Equilibrium existence in infinite horizon economies

Abstract

In sequential economies with finite or infinite-lived real assets in positive net supply, we introduce constraints on the amount of borrowing in terms of the market value of physical endowments. We show that, when utility functions are either unbounded and separable in states of nature or separable in commodities, these borrowing constraints not only preclude Ponzi schemes but also induce endogenous Radner bounds on short-sales. Therefore, we obtain existence of equilibrium. Moreover, equilibrium also exists when both assets are numéraire and utility functions are quasilinear in the commodity used as numéraire.

Appendix

To prove our main result we show, firstly, that there exists equilibrium in finite horizon truncated economies. Then, we find an equilibrium for the original economy as the limit of a sequence of equilibria corresponding to the truncated economies, when the time horizon increases.

Truncated economies

For each T ∈ ℕ, we define a truncated economy, ${\mathcal{E}}^T$, in which agents consume commodities and trade assets in the restricted event-tree $D^T(\xi_0)$.

Let $J^T(\xi)=\{j \in J(\xi): \exists \mu \in D^{T-t(\xi)}(\xi),\,\mu \neq \xi,\,A(\mu,j) \neq 0 \}$ be the set of available securities at $\xi \in D^{T-1}(\xi_0)$. At each ξ ∈ D _T (ξ ₀), we define J ^T(ξ) = ∅. It follows that, given ξ ∈ D, J ^T(ξ) = J(ξ) for every T large enough. Let $D^T(J)=\{(\xi,j) \in D^T(\xi_0) \times J: j \in J^T(\xi)\}$.

Each individual h ∈ H is characterized by her physical, $(w^h(\xi); \xi \in D^T(\xi_0))$, and financial, $(e^h(\xi); \xi \in D^{T-1}(\xi_0))$, endowments. Also, when agent h chooses a consumption plan $(x(\xi))_{\xi \in D^T(\xi_0)}$, her utility is given by $U^{h,T}(x)= \sum_{\xi \in D^T(\xi_0)} u^h(\xi,x(\xi)).$

For each truncated economy ${\mathcal{E}}^T$, we can consider, without loss of generality, prices (p,q) in

$${\mathbb{P}}^T:= \prod_{\xi \in D^{T-1}(\xi_0)} \left(\Delta^L_+ \times {\mathbb{R}}^{J^T(\xi)}_+\right)\,\, \times\,\, \prod_{\xi \in D_T(\xi_0)}\,\Delta^L_+,$$

where $\Delta^L_+:=\{p \in {\mathbb{R}}^L_+: \Vert p \Vert_\Sigma=1\}$. Then, given (p,q) ∈ ℙ^T, agent h ∈ H solves the following optimization problem:

$$ \left(P^{h,T}\right) \left. \begin{array}{cl} \max& \! \! \! \! \! \,\,\,\,\,\,\,\,\,\,\,\,\,\,U^{h,T}(x)\\ \mbox{s.t.}& \,\left\{ \begin{array}{ll} y(\xi)=(x(\xi),\theta(\xi),\varphi(\xi)) \,\,&\geq 0,\ \forall \xi \in D^T(\xi_0),\\ g^{h,T}_{\xi}\left(y(\xi),y(\xi^-); p,q \right)\,\, &\leq 0,\ \forall \xi \in D^{T}(\xi_0),\\ q(\xi) \varphi(\xi)-\kappa p(\xi) w^h(\xi) \,\,&\leq 0 ,\ \forall \xi \in D^{T-1}(\xi_0),\\ (\theta(\xi),\varphi(\xi))\,\, &= 0,\ \forall \xi \in D_T(\xi_0), \end{array} \right. \end{array} \right. $$

where $y(\xi^-_0)=0$ and, for each $\xi \in D^{T}(\xi_0)$,

$$ \begin{array}{rll} g^{h,T}_{\xi}(y(\xi),y(\xi^-);p,q)&:=&p(\xi) \left(x(\xi) -w^h(\xi) \right) \\ &&+ {\displaystyle \sum_{j \in J^T(\xi)}} \,q_j(\xi)\left(\theta_j(\xi)-\varphi_j(\xi) -e^h_j(\xi)\right) \\ &&- {\displaystyle \sum_{j \in J^T(\xi^- )}} (p(\xi)A(\xi,j)+ q_j(\xi))\left(\theta_j(\xi^-)- \varphi_j(\xi^-)\right). \end{array} $$

Let B ^h,T(p,q) be the truncated budget set of agent h, i.e., the set of plans $(y(\xi))_{\xi \in D^T(\xi_0)}$ that satisfy the restrictions of the problem P ^h,T above.

Definition 1

An equilibrium for the economy ${\mathcal{E}}^T$ is given by prices (p ^T,q ^T) ∈ ℙ^T and individual allocations $(y^{h,T}(\xi))_{\xi \in D^T(\xi_0)}\in {\mathbb{E}}^{T}:={\mathbb{R}}^{D^{T}(\xi_{0}) \times L}_+ \times {\mathbb{R}}^{D^{T}(J)}_+ \times {\mathbb{R}}^{D^{T}(J)}_+,$ such that:

• (1) For each h ∈ H, $(y^{h,T}(\xi))_{\xi \in D^T(\xi_0)}$ is an optimal solution for P ^h,T at prices (p ^T,q ^T);

• (2) Physical and financial markets clear at each $\xi \in D^T(\xi_0)$.

Equilibrium existence in the truncated economies

In order to show the existence of equilibria in ${\mathcal{E}}^T$ we follow a generalized game approach. For each $({\mathcal X}, \Theta,\Psi, M) \in{\mathbb{F}}^T:={\mathbb{E}}^{T} \times{\mathbb{R}}^{D^T(J)}_{++},$ consider the convex and compact set ${\mathcal K}({\mathcal X}, \Theta,\Psi)=[0, {\mathcal X}] \times [0, \Theta] \times [0,\Psi] \subset{\mathbb{E}}^{T}$ and define,

$${\mathbb{P}}^T_M = \prod_{\xi \in D^{T-1}(\xi_0)}\, \left(\Delta^L_+ \times [0,M_\xi]\right)\,\times\, \prod_{\xi \in D_T(\xi_0) }\,\Delta^L_+.$$

Let ${\mathcal{G}}^T({\mathcal X}, \Theta,\Psi, M)$ be a generalized game where each consumer is represented by a player h ∈ H and, at each $\xi \in D^T(\xi_0),$ there is also a player who behaves as an auctioneer.

More precisely, in ${\mathcal{G}}^T({\mathcal X}, \Theta,\Psi, M)$ each player h ∈ H behaves as price-taker and, given $(p,q) \in {\mathbb{P}}^T_M$, she chooses strategies in the truncated budget set $B^{h,T}(p,q) \cap {\mathcal K}({\mathcal X}, \Theta,\Psi)$ in order to maximize the function U ^h,T. Also, at each $\xi \in D^{T-1}(\xi_0)$ (resp. ξ ∈ D _T (ξ ₀)) the corresponding auctioneer chooses commodity and asset prices $(p(\xi), q(\xi)) \in \Delta^L_+ \times [0,M_\xi]$ (resp. just commodity prices $p(\xi) \in \Delta^L_+$) in order to maximize the function $\sum_{h \in H}g^{h,T}_{\xi}(y^h(\xi),y^h(\xi^-);p,q),$ where $y^{h}=(y^h(\xi))_{\xi \in D^T(\xi_0)}$ are the strategies selected by player h ∈ H.

Definition 2

A strategy profile $\left[(p^T(\xi),q^T(\xi));(y^{h,T}(\xi))_{h \in H}\right]_{\xi \in D^T(\xi_0)}\in{\mathbb{P}}^T_M \times \left({\mathcal K}({\mathcal X}, \Theta,\Psi)\right)^{H}$ is a Nash equilibrium for ${\mathcal{G}}^T({\mathcal X}, \Theta,\Psi, M)$ if each player maximizes her objective function, given the strategies chosen by the other players, i.e., no player has an incentive to deviate.

Lemma 1

Let T ∈ ℕ and $({\mathcal X}, \Theta, \Psi, M) \in {\mathbb{F}}^T.$ Under Assumptions (A1) and (A3) the set of Nash equilibria for the game ${\mathcal{G}}^T({\mathcal X}, \Theta,\Psi, M)$ is non-empty.

Proof

Note that each player’s strategy set is non-empty, convex and compact. Further, it follows from Assumption (A3) that the objective function of each player is continuous and quasi-concave in her own strategy. Assumption (A1) assures that the correspondences of admissible strategies are continuous, with non-empty, convex and compact values. Therefore, we can find an equilibrium of the generalized game by applying Kakutani Fixed Point Theorem to the correspondence defined as the product of the optimal strategy correspondences.□

Lemma 2

Let T ∈ ℕ. Under Assumptions (A1)–(A4) there exists (Θ^T, Ψ^T) such that, if (Θ, Ψ) ≫ (Θ^T, Ψ^T), then every Nash equilibrium of the game ${\mathcal{G}}^T({\mathcal{X}},\Theta, \Psi, M)$ is an equilibrium of the economy ${\mathcal{E}}^T$ whenever ${\mathcal{X}}$ and M are large enough.

Proof

Let $\left[(p^T(\xi),q^T(\xi));(y^{h,T}(\xi))_{h \in H}\right]_{\xi \in D^T(\xi_0)}$ be a Nash equilibrium for ${\mathcal{G}}^T({\mathcal X}, \Theta,\Psi, M),$ with allocations given by y ^h,T(ξ) = (x ^h,T(ξ), θ ^h,T (ξ), ϕ ^h,T (ξ)). Note that, for each h ∈ H,

$$(y^{h,T}(\xi))_{\xi \in D^T(\xi_0)} \in \mbox{argmax}_{B^{h,T}(p^{T}, q^{T})\cap {\mathcal K}({\mathcal X}, \Theta,\Psi)}\,\,\,U^{h,T}(x).$$

Then, as each auctioneer maximizes his objective function, we have that, at each $\xi \in D^T(\xi_0)$,

$$ \sum_{h \in H} x^{h,T}(\xi) \leq \Upsilon^T(\Theta,\xi):=\sum_{h \in H} \left(w^h(\xi) + \sum_{j \in J^T(\xi^-)} A(\xi,j) \Theta(\xi^{-},j)\right). $$

It follows from Assumptions (A3) and (A4) that, for each $\xi \in D^T(\xi_0)$, there exists a real number $a^T_{\Theta}(\xi)>0$ such that,

$$ \min_{h \in H}\,\,u^h\left(\xi,(a^T_{\Theta}(\xi),\ldots,a^T_{\Theta}(\xi))\right)> \max_{h \in H}\,\,\,U^{h,T}(\Upsilon^T(\Theta)), $$

where $\Upsilon^T(\Theta):=(\Upsilon^T(\Theta,\xi); \xi \in D^T(\xi_0))$.

Suppose that $\mathcal{X}(\xi,l)> a^T_\Theta(\xi)$, for every $(\xi,l) \in D^T(\xi_0) \times L$. As $\Vert p^T(\xi) \Vert_\Sigma=1$, it follows from individual optimality that the value of accumulated individual financial endowments, at any $\xi \in D^T(\xi_0)$, is necessarily less than $p^T(\xi) (a^T_{\Theta}(\xi),\ldots,a^T_{\Theta}(\xi))= a^T_{\Theta}(\xi)$. Therefore, for each j ∈ J ^T(ξ),

$$q^T_j(\xi) \leq M^T_{\Theta}(\xi,j):=\frac{a^T_\Theta(\xi)\,\#H}{\sum_{h \in H}\,\overline{e}^h_j(\xi)}.$$

Let $M^T_\Theta=(M^T_\Theta(\xi,j); (\xi,j) \in D^T(J))$. We conclude that if $M \gg M^T_\Theta$, then in any Nash equilibrium of ${\mathcal{G}}^T({\mathcal{X}},\Theta, \Psi, M)$ the upper bounds of asset prices, which were previously imposed, are non-binding. Along the rest of this proof we assume that this property holds.

• Step 1 Physical markets clear

For each $\xi \in D^{T}(\xi_{0}),$ let

$$ \begin{array}{rll} \Gamma(\xi) &=& \sum\limits_{h \in H} x^{h,T}(\xi) - W(\xi)\,,\\ \Omega(\xi) &=& \sum\limits_{h \in H} \theta^{h,T}(\xi) - \sum\limits_{h \in H}\overline{e}^h(\xi) - \sum_{h \in H} \varphi^{h,T}(\xi). \end{array} $$

Summing up the budget constraints at ξ ₀ we have $p^T(\xi_{0}) \Gamma(\xi_{0}) + q^T(\xi_{0}) \Omega(\xi_{0}) \leq 0 .$ Since the auctioneer at ξ ₀ maximizes p(ξ ₀) Γ(ξ ₀) + q(ξ ₀) Ω(ξ ₀), we obtain that Γ(ξ ₀) ≤ 0. Assume now that Ω(ξ ₀,j) > 0 , for some $j \in J^T(\xi_0).$ By the construction of the plan M, we know that $q^T_{j}(\xi_{0}) < M_{\xi_{0},j},$ which leads us to obtain a contradiction with the optimal behaviour of the auctioneer at ξ ₀. Thus Ω(ξ ₀) ≤ 0. Hence, if ${\mathcal X}(\xi_{0},l) > \max\{W(\xi_{0},l), a^T_\Theta(\xi_0)\}$ for each l ∈ L, then the upper bound on consumption is non-binding at ξ ₀, allowing us to conclude, as a consequence of the monotonicity of preferences, that commodity markets clear at the initial node ξ ₀, i.e., Γ(ξ ₀) = 0. Moreover, $q^T(\xi_{0}) \Omega(\xi_{0})=0$.

Consider now a node ξ with t(ξ) = 1, and recall that the corresponding auctioneer at ξ chooses prices in $\Delta^L_+ \times [0,M_\xi]$ in order to maximize the function $\sum_{h \in H} g^{h,T}_{\xi}(y^{h,T}(\xi),y^{h,T}(\xi_{0});p,q).$ Using the fact that Ω(ξ ₀) ≤ 0, we can deduce that p ^T(ξ) Γ(ξ) + q ^T(ξ) Ω(ξ) ≤ 0, for every ξ with t(ξ) = 1. As before, Γ(ξ) ≤ 0 and Ω(ξ) ≤ 0. Furthermore, if ${\mathcal X}(\xi) > \max\{W(\xi,l), a^T_\Theta(\xi)\}$ for every l ∈ L, then the upper bound on consumption is not binding at ξ, which implies that Γ(ξ) = 0.

By applying successively analogous arguments to the nodes with periods t = 2,...,T, we conclude that Γ(ξ) = 0 for every $\xi \in D^{T}(\xi_{0})$, provided that, for each l ∈ L, ${\mathcal X}(\xi,l) > \max\{W(\xi,l),\,a^T_\Theta(\xi)\}$. That is, physical markets clear in the economy ${\mathcal E}^{T}.$ Furthermore, there is no excess of demand for financial markets, i.e., Ω(ξ) ≤ 0, for every $\xi \in D^{T-1}(\xi_{0}).$

• Step 2 Lower bounds for asset prices

Given (ξ, j) ∈ D ^T(J), fix a node μ(ξ,j) that belongs to the non-empty set $\mbox{\it argmin\,} \{t(\mu)\,:\,\mu \in D^{T-t(\xi)}(\xi),\, \mu \neq \xi,\, A(\mu,j) \neq 0 \}.$

By Assumptions (A1), (A3) and (A4), there exists b(ξ, j) ∈ (0,1), independent of T, such that, for every h ∈ H, the following inequality holds,

$$\label{prices} u^{h}\left(\mu(\xi,j), w^h(\mu(\xi,j)) + {\displaystyle \frac{ A(\mu(\xi,j),j) \min_{l \in L} w^{h}_{l}(\xi)}{b(\xi,j)}} \right) > U^h(W). $$

(1)

Suppose that,

$$ \Theta(\xi,j) > \widehat{\Theta}(\xi,j):=\max_{h \in H}\,{\displaystyle \frac{\min_{l \in L}w^{h}_{l}(\xi)}{b(\xi,j)}}\,, $$

and for every μ ∈ D ^{T − t(ξ)}(ξ) with j ∈ J ^T(μ),

$$ \begin{array}{rll} &&\min_{l \in L}\,{\mathcal X}(\mu,l) > {\mathcal{X}}^T_{\Theta,\xi}(\mu,j) \\ &&{\kern6pt}:=\max_{(l,h) \in H \times L}\,\,\left\{ W(\mu,l),\,a^T_\Theta(\mu),\, w^h_l(\mu) +{\displaystyle \frac{ A_l(\mu,j) \min_{l' \in L} w^{h}_{l'}(\xi)}{b(\xi,j)}}\right\}. \end{array} $$

We claim that $q^T_{j}(\xi) > b(\xi, j)$. In fact, if $q^{T}_{j}(\xi)\leq b(\xi,j)$ then, as by Step 1 x ^h,T(μ) ≤ W(μ) for every $\mu \in D^T(\xi_0)$, it follows from Assumption (A3) and inequality (1) that any agent h ∈ H has an incentive to deviate by choosing any budget feasible strategy (x ^h, θ ^h, ϕ ^h) that satisfies,

$$ \begin{array}{rll} \theta^{h}_{j}(\xi) &=& \frac{\min_{l \in L}w^{h}_{l}(\xi)}{ b(\xi,j)}, \\ x^{h}(\mu) &=& w^h(\mu)+ A(\mu,j)\theta^{h}_{j}(\xi),\,\,\,\,\,\,\,\,\mbox{if}\,\,\mu=\mu(\xi,j). \end{array} $$

Therefore, if for each $\eta \in D^T(\xi_0)$,

$$ \begin{array}{rll} \Theta(\eta,j) &>& \widehat{\Theta}(\eta,j), \ \forall j \in J^T(\eta),\\ {\mathcal{X}}(\eta,l) &>& {\mathcal{X}}^T_{\Theta}(\eta):=\max_{(\xi,j) \in D^T(J): \atop \eta>\xi,\,j \in J^T(\eta)} {\mathcal{X}}^T_{\Theta,\xi}(\eta,j), \ \forall l \in L, \end{array} $$

then equilibrium asset prices have a positive lower bound away from zero. In fact, for each (η,j) ∈ D ^T(J), we have that $q^T_{j}(\eta) > b(\eta, j)$.

• Step 3 Non-binding short-sales constraints

Define $\widehat{\Theta}^T= (\widehat{\Theta}(\eta,j); (\eta,j) \in D^T(J))$ and ${\mathcal{X}}^T_\Theta= ({\mathcal{X}}^T_\Theta(\eta); \eta \in D^T(\xi_0))$. If $\Theta \gg \widehat{\Theta}^T$ and ${\mathcal{X}} \gg {\mathcal X}^T_\Theta$, asset prices are bounded away from zero. Thus, using the borrowing constraints, we conclude that, for every player h ∈ H,

$$ \varphi^{h,T}_{j}(\xi) < \widehat{\Psi}_j(\xi):= \kappa \,\frac{\max_{(h,l) \in H\times L} w^h_{l}(\xi)}{b(\xi,j)},\,\,\,\,\,\,\,\,\,\,\,\forall (\xi,j) \in D^{T}(J). $$

Let $\Psi^T=(\widehat{\Psi}_j(\xi); (\xi,j) \in D^T(J))$. If Ψ ≫ Ψ^T then short-sales restrictions induced by ${\mathcal K}({\mathcal{X}}, \Theta, \Psi, M)$ are non-binding.

• Step 4 Financial markets clear and upper bounds for long-positions are non-binding

Suppose that $(\Theta, \Psi) \gg (\widehat{\Theta}^T, \Psi^T)$ and ${\mathcal{X}} \gg {\mathcal{X}}^T_\Theta$. Now, by Step 1 we have that q ^T(ξ) Ω(ξ) = 0 and Ω(ξ) ≤ 0 , for each $\xi \in D^{T-1}(\xi_0)$. Thus, if for some (ξ,j) ∈ D ^T(J), Ω _j (ξ) < 0, then $q^T_{j}(\xi)=0,$ which is in contradiction with the lower bound on asset prices find in Step 2.

On the other hand, for each $\xi \in D^{T-1}(\xi_0)$, $(\varphi^{h,T}(\xi))_{h \in H}$ is bounded. Thus, as Ω(ξ) ≤ 0, $\sum_{h \in H} \theta^{h,T}(\xi)$ is also bounded. We conclude that there exists $\Theta^T \geq \widehat{\Theta}^T$ such that, if Θ ≫ Θ^T then upper bounds on long positions are non-binding.

• Step 5 Individual optimality

As a consequence of all previous steps, if (Θ, Ψ) ≫ (Θ^T, Ψ^T) and $({\mathcal{X}}, M)\gg ({\mathcal{X}}^T_\Theta, M^T_\Theta)$ then, for each h ∈ H, the optimal allocation y ^h,T belongs to the interior of ${\mathcal{K}}({\mathcal{X}}, \Theta, \Psi, M)$ (relative to ${\mathbb{E}}^T$). As budget correspondences has finite-dimensional convex values, we conclude that,

$$ (y^{h,T}(\xi))_{\xi \in D^T(\xi_0)} \in \mbox{argmax}_{B^{h,T}(p^{T}, q^{T})}\,\,\sum_{\xi \in D^T(\xi_0)} u^h(\xi,x(\xi)). $$

Therefore, since (Θ, Ψ) ≫ (Θ^T, Ψ^T) and $({\mathcal{X}}, M)\gg ({\mathcal{X}}^T_\Theta, M^T_\Theta),$ any Nash equilibrium of the game ${\mathcal{G}}^T({\mathcal{X}},\Theta, \Psi, M)$ is an equilibrium of the truncated economy ${\mathcal{E}}^T.$□

Recall that, given ξ ∈ D, J ^T(ξ) = J(ξ) for T large enough. Thus, by construction, the upper bounds ( Θ^T(ξ), Ψ^T(ξ)) are independent of T > t(ξ), when T is large enough. Therefore, node by node, independently of the truncated horizon T, individual equilibrium allocations are uniformly bounded and commodity prices belong to the simplex.

Moreover, under Assumptions (A2)–(A4) asset prices are uniformly bounded by above, node by node. In fact, as consumption allocations are bounded by the aggregated resources, by analogous arguments to those made in the proof of Lemma 2, we can conclude that,

$$ q^T_j(\xi) \leq \frac{a(\xi)\,\#H}{\sum_{h \in H}\,\overline{e}^h_j(\xi)},\,\,\,\,\,\,\,\,\forall j \in J^T(\xi), $$

where a(ξ) > 0 is independent of T > t(ξ) and is defined implicitly by

$$ \min_{h \in H}\,\,u^h\left(\xi,(a(\xi),\ldots,a(\xi))\right)> \max_{h \in H} U^h(W). $$

Asymptotic equilibria

In order to find an equilibrium of our original economy, we look for an uniform bound (node by node) for the Kuhn–Tucker multipliers associated to the truncated individual problems.

To attempt this aim, for each T ∈ ℕ, consider an equilibrium $\left[p^T(\xi),q^T(\xi);(y^{h,T}(\xi))_{h \in H}\right]_{\xi \in D^T(\xi_0)}$ for the economy ${\mathcal{E}}^T.$ Then. there exist non-negative multipliers $\left((\gamma^{h,T}_\xi)_{\xi \in D^T(\xi_0)};\,(\rho^{h,T}_\xi )_{\xi \in D^{T-1}(\xi_0)}\right)$ such that,

$$ \gamma^{h,T}_{\xi} g^{h,T}_\xi(y^{h,T}(\xi),\,y^{h,T}(\xi^-); p^T,q^T) = 0,\,\,\,\,\,\,\forall \xi \in D^{T}(\xi_0); $$

(2)

$$ \rho^{h,T}_{\xi}\left(\kappa p^T(\xi) w^h(\xi)- q^T(\xi)\varphi^{h,T}(\xi)\right) = 0,\,\,\,\,\,\,\forall \xi \in D^{T-1}(\xi_0). $$

(3)

Moreover, for each plan $(x(\xi), \theta(\xi), \varphi(\xi))_{\xi \in D^T(\xi_0)} \geq 0$, with (θ(η), $\varphi(\eta))_{\eta \in D_T(\xi_0)}=0$, the following saddle point property is satisfied (see Rockafellar 1997, Section 28, Theorem 28.3),

$$ \begin{array}{rll}\label{pontoselau} U^{h,T}(x) &-&\sum\limits_{\xi \in D^T(\xi_0)} \gamma^{h,T}_\xi g^{h,T}_\xi(y(\xi),y(\xi^-);\,p^T,q^T) \\ &+&\sum\limits_{\xi \in D^{T-1}(\xi_0)} \rho^{h,T}_\xi (\kappa p^T(\xi) w^h(\xi)- q^T(\xi)\varphi(\xi))\leq U^{h,T}(x^{h,T}). \end{array} $$

(4)

Let us take $(x(\xi), \theta(\xi), \varphi(\xi))_{\xi \in D^T(\xi_0)}=(0,0,0)$ to obtain,

$${\label{dos}} \sum_{\xi \in D^{T-1}(\xi_0)} p^T(\xi) w^h(\xi) \left[\gamma^{h,T}_\xi+ \rho^{h,T}_\xi\kappa \right]\leq U^h(W)<+\infty. $$

(5)

Since commodity prices are in the simplex, node by node, for every ξ ∈ D and for all T > t(ξ), we conclude that,

$$ 0 \leq \gamma^{h,T}_\xi \leq \frac{U^h(W)}{\underline{w}^h_\xi},\,\,\,\,\,\,\,\,\,\,0 \leq \rho^{h,T}_\xi \leq \frac{U^h(W)}{\kappa\, \underline{w}^h_\xi}, $$

where, by Assumption (A1), $\underline{w}^h_\xi:= \min_{l \in L} w^h_{l}(\xi)>0$.

In short, for each ξ ∈ D, the sequence formed by equilibrium prices, equilibrium allocations and Kuhn–Tucker multipliers, ((p ^T(ξ), q ^T(ξ)); (y ^h,T(ξ), $\gamma^{h,T}_\xi$, $\rho^{h,T}_{\xi})_{h \in H})_{T> t(\xi)}$, is bounded. Applying Tychonoff Theorem we can find a common subsequence (T _k ) _{k ∈ ℕ} ⊂ ℕ such that, for each ξ ∈ D,

$$ \begin{array}{lll} &&\lim_{k \rightarrow +\infty}\,\,\left(\left(p^{T_k}(\xi),q^{T_k}(\xi)\right);\left(y^{h,T_k}(\xi), \gamma^{h,T_k}_\xi, \rho^{h,T_k}_\xi\right)_{h \in H}\right) \\ &&{\kern6pt}= \left(\left(\overline{p}(\xi),\overline{q}(\xi)\right);\left(\overline{y}^{h}(\xi), \overline{\gamma}^{h}_\xi, \overline{\rho}^{h}_\xi \right)_{h \in H}\right). \end{array} $$

Hence, for each h ∈ H, $\left(\overline{y}^{h}(\xi)\right)_{\xi \in D}\in B^h(\overline{p}, \overline{q})$. Moreover, limit allocations are cluster points, node by node, of equilibria in truncated economies and then market clearing follows. Therefore, in order to conclude that $\left[(\overline{p}(\xi),\overline{q}(\xi));(\overline{y}^{h}(\xi))_{h \in H}\right]_{\xi \in D}$ is an equilibrium it remains to show that, for each agent h ∈ H, $(\overline{y}^{h}(\xi))_{\xi \in D}$ is an optimal choice when prices are $(\overline{p},\overline{q})$.

Lemma 3

Under Assumptions (A1)–(A4), $U^h(\tilde{x}) \leq U^h( \overline{x}),$ for every $\tilde{y}:=(\tilde{x}, \tilde{\theta}, \tilde{\varphi}) \in B^h(\overline{p},\overline{q}).$

Proof

Fix a node ξ ∈ D. Let us take T > t(ξ) large enough to assure that J ^T(μ) = J(μ) for each μ ≤ ξ and consider the allocation,

$$ (x(\mu),\theta(\mu),\varphi(\mu))=\left\{ \begin{array}{ll} \left(x^{h,T}(\mu), \theta^{h,T}(\mu), \varphi^{h,T}(\mu)\right),&\,\,\,\,\,\,\,\,\,\mbox{if}\,\,\,\mu \neq \xi,\\[8pt] \left(\tilde{x}(\xi), \tilde{\theta}(\xi), \tilde{\varphi}(\xi)\right),&\,\,\,\,\,\,\,\,\,\mbox{if}\,\,\,\mu =\xi. \end{array} \right. $$

Then, it follows from inequality 4 that, under Assumption (A3),

$$ \begin{array}{rll} u^h\left(\xi, \tilde{x}(\xi)\right) - u^h\left(\xi, x^{h,T}(\xi)\right) &\leq& -\,\rho^{h,T}_\xi \left(\kappa p^T(\xi) w^h(\xi)- q^T(\xi)\tilde{\varphi}(\xi)\right)\\ &&+\,\gamma^{h,T}_\xi g^h_\xi\left(\tilde{y}(\xi),y^{h,T}\left(\xi^-\right);\,p^T,q^T\right) \\ &&+\,\sum_{\mu \in \xi^+}\gamma^{h,T}_\mu g^h_\mu\left(y^{h,T}(\mu),\tilde{y}(\xi);\,p^T,q^T\right) , \end{array} $$

where $g^h_\xi \leq 0$ denotes the budget constraint at ξ ∈ D. As $\tilde{y}$ is budget feasible at prices $(\overline{p}, \overline{q})$, taking the limit as T = T _k goes to infinity, we obtain that,

$$ \begin{array}{rll} u^h(\xi, \tilde{x}(\xi)) - u^h(\xi, \overline{x}(\xi)) &\leq& \overline{\gamma}^{h}_\xi g^h_\xi(\tilde{y}(\xi),\overline{y}^{h}(\xi^-);\,\overline{p},\overline{q})\\ &&+\sum\limits_{\mu \in \xi^+}\overline{\gamma}^{h}_\mu g^h_\mu(\overline{y}^{h}(\mu),\tilde{y}(\xi);\,\overline{p},\overline{q}) . \end{array} $$

As $\tilde{y}$ and $(\overline{y}^h(\xi))_{\xi \in D}$ belongs to $B^h(\overline{p},\overline{q})$, adding previous inequality over the nodes in $D^N(\xi_0)$, with N ∈ ℕ, it follows that,

$$ U^{h,N}\left(\tilde{x}\right) - U^{h,N}\left(\overline{x}\right) \leq \sum_{\mu \in D_{N+1}\left(\xi_0\right)}\overline{\gamma}^{h}_\mu g^h_\mu\left(\overline{y}^{h}(\mu),\tilde{y}\left(\mu^-\right);\,\overline{p},\overline{q}\right). $$

Thus, as $\tilde{y}$ is budget feasible, borrowing constraints imply that,

$$ \begin{array}{rll}\label{eq6} &&{\kern-6pt} U^{h,N}\left(\tilde{x}\right) - U^{h,N}\left(\overline{x}\right)\\ &&\leq \sum\limits_{\mu \in D_{N+1}\left(\xi_0\right)}\overline{\gamma}^{h}_\mu\left(\overline{p}(\mu)\overline{x}^h(\mu) + \overline{q}(\mu) \left(\overline{\theta}^h(\mu)- \overline{\varphi}^h(\xi)\right) +\kappa \overline{p}(\mu) w^h(\mu) \right). \end{array} $$

(6)

Define $L_\xi^{h,T}= p^T(\xi) x^{h,T}(\xi) + q^T(\xi)(\theta^{h,T}(\xi)- \varphi^{h,T}(\xi))$ and consider the allocation,

$$ (x(\mu),\theta(\mu),\varphi(\mu))=\left\{ \begin{array}{ll} (x^{h,T}(\mu), \theta^{h,T}(\mu), \varphi^{h,T}(\mu)),&\,\,\,\,\,\,\,\,\,\mbox{if}\,\,\,\mu \neq \xi\,,\\ (0, 0, 0),&\,\,\,\,\,\,\,\,\,\mbox{if}\,\,\,\mu =\xi. \end{array} \right. $$

Using inequality (4), Assumption (A3) assures that,

$$ \begin{array}{rll} \gamma^{h,T}_\xi L_\xi^{h,T} &\leq&u^h(\xi,x^{h,T}(\xi)) + \sum\limits_{\mu \in \xi^+} \gamma^{h,T}_\mu L_\mu^{h,T} ,\,\,\,\,\,\,\,\,\forall \xi \in D^{T-1}(\xi_0); \\ \gamma^{h,T}_\xi L_\xi^{h,T} &\leq&u^h(\xi,x^{h,T}(\xi)) ,\,\,\,\,\,\,\,\,\forall \xi \in D_{T}(\xi_0). \end{array} $$

Thus, by monotonicity of preferences,

$$ \sum_{\xi \in D_{N+1}(\xi_0)} \gamma^{h,T}_\xi L^{h,T}_\xi \leq \sum_{\mu \in D \setminus D^N(\xi_0)} u^h(\mu, W(\mu)),\,\,\,\,\,\,\,\forall T > N+1. $$

Taking the limit as T goes to infinity we obtain,

$$ \sum_{\xi \in D_{N+1}(\xi_0)} \overline{\gamma}^{h}_\xi \left(\overline{ p}(\xi) \overline{x}^{h}(\xi) + \overline{q}(\xi)(\overline{\theta}^{h}(\xi)- \overline{\varphi}^{h}(\xi)) \right) \leq \sum_{\mu \in D \setminus D^N(\xi_0)} u^h(\mu, W(\mu)). $$

Thus, it follows from inequality (6) that,

$$ U^{h,N}(\tilde{x}) - U^{h,N}(\overline{x})\leq \sum_{\mu \in D \setminus D^N(\xi_0)} u^h(\mu, W(\mu)) +\kappa \,\sum_{\mu \in D_{N+1}(\xi_0)} \overline{\gamma}^{h}_\mu \overline{p}(\mu) w^h(\mu) .$$

Now, inequality (5) assures that,

$$\label{eq7} \sum_{\xi \in D} \overline{\gamma}^h_\xi \overline{p}(\xi) w^h(\xi) <+\infty. $$

(7)

Therefore, it follows from Assumption (A3) that: For each ε > 0 there exists $\overline{N}_\varepsilon> 0$ such that,

$$ \sum_{\xi \in D^N(\xi_0) }u^h(\xi, \tilde{x}(\xi))< \varepsilon + U^h(\overline{x}),\,\,\,\,\,\,\forall N> \overline{N}_\varepsilon $$

Finally, we conclude that, for each ε > 0, $U^h(\tilde{x}) \leq \varepsilon + U^h(\overline{x})$, which ends the proof. □

Proof

Given (ξ,h) ∈ D ×H, define

$$ \begin{array}{rll} \tilde{u}^h(\xi,x)&=& v^h\left(\xi,(x_l)_{l \in L \setminus L(J)}\right) \\ &&+ \sum\limits_{l \in L(J)}\, \left(f^h_l\left(\xi, \min\left\{x_l, 2 W_l(\xi)\right\}\right)+ \rho(\xi,l) \max\left\{x_l - 2W_l(\xi), 0\right\} \right), \end{array} $$

where $x=(x_l; l \in L) \in {\mathbb{R}}^L_+$ and $\rho(\xi,l) \in \partial f^h_l(\xi, 2W_l(\xi))$.⁶ It follows from the separability of the inter-temporal utilities on commodities in L(J) that the functions,

$$ \tilde{U}^h(x):= \sum_{\xi \in D} \tilde{u}^h(\xi, x(\xi)), $$

satisfy Assumptions (A3) and (A4). Therefore, there exists an equilibrium $\left[(\overline{p}(\xi),\overline{q}(\xi));(\overline{y}^{h}(\xi))_{h \in H}\right]_{\xi \in D}$, being $\overline{y}^{h}(\xi)= (\overline{x}^h(\xi),\overline{\theta}^h(\xi),\overline{\varphi}^h(\xi)),$ for the economy in which each h ∈ H has preferences represented by the function $\tilde{U}^h$ instead of U ^h. Moreover, this equilibrium is actually an equilibrium for the original economy. In fact, since agents are restricted to choose bounded consumption plans, if there exists a budget feasible allocation (x ^h, θ ^h, ϕ ^h) such that $U^h(x^h)> U^h(\overline{x}^h)$ then there is λ ∈ (0,1) such that, the consumption plan $x(\lambda):= \lambda x^h +(1-\lambda) \overline{x}^h$, with x(λ) = (x _l (λ, ξ); ξ ∈ D), satisfies x _l (λ, ξ) < 2 W _l (ξ), ∀ l ∈ L(J). Thus,

$$ \tilde{U}^h(x(\lambda))= U^h(x(\lambda))> \lambda U^h(x^h) +(1-\lambda) U^h\left(\overline{x}^h\right) > U^h\left(\overline{x}^h\right)=\tilde{U}^h\left(\overline{x}^h\right), $$

which is a contradiction. □