Classwork 11

Market for Pollution

Question 1

Firms A and B each produce 80 units of pollution. The federal government wants to reduce pollution levels. The marginal costs associated with pollution reduction are: \[ \begin{aligned} MC_A = 50 + 3Q_A \qquad\text{for firm A} \\ MC_B = 20 + 6Q_B \qquad\text{for firm B} \end{aligned} \]

where \(Q_A\) and \(Q_B\) are the quantities of pollution reduced by each firm. Society’s marginal benefit from pollution reduction is given by \[ MB = 590 – 3Q_{T} \]

where \(Q_{T} = Q_A + Q_B\) is the total reduction in pollution.

Part A

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

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At the social optimum, we need cost-effectiveness and efficiency:

1. Cost-effectiveness: \(MC_A = MC_B\)
   
2. Efficiency: \(MC_A = MC_B = MB\)

Solve:

• Set \(MC_A = MC_B\):
  \[ \begin{aligned} 50 + 3Q_A &= 20 + 6Q_B\\ \Rightarrow\quad\quad\;\; 3Q_A &= -30 + 6Q_B\\ \Rightarrow\quad\quad\quad Q_A &= -10 + 2Q_B \end{aligned} \]

• Set \(MC_A = MB\):
  \[ \begin{aligned} 50 + 3Q_A &= 590 - 3(Q_A + Q_B)\\ \Rightarrow\quad 6Q_A + 3Q_B &= 540 \end{aligned} \]

Substitute \(Q_A = -10 + 2Q_B\) into \(6Q_A + 3Q_B = 540\): \[ \begin{aligned} 6(-10 + 2Q_B) + 3Q_B &= 540\\ \Rightarrow\quad -60 + 12Q_B + 3Q_B &= 540\\ \Rightarrow\quad\quad\quad\quad\quad\quad\quad 15Q_B &= 600\\ \Rightarrow\quad\quad\quad\quad\quad\quad\quad\quad Q_B^* &= 40\\ \end{aligned} \] \[ Q_A^* = -10 + 2(40) = 70 \]

Socially optimal reductions:

• Firm A: \(\boxed{Q_A^* = 70}\)
  
• Firm B: \(\boxed{Q_B^* = 40}\)

Total reduction: \(Q_T^* = 70 + 40 = 110\).

Part B

How much total pollution is there in the social optimum?

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Each firm initially emits 80 units, so initial total pollution is: \[ 80 + 80 = 160 \]

At the optimum, total reduction is \(Q_T^* = 110\) units, so remaining pollution is: \[ 160 - 110 = 50 \]

We can also see the distribution:

• Firm A reduces 70 → emits \(80 - 70 = 10\)
  
• Firm B reduces 40 → emits \(80 - 40 = 40\)

So total pollution in the social optimum is: \[ \boxed{50 \text{ units of pollution}} \]

Part C

Explain why it is inefficient to give each firm an equal number of pollution permits (if they are not allowed to trade them).

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If each firm gets the same number of permits and cannot trade, they must reduce the same amount of pollution (say 55 units each if total reduction is 110).

Check marginal costs at \(Q_A = Q_B = 55\):

• \(MC_A(55) = 50 + 3(55) = 215\)
  
• \(MC_B(55) = 20 + 6(55) = 350\)
  
• \(MB(110) = 590 - 3(110) = 260\)

We have: \[ MC_A = 215 < MB = 260 < MC_B = 350 \]

This means:

• Firm B is forced to undertake too much expensive abatement (its \(MC_B > MB\)).
• Firm A does too little abatement (its \(MC_A < MB\)).

We could lower total cost by shifting some abatement from firm B to firm A (e.g., one unit):

• Increasing A’s abatement by 1 costs about 215
  
• Decreasing B’s abatement by 1 saves about 350

Net gain ≈ 350 − 215 = 135.
So equal, non-tradable permits violate the equimarginal principle (MCs are not equal), making the allocation cost-inefficient.

Part D

Explain how the social optimum can be achieved if firms are given equal numbers of pollution permits but are allowed to trade them.

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If total permits are set so that total pollution = 50 (i.e., total reduction = 110), and firms can trade permits:

• A market price for permits will emerge.
  
• Each firm chooses its abatement until: \[ MC_i = \text{permit price} \]
• In equilibrium, both firms have the same marginal cost, which equals the marginal benefit at the optimum: \[ MC_A = MC_B = MB = 260 \]

Solving:

• \(50 + 3Q_A = 260 \Rightarrow Q_A^* = 70\)
  
• \(20 + 6Q_B = 260 \Rightarrow Q_B^* = 40\)

Firm A, with the lower marginal cost, finds it cheaper to abate more and sell permits.
Firm B, with the higher marginal cost, finds it cheaper to buy permits and abate less.

Thus, trading reallocates abatement between firms so that:

• Total emissions still equal the cap (50 units), and
  
• \(MC_A = MC_B\), achieving the socially optimal allocation at minimum total cost.

Part E

Can the social optimum be achieved using a tax on pollution?

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Yes. A Pigouvian tax on pollution can achieve the social optimum.

Set a per-unit tax \(\tau\) equal to the socially optimal marginal damage, which at the optimum equals the common marginal cost and marginal benefit: \[ \tau = MB(Q_T^*) = MC_A(Q_A^*) = MC_B(Q_B^*) = 260 \]

Each firm then chooses its own optimal abatement by setting: \[ MC_i(Q_i) = \tau \]

• Firm A:
  \[ 50 + 3Q_A = 260 \Rightarrow Q_A^* = 70 \]
• Firm B:
  \[ 20 + 6Q_B = 260 \Rightarrow Q_B^* = 40 \]

Total reduction is \(70 + 40 = 110\), and remaining pollution is 50 units — exactly the social optimum.

So, a uniform pollution tax of $260 per unit induces firms to self-select the cost-effective allocation and achieves the same outcome as the optimal cap-and-trade system.