Private property is necessary for markets and the Theorems to do their work. When the law and customs do not guarantee the right to one's property trade will halter. Ill-defined property rights were the main problem for Russia in its transition from a planned to a market economy. Buying a company was risky as long you could not be guaranteed the right to that company. There was always the chance that the government would take the company away from you or make it impossible for you to operate.
Ill-defined property rights easily lead to waste. You take less good care of the rental car than you do your own. People tend to take better care of their apartments and houses when they own the property rights.
An important case of ill-management is that of the public lands in the United States. Because they are owned by everybody, they are owned by nobody in particular. Thus they have to be run by the government or voluntary societies. This does not always produce the best (most efficient) outcomes.
The United States was founded on the principle that land should be private. But in the late 19th century, as part of the movement known as Progressivism, conservationists overturned the old principle. They wanted the government to "hold the land in trust for the people," as it is usually expressed. Most of the public land is in the West - almost all of Alaska and Nevada, for example, and large tracts in Montana, Arizona, and elsewhere. The Eastern and Midwestern land was sold off to private parties in the first century or so of the Republic. The Western lands, therefore, are owned by the people as a whole, not individual ranchers, loggers, and mining companies. It sounds like a good idea. But because everybody owns the public lands, nobody actually does. No market therefore-no advocates for maximizing the value of marginal products-decides how they are used.
For example, the Bureau of Land Management allows local ranchers to lease some of the land for grazing. The head of the Bureau does not literally own the land. Unsurprisingly, therefore, he does not charge a market-clearing price for the grazing rights. He has no incentive to do so, since he personally gets no profit from extracting the market-clearing price. What does he care about the government's income? The income goes to the government treasury. So what incentive does the bureaucrat have? How could he exploit the fact that the demand for the land exceeds the supply? If he is a philosopher he might persuade the ranchers to compete in being public spirited, to satisfy the bureaucrat's notion of public spirit. A less public spirited bureaucrat may be receptive to a case of Johnny Walker whiskey at Christmas, merely as a personal gift you understand.
When property rights are ill-defined or undetermined, externalities occur. The smoke that a factory spits out is a sign of productive activity but may harm the neighbors-far and near. The factory can sell what it produces and thus monetize its productive activity. However, it may not have to pay for the harm that it causes the public with its smoke. The harm caused is an externality of the production because it is not priced in any way. In general:
An externality is a cost or benefit that results from an activity or transaction and is not priced in any market
An externality can also be a benefit as it is in the case of the beautiful violin that a fellow student plays in the room next door. You did not pay anything additional to hear the music and yet you can enjoy the music without leaving the comfort of your room.
In either case property rights are undetermined. In the case of the polluting factory it is the right to affect the air that is ill-defined. If the citizens were to have the right to the air, they could forbid the factory from spitting out smoke, or they could charge a price. In the case of the music your fellow student can not prevent you from enjoying his music because she does not own the rights to the sound waves in the hallway. Accordingly,
Externalities are caused by ill-defined or undetermined or just plain ignored property rights.
The solution could begin with the determination and attribution of property rights. In the case of pollution the government could claim property over the air on behalf of its citizens and sell rights to spit out certain amounts of smoke to the highest bidder. The revenues could subsequently be used to compensate those who suffer from the pollution. The solution meets the criteria of the First Theorem as now beneficial deals are possible without disadvantaging anyone. When a factory wants to increase its production with more pollution as the result, it can do so if it is willing to pay the price for the right to pollute. Those living nearby will have to suffer more pollution but will also receive more compensation. In the case of the beautiful music in the hallway you and your fellow student may wish to maintain the status quo. But what if the music of the fellow student is very special and very expensive as well? First she could make the walls of her apartment soundproof to prevent you from hearing anything. Since she owns the rights to whatever can be heard within her own space, she could subsequently charge you a price if you really would like to hear her music. This is the solution of the concert hall: only after payment of an admission-fee are people able to hear the music. Thus a market is created with the possibility of (voluntary) exchanges.