Optimal monetary policy with interest on reserves and capital over-accumulation

We propose a model where banks with limited liability issue deposits backed by capital, and households demand deposits and currency as means-of-payments for decentralized trades. When financial friction is severe or productivity is low, banks hold excess reserves in equilibrium with a low nominal interest rate, and with capital holdings above the efficient level. In that region, paying a positive interest on reserves financed by money creation is optimal. Under a fixed inflation target, optimal interest rate on reserves is procyclical. The constrained efficient allocation is implemented with a fixed and a proportional liquidity requirement in addition to interest on reserves when productivity is not too high.