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On an approach to the determination of equilibrium in elementary exchange models. (English. Russian original) Zbl 0549.90014
Sov. Math., Dokl. 27, 230-233 (1983); translation from Dokl. Akad. Nauk SSSR, 268, 1062-1066 (1983).
Let $$I=\{1,..,i,...,m\}$$ and $$J=\{1,...,j,...,n\}$$ be two finite sets, and for each $$i\in I$$, $$c^ i\in {\mathbb{R}}^ n_+$$ and $$d^ i\in {\mathbb{R}}^ n_+$$ be two positive vectors, with $$\sum_{i\in I}d^ i=(1,...,1)=v$$. Given $$p\in\sigma =\{p\in {\mathbb{R}}^ n$$; $$p\geq 0$$, $$<p,v>=1\}$$, consider for $$i\in I$$, the program: $$Opt<c^ i,x^ i>$$ subject to $$<p,x^ i>=<p,d^ i>$$, $$x^ i\geq 0$$. Interpreting I as a set of participants, J as a set of products, when Opt is Min, we have a (linear) exchange model and when Opt is Max, it is a cooperative model. A model is said to be with set budgets if $$d^ i=\alpha_ iv$$, $$\alpha_ i>0$$, for all $$i\in I$$; and with variable budgets if not. An equilibrium price vector is $$p\in\sigma$$ such that there exists $$\tilde x{}^ i$$, $$i\in I$$, an optimal solution of the participants problems such that $$\sum_{i\in I}\tilde x^ i=\sum_{i\in I}d^ i$$. The author studies algorithms to determine equilibrium prices in the two models.
One may associate to any model (exchange or cooperation) a parametric transport problem, namely for $$p\in\sigma :$$
(1) Op$$t\sum_{i\in I}\sum_{j\in J}z_{ij}\ln c^ i_ j$$ subject to (2) $$\sum_{j\in J}z_{ij}=<p,d^ i>$$, $$i\in I$$
(3) $$\sum_{i\in I}z_{ij}=p_ i$$, $$j\in J$$, (4) $$z_{ij}\geq 0$$, $$i\in I$$, $$j\in J,$$
(with a condition ensuring that the problem is dually non-degenerate). Let, for such a transport problem, $${\mathfrak B}$$ be the collection of dually admissible basic sets in $$I\times J$$ and all possible i-covering subsets of it (i.e. for $$B\in {\mathfrak B}$$, $$\{$$ $$j\in J$$; (i,j)$$\in B\}\neq\emptyset )$$. To each $$B\in {\mathfrak B}$$ we associate polyhedral triangulations of $$\sigma$$ and $$\sigma^ 0$$ (its relative interior) which are respectively: $$\Omega (B)=\{p\in\sigma$$; there exist $$z_{ij}=z_{ij}(p)$$, (i,j)$$\in I\times J$$, solution of (2)-(4) satisfying $$z_{ij}=0$$, (i,j)$$\not\in B\}$$, $$\Xi (B)=\{p\in\sigma^ 0$$; $$_{k\in J}\frac{c^ k_ k}{p_ k}=\frac{c^ i_ j}{p_ j}$$, for every (i,j)$$\in B\}$$. Equilibrium price vectors of the original model are the fixed points of the multi-valued mapping F on $$\sigma$$ defined by letting for each $$B\in {\mathcal B}$$, and for $$p\in\Omega^ 0(B)$$, $$F(p)=\Xi (B)$$. For the cooperation model with set budgets a procedure following adjacent vertices and a procedure by iteration give equilibrium prices. The proof is based upon the observation that the function over $$\sigma$$ defined by $$\psi (q)=_{p\in\sigma }\{-<p,\ln q>+f(p)\}$$ is strictly increasing. For an exchange model with set budgets, an equilibrium gives a minimum of $$\psi$$ on $$\sigma^ 0$$. Two algorithms are formulated on this basis. Sufficient conditions for an extension to exchange models with variables sets are studied.
Reviewer: L.-A.Gerard-Varet

##### MSC:
 91B50 General equilibrium theory 90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.) 52Bxx Polytopes and polyhedra 54C60 Set-valued maps in general topology