In economics, the overtaking criterion is used to compare infinite streams of outcomes. Mathematically, it is used to properly define a notion of optimality for a problem of optimal control on an unbounded time interval.
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.
formula_1 is the set of possible outcomes. E.g., it may be the set of positive real numbers, representing the possible annual gross domestic product. It is normalized
formula_2 is the set of infinite sequences of possible outcomes. Each element in formula_2 is of the form: formula_4.
formula_5 is a partial order. Given two infinite sequences formula_6, it is possible that formula_7 is weakly better (formula_8) or that formula_9 is weakly better (formula_10) or that they are incomparable.
formula_11 is the strict variant of formula_5, i.e., formula_13 if formula_14 and not formula_15.
Cardinal definition.
formula_11 is called the "overtaking criterion" if there is an infinite sequence of real-valued functions formula_17 such that:
formula_13      iff      formula_19
An alternative condition is:
formula_20      iff      formula_21
Examples:
1. In the following example, formula_13:
formula_23
formula_24
This shows that a difference in a single time period may affect the entire sequence.
2. In the following example, formula_7 and formula_9 are incomparable:
formula_27
formula_28
The partial sums of formula_7 are larger, then smaller, then equal to the partial sums of formula_9, so none of these sequences "overtakes" the other.
This also shows that the overtaking criterion cannot be represented by a single cardinal utility function. I.e, there is no real-valued function formula_31 such that formula_13 iff formula_33 and formula_34
Hence, there is a set of disjoint nonempty segments in formula_35 with a cardinality like the cardinality of formula_36. In contrast, every set of disjoint nonempty segments in formula_37 must be a countable set.
Ordinal definition.
Define formula_38 as the subset of formula_2 in which only the first "T" elements are nonzero. Each element of formula_38 is of the form formula_41.
formula_11 is called the "overtaking criterion" if it satisfies the following axioms:
1. For every formula_43, formula_5 is a complete order on formula_38
2. For every formula_43, formula_5 is a continuous relation in the obvious topology on formula_38.
3. For each formula_49, formula_38 is preferentially-independent (see Debreu theorems#Additivity of ordinal utility function for a definition). Also, for every formula_51, at least three of the factors in formula_38 are essential (have an effect on the preferences).
4. formula_13      iff      formula_54
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 formula_2, and a complete ordering only on formula_38.
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.
References.
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