Motivation
We have introduced the government’s budget identity and we have seen how changes in the government budget determine the increase (in the case of deficits) or decrease (in the case of surpluses) of the stock of public debt. Of concern for economists is not much the level of public debt (i.e. the amount of debt expressed in currency units, e.g. euros) but rather the stock of public debt relative to national income. As the name suggests, the debt-to-GDP ratio is the ratio between a country’s government debt and its gross domestic product (GDP). Why is public debt expressed as a ratio of GDP? Putting in relation the country’s debt with the country’s total production and income, the debt ratio provides an indication of the country’s ability to sustain its debt. If a country has a debt-to-GDP ratio of \(150\%\), it means that the stock of debt is \(50\%\) larger than its GDP of a year.
Figure 2.1 shows us the evolution of the debt ratio from 1995 to 2021 for the four largest economies of the eurozone, France, Germany, Italy and Spain. We can observe a rapid increase in the ratio starting with financial and economic crisis of 2007-09 that follows a prolonged period of decline for Spain and Italy and of moderate increase for France and Germany. In Spain, from 2008 to 2014 the debt ratio increased from about \(40\%\) to \(100\%\) of GDP. An increase of \(60\) percentage points! In the same period, the debt ratio in Italy has increased by around \(30\) percentage points. Another large increase (in all country in this case) is observed between 2019 and 2020, where 2020 is the year marking the tragic beginning of the COVID-19 the pandemic as shown in 2.1 (b). What determines the movements of the debt ratio? Why does it increase so rapidly in years of recession?
The debt dynamic equation
We now want to understand what macroeconomic factors determine the evolution of the debt ratio. As we will see during the rest of the explanation, the debt ratio is a reference point for budget policies and compliance with a certain level of debt (60% of GDP) was one of the main prerequisites for joining the European Economic and Monetary Union (EMU) and still today the reference value of the debt ratio for euro area countries. A rapidly rising debt ratio is also perceived as an element of fragility of the public finances. It is of utmost importance to understand the factors determining the path of the debt ratio. To do so, we must investigate the economic forces that determine the growth of the numerator of the ratio, the stock of debt, and the growth of the denominator of the ratio, GDP. Our starting point is again the government budget identity (1.1). Rewriting \(\Delta B_t\) as \(B_{t} - B_{t-1}\) and rearranging, we obtain the following expression:
\[\begin{equation}
B_t \equiv G - T + (1 + i) B_{t-1}
\tag{2.1}
\end{equation}\]
After some manipulation (the full derivation is available in the appendix A), we obtain the so-called debt dynamic equation which is entirely derived from the government budget identity. In (2.2), \(b_t\) stands for the debt ratio in relation to GDP at the end of period t, \(d\) indicates the primary deficit in relation to GDP, \(i\) is the nominal interest rate, \(\pi\) is the inflation rate, \(g\) is the growth rate of real GDP and \(b_{t-1}\) is the debt ratio in relation to GDP at the end of period \(t-1\).
\[\begin{equation}
b_t = d + \frac{(1 + i)}{(1 + \pi)(1 + g)} b_{t-1}
\tag{2.2}
\end{equation}\]
We can further simplify equation (2.2). We can exploit the fact that the term \((1 + i)/[(1 + \pi)(1 + g)]\) is approximately equal to \(1 + i - \pi - g\) and rewrite (2.2) as:
\[\begin{equation}
b_t = d + (1 + i - \pi - g) b_{t-1}
\tag{2.3}
\end{equation}\]
We can perform an additional manipulation. We can use the Fisher equation, i.e., the relationship between the nominal interest rate, the real interest rate and the rate of inflation, i.e. \(r = i - \pi\), to rewrite the term in parenthesis \((1 + i - \pi - g)\) as \((1 + r - g)\).
\[\begin{equation}
b_t = d + (1 + r - g) b_{t-1}
\tag{2.4}
\end{equation}\]
Technically, equation (2.4) is a first order linear difference equation, an equation in which the solution at time \(t\) depends linearly on the solution at time \(t-1\). For now, this is all we need to know. In the appendix B, we will see how it is possible to solve such an equation. We can also rewrite (2.4) so that it expresses the change in the debt ratio. After all, we are interested in explaining the variation of the debt ratio over time.
\[\begin{equation}
\Delta b_t = d + (r - g) b_{t-1}
\tag{2.5}
\end{equation}\]
We have all elements in place. In equation (2.5), we observe the factors that determine the motion of the debt ratio and how they can contribute to bring and keep the debt ratio on a sustainable path or, conversely, how they can contribute to the “explosion” of the debt ratio. There are four variables determining the evolution of the debt ratio:
- the ratio of the primary deficit to national income (\(d\))
- the real interest rate (\(r\))
- the real growth rate (\(g\))
- the debt ratio inherited from the past (\(b_{t-1}\))
The ratio of the primary deficit ratio to national income (\(d\)) has an increasing effect on the evolution of the debt ratio. Everything else equal, higher budget deficits will increase the debt ratio. On the contrary, higher primary surpluses will decrease the debt ratio as government income, which is tax revenue, is higher than government spending. The real interest rate has a negative effect on the debt ratio (again, everything else constant). This comes as no surprise. A high interest rate corresponds to a higher government outlay for interests payments with obvious negative consequences on the accumulation of debt. Real GDP growth has instead (at parity of other conditions) a dampening effect on the debt ratio. As GDP increases, the relative weight of debt decreases as the capacity of the country to generate income has now increased. In this case, the denominator of the fraction, national income, increases faster than the numerator, debt. The term \((r - g)\) is the interest-growth differential, also known as the “snowball effect”. Assuming a balanced budget position of the government, whenever the interest-growth differential is positive, i.e. the interest rate is higher than the growth rate, the debt ratio will rise. Whenever the interest-growth differential is negative, i.e. the interest rate is lower than the growth rate, the debt ratio will decline. In the next section, we simulate the case where \((r - g)\) is positive and the case where \((r - g)\) is negative under different assumptions about the government budget. Later in the text, we will take a closer look at the role that the government budget can play in stabilizing or reducing the debt ratio.
The evolution of the debt ratio under different assumptions
Goal of this section is to study the evolution of the debt ratio by means of some simple simulations. Before we delve into the simulations, we have to begin with some important caveats. First, we are assuming that the values for \(d\), \(r\), and \(g\) remain constant throughout the time range of the numerical simulation that we are about to perform. For the variable \(b_t\), the debt ratio, we will assume an initial value while the rest of the values will be the output of the simulations. The debt ratio \(b_t\) is therefore our endogenous variable except the initial value \(b_0\) which we define by assumption. Second, we are excluding any possible feedback effects. We are assuming that, for example, the rate of growth of the economy is independent of public spending (and its composition) or that the increase in the debt ratio has no effect on government interest rate. In other words, the variables \(d\), \(r\), and \(g\), as well as the value of the initial debt ratio \(b_0\), are exogenous and fixed by assumption. This is rather a strong set of assumptions. Moreover, our simulations extend over a large number of periods, which makes the assumption of constant exogenous values (such as GDP growth) even more unrealistic. Even though the analysis and the simulations that we are about to perform is highly theoretical, it will help us to understand a number of issues underlying some important economic policy decisions.
First of all, we must introduce the concept of the steady-state. The steady-state is the long-run equilibrium of a variable. At the steady-state, the variable of interest, the debt ratio in our case, remains constant, i.e. it does no change as time goes on.
\[\begin{equation}
b_{t} = b_{t-1} = b_{t-2} = ... = \bar{b}
\tag{2.6}
\end{equation}\]
The steady-state is, however, a “neutral concept”. The steady-state does not tell whether from an economic point of view the value to which the debt ratio converges is desirable or not. We only know that the debt ratio reaches an equilibrium and that it will remain constant over time. Whether this value is “desirable” from an economic perspective has nothing to do with the concept of the steady-state. It could also be the case that the debt ratio does not converge at all. In this case, we say that the variable “diverges” and the steady-state will never be reached. In order to derive the steady-state, we can set the left-hand side of (2.5) equal to zero (\(\Delta b_{t} = 0\)) and rearrange to obtain the steady-state value (\(\bar{b}\)) of the debt ratio:
\[\begin{equation}
\bar{b} = \frac{d}{(g - r)}
\tag{2.7}
\end{equation}\]
The steady-state value of the debt ratio depends on the values of \(r\), \(g\) and \(d\). Whether or not the debt ratio converges towards its steady-state will depend on the values of the difference between \(r\) and \(g\), as we are about to see. We will now set up four possible scenarios. In the first two scenarios, the government will run a primary balance deficit. In the second two scenarios, the government will run a primary budget surplus. For each case, we will look at the situations where the interest-growth differential \((r - g)\) is both positive and negative. We will learn about the conditions under which the debt ratio may converge towards the steady-state or it may diverge. Later, we will see which role the government can play with its budget policy.
Primary deficit
Scenario 1: r - g > 0
In the first scenario, we assume that the government runs a budget deficit of \(1\%\) and that the interest rate (\(3\%\)) exceeds the growth rate of the economy (\(2\%\)). We further assume a positive initial level of the debt ratio (\(b_0\)) equal to \(100\%\). From figure 2.2, we can see how the line that graphically describes the dynamics of the debt ratio expressed in (2.5) has a positive slope and a positive intercept, since \(r - g > 0\) and \((G - T)/Y > 0\). This is the black line (technically called the phase line) in quadrant (a) of 2.2. Quadrant (b) shows instead the evolution of the debt ratio as determined by the debt dynamic equation (2.4). We can observe how the value of \(b_t\) is moving further and further away from the initial level of \(100\%\). This means that, given the particular set of values that we have chosen for \(r\), \(g\) and \(d\), debt will rise faster than GDP leading progressively to the explosion of the debt ratio.
We can also observe how of the debt ratio does not move in the direction of the steady-state, which, given the numerical values that we have chosen for this scenario, is equal to negative \(100\%\), a situation where the government is a net creditor. The steady-state is represented in the graph on the left-hand side of 2.2 by the intersection of the black line with the horizontal axis (red point) and in the graph on the right-hand side by the red dashed line that intercepts the vertical axis at a value equal to negative \(100\%\). We can now make a final and technical point. In this scenario, the equilibrium point \(\bar{b}\) is said to be “unstable”. An unstable equilibrium means that as soon as the variable moves away from the equilibrium, the value will tend to diverge, positively, if it moves to the right along the horizontal axis, and negatively, if it moves to the left.
Scenario 2: r - g < 0
In the second scenario, we still assume that the government generates primary deficits but this time we assume that the real growth rate \(g\) is greater than the real interest rate \(r\). Let’s also assume again that the primary deficit is equal to \(1\%\). This is a very favorable set of conditions for the debt ratio. The debt ratio converges to the steady-state value either assuming a starting value greater than the equilibrium value (first case, blue line) or a lower value (second case, light blue line) as shown in quadrant (b) of figure 2.3. In this case, the equilibrium value is defined as “stable”. In economic terms this means that the growth of the economy is able to cover the growth of interest allowing the government to maintain an expansionary fiscal stance running constant primary deficits.
Differently from the previous scenario where the debt line was sloping upwards, we can now see that in 2.3 (a) the debt line slopes downwards as the growth rate is now exceeding the interest rate (\(r - g < 0\)). The intercept of the debt line with the vertical axis is still in the positive section of the graph as we are still assuming a budget deficit, i.e. \((G - T)/Y > 0\). The intersection of the debt line with the horizontal axis indicates the steady-state value which, given the values of the parameters \(r\), \(g\), and \(d\) that we chose for this particular example, is equal to \(50\%\).
Primary surplus
Scenario 3: r - g > 0
In the third scenario, we assume that the real interest rate (\(3\%\)) is again higher than the real growth rate of the economy (\(1\%\)) but that the government now runs a fiscal surplus of \(1\%\), i.e. \((G - T)/Y < 0\). As in the second scenario, we compare two cases. In the first case, (blue line) where the initial debt ratio is equal to \(100\%\), we see how the debt ratio explodes despite the government registering a primary surplus. This means that a primary surplus of \(1\%\) is not sufficient to stabilize the debt ratio. We will see later what the government can do in this situation. In the second case (light blue line) where the initial debt ratio is equal to to \(25\%\), a primary surplus of \(1\%\) is even excessive and the debt ratio diverges in the negative direction. This constellation of values where \(r - g > 0\) depicts a situation where the equilibrium value is again unstable.
Scenario 4: r - g < 0
In this fourth and last scenario, the interest growth differential is negative with the interest rate equal to \(1\%\), the rate of growth of the economy equal to \(3\%\) and the government assumes a restrictive fiscal stance (\(d = -1\%\)). Given that \(r - g < 0\), the debt ratio converges towards its (negative) long-term equilibrium value. At the steady state, the government is a net creditor. Given the negative interest-growth differential, the government could even afford to run a primary deficit as we have seen with scenario 2.
Table 2.1 summarizes the long-term behavior of the debt ratio, that is whether it converges or it diverges towards the steady-state, under different assumptions about the primary balance of the government and the interest-growth differential. As long as \(r-g<0\), the government can run primary deficits and the debt ratio will not explode but it will rather converge towards the steady-state value.
Table 2.1: Summary table of the long-term behaviour of the debt ratio under different assumptions.
|
|
r - g < 0
|
r - g > 0
|
|
Primary deficit
|
Converging
|
Diverging
|
|
Primary surplus
|
Converging
|
Diverging
|
A few closing words to summarize what we have learned so far. In this section, we have seen a series of scenarios in which we combined different conditions for the primary balance of the government and the interest-growth differential. We have learned the concept of the steady-state and we have seen the conditions under which the debt ratio converge towards the steady-state value and the conditions under which the debt ratio diverges. Even if what we have seen are just some theoretical examples, public debt analyses performed by international institutions such as the International Monetary Fund (IMF) and the European Commission (EC) are based on the framework that we have just presented. It is therefore very important to understand how it works.
We now leave to the reader the possibility of testing the scenarios discussed above using the interactive tool accessible at the following link. In the app, it is possible to set the value of the growth rate (\(g\)), the interest rate (\(r\)), the government’s deficit (\(d\)) as well as the initial value of the debt ratio (\(b_0\)). To the reader is given the possibility of replicating the scenarios discussed in this section or to use the app freely.
The effect of an economic crisis
Until now we have assumed that the values of interest rate and growth remain constant through time. Let’s now assume that from a certain period the growth rate of the economy becomes negative as for example during an economic crisis. We assume the same starting conditions of the second scenario above where the real growth rate (\(3\%\)) is greater than the interest rate (\(1\%\)) and the government is running a budget deficit (\(1\%\)). We now assume that as the result of the crisis, the growth rate drops to \(-3\%\) and it remains negative thereafter. What happens to the debt ratio after that the growth rate has suddenly turned negative? In quadrant (a) of figure 2.6 we can see that the slope of the debt line is now positive (continuous black line) as the interest-growth differential has turned from negative to positive. The debt line rotates around the intersection with the vertical axis from position A to position B but since the fiscal balance has not changed, the debt line will not shift. In quadrant (b) of 2.6 we can observe that from the moment the economic crisis occurs (in period 50) the debt ratio starts suddenly to increase (and to diverge) even though it was initially converging towards the steady-state value (dashed blue line converging towards the dashed red line).