The government budget identity
The government’s taxation and spending decisions are referred to as fiscal policies. Each year the government decide how much to spend and at the same time it must find the necessary resources to finance its spending. The government can decide to finance its expenditures by collecting taxes from businesses and households or it may decide to finance its spending by borrowing from the capital market. If the government decides to finance spending through borrowing, it will have to issue financial liabilities or, more simple, it will have to run into debt. The total amount of outstanding financial liabilities issued by the government is called the government debt or public debt.
The term fiscal policy refers to spending and taxation decisions of the government. Fiscal policies aimed at increasing aggregate demand and employment are called expansionary fiscal policies. Fiscal policies aimed at reducing aggregate demand, for example, to avoid rising inflation, are called restrictive fiscal policies. Fiscal policy acquired a central role in macroeconomics thanks to John Maynard Keynes (1983-1946) who during the Great Depression of the 1930s demonstrated how fiscal policy and in particular deficit spending can be used to stabilize production and employment. Economists from different traditions or “schools of thought” express conflicting views on the role and effectiveness of fiscal policy. Economists in the Keynesian tradition are in favor of deficit spending to stimulate output and employment, especially during economic downturns. Orthodox economists, on the other hand, believe in the self-equilibrating capacity of the market economy. State intervention in the economy is ineffective if not counterproductive. Orthodox economists talk about Ricardian equivalence, a theory according to which forward-looking households base their spending decisions on the government’s budget policy. An increase in government deficit implies higher taxes in the future. Households react by decreasing their current consumption with the result that there is no change in aggregate demand.
We define government spending with the letter (\(G\)) and taxation with the letter (\(T\)), both in nominal terms. With the letter (\(B\)) we represent the nominal stock of public debt and with the letter (\(i\)) we indicate the nominal interest rate (given by the sum of the real interest rate and the inflation rate) that the government pays on outstanding debt. Since we are talking about magnitudes that vary over time, we must include a time subscript. For example (\(G_t\)) refers to government spending in period \(t\) and with (\(i_t B_{t-1}\)) we indicate the interest payment in period t on the stock of debt accumulated until the end of period \(t-1\). For sake of convenience, however, we will report a time subscript only for variables at time other than \(t\) or whenever it is required.
We can now set up the government’s budget identity. On the left-hand side of (1.1), we report the type of expenditure of the government. This will be government spending (\(G\)) and interest payments (\(i B_{t-1}\)). On the right-hand side of (1.1), we report the source of financing of the government. As we have just learned, the government can either receive revenues from taxation (\(T\)) or it can finance its spending by issuing new debt (\(\Delta B\)).
\[\begin{equation}
G + i B_{t-1} \equiv T + \Delta B
\tag{1.1}
\end{equation}\]
In words, we can write the government budget identity (1.1) as:
\[\begin{gather}
\text{government spending} \space\equiv \text{government revenues} \nonumber
\end{gather}\]
The difference between government’s total expenditure (i.e. including interest payments) and tax revenue is called the government’s total budget deficit (\(D'\)). Rearranging the government’s budget identity from above, we observe that the government budget is by definition equal to the variation of public debt:
\[\begin{equation}
D' \equiv G - T + i B_{t-1} \equiv \Delta B
\tag{1.2}
\end{equation}\]
We need to be careful with terminology. Economists talk about a primary deficit for the calculation of the government budget deficit excluding interest payments (\(D\)). In identity (1.2), this corresponds to \(G - T\). Whenever \(G > T\), the government registers a primary budget deficit. Conversely, when \(G < T\) the government registers a primary budget surplus. In words, the government deficit is equal to the primary deficit plus interest expenses on outstanding debt.
\[\begin{gather}
\text{government deficit} \space\equiv \text{primary deficit + interests on debt} \nonumber
\end{gather}\]
From identity (1.2), we notice that every government deficit correspond to a change (i.e. an an increase) in the stock of public debt. When tax revenues are not sufficient to finance spending, the government runs a budget deficit for which it has to issue new debt. The stock of public debt is by definition the sum of all past deficits of the government. The government does not have always to run budget deficits. The government can decide to follow a balanced budget policy by making its spending equal to the tax revenue or it can decide to run budget surplus in the event, for example, that it wants to reduce the stock of public debt. In section 3.1, we will look in detail at the role of the primary balance.
Public debt dynamics
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).
How should the government react in order to prevent the debt ratio from increasing? Based on the debt dynamic equation, we can conclude that, in order to offset the negative effect of a drop in the growth rate on the debt ratio, the government should switch from a budget deficit to a budget surplus. By how much should the government tighten the budget? Is a primary budget surplus of \(1\%\) large enough? In the next section, we will see how the government can use fiscal policy to stabilize the debt ratio. For now, we simply assume that, after the crisis, the government will run a budget surplus of \(2\%\), i.e. \(d = -2\%\). Graphically, this corresponds to a downward shift of the debt line as soon as the growth rate turns negative, as we can observe from figure 2.7 (a). To summarize, following the fall in the growth rate, the slope of the debt line turns negative moving from A to B and the debt ratio begins to diverge. A timely intervention of fiscal policy prevents the increase of the debt ratio. The debt line shift downward from B to C after the fiscal restriction and the debt ratio begins to decline (and diverge in the negative direction). Keep in mind that we are assuming that budget policies have no effect on growth (or the interest rate). It is likely that restrictive fiscal policy put in place in the midst of an economic crisis will put additional recessionary pressure on the economy with undesirable repercussions on the debt ratio. Here we are just applying mechanically the functioning of the debt dynamic equation.
The effect of inflation on the debt-to-GDP ratio
Up to this point, we have assumed that the growth rate of the economy (\(g\)) and the interest rate on debt (\(r\)) were both expressed in real terms. Haven’t we forgotten about inflation? Does inflation have an impact on the dynamics of the debt ratio? Let’s go back to equation (2.3) and express the equation so that we have the change in the debt ratio appearing on the left-hand side. This can be seen with (2.8) below where \(i\) is again the nominal interest rate, \(\pi\) is the inflation rate and \(g\) is the rate of growth of the economy expressed in real terms.
\[\begin{equation}
\Delta b_t = d + (i - \pi - g) b_{t-1}
\tag{2.8}
\end{equation}\]
We can see that the inflation rate enters with a minus sign in the term in parentheses, the interest-growth differential. This implies that, everything else equal, an increase in the inflation rate results in a lower real interest rate and in a decline of the debt ratio. Rising inflation helps to decrease the debt ratio over time reducing the burden of debt in real terms. Three scenarios are shown in figure 2.8. For the three scenarios, we have the same conditions regarding the nominal interest rate (\(i = 3\%\)), the real growth rate of the economy (\(g = 1\%\)), the deficit (\(d = 0\%\)) and the initial value of the debt ratio (\(b_0 = 100\%\)). For each scenario, we have selected instead a different value of inflation . In scenario 1, we assume an inflation rate of \(1\%\). As we can see, this is not sufficient to reverse the negative effect of the interest-differential on the future path of the debt ratio. We have that \((i - \pi - g) > 0\). In scenario 2, we assume an inflation rate of \(2\%\). Given the values that we have chosen for this scenario, the interest-differential is zero and the debt ratio remains constant \((i - \pi - g) = 0\). In the third scenario, we assume an inflation rate of \(3\%\). The interest-differential is now negative \((i - \pi - g) < 0\). In this case, the debt ratio will be set on a declining (and converging) path.
With some simple algebra, we can show that in the debt dynamics equation it is equivalent to report interest rate and growth rate in real or nominal terms. We know that the real interest rate is given by \(r = i - \pi\) and that the growth rate is given by \(g = g^n - \pi\), where \(g^n\) is the nominal growth rate of output. Thus, we have that:
\[(r - g) = (i - \pi) - (g^n - \pi) = (i - g^n) \]
The app accessible at the link below allows to simulate the dynamics of the debt ratio selecting values for the interest rate, the growth rate, the deficit, and the initial debt ratio. In this case, the interest rate is expressed in nominal terms and it is possible to control the value of the rate of inflation. The reader can try to replicate the three scenarios proposed in figure 2.8 or it may use the app freely.
Sustainability and debt dynamics
The issue of debt sustainability is of utmost importance. The failure of a state is a dramatic event with negative repercussions for the whole society. But what exactly does it mean that public debt is sustainable? Is there an unambiguous definition of public debt sustainability? According to the European fiscal rules, public debt sustainability is identified as the “solvency” of the public sector. Solvency means that current debt cannot exceed the present value of all future primary balances. This is based on the idea of the intertemporal budget constraint of the government, meaning that every present or future deficit spending has to be necessarily matched by a present or future tax income. The financing of debt and interest with new additional debt, what is called “Ponzi game”, is ruled out. This concept is based on the assumption that the interest-growth differential is positive. This implies that primary surpluses are necessary to finance deficit spending. What would happen if the interest-growth differential were negative, as observed in some developed countries in recent years, such as Germany? In what follows we take a broader but at the same time pragmatic definition of debt sustainability. A sustainable debt ratio is, broadly speaking, a debt ratio that remains constant or at best decreases. We look at the dynamics of the debt ratio and in particular at the role of the primary balance in both situations when the interest-growth differential is positive and when the interest-growth differential is negative (Priewe 2020a).
The role of the primary balance
The primary balance when r - g < 0
In section 2.3, we saw how the difference between the interest rate of the government and the growth rate of GDP shapes the path of the debt ratio after having made different assumptions about the government balance. In scenario 2 in particular, we saw that when the interest-growth differential is negative, the government can afford to run primary deficits and the debt ratio will still converge. There, we assumed the same level of deficit and interest-growth differential for two different starting values of the debt ratio. The debt ratio was converging to a higher value, in the case when the initial debt ratio was small and to a lower value when the initial debt ratio was high. Let us try to clarify this by starting from the case in which the government wants to keep the debt ratio constant. There is one level of the primary balance of the government that keeps the debt ratio constant over time. To obtain this value, we rearrange the steady-state equation (2.6) from section 2.3 and isolate the value of the deficit on the left-hand side.
\[\begin{equation}
\bar{d} = b (g - r)
\tag{3.1}
\end{equation}\]
Equation (3.1) allows us to find the level of the deficit (\(\bar{d}\)) that if held constant keeps the debt ratio on a stable path given a value of the interest-growth differential (which in this case we assume to be negative, therefore \(g-r<0\)). Assuming a growth rate of \(3\%\), an interest rate of \(2\%\), and a debt ratio of \(100\%\), the government can run permanently a deficit of \(1\%\) and the debt ratio will remain stable at \(100\%\).
\[100(0.03 - 0.02) = 1\%\]
This means that given a negative interest-growth differential, the government would be able to keep the debt ratio constant even running primary deficits continuously. A rather striking result. Deficit values greater than the stabilizing value (\(\bar{d}\)) will let the debt ratio increase. How large can budget deficits be? From a theoretical point of view, there is no limit. Assuming a negative interest-growth differential with a deficit higher than the value that keeps the debt ratio constant, the debt ratio will rise but always converge to a finite value. Deficit values smaller than the value whereby the debt ratio remains constant, will instead let the debt ratio decrease. This means that, with a negative interest-growth differential, the government can run primary deficits forever and the debt ratio will still decrease and converge to a lower level. In figure 3.1, we have assumed an initial debt ratio value of \(100\%\) and an interest-growth differential of \(-1\%\). We then assumed different values of the deficit ranging from a deficit of \(0\%\) (balanced budget) to a deficit of \(3\%\) for a total of 7 deficit values (corresponding 7 scenarios). In the first two scenarios, the deficit is lower than the value that keeps the debt ratio constant (\(0\%\) and \(0.5\%\), respectively). Here, the debt ratio converges to a lower level with respect to the initial one despite the fact that we are assuming a balanced budget and a deficit. In scenario 3, the deficit is \(1\%\). This is the value of the deficit according to which the debt ratio remains constant at the initial value of \(100\%\). In the remaining scenarios, where the deficit is higher the stabilizing value, the debt ratio will increase and converge to a higher but finite value (double click on the image extend the axes of the graph and see the convergence of the debt ratio).
Domar’s pioneering work on public debt
In his work titled “The ‘burden of the debt’ and the national income” (1944), Evsey D. Domar (1914-1997), at the time member of the Board of Governors of the US Federal Reserve System, demonstrated analytically and by means of numerical scenarios the existence of an inverse relationship between GDP growth and the debt ratio and that to reduce the debt ratio, the focus should have been on growth rather than budget cuts. Domar’s conclusions were clearly against the dominant opinion at a time, when most economists concerned about the rising burden of public debt in the United States in the course of World War II argued against the use of deficit spending to finance the reconversion of the national economy after the war effort. The crucial question that Domar asked was then how to sustain the growth of national income. Domar suggests that the government should contribute to raise aggregate demand and income through an increase in productive public spending. But what are productive public expenditures? In this regard, Domar’s words remain as relevant as ever: “Since government is absorbing a part of savings, it is of course desirable that its expenditures be productive. (…) As a matter of fact, the term ‘investment expenditures’ may be misleading, because it is too closely associated with steel and concrete. If healthier people are more productive, expenditures on public health satisfy these requirements. The same holds true for expenditures on education, research, flood control, resource development and so on” (Domar 1944, 820).
A negative interest-growth differential has dramatic implications in terms of economic policy. In the EMU where the treaties were designed at a time when interest rates were higher than growth rates, fiscal rules were built on the assumption that current deficits must have been always offset by future surpluses. At the time, a negative interest-growth differential was thought just as a “theoretical curiosum” (Blanchard et al. 1990, 15). When \(r-g>0\), an increase in government deficit spending, and the payment of the respective interest, inevitably implies an increase in future taxation, whether it is to pay interest only, and thus to keep the debt constant, or to repay the debt entirely. In recent years, however, advanced economies have witnessed a reversal of the sign of the interest-growth differential. An increase in the deficit (temporary or permanent, not a continuous increase) and the increase in the associated interest payments will not have to be financed by a future tax increase because by definition output will grow faster than interest payments (Blanchard, Leandro, and Zettelmeyer 2020).
The primary balance when r - g > 0
In section 2.3, we have learned that a positive interest-growth differential can lead to a divergent the debt ratio. We have also learned that a generic primary surplus may not be sufficient to stop the debt ratio from rising when the interest-growth differential is positive. We now take a closer look at the role of the primary balance when \(r - g > 0\). Since the government has no direct control over the interest rate \(r\) and can not immediately influence the real rate of growth of the economy \(g\), fiscal policy can prevent the debt ratio from rising only by controlling the the primary balance. From a theoretical point of view, we know that, whatever the values of \(r\) and \(g\), and whatever level the debt ratio, there is always a value of the primary surplus that is able to keep the debt ratio constant. The higher the level of the debt ratio, and the higher the interest-growth differential, the higher is the primary surplus required to maintain the debt ratio stable. When we say that there is always a primary surplus that keeps the debt ratio stable, we do not mean that this particular value of the primary balance is at the same time economically and politically feasible. We are assuming a purely accounting perspective here.
Let’s go back to scenario 1 in section 2.3 where we had a positive interest-growth differential and a primary deficit for the government. What would be in this case the primary surplus that keeps the debt ratio constant at the starting value of \(100\%\) assuming again an interest rate \(r\) of \(3\%\) and a growth rate \(g\) at \(2\%\)? We plug in these values in (3.1) and obtain our primary surplus that keeps the debt ratio constant over time.
\[100(0.02 - 0.03) = -1\%\]
In order to maintain the debt ratio constant at \(100\%\), the government must run a budget surplus (the negative of the deficit is a budget surplus) equal to \(1\%\) of GDP. In general, for each level of debt ratio there is a primary balance such that the debt ratio remains constant even in the case when \(r - g > 0\). What if the government were to run an even larger surpluses? And what if the government were to run a budget surplus that was smaller than the surplus needed to stabilize the debt ratio? Assuming again the previous values, a surplus larger than \(1\%\) would cause the debt ratio to decrease (and eventually to diverge towards infinitely negative values). In figure 3.2 we have assumed again an initial debt ratio value of \(100\%\) and an interest-growth differential of \(−1\%\). We then assumed different values of the surplus ranging from a surplus of \(0\%\) (balanced budget) to a surplus of \(3\%\) for a total of 7 scenarios, similarly to what we did in figure 3.1 before. A primary balance smaller than the value keeping the debt ratio constant (scenario 1 and 2) leads to an increase in the debt ratio (positive divergence). A primary balance greater than this value (scenario 4 to 7) leads to a decrease of the debt ratio even with a positive interest-growth differential (negative divergence). In scenario 3, the debt ratio remains constant.
The 3% and 60% reference values
We now have the necessary tools to understand the relationship between two very important reference values in the European economic policy discussion. We are talking about the \(3\%\) and the \(60\%\) reference values of the Maastricht Treaty respectively for deficit- and the debt-to-GDP ratio.
The origin of reference values
On the 7th of February 1992, representatives from 12 European countries signed the “Treaty on European Union”. The treaty, more commonly known as the Maastricht Treaty, gave birth to the today’s European Union (EU) and kick-started the formation of the single currency area, the EMU. Entry into the final stage of the EMU was conditional upon compliance with a series of macroeconomic criteria on inflation, interest rate, exchange rate and government budgetary policy. In order to join the monetary union, the public deficit had to be no greater than \(3\%\) of GDP and the debt ratio had not to exceed the \(60\%\) reference value or, if this was not the case, it had to approach the value at a “satisfactory” rate. But why \(3\%\) and \(60\%\) values? After lengthy negotiations, the values were introduced into the Treaty but only in an annex (Protocol) and no theoretical justification was provided. In retrospect, it became clear that the value of \(60\%\) was chosen as it roughly corresponded to the average debt ratio of the 12 member States of the European Community. Less clear is the origin of the \(3\%\) value. It even seems that the value enters the negotiations for the first time as a suggestion made by a low-rank official in the French Ministry of Finance under the Presidency of François Mitterrand. It also appears that the number was originally intended as an average of the deficit and not a maximum threshold as it was then interpreted. Only later a theoretical justification for the relationship between the two values was provided, as we will see below (Priewe 2020a, 2020b).
As we are going to see, the value of \(3\%\) is the value which keeps the debt-to-GDP ratio constant at the \(60\%\) reference value assuming a nominal growth rate of \(5\%\). An early critique to this argument came from the economist Luigi Pasinetti (Pasinetti 1998). If we assume as definition of debt sustainability a debt ratio that remains constant or that it decreases over time, it can be easily demonstrated that the combination of the values for the deficit and the debt ratio that were chosen for the Maastricht Treaty are actually only one pair of values among numerous pairs of values (and possibly infinite pairs of values) for the deficit and the debt ratio. To understand this argument, we need to use again our debt dynamic equation with interest rate and the growth rate expressed in nominal terms.
\[\begin{equation}
\Delta b_t = d + (i - g^n) b_{t-1}
\tag{3.2}
\end{equation}\]
This time we include the interest rate in the the budget balance so that in the debt equation we now report the total budget balance (\(d'\)) again in relation to GDP. After all, the \(3\%\) rule refers to the total budget balance, i.e. including interest payments.
\[\begin{equation}
\Delta b_t = d' - g^n b_{t-1}
\tag{3.3}
\end{equation}\]
We can now see that given a nominal growth rate of \(5\%\), to keep the debt-to-GDP ratio constant at the level of \(60\%\), the government deficit must be equal to \(3\%\). In other words, if we want to have a constant debt-to-GDP ratio, i.e., \(\Delta b_t = 0\) in (3.3), starting from a debt ratio of \(60\%\) and assuming a nominal growth rate of \(5\%\), the value for the deficit is in this case \(3\%\).
\[0 = d' - 60(0.05)\]
\[d' = 60(0.05)\]
\[d' = 3\]
However, the pair of values \(60\%\) and \(3\%\) is not the only combination of debt ratio and budget deficit that keep the deb ratio constant. If we define the sustainable debt ratio as a debt ratio that remains constant of that decreases over time there are infinite pairs of deficit and debt ratio values that satisfy this condition. We can express this condition with the following inequality.
\[b_t \leq b_{t-1}\]
From which it follows that
\[\Delta b_t \leq 0\]
Conversely, debt can be defined as unsustainable if the debt ratio keeps increasing over time. In this case, the inequalities above would not be satisfied. We can work out the conditions under which this inequality holds. We start again with the debt equation with interest rate and growth rate expressed in nominal terms.
\[\begin{equation}
\Delta b_t = d + (i - g^n)b_{t-1}
\tag{3.4}
\end{equation}\]
In order for the left-hand side of equation (3.4) to be zero, it is necessary for the terms \(d\) and \((i - g^n)b_{t-1}\) to be equal but opposite in sign. Assuming a negative interest-growth differential, i. e. \(i - g^n < 0\), the government can run a budget deficit (\(d\)) equal to \((i - g^n)b_{t-1}\) and the debt ratio is constant. In order for the left-hand side of the equation to be negative, we need the government to run a deficit (\(d\)) smaller than \((i - g^n)b_{t-1}\), assuming again a negative interest-growth differential. Of course, the opposite is true. Given a positive growth interest differential, \(i - g^n > 0\), the government can run a budget surplus (\(-d\)) equal to \((i - g^n)b_{t-1}\) and the debt ratio is constant again. To decrease the debt ratio, the surplus will have to be larger \((i - g^n)b_{t-1}\). We can express these conditions with the following inequality.
\[\begin{equation}
0 \geq d + (i - g^n)b_{t-1}
\tag{3.5}
\end{equation}\]
After few simple manipulations, we obtain our expression for the debt sustainability condition.
\[\begin{equation}
d \leq (g^n - i)b_{t-1}
\tag{3.6}
\end{equation}\]
We can also express this condition in terms of the total budget (\(d'\)):
\[\begin{equation}
d' \leq g^n b_{t-1}
\tag{3.7}
\end{equation}\]
The limit of deficit spending at \(3\%\) of GDP and a debt ratio of \(60\%\) are therefore only one possible combination that keep the debt ratio constant. Assuming again a positive nominal growth rate of \(5\%\), a debt ratio of \(100\%\) would be consistent with a deficit spending of \(5\%\) of GDP.
\[0 = d' - 100(0.05)\]
\[d' = 100(0.05)\]
\[d' = 5\]
For a higher value of the debt ratio, it is would be possible to have more deficit spending and still having a constant debt ratio as long as growth is positive and large enough. Assuming a positive growth rate, a higher debt ratio can be sustained even higher deficits and still remain constant over time. This fact is expressed graphically in 3.3 for the primary balance (a) and the total budget balance (b). The solid line represents the boundary conditions, i.e. all the combinations of debt ratio and budget balance such that the debt ratio remains constant. The colored surface represents the sustainability area, i.e. all those combinations of debt ratio and budget balance for which the debt ratio would instead decrease over time.
The app accessible at the link below allows you to test interactively the concepts that we have just discussed. The user can control the economy’s nominal growth rate (\(g^n\)), the government budget deficit (\(d'\)), and the initial value of the debt ratio (\(b_0\)). Try to assume a value of \(5\%\) of growth and a deficit of \(3\%\) but an initial debt ratio value of \(100\%\). What path does the debt ratio would follow in this case? Also try to select a pair of debt ratio and government deficit values within the sustainability area. How does the path of the debt ratio evolves? What happens to the debt ratio if the values of debt ratio and government deficit lie outside the sustainability area?
Growing rigidity in the European fiscal framework
Over the last decades, the European fiscal rules have become even more restrictive than originally established in the Maastricht Treaty. In June 1997, the European countries signed the Stability and Growth Pact (SGP). The goal of SGP was to provide a set of operational rules to monitor and eventually correct the fiscal conduct of euro countries (in the SGP, so-called “preventive arm” and “corrective arm” are introduced). The \(3\%\) and \(60\%\) rules remained the pillars of the fiscal framework. The treaty underwent some revisions in 2005 when the two largest economies of the euro area, namely Germany and France, breached the \(3\%\) rules to tackle the effects of the recession of the early 2000s caused by the bursting of the bubble in the U.S. financial market (dot.com bubble). The reform of the treaty went in two directions. On the one hand, the reform introduced some aspects, albeit minimal, of flexibility such as the clarification of the escape clauses. On the other hand, it included greater restrictions on government deficit spending. A country-specific medium-term budgetary objective (MTO), formulated in cyclically adjusted terms net of one-off measures, was introduced. Along with the MTO, the revised SGP introduced the obligation for the euro countries to present to European authorities an “adjustment path” toward the MTO in the case the country was distant from the target. Essentially, the SGP required the achievement of a almost balanced budget in structural terms where the \(3\%\) reference value remained just as a ceiling for the deficit.
After the financial and economic crisis of 2007–2009, in response to rising deficits and debt levels, the European countries led by Germany and France pushed for the adoption of an even more rigid fiscal framework. In 2012, the fiscal compact, formally the “Treaty on Stability, Coordination and Governance in the Economic and Monetary Union”, was signed by the heads of State of all EU euro countries. In essence, the fiscal compact required compliance with two main fiscal rules. First, a mandatory balanced budget rule. More specifically, a structural deficit not exceeding \(0.5\%\) of GDP for countries with a debt ratio at or below the \(60\%\) reference value and up to \(1\%\) of GDP for member countries with debt ratios sufficiently below the \(60\%\) reference value. In addition, the fiscal compact required the introduction of a specific balanced budget rule in the national constitutions of the euro countries. Second, for countries with debt ratios exceeding the \(60\%\) threshold, the treaty required compliance with country-specific MTOs, essentially a series of primary surpluses (net of interest payments) in structural terms geared at reducing the debt ratio towards the \(60\%\) reference value. The fiscal compact included a numerical rule for debt reduction. The treaty defined the speed of reduction of the debt ratio, something that in the Maastricht Treaty and in the SGP had not received operational formulation, at a pace of 1/20 per year of difference between the current debt ratio and the \(60\%\) reference value.
In the next section, we want to simulate (in a stylized fashion) the operation of the debt reduction rule included in the fiscal compact. As a result of the COVID-19 pandemic, debt ratios in the euro area have increased considerably. We will simulate the fiscal adjustment that would be required to reduce the debt ratio towards the \(60\%\) reference value starting from different debt levels.
Required primary balance to bring debt to target
During the COVID-19 pandemic, debt levels in the euro zone have risen sharply. In 2021, the debt ratio has reached in Greece and Italy, the two countries with the highest debt-to-GDP ratio in the monetary union, \(193\%\) and \(150\%\) respectively. A considerable increase in the debt ratio has been observed in the large majority of the euro countries so that in 2021 only 7 countries had a debt ratio below the \(60\%\) reference value (figure 3.4).
What would then be the budget required by the current fiscal rules as stated in the fiscal compact? This depends primarily on the level reached by the debt ratio. The higher the debt ratio, the higher the primary surplus required to reduce the debt ratio towards the \(60\%\) target. We formulate three scenarios where we assume three initial levels for the debt ratio, namely \(70\%\), \(160\%\) and \(200\%\). Further, we assume a positive value for the interest-growth differential (a slightly pessimistic situation but something that we cannot completely rule out). Specifically, we assume an interest rate (\(r\)) of \(2\%\) and a growth rate (\(g\)) of \(1\%\). Formula (3.8) gives us the primary balance (\(pb_t^{b^T}\)) needed to reduce the debt ratio toward the desired target (\(b^{T}\)). As stated in the fiscal compact, we want to reduce the debt ratio by one-twentieth (that is, \(5\%\)) per year of the distance between the value of the debt ratio and the desired target, \(60\%\) in our case. The parameter \(N\) takes therefore the value of \(20\).
\[\begin{equation}
pb^{b^T}_{t} = (r - g) b_{t-1} + \frac{1}{N} (b_{t-1} - b^{T})
\tag{3.8}
\end{equation}\]
Figure 3.5 in quadrant (a) shows us the path of debt adjustment. The debt ratio converges toward the value of \(60\%\) in all three scenarios. Quadrant (b) of figure 3.5 shows us instead the average primary balance (for the first twenty years) necessary so that the debt ratio converges towards the desired value. In scenario 1, the average primary balance is \(1\%\), approximately. In scenario 2 and 3, the average required primary balance is significantly higher. In the second scenario, in which the initial debt ratio is \(160\%\), the average primary balance is roughly \(4.6\%\). In the third scenario, where the initial debt ratio is \(200\%\), the average primary balance is higher than \(6\%\).
Fiscal adjustments of this size (even an average primary surplus of \(1\%\) over 20 years is definitely high) would be a significant burden on the economy. This would have significant repercussions on employment and growth, with negative cascading effects on the debt ratio itself. It seems therefore unrealistic to think that the reference value of \(60\%\) for the debt ratio can continue to be the reference value also in the future. If this were the case, a return to the rules of the fiscal compact would make drastic austerity policies unavoidable for many countries in the euro area.
As with the other simulations presented in this text, the app accessible at the link below allows you to test interactively the functioning of the European fiscal rule controlling the values of the parameter in formula (3.8).
Concluding words
We would like to close this text with a quote from the economist Evsey Domar. The message fits perfectly our times.
Now, some economic circles are burning with a desire to reduce the debt burden after the war. They recognize no other method of
achieve their goal but by reducing the absolute size of the debt; that the government must stop borrowing is of course taken for
granted. They should beware, however, lest the policies they advocate exert such a depressing effect on the national income as to
result in an actually heavier debt burden, even though they succeed in paying off a part of the debt (Domar 1944, 815 - 816).
Appendix
Derivation of the debt dynamic equation
To derive the debt dynamic equation, we start with the government’s budget identity as we have learned in section 1 and 2.
\[\begin{align}
B_t \equiv G_t - T_t + (1 + i_t) B_{t-1}
\end{align}\]
Since we are interested in the ratio of debt to gross domestic product, we divide both sides of the equation by nominal GDP which we express as the product of \(P_t\) and \(Y_t\), where \(P_t\) is the general price level and \(Y_t\) is real output or GDP.
\[\begin{equation}
\frac{B_t}{P_t Y_t} = \frac{G_t - T_t}{P_t Y_t} + (1 + i_t) \frac{B_{t-1}}{P_t Y_t}
\tag{A.1}
\end{equation}\]
We then expand the second term on the right-hand side of (A.1) with \(P_{t-1} / P_{t-1}\) and \(Y_{t-1} / Y_{t-1}\) (both fractions are equal to one and this does not change the equation).
\[\begin{equation}
\frac{B_t}{P_t Y_t} = \frac{G_t - T_t}{P_t Y_t} + (1 + i_t) \frac{B_{t-1}}{P_t Y_t} \frac{P_{t-1}}{P_{t-1}} \frac{Y_{t-1}}{Y_{t-1}}
\tag{A.2}
\end{equation}\]
After rearranging terms on the right-hand side of (A.2), we get
\[\begin{equation}
\frac{B_t}{P_t Y_t} = \frac{G_t - T_t}{P_t Y_t} + (1 + i_t) \frac{B_{t-1}} {P_{t-1} Y_{t-1}} \frac{P_{t-1}}{P_t} \frac{Y_{t-1}}{Y_t}
\tag{A.3}
\end{equation}\]
We know that \(P_{t} / P_{t-1} = 1 + \pi_t\) and that \(Y_{t} / Y_{t-1} = 1 + g_t\), where \(\pi\) is the rate of inflation at time \(t\) and \(g\) is the real GDP rate of growth at time \(t\) (note that in (A.3) numerator and denominator of these two fractions appear inverted). Substituting \(1 + \pi_t\) and \(1 + g_t\) in (A.3), we get
\[\begin{equation}
\frac{B_t}{P_t Y_t} = \frac{G_t - T_t}{P_t Y_t} + (1 + i_t) \frac{B_{t-1}} {P_{t-1} Y_{t-1}} \frac{1}{(1 + \pi_t)} \frac{1}{(1 + g_t)}
\tag{A.4}
\end{equation}\]
Rearranging terms on the right-hand side of (A.4) and rewriting \(B_t / P_t Y_t\) and \((G_t - T_t) / P_t Y_t\) as \(b_t\) and \(d_t\), respectively, we finally obtain our debt dynamic equation from section 2.2.
\[\begin{equation}
b_t = d_t + \frac{(1 + i_t)}{(1 + \pi_t)(1 + g_t)} b_{t-1}
\tag{A.5}
\end{equation}\]
Solution of the debt dynamic equation
We now want to derive the analytical solution of the debt dynamic equation for constant values of \(r\), \(g\) and \(d\). Why do we need to know about the solution of a first-difference equation? The solution allows us to calculate the result of the equation at time \(t\) without having to know the result of the equation at time \(t-1\). If we did not know the solution or we were unable to derive it analytically, we would necessarily have to simulate the values of the equation up to the time \(t\) that we are interested in knowing. How to solve a first-difference equation? The equation should be solved recursively, that is, by inserting repeatedly the equation at time \(t-1\) into the equation at time \(t\). This procedure cannot be repeated indefinitely, of course. However, we will reach a point at which we will be able to identify the solution. We start with (2.4) from section 2.2. For convenience, we rewrite \((1+r-g)\) as \((1 + \lambda)\).
\[\begin{equation}
b_t = (1 + \lambda) b_{t-1} + d
\tag{B.1}
\end{equation}\]
We now replace the subscript \(t\) with the reference periods \(1,2,3,...\) and begin to substitute the equation recursively. \(b_0\) is again our starting value.
\[\begin{equation}
\begin{split}
b_1 &= (1 + \lambda) b_0 + d \\
b_2 &= (1 + \lambda)((1 + \lambda) b_0 + d) + d\\
&= (1 + \lambda)^2 b_0 + d((1 + \lambda) + 1) \\
b_3 &= (1 + \lambda)((1 + \lambda)^2 b_0 + d((1 + \lambda) + 1)) + d \\
&= (1 + \lambda)^3 b_0 + d((1 + \lambda)^2 + (1 + \lambda) + 1) \\
b_4 &= (1 + \lambda)((1 + \lambda)^3 b_0 + d((1 + \lambda)^2 + (1 + \lambda) + 1)) + d \\
&= (1 + \lambda)^4 b_0 + d((1 + \lambda)^3 + (1 + \lambda)^2 + (1 + \lambda) + 1) \\
&= \ldots \\
b_N &= (1 + \lambda)^N b_0 + d((1 + \lambda)^{N-1} + (1 + \lambda)^{N-2} + \ldots + 1)
\end{split}
\end{equation}\]
After rewriting \(d((1 + \lambda)^{N-1} + (1 + \lambda)^{N-2} + \ldots + 1)\) using sigma notation as \(d \sum_{t=1}^N (1 + \lambda)^{N-t}\), we finally get to the solution of our first-difference linear equation.
\[\begin{equation}
b_N = b_0 (1 + \lambda)^N + d \sum_{t=1}^N (1 + \lambda)^{N-t}
\tag{B.2}
\end{equation}\]
At the end, the analytical solution (B.2) will not contain the lagged values of the debt ratio (as it is the case with the (B.1)) but only the coefficients, \(d\) and \((1 + \lambda)\), the initial condition \(b_0\) and the time variable \(N\). The solution of the debt dynamic equation is also presented in Escolano (2010) for both constant and time varying values of \(\lambda\).
Stability condition
In section 2.3, we have introduced the concept of the steady-state. We have also investigated the conditions under which the debt ratio converges to its long-run equilibrium value and the conditions under which the debt ratio diverges. We now want to try to derive the stability condition for our debt dynamic equationusing the solution of the dynamic debt equation. Our starting point is (B.3).
\[\begin{equation}
b_N = b_0 (1 + \lambda)^N + d \sum_{t=1}^N (1 + \lambda)^{N-t}
\tag{B.3}
\end{equation}\]
We need to perform some manipulations. First, we rewrite the sum on the right-hand side of (B.3) (technically a geometric series) as
\[\begin{equation}
\sum_{t=1}^N (1 + \lambda)^{N-t} = \frac{1 - (1 + \lambda)^N}{1 - (1 + \lambda)} = \frac{(1 + \lambda)^N - 1}{\lambda}
\tag{B.4}
\end{equation}\]
Substituting (B.4) into (B.3) we obtain
\[\begin{equation}
b_N = (b_0 + \frac{d}{\lambda}) (1 + \lambda)^N - \frac{d}{\lambda}
\tag{B.5}
\end{equation}\]
To simplify the notation, we rewrite \((b_0 + d/\lambda)\) as \(A\) and \(-d/\lambda\) as \(c\)
\[\begin{equation}
b_N = A(1 + \lambda)^N + c
\tag{B.6}
\end{equation}\]
Equation (B.6) is dynamically stable, i.e. the debt ratio converges towards a finite value, only if the term \(A(1 + \lambda)^N\) tends towards zero as time increases, \(A(1 + \lambda)^N \rightarrow 0\) as \(N \rightarrow \infty\). As the term \(A(1 + \lambda)^N\) disappears, the values of \(b\) approaches the value of the constant term \(c\). The constant term \(c\) is simply the steady-state value that we have seen above with (2.7). Whether the value of \(b\) converges or not towards equilibrium, it depends on the value of the coefficient \((1 + \lambda)\). If the interest-growth differential is positive, i.e. \(\lambda > 0\) and \((1 + \lambda) > 1\), the debt ratio will diverge as the term \(A(1 + \lambda)^N\) becomes larger and larger as \(N \rightarrow \infty\). If the interest-growth differential is negative, i.e. \(\lambda < 0\) and \((1 + \lambda) < 1\), the debt ratio will converge towards the steady-state value \(c\) as the term \(A(1 + \lambda)^N\) becomes smaller and smaller as \(N \rightarrow \infty\). For those who would like to explore the field of first difference equations further, we recommended looking at Gandolfo (2009, chap. 1) and Chiang (1984, chap. 16).
Required primary balance to bring debt to target in a finite number of year
To obtain the primary balance that allows to reach a predefined target for the debt ratio in a finite number of years, we have to start with the solution of the debt dynamic equation that we have seen with equation (B.2). This time we replace the budget deficit (\(d\)) with the primary balance (\(pb\)). Pay attention to the sign.
\[\begin{equation}
b_N = b_0 (1 + \lambda)^N - pb \sum_{t=1}^N (1 + \lambda)^{N-t}
\tag{C.1}
\end{equation}\]
We bring the term \(b_0 (1 + \lambda)^N\) to the left-hand side and multiply both sides by \((1 + \lambda)^{-N}\).
\[\begin{equation}
-b_0 + b_N (1 + \lambda)^{-N} = -pb \sum_{t=1}^N (1 + \lambda)^{-t}
\tag{C.2}
\end{equation}\]
We can rewrite the sum on the right-hand side of (C.2) (technically a geometric series) as
\[\begin{equation}
\sum_{t=1}^N (1 + \lambda)^{-t} = \frac{1 - (1 + \lambda)^{-N}}{(1 + \lambda) - 1} = \frac{1 - (1 + \lambda)^{-N}}{\lambda}
\tag{C.3}
\end{equation}\]
Substituting (C.3) into (C.2) we obtain
\[\begin{equation}
-b_0 + b_N (1 + \lambda)^{-N} = -pb \frac{1 - (1 + \lambda)^{-N}}{\lambda}
\tag{C.4}
\end{equation}\]
After changing the sign and multiplying both sides by the inverse of \(\frac{1 - (1 + \lambda)^{-N}}{\lambda}\), we finally obtain the formula which is needed to find the required primary balance (\(pb^{b^T}\)) necessary to bring the debt ratio to target (\(b^{T}\)) in a finite number of years (\(N\)).
\[\begin{equation}
pb^{b^T} = \frac{\lambda}{1 - (1 + \lambda)^{-N}} (b_0 - b^{T} (1 + \lambda)^{-N})
\tag{C.5}
\end{equation}\]
If the primary balance in (C.5) is held constant, the debt ratio will hit the desired target exactly after \(N\) years. This formula can be tested interactively. Click here to access the web application.
List of variables
Table C.1: Variables and parameters in the text
|
Abbreviation
|
Name
|
Color*
|
|
\(\lambda\)
|
Interest-growth differential
|
#
|
|
\(B\)
|
Stock of public debt
|
#
|
|
\(b\)
|
Debt-to-GDP ratio
|
#
|
|
\(\bar{b}\)
|
Steady-state debt-to-GDP ratio
|
#
|
|
\(b^T\)
|
Target debt ratio
|
#
|
|
\(D\)
|
Government primary deficit
|
#
|
|
\(D'\)
|
Government total deficit
|
#
|
|
\(d\)
|
Government primary deficit in percent of GDP
|
#
|
|
\(d'\)
|
Government total deficit in percent of GDP
|
#
|
|
\(G\)
|
Government spending
|
#
|
|
\(g^n\)
|
Nominal GDP growth rate
|
#
|
|
\(g\)
|
Real GDP growth rate
|
#
|
|
\(i\)
|
Nominal interest rate
|
#
|
|
\(N\)
|
Numbers of years
|
#
|
|
\(r\)
|
Real interest rate
|
#
|
|
\(\pi\)
|
Inflation rate
|
#
|
|
\(P\)
|
Price level
|
#
|
|
\(pb\)
|
Government primary balance in percent of GDP
|
#
|
|
\(pb^{b^T}\)
|
Required primary balance to bring debt to target
|
#
|
|
\(T\)
|
Taxes
|
#
|
|
\(Y\)
|
Total income
|
#
|
|
* Color in pictures (if applicable)
|
References
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———. 2020b. “Why 3 and 60 Per Cent? The Rationale of the Reference Values for Fiscal Deficits and Debt in the European Economic and Monetary Union.” European Journal of Economics and Economic Policies: Intervention 17 (2): 111–26.
Rietzler, K., and A. Truger. 2019. “Is the "Debt Brake" Behind Germany’s Successful Fiscal Consolidation: A Comparative Analysis of the "Structural" Consolidation of the Government Subsector Budgets from 1991 to 2017.” Revue de l’OFCE 2: 11–30.