In economics, the overtaking criterion is used to compare infinite streams of outcomes.
Contents
Often, the decisions of a policy-maker may have influences that extend to the far future. Economic decisions made today may influence the economic growth of a nation for an unknown number of years into the future. In such cases, it is often convenient to model the future outcomes as an infinite stream. Then, it may be required to compare two infinite streams and decide which one of them is better (for example, in order to decide on a policy). The overtaking criterion is one option to do this comparison.
Notation
Cardinal definition
An alternative condition is:
Examples:
1. In the following example,
This shows that a difference in a single time period may affect the entire sequence.
2. In the following example,
The partial sums of
This also shows that the overtaking criterion cannot be represented by a single cardinal utility function. I.e, there is no real-valued function
Hence, there is a set of disjoint nonempty segments in
Ordinal definition
Define
1. For every
2. For every
3. For each
4.
Every partial order that satisfies these axioms, also satisfies the first cardinal definition.
As explained above, some sequences may be incomparable by the overtaking criterion. This is why the overtaking criterion is defined as a partial ordering on
Applications
The overtaking criterion is used in economic growth theory.
It is also used in repeated games theory, as an alternative to the limit-of-means criterion and the discounted-sum criterion. See Folk theorem (game theory)#Overtaking.