Subaddivity

Abstract

In the economics literature ‘subadditivity’ is a mathematical representation of the concept of natural monopoly. An industry is a natural monopoly if total output can be produced at lower cost by a single firm than by any collection of two or more firms. If all potentially active firms in the industry have access to the same technology, which is represented by a cost function c, then at aggregate output x, the industry is a natural monopoly if c(x) ≤ c(x¹) + ⋯ + c(x′) for any set of outputs x¹,…,x^t such that

$$ \sum_{i=1}^t{x}^i=x. $$

A cost function c is globally subadditive if for any non-negative output vectors x and y,

$$ c\left({x}_1+{y}_1,\dots, {x}_n+{y}_n\right)\le c\left({x}_1,\dots, {x}_n\right)+c\left({y}_1,\dots, {y}_n\right). $$

In the production of a set N = {1,…,n} of indivisible objects the cost function is subadditive if c(S ∪ T) ≤ c(S) + c(T) for any disjoint subsets S and R. While this ‘economic’ definition of subadditivity is intuitively appealing, it is generally not obvious whether or not a particular cost function is subadditive. It is therefore of interest to determine both necessary and sufficient conditions for subadditivity in order to formulate empirical tests for natural monopoly. Subadditivity is closely associated with the concepts of ‘economies of scale’ and ‘economies of scope’. A cost function exhibits economies of scale if c(λx) ≤ λc(x) for 1 ≤ λ ≤ 1 + ∈, for small positive ε. A cost function exhibits economies of scope if the subadditivity condition is applied only for orthogonal output vectors. For example, the cost function c(x₁, x₂) = 1 + (x₁ + x₂)² + (x₁x₂)^1/2 exhibits economies of scale whenever x₁ + x₂ ≤ 1, economies of scope whenever x₁x₂ ≤ 1/4, and is subadditive whenever x₁ + x₂ ≤ 2 and x₁x₂ ≤ 1/4. While economies of scope are clearly necessary for subadditivity, economies of scale are neither necessary nor sufficient for subadditivity of a function of two or more variables. Therefore a valid empirical test for multiproduct natural monopoly, based on subadditivity, should not depend entirely on a test for economies of scale.