Economics Tutor Milan – Best Microeconomics, Macroeconomics, Econometrics tutors in Milan, Rome, Barcelona, Paris

A government can run budget surpluses for years and still watch its debt-to-GDP ratio climb. Whether that debt is sustainable turns on one comparison — the interest rate against the growth rate — not on any headline threshold. That mechanism, and the law of motion behind it, is what second-years most often want to pin down with an economics tutor in Milan.

1 · The government budget identity and the debt ratio

Start with the government’s books. Debt at the end of this year is last year’s debt, plus the interest owed on it, minus the primary balance — tax revenue less non-interest spending. Write B for debt, r for the interest rate and PB for the primary balance:

   

That is a stock, B, driven by a flow, PB, and the interest rate. But a debt level in isolation says little: the same £2 trillion is trivial for one economy and fatal for another. What matters is debt relative to the income that services it. So divide through by GDP.

Let Y be GDP, growing at rate g, so Y_t = (1 + g)Y_t−1. Write b = B/Y for the debt ratio and ps = PB/Y for the primary balance as a share of GDP. Dividing the identity by Y_t gives the debt-ratio law of motion:

   

Everything on this page is that one line, read carefully.

2 · Why the sign of r − g is the whole game

Subtract b_t−1 from both sides to expose the change in the ratio:

b_t − b_t−1 = ( (r − g) / (1 + g) ) b_t−1 − ps

Now the driver is naked. The multiplier (1 + r)/(1 + g) sits above 1 exactly when r exceeds g. If r > g, interest compounds the debt faster than growth dilutes it, so last year’s stock rolls forward heavier relative to the economy that has to carry it. If r < g, the economy outgrows the debt and the ratio drifts down on its own, before any surplus is struck.

Two forces, then. The primary balance ps is the lever a finance ministry controls. The gap r − g is the wind, and it is mostly not in the ministry’s gift. Italy is the standing example: it has often run primary surpluses, yet its debt ratio has stayed high, because r − g has so frequently worked against it. The surpluses were doing real work. The wind was simply stronger.

3 · The primary balance that stabilises the debt

Set the change to zero and solve for the surplus that exactly holds the ratio steady. Call it ps*:

ps* = ( (r − g) / (1 + g) ) b

Read the three cases straight off it. When r > g, ps* is positive: you must run a primary surplus just to stand still, and a larger one the higher the debt. When r < g, ps* is negative: you can run a primary deficit and the ratio still falls. When r = g, the multiplier is exactly 1 and ps* is zero — a balanced primary budget freezes the ratio wherever it sits.

The feature examiners test is that ps* rises with b — a country at 120% needs a bigger surplus than the same country at 60% facing the same r − g. High debt raises the price of standing still, the first hint of why a snowball, once rolling, is so hard to stop.

4 · Risk premia, the doom loop, and sustainability as a path

So far r has been fixed. It is not. Investors who doubt they will be repaid demand a higher yield, so r rises with the debt ratio itself. That closes a loop: a higher ratio lifts r, a higher r enlarges the interest bill, the bill enlarges the ratio, round again. This is the doom loop — it can turn a calm debt path sharply upward, and the rate shock in the worked example below is exactly what it delivers.

The eurozone sharpens it. A government inside the euro borrows in a currency it cannot print, so no domestic central bank stands behind its bonds by default. That leaves room for a self-fulfilling panic — unlike a government with its own floating currency — and is why euro-area spreads can widen fast.

None of this makes a fixed threshold — 60%, 90% — the test. Sustainability is a path condition, not a magic number. The question is whether the ratio is on a trajectory that stabilises under policy the government can credibly deliver. A country at 130% with r < g and a steady surplus can be on firmer ground than one at 70% with r > g and a widening deficit. Direction beats level.

Worked example — a government at 120% of GDP

A government enters the year with debt at 120% of GDP (b₀ = 1.20). It runs a primary surplus of 2% of GDP (ps = 0.02) and keeps it there. Real GDP grows at g = 3%.

Step 1 — The law of motion, base case. Suppose the interest rate is r = 1%, comfortably below growth. The multiplier is (1 + 0.01)/(1 + 0.03) = 1.01/1.03 = 0.9806. Each year the ratio becomes 0.9806 of last year’s, minus the 2-point surplus.

Step 2 — Iterate it. b₁ = 0.9806 × 1.20 − 0.02 = 1.157 (115.7%). Then b₂ = 1.114 (111.4%), b₃ = 1.073 (107.3%). The ratio falls — and it falls even though the surplus is only 2%, because r < g is doing half the work.

Step 3 — The stabilising check. How hard is stabilisation here? The stabilising surplus is ps* = ((0.01 − 0.03)/1.03) × 1.20 = −2.3%. It is negative: this government could run a primary deficit of 2.3% of GDP and still hold 120%. Running a 2% surplus instead, it clears the bar comfortably, so debt declines.

Step 4 — The perturbation: a rate shock. Now the markets turn. A risk premium pushes the interest rate from 1% to r = 6%, above growth, with g and the 2% surplus unchanged. The multiplier flips to 1.06/1.03 = 1.0291 — now above 1.

Step 5 — Iterate the shock from the same start. From the same b₀ = 1.20: b₁ = 1.0291 × 1.20 − 0.02 = 1.215 (121.5%). Then b₂ = 1.230 (123.0%), b₃ = 1.246 (124.6%) — and each yearly rise is larger than the last. The path bends upward. A snowball.

Step 6 — Why the same surplus cannot stop it. The stabilising surplus is now ps* = ((0.06 − 0.03)/1.03) × 1.20 = 3.5%. The 2% surplus is 1.5 points short, so the ratio climbs. That same 2% surplus would have frozen a ratio of only about 69% (b* = 0.02 ⁄ (1.0291 − 1)) — but 120% sits far above 69%, and above that point the surplus can no longer hold the line.

Step 7 — Interpretation. Run both paths out twenty years. The low-rate economy drifts down to about 48% of GDP; the shocked economy climbs to about 160% and is still accelerating. Same debt, same surplus, opposite fates — the only thing that changed was r relative to g. That comparison, plotted, is the diagram below.

Can you say why the same 2% surplus rescues one government and sinks another, with nothing changed but the interest rate against growth? Reading the debt path off the sign of r − g, rather than reaching for a 60% or 90% threshold, is the distinction examiners test on sustainability. A one-on-one economics tutor works the law of motion and the stabilising surplus with you until you can derive the path, not just name a number. Book a trial session.

Practice

Q1. A country starts at b₀ = 100% of GDP, runs a 3% primary surplus (ps = 0.03), and faces r = 7% with g = 2%. Find b₁. Is the ratio rising or falling, and what surplus would have stabilised it?

Q2. A stagnant economy has no growth (g = 0), an interest rate r = 3%, and a debt ratio of 150%. What primary balance exactly stabilises the ratio?

Q3. A country has r = g = 4% and a debt ratio of 100%. (a) With a balanced primary budget, what happens to the ratio over time? (b) With a primary deficit of 2% of GDP each year, what is the ratio after one year and after two?

Answers. Q1: the multiplier is 1.07/1.02 = 1.0490, so b₁ = 1.0490 × 1.00 − 0.03 = 1.019 (101.9%) — the ratio is rising. The stabilising surplus is ((0.07 − 0.02)/1.02) × 1.00 = 4.9%; the 3% surplus falls short, which is exactly why debt climbs. Q2: with g = 0, ps* = r × b = 0.03 × 1.50 = 4.5% of GDP — a surplus equal to the entire interest bill. Q3: (a) at r = g the multiplier is 1, so a balanced primary budget holds the ratio at 100% forever; (b) the deficit adds 2 points a year — 102% after one year, 104% after two — a straight-line climb, not yet a snowball, because that needs r > g.

Key takeaways

• The law of motion is the whole topic: b_t = ((1+r)/(1+g)) b_t−1 − ps. Everything else is reading that one line.
• The sign of r − g sets the direction. Above zero, rolling debt over compounds it; below zero, growth dilutes it. The primary balance is the lever; r − g is the wind.
• The stabilising surplus is ps* = ((r − g)/(1+g)) b, and it rises with the debt ratio — high debt raises the price of merely standing still.
• The doom loop makes r endogenous: high debt lifts the risk premium, which lifts the interest bill, which lifts the debt. That feedback, not any threshold, is the real danger.
• Sustainability is a path condition, not a number. Ask where the ratio is heading under credible policy, not whether it has crossed 60% or 90%.

Why Milan students choose our economics tutoring

• Models built from the identity, not memorised: you derive the law of motion from the budget constraint, so you can rebuild and adapt it under exam pressure instead of quoting a formula you half-remember.
• The distinctions examiners reward, drilled: r versus g, the stabilising balance versus a threshold, stock versus flow, benchmark versus description — the differences that separate a first from a 2:1.
• In person across four cities, or online: sessions also run in Rome, Barcelona and Paris, or online wherever your course is taught.
• One-on-one, matched to your syllabus: tutors work from your own notation and past papers, whether your macro module follows Blanchard, Burda and Wyplosz, or Carlin and Soskice.

FAQ

Q: What does “r greater than g” actually mean for debt?
A: It means the interest rate on government debt exceeds the economy’s growth rate. When that holds, the debt rolled over each year grows faster than the GDP that services it, so the debt-to-GDP ratio tends to rise unless a primary surplus offsets it. When growth beats the rate, the ratio drifts down on its own.

Q: Is there a debt-to-GDP level that is automatically dangerous?
A: No. Thresholds like 60% or 90% are reference points, not cliffs. What matters is the path: a high ratio that is stable or falling under credible policy is safer than a lower one that is climbing. Sustainability is about direction and the sign of r − g, not a single number.

Q: How can a country run surpluses and still see its debt ratio rise?
A: Because the surplus is only one of two forces. If r − g is positive and large enough, the interest burden on existing debt outruns both growth and the surplus. Italy is the classic case: repeated primary surpluses, yet a persistently high ratio, because the interest-growth gap kept working against it.

Q: What is the doom loop?
A: A feedback between debt and interest rates. As the ratio rises, investors demand a higher yield, which raises the interest bill, which raises the ratio, which raises the yield again. It can turn a slow drift into a fast spiral, and it bites hardest where a government cannot print the currency it borrows in.

Q: Why is sovereign debt riskier inside the eurozone?
A: A eurozone government borrows in euros, a currency it cannot issue, so no national central bank automatically backs its bonds. That leaves the door open to a self-fulfilling panic: investors fear default, demand higher yields, and thereby help cause the trouble they feared.

Book an economics tutor in Milan or online

Sovereign debt rewards the student who can derive the path, not just recite a threshold — the law of motion, the r − g condition, the stabilising surplus, and the honest reason a fixed number is the wrong test. One-on-one sessions build that fluency on your own course and past papers. Tell us your module and exam date, and we will match you with the right tutor this week.