Fiscal policy and the debt-to-GDP ratio. A seemingly counterintuitive effect?
Alessandro Bramucci
30 Jun 2023
1 Motivation
According to the conventional view, a government that wants to keep stable or to reduce the debt-to-GDP ratio (debt ratio, for short) has no choice but to realize primary surpluses. This interpretation relies on the assumption that a fiscal contraction has no effect on output. A reduction in public demand and public investment, for example, would have no direct effect on the economy in terms of production and employment. Proponents of austerity policies went even further theorizing the so-called “expansive austerity” according to which the reduction in public spending would not have recessionary effects on the economy but it would rather stimulate demand and production.1
The goal of this short text is to challenge the conventional interpretation. Using some simple “Keynesian arithmetic”, we will analyze the effects of fiscal policy on the debt ratio. In particular, we will show the conditions under which fiscal policy could have an undesirable or “perverse” effect on the debt ratio. We will analyze the conditions under which, for example, an increase in deficit spending may lead to a decrease in the debt ratio and, in a similar counterintuitive fashion, the conditions under which a reduction in deficit spending may lead to an increase in the debt ratio. To conclude, we will evaluate our results from a critical perspective also in light of the academic literature on the issue.
2 Fiscal policy can have a “perverse” effect
To understand why fiscal policy can have perverse effects on the debt ratio, we need to think about the two parts of debt the ratio separately. At the numerator of the debt ratio (\(b\)) we have the stock of debt (\(B\)) while at the numerator of the ratio we have gross domestic product (\(Y\)). Both quantities are expressed in monetary terms, for example in euros.
\[\text{Debt ratio}\; (b) = \text{Stock of debt}\; (B)\; /\; \text{Gross domestic product}\; (Y)\]
Assume now that the government wants to reduce the debt ratio. This was the case in the euro zone with the so-called “sovereign debt crisis”.2 To reduce the debt ratio, the government will seek to reduce the numerator of the ratio, the stock of debt, by running primary surpluses. Assuming that GDP, the denominator of the ratio, remains unchanged, the ratio will decrease. What happens instead if we assume that GDP decreases as a result of the fiscal crunch? Contrary to what expected, the debt ratio could increase if the reduction in GDP is larger than the reduction in debt. What does it all mean? The fiscal tightening has simply reduced the stock of debt (the numerator of the ratio) less than GDP (the denominator of the ratio) so that the total effect on the fraction is unexpected or one could say “perverse”.
Figure 2.1: Austerity leads to a fall in GDP which in turn leads to an increase in the debt ratio. The size of the fiscal multiplier contributes to determining how large the fall in GDP will be.
It turns out that fiscal policy has an uncertain effect on the ratio. By imagining the effect of fiscal policy on GDP, we have implicitly assumed the effect of the so-called fiscal multiplier. The effect of fiscal tightening on GDP therefore depends on the magnitude of the fiscal multiplier and, as we shall see in the next section, on the value of the debt ratio itself. What is the fiscal multiplier? In a nutshell, the (short-run) fiscal multiplier is represented by the change in a country’s GDP due to a discretionary change in fiscal policy.3
The size of fiscal multipliers is not directly measurable and it is subject of intense debate among economists. While the mainstream approach has long underestimated the magnitude of the fiscal multipliers, an influential paper by Olivier Blanchard and Daniel Leigh, both working at the time at the International Monetary Fund (IMF) showed that the fiscal multipliers used by international institutions for their forecasts (less than or slightly greater than unity) were substantially higher than previously assumed (Blanchard and Leigh 2013). This implied that the restrictive fiscal policies implemented in many countries following the “great recession” and the debt crisis in the euro zone had clearly underestimated the impact of budget consolidation policies on output leading to a further increase in the debt ratio.
The analysis of Blanchard and Leigh
Blanchard and Leigh’s analysis related errors on economic growth forecasts and countries’ fiscal consolidation plan during the crisis (Blanchard and Leigh 2013).4 If the value of the fiscal multiplier applied in the models used to calculate growth forecasts had been correct, there should have been no relationship between growth forecast errors (i.e. the difference between the growth that was expected and the growth that was actually realized) and planned fiscal consolidation. The empirical analysis conducted by the authors found instead a negative and strong relationship between planned fiscal consolidation and growth forecast errors. This means that, as the authors suggested, fiscal multipliers were greater than assumed in growth forecasts. The authors tested their results with numerous control variables and robustness checks confirming their findings. As we can observe from figure 2.2 the relationship between planned fiscal consolidation and growth forecast error is clearly negative. On the horizontal axis, we show the forecast of the adjustment in the structural fiscal balance of the government as a percentage of GDP. On the vertical axis, we show instead the error in the forecast of the (cumulative) change in real GDP. Both variables are reported over the two year period 2010-2011.5 The slope of the black line in 2.2 is equal to 1% approximately.6 According to the authors, this implied that the multipliers were underestimated by approximately 1%.
Figure 2.2: Blanchard and Leigh’s test of the underestimation of fiscal multipliers in the period 2010-2011 for European countries. Source: Brancaccio and De Cristofaro (2019).
In the next section, we work out the conditions under which fiscal policy may have a perverse effect on the debt ratio. Using some fairly simple “Keynesian arithmetic”, it is possible to prove how, under certain circumstances, a fiscal tightening aimed at reducing the debt ratio could have exactly the opposite effect of the one desired. This condition is widely known to economists. Among economists of a heterodox tradition, this condition has been studied and debated in the works of Nuti (2013), Leão (2013) and Di Bucchianico (2019). This condition is widely known even among researchers of mainstream inspiration like for example Gros and Maurer (2012) and Codogno and Galli (2017) who, however, tend to emphasize its limitations (which we will cover in the next section).7
2.1 The condition in the short run
Let us now focus on understanding how this condition works from a purely algebraic point of view. How is this apparently counter intuitive result possible? What are the circumstances for this condition to apply? Does this condition hold over time? In this section, we are going to derive the condition in the short run, where for short run we mean two consecutive periods (which we can imagine to be years), such as \(t = 0\), the initial condition, and \(t = 1\), the period of the policy change.
We start with a very simple Keynesian model, where the variation in output or GDP (\(\Delta Y_t\)) is determined by the product of the fiscal multiplier (\(m\)) and deficit spending of the government (\(D_t\)). Both quantities \(\Delta Y_t\) and \(D_t\) are expressed in monetary terms, for example euros. Assuming that the change in spending represents the entire deficit is a strong simplification, of course. We are also assuming that there are no borrowing costs for the government therefore no interest payments. We will go on to remove some of these assumptions later.
\[\begin{equation} \Delta Y_t = m D_t \tag{2.1} \end{equation}\]
The stock of debt (\(B\)) at time \(t\) is given by the sum of the stock of debt accumulated up to the previous period, time \(t-1\), and the deficit (\(D\)) incurred at time \(t\). If instead of a deficit we had a budget surplus, the quantity \(D_t\) would be negative and it would decrease the amount of debt.
\[\begin{equation} B_{t} = B_{t-1} + D_t \tag{2.2} \end{equation}\]
By definition, the change in debt at time \(t\) is equal to the deficit.
\[\begin{equation} \Delta B_t = D_t \tag{2.3} \end{equation}\]
As we already know, the debt ratio (\(b_t\)) is expressed as follows:
\[\begin{equation} b_{t} = \frac{B_t}{Y_t} \tag{2.4} \end{equation}\]
We can now derive the conditions under which a fiscal consolidation (i.e. a budget surplus) may have a perverse effect on the debt ratio, which is to say the situations where a budget surplus may increase the debt ratio. As said, we first derive the condition in the short run, i.e. between the first period (\(t = 1\)) and the initial condition (\(t = 0\)). We are interested in the change in the debt ratio, so we need to look at what happens to the debt ratio between the two consecutive periods. We begin then by breaking down the variation of the debt ratio \(\Delta b_1\) between time \(0\) and time \(1\).
\[\begin{equation} \Delta b_1 = b_1 - b_0 = \frac{(B_1 - B_0)Y_0 - (Y_1 - Y_0)B_0}{Y_0 Y_1} \tag{2.5} \end{equation}\]
We now substitute \(B_1 - B_0 = \Delta B_1\) and that \(\Delta B_1 = D_1\) in (2.5). We also know that \(Y_1 - Y_0 = \Delta Y_1\) is equal to \(m D_1\) as supposed at the beginning with (2.1). Substituting in (2.5), we obtain:
\[\begin{equation} \Delta b_1 = \frac{\Delta B_1}{Y_1} - \frac{\Delta Y_1}{Y_1}b_0 = \frac{D_1}{Y_1} - \frac{m D_1}{Y_1}b_0 \tag{2.6} \end{equation}\]
Collecting terms, we obtain:
\[\begin{equation} \Delta b_1 = \frac{D_1}{Y_1}(1 - m b_0) \tag{2.7} \end{equation}\]
In order to understand the effect of the deficit (\(D_1\)) on the debt ratio (\(b_1\)), we have to investigate the sign of the term in parenthesis in (2.7). If the term is negative, i.e. \((1 - m b_0) < 0\), a budget consolidation \((D < 0)\) would increase the debt ratio while a budget expansion \((D > 0)\) would decrease the debt ratio. In either case, the result will be counterintuitive.8 Rearranging \((1 - m b_0) < 0\), we obtain:
\[\begin{equation} m > \frac{1}{b_0} \tag{2.8} \end{equation}\]
Equation (2.8) gives us a very important condition. If the spending multiplier (\(m\)) is larger than the inverse of the debt ratio in the initial period (\(b_0\)), a fiscal consolidation has a perverse effect on the debt ratio in the following period (\(b_1\)). It is easy to see that the inverse of the debt ratio is smaller the higher the ratio itself. We now make a numerical example to fix ideas. We assume, for pure hypothesis, a value of the spending multiplier equal to \(1\). Let’s also assume a rather high debt ratio equal to \(160\%\). More specifically we assume that \(Y_0 = 100\) and \(B_0 = 160\) billion of euros. The standard economic policy prescription in this case would be to reduce the debt ratio by means of primary surpluses. So let’s assume that the government commits to a budget surplus of \(5\) billion of euros (i.e. \(D = -5\)). Given the starting conditions that we have assumed here, we can observe that the condition expressed in (2.8) is met, i.e. the value of the spending multiplier (\(m = 1\)) is greater than the inverse of the debt ratio (\(1/b_0 = 1/1.6\)). What happens to the debt ratio in period \(1\) after the fiscal consolidation? Using formula (2.7) we can see that the debt ratio will actually increase.
\[\begin{equation} -\frac{5}{95}(1 - 1 * 1.6) \approx 0.032 \; \text{or} \; 3.2\% \end{equation}\]
With a budget surplus of \(5\) billion of euros, the debt ratio will increase by \(3.2\) percentage points, approximately. This effect is also displayed in figure 2.3. On the left-hand side of 2.3, the black line tells us the change in the debt ratio between time \(t=0\) and \(t=1\) given the debt ratio in period \(t=0\). The equation of the line is described by formula (2.7). The initial value of the debt ratio (\(160\%\)) appears on the right of the intersection of the line with the horizontal axis meaning an increase in the debt ratio. Note that in the graph on the right-hand side of 2.3 the debt ratio bar for period \(1\) appears in orange. This shall emphasizes that the restrictive budget policy has had a counterintuitive effect on the debt ratio.
Figure 2.3: The perverse effect of a restrictive fiscal policy in the short run with a multiplier value of 1.
Why is this effect occurring? After all, GDP will be reduced by \(5\) billion of euros and debt will also be reduced by the same amount. The fact is that the debt ratio is high, so the stock of debt (\(160\) billion of euros) is greater than GDP (\(100\) billion of euros) and a deficit reduction of \(5\) billion of euros will weigh much more on GDP than debt. In other words, in relative terms, the numerator decreases less than the denominator.
Using (2.7), we can derive another important information. We can calculate the threshold value of the debt ratio below which the perverse effect of fiscal tightening does not materialize. Given the same value of the spending multiplier of \(1\), we can easily observe that for values of the debt ratio below \(1\), or better \(100\%\), the unexpected effect of fiscal policy is not realized. In this case, a fiscal tightening leads to a reduction in the debt ratio. This is becomes evident by looking again at the quadrant on the left-hand side of figure 2.3. There we can see that the debt ratio in the period \(t = 0\) (the red dot) ought to be less than \(100\%\) for the fiscal restriction to reduce the debt ratio. \(100\%\) is in fact the value where the debt line is intersecting the horizontal axis.
In the previous example, we assumed a spending multiplier value of \(1\). Much empirical work on estimating multipliers has concluded that multipliers tend to be larger in phases of recessions (see for example Gechert and Rannenberg 2018). We can conclude that a fiscal tightening has a negative effects in particular in those countries experiencing an economic crisis and with high levels of public debt. Let’s repeat the previous example of a fiscal consolidation corresponding to a budget surplus of \(5\) billion of euros. This time we will consider a larger value of the fiscal multiplier, for instance equal to \(2\).
\[\begin{equation} -\frac{5}{90}(1 - 2 * 1.6) \approx 0.12 \; \text{or}\; 12\% \end{equation}\]
Figure 2.4: The perverse effect of a restrictive fiscal policy in the short run with a multiplier value of 2.
With a higher value of the spending multiplier, the negative effect of fiscal tightening is even stronger. The debt ratio increases by approximately \(12\) percentage points. Given the same values of the multiplier and of the initial debt ratio used in the initial example (\(m = 1; b_0 = 160\%\)), what would then happen to debt ratio with a fiscal expansion of \(5\) billion of euros (\(D = 5\))? The reader can try to solve this exercise using the app accessible at the link below. The reader can also try repeating the exercises presented in this chapter or try new ones, for example experimenting with the value of the spending multiplier and the value of the debt ratio to test the condition under which the counterintuitive effect of the budget policies may or may not hold.
2.2 The condition in the long run
We now want to test whether the condition that we have studied in section 2.1 holds for more than one period. In the previous section, we have studied the effect of the deficit on the debt ratio at time \(t = 1\) given a value of the debt ratio at time \(t = 0\). We now investigate what happens once we assume that the change in fiscal policy, whether deficit or surplus, is permanent. That is, that the change in deficit will be the same from period \(t = 1\) through period \(t = n\). By doing so, we must take into account the cumulative effect of the deficit (or surplus) on the stock of debt over time. A series of budget deficits will increase the debt stock while a series of budget surpluses will decrease it. All other variables will remain constant after the policy change. Since we want to study the effect of a change in budget policy between \(t = 0\) and \(t = n\), we rewrite (2.2) taking into account the cumulative effect of the deficit until period \(n\).
\[\begin{equation} B_n = B_0 + \sum_{t=1}^{n} D \tag{2.9} \end{equation}\]
As in the previous section 2.1, we decompose the change in the debt ratio between the debt ratio in the initial period (\(b_0\)) and the debt ratio in period \(n\) (\(b_n\)).
\[\begin{equation} \Delta b_n = b_n - b_0 = \frac{(B_n - B_0)Y_0 - (Y_n - Y_0)B_0}{Y_0 Y_n} \tag{2.10} \end{equation}\]
Note that in equation (2.9) we used the fact that the change in the stock of debt between \(t = 0\) and \(t = n\) is nothing more than the sum of all past deficits (or surpluses). Since we are assuming that \(D\) stays constant, we can rewrite the summation simply as the product of the deficit and the number of periods \(n\), the upper boundary of the summation, therefore \(\sum_{t=1}^{n} D = n D\). Simplifying, we get the following expression:
\[\begin{equation} \Delta b_n = \frac{\sum_{t=1}^{n} D}{Y_n} - \frac{m D}{Y_n}b_0 = \frac{D}{Y_n} (n - m b_0) \tag{2.11} \end{equation}\]
As before, we have to investigate the sign of the term in parenthesis. If \((n - mb_0) < 0\), a budget consolidation (\(D < 0\)) increases the debt ratio. Rearranging the inequality \((n - mb_0) < 0\), we obtain the following condition:
\[\begin{equation} m > \frac{n}{b_0} \tag{2.12} \end{equation}\]
If condition (2.12) is satisfied, a fiscal consolidation has a perverse effect on the debt ratio. Note that if \(n = 1\), the condition corresponds to the one seen above with (2.6). As long as the spending multiplier is greater than the period number divided by the debt ratio, a deficit will have a perverse effect on the debt ratio. We must not forget that \(n\) is all the greater the more time passes. This means that the perverse effect will not last over time. This situation is visualized with figure 2.5.
Figure 2.5: The effect of a restrictive fiscal policy in the long run with a multiplier value of 1.
What do we observe from figure 2.5? We see that the perverse effect of the budget policy lasts for period 1 only (the orange bar indicates the perverse effect of restrictive fiscal policy). In period 1, the debt ratio has increased by approximately \(3.2\) percentage points. Starting in period 2, the debt ratio starts to decline as a result of constant budget surpluses that gradually decrease the stock of debt. The numerator of the debt ratio remains constant from period 1 when the new fiscal policy is implemented (we are assuming no change in the other components of aggregate demand). We can obtain analytically the same result using (2.12). In period 1, we can see that the condition is satisfied, as much as it was in the previous section:
\[ 1 > 1 / 1.6 \]
In period 2, the condition is no longer satisfied nor is it in period 3 and in any of the following periods as the time variable \(n\) in (2.12) becomes larger as time increases.
\[ 1 < 2 / 1.6 \] \[ 1 < 3 / 1.6 \]
\[ 1 < 4 / 1.6 \] \[ ... \]
We can now repeat our exercise by increasing the value of the spending multiplier. This time, we will use a multiplier value of \(2\). Results are shown in figure 2.6.
Figure 2.6: The effect of a restrictive fiscal policy in the long run with a multiplier value of 2.
With a higher value of the spending multiplier, the perverse effect of restrictive fiscal policy will last until period 3 (orange bars). In period 3 the debt ratio is still higher than the initial value. From period 4 onwards, condition (2.12) is no longer satisfied.
\[2 < 3/1.6\] \[2 > 4/1.6\]
\[2 > 5/1.6\]
\[ ... \]
Our app accessible at the link below allows to test interactively the functioning of our condition in the long run. Again, we recommend to experiment with different initial values of debt ratio and of the spending multiplier.
3 Changing government spending
3.1 The condition in the short run
Until now, we have assumed that the government could directly control budgetary decisions. In reality, we know that fiscal policy does not work like this. The government can directly control the amount of public spending. At the same time, the government will be able to decide the rate of taxation.9 The dynamic debt equation therefore becomes as shown in equation (3.1) where the government deficit is defined this time as the amount of public spending (\(G\)) minus tax revenues (\(T\)) net of transfer (just tax revenues or tax income, from now on) where the amount of tax income will depend on the general level of income of the economy, as shown with (3.2).
\[\begin{equation} B_{t} = B_{t-1} + [G_t - T(Y_t)] \tag{3.1} \end{equation}\]
\[\begin{equation} T(Y_t) = t Y_t \tag{3.2} \end{equation}\]
To see if, again, a budget restriction (or expansion) could have a perverse effect on the debt ratio, we need to repeat the same procedure as before.10 Let’s start by pointing out that now the change in income is given by the product of the Keynesian spending multiplier and the change in government spending.
\[\begin{equation} \Delta Y_t = m \Delta G_t \tag{3.3} \end{equation}\]
Given the change in income, there will be a change in tax revenues of the government.
\[\begin{equation} \Delta T_t = t \Delta Y_t \tag{3.4} \end{equation}\]
Assuming that we start from a condition of equilibrium in which the budget is balanced (\(G_0 - T_0 = 0\)), we can rewrite the deficit, i.e. the change in debt, as:11
\[\begin{equation} \Delta B_t = \Delta G_t - \Delta T_t \tag{3.5} \end{equation}\]
At this point it is easy to see that the increase in debt will not equal the increase in government spending. The debt will increase less due to the fact that income has increased and with income tax revenues. Just as in the previous sections, we now need to look at what happens to the debt ratio between period \(t = 0\) and period \(t = 1\) after the change in government spending.
\[\begin{equation} \Delta b_1 = b_1 - b_0 = \frac{(B_1 - B_0)Y_0 - (Y_1 - Y_0)B_0}{Y_0 Y_1} \tag{3.6} \end{equation}\]
First, we plug in (3.3) and (3.5) in (3.6).
\[\begin{equation} \Delta b_1 = \frac{\Delta B_1}{Y_1} - \frac{\Delta Y_1}{Y_1}b_0 = \frac{\Delta G_1 - t m \Delta G_1}{Y_1} - \frac{m \Delta G_1}{Y_1}b_0 \tag{3.7} \end{equation}\]
Collecting the common terms on the right-hand side of (3.7), we obtain:
\[\begin{equation} \Delta b_1 = \frac{\Delta G_1}{Y_1}[1 - m (t + b_0)] \tag{3.8} \end{equation}\]
To see if a change in government spending has a perverse effect on the change in the debt ratio, we need to check the sign of the term inside the square brackets in (3.8). If \([1 - m (t + b_0)] < 0\), a reduction in government spending (\(\Delta G < 0\)) has a perverse effect on the debt ratio. Rearranging the inequality in brackets, we obtain the following condition:12
\[\begin{equation} m > \frac{1}{t + b_0} \tag{3.9} \end{equation}\]
We now simulate the case of a fiscal consolidation. We assume that government spending decreases by \(5\) billion euros, as we did before. Similarly, we assume a value of the multiplier equal to \(1\) and a value of the initial debt ratio equal to \(160\%\). This time we have to make an assumption for the value of the average tax rate (\(t\)). We assume for the moment a value of \(0.4\) or \(40\%\), a rather reasonable value for developed economies.13 What should we expect this time? Using formula (3.8) we can see the debt ratio will increase by \(5.3\) percentage points.
\[-\frac{5}{95} [1 - 1 * (0.4 + 1.6)] \approx 0.053 \; \text{or} \; 5.3\%\]Figure 3.1: The perverse effect of a reduction in government spending in the short run.
Looking at the right-hand side in figure 3.1, we see that public spending has decreased by \(5\) billion euros (green bar) and that GDP has decreased by an equal amount (black bar), as we have assumed a value of the spending multiplier equal to \(1\). However, we see that tax income has decreased by \(2\) billion euros (the red bar). Since the tax rate is set at \(40\) percent (\(0.4 * 5 = 2\)), the stock of debt has decreased by only \(3\) billion (purple bar) and not by the same amount of the initial reduction in government spending.14 Therefore, a reduction in government spending translates only partially into a reduction in debt since tax revenues are reduced along with the fall in income. This is what is observed during a recession. During a recession the deficit automatically increases. As income decreases, tax revenues decrease and transfers (such as unemployment benefits) increase. As a result, public debt can only rise.
In the previous section we have seen that the perverse effect of fiscal policy may work only for certain values of the debt ratio. This is true also in this case. The higher the debt ratio the greater the perverse effect of fiscal consolidation. For values of the debt ratio that lie on the left side of the intersection of the debt line with the horizontal axis (\(60\%\) in this case) in figure 3.1, the counterintuitive effect is not realized. However, given the value of the debt ratio and the tax rate, we can ask an even more interesting question. What is the minimum value of the multiplier for which an increase in government spending would reduce the debt ratio? In table 3.1 we make a back of the envelope calculation assuming different values for the initial debt ratio and the tax rate. The higher the debt ratio (and the tax rate) the lower the value of the spending multiplier for which the fiscal consolidation would be self-defeating.15
| b0 (in %) | t = 30% | t = 40% | t = 50% |
|---|---|---|---|
| 50 | 1.25 | 1.11 | 1.00 |
| 70 | 1.00 | 0.91 | 0.83 |
| 90 | 0.83 | 0.77 | 0.71 |
| 110 | 0.71 | 0.67 | 0.62 |
| 130 | 0.62 | 0.59 | 0.56 |
| 150 | 0.56 | 0.53 | 0.50 |
| 170 | 0.50 | 0.48 | 0.45 |
| 190 | 0.45 | 0.43 | 0.42 |
| 210 | 0.42 | 0.40 | 0.38 |
This leads us to ask ourselves how large are fiscal multipliers in reality. An expenditure multiplier greater than the values shown in table 3.1 would result in a counterintuitive situation where a fiscal tightening turns out to be self-defeating. This has nontrivial implications in terms of economic policy. There has been an heated debate among economists on these issues during the euro zone crisis. The following box offers a brief introduction to the academic literature on multipliers. In the next section instead we extend our model to the multiple periods case.
How large are fiscal multipliers?
The value of fiscal multipliers is not directly observable and determining their magnitude is a complex task. The size of fiscal multipliers depends on several factors. The values of the multipliers depend first of all on the composition of the fiscal stimulus (spending increase or tax reduction) as well as on the spending categories (public investment, public consumption, etc). The business cycle, the monetary and exchange rate policy regime of the country have also a crucial influence on the size of the multiplier. The way multipliers are calculated also plays a key role. Estimates of multiplier values are obtained with economic models like for example DSGE models. Other estimates are obtained using empirical methods, like VAR estimation techniques. A survey of the literature prepared by the IMF (see Batini et al. 2014) suggests that in advanced countries and in “normal times” the value of the multiplier is between 0 and 1. Spending multipliers are also found to be larger than tax multipliers as a reduction in taxes does not translate entirely into a demand increase. The value of the expenditure multiplier is particularly high when monetary policy reaches the so-called zero lower bound (ZLB). The model by Christiano, Eichenbaum, and Rebelo (2011) finds a value of 3.7 for the impact multiplier (i.e. immediately following the fiscal stimulus). Empirical studies have found that fiscal multipliers are larger during periods of economic slack. The meta-regression study by Gechert and Rannenberg (2018) found for a sample total of 1914 estimates that multipliers are significantly larger than 1 during phases of downturns for all spending categories other than government consumption. The study confirms that tax multiplier tend to be generally lower than spending multiplier and insensitive to the economic regime.
The app accessible at the link below allows to test interactively how the condition works in the short run.
3.2 The condition in the long run
We can continue to develop our simple Keynesian model with government spending and tax revenues to test whether the counterintuitive effect of fiscal policy lasts over time. Analogous to section 2.2, we derive the condition that tells how long the perverse effect of government spending will last. The debt dynamic equation is now defined as follows:
\[\begin{equation} B_n = B_0 + \sum_{t=1}^{n} [G_t - T(Y_t)] \tag{3.10} \end{equation}\]
The summation \(\sum_{t=1}^{n} [G_t - T(Y_t)]\) is the sum of all deficits from \(t = 1\) to \(t = n\). Assuming we start from a condition in \(t = 0\) where the budget is balanced, we can rewrite the previous sum as \(\sum_{t=1}^{n} (\Delta G_t - \Delta T_t)\). In addition, since we assume that the change in public spending remains constant in every period after the policy change, we can rewrite the sum as \(n (\Delta G - \Delta T)\), where \(n\) is the upper boundary of the sum. Plugging \(n (\Delta G - \Delta T)\) in (2.10), and remembering that \(\Delta T = t m \Delta G\), we obtain:
\[\begin{equation} \Delta b_n = \frac{\Delta G}{Y_n}[n - m (n t + b_0)] \tag{3.11} \end{equation}\]
If the term in the square brackets is negative, \([n - m (n t + b_0)] < 0\), a reduction in government spending will have a perverse effect on the debt ratio. Rearranging the inequality, we obtain the following condition:
\[\begin{equation} m > \frac{1}{\frac{b_0}{n} + t} \tag{3.12} \end{equation}\]
Assuming, as we did before, an initial value of the debt ratio equal to \(160\%\), a tax rate of \(40\%\) and a value of the spending multiplier equal to \(2\), we can see that the perverse effect of a fiscal tightening will last for several periods. This situation is depicted in figure 3.2. This is due to the fact that the high value of the multiplier (\(2\) in this case) has reduced total income by twice the reduction in government spending (10 billion in spite of the spending reduction of 5 billion). At the same time, tax revenues of the government have fallen (4 billion) so that, at the end, the deficit has decreased less than the initial reduction of government spending (debt is reduced by only 1 billion).
Figure 3.2: The perverse effect of a reduction in government spending in the long run with a multiplier value of 2. The orange bars indicates the perverse effect of the restrictive fiscal policy
This time, the perverse effect of the fiscal consolidation on the debt ratio lasts longer. We use condition (3.12) to check that the perverse effect will last until period 15 and will disappear from period 17.
\[2 > \frac{1}{\frac{1.6}{15} + 0.4}\]
In period 16, the value of the debt ratio is the same as in the initial period (simply because of the combination of values that we have chosen for this example).
\[2 = \frac{1}{\frac{1.6}{16} + 0.4}\]
From period 17, the debt ratio is lower than in the initial period.
\[2 < \frac{1}{\frac{1.6}{17} + 0.4}\]
The app accessible at the link below allows to test interactively how the condition works in the long run. The user can repeat the exercise proposed in the text or use the app freely.
4 Can austerity be self-defeating?
Can austerity be self-defeating? Using some simple “Keynesian arithmetic” we have come to a rather striking conclusion. Under certain conditions a fiscal consolidation aimed at reducing the debt ratio can have the opposite effect. The fall in GDP is so large that the debt ratio gets actually worse. The same holds true for a fiscal expansion. Under certain circumstances an increase in the deficit leads to a boost in economics activity and to a reduction of the debt ratio. The initial value of the debt ratio plays a key role. The unexpected effect is realized for high values of the debt ratio. The value of the spending multiplier also plays a key role. The higher the value of the multiplier, the greater the counterintuitive effect. This led us to ask how large fiscal multipliers actually are. There is no clear consensus in the macroeconomic literature on the size of fiscal multipliers. However by doing some simple back of the envelope calculations and assuming different values of the debt ratio we have seen that the critical value of the multiplier for which the unexpected effect is realized is quite low, especially for large values of the debt ratio. This means that small multipliers are enough for fiscal consolidation to be self-defeating. We must also not forget a simple fact. Government spending is part of GDP (we refer obviously only to government demand and not all public spending). Any reduction in government demand (or increase) results in an immediate decrease (or increase) in GDP by the same amount. In figure 4.1 we report the growth rate of public demand (horizontal axis) and the growth rate of GDP (same time period, vertical axis) for euro zone countries during the apical years of the euro crisis (2010-2013). Countries where public demand has been reduced the most are those countries where GDP has recorded the largest drop.16
Figure 4.1: Positive relationship between real growth rate of public demand and real GDP growth rate between 2010 and 2013 for euro zone countries. The size of the bubble corresponds to the debt ratio in 2013. Source: AMECO.
Does austerity reduce debt in the long run? Expanding our Keynesian arithmetic to the case of multiple periods, we have seen that the counterintuitive effect tends to vanish over time. We have assumed that the change in fiscal policy (a deficit or a surplus) remains constant over time. So while GDP remains constant (the effect of fiscal policy is the same in all periods so that GDP does not change anymore after the new fiscal policy) the stock of debt decreases leading to a reduction of the debt ratio. This is the conclusion of the paper by Codogno and Galli (2017) (on which our exposition is partially drawn). To this same conclusion comes also a study by some economists of the European Commission (Boussard et al.’s 2012). The authors proposed a theoretical formulation17 according to which the undesirable effect of fiscal consolidation may last only a few years depending crucially on the value of the fiscal multiplier. Differently from what we have done here, the authors introduce the interest rate on public debt. When the interest rate depends endogenously on the debt ratio, the undesirable effect of the fiscal consolidation is only temporary and tends to fade away after 2-3 years. The counterintuitive effect is prolonged over time only when it is assumed that the value of the fiscal multiplier is high, as during phases of economic slack, and that financial markets are myopic, i.e. they change the interest rate only by looking at debt of the period immediately following the fiscal consolidation (i.e. when the unexpected effect occurs). This is the only case in Boussard et al.’s (2012) model when a fiscal consolidation could have a long lasting self-defeating effect. A number of conditions that are unlikely to occur together, the authors conclude. Other theoretical models of heterodox inspiration suggest different results. The model proposed by Ciccone (2011), for example, comes to the opposite conclusion. Ciccone (2013) includes private investment in the model, something from which we have abstracted in our discussion. In the model, government spending influences demand, which in turn influences private investment. In this way, a fiscal consolidation is assumed to have a negative feedback on investment, with permanent self-defeating effects on the debt ratio. In this direction also goes the interesting exercise proposed by Garbellini (2016) where the policy variable become the growth rate of public spending. Here, is the rate of growth of government spending which if appropriately chosen (within certain conditions) would stabilize the debt ratio. This reminds us of the simple stock-flow consistent (SFC) model called SIM developed by Godley and Lavoie (2012) (so simple that it was called SIM, for simple) where an exogenously determined rate of growth of government spending reduces the debt ratio towards a new steady-state below the initial one.18
To conclude. Does austerity increase the debt-to-GDP ratio? In the short run, the answer is clearly yes. Given the conditions that we have discussed above, a reduction in spending has a negative effects on GDP, which in turn worsens the deficit and ultimately the debt ratio. The same conclusion is shared by numerous studies and models, as well as by the experience for some euro zone countries.19 Is the effect permanent? Here views are divided. In the long-run model that we have presented in this text, the debt ratio decreases after an initial increase that will last, however, only for some periods (depending on the values of the tax rate, the spending multiplier and the initial debt ratio). We, however, believe otherwise. Austerity has long-term effects on output and growth with negative effects on public finances. Reducing public investment programs, cutting health and education spending have clear long-term effects on the society. A less educated society with crumbling public infrastructure cannot be a driver of long-term (and environmentally sustainable) economic growth. In an context of low growth and stagnation, public finances can only deteriorate. Is austerity needed to keep interest rates on government debt low? The COVID crisis has shown that the central bank, the ECB in the European situation, is able to keep interest rates low in spite of the dramatic increase in the deficits and debts.
List of variables
| Abbreviation | Name | Color* |
|---|---|---|
| \(B\) | Stock of public debt | # |
| \(b\) | Debt-to-GDP ratio | # |
| \(D\) | Government deficit | # |
| \(G\) | Government spending | # |
| \(m\) | Spending multiplier | # |
| \(n\) | Numbers of periods | # |
| \(T\) | Taxes | # |
| \(t\) | Tax rate | # |
| \(Y\) | Total income | # |
| * Color in pictures (if applicable) |
References
See, for example, Giavazzi and Pagano (1990) and more recently Alesina, Favero, and Giavazzi (2019).↩︎
In reality, we know that the crisis had quite different causes. With “sovereign debt crisis”, we refer to the mainstream interpretation.↩︎
For an introduction to the idea and functioning of the fiscal multiplier as well as its algebraic derivation see our earlier work Prante et al. (2022) and in particular chapter 6. For a more advanced introduction see Batini et al. (2014).↩︎
The analysis is initially presented in a box in the 2012 World Economic Outlook and then extended in an IMF working paper published in 2013. In the same year, the research then appears in the journal American Economic Review: Papers and Proceedings.↩︎
The sample include member countries of the European Union (United Kingdom included but without Estonia and Latvia) plus Iceland, Norway and Switzerland.↩︎
The regression coefficient is statistically significant below the 1% significance level. The \(R^2\) of the regression is approximately 0.5 (50%).↩︎
This condition is also discussed in a box in an early version of a paper by Blanchard, Leandro, and Zettelmeyer (2020) titled Revisiting the EU fiscal framework in an era of low interest rates but it disappears from following editions.↩︎
In the first case we have the product between two negative terms whose result is obviously positive, meaning an increase in \(b_t\). In the second case we have the product between a negative term and a positive term whose result will be negative, that is a reduction of \(b_t\).↩︎
We are assuming an average tax rate for the entire economy.↩︎
Our explanation here is inspired by the algebra in Codogno and Galli (2017). For sake of simplicity we abstracts from the interest rate.↩︎
We start with the fact that \(\Delta G_1 = G_1 - G_0\) and \(\Delta T_1 = T_1 - T_0\). Now we see that the deficit at time 1 is equal to the change in the deficit, i.e. \(G_1 - T_1 = \Delta G_1 - \Delta T_1\) which in turn is equal to \(G_1 - T_1 = G_1 - G_0 - (T_1 - T_0)\), only in the case where \(G_0 - T_0 = 0\) i.e. a starting situation where the deficit is balanced.↩︎
This is the same condition presented in Leão (2013).↩︎
See for example the Global Revenue Statistics Database of the OECD (2021).↩︎
The equation of the black line in the first quadrant of figure 3.1 is given by equation (3.8).↩︎
This type of exercise applied to different countries is done more rigorously (in particular regarding the estimation of the tax parameter) in many of the articles on which the model and discussion presented in this text is based. See as example Leão (2013), Di Bucchianico (2019) and also Boussard et al.’s 2012 and Garbellini (2016).↩︎
We also observe that the reduction in public spending has been the largest in those countries with the highest debt ratio (size of the bubble in the graph).↩︎
In essence, similar to ours but a bit more sophisticated.↩︎
We cannot go into the details of the models. Therefore, personal study is recommended. The SIM model in particular is very interesting. A recommended reading for anyone who plans to venture into the post-Keynesian SFC literature.↩︎
See in this regard the very interesting article by Storm (2019) on the Italian case.↩︎