Sustainable Public Debt: Theory and Empirical Evidence

Abstract

Modern research on sustainability of debt policies that applies statistical tests has started with the contribution by Hamilton and Flavin (1986) who analyzed whether the series of public debt in the USA contains a bubble term. Since then a great many papers have been written that try to answer the question of whether given debt policies can be considered as sustainable. The interest in that question is in part due to the fact that the latter question is not only of academic interest but that it has practical relevance, too. Hence, if tests reach the conclusion that given debt policies cannot be considered as sustainable governments should undertake corrective actions.

Appendix

2.1.1 Proof of Proposition 1

To prove that proposition we note that the evolution of public debt is given by Eq. (2.3). Integrating that equation and multiplying the resulting expression by $e^{-\int _{0}^{t}r(\mu )d\mu }$ to get present values gives,⁶²

$$\displaystyle{ e^{-C_{1}(t)}B(t) = e^{-C_{3}(t)}B(0) - Y (0)e^{-C_{3}(t)}\int _{ 0}^{t}e^{-C_{1}(\mu )+C_{2}(\mu )+C_{3}(\mu )}\phi (\mu )d\mu,\; }$$

(2.18)

with

$$\displaystyle\begin{array}{rcl} & & \int _{0}^{t}r(\mu )d\mu =: C_{ 1}(t),\int _{0}^{\mu }r(\nu )d\nu =: C_{ 1}(\mu ),\int _{0}^{\mu }g(\nu )d\nu =: C_{ 2}(\mu ), {}\\ & & \qquad \int _{0}^{\mu }\psi (\nu )d\nu =: C_{ 3}(\mu ). {}\\ \end{array}$$

For $\lim _{t\rightarrow \infty }C_{3}(t) =\lim _{t\rightarrow \infty }\int _{0}^{t}\psi (\nu )d\nu = \infty$, the first term on the right hand side in (2.18), that is $e^{-C_{3}(t)}\,B(0)$, converges to zero.

The second term on the right hand side in (2.18) can be written as

$$\displaystyle{\frac{Y (0)\int _{0}^{t}e^{-C_{1}(\mu )+C_{2}(\mu )+C_{3}(\mu )}\phi (\mu )d\mu } {e^{C_{3}(t)}} =: K_{1}(t).}$$

Since | ϕ | < ∞ we can set ϕ Y (0) = 1. If $\int _{0}^{\infty }e^{-C_{1}(\mu )+C_{2}(\mu )+C_{3}(\mu )}d\mu$ remains bounded $\lim _{t\rightarrow \infty }C_{3}(t) = \infty$ guarantees that K ₁ converges to zero. If $\lim _{t\rightarrow \infty }\int _{0}^{t}e^{-C_{1}(\mu )+C_{2}(\mu )+C_{3}(\mu )}$ dμ = ∞, applying l’Hôpital gives the limit of K ₁ as

$$\displaystyle{\lim _{t\rightarrow \infty }K_{1}(t) =\lim _{t\rightarrow \infty }\frac{e^{-C_{1}(t)+C_{2}(t)}} {\psi (t)} \,,}$$

where we have set ϕ Y (0) = 1 since | ϕ(t) | is bounded. Since $-C_{1}(t) + C_{2}(t) <0$ we can find a constant k > 0 such that K ₁ ≤ e ^−kt∕ψ(t). The right hand side in the former inequality does not converge to zero if ψ(t) converged to zero exponentially. However, in that case $\lim _{t\rightarrow \infty }\int _{0}^{t}\psi (\mu )d\mu <\infty$ would hold. Consequently, in case that $\lim _{t\rightarrow \infty }\int _{0}^{t}\psi (\mu )d\mu = \infty$ holds, ψ(t) cannot decline exponentially, and K ₁(t) converges to zero.

These considerations demonstrate that the intertemporal budget constraint holds for $\lim _{t\rightarrow \infty }\int _{0}^{t}\psi (\mu )d\mu = \infty$ which means that the reaction coefficient ψ(t) is positive on average.

The debt ratio is obtained from (2.5) as

$$\displaystyle{b(t) = e^{(C_{1}(t)-C_{2}(t)-C_{3}(t))}b(0) - e^{(C_{1}(t)-C_{2}(t)-C_{3}(t))}\int _{ 0}^{t}e^{-(C_{1}(\mu )-C_{2}(\mu )-C_{3}(\mu ))}\phi (\mu )d\mu.}$$

That expression shows that the debt to GDP ratio diverges to plus or minus infinity in the case of $\int _{0}^{t}\psi (\mu )d\mu \leq \int _{0}^{t}\left (r(\mu ) - g(\mu )\right )d\mu,$ while it remains constant or converges in all other cases. □ 

2.1.1.1 Proof of Proposition 3

To prove that proposition we note that the present value of public debt is now obtained from (2.6) as

$$\displaystyle{e^{-C_{1}(t)}B(t) = B(0) - mY (0)\int _{ 0}^{t}e^{-(C_{1}(\mu )-C_{2}(\mu ))}d\mu.}$$

The intertemporal budget constraint is fulfilled for $\lim _{t\rightarrow \infty }e^{-C_{1}(t)}B(t) = 0$ which implies $b(0) = m\int _{0}^{\infty }e^{-(C_{1}(\mu )-C_{2}(\mu ))}d\mu$. If the initial debt to GDP ratio, b(0), is larger than $m\int _{0}^{\infty }e^{-(C_{1}(\mu )-C_{2}(\mu ))}d\mu$ sustainability of public debt is excluded.

The debt to GDP ratio is obtained from (2.7) as

$$\displaystyle{b(t) = e^{(C_{1}(t)-C_{2}(t))}\left (b(0) - m\int _{ 0}^{t}e^{-(C_{1}(\mu )-C_{2}(\mu ))}d\mu \right ).}$$

If the intertemporal budget constraint holds we have $b(0)\,=\,m\int _{0}^{\infty }e^{-(C_{1}(\mu )-C_{2}(\mu ))}d\mu$ giving $b(t) = -m\int _{\infty }^{t}e^{-(C_{1}(\mu )-C_{2}(\mu ))}d\mu /e^{-(C_{1}(t)-C_{2}(t))}$. Using l’Hôpital gives $\lim _{t\rightarrow \infty }b(t) = m/(r - g)$ for asymptotically constant values of r and g. □ 

2.1.1.2 Public Debt Accumulation with a Stochastic Disturbance

Assume that the evolution of public debt is described by a stochastic differential equation with an additive noise. Equation (2.3), then, can be written as,

$$\displaystyle{\mathit{dB}_{t} = \left (h(t)B_{t} -\phi (t)Y (t)\right )\mathit{dt} +\sigma _{d}\,\mathit{dW }_{t},}$$

with h(t): = r(t) −ψ(t) and W is a Wiener process with constant diffusion σ _d which is set equal to one, σ _d  = 1. Solving that equation yields

$$\displaystyle{B_{t} = e^{\int _{0}^{t}h(\tau )d\tau }\left (B_{0} -\int _{0}^{t}e^{-\int _{0}^{\tau }h(\mu )d\mu }\phi (\tau )Y (\tau )d\tau +\int _{ 0}^{t}e^{-\int _{0}^{\tau }h(\mu )d\mu }\mathit{dW }_{\tau }\right )}$$

with B ₀ public debt at time t = 0. Multiplying both sides by the discount factor $e^{-\int _{0}^{t}r(\tau )d\tau }$ and rewriting gives

$$\displaystyle\begin{array}{rcl} e^{-C_{1}(t)}B_{ t}& =\ & e^{-C_{3}(t)}B_{ 0} - Y _{0}e^{-C_{3}(t)}\int _{ 0}^{t}e^{C_{3}(\tau )}e^{C_{2}(\tau )}e^{-C_{1}(\tau )}\phi (\tau )d\tau + \\ & & e^{-C_{3}(t)}\int _{ 0}^{t}e^{C_{3}(\tau )}e^{-C_{1}(\tau )}\mathit{dW }_{\tau } {}\end{array}$$

(2.19)

with

$$\displaystyle\begin{array}{rcl} & & \int _{0}^{t}\psi (\tau )d\tau =: C_{ 3}(t),\int _{0}^{\tau }\psi (\mu )d\mu =: C_{ 3}(\tau ),\int _{0}^{\tau }g(\mu )d\mu =: C_{ 2}(\tau ), {}\\ & & \qquad \int _{0}^{\tau }r(\mu )d\mu =: C_{ 1}(\tau ), {}\\ \end{array}$$

where g gives the growth rate of Y. The first two terms are as in Eq. (2.18) above so that we do not have to consider them again.

The third term on the right hand side in (2.19) is stochastic with the expected value equal to zero. Defining the third term as $X_{t}(\omega ):= e^{-C_{3}(t)}\int _{0}^{t}e^{C_{3}(\tau )}e^{-C_{1}(\tau )}\mathit{dW }_{\tau }(\omega ),$ the second moment can be written as

$$\displaystyle{E[X_{t}^{2}(\omega )] = E\left [\left (\frac{\int _{0}^{t}e^{C_{3}(\tau )}e^{-C_{1}(\tau )}\mathit{dW }_{\tau }(\omega )} {e^{C_{3}(t)}} \right )^{2}\right ] = \left (\frac{\int _{0}^{t}E\left [e^{2C_{3}(\tau )}e^{-2C_{1}(\tau )}\right ]d\tau } {e^{2C_{3}(t)}} \right ),}$$

because $E\left [\left (\int _{0}^{t}e^{C_{3}(\tau )}e^{-C_{1}(\tau )}\mathit{dW }_{\tau }(\omega )\right )^{2}\right ] =\int _{ 0}^{t}E\left [\left (e^{C_{3}(\tau )}e^{-C_{1}(\tau )}\right )^{2}\right ]d\tau.$

Since the mean of the realized real interest rate is strictly positive, we can find a constant $\bar{r}> 0$ so that $-C_{1}(\tau ) = -\int _{0}^{\tau }r(\mu )d\mu \leq -\int _{0}^{\tau }\bar{r}\,d\mu.$ Then, we can write

$$\displaystyle{\frac{\int _{0}^{t}e^{2C_{3}(\tau )}E\left [e^{-2C_{1}(\tau )}\right ]d\tau } {e^{2C_{3}(t)}} \leq \frac{\int _{0}^{t}e^{2C_{3}(\tau )}e^{-2\,\bar{r}\,\tau }d\tau } {e^{2C_{3}(t)}} \,.}$$

If $\int _{0}^{t}e^{2C_{3}(\tau )}e^{-2\,\bar{r}\,\tau }d\tau$ remains bounded and if $\lim _{t\rightarrow \infty }C_{3}(t) = \infty$ holds, the expression converges to zero. If $\int _{0}^{t}e^{2C_{3}(\tau )}e^{-2\,\bar{r}\,\tau }d\tau$ diverges, applying l’Hôpital gives the right hand side as $e^{-2\,\bar{r}\,t}/2\psi (t)$ showing that ψ(t) must not converge to zero faster than $e^{-2\,\bar{r}\,t}$ if that term is to converge to zero asymptotically. Now, assume that ψ(t) declines exponentially. This would imply that $\lim _{t\rightarrow \infty }C_{3}(t) =\lim _{t\rightarrow \infty }\int _{0}^{t}\psi (\tau )d\tau <\infty$ holds. Consequently, if $\lim _{t\rightarrow \infty }C_{3}(t) = \infty$ holds, ψ(t) cannot decline exponentially so that the expression E[X _t ²(ω)] converges to zero. □ 

2.1.1.3 Time-Varying Reaction Coefficients and Plots of the Data Used

The following figures show plots of the variables used in the estimations as well as the smooth term sm(t) giving the deviation of the reaction coefficient from its average value (Figs. 2.41–2.46).

Fig. 2.41

Plot of variables and smooth term sm(t) for France

Fig. 2.42

Plot of variables and smooth term sm(t) for Ireland

Fig. 2.43

Plot of variables and smooth term sm(t) for Portugal

Fig. 2.44

Plot of variables and smooth term sm(t) for Spain

Fig. 2.45

Plot of variables and smooth term sm(t) for Greece

Fig. 2.46

Plot of variables and smooth term sm(t) for Italy

2.1.1.4 Additional Regressions for Spain

The subsequent tables summarize the regression outcomes for Eq. (2.15) for Spain once the sample only consists of observations until 2010 and 2011, respectively.

As these tables show, for a shorter time period the coefficient of interest, i.e. the reaction coefficient in the second line, shows a statistically significant and positive value indicating fiscal sustainability for Spain. The other significant variables possess the expected signs and the diagnostics suggest that the model is suited to replicate the data generating process. Concerning the plots of the smooth term, their shape remains similar as depicted in Fig. 2.15.