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Cap and trade

October 18, 2011

“Cap and trade” systems are market based solutions to negative externalities like pollution. They work by creating a market for the externality (and solving the issue of missing markets for externalities). A typical cap and trade system will use permits that allow firms to produce a certain amount of pollution, and if they want to produce a level of output that will mean they end up producing more pollution than they have permits, they have to buy some permits from other firms.

The market solves the issue of firms having different costs of abatement. These are the costs incurred by a firm when it reduces its pollution by a certain amount. Some firms produce more efficiently than others (they produce more units of output per unit of pollution) and so they have lower costs of abatement. It makes sense for them to purchase extra permits from less efficient firms as they will be able to get more profitable output from being allowed to pollute more than the less efficient firms. Similarly, it is better for society if the less efficient firms produce less and the more efficient firms produce more as part of an overall pollution reduction strategy. Under a cap and trade system, the less efficient firms are incentivised to produce less because they can sell some of their pollution permits to more efficient firms. By creating a market for the externality, the more efficient firms will naturally end up producing more and the less efficient firms will produce less, which would not be the case if the government just imposed restrictions on the amount each firm could produce – unless government knew EXACTLY how efficient every firm was, which would be difficult to achieve in practice.

It may seem less complicated if we look at an example.

We can show this by setting up a model where there are three firms, A, B and C, which are all producing pollution and all have constant (but different for each firm) costs of abatement.

Firm A is initially producing 350 units of pollution, and has a cost of abatement of £40 per unit.
Firm B is initially producing 400 units of pollution, and has a cost of abatement of £50 per unit.
Firm C is initially producing 250 units of pollution, and has a cost of abatement of £20 per unit.

The initial total amount of pollution is 1000 units. Suppose the government wants to reduce total pollution levels by 400 units. This means total pollution will have to be no more than 600 units.

We can consider three options here:
1) limiting each firm to producing 200 units of pollution
2) giving each firm 200 ‘pollution permits’ (1 permit allowing production of 1 unit of pollution), and allowing firms to buy and sell the permits from each other
3) auctioning off 600 ‘pollution permits’ in a second price auction (like ebay, the winner pays the second highest bid price)

Option 1: limiting each firm to producing 200 units of pollution.

Firm A would have to reduce its pollution by 150 units, costing £40 x 150 = £6000
Firm B would have to reduce its pollution by 200 units, costing £50 x 200 = £10000
Firm C would have to reduce its pollution by 50 units, costing £20 x 100 = £2000

The total reduction of pollution is 400 units, hitting the government’s target, and the total cost of abatement is £18000.

Option 2: issuing each firm with 200 ‘pollution permits’ and allowing them to trade the permits between them. If any firms want to produce more than 200 units, they need to buy permits from another firm (so the other firm has to produce less). This creates a market for pollution, and there will be a market price for a unit pollution at which these permits will trade.

We can analyse this market by looking at the demand and supply for permits and see if we can determine an equilibrium market price.

First look at the demand. Any firm will be willing to pay up to its cost of abatement for a permit. For Firm A, for instance, the cost of abatement is £40, so if it can buy a permit for £39 it may as well do that and be allowed produce an extra unit of pollution rather than lose £40 in abatement costs by not producing it.

So firm A will be willing to pay up to £40 for a permit, firm B will be willing to pay up to £50 for a permit and firm C will be willing to pay up to £20 for a permit.

This means that if the price, P < 20, then all firms will want to buy permits and demand for them at that price will be 600.
If 20 < P < 40 then firms A and B will want to buy permits but C will not. A was originally producing 350 units of pollution and has 200 permits from the government so needs another 150 permits to produce its original level. B was originally producing 400 units and has 200 permits from the government so needs another 200 permits.The total demand for permits will be the 550 permits.
If 40 < P 50 then none of the firms will want to buy permits.

Now consider the supply. Any firm will be willing to sell a permit if it is worth more than the cost of abatement. For Firm A, for instance, the cost of abatement is £40, so if it can sell a permit for £41 it may as well do that and pay the £40 abatement costs for reducing a unit of pollution.

So firm A will be willing to sell permits at any price over £40, firm B will be willing to sell at anything over £50 and firm C will be sell at anything over £20.

This means that if the price, P < 20, then no firms will be willing to sell.
If 20 < P < 40 then firm C will be willing to sell but A and B won't, so the total supply of permits will be the 200 that firm C were given.
If 40 < P 50 then all firms will be willing to sell so the total supply of permits will be 600.

The supply and demand curves here are stacked, there is a zigzagging upward sloping supply curve and zigzagging downward sloping demand curve. The price of £40 is the tipping point at which there will be possibility of market equilibrium. At £39, demand will exceed supply as A and B want to buy permits meaning a total demand of 550, but only C will be willing to sell meaning a total supply of 200. The excess of demand over supply will push the price up. At £41, only B wants to buy, meaning total demand is 200, while A and C want to sell, meaning total supply is 400. This excess of supply over demand will push the price down.

So if the market price of permits was £40, what would happen? A would be indifferent about buying, selling or doing nothing. B wants to buy 200 permits. C wants to sell 200 permits. So the market trade is that B buys 200 permits from C.

Now the total pollution and abatement would be that:
Firm A would have to reduce its pollution by 150 units, costing £40 x 150 = £6000
Firm B does not have to reduce its pollution at all as it buys enough permits from C to produce its original level of pollution, costing £0.
Firm C doesn’t have any permits to allow it to pollute so would have to reduce its pollution by 250 units, costing £20 x 250 = £5000.

The total reduction of pollution is 400 units, hitting the government’s target, and the total cost of abatement is £11000. This is less than the £18000 that the original plan cost.

What has actually happened there is firm C has stopped producing, it has made £40 x 200 = £8000 selling permits to B and so is better off not producing and selling permits.

Option 3: auctioning off the ‘pollution permits’ in a second price auction.

Suppose the 600 permits just went straight to a government auction. Firm B would be the highest bidder as it was willing to pay £50 per unit, with firm A the next highest bidder, willing to pay £40 per unit. So being a second price auction, B would win this at a price of £40 (technically there would be a slight amount over eg if this was on ebay B would win it at £40.50 but I am discarding that here). So firm B would get the 400 permits it wanted to allow it to produce its original level of 400 units of pollution, paying £40 each (total price to government of £16000).

That means there are 200 permits left and A and C still to bid. A would be willing to pay £40 per permit and C willing to pay £20 so as it’s a second price auction, A wins at price £20. A wants 350 permits but there are only 200 so it just gets the remaining 200 and the government gets £4000).

Now the total pollution and abatement would be that:
Firm A would have to reduce its pollution by 150 units, costing £40 x 150 = £6000, it has also just paid £4000 for its permits, so the total cost to A is £10000.
Firm B does not have to reduce its pollution at all as it buys enough permits from the government to produce its original level of pollution, costing £0, it has paid £16000 for its permits so total cost to B is £16000.
Firm C doesn’t have any permits to allow it to pollute so would have to reduce its pollution by 250 units, costing £20 x 250 = £5000.

The total reduction in pollution is again 400 which hits the government target. The total costs for the three firms are now £31000 but the government has received £20000 so the total net cost to society of the pollution reduction is £11000.

So comparing the three options, all three deliver the government’s pollution reduction targets but option 1 – stipulating a set and equal amount that each firm can produce – is the most expensive. Option 2 – distributing pollution permits and allowing firms to trade them, and option 3 – auctioning off pollution permits – both have the same net cost to society which is lower than option 1. The difference between them is that distributing the permits is better for the firms and auctioning off the permits is better for government.

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