Public Goods and Free Riders explained why markets supply too little of goods that no one can be kept from and that everyone uses at once. With The Marginal Rate of Substitution we can state exactly how much is efficient, a rule Paul Samuelson gave in 1954.
Key idea
For a public good $G$ with marginal cost $MC$, the efficient amount satisfies the Samuelson condition $$\sum_i MB_i(G) = MC(G).$$ Because everyone uses the same units, individual marginal benefits are added vertically (at each quantity), not horizontally as for private goods.
Why it works
One more unit of a streetlight program lights the street for everyone at once. Its value to society is the sum of what each person would pay for that unit: $MB_1 + MB_2 + \cdots$. Efficiency adds units while that sum exceeds the cost. Compare a private good, where one more unit goes to one person and markets add quantities at a price (Adding Up to Market Demand). If each person decides alone how much to contribute, she stops where her own marginal benefit equals the cost, ignoring the benefit to others, so voluntary contributions fall short: free riding.
Worked example: The efficient amount
$MB_1 = 20 - G$, $MB_2 = 30 - 2G$, $MC = 23$.
- Both are positive for $G < 15$, so there $\sum MB = 50 - 3G$.
- $50 - 3G = 23$ gives $G^* = 9$.
- At 9: $MB_1 = 11$, $MB_2 = 12$, summing to 23. Charging each person her own marginal benefit (a Lindahl price) would cover the cost exactly.
Answer: $G^* = 9$.
Worked example: Free riding
Same people, but each decides alone how much to buy.
- Person 1: $20 - G \ge 23$ never holds, so she buys nothing on her own.
- Person 2: $30 - 2G = 23$ at $G = 3.5$, so he provides 3.5.
- At 3.5, person 1's benefit (16.5 per unit) is below the cost, so she adds nothing and rides free. Total: 3.5, far below 9.
Answer: 3.5 units.
Common mistake
Adding demand curves horizontally, as for private goods. That would answer "how many units would they buy in total at a price of \$23," treating each unit as consumed by one person. For a public good every unit serves both, so add the heights (willingness to pay) at each quantity. Check: at your answer, the marginal benefits of all users should add up to the marginal cost.
The hard part in practice is learning the $MB_i$: if a person's share of the bill depends on what she reports, she has a reason to understate her benefit. Governments fund public goods through taxes, and economists study mechanisms that make honest reporting worthwhile.