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.

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 |