Private digital payment systems, e.g., Bitcoin or Ethereum, allow for transactions without the need for a financial institution. These institutions, central and retail banks, may thus observe a decline in the demand for their own payment systems, i.e., cash and deposits. Several questions arise: do digital payment systems affect the value of analog currencies such as cash? Are digital payment systems able to increase welfare? On the other hand, are central banks able to control providers of private digital payment systems? Do retail banks survive if central banks issue their own digital currency? To answer these questions, monetary search models are extended by digital money. Even if there are several studies which extend monetary search models by a secondary currency, the currency competition and the welfare effects are less well explored. It turns out that digital currencies reduce the value of analog currencies such as cash as long as digital currencies have a positive return. Nevertheless, digital currencies are able to increase welfare if the share of users is limited. In addition, central banks are able to tilt the playing field by providing an interest-bearing central bank digital currency. In equilibrium, retail banks face a lower profit but survive. Providers of private digital payment systems such as miners, on the other hand, go bankrupt.