Private ownership economies with externalities and existence of competitive equilibria: a differentiable approach

Abstract

We consider a general equilibrium model of a private ownership economy with consumption and production externalities. Utility functions and production technologies may be affected by the consumption and production activities of all other agents in the economy. We use homotopy techniques to show that the set of competitive equilibria is non-empty and compact. Fixing the externalities, the assumptions on utility functions and production technologies are standard in a differentiable framework. Competitive equilibria are written in terms of first order conditions associated with agents’ behavior and market clearing conditions, following the seminal paper of Smale (J Math Econ 1:1–14, 1974). The work of adapting the homotopy approach to economies with externalities on the production side is non-trivial and it requires some ingenious adjustments, because the production technologies are not required to be convex with respect to the consumption and production activities of all agents.

Appendix

Proof of Lemma 5

Let $(x^{\prime },y^{\prime }) \in A(x,y;r)$. Since $\sum \nolimits _{h\in {\mathcal {H}}} x_{h}^{\prime } \gg 0, y^{\prime }$ belongs to the bounded set C(r) given by Assumption 3. Furthermore, for every $h\in {\mathcal {H}}, 0\ll x_{h}^{\prime }\ll \sum \nolimits _{h\in {\mathcal {H}}} x_{h}^{\prime } \le \sum \nolimits _{j\in {\mathcal {J}}} y_{j}^{\prime } + r$. Thus, there exists a bounded set $K(r) \subseteq {\mathbb {R}}^{\textit{CH}}_{++}\times {\mathbb {R}}^{\textit{CJ}}$ such that for every $(x,y) \in {\mathbb {R}}^{\textit{CH}}_{++}\times {\mathbb {R}}^{\textit{CJ}}, A(x,y;r) \subseteq K(r)$. $\square$

Proof of Proposition 13

Let ${\overline{E}}$ be the production economy defined in Section 5.1. We remind that $A({\overline{x}},{\overline{y}};r):=\{(x^{\prime },y^{\prime })\in {\mathbb {R}}_{++}^{CH}\times {\mathbb {R}}^{\textit{CJ}} : {\overline{t}}_{j}(y_{j}^{\prime })\le 0, \forall j \in {\mathcal {J}}\,\text { and}\,\sum \nolimits _{h\in {\mathcal {H}}} x_{h}^{\prime }-\sum \nolimits _{j\in {\mathcal {J}}} y_{j}^{\prime }\le r \}$. Consider the set $U(r):= \{( u^{\prime }_{h})_{h\in {\mathcal {H}}} \in { \prod \nolimits _{h\in {\mathcal {H}}} } \mathrm{Im}\,{\overline{u}}_{h} \vert \exists (x^{\prime },y^{\prime })\in A({\overline{x}},{\overline{y}};r) :{\overline{u}}_{h}(x^{\prime }_{h}) \ge u^{\prime }_{h}, \forall h\in {\mathcal {H}}\}$. By Point 2 of Assumption 1, the set $U_{r}$ is non-empty. Fix $( u^{\prime }_{h})_{h\in {\mathcal {H}}} \in U(r)$ and consider the maximization problem

$$\begin{aligned} \begin{array}{ll} {\max \limits _{(x,y) \in {\mathbb {R}}_{++}^{\textit{CH}}\times {\mathbb {R}}^{\textit{CJ}}}} &{} {\overline{u}}_{1}(x_{1}) \\ \text {subject to } &{} \left\{ \begin{array}{l} {\overline{t}}_{j}(y_{j})\le 0, \quad \forall j \in {\mathcal {J}} \\ {\overline{u}}_{h}(x_{h}) \ge u^{\prime }_{h}, \quad \forall h \in {\mathcal {H}} \\ \sum _{h\in {\mathcal {H}}}x_{h}-\sum \limits _{j\in {\mathcal {J}}}y_{j} \le r \end{array} \right. \end{array} \end{aligned}$$

This problem has at least one solution $({\widetilde{x}},{\widetilde{y}}) \in {\mathbb {R}}_{++}^{\textit{CH}}\times {\mathbb {R}}^{\textit{CJ}}$. Then, $({\widetilde{x}},{\widetilde{y}})$ solves the maximization problem given below and it is a Pareto optimal allocation of the economy ${\overline{E}}$.²⁶

$$\begin{aligned} \begin{array}{ll} {\max \limits _{(x,y)\in {\mathbb {R}}_{++}^{\textit{CH}}\times {\mathbb {R}}^{\textit{CJ}}}} &{} {\overline{u}}_{1}(x_{1}) \\ \text {subject to} &{} \left\{ \begin{array}{l} - {\overline{t}}_{j}(y_{j})\ge 0, \quad \forall j \in {\mathcal {J}} \\ {\overline{u}}_{h}(x_{h}) - {\overline{u}}_{h}\left( {\widetilde{x}}_{h}\right) \ge 0, \quad \forall h\ne 1 \\ r - \sum \limits _{h\in {\mathcal {H}}}x_{h} + \sum _{j\in {\mathcal {J}}}y_{j} \ge 0 \end{array} \right. \end{array} \end{aligned}$$

(14)

We now claim that there exists $({\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }}):=(({\widetilde{\beta }}_{j})_{j\in {\mathcal {J}}},( {\widetilde{\theta }} _{h}) _{h \ne 1},{\widetilde{\gamma }}) \in {\mathbb {R}}_{++}^{J}\times {\mathbb {R}}_{++}^{H-1}\times {\mathbb {R}}_{++}^{C}$ such that $({\widetilde{x}},{\widetilde{y}},{\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }})$ is the unique solution to system (6). We first prove the existence of $({\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }})$, afterwards we show the uniqueness of $({\widetilde{x}},{\widetilde{y}},{\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }})$.

Existence of $({\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }}) \gg 0$. We know that $({\widetilde{x}},{\widetilde{y}})$ solves problem (14). The KKT conditions associated with problem (14) are given by

$$\begin{aligned} \left\{ \begin{array}{l} D_{x_1} {\overline{u}}_{1}( x_{1}) = \gamma , \quad \forall h \ne 1: \theta _{h}D_{x_h}{\overline{u}}_{h}( x_{h}) = \gamma \quad \text {and}\quad \theta _{h} \left( {\overline{u}}_{h}( x_{h}) - {\overline{u}}_{h}\left( {\widetilde{x}}_{h}\right) \right) =0, \\ \forall j \in {\mathcal {J}} : \gamma = \beta _{j}D_{y_{j}}{\overline{t}}_{j}(y_{j})\quad \text {and}\quad \beta _{j}\left( - {\overline{t}}_{j}(y_{j})\right) =0, \quad \forall c \in {\mathcal {C}} : \gamma ^{c}\left( r^{c} - {\sum \limits _{h\in {\mathcal {H}}}}x_{h}^{c}+\sum \limits _{j\in {\mathcal {J}}}y_{j}^{c}\right) =0\\ \end{array} \right. \nonumber \\ \end{aligned}$$

(15)

where $(\beta ,\theta , \gamma ):=((\beta _{j})_{j\in {\mathcal {J}}},( \theta _{h}) _{h\ne 1},(\gamma ^{c})_{c \in {\mathcal {C}}}) \in {\mathbb {R}}_{+}^{J}\times {\mathbb {R}}_{+}^{H-1}\times {\mathbb {R}}^{C}_{+}$ are the Lagrange multipliers associated with the constraint functions of problem (14). KKT conditions are obviously necessary conditions to solve problem (14). Indeed, it is easy to verify that the Jacobian matrix associated with the constraint functions of problem (14) has full row rank equal to $J+(H-1)+C$. This property is standard because the externalities are fixed. Therefore, there exists $({\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }}) \ge 0$ such that $({\widetilde{x}},{\widetilde{y}},{\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }})$ solves system (15). Furthermore, Point 4 of Assumption 1 and Point 2 of Assumption 6 imply that all the Lagrange multipliers $({\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }})$ must be strictly positive. Consequently, all the constraints of problem (14) are binding, and then $({\widetilde{x}},{\widetilde{y}},{\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }})$ is a solution to system (6).

Uniqueness of $({\widetilde{x}},{\widetilde{y}},{\widetilde{\beta }}, {\widetilde{\theta }},{\widetilde{\gamma }})$. Define ${\widetilde{\theta }}_{1}:=1$, by equations (1) and (2) of system (6), for all h one gets $D_{x_h} {\overline{u}}_{h}({\widetilde{x}}_{h}) = \frac{{\widetilde{\gamma }}}{{\widetilde{\theta }}_{h}}$. Then, for every $h, {\widetilde{x}}_{h}$ solves the maximization problem: ${\max \nolimits _{x_{h} \in {\mathbb {R}}^{C}_{++}}}{\overline{u}}_{h}(x_{h})$ subject to $\frac{{\widetilde{\gamma }}}{{\widetilde{\theta }}_{h}} \cdot x_{h} \le \frac{{\widetilde{\gamma }}}{{\widetilde{\theta }}_{h}} \cdot {\widetilde{x}}_{h}$ because KKT are sufficient conditions to solve this problem. Thus, the uniqueness of ${\widetilde{x}}_{h}$ follows from the strict quasi-concavity of ${\overline{u}}_{h}$. Analogously, by Eqs. (4) and (5) of system (6), ${\widetilde{y}}_{j}$ solves the maximization problem: ${\max \nolimits _{y_{j} \in {\mathbb {R}}^{C}}} \frac{{\widetilde{\gamma }}}{{\widetilde{\beta }}_{j}} \cdot y_{j}$ subject to ${\overline{t}}_{j}(y_{j})\le 0$ for every j. Thus, the uniqueness of ${\widetilde{y}}_{j}$ follows from the continuity and the strict quasi-convexity of ${\overline{t}}_{j}$. Therefore, $({\widetilde{x}},{\widetilde{y}})$ is unique and consequently, the uniqueness of $({\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }})$ follows from equations (1), (2) and (4) of system (6). $\square$

Proof of Proposition 14

We use the functions ${\overline{u}}_{h}$ and ${\overline{t}}_{j}$ defined in Subsection 5.1. We have already pointed out that $G({\widetilde{\xi }})=0$. Let $\xi ^{\prime }=(x^{\prime },\lambda ^{\prime },y^{\prime },\alpha ^{\prime }, p^{\prime \backslash }) \in \Xi$ be such that $G(\xi ^{\prime })=0$, we show that ${\widetilde{\xi }}=\xi ^{\prime }$.

First, notice that

$$\begin{aligned} \sum _{h\in {\mathcal {H}}}x_{h}^{\prime } - \sum _{j\in {\mathcal {J}}} y_{j}^{\prime } ={ \sum _{h\in {\mathcal {H}}}} e_{h} \end{aligned}$$

(16)

Indeed, summing $G^{h.2}(\xi ^{\prime })=0$ over h, one gets $\sum \nolimits _{h\in {\mathcal {H}}}x_{h}^{\prime } - \sum \nolimits _{j\in {\mathcal {J}}} y_{j}^{\prime } ={ \sum \nolimits _{h\in {\mathcal {H}}}} {\widetilde{e}}_h$ by $G^{M}(\xi ^{\prime })=0$. Using the definition of ${\widetilde{e}}_h$ given in (7) and Proposition 13, one deduces (16).

Second, we show that

$$\begin{aligned} {\overline{u}}_{h}\left( x_{h}^{\prime }\right) = {\overline{u}}_{h}\left( {\widetilde{x}}_{h}\right) ,\quad \forall h \in {\mathcal {H}} \end{aligned}$$

(17)

Using the definition of ${\widetilde{e}}_h$ given in (7) and $G^{h.1}(\xi ^{\prime })=G^{h.2}(\xi ^{\prime })=0, x^{\prime }_{h}$ solves the following maximization problem

$$\begin{aligned} \begin{array}{l} {\max \limits _{x_{h}\in {\mathbb {R}}_{++}^{C}}}\; {\overline{u}}_{h}(x_{h}) \\ \text {subject to}\;\;p^{\prime }\cdot x_{h}\le p^{\prime }\cdot {\widetilde{x}}_{h}+ \sum \limits _{j\in {\mathcal {J}}}s_{jh} p^{\prime }\cdot \left( y^{\prime }_{j} -{\widetilde{y}}_{j}\right) \\ \end{array} \end{aligned}$$

(18)

because KKT are sufficient conditions to solve this problem. Analogously, from $G^{j.1}(\xi ^{\prime })=G^{j.2}(\xi ^{\prime })=0, y^{\prime }_{j}$ solves the maximization problem: ${ \max \nolimits _{y_{j} \in {\mathbb {R}}^{C}}} p^{\prime }\cdot y_{j}$ subject to ${\overline{t}}_{j}(y_{j})\le 0$. Notice that ${\widetilde{y}}_{j}$ satisfies the constraint of this problem because $G^{j.2}({\widetilde{\xi }})=0$. Thus, $p^{\prime } \cdot (y^{\prime }_{j}-{\widetilde{y}}_{j}) \ge 0$ for all j, and consequently ${\widetilde{x}}_{h}$ belongs to the budget constraint of problem (18). Then, ${\overline{u}}_{h}(x_{h}^{\prime }) \ge {\overline{u}}_{h}({\widetilde{x}}_{h})$ for all h. Now, suppose that ${\overline{u}}_{k}(x_{k}^{\prime }) > {\overline{u}}_{k}({\widetilde{x}}_{k})$ for some $k \in {\mathcal {H}}$. From (16) and $G^{j.2}(\xi ^{\prime })=0$ for all j, one deduces that $(x^{\prime },y^{\prime })$ is a feasible allocation of the production economy ${\overline{E}}$, and then one gets a contradiction since $({\widetilde{x}},{\widetilde{y}})$ is a Pareto optimal allocation of ${\overline{E}}$ by Proposition 13. Thus, (17) is completely proved.

Now, define $\beta ^{\prime }:=(\beta _{j}^{\prime })_{j\in {\mathcal {J}}}$ where $\beta _{j}^{\prime }:= \lambda _{1}^{\prime } \alpha _{j}^{\prime }$ for all $j, \theta ^{\prime }:=(\theta _{h}^{\prime }) _{h \ne 1}$ where $\theta _{h}^{\prime }:= \frac{\lambda _{1}^{\prime } }{\lambda _{h}^{\prime } }$ for all $h \ne 1$ and $\gamma ^{\prime }:= \lambda _{1}^{\prime } p^{\prime }$. From $G^{h.1}(\xi ^{\prime })=0$ for all $h, G^{j.1}(\xi ^{\prime })=G^{j.2}(\xi ^{\prime })=0$ for all j, (16) and (17), it is an easy matter to check that $(x^{\prime },y^{\prime }, \beta ^{\prime },\theta ^{\prime },\gamma ^{\prime })$ solves system (6). Then, Proposition 13 implies that $({\widetilde{x}},{\widetilde{y}},{\widetilde{\beta }},{\widetilde{\theta }},{\widetilde{\gamma }})=(x^{\prime },y^{\prime }, \beta ^{\prime },\theta ^{\prime },\gamma ^{\prime })$, and consequently, one deduces that ${\widetilde{\xi }}=\xi ^{\prime }$.

We remark that G is $C^{1}$ by Point 1 of Assumptions 1 and 6. Finally, in order to show that 0 is a regular value for G, one proves that $D_{\xi }G( {\widetilde{\xi }})$ has full row rank. In this regard, we show that if $\Delta D_{\xi }G( {\widetilde{\xi }})=0$, then $\Delta =0$ where $\Delta :=((\Delta x_{h},\Delta \lambda _{h})_{h\in {\mathcal {H}}},(\Delta y_{j},\Delta \alpha _{j})_{j\in {\mathcal {J}}},\Delta p^{\backslash })\in {\mathbb {R}}^{\dim \Xi }$. The system $\Delta D_{\xi }G( {\widetilde{\xi }})=0$ is given below.

$$\begin{aligned} \left\{ \begin{array}{l} (h.1)\;\;\Delta x_{h} D_{x_{h}}^{2}{\overline{u}}_{h}\left( {\widetilde{x}}_{h}\right) - \Delta \lambda _{h} {\widetilde{p}} + \Delta p^{\backslash } \left[ I_{C-1}|0\right] =0, \quad \forall \;h\in {\mathcal {H}} \\ (h.2) \;\; - \Delta x_{h} \cdot {\widetilde{p}}=0, \forall \;h\in {\mathcal {H}} \\ (j.1) \;\; \sum \limits _{h\in {\mathcal {H}}} \Delta \lambda _{h} s_{jh} {\widetilde{p}} - {\widetilde{\alpha }}_{j} \Delta y_{j} D_{y_{j}}^{2}{\overline{t}}_{j}\left( {\widetilde{y}}_{j}\right) -\Delta \alpha _{j} D_{y_{j}} {\overline{t}}_{j}\left( {\widetilde{y}}_{j}\right) - \Delta p^{\backslash } \left[ I_{C-1}|0\right] =0, \quad \forall \;j\in {\mathcal {J}} \\ (j.2) \;\; - \Delta y_{j} \cdot D_{y_{j}} {\overline{t}}_{j}\left( {\widetilde{y}}_{j}\right) =0, \quad \forall \;j\in {\mathcal {J}} \\ (M) \;\; -{\displaystyle \sum _{h \in {\mathcal {H}}}} {\widetilde{\lambda }}_{h} \Delta x_{h}^{\backslash } + \sum _{j\in {\mathcal {J}}}\Delta y_{j}^{ \backslash }=0 \end{array} \right. \end{aligned}$$

We first prove that $\Delta x_{h}=0$ for all $h\in {\mathcal {H}}$. Otherwise, suppose that there is ${\overline{h}}\in {\mathcal {H}}$ such that $\Delta x_{{\overline{h}}} \ne 0$. The proof goes through the two following claims that contradict each others.

We first claim that $\Delta p^{\backslash }\cdot (\sum \nolimits _{h\in {\mathcal {H}}} {\widetilde{\lambda }}_{h} \Delta x_{h}^{\backslash })>0$. Multiplying (h.1) by ${\widetilde{\lambda }}_{h} \Delta x_{h}$ and summing over h, from (h.2) we get $\sum \nolimits _{h\in {\mathcal {H}}} {\widetilde{\lambda }}_{h} \Delta x_{h}D_{x_{h}}^{2}{\overline{u}}_{h}({\widetilde{x}}_{h}) (\Delta x_{h})=-\Delta p^{\backslash }\cdot ( \sum \nolimits _{h\in {\mathcal {H}}} {\widetilde{\lambda }}_{h} \Delta x_{h}^{\backslash })$. Multiplying $G^{h.1}({\widetilde{\xi }})=0$ by $\Delta x_{h}$ and using (h.2), we get $\Delta x_{h} \cdot D_{x_{h}} {\overline{u}}_{h}({\widetilde{x}}_{h})=0$ for all h. Therefore, Point 3 of Assumption 6 completes the proof of the claim since ${\widetilde{\lambda }}_{h}>0$ for all h and $\Delta x_{{\overline{h}}}\ne 0$.

Second, we claim that $\Delta p^{\backslash }\cdot (\sum \nolimits _{h\in {\mathcal {H}}} {\widetilde{\lambda }}_{h} \Delta x_{h}^{\backslash })\le 0$. Multiplying both sides of $G^{j.1}({\widetilde{\xi }})=0$ by $\Delta y_{j}$ and using (j.2), we get $\Delta y_{j} \cdot {\widetilde{p}} =0$. Then, multiplying (j.1) by $\Delta y_{j}$ and summing over j, from (j.2) we get $- \sum \nolimits _{j\in {\mathcal {J}}} {\widetilde{\alpha }}_{j} \Delta y_{j} D_{y_{j}}^{2} {\overline{t}}_{j}({\widetilde{y}}_{j})(\Delta y_{j})=\Delta p^{\backslash }\cdot {\sum \nolimits _{j\in {\mathcal {J}}}}\Delta y_{j}^{\backslash }$. Since, ${\widetilde{\alpha }}_{j}>0$ for all j, Point 3 of Assumption 1 and (j.2) imply that $\Delta p^{\backslash }\cdot {\sum \nolimits _{j\in {\mathcal {J}}}}\Delta y_{j}^{\backslash } \le 0.$ Using (M), the claim is completely proved.

Since ${\widetilde{p}}^{C}=1$ and $\Delta x_{h} = 0$ for all $h\in {\mathcal {H}}$, from (h.1) we get $\Delta \lambda _{h}=0$ for all h, and then $\Delta p^{\backslash }=0$. Thus, multiplying (j.1) by $\Delta y_{j}$, Point 3 of Assumption 1 and (j.2) imply that $\Delta y_{j}=0$. Therefore, using once again (j.1), we get $\Delta \alpha _{j}=0$ by Point 4 of Assumption 1. Thus, we get $\Delta = 0$. $\square$

Proof of Proposition 16

Observe that $H^{-1}(0)=\Phi ^{-1}(0)\cup \Gamma ^{-1}(0)$. Since the union of a finite number of compact sets is compact, it is enough to show that $\Phi ^{-1}(0)$ and $\Gamma ^{-1}(0)$ are compact. $\square$

Claim 1

$\Phi ^{-1}(0)$ is compact.

We prove that, up to a subsequence, every sequence $( \xi ^{\nu }, \tau ^{\nu }) _{\nu \in {\mathbb {N}}} \subseteq \Phi ^{-1}(0)$ converges to an element of $\Phi ^{-1}( 0)$, where $\xi ^{\nu }:=(x^{\nu },\lambda ^{\nu }, y^{\nu },\alpha ^{\nu },p^{\nu \;\backslash })_{\nu \in {\mathbb {N}}}$. Since $\{\tau ^{\nu }: \nu \in {\mathbb {N}}\}\subseteq [0,1]$, up to a subsequence, $( \tau ^{\nu }) _{\nu \in {\mathbb {N}}}$ converges to some $\tau ^{*} \in [0,1]$. From Steps 1.1–1.4 below, up to a subsequence, $(\xi ^{\nu })_{\nu \in {\mathbb {N}}}$ converges to some $\xi ^{*}:=(x^{*},\lambda ^{*}, y^{*},\alpha ^{*},p^{*\;\backslash })\in \Xi$. Since $\Phi$ is continuous, taking the limit, one gets $(\xi ^{*},\tau ^{*})\in \Phi ^{-1}( 0)$.

We remind that for every $\tau \in [0,1], e_{h}({\tau })$ defined in (11) is given by

$$\begin{aligned} e_{h}({\tau })= \tau e_{h}+(1-\tau ) {\widetilde{e}}_{h} \end{aligned}$$

Step 1.1. Up to a subsequence, $(x^{\nu },y^{\nu })_{\nu \in {\mathbb {N}}}$ converges to some $(x^{*},y^{*})\in {\mathbb {R}}_{+}^{\textit{CH}}\times {\mathbb {R}}^{\textit{CJ}}$. We show that for $r=\sum \nolimits _{h\in {\mathcal {H}}}e_{h}$, the sequence $(x^{\nu },y^{\nu })_{\nu \in {\mathbb {N}}}$ is included in the bounded set K(r) given by Lemma 5. By $\Phi ^{j.2}(\xi ^{\nu },\tau ^{\nu })=0$, for every j we get

$$\begin{aligned} t_{j}\left( y_{j}^{\nu },{\overline{y}}_{-j},{\overline{x}}\right) =0, \quad \forall \; \nu \in {\mathbb {N}} \end{aligned}$$

Thus, the sequence $(y^{\nu })_{\nu \in {\mathbb {N}}}$ is included in the set $Y({\overline{x}},{\overline{y}})$ given by (1). Summing $\Phi ^{h.2}(\xi ^{\nu },\tau ^{\nu })=0$ over h, by $\Phi ^{M}(\xi ^{\nu },\tau ^{\nu })=0$ we have $\sum \nolimits _{h\in {\mathcal {H}}} x_{h}^{\nu } - \sum \nolimits _{j\in {\mathcal {J}}} y_{j}^{\nu }= \sum \nolimits _{h\in {\mathcal {H}}} e_{h}( \tau ^{\nu })$ for every $\nu \in {\mathbb {N}}$. Using the definition of ${\widetilde{e}}_h$ given by (7) and Proposition 13, one gets $\sum \nolimits _{h\in {\mathcal {H}}} e_{h}( \tau ) = r$ for every $\tau \in [0,1]$, and then $\sum \nolimits _{h\in {\mathcal {H}}} x_{h}^{\nu } - \sum \nolimits _{j\in {\mathcal {J}}} y_{j}^{\nu } = r$ for every $\nu \in {\mathbb {N}}$. Thus, $(x^{\nu },y^{\nu })_{\nu \in {\mathbb {N}}} \subseteq A({\overline{x}},{\overline{y}};r) \subseteq K(r)$. Consequently, $(x^{\nu },y^{\nu })_{\nu \in {\mathbb {N}}}$ is included in $\mathrm{cl}\,K(r)$ which is a compact set. Therefore, up to a subsequence, $(x^{\nu },y^{\nu })_{\nu \in {\mathbb {N}}}$ converges to some $(x^{*},y^{*})\in \mathrm{cl}\,K(r) \subseteq {\mathbb {R}}_{+}^{\textit{CH}}\times {\mathbb {R}}^{\textit{CJ}}$, and then $(x^{*},y^{*})\in {\mathbb {R}}_{+}^{CH}\times {\mathbb {R}}^{\textit{CJ}}$.

Step 1.2. The consumption allocation $x^{*}$ is strictly positive, i.e. $x_{h}^{*} \gg 0$ for every $h\in {\mathcal {H}}$. By $\Phi ^{h.1}(\xi ^{\nu },\tau ^{\nu })=\Phi ^{h.2}(\xi ^{\nu },\tau ^{\nu })=0$ and KKT sufficient conditions, $x_{h}^{\nu }$ solves the following problem for every $\nu \in {\mathbb {N}}$.

$$\begin{aligned} \begin{array}{ll} \max \limits _{x_{h}\in {\mathbb {R}}_{++}^{C}} \; u_{h}\left( x_{h}, {\overline{x}}_{- h}, {\overline{y}}\right) \\ \begin{aligned} \text {subject to } &{} p^{\nu }\cdot x_{h} \le p^{\nu }\cdot \left[ \tau ^{\nu } e_{ h}+\left( 1-\tau ^{\nu }\right) {\widetilde{x}}_{h}\right] + p^{\nu }\cdot \sum _{j\in {\mathcal {J}}}s_{jh}\left( y_{j}^{\nu }- \left( 1-\tau ^{\nu }\right) {\widetilde{y}}_{j}\right) \end{aligned} \end{array}\nonumber \\ \end{aligned}$$

(19)

We first claim that for every $\nu \in {\mathbb {N}}$, the following vector

$$\begin{aligned} {\widehat{e}}_{h}\left( \tau ^{\nu }\right) :=\tau ^{\nu }e_{ h}+\left( 1-\tau ^{\nu }\right) {\widetilde{x}}_{h} \end{aligned}$$

(20)

belongs to the budget constraint of the problem above. By $\Phi ^{j.1}(\xi ^{\nu },\tau ^{\nu })=\Phi ^{j.2}(\xi ^{\nu },\tau ^{\nu })=0$ and KKT sufficient conditions, $y_{j}^{\nu }$ solves the following problem for every $\nu \in {\mathbb {N}}$.

$$\begin{aligned} \begin{array}{l} \max \limits _{y_{j}\in {\mathbb {R}}^{C}}p^{\nu }\cdot y_{j} \\ \text {subject to } t_{j}\left( y_{j},{\overline{y}}_{- j},{\overline{x}}\right) \le 0 \end{array} \end{aligned}$$

(21)

$t_{j}({\widetilde{y}}_{j},{\overline{y}}_{- j},{\overline{x}})= 0$ since $G^{j.2}({\widetilde{\xi }})=0$, see (8). By Point 2 of Assumption 1, $t_{j}(0,{\overline{y}}_{- j},{\overline{x}})= 0$. Then, we get $t_{j}((1-\tau ^{\nu }){\widetilde{y}}_{j},{\overline{y}}_{-j},{\overline{x}})< 0$ since $t_{j}(\cdot ,{\overline{y}}_{-j},{\overline{x}})$ is strictly quasi-convex, that is, the production plan $(1-\tau ^{\nu }){\widetilde{y}}_{j}$ belongs to the constraint set of problem (21). Thus, $p^{\nu }\cdot (y_{j}^{\nu } -(1-\tau ^{\nu }){\widetilde{y}}_{j})\ge 0$ for every j, and then $p^{\nu }\cdot \sum \nolimits _{j\in {\mathcal {J}}}s_{jh}( y_{j}^{\nu }- (1-\tau ^{\nu }){\widetilde{y}}_{j}) \ge 0$ which completes the proof of the claim.

Therefore, for every $\nu \in {\mathbb {N}}, u_{h}(x_{h}^{\nu }, {\overline{x}}_{- h}, {\overline{y}}) \ge u_{h}( {\widehat{e}}_{h}(\tau ^{\nu }), {\overline{x}}_{- h}, {\overline{y}})$. By Point 2 of Assumption 6, for every $\varepsilon >0$ we get $u_{h}(x_{h}^{\nu }+\varepsilon \mathbf{1}, {\overline{x}}_{- h}, {\overline{y}}) > u_{h}( {\widehat{e}}_{h}(\tau ^{\nu }), {\overline{x}}_{- h}, {\overline{y}})$ where $\mathbf{1}:=(1,\dots ,1)\in {\mathbb {R}}_{++}^{C}$. Taking the limit over $\nu$ and using Point 1 of Assumption 6, we get $u_{h}(x_{h}^{*} +\varepsilon \mathbf{1}, {\overline{x}}_{- h}, {\overline{y}}) \ge u_{h}( {\widehat{e}}_{h}(\tau ^{*}), {\overline{x}}_{- h}, {\overline{y}})$ for every $\varepsilon >0$. Then, $x_{h}^{*}$ belongs to the closure of the upper counter set of $({\widehat{e}}_{h}(\tau ^{*}), {\overline{x}}_{- h}, {\overline{y}})$, which is included in ${\mathbb {R}}_{++}^{C}$ by Point 4 of Assumption 6. Thus, $x_{h}^{*} \gg 0$.

Step 1.3. Up to a subsequence, $(\lambda ^{\nu },p^{\nu \;\backslash }) _{\nu \in {\mathbb {N}}}$ converges to some $\lambda ^{*} \in {\mathbb {R}}_{++}^{H}\times {\mathbb {R}}_{++}^{C-1}$. The proof is similar to the proof of Step 2.3.

Step 1.4. Up to a subsequence, $(\alpha ^{\nu })_{\nu \in {\mathbb {N}}}$ converges to some $\alpha ^{*} \in {\mathbb {R}}_{++}^{J}$. The proof is similar to the proof of Step 2.4.

Claim 2

$\Gamma ^{-1}(0)$ is compact.

Let $( \xi ^{\nu }, \tau ^{\nu }) _{\nu \in {\mathbb {N}}}$ be a sequences in $\Gamma ^{-1}( 0)$. As in Claim $1, ( \tau ^{\nu }) _{\nu \in {\mathbb {N}}}$ converges to $\tau ^{*}\in [0,1]$. From Steps 2.1–2.4 below, up to a subsequence, $(\xi ^{\nu })_{\nu \in {\mathbb {N}}}$ converges to an element $\xi ^{*}:=( x^{*},\lambda ^{*}, y^{*},\alpha ^{*},p^{*\backslash }) \in \Xi$. Since $\Gamma$ is a continuous function, taking limit, we get $(\xi ^{*},\tau ^{*})\in \Gamma ^{-1}( 0)$.

We remind that for every $\tau \in [0,1], x({\tau })$ and $y({\tau })$ defined in (11) are given by

$$\begin{aligned} x({\tau })= \tau x+(1-\tau ) {\overline{x}} \quad \text { and }\quad y({\tau })=\tau y+(1-\tau ) {\overline{y}} \end{aligned}$$

Step 2.1. Up to a subsequence, $(x^{\nu },y^{\nu })_{\nu \in {\mathbb {N}}}$ converges to some $(x^{*},y^{*})\in {\mathbb {R}}_{+}^{CH}\times {\mathbb {R}}^{CJ}$. We show that for $r=\sum \nolimits _{h\in {\mathcal {H}}}e_{h}$, the sequence $(x^{\nu },y^{\nu })_{\nu \in {\mathbb {N}}}$ is included in the bounded set K(r) given by Lemma 5. Then, one completes the proof as in Step 1.1. By $\Gamma ^{j.2}(\xi ^{\nu },\tau ^{\nu })=0$, for every j we have that

$$\begin{aligned} t_{j}\left( y_{j}^{\nu },y_{-j}^{\nu }\left( \tau ^{\nu }\right) ,x^{\nu }\left( \tau ^{\nu }\right) \right) =0, \quad \forall \nu \in {\mathbb {N}} \end{aligned}$$

Thus, for every $\nu \in {\mathbb {N}}$, the production allocation $y^{\nu }$ belongs to the set $Y(x^{\nu }(\tau ^{\nu }),y^{\nu }(\tau ^{\nu }))$ given by (1). Now, summing $\Gamma ^{h.2}(\xi ^{\nu },\tau ^{\nu })=0$ over h, by $\Gamma ^{M}(\xi ^{\nu },\tau ^{\nu })=0$, we get $\sum \nolimits _{h\in {\mathcal {H}}}x_{h}^{\nu }- \sum \nolimits _{j\in {\mathcal {J}}} y_{j}^{\nu }=r$. Then, for every $\nu \in {\mathbb {N}}$, the allocation $(x^{\nu },y^{\nu })$ belongs to the set $A(x^{\nu }(\tau ^{\nu }),y^{\nu }(\tau ^{\nu });r) \subseteq K(r)$, and consequently, $(x^{\nu },y^{\nu })_{\nu \in {\mathbb {N}}} \subseteq K(r)$.

Step 2.2. The consumption allocation $x^{*}$ is strictly positive, i.e. $x_{h}^{*} \gg 0$ for every $h\in {\mathcal {H}}$. The argument is similar to the one used in Step 1.2 except for the last part which is quite different due to the presence of consumption externalities in the utility functions. For this reason, at the end of this step we need Assumption 7 or, alternatively, Assumption 8.

First, according to $\Gamma ^{h.1}(\xi ^{\nu },\tau ^{\nu })=\Gamma ^{h.2}(\xi ^{\nu },\tau ^{\nu })=0$, replace problem (19) with the following problem

$$\begin{aligned} \begin{array}{ll} \max \limits _{x_{h}\in {\mathbb {R}}_{++}^{C}} u_{h}\left( x_{h}, x^{\nu }_{- h}\left( \tau ^{\nu }\right) , y^{\nu }\left( \tau ^{\nu }\right) \right) \\ \begin{aligned} \text {subject to } &{}p^{\nu }\cdot x_{h} \le p^{\nu }\cdot e_{h}+p^{\nu }\cdot \sum _{j\in {\mathcal {J}}}s_{jh} y_{j}^{\nu } \end{aligned} \end{array} \end{aligned}$$

(22)

and replace ${\widehat{e}}_h(\tau ^{\nu })$ given by (20) with ${\widehat{e}}_h(\tau ^{\nu }):=e_h$. Second, according to $\Gamma ^{j.1}(\xi ^{\nu },\tau ^{\nu })=\Gamma ^{j.2}(\xi ^{\nu },\tau ^{\nu })=0$, replace problem (21) with the following problem

$$\begin{aligned} \begin{array}{l} \max \limits _{y_{j}\in {\mathbb {R}}^{C}}p^{\nu }\cdot y_{j} \\ \text {subject to } t_{j}\left( y_{j},y_{-j}^{\nu }\left( {\tau ^{\nu }}\right) ,x^{\nu }\left( {\tau ^{\nu }}\right) \right) \le 0 \end{array} \end{aligned}$$

Third, one follows the same strategy as for Step 1.2. For every $\nu \in {\mathbb {N}}, u_{h}(x_{h}^{\nu }, x^{\nu }_{- h}(\tau ^{\nu }), y^{\nu }(\tau ^{\nu })) \ge u_{h}( e_{h}, x^{\nu }_{- h}(\tau ^{\nu }), y^{\nu }(\tau ^{\nu }))$. Notice that $x^{\nu }_{- h}(\tau ^{\nu })$ belongs to ${\mathbb {R}}_{++}^{C(H-1)}$, because $x^{\nu }_{- h}(\tau ^{\nu })= \tau ^{\nu } x^{\nu }_{- h}+(1-\tau ^{\nu }) {\overline{x}}_{- h}$, and $x^{\nu }_{- h}$ and ${\overline{x}}_{- h}$ are both strictly positive. Then, by Point 2 of Assumption 6, for every $\nu \in {\mathbb {N}}$ and for every $\varepsilon >0$ we get

$$\begin{aligned} u_{h}\left( x_{h}^{\nu }+\varepsilon \mathbf{1}, x^{\nu }_{- h}\left( \tau ^{\nu }\right) , y^{\nu }\left( \tau ^{\nu }\right) \right) > u_{h}\left( e_{h}, x^{\nu }_{- h}\left( \tau ^{\nu }\right) , y^{\nu }\left( \tau ^{\nu }\right) \right) \end{aligned}$$

(23)

Now, take the limit over $\nu$. Differently from Step 1.2, we have the two following cases: $x^{*}_{-h}(\tau ^{*}) \gg 0$ or $x^{*}_{-h}(\tau ^{*}) \in \mathrm{Bd}\,( {\mathbb {R}}_{++}^{C(H-1)})$. The second case may happen here, but not in Step 1.2, because now there are consumption externalities in the utility functions. Indeed, if $( \tau ^{\nu }) _{\nu \in {\mathbb {N}}}$ converges to $\tau ^{*}=1$, then $x^{*}_{- h}(\tau ^{*})=x^{*}_{-h}$ which a priori is not necessarily strictly positive.

Suppose that $x^{*}_{-h}(\tau ^{*}) \gg 0$. Using (23) and Point 1 of Assumption 6, we get $u_{h}(x_{h}^{*} +\varepsilon \mathbf{1}, x^{*}_{-h}(\tau ^{*}), y^{*}(\tau ^{*})) \ge u_{h}( e_{h}, x^{*}_{-h}(\tau ^{*}), y^{*}(\tau ^{*}))$ for every $\varepsilon >0$. Then, $x_{h}^{*}$ is strictly positive, because it belongs to the closure of the upper counter set of $(e_{h}, x^{*}_{-h}(\tau ^{*}), y^{*}(\tau ^{*}))$, which is included in ${\mathbb {R}}_{++}^{C}$ by Point 4 of Assumption 6.

Suppose that $x^{*}_{-h}(\tau ^{*}) \in \mathrm{Bd}\,( {\mathbb {R}}_{++}^{C(H-1)})$. Under Assumption 7, the previous argument still holds true.

Alternatively, if Assumption 7 is not satisfied, one uses Assumption 8 as follows. By Assumption 8 and (23), there exists $\delta >0$ such that for every $\nu \in {\mathbb {N}}$ and for every $\varepsilon >0$ one gets

$$\begin{aligned} u_{h}\left( x_{h}^{\nu }+\varepsilon \mathbf{1}, x^{\nu }_{- h}\left( \tau ^{\nu }\right) +\delta \mathbf{1}, y^{\nu }\left( \tau ^{\nu }\right) \right) \ge u_{h}\left( e_{h}, x^{\nu }_{- h}\left( \tau ^{\nu }\right) +\delta \mathbf{1}, y^{\nu }\left( \tau ^{\nu }\right) \right) \end{aligned}$$

Take the limit over $\nu$ and remark that $x^{*}_{-h}(\tau ^{*}) +\delta \mathbf{1} \gg 0$. Consequently, from Point 1 of Assumption 6 one gets $u_{h}(x_{h}^{*} +\varepsilon \mathbf{1}, x^{*}_{-h}(\tau ^{*}) +\delta \mathbf{1}, y^{*}(\tau ^{*})) \ge u_{h}( e_{h}, x^{*}_{-h}(\tau ^{*}) +\delta \mathbf{1}, y^{*}(\tau ^{*}))$ for every $\varepsilon >0$. Then, $x_{h}^{*}$ is strictly positive, because it belongs to the closure of the upper counter set of $(e_{h}, x^{*}_{-h}(\tau ^{*}) +\delta \mathbf{1}, y^{*}(\tau ^{*}))$, which is included in ${\mathbb {R}}_{++}^{C}$ by Point 4 of Assumption 6.

Step 2.3. Up to a subsequence, $(\lambda ^{\nu },p^{\nu \;\backslash }) _{\nu \in {\mathbb {N}}}$ converges to some $(\lambda ^{*}, p^{*\;\backslash }) \in {\mathbb {R}}_{++}^{H} \times {\mathbb {R}}_{++}^{C-1}$. By $\Gamma ^{h.1}(\xi ^{\nu },\tau ^{\nu })=0$, fixing commodity C, for every $\nu \in {\mathbb {N}}$ we have $\lambda _{h}^{\nu }=D_{x_{h}^{C}}u_{h}(x_{h}^{\nu }, x_{-h}^{\nu }(\tau ^{\nu }),y^{\nu }(\tau ^{\nu }))$. Taking the limit, by Points 1 and 2 of Assumption 6, we get $\lambda _{h}^{*}:=D_{x_{h}^{C}}u_{h}(x_{h}^{*}, x_{-h}^{*}(\tau ^{*}),y^{*}(\tau ^{*})) >0$.

By $\Gamma ^{h.1}(\xi ^{\nu },\tau ^{\nu })=0$, for all commodity $c\ne C$ and for all $\nu \in {\mathbb {N}}$ we have $p^{\nu \;c}=\frac{D_{x_{h}^{c}}u_{h}(x_{h}^{\nu }, x_{-h}^{\nu }(\tau ^{\nu }),y^{\nu }(\tau ^{\nu }))}{\lambda _{h}^{\nu }}$. Taking the limit, by Points 1 and 2 of Assumption 6, we get $p^{*\;c}:=\frac{D_{x_{h}^{c}}u_{h}(x_{h}^{*}, x_{-h}^{*}(\tau ^{*}),y^{*}(\tau ^{*}))}{\lambda _{h}^{*}} >0$. Therefore, $p^{*\;\backslash } \gg 0$.

Step 2.4. Up to a subsequence, $(\alpha ^{\nu })_{\nu \in {\mathbb {N}}}$ converges to some $\alpha ^{*} \in {\mathbb {R}}_{++}^{J}$. For every firm j, fix a commodity $c(j) \in {\mathcal {C}}$. By $\Gamma ^{j.1}(\xi ^{\nu },\tau ^{\nu })=0$, for every $\nu \in {\mathbb {N}}$ we have that

$$\begin{aligned} \alpha _{j}^{\nu }= \frac{p^{\nu c(j)}}{ D_{y_{j}^{c(j)}}t_{j}\left( y_{j}^{\nu },y_{-j}^{\nu }\left( \tau ^{\nu }\right) , x^{\nu }\left( \tau ^{\nu }\right) \right) } \end{aligned}$$

which is strictly positive by Point 4 of Assumption 1. Taking the limit, by Points 1 and 4 of Assumption 1, we get $\alpha _{j}^{*}:= \frac{p^{*\; c(j)}}{D_{y_{j}^{c(j)}}t_{j}(y_{j}^{*},y_{-j}^{*}(\tau ^{*}), x^{*}(\tau ^{*}))}>0$. $\square$