For public intervention in form of asset purchases, see J. Tirole, “Overcoming Adverse Selection: How Public Intervention Can Restore Market Functioning,” American Economic Review, 2012; and T. Philippon and V. Skreta, “Efficient Interventions in Markets with Adverse Selection,” American Economic Review, 2012. 8  See D. Diamond and P. Dybvig, “Bank Runs, Deposit Insurance, and Liquidity”, Journal of Political Economy, 1983.
In fact, it is optimal for ex-ante efficiency to expose banks to liquidity risk, see C. Calomiris and C. Kahn, “The Role of Demandable Debt in Structuring Optimal Banking Arrangements,” American Economic Review,  BIS central bankers’ speeches  3  of last resort, but to perform that function at a penalty rate, illustrates both the need for (liquidity) insurance and the need to safeguard incentives.