A village has 10 fishermen/fisherwomen. Together, they must decide how bright to make the village lighthouse.

Let x be the brightness of the lighthouse, where x ≥ 0.

Each of them gets a benefit, measured in pounds, of:

B(x) = 500 ln(1 + x)

Brightness costs £250 per unit, so the total cost is:

C(x) = 250x

The lighthouse is a public good. It is:

So if one fisherman or fisherwoman pays for extra brightness, everyone else benefits too.

How bright should the lighthouse be, and why might the village end up with too little?

Q1: Why might B(x) = 500 ln(1 + x) be a sensible way to model the benefit of lighthouse brightness?

Q2: Write down an expression for the total benefit to the whole village, and hence the social surplus (total benefit minus total cost).

Q3: What brightness x maximises social surplus?

Q4: Suppose one fisherman or fisherwoman had to pay the full cost alone. What brightness would they choose to maximise their own surplus?

Q5: Why is the privately chosen brightness so much lower than the social optimum? What does this reveal about public goods?

Bonus: Suppose instead that each of them can contribute voluntarily towards the lighthouse. How bright would you expect it to end up, and would adding more fishermen and fisherwomen help?

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!)

£2.464 bn

This week’s marginal number is £2.464 billion.

That is how much unsecured lending to UK consumers rose in August, according to Bank of England data reported by Reuters.

It was the biggest monthly rise since the series began in 1993.

Source: Reuters, 29 September 2026.

Why does this matter?

Because borrowing is ambiguous.

On one reading, it is a sign of confidence. Households borrow more when they feel secure enough to spend, when they expect income to hold up, and when they are willing to bring future consumption into the present.

But there is another reading.

Borrowing can also be a sign of pressure. If prices are high, savings are thin, and wages are not keeping up with the cost of living, households may turn to credit not because they feel optimistic, but because they are trying to get through the month.

That is the useful economics lesson.

The same number can tell two very different stories.

More credit can mean stronger demand.

Or it can mean stretched household balance sheets.

So the question is not simply:

Are people borrowing more?

The better question is:

Why are people borrowing more?

That distinction matters.

A consumer boom and a household squeeze can look surprisingly similar in the data, at least at first glance.

The margin is where we ask what sits underneath the headline number.

When YouGov asked 4,598 adults in Great Britain, 73% chose the guaranteed £50,000, 21% chose the coin flip, and 6% did not know.

So were the 73% right, or the 21%? It depends on what we mean by "right".

There is no single correct answer until we know the person's circumstances. But there are two natural instincts.

The first is to take the £50,000. It is real, immediate and useful. There is no chance of walking away with nothing. For many people, £50,000 could clear debts, fund a house deposit or change their financial position overnight.

The second is to flip the coin. The upside is enormous: a genuine, even-money chance of becoming a millionaire.

So the first question is not really about maths. It is about what you value more, certainty or upside, and that is why it is so revealing. A simple question immediately exposes your attitude to risk.

Because certainty has value.

For someone with debts, no savings, insecure income, high rent or dependants, £50,000 means security. It might pay off expensive debt, build an emergency fund, provide a housing deposit, or simply remove the risk of ending up with nothing.

For that person, the jump from £0 to £50,000 may matter more than the jump from £50,000 to £1 million. This is the key economic idea: the value of money is not linear. The first £50,000 can transform a life. The extra £950,000 would be wonderful, but it is unlikely to be nineteen times as valuable in terms of wellbeing.

So taking the £50,000 can be perfectly rational. The person is not maximising expected money. They are maximising security, welfare and peace of mind.

Now look at it from the other side. Compare the expected values.

Option A: £50,000 for certain, so its expected value is £50,000.

Option B: a 50% chance of £1 million and a 50% chance of nothing:

The coin flip is worth, on average, ten times the guaranteed amount. If you could take this gamble many times, you would certainly want to: over many flips, you would average £500,000 each time. Taking £50,000 instead means accepting a tenth of the gamble's expected value.

So if expected money is all you care about, taking the £50,000 is irrational. But that is a big "if". This is a one-off decision, not a repeated game. You cannot average out a single coin flip: you either win £1 million or you get nothing. That is exactly why we need a better tool than expected value, which is where Q5 comes in.

Not everyone faces this choice in the same way. The answer depends on:

The same gamble can be sensible for one person and reckless for another. Rational choices depend on circumstances, not just on the numbers.

Suppose utility from wealth is U(W) = √W, where W is wealth after the choice and you start with nothing.

Option A:

Option B (expected utility):

The coin flip gives the higher expected utility. Here is the twist: U(W) = √W describes someone who is risk-averse (each extra pound adds less satisfaction than the last), yet they still flip the coin. The gamble is so favourable that this level of risk aversion is not enough to turn it down.

So what explains the 73%? Several possibilities:

Economists' models of rational choice are powerful, but real people are not always expected-utility calculators. Knowing where the models fit, and where they don't, is part of thinking like an economist.

We want the certainty equivalent: the guaranteed amount C that gives the same utility as the gamble.

So a person with U(W) = √W would need at least £250,000 to give up the coin flip, five times the £50,000 on offer.

The difference between the gamble's expected value and its certainty equivalent is the risk premium:

That is how much expected money this person is willing to give up to avoid risk. It is large, yet still nowhere near enough to make £50,000 attractive, which is exactly why the 73% need a richer explanation than √W alone.

EXTENSION READING:

Start here: The 2002 Nobel Prize in Economic Sciences: popular information (NobelPrize.org). A clear, non-technical account of Kahneman's work and prospect theory.

Loss aversion: Prospect Theory (The Decision Lab). A short, readable explanation of why losses loom larger than gains.

The certainty effect: Certainty effect (Cornell Law School's Wex). A one-page definition with examples.

For the ambitious: Advanced information on the 2002 Prize (NobelPrize.org, PDF). The technical version, including the maths behind prospect theory's S-shaped value function. Excellent preparation for university-level reading.

The book: Thinking, Fast and Slow by Daniel Kahneman (2011). Kahneman's own accessible summary of his life's work.

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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