Variation Principles in Models of Economic Equilibrium
There are two approaches to modeling the price structure in mathematical economics. On the one hand, the Lagrange multipliers are conventionally interpreted as prices in optimization problems, in which the constraints represent balance relations describin
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Abstract. There are two approaches to modeling the price structure in mathematical economics. On the one hand, the Lagrange multipliers are conventionally interpreted as prices in optimization problems, in which the constraints represent balance relations describing the production capacity of the economic system. In case of linear optimization models of interindustry balance, prices are determined by the solution of a dual linear programming problem. On the other hand, prices in economic equilibrium models are determined from the equilibrium between supply and demand. We suggest variation principles for economic equilibrium models and a scheme for the construction of dual problems to determine equilibrium prices. These principles are formulated in the form of variational inequalities (Aubin 1983). If the total consumer demand functions satisfy the Frobenius integrability conditions and obey the Hicks law, then the variation principles degenerate into an ordinary pair of dual optimization problems corresponding to the first approach to modeling the price structure. Keywords: comptitive equilibrium; Arrow-Debreu model, variational principle, variational inequality, dual optimization problem.
1
Preliminary considerations
Consider the Arrow-Debreu competitive equilibrium model. Assume that the economy involves m commodity types, n consumers, and k producers. The production capacity of the jth producer (j = 1,2, ... , k) is described by the production set Dj C]RID that contains O. The producers maximize their profits 71'j(P) = maxpx, where x is the input-output vector of the jth producer, p E ZED;
the price vector for the commodities under consideration, and 7rj(P) is the profit function of the j-th producer. The supply of commodities on the market is defined by the set-valued mapping ]R~\{O} is
~(P)
where D
11
= E Dj j=l
= argmax {px Ix ED}
,
is the total production set of the producers.
* This work was supported by the Russian Foundation for Basic Research (project No. 99-01-01238). A. S. Tangian et al. (eds.), Constructing and Applying Objective Functions © Springer-Verlag Berlin Heidelberg 2002
454
A. Shananin
The ith consumer (i = 1,2, ... , n) is assumed to have the initial vector of commodities wi E intJR+ and receives a fraction (Xii ~ a of the profit of the jth producer, where
n
L
(Xii
i=l
= 1, (j = 1, ... , k).
The preferences of the ith
consumer are specified by the utility function Ui (x), which is defined on JR+, is continuous, concave, and monotonically nondecreasing on JR+, and has no maxima. The demand of the ith consumer on the market is described by the set-valued mapping
The aggregate consumer demand is defined by the demand function cP(p) n
L CPi(P).
Obviously, cp(..\p)
i=l
= cP(P) for all ..\ > a,p E JR+.
=
The aggregate supply
is formed from the supplies of the producers and from the initial endowments of the consumers and is described by the set-valued mapping 'ljJ(P) = ~(P) + n
L
i=l
wi . The equilibrium price vector is a vector
('ljJ(P) - JR+)
We set E
# 0. n
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