A Market Approach to Dealing with Externalities

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…

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 $x=1$ , $\text{total cost}=26$

For Firm 2, if $x=3$ , $\text{total cost}=18$ , $18+26=44$ . 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

Table 7.8 Amount 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:

Table 7.9 Amount of Pollution

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:

Table 7.10 Marginal Pollution Control Costs Ranking

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?

Table 7.11 Firm 1:

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?

Table 7.12 Firm 2:

Emissions	0	1	2	3
MC	8	6.5	4	2

Before trading cost = 6. After trading cost = 6