Double moral hazard and banks' risk taking

There are multiple papers related to banks' risk-taking behaviour and competition, yet they have different conclusions.

Keeley (1990), Repullo (2004) and Allen and Gale (2004) focus on franchise value and their results are that as the number of banks increases, the equilibrium level of their risk-shifting behaviour will also increase. As deposit markets become more concentrated, banks will be more profitable and their franchise value will be larger. They will be reluctant to seek high return but low probability projects since those risk-taking behaviours may induce them to lose their franchise. The key assumptions of their model are that banks invest in assets with exogenous distributions of returns and the risk exists only in the banks themselves, which means no firms' participation in risk.