Economic equilibrium theorems of Shafer-Sonnenschein version and nonempty intersection theorems in interval spaces
The main concerns of this paper are (I) the existence of equilibria in an abstract economy, and (II) the nonempty intersection theorem for sets with convex sections. Usually both problems heavily depend upon the concept of convexity. The authors try to extend known results to the case formulated in topological spaces without linear structure making use of a few newly designed concepts such as interval space (a topological space in which something analogous to a segment in a vector space is defined, so that a \(W\)-convex set in an interval space is analogous to a convex set in a linear space). (I) Let \(I\) be a set of agents, \(X_i\) a topological space (choice set), \(A_i\), \(B_i: \prod_{k\in I} X_k\to 2^{X_i}\) set- valued mappings which express certain constraints, and \(P_i: \prod_{k\in I} X_k\to 2^{X_i}\) a set-valued mapping which represents a preference of the agent \(i\) \((i\in I)\). A family \({\mathcal E}= \{(X_i, A_i, B_i, P_i), i\in I\}\) is called an abstract economy. Finally, a point \(x^*\in X\equiv \prod_{i\in I} X_i\) is called an equilibrium of \(\mathcal E\) if \(x^*_i\in B_i(x^*)\) and \(P_i(x^*)\cap A_i(x^*_i)= \emptyset\) for each \(i\in I\). Then we can state the first problem: under what assumptions does an equilibrium exist? Several versions of the full answer to this question in the case each \(X_i\) is an interval space are given. (II) Let \(\{X_i\mid i\in I\}\) be a family of Hausdorff compact interval spaces and let \(X= \prod_{i\in I} X_i\) be strongly Dedekind complete (new concept to be defined). We denote \(X_{- i}= \prod_{k\neq i} X_k\). For any \(x\in X\), we also use the notations \(x_{- i}= \text{proj}_{x_{-i}}x\), \(x_i= \text{proj}_{x_i} x\), and \(x= x_i\otimes x_{-i}\). Given \(A\subset X\), \(x_i\in X_i\) and \(y_{-i}\in X_{-i}\), the sets \(A(x_i)\) and \(A(y_{-i})\) are defined by \(A(x_i)= \{x_{- i}\in X_{-i}\mid x_i\otimes x_{- i}\in A\}\) and \(A(y_{-i})= \{y_i\in X_i\mid y_i\otimes y_{- i}\in A\}\). Let \(\{A_i\mid i\in I\}\) and \(\{B_i\mid i\in I\}\) be families of subsets in \(X\) which satisfy the following conditions: (i) the set \(\{x\in X\mid y_i\in W\text{-}\overline{\text{co}}A_i(x_{- i})\}\) is closed for each \(i\in I\) and \(y_i\in X_i\); (ii) \(A_i(x_{- i})\neq \emptyset\) and \(W\text{-}\overline{\text{co}}A_i(x_{- i})\subset B_i(x_{-i})\) for any \(i\in I\) and \(x_{-i}\in X_{- i}\); (iii) for any \(y\in X\), \(x_1,x_2\in X\) and \(x\in [x_1,x_2]\backslash \{x_1, x_2\}\), there exists an \(i_0\in I\) such that \(y_{i_0}\not\in W\text{-}\overline{\text{co}} A_{i_0}(x_{- i_0})\). Then \(\bigcap_{i\in I} B_i\neq \emptyset\). Here \(W\text{-}\overline{\text{co}}\) (resp. \([x_1, x_2]\)) denotes the generalized closed convex hull (resp. the generalized interval joining \(x_1\) and \(x_2\)) in an interval space.
existence of equilibria in an abstract economy
scientific article; zbMATH DE number 743899