# Poverty and Famines: An Essay on Entitlement and Deprivation

By Amartya Sen | Go to book overview

Appendix A
Exchange Entitlement

A.1
FIXED PRICE EXCHANGES

X is the non-negative orthant of n-dimensional real space, representing the amounts of n commodities; it is the set of all non-negative vectors of all commodities. γ is the power-set of X, i.e., the set of all subsets of X. Let x be the vector of commodities (including 'labour power') that the person owns, and p is the n-vector of prices faced by him.

Given his ownership vector x, his exchange entitlement set E(x) is the set of vectors any one of which he can acquire by exchanging x.

 (A1) E (x) = {y∣y ∈ X & py ≤ p x}.]

The function E (.) from X to γ is his 'exchange entitlement mapping', or E-mapping, for short.

Two explanatory points. First, clearly x E(x). Second, the exchanges covered by (A1) are not, of course, confined to selling all of x, and a part of it can be retained (since this will not affect the exchangepossibility of the remainder, as given by (A1)

Let the set of commodity vectors that satisfy the specified minimum food requirement be given by FX. Starvation must occur, in the absence of non-entitlement transfers (such as looting), if E(x) F = ϕ. The 'starvation set' S of ownership vectors consists of those vectors x in X such that the exchange entitlement set E(X) contains no vector satisfying the minimum food requirements. Obviously, S depends on F and the E-mapping.

 (A2) S = {x∣x ∈ X & E(x) ∈ F = ϕ}.]

To illustrate consider a simple two-commodity case with commodity I standing for food, and let OA in Figure A1 represent the minimum food requirement. The price ratio is given by p. The starvation set S is given by the region OAB.

More generally, when food is not one commodity but many and the 'food requirements' can be met in many different ways, let the minimum cost of meeting the food requirements, i.e. for attaining any vector in F, be m(p, F).

 (A3) m(p, F) == min px∣x ∈ F.

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