Pareto efficiency

Definition

An allocation is Pareto efficient if there is no other allocation in which some other individual is better off and no individual is worse off.

Notes:

There is no connection between Pareto efficiency and equity! In particular, a Pareto efficient outcome may be very inequitable. For example, the outcome in which I have all the goods in the world is Pareto efficient (since there is no way to make someone better off without making me worse off).
Pareto efficiency is an absolute notion: an allocation is either Pareto efficient or it is not. If in the allocation x someone is better off and no one is worse off than in the allocation y then we say that x Pareto dominates y. The allocation x in this case may of course not be Pareto efficient: there may be some other allocation that Pareto dominates it.

Examples and exercises on Pareto efficiency

Pareto efficiency and competitive equilibrium in an exchange economy

We can show the following result.

If every trader cares only about the bundle she has (not the bundle any other trader has) then a competitive equilibrium allocation is Pareto efficient.

Notes

Since the notion of Pareto efficiency is not connected with the notion of equity, this result does not imply that a competitive equilibrium is equitable! Indeed, in general there is no reason to think that a competitive equilibrium is equitable in any sense.
The result does not claim that the competitive equilibrium outcome is the only Pareto efficient outcome. Indeed, in general there are many Pareto efficient allocations; the competitive equilibrium allocation is one of these allocations.
Any outcome in which the buyers and sellers who trade are the same as the ones who trade in a competitive equilibrium is Pareto efficient, regardless of the prices at which the transactions take place.
The assumption that everyone cares only about the bundle she consumes is essential. If a person's welfare depends on whether or not some other person buys a good, then a competitive equilibrium in general is not Pareto efficient.

This result is the main economic argument in favor of using markets to allocate goods. Thus if you wish to make an economic argument against the use of markets you need either to question the assumptions behind the result or to point to the weakness of the conclusion. The main assumption is that all goods are "private": each person's welfare depends only on her own consumption bundle, not on anyone else's. Two reasons why the conclusion is weak are:

It does not imply that if there is a constraint of some sort---some market is not competitive, for example---then the outcome when the remaining markets are competitive is Pareto efficient. Indeed, in the presence of a constraint a competitive equilibrium is not in general Pareto efficient.
In general there are very many Pareto efficient allocations, some of which are very bad from the point of view of equity.

Argument for the result. Why is the result true? For simplicity, think of a situation in which there is a single good, each seller has one unit of the good, and each buyer can buy either nothing or one unit of the good. Suppose that the buyers and sellers differ in how they value the good. Then the demand Q_d(p) at the price p is the number of buyers whose value for the good is at least p. Similarly, the supply Q_s(p) at the price p is the number of sellers whose value for the good is at most p.

In a competitive equilibrium in this market the buyers who trade are those who value the good most highly, and the sellers who trade are those who value the good the least. Is there any other pattern of trade that results in someone being better off without anyone being worse off?

If an additional transaction between a buyer and a seller who are currently not trading takes place then, depending on the price of the transaction, either the buyer must pay more than her value or the seller must receive a price less than her value. Thus at least one of the traders is worse off than before.
If one fewer transaction takes place, then either a seller who sold at a price above her value and/or a buyer who paid a price below her value no longer trades, and hence is worse off.

Examples and exercises on Pareto efficiency and competitive equilibrium