# Fiscal calculations on the Greek deficit

In previous Notes for Discussion, we addressed a possible path for real GDP, the deflator as a measure of prices and inflation, and the path for nominal GDP. We concluded that the projection for nominal GDP yielded an average growth rate between 2018-2080 of around 2.5 percent a year, with somewhat higher nominal growth in the 2020s as unemployment is coming down, weaker growth in 2030s-2040s as the demographic pull of a declining population restricts employment, and then a transition to around the long-term average again from 2050 onward.

Needless to say, these numbers come from considering population projections, demographic transition toward an elderly population and aspirational assumptions about productivity growth and future inflation, so the projections are not predictions or guarantees of what will happen, but a reasoned narrative of what is most likely to happen, given the assumptions we have stated.

It is now possible to demonstrate some logical implications of this nominal growth analysis for fiscal policies. We will proceed from the most general to the more specific. The most general is the implication for the path of the fiscal balance and the debt. The more specific is all about how to organize this path for a fiscal balance and considering the infinite possible combinations of revenue and expenditure that may get the country to this path. Obviously, only a small subset of revenue and expenditure combinations will be realistic from a political and social point of view, so we need not worry excessively about sets of unbounded possibilities.

Let us describe nominal GDP with the letter Y and the stock of public debt with the letter B (reminiscent of “bonds”). Then the debt ratio to GDP can be reflected as B/Y, which we may describe by the lower-case letter b = B/Y. The debt ratio in Greece at end-2017 was b = 179 percent of GDP (roughly 317 billion euros), and the Maastricht criterion on debt as agreed between the euro-area partner countries is a ceiling of b = 60 percent of GDP, so there is a lot of work to do to get the Greek ratio down – this will take time. And following up on a previous Note on what good policymaking and good communication is all about, the government might wish to indicate how this can be done, and how much time this could take.

If b is the debt ratio, then what is the rate of change in the debt ratio? After all, we are interested in knowing the level of the debt ratio, but also whether it is going up or down. If the debt ratio is b, then I am going to describe the rate of change in the debt ratio as “delta b” or Δb. Further, using some logarithmic mathematics, we can show that the rate of change in the debt ratio is equal to the rate of change in the debt minus the rate of change in nominal GDP: Δb = ΔB – ΔY.

This equation has intuitive appeal. If you have a ratio of two variables, then, upon reflection, it must be true that the ratio will go up if the rate of change of the numerator exceeds the rate of change in the denominator, and vice-versa. If debt increases by 5 percent and income grows at 2 percent, then the debt is rising at a faster rate than income and the debt ratio must increase.

Let us go back to nominal GDP growth in Greece, as projected, which we suggested was about 2.5 percent a year over the long run. For the debt ratio to come down, this means that the stock of debt must grow slower than 2.5 percent a year. The stock of debt is about 317 billion euros. This means that the debt stock should increase by less than 0.025 * 317 = 7.9 billion euros in 2018 for the debt ratio to come down (this number varies over time as GDP and the debt stock themselves vary, but we are using long term averages to get an idea how this debt reduction would work).

So, if the debt should increase by less than 7.9 billion euros, then the “true” fiscal deficit should be less than 7.9 billion euros, because the increase in the debt is what I call the “true” deficit. It is the amount of new debt that needs to be placed to finance the deficit. Further, if GDP is around 182 billion euros in 2018, then the (local) deficit (= the increase in the debt) should not be larger than 7.9/182 = 4.3 percent of GDP for the debt ratio to start coming down. Forecasts from a number of institutions on the general government balance for Greece suggest a range from a small deficit to a small surplus (let us say between -0.5 to +1 percent of GDP). Since this remains below the (local) threshold value of 4.3 percent of GDP, we should expect the debt ratio in 2018 to decline.

One interesting aspect of debt reduction is that over time, as the debt ratio declines, it becomes in some way more difficult to keep it on a downward trajectory. Why is this so? Consider above that with a debt stock of 317 billion euros, or some 179 percent of GDP, it took a budget deficit below 4.3 percent of GDP to lower the debt ratio. Now imagine that the debt ratio had been 70 percent (127 billion euros) instead – what “true” deficit could be afforded this time to bring the debt ratio down? Well, 2.5 percent of 127 = 3.2 billion euros, or the equivalent of 1.8 percent of GDP. Thus, as the debt ratio comes down, the tolerated “true” deficit to continue to bring the debt ratio down becomes smaller. One aspect that is helpful is that with less debt, the government has to pay less interest, which makes the deficit objective easier to reach, but yet, many political systems find it difficult to keep lowering the deficit as the debt ratio improves.

There is a second aspect of great interest in these debt ratio and deficit calculations. Some countries report deficits that are smaller than the increase in their debt would suggest. The debt often increases by amounts that are bigger than the reported (general government) deficit. Why might that be the case? It is so, because accounting is intricate and communication is not always good or complete. The government may not include in the deficit all operations that lead to debt, but that are not reported as a “deficit.” For instance, bank bailouts are often excluded from the deficit. Sometimes one-time operations to help state enterprises are excluded from the deficit. Indeed, the perimeter of calculating the deficit may not include all state-owned enterprises or other off-budget entities that each generate debt and should get reflected in the overall “true” public deficit and debt.

Thus, to keep track of these calculations, it is crucial to cast the broadest possible fiscal perimeter and include all entities that are owned by the tax payers via the public sector (to give the “true” deficit), not just the budget of the general government (the “reported” deficit). Conversely, it is important to check whether the amount of deficit that gets reported is equal to the amount of increase in the debt. If not, the ministry of finance should clarify this discrepancy.

Exploring differential calculus a bit further, we can report another interesting relationship between the deficit, the growth rate of the economy, and the debt ratio. We can ask if the deficit, in percent of GDP, is a constant number, and the nominal growth rate of the economy is a constant number, will the debt ratio eventually become unbounded, or is there some limit where the debt ratio will stop? The answer to this question is that the debt ratio will reach an asymptote – it gets closer and closer to a certain fixed value as the time series is longer and longer. Thus, it generates a bounded, and not an unbounded, debt ratio. The value of this asymptotic number is a simple expression: b = d/y, whereby b is again the debt ratio; d is the deficit in percent of GDP, and y is the nominal growth rate of the economy. I will give two examples of why this calculation is interesting.

One, consider the Maastricht criteria that call on countries to lower the debt ratio to below 60 percent of GDP. They also place a limit on the deficit of 3 percent of GDP. What is less known is that they also assume a nominal growth rate for GDP in EU countries around 5 percent of GDP (an aspirational growth rate of 3 percent combined with target inflation of 2 percent). With the above formula for our asymptotic debt ratio as d/y we can see that 3/5 = 0.6 or 60 percent of GDP. Thus, the Maastricht criteria have an interior logic that is not always appreciated, or known by the general public.

Two, let us consider what this insight means for Greek fiscal policies over the long run. We have suggested that nominal GDP growth will be around 2.5 percent a year. If Greece wants to lower the debt ratio to 60 percent of GDP, what is the steady state deficit that would do the trick? It is 2.5 * 0.6 = 1.5 percent of GDP. Thus, if the “true” deficit in Greece is kept at 1.5 percent of GDP or below, and nominal GDP growth over time averages some 2.5 percent a year, then the debt ratio will gravitate in the limit to 60 percent of GDP without the need for debt forgiveness. We now arrive at a new fiscal aspirational objective: keep the overall “true” deficit of the public sector below 1.5 percent of GDP. Indeed, if nominal growth in Greece is only half of what is assumed in the Maastricht criteria, then the deficit ceiling will also be half of what the Maastricht criterion suggests, to reach the same objective debt ratio of 60 percent of GDP.

**Closing thoughts**

– Given the assumptions we have made, it is possible to derive an aspirational objective or ceiling for the overall public sector “true” deficit – it is 1.5 percent of GDP or below.

– This deficit is a ceiling that needs to be observed permanently.

– Our calculations are deterministic; this means that we assume that the world moves in nice gradual fashion with hardly any volatility in sight. This is where the devil is in the details. There will be volatility. Thus, when times are good, the government may wish to have a public sector deficit that is well below 1.5 percent of GDP; when times are difficult and it can be established with some confidence that the economy is operating below potential, then some countercyclical stimulus with slightly higher deficits can be contemplated, if they can be financed at reasonable cost.

– Watch the increase in the debt – the “true” deficit – and compare this number to the “reported” deficit; is there a difference between the reported and true deficits?

– The above only refer to the “what” of deficit and debt dynamics; it says nothing about “how” a government can manage its fiscal affairs in a complex political setting to get there. The Swiss mathematical genius, Leonard Euler (1707-1783), has given us beautiful equations for dynamic analysis (Euler equations and transversality, or optimality, conditions underpin the formulas above); he has not solved human nature…Therefore, sound and mutually supportive politics remains essential.

– In the next three Notes for Discussion, we are going to look at potential mismeasurement of the fiscal deficit (which is not the “true deficit” that we need to measure), and two factors that are likely to impart autonomous pressure on the debt in the future. All three factors have proven highly relevant for the case of Greece in the past.

* Bob Traa is an independent economist. This is the ninth in a series of articles by him for Kathimerini titled “Notes for Discussion – Essays on the Greek Macroeconomy.”