Subadditivity

In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive functions are special cases of subadditive functions.

1 Definitions
2 Properties
3 Economics
4 See also
5 Notes
6 References
7 External links

Definitions

A subadditive function is a function $f \colon A \to B$ , having a domain A and an ordered codomain B that are both closed under addition, with the following property:

$\forall x, y \in A, f(x%2By)\leq f(x)%2Bf(y)$ .

An example is the square root function, having the non-negative real numbers as domain and codomain, since $\forall x, y \geq 0$ we have:

$\sqrt{x%2By}\leq \sqrt{x}%2B\sqrt{y}.$

A sequence $\left \{ a_n \right \}, n \geq 1$ , is called subadditive if it satisfies the inequality

$(1) \qquad a_{n%2Bm}\leq a_n%2Ba_m$

for all m and n.

Properties

A useful result pertaining to subadditive sequences is the following lemma due to Michael Fekete.^[1]

Fekete's Subadditive Lemma: For every subadditive sequence ${\left \{ a_n \right \}}_{n=1}^\infty$ , the limit $\lim_{n \to \infty} \frac{a_n}{n}$ exists and is equal to $\inf \frac{a_n}{n}$ . (The limit may be $-\infty$ .)

The analogue of Fekete's lemma holds for superadditive functions as well, that is: $a_{n%2Bm}\geq a_n %2B a_m.$ (The limit then may be positive infinity: consider the sequence $a_n = \log n!$ .)

There are extensions of Fekete's lemma that do not require equation (1) to hold for all m and n. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present.^[2]

If f is a subadditive function, and if 0 is in its domain, then f(0) ≥ 0. To see this, take the inequality at the top. $f(x) \ge f(x%2By) - f(y)$ . Hence $f(0) \ge f(0%2By) - f(y) = 0$

A concave function $f: [0,\infty) \to [0,\infty)$ with f(0) = 0 is also subadditive. To see this, one first observes that $f(x) \ge 0 f(0) %2B \textstyle{\frac{x}{x%2By}} f(x%2By)$ . A properly weighted sum of this bound for $f(x)$ and $f(y)$ shows that f is subadditive.^[3]

The negative of a subadditive function is superadditive.

Economics

Subadditivity is an essential property of some particular cost functions. It is, generally, a necessary and sufficient condition for the verification of a natural monopoly. It implies that production from only one firm is socially less expensive (in terms of average costs) than production of a fraction of the original quantity by an equal number of firms.

Economies of scale are represented by subadditive average cost functions.

In general, the price of goods (as a function of quantity) must be subadditive. Otherwise, if the sum of the cost of two items is cheaper than the cost of the bundle of two of them together, then nobody would ever buy the bundle, effectively causing the price of the bundle to "become" the sum of the prices of the two separate items.

Notes

^ Fekete, M. "Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit. ganzzahligen Koeffizienten." Mathematische Zeitschrift 17 (1923), pp. 228–249.
^ Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997). ISBN 0-89871-380-3.
^ Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press. ISBN 126227608. , p.314,12.25

References

György Pólya and Gábor Szegő. "Problems and theorems in analysis, volume 1". Springer-Verlag, New York (1976). ISBN 0-387-05672-6.

External links

This article incorporates material from subadditivity on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.