Why Private Provision Can Fall Short
A public good can benefit people who do not pay for it. When provision depends on voluntary contributions, each person weighs their own benefit against their own contribution cost, even though the good’s benefits may be shared by others. The result can be less provision than would maximize the total net benefit.
The size of the gap is not determined by the phrase “free rider” alone. It depends on everyone’s marginal benefit, the cost of provision, and the form of government or charitable support. In particular, a direct public contribution can replace private gifts rather than add to them. A matching grant changes the incentive to give and can produce a different response.
These conditions describe a voluntary-contribution equilibrium: each person chooses their contribution while taking the contributions of others, and any fixed outside provision, as given. They assume that the public good is nonrival, that each person benefits from the same total quantity, and that contributions are not limited by a binding income constraint. For clarity, examples below use quasi-linear utility: a person’s payoff is their benefit from \(G\), plus their remaining income. The conditions do not automatically hold if the good is congestible, if contributions are constrained, or if a change in public provision also changes contributors’ disposable income.
The Private Equilibrium and the Efficient Benchmark
A private contributor compares their own marginal benefit with the full cost of contributing. The efficient benchmark instead compares the sum of everyone’s marginal benefits with the marginal resource cost. With unit cost 1, the efficient condition is \(\sum_i MB_i(G)=1\) for an interior efficient quantity. This is the Samuelson condition introduced in the earlier material on public-good provision.
The difference between the conditions explains the usual underprovision result. If one person is an interior contributor, that person’s marginal benefit equals 1. Any other person with a positive marginal benefit adds to the total benefit from one more unit, but does not affect the contributor’s private cost. Thus, if other people’s marginal benefits are positive at the private equilibrium, the sum of marginal benefits exceeds 1. A larger quantity can raise total net benefit.
Underprovision is a common result, not a mechanical guarantee in every case. If only one person values the good, that person’s private condition can coincide with the efficient condition. At a zero-contribution equilibrium, each person may have an individual marginal benefit no greater than 1 while the sum of those benefits exceeds 1; the efficient condition must still be checked. The relevant comparison is total marginal benefit against marginal cost, not whether anyone gives.
Worked Example: Measuring the Underprovision
Two people value a public-good project. Their benefit functions are \(B_1(G)=10G-\frac{1}{2}G^2\) and \(B_2(G)=9.5G-\frac{1}{2}G^2\), over the quantities considered. Provision costs 1 per unit. Find the voluntary-contribution equilibrium when there is no outside provision, then find the efficient quantity and compare net benefits.
Method. First apply each person’s private marginal-benefit condition, taking the other person’s contribution as fixed. Then apply the aggregate efficient condition. This avoids the mistake of using one person’s marginal benefit as if it were the total social marginal benefit.
The marginal benefits are \(MB_1(G)=10-G\) and \(MB_2(G)=9.5-G\). If person 1 contributes and person 2 gives zero, person 1 chooses \(G\) so that \(10-G=1\), giving \(G=9\) and \(c_1=9\). At this quantity, person 2’s marginal benefit is \(9.5-9=0.5\), which is below the unit contribution cost. Person 2 optimally contributes zero. Thus the proposed choices are mutually consistent: the private equilibrium is \(G=9\), \(c_1=9\), and \(c_2=0\).
For efficiency, add the marginal benefits and set their sum equal to the unit cost:
This gives \(19.5-2G=1\), so \(2G=18.5\) and \(G^*=9.25\). At \(G=9\), the sum of marginal benefits is \(1+0.5=1.5\), exceeding the marginal cost of 1. That excess indicates that increasing provision from the private equilibrium can raise total net benefit.
As a check, at \(G=9\) total benefit is \(49.5+45=94.5\), and net benefit after the provision cost is \(94.5-9=85.5\). At \(G=9.25\), the squared term is \(9.25^2/2=42.78125\). The two benefits are \(92.5-42.78125=49.71875\) and \(87.875-42.78125=45.09375\). Total benefit is \(94.8125\), and net benefit is \(94.8125-9.25=85.5625\). Efficiency raises net benefit by \(85.5625-85.5=0.0625\), consistent with the positive combined marginal benefit in the interval. The method avoids assuming that the private equilibrium is efficient just because the active donor has set their own marginal benefit equal to cost.
Direct Government Provision and Full Crowding Out
Suppose government supplies some of the same public good that private donors would otherwise fund. In the standard quasi-linear model, direct provision does not change private contributors’ disposable income, and donors care about total \(G\), not whether a unit came from a private gift or government. If at least one donor remains active, a fixed increase in government provision can be offset by an equal reduction in private contributions.
The reason is that an active donor continues to choose a total quantity at which their own marginal benefit equals the unit cost. If government raises \(g\), the donor reduces their contribution until total provision returns to that preferred level. This is full crowding out over the range where the donor remains willing and able to contribute. It stops when private contributions reach zero.
The result follows directly from the donor’s choice condition. If donor \(i\) remains active, \(MB_i(G)=1\). Differentiating with respect to \(g\), and assuming diminishing marginal benefit, gives the following implication:
The first implication uses \(MB_i'(G)<0\), so the equation can hold only if total provision does not change. Since \(G=g+\sum_j c_j\), the final implication follows. This is a local result: it applies while the donor remains interior, not after their contribution has fallen to zero.
Worked Example: Direct Provision That Replaces Private Gifts
Suppose two people have marginal benefits \(MB_1(G)=10-G\) and \(MB_2(G)=9.5-G\). A government agency initially supplies zero units. It then supplies \(g=3\) units, financed without changing either person’s disposable income. Find the private contributions and total provision before and after the intervention.
Method. Use the active donor’s private condition to find the preferred total provision, then subtract government provision to find the remaining private contribution. This avoids counting government units as if they changed the donor’s preferred total quantity.
Before the intervention, person 1 can be the active donor. Their condition is \(10-G=1\), so \(G=9\). Person 2’s marginal benefit is \(9.5-9=0.5\), below the unit cost, so person 2 contributes zero. The baseline is \(c_1=9\), \(c_2=0\), and total provision \(G=9\).
With government provision \(g=3\), person 1 still chooses total provision of 9 while their private contribution remains positive. Therefore \(c_1=9-3=6\). Person 2 still faces marginal benefit \(9.5-9=0.5\) and contributes zero. Total provision is \(G=3+6+0=9\).
Private contributions have fallen from 9 to 6, a decrease of 3, exactly matching the government’s contribution. Total provision has not changed. This is full crowding out under the stated assumptions. The calculation would need to be reconsidered if the government contribution were financed by a tax that reduced the donor’s income or if the public provision were not a perfect substitute for private provision.
Partial Crowding Out and Matching Grants
Full crowding out is a clear benchmark, not a universal prediction. A government contribution may only partly displace private giving if donations respond less than dollar for dollar. For example, donors may not regard the public and private versions of a service as interchangeable, or the funding arrangement may affect their budgets and incentives. A response rule that specifies how private giving changes can be used to quantify this partial displacement, but its parameters must be supplied or estimated; they do not follow from the full-crowding-out model.
A matching grant works differently from direct provision. The government adds a specified amount for each dollar contributed by an individual, so each private dollar generates more total provision. The donor compares the benefit from that extra total provision with the donor’s own contribution cost. A matching grant therefore changes the private first-order condition. The government match itself still has a fiscal cost, which matters when evaluating the policy’s overall efficiency.
Worked Example: Calculating Partial Crowding Out
A hypothetical community has baseline private donations of 10 units. For the range being considered, suppose donations respond to direct government provision according to \(P(g)=\max(0,10-0.25g)\), where \(P\) is total private giving and \(g\) is government provision. Find private giving and total provision when \(g=8\), and calculate the crowd-out rate.
Method. Substitute the government contribution into the stated response rule, then compare both private giving and total provision with the baseline. This avoids treating a partial response as if the full-crowding-out result must apply.
At \(g=8\), the response rule gives \(P(8)=\max(0,10-0.25(8))=\max(0,10-2)=8\). Total provision is \(G=8+8=16\). With no government provision, private giving is \(P(0)=10\), so baseline total provision is 10.
Private giving has fallen by \(10-8=2\) units while government provision has risen by 8 units. The crowd-out rate, defined here as the fall in private giving divided by the increase in government provision, is \(2/8=0.25\), or 25%. Total provision rises by \(16-10=6\) units. These results check against the stated response rule: private gifts replace one quarter of the government contribution, and the remaining three quarters add to provision. The rule is an assumption of this example, not a general law of donor behavior.
Worked Example: A Matching Grant Changes the Giving Incentive
A single donor values total provision according to the marginal-benefit function \(MB(G)=15-G\), and pays 1 per unit of their own contribution. With no grant, find the privately chosen quantity. Then suppose the government adds 0.50 units for each unit the donor contributes. Find the donor’s contribution, the government match, and total provision.
Method. Write total provision as a multiple of the donor’s contribution and equate the donor’s marginal benefit from an extra contribution with its private cost. This avoids applying the no-match condition to a policy that changes how much total provision each donated unit creates.
Without a match, each private unit adds one unit to \(G\). The donor’s condition is \(15-G=1\), giving \(G=14\) and a private contribution of 14 units.
With a match rate \(m=0.50\), a private contribution \(c\) produces total provision \(G=(1+m)c=1.5c\). An additional unit of private contribution now creates 1.5 units of total provision, so its marginal benefit to the donor is \(1.5MB(G)\). The choice condition is \(1.5(15-G)=1\). Thus \(15-G=2/3\), and \(G=43/3\), or about 14.3333 units.
The private contribution is \(c=G/1.5=(43/3)(2/3)=86/9\), or about 9.5556 units. The government match is \(0.50c=43/9\), or about 4.7778 units. The check is \(86/9+43/9=129/9=43/3\), the total provision found above. Compared with the no-match baseline, private giving falls by \(14-86/9=40/9\), while the government adds \(43/9\). Total provision consequently rises by \(43/3-14=1/3\). This example isolates the donor’s incentive response; judging whether the grant is worthwhile also requires accounting for its fiscal cost and financing.
Interpreting the Policy Response
When evaluating a proposed contribution program, separate three quantities: the change in government provision, the change in private contributions, and the resulting change in total provision. A rise in public spending does not by itself show how much more of the good people receive. The private response determines whether public funds mostly replace gifts or largely add to them.
- Full crowding out: private contributions fall by the full increase in direct provision, so total provision is unchanged. This is the standard result only under the assumptions stated above and while an active donor remains.
- Partial crowding out: private contributions fall, but by less than the increase in public provision. Total provision rises by the difference.
- No crowding out: private contributions do not fall as public provision rises, so total provision rises by the full public contribution.
- Crowding in: private contributions rise alongside public support, so total provision rises by more than the direct public contribution. This requires a behavioral response that makes private giving complementary to the policy; it is not predicted by the basic full-crowding-out model.
These labels describe changes in quantities, not welfare conclusions. A policy that increases total provision may still have costs, including the resources used to provide the good and the cost of raising public funds. Conversely, a policy that displaces private contributions may still change who pays or how the good is delivered. For a complete assessment, compare total marginal benefits with marginal resource and financing costs, rather than treating the size of the public budget as the objective.
Check Your Understanding
For each problem, identify the relevant private response before comparing it with total provision.
- Two people have marginal benefits \(MB_1(G)=14-G\) and \(MB_2(G)=13.5-G\), and provision costs 1 per unit. Assume no outside provision and no binding contribution constraints. Find the voluntary equilibrium if person 1 is the donor, check person 2’s choice, and find the efficient quantity. (Answer: Private provision \(G=13\), with person 1 contributing 13 and person 2 contributing 0; person 2’s marginal benefit is 0.5. Efficient provision is \(G^*=13.25\).)
- A donor has marginal benefit \(MB_1(G)=10-G\). Another person has \(MB_2(G)=9.5-G\). With unit cost and unchanged disposable income, government provision rises from 0 to 4. Assume the first person remains an active donor. What are private contributions and total provision after the change? (Answer: The donor contributes 5, the other person contributes 0, and total provision remains \(G=9\); private contributions fall by 4.)
- Private giving follows \(P(g)=\max(0,10-0.25g)\), where \(g\) is government provision. At \(g=8\), calculate private giving, total provision, and the crowd-out rate relative to \(g=0\). (Answer: Private giving is 8, total provision is 16, and the crowd-out rate is 25%.)
- A donor’s marginal benefit is \(MB(G)=15-G\). A matching program supplies 0.50 units for each private unit contributed, and private contribution cost is 1 per unit. Find the donor’s contribution, the match, and total provision, assuming the donor chooses an interior contribution. (Answer: Private contribution \(86/9\approx9.5556\), government match \(43/9\approx4.7778\), and total provision \(43/3\approx14.3333\).)
- Two people have marginal benefits \(MB_1(G)=8-G\) and \(MB_2(G)=7.4-G\). Provision costs 1 per unit. If person 1 contributes alone in the voluntary equilibrium, find that equilibrium and the efficient quantity, and state whether the private amount is below or above the efficient amount. (Answer: The private equilibrium is \(G=7\), with person 2’s marginal benefit equal to 0.4; efficient provision is \(G^*=7.2\), so private provision is below the efficient amount.)