# 8 The effects of inflation and interest payments on public indebtedness

So far, we have kept our analysis very simple. We have used a very basic decomposition of public debt, which ultimately only separated the effects of the overall deficit as a % of GDP and the growth rate of nominal GDP. Let us now turn to a slightly more complex decomposition of public debt and the evolution of the debt ratio. To do this, we need to introduce two other important macroeconomic determinants of public indebtedness: the inflation rate and the interest rate.

## The rate of inflation

As we have already mentioned, and as you probably already know, the growth rate of nominal GDP is the result of two different processes. On the one hand, actual economic activity can increase, which leads to more goods being produced. On the other hand, the prices of these goods can rise, which is the process of inflation. Together, real economic growth and inflation determine the growth rate of nominal GDP. A commonly used approximatino of the rate of growth of nominal GDP is this:^{10}

\[\text{Growth rate of nominal GDP} \approx \text{Growth rate of real GDP} + \text{Inflation}\]

Recalling the definition of the debt ratio (\(\text{Debt-to-GDP ratio} = \frac{\text{Total public debt}}{\text{GDP}}\)), one immediately sees that if we assume that only the inflation rate changes and everything else remains constant, a higher inflation rate would mean a higher growth rate of nominal GDP and the debt ratio would thus tend to fall. The same, of course, applies to a higher real GDP growth rate in the absence of other changes.

## The interest expenditure of the government

So far we have not distinguished between the government’s interest payments and its other expenditures. However, interest payments are conceptually somewhat distinct from the other objectives of fiscal policy (provision of public goods, income and wealth redistribution, resource allocation, stabilisation, as briefly described in section 1). In a sense, we could view interest payments as the result of the state’s past efforts to fulfil these other primary objectives of fiscal policy.

However, interest payments are an important determinant of the change in public debt. To take them explicitly into account, we can distinguish between the total deficit and the so-called **primary deficit** of the government. The total public deficit, which we used so far in our contemplations, was already defined at the beginning of the course and it includes interest payments. In contrast, the primary deficit is simply the total deficit excluding interest payments:

\[\text{Primary deficit} = \text{Total deficit} - \text{interest payments}\]

As we did with the total deficit, we can of course also state the primary deficit in relative terms to the size of the economy:^{11}

\[\text{Primary deficit-to-GDP ratio} = \frac{\text{Primary deficit}}{\text{GDP}}\]

The interest rate that the government pays on its debt obligations, but which is not directly controlled by the government, affects the difference between the total deficit and the primary deficit. Changes in the interest rate can therefore have a large impact on the overall deficit, even if the government’s stance on primary expenditure categories and tax revenues does not change. If the interest rate rises, public debt will increase in the absence of other changes.

## An extended decomposition of the long-run debt ratio

With the explicit accounting for inflation and interest rate, we can also account for both of them in the equation for the long-run debt-ratio.

\[\text{Debt-to-GDP ratio} = \frac{\text{Primary deficit}}{\text{Real growth rate} - \text{real interest rate}}\]
You might wonder where inflation is in this equation. It enters through the **real interest rate**, since the nominal interest rate in monetary terms and the inflation rate together determine the real interest rate. The nominal interest rate indicates the monetary value that must be paid for a loan (per time period). The real interest rate instead indicates the value of this payment in terms of actual goods or, simply put, in real terms.

To make things a little more concise, we can also denote the debt ratio by \(b\), the primary deficit by \(pd\), the real GDP growth rate by \(g\), the nominal interest rate by \(i\) and the inflation rate by \(\pi\). Let us also define the real interest rate as \(r = i - \pi\).^{12} The equation for the long-term debt ratio then reads:

\[b = \frac{pd}{g - r}\]
However, convergence to this long-run value is only possible if the real growth rate exceeds the real interest rate. You might have heard about the importance of the r-g-differential – the **differential between the real interest rate and the real growth rate** – in the context of the debate about public debt sustainability and fiscal policy. The equation above already reveals why this differential is important. To see this even more clearly, we can make use of the decomposition of the change in the debt ratio. The movement of the debt-ratio over time, which we denote by \(∆b\), can be summarised by this equation:^{13}

\[∆b = pd + (r – g)b\] In order to stabilise the debt ratio (its change should be zero: ∆b = 0), the following must apply:

\[b \leq \frac{-pd}{r-g}\]
That is, if the real interest rate is greater than the real growth rate (\(r > g\)), a primary surplus (\(pd < 0\)) is required to prevent the debt ratio \(b\) from increasing forever. If, on the other hand, the real growth rate exceeds the real interest rate (\(r < g\)), the government can run permanent primary deficits (\(pd > 0\)) without increasing the debt ratio.^{14}

**Inflation, interest and the long-run debt-ratio**: In this app, you can insert values for the primary deficit, the real growth rate, the inflation rate and the nominal interest rate and observe to which value the debt-ratio converges.

**The debt-ratio with inflation, interest and simple feedbacks to GDP**: In this app, you can insert shocks to real GDP and inflation and also choose the effects of fiscal policy on real GDP and inflation. Again this is connected to the data of actual countries. Simulations are for illustrative purposes only.

**Important assumptions and limitations**: There are no cyclical feedback effects on government revenue and expenditure. We will change this in the next section.