Public goods are nonrival and nonexcludable (Rival and Excludable Goods), and like other goods with spillovers (Positive Externalities), markets tend to provide too little of them. The reason is a game, played out in the Game Theory course's Dominance and the Prisoner's Dilemma chapter, where a public-goods game unravels.
Key idea
Because no one can be kept from using a public good, each person can free ride: enjoy it without paying. So private sellers can't collect enough to cover the cost, and markets supply too little. A public good is worth providing when the sum of everyone's benefits is at least its cost; for each extra unit, compare the sum of the marginal benefits with the marginal cost.
Why it works
Each person reasons: "If others pay, I get it free; if they don't, my small payment won't buy it alone." Not paying is a dominant strategy, as in a prisoner's dilemma, so the good may not get provided even when everyone would be better off if all paid. And because one person's use doesn't reduce anyone else's, the value of a unit is everyone's benefit added together, not one person's. That's why taxes, which everyone pays, often fund public goods.
Worked example: A streetlight
A streetlight costs \$75. Three neighbors value it at \$40, \$30 and \$20. Should it be built? Will any one neighbor buy it alone?
- Total benefit: $\$40 + \$30 + \$20 = \$90 > \$75$: it's worth building.
- No single neighbor values it at \$75 or more, so none buys it alone.
- And each hopes the others pay: a free-rider problem. Splitting the cost by a rule, such as a tax of \$25 each, gets it built.
Answer: Yes, it should be built (\$90 > \$75), but no one will buy it alone.
Worked example: How many units?
A town is deciding how many benches to put in its park. For benches 1 to 4, the marginal benefit to Group A is \$40, \$30, \$20 and \$10, and to Group B \$35, \$25, \$15 and \$5. Each bench costs \$50. How many should it build?
- Everyone can use every bench, so a bench's value is the sum of both groups' benefits.
- Add vertically for each bench: \$75, \$55, \$35, \$15.
- Benches 1 and 2 are worth more than \$50; bench 3 (\$35) isn't. Build 2.
Answer: 2 benches.
Common mistake
Comparing the cost with one person's benefit instead of the sum of everyone's. By that test the streetlight (worth at most \$40 to any one neighbor) would fail, though it's worth \$90 to the street. Check: for a public good, add the benefits of everyone who uses it.