What makes a good public — non-rival and non-excludable — Economics

Before this

What is an externality
Positive externalities
What governments spend money on

This leads to

How much defence did you buy today?

Think about the last thing you bought: a snack, a bus ticket, a pen. You paid, and you got it. If you had not paid, the seller would have kept it. The price decided who got the good.

Now think about your country's defence. Ask three questions about it.

How much of it did you use today? There is no way to say. You did not use "an amount".
Could you be left out of it if you refused to pay? No. If your country is defended, your house is defended with it.
Does your protection leave less protection for your neighbour? No. Your neighbour is exactly as protected as you are.

These three answers break the way a market works. A market needs a price, and a price only works if people who do not pay can be left out. Defence cannot leave anyone out. So who would pay for it?

The lecture on positive externalities ended with the lighthouse. Once its light shines, every ship in range benefits, and no ship can be stopped from benefiting. This lecture makes that idea precise. It asks what makes a good "public", and why markets struggle to supply such goods.

Two questions to ask about any good

Economists sort goods with two questions. Each question has a yes-or-no answer.

Question 1: is the good rival? A good is rival when one person's use leaves less of it for others. A bowl of rice is rival: if you eat it, nobody else can. A seat on a full train is rival: if you sit in it, the next passenger stands. A good is non-rival when one person's use leaves just as much for everyone else. A weather forecast on the radio is non-rival: a million listeners do not use it up. A theorem in mathematics is non-rival: you can use Pythagoras' theorem, and it is still there for everyone else.

Question 2: is the good excludable? A good is excludable when it is possible, at a reasonable cost, to stop people who do not pay from using it. A cinema is excludable: there is a door and a ticket check. A good is non-excludable when stopping non-payers is impossible, or costs far too much. A flood barrier that protects a city is non-excludable: it cannot hold back the water from one house and let it into the house next door.

Notice the words "at a reasonable cost". Excludability is not about whether people are charged. It is about whether they could be charged. A park with free entry is still excludable if a fence and a gate could be built at a sensible cost. The question is about what is possible, not about what is chosen.

Two questions with two answers each give four kinds of good.

The two questions sort every good into four boxes. Public goods are in the bottom-right box: non-rival and non-excludable.

Private goods (rival, excludable): most of what people buy in shops. The market handles these well, which is why the earlier lectures on demand and supply worked.
Common resources (rival, non-excludable): anyone can take them, but what one person takes is gone. Fish in the open sea are the classic case. Because nobody can be excluded, these tend to be overused.
Club goods (non-rival, excludable): one more user costs almost nothing, but access can be sold. A streaming service can let one more person watch a film without taking it from anyone, and it can still demand a password.
Public goods (non-rival, non-excludable): one person's use takes nothing from anyone, and nobody can be left out.

Public good

A public good is a good that is both non-rival and non-excludable.

Non-rival: one person's use does not reduce what is available to others.

Non-excludable: people who do not pay cannot be stopped from benefiting, at any reasonable cost.

Practise the two questions before going on. For each card, ask "Does one person's use leave less for others?" and then "Could non-payers be kept out at a reasonable cost?"

Sorter — which box does each good belong in?

Put each card in a box. Ask the two questions for each one. Every card explains itself when you place it.

Nine goods sorted with the two questions.

Good	Box	Reason
A cup of coffee at a café	Private	You drink it, so nobody else can; no payment, no coffee.
A pair of shoes	Private	Only one person can wear them; the shop keeps them until you pay.
Grass on a village field open to every herd	Common resource	Grass eaten by one herd is gone; nobody is kept off the field.
Groundwater that any farm may pump	Common resource	Water one farm pumps is gone; no farm is kept out.
An online newspaper behind a paywall	Club good	One more reader takes nothing away; the paywall keeps non-payers out.
A cable TV channel	Club good	Another viewer takes nothing away; the signal goes only to subscribers.
Street lighting	Public good	One walker's light does not dim it; nobody on the street can be kept in the dark.
A radio weather forecast	Public good	A million listeners do not use it up; anyone with a radio can hear it.
A fireworks display in an open park	Public good	Thousands can watch at once; anyone nearby can look up.

The rule: rivalry asks whether use takes something away; excludability asks whether non-payers can be kept out. A public good answers "no" to both.

Look at the two cards in the club box. A cable channel and a weather forecast are both non-rival. The only difference is that the cable company can switch off the signal to a home that does not pay. That one difference decides whether a firm can sell the good.

Stop and think

A government hospital treats patients for free. Is it a public good in the economist's sense?

Show the answer

No. Ask the two questions. A hospital bed used by one patient cannot be used by another at the same time, so treatment is rival. And a hospital could charge at the door, so it is excludable. Treatment is a private good that the government chooses to provide free. "Public good" does not mean "provided by the government", and it does not mean "good for the public". It is a technical name for goods with two properties. Many things a government supplies are not public goods, and a few public goods are supplied without a government.

Why markets struggle: the free rider

The lecture on externalities defined a free rider: someone who enjoys a benefit without paying for it. With a public good, free riding is not a rare bad habit. It is the sensible choice for each person. The worked example shows why.

Worked Example 1 — A flood wall for a village (invented numbers)

A river village has 200 households. A flood wall would cost $20 000. Each household values the protection at $150.

Is the wall worth building? The wall is non-rival: it protects every house at once. So its total benefit is the sum over all households: $200 \times $150 = $30 000$ . That is more than the cost of $20 000. Society gains $10 000 by building it.

Will one household build it alone? No. One household gains $150 and would pay $20 000.

Will the village raise the money by asking for voluntary payments? Probably not enough. Think like one household. If the wall is built, your house is protected whether you paid or not, because the wall is non-excludable. If you pay, the wall is only slightly more likely to be built, because your payment is a small part of $20 000. So the sensible choice for each household is to keep its money and hope the others pay. If many households reason this way, the collection falls short and the wall is never built.

What does a compulsory tax do? Suppose the village council charges every household $100. That raises $200 \times $100 = $20 000$ , exactly the cost. Each household pays $100 and gets protection worth $150, so each household is $50 better off. Nobody can free ride, because the payment is no longer a choice.

Check: 200 households × $50 = $10 000, the same gain to society found above.

That is it. That is the whole idea. Because nobody can be excluded, nobody needs to pay; and because nobody needs to pay, the good may not be supplied, even when everyone would be better off with it. This is why public goods are a market failure. The problem is not that the good costs too much. The problem is that the price system cannot collect payment for it.

The model below lets you sit in one household's chair. It is invented, but the reasoning is the same as the flood wall's.

Real-world model — street lamps and the free rider

An invented street of 8 households. Each household may pay $30 to buy one lamp. Every lamp lights the whole street, so each lamp gives every household $8 of benefit, whoever paid for it. Choose how many of your 7 neighbours pay. Then compare your result when you pay and when you do not.

Neighbours who pay: 4 of 7

An invented street of 8 households. Each household may pay $30 for one lamp, and each lamp gives every household $8 of benefit. Your net result, for three choices by your 7 neighbours:

Others pay	You pay	You don't
0	−$22	$0
4	$10	$32
7	$34	$56

For example, when 4 neighbours pay and you pay too, there are 5 lamps: you get 5 × $8 = $40 of benefit and pay $30, so your net result is $10. If you do not pay, there are 4 lamps and you keep $32.

Whatever the neighbours do, not paying leaves you $22 better off. If everyone reasons this way, there are no lamps and everyone gets $0. If everyone pays, there are 8 lamps and everyone gets $34.

Look at what the model shows. Whatever your neighbours do, not paying leaves you exactly $22 better off. This is because your $30 buys one lamp, and that lamp gives you only $8 of benefit. The other $56 of its benefit goes to your neighbours. So each household, thinking only about itself, keeps its money. If all eight reason this way, the street stays dark and everyone gets $0. If all eight pay, everyone gets $34. Every household prefers the lit street, yet each one's own choice leads to the dark one.

The model leaves out a great deal. Neighbours talk to each other. People feel shame when they do not pay their share, and some people simply enjoy contributing. Laboratory experiments test exactly this kind of game with real money. A well-known survey of these experiments was written by John Ledyard (1995). It found that people at first put in between about 40% and 60% of the money they are given to play with. The simple model predicts 0%. But when the game is repeated, contributions usually fall towards the free-riding level. So the free-rider problem is real, but it is a pull, not an iron law. Voluntary giving supplies some public goods, but usually less than people would choose together.

A real case: who paid for Britain's lighthouses?

Worked Example 2 — Lighthouses in Britain (1600s–1836)

The textbook view. For more than a century, economists used the lighthouse as the perfect public good. In 1848 John Stuart Mill wrote that building lighthouses was "a proper office of government". Paul Samuelson's famous textbook said that a businessman "could not build it for a profit, since he cannot claim a price from each user."

The prediction. If this is right, private owners should never have built lighthouses. Only governments should have done it.

What happened. In 1974 the economist Ronald Coase studied the history of British lighthouses in an article called "The Lighthouse in Economics" (Journal of Law and Economics). He found that many were built and run by private owners. In 1820, Trinity House, the official lighthouse authority, ran 24 lighthouses, and private individuals or organisations ran 22. How did private owners get paid? They held a patent from the Crown, which gave them the right to charge "light dues". The dues were collected by customs officers at the ports, as a charge per ton of each ship arriving at or leaving a British port. In 1836, an Act of Parliament gave all lighthouses in England to Trinity House, which bought out the remaining private owners for just under £1 200 000.

The verdict. The textbook view was too simple, but the public-good idea survives. The light itself was still non-excludable at sea. What the owners found was a different thing that could be excluded: entry to a port. A ship that refused to pay could be stopped in harbour, not in the dark. And the payment only worked because the state granted the right to charge and used customs officers to collect it. Later economists (for example Van Zandt in 1993 and Bertrand in 2006) pointed out that lights which depended on voluntary payment did not last. So the case shows two things. Public goods can sometimes be financed without direct government provision. But it takes a way to tie payment to something excludable, backed by law.

The same problem on the internet

Information has the first property of a public good in a very strong form. Once an article, a song or a piece of software exists, one more copy costs almost nothing, and copying it takes nothing from anyone else. It is non-rival. If it is also free for anyone to reach, it is non-excludable too. Then the free-rider problem appears: why would anyone pay to produce it?

Look at the solutions that have grown up, and notice that each one works on one of the two questions. Paywalls and licence keys create excludability, turning the information into a club good. Advertising lets a third party pay, so the reader can free ride while the producer is still paid. Public funding through taxes, as with many public broadcasters, uses the flood-wall solution: payment is compulsory. And volunteer projects, such as online encyclopaedias and open-source software, rely on the kind of willing contribution the experiments found.

Three classic mistakes

Trap 1 — "A public good is anything the government provides"

Free school meals, public hospitals and state housing are provided by governments, but they are rival and excludable. They are private goods supplied publicly. Always classify a good by the two questions, never by who supplies it.

Trap 2 — "A public good is a good thing for the public"

"Public good" is a technical term, not praise. National defence is a public good even for citizens who disagree with their country's army. And many valuable things, such as education, are mostly rival and excludable. Value to society is a different question from the two properties.

Trap 3 — Confusing "free" with "non-excludable"

A park with free entry is not non-excludable just because nobody is charged. If a fence and a gate could keep non-payers out at a reasonable cost, it is excludable. Non-excludable means that exclusion is impossible or far too expensive, as with defence or clean air over a city.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

Classify each good as private, common resource, club or public.

(a) A sandwich.
(b) A television signal sent free over the air.
(c) Wild mushrooms in a forest open to all.
(d) A gym that only members can enter, at a quiet time of day.

Hint 1
For each one, ask: "Does one person's use leave less for others?"

Hint 2
Then ask: "Could people who do not pay be kept out at a reasonable cost?"

Full answer
(a) Private: rival (eaten once) and excludable (sold). (b) Public: non-rival (one more viewer takes nothing) and non-excludable (anyone with an aerial receives it). (c) Common resource: rival (picked once) and non-excludable (nobody is kept out). (d) Club good: non-rival while the gym is quiet, and excludable by membership. (When the gym becomes crowded, one person's use does take space from others. The next lecture looks at goods like this.)
A building has 50 flats. A security camera system for the entrance costs $4000 a year. Each flat values the extra safety at $120 a year. Is the system worth having? Why might a voluntary collection fail? What happens if the building charges every flat the same compulsory fee to cover the cost?

Hint 1
The cameras protect every flat at once. Add up the benefit across all 50 flats.

Hint 2
The fee that exactly covers the cost is $4000 ÷ 50.

Full answer
Total benefit = 50 × $120 = $6000, more than the $4000 cost, so the system is worth having. A voluntary collection may fail because the cameras protect every flat whether it paid or not, so each flat is tempted to free ride. A compulsory fee of $4000 ÷ 50 = $80 covers the cost exactly. Each flat pays $80 and gains $120, so each is $40 better off. Check: 50 × $40 = $2000 = $6000 − $4000.
A coastal town has a tsunami warning siren. A company also offers a warning text message service for $2 a month. Classify each, and explain why a company can sell one but not the other.

Hint 1
Both are non-rival: one more person warned does not weaken the warning for others.

Hint 2
Can the siren be heard only by people who paid? Can the text be sent only to people who paid?

Full answer
The siren is a public good: non-rival and non-excludable, since everyone within earshot hears it. The text service is a club good: non-rival, but excludable, since messages go only to subscribers. A company can sell the text service because it can leave out non-payers. It cannot sell the siren because it cannot. The information is the same; technology changed the answer to the excludability question. This is why the siren is usually paid for from taxes.
Go back to the street of 8 households. Each lamp costs $30. Suppose each lamp gives every household a benefit of $b$ dollars. (a) For which values of $b$ is a lamp worth building for the street as a whole? (b) For which values will one household buy a lamp on its own? (c) For which values of $b$ does the free-rider problem stop a lamp that the street would want? (d) With $b = 8$ , what does a compulsory charge for one lamp, shared equally, do to each household?

Hint 1
The lamp is non-rival, so its total benefit to the street is $8 b$ . One household's own benefit is only $b$ .

Hint 2
Compare $8 b$ with $30, and compare $b$ with $30.

Full answer
(a) Worth building when $8 b > 30$ , that is, $b > 3.75$ . (b) One household buys alone only when $b > 30$ . (c) The free-rider problem bites when $3.75 < b < 30$ : the street gains, but no single household gains enough to pay. (d) Shared equally, the charge is $30 ÷ 8 = $3.75 per household. Each household gains $8, so each is $8 − $3.75 = $4.25 better off. Check: 8 × $4.25 = $34, the street's gain per lamp from the model. Notice how wide the problem range is: the more households share a good, the smaller each one's share of the benefit, and the stronger the pull to free ride.

The bigger idea

The price system needs exclusion to work. A price is a gate: pay, and you pass; refuse, and you do not. Every market in the earlier lectures, from coffee to carbon permits, quietly relied on that gate. Public goods are the case with no gate, so the price system has nothing to hold on to.

A public good is also the extreme case of a positive externality. With a vaccine or a training course, part of the benefit spills over to others. With a lighthouse or a flood wall, almost all of the benefit goes to people other than the payer, so the private incentive to pay almost disappears. That is why the usual answer is collective payment through taxes. But taxes raise a new problem: without prices, how does a government know how much defence, or how many lamps, people really want? The lecture on government failure returns to that question. First, the next lecture shows that the four boxes are not as tidy as they look, because many real goods sit between them.

Key concepts in this lesson

A good is rival when one person's use leaves less for others, and excludable when non-payers can be kept out at a reasonable cost.
The two questions sort goods into four kinds: private goods, common resources, club goods and public goods.
A public good is both non-rival and non-excludable, such as national defence, a flood barrier or street lighting.
Because nobody can be excluded, each person is tempted to free ride, so markets and voluntary giving tend to supply too little of a public good, or none.
Public goods are usually paid for through compulsory taxes; British lighthouses show that payment can sometimes be tied to something excludable instead, with the help of the law.
"Public good" is a technical term about two properties, not a name for anything the government provides or anything that benefits the public.