Liquidity provision vs. deposit insurance: preventing bank panics without moral hazard

Equilibrium bidding strategies under most multi-unit auction rules cannot be obtained as closed form expressions. Research in multi-unit auctions has, therefore, depended on implicit characterization of equilibrium strategies using the first-order conditions of the bidders’ expected payoff maximization problem. In this paper we consider the pay-as-bid auction with diminishing marginal values for two units and show that any symmetric equilibrium in continuous strategies has the necessary properties to allow such a characterization. Moreover, any increasing solution to the system of differential equations that is used to characterize the equilibrium strategies describes an equilibrium strategy.Generous suggestions and comments from Richard Engelbrecht-Wiggans, Charles M. Kahn, and an anonymous referee are gratefully acknowledged.