Why is any government needed at all?

What differentiates government services from those that may be provided by the private sector is that government services involve the use of (retaliatory) force: police, military, and courts of law.

There can be no "market" involved, and therefore no private sector, when the underlying action involves force, since a market transaction is one in which all parties participate willingly. For example, a police officer arresting someone for fraud is not involved in a transaction to mutual benefit with the accused.

Therefore, the government, as its only moral purpose, can and should evaluate when the use of retaliatory force is appropriate, via objectively defined laws.

See The Nature of Government.

Often the purpose of this question is the particular challenge: "Why do we need a single institution called 'the government' to have the privilege of using retaliatory force?" Or: "Why should one institution have a monopoly on force?"

Harry Binswanger has pointed out that this question, "Why should anyone have a monopoly on force?" is based upon a mistaken premise: that the monopoly is incidental. But the truth is that any use of force -- just or unjust -- is inherently monopolizing in its purpose. A person using force wants things done his way, or else. He is not tolerating challenge or dissent. He is not looking to make a deal. And he is most certainly not going to coexist peacefully with others who wish to use force against him.

If people create an institution to use retaliatory force in their defense, that institution must be able to use force without being thwarted by rival force-wielding agencies. It could not be effective in its goal otherwise. And if there is organized resistance, then the situation is a state of civil war in which one side (or some opportunistic third party) will eventually be victorious -- and then assert its sole authority to use force.

The Objectivist defense of government over anarchism is based upon an inductive understanding of the nature of force, which is readily observable in the current world and in history.

(People who believe that there should be multiple force-wielding agencies in the same jurisdiction call themselves "anarchocapitalists," a particular flavor of libertarian. This is one more reason to avoid the L word in describing Objectivist politics.)

Because without government—that is, under anarchy—the result is bloody tribal warfare. Just look at Europe during the Dark Ages, after the fall of the Roman Empire. A strong central government performs the crucial function of maintaining law and order, and preventing attack from the outside. In Ayn Rand's identification, it protects individual rights.

These "services" can't be provided by "the private sector" because they involve and require the use of force. Truly private services, such as restaurants or phone companies, can co-exist, competing peacefully. But governments, by their nature, cannot. To maintain law and order, a government needs a monopoly on the use of force in a region. It needs jurisdiction, sovereignty. When two governments disagree on issues of jurisdiction and sovereignty, their competition is not the peaceful competition of private companies. It is war.

In the end, a strong government, maintaining order and protecting rights, is the precondition of any kind of private sector.

The idea that private companies could provide police services is actually guilty of a particularly gross form of stolen concept. The concept they steal is "contract." A contract presupposes an agency--the courts and police--that will enforce that contract. Outside that context, a "contract" is wishful thinking.

This relates to the question only regarding private police/military/courts.

From a more anthropological perspective: man needs governments in order to subordinate might to right. If man is to live rationally, to live by his own knowledge and judgment, he has to be free from coercion. If man is to live productively, he must be able to count on keeping the products of his effort. So, in his person and in his possessions, man needs security, he needs to be free from forceful interference. Civilization begins when men agree to respect one another's lives and property, with the weight of the whole group standing behind any victims, against transgressors. At least some rules and laws defining the transgressions and setting up a process for redress must be instituted to achieve the practical implementation of this, and we call that government. It follows that man needs governments in order to live as a rational being. In instituting governments, men are consciously choosing the "rational" over the "animal," and production over predation. When and only when they do, they achieve at once both peace and prosperity.

Seconding to what has been posted, I think this quote might be apt. "Government is like fire, it is a dangerous servant and a fearful master". So, its better to limit the scope of that servant than fear for his out-of-scope work by writing it in the Constitution.