Pareto efficiency, or Pareto optimality, is a central theory in economics with broad applications in game theory, engineering and the social sciences. Given a set of alternative allocations and a set of individuals, a movement from one alternative allocation to another that can make at least one individual better off, without making any other individual worse off is called a Pareto improvement or Pareto optimization. An allocation of resources is Pareto efficient or Pareto optimal when no further Pareto improvements can be made.
Thus in a Pareto Efficient outcome, we cannot make any person better off without making someone worse off. In the case of this question, giving Joe $1 and Martha $1 would not be a Pareto Efficient outcome, because we would have $98 left over. We could just as easily give each of them $2, making them both better off without making anyone worse off. Thus in this case the set of Pareto efficient outcomes are ones where there are no waste - Joe and Martha's combined allocation is equal to $100. This includes situations where Joe gets $100 and Martha gets nothing. We cannot make Martha better off without taking money away from Joe; thus we have a Pareto Efficient outcome. The example above is at the heart of the biggest criticism of the concept of Pareto Efficiency - it says nothing about equality. The outcome where they each get $50 and the outcome where one party gets all the money are Pareto Efficient.
This example of a Production-possibility frontier provides a simple example for illustrating Pareto efficiency. Suppose that there are two agents in an economy, one that only values guns and one that only values butter. Point A is not Pareto efficient because it is possible to produce more of either one or both goods (Butter and Guns) without producing less of the other. Thus, moving from A to D enables you to make one person better off without making anyone else worse off (Pareto improvement). Moving to point B from point A, however,...
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