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Variational Inequalities and General Equilibrium Models

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Mathematical Analysis in Interdisciplinary Research

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 179))

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Abstract

We deal with the study of several general equilibrium models by using the variational inequality theory. The theory of variational inequalities was introduced in the sixties of the past century by Fichera (1964), and Lions and Stampacchia (1965), as an innovative and effective method to solve equilibrium problems arising in mathematical physics. Afterward this theory turned out as a powerful tool, and it was used to analyze different kinds of equilibrium problems.

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Notes

  1. 1.

    Recall that \(\mathcal {M}_{S, A}\) is the set of all S × A dimensional matrices of real numbers.

  2. 2.

    Following standard notation, for vectors \(y:=\left ( y_{i}\right ) _{i=1}^{n}, z:=\left ( z_{i}\right ) _{i=1}^{n}\in \mathbb {R}^{n}\); y ≥ z means that for \(i\in \left \{ 1, \ldots , n\right \}, \) y i ≥ z i; y >> z means that for \(i\in \left \{ 1, \ldots , n\right \}, \) y i > z i, and y > z means that y ≥ z but y ≠ z.

  3. 3.

    In the symbol Qu, the superscript u stays for “unrestricted.”

  4. 4.

    For vectors \(y, z\in \mathbb {R}^{n}\), y ≥ z means that for i = 1, …, n, y i ≥ z i; y >> z means that for i = 1, …, n, y i > z i, and y > z means that y ≥ z but y ≠ z.

  5. 5.

    rec B h is the recession cone of B h, which is defined as follows \(\text{rec}B_h=\left \{ y\in \mathbb {R}^{A}:\forall x^{0}\in B_h, \forall \lambda \geq 0, x^{0}+\lambda y\in B_h\right \}.\)

  6. 6.

    \(\mathcal {F}\) stays for financial structure.

  7. 7.

    Recall that, from Remark 5, if \(\left ( p^{sC}\right ) _{s\in \mathcal {S}}\in \mathbb {R}_{++}^{S}\), then Q h(Y, B h) = Q h(P, Y, B h).

  8. 8.

    Observe that \(\frac {1}{C}\geq \frac {1}{n}\) since by assumption n ≥ C.

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Donato, M.B., Maugeri, A., Milasi, M., Villanacci, A. (2021). Variational Inequalities and General Equilibrium Models. In: Parasidis, I.N., Providas, E., Rassias, T.M. (eds) Mathematical Analysis in Interdisciplinary Research. Springer Optimization and Its Applications, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-030-84721-0_11

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