Empirical evidence shows an inverted U-shaped relationship between public debt-to-GDP ratio and economic growth for many advanced economies. Using a simple endogenous growth model with public debt under the Golden Rule of Public Finance (GRPF), which allows the government to issue bonds only to finance public investment, this paper explains the relationship. Although Greiner [1] explains it in the similar model, he introduces a more restrictive assumption than GRPF that the amount of public investment must be always equal to that of newly issued bonds, i.e., public investment must be financed only by newly issued bonds. This paper shows that the assumption is not needed. In other words, the inverted U-shaped relationship emerges in a more realistic case when public investment is partly financed by other sources than government bonds such as taxes.
For many advanced economies such as the United States, Japan, and European countries, the accumulated public debt has been one of the biggest concerns. It not only increases the risk for the fiscal sustainability but also decreases the freedom of fiscal policies. If a high debt-GDP ratio itself has bad impacts on the economy, especially on the growth performance, then the economy has difficulty of getting out of the high debt-GDP ratio trap once it has accumulated a certain level of debt. Therefore, the relationship between public debt and economic growth is of crucial importance in exploring the future of the economy.
Quite naturally, therefore, many theoretical and empirical studies have analyzed the economic effects of public debt and fiscal deficits on growth performance. Reinhart and Rogoff [
Checherita-Westphal and Rother [
In sharp contrast to the aforementioned empirical results, many of theoretical studies at an early stage suggest that relationship between public debt and economic growth is monotonically negative. For example, Saint-Paul [
growth effects in the endogenous growth literature1.
In analyzing the long-run relationship between public debt and economic growth, the fiscal sustainability must be considered. One of the most plausible ways to ensure the sustainability is the introduction of the Golden Rule of Public Finance (GRPF), under which public bonds can be exclusively used for the productive purposes, i.e., public investment. The amount of newly issued bonds cannot therefore exceed that of public investment. Checherita-Westphal et al. [
Greiner [
In this paper, therefore, we reexamine the conditions under which an inverted U-shaped relationship between debt-GDP ratio and economic growth emerges in an endogenous growth model. To be more concrete, we consider the more realistic case in which public investment is financed not only by newly issued bonds but also by other sources, i.e., a part of public investment is financed by taxes. As a result, it is shown that the assumption employed in Greiner [
The rest of paper is organized as follows. Section 2 sets up the endogenous growth model with public debt and productive public capital. Section 3 analyzes the model to derive the main propositions. Section 4 shows findings in the analysis. Section 5 concludes the paper.
Since the framework of the model is basically the same as Greiner [
The number of households is constant over time and normalized to unity. Each household maximizes the discounted sum of instantaneous utilities3:
max c ( t ) ∫ 0 ∞ e − ρ t ln C ( t ) d t , (1)
subject to the following flow budget constraint:
K ˙ ( t ) + B ˙ ( t ) = r ( t ) B ( t ) + ( 1 − τ ) Y ( t ) − C ( t ) , (2)
where ρ is the time preference rate, C ( t ) is consumption, K ( t ) is private capital, B ( t ) is public debt, r ( t ) is the interest rate, τ ∈ ( 0 , 1 ) is a constant tax rate on output, and Y ( t ) is output4. Since we assume that the population is constant at unity through time, each variable represents its per-capita value.
The production function is given by
Y ( t ) = A K ( t ) 1 − α G ( t ) α , (3)
where G ( t ) is public capital, A is a technology parameter, and α ∈ ( 0 , 1 ) is the elasticity of output with respect to public capital. In equilibrium, the interest rate is equal to the marginal product of private capital net of taxes:
r ( t ) = ( 1 − τ ) ( 1 − α ) A K ( t ) − α G ( t ) α . (4)
The government levies taxes on output and issues bonds in order to finance the expenditures. To simply the analysis, suppose that public capital does not depreciate. Then the government flow budget constraint is
B ˙ ( t ) = r ( t ) B ( t ) − τ Y ( t ) + G ˙ ( t ) . (5)
The government must also satisfy the following intertemporal budget constraint:
lim e − r t B ( t ) = 0 .
Let us introduce the following fiscal rule, i.e., the rule on the bond issue:
B ˙ ( t ) = ψ G ˙ ( t ) with ψ ∈ ( 0 , 1 ) . (6)
Equation (6) means that the amount of newly issued bonds B ˙ ( t ) cannot exceed that of public investment G ˙ ( t ) . Putting it differently, the government can issue bonds only to finance public investment, i.e., B ˙ ( t ) = 0 when G ˙ ( t ) = 0 . This rule, which restricts the bond issue to productive purposes of public investment, is referred to as the “Golden Rule of Public Finance (GRPF)”.
For the government to follow GRPF, ψ must be less than or equal to one. However, it needs not to be necessarily equal to one. To simplify the analysis, Greiner [
Solving the optimization of the household, the consumption growth is given by the following equation:
g C ( t ) = C ˙ ( t ) C ( t ) = − ρ + ( 1 − τ ) ( 1 − α ) A ( G ( t ) K ( t ) ) α . (7)
Combining the household’s budget constraint (2) and government’s budget constraint (5), the growth of private capital becomes
g K ( t ) = K ˙ ( t ) K ( t ) = A ( G ( t ) K ( t ) ) α − C ( t ) K ( t ) − ( G ˙ ( t ) G ( t ) ) ( G ( t ) K ( t ) ) . (8)
In use of equation (5) and (6), we obtain the growth of public debt and public capital as follows5:
g B ( t ) = B ˙ ( t ) B ( t ) = ψ ( G ( t ) K ( t ) ) ( K ( t ) B ( t ) ) ( G ˙ ( t ) G ( t ) ) , (9)
g G ( t ) = G ˙ ( t ) G ( t ) = ( 1 − ψ ) − 1 ( τ A ( G ( t ) K ( t ) ) α − 1 − ( 1 − τ ) ( 1 − α ) A ( G ( t ) K ( t ) ) α − 1 ⋅ B ( t ) K ( t ) ) . (10)
Turning to the steady state of the economy in which all variables grow at the same constant rate g ∗ , and hence the ratios of two variables are constant through time, the Equations (7) to (10) become as follows:
g C = C ˙ C = g ∗ = − ρ + ( 1 − τ ) ( 1 − α ) A z ∗ α , (11)
g K = K ˙ K = g ∗ = A z ∗ α − c ∗ − g ∗ z ∗ , (12)
g B = B ˙ B = g ∗ = ψ z ∗ b ∗ g ∗ , (13)
g G = G ˙ G = g ∗ = ( 1 − ψ ) − 1 ( τ − ( 1 − τ ) ( 1 − α ) b ∗ ) A z ∗ α − 1 , (14)
where each lowercase letter represents the per-capital value of the corresponding uppercase letter, i.e., c ≡ C / K , z ≡ G / K , and b ≡ B / K .
Finally, the following condition must be imposed for the non-degenerate steady state6:
ψ ≤ τ [ ( 1 − α ) ( 1 − τ ) ] 1 − α α ( A ρ ) 1 α .
Equation (13) implies
ψ z * / b * = 1 or b ∗ = ψ z * . (15)
Substitution of Equation (15) into Equation (14) gives
g ∗ = ( 1 − ψ ) − 1 ( τ − ( 1 − τ ) ( 1 − α ) ψ z ∗ ) A z ∗ α − 1 . (16)
From Equations (11) and (16), we have
− ρ + ( 1 − τ ) ( 1 − α ) A z ∗ α = ( 1 − ψ ) − 1 ( τ − ( 1 − τ ) ( 1 − α ) ψ z ∗ ) A z ∗ α − 1 .
This equation can be rewritten as
( LHS = ) ( 1 − τ ) ( 1 − α ) A z ∗ α = τ A z ∗ α − 1 + ( 1 − ψ ) ρ ( = RHS ) . (17)
The LHS and RHS of Equation (17) can be depicted as in
Taking advantage of
d z ∗ d τ = [ ( 1 − α ) z ∗ + 1 ] z ∗ ( 1 − α ) [ ( 1 − τ ) α z ∗ + τ ] > 0 . (18)
Differentiating the growth rate in Equation (11) with respect to τ , we obtain:
d g ∗ d τ = − ( 1 − α ) A z ∗ α + ( 1 − τ ) ( 1 − α ) A z ∗ α − 1 d z ∗ d τ .
Substitution of Equation (18) into the above gives,
d g ∗ d τ = ( α − τ ) A z ∗ α ( 1 − τ ) α z ∗ + τ .
Hence, the relationship between the tax rate and growth rate is given by:
s i g n ( d g ∗ d τ ) = s i g n ( α − τ ) . (19)
Equation (19) shows that the growth maximizing tax rate is equal to the elasticity of output with respect to public capital α . Also, the long-run growth rate g ∗ increases with tax rate τ when τ is below α , and vice versa. These observations lead to the following proposition.
Proposition 1
Under the Golden Rule of Public Finance there exists a unique tax rate that maximizes the long-run growth rate. In other words, the relationship between tax rate and growth is inverted U-shaped in the steady state.
The above proposition is the same as in Barro [
Taking the conditions held in the steady state into account, we can rewrite the production function (3) as follows:
d ∗ = ( ψ α / A ) b ∗ 1 − α ,
where d ∗ = ( B / Y ) * is the debt-GDP ratio in the steady state. Using Equation (15), the above equation can be further rewritten as:
d ∗ = ( ψ / A ) z ∗ 1 − α . (20)
As equation (18) shows, z ∗ increases with τ . In other words, other things being equal, z ∗ moves in the same direction as τ . Since, as Proposition 1 states, the relationship between τ and g ∗ is inverted U-shaped, the relationship between z ∗ and g ∗ is also inverted U-shaped. Now we come to the following proposition.
Proposition 2
Under the Golden Rule of Public Finance the relationship between debt-GDP ratio and the long-run growth is inverted U-shaped regardless of the share of public investment financed by government bonds.
The analysis of Subsection 3.1 - 3.2 implies that the relationship between tax rate and growth is inverted U-shaped in the steady state. From Subsection 3.3, which shows debt-GDP ratio increases with tax rate, we derive the relationship between debt-GDP ratio and growth is inverted U-shaped.
It should be noted that Proposition 1 holds regardless of the value of ψ , although, as Appendix shows, it surely affects the long-run growth. So does Proposition 2. Greiner [
One of the key mechanisms to derive the inverted U-shaped relationship is that the long-run debt-GDP ratio d ∗ increases with tax rate τ . This seems counterintuitive because a tax rate hike increases the government revenues and hence seems to reduce the amount of newly issued bonds. Looking only at the instantaneous budget constraint, it is the fact in the short-run that an increase in the government revenues reduces newly issued bonds. The government, however, follows the intertemporal or long-run budget constraint. A tax rate hike implies increases not only in current revenues bud also in the future revenues. In other words, the government can increase public debt because it will be able to bear the increased burden thanks to the tax revenue increases in the future. As a result, the debt-GDP ratio in the steady state increases with tax rate under GRPF8.
This paper investigates an endogenous growth model with public debt as well as with private and public capital to analyse the long-run relationship between debt-to-GDP ratio and economic growth. As a result, it is shown that the inverted U-shaped relationship emerges under the Golden Rule of Public Finance (GRPF), which allows the government to issue bonds only to finance public investment, i.e., the amount of newly issued bonds must be less than or equal to the amount of public investment.
The inverted U-shaped relationship, which is consistent with empirical evidence, has already been explained by Greiner [
GRPF itself might be a restrict assumption because the government issues new bonds to finance other expenditures than public investment such as social security expenses. In reality, public debt has been increasing mainly due to an increase in social security expenses caused by low birth rates and aging in many advanced economies. It deserves future research to analyse the relationship taking the important aspects into account.
The authors would like to thank Kenichiro Ikeshita, Toshiyuki Kawai, Takeshi Koba, and participants at the 2015 Fall Meeting of the Japanese Association of Applied Economics, and an anonymous referee for their valuable comments. The second author gratefully acknowledges the financial support of Grants-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science (15K03431). The usual disclaimer applies.
The authors declare no conflicts of interest regarding the publication of this paper.
Ueshina, M. and Nakamura, T. (2019) An Inverted U-Shaped Relationship between Public Debt and Economic Growth under the Golden Rule of Public Finance. Theoretical Economics Letters, 9, 1792-1803. https://doi.org/10.4236/tel.2019.96114
Equation (16) can be rewritten as
( 1 − ψ ) g ∗ = [ τ − ( 1 − τ ) ( 1 − α ) ψ z ∗ ] A z ∗ α − 1 . (A.1)
Substitution for z ∗ from Equation (11) into Equation (A.1) implies
( 1 − ψ ) g ∗ = τ A [ g ∗ + ρ ( 1 − τ ) ( 1 − α ) A ] 1 − α α − ψ ( g ∗ + ρ ) .
After some manipulation, the above equation can be rewritten in the following form:
( LH S ′ = ) g ∗ + ψ ρ = τ A [ ( 1 − τ ) ( 1 − α ) A g ∗ + ρ ] 1 − α α ( = RH S ′ ) . (A.2)
The LHS curve and the RHS curve are depicted in
As
τ A [ ( 1 − τ ) ( 1 − α ) A g ∗ + ρ ] 1 − α α ≥ ψ ρ .
The above inequality can be rewritten as
ψ ≤ τ [ ( 1 − α ) ( 1 − τ ) ] 1 − α α ( A / ρ ) 1 α ( = ψ ¯ ) . (A.3)
The long-run growth is negative if Inequality (A.3) is not met. In order to examine the plausibility, let us present the numerical example in which it is assumed that ρ = 0.05 , A = 0.15 , α = 0.25 10. As
τ | ψ ¯ | g ∗ × 100 |
---|---|---|
0.05 | 1.46 | 0.50% |
0.1 | 2.49 | 1.28% |
0.2 | 3.50 | 1.84% |
(0.25) | (3.60) | (1.89%) |
0.3 | 3.52 | 1.85% |
0.4 | 2.95 | 1.55% |
0.5 | 2.14 | 1.04% |
0.6 | 1.31 | 0.35% |
0.7 | −0.65 | -0.52% |
Note: Numbers in parenthesis are those under maximum growth & ψ ¯ ≡ τ [ ( 1 − α ) ( 1 − τ ) ] ( 1 − α ) / α ( A / ρ ) 1 α .