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Externalities and the Coase Theorem

October 17, 2011

The Coase theorem is a method of tackling the inefficiency caused by an externality, by awarding property rights to the externality to one party and allowing the parties concerned to bargain their way to an efficient solution. In the case of a negative externality such as pollution, the property right to pollute could be awarded to the polluter, and the victim of the pollution would have to offer to pay to get the polluter to reduce their output. Alternatively, the property right to clean air/water/whatever else is being polluted, could be awarded to the victim, so the polluter would have to offer to pay the victim to be allowed to pollute.

Consider the situation where we have a firm, firm A, producing some good in a competitive market that causes some pollution as a result of its output. Another firm, firm B, which produces a different good in a different market, is directly harmed by this pollution. It could be something like A being a factory and B being fisheries, and the factory pollutes the water and reduces the number of fish.

Anyway we consider the market for firm A. We said that it was a competitive market so it is a price taker , it faces a horizontal demand curve at the market price, Pc, and can produce as much output as it likes at this price. As the market price is constant and does not vary depending on the output the firm produces, the marginal revenue is also equal to the price. Assume as well that firm A faces an upward sloping (private) marginal cost curve, the more output it produces, the greater its marginal costs.

Firm A will produce up to the point where its marginal revenue equals its private marginal costs, here MR=PMC at Qc, the competitive output quantity.

However also on that diagram is a social marginal cost curve (here ‘society’ includes firm A and firm B), this is higher than A’s PMC curve because it includes the cost of the externality (the pollution) on firm B. By producing at Qc, firm A is producing output above the social optimum. This is because the joint benefit as a result of this output is simply the total revenue gained by firm A (Pc x Qc), firm B does not get any benefit. But the joint social cost is greater. The cost to firm A is (Pc x Qc), and the cost to firm B caused by the externality is ((SCc – PC) x Qc), making the total social cost (SCc x Qc). As (SCc x Qc) > (Pc x Qc) then there is a net loss incurred by this level of production.

The Coase solution offers us two options, either we could
i) award firm A the right to pollute or
ii) award firm B the right to clean resources.

Consider situation i) first, awarding firm A the right to pollute.

Here if A has the right to pollute, then firm B must offer some form of bribe payment to A in order to get A to reduce its output, ie B has to pay A not to produce. Lets call this bribe payment x.

At any output level, the marginal gain to firm A to produce another unit of output is MR – PMC. When the marginal gain is positive it is going to produce another unit of output so it will keep producing until its marginal gain is 0. But each unit that A produces delivers a marginal loss to firm B of MD, the marginal damage produced by the externality for that unit of A’s output.

Now if firm B offers firm A a payment of x in order to stop production at that output, then the marginal gain to firm A is x instead of being MR – PMC, so as long as x > MR – PMC, then A will stop producing.

From firm B’s perspective, if the value of the payment, x, is less than the value of the marginal damage done by the externality for producing that extra unit of output, then it makes sense for firm B to pay x to firm A not to produce it. So by thinking at the margin, firm B will be willing to increase its value of x up to a point where x = MD, the payment equals the amount of marginal damage that the pollution would do to it.

If x = MD, then we can say that as long as MD > MR – PMC, then there is the opportunity for a bargain to stop A producing more output. Even if the MD to firm B is just marginally greater than MR – PMC for firm A, it is worth B making the offer of payment, so we can effectively say they will bargain up to the point where MD = MR – PMC.

Now remember the definition of the social marginal cost is SMC = PMC + MD, it is the private marginal cost of production to firm A plus the marginal damage of the externality to firm B. So this means that we can rewrite that as MD = SMC – PMC.

Substituting this into our equation above we can say SMC – PMC = MR – PMC, hence SMC = MR. That means that after bargaining, production will take place at the point where SMC = MR, the socially efficient level of output.

Now consider situation ii) where firm B is awarded the right to clean resources

This time A does not have the right to pollute, it has to instead compensate B for producing an extra unit of output. This time the payment, x, will be made by A to B.

Now at any output level, the marginal gain to firm A to produce another unit of output is MR – PMC – x. To firm B the marginal loss suffered as a result of firm A producing another unit of output is MD – x. So firm B will accept payment x if x > MD. At any level where x is marginally greater than MD, there exists the opportunity to bargain and A can produce another unit by paying x to B, so they will be able to bargain up to the point where x = MD.

Firm A will produce up to the point where its marginal gain is 0, ie MR – PMC – x = 0. If x = MD then MR – PMC – MD = 0, so MR = PMC + MD. As we know that MD = SMC – PMC, this means MR = PMC + SMC – PMC so MR = SMC. Again we have reached a situation where firm A produces up till the level where MR = SMC, the socially efficient output.

The conclusion is that with the Coase theorem, it does not matter whether the property rights are allocated to the polluter or to the victim of the pollution, the socially efficient output level will result either way.

However there are some problems with the Coase solution to externalities. Whilst an equilibrium reached through Coasian bargaining may be good in terms of efficiency there may be other concerns in terms of equity. In the case above, if firm A was awarded the property right to pollute then firm B is in an undesirable situation either way, it either has to suffer the pollution or pay A not to pollute. Alternatively if firm B was awarded the property right to clean resources then firm A will be less well off because B is now insulated against damage, either it gets compensated for the pollution produced or it doesn’t produce and pollute.

Instinctively most people would probably say it was more fair to award the property rights to clean resources to firm B. One advantage of this is that by making firm A face the full social costs of production, rather than just the private costs, it is having to internalise the externality and this may encourage it to invest in cleaner production techniques. There is no incentive to invest money in cleaner production techniques if the costs caused by the dirty production are incurred by someone else. However the Coase theorem doesn’t make these sorts of judgements, it simply says that awarding the property rights to one party will lead to the efficient output through bargaining.

There are other problems with the Coase solution:
Bargaining costs may be too high. It is easier when there are just two parties involved but what if there were many parties, for instance if a firm was polluting a coastline and there were hundreds of fishermen and residents that could be harmed by it. It may be difficult and costly to get all of the fishermen and residents together to actually bargain a solution. Some people may also not bother taking part, hoping that their fellow residents will put the case forward (free riding), and this may mean proper negotiations don’t take place.
– There may be difficulties obtaining information on the relative source and size of the damages, and there may be asymmetric information where each party does not know how much the other party really values the relative MD and MB.

To conclude, the things to remember about the Coase solution are:
– If property rights are clearly defined then bargaining can lead to the optimal levels of pollution and output
– Efficiency is achieved regardless of who gets the property rights, but the income distribution is affected by who gets the property rights
– To work it relies on there being no impediments to bargaining, the costs of bargaining can’t be prohibitive, all parties need to have full information and not engage in strategic bargaining behaviour

In reality a fully efficient Coasian solution will not be possible in most situations but the theory is relevant to things like ‘cap & trade’ systems of managing pollution.

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