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.
2.1. Production of the tradable good
It is assumed that the production function of the tradable sector is Cobb-Douglas:
where Y T is the production of the tradable good, A T is a positive parameter of efficiency, K T is the stock of physical capital accumulated of the tradable good, L T is the labor employed in the sector, α and 1−α are the shares of K T and L T , respectively, with 0 < α < 1, and E 1 is a learning externality. It is assumed that K T is used only in the tradable sector.
Domestic technological knowledge is created through learning by doing in the sector. Therefore, E
1 is the external effect of K
T
in the production function of the tradable sector. In order to generate endogenous growth, it is assumed that
where η is a positive parameter that reflects country specific factors (see Eicher and Turnovsky, 1999, Eicher and Hull, 2004). Considering that the rate of depreciation of K
T
is zero, we obtain
Equation (3) establishes that the wage is equal to the value of the marginal product of labor in the tradable good sector. Equation (4) states that the rental price of K T is equal to the marginal product of K T .
2.2. Production of the non-tradable good
With respect to the non-tradable sector, the production function is Cobb-Douglas:
where Y N is the production of the non-tradable good, A N is a positive parameter of efficiency, K N is the stock of physical capital accumulated from the non-tradable good, L N is the labor employed in the sector, β and 1 − β are the shares of K N and L N , respectively, with 0 < β < 1, and E 2 is a learning externality. The stock of K N is used only in the non-tradable sector.
As the knowledge generated in the tradable sector is a public good, there is a spillover effect of knowledge between sectors. Thus, E
2 is the contribution of domestic technological knowledge in the production of the non-tradable good. Additionally, in order to have constant returns for a broad measure of capital in the sector, it is assumed that
The variable p N is defined as the relative price of the non-tradable to the tradable good. Considering that the rate of depreciation of K N is zero, the rental price of K N is R N = p N (r−ṗ N /p N ), where ṗ N /p N is the growth rate of p N , or the capital gains of K N . Non-tradable firms maximize profits taking the externality as given. The first order conditions are:
Equation (6) states that the wage equals the value of the marginal product of labor in the non-tradable sector. Equation (7) is the dynamic equilibrium condition for K N . Thus, the equation says that the rental price of K N is equal to the value of the marginal product of K N .
In models with tradable and non-tradable goods, the real exchange rate is defined as the level of relative prices of non-tradable goods in the foreign country in physical terms divided by the level of relative prices of non-tradable goods in the domestic country in physical terms. Considering that the level of relative prices of the foreign country is constant, the real exchange rate is inversely related to the level of relative prices of non-tradable goods in the domestic country in physical terms. Therefore, an increase in p N corresponds to an appreciation of the real exchange rate.
2.3. Government
Regarding the tradable goods, the government spends a certain sum on consumption and interest payment on its external debt. Its spending is financed by a lump sum tax levied on households and by foreign loans. Consequently, the government budget constraint on the tradable good is:
where D G is the external public debt, Ḋ G is the increase in public debt over time, rD G is interest payment on the public debt, G T is spending on consumption on the tradable good, and T T is a lump sum tax. The level of external public debt is measured as a constant fraction, θ G , of Y T , that is, D G = θ G Y T , where θ G > 0 and Ḋ G = θ G Ẏ T . Furthermore, it is assumed that G T = φ T Y T ; that is, public spending on tradable goods is a constant fraction, φ T , of the product of the tradable sector, where 0 < φ T < 1. Given that the inter-temporal budget constraint of the government, deducible from Equation (8), is met by the appropriate adjustment of some residual fiscal variable (see Serven, 2007), I assume that the level of T T is adjusted residually.5 Considering the prior definitions given, I find that:
Regarding the non-tradable goods, the government has an expenditure on consumption and this consumption is financed by a lump sum tax charged to households. Therefore, the government budget constraint in the non-tradable good is T N = p N G N , where T N is a lump sum tax and p N G N is consumption spending in the non-tradable good. I assume that p N G N = φ N p N Y N ; that is, public spending on non-tradable goods is a constant fraction, φ N , of the product of the non-tradable sector, where 0 < φ N < 1. Considering the above definitions, we have:
Therefore, the government only borrows from the rest of the world for the purchase of tradable goods.
2.4. Households
Households own K T and K N , and foreigners own the external debt of the households. The household budget constraint is:
where
I assume that only a constant and exogenous fraction, θ
H
, of K
T
can be used as collateral for loans in the world market, with 0 < θ
H
< 1. Therefore, the borrowing constraint is D
H
= θ
H
K
T
. Thus, domestic residents own the entire stock of K
T
, which is partially funded by the world market, and external residents own the debt of K
T
(see Barro, Mankiw and Sala-i-Martin, 1995). Moreover, given that D
H
=θ
H
K
T
, one obtains
Next, I deduce the consumption demands for the tradable and non-tradable goods. I assume that consumption demands result from the maximization of the utility function
For simplicity, I assume that households choose the level of aggregate consumption as a constant fraction of disposable income, w T L T +w N L N +R T K T +R N K N −T T −T N −rD H (there is no possibility for inter-temporal choice, which is a limitation of the model). Consequently, I obtain:
where s is the savings rate and (1 − s) is the consumption rate, s is constant and exogenous, with 0 < s < 1. Note that since T T , represented by Equation (9), is a residual tax, the disposable income of the households is decreased by the interest payments on the external public debt and by public spending on the tradable good, but increased by Ḋ G , along with public spending on the non-tradable good, represented by Equation (10).
2.5. Equilibrium
First, I deduce the aggregate condition of savings being equal to investment. Substituting w T , w N , R T and R N , Equations (3), (4), (6) and (7), in Equation (11), I obtain the resource constraint of the economy:
where Y T + p N Y N − T T − T N − rD H is equivalent to household disposable income. Thus, the aggregated consumption of households is:
Substituting Equation (14) in (13), with C = C T + p N C N , one obtains:
Equation (15) says that household savings plus the foreign credit extended to households serve to finance capital accumulation.
Next, I obtain the equilibrium conditions of the market of the tradable and non-tradable goods. Since the relative price of the non-tradable good is flexible, the supply of the non-tradable good is always equal to its demand. Therefore, the equilibrium condition for the market for the non-tradable good is:
where p N G N = T N . In order to obtain the equilibrium condition for the market of the tradable good, Equation (16) is substituted into (13), yielding:
Substituting the government budget constraint, Equation (8), in the above equation results in:
Considering that D = D H + D G , where D is the total external debt, and that Ḋ= Ḋ H + Ḋ G , the current account is defined as:
where NX is the trade balance. Finally, substituting (19) into (18), one obtains:
The previous equation shows the equilibrium condition for the market of the tradable good. Regarding the labor market, I assume that the total labor supply, L, is constant. The equilibrium condition in the labor market is L=L T +L N .
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
=
Now, I determine
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 >
Finally, substituting D
H
= θ
H
K
T
and the production functions, Equations (21) and (22), in Equation (35) and solving for
where r is defined by Equation (23). Similarly, I obtain the growth rate of K
N
in stationary variables. Substituting I
T
=
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
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
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
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
As
As
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
At point A, in Figure 1, θ
G
= 0.0001 and
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
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
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
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
where
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.