Private and social welfare gains in the Diamond-Dybvig model: A rationale for the existence of banks

In this note, I evaluate the private and the social welfare gains that in the Diamond-Dybvig model of bank runs characterize the switch from a decentralized to a centralized equilibrium that may hold even in an atomistic environment with banking intermediation. Specifically, relying on logarithmic preferences, I show that such a social welfare gain is an increasing function of the discount rate of more patient agents. Moreover, I show that for each level of the discount rate of patient agents, there is an optimal value of the proportion of these agents in the economy that maximizes the social welfare gain.