Some applications of set-valued mappings in mathematical economics
By a preference on a separable metrizable space X we mean any set-valued mapping R: \(X\to 2^ X\) (y is said to be preferred to x iff \(y\in R(x))\). A real-valued function u on X is said to be a utility function for R if \(y\in R(x)\Rightarrow u(y)\geq u(x)\) and \(y\in R(x)\), \(x\not\in R(y)\Rightarrow u(y)>u(x)\). A characterization of preferences that admit d-Lipschitz utility functions is given, where d is a metric on X. Also characterized are choice functions that can be rationalized by d- Lipschitz or continuous utility functions. The asymptotic behavior of a dynamic system determined by R is another subject studied in the paper. The trajectories of such a system are sequences \(\chi =(\chi (t))^{\infty}_{t=0}\) with \(\chi\) (t)\(\in R(\chi (t-1))\), \(t=1,2,...\), where R is supposed to be non-void-compact- valued. Let H(R) denote the set of continuous isotone functions on X, \[ W_ u:=\{x:\;u(x)=\min_{y\in R(x)}u(y)\},\text{ and } W:=\cap_{u\in H(R)}W_ u. \] The notion of a d-quasi-periodic point is defined and the set W is characterized by the equality \(W=\cap_{d\in D}Q_ d\), where \(Q_ d\) is the set of all d-quasi-periodic points, and D is the set of all bounded metrics compatible with the topology of X. Yet another characterization of W as a global structure of the dynamic system is given.
scientific article; zbMATH DE number 4176769
scientific article