exam set 2022 Flashcards
Problem 1: Tradable permits (weight 33%)
Two polluting firms can control emission (𝐸) of a pollutant by incurring the following marginal
abatement costs:
𝑀𝐴𝐶1 = 300 − 10𝐸1
𝑀𝐴𝐶2 = 90 − 5𝐸2
Assume that the target level of pollution is 30 units. We do not know if this is the socially efficient level or not.
Quesiton 1.a)
Compute the level of emission per firm that is cost-effective for society.
Answer : Cost-efficiency requires 𝑀𝐴𝐶1 = 𝑀𝐴𝐶2. The target level is 30 units of pollution. Define 𝐸2 = 30 − 𝐸1 and solve for E1, hence
This gives 𝐸1 = 24 and 𝐸2 = 30 − 24 = 6. Now insert E1 and E2 in MAC1 and MAC2
This furfill 𝑀𝐴𝐶1 = 𝑀𝐴𝐶2 = 60, both firm now have eqaul marginal abatement costs of one extra unit of emission. When MACs are equal, the cost of reducing an additional unit of pollution is the same across all firms, leading to the most cost-effective distribution of pollution reduction efforts. Notice that they are using different level of emissions to achieve the cost-efficiency for the society. It is cheaper for firm 2 to abate emissions then firm2.
Question 1.b)
Explain how a tradable emission permits scheme could be applied to achieve the target level
of emissions. Assume that the regulator initially assigns 15 permits to each polluter. The
government gives these permits to the firms without charge. Solve for: (i) the number of
permits each firm holds after a permit market operates, (ii) the equilibrium permit price, and
(ii) total private cost of the permit system.
Answer: For a given target level of emissions, a tradable emissions scheme help equalize MAC between the firms. The two firms initially hold 15 permits each. For the case where the government give these free of charge:
At 15 permits the MACs are
This means that the permit price p would initially be between 150 and 15. Firm 1 (with a high MAC of 150) finds it expensive to reduce pollution and would prefer to buy permits to emit more. Firm 2 (with a low MAC of 15) finds it relatively cheap to reduce pollution and would prefer to sell permits.
Provided that the firms trade with each other, firm 1 would buy permits from firm 2 as long as MAC1 > p. Firm 2 would also be willing to sell as long as p > MAC2. Trade will continue until both firms have the same MAC1 = MAC2 = 60, at this pont the price is p = 60.
For MAC1 = MAC2 = 60, then we have
Firm 1 now holds 24 permits, meaning it emits 24 units and abates 30−24=630−24=6 units.
Firm 2 now holds 6 permits, meaning it emits 6 units and abates 30−6=2430−6=24 units.
The emissions after trading are E1 = 24 and E2 = 6. This means that the firm 1 holds 24 permits and firm 2 holds 6 permits.
Firm 1 originally had 15 permits but now holds 24, so it bought 24−15=9 permits.
Firm 2 originally had 15 permits but now holds 6, so it sold 15−6=9 permits.
Therefore Firm 1 pays Firm 2, this is a transfer payment, not an additional cost. The overall private cost in the system remains zero because the permits were initially allocated for free. To see this observe that a tradable emissions scheme when permits are allocated for free involves a transfer of
Firm 1 pays Firm 2 for the 9 permits at the equilibrium price of 60 per permit.
Total transfer: 9×60=540
By allowing firms to trade permits, the overall cost of pollution reduction is minimized. The firm with a lower MAC (Firm 2) abates more pollution because it can do so more cheaply. The firm with a higher MAC (Firm 1) buys permits because it is more expensive for it to reduce pollution. The trading process equalizes the MACs between the firms, achieving the most cost-effective distribution of abatement efforts.
The total targeted pollution reduction of 30 units is achieved in the most economically efficient way, with Firm 1 reducing less and Firm 2 reducing more, according to their respective costs.
Question 1.c
How would the private costs to each polluter change if the government initially auctioned the permits off to the polluters?
Answer: If the government initially auctioned the permits off to the polluters, then there would be a positive private cost of the permit system to each firm as it involves transfers from firms to the government.
Problem 2: Taxes and subsidies (weight 33%)
The Fireyear and Goodstone rubber companies are two firms located in the rubber capital of the
world. These factories produce finished rubber and sell that rubber into a highly competitive world
market at the fixed price of $60 per ton.
The process of producing a ton of rubber also results in a ton of air pollution that affects the rubber
capital of the world. This 1:1 relationship between rubber output and pollution is fixed and
immutable (in the sense that it cannot be changed) at both factories. Consider the following
information regarding the cost (in $) of producing rubber at the two factories (𝑄_𝐹 and 𝑄_𝐺):
- Fireyear costs: 300 + 2(𝑄_𝐹)^2 - Goodstone costs: 500 + (𝑄_𝐺)^2
Total pollution emissions generated are 𝐸_𝐹 + 𝐸_𝐺 = 𝑄_𝐹 + 𝑄_𝐺. Marginal damage from pollution is
equal to $12 per ton of pollution.
2.a) In the absence of regulation, how much rubber would be produced by each firm? What is
the profit for each firm?
Answer: First we find the profitfunction for Fireyear
math
Where they gain income from a fixed price of $60 per ton subtracted the Fireyear costs. To find the amount of rubber Fireyear produce, we take the FOC, hence
math
To find the Fireyear profit we insert the optimal amount into the profit function.
math
We do the same for Goodstone
math
Insert the optimal amount in the profit function, hence
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Goodstone profits are 400 dollars which make them more profitable than Fireyear in the case without regulation.
2.b) The local government decides to impose a Pigouvian tax on pollution in the community.
What is the efficient amount of such a tax per unit of emissions? What are the post-
regulation levels of rubber output and profits for each firm?
Answer: The efficient amount of such a tax per unit of emissions the marginal damage from pollution of 12$, given in the description of problem 2. In this way we can internalize the damage form the pollution in theory. For Fireyear the, the profit function with respect to rubber output would look like
math
Taking the FOC, give us the new optimal rubber output
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Plug it into the profit funtion to find the profits with the tax
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Profits for Firstyear is now -12 $ due to the tax on pollution.
Due exactly the same for Goodstone, hence
math
The profit for Goodstone is 76 $, so they are still able to produce af positive profit evnethough the introduction on a tax on pollution.
2.c) Suppose instead of an emission tax, the government observes the outcome in part (a) and
decides to offer a subsidy to each firm for each unit of pollution abated. What is the efficient
per unit amount of such a subsidy? Again calculate the levels of output and profit for each
firm.
To find the amount the firms abate, we take the level of optimal emission under an emission tax, e.g. Q_F, and subtract that from the optimal level without any regulation, Qhat_F. The efficient per unit amount of such a subsidy equals the MD of pollution 12$, hence
math
For Fireyear we now have the profit function
math
We know the optimal quantity of rubber under taxation is Q_F =12 given the question 2.B and the optimal quantity without any regulation is Qhat_F = 15. Plugging this into the profit function gives us, 168 dollars in profit
math
Due the same for the Goodstone
math
Q_G =24 given the question 2.B and the optimal quantity without any regulation is Qhat_G = 30, giving the 436 dollars in profit
math
2.d) Compare the output and profits for the two firms in part (a) through (c) and comment on the
results.
In the short run, output levels are equal under a tax or subsidy, but not in the case with no regulation. However, profit is lower under a tax. Fireyear may even exit the market, beacue they get negative profits when the tax is introduced.
In the long run, more firms enter under a subsidy compared to a tax. Thus, more pollution occurs in the long run, which would argue that subsidizing is not the way forward in lowering pollution.
