Unsustainable sovereign debt—is the Euro crisis only the tip of the iceberg?

Abstract

As a direct effect of the financial crisis in 2008, public debt began to accumulate rapidly, eventually leading to the European sovereign debt crisis. However, the dramatic increase in government debt is not only happening in European countries. All major G7 countries are experiencing similar developments. What are the implications of this kind of massive deficit and debt policy for the long term stability of these economies? Are there limits in debt-ratios that qualitatively change policy options? While theory can easily illustrate these limits, where are these limits in real economies? This paper examines the relationship between sovereign debt dynamics and capital formation, and accounts for the effects of the 2008 financial crisis on debt sustainability for the four largest advanced economies. We contribute to the literature on fiscal sustainability by framing the problem in an OLG model with government debt, physical capital, endogenous interest rates, and exogenous growth. For the calibration exercise we extract data from the OECD for Germany as a stabilization anchor in Europe, the US, the UK, and Japan for almost two decades before the 2008 crisis. Except for intertemporal preferences, all parameters are drawn or directly derived from the OECD database, or endogenously determined within the model. The results of the calibration exercise are alarming for all four countries under consideration. We identify debt ceilings that indicate a sustainable and unsustainable regime. For 2011 all four economies are either close to, or have already passed the ceiling. The results call for a dramatic readjustment in budget policies for a consolidation period and long-term fiscal rules that make it possible to sustain sufficient capital intensity so that these economies can maintain their high income levels. Current conditions are already starting to restrict policy choices. However, the results also make it very clear that none of these economies would survive a second financial crisis such as the one in 2008.

Appendices

Appendix

This is an extended appendix for the working paper version:

1.1 Notation

α :

share of physical capital

1 − β :

the future discount factor

γ :

bequest share of sovereign debt

A :

exogenous scale parameter reflecting technological level

n :

population growth rate

ε _t :

labor efficiency growth

η :

total growth factor

$c_{t}^{1}$ :

individual’s consumption per capita of the young generation at time t

$c_{t+i}^{2}$ :

individual’s consumption per capita of the old generation in period t + i of life

B _t :

stock of government net financial liabilities at the start of period t

D _t :

principal of debt issued at the start of period t

b _t :

government net financial liabilities as a share of GDP at the start of period t

k _t :

capital efficiency-labour ratio at the start of period t

g _t :

government expenditure as a share of GDP at time t

τ _t :

tax revenues as share of wage income at the time t

d _t :

primary balance as share of GDP at time t

i _t :

real interest rate on public debt

r _t :

net real physical rate of return on physical capital at time t

δ _t :

depreciation rate for real capital stock

1.2 Appendix 1: Solving the model

1.2.1 Production per labor in efficiency units

The output of a firm is defined by the Cobb-Douglas production function

$$ Y_{t}=A\left( K_{t}\right) ^{\alpha }\left( H_{t}\right) ^{1-\alpha }. $$

In order to determine the production in efficiency units we have to compute the ratio each efficiency-weighted labour input provides. Hence, we obtain

$$ \tilde{y}:=\frac{Y_{t}}{H_{t}}=A^{\frac{1}{1-\alpha }}k_{t}^{\frac{\alpha }{ 1-\alpha }}\textrm{with} \,k_{t}:=K_{t}/Y_{t}. \label{y_schlange} $$

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1.2.2 Profit maximization of firms

In order to determine the input factor prices we consider the profit function

$$ \Pi =A(K_{t})^{\alpha }(H_{t})^{1-\alpha }-\left( r_{t}+\delta \right) (K_{t})-\tilde{w}_{t}(H_{t}). $$

Maximizing leads to the first order conditions

$$ \tilde{w}_{t}=\left( 1-\alpha \right) A^{\frac{\alpha }{1-\alpha }}k_{t}^{ \frac{\alpha }{1-\alpha }}\textrm{and}\,r_{t}=\alpha k_{t}^{-1}-\delta . $$

1.2.3 Lifetime consumption maximization

Using the Langrangeian function for the consumer’s maximization problem

$$ L=\left( \beta \left( \tilde{c}_{t}^{1}\right) ^{-\rho }+(1-\beta )\left( \tilde{c}_{t+1}^{2}\right) ^{-\rho }\right) ^{\frac{-1}{\rho }}+\lambda _{1}\left( \tilde{w}_{t}\left( 1-\tau _{t}\right) -\left( \tilde{c}_{t}^{1}+ \frac{1}{1+i_{t}}\tilde{c}_{t+1}^{2}\right) \right) $$

we obtain the condition

$$ 1+i_{t}=\frac{\beta }{1-\beta }\left( \frac{\tilde{c}_{t+1}^{2}}{\tilde{c} _{t}^{1}}\right) ^{\rho +1}. \label{con_opt} $$

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Using the derivative with respect to λ ₁ and plugging in we obtain the household’s savings rule in efficiency units

$$ \tilde{s}_{t}=\frac{\tilde{w}_{t}\left( 1-\tau _{t}\right) \left( \frac{ 1-\beta }{\beta }\right) ^{\varphi }}{\left[ \left( \frac{1-\beta }{\beta } \right) ^{\varphi }+(1+i_{t})^{1-\varphi }\right] }\textrm{with}\,\varphi = \frac{1}{\rho +1}. $$

Hence, for the savings rule in terms of the savings rate (savings per income) it holds

$$ s_{t}:=\frac{\left( 1-\alpha \right) \left( 1-\tau _{t}\right) \left( \frac{ 1-\beta }{\beta }\right) ^{\varphi }}{\left[ \left( 1+i_{t}\right) ^{1-\varphi }+\left( \frac{1-\beta }{\beta }\right) ^{\varphi }\right] }. $$

We can show that the derivative of s _t according to i _t is increasing

$$ \frac{\partial s_{t}}{\partial i_{t}}=-\frac{\left( 1-\varphi \right) \left( 1-\alpha \right) \left( 1-\tau _{t}\right) \left( \frac{1-\beta }{\beta } \right) ^{\varphi }}{\left[ \left( 1+i_{t}\right) +\left( 1+i_{t}\right) ^{\varphi }\left( \frac{1-\beta }{\beta }\right) ^{\varphi }\right] ^{2}}>0 \textrm{for}\, \varphi >1 $$

1.2.4 The public sector

Each period debt changes by

$$ \begin{array}{lll} B_{t+1}-B_{t} &=&D_{t}+i_{t}B_{t}=G_{t}-T_{t}+i_{t}B_{t}. \\[4pt] &\Leftrightarrow &\frac{B_{t+1}\eta _{t}}{Y_{t}\eta _{t}}=\frac{D_{t}}{Y_{t}} +\left( 1+i_{t}\right) \frac{B_{t}}{Y_{t}}=\frac{G_{t}}{Y_{t}}-\frac{T_{t}}{ Y_{t}}+\left( 1+i_{t}\right) \frac{B_{t}}{Y_{t}}. \end{array} $$

In order to obtain the total debt accumulation in ‘debt ratio’ b _t  = :B _t /Y _t substitute $d_{t}=\frac{D_{t}}{Y_{t}},g_{t}=\frac{G_{t}}{ Y_{t}}$ and $\tau _{t}=\frac{T_{t}}{Y_{t}}.$ Hence, we obtain

$$ \eta _{t}b_{t+1}=d_{t}+(1+i_{t})b_{t}=g_{t}-\tau _{t}+(1+i_{t})b_{t} $$

with η _t denoting the growth factor of output which in a potential steady state becomes equal to the product of the population growth factor (1 + n _t ) and the growth factor of labor productivity $\left( 1+\varepsilon _{t}\right)$, such that $\eta _{t}=\left( 1+\varepsilon _{t}\right) (1+n_{t})$.

1.2.5 Equilibrium and stationarity conditions

Pure transmission of capital and bonds from generation to generation:

We assume that the old generation bequeaths the total capital K _t and bonds B _t with the share γ if the restriction is fulfilled that total bequest does not restrict full capital transfer. Therefore the income and expenditure equation becomes:

$$ \begin{array}{lll} wH&+&K_{t}+\gamma B_{t}-T-C^{y}+K_{t}r_{t}+B_{t}r_{t}+\left( 1-\gamma \right) B_{t}\\ &=&C^{o}+K_{t+1}+B_{t+1}-F\\ &\Rightarrow &s_{t}+f_{t}+k_{t}+\gamma b_{t}=\eta _{t}k_{t+1}+\eta _{t}b_{t+1} \end{array} $$

General stationarity curve:

For the general stationarity curve we use that b _t + 1 = b _t and k _t + 1 = k _t :

$$ \Rightarrow b_{t}^{s}=:b_{t}=\frac{s_{t}}{\left( \eta _{t}-\gamma \right) }+ \frac{f_{t}}{\left( \eta _{t}-\gamma \right) }+\frac{\left( 1-\eta _{t}\right) }{\left( \eta _{t}-\gamma \right) }k_{t} $$

Properties of the general stationarity curve:

$$ \begin{array}{cc} \text{Case} & \underset{k\rightarrow 0}{\lim }b_{t}^{s}(k_{t}) \\ \varphi >1: & \frac{\left( 1-\alpha \right) \left( 1-\tau _{t}\right) }{ \left( \eta _{t}-\gamma \right) }+\frac{f_{t}}{\left( \eta _{t}-\gamma \right) } \\ \varphi =1: & \frac{\left( 1-\alpha \right) \left( 1-\tau _{t}\right) }{ \left[ \left( \frac{1-\beta }{\beta }\right) ^{-1}+1\right] \left( \eta _{t}-\gamma \right) }+\frac{f_{t}}{\left( \eta _{t}-\gamma \right) } \\ \varphi <1: & \frac{f_{t}}{\left( \eta _{t}-\gamma \right) } \end{array} $$

For k→ ∞ we obtain

$$ \underset{k\rightarrow \infty }{\lim }b_{t}^{s}(k_{t})=-\infty . $$

For the derivative we obtain

$$ \begin{array}{lll} \frac{\partial b_{t}^{s}(k_{t})}{\partial k_{t}} &=&\frac{\left( 1-\alpha \right) \left( 1-\tau _{t}\right) }{\left( \eta _{t}-\gamma \right) }\\ &&\times\left[ \frac{\alpha \left( \frac{1-\beta }{\beta }\right) ^{-\varphi }}{ k_{t}^{2}\left( 1+\alpha k_{t}^{-1}-\delta \right) ^{\varphi }\left[ \left( 1+\alpha k_{t}^{-1}-\delta \right) ^{1-\varphi }\left( \frac{1-\beta }{\beta }\right) ^{-\varphi }+1\right] ^{2}}\right] \\ &&+\frac{(1-\eta )}{\left( \eta _{t}-\gamma \right) } \end{array} $$

$$ \underset{k\rightarrow 0}{\lim }\frac{\partial b_{t}^{s}(k_{t})}{\partial k_{t}}=\infty . $$

$$ \underset{k\rightarrow \infty }{\lim }\frac{\partial b_{t}^{s}(k_{t})}{ \partial k_{t}}=\frac{(1-\eta )}{\left( \eta _{t}-\gamma \right) }<0. $$

Since we have parts where the derivative of the $b_{t}^{s}-$curve is positive and parts where it is negative we can state that with the intermediate value theorem there is at least one point where the derivative is 0, hence a maximum.

Non-stationary dynamics:

From Eqs. 10 and 11 it is apparent that below the $b_{t}^{s}-curve$ capital intensity will increase and above the $b_{t}^{s}-curve$ capital intensity will decrease.

$$ k_{t+1}\geq k_{t}\textrm{if}\,b_{t}\leq b_{t}^{s}\qquad \textrm{and}\, \qquad k_{t+1}\leq k_{t}\textrm{if}\,b_{t}\geq b_{t}^{s} $$

1.2.6 Relative price changes of real assets:

In order to include the relative price change of real assets into the budget constraint we assume the price change to be

$$ p_{t+1}^{K}=\underset{:=\pi _{t}^{K}}{\underbrace{\left( \frac{ (p_{t+1}^{K}-p_{t}^{K})}{p_{t}^{K}}+1\right) }}p_{t}^{K}=\pi _{t}^{K}p_{t}^{K}. $$

Then the income and expenditure equation becomes

$$\begin{array}{lll} wH&+&p_{t}K_{t}+\gamma B_{t}-T-C^{y}+K_{t}r_{t}+B_{t}r_{t}+\left( 1-\gamma \right) B_{t}\\ &=&C^{o}+p_{t+1}K_{t+1}+B_{t+1}-F\\ &\Rightarrow & s_{t}+f_{t}+p_{t}k_{t}+\gamma b_{t}=\eta _{t}\pi _{t}^{K}p_{t}^{K}k_{t+1}+\eta _{t}b_{t+1} \end{array} $$

With the same procedure we obtain for the general stationarity curve:

$$\begin{array}{lll} b_{t}^{s}&:=&b_{t}=\frac{\left( 1-\alpha \right) \left( 1-\tau _{t}\right) }{ \left[ \left( 1+\alpha p_{t}k_{t}^{-1}-\delta \right) ^{1-\varphi }\left( \frac{1-\beta }{\beta }\right) ^{-\varphi }+1\right] \left( \eta _{t}-\gamma \right) }\\[4pt] &&+\,\frac{f_{t}}{\left( \eta _{t}-\gamma \right) }+\frac{(1-\eta \pi _{t}^{K}p_{t}^{K})}{\left( \eta _{t}-\gamma \right) }k_{t}\end{array} $$

Note that the price change could also be implemented before deriving the savings rule. However, since the total capital stock is bequeathed an implementation would not change the interpretation.

Appendix 2: A calibration exercise

Definition of extracted input-data

Table 6 All extracted variables are taken as a simple averages for the respective country and base line period

Table 7 All extracted variables are taken as a simple averages for the respective country and base line period

Results stationarity tests

Table 8 Stationarity tests

Determining values for the baseline scenario from the model

Table 9 Determining values for baseline scenario from the model