Homework 4

Climate Policy and Carbon Market Analysis

Author

Byeong-Hak Choe

Published

November 13, 2025

Modified

November 19, 2025

Homework Instructions

No Generative AI: You are not allowed to use generative AI tools for Homework Assignment 4, except for part d2 in Question 1
Deadlines: Wednesday, November 19, 2025 at 10:30 A.M.
- Hand in your written answers on paper to Prof. Choe at the beginning of class.

Question 1. Evaluating Climate Adaptation Benefits: Heatwave Mortality

A proposed climate adaptation policy—such as expanding cooling centers and improving heat-alert systems—is expected to reduce the annual mortality risk from extreme heat exposure from 6 in 100,000 to 2 in 100,000 for a population of 4 million residents.

Part A

How many statistical lives per year would this policy save?

Show answer

Change in individual annual risk:

$\Delta \text{risk} = \dfrac{6}{100000} - \dfrac{2}{100000} = \dfrac{4}{100000} = 4 \times 10^{-5}$

Population size: $4{,}000{,}000$

Expected number of statistical lives saved per year:

$\text{Lives saved} = 4{,}000{,}000 \times 4 \times 10^{-5} = 160$

Answer: The policy saves 160 statistical lives per year.

Part B

Using a Value of a Statistical Life (VSL) of $7 million, what is the maximum annual cost society should be willing to pay for this policy based on mortality-risk reduction alone?

Show answer

VSL: $\$7{,}000{,}000$ per statistical life

Annual benefit from reduced mortality:

$\text{Annual benefit} = 160 \times 7{,}000{,}000 = 1{,}120{,}000{,}000$

So the benefit is $\$1.12$ billion per year.

Answer: The maximum annual cost that can be justified based on mortality-risk reduction alone is $\$1.12$ billion per year.

Question 2. Linking International Carbon Markets

Two countries operate their own domestic cap-and-trade systems to limit carbon dioxide emissions.

Country A
- Uncontrolled CO₂ emissions: 60 tons
- Emissions cap: 20 tons
- Marginal abatement cost: $MC_A = 10$
Country B
- Uncontrolled CO₂ emissions: 80 tons
- Emissions cap: 40 tons
- Marginal abatement cost: $MC_B(q) = q$, where where $q$ is the tons of emission abatement.

Note

Uncontrolled CO₂ emissions refer to the amount of CO₂ a country or firm would emit in the absence of any climate policy, such as taxes, cap-and-trade, or regulations. It represents the baseline level of emissions if no abatement efforts were taken.

The two countries are considering whether to link their carbon markets to improve cost-effectiveness and enhance cooperation on climate mitigation.

Part A

Before linking the two carbon markets, what would be the permit prices in each country, and how much CO₂ abatement would each country undertake domestically?

Show answer

Country A

Required abatement:

$q_A = \text{uncontrolled emissions} - \text{cap} = 60 - 20 = 40 \text{ tons}$

Marginal abatement cost is constant: $MC_A = 10$.

In a competitive permit market, the permit price equals marginal abatement cost at the chosen abatement level:

$p_A = 10$

So:

Country A abatement: $q_A = 40$ tons
Country A permit price: $p_A = \$10$ per ton

Country B

Required abatement:

$q_B = 80 - 40 = 40 \text{ tons}$

Marginal abatement cost: $MC_B(q_B) = q_B$.

At the required abatement:

$p_B = MC_B(40) = 40$

So:

Country B abatement: $q_B = 40$ tons
Country B permit price: $p_B = \$40$ per ton

Answer:

Country A: $q_A = 40$ tons, $p_A = \$10$ per ton
Country B: $q_B = 40$ tons, $p_B = \$40$ per ton

Part B

If the two countries link their cap-and-trade systems, carbon allowances can trade freely across borders. In the linked system:

Total Uncontrolled CO₂ Emissions:
$140$ tons = $60 + 80$
Total Emissions Cap:
$60$ tons = $20 + 40$
Initial Allowance Allocation:
- Country A: 20 allowances
- Country B: 40 allowances

With linkage, both countries face a single, common permit price. Each country buys or sells allowances to the other depending on whether they emit more or less than their initial allocation.

What is the common permit price in the integrated carbon market?
How much CO₂ does each country abate under the linked system?
Does allowance trading occur between countries?
- If so, describe which country buys or sells allowances and the number of allowances traded.

Tip

Let $p$ be the common permit price. If $p$ > 10, it is profitable for Country A to reduce emissions as much as possible (up to 60 tons) and sell extra allowances to Country B.

Show answer

Step 1: Total required abatement

Total uncontrolled emissions:

$E_{\text{uncontrolled}} = 60 + 80 = 140 \text{ tons}$

Total cap (total allowances):

$E_{\text{cap}} = 20 + 40 = 60 \text{ tons}$

Total abatement required:

$q_{\text{total}} = 140 - 60 = 80 \text{ tons}$

Step 2: Abatement behavior in the linked market

Let $p$ be the common permit price.

Country A:
$MC_A = 10$ (constant).
If $p > 10$, it is profitable for A to abate as much as possible (up to 60 tons) and sell extra allowances.
Country B:
$MC_B(q_B) = q_B$.
B chooses abatement $q_B$ such that $MC_B = p$, so: $q_B = p$.

Step 3: Allowance trading and market equilibrium (supply = demand)

Initial allowances:

Country A: 20 allowances
Country B: 40 allowances

Assume $p > 10$ so that A fully abates:

Country A:

Abatement: $q_A = 60$
Emissions: $E_A = 60 - 60 = 0$
- Allowances used: 0
- Allowances available to sell: $20$ (Supply of allowances)

Country B:

Abatement: $q_B = p$
Emissions: $E_B = 80 - q_B = 80 - p$
- Allowances needed: $80 - p$

Country B starts with 40 allowances, so its net demand for allowances is:

\[ \begin{aligned} \text{Net demand}_B &= (80 - p) - 40 \\ &= 40 - p \end{aligned} \]

At the market equilibrium, quantity supplied is equal to quantity demanded: \[ \begin{aligned} \text{Allowances supplied by A} &= \text{Net demand by B} \\ 20 &= 40 - p \end{aligned} \] Solve for $p$:

\[ p = 20 \]

Then:

\[ \begin{aligned} q_B &= p \\ &= 20 \text{ (tons of abatement)} \end{aligned} \]

Check totals:

Total abatement: $q_A + q_B = 60 + 20 = 80$ (matches required total abatement)
Total emissions: $E_A + E_B = 0 + (80 - 20) = 60$ (matches total cap)

Summary

Common permit price:
$p = \$20$ per ton
Abatement:
- Country A: $q_A = 60$ tons (emits 0).
- Country B: $q_B = 20$ tons (emits 60 tons).
Allowance trading:
- Country A uses 0 allowances and sells 20 allowances
- Country B buys 20 allowances to reach its needed 60 allowances

Question 3. Class Participation (Not Graded for Homework 4)

Please write one sentence for each time you participated in class (any public speaking during class time) between October 13, 2025 and November 10, 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.”