Homework 4 - Example Answer

Question 1.

Q1a

Suppose that hedonic wage studies indicate a willingness to pay $50 per person for a reduction in the risk of a premature death from an environmental hazard of \(\frac{1}{100,000}\). If the exposed population is 4 million people, what is the implied value of a statistical life?

Answer:

The total willingness to pay for this risk reduction is $200 million: \[ \$200 \text{million} \,=\, \$50\; \text{per person} × 4\; \text{million exposed people}. \]

The expected number of lives saved would be 40: \[ 40 \,=\,\frac{1}{100,000}\; \text{risk of premature death} \times 4,000,000\; \text{exposed population}. \]

The implied value of a statistical life is $5,000,000: \[ \$5,000,000\;\text{per life saved} \,=\, \frac{\$200,000,000\; \text{total willingness to pay}}{40\; \text{lives saved}}. \]

Q1b

Suppose that an impending environmental regulation to control that hazard is expected to reduce the risk of premature death from \(\frac{6}{100,000}\) to \(\frac{2}{100,000}\) per year in that exposed population of 4 million people. Your boss asks you to tell them what is the maximum this regulation could cost and still have the benefits be at least as large as the costs. What is your answer?

Answer:

The program is expected to save 160 lives: \[ 160 \,=\, \left(\frac{6}{100,000} – \frac{2}{100,000}\right) \times 4,000,000. \] According to the value of a statistical life in Q1a, the program will have more benefits than costs as long as it costs no more than $800,000,000:

\[ $800,000,000 \,=\, \$5,000,000\; \text{value per life} \times 160\; \text{lives saved} \]

Question 2.

Consider two major economies, Country A and Country B, deciding whether to implement strict mitigation policies. Each country can choose to either “Reduce Emissions” or “Continue Business as Usual (BAU).” The payoff matrix below shows the economic benefits (in billions of dollars) for each country over a 10-year period.

		Country B
		Reduce	BAU
Country A	Reduce	(5,5)	(-2,8)
	BAU	(8,-2)	(1,1)

Q2a.

What is the Nash equilibrium (or equilibria) in this game? Show your work.

Answer:

For Country A:

• If B chooses “Reduce”: 5 vs 8 (BAU gives higher payoff)
• If B chooses “BAU”: -2 vs 1 (BAU gives higher payoff)
• Conclusion: BAU is dominant strategy for A

For Country B:

• If A chooses “Reduce”: 5 vs 8 (BAU gives higher payoff)
• If A chooses “BAU”: -2 vs 1 (BAU gives higher payoff)
• Conclusion: BAU is dominant strategy for B

Nash Equilibrium: (BAU, BAU) with payoff (1,1)

Q2b

Explain why this game represents a Prisoner’s Dilemma in the context of climate change policy.

Answer:

• The game mirrors the classic Prisoner’s Dilemma.
• Rational self-interest leads to mutual defection, resulting in lower payoffs for both.
  • Mutual cooperation (Reduce, Reduce) yields better outcome (5,5)
  • Mutual defection (BAU, BAU) yields worse outcome (1,1)
• Highlights the conflict between individual rationality and collective welfare.
  • Individual incentive to defect leads to suboptimal collective outcome
• Represents real-world challenge in climate agreements

Q2c

If the payoffs represent only economic benefits and ignore environmental costs:

1. What would be the socially optimal outcome?
2. Why do countries tend to deviate from this outcome?

Answer:

1. Socially optimal outcome is (Reduce, Reduce) because:
   • Total payoff is highest (5+5 = 10)
   • Environmental benefits (not included in payoffs) would make this even more favorable
2. Countries deviate because:
   • Individual incentive to free-ride
   • Lack of enforcement mechanisms
   • Short-term economic gains prioritized over long-term benefits

Q2d

Consider implementing a policy mechanism to modify the payoff structure of this climate agreement game.

1. Suggest and explain a specific policy mechanism (e.g., carbon tax, trade sanctions, or technology subsidies) that could alter the payoffs.
2. Provide a new payoff matrix that incorporates your suggested policy mechanism. Show all calculations for the modified payoffs.
3. Prove that under your proposed policy:

• The socially optimal outcome (Reduce, Reduce) becomes a Nash equilibrium

4. Discuss one potential challenge in implementing your proposed policy mechanism in the real world.

Answer:

New Payoff Matrix with Policy

		Country B
		Reduce	BAU
Country A	Reduce	(5,5)	(0,4)
	BAU	(4,0)	(-3,-3)

Policy Components:

1. International carbon tax of 4 billion on BAU choice
2. Tax redistribution to countries choosing “Reduce”
3. International monitoring and enforcement
4. Binding legal framework

New Nash Equilibrium Analysis:

For Country A: - If B chooses “Reduce”: 5 vs 4 (Reduce is better) - If B chooses “BAU”: 0 vs -3 (Reduce is better)

For Country B: - If A chooses “Reduce”: 5 vs 4 (Reduce is better) - If A chooses “BAU”: 0 vs -3 (Reduce is better)

Result: (Reduce, Reduce) becomes the only Nash equilibrium

Implementation Requirements

1. Enforcement Mechanism
   • International monitoring body
   • Transparent emissions reporting
   • Third-party verification
2. Collection System
   • Automatic deduction from international trade
   • World Bank or IMF oversight
   • Binding legal framework
3. Distribution Method
   • Clear formula for redistributing penalties
   • Technology transfer funds
   • Green infrastructure support

Question 3

Two firms are ordered by the U.S. government to reduce their greenhouse gas (GHG) emission levels. Firm A’s marginal costs associated with GHG emission reduction is \(MC_{A} = 20 + 4Q\). Firm B’s marginal costs associated with GHG emission reduction is \(MC_{B} = 10 + 8Q\). The marginal benefit of GHG emission reduction is \(MB = 400 - 4Q\).

Q3a

What is the socially optimal level of each firm’s GHG reduction?

Answer:

There are two ways of interpreting this question: the marginal benefit of GHG emission reduction applies either to the overall level of pollution reduction (for the two firms combined) or to each firm separately.

The First Interpretation

In the first interpretation, the social optimum must solve: \[ \begin{align} MC_{A} &\,\overbrace{=}^{(1)}\, MB \overbrace{=}^{(2)} MC_{B}\\ (1):\quad 20 + 4Q_{A} &\,\,\,=\, 400 - 4(Q_{A} + Q_{B})\\ (2):\quad 10 + 8Q_{B} &\,\,\,=\, 400 - 4(Q_{A} + Q_{B}) \end{align} \] Equation (1) simplifies to \[ Q_{A} \,=\, \frac{95}{2} - \frac{1}{2}Q_{B}.\tag{3} \]

Substituting out \(Q_{A}\) in equation (2) using equation (3) gives: \[ \begin{align} 10 + 8Q_{B} \,=\, 400 - 4\left(\frac{95}{2} - \frac{1}{2}Q_{B} + Q_{B} \right)\tag{4} \end{align} \] Solving equation (4) gives: \[ \begin{align} Q_{B}^{o} &\,=\, 20\\ Q_{A}^{o} &\,=\, \frac{75}{2} \end{align} \]

The Second Interpretation

In the second interpretation, we set \(MC = MB\) separately for each firm: \[ \begin{align} MC_{A} &\,=\, MB\\ 20 + 4Q_{A} &\,\,\,=\, 400 - 4Q_{A}\tag{5} \end{align} \]

\[ \begin{align} MC_{B} &\,=\, MB\\ 10 + 8Q_{B} &\,\,\,=\, 400 - 4Q_{B}\tag{6} \end{align} \]

Solving equations (5) and (6) give: \[ \begin{align} Q_{A}^{oo} &\,=\, \frac{95}{2}\\ Q_{B}^{oo} &\,=\, \frac{65}{2} \end{align} \]

Q3b

Compare the social efficiency of three potential outcomes: (1) mandating equal GHG emission reductions across all firms; (2) imposing a uniform tax per unit of GHG emissions; or (3) mandating equal GHG emission reductions across all firms but allowing the trading of GHG emission permits.

Answer:

The First Interpretation

1. The same total reduction could be achieved by requiring each firm to reduce pollution by 28.75 units. This would be less efficient than the social optimum, since it would be less costly for firm A to reduce pollution by more and for firm B to reduce pollution by less (since \(MC_A < MC_B\) at 27.25 units).

2. A common tax could be used to achieve the social optimum. Setting a tax of 170 would lead firm A (respectively, B) to reduce pollution to the point where \(MC_A = 170\) (respectively \(MC_B = 170\)). Solving gives \(Q_A = 37.5\) and \(Q_B = 20\).

3. Requiring both firms to reduce pollution by 27.25 units but allowing them to trade pollution permits can also be used to achieve the social optimum. The value to firm B of being able to produce 1 more unit of pollution (i.e., \(MC_B\)) is higher than the cost to firm A of reducing pollution by one unit (i.e., \(MC_B\)) when \(Q_B = 27.25 = Q_A\), so both can gain by trading a unit of pollution permits. This continues to be true as long as \(Q_A < 37.5\) and \(Q_B > 20\)—so they will trade until \(Q_A = 37.5\) and \(Q_B = 20\).

The Second Interpretation

1. The same level of pollution reduction could be achieved by requiring both firms to reduce pollution by 40 units. Firm A stops reducing pollution before it has exhausted all reduction steps for which the marginal cost is less than the marginal benefit, but firm B takes some pollution reduction steps for which the marginal cost exceeds marginal benefit. This is not socially efficient.

2. A common tax would yield the same result: a tax designed to be optimal for firm A would be too low to induce firm B to reduce to the efficient quantity, and a tax designed to be optimal for firm B would induce firm A to reduce by too much.

3. If the firms started at the pollution reduction levels suggested in part (1), a pollution permit market would allow firm A to reduce its pollution by 7.5 more units and sell the permits to firm B, yielding the same result as in Q3a.