Savings and default

Abstract

In the presence of uninsurable idiosyncratic risk, the optimal credit contract allows for the possibility of default. In addition, the optimal contract incorporates a precautionary savings motive over and above what agents would otherwise save. When default is sufficiently high, credit markets may collapse. A regulatory requirement on the level of savings can increase risk sharing and improve welfare by increasing the gains to trade in credit exchange. Under the appropriate verifiability condition on the level of savings, an appropriate market structure, agents voluntarily increase their level of storage such that trade and welfare improve.

Appendix

1.1 Proof of Proposition 1

Proof

Assume that the equilibrium for a certain region of $\lambda$ is autarkic, that is, the bond is not traded and agents optimise only over storage, $\sigma$.

Under autarky, we obtain the following first-order condition (FOC): $\sum _{s \in S}\mu _s -\mu _0^h=0$. Solving for storage we find $\sigma = \frac{-1+\sqrt{17}}{8}$. In this equilibrium we need to check whether the bond should be traded or not.

The marginal utilities for this equilibrium are $\mu _1=0.72$ and $\mu _2=2.56$ in states 1 and 2 respectively. For $\lambda <0.72$ the bond would definitely not be traded, since agents would default completely in both states. Thus, we consider $\lambda >0.72$ for our analysis. The price offered by the on the verge seller here is $\frac{\min [\frac{1}{2}\lambda ,\mu _1]+\min [\frac{1}{2}\lambda ,\mu _2]}{\mu _0}$. For $1\le \lambda$ this price would be at least $0.524$ and increasing on $\lambda$. The price offered by the on the verge buyer would be $\frac{K_1\mu _1+K_2\mu _2}{\mu _0}$. However, at $1\le \lambda <4/3$, the seller would deliver fully in his good state as his marginal utility is strictly less than $\frac{1}{2}\lambda$ and delivery nothing in the other state. Given that there are two agents, the delivery rate would be $\frac{1}{2}$. In this range of $\lambda$, the price offered by the buyer would be $\frac{1}{2}$ which is less than that demanded by the buyer.

We leave it to the reader to show that an equilibrium, which involves trade in the bond and agents optimally choose not to save, that is, $\sigma =0$, does not exist for this region of $\lambda ^{\prime }s$. The proof starts with the assumption that agents trade the bond and considers two subcases: one where agents deliver partially and the other where they deliver fully. Under this assumption, the bond market fails when $\lambda <1$, while full delivery occurs for $\lambda \ge 4/3$. The last step of the proof is to calculate the intertemporal marginal utilities and show that for $\lambda \ge 1$ there is excess demand for storage, which leads to a contradiction given the assumption that $\sigma =0$.

Let us now consider the range $0.72\le \lambda <1$ and conjecture that the equilibrium is autarkic. In this case, the price offered by an on the verge seller is $\frac{\mu _1+\frac{1}{2}\lambda }{\mu _0}$, since $\lambda >\mu _1$ as they would deliver fully in their good state and nothing in their bad state. Therefore, the price offered by an on the verge buyer is $\frac{\frac{1}{2}\mu _1+\frac{1}{2}\mu _2}{\mu _0}>\frac{\mu _1+\frac{1}{2}\lambda }{\mu _0}$ indicating that agents have an excess demand for the bond in this region of $\lambda$. Moreover, trade in the bond alone is not possible since the marginal disutility of delivering in the good state is 1 and the marginal disutility of default is less than 1 so agents would default completely in all states. The only other possible candidate equilibrium is one where both agents both trade the bond and save.

We now examine the equilibrium where agents optimise over the bond and $\sigma$. We first look at the case that agents partially default on the bond and then on the case that they deliver fully.

The first-order equation we get when agents optimise with respect to $\sigma$ is

$$\begin{aligned} \frac{1}{1-\sigma }=\frac{1}{2}\lambda +\frac{1}{2}\frac{2}{2\sigma +D}. \end{aligned}$$

But $D=2\sigma +2-\frac{2}{\lambda }$, thus $\sigma =\frac{-4+\lambda +\sqrt{9\lambda ^2-24\lambda +32}}{4\lambda }$. Using the first-order equations of buying and selling the bond, we find that $K=\frac{4\sigma \lambda +2\lambda -2}{2\sigma \lambda +\lambda }$. Under the assumption of partial delivery, we need $K<0.5$, which yields $\sigma \le \frac{4-3\lambda }{6\lambda }$ and $\lambda <7/9$ using the equations above. Up to $\lambda =\frac{7}{9}$ we get trade in both bonds and agents partially deliver on the bond in their good states while completely defaulting in their bad. After this $\lambda$ we have to consider full delivery of the bond. Adjusting the above first-order conditions for saving and trading the bond, we get that agents both save and trade the bond, while they deliver fully in the good state, for $7/9\le \lambda <0.92$. Autarky prevails for $\lambda \ge 0.92$.$\square$

1.2 Proof of Proposition 2

Proof

Consider, first, that $\lambda <0.72$. Under partial default the marginal utility for the good state will be $\frac{1}{2}\lambda$ and for the bad $\frac{1}{2}\frac{1}{1+2\rho ^*-\frac{1}{\lambda }}$. The delivery rate will be $K=\frac{2\lambda }{\lambda +\frac{1}{1+2\rho ^*-\frac{1}{\lambda }}}$.

For this area of default penalties $\rho ^*>\rho ^{**}$. Thus, agents have to hold at least $\rho ^*$. In addition, both D and K have to be positive and $K<1/2$. This is satisfied for $0.66\le \lambda <0.72$. Obviously, for $0<\lambda <0.66$ the equilibrium is autarkic.

For $0.72\le \lambda <0.92$, $\rho ^*$ is always less than $\rho ^{**}$. Thus, agents have to hold at least $\rho ^{**}$ to guarantee trade in the bond. Up to $\lambda =\frac{7}{9}$ agents default partially and the delivery rate as well as the actual delivery are given by the formulas above. By holding more the required storage requirement agents can increase their delivery up to the point that they start delivering fully.

For $\lambda >\frac{7}{9}$ agents deliver fully in their good states even if they are required to hold the minimum required, that is, $\rho ^{**}$. Thus, the final delivery for the bond is given by:

$$\begin{aligned} 4\lambda \rho ^{{**}2}+4\lambda \rho ^{**}+2\lambda D-\lambda D^2+2D-2=0 \end{aligned}$$

As long as it yields a positive D under the relevant constraints, the equilibrium in which agents sell the bond and deliver fully in their good states exists. It turns out that the maximum level of $\lambda$ for which the bond will be traded under storage requirements is 0.92.

Assume now that the default penalty is higher than 0.92. The bond should be traded, since agents would not default completely in their good states. Nevertheless, we showed in Propositions 1 that agents would not actually prefer to trade the bond and would rather save some of their initial wealth. As before the following two relations have to hold:

      $4\lambda \rho ^{**2}+4\lambda \rho ^{**}+2\lambda D-\lambda D^2+2D-2=0$ and $8\rho ^{**2}+2D-D^2+2\rho ^{**}-2\ge 0$. The latter condition requires that there is no excess demand for storage. As long as these two relations yield positive D and $\rho ^{**}$ equilibrium exist. Manipulating and adding them we get

$$\begin{aligned} -2\lambda D-6\lambda \rho ^{**}+2\lambda D^2-4D+4-\lambda D^2-2\lambda \ge 0 \end{aligned}$$

Assume that $\rho ^{**}>0$ and $D>0$, then the last equation becomes

$$\begin{aligned} \lambda D^2-4D+4>K>0 \end{aligned}$$

where K is a positive number. The above relation gives no real solution for $D$ for $\lambda \ge 0.92$, that is, a contradiction.

We, thus, get that for $\lambda \ge 0.92$ the equilibrium is autarkic even under storage requirements $\square$

1.3 Proof of Proposition 3

Proof

The proof for trade, when $0<\lambda <0.72$, is straightforward from Proposition 2. The marginal utility in the bad state (for the first agent state 2) is higher than the marginal utility in the good state, which is equal to $\frac{\lambda }{2}$, since agent default completely. Thus, $\mu _2>\mu _1$ and $\frac{\partial w}{\partial \rho }<-\mu _0+2\mu _2(1+D\mu _2)$. But, $D\mu _2=\frac{1+\rho -\frac{1}{\lambda }}{1+2\rho -\frac{1}{\lambda }}<1$. Consequently, $\frac{\partial w}{\partial \rho }<-\mu _0+4\mu _2=-\frac{1}{1-\rho }+\frac{2}{1+2\rho -\frac{1}{\lambda }}.$ From Lemma 1, $\rho (\epsilon )=\frac{1}{\lambda }-1+\epsilon ,\mathrm { \ }\epsilon >0$ for this are of default penalties. Thus,

$$\begin{aligned} \frac{\partial w}{\partial \rho }<-\frac{1}{2-\frac{1}{\lambda }-\epsilon }+\frac{2}{\frac{1}{\lambda }+2\epsilon -1}. \end{aligned}$$

As $\epsilon \rightarrow 0$, $\lim _{\epsilon \rightarrow 0}{-\frac{1}{2-\frac{1}{\lambda }-\epsilon }+\frac{2}{\frac{1}{\lambda }+2\epsilon -1}}<0$ and the first derivative of this expression with respect to $\epsilon$ is negative. As a result, $\frac{\partial w}{\partial \rho }<0$.

Consider, now, an equilibrium with partial default and some storage (optimised or not), that is, $0.72\le \lambda <7/9$. The delivery is given by $2(1+\rho - 1/\lambda )$ and the delivery rate by $2 - \frac{2}{\lambda (1+2\rho )}$. The quantity traded is $\phi = D/(2K) = \frac{1+\rho }{2}\frac{\lambda - \frac{1}{1+\rho }}{\lambda - \frac{1}{1+2\rho }}$. As agents are delivering partially, $\lambda - \frac{1}{1+\rho }>0$. Differentiating, we find that the quantity of trade will increase with the storage requirement for $\lambda <1$.

We now turn to the analysis of welfare. For this region of default penalties $\rho (\epsilon )=\frac{\lambda -4+\sqrt{9\lambda ^2-24\lambda +32}}{4\lambda }+\epsilon$. For $\epsilon \rightarrow 0$, it is easy to show that $\lim _{\epsilon \rightarrow 0}{\frac{\partial w}{\partial \rho }}>0$. Also, $\lim _{\epsilon \rightarrow \bar{\epsilon }}{\frac{\partial w}{\partial \rho }}>0$, where $\rho (\bar{\epsilon })=\frac{4-3\lambda }{6\lambda }$, which is the level of storage at which agents deliver fully in their good states (see Proposition 2). Finally, $\frac{\partial ^2 w}{\partial \rho ^2}=-\frac{1}{(1-x)^2}-\frac{4 \left(1+x-\frac{1}{\lambda }\right)}{\left(1+2 x-\frac{1}{\lambda }\right)^3}<0$, since $1+x-\frac{1}{\lambda }>0$. Hence, $\frac{\partial w}{\partial \rho }$ is monotonic and always positive in this region of default penalties.

Finally, under full delivery, that is, for $7/9\le \lambda <0.92$, we can take the derivative of the on the verge for trading the bond which is

$$\begin{aligned} \lambda = \frac{1}{2}\frac{1}{\frac{D}{2}+\rho } -\frac{1}{2} \frac{1}{1-\frac{D}{2}+\rho }. \end{aligned}$$

Solving for $\frac{\partial D}{\partial \rho } = 2\frac{\mu _1^2 - \mu _2^2}{\mu _1^2 + \mu _2^2}$ we can see that the marginal utility in the state 1 is strictly less than that in state 2 and as the denominator is positive, then the delivery of agents is falling on $\rho$. When there is full delivery, then the quantity of bonds sold will equal the delivery on the bonds, and hence, the trade in bonds falls on $\rho$ for this region of $\lambda$.

Turning to welfare analysis for this region of default penalties, observe that the consumption in the good state for each agent is now $1+\rho - D/2$ and in the bad state $\rho +D/2$. The delivery rate is given by $K= \frac{\mu _1 + 1/2\lambda }{\mu _1+\mu _2} = 1/2$ where we take state 1 to be the good state and state 2 to be the bad state. Rearranging this expression we get $\mu _1 + \lambda = \mu _2$. Now take the derivative of this equation with respect to $\rho$ and we get that $\frac{\partial D}{\partial \rho } = 2\frac{\mu _1^2 - \mu _2^2}{\mu _1^2 + \mu _2^2}$ which is negative as the marginal utility in the state 1 is strictly less than that in state 2.

The welfare of this agent is given by

$$\begin{aligned} ln(1-\rho ) + 1/2ln(1+\rho -D/2) + 1/2ln(\rho +D/2) - \lambda /2 D. \end{aligned}$$

since $\phi = D$. Taking the derivative of this expression with respect to $\rho$ we get

$$\begin{aligned} -\mu _0 + \mu _1 + \mu _2 + 1/2\frac{\partial D}{\partial \rho } [\mu _2 -\mu _1 - \lambda ]. \end{aligned}$$

We know that the last part of this expression is zero. The first part of the expression is zero when agents optimise over the bond and negative when they are forced to hold more liquidity than this. Under a storage requirements the whole expression is negative. Hence, when there is already full delivery, welfare is falling when the storage requirement is increasing. $\square$

1.4 Proof of Lemma 2

Proof

Suppose $(x,\sigma _v,\phi _v,\theta _v,\phi _v,D_v)\in B^h(q_v,K_v,\rho _v, \hat{\rho })$. Assume that $\rho -\rho _v=\epsilon >0$. Since there is a continuum of contracts $\exists \mathrm { \ }\epsilon$ such that $\epsilon =\hat{\rho }\phi ^h_v$. Also assume that there is partial default in equilibrium. Then, $q_v=\frac{\lambda }{\mu _0^v}$ and $q=\frac{\lambda }{\mu _0}$, where $\mu _0^v$ is then marginal utility of agent $h$ at $t=0$ in the equilibrium with variable storage requirements and $\mu _0$ the marginal utility of agent $h$ at $t=0$ in the equilibrium with a flat storage requirement. Finally, $\exists \phi ,\theta \mu _0>0$ such that $\phi =\frac{\mu _0}{\mu _0^v}\phi _v$ and $\theta =\frac{\mu _0}{\mu _0^v}\theta _v$, since $\mu _0^v,\mu _0,\phi _v,\theta _v>0$.

The budget constraint for GE$(R,\lambda , \rho )$ at $t=0$, under the conditions in Lemma 1 $\tilde{\sigma }^h=\sigma ^h_v=0$, can be written as:

$$\begin{aligned} x_0-e^h_0+\rho +q(\theta -\phi )&= x_0-e^h_0+\rho _v+\hat{\rho }\phi ^h_v+q_v\left(\frac{\mu _0^v}{\mu _0}\theta - \frac{\mu _0^v}{\mu _0}\phi \right)\\&= x_0-e^h_0+\rho _v+\hat{\rho }\phi ^h_v+q_v(\theta _v-\phi _v)=0 \end{aligned}$$

The budget constraint for GE$(R,\lambda , \rho )$ in state $s$ that agent $h$ chooses to partially deliver on his promise (an equivalent argument holds for the state that he chooses to default completely) can be written as:

$$\begin{aligned} x_s-e^h_s+D_s-\theta ^h K_s-\rho&= x_s-e^h_s-\frac{D_s}{2}-\rho \\&= x_s-e^h_s-\frac{D_{sv}}{2}-\rho _v-\hat{\rho }\phi ^h_v\\&= 0 \end{aligned}$$

Note the spans of GE$(R,\lambda , \rho _v, \hat{\rho })$ and GE$(R,\lambda , \rho )$ are the same, since $D=D_v$ and $\rho =\rho _v+\hat{\rho }\phi ^h_v$.

Thus, there exist $\phi ^h,\theta ^h,D^h$ such that $(x^h,\phi ^h,\theta ^h,\phi ^h,D^h)\in B^h(q,K,\rho )$ and are optimal due to strictly concave utility functions.

The converse argument follows easily from the analysis above. $\square$

1.5 Proof of Proposition 5

Proof

Define by $r$ the yield on the bond. Then, $r=\frac{1}{q}-1$. As the risk-free interest rate in our economy is zero (agents have the option of costlessly storing wealth), a higher yield translates into a higher credit spread. Equivalently, a lower price translates into a higher credit spread. The price of the bond in equilibrium (from the first-order condition of the seller) is: $q=\lambda (1-\rho )$. Differentiating the price with respect to the cash holdings we get $\frac{\partial q}{\partial \rho }=-\lambda <0$.

To prove the second point we use the fact that, in equilibrium, agents will increase their cash holdings until they deliver fully in their good states or equivalently until $K=\frac{1}{2}$(from Sect. 5.1). Recall that the delivery rate is given by $K=\frac{4\rho \lambda +2\lambda -2}{2\rho \lambda +\lambda }$. Thus, $\rho =\frac{4-3\lambda }{6\lambda }$ for $K=1/2$. Substituting the above expression for $\rho$ into that for $q$ and differentiating with respect to $\lambda$ we get $q=\lambda \left(1-\frac{4-3\lambda }{6\lambda }\right)$. Thus, $\frac{\partial q}{\partial \lambda }=\frac{3}{2}>0$. A higher default penalty given the endowment of the agent in the state that he delivers on his promise is qualitatively the same as a higher endowment in that state given the level of punishment. The credit spread is a decreasing function of the future endowment as well. $\square$