Government debt is usually discussed as a big number.

US government debt has passed $40 trillion. French government debt stands at around €3.6 trillion.

But economists usually judge government debt as a share of the economy: the debt-to-GDP ratio. US gross debt is around 125% of GDP; France’s is around 119%.

So why have bond markets recently been nervous about France? Reuters reported that stress in French government bonds had rattled the euro, with the premium investors demand to hold French rather than German government bonds at its highest since the eurozone debt crisis. The Financial Times describes France as caught between bond-market pressure and political resistance to cutting its deficit.

A ratio has two parts, and that means it can move in surprising ways.

So when does debt really become dangerous?

Q1: Can you work out what happens to the debt ratio?

A country starts with debt equal to 120% of GDP.

What happens to the debt-to-GDP ratio after one year?

Q2: What does the debt dynamics formula mean?

Show that your working in Q1 can be written as:

d₁ = d₀ × (1 + r) / (1 + g) + p

where:

Explain the formula intuitively.

Q3: Can fear of debt create more debt?

Now investors become nervous, and r rises from 3% to 7%. Everything else stays as in Q1.

What happens to the debt-to-GDP ratio now? What feedback loop could this create?

Q4: When does the debt doom spiral stop?

With d₀ = 120%, r = 7% and g = 4%, find the value of p that stops the debt ratio rising.

In other words, what primary balance would keep debt at 120% of GDP?

Q5: How can cutting the deficit make debt worse?

Stay with the country in Q3 (d₀ = 120%, r = 7%, g = 4%, p = 0%). Worried about a doom spiral, the government cuts spending by 1% of GDP, so p falls from 0% to −1%.

But spending cuts also shrink the economy. Suppose the fiscal multiplier (remember #002?) is 1.5: every £1 of spending cut reduces GDP by £1.50. So nominal growth falls from 4% to 2.5%.

(a) What happens to the debt-to-GDP ratio? Compare your answer with Q3.

(b) How large would the multiplier need to be for the spending cut to make the debt ratio worse after one year?

(c) Does this mean governments should never cut their deficits?

Q6: Can inflation make debt disappear?

Inflation is often said to “erode” government debt. Using the formula from Q2, show how higher inflation could reduce the debt-to-GDP ratio. Then explain why it might not work, or could even make things worse.

Bonus: Why might the same debt ratio worry investors in one country but not another?

Two countries both have debt of 100% of GDP.

In one country, investors barely worry.

In another, they demand much higher interest rates.

For reasons beyond what has already been discussed, why might this be the case? See how many factors you can come up with.

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!)

45%

Marginal number: 45%

That's the share of the world's 8.2 billion people now living in cities, more than double the 20% in 1950.

Source: UN DESA, World Urbanization Prospects 2025

But how urban is the world, really? That depends on where you draw the line. The UN now sorts every settlement into cities, towns or rural areas using a single global method. On that measure, 45% of people live in cities, 36% in towns and just 19% in rural areas. Add up each country's own definitions of "urban", though, and you get 58%. Count cities and towns, and it's 81%. Same world, three very different answers.

Why does this matter? Because cities are where opportunity concentrates. They create agglomeration effects: thicker labour markets, more customers, more specialisation, more ideas and more chance encounters. That's why cities can be engines of productivity, and why two-thirds of the world's population growth to 2050 is expected to happen in them.

But density has costs too. Housing becomes scarce, land prices rise, roads and public transport become congested, and inequality within a city can widen even as the city as a whole grows richer. The same density that makes cities productive also makes them expensive, crowded and hard to manage.

Geography shapes opportunity. But when people cluster in the same places, the gains from being close together have to be weighed against the costs of crowding.

Last time, we asked whether a train carriage should be quiet or not. 30 quiet passengers would each pay £4 for silence, while 20 talkative passengers would each pay £3 to talk freely.

Add up the total value on each side:

The efficient outcome is a quiet carriage. The quiet passengers gain more from silence (£120) than the talkative passengers lose from not talking (£60), so silence creates a net surplus of £60.

Suppose the carriage starts as "quiet". The quiet passengers already have the right to silence, so the talkative passengers would have to buy permission to talk. But talking is worth only £60 to them, and silence is worth £120 to the quiet passengers. No price works, so the carriage stays quiet.

Now suppose it starts as "normal". The talkative passengers have the right to talk. But silence is worth £120 to the quiet passengers, and talking only £60 to the talkative ones, so the quiet passengers could pay them to stop. Any total payment between £60 and £120 would leave both groups better off, so the carriage becomes quiet.

Either way, the carriage ends up quiet. That's the Coase theorem, from the economist Ronald Coase, whom you met in #008: if property rights are clear and bargaining is costless, people will trade rights until they reach the efficient outcome, whoever starts with them.

But the starting rule still matters, for distribution. If the quiet passengers start with the right, they pay nothing. If the talkative passengers start with it, the quiet passengers must compensate them. The Coase theorem doesn't say property rights don't matter. It says they don't affect the efficient outcome, under ideal conditions. They do affect who gains.

Because real trains are full of transaction costs. Passengers would have to work out who values silence, who values talking and by how much, negotiate a price, collect payments, make sure everyone sticks to the deal, and start again every time someone gets on or off. On top of that, it's awkward: few people want to offer a stranger money to stop talking.

Bargaining can also break down when some passengers hold out, demanding more because they know the deal depends on them. So even when an efficient solution clearly exists, people may never reach it. That's the practical limit of the Coase theorem, and it's why institutions matter.

Not in a single shared carriage. Silence there is non-excludable: once the carriage is quiet, everyone in it enjoys the silence, whether they paid or not. So if the company tried to charge each quiet passenger £4, any quiet passenger who refused to pay would still get the quiet for free. Each has an incentive to let the others pay, and the company can't stop them.

The company could try charging everyone who uses the carriage. But then the talkative passengers would be paying for silence they don't want, on top of losing the chance to talk. Unsurprisingly, they'd look for another seat.

To make charging work, the company has to make the quiet excludable. That means selling access to it: a separate quiet carriage, with its own tickets. Only people with a quiet-carriage ticket get in, so only people who pay enjoy the silence.

And that changes everything for the talkative passengers. With a separate quiet carriage, they no longer have to give anything up: they simply sit elsewhere and talk freely. The conflict largely disappears, because the two groups are sorted into different spaces. Quiet passengers get silence, talkative passengers get to talk, and the company can charge a premium of up to £4 for the quiet.

Economists call a good that is non-rival but excludable, where you can only enjoy it if you pay to get in, a club good. By selling quiet-carriage tickets, the company turns silence from a public good into a club good. That's exactly what real train companies do with quiet coaches.

Not necessarily.

With five carriages, the company can make some carriages quiet and sell tickets for them. Because each quiet carriage has its own tickets, the quiet is now excludable: only passengers who pay get in, so the company can sell it.

In our original example, every quiet passenger valued silence at exactly £4. But in reality, passengers value quiet differently: some would pay a lot, others only a little. And to fill more quiet seats, the company would have to lower the price for everyone.

A simple example shows the problem. Suppose three passengers would pay £4, £3 and £2 for a seat in a quiet carriage, and there's spare room in it, so an extra quiet-carriage passenger costs the company nothing:

A profit-maximising company might well charge £3 and sell two tickets. But the third passenger values a quiet seat at £2, and it would cost nothing to provide. Leaving them out wastes £2 of value. That's a deadweight loss: profitable for the company, inefficient for society.

This is the classic monopoly problem. A firm with pricing power restricts how much it sells to keep its price high. It's the mirror image of the monopsony in #009, which restricted how many workers it hired to keep wages low.

Quiet space also comes in whole carriages, not single seats. The company chooses between, say, one or two quiet carriages, so it can easily end up with too little quiet space, or with quiet seats sitting empty. And a company that wants to keep its premium high has a reason to keep quiet carriages scarce.

Two things limit the problem in practice. If passengers can choose other train companies, or drive instead, the company's pricing power shrinks. And if it could charge each passenger exactly what they're willing to pay (price discrimination), it would sell all three tickets, though it would then capture all the value itself.

So a profit-maximising company may provide less quiet space than would be efficient. Profit and efficiency point in the same direction in our single-carriage example, but once the company chooses how much quiet to sell, they can pull apart.

"Shhhh" (The Economist, 12 May 2012): The Economist's take on the economics of silence.

Latest nuisance: Zoom calls on an airplane!

Noise is an externality: talking brings a private benefit to the talker but imposes a cost on others. And silence is fragile: one loud phone call can ruin it for everyone nearby.

When rights are clear and bargaining is cheap, people can reach the efficient outcome themselves. When bargaining is awkward, costly or plagued by free riders, a simple rule can deliver what bargaining would have achieved in theory.

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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