Expected Utility for an n-member Market

For each endowment realization, a Walrasian equilibrium consists of a price–allocation pair $〈p=(p1,p2),{x=(x1i,x2i)}i−1n〉$ for which (1) preferences are maximized and (2) markets clear. More formally,

For a given endowment realization there are basically two types of agents in this economy: those with endowment (1, 0) and those with endowment (0, 1). Let n₁ and n₂ be the number of individuals with endowment vectors (1, 0) and (0, 1), respectively (n₁ + n₂ = n). Let $x¯=(x¯1,x¯2)$ denote the consumption of (1, 0)-individuals and $x¯=(x¯1,x¯2)$ represent the bundle consumed by (0, 1) individuals. The two conditions above can then be translated into: