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A pure public good is consumed by everyone at once, so one unit benefits every person. Paul Samuelson showed in 1954 that Pareto-efficient provision requires the sum of all individuals' marginal rates of substitution between the public good and a private good to equal the marginal rate of transformation, Σᵢ MRSᵢ = MRT. For a private good, by contrast, each person's MRS equals the MRT separately. When people contribute voluntarily, each one compares only their own marginal benefit with the cost, so the Nash equilibrium supplies too little. Lindahl equilibrium, with personalized prices that sum to marginal cost, reaches the efficient level but relies on people reporting their benefits truthfully.
Use a small town with three residents and linear marginal-benefit schedules for a public park. Sum the schedules vertically, because every resident uses the same park, and set the sum equal to marginal cost. Then compute what would be built if residents contributed on their own, and finally find Lindahl prices. The gap between the voluntary and efficient levels makes the free-rider problem quantitative.
You know that a pure public good is non-rival and non-excludable: one person's use does not reduce what others get, and no one can easily be kept out. National defense and a lighthouse are textbook cases. The question here is how much of such a good an efficient economy should produce.
Start with a private good. If an extra apple goes to Ana, nobody else eats it. At a Pareto-efficient allocation, Ana's marginal rate of substitution between apples and other goods must equal the rate at which the economy can transform other goods into apples, the MRT. The same holds for every consumer separately.
A public good differs because an extra unit is enjoyed by everyone. Its value to society is the sum of what each person would give up for it. Paul Samuelson's 1954 paper "The Pure Theory of Public Expenditure" stated the condition: provide the public good up to the point where Σᵢ MRSᵢ = MRT. In money terms, the sum of individual marginal benefits equals marginal cost. Graphically, individual demand curves are added vertically, not horizontally.
Take three identical residents, each with marginal benefit 10 − G, and a marginal cost of 6. The vertical sum is 30 − 3G, so the efficient quantity is 8.
Now let residents contribute on their own. Each one gains only their own marginal benefit from an extra unit but pays its full cost. So each stops contributing once their own MB reaches 6, at G = 4. If the total were below 4, someone would add more; above 4, nobody would. Voluntary provision gives 4, half the efficient amount. Each resident free-rides on the others' contributions. This underprovision is a general result of private provision of public goods.
The Swedish economist Erik Lindahl described an allocation that fixes the problem in principle. Each person pays a personalized price per unit equal to their own marginal benefit at the efficient quantity, and these prices sum to marginal cost. In the example each resident pays 2 per unit. At a price of 2, each resident's demand is 10 − G = 2, or G = 8, so all agree on the efficient quantity and costs are covered.
The difficulty is information. Lindahl prices depend on preferences that only individuals know, and a person who understates their benefit pays less while still enjoying the good. Mechanism design asks whether any procedure can get people to reveal their values; the Vickrey-Clarke-Groves mechanism is one answer, at a cost in budget balance.