Homework 2

Externalities; Common Pool Resources; Public Goods

Homework Instructions

• No Generative AI: You are not allowed to use generative AI tools for Homework Assignment 1, except for part d2 in Question 1

• Deadlines: Wednesday, October 8, 2025 at 10:30 A.M.

  • Hand in your written answers on paper to Prof. Choe at the beginning of class.

Question 1. Optimal Logging, Externalities, and REDD+ (Points: 50)

A forest jurisdiction evaluates logging activity \(L\) under two private market components and one external cost:

• Timber benefit (private): \(TB(L)=2L - L^{2}\)
  
• Harvesting cost (private): \(TC(L)=L\)
  
• Carbon damage (external): \(TD(L)=\delta\,L\), where \(\delta = 0.5>0\) is the monetized climate damage per unit of logging effort \(L\).

Here, \(L\) is logging effort (e.g., worker-days). Private loggers ignore \(TD\).

Notes:
- \(MB(L)=\frac{dTB}{dL}=2-2L\)
- \(MC(L)=\frac{dTC}{dL}=1\)
- \(AB(L)=TB/L=2-L\) (for \(L>0\))
- \(AC(L)=TC/L=1\) (for \(L>0\))
- \(MD(L)=\frac{dTD}{dL}=\delta=0.5\)

a — Total Curves (Top Panel)

Plot TB, TC, TD, and Total Social Cost \(TSC(L)=TC(L)+TD(L)\) versus \(L\).

b — Marginal Curves (Bottom Panel)

Plot marginals:
- \(MB(L)=2-2L\)
- \(MC(L)=1\)
- \(MD(L)=\delta=0.5\) (External Marginal Cost)

Also plot averages:
- \(AB(L)=2-L\)
- \(AC(L)=1\)

c — Benchmarks (align vertical guide lines across panels)

For each benchmark below, (i) denote private optimum \(L^{priv}\), socially efficient optimum \(L^{*}\), and open-access level \(L_{OA}\), (ii) write the condition to solve each, and (iii) compute the numeric value for each using \(\delta=0.5\).
Then draw vertical guide lines at the three effort levels on both panels so they align.

1. Private optimum \(L^{priv}\) — Privately optimal with no externality considered.
   • Definition: A single logging firm (a private decision-maker with exclusive rights to the forest) maximizing their own net private benefit, ignoring the external damage.
2. Socially efficient optimum \(L^{*}\) — Maximizes total net social benefit (accounts for the external damage).
   • Definition: Chosen to internalize the carbon damage at the margin.
3. Open-access level \(L_{OA}\) — Free entry expands effort until profit is zero, ignoring externalities.
   • Definition: With many competing loggers, free entry expands effort until each logger earns zero economic profit and all resource rents are dissipated.

d — What is REDD+ and how can it help here?

REDD+ (Reducing Emissions from Deforestation and forest Degradation) is an international climate policy framework that provides payments when countries or jurisdictions reduce logging and therefore avoid CO\(_2\) emissions.

For simplicity, assume the climate damage from one unit of logging is valued at \(\delta = 0.5\) dollars. Under REDD+, loggers (or local governments) receive a payment \(\tau\) for each unit of logging not carried out, relative to a baseline.

From the logger’s perspective, this acts like an extra cost of logging: cutting down one more unit of forest means losing \(\tau\) dollars of REDD+ payment. In economic terms, the decision rule becomes

\[ MB(L) = MC(L) + \tau. \]

If \(\tau\) is set equal to the external damage per unit of logging (\(\delta\)), then the logging decision takes into account the climate cost, and effort is reduced to the efficient level \(L^*\).

d1 — Pigouvian Alignment

Assume \(\delta=0.5\). Show that if a single logging firm faces a REDD+ payment set at \(\tau=\delta\), it will choose the socially efficient effort \(L^*\).

d2 — Practical Challenges of REDD+

Briefly discuss the following implementation issues:

• Additionality: How can we verify that emission reductions are truly extra (would not have happened without REDD+)?
• Leakage: Could logging or deforestation simply shift to another location, offsetting the gains?
• Permanence: Even if forests are protected now, what ensures the carbon remains stored long term (e.g., risk of future logging or fires)?

Question 2. Expansion of Data Centers and the Environment (Points: 50)

Business Insider - Exposing The Dark Side of America’s AI Data Center Explosion, September 18, 2025

Watch the video above. Then solve the question:

Two representative households, A and B, both benefit from grid reliability \(G\).
Grid reliability refers to the stability and resilience of the electricity system—fewer blackouts, quicker recovery from outages, and greater assurance that power is available when needed.
This reliability is a public good: once provided, it benefits everyone simultaneously, and no household can be excluded from enjoying it.

Each household’s marginal benefit (MB) function is:

• \(MB_A(G) = 40 - 2G\)
  
• \(MB_B(G) = 30 - G\)

The marginal cost (MC) of providing additional reliability, such as building more transmission lines, adding backup generators, or investing in grid hardening, is given by:

• \(MC(G) = 10 + G\)

a — Efficient Reliability Level

Find the efficient level \(G^{*}\).

b — Private vs. Social Choice

Compare this to what might happen if households acted alone and only equated their own \(MB\) to \(MC\).

c — Underprovision of Public Goods

Explain why underprovision arises.

d — Future Reliability with Data-Center Growth

Suppose data center growth increases society’s valuation of grid reliability and alters the cost of providing it. In the future:

• \(MB_{A}^{Future}(G) = 60 - 2G\)
  
• \(MB_{B}^{Future}(G) = 50 - G\)
  
• \(MC^{Future}(G) = 10 + 2G\)

d1 — Efficient Future Level

Compute the efficient future level \(G^{*Future}\).

d2 — Interpretation

Briefly interpret what the change in the optimal reliability level implies for society, considering:

• Investment needs: what new infrastructure or grid upgrades might be required?
• Costs: how do higher marginal costs affect who pays (households vs. data centers)?
• Benefits: how does increased reliability reduce risks (e.g., outages, health costs, lost data) for both residents and businesses?

f — Cost Sharing Debate

Who should bear the costs of grid expansion — households, data centers, or government? Justify.

g — Local Government Incentives

Why might local governments still provide subsidies to large technology companies, or approve land-use changes that allow data centers to be built, even when these projects create negative externalities?

Question 3. Class Participation (Not Graded for Homework 2)

Please write one sentence for each time you participated in class (any public speaking during class time) between September 24, 2025 and October 6, 2025.

Examples:
- “I asked a question to clarify the common-pool resource”
- “I asked how market instruments affect consumer behavior.”
- “I gave plastic pollution as an example of a negative externality.”
- “I pointed out a calculation error during the lecture.”