Chapter 68: Public Goods, Free Riding, Clubs, and Collective Funding. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Sum the marginal values, then compare to cost

Should four districts buy one more storm-alert package that none of them would buy alone?

A public good is shared, so one more unit is worth the sum of everyone's marginal values. The Samuelson condition compares that vertical sum with marginal cost, which is why a unit can be efficient even though no single user would pay for it.

Equation, written in LaTeX: MB_3=22+15+11+8=56.

Equation, written in LaTeX: MB_4=16+12+9+6=43<50.

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Each district's marginal willingness to pay for one more alert package is in dollars per storm. MB is their sum, because every district receives the same alerts. MC is the marginal cost of the package. An inland spillover adds a benefit the first boundary left out.

Predict first. Does any single district value the third package enough to buy it alone at a cost of 50?

Your prediction Yes No

Package considered

Marginal cost

Inland spillover

Figure: Sum the marginal values, then compare to cost. Stacked bar of the marginal values for the third package summing to 56, a bar for the largest single value 22, and the marginal cost line at 50. — Package considered: Third, Marginal cost: 50, Inland spillover: 0
Constructed example: the chapter's hypothetical coastal districts (values 22, 15, 11, 8 and 16, 12, 9, 6, cost 50 or 58, spillover 5); a marginal cost of 42 is added for comparison.

Sum of marginal values: 56
Marginal cost: 50
Surplus: 6
Largest single value: 22
Package: passes

The districts' marginal values sum to 22 + 15 + 11 + 8 = 56. 56 > 50, so the third package passes with surplus 6. No district would pay 50 alone: the largest single value is 22.

Worked steps

MB = 22 + 15 + 11 + 8 = 56
MB - MC = 56 - 50 = 6
Largest single value 22 < 50: no district buys alone

Use the idea

For a shared service, collect each group's marginal value for the next unit, add them and compare with marginal cost; include spillovers to people outside the paying group.

Where the conclusion applies

Values are known and truthfully reported, and units are compared one at a time. Who pays is a separate question the test does not answer.

Check your understanding: With finance costs raising marginal cost to 58 and the 5 spillover included, does the third package pass?

56 + 5 = 61 > 58, so it passes; without the spillover 56 < 58 and it fails. The fourth package fails at a cost of 50 either way: 43 < 50 and 43 + 5 = 48 < 50.

Chapter 68 source: section "Samuelson condition for public goods".

Demonstration 2 of 4

Quadratic funding rewards breadth

With the same 144 of private money, how much does a matching fund add when more people give?

Square roots reward many small gifts: splitting the same money among more people raises the sum of roots, and squaring it adds cross terms between every pair of donors. A fixed pool then scales every desired match by the same factor.

$Equation, written in LaTeX: F_B=(4\sqrt{36})^2=(4\times6)^2=576.$

Equation, written in LaTeX: M_B=576-144=432.

$Equation, written in LaTeX: F_C=(9\times4)^2=1{,}296,$

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Each donor gives c_i. The funding target is F = (sum of sqrt c_i)^2 and the match is F minus the private total. Project B has four donors of 36; the other project splits 144 equally among the chosen number of donors. When desired matches exceed the pool, each is scaled by beta = pool / desired total.

Predict first. Holding the 144 fixed, how does the funding target grow with the number of equal donors?

Your prediction In proportion to the number of donors With its square root It stays at 144

Number of equal donors (144 total)

Match pool for B and the other project

Figure: Quadratic funding rewards breadth. Left: the funding target 144 n rises in a straight line with the number of donors; at 4 it is 576. Right: desired and paid matches for B (300.00 paid) and the 4-donor project (300.00 paid) from a pool of 600. — Number of equal donors (144 total): 4, Match pool for B and the other project: 600
Constructed example: the chapter's hypothetical projects A, B and C (144 from 1, 4 or 9 donors, pool 600); 16 donors of 9 each and a pool of 1,000 are added, and 1,584 is the book's unconstrained total.

Funding target: 576
Match: 432
Scaling factor beta: 0.6944
B receives: 300.00
4-donor project receives: 300.00
Pool left over: 0.00

4 donors give 36 each for 144 in total. The target is (4 x 6)^2 = 576, so the match is 432: holding private money fixed, the target grows in proportion to the number of donors. The desired matches total 864, more than the pool of 600, so both are scaled by 0.6944: B receives 300.00 and the 4-donor project 300.00. The 4-donor project is identical to B (four donors of 36), so the two tie and receive the same 300.00.

Worked steps

Each of 4 gives 36; sqrt(36) = 6
F = (4 x 6)^2 = 576; match = 576 - 144 = 432
Desired matches: 432 + 432 = 864
beta = 600 / 864 = 0.694444
B: 432 x 0.694444 = 300.00; other: 432 x 0.694444 = 300.00

Use the idea

When designing a matching fund, budget for the unconstrained matches, decide in advance how they will be scaled, and verify that donors are distinct people.

Where the conclusion applies

Independent donors with genuine valuations and proportional scaling. The chapter warns that one donor splitting money across controlled accounts produces the same arithmetic.

Check your understanding: With nine donors, what match is implied before the pool limit, and what does a 600 pool pay?

(9 x 4)^2 = 1,296, so the match is 1,296 - 144 = 1,152. With the 600 pool, beta = 600/1,584 = 0.3788; B receives 163.64 and the nine-donor project 436.36.

Chapter 68 source: section "Quadratic funding".

Demonstration 3 of 4

Crowding versus cost sharing sets club size

How many members should a club admit when each newcomer shares the fixed cost but adds crowding?

Adding a member spreads the fixed cost more thinly but crowds everyone. The best size balances the marginal congestion loss 2an against the marginal cost-sharing gain C/n^2; because members are whole people, the two neighbouring integers are compared.

$Equation, written in LaTeX: u(n)=2{,}100-0.12n^2-\frac{72{,}000}{n}.$

$Equation, written in LaTeX: 0.24n=\frac{72{,}000}{n^2},$

$Equation, written in LaTeX: n^3=300{,}000.$

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n is membership. Each member's gross benefit is 2,100, crowding lowers it by a n^2 (a = 0.12 in the book), and the fixed facility cost C (72,000 in the book) is split equally. u(n) is net benefit per member; n* is the continuous optimum.

Predict first. If the facility cost rises, does the optimal club get bigger or smaller?

Your prediction Bigger Smaller

Fixed facility cost

Congestion coefficient

Figure: Crowding versus cost sharing sets club size. Net benefit per member u(n) = 2,100 - 0.12n^2 - 72,000/n for 30 to 110 members, peaking near n* = 66.94; the best whole club has 67 members with u = 486.69. — Fixed facility cost: 72,000, Congestion coefficient: 0.12
Constructed example: the chapter's hypothetical recreation club (cost 72,000, benefit 2,100, crowding 0.12); costs 48,000 and 96,000 and crowding coefficients 0.08 and 0.16 are added for comparison, and one added state shows negative net benefit.

n^3 = C / 2a: 300,000
Continuous optimum n*: 66.94
Best whole club: 67 members
Member net benefit u: 486.69
Worth joining: yes

Congestion costs 0.12n^2 per member and cost sharing gives 72,000/n. Setting the marginal congestion loss 0.24n equal to the cost-sharing gain 72,000/n^2 gives n^3 = 300,000 and n* = 66.94. Comparing 66 and 67 members, the best whole club has 67, where u = 2100 - 538.68 - 1074.63 = 486.69. Each member nets a positive benefit, so the club is worth forming.

Worked steps

2(0.12)n = 72,000/n^2 gives n^3 = 72,000 / 0.24 = 300,000
n* = 300,000^(1/3) = 66.94
u(66) = 2,100 - 522.72 - 1,090.91 = 486.37
u(67) = 2,100 - 538.68 - 1,074.63 = 486.69
Best whole club: 67 members

Use the idea

For a gym, co-working space or shared facility, estimate how crowding grows with membership and compare it with the cost each new member takes off the others.

Where the conclusion applies

Identical members, a fixed facility and quadratic crowding. A larger facility would change both the cost and the crowding term.

Check your understanding: With cost 96,000 and coefficient 0.16, is the best club still worth joining?

n^3 = 96,000/0.32 = 300,000, so n* = 66.94 and the best club has 67 members, but u(67) = 2,100 - 718.24 - 1,432.84 = -51.08, below zero: not worth forming.

Chapter 68 source: section "Optimal club size".

Demonstration 4 of 4

Grants for spillovers, net of diffusion costs

When should a public grant fund a discovery whose benefits the developer cannot collect?

A grant equal to the financing gap makes the project privately viable, but it is justified only if the social value, net of what it costs to use the spillover, exceeds the cost. The grant moves who pays; it does not create the spillover.

Equation, written in LaTeX: P=6(45{,}000)=270{,}000.

Equation, written in LaTeX: V=270{,}000+230{,}000=500{,}000.

Equation, written in LaTeX: C-P=420{,}000-270{,}000=150{,}000.

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C is the development cost, P the payoff the developer can collect (six firms at 45,000) and the spillover the benefit to later users that cannot be charged. Diffusion cost is the replication, training and validation expense of using the spillover. The grant C - P closes the private gap.

Predict first. Once 110,000 of diffusion cost is counted, should the 150,000 grant still be made?

Your prediction Yes No

Uncollectible spillover

Diffusion cost

Private net: -$150,000
Social value: $500,000
Social net: $80,000
Grant to close the gap: $150,000
Verdict: grant justified

The developer can collect 270,000 against a cost of 420,000, so it does not invest without help; a grant of 150,000 closes the gap. Society also gets 230,000 of spillover, for a social value of 500,000. In thousands, social net = 270 + 230 - 420 = 80. Social net value is $80,000, so the grant funds a project worth doing.

Worked steps

Private net = 270,000 - 420,000 = -150,000
Spillover net of diffusion = 230,000 - 0 = 230,000
Social value = 270,000 + 230,000 = 500,000
Social net = 500,000 - 420,000 = 80,000
Grant = 420,000 - 270,000 = 150,000

Use the idea

Before funding a project for its spillovers, subtract the costs users bear to benefit and check that the social net value is positive.

Where the conclusion applies

Expected values are known, the grant is exactly the gap, and the open-publication condition preserves the spillover.

Check your understanding: With diffusion cost 110,000, what is social net value?

270,000 + (230,000 - 110,000) - 420,000 = -30,000, so the grant would fund a project with negative expected social value.

Chapter 68 source: section "Public Goods, Free-Riding, and Knowledge Spillovers".

Sum the marginal values, then compare to cost

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Quadratic funding rewards breadth

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Crowding versus cost sharing sets club size

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Grants for spillovers, net of diffusion costs

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies