Classwork 5

Public Goods in Action

Author

Byeong-Hak Choe

Published

October 3, 2025

Modified

October 8, 2025


Part 1 — Conceptual Warm-Up

  1. Explain in your own words why public goods require vertical summation of marginal benefits.

Because everyone consumes the same quantity \(Q\) of a public good. The social marginal benefit (SMB) at that \(Q\) is the sum of each individual’s marginal benefit at that same quantity, so we add vertically the MB curves:
\[ \text{SMB}(Q) = \sum_i \text{MB}_i(Q). \]

  1. In the Doug–Sasha forest preservation example, what condition makes 10 acres optimal?

At \(Q=10\), Doug’s MB \(=\$5\) and Sasha’s MB \(=\$2\), so
\[ \text{SMB}(10) = 5 + 2 = \$7 = \text{MC}. \]

SMB = MC holds, making \(Q=10\) optimal.

  1. Give one local and one global example of an environmental public good not listed in the lecture.
  • Local: Urban street lighting; city flood siren system.
  • Global: Global disease surveillance networks; open satellite climate data.

Part 2 — Applied Problem

Suppose two people, Alex and Jamie, value urban tree planting as follows:

\[ MB_{Alex}(Q) = 16 - Q \quad \quad MB_{Jamie}(Q) = 12 - 2Q \]

(Values are in dollars; truncate at zero.)
The marginal cost of planting trees is constant at $10 per tree.

Tasks:
1. Draw the MB curves for Alex and Jamie separately.
2. Construct the social MB curve by vertical summation.
3. Find the optimal number of trees Q* where SMB = MC.
4. Compare the total benefits received by Alex vs. Jamie at Q*.

  • Social MB by vertical summation (piecewise)

For \(0 \le Q \le 6\): both positive, \[ \text{SMB}(Q) = (16 - Q) + (12 - 2Q) = 28 - 3Q. \] For \(6 < Q \le 16\): Jamie’s MB is 0, so \[ \text{SMB}(Q) = 16 - Q. \] For \(Q > 16\): \(\text{SMB}(Q)=0\).

  • Optimal quantity \(Q^*\) where SMB = MC

Solve in the relevant segment: \[ 28 - 3Q = 10 \;\Rightarrow\; 3Q = 18 \;\Rightarrow\; Q^* = 6. \]

  • Total benefits at \(Q^*\) and comparison

Alex: \[ TB_{Alex}(Q^{*}) = \frac{1}{2}\times(16+10)\times 6 = 78 \]

Jamie: \[ TB_{Jamie}(Q^{*}) = \frac{1}{2}\times12 \times 6 = 36 \]

  • Total benefit at \(Q^*\):

\[ TB_{Alex}(Q^{*}) + TB_{Jamie}(Q^{*}) = 114 \]

  • Total cost at \(Q^*\):

\[ TC(Q^{*}) = 10 \times 6 = 60 \]

  • Net surplus: \(114 - 60 = \boxed{54}\).
Q* = 6.00
Alex benefit = 78.00
Jamie benefit = 36.00
Total benefit = 114.00
Total cost = 60.00
Net surplus = 54.00

  1. Propose two possible cost-sharing rules (e.g., equal split, benefit-based) and discuss whether each is fair and/or efficient.
  • Equal split: Each pays $30. Simple and transparent; ignores heterogeneous benefits.
  • Benefit-based: Share proportional to benefits:
    • Alex: \(78/114 \approx 0.684 \Rightarrow \$41.05\).
    • Jamie: \(36/114 \approx 0.316 \Rightarrow \$18.95\).
      Efficient in the sense that each pays ≤ benefit received; equity depends on norms and ability to pay.

Part 3 — Sensitivity

  1. Suppose the cost per tree rises from $10 to $12. Recalculate Q*.
  2. Suppose Jamie’s valuation increases to \(MB_{Jamie}(Q) = 15 - 2Q\). How does this change Q*?
  3. Which change (higher cost vs. higher demand) has a larger effect on optimal provision? Why?

  1. Higher cost: \(MC=12\)
    On \(0\le Q\le 6\): \(28 - 3Q = 12 \Rightarrow 3Q = 16 \Rightarrow Q^* = \boxed{16/3 \approx 5.33}\).
    (Jamie’s MB at \(5.33\) is \(12 - 2\cdot 5.33 \approx 1.33 > 0\), so the piecewise region is valid.)

  2. Higher demand for Jamie: \(MB_{Jamie}(Q)=15 - 2Q\)
    For \(0 \le Q \le 7.5\):
    \[ \text{SMB}(Q) = (16 - Q) + (15 - 2Q) = 31 - 3Q. \]
    Set equal to MC: \(31 - 3Q = 10 \Rightarrow 3Q = 21 \Rightarrow Q^* = \boxed{7}\).

  3. Which change has a larger effect? Why?

  • Baseline \(Q^*=6\).
  • With \(MC=12\): \(Q^*\approx 5.33\) (down by ~0.67).
  • With higher Jamie MB: \(Q^*=7\) (up by 1.0).
    Demand increase has the larger effect here because it raises SMB across relevant \(Q\), moving the SMB=MC intersection more than the modest MC increase reduces it.



Part 4 — Reflection

  • How does this exercise illustrate the free-rider problem?
  • What role should government play when preferences and benefits differ across individuals?

Understanding the Free-Rider Problem and the Government’s Role

The Free-Rider Problem — Why Public Goods Are Underprovided

1. What is “free riding”?
A free rider is someone who enjoys the benefits of a public good without paying for it.
Because public goods are nonexcludable, once they are provided, everyone benefits — whether they contribute or not.

Example: Once a city builds flood barriers, all residents are protected. Even those who did not contribute still enjoy the same protection.

2. Why is \(WTP < MB\)?
For private goods, people reveal their preferences by purchasing — they pay if \(P \le MB\).
But for public goods, individuals can benefit without paying, so their willingness to pay (WTP) is less than their true marginal benefit (MB):

\[ WTP_i < MB_i \]

This causes the aggregate demand curve to lie below the true social marginal benefit (SMB) curve, leading to underprovision when relying solely on voluntary contributions.

3. Why the market fails
Firms cannot collect enough payment to cover costs because of free riding.
As a result, markets stop short of the efficient provision level \(Q^*\), where:

\[ SMB = SMC \]

Hence, too little of the public good is produced relative to the socially optimal level.

4. Policy tools to narrow the gap

  • Matching grants / subsidies: Encourage private contributions by adding public funds (e.g., government donation matching).
  • Direct public provision: Government directly supplies the good (e.g., national defense, public health).
  • Mandatory taxes or fees: Ensure everyone contributes to a good they all benefit from (e.g., roads, national parks).

These tools realign private incentives with social benefits, helping the economy move toward the efficient \(Q^*\).

Government’s Role — Balancing Efficiency and Equity

1. Efficiency
The optimal level of a public good occurs when the sum of all individuals’ marginal benefits equals the marginal cost of provision:

\[ \sum_i MB_i(Q^*) = MC(Q^*) \]

At this point, society’s total willingness to pay for the last unit just equals its cost — the condition for Pareto efficiency in public goods provision.

2. Financing mechanisms
Once \(Q^*\) is determined, the challenge becomes how to finance it:

  • Taxation: Everyone contributes through general taxes.
  • Subsidies / matching: Encourage private provision or local co-financing.
  • Earmarked levies: Tie specific taxes to related benefits (e.g., gas tax for road repair).
  • Public–private partnerships: Share costs between government and private sector.

3. The equity challenge
Efficiency tells us how much to produce; equity determines who should pay.

Two fairness principles guide policy: - Benefit-based fairness: Those who benefit more should contribute more.
- Ability-to-pay fairness: Those with higher incomes contribute more, even if benefits are similar.

The government must balance these principles, designing taxes or cost-sharing rules that achieve efficiency (\(SMB = SMC\)) while maintaining equity.

Summary

Problem Mechanism Policy Response
Free riding → underprovision \(WTP < MB\) Government funding or intervention
Market failure No price mechanism for shared benefits Taxes, subsidies, or direct provision
Unequal benefits/incomes Efficiency–equity tradeoff Mixed cost-sharing rules

Bottom line:
- Without government: free riding causes \(WTP < MB\) and underprovision.
- With government: collective financing (taxes, subsidies, matching) enables provision at the efficient level \(Q^*\).
- The policy goal is to achieve both efficiency (produce where \(SMB = SMC\)) and equity (fair cost-sharing across individuals).

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