Theorem (1960) which states -- in opposition to Pigou's theory that "only governments, by means of taxes and subsidies, can "internalize" externalities in economic exchange or production -- that in a bilateral agreeement with externalities, the parties can "internalize" them through negotiation without external influence when one considers opportunity cost fully.

What this all means is that if person A is negatively affecting person B (said effect is an "externality", for example a rancher's cattle are eating the protective plants at the edge of a farmer's crops which degrade his outputs), then it is not necessary that the government step in and force the parties to act properly. Instead, when they examine the possible costs to them, they will be able to find an agreement which is more Pareto efficient ("Pareto efficient" implies that in a given allocation no individual can be improved in standing without injuring someone else's standing)

Here's an example to make it all clearer: the farmer would normally receive \$100/day/acre but receives only \$50/day/acre due to the cattle, while the rancher receives \$150/day/head without any negative influences from the farmer. Now, the government could step in and attempt to protect the protective plants which are a natural capital of sorts, but in general the outcome will still end up being inefficient (at least in the Pareto sense) However, if the farmer and the rancher talk, the farmer might suggest that the rancher take his cattle to the other end of the field which is not as important to the farmer. In doing so, the farmer's income increases to \$75/day/acre but in walking his cattle farther the rancher loses and afterward receives only \$130/day/head. What the farmer must do to make such a scheme attractive for the rancher is to offer to pay the difference. The farmer now receives: \$75 - \$20 = \$55 (\$20 is (the difference=\$150-\$130) or \$5 more, and the rancher receives the same amount (\$130 earned and \$20 provided by the farmer).1 All are happier.

The theorem holds true until the transactional costs become to high. For example the price of the lawyer to write up the agreement between the rancher and farmer is higher than the price earned. Or another example would be if Person A were a power plant or industry. In such cases, the transaction costs for an individual are simply too high to enter into agreement. In these cases, the government would need to step in and use incentives, taxes, or other such measures to balance the inequality.

1  I have no idea about farming and much less about ranching, so please no comments on how ridculously outlandish these incomes are.