appropriate exogenous solvency function. We will assume that once a firm is deemed insolvent it can never recover to solvency again. Two possible systems for considering insolvent firms are:

1.  [**Receivership:**]{} In such a system, when a firm is deemed insolvent it is placed in receivership so that obligations are payed out on a first-come first-serve basis.

2.  [**Auctions:**]{} In such a system, when a firm is deemed insolvent its future assets are auctioned off in order to pay the future liabilities (in a proportional scheme) at the next time point. This will then affect the cash flows $c$ and nominal liabilities $L$, as such we would need to consider $c(t,V_{t-1})$ and $L(t,V_{t-1})$ to truly consider this case. We refer to [@CC15] for a detailed discussion of the auction model for insolvency. The auction system can be interpreted as an internal mechanism for determining bankruptcy costs in contrast to the exogenous parameter in, e.g., [@RV13].

The existence and uniqueness of the clearing solutions in these scenarios require an additional monotonicity property; we can use the notion a speculative system from [@banerjee2017insurance] to get the desired results. This condition encodes the notion that a firm does not benefit from any firm’s 