Public debt

5 Analyzing debt dynamics

A simple equation can help us understand how the debt-to-GDP ratio changes over time as a result of economic conditions and policies. We start with the numerator of the ratio, debt. The change in debt is the difference between government expenditure and government revenues, as explained in Section 2. When a government spends more than it earns, it finances the difference by issuing debt. Conversely, when a government spends less than it accrues in revenue, it uses the surplus to extinguish debt or accumulate assets. Thus, debt today (\(\text{B}\)) is equal to debt in the previous period \((\text{B}_{t−1})\) plus government expenditure (\(\text{TE}\)) minus government revenues (\(\text{R}\)). The letter \(\text{B}\) stands for bonds, which is the most frequent form of government debt (although \(\text{B}\) here includes all types of government debt, not only bonds). Debt today is calculated as:

\[\begin{align*} \text{B} &= \text{B}_{t − 1} + \text{TE} − \text{R} &&(1) \end{align*}\]

Note that variables without a subscript refer to the current period. While formal notation uses a subscript of t for all variables in the current period, this can look cluttered so we have chosen to exclude it.

primary deficit

The government deficit (its revenue minus its expenditure) excluding interest payments on its debt. See also: government debt.

It is useful to distinguish interest payments from other expenditures such as public employee salaries, social transfers, and military procurement. The non-interest part is referred to as ‘primary’ expenditure. Correspondingly, the difference between primary expenditure and total revenues is the primary budget deficit (also known as the primary deficit). As total expenditure is equal to primary expenditure (\(\text{E}\)) plus interest payments (\(\text{INT}\)), we can write that debt is equal to:

\[\begin{align*} \text{B} &=\text{B}_{t−1}+\text{INT}+\text{E}−\text{R} &&(2) \end{align*}\]

Since the primary deficit (denoted by \(\text{D}\)) is equal to \(\text{E}−\text{R}\), the equation can be rewritten as:

\[\begin{align*} \text{B} &=\text{B}_{t−1}+\text{INT}+\text{D} &&(3) \end{align*}\]

We distinguish the primary deficit because policymakers have the capacity to adjust it by changing either taxes or primary expenditure. Interest payments, in contrast, are largely determined by the size of the debt (which is due to past decisions) and by the interest rates demanded by investors in order to hold government debt securities.

real interest rate

The interest rate corrected for inflation (that is, the nominal interest rate minus the rate of inflation). It represents how many goods in the future one gets for the goods not consumed now.

We now define \(b\) as the debt-to-GDP ratio \((b=\text{B}/Y\), where \(Y\) refers to GDP), and \(d\) as the primary deficit-to-GDP ratio \((d=\text{D}/Y)\), where we use lowercase letters to denote variables that have been divided by GDP.

Real (net of inflation) interest expenditure as a share of GDP is equal to the real interest rate (the interest rate net of inflation), denoted \(r\), multiplied by the debt-to-GDP ratio \(b\). The overall deficit (as a share of GDP) is then the primary deficit plus interest payments: \(d+rb_{t−1}\). Note the use here of \(b_{t−1}\) instead of \(b\). This is because the government pays interest on the debt contracted in the previous period.

So far, we have seen that the debt-to-GDP ratio will change when the overall deficit (as a share of GDP) changes. If, for the moment, we assume that the economy is not growing then:

\[\begin{align*} \Delta b &=d+rb_{t−1} &&(4) \end{align*}\]

However, the equation above ignores the rate at which the economy grows, which also affects how the debt-to-GDP ratio changes. A given size of debt is a smaller proportion of GDP if the economy grows. Equivalently, the government will have greater resources available to pay back its debt.

If \(g\) is the rate of growth of GDP, where \(g\) is defined like this: \(Y_t=Y_{t−1}+gY_{t−1}\), then, as set out in more detail in ‘Find out more: Debt restructuring’, when GDP changes, we can write the change in the debt-to-GDP ratio as:

\[\begin{align*} ∆b &=d+rb_{t-1}-gb_{t-1} \\ ∆b &=d+(r-g) b_{t-1} &&(5) \end{align*}\]

The steps to derive Equation 5 are set out in ‘Find out more: Derivation of the basic equation for debt dynamics’. If the initial debt-to-GDP ratio is, say, 100%, \(r\) is 4%, \(g\) is 3% and \(d\) is 1%, the debt-to-GDP ratio will increase by 2 per cent:

\[\begin{align*} ∆b=0.01+(0.04-0.03)×1=0.02 \end{align*}\]

If, however, \(r\) is 3% and \(g\) is 4%, the debt-to-GDP ratio will remain constant, even if the country runs a deficit equal to 1% of GDP. Use Equation 5 to verify this for yourself.

Find out more Derivation of the basic equation for debt dynamics

Here we use some basic arithmetic manipulations to derive Equation 5. We want to show that \(b−b_{t−1}≡ ∆b \approx d+(r−g)b_{t−1}\), where \(\text{B}\) is the stock of debt, \(d\) is the primary deficit, \(Y\) is GDP, \(b\) and \(d\) are the debt-to-GDP ratio and the primary deficit-to-GDP ratio (that is, \(b=\text{B}/Y\) and \(d=\text{D}/Y\)), and \(r\) and \(g\) are the real interest rate and real growth rate respectively.

Start by writing:

\[\begin{align*} \text{Δ}b \equiv b − b_{t − 1} \equiv \frac{\text{B}}{Y} − \frac{\text{B}_{t − 1}}{Y_{t − 1}} &&(A1) \end{align*}\]

The stock of debt today is equal to the stock of debt in the previous year plus primary expenditure (\(\text{E}\)), plus interest payments, minus government revenues (\(\text{R}\)).

\[\begin{align*} \text{B} &= \text{B}_{t-1}+\text{E}-\text{R}+r\text{B}_{t-1}\\ \text{B} &= \text{B}_{t-1}+\text{D}+r\text{B}_{t-1} &&(A2) \end{align*}\]

Analogously, real GDP is equal to real GDP in the previous year plus real GDP growth: \(Y = Y_{t−1}\)(1 + \(g\)). Substituting A2 into A1:

\[\text{Δ}b = \frac{\text{B}_{t − 1} + \text{D} + \text{rB}_{t − 1}}{Y_{t − 1}(1 + g)} − \frac{\text{B}_{t − 1}}{Y_{t − 1}}\]

Using \(Y_{t-1} (1+g)\) as the denominator:

\[\mathrm{\Delta}b = \frac{\left( \text{B}_{t - 1} + \text{D} + r\text{B}_{t - 1} \right) - \text{B}_{t - 1}(1 + g)}{Y_{t - 1}(1 + g)}\]

Or, simplifying:

\[\mathrm{\Delta}b =\frac{\text{D} + r\text{B}_{t - 1} - g\text{B}_{t - 1}}{Y_{t - 1}(1 + g)}\] \[=\frac{\text{D}}{Y_{t - 1}(1 + g)} + \frac{r\text{B}_{t - 1} - g\text{B}_{t - 1}}{Y_{t - 1}(1 + g)}\]

And, using the fact that \(Y=(1+g)Y_{t-1}\),

\[\text{Δ}b = \frac{\text{D}}{Y} + \frac{\text{rB}_{t − 1} − \text{gB}_{t − 1}}{Y_{t − 1}(1 + g)}\] \[= d + \frac{(r − g)}{(1 + g)}\frac{\text{B}_{t-1}}{Y_{t-1}}\] \[= d + \frac{(r − g)}{(1 + g)}b_{t − 1}\]

Since \(g\) tends to be small, \(\frac{r − g}{(1 + g)} \approx (r − g)\) and \(\text{Δ}b \approx d + (r − g)b_{t − 1}\).

Note that the standard formula described above in the text, which is Equation 5, is an approximation. We should have written \(∆b≈d+(r−g) b_{t−1}\).

We can now put Equation 5 to work. Say government officials want to calculate the primary deficit or surplus necessary to keep the country’s debt-to-GDP ratio constant. They then set \(∆b\) = 0, and solve Equation 5 to obtain:

\[\begin{align*} d &=(g − r)b_{t−1} &&(6) \end{align*}\]

Thus, the primary deficit compatible with keeping the debt-to-GDP ratio stable will rise with the GDP growth rate \(g\), but fall with the interest rate on government securities \(r\). To understand this, notice that faster GDP growth reduces the debt burden by raising the denominator of \(b\) over time, and a higher interest rate means that there are fewer resources to devote to debt retirement (or the need to issue additional debt to meet interest payments). Thus, \(r-g\) is critically important for debt dynamics. ‘Find out more: The importance of r − g in the history of debt dynamics’ summarizes the global historical record.

Find out more The importance of r − g in the history of debt dynamics

As we’ve seen, the dynamics of public debt depend importantly on the difference between the real (inflation-adjusted) interest rate \(r\) and the economy-wide rate of economic growth \(g\). The financial historian Paul Schmelzing has heroically assembled information on government bond yields for the now-advanced (high-income) economies (which include Europe and, in the subperiod from when it existed as a sovereign nation, the U.S.) and estimated growth rates over the last eight centuries (Figure 5). There’s lots of volatility, but \(r-g\) shows a tendency to trend downward over the long sweep of historical time.

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https://books.core-econ.org/insights/public-debt/05-analyzing-debt-dynamics.html#figure-5

Figure 5 700 years of global \(r-g\) for the now-advanced (high-income) economies.

Own elaborations based on data from Schmelzing, 2020.

Several factors contributed to this development. The cost of borrowing declined as sovereign debt came to be recognized as an obligation of the state rather than of the individual occupying the throne, with the establishment of parliaments and legislatures in which creditors were represented, and with the creation of markets in which debt securities could be bought and sold. Meanwhile, growth rates went up with the industrial revolution and the transition to modern economic growth in the nineteenth and twentieth centuries.

There has been a further fall in \(r-g\) in high-income economies in the last four decades, prompting debate about whether prevailing levels of debt are becoming less worrisome. To some extent, current low interest rates on government debt reflect high savings rates in emerging economies such as China and the high savings propensities of the wealthy, who have enjoyed disproportionate income gains in high-income economies. In addition, they reflect a safe-asset shortage due to the value investors attach to safe and liquid government bonds in a volatile economic and financial environment. Whether these favorable conditions for public-debt management will continue to prevail going forward is anyone’s guess.

Imagine now that the growth rate of the economy is larger than the interest rate. The country then can run a primary deficit (a positive value of \(d\)) and still keep its debt-to-GDP ratio from rising. For instance, if \(g\) is 3%, \(r\) is 1% and the initial debt-to-GDP ratio is 50%, the country can run a primary deficit of 1% of GDP without its debt-to-GDP ratio rising.

The opposite is true, however, if the interest rate is higher than the growth rate. If \(g\) = 1%, \(r\) = 3%, and \(b_{t−1}\) is again 50%, then keeping the debt ratio stable will require a primary (budget) surplus of 1% of GDP (\(d\) will have to be −1%). The surplus required to stabilize the debt will increase with the size of the initial debt ratio. For instance, with \(g\) = 1%, \(r\) = 3%, and \(b\) = 100%, debt stabilization now requires a primary surplus of 2% of GDP. A primary surplus smaller than that will cause the debt ratio to rise without limit.

Another special case is when \(r\) = \(g\). Then, as Equation 5 shows, the increase in the debt-to-GDP ratio is equal to the primary deficit ratio \(d\). In other words, the debt ratio remains constant when primary expenditure (non-interest spending) is limited to tax revenues.

Using the public debt simulator

In the public debt simulator, in SIM1, you can choose your own values of \(b\), \(g\) and \(r\) to compute the debt stabilizing primary deficit.

Follow the steps in Figure 6 to understand what happens to government debt in four different scenarios.

Exercise 2 Plotting the dynamics of debt

For this exercise, you will need sheet SIM1 of the public debt simulator.

1. Choose two countries from the list provided, such that the debt-to-GDP ratio continually increases for one and stabilizes for the other. Calculate the debt-stabilizing primary deficit for each country using the spreadsheet.
2. Draw by hand the geometric representation of the debt dynamics for each country, as in Figure 6.
3. Use your diagram to explain the dynamics of debt in each country. (Hint. You can check your working by using sheet SIM2 of the public debt simulator.)

Figure 7 uses this setup to analyze the evolution of the debt of a country that starts the year 2020 with a 50% debt-to-GDP ratio (SIM3 of the public debt simulator).

The purple line plots a situation in which \(d\) = \(g\) = 0% and \(r\) = 2%. The economy is not growing and the government is not running a primary deficit, but the interest rate on the debt is positive. Given that \(g < r\), and \(d\) = 0%, the debt-to-GDP ratio will rise to 51% in 2021, 52.02% in 2022, and 80.4% in 2045. The debt ratio rises exponentially (it rises by larger amounts in later years) because in every successive year the difference between \(r\) and \(g\) is applied to a higher level of debt.

The orange line in Figure 7 depicts the situation where \(g < r\) and the country runs a deficit (\(d\) = 1%, \(g\) = 0%, \(r\) = 2%). With primary spending now exceeding revenue, the debt ratio rises even faster than when \(d\) = 0%.

The red line (\(d\) = 1%, \(g\) = 2%, \(r\) = 1%) is when \(g > r\). Since the country is growing and \(g > r\), the denominator of the debt-to-GDP ratio grows faster than it accumulates interest obligations, so the government can run a primary deficit and still maintain a constant debt ratio. When \(g-r\) = 1% and \(b_{t−1}\) is 50%, as here, the debt stabilizing primary deficit is 0.5%. Since the country runs a primary deficit of 1% of GDP in the case shown, debt will increase until it reaches 100% of GDP. At this point it will stabilize, since the condition \(d\) = (\(g-r\)) \(b_{t−1}\) is finally met. But when \(g > r\), the debt-to-GDP ratio never grows to infinity (see also Panel C of Figure 6).

The blue line (\(d\) = 1%, \(g\) = 3%, \(r\) = 1%) is when \(d\) = (\(g-r\))\(b_{t−1}\). In this case, the debt remains constant at its initial value of 50%.

The green line (\(d\) = −2%, \(g\) = 2%, \(r\) = 3%) is when \(g < r\), but the country runs a surplus that more than compensates for this difference, causing the debt-to-GDP ratio to decrease rapidly.

SIM3 allows you to enter your chosen values for \(d\), \(r\) and \(g\) and simulate the path of the debt-to-GDP ratio over a 50-year period. The same simulation sheet also provides values for these variables for a sample of emerging and high-income economies.

Factoring in inflation

So far, we worked with real variables and did not consider the role of inflation. However, in some circumstances, inflation can play a key role in the evolution of the debt-to-GDP ratio. Fortunately, it is straightforward to rewrite Equation 5 in terms of inflation (\(π\)) and the nominal interest rate (\(i\)) instead of the real interest rate:

\[\begin{align*} \Delta b &=d+(i−π−g)b_{t−1} &&(7) \end{align*}\]

The relationship between the nominal interest rate and expected inflation is described by the Fisher equation, which states that \(i=r+π^e\), where \(π^e\) is expected inflation. For more details, see Section 15.1 of The Economy 1.0.

Fisher equation

The relation that gives the real interest rate as the difference between the nominal interest rate and expected inflation: real interest rate = nominal interest rate – expected inflation.

Equation 7 shows that, other things being equal, an increase in inflation will reduce the debt-to-GDP ratio. This is because when we compute the debt-to-GDP ratio, we use nominal GDP which is affected by both real growth (\(g\)) and inflation (\(π\)). However, other things are rarely equal. Higher than expected inflation will lead to an increase in the nominal interest rate. This increase in the nominal rate will compensate for the debt reduction effect of higher inflation. In fact, if we define unexpected inflation as \(π\)^u as the difference between expected inflation (\(π\)^e), and actual inflation (\(π\)), we can use the Fisher equation to rewrite Equation 7 as \(∆b=d\) + (\(r\) − \(π\)^u−\(g\)) \(b_{t−1}\). This clarifies that, in the absence of financial repression (see Section 7), only unexpected inflation can reduce the debt-to-GDP ratio.

A higher deficit can be driven by higher expenditure or lower taxes (or both). The belief that tax cuts can pay by themselves is one of the basic tenets of supply side economics, as you can read in Foundations of Supply-Side Economics by Victor Canto, Douglas Joines, and Arthur Laffer. For a critical evaluation, see ‘Evidence on the High-Income Laffer Curve from Six Decades of Tax Reform’ by Austan Goolsbee, or ‘The Elasticity of Taxable Income with Respect to Marginal Tax Rates: A Critical Review’ by Emmanuel Saez, Joel Slemrod, and Seth Giertz.

An important complication concerns the link between deficit spending and GDP growth. Issuing debt to finance productive investment expenditure can have a positive effect on long-run GDP growth. Similarly, countercyclical deficit spending can have a positive effect on GDP growth in the short run. In Equation 7, a larger deficit will lead to a proportional increase in the debt-to-GDP ratio, other things being equal. But when debt is issued to undertake actions that affect the evolution of GDP, other things are no longer equal. Those actions will raise the denominator of the debt-to-GDP ratio. Faster GDP growth will lead to higher tax revenues. In theory, there are even conditions in which a higher deficit could reduce the debt-to-GDP ratio.

SIM4 of the public debt simulator allows you to enter values for \(d\), \(i\), \(π\), and \(g\) and simulate the debt-to-GDP ratio over a 50-year period.

Exercise 3 Using the debt dynamics equation

Go to the OECD Data website. For this exercise, make sure you download the data in yearly frequency and as a percentage of GDP when necessary. For this exercise, you will need to download data for the following variables:

• nominal long-term interest rates
• inflation rates
• real GDP growth (hint: look for quarterly GDP and select a yearly frequency)
• primary deficit and debt-to-GDP ratios (hint: look for general government deficit and general government debt, respectively); note that on the OECD Data website, a positive value of the primary deficit ratio means that the government is running a surplus.

1. Find and download data for the long-term interest rate and inflation rate of a country of your choice, over the longest time span for which data is available. Using the Fisher equation, calculate the real interest rate for each year observed.
2. Plot the real interest rate on a line chart. How does the trend you observe in the real interest rate compare with that described in Figure 4?
3. Now add the growth rate of real GDP to your chart. From a quick look at your chart, in which periods would you expect debt to expand? In which periods would you expect it to contract?
4. Finally, add to your plot the primary deficit and the debt-to-GDP ratio. Does the debt-to-GDP ratio follow your predictions from Question 3? (Hint: For clarity, you can plot the debt-to-GDP ratio adding a secondary axis.)
5. Take the average values of all the variables you plotted over the most recent available 10 years in the data. Use these values to predict how debt for your chosen country will behave in the future. You can use Figure 6 to help you comment on its likely trend over time. (Hint: You can check your prediction by using the public debt simulator, sheet SIM3.)
6. Discuss whether you think that your prediction is reasonable.