Market Failures and Equilibria in Banach Lattices: New Tangent and Normal Cones

Abstract

In this paper, we consider an economy with infinitely many commodities and market failures such as increasing returns to scale and external effects or other regarding preferences. The commodity space is a Banach lattice possibly without interior points in the positive cone in order to include most of the relevant commodity spaces in economics. We propose a new definition of the marginal pricing rule through a new tangent cone to the production set at a point of its (non-smooth) boundary. The major contribution is the unification of many previous works with convex or non-convex production sets, smooth or non-smooth, for the competitive equilibria and for the marginal pricing equilibria, with or without external effects, in finite-dimensional spaces as well as in infinite-dimensional spaces. In order to prove the existence of a marginal pricing equilibria, we also provide a suitable properness condition on non-convex technologies to deal with the emptiness of the interior of the positive cone.

Appendices

Appendix

A Proof of Propositions

Proposition 3.1

We first recall the definition of the Clarke’s tangent and normal cones to the set $Y_{j}({z})$ at the point ${\bar{y}_{j}}$

$$\begin{aligned} {T}_{Y_{j}({z})}({ \bar{y}_{j}}):= & {} \left\{ \nu \in {L} : \begin{array}{l} \forall {r}>0\ \exists \ \varepsilon >0 : \forall {y^\prime _j}\in {B}( \bar{y}_{j},\varepsilon )\cap {Y_{j}({z})}, \forall {t} \in ]0,\varepsilon [\\ {[}{y^\prime _j}+t{B}(\nu ,{r})]\cap {Y({z})}\not =\emptyset \\ \end{array} \right\} \\ {N}_{Y_{j}({z})}({\bar{y}_{j}})= & {} [{T}_{Y_{j}({z})}({ \bar{y}_{j}})]^o=\{{p}\in {L^*} : {p({\nu })}\le 0\ \forall {\nu }\in {T}_{Y_{j}({z})}( \bar{y}_{j})\} \end{aligned}$$

$\hat{T}_{Y_{j}({z})}({\bar{y}_{j}})$ is non-empty since the free-disposal condition implies that $-L_+ \subset \hat{T}_{Y_{j}({z})}({ \bar{y}_{j}})$. Now we show that ${\hat{T}}_{Y_j(z)}(\bar{y}_j)$ is a cone. Let $\nu \in {\hat{T}}_{Y_j(z)}(\bar{y}_j)$ and $\tau >0$. Let $r>0$ and $\varepsilon$ be the parameter associated by the definition of ${\hat{T}}_{Y_j(z)}( \bar{y}_j)$ to $\frac{r}{\tau }$. Consequently, for all $z^{\prime }\in B(z, \varepsilon )$, for all $\bar{y}^{\prime }_j\in B(\bar{y}_j, \varepsilon )\cap Y_j(z^\prime )$ and for all $t\in ]0, \varepsilon [$ there exists $\xi \in B(0, \frac{r}{\tau })$ such that $\bar{y}^{\prime }_j + t(\nu + \xi )\in Y_{j}(z^{\prime })$. Let $\varepsilon ^{\prime }$ strictly smaller than $\varepsilon$ and $\frac{\varepsilon }{\tau }$. Hence, for $z^{\prime }\in B(z, \varepsilon ^{\prime })$ for every $\bar{y}^{\prime }_j\in B(\bar{y}_j, \varepsilon ^{\prime })$ and for every $t\in ]0, \varepsilon ^{\prime }[$, since $t \tau <\varepsilon$, there exists $\xi \in B(0,\frac{r}{\tau })$ such that $\bar{y}^{\prime }_j + \tau t(\nu + \xi )=\bar{y}^{\prime }_j + t(\tau \nu + \tau \xi )\in Y_{j}(z^{\prime })$. As $\tau \xi \in B(0, r)$, we have proved that $\tau \nu \in {\hat{T}}_{Y_j(z)}( y_j)$ by associating with r the parameter $\varepsilon ^\prime$, and thus, $\hat{T}_{Y_{j}({z})}({ \bar{y}_{j}})$ is a cone.

We now show that ${\hat{T}}_{Y_j(z)}( \bar{y}_j) + {\hat{T}}_{Y_j(z)}(\bar{y}_j) \subset {\hat{T}}_{Y_j(z)}(\bar{y}_j)$. Let $\nu$ and $\nu ^\prime$ be two vectors in ${\hat{T}}_{Y_j(z)}( \bar{y}_j)$. For $r>0$, there exist two nonnegative real numbers $\varepsilon$ and $\varepsilon ^{\prime }$ associated by the definition of ${\hat{T}}_{Y_j(z)}(\bar{y}_j)$ to $\frac{r}{2}$. Let $\varepsilon _1>0$ smaller than $\varepsilon$ and $\frac{\varepsilon ^\prime }{1+\Vert \nu \Vert +\frac{r}{2}}$. Hence, for all $z^{\prime }\in B(z, \varepsilon _1)$, for all $\bar{y}^{\prime }_j \in B(\bar{y}_{j},\varepsilon _1)\cap {Y_j(z^{\prime })}$ and for all $t\in ]0, \varepsilon _1[$, there exists $\xi \in B(0, \frac{r}{2})$ such that $\bar{y}^{\prime \prime }_j= \bar{y}^{\prime }_j+t(\nu + \xi )\in Y_j(z^{\prime })$. We remark that $\Vert \bar{y}^{\prime \prime }_j - y_j\Vert \le \Vert \bar{y}^\prime _j - \bar{y}_j \Vert + t (\Vert \nu \Vert +\Vert \xi \Vert )$$< \varepsilon _1+ \varepsilon _1(\Vert \nu \Vert +\frac{r}{2})\le \varepsilon ^\prime$. Consequently, since $\varepsilon _1 < \varepsilon ^\prime$, there exists $\xi ^\prime \in B(0, \frac{r}{2})$ such that $\bar{y}^{\prime \prime }_j+t(\nu ^\prime + \xi ^\prime ) \in Y_j(z^{\prime })$. Hence, $\bar{y}^{\prime }_j+t(\nu +\nu ^\prime + \xi +\xi ^{\prime })\in Y_j(z^{\prime })$ and $\xi + \xi ^\prime \in B(0,r)$. So $\varepsilon _1$ associated with r satisfies the definition of ${\hat{T}}_{Y_j(z)}(\bar{y}_j)$ and we have shown that ${\hat{T}}_{Y_j(z)}( \bar{y}_j)$ is closed under addition.

As for the proof of the second part, this is trivial given the definition of the Clarke’s normal cone. $\square$

Proposition 3.2

1. $\hat{\mathcal {T}}_{Y_{j}({z})}({\bar{y}_{j}})$ is non-empty by Assumption TC. We now prove that $\hat{\mathcal {T}}_{Y_{j}(z)}(\bar{y}_{j})$ is a cone. Let $\rho >0$, $\tau >0$ and $\nu \in \cap _{\rho >0}\hat{\mathcal {T}}^{\rho }_{Y_{j}({z})}({ \bar{y}_{j}})\subset \hat{\mathcal {T}}^{\rho }_{Y_{j}(z)}( \bar{y}_{j})$. Then, there exists $\eta >0$ such that for all $r>0$ there exist $V\in \mathcal {V}_{\prod _{L^{I+J}}\tau }(z)$, $U \in \mathcal {V}_{\tau }(\bar{y}_j)$ and $\varepsilon >0$ such that for all $z^\prime \in B(z, \rho ) \cap V$, for all $\bar{y}^\prime _j\in B(\bar{y}_j, \rho ) \cap U\cap {Y_j(z^\prime )}$ and for all $t\in ]0,\varepsilon [$ it follows that $\bar{y}^\prime _j+t(\nu +\eta ( \bar{y}_{j}-\bar{y}^\prime _j) + \xi )\in Y_{j}(z^\prime )$ for some $\xi \in r[-e, e]$. Let $\eta ^\prime =\tau \eta$. Let $r>0$. Let V, U and $\varepsilon$ the parameter and the open neighborhoods associated for $\nu$ to $\frac{r}{\tau }$. Let $\varepsilon ^\prime =\frac{\varepsilon }{\tau }$ and note that $\tau t\in ]0,\varepsilon [$ is equivalent to $t \in ]0,\varepsilon ^{\prime }[$. Hence, for all $z^\prime \in B(z, \rho ) \cap V$, for all $\bar{y}^\prime _j\in B(\bar{y}_j + \rho )\cap U\cap {Y_j(z^\prime )}$ and for all $\tau t \in ]0,\varepsilon [$, there exists $\xi \in \frac{r}{\tau }[-e, e]$ such that $\bar{y}^\prime _j+\tau t(\nu +\eta (\bar{y}_{j}-\bar{y}^\prime _j)+\xi ) \ {= \bar{y}^\prime _j+ t(\tau \nu +\eta ^{\prime }( \bar{y}_{j}-\bar{y}^\prime _j)+\tau \xi ) \in Y_{j}(z^\prime )}$ with $\tau \xi \in r[-e, e]$. Hence, $\tau \nu \in \hat{\mathcal {T}}^{\rho }_{Y_{j}(z)}( \bar{y}_{j})$. Since this holds for all $\rho >0$, $\tau \nu \in \hat{\mathcal {T}}_{Y_{j}(z)}( \bar{y}_{j})$

The normal cone $\hat{\mathcal {N}}_{Y_{j}(z)}( \bar{y}_{j})$ is weak*-closed since it is the intersection of weak*-closed half spaces $\{\pi \in L^*: \pi (\nu ) \le 0\}$ over the $\nu$ in $\hat{\mathcal {T}}^{\rho }_{Y_{j}(z)}( \bar{y}_j)$.

2. From the definition of $\hat{\mathcal {T}}$, it is enough to prove that $\hat{\mathcal {T}}^\rho _{Y_{j}({z})}({ \bar{y}_{j}}) \subset \hat{T}_{Y_{j}({z})}({ \bar{y}_{j}})$ for all $\rho >0$. Let $\rho >0$ and $\nu \in \hat{\mathcal {T}}^{\rho }_{Y_{j}(z)}(\bar{y}_j)$. Consequently, there exists $\eta >0$ such that for all $r>0$ there exist $V\in \mathcal {V}_{\prod _{L^{I+J}}\tau }(z)$, $U \in \mathcal {V}_{\tau }(\bar{y}_j)$ and $\varepsilon >0$ associated with $\frac{r}{2}$. Let us fix $\varepsilon ^{\prime }>0$ strictly smaller than $\varepsilon , \frac{r}{2\eta }, \rho$ and such that ${B(z, \varepsilon ^{\prime })\subset B(z, \rho )\cap V}$ and ${B(\bar{y}_j, \varepsilon ^{\prime })\subset B( \bar{y}_j, \rho )\cap U}$. Thus, for all $z^{\prime }\in B(z, \varepsilon ^{\prime })$, for all ${\bar{y}^{\prime }_j\in B( \bar{y}_j, \varepsilon ^{\prime })\cap Y_j(z^{\prime })}$ and for all $t\in ]0,\varepsilon ^{\prime }[$, we get the existence of a vector $\xi \in \frac{r}{2}[-e, e]$ such that $\bar{y}^{\prime }_j+t(\nu + \eta ( \bar{y}_{j}-\bar{y}_j^{\prime }) + \xi )\in Y_{j}(z^{\prime })$. Note that $\Arrowvert \eta ( \bar{y}_{j}-\bar{y}^{\prime }_j)\Arrowvert \le \frac{r}{2}$ and thus ${\Vert \xi ^{\prime }=\xi + \eta ( \bar{y}_{j}-\bar{y}^{\prime }_j)\Vert \le r}$ and $\bar{y}^{\prime }_j+t(\nu +\xi ^{\prime })\in Y_{j}(z^{\prime })$. Consequently, ${\hat{\mathcal {T}}^\rho _{Y_{j}({z})}({ \bar{y}_{j}}) \subset \hat{T}_{Y_{j}({z})}({ \bar{y}_{j}})}$.

By polarity we obtain that $\hat{N}_{Y_{j}(z)}(\bar{y}_j)\subset \hat{\mathcal {N}}_{Y_{j}(z)}( \bar{y}_j)$.

3. $\hat{\mathcal {T}}_{Y_j(z)}( \bar{y}_j)\subset {\mathcal { T}}_{Y_j(z)}(\bar{y}_j)$ since order intervals are norm-bounded thanks to the fact that the norm is a lattice norm.

4. Since $e \in \hbox {int}L_+$, as already mentioned in Sect. 3, without any loss of generality, we choose the norm as the lattice norm associated with e. So ${\bar{B}(0, 1) = [-e, e]}$ and we can replace it in the definition of $\hat{\mathcal {T}}_{Y_j(z)}(\bar{y}_j)$, $\xi \in r[-e,e]$ by $\xi \in B(0,r)$.

We show that $\hat{\mathcal {T}}_{Y_{j}(z)}(\bar{y}_{j}) + \hat{\mathcal {T}}_{Y_{j}(z)}( \bar{y}_{j})\subset \hat{\mathcal {T}}_{Y_{j}(z)}( \bar{y}_{j})$ in order to prove the convexity. Let $\nu$ and $\nu ^\prime$ be in $\hat{\mathcal {T}}_{Y_{j}(z)}( \bar{y}_{j})$. Hence they belong to $\hat{\mathcal {T}}^{\rho }_{Y_{j}(z)}( \bar{y}_{j})$ for every $\rho >0$. Let us consider the set $\hat{\mathcal {T}}^{\rho +1}_{Y_{j}(z)}( \bar{y}_{j})$. There exist $\eta$ and $\eta ^{\prime }$ such that for every $r>0$ there exist neighborhoods $\varepsilon , \varepsilon ^{\prime }, U, U^{\prime }, V$ and $V^{\prime }$ associated with the real number $\frac{r}{3}$, such that for all $z^{\prime }\in B(z,\rho +1)\cap V$, for all $\bar{y}^{\prime }_j\in B(\bar{y}_j,\rho +1)\cap U\cap Y_{j}(z^{\prime })$ and for all $t\in ]0,\varepsilon [$, it follows that $\left[ \{\bar{y}^\prime _j\}+tB(\nu +\eta ( \bar{y}_{j}-\bar{y}^\prime _j), \frac{r}{3})\right] \cap Y_{j}(z^\prime )\not =\emptyset$ and for all $z^{\prime }\in B(z,\rho +1)\cap V^{\prime }$, for all ${\bar{y}^{\prime }_j\in B(\bar{y}_j,\rho +1)\cap U^{\prime }\cap Y_{j}(z^{\prime })}$ and all $t\in ]0,\varepsilon ^{\prime }[$, ${[\{\bar{y}^\prime _j\}+tB(\nu +\eta ^{\prime }( \bar{y}_{j}-\bar{y}^\prime _j), \frac{r}{3})]\cap Y_{j}(z^\prime )\not =\emptyset }$. There exist $\alpha \in ]0,1[$, $V^{\prime \prime }$ and $U^{\prime \prime }$ such that $V^{\prime \prime }+B(0, \alpha )\subset V\cap V^{\prime }$ and $U^{\prime \prime }+B(0, \alpha )\subset U\cap U^{\prime }$. Let $\varepsilon ^{\prime \prime }>0$ strictly smaller than $\varepsilon , \varepsilon ^{\prime }, \frac{r}{\eta ^{\prime }(3\Arrowvert \nu \Arrowvert + 3\rho \eta + r)}$ and $\frac{3\alpha }{3\Arrowvert \nu \Arrowvert + 3\rho \eta + r}$. Hence, for every $z^{\prime }$ in $B(z,\rho )\cap V^{\prime \prime } \subset B(z,\rho + 1)\cap V$, for every $\bar{y}^{\prime }_j$ in $B(y,\rho )\cap U^{\prime \prime }\cap Y_{j}(z^{\prime }) {\subset B(\bar{y}_{j},\rho + 1)\cap U \cap Y_{j}(z^{\prime })}$ and for every t in $]0,\varepsilon ^{\prime \prime }[ \ \subset \ ]0,\varepsilon [$, there exists $\xi$ in $B(0,\frac{r}{3})$ in such a way that the vector $\zeta _j= \bar{y}^\prime _j+t(\nu +\eta ( \bar{y}_{j}-\bar{y}^\prime _j)+\xi )$ belongs to $Y_{j}(z^\prime )$. From the definition of $\varepsilon ^{\prime \prime }$, one easily checks that $\Arrowvert \zeta _j-\bar{y}^{\prime }_j \Arrowvert< \alpha <1$ and $\eta ^\prime \Vert \zeta _j - \bar{y}^\prime _j\Vert \le \frac{r}{3}$. So, $\zeta _j\in B( \bar{y}_{j}, \rho + 1)$ and $\Arrowvert t(\nu +\eta ( \bar{y}_{j}-\bar{y}^\prime _j)+\xi \Arrowvert < \alpha$. Consequently, since $\bar{y}_j^\prime \in U^{\prime \prime }$ and $z^{\prime }\in B(z,\rho + 1)\cap V^{\prime }$, there exists $\xi ^{\prime }\in L$ such that ${\Arrowvert \xi ^{\prime } \Arrowvert < \frac{r}{3}}$ and $\zeta _j^{\prime }=\zeta _j+t(\nu ^\prime + \eta ^{\prime }( \bar{y}_{j}-\zeta _j)+\zeta _j^{\prime }) \in Y_{j}(z^{\prime })$. We note that the vector $\zeta _j^{\prime }$ equals $\bar{y}^{\prime }_j+t(\nu + \nu ^\prime + (\eta + \eta ^{\prime })( \bar{y}_{j} -\bar{y}_j^{\prime }) + \xi + \xi ^{\prime } + \eta ^{\prime }(\bar{y}_j^{\prime }-\zeta _j))$ and $\Arrowvert \eta ^{\prime }(\bar{y}^{\prime }_j-\zeta _j) \Arrowvert$ is strictly smaller than $\frac{r}{3}$. Hence, ${\left[ \{\bar{y}^\prime _j\}+tB(\nu +\nu ^\prime +(\eta + \eta ^{\prime })(\bar{y}_{j}-\bar{y}^\prime _j), r)\right] \cap Y_{j}(z^\prime )\not =\emptyset }$ and thus $\nu + \nu ^\prime \in \hat{\mathcal {T}}^{\rho }_{Y_{j}(z)}( \bar{y}_{j})$. Since this is true for all $\rho > 0$, we conclude that $\hat{\mathcal {T}}_{Y_{j}(z)}( \bar{y}_{j})$ is stable under summation. Since $\hat{\mathcal {T}}_{Y_{j}(z)}( \bar{y}_{j})$ is a cone, we get it is convex.

5. We first show that when L is finite-dimensional $\hat{\mathcal {T}}_{Y_{j}(z)}( \bar{y}_{j})=\hat{T}_{Y_{j}(z)}(\bar{y}_{j})$. Let $r>0$ and $\varepsilon$ associated with $\frac{r}{2}$ such that for all $z^\prime \in B(z, \varepsilon )$, for all ${\bar{y}_j^\prime \in B( \bar{y}_j, \varepsilon )\cap Y_j(z^\prime )}$ and for all $t\in ]0, \varepsilon [$ there exists $\xi \in B(0, \frac{r}{2})$ such that $\bar{y}^\prime _j + t(\nu + \xi ) \in Y_j(z^\prime )$. Let $\rho > 0$ and $\eta >0$. Since in this case all topologies are equivalents and the interior of the positive cone is non-empty, we can choose the closed unit ball equal to the order interval $[-e, e]$. We now choose $\varepsilon ^{\prime }>0$ smaller than $\varepsilon$, $\frac{r}{2\eta }$ and $\rho$. Let $V=B(z, \varepsilon ^{\prime })$ and $U=B(\bar{y}_j, \varepsilon ^{\prime })$. Then, for all $\bar{y}^{\prime }_j \in U$ one has that $\Arrowvert \xi - \eta ( \bar{y}_j -\bar{y}^{\prime }_j) \Arrowvert < r$ for all $\xi \in B(\nu , \frac{r}{2})$. This implies that for all $z^{\prime } \in V\cap B(z,\rho )$, for all $\bar{y}^{\prime }_j \in U \cap B( \bar{y}_j,\rho ) \cap Y_{j}(z^{\prime })$ and all $t\in ]0, \varepsilon ^{\prime }[$, there exists $\xi \in B(0, \frac{r}{2})$ such that $\bar{y}^{\prime }_j + t(\nu + \xi ) \in Y_j(z^{\prime })$. But $\bar{y}_j^{\prime } + t(\nu + \xi )=\bar{y}_j^{\prime } + t(\nu + \eta ( \bar{y}_j - \bar{y}^{\prime }_j) -\eta ( \bar{y}_j -\bar{y}^{\prime }_j) + \xi )$. Since $\xi -\eta ( \bar{y}_j -\bar{y}_j^{\prime }) \in B(0, r)$, $\nu \in \hat{\mathcal {T}}^\rho _{Y_{j}(z)}( \bar{y}_{j})$, so $\nu \in \hat{\mathcal {T}}_{Y_{j}(z)}( \bar{y}_{j})$ since this is true for all $\rho >0$.

Let us now consider the following sequential characterization of $\hat{T}_{Y_{j}(z)}( \bar{y}_{j})$ when L is a finite-dimensional vector space:

$$\begin{aligned} \hat{T}_{Y_{j}(z)}(\bar{y}_{j}):= \left\{ \nu \in {L} : \begin{array}{c} \forall z^{n}\rightarrow {z}, \forall y^n_{j}\rightarrow { \bar{y}_j}\ \text { such that} \ y^n_j\in \partial Y_j(z^n)\\ \text { and} \ \forall {t^n}\downarrow 0\ \exists \nu ^{n}\rightarrow {\nu }\ \text { such that} \ y^{n}_{j}+t^{n}\nu ^{n}\in Y_{j}(z^n) \end{array} \right\} \end{aligned}$$

Let us recall the marginal pricing rule of [16] when the commodity space is of finite dimension:

$$\begin{aligned} MP( \bar{y}_j, z)= \text {conv}\left\{ p\in {S} : \begin{array}{c} \exists \ z^{n}\subset \mathbb {R}^{L(I+J)}, z^n\rightarrow {z},\\ \exists \ \bar{y}^n_{j}\subset \mathbb {R}^L, \bar{y}^n_j\rightarrow { \bar{y}_j},\ \bar{y}^n_j\in \partial Y_j(z^n)\\ \exists \ p^n\subset \mathbb {R}^L, p^n\rightarrow p, \ p^n\in {N}_{Y_{j}({z^n})}({\bar{y}^n_{j}})\cap S\\ \end{array} \right\} \end{aligned}$$

To prove that $\hat{N}_{Y_{j}(z)}( \bar{y}_{j})\cap S = MP(\bar{y}_j,z)$, we first prove the following lemma:

Lemma A.1

(i) Let $\nu \in \hat{T}_{Y_{j}(z)}( \bar{y}_{j})$ and let $r>0$ and $\varepsilon >0$ as given by the definition of $\hat{T}_{Y_{j}(z)}( \bar{y}_{j})$. For all $z^\prime \in B(z, \varepsilon )$ and all $\bar{y}^\prime _j\in B( \bar{y}_j, \varepsilon ) \cap Y_{j}(z^\prime )$, $\nu -re\in \hat{T}_{Y_{j}(z^\prime )}(\bar{y}^\prime _j)$.

(ii) Let $\nu \in MP( \bar{y}_j,z)^\circ$, for all $\delta >0$, there exists $\varepsilon >0$ such that for all $z^\prime \in B(z, \varepsilon )$ and all $\bar{y}^\prime _j\in B( \bar{y}_j, \varepsilon ) \cap Y_{j}(z^\prime )$, $\nu -\delta e\in \hbox {int}{T}_{Y_{j}(z^\prime )}(\bar{y}^\prime _j)$.

Proof

(i) Let $\nu \in {\hat{T}}_{Y{j}(z)}(\bar{y}_j)$. Let us choose $\varepsilon ^{\prime }<\varepsilon$ such that $B(z^{\prime }, \varepsilon ^{\prime }) \subset B(z, \varepsilon )$ and $B(\bar{y}^{\prime }_j, \varepsilon ^{\prime }) \subset B( \bar{y}_j, \varepsilon )$. Then, for all $z^{\prime \prime }\in B(z^{\prime }, \varepsilon ^{\prime })$ for all $\bar{y}_j^{\prime \prime }\in B(\bar{y}^{\prime }_j, \varepsilon ^{\prime })\cap Y_j( z^{\prime \prime })$ and all $t\in ( 0,\varepsilon ^{\prime })$ there exists $\xi \in B(0,r)$ such that $\bar{y}_j^{\prime \prime }+t( \nu + \xi ) \in Y_j( z^{\prime \prime })$. By the free-disposal condition, $\bar{y}_j^{\prime \prime }+t( \nu -re) \in Y( z^{\prime \prime })$. From the definition of ${\hat{T}}_{Y_j(\cdot )}$, we have $\nu -re\in {\hat{T}}_{Y_j(z^{\prime })}(\bar{y}_j^{\prime })$.

(ii) Let $\nu \in MP( \bar{y}_j,z)^\circ$ and $\delta >0$. By contraposition, if for all $\varepsilon >0$, there exist vectors $z^\prime \in B(z,\varepsilon )$ and $\bar{y}_j^\prime \in B(\bar{y}_j, \varepsilon ) \cap Y_j(z^\prime )$ such that $\nu - \delta e \notin \hbox {int}T_{Y_j(z^\prime )} (\bar{y}_j^\prime )$, we can build a sequence $(z^n, \bar{y}_j^n, p^n)$ converging to $(z, \bar{y}_j, p)$ such that for each n, $\bar{y}_j^n \in \partial Y_j(z^n)$, $p^n \in N_{Y_j(z^n)} (\bar{y}_j^n)\cap S$ and $\langle p^n , \nu - \delta e \rangle \ge 0$. So, at the limit, from the definition of $MP( \bar{y}_j, z)$, we get $p \in MP( \bar{y}_j, z)$ and $\langle p , \nu - \delta e \rangle \ge 0$, which implies ${\langle p , \nu \rangle \ge \delta >0}$, which contradicts $\nu \in MP( \bar{y}_j, z)^\circ$. $\square$

We now proceed with the proof. Let $\pi \in MP(\bar{y}_j,z)$, then by Carathéodory’s Theorem $\pi =\sum _{k \in K}\lambda _{k}p_{k}$ such that $\lambda _{k}\ge 0$, $\sum _{k \in K}\lambda _{k}=1$, and $(p_{k}) \in S$ for all k. Hence, for all $\delta >0$, ${\langle \pi , \nu - \delta e\rangle } =\sum _{k\in L}\lambda _{k}{\langle p_{k}, \nu -\delta e\rangle }$. By definition, every $p_{k}$ is the limit of a sequence $p_{k}^{n}\in N_{Y( z^{n}) }( \bar{y}_j^{n}) \cap S\subset \hat{N}_{Y( z^{n}) }( \bar{y}_j^{n}) \cap S$ from Proposition 3.1. Let $\nu \in {\hat{T}}_{Y_j(z)} (\bar{y}_j)$. For all $n\ge n_{0}$$\sum _{k\in K}\lambda _{k}{\langle p^{n}_{k}, \nu -\delta e\rangle } \le 0$ by the above lemma. Taking limits, one has ${\langle \pi , \nu -\delta e \rangle } =\sum _{k\in K}\lambda _{l} \langle p_k , \nu -\delta e \rangle \le 0$. Hence, $\langle \pi , \nu \rangle \le \delta$. Since this is true for all $r>0$, we conclude that $\langle \pi , \nu \rangle \le 0$ and then $\pi \in \hat{N}_{Y_j( z)}(\bar{y}_j) \cap S$.

To prove the converse inclusion, we use the duality between closed convex cones and we actually prove that $MP(\bar{y}_j, z)^\circ \subset {\hat{T}}_{Y_j(z)} (\bar{y}_j)$. Let $\nu \in MP( \bar{y}_j, z)^\circ$. To prove $\nu \in {\hat{T}}_{Y_j(z)} (\bar{y}_j)$, it suffices to show that $\nu - \delta e \in {\hat{T}}_{Y_j(z)} (\bar{y}_j)$ for all $\delta >0$. From the above lemma, for all $\delta >0$, there exists $\varepsilon >0$ such that for all $z^\prime \in B(z, \varepsilon )$ and all $\bar{y}^\prime _j\in B(\bar{y}_j, \varepsilon ) \cap Y_{j}(z^\prime )$, $\nu -\delta e\in \hbox {int}{T}_{Y_{j}(z^\prime )}(\bar{y}^\prime _j)$. So, from the characterization of the interior of the Clarke’ tangent cone, there exists $\tau (\bar{y}^\prime _j, z^\prime ) >0$ such that for all $t \in [0, \tau (\bar{y}^\prime _j, z)]$, $\bar{y}_j^\prime + t (\nu -\delta e) \in Y_j(z^\prime )$. Let $z^{n}\rightarrow {z}$ and $\bar{y}^n_{j}\rightarrow { \bar{y}_j}$ such that $\bar{y}^n_j\in \partial Y_j(z^n)$. For n large enough, $z^n \in B(z, \varepsilon )$ and $\bar{y}^n_j \in B( \bar{y}_j, \varepsilon ) \cap Y_{j}(z^\prime )$. So, we can build a sequence $t^n \downarrow 0$ such that $t^n < \tau (\bar{y}_j^n, z^n)$. Hence, $\bar{y}_j^n + t^n (\nu -\delta e) \in Y_j(z^n)$ for all n large enough, which implies that $\nu - \delta e \in {\hat{T}}_{Y_j(z)} ( \bar{y}_j)$.

6. Let $Y_j$ be a convex-valued correspondence. Let

$$\begin{aligned} PM( \bar{y}_j, z)=\{\pi \in L^* : \pi ( \bar{y}_j)\ge \pi (y_j^{\prime }) \forall \ y_j^{\prime }\in Y_j(z) \} \end{aligned}$$

be the profit maximization behavior. Let $\zeta _j - \bar{y}_j \in (Y_{j}(z)-\{ \bar{y}_{j}\})$. Let ${\rho >0}$, $\eta =1$, $r>0$, $0<\delta < r$, $\varepsilon =1$ and $U=L$. By Assumption P(3), there exists $V \in \mathcal {V}_{\prod _{L^{M}}\tau }(z)$ such that $\zeta _j -\delta e \in Y_j(z^{\prime })$ for all $z^{\prime } \in V$. Let ${z^{\prime }\in B(z, \rho )\cap V}$ and $\zeta _j-\delta e \in Y_{j}(z^{\prime })$. Then, for $\bar{y}_j^{\prime }\in B(\bar{y}_j, \rho )\cap Y_{j}(z^{\prime })$ and $t\in ]0, 1[$, ${t(\zeta _j-\delta e) + (1-t)\bar{y}_j^{\prime } \in Y_{j}(z^{\prime })}$ since $Y_{j}(z^{\prime })$ is convex. But this means that $\bar{y}_j^{\prime } + t(\zeta _j-\delta e -\bar{y}_j + (\bar{y}_j -\bar{y}_j^{\prime })) \in Y_{j}(z^{\prime })$. Since $-\delta e \in r[-e, \ e]$ we have that $\zeta _j-\bar{y}_j \in \hat{\mathcal {T}}^{\rho }_{Y_{j}(z)}(\bar{y}_j)$. Since this is true for all $\rho >0$, $\zeta _j- \bar{y}_j \in \hat{\mathcal {T}}_{Y_{j}(z)}(\bar{y}_j)$ and thus $\hat{\mathcal {N}}_{Y_{j}(z)}( \bar{y}_j)\subset (Y_{j}(z)-\{\bar{y}_{j}\})^{\circ }=PM_j( \bar{y}_j, z)$. The converse is immediate since $PM( \bar{y}_j, z)={N}_{Y_j(z)}(\bar{y}_j)\subset \hat{N}_{Y_j(z)}(\bar{y}_j)\subset \hat{\mathcal {N}}_{Y_j(z)}( \bar{y}_j)$. $\square$

B Proof of Lemmata

Lemma 3.1

By Assumption SB (2), for all $\zeta _j \in \partial Y_j(z)$, $T_{Y_j (z)} (\zeta _j) = \{\nu \in L\vert \nabla _1 f_j(\zeta _j, z)(\nu )\le 0\}$ is the Clarke’s tangent cone to $Y_j(z)$ at $\zeta _j$ ([14], Theorem 2.4.7, Corollary 2, p. 57).

Since e is in the quasi-interior of $L_+$ and $\nabla _1 f_j(\zeta _j,z) \in L^*_+{\setminus }\{0\}$, $\nabla _1 f_j(\zeta _j, z)(e)$ is strictly positive. Let $\nu \in L$ such that $\nabla _1 f_j(\zeta _j, z)(\nu )\le 0$, let $\rho > 0$, $\eta = 1$ and $r>0$. Let $\alpha > 0$ such that $\alpha < \frac{r \beta }{2(2\Vert \nu \Vert + 4\rho + r)}$ where $\beta =\nabla _1 f_j(\zeta _j, z)(e)$. Since $\nabla _1$ is continuous, there exist neighborhoods $\ U \in \mathcal {V}_\tau ( \zeta _j)$ and $V \in \mathcal {V}_{\prod _{L^{I+J}}\tau }(z)$ such that $\Vert \nabla _1 f_j(\zeta _j, z) - \nabla _1 f_j(\zeta _j^\prime , z^\prime )\Vert < \alpha$ for all $( \zeta _j^\prime , z') \in U\times V$.

Let $U^\prime = \{\zeta _j^\prime \in L \mid \nabla _1 f_j(\zeta _j, z)(\zeta _j^\prime - \zeta _j) < \frac{r\beta }{4}\}$ be a weak neighborhood of $\zeta _j$. There exists another convex neighborhood $U^{\prime \prime }$ of $\zeta _j$ and $\delta >0$ such that $U^{\prime \prime } + B(0, \delta ) \subset U'\cap U$. Let $\varepsilon > 0$ such that $\varepsilon < \frac{2\delta }{2(2\Vert \nu \Vert + 2\rho + r)}$. From the mean value theorem, for all $z^\prime \in V\cap B(z, \rho )$, for all $\zeta _j^\prime \in U^{\prime \prime }\cap B(\zeta _j, \rho ) \cap Y_j(z^\prime )$ and for all $t\in (0, \varepsilon )$, there exists $\zeta _j^{\prime \prime }$ in the segment $[\zeta _j^\prime , \zeta _j^\prime + t(\nu + \zeta _j - \zeta ^\prime _j -\frac{r}{2}e)]$ such that

$$\begin{aligned} f\left( \zeta _j^\prime + t\left( \nu + \zeta _j - \zeta _j^\prime -\frac{r}{2}e\right) , z^\prime \right) = f(\zeta _j^\prime , z^\prime ) + t\nabla _1 f_j(\zeta _j^{\prime \prime }, z^\prime )\left( \nu + \zeta _j - \zeta _j^\prime -\frac{r}{2}e\right) \end{aligned}$$

From our choice of $\varepsilon$, it follows that $\zeta _j^{\prime \prime } \in U$, hence $\Vert \nabla _1 f_j(\zeta _j^{\prime \prime }, z^\prime )- \nabla _1 f_j(\zeta _j, z)\Vert \le \alpha$. Since $\nabla _1 f_j(\zeta _j^{\prime \prime }, z^\prime )( \nu + \zeta _j - \zeta _j^\prime -\frac{r}{2}e) = (\nabla _1 f_j(\zeta _j^{\prime \prime }, z^\prime ) - \nabla _1 f_j(\zeta _j, z))( \nu + \zeta _j - \zeta _j^\prime -\frac{r}{2}e) + \nabla _1 f_j(\zeta _j, z)( \nu + \zeta _j - \zeta _j^\prime -\frac{r}{2}e)$, we deduce from the previous definitions and inequalities that $\nabla _1 f_j(\zeta _j^{\prime \prime }, z^\prime )( \nu + \zeta _j - \zeta _j^\prime -\frac{r}{2}e) < 0$. Since $f(\zeta _j^\prime , z^\prime )\le 0$, we get $f(\zeta _j^\prime + t(\nu + \zeta _j - \zeta _j^\prime -\frac{r}{2}e), z^\prime ) \le 0$, that is, $\zeta _j^\prime + t(\nu + \zeta _j - \zeta _j^\prime -\frac{r}{2}e) \in Y_j(z^\prime )$. Since $-\frac{r}{2}e\in [-re, re]$, we obtain that $\nu \in \hat{\mathcal {T}^{\rho }}_{Y_{j}(z)}(\zeta _j)$. Since this is true for all $\rho > 0$, we have that $\nu \in \hat{\mathcal {T}}_{Y_{j}(z)}(\zeta _j)$$\square$

Lemma 4.2

• $\hbox {WSA}^F$

Suppose that ${\mathcal {E}} ^F$ does no satisfy $\hbox {WSA}^F$, that is, for all F, $(p^F, z^F, \lambda ^F)$ in $PE^F\times [0, {\bar{\lambda }}]$, $(y_j^F)_{j\in J}\in A(\omega + \lambda ^F e, z^F)$ and ${p^F (\sum _{j \in J} y^F_j + \omega + \lambda ^F e) = 0}$. By Lemma 4.3 there exists $\pi _j^F\in {\hat{N}} _{Y_j(z)}(\bar{y} _j)\cap S$ such that $\pi _{j\vert F} = p^F$ for all j. Since $y_j^F\in A(\omega + {\bar{\lambda }} e, z^F)$, Assumption (B) says that the net $(z^F, (\pi ^F_j), (\pi ^F_j(y_j^F)), \lambda ^F)$ belongs to a weak-compact set such that the subnet $(z^{F(\gamma )}, (\pi ^{F(\gamma )}), (\pi ^{F(\gamma )}_j(y_j^{F(\gamma )})), \lambda ^{F(\gamma )})_{\gamma \in \varGamma }$ converges for the product topology to $(\bar{z}, ({\bar{\pi }} _j), \text {lim}(\pi ^{F(\gamma )}_j(y_j^{F(\gamma )})), {\bar{\lambda }})$.

Since $z^{F(\gamma )}\in A(\omega + \lambda ^{F(\gamma )} e)$, $\sum _{j\in J} y_j^{F(\gamma )} + \omega + \lambda ^{F(\gamma )}e \ge \sum _{i\in I}x_i^{F(\gamma )}$ , and since $L_+$ is $\tau$-closed, it follows $\sum _{j\in J} \bar{y}_j + \omega + \lambda e s\ge \sum _{i\in I}\bar{x}_i$. By Assumptions C(1) and P(1) we get $\bar{z}\in \prod _{i\in I} X_i(\bar{z})\times \prod _{j\in J}Y_j(\bar{z})$ and by repeating the arguments of Claim 1 in Sect. 3.2, ${\bar{\pi }}_j = {\bar{\pi }} > 0$ for all $j\in J$. Consequently, $\sum _{j\in J} {\bar{\pi }}_j(\bar{y}_j ) + {\bar{\pi }}_j(\omega ) + \lambda \ge 0$. Given that ${p^F (\sum _{j\in J} y_j^{F} + \omega + \lambda ^{F}e)}$$= \pi ^F_j(\sum _{j\in J} y_j^{F} + \omega + \lambda ^{F}e) = 0$ for all $F\in {\mathcal {F}}$ and $j\in J$, we obtain that $\text {lim}\pi _j^{F(\gamma )}(y_j^{F(\gamma )}) + {\bar{\pi }}_j(\omega ) + \lambda = 0$. By Proposition 4.1$\text {lim}\pi _j^{F(\gamma )}(y_j^{F(\gamma )})\ge {\bar{\pi }}(\bar{y}_j)$, so we deduce that ${\bar{\pi }}(\sum _{j\in J} \bar{y}_j + \omega + \lambda e ) = \sum _{j\in J}\text {lim}\pi _j^{F(\gamma )}(y_j^{F(\gamma )} )+ {\bar{\pi }}(\omega ) + \lambda = 0$. Hence, $\text {lim}\pi _j^{F(\gamma )}(y_j^{F(\gamma )}) = {\bar{\pi }} (\bar{y}_j)$ for all j. From Proposition 4.1 b) ${\bar{\pi }}\in \hat{\mathcal {N}}_{Y_{j}(\bar{z})}({\bar{y}_{j}})\cap {S}$ and by Assumption (WSA) we get ${\bar{\pi }}(\sum _{j\in J} \bar{y}_j + \omega + \lambda e ) > 0$, which contradicts the above equality.

• ($\hbox {LNS}^F$)

We show that there exists ${\hat{F}}\in {\mathcal {F}}$ such that for all $F\in {\mathcal {F}}$ with ${\hat{F}}\subset F$, the economy ${\mathcal {E}}^F$ satisfies ($\hbox {LNS}^F$). We first prove that preferences are non-satiated. Suppose, on the contrary, that for all $F\in \mathcal F$ preferences are not satiated on the attainable allocations, i.e., for all $F\in {\mathcal {F}}$m there exists $z^F\in A^F(\omega )$ such that for some $i_0$, there does not exist $\xi ^F_{i_0}\in X_{i_0}(z^F )\cap F$ such that $\xi ^F_{i_0}\succ ^F_{i_0, z^F} x^F_{i_0}$. Since $A^F(\omega )\subset A(\omega )$ for all F, there exists a subnet $(z^{F(\gamma )})_{\gamma \in \varGamma }$ converging weakly to $\bar{z} = ((\bar{x}_i)_{i\in I}, (\bar{y}_j)_{j\in J})$. From Assumptions C(1) and P(1) we deduce that $\bar{z} \in \prod _{i\in I}X_i (\bar{z})\times \prod _{j\in J}Y_j(\bar{z})$ and from Assumption C(3) there exists a vector $(\xi _i)_{i\in I}\in \prod _{i\in I}X_i(\bar{z})$ such that $\xi _i\succ _{i, \bar{z}} \bar{x}_i$ for all i. Since the net $(z^{F(\gamma )})_{\gamma \in \varGamma }$ converges weakly to $\bar{z}$ and because of strong lower hemi-continuity of $X_i$ (Assumption C(2)), for $\delta > 0$, there exists $\gamma _0\in \varGamma$ such that $\xi _i + \delta e$ belongs to $X_i(z^{F(\gamma )})$ for $\gamma > \gamma _0$. Further, there exists $F\in {\mathcal {F}}$ such that $\xi _i + \delta e\in F$ and $\gamma _1\in \varGamma$ such that $F\subset F(\gamma )$ for all $\gamma > \gamma _1$. Consequently, for all $\gamma$ larger than $\gamma _0$ and $\gamma _1$, $\xi _i + \delta e\in X_i(z^{F(\gamma )})\cap F(\gamma )$ for all $i\in I$.

Note that $(\bar{z}, \xi _i + \delta e, \bar{x}_i)\not \in G_i$. Accordingly to Assumption C(4), there exists $\gamma _2\in \varGamma$ such that for all $\gamma > \gamma _2$, $(z^{F(\gamma )}, \xi _i + \delta e, x_i^{F(\gamma )})\not \in G_i$. Consequently, for $\gamma$ large enough both $\xi _i + \delta e$ and $x_i^{F(\gamma )}$ belong to $X_i(z^{F(\gamma )})\cap F(\gamma )$ and $\xi + \delta e \succ ^{F(\gamma )}_{i, z^{F(\gamma )}} x_i^{F(\gamma )}$ for all $i\in I$. This contradicts our previous claim that for some $i_0$, there does not exist $\xi ^F_{i_0}\in X_{i_0}(z^F )\cap F$ such that $\xi _{i_0}\succ ^F_{i_0, z^F} x^F_{i_0}$.

Since preferences are also convex (Assumption C(3)) we deduce that they are locally non-satiated. $\square$

Lemma 5.1

We want to prove that for all $(p, z, t) \in S^e\times Z^e \times \mathbb {R}_+$, if $(y_j)\in A^{e}(\omega + t e, z)$ it follows that $p (\sum _{j\in J}y_j + \omega + te)>0$. By Propositions 5.1 and 5.2, there exists a continuous, positive, linear functional $\pi$ which extends p to the whole space L and ${\pi \in \cap _{j\in J} \hat{\mathcal {N}}_{Y_j (z)}(y_j)}$. Hence, $(\pi , z, t) \in S\times Z \times \mathbb {R}_+$ and ${(y_j)\in A(\omega + t e, z)}$. Since $\sum _{j\in J}y_j + \omega + te$ belongs to $L(e)_+$, we get by Assumption SA $0< \pi (\sum _{j\in J}y_j + \omega + te)=p(\sum _{j\in J}y_j + \omega + te)$. Hence, the lemma is proved. $\square$