Walras and dividends equilibrium with possibly satiated consumers
The authors study an economy \({\mathcal E} = \{(X_i, u_i,e_i)_{i\in I}\}\) with a finite number \(l\) of goods and a finite number \(I\) of consumers, where \(X_i\in \mathbb{R}^l\) is the set of consumption goods, \(u_i: X_i \rightarrow \mathbb{R}\) is the utility function and \(e_i\in X_i\) is the \textit{initial endowment} of consumer \(i\in I\). The following four assumptions are considered: (H1) for \(i\in I\), \(X_i\) is closed and convex, (H2) for \(i\in I\), \(u_i\) is strictly quasi-concave and upper semicontinuous, (H3) the utility set \({\mathcal U}\) is compact, and (H4) for \(i\in I\) \(\exists x_i^{'}\in {\mathcal A}^c_i\) s.t. \(u(x_i{'}) \geq \max \{u_i(x_i)| x_i\in {\mathcal A}_i\}\) (here \({\mathcal A}_i\) is the projection of the set \({\mathcal A} = \{(x_i)_{i\in I}| \sum_{i\in I}x_i = \sum_{i\in I}e_i\) and \(u(x_i) \geq u_i(e_i)\}\) on \(X_i\), and \({\mathcal U} = \{(v_i)_{i\in I}| v_i\in R_{+}\) and \(\forall i\in I\;\; \exists x\in {\mathcal A}\) s.t. \(u(e_i)\leq v_i \leq u_i(x_i)\}\). Just the authors' contribution is (H4) as a weaker nonsatiation assumption than the one commonly used in the literature. It is shown in the paper that under (H1)-(H3) or under (H1)-(H2) with the assumption about the compactness of all the sets \(X_i\), there exists a quasi-equilibrium with dividends. The third found result says that assumptions (H1)-(H4) imply the existence of a Walras quasi-equilibrium. Next, it is also shown that the last result remains true when assumptions (H3)-(H4) are replaced by three other ones, called uniformity, Werner's nonsatiation and the third one saying about the existence of a price system at which the value of all useful portfolios is positive.
scientific article; zbMATH DE number 5300736