In short, tradeable permits solve the problem of pollution at the least cost, because the producers that could reduce pollution at low cost would do so, and sell their permits to the producers who could only reduce pollution at high cost.
Anyway, in this post I wanted to compare taxes and permits. Alice Lepissier and Owen Barder at CGD wrote a quite detailed post on this comparison last year. I want to take a different tack to them, and think about what happens when clean technology becomes available.
The diagram below describes the simple model for the optimal quantity of pollution. You might think that the 'optimal' quantity of pollution is zero, but with no pollution we would essentially have no production which wouldn't make us better off at all (or at the extreme, no pollution means no breathing, since we all exhale carbon dioxide). Anyway, the MDC curve is the marginal damage cost (the cost to the environment of each additional unit of pollution) and is upward sloping - this is because at low levels of pollution, there is relatively less damage because the environment is able to absorb it. The capacity for the environment to do this is limited, so as pollution increases the damage increases at an accelerated rate. The MAC curve is the marginal abatement cost (the cost to society of each unit of pollution abated, or reduced) and is upward sloping from right to left. This is because, as more resources get applied to reducing pollution, the opportunity costs increase. Also, less suitable resources (meaning more costly resources) have to begin to be applied to pollution reduction. The optimal quantity of pollution occurs where the MDC and MAC curves intersect - at Q*. Having less pollution than Q* (such as at Q1) means that MAC is greater than MDC. In other words, the cost to society of reducing that last unit of pollution was greater than the cost in terms of environmental damage. Having pollution at Q1 must make us worse off when compared with Q*.
Now consider Pigovian taxes. With a Pigovian tax every firm must pay the government for each unit of pollution they generate. This effectively sets a price for pollution. The optimal price of pollution (which would lead to exactly Q* units of pollution in the diagram above) is P*. Firms would not pollute more than Q*, because the tax (P*) is greater than the MAC - it is cheaper to reduce pollution than it is to pay the tax. On the other hand, firms would not pollute less than Q* either, because the MAC is greater than the tax (P*) - it would be cheaper to pay the tax than to reduce pollution further than Q*.
What about tradeable pollution permits? With tradeable permits, the government sets the number of permits (pollution rights) that are available to the market - one permit allows a firm to emit one unit of pollution. The optimal quantity of permits is Q*, and the market will set the price exactly equal to P*. The price will not rise higher than P*, because then MAC would be less than P* and firms could reduce pollution for less cost than the price of a permit.
So, it is easy to see that both taxes and permits are theoretically equivalent in terms of the market for pollution. Taxes set the market prices, which (if the price is set correctly) results in the optimal quantity of pollution. Permits set the optimal quantity, which leads to the market price. At this point, the argument becomes which system (taxes or permits) would be less costly to administer and which would provide better incentives to adopt cleaner technology - possibly taxes in both cases. Or perhaps the argument is about which system is less risky, which Lepissier and Barder argue is permits.
Now, think about what happens if a new clean technology becomes available that makes it cheaper to reduce pollution. That lowers the marginal abatement cost, so MAC moves to MAC1 in the diagram below. The diagram demonstrates what happens with a Pigovian tax. The price of pollution is fixed at P*, so when MAC decreases to MAC1, the quantity of pollution falls greatly (to Q2). However, Q2 is too little pollution (relative to the new optimal quantity Q1), and at that point MAC is greater than MDC - the cost to society of reducing the last unit of pollution was greater than the cost in terms of environmental damage. We reduce pollution too much, leading to a deadweight loss (the area BEF in the diagram).
The next diagram demonstrates what happens with pollution permits. The quantity of pollution is fixed at Q*, so when MAC decreases to MAC1, the price of pollution permits falls (to P2). Now Q* is too much pollution (relative to the new optimal quantity Q1), and at that point MAC is less than MDC - the cost to society of reducing one more unit of pollution would be less than the cost in terms of environmental damage. We don't reduce pollution enough, leading to a deadweight loss (the area GHJ in the diagram).
So in both cases, when a new clean technology becomes available we end up with a deadweight loss. In this case though, you probably want the clean technology to lead to less pollution rather than the same quantity, so I would argue that this favours taxes over permits, unless it is easy for the government to reduce the number of permits. And when you consider climate risk, it is probably better to over-shoot on pollution reduction, rather than under-shoot.
[Update: Replaced diagrams to fix x-axis labels]