Determinants of Maximum Sustainable Government Debt

Abstract

Debt ratios around the globe differ quite radically between countries. Whereas Japan has maintained a debt–GDP ratio of more than 200 % for some years, significantly lower debt ratios in Europe have caused substantial difficulties for some countries, such as Greece. Therefore, one can suspect that the reasons for this difference in sustainable debt ratios are varying magnitudes of economic variables, such as savings rates, public expenditure shares, the output elasticity of capital, and the debt ratios of main trading partners.

Farmer and Zotti (Int Rev Econ 57:289–316, 2010) showed that debt limits exist in a two-country overlapping generations (OLG) model with trade in commodities. They discovered a negative relationship between one country’s maximum government debt limits and foreign debt ratios. However, little research has been conducted on the linkage between maximum sustainable government debt limits and other economic variables, such as saving rates, productive and unproductive public expenditures, and the output elasticity of capital. Hence, the purpose of our research is to analyze the impact of these variables on maximum sustainable government debt.

Our main findings are that higher savings rates induce an increase in debt limits and we confirm that domestic debt limits are negatively related to foreign debt ratios. Both productive and unproductive government expenditures influence the maximum sustainable debt ratio negatively and, surprisingly, to the same extent. An increase in the output elasticity of private capital decreases the limit of government debt.

Appendix

We seek the debt ratio, $\overline{b}$, for which only one steady state equilibrium exists. This debt ratio is called the maximum sustainable debt ratio as for a higher debt ratio there is no steady state equilibrium. Both, Eqs. (26) and (27) have to be fulfilled simultaneously. Through rearranging Eqs. (26) and (27), we obtain:

$$ {S}^1\left(k,\overline{b}\right)\equiv {A}^LF\left(k,\overline{b}\right)-{A}^Lk=\frac{1-\zeta }{\zeta}\frac{H}{H^{*}}{\varPhi}^{*}+\varPhi =0 $$

(29)

and

$$ {S}^2\left(k,\overline{b}\right)\equiv {F}_k^{\prime}\left(k,\overline{b}\right)-1=0. $$

(30)

Equation (30) equals:

$$ \left[\frac{\left(1-\zeta \right)}{\zeta }H\left({A}^L-\frac{\sigma^{*}{b}^{*}q{}^2}{\alpha^{*}}\right){k}^{\eta }+{H}^{*}k\left({A}^L-\frac{\sigma \overline{b}q{}^2}{\alpha}\right)\right]=\frac{\varPhi }{H}{A}^L\left(k{H}^{*}-{k}^{\eta }H\right), $$

(31)

which can be derived by calculating the partial derivatives F ^′ _k (k, b) and using Eq. (29) to simplify terms.

Equations (29) and (30) represent a system of two equations with two unknowns, k and $\overline{b}$. We are interested in the change in the maximum sustainable debt ratio if a parameter z, $z\in \left\{{b}^{*},\sigma, {\sigma}^{*},\alpha, {\alpha}^{*},g,{g}^{*},u,{u}^{*}\right\}$, changes. Thus, by totally differentiating Eqs. (29) and (30), we obtain:

$$ \left[\begin{array}{cc}\hfill \frac{\partial {S}^1}{\partial k}\hfill & \hfill \frac{\partial {S}^1}{\partial \overline{b}}\hfill \\ {}\hfill \frac{\partial {S}^2}{\partial k}\hfill & \hfill \frac{\partial {S}^2}{\partial \overline{b}}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \frac{\partial k}{\partial z}\hfill \\ {}\hfill \frac{\partial \overline{b}}{\partial z}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill -\frac{\partial {S}^1}{\partial z}\hfill \\ {}\hfill -\frac{\partial {S}^2}{\partial z}\hfill \end{array}\right]. $$

(32)

Note that $\partial {S}^1/\partial k={A}^L\left({F}_k^{\prime}-1\right)=0$ because of Eq. (30). By using this fact and Cramer’s rule, the following equation then follows from Eq. (32):

$$ \frac{\partial \overline{b}}{\partial z}=-\frac{\partial {S}^1}{\partial z}/\frac{\partial {S}^1}{\partial \overline{b}}. $$

(33)

The denominator of Eq. (33) is negative and given by:

$$ \frac{\partial {S}^1}{\partial \overline{b}}=\left(-1\right)\left(\sigma q+\left(1-\sigma \right){A}^L\right)\tilde{g}{k}^{\frac{\alpha }{1-{\alpha}^G}}<0. $$

(34)

Hence, whether the maximum sustainable debt ratio decreases or increases depends on whether $\partial {S}^1/\partial z$ is smaller or larger than zero.

First, we consider the effect on the maximum sustainable debt ratio if Foreign’s debt ratio increases. Thus, we have to calculate the partial derivative $\partial {S}^1/\partial {b}^{*}$, which is negative:

$$ \frac{\partial {S}^1}{\partial {b}^{*}}=\frac{1-\zeta }{\zeta}\frac{H}{H^{*}}\left(-1\right)\left(\left({\sigma}^{*}q+\left(1-{\sigma}^{*}\right){A}^L\right)\frac{\alpha }{\alpha^{*}}\tilde{g}{k}^{\eta \frac{\alpha }{1-{\alpha}^G}}\right)<0. $$

(35)

Next, we consider the effects of a change in the time preference of Home’s consumption: an increase in β. If β increases, σ also increases and vice versa.

$$ \frac{\partial {S}^1}{\partial \sigma }=\left(\left(1-\alpha -\left(g+u\right)\right)-b\left(q-{A}^L\right)\right)\tilde{g}{k}^{\frac{\alpha }{1-{\alpha}^G}}>0 $$

(36)

The inequality in (36) follows from the condition that the tax rate is smaller than one. The government cannot collect more money than the workers earn. The tax rate is given by Eq. (21) and can be simplified to: $\tau ={\left(1-\alpha \right)}^{-1}\left[b\left(q-{A}^L\right)+\left(g+u\right)\right]$ in the steady state equilibrium. From $\tau <1$, through rearranging terms, we obtain: $\left(1-\alpha -\left(g+u\right)\right)-b\left(q-{A}^L\right)>0$.

Similar argumentation leads to the following inequality:

$$ \frac{\partial {S}^1}{\partial {\sigma}^{*}}=\frac{1-\zeta }{\zeta}\frac{H}{H^{*}}\left(\left(1-{\alpha}^{*}-\left({g}^{*}+{u}^{*}\right)\right)-{b}^{*}\left({q}^{*}-{A}^L\right)\right)\frac{\alpha }{\alpha^{*}}\tilde{g}{k}^{\eta \frac{\alpha }{1-{\alpha}^G}}>0. $$

(37)

Next, we are interested in how a change in g influences the maximum sustainable debt. Thus, we need to calculate the following partial derivative:

$$ \frac{\partial {S}^1}{\partial g}=\left[\frac{\left(1-\zeta \right)}{\zeta}\frac{H}{H^{*}}{\varPhi^{*}}_g^{\prime}+\frac{\left(1-\zeta \right)}{\zeta}\frac{H_g^{\prime}{H}^{*}-H{H^{*}}_g^{\prime}}{{\left({H}^{*}\right)}^2}{\varPhi}^{*}+{\varPhi}_g^{\prime}\right]. $$

(38)

This can be rearranged by using Eq. (29) to:

$$ \begin{array}{l}\frac{\partial {S}^1}{\partial g}=\frac{1}{H^{*}}\frac{\alpha^G}{1-{\alpha}^G}\frac{1}{g}\left(\frac{\left(1-\zeta \right)}{\zeta }H\left({A}^L-\frac{\sigma^{*}}{\alpha^{*}}{b}^{*}{q}^2\right){k}^{\eta }+{H}^{*}\left({A}^L-\frac{\sigma }{\alpha }b{q}^2\right)k\right.\\ {}-\left.\frac{\varPhi }{H}{A}^L\left[k{H}^{*}-{k}^{\eta }H\right]\right)+\left(\frac{\varPhi }{H}-\sigma \right)\frac{1}{\alpha }qk.\end{array} $$

(39)

Expression (39) can be simplified using Eqs. (29) and (31):

$$ \frac{\partial {S}^1}{\partial g}=\frac{1}{\alpha }qk\left(\frac{\phi }{H}-\sigma \right). $$

(40)

Hence, the change in maximum sustainable debt caused by a change in g is given by:

$$ \frac{\partial \overline{b}}{\partial g}=-\frac{\partial {S}^1}{\partial g}/\frac{\partial {S}^1}{\partial b}=-\frac{\alpha^{-1}qk\left(\left(\phi /H\right)-\sigma \right)}{\left(-1\right)\left(\sigma q+\left(1-\sigma \right){A}^L\right)qk{\alpha}^{-1}}=\frac{\left(\left(\phi /H\right)-\sigma \right)}{\left(\sigma q+\left(1-\sigma \right){A}^L\right)}. $$

(41)

Differentiating Eq. (29) with respect to g* und using a similar simplification approach leads to:

$$ \begin{array}{l}\frac{\partial {S}^1}{\partial {g}^{*}}=\frac{1-\zeta }{\zeta}\frac{1}{\alpha^{*}}q{k}^{\eta}\frac{1}{H^{*}}\left(\frac{\phi^{*}}{H^{*}}-{\sigma}^{*}\right)=\\[6pt] {}=\frac{1-\zeta }{\zeta}\frac{1}{\alpha^{*}}q{k}^{\eta}\frac{1}{H^{*2}}\left(-\left[\frac{\sigma^{*}q+\left(1-{\sigma}^{*}\right){A}^L}{\alpha^{*}}\right]{k}^{\eta}\left({\alpha}^{*}+{b}^{*}q\right)\right)<0.\end{array} $$

(42)

The inequality holds for all non-negative debt ratios, b*.

Likewise, the partial derivative of S ¹ with respect to u and u* is given by:

$$ \begin{array}{ll}\displaystyle \frac{\partial {S}^1}{\partial u}&=\left[\frac{\left(1-\zeta \right)}{\zeta}\frac{H}{H^{*}}{\varPhi^{*}}_u^{\prime}+\frac{\left(1-\zeta \right)}{\zeta}\frac{H_u^{\prime}{H}^{*}-H{H^{*}}_u^{\prime}}{{\left({H}^{*}\right)}^2}{\varPhi}^{*}+{\varPhi}_u^{\prime}\right]=\left[\frac{\varPhi }{H}-\sigma \right]\\&\quad \displaystyle \tilde{g}{k}^{\alpha /\left(1-{\alpha}^G\right)} \end{array}$$

(43)

and

$$ \begin{array}{l}\frac{\partial {S}^1}{\partial {u}^{*}}=\left[\frac{\left(1-\zeta \right)}{\zeta}\frac{H}{H^{*}}{\varPhi^{*}}_{u*}^{\prime}+\frac{\left(1-\zeta \right)}{\zeta}\frac{H_{u*}^{\prime}{H}^{*}-H{H^{*}}_{u*}^{\prime}}{{\left({H}^{*}\right)}^2}{\varPhi}^{*}+{\varPhi}_{u*}^{\prime}\right]\\ [6pt]{}=\frac{\left(1-\zeta \right)}{\zeta}\frac{H}{H^{*}}\left[\frac{\varPhi^{*}}{H^{*}}-{\sigma}^{*}\right]\frac{\alpha }{\alpha^{*}}\tilde{g}{k}^{\eta {\alpha}^{*}/\left(1-{\alpha}^{*G}\right)}.\end{array} $$

(44)

Next, we are interested in the change in maximum sustainable debt ratio if the production elasticity changes. Thus, we have to calculate the derivative of S ¹ with respect to α:

$$ \begin{array}{l}\frac{\partial {S}^1}{\partial \alpha }=\left[\frac{\left(1-\zeta \right)}{\zeta}\frac{H}{H^{*}}{\varPhi^{*}}_{\alpha}^{\prime}+\frac{\left(1-\zeta \right)}{\zeta}\frac{H_{\alpha}^{\prime}{H}^{*}-H{H^{*}}_{\alpha}^{\prime}}{{\left({H}^{*}\right)}^2}{\varPhi}^{*}+{\varPhi}_{\alpha}^{\prime}\right]=\frac{1}{\alpha}\frac{1}{H}\left[\varPhi {A}^L-H{A}^L-H\sigma q\right]k\\ {}=-\frac{1}{\alpha H}\frac{qk}{\alpha}\left[\left(\sigma q+\left(1-\sigma \right){A}^L\right)\left(\left(1-\left(g+u\right)\right)+b{A}^L\right)\right]k<0\end{array} $$

(45)

The second equality follows from calculating the partial derivatives with respect to α and simplifying the term using the functional relations given in (29) and (31). This partial derivative is negative for non-negative debt ratios in Home, $b\ge 0$. Hence, the change in the maximum sustainable debt ratio caused by a change in the production elasticity is negative, $\partial \overline{b}/\partial \alpha =-\left(\partial {S}^1/\partial \alpha \right)/\left(\partial {S}^1/\partial b\right)$ as both denominator and nominator are negative.

A similar approach can be used to calculate a change in the maximum sustainable debt ratio caused by a change in Foreign’s production elasticity:

$$ \begin{array}{ll} \displaystyle \frac{\partial {S}^1}{\partial {\alpha}^{*}}&=-\frac{\left(1-\zeta \right)}{\zeta}\frac{H}{H^{*2}}\frac{1}{\alpha^{*}}\frac{q{k}^{\eta }}{\alpha^{*}}\left[\left(\left(1-\left({g}^{*}+{u}^{*}\right)\right)+{b}^{*}{A}^L\right)\left({\sigma}^{*}q+\left(1-{\sigma}^{*}\right){A}^L\right)\right]\\& \quad \displaystyle {k}^{\eta }<0,\end{array} $$

(46)

where the inequality follows from a non-negative debt ratio in Foreign.

Finally, we have to show that the effects on the capital intensity are different depending on whether the government increases productive or unproductive expenditures. By using Cramer’s rule as well as (29), (31), (41), and (43), we obtain:

$$ \frac{\partial k}{\partial g}-\frac{\partial k}{\partial u}=\frac{\partial {S}^2/\partial g-\partial {S}^2/\partial u\ }{-\partial {S}^2/\partial k}=\frac{\alpha^Gk}{\left(1-{\alpha}^G-\alpha \right)g}>0. $$

(47)

Thus, in the case that the maximum sustainable debt ratio is reached, capital intensity is higher if the government invests in productive expenditures.