This paper constructs a searching model to study the impact of monetary policy, fiscal policy, transaction costs of private cryptocurrency and regulatory uncertainty on the competitive equilibrium between Central Bank Digital Currency and private cryptocurrency. The results show: (1) Central Bank Digital Currency and private cryptocurrency can conditionally coexist. (2) The level of economic welfare is related to monetary policy and fiscal policy but independent from the issuance and transaction of private cryptocurrency. (3) Under complete competition of private cryptocurrency mining, the mining cost becomes deadweight loss for whole society.