The theory of the firm and industry equilibrium

Martin J. Osborne
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4.2 Pareto stability

Definition
An allocation is Pareto stable if no other allocation exists in which some other individual is better off and no individual is worse off.

Notes:

  • The notion of Pareto stability is called “Pareto efficiency” by many authors. This terminology is misleading, because the word “efficiency” suggests desirability, and Pareto stable allocations are not in general desirable. (In the past, the term “Pareto optimality” was also used. This terminology is even more misleading than “Pareto efficiency”.)
  • There is no connection between Pareto stability and equity. In particular, a Pareto stable outcome may be highly inequitable. For example, the outcome in which all the goods in the world are allocated to one individual is Pareto stable (since there is no way to make someone better off without making the individual worse off).
  • Pareto stability is an absolute notion: an allocation is either Pareto stable 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 stable: there may be some other allocation that Pareto dominates it.

Example
Consider an economy that contains only one good, which everyone likes. Then every allocation is Pareto stable: the only way to make someone better off is to give them more of the good, in which case someone else will have less of the good, and hence be worse off.
Example
An economy contains two people and two goods, apples and bananas. Person 1 likes apples and dislikes bananas (the more bananas she has, the worse off she is), and person 2 likes bananas and dislikes apples. There are 100 apples and 100 bananas available.

The only allocation that is Pareto stable is that in which person 1 has all the applies and person 2 has all the bananas. For any other allocation, one of the persons has some units of the good she does not like, and would be better off if the other person had those units.

Example
An economy contains two people and two goods, apples and bananas. Person 1 likes apples and doesn't care one way or the other about bananas (she is indifferent between any bundles (a,b) and (a,b'), where a is some number of apples and b and b' are numbers of bananas). Person 2 likes bananas and doesn't care one way or the other about apples. There are 100 apples and 100 bananas available.

The only allocation that is Pareto stable is that in which person 1 has all the apples and person 2 has all the bananas. For any other allocation, one of the persons has some units of the good about which she doesn't care; transferring those units to the other person would have no effect her and would make the other person better off.

Example
An economy contains two people and two goods, apples and bananas. Both people like both goods, but value them differently. For person 1, 1 apple is exactly equivalent to 2 bananas: she is indifferent between any bundles (a, b) and (a − n, b + 2n), where a is some number of apples, b is some number of bananas, and n is some number). For person 2, 2 apples are exactly equivalent to 1 banana.

An allocation is Pareto stable if and only if

  • either person 1 has no bananas
  • or person 2 has no apples.
Why? Suppose person 1 has some bananas and person 2 has some apples. Then by transferring one banana from person 1 to person 2 and one apple from person 2 to person 1 we make both of them better off. On the other hand, if person 1 has no bananas then any trade that makes her better off must involve her getting at least twice as many bananas as she gives up in apples, which results in person 2 being worse off. Similarly, if person 2 has no apples then any trade that makes her better off must involve her getting at least twice as many apples as she gives up in bananas, which results in person 1 being worse off.

Three of the allocations that are Pareto stable are those in which

  • person 1 has all the apples and person 2 has all the bananas
  • person 1 has all the apples and all the bananas
  • person 2 has all the apples and all the bananas.

Pareto stability and competitive equilibrium in an exchange economy

We can show the following results.
1. If every individual cares only about the bundle she consumes, not the bundle any other individual consumes, then a competitive equilibrium allocation is Pareto stable.

2. If individuals care not only about the bundles they consume but also about the bundles that other individuals consume, then a competitive equilibrium allocation is not in general Pareto stable.

Notes
  • Since the notion of Pareto stability is not connected with the notion of equity, the first 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 first result does not claim that the competitive equilibrium outcome is the only Pareto stable outcome. Indeed, in general there are many Pareto stable allocations; the competitive equilibrium allocation is one of these allocations.
  • If every individual cares only about the bundle she consumes, any outcome in which the buyers and sellers who trade are the same as the ones who trade in a competitive equilibrium is Pareto stable, regardless of the prices at which the transactions take place.
The first 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 stable. Indeed, in the presence of a constraint a competitive equilibrium is not in general Pareto stable.
  • In general there are many Pareto stable allocations, some of which are highly inequitable.

Argument for the first result. Why is the first 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 Qd(p) at the price p is the number of buyers whose value for the good is at least p. Similarly, the supply Qs(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.