Collateral Versus Default History

Abstract

This paper deals with equilibrium existence for incomplete markets economies with finitely-lived agents and infinitely-lived agents when default is allowed and borrowers have to constitute collateral in terms of durable goods. In the first model, lenders are protected by an exogenous personalized collateral. In the second model, the personalized collateral requirements are endogenously determined by a financial institution whose objective is to minimize the default rate taking into account agent’s default history.

Appendices

Appendix

Proof of Proposition 1

The proof of this proposition is close to the proof of Araujo–Páscoa–Torres-Martínez [3]. However, because in our model the number of alive agents depend on the nodes, the uniform bound on the debt values in the former paper no longer holds. However, we obtain bounds on the short sales using similar tricks. Since the stochastic structure of our model is characterized by a finite number of immediate successors at each node, these bounds rule out the possibility of Ponzi games. Moreover, the presence of finitely-lived agents in our model leads to a new technical difficulty, namely that the interior of the budget sets of these agents could be empty, and therefore the budget correspondences may fail to be lower semicontinuous. To overcome this problem, one can define modified budget correspondences that are lower semicontinuous, and applying the Gale and Mas-Colell fixed point theorem [10, 11], we guarantee that each truncated economy has an equilibrium. These modified correspondences are close to the correspondences defined below (in the following proof of Proposition 2) but obviously they do not depend on the collateral since the personalized collateral are exogenous in our first model.

Proof of Proposition 2

3.1 Equilibria in the Truncated Economies

Let ${\mathcal{E}^{\prime}}^{T}$ be the truncated economy associated with the original economy $\mathcal{E}^{{\prime}},$ which has the same characteristics than $\mathcal{E}^{{\prime}},$ but where we suppose that agents are constrained to stop their exchange of goods at period T and their exchange of assets at period T − 1. Formally, for each T > 0, let us define the following sets:

$$\displaystyle{\pi ^{T-1}\,:=\,\left \{(p,q) \in \mathbb{R}_{ +}^{D^{T}\times G }\times \prod \limits _{\xi \in D^{T}}\mathbb{R}^{J(\xi )}\left \vert \begin{array}{rl} & \forall \xi \,:\,\ t(\xi ) < T,\ \Vert p(\xi )\Vert _{1}\,+\,\Vert q(\xi )\Vert _{1} = 1, \\ &\forall \xi \,:\,\ t(\xi ) = T,\ \Vert p(\xi )\Vert _{1} = 1.\\ \end{array} \right \},\right.}$$

Let us recall that for each node ξ ∈ D ^T, for each asset j ∈ J(ξ ⁻), one has $R^{j}(\xi )\leqslant \Vert A^{j}(\xi )\Vert _{1}.$ Let us denote by:

$$\displaystyle{R^{T}\,:=\,\{R=(R^{j}(\xi ),\ \xi \in D^{T},\ j \in (\xi ^{-}))\mid \forall \xi \in D^{T},\ \forall j \in \left.\left (\xi ^{-}\right )\right )\},\ R^{j}(\xi )\,\leqslant \,\Vert A^{j}(\xi )\left.\Vert _{ 1}\right \}.}$$

Moreover, let us denote by $\mathcal{M}:=\{\alpha \in \mathbb{R}_{+}^{G}\mid \beta \leqslant \alpha \leqslant W\}.$

For each h ∈ H, 

$$\displaystyle\begin{array}{rcl} & X^{hT}:=\{ (x^{h}(\xi ),\ \xi \in D) \in X^{h}\mid \forall \xi: t(\xi ) > T,\ x^{h}(\xi ) = 0\},& {}\\ & Z^{hT}:=\{ (z^{h}(\xi ),\ \xi \in D) \in X^{h}\mid \forall \xi: t(\xi )\geqslant T,\ z^{h}(\xi ) = 0\}, & {}\\ & \mbox{ and}\ \forall \xi: t(\xi ) = T,\ \forall j \in J(\xi ),\ M_{j}^{h}(\xi ) = 0. & {}\\ \end{array}$$

Moreover, the budget set of an agent h ∈ H for the truncated economy can be defined as follows:

$$\displaystyle{ B^{hT}(p,q,R,M^{h}) = \left \{(x,\theta,\phi,\varDelta )\left \vert \begin{array}{rl} &p(\xi ^{h})x^{h}(\xi ^{h}) + p(\xi ^{h})M^{h}(\xi ^{h})\phi ^{h}(\xi ^{h}) + q(\xi ^{h})z^{h}(\xi ^{h}) \\ &\leqslant p(\xi ^{h})\omega ^{h}(\xi ^{h}),\ \xi ^{h} \in D^{T-1}, \\ &p(\xi ) \cdot (x^{h}(\xi ) -\omega ^{h}(\xi )) + p(\xi )M^{h}(\xi )\phi ^{h}(\xi ) + q(\xi ) \cdot z^{h}(\xi ) \\ & +\sum \limits _{j\in J(\xi ^{-})}\varDelta _{j}^{h}(\xi )\leqslant p(\xi )[Y (\xi )x^{h}(\xi ^{-}) + Y (\xi )M^{h}(\xi ^{-})\phi ^{h}(\xi ^{-})]+ \\ &\sum \limits _{j\in J(\xi ^{-})}R^{j}(\xi )\theta _{j}^{h}(\xi ^{-}),\ \forall \xi \in (D^{h} -\delta D^{h}) \cap D^{T-1},\ \xi \neq \xi ^{h}, \\ &p(\xi ) \cdot (x^{h}(\xi ) -\omega ^{h}(\xi )) +\sum \limits _{j\in J(\xi ^{-})}\varDelta _{j}^{h}(\xi )\leqslant \\ &p(\xi )[Y (\xi )x^{h}(\xi ^{-}) + Y (\xi )M^{h}(\xi ^{-})\phi ^{h}(\xi ^{-})]+ \\ &\sum \limits _{j\in J(\xi ^{-})}R^{j}(\xi )\theta _{j}^{h}(\xi ^{-}),\forall \xi \in \delta D^{h} \cup D^{T},\ \xi \neq \xi ^{h}, \end{array} \right \}\right. }$$

Moreover, for each agent h ∈ H, the utility function U ^hT for each truncated economy ${\mathcal{E}^{\prime}}^{T}$ is defined as follows:

$$\displaystyle{ U^{hT}(x^{h},\theta ^{h},\phi ^{h},\varDelta ^{h}):=\sum \limits _{\xi \in D^{T}}v_{\xi }^{h}(\tilde{x}^{h}(\xi )). }$$

(15)

Lemma 1

Under the assumptions stated above, an allocation (x,θ,ϕ,D) which satisfies the conditions (ii), (iii), (iv) and (v) of Definition  4 is bounded.

Proof of Lemma  1 Let (x, θ, ϕ, Δ) be an allocation which satisfies the conditions (ii), (iii), (iv) and (v) of Definition 4. The bounds on x, θ and ϕ are obtained as in Araujo–Páscoa–Torres-Martínez [3]. More precisely, it follows from (ii) that:

$$\displaystyle{ \sum \limits _{(h,g)\in H(0)\times G}[x^{h}(0,g) +\sum \limits _{ j\in J(0)}M_{j}^{h}(0)\phi _{ j}^{h}(0)] =\sum \limits _{ (h,g)\in H\times G}\omega ^{h}(0,g)\leqslant WH(0). }$$

(16)

Let $\overline{Y }:=\max \limits \{ (Y (\xi ))_{g,g^{{\prime}}},(\xi,g,g^{{\prime}}) \in \times D^{T} \times G \times G\}.$ Then, ∀ξ ∈ D ^T ∖{0}, one has:

$$\displaystyle{ \begin{array}{ccc} \sum \limits _{(h,g)\in H(\xi )\times G}[x^{h}(\xi,g) +\sum \limits _{j\in J(\xi )}M_{jg}^{h}(\xi )\phi _{j}^{h}(\xi )]\leqslant && \\ WH(\xi ) + \overline{Y }G +\sum \limits _{(h,g)\in H(\xi )\times G}[x^{h}(\xi ^{-},g) +\sum \limits _{j\in J(\xi ^{-})}M_{jg}^{h}(\xi ^{-})\phi _{j}^{h}(\xi ^{-})]. \end{array} }$$

(17)

It then follows from Eqs. (16) and (17) that for each node ξ ∈ D ^T:  t(ξ) = t that:

$$\displaystyle{ \sum \limits _{(h,g)\in H(\xi )\times G}[\overline{x}^{h}(\xi,g) +\sum \limits _{ j\in J(\xi )}M_{jg}^{h}(\xi )\overline{\phi }_{ j}^{h}(\xi )]\leqslant WH(\xi )\sum \limits _{ n=0}^{t}(\overline{Y }G)^{n}. }$$

(18)

By definition of the personalized collateral, one has that $m^{h}(\xi ) =\min \limits _{j\in J(\xi )}\Vert M_{j}^{h}(\xi )\Vert _{1} > 0,$ and hence ∀h ∈ H one gets:

$$\displaystyle{ x^{h}(\xi,g)\leqslant WH(0)\sum \limits _{ n=0}^{t}(\overline{Y }G)^{n}:=\chi (\xi ) < +\infty, }$$

(19)

$$\displaystyle{ \phi _{j}^{h}(\xi )\leqslant \frac{\chi (\xi )} {m^{h}(\xi )}:= \alpha ^{h}(\xi ) < +\infty,\ \forall j \in J(\xi ), }$$

(20)

$$\displaystyle{ \theta _{j}^{h}(\xi )\leqslant \alpha ^{h}(\xi ) < +\infty,\ \forall j \in J(\xi ), }$$

(21)

On the other hand, since $\forall j \in J(\xi ),\ \varDelta _{j}^{h}(\xi )\leqslant p(\xi )A^{j}(\xi )\phi _{j}^{h}(\xi ^{-})$ and in view of our normalization, one gets:

$$\displaystyle{ \varDelta _{j}^{h}(\xi )\leqslant \Vert A^{j}(\xi )\Vert _{ 1}\alpha ^{h}(\xi ^{-}):= \gamma ^{h}(\xi ) < +\infty. }$$

(22)

We will denote by $\alpha (\xi ):=\sup \limits _{h\in H(\xi )}\alpha ^{h}(\xi )$ and by $\gamma (\xi ):=\sup \limits _{h\in H(\xi )}\gamma ^{h}(\xi ).$

For each h ∈ H, let us define:

$$\displaystyle{B^{hT}(p,q,R,M^{h},\chi,\alpha,\gamma ) = \left \{(x,\theta,\phi,\varDelta ) \in B^{hT}(p,q,R,M^{h})\left \vert \begin{array}{rl} &x^{h}(\xi,g)\leqslant 2\chi, \\ &\theta _{j}^{h}(\xi )\leqslant 2\alpha (\xi ), \\ &\phi _{j}^{h}(\xi )\leqslant 2\alpha (\xi ), \\ &\varDelta _{j}^{h}(\xi )\leqslant 2\gamma (\xi ), \end{array} \right \}\right.}$$

Let ${\mathcal{E}^{\prime}}^{T}(\chi,\alpha,\gamma )$ be an economy with the same characteristics as ${\mathcal{E}^{\prime}}^{T}$ but in which the budget constraints are defined by the set B ^hT(p, q, R, M ^h, χ, α, γ). 

Lemma 2

The truncated and compactified economy ${\mathcal{E}^{\prime}}^{T}(\chi,\alpha,\gamma )$ has an equilibrium $(\overline{p}^{T},\overline{q}^{T},\overline{R}^{T},(\overline{x}^{hT},\overline{\theta }^{hT},\overline{\phi }^{hT},\overline{\varDelta }^{hT})_{h\in H})$ such that $\forall h \in H,\ \forall \xi \in D^{T-1},\ \forall j \in J(\xi ^{-}),$ one has $\overline{M}_{j}^{hT}(\xi ):= F_{\xi,j}^{h}(\overline{p}^{T}(\xi ),\overline{q}^{T}(\xi ),\overline{\kappa }_{\xi }^{hT}).$

Proof of Lemma  2 The first new technical difficulty in our model in comparison with Araujo–Páscoa–Torres-Martínez [3] is that the budget set correspondences of the finitely-lived agents may not be lower semicontinuous (since their interior can be an empty set). Let us consider an agent h ∈ H and define the set B ^{′ hT}(p, q, R, M ^h, χ, α, γ) by replacing all the inequalities in B ^hT(p, q, R, M ^h, χ, α, γ) by strict inequalities. Moreover, let us define the correspondence

$$\displaystyle{B''^{hT}(p,q,R,M^{h},\chi,\alpha,\gamma ) = \left \{\begin{array}{rl} \{(\omega ^{h},0,0,0)\}&\mbox{ if}\ {B^{\prime}}^{hT}(p,q,R,M^{h},\chi,\alpha,\gamma ) =\emptyset \\ B^{hT}(p,q,R,\chi,\alpha,\gamma )&\mbox{ if}\ {B^{\prime}}^{hT}(p,q,R,M^{h},\chi,\alpha,\gamma )\neq \emptyset \end{array}.\right.}$$

Remark 2

$\forall h \in H,\ \forall (p,q) \in \pi ^{T},\ \forall R \in \mathbb{R}^{T},\ \forall M^{h} \in \mathcal{M},\ B''^{hT}(p,q,R,M^{h})\neq \emptyset$ since it always contains (ω ⁱ, 0). 

Moreover, one can easily prove that B″^hT is lower semicontinuous. To simplify the notations, we define v: = (p, q, R, M), and w: = (x, θ, ϕ, Δ). 

For each agent h ∈ H, let us define the following correspondence:

$$\displaystyle{\varPsi ^{hT}(v,w) = \left \{\begin{array}{rl} B^{hT}(v,\chi,\alpha,\gamma )&\mbox{ if}\ w\notin B^{hT}(v,\chi,\alpha,\gamma ) \\ {B^{\prime}}^{hT}(v,\chi,\alpha,\gamma ) \cap P^{h}(w)&\mbox{ if}\ w \in B^{hT}(v,\chi,\alpha,\gamma ) \end{array},\right.}$$

We also define the correspondence:

$$\displaystyle{\varPsi ^{0T}(v,w) = \left \{(p^{{\prime}},q^{{\prime}}) \in \pi ^{T}\left \vert \begin{array}{rl} &\forall \xi \in D^{T}, \\ &(p^{{\prime}}(\xi ) - p(\xi )) \cdot \sum \limits _{h\in H}[x^{h}(\xi ) + M^{h}(\xi )\phi ^{h}(\xi ) + Y (\xi )x^{h}(\xi ^{-}) \\ & - Y (\xi )M^{h}(\xi ^{-})\phi ^{h}(\xi ^{-}) -\omega ^{h}(\xi )] + (q^{{\prime}}(\xi ) - q(\xi )) \cdot \sum \limits _{h\in H}z^{h}(\xi ) > 0. \end{array} \right \}\right.}$$

where P ^h(w): = { w ^′∣U ^h(w ^′) > U ^h(w)}. 

Moreover, we add the following players to this generalized game:

• Given an allocation (x, θ, ϕ, Δ), at each node ξ ∈ D ^T−1, for each asset j ∈ J(ξ), a financial institution chooses M _j ^h(ξ) in order to solve the following problem:

  $$\displaystyle{\min \limits _{M_{j}^{h}(\xi )\in \mathcal{M}}[M_{j}^{h}(\xi ) - F_{\xi,j}^{h}(p(\xi ),q(\xi ),\kappa _{\xi }^{h})]^{2},}$$

• Given an allocation (x, θ, ϕ, Δ), at each node ξ ∈ D ^T ∖{0}, for each j ∈ J(ξ ⁻), an auctioneer chooses $R^{j}(\xi )\leqslant \Vert A^{j}(\xi )\Vert _{1}$ in order to maximize:

  $$\displaystyle{[R^{j}(\xi )\sum \limits _{ h\in H}\theta ^{h}(\xi ^{-}) -\sum \limits _{ h\in H}D_{j}^{h}(\xi )]^{2}.}$$

Since, $\forall h \in H \cup \{ 0\},\ \varPsi ^{hT}$ is lower semicontinuous and by definition of Ψ ^hT,  w ∉ Ψ ^hT(v, w), it follows from the Gale and Mas-Colell fixed point theorem [10, 11] that there exists $(\overline{p}^{T},\overline{q}^{T},\overline{R}^{T},(\overline{M}^{hT})_{h\in H}(\overline{x}^{hT},\overline{\theta }^{hT},\overline{\phi }^{hT},\overline{\varDelta }^{hT})_{h\in H}):= (\overline{v},\overline{w})$ such that:

$$\displaystyle{\forall h \in H \cup \{ 0\},\ \varPsi ^{hT}(\overline{v},\overline{w}) =\emptyset.}$$

That is, $\forall h \in H,\ {B^{\prime}}^{hT}(v,\chi,\alpha,\gamma ) \cap P^{h}(w) =\emptyset$ and

$$\displaystyle{ \begin{array}{ccc} & (p(\xi ) -\overline{p}^{T}(\xi )) \cdot \sum \limits _{h\in H}[\overline{x}^{iT}(\xi ) + M^{h}(\xi )\overline{\phi }^{hT}(\xi ) - Y (\xi )\overline{x}^{hT}(\xi ^{-}) \\ & - Y (\xi )M^{h}(\xi ^{-})\overline{\phi }^{hT}(\xi ^{-}) -\omega ^{h}(\xi )] + (q(\xi ) -\overline{q}^{T}(\xi )) \cdot \sum \limits _{h\in H}\overline{z}^{hT}(\xi )\leqslant 0, \end{array} }$$

(23)

On the other hand, the game played by the financial institution and the auctioneers yield to M _j ^h(ξ) = F _ξ, j ^h(p(ξ), q(ξ), κ _ξ ^h) and $R^{j}(\xi )\sum \limits _{h\in H}\theta _{j}^{h}(\xi ^{-}) =\sum \limits _{h\in H}D_{j}^{h}(\xi ).$ The feasibility conditions can be easily obtained using Eq. (23). □ 

Lemma 3

The truncated economy ${\mathcal{E}^{\prime}}^{T}$ has an equilibrium $(\overline{p}^{T},\overline{q}^{T},\overline{R}^{T},(\overline{M}^{hT})_{h\in H},(\overline{x}^{hT},\overline{\theta }^{hT},\overline{\phi }^{hT},\overline{\varDelta }^{hT})_{h\in H}).$

Proof of Lemma  3 We have already proved that $\forall h \in H,\ {B^{\prime}}^{hT}(v,\chi,\alpha,\gamma ) \cap P^{h}(w) =\emptyset.$ It then remains to prove that $\forall h \in H,\ B^{hT}(v,\chi,\alpha,\gamma ) \cap P^{h}(w) =\emptyset.$ This follows from a classical convexity argument. □ 

Asymptotic Results The techniques used in Araujo–Páscoa–Torres-Martínez [3] can be easily adapted to the case of incomplete participation and personalized collateral to show that the cluster point is an equilibrium of the original economy.