Formation of Optimal Interbank Networks under Liquidity Shocks

We study the formation of an optimal interbank network in a model where banks control both their supply of liquidity, through cash reserves, and their exposures to other banks' risky projects. The value of each bank's project may suddenly decline depending on their cash reserves and both the occurence and magnitude of liquidity shocks. In two distinct settings, we solve the system-wide optimal control problem and obtain explicit formulas for the unique optimal allocations of capital. In the first decentralized setting, banks seek only to maximize their own expected utility. Second, a central planner aims to maximize the sum of all banks' expected utilities. Both of the resulting networks exhibit a `core-periphery' structure in equilibrium. However, in the decentralized setting, banks elect to hold less cash reserves compared to the centralized setting, leading to greater susceptibility to liquidity shortages. We characterize the behavior of the planner's optimal allocation as the size of the system grows. Surprisingly, the relative welfare gap is of constant order. Finally, we derive co-investment requirements that allow the decentralized system to achieve the planner's optimal level of risk. In doing so, we find that banks in the network's core are subjected to the highest co-investment requirements -- ensuring that they are sufficiently incentivized to hold significant cash reserves. Our analysis may inform regulators' requirements on banks' liquidity reserves, as have been debated in the wake of the 2023 regional banking crisis in the US.