Public Debt Dynamics

Public debt dynamics provides a framework for examining the evidence and implications of current policies by sovereigns and central banks, in dealing with the debt abyss.

Most governments post significant public debt Opens in new window. Few record positive net financial wealth: this mainly applies to countries that hold foreign exchange reserves as an insurance against sudden capital outflows or to countries that produce oil or raw material and intend to spread the benefits from the extraction over several generations.


There can be good as well as bad reasons for a government to go into debt.

Debt may fluctuate as a consequence of automatic or discretionary stabilization (but this is not a motive for sustained increase as the effects of booms normally offset those of recessions).

For a country that must fight a war, ward off catastrophes, or undertake massive public investments, public borrowing is a way to spread the corresponding burden over several generations.

In the case of war, it would be morally disputable to require the same generation that pays the price of blood to defend the nation against an aggressor to shoulder the corresponding financial effort.

In the same way, a case could be made for partially financing action against climate change through debt: it would be a way to make the future generations pay for the milder climate they will benefit from.

But public borrowing that repeatedly finances current expenditures is highly disputable.

It is essentially a way to make the future generations pay for the consumption of the current one, and there are no plausible equity arguments to justify such a behavior.

Debt, however, does not need to be repaid entirely because, unlike households, the public sector is not expected to die.

Debt can be passed from one generation to the next.

Furthermore, debt must be assessed in comparison to the country’s national income. It is not the nominal amount of debt that matters, but its size in proportion to GDP as a broad measure of the tax base.

An identical rate of debt accumulation does not have the same consequences in a country whose nominal GDP grows at a fast pace and in a country where it stagnates.

Nominal GDP growth resulting from real growth and inflation diminishes the burden of past debt and therefore makes deficits more sustainable. This can be illustrated by a comparison between the United States and three European countries: France, Italy, and Greece over 2007 – 2014.

While the increase in nominal debt was higher in the United States than in the three European countries (Greece included!), it was in significant part offset by nominal GDP growth.

In contrast, Italy did not record any nominal GDP growth effect, and, in Greece, the decline in nominal GDP was so large that its adverse impact on the debt ratio was almost twice as large as the opposite impact of the 2012 debt restructuring.

The dynamics of the public debt ratio can be described as a horse race between debt accumulation and nominal GDP growth:

if the financial deficit remains below a threshold called the debt-stabilizing deficit, which is equal to the debt ratio times the nominal grow rate, the debt ratio diminishes; it is above this threshold, it increases.

This sheds light on the debt limit (60% of GDP) and the deficit limit (3% of GDP) set by the European treaty of Maastricht Opens in new window. It was at the time (1992) assumed that countries would grow at 5% annually (3% real growth and 2% inflation), which implied that 3% = 5% x 60% was the corresponding debt-stabilizing deficit.

The financial deficit ratio is, however, not the best variable to assess debt sustainability because it results in part from the interest payments on public debt, which are largely beyond the control of the current government. It is therefore preferable to write down the debt stabilization condition for the primary balance.

Other things being equal, primary deficits and interest payments on the outstanding stock of public debt increase the debt ratio, whereas primary surpluses and positive nominal GDP growth reduce it.

When the nominal GDP growth rate is higher than the interest rate (or, equivalently, when the real GDP growth rate is higher than the real interest rate), a stable debt ratio is compatible with a permanent primary deficit.

Conversely, when the interest rate is higher than the growth rate, there must be a primary surplus to stabilize the ratio of debt to GDP, and the larger the (positive) difference between the interest rate and the growth rate, the larger the necessary primary surplus.

Public Debt Dynamics Equation

Let ∆ be the primary public deficit, B the public debt at year-end, both in euro, and i the nominal interest rate. We neglect cash revenues or disbursements (such as asset sales and purchases) that may affect public debt for a given public deficit.

We also suppose that debt is measured at face value and not at current market value, thus ignoring valuation effects. Such assumptions are common but not innocuous: in emerging countries, part of the public debt is US dollar-denominated, and exchange rate movements also influence debt dynamics.

The debt dynamics can be written as (with the –1 subscript denoting the end of the previous year):
B = (1+i)B–1 + ∆

If lowercase letters denote the ratios of the corresponding variables to GDP, n is the nominal growth rate (growth in volume + inflation), the real growth rate, the rate of inflation, and r the real ℯ𝑥 post interest rate, we have:
= ℊ + ℼ
i = r + ℼ

Debt dynamics can thus be expressed as: bb–1 ≅ (i)b–1 + ∂

The debt ratio increases if the interest rate is higher than the nominal GDP growth rate and there is not a sufficient offsetting primary surplus.

A country whose nominal growth rate exceeds its nominal interest rate can run a primary deficit without seeing its debt-to-GDP ratio increase.


The debt dynamics equations help explain why, in the second half of the 1980s, public debt became an issue for advanced countries:

the combination of disinflation, lower growth, and higher real interest rates suddenly brought it to the fore after two decades during which governments could afford to overlook debts and deficits.

In the same way, the no-growth-no inflation environment of the mid-2010s resulted in public debt becoming a serious cause for concern in spite of the record low interest rates.

The approach presented here can be tweaked for assessing the debt situation of developing and emerging countries that are often obliged to borrow in foreign currency.

This original sin of emerging-country financing adds a currency risk dimension, and, for this reason, it is generally assessed that the limits to their debt are significantly lower than for countries that borrow in their own currency.