In a Polluted Environment who holds the property rights ?
For ex. If a Steel mill is setup near a Laundry then laundry's cost will go up because of the smoke and dust produced by the Steel Mill. So is the Laundry liable to receive a compensation from the Steel Mill ?
Property Rights deals with the transaction occurring due to presence of Externalities, and is paid so as to maximize the profit of the firms producing externalities/affecting from externalities, or to achieve Pareto Optimal Condition. It is a compensation paid to one of the party by other depending upon who holds the property rights.
For ex. If steel mill holds the property rights then there is a pollution cost that will be incurred in the Laundrys original cost. So Laundry may pay steel mill some of its profit for steel mill to decrease its production by some units which will reduce the pollution and reduce the cost for Laundry. Here Pareto Optimal Condition is to be calculated which will predict what amount Laundry needed to pay to Steel Mill so as to Maximize to profit of both of them.
Now if Laundry hold the property rights, Laundry will object on the dust and smoke produced by steel mill. So in order to continue producing steel, Steel Mill will offer a compensation which will reduce the cost of Laundry.
A condition may come up that paying a compensation is not feasible or profitable for a firm. In that case no transaction can occur and the one who has the Property Rights will continue its production only.
Lets view it in another way. Suppose you a farm near a railway track. The dust from railway is affecting your crop. So you may be willing to pay to Railways a part of your profit to decrease the frequency so that the pollution can be reduced and your crop production can be increased.
Sometimes opportunity cost also plays a major role in determining the property rights. If you have a house near beach having a nice view and someone who has a plot in between you house and the nice view. So depending upon the utility you may get with the view you may offer a compensation to the owner of the other plot for not constructing any such building that diminishes your view.
Further Readings: http://en.wikipedia.org/wiki/Coase_theorem