Consumer default with complete markets: default-based pricing and finite punishment

Abstract

This paper studies economies with complete markets where there is positive default on consumer debt. In a simple tractable two-period model, households can default partially, at a finite punishment cost, and competitive intermediaries price loans of different sizes separately. This environment yields only partial insurance. The default-based pricing of debt makes it too costly for the borrower to achieve full insurance, and there is too little trade in securities. This framework is in contrast to existing literature. Unlike the literature with default, there are no restrictions on the set of state contingent securities that are issued. Unlike the literature on lack of commitment, limited trade arises without need of debt constraints that rule default out. Compared with the latter, the present approach appears to imply more consumption inequality. An extended model with an infinite horizon, idiosyncratic risk and more realistic assumptions is used to demonstrate the general validity of this approach and its main implications.

Appendices

Appendix A: Propositions 1, 2 and 3

Proof of Proposition 1

The problem is $\max _{d} W(d)\equiv u(y_{s}^{i}-(1-d)l)-\eta d^{\gamma }$ subject to $d\in [0,1]$ and $y_{s}^{i}-(1-d)l\ge 0$, for some $l>0$. The objective is continuous and the choice set is compact, so a solution exists. The objective is differentiable in the interior of the choice set, with the first and second derivatives $W'(d)=l u'(y_{s}^{i}-(1-d)l)-z'(d)$ and $W''(d)=l^{2} u''(y_{s}^{i}-(1-d)l)-z''(d)$. Given $\gamma >1$, the objective is thus strictly concave and the solution is unique. The derivative of the objective at the lowest feasible value is positive: If $y_{s}^{i}>l$, then $W'(0)=l/(y_{s}^{i}-l)>0$; otherwise, $W'(1-y_{s}^{i}/l)=+\infty$. At the other extreme, $W'(1)=l/y_{s}^{i}-\eta \gamma$. Therefore, if $l<y_{s}^{i}\eta \gamma 0$, the solution is interior and given by $W'(d)=0$. Otherwise, if $l/y_{s}^{i}-\eta \gamma <0$, the solution is at the corner with $d=1$.

In an interior solution, $W'(d)$ increases with $l$; concavity implies that it decreases with $d$. Therefore, the optimal $d$ rises with $l$. As $l$ approaches $l/y_{s}^{i}-\eta \gamma$ from the left, the optimal $d$ approaches 1. So $d$ is continuous in $l$, and differentiable except at $l=y \eta \gamma$.

Proof of Proposition 2

The first part is a direct implication of the properties of $D_{s}^{i}(.)$ in the previous proposition and the determination of the price schedule in Eq. (6). Using (6) and (4) to differentiate implicitly $Q_{s}^{i}(l)l$, the resulting derivative is

$$\begin{aligned} {Q_{s}^{i}}'(l)l+Q_{s}^{i}(l)&= p_{s}{{\gamma \eta (\gamma -1){D_{s}^{i}(l)}^{-1}-\gamma \eta \gamma } \over {\gamma \eta (\gamma -1){D_{s}^{i}(l)}^{-1}+\gamma \eta l/(y_{s}^{i}-l(1-D_{s}^{i}(l)))}}\\&= p_{s}{{\gamma \eta (\gamma -1){D_{s}^{i}(l)}^{-1}- \gamma \eta \gamma }\over {\gamma \eta (\gamma -1){D_{s}^{i}(l)}^{-1} +\gamma \eta \gamma \eta {D_{s}^{i}(l)}^{\gamma -1}}}, \end{aligned}$$

where the equality uses (4) again. This shows that $Q_{s}^{i}(l)l$ is increasing only when $l$ is such that $D_{s}^{i}(l)<(\gamma -1)/\gamma$; that $D_{s}^{i}(.)$ is increasing implies the slope result. Concavity follows from the fact that as $D_{s}^{i}(.)$ is increasing, the value ${Q_{s}^{i}}'(l)l+Q_{s}^{i}(l)$ decreases with $l$.

Proof of Proposition 3

The shape of the pricing schedule means that in equilibrium the household will only choose $l_{s}^{i}<y_{s}^{i} \eta \gamma$. Therefore, default is less than full, and the default and pricing functions are differentiable. Then condition (5) holds because, by proposition 1, (5) is satisfied and point (iv) of the definition holds. Consider the household’s utility as a function of $l_{s}^{i}$ with default determined by $D_{s}^{i}(l_{s}^{i})$. Calculate the first and second derivatives, using the envelope property and second-order condition on default $D(l)$. Because $Q_{s}^{i}(l)l_{s}^{i}$ is concave from proposition 2, this objective is also concave. That $l_{s}^{i}$ is positive means the derivative is initially positive; that $l_{s}^{i}$ eventually leads to full default and zero price means the derivative becomes negative for $l_{s}^{i}$ large enough, and the unique solution is given by the first-order condition (2). Note that this optimal solution requires $q_{s}^{i}+{Q_{s}^{i}}'(l_{s}^{i})l_{s}^{i}$ to be positive which, by proposition 2, implies the upper bound for default.

Appendix B: Characterization of the equilibrium in Sect. 5.1

Symmetric fundamentals mean $y_{1}^{A}=y_{2}^{B}$ and $y_{1}^{B}=y_{2}^{A}$. We can suppose, without any loss of generality, that $y_{1}^{A}=y_{2}^{B}>y_{1}^{B}=y_{2}^{A}$. We want to argue first that debts are held in the good state so $l_{1}^{A}>0$ and $l_{2}^{B}>0$, that an equilibrium can be characterized by symmetric allocations and that aggregate variables are then state independent. We consider only equilibria where there is some trade.

Debts held in high-income states. One can first argue that consistency with consumer constraints and market clearing in (8) and savers’ optimality (3) requires that debts are held either in the individual bad states or in the good states. That is, either $l_{1}^{A}>0$ and $l_{2}^{B}>0$ or $l_{2}^{A}>0$ and $l_{1}^{B}>0$. For example, suppose, by way of contradiction, $l_{1}^{A}>0$ and $l_{2}^{B}=0$. Then (8b) implies $c_{1}^{A}<y_{1}^{A}$, and thus, market clearing requires $c_{1}^{B}>y_{1}^{B}$. That $l_{2}^{B}=0$ implies, by (8b), $c_{2}^{A}=y_{2}^{A}$ hence, by market clearing, $c_{2}^{B}=y_{2}^{B}$, and, by (8b), $l_{2}^{A}=0$. By (3), $c_{0}^{A}=c_{0}^{B}$. But then, the constraints in (8a) imply $c_{0}^{A}>c_{0}^{B}$, a contradiction. Specifically, we can establish that $l_{1}^{A}>0$ and $l_{2}^{B}>0$ will be the case. By (8), this amounts to showing that $c_{1}^{A}<y_{1}^{A}$ and $c_{2}^{B}<y_{2}^{B}$. Suppose, by way of contradiction and given the first result above, that $c_{1}^{A}>y_{1}^{A}$ and $c_{2}^{B}>y_{2}^{B}$. By market clearing in (8) and (7), it follows that $c_{1}^{B}<y_{1}^{B}$ and $c_{2}^{A}<y_{2}^{A}$. Therefore, $c_{2}^{A}<c_{1}^{A}$ and then $u(c_{2}^{A})/u(c_{1}^{A})>[q_{2}^{A}+{Q_{2}^{A}}'(l_{2}^{A})l_{2}^{A}]/(p_{2}(1-d_{2}^{A})$, which violates optimality conditions (2) and (3), a contradiction.

Symmetric allocations. Suppose $p_{1}=p_{2}$, then (4) and (6) imply $Q_{1}^{A}(.)=Q_{2}^{B}(.)$. Suppose also $q_{1}^{A}=q_{2}^{B}$. Now consider the optimality conditions (2), (3), (5), (7), (8a) and (8b) for $i=A$, involving $c_{0}^{A}, c_{1}^{A}, c_{2}^{A}, l_{1}^{A}$ and $d_{1}^{A}$. Given the symmetry of prices, conditions (2), (3), (5), (7), (8a) and (8b) for $i=B$ are satisfied for $c_{0}^{B}, c_{2}^{B}, c_{1}^{B}, l_{2}^{B}$ and $d_{2}^{B}$ equating, respectively, $c_{0}^{A}, c_{1}^{A}, c_{2}^{A}, l_{1}^{A}$ and $d_{1}^{A}$. Finally, as the equilibrium condition (8c) holds for $c_{2}^{A}$, symmetric allocations also imply that it holds for $c_{1}^{B}$. So the equalization of prices assumed $p_{1}=p_{2}$ and $q_{1}^{A}=q_{2}^{B}$ satisfies all the equilibrium conditions.

Constant values. Given the symmetric allocations, aggregate variables in Sect. 5.1 are labeled as $y_{l}=y_{1}^{A}=y_{2}^{B}, c_{l}=c_{1}^{A}=c_{2}^{B}, y_{a}=y_{2}^{A}=y_{1}^{B}, c_{a}=c_{2}^{A}=c_{1}^{B}, l=l_{1}^{A}=l_{2}^{B}>0$ and $d=d_{1}^{A}=d_{2}^{B}>0$, and $p=p_{1}=p_{2}$ and $q=q_{1}^{A}=q_{2}^{B}$. Clearly, these values remain invariant to the state $s=1,2$.