Public Goods & Commons – Exam Study Notes

Perfect Competition & Preconditions

• Recall from Microeconomics I:

  • Under the assumptions of perfect competition, market equilibria are Pareto-efficient.

  • Key prerequisite: all goods are private goods (both rivalrous and excludable).

• If the good is not rivalrous and/or excludable, the standard invisible-hand result can fail, leading to inefficiencies.

Public Goods: Core Definitions & Intuition

• Non-rivalry (non-diminishability)

  • Consumption by one individual does not reduce utility available to others.

  • Example: national defence – my protection does not diminish yours.

• Non-excludability

  • Either impossible or prohibitively costly to exclude someone from consumption.

  • Example: once the country is protected, you cannot cheaply keep one citizen unprotected.

• A good satisfying both traits fully is a pure public good.

  • Important distinction from a publicly provided good (simply financed/produced by government; may still be private in nature).

• Provision can come from private sector or government; problem lies in incentives, not producer identity.

Taxonomy of Goods (Rivalry × Excludability Matrix)

• Classification practice questions:

  • Clock tower → non-rival, non-excludable ⇒ pure public good.

  • Streets → depends on congestion; empty street acts like a club good, congested street becomes rivalrous.

  • Fundamental non-patentable research → pure public good.

  • Concert → excludable (ticketing) & rivalrous (seat scarcity) ⇒ private/club mix.

Inefficiency of Private Provision: Two-Person Diagram

• Individuals A and B each have downward-sloping inverse demand (WTP) curves for a public good.

• For public goods, social demand is the vertical sum (add WTPs at each quantity).

  • Efficient output where $\sum WTP = MC$ (Samuelson condition).

• Market outcome with private sellers:

  • Each behaves as if good is private ⇒ horizontal sum of demands; generally yields too low quantity.

  • If WTP_A < MC for all quantities, A buys none and free-rides on B.

• Result: Q{private} < Q^{*}{Pareto}.

Samuelson Condition (Formal)

• Efficiency requires
  $\sum<em>{i=1}^{n} MB</em>i(Q) = MC(Q)$
  where $MB<em>i$ equals marginal willingness to pay or marginal rate of substitution $MRS</em>{i}^{(G,X)}$ between public good G and private composite good X.

• Equivalently, $\sum<em>{i} MRS</em>{i}^{(G,X)} = MRT_{(G,X)}$.

• If condition not met, reallocating spending between private & public goods raises aggregate utility.

Free-Riding Dynamics

• When benefits are non-excludable, individuals misreport or withhold contributions.

• Scenarios:

  1. Both A and B have WTP < MC ⇒ zero voluntary supply despite positive social value.

  2. Both have WTP > MC but act independently ⇒ each hopes other will pay, total still < efficient.

• Link to externalities: under-provision of public good is symmetric to positive externality (each unit confers uncompensated benefits on others).

Tragedy of the Commons (Common-Pool Resources)

• Thought experiment: 6 villagers, each with 100 €; choice:

  1. Buy government bond (12 % return).

  2. Buy a steer grazing on shared commons.

• Grazing density lowers individual steer weight → lower sale price (Table 16.7):
  1 steer → 120 €; … ; 6 steers → 105 €.

• Individual decisions ignore negative externality on existing cattle; outcome: over-grazing (over-use).

• Remedies:

  • Assign property rights (auction pastureland; Coasean bargaining).

  • Practical difficulties at large scale: whales in international waters, global warming, orbital satellites.

Government Intervention: Challenges & Tools

• Non-rivalry/excludability imply private provision is costly & information-constrained.

• Government needs individuals’ true WTP to set efficient quantity, but faces incentive-compatibility problem (people understate).

• Broad categories of intervention:

  • Direct public provision & taxation.

  • Lindahl pricing (personalized prices; requires preference revelation).

  • Voting on discrete provision levels.

  • Vickrey-Clarke-Groves (VCG)/Clarke tax mechanism.

  • Provision-point (Bagnoli–Lipman) mechanisms.

VCG / Clarke Tax Mechanism (Summary)

• Steps:

  1. Individuals state maximum WTP for each alternative.

  2. Alternative with highest aggregate stated value is chosen.

  3. Each individual pays a tax only if her report is pivotal (decisive); tax equals net external effect on others.

• Key properties:

  • Payment never exceeds own stated value.

  • No incentive to misrepresent preferences (dominant-strategy truth-telling).

• Numerical Illustration (3 voters A, B, C; alternatives X vs Y):

  • Truthful values (in €): $A: (1,3),\ B:(3,2),\ C:(3.5,2)$ ⇒ $\sum<em>{X}=7.5, \sum</em>{Y}=7$ ⇒ X wins.

  • A is not pivotal ⇒ tax 0.

  • Remove B ⇒ sums 4.5 vs 5 → Y would win; hence B pivotal, pays tax $5-4.5 = 0.5$.

  • Remove C ⇒ sums 4 vs 5 → Y wins; C pivotal, pays tax $5-4 = 1$.

  • Misreporting (A pretending Y = 4) flips choice; now A pivotal & pays tax 2.5 > personal gain ⇒ lying unprofitable.

Provision-Point (Crowdfunding-Style) Mechanism

• Specify cost/target for public good.

• Collect voluntary pledges $C_i$.

• If $\sum C_i \ge \text{target}$ → good provided; if not → money-back guarantee.

• Eliminates risk of being sole contributor; minimal strategic complexity.

• Real-world use: Kickstarter campaigns, minimum-signatory climate treaties, charity drives.

Behavioural Economics Evidence: Public Good Games

• Standard lab design (3 players):

  • Each endowed with 5 €; chooses contribution $C_i \in [0,5]$ to a common pool.

  • Pool is doubled and evenly split.

  • Monetary payoff: $Y<em>i = 5 - \frac{1}{3}C</em>i + \frac{2}{3}(C<em>j+C</em>k)$.

  • Self-interest prediction: $C<em>i = 0$; Pareto-efficient: $C</em>i=5$ for all.

• Classroom data (single shot): average contribution 3.57 € (≈ 71 % endowment); 43 % fully cooperative.

• Robust experimental regularities:

  • One-shot mean ≈ 50 % of endowment.

  • Contributions decay across rounds in repeated games (learning to free-ride).

  • Stable groups (fixed partners) sustain higher cooperation.

  • Introducing costly punishment opportunities markedly raises contributions.

  • Large share of participants are conditional cooperators – contribute proportionally to others’ average.

  • Conditional cooperation documented as early as age 6.

Implications & Open Problems

• Although theory predicts zero voluntary provision, human behaviour features altruism, norms, reciprocity & punishment.

• Free-riding remains substantial; private mechanisms rarely reach full efficiency.

• Central policy challenge: preference aggregation – obtaining reliable WTP data to meet Samuelson condition.

• Clarke tax & related mechanisms are elegant but prone to:

  • High informational demands, complexity, administrative cost, potential collusion.

• No universal solution; mix of regulation, property rights, taxation, and institution design needed.

Numerical & Formal References

• Steer price schedule: $P(N) = 120 - 2(N-1)$ for $N=1,\ldots,6$ (implied by table).

• Government bond return: $r = 0.12$ per annum.