Subadditivity

Definitions

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

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

A sequence { a n } , n ≥ 1 , is called subadditive if it satisfies the inequality

for all m and n. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers.

Sequences

A useful result pertaining to subadditive sequences is the following lemma due to Michael Fekete.

Fekete's Subadditive Lemma: For every subadditive sequence { a n } n = 1 ∞ , the limit lim n → ∞ a n n exists and is equal to inf a n n . (The limit may be − ∞ .)

The analogue of Fekete's lemma holds for superadditive functions as well, that is: a n + m ≥ a n + 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 the inequality (1) to hold for all m and n, but only for m and n such that 1 2 ≤ m n ≤ 2. Moreover, the condition a n + m ≤ a n + a m may be weaken as follows: a n + m ≤ a n + a m + ϕ ( n + m ) provided that ϕ is an increasing function such that the integral ∫ ϕ ( t ) t − 2 d t converges (near the infinity).

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.

Functions

Theorem: For every measurable subadditive function f : ( 0 , ∞ ) → R , the limit lim t → ∞ f ( t ) t exists and is equal to inf t > 0 f ( t ) t . (The limit may be − ∞ . )

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 ) ≥ f ( x + y ) − f ( y ) . Hence f ( 0 ) ≥ f ( 0 + y ) − f ( y ) = 0

A concave function f : [ 0 , ∞ ) → [ 0 , ∞ ) with f ( 0 ) ≥ 0 is also subadditive. To see this, one first observes that f ( x ) ≥ y x + y f ( 0 ) + x x + y f ( x + y ) . Then looking at the sum of this bound for f ( x ) and f ( y ) , will finally verify that f is subadditive.

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.

Except in the case of complementary goods, 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. Thus proving that it is not a sufficient condition for a natural monopoly; since the unit of exchange may not be the actual cost of an item. This situation is familiar to everyone in the political arena where some minority asserts that the loss of some particular freedom at some particular level of government means that many governments are better; whereas the majority assert that there is some other correct unit of cost.

Thermodynamics

Subadditivity occurs in the thermodynamic properties of non-ideal solutions and mixtures like the excess molar volume and heat of mixing or excess enthalpy.