How Voluntary Contributions Determine Provision
A public good is nonrival: one person’s enjoyment of its provision does not prevent others from benefiting. When provision depends on voluntary contributions, each person chooses how much to contribute while taking the contributions of others as given. The resulting equilibrium reflects individual incentives, not a coordinated choice to maximize total benefit.
The central calculation is an individual best response. A person compares the extra benefit they receive when total provision rises with the extra cost of their own contribution. A contribution profile is an equilibrium only if no person can improve their payoff by changing their contribution alone.
For the standard continuous model used here, assume benefits are differentiable and concave in \(G\), costs are differentiable and convex in a person's own contribution, and contributions cannot be negative. These assumptions make each person's payoff concave in their own contribution, so the marginal conditions below identify a best response. Let \(G_{-i}=\sum_{j\ne i}c_j\) denote provision from everyone other than person \(i\). Person \(i\) chooses \(c_i\) to maximize \(B_i(G_{-i}+c_i)-C_i(c_i)\).
Best Responses and Equilibrium Conditions
If a person contributes a positive amount, a small increase in their contribution raises total provision by the same amount. At an interior best response, the marginal benefit of that increase must equal its marginal cost. If the person contributes zero, the relevant question is whether starting to contribute would raise their payoff; the marginal benefit at the existing provision must then be no greater than the marginal cost of a small contribution.
These conditions are individual conditions: \(B_i'(G)\) is person \(i\)'s marginal benefit, not the sum of everyone’s marginal benefits. The zero-contribution inequality is important. It allows a person to contribute nothing even when the public good has value to them, provided the value of a small increase is no greater than the cost they would personally bear. With concave own payoff, satisfying the applicable condition gives a best response. An equilibrium must satisfy the condition for every person at once.
Set \(G=\sum_i c_i\), and identify the contributions held fixed when one person considers changing their choice.
Differentiate that person's payoff with respect to their own contribution. Check whether the optimum is positive or at the zero-contribution boundary.
Verify that every positive contribution satisfies marginal benefit equals marginal cost and every zero contribution satisfies the boundary inequality.
Maximize total benefit less the least resource cost of providing \(G\); do not use an individual's marginal benefit in place of total marginal benefit.
Why the Efficient Condition Is Different
For an efficient provision decision, the relevant benefit is the sum of all people’s benefits, because an extra unit of the public good benefits everyone. If total resource cost is \(K(G)\), total surplus is \(\sum_i B_i(G)-K(G)\). For an interior efficient quantity, differentiating total surplus gives the condition that the sum of marginal benefits equals the marginal resource cost.
This follows because changing \(G\) by a small amount changes every person's benefit, so the total benefit change is the sum of their individual changes. It does not follow that each person voluntarily accounts for all those benefits when choosing a private contribution. In the special case of a constant unit resource cost \(m\), efficient provision satisfies \(\sum_i B_i'(G)=m\), provided the optimum is interior. At a boundary, the corresponding one-sided condition applies.
If people have different contribution costs, efficient provision also requires choosing who supplies it. In that case, \(K(G)\) is the minimum total resource cost of producing \(G\), not automatically the cost schedule of whichever person happens to contribute in equilibrium. The examples make the cost assumptions explicit.
Worked Example: A Sole Contributor with Logarithmic Benefits
Two residents choose contributions \(c_1,c_2\geq0\), and provision is \(G=c_1+c_2\). Their benefits are \(B_1(G)=12\ln(1+G)\) and \(B_2(G)=8\ln(1+G)\). Each unit of contribution costs its contributor one unit, so \(C_i(c_i)=c_i\). Find an equilibrium in which resident 1 contributes alone and calculate efficient provision, assuming one unit of public-good provision uses one unit of resources.
Method. Find resident 1’s best response when resident 2 contributes zero, then check whether resident 2 prefers to remain at zero. The logarithmic benefits are strictly concave, so the marginal conditions identify the best responses.
Resident 1’s payoff when \(c_2=0\) is \(12\ln(1+c_1)-c_1\). Its derivative with respect to \(c_1\) is \(12/(1+c_1)-1\). Setting this equal to zero gives \(1+c_1=12\), so \(c_1=11\) and \(G=11\). The second derivative is \(-12/(1+c_1)^2<0\), confirming a maximum.
At \(G=11\), resident 2’s marginal benefit is \(8/(1+11)=8/12=2/3\). Their marginal contribution cost is 1, so \(2/3<1\); zero is a best response. Thus \((c_1,c_2)=(11,0)\) is an equilibrium.
For efficiency, total marginal benefit is \(12/(1+G)+8/(1+G)=20/(1+G)\). Equating it to the unit resource cost gives \(20/(1+G)=1\), hence \(G=19\). The equilibrium quantity, 11, is below the efficient quantity, 19.
Common mistake avoided: Using resident 1’s marginal benefit alone to find the efficient quantity would give 11. That is the sole contributor’s best response, not the efficient quantity; efficiency uses both residents’ marginal benefits.
Worked Example: Checking a Profile with Identical Residents
Two residents each have benefit \(B_i(G)=8\sqrt{G}\), and each unit contributed costs 1. Verify whether \((c_1,c_2)=(6,10)\) is an equilibrium, and find the efficient quantity. Assume the contribution costs are real resource costs.
Method. Check each resident’s marginal payoff at the proposed total, then use the sum of marginal benefits for the efficient quantity.
Total provision is \(G=6+10=16\). For either resident, marginal benefit is \(B_i'(G)=4/\sqrt{G}\). At \(G=16\), this is \(4/4=1\), equal to that resident’s unit marginal cost. Both contributions are positive, so both satisfy the interior condition. Since \(8\sqrt{G}\) is concave, each resident’s payoff is concave in their own contribution; the first-order condition gives a best response. The profile is therefore an equilibrium.
For efficiency, total marginal benefit is \(2(4/\sqrt{G})=8/\sqrt{G}\). Setting this equal to the unit resource cost gives \(8/\sqrt{G}=1\), so \(\sqrt{G}=8\) and \(G=64\). The efficient quantity is 64, not 16.
Common mistake avoided: The equilibrium condition for each person is \(4/\sqrt{G}=1\), which gives an equilibrium total of 16. The efficient condition is \(8/\sqrt{G}=1\), because it sums the two marginal benefits; squaring \(8/\sqrt{G}=1\) gives 64.
Worked Example: Different Contribution Costs and a Positive-Contribution Equilibrium
Two residents have benefits \(B_1(G)=10G-G^2/2\) and \(B_2(G)=9G-G^2/2\). Resident 1 pays 2 per unit contributed, while resident 2 pays 1 per unit: \(C_1(c_1)=2c_1\) and \(C_2(c_2)=c_2\). Check whether \((c_1,c_2)=(3,5)\) is an equilibrium. Then find the efficient quantity and the least-cost way to provide it.
Method. For each resident, differentiate their payoff with respect to their own contribution, treating the other person’s contribution as fixed. For efficiency, minimize resource cost for each proposed total provision before maximizing total surplus.
At the proposed profile, \(G=3+5=8\). Resident 1’s marginal benefit is \(B_1'(G)=10-G=2\), equal to their marginal cost of 2. Resident 2’s marginal benefit is \(B_2'(G)=9-G=1\), equal to their marginal cost of 1. Both contribute positively. Each benefit function is strictly concave, since its second derivative is \(-1\), so these conditions establish best responses. The profile is an equilibrium.
For a given total \(G\), the resource cost is \(2c_1+c_2\), with \(c_1+c_2=G\). Since resident 2’s unit cost is lower, the least-cost provision has \(c_1=0\), \(c_2=G\), and total cost \(K(G)=G\). Combined benefits are \(B_1(G)+B_2(G)=19G-G^2\). Total surplus is therefore \(19G-G^2-G=18G-G^2\). Differentiating gives \(18-2G=0\), so efficient provision is \(G=9\). The equilibrium quantity is 8.
Common mistake avoided: Setting each resident’s marginal benefit equal to the other resident’s cost would use the wrong incentives. Each person’s equilibrium condition compares their own marginal benefit with their own contribution cost; efficient provision instead accounts for total benefits and least resource cost.
Interpreting and Checking an Equilibrium
A proposed profile should be checked person by person. A positive contributor who faces marginal benefit above marginal cost would gain by increasing their contribution; one whose marginal benefit is below marginal cost would gain by reducing it. A person at zero must not have an incentive to begin contributing. When benefits are concave and costs convex, these marginal comparisons are sufficient; without those assumptions, a stationary point need not be a global best response.
It is also useful to keep the equilibrium and efficient calculations separate. In a voluntary-contribution equilibrium, each person’s decision responds to their own benefit and cost. In the efficient calculation, the decision-maker accounts for the total benefit to all people and the resource cost of provision. Consequently, equilibrium provision need not equal the efficient quantity. The exact difference depends on the benefit and cost schedules; it should be calculated rather than assumed.
Check Your Understanding
For each problem, use the stated benefit and cost schedules and check the equilibrium conditions before comparing with efficiency.
- Two residents each have \(B_i(G)=5\ln(1+G)\), with unit contribution cost. Is \((c_1,c_2)=(1.5,2.5)\) an equilibrium, and what is efficient provision? (Answer: Yes; \(G=4\) in equilibrium and \(G=9\) is efficient.)
- Resident 1 has \(B_1(G)=15\ln(1+G)\) and pays 3 per unit; resident 2 has \(B_2(G)=6\ln(1+G)\) and pays 1 per unit. Is \((4,0)\) an equilibrium? Find an equilibrium with resident 2 contributing alone. (Answer: No; resident 2 would gain from contributing at \(G=4\). The sole-contributor equilibrium is \((0,5)\).)
- Benefits are \(B_1(G)=7G-G^2/2\) and \(B_2(G)=6G-G^2/2\). Contribution costs are \(2c_1\) and \(c_2\). Verify \((c_1,c_2)=(2,3)\) as an equilibrium, then find efficient provision. (Answer: Equilibrium, \(G=5\); efficient provision \(G=6\), supplied at least cost by resident 2.)
- Two residents have benefits \(B_1(G)=9\ln(1+G)\) and \(B_2(G)=3\ln(1+G)\), with unit contribution cost. Check a profile in which resident 1 contributes alone, and compare its provision with efficiency. (Answer: \((c_1,c_2)=(8,0)\) is an equilibrium; \(G=8\) in equilibrium and \(G=11\) is efficient.)
- Two residents each have \(B_i(G)=4\sqrt{G}\), with unit contribution cost. Find the equilibrium total provision if both contribute positively, and the efficient quantity. (Answer: Any positive contributions summing to \(G=4\) form an equilibrium; efficient provision is \(G=16\).)