Endogenous differential information

Abstract

We include endogenous differential information in a model with sequential trade and incomplete financial participation. Agents update information through market signals given by commodity prices and asset deliveries. Information acts over admissible strategies and consumption tastes, allowing discontinuities in preferences and choice sets. Therefore, equilibrium may cease to exist. However, internalizing the compatibility between information and consumption through preferences, and without requiring either financial survival assumptions or fully revealing prices, equilibrium existence can be ensured.

Appendix: Proof of Theorem 1

To prove equilibrium existence, we first define a generalized game in which agents maximize utility functions in truncated budget sets. Auctioneers choose prices in order to maximize the value of the excess of demand in commodity and financial markets. We prove that this generalized game has a Cournot–Nash equilibrium and also that when the upper bounds on allocations are high enough, any equilibrium of the generalized game will be an equilibrium of our economy.

The generalized game ${{\mathcal {G}}\left( Q,X,\varTheta \right) .}$ Given any vector $\left( Q,X,\varTheta \right) \in {{\mathbb {R}}}^3_{++}$, we define a game characterized by the following set of players and strategies.

Set of players. There is a finite set of players constituted by

• (i) The set of agents of the economy, I.

• (ii) An auctioneer, h(s), for each $s \in S^*$.

We denote the set of players by $H=I\cup H(S^*)$ where $H(S^*):=\{h(s):s\in S^*\}$.

Sets of strategies. Given $\overline{W}:=\max \limits _{(s,l)\in S^* \times L} \sum _{i \in I} w^i_{s,l}$, define for any $i \in I$,

$$\begin{aligned} K^i(X,\varTheta )=[0,X]^{L}\times [0,2\overline{W}]^{S\times L}\times [-\varTheta ,\varTheta ]^{J^i}, \end{aligned}$$

and recall ${{\mathcal {P}}}_\zeta =\{p\in {{\mathbb {R}}}^L_{+}: p \cdot \zeta =1\}$. The set of strategies for the players in the generalized game, $(\overline{\varGamma }^h; h \in H)$, are given by,

• (i) For each $h\in I$, $\overline{\varGamma }^h= K^h(X,\varTheta ).$

• (ii) For $h=h(0)$, $\overline{\varGamma }^h={{\mathcal {P}}_\zeta }\times [0,Q]^{\#J}$

• (iii) For $h=h(s)$, with $s \in S$, $\overline{\varGamma }^h={{\mathcal {P}}_\zeta }$.

For simplicity, let $\eta ^h=\left( x^h,\theta ^h\right) \in \overline{\varGamma }^h$ be a generic vector of strategies for a player $h\in I$; $(p_0,q)$ will denote a generic strategy for the player h(0); and $p_s$ a generic strategy for a player h(s), with $s \in S$. Finally, let $\overline{\varGamma }=\prod _{h\in H}\overline{\varGamma }^h$ be the space of strategies of ${\mathcal {G}}\left( Q,X,\varTheta \right) .$ A generic element of $\overline{\varGamma }$ is denoted by $\left( p,q,\eta \right)$, where $\eta :=\left( \eta ^h; h \in I\right)$ is a generic element of $\prod _{i\in I}\overline{\varGamma }^i$.

Admissible strategies. Strategies effectively chosen for players depend on the actions taken by other players, through a correspondence of admissible strategies $\phi ^h:\overline{\varGamma }_{-h}\twoheadrightarrow \overline{\varGamma }^h$, where $\overline{\varGamma }_{-h}=\prod _{h'\ne h}\overline{\varGamma }^{h'}$. Let $\left( p,q,\eta \right) _{-h}$ be a generic element of $\overline{\varGamma }_{-h}$. We suppose that

• (i) If $h\in I$, $\phi ^h\left[ \left( p,q,\eta \right) _{-h}\right] =B^h(p,q)\cap \overline{\varGamma }^h$.

• (ii) If $h\in H(S^*)$, $\phi ^h\left[ \left( p,q,\eta \right) _{-h}\right] =\overline{\varGamma }^h$.

Objective functions. Each player is also characterized by an objective function $F^h:\overline{\varGamma }^h\times \overline{\varGamma }_{-h}\rightarrow {\mathbb {R}}_+$. We assume that

• (i) When $h\in I$ and $\eta ^h=\left( x^h,\theta ^h\right) \in \overline{\varGamma }^h$, then $F^h\left( \eta ^h;\left( p,q,\eta \right) _{-h}\right) =V^i\left( p, x^h\right) .$

• (ii) If $h=h(0)$ and $\left( p_0,q\right) \in \overline{\varGamma }^h$, then

  $$\begin{aligned} F^h\left( \left( p_0,q\right) ;\left( p,q,\eta \right) _{-h}\right) :=p_0\sum _{i\in I}\left( x^i_0-w_0^i\right) +\sum _{i\in I}\sum _{j\in J^i}q_j\theta ^i_j. \end{aligned}$$

• (iii) If $h(s)\in H(S^*)\setminus \{h(0)\}$ and $p_s\in \overline{\varGamma }^h$, then $F^h\left( p_s;\left( p,q,\eta \right) _{-h}\right) :=p_s\sum _{i\in I}\left( x^i_s-w^i_s\right)$.

We define the correspondence of optimal strategies for each $h\in H$, $\varPsi ^h: \overline{\varGamma }_{-h} \twoheadrightarrow \overline{\varGamma }^h$ as

$$\begin{aligned} \varPsi ^h\left( \left( p,q,\eta \right) _{-h}\right) := \mathop {{{\mathrm{arg\,max}}}}\limits _{y\in \phi ^h\left( \left( p,q,\eta \right) _{-h}\right) }F^h\left( y;\left( p,q,\eta \right) _{-h}\right) . \end{aligned}$$

Finally, let $\varPsi :\overline{\varGamma }\twoheadrightarrow \overline{\varGamma }$ be the correspondence of optimal game response, which is given by $\varPsi \left( p,q,\eta \right) =\prod _{h\in H}\varPsi ^h\left( \left( p,q,\eta \right) _{-h}\right)$.

Definition 2

A Cournot–Nash equilibrium for the generalized game ${\mathcal {G}}\left( Q,X,\varTheta \right)$ is given by a strategy profile $\left( \overline{p},\overline{q},\overline{\eta }\right) \in \overline{\varGamma }$ such that $\left( \overline{p},\overline{q},\overline{\eta }\right) \in \varPsi \left( \overline{p},\overline{q},\overline{\eta }\right)$.

In order to prove the existence of equilibrium in the generalized game, we need some properties of the admissible strategy correspondence which the following lemma provides.

Lemma 1

For any $h \in H$, $\phi ^h$ is continuous and has nonempty, compact, and convex values.

Proof

For each player $h\in H(S^*)$, the correspondence of admissible strategies is constant and, therefore, it is continuous and nonempty. Also, by definition, its values are compact and convex.

On the other hand, for each player $h\in I$, it follows from the definition of the budget set that the correspondence of admissible strategies $\phi ^h$ has nonempty, compact, and convex values. Since the graph of this correspondence is closed, we obtain upper hemicontinuity. To assure the lower hemicontinuity of $\phi ^h$, we consider the correspondence $\mathring{\phi }^h\left( (p,q,\eta )_{-h}\right) :=int_{K^h(X,\varTheta )}\;B^h(p,q)$, which associates with a vector of commodity and asset prices the set of allocations in $K^h(X,\varTheta )$ that satisfy all the budget restrictions of agent h as strict inequalities. Note that this correspondence has nonempty values and open graph. Therefore, it is lower hemicontinuous. We know that the closure of $\mathring{\phi }^h\left( (p,q,\eta )_{-h}\right)$, which is equal to $\phi ^h\left( (p,q,\eta )_{-h}\right)$, is also lower hemicontinuous. Therefore, correspondences of admissible strategies $(\phi ^h; h \in I)$ are continuous. $\square$

Lemma 2

Under (A) and (B), the set of Cournot–Nash equilibria of ${\mathcal {G}}\left( Q,X,\varTheta \right)$ is nonempty.

Proof

By Assumption (A) and (B), each objective function in the game is continuous in all variables and quasi-concave in its own strategy. Also, the sets of strategies are nonempty, compact, and convex. By Lemma 1, admissible correspondence is continuous with nonempty, convex, and compact values. Thus, we can apply Berge’s maximum theorem to assure that for each player $h\in H$, the correspondence of optimal strategies, $\varPsi ^h$, is upper hemicontinuous with nonempty, convex, and compact values. Therefore, the correspondence $\varPsi$ has closed graph with nonempty, compact, and convex values. Applying Kakutani’s fixed-point theorem to $\varPsi$, we conclude the proof.

We will prove that for vectors $\left( Q,X,\varTheta \right) \in {{\mathbb {R}}}^3_{++}$ for which coordinates are high enough, any equilibrium of the generalized game is an equilibrium for our economy. However, we need to previously find endogenous upper bounds for equilibrium variables.

Lemma 3

For each $(i,s) \in I\times S$, fix a vector $(p_s, w_s^i, x_s^i) \in {{\mathcal {P}}_\zeta }\times {{\mathbb {R}}}^L_+ \times {{\mathbb {R}}}^L_+$, with $x_s^i<\overline{W}$. Then, there exists $A>0$ such that any vector $\left( \kappa _j^i; j \in J^i\right) \in {\mathbb {R}}^{J^i}$ satisfying

$$\begin{aligned} p_s x_s^i = p_s w_s^i +\sum _{j \in J^i} R_{s,j}\kappa _j^i,\quad \forall (i,s)\in I\times S; \end{aligned}$$

belongs to $[-A,A]^{\#J^i}$. Also, A only depends on $((\overline{W}, w_s, R_{s,j}); (s,j) \in S\times J).$

Proof

Fix $i\in I$. Note that as S (respectively, J) is a finite set, by abusing of the notation and identifying it with $\{1,\ldots ,S\}$ (respectively, $\{1,\ldots ,J^i\}$) we can rewrite the conditions in the statement of the lemma in a matricial form:

$$\begin{aligned} \left[ \begin{array}{c} p_1(x_1^i-w_1^i)\\ \vdots \\ p_{S}(x_{S}^i-w_{S}^i) \end{array}\right] = \left[ \begin{array}{ccc} R_{1,1}&{}\cdots &{} R_{1,J^i}\\ \vdots &{}\ddots &{}\vdots \\ R_{S,1}&{}\cdots &{}R_{S,J^i} \end{array}\right] \left[ \begin{array}{c} \kappa _1^i\\ \vdots \\ \kappa _{J^i}^i \end{array}\right] \end{aligned}$$

Since for $i\in I$ there are no redundant assets in $J^i$, we have that $J^i\le S$. Moreover, we can find a nonsingular submatrix of dimension $J^i\times J^i$. Specifically, we may assume, without loss of generality, that this matrix is given by

$$\begin{aligned} B^i=\left| \begin{array}{ccc} R_{1,1}&{}\cdots &{}R_{1,J^i}\\ \vdots &{}\vdots &{}\vdots \\ R_{J^i,1}&{}\cdots &{}R_{J^i,J^i} \end{array}\right| . \end{aligned}$$

Thus, we have that

$$\begin{aligned} \left[ \begin{array}{c} p_1(x_1^i-w_1^i)\\ \vdots \\ p_{J}\left( x_{J^i}^i-w_{J^i}^i\right) \end{array}\right] = B \left[ \begin{array}{c} \kappa _1\\ \vdots \\ \kappa _{J^i}^i \end{array}\right] \end{aligned}$$

By Cramer’s Rule,

$$\begin{aligned} \kappa _j^i=\frac{det(B^i(y^i,j))}{\det (B^i)}, \quad \forall j \in \{1,\ldots , J^i\}, \end{aligned}$$

where $y^i=(p_1(x_1^i-w_1^i), \ldots , p_{J}(x_{J^i}^i-w_{J^i}^i))$ and $B^i(y^i,j)$ is the matrix obtained by change, in the matrix $B^i$, the j-ith column for the vector $y^i$. Since (i) the determinant is a continuous function, (ii) the vector $y^i$ depends continuously on $((p_s,x_s^i); s \in S)$, and (iii) vectors $((p_s, x_s^i, w_s^i); s \in S)$ are in a compact space, it follows that vector $(k_j^i; j \in J^i)$ is bounded, independently of the value of $((p_s, x_s^i, w_s^i); s \in S)$. Thus, there exists $A>0$, the maximum across the bounds of $(k_j^i; j \in J^i)$ for each $i\in I$, which satisfies the conditions of the lemma and depends on $((\overline{W}, w_s, R_{s,j}); (s,j) \in S\times J).$ $\Box$

Following the notation of the previous lemma, let $\overline{\varTheta }:=2A.$

The next two lemmas are used to prove that equilibrium asset prices of the generalized game are uniformly bounded. For convenience of notations, let $W_0=(W_{0,l}; l \in L)$ be the vector of aggregated physical resources at $t=0$, where $W_{0,l}:=\sum _{i \in I} w^i_{0,l}$.

Lemma 4

Under Assumptions (B) and (C), fix $\left( \overline{p},\overline{q}\right) \in {{\mathcal {P}}}\times {{\mathbb {R}}}^{J}_+$ and suppose that for any agent $i \in I$, there is an optimal solution $(\overline{x}^i,\overline{\theta }^i)\in \overline{\varGamma }^i$ for his individual problem such that $\overline{x}^i_0\le W_0$ and $\overline{x}^i_{s,l}\le 2\overline{W},\,\forall (s,l) \in S\times L$. Then, there exists $\overline{Q}>0$, independent of prices, such that $\max \limits _{j\in J}\overline{q}_j<\overline{Q}$.

Proof

Fix $j \in J$ and let $\mu :=\frac{\min \limits _{(k,l,h)\in S\times L \times I} \,\,w_{k,l}^h}{2}>0$. Suppose that an agent $i \in I$ for which $j \in J^i$ borrows a quantity $\tilde{\theta }_j>0$ of asset j such that $R_{s,j}\tilde{\theta }_j\le \mu$, for any $s \in S$.¹⁷ This position on asset j reports a quantity of resources which allow agent i to consume at the first period the bundle $w_0^{i}+(\overline{q}_j \tilde{\theta }_j)\zeta$ and, therefore,

$$\begin{aligned} V^i\left( \overline{p}, w_0^{i}+(\overline{q}_j \tilde{\theta }_j)\zeta ,(0.5 w_{s}^{i}; s \in S)\right) \le V^i(\overline{p}, \overline{x}^i) < V^i\left( \overline{p}, W_0,(2\overline{W}(1,\ldots ,1))_{s\in S}\right) . \end{aligned}$$

On the other hand, Assumption (C) guarantees that there exists $\overline{r}(\overline{p})\in {{\mathbb {R}}}^L_+$ such that

$$\begin{aligned} V^i\left( \overline{p}, W_0,(2\overline{W}(1,\ldots ,1))_{s\in S}\right) < V^i(\overline{p}, w^i_0 +\overline{r}(\overline{p}), (0.5 w_{s}^{i}; s \in S)). \end{aligned}$$

Indeed, following the notation of Assumption (C), the inequality above follows from

$$\begin{aligned} \overline{r}(\overline{p})=\overline{r}_{\tilde{\sigma }}\left( \overline{p},(W_0,(2\overline{W}(1,\ldots ,1))_{s\in S})\right) +W_0-w^i_0 \in {{\mathbb {R}}}^L_+, \end{aligned}$$

where $\tilde{\sigma }\in (0,1)$ is chosen to satisfy $2\overline{W} \tilde{\sigma } < \mu .$

We conclude that

$$\begin{aligned} \overline{q}_j < Q_j(\overline{p}):= \frac{\Vert \overline{r}(\overline{p})\Vert _\varSigma }{\tilde{\theta }_j \Vert \zeta \Vert _\varSigma }. \end{aligned}$$

Moreover, the upper bound $Q_j(p)$ is well defined for any $p \in {{\mathcal {P}}}$, and it follows from Assumption (C) that it varies continuously with commodity prices. Thus, the function $Q:{{\mathcal {P}}}\rightarrow {{\mathbb {R}}}$ defined by $Q(p)= \max \nolimits _{j \in J} Q_j(p)$ is continuous. Since ${{\mathcal {P}}}$ is compact, we conclude that there exists $\overline{Q}>0$ such that $\max \nolimits _{j\in J}\overline{q}_j<\overline{Q}$. $\square$

We define $\overline{X}= 2(1+\overline{Q})\overline{W}.$

Note that for any $X>\overline{X}$ and $Q>\overline{Q}$, in the associated generalized game ${\mathcal {G}}\left( Q,X,\varTheta \right)$ any player $h \in I$ may demand in the first period the bundle used in the proof of Lemma 4. Thus, in this type of generalized game, the existence of an optimal plan satisfying the conditions of lemma above will imply that the unitary prices of assets are bounded from above by $\overline{Q}$.

The existence of equilibria in our economy is a consequence of the following result.

Lemma 5

Under Assumptions (A), (B) and (C), if $(Q,X,\varTheta )\gg (\overline{Q}, \overline{X},\overline{\varTheta })$, then every Cournot–Nash equilibrium for ${\mathcal {G}}\left( Q,X,\varTheta \right)$ is an equilibrium of the original economy.

Proof

Let $\left( \overline{p},\overline{q},(\overline{\eta }^i; i \in I)\right)$, where $\overline{\eta }^i=\left( \overline{x}^i,\overline{\theta }^i\right) \in \overline{\varGamma }^i$, be a equilibrium for the generalized game ${\mathcal {G}}\left( Q,X,\varTheta \right)$, with $(Q,X,\varTheta )\gg ( \overline{Q}, \overline{X},\overline{\varTheta })$.

Step I: Market feasibility. Aggregating agent’s first period budget constraints, we have

$$\begin{aligned} \overline{p}_0\sum _{i\in I}\left( \overline{x}^i_0-w_0^i\right) +\sum _{i\in I}\sum _{j\in J^i}\overline{q}_j\overline{\theta }^i_j\le 0. \end{aligned}$$

It follows that if $\sum \nolimits _{i\in I}\left( \overline{x}^i_{0,l}-w_{0,l}^i\right) > 0$, then the auctioneer h(0) will choose the greater price for this good, $\overline{p}_l=1$, and zero prices for the other goods and assets, making his objective function positive, which contradicts the inequality above. Therefore, $\sum \nolimits _{i\in I}\overline{x}^i_{0}\le \sum \nolimits _{i\in I}w_{0}^i<W_{0}$. Analogously, if $\sum \nolimits _{i\in I(j)}\overline{\theta }_j^i>0$, then the auctioneer h(0) would choose the maximum price possible for this asset, i.e., $\overline{q}_j=Q>\overline{Q}$, which is a contradiction with the result of Lemma 4. Thus, for any $j\in J$, $\sum \nolimits _{i\in I(j)}\overline{\theta }_j^i\le 0$.

Since first period consumption is bounded from above by the aggregate endowment, which is less than X, it follows that budget constraints at $t=0$ are satisfied with equality. Hence, the auctioneer h(0) has an optimal value equal to zero. As a consequence, if $\sum \nolimits _{i\in I}\left( \overline{x}^i_{0,l}-w_{0,l}^i\right) < 0$, the auctioneer h(0) would choose a zero price for the good l, a contradiction with the strict monotonicity of preferences for some $i\in I$ (Assumption B). Therefore, $\sum \nolimits _{i \in I} \overline{x}^i_0= W_0$. Furthermore, if $\sum \nolimits _{i\in I(j)}\overline{\theta }_j^i<0$, the auctioneer would choose $\overline{q}_j=0$, a contradiction with the monotonicity of preferences. Then, market feasibility conditions hold at $t=0$ in both physical and financial markets.

Using the market feasibility of $\left( \left( \overline{x}^i,\overline{\theta }^i\right) ; i \in I\right)$ at $t=0$, and aggregating budget constraints at $s \in S$, we obtain that $\overline{p}_s\sum \nolimits _{i\in I}\left( \overline{x}^i_s-w^i_s\right) \le 0$. Therefore, analogous arguments to those made above ensure that $\sum \nolimits _{i\in I}\left( \overline{x}^i_{s}-w^i_{s}\right) \le 0$. This last property guarantees that budget constraints are satisfied as an equality in the state of nature s. Finally, if $\sum \nolimits _{i\in I}\left( \overline{x}^i_{s,l}-w_{s,l}^i\right) <0$, then the auctioneer h(s) would choose a zero price for the good $l\in L$, which contradicts individual optimality under strictly monotonic preferences for some $i\in I$. We conclude that market feasibility also holds at each state of nature $s \in S$.

Step II. Optimality of individual allocations. Since market feasibility holds in physical markets, it follows that $\overline{x}^i_{0,l}< X$ and $\overline{x}^i_{s,l}< 2\overline{W}$, for any $(i,s,l) \in I \times S \times L$. Using Lemma 3, we have that for any $i \in I$ and $j \in J^i$, $\vert \theta ^i_j\vert < \varTheta$. Thus, for any $i \in I$, $\overline{\eta }^i$ belongs to the interior of $K^i(X,\varTheta )$.

Suppose that there exists another allocation $\eta ^i\in {\mathbb {R}}^{L\times S^*}_+\times {\mathbb {R}}^{J^i}$ such that $V^i(\overline{p}, \eta ^i)>V^i(\overline{p}, \overline{\eta }^i)$. Since for $\lambda \in (0,1)$ sufficiently small, $\eta ^i(\lambda ):=\lambda \eta ^i + (1-\lambda )\overline{\eta }^i \in K^i(X,\varTheta )$, the strictly concavity of $V^i(\overline{p},\cdot )$ implies that $V^i(\overline{p}, \eta ^i(\lambda ))> V^i(\overline{p}, \overline{\eta }^i)$, a contradiction with the optimality of $\overline{\eta }^i \in \overline{\varGamma }^i$. Therefore, for any $\eta ^i\in {\mathbb {R}}^{L\times S^*}_+\times {\mathbb {R}}^{J^i}$, $V^i(\overline{p}, \eta ^i)\le V^i(\overline{p}, \overline{\eta }^i)$, which proves the optimality of $\overline{\eta }^i\in B^i(\overline{p}, \overline{q})$ among the allocations in the agent i’s budget set. Notice that as was proved in Sect. 3, informational compatibility of consumption allocations follows from Assumption (A). $\square$