Credit risk in general equilibrium

Abstract

This paper contributes to the literature on default in general equilibrium. Borrowing and lending takes place via a clearing house (bank) that monitors agents and enforces contracts. Our model develops a concept of bankruptcy equilibrium that is a direct generalization of the standard general equilibrium model with financial markets. Borrowers may default in equilibrium and returns on loans are determined endogenously. Restricted to a special form of mean variance preferences, we derive a version of the capital asset pricing model with bankruptcy. In this case, we can characterize equilibrium prices and allocations and discuss implications for credit risk modeling.

Appendix

1.1 Proof of Proposition 1

Our proof consists of three steps:¹⁶ In Step (i), we prove some preliminary results. In particular, we show that asset returns induce future state-contingent consumption, which can be evaluated by an indirect utility function. Moreover, a consumer will not buy and sell bonds simultaneously, the lower bound on borrowing implies a bounded set of feasible asset trades, and the bankruptcy clearing mechanism implies a strictly positive return of the bond in all states. Step (ii) provides an existence proof for an asset market equilibrium with a bankruptcy clearing mechanism in period $t=0.$ For this proof, we can adapt standard arguments of general equilibrium theory. Finally, in Step (iii), we show that the bankruptcy clearing mechanism guarantees nonnegative consumption in all states.

1.1.1 Preliminary results

The indirect utility function Consider the price simplex in $\mathbb {R}_{+}^{J+2},$

$$\begin{aligned} Q:=\left\{ (p_{0},q_{b},q_{e})\in \mathbb {R}_{+}^{J+2}|\ p_{0}+q_{b}+q_{e} \mathbf {1}_{J}=1\right\} \!. \end{aligned}$$

For given $r_{{\varvec{1}}}\in$ $(0,1]^{S}$and $\left( p_{0},q_{b},q_{e}\right) \in Q,$ each consumer $i\in I$ maximizes $u^{i}(x_{0}^{i},x_{\mathbf {1}}^{i})$ by choosing asset trades $(z_{b+}^{i},z_{b-}^{i},z_{e}^{i})$ and a consumption allocation $(x_{0}^{i},x_{{\varvec{1}}}^{i})\in X^{i}$ subject to¹⁷

$$\begin{aligned} \begin{array}{llll} (a) &{} p_{0}\left( x_{0}^{i}-\omega _{0}^{i}\right) &{} \le &{} -q_{b}z_{b+}^{i}+q_{b}z_{b-}^{i}-q_{e}z_{e}^{i}, \\ (b) &{} x_{{\varvec{1}}}^{i} - \omega _{{\varvec{1}}}^{i}-Y\delta ^{i} &{} \le &{} r_{{\varvec{1}}}z_{b+}^{i} - \mathbb {1}z_{b-}^{i}+Yz_{e}^{i},\\ (c) &{} z_{b+}^i&{} \ge &{} 0 \\ (d) &{} z_{b-}^{i} &{} \ge &{} 0, \\ (e) &{} z_{b-}^{i} &{} \le &{} \kappa , \\ (f) &{} z_{e}^{i} &{} \ge &{} -\delta ^{i}, \end{array} \end{aligned}$$

where the constraint $(e)$ follows from A5 in Proposition 1.

Since $u^{i}$ is strictly increasing in its arguments (Proposition 1, A2), condition $(b)$ is binding for an optimal choice $(x_{0}^{i},x_{{\varvec{1}}}^{i},z_{b+}^{i}, z_{b-}^{i}, z_{e}^{i}),$ i.e.,

$$\begin{aligned} x_{{\varvec{1}}}^{i}=\omega _{{\varvec{1}}}^{i} + Y\left( \delta ^{i}+z_{e}^{i}\right) +r_{{\varvec{1}}}z_{b+}^{i} - \mathbb {1}z_{b-}^{i}. \end{aligned}$$

(8)

Moreover, for $r_{{\varvec{1}}}\ne 1\!\!1,$ the consumer will never take a short position $z_{b-}^{i}>0$ and a long position $z_{b+}^{i}>0$ simultaneously. These observations allow us to restrict attention to net trades in bonds $z_{b}^{i}:= \left( z_{b+}^{i}-z_{b-}^{i}\right) =\left( z_{b}^{i}\vee 0\right) +\left( z_{b}^{i}\wedge 0\right) \in \mathbb {R}.$ Hence, we can write $z^{i}=(z_{b}^{i},z_{e}^{i})$ for the net asset trades of a consumer $i\in I.$

For each consumer $i\in I,$ consider the consumption set in period 0, $X_{0}^{i}:=\mathbb {R}_{+},$ and the set of feasible net asset trades,$\ Z^{i}:=[-\kappa ,\infty )\times \left( \mathbb {R}_{+}^{J} -\left\{ \delta ^{i}\right\} \right)$.

For any $r_{1}\in (0,1]^{S}$, Eq. (8) allows us to define the following indirect utility function $v^{i}:X_{0}^{i}\times Z^{i}\rightarrow \mathbb {R}$

$$\begin{aligned} v^{i}(x_{0}^{i},z_{b}^{i},z_{e}^{i}; r_{{\varvec{1}}}):=u^{i}\left( x_{0}^{i}, \omega _{{\varvec{1}}}^{i}+ Y\left( \delta ^{i}+z_{e}^{i}\right) +r_{{\varvec{1}}}\left( z_{b}^{i}\vee 0\right) +\mathbb {1}\left( z_{b}^{i}\wedge 0\right) \right) \end{aligned}$$

By Assumption A2, the indirect utility function $v^{i}$ is a continuous function. Moreover, $v^{i}$ is strictly increasing and concave in $\left( x_{0}^{i},z_{b}^{i},z_{e}^{i}\right)$.

Feasible allocations An allocation for a consumer $i\in I\$in period $t=0$ is a vector of period 0 consumption and of asset trades $(x_{0}^{i},z^{i}) \in X_{0}^{i}\times Z^{i}.$ Denote by $A:=\prod _{i\in I}\left( X_{0}^{i}\times Z^{i}\right) \subset \mathbb {R}^{J+2}$ the set of allocations for the economy. Denote $(x_{0}^{i},z^{i})_{i\in I}$ by $(x_0,z)$. An allocation $(x_0,z) \in A$ is weakly feasible if $\sum _{i\in I}\left( x_{0}^{i}-\omega _{0}^{i},z^{i}\right) \le \left( 0,0\right)$ holds. Since $X_{0}^{i}\times Z^{i}$ is bounded below for all $i\in I$, the set of weakly feasible allocations

$$\begin{aligned} F:=\left\{ (x_{0}^{i},z^{i})_{i\in I}\in A\ |\ \sum _{i\in I}\left( x_{0}^{i}-\omega _{0}^{i},z^{i}\right) \le \left( 0,0\right) \right\} \end{aligned}$$

is a compact set.

The bankruptcy clearing mechanism The bankruptcy clearing mechanism (compare Eq. 4) is a function $\rho :A\rightarrow (0,1]^{S}$ defined by its component functions

$$\begin{aligned} \rho _{s}(x_{0},z)=\left\{ \begin{array}{lll} \frac{\sum \nolimits _{i=1}^{I}\left( -z_{b}^{i}\vee 0\right) \wedge (\omega _{s}^{i}+Y_{s}(\delta ^{i}+z_{e}^{i}))}{\sum \nolimits _{i=1}^{I}\left( -z_{b}^{i}\vee 0\right) }. &{} \hbox {for} &{} \sum \nolimits _{i=1}^{I}\left( -z_{b}^{i} \vee 0\right) >0\\ 1 &{} \hbox {for} &{} \sum \nolimits _{i=1}^{I}\left( -z_{b}^{i}\vee 0\right) =0 \end{array}\right. \! , \end{aligned}$$

$\rho _{s}(x_{0},z)$ is a continuous function on $A.$ Notice that, by Assumption A3, $\omega _{s}^{i}>0$ for all $s\in S$ and all $i\in I.$ Hence, $\left( -z_{b}^{i}\vee 0\right) \wedge (\omega _{s}^{i}+Y_{s}(\delta ^{i}+z_{e}^{i}))=-z_{b}^{i}$ for $0<-z_{b}^{i}<\omega _{s}^{i}.$ This implies that $\rho _{s}(x_{0},z)=1$ in a neighborhood of $z_{b}=0.$ Thus, $\rho _{s}(x_{0},z)$ is well defined and continuous at $z_{b}=0.$ Moreover, $\frac{\left( -z_{b}^{i}\vee 0\right) \wedge (\omega _{s}^{i}+Y_{s}(\delta ^{i}+z_{e}^{i}))}{\left( -z_{b}^{i}\vee 0\right) }$ is an increasing function of $z_{b}^{i}$ with a minimum $\frac{\kappa \wedge (\omega _{s}^{i}+Y_{s}(\delta ^{i} +z_{e}^{i}))}{\kappa } \ge \frac{\kappa \wedge \omega _{s}^{i}}{\kappa }\$at $z_{b}^{i}=-\kappa .$ Therefore, we have

$$\begin{aligned} \rho _{s}(x_{0},z)&= \frac{\sum _{i=1}^{I}\left( -z_{b}^{i} \vee 0\right) \wedge (\omega _{s}^{i}+Y_{s} (\delta ^{i} +z_{e}^{i}))}{\sum _{i=1}^{I}\left( -z_{b}^{i}\vee 0\right) } \\&= \sum _{i=1}^{I}\left[ \frac{\left( -z_{b}^{i} \vee 0\right) \wedge (\omega _{s}^{i}+Y_{s}(\delta ^{i}+z_{e}^{i}))}{\left( -z_{b}^{i}\vee 0\right) } \right] \frac{\left( -z_{b}^{i}\vee 0\right) }{\sum _{i=1}^{I}\left( -z_{b}^{i}\vee 0\right) } \\&\ge \sum _{i=1}^{I}\left[ \frac{\kappa \wedge \omega _{s}^{i}}{\kappa } \right] \frac{\left( -z_{b}^{i}\vee 0\right) }{\sum _{i=1}^{I}\left( -z_{b}^{i}\vee 0\right) } \\&\ge \underset{i\in I}{\min } \left[ \frac{\kappa \wedge \omega _{s}^{i}}{ \kappa }\right] \underset{=1}{\underbrace{\sum _{i=1}^{I}\frac{\left( -z_{b}^{i}\vee 0\right) }{\sum _{i=1}^{I} \left( -z_{b}^{i}\vee 0\right) }}}\\&= \underset{i\in I}{\min } \left[ \frac{\kappa \wedge \omega _{s}^{i}}{ \kappa }\right] =:\underline{\rho }_{s}>0. \end{aligned}$$

Hence, for all $s\in S,\ \ \rho _{s}(x_{0},z)$ is bounded below by a strictly positive value $\underline{\rho }_{s}$. Denote by $R:=\prod _{s\in S} \left[ \underline{\rho }_{s},1\right] \subset (0,1]^{S}$ the set of feasible return rates for the bond.

1.1.2 Existence of equilibrium in $\mathbf {t=0}$

Choose a compact and convex set $K\subset \mathbb {R}^{J+2}$ such that $(x_{0}^{i},z^{i})_{i\in I}\in F$ implies $(x_{0}^{i},z^{i}) \in \text {int}K$ for all $i\in I.$

For any $\left( p_{0},q_{b},q_{e}\right) \in Q,$ denote by

$$\begin{aligned} \mathbb {B}_{c}^{i}(p_{0},q_{b},q_{e}):=\left\{ (x_{0}^{i},z_{b}^{i},z_{e}^{i}) \in \left( X_{0}^{i}\times Z^{i}\right) \cap K|\ p_{0}\left( x_{0}^{i}\!-\!\omega _{0}^{i}\right) \!+\!q_{b}z_{b}^{i}\!+\!q_{e}z_{e}^{i}\le 0\right\} \end{aligned}$$

the bounded budget correspondence for $t=0\$ restricted to the compact set $K.$

Lemma 1

$\mathbb {B}_{c}^{i}: Q\rightarrow \left( X_{0}^{i}\times Z^{i}\right) \cap K,$ is a compact- , convex-valued, and continuous correspondence.

Proof

Since $\mathbb {B}_{c}^{i}(p_{0},q_{b},q_{e}) \subset K$ for all $(p_{0},q_{b},q_{e}) \in Q,$ the budget set is compact. It is obviously convex. Since $\omega _{0}^{i}>0,$ $\kappa >0$ and $\delta ^{i}\in \mathbb {R}_{++}^{J},$ continuity follows by standard arguments, e.g., Debreu (1956), p. 63. $\square$

For $r_{{\varvec{1}}}\in R$ and $\left( p_{0},q_{b},q_{e}\right) \in Q$, define the bounded demand correspondence $f_{c}^{i}:Q\times R\rightarrow \mathbb {R}^{J+2}$ by

$$\begin{aligned} f_{c}^{i}(p_{0},q_{b},q_{e};r_{{\varvec{1}}}):=\arg \max \left\{ v^{i}(x_{0}^{i},z_{b}^{i},z_{e}^{i};r_{{\varvec{1}}})|\ \left( x_{0}^{i},z_{b}^{i},z_{e}^{i}\right) \in \mathbb {B} _{c}^{i}(p_{0},q_{b},q_{e})\right\} . \end{aligned}$$

Lemma 2

$f_{c}^{i}(p_{0},q_{b},q_{e};r_{{\varvec{1}}})$ is a nonempty, compact- and convex-valued, u.h.c. correspondence from $Q\times R$ into $\mathbb {R}^{J+2}$.

Proof

The indirect utility $v^{i}$ is a continuous function on $\mathbb {R}_{+}\times \mathbb {R}^{J+1}\times R.$ By Lemma 1, $\mathbb {B}_{c}^{i}(p_{0},q_{b},q_{e})$ is a compact-, convex-valued, and continuous correspondence on $Q$. Hence, by the maximum theorem, the demand correspondence $f_{c}^{i}(p_{0},q_{b},q_{e}; r_{{\varvec{1}}})$ is nonempty, compact-valued, and u.h.c. for all $(p_{0},q;r_{{\varvec{1}}})\in Q\times R$. Since $v^{i}$ is concave, it follows by standard arguments that $f_{c}^{i}(p_{0},q_{b},q_{e};r_{{ \varvec{1}}})$ is convex-valued. $\square$

Lemma 3

Consider a vector of prices $(p_{0},q_{b},q_{e})\in Q$ and an allocation of optimal choices $(x_{0}^{i},z^{i})\in f_{c}^{i}(p_{0},q_{b},q_{e};r_{{\varvec{1}}})$ for all $i\in I$ which is weakly feasible, $(x_{0}^{i},z^{i})_{i\in I}\in F.$ Then, all prices must be strictly positive, $(p_{0},q_{b},q_{e})>>0.$

Proof

Suppose there was a price $q_{e}^{j}=0,$ since the indirect utility is strictly increasing in $(x_{0}^{i},z^{i}),$ $z_{e_{j}}^{i}$ must be in the boundary of $K.$ Since $(x_{0}^{i},z^{i})_{i\in I}\in F$ implies $(x_{0}^{i},z^{i})\in \text {int}K$ for all $i\in I,$ this contradicts $(x_{0}^{i},z^{i})_{i\in I}\in F.$ A similar argument holds for $p_{0}=0$ and $q_{b}=0.$ $\square$

Lemma 3 shows also that a weakly feasible allocation of optimal choices cannot be in the boundary of $K.$ Hence, for a feasible allocation, the constraints of $K$ will never be binding.

Define the individual excess demand correspondence $\zeta ^{i}:Q\times R\rightarrow \mathbb {R}^{J+2}$ as

$$\begin{aligned} \zeta ^{i}(p_{0},q;r_{{{\varvec{1}}}}):= f_{c}^{i}(p_{0},q;r_{{{ \varvec{1}}}})- \left\{ (\omega _{0}^{i},0,0)\right\} . \end{aligned}$$

Obviously, $\zeta ^{i}$ inherits all relevant properties of $f_{c}^{i},$ in particular, $\zeta ^{i}$ is a nonempty, compact- and convex-valued u.h.c. correspondence. Since $f_{c}^{i}(p_{0},q;r_{{{\varvec{1}}}})\subseteq \mathbb {B}_{c}^{i}(p_{0},q),$ individual excess demand correspondences $\zeta ^{i}$ are bounded below by $(-\omega _{0}^{i},-\kappa ,-\delta ^{i}).$ Consequently, the aggregate excess demand correspondence $\sum _{i\in I}\zeta ^{i}(p_{0},q;r_{{{\varvec{1}}}})$ is bounded below by $\left( -\sum _{i\in I}\omega _{0}^{i},-|I|\kappa , -\sum _{i\in I}\delta ^{i}\right) .$

We denote by $\zeta :Q\times R\rightarrow K^{I},$

$$\begin{aligned} \zeta (p_{0},q;r_{{\varvec{1}}}):=\prod \limits _{i\in I}\zeta ^{i}(p_{0},q;r_{{\varvec{1}}}), \end{aligned}$$

the Cartesian product of the individual excess demand correspondences. By Lemma 2, $\zeta (p_{0},q;r_{{\varvec{1}}})$ is a nonempty, compact-, and convex-valued u.h.c. correspondences on $Q\times R$ as a product of correspondences with these properties.

For any excess demand vector $(x_{0},z):=(x_{0}^{i},z^{i})_{i\in I}\in K^{|I|},$ let

$$\begin{aligned} \mu (x_{0},z):=\arg \max \left\{ p_{0} \sum _{i=1}^{I}\left( x_{0}^{i}-\omega _{0}^{i}\right) +q_{b}\sum _{i=1}^{I}z_{b}^{i}+q_{e} \sum _{i=1}^{I}z_{e}^{i}|~(p_{0},q_{b},q_{e})\in Q\right\} \end{aligned}$$

be the set of prices in $Q$ that maximize the value of the excess demand$.$ Since $Q$ is compact and $p_{0} \sum _{i=1}^{I}\left( x_{0}^{i} -\omega _{0}^{i}\right) +q_{b}\sum _{i=1}^{I}z_{b}^{i}+q_{e} \sum _{i=1}^{I}z_{e}^{i}$ is a continuous function on $Q,\ \mu (x_{0},z)$ is not empty. By the maximum theorem, $\mu :K^{|I|}$ $\rightarrow Q$ is a compact-, convex-valued, and u.h.c. correspondence on $K^{|I|}.$

For each $(p_{0},q_{b},q_{e},r_{{\varvec{1}}},(x_{0},z))\in Q\times R\times K^{|I|}$, define the correspondence

$$\begin{aligned} \varPhi (p_{0},q_{b},q_{e},r_{{\varvec{1}}}, (x_{0},z)):=\mu (x_{0},z)\times \left\{ \rho (x_{0},z)\right\} \times \zeta (p_{0},q_{b},q_{e},r_{{\varvec{1}}}). \end{aligned}$$

The correspondence $\varPhi$ is a mapping $Q\times R\times K^{|I|}\rightarrow$ $Q\times R\times K^{|I|}.$ The correspondence $\varPhi$ is nonempty, compact-valued, convex-valued, and u.h.c., as a Cartesian product of correspondences with these properties. Hence, by Kakutani’s fixed point theorem, there is $(p_{0}^{*},q_{b}^{*},q_{e}^{*},r_{{\ \varvec{1}}}^{*},\left( x_{0}^{*},z^{*}\right) ) \in \varPhi (p_{0}^{*},q_{b}^{*},q_{e}^{*},r_{{\varvec{1}}}^{*},\left( x_{0}^{*},z^{*}\right) ).$

By construction of $\mu ,$

$$\begin{aligned}&p_{0}^{*}\sum _{i=1}^{I} \left( x_{0}^{*i} -\omega _{0}^{i}\right) +q_{b}^{*}\sum _{i=1}^{I}z_{b}^{*i}+q_{e}^{*}\sum _{i=1}^{I}z_{e}^{*i}\ge p_{0} \sum _{i=1}^{I}\left( x_{0}^{*i}-\omega _{0}^{i}\right) \\&\quad +q_{b}\sum _{i=1}^{I}z_{b}^{*i}+q_{e}\sum _{i=1}^{I}z_{e}^{*i} \end{aligned}$$

for all $(p_{0},q_{b},q_{e})\in Q.$ Moreover, by the budget constraint,

$$\begin{aligned} p_{0}^{*}\left( x_{0}^{*i} -\omega _{0}^{i}\right) +q_{b}^{*}z_{b}^{*i}+q_{e}^{*}z_{e}^{*i}\le 0 \end{aligned}$$

for all $i\in I.$ Hence,

$$\begin{aligned} p_{0}\sum _{i=1}^{I}\left( x_{0}^{*i} -\omega _{0}^{i}\right) +q_{b}\sum _{i=1}^{I}z_{b}^{*i}+q_{e} \sum _{i=1}^{I}z_{e}^{*i}\le 0 \end{aligned}$$

for all $(p_{0},q_{b},q_{e})\in Q.$ This implies

$$\begin{aligned} \sum _{i\in I}\left( x_{0}^{*i}-\omega _{0}^{i}\right) \le 0,\quad \sum _{i\in I}z^{*i}\le 0. \end{aligned}$$

Therefore, the allocation $(x_{0}^{*i},z^{*i})_{i\in I}$ is weakly feasible, i.e., $(x_{0}^{*},z^{*})\in F.$

Since $(x_{0}^{*i},z^{*i})\in f_{c}^{i}(p_{0}^{*},q_{b}^{*},q_{e}^{*},r_{{\varvec{1}}}^{*})$ for all $i\in I$ and $(x_{0}^{*i},z^{*i})_{i\in I}$ is a weakly feasible allocation, it follows from Lemma 3 that all prices must be strictly positive, $\left( p_{0}^{*},q_{b}^{*}, q_{e}^{*}\right) >>0.$ Moreover, since the indirect utility is strictly increasing, the budget constraint must be binding for all $i\in I,$ i.e., $p_{0}^{*}\left( x_{0}^{^{*i}}-\omega _{0}^{i}\right) +q_{b}^{*}z_{b}^{*i}+q_{e}^{*}z_{e}^{*i}=0.$ Together with weak feasibility, this implies market clearing,

$$\begin{aligned} \sum _{i\in I} \left( x_{0}^{*i} -\omega _{0}^{i}\right) =0,\quad \sum _{i\in I}z^{*i}=0. \end{aligned}$$

1.1.3 Existence of equilibrium in $t=1$

It remains to show that in a bankruptcy equilibrium

• (i) $r_{{\varvec{1}}}^{*}=\rho (x_{0}^{*},z^{*})\$ and

• (ii) $(x_{0}^{*i},z^{*i})\in f_{c}^{i}(p_{0}^{*},q_{b}^{*},q_{e}^{*}, r_{{\varvec{1}}}^{*})\$for all $i\in I$ with $\sum _{i\in I}z^{*i}=0,$

consumption will be nonnegative in every state $s\in S$, i.e., condition (iii) of the definition of a bankruptcy equilibrium

$$\begin{aligned} \sum _{i=1}^{I}\left( x_{s}^{*i}\vee 0\right) = \sum _{i=1}^{I}(\omega _{s}^{i}+Y_{s}\delta ^{i}) \end{aligned}$$

for all $s\in S$ is also satisfied.

Recall that, for an equilibrium price system $(p_{0}^{*},q_{b}^{*},q_{e}^{*},r_{{\varvec{1}}}^{*})$ and an equilibrium allocation $(x_{0}^{*i},z^{*i})\in f_{c}^{i}(p_{0}^{*},q_{b}^{*},q_{e}^{*},r_{{\varvec{1}}}^{*}),$ by Eq. 8,

$$\begin{aligned} x_{s}^{*i}=\omega _{s}^{i} +Y_{s}\left( \delta ^{i}+z_{e}^{*i}\right) +r_{s}(z_{b}^{*i}\vee 0)+(z_{b}^{*i}\wedge 0) \end{aligned}$$

and, by Eq. 4,

$$\begin{aligned} r_{{\varvec{1}}}^{*}=\rho (x_{0}^{*},z^{*}):=\frac{ \sum _{i=1}^{I}\left( -z_{b}^{*i}\vee 0\right) \wedge (\omega _{s}^{i}+Y_{s} (\delta ^{i}+z_{e}^{*i}))}{\sum _{i=1}^{I}(-z_{b}^{*i}\vee 0)} \end{aligned}$$

for all $s\in S$ hold. These conditions imply nonnegative consumption in all states.

Lemma 4

If for all $i\in I\$ and all $s\in S$ Eqs. 8 and 4 hold, then

$$\begin{aligned} \sum _{i=1}^{I}z_{b}^{*i}=0\quad \text { and } \quad \sum _{i=1}^{I}z_{e}^{ *i}=0 \end{aligned}$$

imply

$$\begin{aligned} \sum _{i=1}^{I}\left( x_{s}^{*i}\vee 0\right) = \sum _{i=1}^{I}(\omega _{s}^{i}+Y_{s}\delta ^{i}) \end{aligned}$$

for all $s\in S.$

Proof

For notational convenience, we drop the reference $^{*}$ to the equilibrium values. Consider an arbitrary $s\in S.$ By Eq. 8, summing over all consumers $i\in I,$ one obtains

$$\begin{aligned} \sum _{i=1}^{I}\left( x_{s}^{i} -\omega _{s}^{i}- Y_{s}\delta ^{i}\right) =\sum _{i=1}^{I}\left[ r_{s}(z_{b}^{i}\vee 0)+(z_{b}^{i}\wedge 0)+Y_{s}z_{e}^{i}\right] , \end{aligned}$$

which is equivalent to

$$\begin{aligned} \sum _{i=1}^{I}\left[ (x_{s}^{i}\vee 0) - \omega _{s}^{i}-Y_{s}\delta ^{i} \right] =\sum _{i=1}^{I}\left[ r_{s}(z_{b}^{i}\vee 0)+(z_{b}^{i}\wedge 0)+Y_{s}z_{e}^{i}-(0\wedge x_{s}^{i})\right] . \end{aligned}$$

The asset market equilibrium conditions

$$\begin{aligned} \sum _{i=1}^{I}z_{b}^{i}=0 \quad \text { and } \quad \sum _{i=1}^{I}z_{e}^{i}=0 \end{aligned}$$

imply

$$\begin{aligned}&\sum _{i=1}^{I}\left[ r_{s}(z_{b}^{i}\vee 0)+(z_{b}^{i}\wedge 0)+Y_{s}z_{e}^{i}-(x_{s}^{i}\wedge 0)\right] \\&\quad =r_{s}\underset{=-\sum _{i=1}^{I}(z_{b}^{i}\wedge 0)}{\underbrace{ \sum _{i=1}^{I}(z_{b}^{i}\vee 0)}}+\sum _{i=1}^{I}(z_{b}^{i}\wedge 0)-\sum _{i=1}^{I}(x_{s}^{i}\wedge 0)+Y_{s}\underset{=0}{\underbrace{ \sum _{i=1}^{I}z_{e}^{i}}} \\&\quad =r_{s}\sum _{i=1}^{I}(-z_{b}^{i}\vee 0) +\sum _{i=1}^{I}(z_{b}^{i}\wedge 0)-\sum _{i=1}^{I}(x_{s}^{i}\wedge 0). \end{aligned}$$

Denote by $I^{+}:=\{i\in I|\ z_{b}^{i}\ge 0\}\$ the set of consumers who are not borrowing, by $I_{s}^{-}:=\{i\in I|\ z_{b}^{i}<0,\ x_{s}^{i}\ge 0\}$ the set of loan takers who are solvent, and by $I_{i}^{-}:\{i\in I|\ z_{b}^{i}<0,\ x_{s}^{i}<0\}$ consumers who are insolvent, then substituting Eq. 4 for $r_{s}$ yields

$$\begin{aligned}&r_{s}\sum _{i=1}^{I}(-z_{b}^{i}\vee 0) +\sum _{i=1}^{I}(z_{b}^{i}\wedge 0)-\sum _{i=1}^{I}(x_{s}^{i}\wedge 0) \\&\quad =r_{s}\sum _{i\in I^{-}}(-z_{b}^{i}\vee 0)+\sum _{i\in I^{-}}(z_{b}^{i}\wedge 0)-\sum _{i\in I^{-}}(x_{s}^{i}\wedge 0) \\&\quad =\sum _{i\in I^{-}}\left[ \left( -z_{b}^{i}\right) \wedge (\omega _{s}^{i}+Y_{s}(\delta ^{i}+z_{e}^{i}))\right] -\sum _{i\in I^{-}}(-z_{b}^{i})-\sum _{i\in I^{-}}(x_{s}^{i}\wedge 0) \\&\quad =\sum _{i\in I_{i}^{-}}\left[ \omega _{s}^{i} +Y_{s}(\delta ^{i}+z_{e}^{i})) \right] -\sum _{i\in I_{i}^{-}}(-z_{b}^{i}) -\sum _{i\in I_{i}^{-}}(x_{s}^{i}) \\&\quad =\sum _{i\in I_{i}^{-}} \left[ \left( \omega _{s}^{i}+Y_{s}\left( \delta ^{i}+z_{e}^{i}\right) +z_{b}^{i}\right) -x_{s}^{i})\right] =0, \end{aligned}$$

since $x_{s}^{i}<0$ implies $x_{s}^{i}= \omega _{s}^{i}+Y_{s}\left( \delta ^{i}+z_{e}^{i}\right) +z_{b}^{i}<0.\$ Hence, $\sum _{i=1}^{I}\left( x_{s}^{i}\vee 0\right) =\sum _{i=1}^{I}(\omega _{s}^{i} +Y_{s}\delta ^{i})$. $\square$

1.2 Proof of Proposition 2

We partition the matrix

$$\begin{aligned} T= \left[ \begin{array}{l@{\quad }l@{\quad }l} -q_b &{} q_b &{} -q_{e}\\ r_{\varvec{1}} &{} - 1\!\!1 &{} Y \end{array}\right] \end{aligned}$$

and the portfolio vector $z^i \in Z^i$ into long-bond, short-bond, and equity trades by $T=(T_{b+},T_{b-},T_e)$ and $z^i=(z^i_{b+},z^i_{b-},z^i_e)^T$, respectively. Using Lagrange multipliers $\pi ^i\in \mathbb {R}^{S+1}$, $\sigma ^i_{b+} \ge 0$, and $\sigma ^i_{b-}\ge 0$, and assuming we are at an interior solution ($\delta ^i$ are such that short sales constraints on equity are not binding at equilibrium, and parameters are such that $x^i_0 > 0$ at equilibrium), the KKT conditions for the minimization of the Lagrange function

$$\begin{aligned} L^i(x^i, z^i, \pi ^i, \sigma ^i_{b+}, \sigma ^i_{b-})= -u^i(x^i)+ \langle \pi ^i,x^i-\omega ^i- e^i-Tz^i\rangle -\sigma ^i_{b+} z^i_{b+} -\sigma ^i_{b-} z^i_{b-} \end{aligned}$$

imply that

$$\begin{aligned} \langle \nabla u^i(\bar{x}^i),T_e \rangle&= (0,\ldots ,0),\\ \langle \nabla u^i(\bar{x}^i),T_{b+} \rangle&= -\sigma ^i_{b+}\le 0,\text {and}\\ \langle \nabla u^i(\bar{x}^i),T_{b-} \rangle&= -\sigma ^i_{b-}\le 0, \end{aligned}$$

Since the optimization problem is convex, the KKT conditions are also sufficient for an optimal solution to the agent’s problem.

The gradient of the linear quadratic utility function fulfills in the equilibrium allocation $\bar{x}^i$

$$\begin{aligned} \langle \nabla u^i(\bar{x}^i),\tau \rangle \le 0 \quad \forall \tau \in \mathcal {C }. \end{aligned}$$

Summing up all agent’s equilibrium gradients, we define the vector

$$\begin{aligned} \bar{\gamma }:= \sum _{i=1}^I \nabla u^i(\bar{x}^i)= (\alpha _0, \alpha _1 1\!\!1-((\omega _{\varvec{1}}+Y\delta )-d_{\varvec{1}}))^T, \end{aligned}$$

where $d_{\varvec{1}}=\sum _{i=1}^I d_{\varvec{1}}^i := \sum _{i=1}^I (1\!\!1 - r_{{ \varvec{1}}})z^i_{b-}$ is the aggregate shortfall from promises on the bond. Still we have $\langle \bar{\gamma },\tau \rangle \le 0 \quad \forall \tau \in \mathcal {C}.$ Since any trade $\tau \in \mathcal {C}$ can be decomposed as $\tau =(-c(m),m)$, we get for $\frac{1}{\alpha _0}{\bar{\gamma }}$ that $\frac{\bar{\gamma }_{\varvec{1}}}{\alpha _0}= \frac{\alpha _1}{\alpha _0}1\!\!1- \frac{1}{\alpha _0}\tilde{ \omega }_{\varvec{1}}$ and $c(m)\ge \left\langle \frac{ \bar{\gamma }_{\varvec{1}}}{\alpha _0}, m \right\rangle$, which proves the proposition. $\square$

1.3 Proof of Proposition 3

Suppose agent $i$ goes long in the bond. Define the matrix by $T_\text {long} = \begin{pmatrix} -q_b &{}\quad q_e \\ r_{\varvec{1}} &{}\quad Y \end{pmatrix}$. We know from proposition 2 that

$$\begin{aligned} \langle T_\text {long}^T, \nabla u^i(\bar{x}^i) \rangle&= \begin{pmatrix} 0\\ 0\end{pmatrix} \text { and} \end{aligned}$$

(9)

$$\begin{aligned} \langle T_\text {long}^T, \bar{\gamma } \rangle&= \begin{pmatrix} -\sigma _{b+} \\ 0\end{pmatrix} , \end{aligned}$$

(10)

where $\nabla u^i(x^i)= (\alpha _0^i, \alpha _1^i-x^i_{\varvec{1}})^T$ and $\bar{ \gamma }= (\alpha _0, \alpha _11\!\!1 -\tilde{\omega }_{\varvec{1}})^T$. Divide Eq. (9) by $\alpha _0^i$ and Eq. (10) by $\alpha _0$ , subtract the equations and multiply the result by $\alpha _0^i$ again. With $\bar{\tau }_{\varvec{1}}^i:=\bar{x}_{\varvec{1}}^i -\omega _{\varvec{1}}^i - Y \delta ^i$ this gives

$$\begin{aligned} \langle Y_{b+}^T, \bar{\tau }_{\varvec{1}}^i \rangle = \left\langle Y_{b+}^T, \left( \alpha _1^i-\frac{\alpha _0^i}{\alpha _0}\alpha _1\right) 1\!\!1 -\left( \left( \omega _{\varvec{1}} ^i+ Y \delta ^i\right) - \frac{\alpha _0^i}{\alpha _0} \tilde{\omega }_{\varvec{1}}\right) \right\rangle - \frac{\alpha _0^i}{\alpha _0} \begin{pmatrix} \sigma _{b+}\\ 0 \end{pmatrix} . \end{aligned}$$

We can write

$$\begin{aligned} \begin{pmatrix} \sigma _{b+} \\ 0 \end{pmatrix} = \left\langle Y_{b+}^T, \sigma _{b+}\frac{r_{\varvec{1}}- P_Y(r_{\varvec{1}})}{||r_{\varvec{1}} -P_Y(r_{\varvec{1}})||^2} \right\rangle = \langle Y_{b+}^T, \sigma _{b+} r_{{\varvec{1} }e} \rangle . \end{aligned}$$

Now, since $\langle Y_{b+}^T, v_{\varvec{1}} \rangle =\langle Y_{b+}^T, P_{Y_{b+}}(v_{\varvec{1}}) \rangle$ for any vector $v_{\varvec{1}}\in \mathbb {R}^{S}$, it follows that

$$\begin{aligned} \bar{\tau }_{\varvec{1}}^i = P_{Y_{b+}} \left( \left( \alpha _1^i-\frac{\alpha _0^i}{\alpha _0} \alpha _1\right) 1\!\!1 -\left( \left( \omega _{\varvec{1}}^i + Y \delta ^i\right) -\frac{\alpha _0^i}{ \alpha _0}\tilde{\omega }_{\varvec{1}}\right) - \sigma _{b+} \frac{\alpha _0^i}{\alpha _0} r_{ {\varvec{1}}e} \right) . \end{aligned}$$

Finally, since $r_{\varvec{1}}-P_Y(r_{\varvec{1}})\in \text {span}(Y_{b+})$, the result follows.

The results for agents going short and for agents that do not trade in the bond are proved similarly. $\square$