Optimality and the theory of value
Let \(X_i\subset \mathbb{R}^q\) be the consumption set of the consumer \(i\) \((= 1,2,\dots, m)\). The complete and transitive preference relation \(\lesssim_i\) of the consumer \(i\) is represented by a continuous utility function \(u_i: X_i\to \mathbb{R}\). \(Y\) denotes the total production set of the economy. A pair \((x^*, p^*)\in \left(\prod^m_{i= 1} X_i\right)\times (\mathbb{R}^q\backslash \{0\})\) is said to be an equilibrium of the economy if (a) \(\sum^m_{i= 1} x_i\in Y\), (b) \(x_i\in X_i\), \(u_i(x_i)> u_i(x^*_i)\Rightarrow p^* x_i> p^* x^*_i\) for all \(i\), (c) \(p^*\cdot \sum^m_{i= 1} x^*_i\geq p^*\cdot y\) for all \(y\in Y\). Define a family of functions \(b_i: \mathbb{R}^q_+\times X_i\times u_i(X_i)\to \mathbb{R}\cup \{- \infty\}\), \(e_i: \mathbb{R}^q\times u_i(X_i)\to \mathbb{R}\) \((i= 1,2,\dots, m)\) by \(b_i(g, x_i, v_i)= \sup\{\beta\mid x_i- \beta g\in X_i, u_i(x_i- \beta g)\geq v_i\}\) and \(e_i(p, v_i)= \inf\{p\cdot x_i\mid u_i(x_i)\geq v_i, x_i\in X_i\}\). We also define: \(B(g, x, v)= \sum^m_{i= 1} b_i(g, x_i, v_i)\), \(S(p, v)= \pi(p)- \sum^m_{i= 1} e_i(p, v_i)\), where \(\pi(p)= \sup\{p\cdot y\mid y\in Y\}\), \(x= (x_1, x_2,\dots, x_m)\), \(v= (v_1, v_2,\dots, v_m)\). Roughly speeking the equilibrium can be completely characterized by the following relations under some assumptions: \((x^*, p^*)\) is an equilibrium \(\Leftrightarrow \text{Max}\left\{B(g, x, v^*)\mid x\in \prod^m_{i= 1} X_i,\;\sum^m_{i= 1} x_i\in Y\right\}= B(g, x^*, v^*)= 0\) (\(g\) is chosen suitably), \(\Leftrightarrow \text{Min}\{S(p, v^*)\mid p\geq 0, p\cdot g= 1\}= S(p^*, v^*)= 0\), where \(v^*= (u_1(x^*_1),\dots, u_m(x^*_m))\). Making use of this characterization the author obtains an alternative proof for the existence theorem of general equilibria of the Arrow-Debreu private ownership economy. This proof provides us with a new interpretation of the role of price mechanism and suggests a way of computation of equilibria.
scientific article; zbMATH DE number 633441
scientific article