the growth rate of K _N in stationary variables. Substituting I _T = K˙T, I _N = K˙N, and Ḋ _H =θ _H K˙T in to Equation (15), dividing by K _N , using K˙T KT, Equation (33), substituting T _T , Equation (9) and T _N , Equation (10), and using the production functions, Equations (21) and (22), I obtain:

As n is always in a steady state and is constant, it is possible to show that the rate of growth of national income, Y = Y _T + p _N Y _N - r (θ _H K _T +θ _G A _T K _T n ^1−α ), is:

where p _N Y _N /Y is the participation of p _N Y _N in the national income. In the next section, the steady state solution is shown.

4. The steady state and the relationship between external public debt and growth

The steady state solution implies the existence of the equilibrium. Therefore, the growth rates of z, n and p _N must be zero in the steady state, so their levels remain constant. Furthermore, the growth rates of K _T , K _N , Y _T , Y _N and Y must be equal to a constant rate in the steady state.

Therefore, with Equations (23) and (25), I have:

Given that r ^w , η, θ _G , A _T and α are constant, the level of n ^∗ is constant in the steady state (steady state levels are denoted with *). It can be shown that ∂n ^∗/∂θ _G > 0; that is, an increase in θ _G increases n ^∗. Also, given an increase in θ _G , the level of n ^∗ will jump immediately to the new steady state level and, thus, n ^∗ will always be in a steady state, as mentioned above. With Equation (23), the interest rate in the steady state is r ^∗ = r ^w + ηθ _G A _T n ^∗(1−α) .

Using Equation (27), with p˙N = 0 and r ^∗ = A _T αn ^∗(1−α) , I obtain:

As z ^* depends on parameters, the level of z ^∗ is constant in the steady state. It can be shown that ∂z ^∗/∂θ _G = (∂z ^∗ /∂n ^∗)(∂n ^∗/∂θ _G ) < 0; that is, an increase in θ _G decreases z ^∗.

Using Equation (29), and substituting the level of z ^∗, the relative price of the non-tradable good, p _N , in the steady state is:

As pN*depends on parameters, the level of pN* is constant in the steady state. Given that α > β, it is possible to demonstrate that ∂pN*/∂θ _G = (∂pN*/∂n ^∗) (∂n ^∗/∂θ _G ) < 0; that is to say that an increase in θ _G , causes pN* to decrease. Thus, for the stationary variables, the existence of the steady state has been analytically shown.

As in the steady state ṗ _N = 0, the growth rate of K _T in the steady state, Equation (36), equals the growth rate of K _N in the steady state, Equation (37). Similarly, as in the steady state ż = 0 and ṗ _N = 0, the rate of growth of national income, Equation (38), is equal to K˙T/K _T . Also, the growth rate of Y _T and Y _N is equal to K˙T/K _T . Therefore, the steady state growth rate of the economy, g ^∗, is:

Given that g ^∗ depends only on parameters, the level of g ^∗ is constant in the steady state.

In order to study the relationship between public debt and growth, I define the proportions of external public debt to GDP and savings to GDP in the steady state. Since the values of n ^∗, z ^∗ and pN* are known, the ratio of external public debt to GDP in the steady state, DG, GDP *, is

As DG, GDP * depends on parameters, its level is constant in the steady state. Moreover, since g ^∗ is known, the proportion of savings to GDP in the steady state, SH, GDP *, is:

As SH, GDP* depends on parameters, its level is constant in the steady state. Thus for g ^∗, DG, GDP* and SH, GDP*, the existence of the steady state has also been analytically demonstrated.

Next, I numerically analyze, in the steady state, the relationship between the proportion of external public debt to GDP and the rate of growth of the economy. The values of the parameters α and β are taken from ^{Valentinyi and Herrendorf (2008)} where, for the US economy, they show that the tradable sector is more capital intensive, α = 0.37, than the non-tradable sector, β = 0.32. Given that there is only net investment, the level of s is the rate of net national saving for OECD economies of 8% relative to national income, s = 0.08 (1980-2012 average, calculated using the world development indicators of the World Bank). I calculated the levels of public expenditure, the final consumption expenditure of the general government sector of OECD economies, to be 18% of their GDP (1980-2013 average, calculated using the world development indicators of the World Bank). Given the lack of disaggregated data for expenditure on final consumption of the general government sector in tradable and non-tradable goods, I opted for φ _T = 0.10 and φ _N =0.18. The level of r ^w =0.031is the average level of the 10-year Treasury Bonds in the United States (2007-2013 average). As the levels of θ _H , η, A _T and A _N depend on the unique characteristics of each economy, they are set only for explanatory purposes.

Therefore, I present a representative simulation, where the parameter values are: α = 0.37, β = 0.32, s = 0.08, φ _T = 0.18, φ _N =0.18, r ^w =0.031,θ _H =0.1, η =0.09, A _T =1.5 and A _N =0.3. In the simulation, I used increasing levels of θ _G and DG, GDP* to clearly illustrate the effect of the increasing external public debt on economic growth. Thus, in Figure 1, the relationship between the proportion of external public debt to GDP and the growth rate in the steady state is shown. As shown in Figure 1, there is a nonlinear relationship between the external public debt to GDP ratio and the growth rate. That is to say, there is an inverted U-shaped curve between the proportion of external public debt to GDP and growth. A large number of simulations were carried out and the result holds. This nonlinearity is explained by two opposite effects on the rate of growth of the economy when θ _G and DG, GDP* increase. The positive effect is as follows: when θ _G andDG, GDP* increase, pN* decreases; that is to say, the real exchange rate depreciates. Therefore, the tradable sector employs more labor and accumulates relatively more capital than the non-tradable sector; n ^∗ increases and z ^∗ decreases. Given that the tradable sector is the technological leader, the growth rate of the economy is benefited. The negative effect is as follows: when θ _G and DG, GDP* increase, the country risk premium increases, and interest payments on private and public debt increase. Thus, national income and the savings to GDP ratio decrease, and the resources for capital accumulation decrease. Therefore, the growth rate of the economy is damaged.

At point A, in Figure 1, θ _G = 0.0001 and DG, GDP* = 0%. The levels of the stationary variables are: n ^∗ = 0.0102, z ^∗ = 5.217 and pN* = 14.812. The growth rate in the steady state is 0.63% per annum. The country risk premium is zero and r ^∗ is 3.1%. The level of savings to GDP ratio is 6.55%.

Point B, in Figure 1, is where the maximum rate of growth is reached by increasing the external public debt to GDP ratio. The level of θ _G is 3.5 and the levels of the stationary variables are: n ^∗ = 0.2642, z ^∗ = 0.191 and pN* = 11.665. Now, the level of DG, GDP* is 100% of the GDP and the growth rate is 3.59% per annum. If this case is compared with the previous one, it can be seen that the stationary variables move in the direction analytically predicted. In particular, pN* decreases from 14.812 to 11.665 (the real exchange rate depreciates), so the tradable sector attracts resources and that n ^∗ increases from 0.0102 to 0.2642 and z ^∗ = K _N /K _T declines from 5.217 to 0.191. Given that the tradable sector is the sector that generates technical progress, this flow of resources to the tradable sector stimulates economic growth (positive effect). However, the negative effect is that the country risk premium increases 2089 basic points, and r ^∗ is 23.99%. Thus, the payment of interest on external private and public debt causes the ratio of savings to GDP to decrease from 6.55 to 4.84%. It is important to note that at country risk levels of 2089 basic points, the probability of default is very high and the process of accumulation of external debt could stop abruptly, although this mechanism is not present in this model.

Point C, in Figure 1, is where the increase in the external public debt to GDP ratio produces the same rate of growth as the case without external public debt, g ^∗ = 0.63%. The corresponding level of θ _G is 3.7638 and the levels of the stationary variables are: n ^∗ = 0.5185, z ^∗ = 0.067 and pN* = 11.101. In this case the level of DG, GDP* is 202.3% of GDP and the growth rate is 0.63% per annum. If this case is compared with the case θ _G = 3.5, it is observed that n ^∗, z ^∗ and pN* move in the direction predicted analytically. In particular, pN* decreases from 11.665 to 11.101 (the real exchange rate depreciates even further). Therefore, the tradable sector attracts more resources, n ^∗ increases from 0.2642 to 0.5185, and z ^∗ = K _N /K _T decreases from 0.191 to 0.067. This flow of resources to the tradable sector stimulates economic growth (positive effect). However, the country risk premium is 3359 basic points and r ^∗ is 36.6%. Thus the interest payments on private and public external debt reduce disposable income and the savings to GDP ratio goes from 4.84 to 0.56% (negative effect). Finally, when the level of DG, GDP* is 211% of GDP, the growth rate is zero.

In sum, an inverted U-shaped curve has been presented here, relating the external public debt with economic growth. In the upward segment of the curve, an increase in the external public debt to GDP ratio increases growth. However, in the downward segment of the curve, an increase in the external public debt to GDP ratio decreases growth. Depending on the levels of θ _H , η, A _T and A _N (whose values depend on specific factors in each country), the maximum level of DG, GDP* may increase or decrease.

In order to study the stability and the transitional dynamics, Equations (32) and (27) are used, yielding:

ż + (1/β) (r ^w + ηθ _G A _T n ^1−α ) z = A _N (1−n) ^1−β z ^β

The above equation is a Bernoulli equation, which can be solved by a change of variable v = z ^1−β and calculating v˙= dv/dz dz/dt, we obtain:

where 1-β/β rw+θGATn1-α is a positive constant. Also, I find that 1-βAN1-n1-β is a positive constant. Thus, when θ _G increases, z slowly decreases to its new lower steady state level. Using Equation (29), one can see that at t = 0, p _N decreases instantaneously when n jumps, while z stays at the same level. Using Equation (32), and since ż/z < 0, it can be seen that ṗ _N /p _N > 0. By all of the above, at t = 0, it follows that p _N decreases and overshoots its new steady state level. Thus, in the transition, p _N rises slowly to its new steady state level.

5. Conclusions

A model of endogenous growth was developed with two productive sectors, where the tradable sector is the only one that generates domestic technical progress. The knowledge generated in the tradable sector is used in the non-tradable sector. The tradable sector and the non-tradable sector can accumulate physical capital. The government spends on tradable goods and on interest payments on its external debt. This expenditure is financed by a lump sum tax levied on households, and by external public debt. The government also spends on non-tradable goods, which are financed by a lump sum tax. I assume that the country risk premium increases with the level of external public debt, and that households borrow from abroad, and face a restriction on foreign credit.

I demonstrate analytically that an increase in the external public debt to GDP ratio has a positive impact on the tradable sector by reducing the relative price of the non-tradable good. Thus, with the depreciation of the real exchange rate, the fraction of labor employed in the tradable sector increases and the proportion of non-tradable capital to tradable capital decreases The relationship between external public debt and economic growth is shown to have an inverted U-shape. Two opposite effects on the growth rate of the economy explain this nonlinearity between the external public debt to GDP ratio and growth. The positive effect is that, when the external public debt increases, the relative price of the non-tradable good decreases, so the tradable sector attracts resources, and since the tradable sector is the technological leader, the growth rate benefits. The negative effect is that, when the external public debt increases, the country risk premium increases, and interest payments on the private and public debt increase. Thus, the household disposable income and the savings to GDP ratio decrease, and the resources for capital accumulation are reduced; consequently, the growth rate is damaged.

Thus, it has been shown that at low levels of indebtedness, an increase in the external debt to GDP ratio could promote growth; however, with high levels of indebtedness, an increase in the external debt to GDP ratio could hurt economic growth. This theoretical result resembles certain empirical results that have demonstrated a nonlinear relationship between debt and growth.

Furthermore, the inverted U-shaped relationship between external debt and economic growth indicates the existence of a maximum level of external debt, which the policy makers should avoid reaching, in order to prevent situations such as Latin America in the eighties and the European periphery in recent years.