Existence for Competitive Equilibrium by Means of Generalized Quasivariational Inequalities

and Applied Analysis 3 3. Equilibrium Model We consider a marketplace consisting of two types of agents: n consumers, indexed by a, and m producers, indexed by b. We denote with A = {1, . . . , n}, B = {1, . . . , m}, and J = {1, . . . , l}, respectively, sets of consumers, producers, and goods. We denote by e a and x a the nonnegative quantities of commodity j, respectively, owned and consumed by agent a. The vectors e a = (e 1 a , . . . , e l a ) ∈ R + and x a = (x 1 a , . . . , x l a ) ∈ R + represent, respectively, the initial endowment and the consumption of agent a and x = (x 1 , . . . , x n ) ∈ R represents the consumption of the market. For each a ∈ A we denote by I a the set of indexes corresponding to initial holdings, namely, I a = {j ∈ J : e j a > 0} and we assume that he is endowed with at least one positive commodity, then I a ̸ = 0. We denote by yj b the quantity of commodity j produced by producer b. We note that the commodity yj b can also assume negative values: the positive quantity yj b represents the commodity offered in the market by producer b, the negative quantity yj b represents the demand required by the market but not satisfied by producer b, and yj b equals zero which means that the producer b does not produce the commodity j. The vector y b = (y 1 b , . . . , y l b ) ∈ R and the matrix y = (y 1 , . . . , y m ) ∈ R represent the productions, respectively, of producer b and of the market. To each commodity j ∈ J is a fixed minimum price q such that 0 ≤ q < 1/l. More precisely, each commodity j has a positive price p, which we suppose p ≥ q for all j ∈ J. We denote by p = (p, . . . , p) ∈ R + the price vector and we suppose that prices belong to the set