From Assumption (H1’) (d), let us fix ${\underset{\xaf}{x}}_{i}\in {X}_{i}$ and ${\underset{\xaf}{y}}_{i,j}\in {Y}_{j}^{\prime}$ such that

${\underset{\xaf}{x}}_{i}={\omega}_{i}+\sum _{j\in J}{\theta}_{i,j}{\underset{\xaf}{y}}_{i,j}$

for every $i\in I$. Let ${\overline{B}}^{\nu}$ be the closed ball with center 0 and radius ν with ν large enough so that ω, ${\underset{\xaf}{x}}_{i}$, ${\underset{\xaf}{y}}_{i,j}$ and ${\omega}_{i}$ belong to ${B}^{\nu}$, the interior of ${\overline{B}}^{\nu}$, for all $i,j$. We consider the truncated economy obtained by replacing agent’s consumption sets by ${X}_{i}^{\nu}={X}_{i}\cap {\overline{B}}^{\nu}$ for all $i\ne {i}_{0}$, and

${X}_{{i}_{0}}^{\nu}={X}_{{i}_{0}}\cap {\overline{B}}^{(\mathrm{\u266f}I+\mathrm{\u266f}J)\nu}.$

The production set becomes ${Y}_{j}^{\nu}={Y}_{j}^{\prime}\cap {\overline{B}}^{\nu}$ and the augmented preference correspondences are ${\widehat{P}}_{i}^{\nu}={\widehat{P}}_{i}\cap {B}^{\nu}$ for $i\ne {i}_{0}$ and

${\widehat{P}}_{{i}_{0}}^{\nu}={\widehat{P}}_{{i}_{0}}\cap {B}^{(\mathrm{\u266f}I+\mathrm{\u266f}J)\nu}.$

The closed unit ball $\overline{B}=\{x\in {\mathbb{R}}^{L}:\parallel x\parallel \le 1\}$ will be the price set. The truncation of ${X}_{{i}_{0}}$ is chosen in such a way that if $(x,y)\in {\prod}_{i\in I}{X}_{i}^{\nu}\times {\prod}_{j\in J}{Y}_{j}^{\nu}$ is feasible, that is, ${\sum}_{i\in I}{x}_{i}=\omega +{\sum}_{j\in J}{y}_{j}$, then ${x}_{{i}_{0}}$ belongs to the open ball ${B}^{(\mathrm{\u266f}I+\mathrm{\u266f}J)\nu}$.

We now consider the economy

${\mathcal{\mathcal{E}}}^{\nu}=({\mathbb{R}}^{L},{({X}_{i}^{\nu},{\widehat{P}}_{i}^{\nu},{\omega}_{i})}_{i\in I},{({Y}_{j}^{\nu})}_{j\in J},{({\theta}_{i,j})}_{(i\in I,j\in J)}),$

where the consumption and production sets are compact.

Since each ${Y}_{j}^{\nu}$ is compact, we can define for every $p\in \overline{B}$ the profit function

${\pi}_{j}^{\nu}(p)=supp.{Y}_{j}^{\nu}=sup\{p.{y}_{j}:{y}_{j}\in {Y}_{j}^{\nu}\},$

and the wealth of consumer *i* is defined by

${\gamma}_{i}^{\nu}(p)=p.{\omega}_{i}+\sum _{j\in J}{\theta}_{ij}{\pi}_{j}^{\nu}(p).$

Note that the function ${\pi}_{j}^{\nu}:\overline{B}\to \mathbb{R}$ is continuous since it is finite and convex.

In what follows, we will use the following notations for simplicity

${Z}^{\nu}={\displaystyle \prod _{i\in I}}{X}_{i}^{\nu}\times {\displaystyle \prod _{j\in J}}{Y}_{j}^{\nu}\times \overline{B}\hspace{1em}\text{and}z=(x,y,p)\text{denotes a typical element of}{Z}^{\nu},$${\widehat{\gamma}}_{i}^{\nu}(z)={\gamma}_{i}^{\nu}(p)+{\displaystyle \frac{1-\parallel p\parallel}{\mathrm{\u266f}I}},$${\stackrel{~}{\gamma}}_{i}^{\nu}(z)=\mathrm{max}\{{\widehat{\gamma}}_{i}^{\nu}(z),{\displaystyle \frac{1}{2}}\left[{\widehat{\gamma}}_{i}^{\nu}(p)+p\cdot {x}_{i}\right]\}.$

Let now $N=I\cup J\cup \{0\}$ be the union of the set of consumers *I* indexed by *i*, the set of producers *J* indexed by *j*, and an additional agent 0 whose function is to react with prices to a given total excess demand.

For all $i\in I$, we define the correspondences ${\alpha}_{i}^{\nu}:{Z}^{\nu}\to {X}_{i}^{\nu}$ and ${\stackrel{~}{\beta}}_{i}^{\nu}:{Z}^{\nu}\to {X}_{i}^{\nu}$ as follows:

${\alpha}_{i}^{\nu}(z)=\{{\xi}_{i}\in {X}_{i}^{\nu}:p\cdot {\xi}_{i}\le {\widehat{\gamma}}_{i}^{\nu}(z)\},$${\stackrel{~}{{\beta}_{i}}}^{\nu}(z)=\{{\xi}_{i}\in {X}_{i}^{\nu}:p\cdot {\xi}_{i}<{\stackrel{~}{\gamma}}_{i}^{\nu}(z)\}.$

From the construction of the extended budget set, one checks that for all *i* the consumption ${\underset{\xaf}{x}}_{i}$ belongs to ${\stackrel{~}{{\beta}_{i}}}^{\nu}(z)$ if ${x}_{i}\notin {\alpha}_{i}^{\nu}(z)$. Indeed, from (H1’) (d),

${\underset{\xaf}{x}}_{i}={\omega}_{i}+\sum _{j\in J}{\theta}_{i,j}{\underset{\xaf}{y}}_{i,j}$

since ${x}_{i}\notin \alpha _{i}{}^{\nu}(z)$, $p\cdot {x}_{i}>{\widehat{\gamma}}_{i}^{\nu}(z)$ and ${\stackrel{~}{\gamma}}_{i}^{\nu}(z)>{\widehat{\gamma}}_{i}^{\nu}(z)$. Furthermore,

$p\cdot {\underset{\xaf}{x}}_{i}=p\cdot {\omega}_{i}+\sum _{j\in J}{\theta}_{i,j}p\cdot {\underset{\xaf}{y}}_{i,j}\le p\cdot {\omega}_{i}+\sum _{j\in J}{\theta}_{i,j}{\pi}_{j}^{\nu}(p)\le {\widehat{\gamma}}_{i}^{\nu}(z)<{\stackrel{~}{\gamma}}_{i}^{\nu}(z),$

which means that ${\underset{\xaf}{x}}_{i}$ belongs to ${\stackrel{~}{{\beta}_{i}}}^{\nu}(z)$. Moreover, since ${\stackrel{~}{\gamma}}_{i}^{\nu}$ is continuous, the correspondence ${\stackrel{~}{{\beta}_{i}}}^{\nu}$ has an open graph in ${Z}^{\nu}\times {X}_{i}^{\nu}$.

Now, for $i\in I$, we consider the mapping ${\varphi}_{i}^{\nu}$ defined from ${Z}^{\nu}$ to ${X}_{i}^{\nu}$ by

${\varphi}_{i}^{\nu}(z)=\{\begin{array}{cc}{\stackrel{~}{\beta}}_{i}^{\nu}(z)\hfill & \text{if}{x}_{i}\notin {\alpha}_{i}^{\nu}(z),\hfill \\ {\stackrel{~}{\beta}}_{i}^{\nu}(z)\cap {\widehat{P}}_{i}^{\nu}(x)\hfill & \text{if}{x}_{i}\in {\alpha}_{i}^{\nu}(z).\hfill \end{array}$

For $j\in J$, we define ${\varphi}_{j}^{\nu}$ from ${Z}^{\nu}$ to ${Y}_{j}^{\nu}$ by

${\varphi}_{j}^{\nu}(z)=\{{y}_{j}^{\prime}\in {Y}_{j}^{\nu}\mid p\cdot {y}_{j}<p\cdot {y}_{j}^{\prime}\},$

and the mapping ${\varphi}_{0}^{\nu}$ from ${Z}^{\nu}$ to $\overline{B}$ is defined by

${\varphi}_{0}^{\nu}(z)=\left\{q\in \overline{B}\right|(q-p)\cdot \left(\sum _{i\in I}{x}_{i}-\omega -\sum _{j\in J}{y}_{j}\right)>0\}.$

Now we will apply to ${Z}^{\nu}$ and the correspondences ${({\varphi}_{i})}_{i\in I}^{\nu}$, ${({\varphi}_{j})}_{j\in J}^{\nu}$ and ${\varphi}_{0}^{\nu}$ the well-known theorem of Gale and Mas-Colell [11]. We will actually use the Bergstrom version of this theorem in [3], which is more adapted to our setting.

#### Theorem 4.5 ([11, 3]).

*For each $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}\overline{k}$, let ${Z}_{k}$ be a non-empty, compact, convex subset of some finite-dimensional Euclidean space.
Given $Z\mathrm{=}{\mathrm{\prod}}_{k\mathrm{=}\mathrm{1}}^{\overline{k}}{Z}_{k}$, for each **k*, let ${\varphi}_{k}\mathrm{:}Z\mathrm{\to}{Z}_{k}$ be a lower semicontinuous correspondence satisfying ${z}_{k}\mathrm{\notin}\mathrm{co}\mathit{}{\varphi}_{k}\mathit{}\mathrm{(}z\mathrm{)}$ for all $z\mathrm{\in}Z$. Then there exists $\overline{z}\mathrm{\in}Z$ such that for each $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}\overline{k}$, one has

${\varphi}_{k}(\overline{z})=\mathrm{\varnothing}.$

For the correspondences ${({\varphi}_{j}^{\nu})}_{j\in J}$ and ${\varphi}_{0}^{\nu}$, one easily checks that they are convex-valued, irreflexive and lower semicontinuous since they have an open graph.

We now check that for all $i\in I$ the correspondence ${\varphi}_{i}^{\nu}$ satisfies the assumptions of Theorem 4.5. We first remark that ${\varphi}_{i}^{\nu}$ is convex-valued since ${\stackrel{~}{\beta}}_{i}^{\nu}$ and ${\widehat{P}}_{i}$ are convex. We now check the irreflexivity. If ${x}_{i}\in {\alpha}_{i}^{\nu}(z)$, then, from Assumption (H1’) (b), ${x}_{i}\notin {\widehat{P}}_{i}(x)$, so ${x}_{i}\notin {\varphi}_{i}^{\nu}(x)$ since ${\varphi}_{i}^{\nu}(x)\subset {\widehat{P}}_{i}(x)$. If ${x}_{i}\notin {\alpha}_{i}^{\nu}(z)$, then from Remark 4.4, we obtain $p\cdot {x}_{i}>{\stackrel{~}{\gamma}}_{i}^{\nu}(z)$, so ${x}_{i}\notin {\stackrel{~}{\beta}}_{i}^{\nu}(z)={\varphi}_{i}^{\nu}(z)$.

For the lower semicontinuity, let *V* be an open set and let *z* be such that ${\varphi}_{i}^{\nu}(z)\cap V\ne \mathrm{\varnothing}$. If ${x}_{i}\notin {\alpha}_{i}^{\nu}(z)$, then $p\cdot {x}_{i}>{\widehat{\gamma}}_{i}^{\nu}(z)$. Since ${\widehat{\gamma}}_{i}^{\nu}$ is continuous, there exists a neighborhood *W* of *z* such that ${p}^{\prime}\cdot {x}_{i}^{\prime}>{\widehat{\gamma}}_{i}^{\nu}({z}^{\prime})$ for all ${z}^{\prime}\in W$. Since ${\stackrel{~}{\beta}}_{i}^{\nu}$ has an open graph, there exists a neighborhood ${W}^{\prime}$ of *z* such that ${\stackrel{~}{\beta}}_{i}^{\nu}({z}^{\prime})\cap V\ne \mathrm{\varnothing}$ for all ${z}^{\prime}\in {W}^{\prime}$. So, ${\varphi}_{i}^{\nu}({z}^{\prime})\cap V\ne \mathrm{\varnothing}$ for all ${z}^{\prime}\in W\cap {W}^{\prime}$, and consequently ${\varphi}_{i}^{\nu}$ is lower semicontinuous at *z*. If ${x}_{i}\in {\alpha}_{i}^{\nu}(z)$, we first remark that ${\stackrel{~}{{\beta}_{i}}}^{\nu}\cap {\widehat{P}}_{i}^{\nu}$ is lower semicontinuous as an intersection of a lower semicontinuous correspondence with an open graph correspondence. So, there exists a neighborhood *W* of *z* such that ${\stackrel{~}{{\beta}_{i}}}^{\nu}({z}^{\prime})\cap {\widehat{P}}_{i}^{\nu}({x}^{\prime})\cap V\ne \mathrm{\varnothing}$ for all ${z}^{\prime}\in W$. This implies that ${\stackrel{~}{{\beta}_{i}}}^{\nu}({z}^{\prime})\cap V\ne \mathrm{\varnothing}$. Hence, in both cases ${x}_{i}^{\prime}\in {\alpha}_{i}^{\nu}({z}^{\prime})$ or ${x}_{i}^{\prime}\notin {\alpha}_{i}^{\nu}({z}^{\prime})$, we obtain ${\varphi}_{i}^{\nu}({z}^{\prime})\cap V\ne \mathrm{\varnothing}$ from the definition of ${\varphi}_{i}^{\nu}$. Thus ${\varphi}_{i}^{\nu}$ is also lower semicontinuous at *z* in this case.

From Theorem 4.5 there exists ${\overline{z}}^{\nu}=({\overline{x}}^{\nu},{\overline{y}}^{\nu},{\overline{p}}^{\nu})\in {Z}^{\nu}$ such that, for all $k\in N$,

${\varphi}_{k}^{\nu}({\overline{z}}^{\nu})=\mathrm{\varnothing}.$

As already noticed, since ${\underset{\xaf}{x}}_{i}\in {\stackrel{~}{\beta}}_{i}^{\nu}({\overline{z}}^{\nu})$ and ${\varphi}_{i}^{\nu}({\overline{z}}^{\nu})=\mathrm{\varnothing}$ for all $i\in I$, we conclude from the definition of ${\varphi}_{i}^{\nu}$ that

$\{\begin{array}{cc}& {\overline{p}}^{\nu}\cdot {\overline{x}}_{i}^{\nu}\le {\widehat{\gamma}}_{i}^{\nu}({\overline{z}}^{\nu}),\hfill \\ & {\stackrel{~}{\beta}}_{i}^{\nu}({\overline{z}}^{\nu})\cap {\widehat{P}}_{i}^{\nu}({\overline{x}}^{\nu})=\mathrm{\varnothing}.\hfill \end{array}$(4.1)

Furthermore, from Remark 4.4 one deduces that ${\stackrel{~}{\gamma}}_{i}^{\nu}({\overline{z}}^{\nu})={\widehat{\gamma}}_{i}^{\nu}({\overline{z}}^{\nu})$.

In addition, for all $j\in J$, since ${\varphi}_{j}^{\nu}({\overline{z}}^{\nu})=\mathrm{\varnothing}$, we deduce that

${\overline{p}}^{\nu}\cdot {y}_{j}\le {\overline{p}}^{\nu}\cdot {\overline{y}}_{j}^{\nu}={\pi}_{j}^{\nu}({\overline{p}}^{\nu})\mathit{\hspace{1em}}\text{for all}{y}_{j}\in {Y}_{j}^{\nu},$(4.2)

and since ${\varphi}_{0}^{\nu}({\overline{z}}^{\nu})=\mathrm{\varnothing}$, we have

$p\cdot \left(\sum _{i\in I}{\overline{x}}_{i}^{\nu}-\omega -\sum _{j\in J}{\overline{y}}_{j}^{\nu}\right)\le {\overline{p}}^{\nu}\cdot \left(\sum _{i\in I}{\overline{x}}_{i}^{\nu}-\omega -\sum _{j\in J}{\overline{y}}_{j}^{\nu}\right)\mathit{\hspace{1em}}\text{for all}p\in \overline{B}.$(4.3)

We now prove that $({\sum}_{i\in I}{\overline{x}}_{i}^{\nu}-\omega -{\sum}_{j\in J}{\overline{y}}_{j}^{\nu})=0$. Indeed, if not, it follows from (4.3) that ${\overline{p}}^{\nu}$ belongs to the boundary of $\overline{B}$, that is,

$\parallel {\overline{p}}^{\nu}\parallel =1\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\overline{p}}^{\nu}\cdot \left(\sum _{i\in I}{\overline{x}}_{i}^{\nu}-\omega -\sum _{j\in J}{\overline{y}}_{j}^{\nu}\right)>0.$

Now, by (4.1) and (4.2), ${\overline{p}}^{\nu}\cdot {\overline{x}}_{i}^{\nu}\le {\widehat{\gamma}}_{i}^{\nu}({\overline{z}}^{\nu})={\gamma}_{i}^{\nu}({\overline{z}}^{\nu})={\overline{p}}^{\nu}\cdot {\omega}_{i}+{\sum}_{j\in J}{\theta}_{i,j}{\overline{p}}^{\nu}\cdot {\overline{y}}_{j}^{\nu}$ for all *i*. Summing these inequalities over $i\in I$, we get

${\overline{p}}^{\nu}\cdot \left(\sum _{i\in I}{\overline{x}}_{i}^{\nu}-\omega -\sum _{j\in J}{\overline{y}}_{j}^{\nu}\right)\le 0,$

which yields a contradiction. We thus have proved that $({\overline{x}}^{\nu},{\overline{y}}^{\nu})\in \mathcal{\mathcal{A}}({\mathcal{\mathcal{E}}}^{\nu})$.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.