In 1960, the world’s poorest countries had a lot of catching up to do.

Q1: What is this graph showing?

Look carefully at the title, the axes, the line and the size of the bubbles. Does anything surprise you?

If poorer countries automatically caught up with richer ones, what would you expect the graph to look like?

Q2: Why might poorer countries grow faster than rich ones?

Suppose output per worker depends on capital per worker:

y = √k

where:

Find dy/dk. What does it tell you, and how might it help explain why poorer countries could grow faster?

Q3: Why can growth slow down even when a country keeps investing?

Still using y = √k, suppose a country saves and invests a share s of its output, so investment per worker is s × y.

Capital also wears out at rate δ, so depreciation per worker is δk.

Each year, capital per worker changes by:

Δk = s × y − δk

A steady state is where capital per worker stops changing.

Explain why a country that keeps investing will eventually reach a steady state.

Q4: Does saving more make a country richer?

Suppose δ = 10%.

Find each country’s steady-state k and y.

Does saving more make a country richer? Does it make it grow faster forever?

Q5: Where does productivity come in?

Now add productivity, A:

y = A√k

How does A affect output per worker, and the return from adding more capital?

Find steady-state y in terms of A, s and δ. What does this suggest about catch-up growth?

Q6: What is “A”, really?

In the model, A is just a letter, often called “productivity” or “technology”.

In the real world, what might sit inside it? Why might two countries with the same capital per worker produce very different amounts?

Bonus: So is the model wrong?

Countries that were poor in 1960 ended up all over the graph: some grew very fast, others barely grew at all, or shrank.

Does that prove the model wrong?

P.S roll up your sleeves and explore the question… once you’ve grappled with it for a while, if you are still stuck…Graduated hints below — but challenge yourself first! (Answer comes in the next edition! Subscribe to receive it directly to your inbox!)

This week’s marginal number is 6.036%.

That is the level the UK 30-year gilt yield briefly reached on 7 October 2026, its highest since January 1998.

A gilt is UK government debt. The yield is the return investors demand to lend to the government.

Why does this matter?

Because long-term government borrowing costs are not just a financial-market detail.

They affect the whole economy.

The Guardian reported that UK long-term borrowing costs had risen to their highest level since 1998, adding pressure on the Chancellor ahead of the Budget.

Higher gilt yields raise the cost of government borrowing. That makes debt interest more expensive and puts pressure on the Chancellor’s Budget arithmetic.

They also affect mortgage rates, company borrowing costs, pension funds and asset prices.

In other words, when the government’s long-term borrowing cost rises, everyone else’s discount rate starts to move too.

The economics lesson is simple:

Debt sustainability is not only about how much a government owes.

It is also about the price investors charge to keep lending.

Last time, we asked when government debt becomes dangerous.

Debt is normally judged relative to GDP, and a ratio has two parts. So the debt ratio can change because debt changes, because GDP changes, or both.

Debt starts at 120% of GDP. Interest adds 3%, so debt becomes 120 × 1.03 = 123.6. But GDP grows by 4%, so we divide by 1.04:

123.6 / 1.04 = 118.85%

The debt ratio falls, even though the government hasn’t repaid a penny or run a primary surplus. That’s because nominal GDP is growing faster than the interest rate on the debt: g > r. The economy that carries the debt is growing faster than the debt itself.

The first lesson: high debt isn’t automatically dangerous, if growth outpaces the interest rate.

Call debt D and nominal GDP Y, so the debt ratio is d₀ = D₀ / Y₀.

Over the year, interest is added to the debt, and the government borrows to cover any primary deficit P:

D₁ = D₀ × (1 + r) + P

Meanwhile, GDP grows: Y₁ = Y₀ × (1 + g)

Dividing one by the other:

d₁ = D₁ / Y₁ = d₀ × (1 + r) / (1 + g) + p

where p = P / Y₁ is the primary deficit as a share of next year’s GDP.

Intuitively: the old debt grows at rate r, the economy that carries it grows at rate g, and new borrowing is added on top.

A handy shortcut: d₁ − d₀ ≈ (r − g) × d₀ + p. The change in the debt ratio depends on the gap between r and g, scaled up by how much debt there already is, plus any new borrowing.

With r now at 7%:

d₁ = 120 × 1.07 / 1.04 = 123.46%

In Q1, the ratio fell to 118.85% because g > r. Now it rises, because r > g.

This is how a doom spiral begins:

fear of debt → higher interest rates → rising debt ratio → more fear of debt

Notice what changed between Q1 and Q3: nothing about the country itself. Same debt, same growth, same budget. Only investors’ beliefs changed, and those beliefs made the debt less sustainable. If investors trust a country, it borrows cheaply and its debt looks safe. If they don’t, it borrows expensively and its debt looks dangerous. Both can be true of the same country: a debt crisis can be self-fulfilling.

In practice, the spiral is slower than in our example. r is the average interest rate on all existing debt, and most government debt is long-term, so higher market rates only bite as old bonds are repaid and refinanced. That buys time, but not forever.

To keep the ratio at 120%, set d₁ = d₀:

120 = 120 × 1.07 / 1.04 + p

120 = 123.46 + p

p = −3.46% of GDP

A negative primary deficit is a primary surplus: the government must raise about 3.5% of GDP more in taxes than it spends (excluding interest) just to stop the ratio rising. The shortcut gives almost the same answer: (7% − 4%) × 120% = 3.6%.

That’s a large adjustment. For France, it would mean finding over €100 billion a year. When r > g, a balanced primary budget isn’t enough.

The spiral stops only if r falls, g rises, or the government runs a large enough primary surplus. The trouble is that these are connected: tightening may restore confidence and lower r, but it may also slow growth.

Before the cut (from Q3): 123.46%

After the cut: d₁ = 120 × 1.07 / 1.025 − 1 = 125.27 − 1 = 124.27%

The government has cut its deficit, and the debt ratio is higher. The numerator improves, because the government borrows 1% of GDP less. But the denominator worsens, because GDP is smaller than it would have been. Here, the denominator effect wins.

Austerity can improve the deficit but worsen the debt ratio, if the hit to growth is big enough.

With multiplier m, growth falls to (4 − m)%. The cut backfires if:

120 × 1.07 / (1.04 − 0.01m) − 1 > 123.46

Rearranging: 1.04 − 0.01m < 128.4 / 124.46 = 1.0316, so m > 0.84.

So any multiplier above about 0.84 makes the debt ratio worse after one year. At 1.5, the cut backfires comfortably.

There’s a neat intuition. The cut lowers debt by 1% of GDP. But it lowers GDP by m%, which raises the ratio by roughly d × m. So the cut backfires when d × m > 1, i.e. m > 1/d. With d = 1.2, that’s m > 0.83. The more indebted the country, the smaller the multiplier needed for austerity to backfire.

Multipliers tend to be largest when the economy is weak, when the central bank can’t cut interest rates to cushion the blow, and when many economies tighten at once. That’s exactly when governments come under most pressure to cut.

No. The maths shows that cutting the deficit can backfire. It doesn’t show that it always does.

First, timing. The hit to growth is mostly a one-off: GDP is lower, but growth recovers. The lower deficit, though, keeps going year after year. So cuts that raise the debt ratio in year one can still lower it over several years.

Second, confidence. If cuts reassure investors and r falls, the picture changes. If r falls from 7% to 6%: 120 × 1.06 / 1.025 − 1 = 123.10%, better than doing nothing (123.46%).

Third, the central bank. A central bank that cuts interest rates can offset some of the hit to growth. A eurozone country like France can’t count on this: the European Central Bank sets one interest rate for the whole euro area.

So the right answer isn’t “always cut” or “never cut”. It’s: ask what happens to both the numerator and the denominator, and over what time frame.

Inflation raises nominal GDP growth, which is g in the formula. So it can help. Start again from the Q3 country (r = 7%, g = 4%: debt rises to 123.46%).

Suppose inflation rises by 3 points, so g = 7%, and r stays at 7%:

d₁ = 120 × 1.07 / 1.07 = 120.00%

The debt ratio stops rising. This works if the debt was borrowed at fixed rates before anyone expected the inflation: the government repays debts with money worth less than lenders expected.

But investors catch on. If they demand compensation, r rises to 10%:

d₁ = 120 × 1.10 / 1.07 = 123.36%

Almost exactly where we started. If inflation raises r and g by the same amount, it achieves almost nothing.

And if investors demand extra compensation, because they fear inflation will keep rising, r might rise to 11%:

d₁ = 120 × 1.11 / 1.07 = 124.49%

Now inflation has made things worse.

How much inflation helps depends on the debt:

So inflation only erodes debt if it’s a surprise, and only for as long as investors are locked in. It’s a trick that works once: a government that uses it loses credibility, and pays for it in higher interest rates afterwards.

100% debt is not always 100% debt. Beyond r, g and p, investors care about:

1. Who controls the currency. This is the big one. The US, UK and Japan borrow in currencies their own central banks create, so they can always pay in nominal terms; the risk is inflation, not default. France borrows in euros, which it can’t create. In that respect, a eurozone government is more like a household or a company: it can, in principle, run out of money. That’s a large part of why markets worry more about France at 119% than the US at 125%, and why Japan, with debt well above 200% of GDP, still borrows cheaply.

2. Debt in foreign currency. A country that borrows in dollars but collects taxes in its own currency faces a double hit if its currency falls.

3. Who holds the debt. Domestic savers, pension funds and banks tend to be steadier holders than foreign investors, who can sell and leave. Most Japanese government debt is held at home.

4. Maturity. Long-term debt gives a government time; short-term debt must be refinanced at whatever rate the market demands.

5. Reserve-currency status. Global demand for dollars and US Treasuries lets the US borrow more cheaply than its debt alone would suggest.

6. Tax capacity. Can the government realistically raise more revenue? Countries with already high taxes, like France, have less room.

7. Political stability. Can the government actually pass a budget? France’s recent difficulties are a large part of the market’s concern.

8. Institutions and track record. Past defaults, independent central banks, reliable statistics and credible fiscal rules all matter.

9. Reliance on foreign lenders. A country that saves less than it invests (a current-account deficit) needs foreigners to keep lending.

10. Risk of fiscal dominance. If investors suspect the central bank will be pressured to print money to fund the government, they’ll demand compensation for inflation risk.

Not at any particular number. Debt becomes dangerous when r exceeds g, when confidence starts to crack, and when politics makes the fix hard to deliver. The danger is that these reinforce each other: high debt raises interest rates, higher rates raise debt, and the politics of cutting gets harder at exactly the wrong moment.

Debt sustainability isn’t one number. It’s a judgement about maths, markets and institutions.

Marginal gains… Answer for today’s question is out in two days… Come back then or subscribe to get it direct to your inbox!

P.P.S.

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