Restricted Participation on Financial Markets: A General Equilibrium Approach Using Variational Inequality Methods

Abstract

We deal with the analysis of a general equilibrium model with restricted participation in financial markets and with numeraire assets. We consider an exchange economy and assume that there are two periods of time and S possible states of nature in the second period. Markets may in principle be complete, but each household has her own specific restricted way to access to it. In particular, we assume that households are allowed to choose portfolios in a closed and convex set containing zero. Our main goal in this work is to provide a proof of existence of equilibria under relatively general assumptions, by assuming that the households may have non-complete or non-transitive preferences, and by using a variational inequality approach. More precisely, we introduce a sequence of generalized quasi-variational inequalities and we show that an associated sequence of solutions converges to an equilibrium.

Appendix

Definition A.1

Let A be a nonempty convex set in $\mathbb {R} ^{n}$. The recession cone of A is denoted and defined as follows

$$ \text{rec}A=\left\{ y\in \mathbb{R}^{n}:\forall x^{0}\in A,\forall \lambda \geq 0,x^{0}+\lambda y\in A\right\} . $$

Proposition A.1

(see Soltan (2015)) Let A be a convex set in $\mathbb {R} ^{n}$. Then

• 1. if A is nonempty, closed and 0_n ∈ A,

  $$ \text{rec}A=\{z\in \mathbb{R}^{n}:\forall \lambda \geq 0,\lambda z\in A\}\subseteq A; $$

• 2. recA is a convex cone;

• 3. if A is closed, then recA is closed;

• 4. for any $a\in \mathbb {R}^{n}$ and $\mu \in \mathbb {R}$,

  $$ \text{rec}(a+\mu A)=\mu \text{rec}A=sgn(\mu )\text{rec}A. $$

Definition A.2

Let $C\subseteq \mathbb {R}^{n}$ be a nonempty, closed and convex set and let $S:C\rightrightarrows \mathbb {R}^{n}$ and $\varPhi :C\rightrightarrows \mathbb {R}^{n}$ be set-valued maps. A Generalized Quasi-Variational Inequality associated with C, S, Φ, denoted by GQVI, is the following problem:

$$ \text{Find } \overline{x}\in S\left( \overline{x}\right) \text{ such that there exists } \varphi \in \varPhi \left( \overline{x}\right) \text{ with }\left\langle \varphi ,x- \overline{x}\right\rangle \geq 0, \ \forall x\in S\left( \overline{x} \right). $$

(36)

In particular, when S(x) = C for any x ∈ C, (36) is a Generalized Variational Inequality, GVI; when Φ is single-valued, (36) reduces to the Quasi-Variational Inequality, QVI. When both Φ(x) is singleton and S(x) = C, for any x ∈ C, we have the classical Stampacchia Variational Inequality, VI.

Theorem 4

(seeTan (1985)) Let C be a nonempty, convex and compact subset of $\mathbb {R}^{n}$. Let $\varPhi :C\rightrightarrows \mathbb {R}^{n}$ and $S:C\rightrightarrows C$ be two set-valued maps satisfying the following properties:

• (i) S is closed, lower semicontinuous and with nonempty, convex and compact values;

• (ii) Φ is upper semicontinuous with nonempty, convex and compact values.

Then the GQVI (36) admits at least a solution.

The above theorem, which establishes the existence of a solution for variational inequalities, is a consequence of a Kakutani’s fixed point Theorem.

$$ N^{>}(x):=\{h\in \mathbb{R}^{n}:\langle h,z-x\rangle \leq 0\quad \forall z\in P(x)\}. $$

Let us define $\mathcal {\widetilde {G}}:\mathbb {R}^{n}\rightrightarrows \mathbb {R}^{n}$ such that

$$ \mathcal{\widetilde{G}}(x):=\left\{ \begin{array}{ll} \overline{\mathrm{B}}(0,1) & if\ P(x)=\emptyset \\ & \\ conv\left( N^{>}(x)\cap S(0,1)\right) & if\ P(x)\neq\emptyset \end{array} \right. $$

(37)

where $\overline {\mathrm {B}}(0,1)=\{x\in \mathbb {R}^{n}:\Vert x\Vert \leq 1\}$ and $S(0,1)=\{x\in \mathbb {R}^{n}:\Vert x\Vert = 1\}$ are, respectively, the closed unit ball and the unit sphere of $\mathbb {R}^{n}$.

Theorem 5

Let P be lower semicontinuous with open and convex valued. Then the set-valued map $\mathcal {\widetilde {G}}$, defined in Eq. 37 is with nonempty and convex and compact values, and upper semicontinuous.

Proof

Firstly, we prove that N^> is a closed map. Let $\{x_{n}\}\subseteq \mathbb {R}^{n}$ and $\{y_{n}\}\subseteq \mathbb {R}^{n}$ be sequences such that $\lim\limits _{n\rightarrow +\infty }x_{n}=x$ and $\lim\limits _{n\rightarrow +\infty }y_{n}=y$ with y_n ∈ N^>(x_n). We have to prove that y ∈ N^>(x), that is 〈y, z − x_h〉≤ 0 ∀z ∈ P(x). We fix z ∈ P(x_h) and since P is lower semicontinuous, there exists {z_n} such that z_n ∈ P(x_n) with $\lim\limits _{n\rightarrow +\infty }z_{n}=z$. From z_n ∈ P(x_{h, n}) and y_n ∈ N^>(x_n), it follows 〈y_n,z_n − x_n〉≤ 0. Passing to the limits 〈y, z − x〉≤ 0, so y ∈ N^>(x). All Properties follows from the closedness of N^> and the proof of Theorem 3.2 in Milasi et al. (2019). □

Definition A.3

Let C be a convex set of $\mathbb {R}^{n}$, call (GVI) the following problem:

$$ \text{\textquotedblleft\ Find}\widetilde{x}\in C\text{such that} \exists \ g\in \mathcal{G}(\widetilde{x})\text{with}\langle g,x- \widetilde{x}\rangle \geq 0\quad \forall x\in C\text{.\ \textquotedblright } $$

(38)