A game between central banks and households involving central bank digital currencies, other digital currencies and negative interest rates

Central Bank Digital Currencies (CBDCs) enable negative interest rates. A game is analyzed between a central bank (accounting for the government's interest) and a representative household choosing to consume, hold CBDC, or hold non-CBDC. The central bank chooses negative interest rate when it realizes that the household is willing to pay the central bank for holding CBDC. The household pays the negative interest rate because of its Cobb Douglas preferences whereby it values holding CBDC while simultaneously holding the competitive non-CBDC with a given interest rate, consuming with various output elasticities, and accounting for transaction efficiencies and costs. More explicitly, intuition and how the players benefit are provided for the following results: The central bank chooses more negative interest rate when the household's output elasticity for consumption increases, the household's output elasticity for holding CBDC decreases, the CBDC and non-CBDC transaction efficiencies increase, the household's transaction efficiency for consumption decreases, the household's scaling of the transaction cost increases, the scaling parameter for the central bank's profit per household decreases, the household's monetary energy decreases, and the non-CBDC interest rate decreases. The results are determined analytically and illustrated numerically where each of nine parameter values is varied relative to a benchmark.