Bank Runs : The Pre-Deposit Game

We analyze in some detail the full pre-deposit game in a simple, tractable, yet very rich, banking environment. How does run-risk affect the optimal deposit contract? If there is a run equilibrium in the post-deposit game, then the optimal contract in the pre-deposit game tolerates runs for small probabilities. In some cases, the optimal contract becomes -- as one would expect -- strictly more conservative as the run-probability increases (until it switches to the best run-proof contract), and the equilibrium allocation is not a mere randomization over the equilibrium allocations from the post- deposit game. In the other cases, the allocation is a mere randomization over the equilibria from the post-deposit game. In first cases (the more intuitive ones), the incentive constraint does not bind. In the second cases, the incentive constraint does bind