Marginal Pricing and Marginal Cost Pricing Equilibria in Economies with Externalities and Infinitely Many Commodities

Abstract

This paper considers a general equilibrium model of an economy in which some firms may exhibit various types of non-convexities in production, there are external effects among agents and the commodity space is infinite dimensional. The consumption sets, the preferences of the consumers and the production possibilities are represented by correspondences in order to take into account the external effects. The firms are instructed to follow the marginal pricing rule from which we obtain an existence theorem. Then, the existence of a marginal cost pricing equilibrium is proved by adding additional assumptions. The simultaneous presence of externalities and infinitely many commodities are sources of technical difficulties when attempting to generalize previous existence results in the literature.

Appendix

7.1.1 Proof of Proposition 1

Before proving Proposition 1, we show that, given Assumption P, for every z ∈ L ^m+n and every j, $\overline{\mathrm{int}Y _{j}\left (z\right )} = Y _{j}\left (z\right )$. Let y belong to $Y _{j}\left (z\right )$. If y belongs to $\mathrm{int}Y _{j}\left (z\right )$, then y belongs to $\overline{\mathrm{int}Y _{j}\left (z\right )}$. If y belongs to $\partial _{\infty }Y _{j}\left (z\right )$, then for all ɛ > 0, $y - \frac{\varepsilon } {2}\chi _{M}$ belongs to $\mathrm{int}Y _{j}\left (z\right )$ by free disposal. Consequently, $B\left (y,\varepsilon \right ) \cap \mathrm{int}Y _{j}\left (z\right )\neq \varnothing$ for all ɛ > 0, and thus, y belongs to $\overline{\mathrm{int}Y _{j}\left (z\right )}.$

We now prove that the correspondence Y _j : L ^m+n ↦ L is $\left (\prod \nolimits _{L^{m+n}}\sigma ^{\infty },\mathcal{T}\right )$-l.h.c. From the above result and Lemma 14.21 in Aliprantis and Border (1994), it is enough to prove that intY _j : L ^m+n ↦ L is $\left (\prod \nolimits _{L^{m+n}}\sigma ^{\infty },\mathcal{T}\right )$-l.h.c. Let y ∈ $\mathrm{int}Y _{j}\left (z\right )$ and let z ^α be a net which $\prod \nolimits _{L^{m+n}}\sigma ^{\infty }-$ converges to z. Since f _j is $\sigma ^{\infty }\times \prod \nolimits _{L^{m+n}}\sigma ^{\infty }$-continuous, there exists α ₀ ∈ Γ such that α > α ₀ implies $f_{j}\left (y,z^{\alpha }\right ) < 0$. Hence, there exists a net y ^α $\left (= y\right ) \in \mathrm{int}Y _{j}\left (z^{\alpha }\right )$ for all α and y ^α → y. 

The weak* closeness of Y _j is immediate from Assumption P(ii).

Remark 3.

Given Assumption P, if the correspondence Y _j is convex valued, then $\overline{\mathrm{int}Y _{j}\left (z\right )} = Y _{j}\left (z\right )$ without free disposal requirement (Schaefer and Wolf 1999, p. 38, 1.3). On the other hand, we can repeat the argument made above to show that the correspondence Y _j ^F: F ^m+n ↦ F is l.h.c.

7.1.2 Proof of Proposition 2

We omit the index j in order to simplify the notation. We first state the following Lemma:

Lemma 4.

For a given $\overline{z} \in L^{m+n}$ , let $T_{Y \left (\overline{z}\right )}\left (\overline{y}\right )$ be the Clarke tangent cone of $Y \left (\overline{z}\right )$ at $\overline{y}$ . Let $v \in T_{Y \left (\overline{z}\right )}\left (\overline{y}\right )$ and δ > 0. There exist weak* open neighbourhoods of $\overline{z}$ and $\overline{y}$, $W^{\overline{z}}$ and $W\overline{^{y}}$ , respectively, such that for all ɛ > 0, for all $z \in W^{\overline{z}}$ and for all $y \in W\overline{^{y}} \cap B\left (\overline{y},\varepsilon \right ),$ $v + \overline{y} - y -\delta \chi _{M} \in T_{Y \left (z\right )}\left (y\right )$.

Proof.

Given $\overline{z} \in Z,$ we have to prove that $\nabla _{1}f\left (\overline{y},\overline{z}\right )\left (v + \overline{y} - y -\delta \chi _{M}\right ) \leq 0$. Let $0 <\alpha < \frac{\delta \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (\chi _{M}\right )} {2\left (\left \Vert v\right \Vert +\varepsilon +\delta \right )}.$ From Assumption P(v), there exists a $\prod \nolimits _{L^{m+n+1}}\sigma ^{\infty }-$ open neighbourhood of $\left (\overline{z},\overline{y}\right ),$ $U^{\overline{z}} \times U\overline{^{y}},$ such that for all $\left (z,y\right ) \in U^{\overline{z}} \times U\overline{^{y}}$, $\left \vert \nabla _{1}f\left (y,z\right ) -\nabla _{1}f\left (\overline{y},\overline{z}\right )\right \vert <\alpha$. Let us consider the following σ ^∞−open neighbourhood of $\overline{y,}$

$$\displaystyle{V \overline{^{y}} = \left \{y \in L: \left \vert \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (\overline{y} - y\right )\right \vert < \frac{\delta \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (\chi _{M}\right )} {2} \right \}}$$

Let $W\overline{^{y}} = U\overline{^{y}} \cap V \overline{^{y}}$, $W^{\overline{z}} = U^{\overline{z}}$ and ɛ > 0. For all $\left (y,z\right ) \in W\overline{^{y}} \cap B\left (\overline{y},\varepsilon \right ) \times W^{\overline{z}}$,

$$\displaystyle\begin{array}{rcl} \nabla _{1}f\left (y,z\right )\left (v + \overline{y} - y -\delta \chi _{M}\right )& =& \left (\nabla _{1}f\left (y,z\right ) -\nabla _{1}f\left (\overline{y},\overline{z}\right )\right )\left (v + \overline{y} - y -\delta \chi _{M}\right ) {}\\ & & +\nabla _{1}f\left (\overline{y},\overline{z}\right )\left (v + \overline{y} - y -\delta \chi _{M}\right ) {}\\ & & <\alpha \left (\left \Vert v\right \Vert +\varepsilon +\delta \right ) {}\\ & +& \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (v\right ) + \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (\overline{y} - y\right ) -\delta \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (\chi _{M}\right ) {}\\ & & < \frac{\delta \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (\chi _{M}\right )} {2} + \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (v\right ) {}\\ & & +\frac{\delta \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (\chi _{M}\right )} {2} -\delta \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (\chi _{M}\right ) {}\\ & =& \nabla _{1}f\left (\overline{y},\overline{z}\right )\left (v\right ) \leq 0 {}\\ \end{array}$$

We now proceed to the proof of Proposition 2. Let $\left (z^{\alpha },\pi ^{\alpha }\right )_{\left (\varGamma,\leq \right )}$ be a net of Z × S, $\prod \nolimits _{L^{m+n}}\sigma ^{\infty }\times \sigma ^{ba}-$ converging to $\left (\overline{z},\overline{\pi }\right )$. Let $v \in T_{Y \left (\overline{z}\right )}\left (\overline{y}\right )$ and δ > 0. There exist ɛ > 0 and α ₀ ∈ Γ such that for all α > α ₀, $y^{\alpha } \in B\left (0,\varepsilon \right )$. We note that $\left \Vert y^{\alpha } -\overline{y}\right \Vert <\varepsilon +\left \Vert \overline{y}\right \Vert =\varepsilon ^{{\prime}}$. Hence, for all α > α ₀, $y^{\alpha } \in B\left (\overline{y},\varepsilon ^{{\prime}}\right )$. From the above lemma, there exist weak*-open neighbourhoods of $\overline{z}$ and $\overline{y}$, $W^{\overline{z}}$ and $W\overline{^{y}}$, respectively, such that for ɛ ^′ > 0 and all α > α ₀, $\left (y^{\alpha },z^{\alpha }\right ) \in W\overline{^{y}} \cap B\left (\overline{y},\varepsilon ^{{\prime}}\right ) \times W^{\overline{z}}$ and $v + \overline{y} - y^{\alpha } -\delta \chi _{M} \in T_{Y \left (z^{\alpha }\right )}\left (y^{\alpha }\right )$.

Since $\pi ^{\alpha } \in N_{Y \left (z^{\alpha }\right )}\left (y^{\alpha }\right )$, $\pi ^{\alpha }\left (v + \overline{y} - y^{\alpha } -\delta \chi _{M}\right ) \leq 0$ for all α > α ₀. Passing to the limit, we obtain $\overline{\pi }\left (v\right ) + \overline{\pi }\left (\overline{y}\right ) -\mathop{\lim }\limits_{\alpha }\pi ^{\alpha }\left (y^{\alpha }\right )-\delta \leq 0$. Since 0 ∈ $T_{Y \left (\overline{z}\right )}\left (\overline{y}\right )$, $\overline{\pi }\left (\overline{y}\right ) \leq \mathop{\lim }\limits_{\alpha }\pi ^{\alpha }\left (y^{\alpha }\right )+\delta$, and since this inequality holds true for all δ > 0, we have $\overline{\pi }\left (\overline{y}\right ) \leq \mathop{\lim }\limits_{\alpha }\pi ^{\alpha }\left (y^{\alpha }\right )$.

Let $v \in T_{Y \left (\overline{z}\right )}\left (\overline{y}\right ).$ If $\mathop{\lim }\limits_{\alpha }\pi ^{\alpha }\left (y^{\alpha }\right ) = \overline{\pi }\left (\overline{y}\right ),$ then $\overline{\pi }\left (v\right ) \leq 0$. Consequently, $\overline{\pi } \in N_{Y \left (\overline{z}\right )}\left (\overline{y}\right ) \cap S$ since π ^α ∈ S for all α.

7.1.3 Proof of Proposition 4

We first state and prove the following lemma, which is used in the proof of Proposition 4. To simplify, we suppress index j.

Lemma 5.

Let $p_{I} \in \left (L^{I}\right )_{+}^{{\ast}},$ then there exists $\hat{p}_{I} \in L_{+}^{{\ast}}$ such that $\hat{p}_{I}\left (x\right ) = p_{I}\left (x^{I}\right )$ if x ∉ L ^O and $\hat{p}_{I}\left (x\right ) = 0$ if x ∈ L ^O.

Proof.

Let $p_{I} \in \left (L^{I}\right )_{+}^{{\ast}}$. By a classical extension theorem, there exists a functional $\tilde{p}_{I} \in L_{+}^{{\ast}}$, and hence, a measure $\tilde{v}_{I}$ ∈ ba ⁺ M, ℳ, μ such that $\tilde{p}_{I}\left (x\right ) =\int _{m\in M}x\left (m\right )d\tilde{v}_{I}\left (m\right )$ and $p_{I}\left (x\right ) =\tilde{ p}_{I}\left (x\right )$ for all x ∈ L ^I, since L ^∗ and $ba\left (M,\mathcal{M},\mu \right )$ are isometrically isomorphic (Dunford and Schwarz 1958). We now define the measure $\hat{v}_{I}$ as:

$\hat{v}_{I}\left (A\right ) = \left \{\begin{array}{c} \tilde{v}_{I}\left (A^{I}\right )\quad \text{ if }A \subsetneq O \\ 0\quad \text{ otherwise} \end{array} \right.$.

One easily checks that $\hat{v}_{I} \in ba^{+}\left (M,\mathcal{M},\mu \right )$ which is identified with a functional $\hat{p}_{I} \in L_{+}^{{\ast}}$. Take x ∉ L ^O. There exists M ^′ ⊂ I such that $\mu \left (M^{{\prime}}\right )\neq 0$ and $x^{I}\left (m\right )\neq 0$ for all m ∈ M ^′. Consequently, $\hat{p}_{I}\left (x\right ) =\hat{ p}_{I}\left (x^{I}\right )+\hat{p}_{I}\left (x^{O}\right ) =\int _{m\in M}x^{I}\left (m\right )d\hat{v}_{I}\left (m\right )+\int _{m\in M}x^{O}\left (m\right )d\hat{v}_{I}\left (m\right ) =\int _{m\in I}x^{I}\left (m\right )d\hat{v}_{I}\left (m\right ) =\int _{m\in I}x^{I}\left (m\right )d\tilde{v}_{I}\left (m\right ) =\tilde{ p}_{I}\left (x^{I}\right ) = p_{I}\left (x^{I}\right ).$ If x ∈ L ^O, $\hat{p}_{I}\left (x\right ) =\int _{m\in M}x^{O}\left (m\right )d\hat{v}_{I}\left (m\right ) = 0$.

Remark 4.

The above lemma can be rewritten in terms of the subspace $\left (L^{O}\right )^{{\ast}}$ as follows: for every $p_{O} \in \left (L^{O}\right )_{+}^{{\ast}},$ there exists a functional $\hat{p}_{O} \in L_{+}^{{\ast}}$ such that $\hat{p}_{O}\left (x\right ) = p_{O}\left (x^{O}\right )$ if x ∉ L ^I and $\hat{p}_{O}\left (x\right ) = 0$ if x ∈ L ^I.

First, we claim that for all t > 0, $-y_{I^{j}}$ does not belong to the relative interior of $Y _{j}\left (y_{O^{j}} + t\chi _{O^{j}},z\right )$. Otherwise, $y_{j} \in \mathrm{int}Y _{j}\left (z\right )$. We also note that for all t > 0, the relative interior of $Y _{j}\left (y_{O^{j}} + t\chi _{O^{j}},z\right )$ is non-empty. Finally, since for all t > 0, $Y _{j}\left (y_{O^{j}} + t\chi _{O^{j}},z\right )$ is convex, $\cup _{t>0}\mathrm{int}Y _{j}\left (y_{O^{j}} + t\chi _{O^{j}},z\right )$ is open, non-empty and convex (Schaefer and Wolf 1999, p. 38, 1.2).

Since $-y_{I^{j}}\notin \cup _{t>0}\mathrm{int}Y _{j}\left (y_{O^{j}} + t\chi _{O^{j}},z\right )$, there exists a continuous linear functional $p_{I^{j}} \in \left (L^{I^{_{j}} }\right )_{+}^{{\ast}}$ such that $p_{I^{j}}\left (-y_{I^{j}}\right ) \leq p_{I^{j}}\left (a\right )\forall a \in \cup _{t>0}\mathrm{int}Y _{j}\left (y_{O^{j}} + t\chi _{O^{j}},z\right )$,⁸ whence $p_{I^{j}}\left (-y_{I^{j}}\right ) \leq p_{I^{j}}\left (a^{{\prime}}\right )$ for all a ^′ ∈ ∪ _t > 0 $Y _{j}\left (y_{O^{j}}+\right.$ $\left.t\chi _{O^{j}},z\right )$. Consequently, $p_{I^{j}}\left (-y_{I^{j}}\right ) = c_{j}\left (p_{I^{j}},y_{O^{j}} + t\chi _{O^{j}},z\right )$ since $-y_{I^{j}} \in Y _{j}\left (y_{O^{j}} + t\chi _{O^{j}},z\right )$ for all t > 0. By the above lemma, we can extend the functional $p_{I^{j}}$ to an element of L ₊ ^∗—denoted by $p_{I^{j}}$ as well—such that $p_{I^{j}}\left (\xi \right ) = 0$ for all $\xi \in L^{O^{j} }$. Let $p_{O^{j}} = \bigtriangledown _{O^{j}}c_{j}\left (p_{I^{j}},y_{O^{j}},z\right ) \in \left (L_{+}^{O^{j} }\right )^{{\ast}}$. We also extend $p_{O^{j}}$ to L ₊ ^∗—denoted by $p_{O^{j}}$ as well—such that $p_{O^{j}}\left (\xi \right ) = 0$ for all $\xi \in L^{I^{j} }$. Consequently, by Lemma 2, $p_{j} = p_{I^{j}} + p_{O^{j}} \in N_{Y }{}_{j}\left (z\right )\left (y_{j}\right )$. Since, $\left (z,\pi \right )$ is a marginal pricing equilibrium, π = λ p _j for some λ > 0. Hence, $\pi _{I^{j}} =\lambda p_{I^{j}}$ and $\pi _{O^{j}} =\lambda p_{O^{j}} =\lambda \bigtriangledown _{O^{j}}c_{j}\left (p_{I^{j}},y_{O^{j}},z\right ) = \bigtriangledown _{O^{j}}c_{j}\left (\lambda p_{I^{j}},y_{O^{j}},z\right ) = \bigtriangledown _{O^{j}}c_{j}\left (\pi _{I^{j}},y_{O^{j}},z\right )$. Consequently, $\pi _{I^{j}}\left (-y_{I^{j}}\right ) = c_{j}\left (\pi _{I^{j}},y_{O^{j}},z\right )$ and $\pi =\pi _{I^{j}} + \bigtriangledown _{O^{j}}c_{j}\left (\pi _{I^{j}},y_{O^{j}},z\right ) \in N_{Y _{j}\left (z\right )}\left (y_{j}\right )$. Hence, conditions a., b’. and c’. of Definition 2 are satisfied.

Remark.

We point out that Bonnisseau and Cornet show that if $\left (z,\pi \right )$ is a marginal pricing equilibrium, then there exists a vector $\left (w_{j}\right )_{j=1}^{n}$ ∈ L ⁿ (our notation) defined as $w_{j} = y_{I^{j}} + y_{O^{j}}^{+}$, such that $\left (\left (x_{i}\right )_{i=1}^{m},\left (w_{j}\right )_{j=1}^{n},\pi \right )$ is a marginal cost pricing equilibrium. A significant difference between our approach and theirs is that in their case, $\left (\left (x_{i}\right )_{i=1}^{m},\left (w_{j}\right )_{j=1}^{n}\right ) \in \varPi _{i=1}^{m}X_{i} \times \varPi _{j=1}^{n}Y _{j}$, while in ours, if z ∈ Z, x _{i i = 1} ^m, w _{j j = 1} ⁿ may not be in $\varPi _{i=1}^{m}X_{i}\left (\left (\left (x_{i}\right )_{i=1}^{m},\left (w_{j}\right )_{j=1}^{n}\right )\right ) \times \varPi _{j=1}^{n}Y _{j}\left (\left (\left (x_{i}\right )_{i=1}^{m},\left (w_{j}\right )_{j=1}^{n}\right )\right )$ since the sets are not comparable. This justifies Assumption SPP.

Another important difference with the above paper is that, even if ​x _{i i = 1} ^m, w _{j j = 1} ⁿ belongs to $\varPi _{i=1}^{m}X_{i}\left (\left (x_{i}\right )_{i=1}^{m},\left (z_{j}\right )_{j=1}^{n}\right ) \times \varPi _{j=1}^{n}Y _{j}\left (\left (\left (x_{i}\right )_{i=1}^{m},\left (z_{j}\right )_{j=1}^{n}\right )\right )$, we cannot prove that $\bigtriangledown _{O^{j}}c_{j}\left (\pi _{I^{j}},y_{O^{j}},z\right ) = \bigtriangledown _{O^{j}}c_{j}\left (\pi _{I^{j}},y_{O^{j}}^{+},\right.$   $\left.z\right )$ as they did, since the argument they constructed does not work in Fréchet derivatives in infinite-dimensional spaces. Consequently, in the present context, $\bigtriangledown _{O^{j}}c_{j}\left (\pi _{I^{j}},y_{O^{j}},z\right ) = \bigtriangledown _{O^{j}}c_{j}\left (\pi _{I^{j}},y_{O^{j}}^{+},z\right )$ whenever $y_{O^{j}} = y_{O^{j}}^{+}$ which also justifies the Assumption SPP.