Let's assume 2 firms, and no environmental regulation. Each firm pollutes 4 "units" (say tons) of "guck," an environmental bad. Abatement is costly.

Scenario 1) The Environmental Protection Agency (EPA) announces each firm must reduce pollution 2 units. So, each firm gets a non-tradeable right to pollute 2 units.

Scenario 2) EPA gives each firm tradeable rights to pollute 2 units (in our example, 10 in all).

So how much does it cost firms?

A small detour:

$MC\left(X\right)=TC\left(X\right)-TC\left(X+1\right)$ ;

But…let Z=the “no regulation” state. This implies $TC\left(Z\right)=0$ .

So, $MC\left(Z-1\right)=TC\left(Z-1\right)-TC\left(Z\right)=TC\left(Z-1\right)$ ;

$TC\left(Z-1\right)=MC\left(Z-1\right)$

$MC\left(Z-2\right)=TC\left(Z-2\right)-TC\left(Z-1\right)=TC\left(Z-2\right)-MC\left(Z-1\right)$ ;

$TC\left(Z-2\right)=MC\left(Z-2\right)+MC\left(Z-1\right)$ ;

We can show that:

$TC\left(Z-3\right)=MC\left(Z-3\right)+MC\left(Z-2\right)+MC\left(Z-1\right)$ and so on.

This all implies that we can calculate total costs by simply adding up the marginal costs from right to left.

Back to the problem…

### Practice Exercise

Given the data below, calculate the marginal cost of pollution abatement.

Amount of Firm 1 Pollution | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

TC of Abatement | 35 | 26 | 18 | 11 | 5 | 0 |

Amount of Firm 2 Pollution | 0 | 1 | 2 | 3 | 4 | 5 |

TC of Abatement | 60 | 44 | 30 | 18 | 8 | 0 |

Now, the EPA decides that each firm must reduce its pollution to 2. How much will this cost?

Reading off the total cost tables, we get $18+30=48$ .

#### Allowing Trading Between Firms

Or: The EPA gives each firm 2 pollution “credits,” which are tradeable. To pollute x units, each firm must own x credits. Since Firm 2 is the “high cost” firm, we’ll ask: Should Firm 2 buy a credit from Firm 1?

Firm 2's marginal cost of abatement (going from 2 to 3 units of pollution) is 12. Firm 1's marginal cost of abatement (going from 2 to 1) is 8. So, if Firm 1 sells a credit to Firm 2, abatement costs will go down by 4.

Check: For Firm 1, if $x=1$ , $\text{totalcost}=26$

For Firm 2, if $x=3$ , $\text{totalcost}=18$ , $18+26=44$ . Cost went down by 4.

What price should they trade at? It costs Firm 1 8 “units” to abate one more unit, so that is the lowest they should accept. Firm 2 gains 12, so that is the most they should be willing to pay. So, the price of this credit will be in the range [8, 12].

Should Firm 2 buy a second credit from Firm 1? Firm 2's marginal cost of abatement (going from 4 to 3) is 10. Firm 1's marginal cost of abatement (going from 1 to 0) is 9. So, they should trade, reducing total costs by 1 (you’ll want to check this), at a price in the range [9,10].

#### Market Trading of Permits to Pollute

Let's assume 5 firms, and no environmental regulation. Each firm pollutes 4 "units" (say tons) of "guck," an environmental bad. Abatement is costly.

Scenario 1) EPA announces each firm must reduce pollution 2 units. So, each firm gets a non-tradeable right to pollute 2 units.

Scenario 2) EPA gives each firm tradeable rights to pollute 2 units (in our example, 10 in all).

So, how much does it cost firms? Assume 5 firms, as in the next table.

#### Marginal Cost of Pollution

Firm | 0 | 1 | 2 | 3 |
---|---|---|---|---|

1 | 4 | 3 | 2 | 1 |

2 | 8 | 6.5 | 4 | 2 |

3 | 6 | 3.5 | 2.5 | 0.5 |

4 | 12 | 9 | 6 | 2.5 |

5 | 8.5 | 7 | 5.5 | 3.5 |

From this, we need to calculate a total cost table. To do this, just add up the marginal costs from right to left. Thus, the total cost for firm 1 of 3 units of emission is 1. The total cost of 2 emissions is $1+2=3$ . The total cost of 1 emission is $1+2+6=9$ , and so on.

This results in a total cost table:

Firm | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

1 | 10 | 6 | 3 | 1 | 0 |

2 | 20.5 | 12.5 | 6 | 2 | 0 |

3 | 12.5 | 6.5 | 3 | 0.5 | 0 |

4 | 29.5 | 17.5 | 8.5 | 2.5 | 0 |

5 | 24.5 | 16 | 9 | 3.5 | 0 |

So, the total cost of Scenario 1 (no trading, 2 units guck emissions per firm) is the sum of the total costs at 2 units of pollution:

$3+6+3+8.5+9=29.5$

#### The Benefits of Allowing Trading

Now, let’s go to scenario 2, where trading is allowed:

We need to derive supply and demand curves for permits.

What does our supply curve look like? $Q=10$ . There is always a supply of 10 in the market.

What does the demand curve look like? Simply rank marginal pollution control costs from high to low:

Number | Marginal Cost |
---|---|

1 | 12 |

2 | 9 |

3 | 8.5 |

4 | 8 |

5 | 7 |

6 | 6.5 |

7 | 6 |

8 | 6 |

9 | 5.5 |

10 | 4 |

11 | 4 |

12 | 3.5 |

13 | 3.5 |

14 | 3 |

15 | 2.5 |

16 | 2.5 |

17 | 2 |

18 | 2 |

19 | 1 |

20 | 0.5 |

With 10 permits available, the market price will be the average of the 10^{th} and the 11^{th} value. Here, that is 4.

The top 10 demanders will get the pollution units—at a market price of 4.

**How much will each firm make from the market?**

Emissions | 0 | 1 | 2 | 3 |
---|---|---|---|---|

MC | 4 | 3 | 2 | 1 |

If this firm could not trade, it would have costs = 3.

With a market price of 4, it sells 1 (or 2) pollution rights, has abatement costs $1+2+3=6\text{or}1+2+3+4=10$ and sells 1 or 2, revenues 4 (8)

$TC=6-4=2\text{or}TC=10-8=2$ . So, Firm 1 makes 1 off the market.

**How much does Firm 2 make?**

Emissions | 0 | 1 | 2 | 3 |
---|---|---|---|---|

MC | 8 | 6.5 | 4 | 2 |

Before trading cost = 6. After trading cost = 6

### Practice Exercise

Calculate the net wealth increase (across all 5 firms) created by the market.

### Practice Exercise

You are the incredibly greedy owner of Guck, Unlimited, a major polluter. Currently, in the “free” state of the world, you emit 5 units of pollution. Your costs of pollution abatement are below:

A) Fill in the marginal cost portion of the table below.

Units of Pollution | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

Total Cost | 119 | 72 | 44 | 24 | 10 | 0 |

Marginal Cost | ? | ? | ? | ? | ? | ? |

B) Explain what Guck’s net costs (costs of abatement plus the costs of permits bought, minus the cost of permits sold) would be if

- Pollution permits cost $30/unit, and you must buy any you want from the market; you are allocated none;
- Pollution permits cost $40/unit on the market, and the EPA allocates you 4.

### Practice Exercise

There are four firms in an industry, with total costs of pollution abatement as described below. The government decides that to pollute, a firm requires one permit per unit of pollution. The government also will auction off 7 permits. Given this, what is the market price of permits, and which firms will buy how many permits?

Firm # | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

1 | 580 | 430 | 290 | 160 | 70 |

2 | 770 | 470 | 250 | 100 | 30 |

3 | 535 | 285 | 150 | 70 | 10 |

4 | 630 | 450 | 290 | 150 | 50 |

### Practice Exercise

There are four firms in an industry, with total costs of pollution abatement as described to the right. The government decides that to pollute, a firm requires one permit per unit of pollution. The government also will auction off 7 permits.

Given this, what is the market price of permits, and which firms will buy how many permits? What is the total cost of abatement?

Firm # | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

1 | 56 | 36 | 20 | 8 | 0 |

2 | 62 | 41 | 24 | 9 | 0 |

3 | 45 | 30 | 17 | 7 | 0 |

4 | 60 | 40 | 24 | 10 | 0 |

### Practice Exercise

- Calculate MCs.
- Rank order MCs from high to low.
- To find the market price, take a number halfway between the 7
^{th}and 8^{th}MC. - Allocate the permits to the firms with the 7 highest MCs.
- Figure out the total cost of abatement by firm.
- Add up TCs across all firms.