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The dynamics of public debt
When does debt run away on its own — and when is growth enough to contain it?
Δb = (r − g)/(1 + g) · b − sProblem / motivation
Two countries start from the same debt — 110% of GDP — and run the same primary deficit of 1% of GDP every year. Ten years later, one is back down to 100%, the other has climbed to 159%. Neither their starting debt nor their budgetary effort separates them: a single gap tells them apart.
That thing is the race between two speeds. On one side the interest rate r the state pays on its debt, which makes it swell all by itself. On the other the growth rate g of GDP, which inflates the denominator of the ratio and dilutes it. The country back down to 100% had g above r; the one that shot up to 159% had r above g. This course builds the equation linking those three forces — interest, growth, budgetary effort — then runs it year after year, because a runaway is never visible in a single financial year.
Two markers carried over from the course “Deficit and the public balance”, on which this one rests directly: b is the debt/GDP ratio (115.6% for France at the end of 2025) and the PRIMARY BALANCE is the public balance excluding debt interest (−€87.8bn, that is −2.9% of GDP in 2025). We shall start from the identity that course established — end-of-year debt is start-of-year debt plus the deficit — and do only one thing with it: divide it by a GDP that has also moved.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
“Deficit and the public balance” established that end-of-year debt is start-of-year debt plus the year's deficit: Debt₁ = Debt₀ + Deficit. Now open that deficit into two pieces. First there is the INTEREST on inherited debt — the state pays r on what it already owes, that is r × Debt₀. Then there is everything else, revenue minus spending excluding interest: that is the primary balance S. A primary deficit adds to the hole, a primary surplus fills it in. Substituting: Debt₁ = Debt₀ + r × Debt₀ − S, that is Debt₁ = Debt₀ × (1 + r) − S.
We want the RATIO, not the amount. So divide both sides by year 1's GDP — which is no longer year 0's: GDP₁ = GDP₀ × (1 + g). On the left, Debt₁/GDP₁ = b₁. On the right, Debt₀ × (1 + r) / [GDP₀ × (1 + g)] = b₀ × (1 + r)/(1 + g), and the primary balance becomes s, related to GDP as well. We get b₁ = b₀ × (1 + r)/(1 + g) − s. All that is left is to subtract b₀ from both sides to bring out the CHANGE: Δb = b₀ × [(1 + r)/(1 + g) − 1] − s. The bracket simplifies once the 1 is put over the same denominator: (1 + r)/(1 + g) − (1 + g)/(1 + g) = [(1 + r) − (1 + g)]/(1 + g) = (r − g)/(1 + g), the two 1s cancelling. The equation is born — in four lines, from an identity already known.
Read it as a confrontation. The term (r − g)/(1 + g) · b is the SNOWBALL effect: it does not depend on this year's budget, only on inherited debt and on the race between the two speeds. The term −s is the EFFORT: a primary surplus (s > 0) subtracts from it, a primary deficit adds to it. One detail that matters at this level: the first term is often written simply (r − g)·b, forgetting the divisor (1 + g). The shortcut overstates the snowball by g/(1 + g), that is 1% when g is 1%, but 7.4% when g is 8%. Over one year that is trivial; over a ten-year path, the error accumulates. We shall keep the exact form.
Should the rates be real or nominal? Both work, and here is the reason in two lines. If π is inflation, the nominal rate satisfies (1 + i) = (1 + r)(1 + π) and nominal growth (1 + γ) = (1 + g)(1 + π). Form their ratio: (1 + i)/(1 + γ) = [(1 + r)(1 + π)] / [(1 + g)(1 + π)]. The same factor (1 + π) appears above and below: it CANCELS, leaving (1 + r)/(1 + g). The ratio (1 + r)/(1 + g), which is all the equation needs, is therefore identical in both units. Careful: this does not mean inflation is neutral for an indebted state — surprise inflation does erode the ratio, because nominal GDP rises at once while the apparent rate takes years to follow (limits). What the computation says is only that the units must not be MIXED. Click each term:
Solving / calculation
Let's run the equation on France, with the real 2025 figures (INSEE) — then look at what it announces for what follows. The result will contradict a very widespread intuition.
- The three ingredients, all measured
b = 112.6% (end 2024); r = 64.7 / 3,306.1 = 1.96%; g = +2.0% (GDP in value); s = −87.8 / 2,994.7 = −2.93%r − g = −0.04 point: the two speeds are level - The snowball effect
112.6 × (1.96% − 2.00%) / 1.02−0.05 point: the snowball is AT A STANDSTILL in 2025 - The primary effort
− s = − (−2.93)+2.93 points: it alone is pushing the ratio - The change predicted by the equation (unrounded values: −0.047 and −2.932)
Δb = −0.047 + 2.932+2.88 points - Against reality: the ratio went from 112.6% to 115.6%
+3.0 points observed against +2.88 predicteda gap of +0.12 point — the deficit-debt adjustment and rounding (course “Deficit and the public balance”) - The primary balance that would have stabilised the ratio
s* = b · (r − g)/(1 + g) = −0.05s* = −0.05% of GDP: it would have taken a move from −2.93 to −0.05, that is 2.88 points of effort - The same result in euros — matching the course “Deficit and the public balance”
s* = −0.05% of GDP, that is a primary deficit of €1.4bn; now TOTAL deficit = interest − primary balance = 64.7 − (−1.4)€66.1bn — exactly the “stabilising deficit” Debt₀ × g of the other course: the two ways of saying it coincide - And tomorrow? The apparent rate gradually catches up with the market (b = 115.6 at end-2025)
at r = 3.0%: 115.6 × (3.0 − 2.0)/1.02 · at r = 3.5%: 115.6 × (3.5 − 2.0)/1.02+1.13 then +1.70 point a year — the snowball starts up again
Here is the surprising result: in 2025, France is NOT in a snowball effect. Its apparent rate (1.96%) and its nominal growth (2.0%) are level, so that inherited debt costs nothing in points of ratio. The three points the ratio gained come almost entirely from the primary deficit. So it is not “interest choking the country” — not yet: it is the gap between current spending and current revenue. But step 7 shows where the risk lies: every old loan contracted at 0 or 1% is replaced by a loan at a markedly higher market rate, and the apparent rate climbs slowly towards it (it has already gone from 1.8% in 2024 to 1.96% in 2025). If r reaches 3.5% with unchanged growth, the snowball alone adds 1.7 point a year — without any budgetary decision having been taken.
Δb = b · (r − g)/(1 + g) − s, repeated year after yearThe starting settings are France's at end-2025. Vary r and g to find the frontier, then push the HORIZON: it is by projecting over ten or twenty years that a runaway becomes visible — over one year, it never shows. Watch the “Regime” line too: it changes in nature depending on whether r exceeds g or not. And you can replay the two countries of the opening here: keep b = 110 and s = −1, then compare r = 1% / g = 3% with r = 4% / g = 1% over 10 years.
Economic interpretation
Three readings, two of which contradict very widespread shortcuts — including among serious commentators.
The repeated equation is a multiplication by λ = (1 + r)/(1 + g) each year. With r = 2% and g = 1%, λ is 1.0099: with zero primary effort, it would take about 70 years for the ratio to double. A century-scale snowball, then — the image of a sudden runaway is misleading. The two countries of the opening show it: λ = 1.01/1.03 = 0.981 for the first, whose ratio melts by nearly 2% a year before any effort at all, against λ = 1.04/1.01 = 1.030 for the second. Repeated ten times, that small gap between two λs is enough to open the 59 points that separate them — but it did take ten years. Real derailments never come from a one-point gap: they come from a collapse in g (Greece lost more than a quarter of its GDP after 2008), from a market rate that explodes, or from a massive primary deficit — often all three together. Greek debt went from 125% of GDP in 2009 to more than 176% at the end of 2013 — and that DESPITE the 2012 restructuring, which had already wiped out part of the stock. Fifty points in four years when λ alone would have taken decades: this is not the slow arithmetic of the snowball, it is a shock.
This is the commonest shortcut, and it is wrong. Take the equation again: g > r only cancels the first term, and −s remains. A country with r = 2%, g = 3% and a primary deficit of 2% of GDP sees its ratio RISE, year after year. What g > r really guarantees is something else: the path CONVERGES instead of diverging. Towards what? Towards the debt level that makes Δb zero — and you find it by solving the same equation, this time for b instead of s: b·(r − g)/(1 + g) = s gives b* = s(1 + g)/(r − g). With r = 2%, g = 3% and s = −2%, that comes to 206% of GDP. The country does settle down… twice as indebted as today, and after decades. “Not exploding” and “staying reasonable” are two different things — exactly the nuance the course “Deficit and the public balance” announced.
Stabilising means aiming for Δb = 0 this year. Sustainability is a broader notion: a path is sustainable if the state can go on funding itself without being forced into a brutal adjustment — which perfectly allows the ratio to rise for a while, during a war or a pandemic for instance, then come back down. There is no “right” level of debt demonstrated by theory: Japan has lived above 200% of GDP for years without a funding crisis, while Greece had one with debt of the order of 130%. What separates them has to do with who holds the debt, in what currency it is denominated, and the credibility of the path. As for the European treaty's 60%, it comes from no model: the course “Deficit and the public balance” shows that it simply forms a coherent pair with the 3% deficit, under an assumption of 5% nominal growth a year.
Limits / critiques
Exercises
Debt b = 100% of GDP, r = 3%, g = 1%, primary balance s = 0. By how much does the debt/GDP ratio change in the first year (in points, two decimals)?
The same country (b = 120%, r = 2%, g = 3%), but its primary deficit is 3% of GDP (s = −3). What does the ratio do in the first year (in points, two decimals)?
A country has b = 115% of GDP, growth g = 2% and a primary deficit s = −1% of GDP. What gap r − g (in points, two decimals) would leave its ratio stable?
In 2025, the French debt ratio went from 112.6% to 115.6% of GDP. What is the main cause?
Debt b = 120% of GDP, r = 2%, g = 3%. What primary balance s* (in % of GDP, two decimals) would exactly stabilise the ratio?
That same country keeps these parameters indefinitely. Towards what debt level (in % of GDP, to the nearest unit) does its ratio settle in the long run?
Debt b = 100% of GDP, r = 4%, g = 2%, primary balance s = 0, parameters constant. Where is the ratio after 5 years (in % of GDP, one decimal)?
True or false: as long as growth exceeds the interest rate, a state can run whatever primary deficit it likes without its debt derailing.